Practical Approaches To Biological Inorganic Chemistry [2 ed.] 9780444642257, 0444642257

Table of contents :
Cover
Practical Approaches to Biological Inorganic Chemistry
Copyright
List of Contributors
1 An overview of the role of metals in biology
Introduction
Essential metal ions and their functions
Toxic metals
Metals in diagnosis and therapeutics
References
Further reading
2 Introduction to ligand field theory and computational chemistry
Introduction
Introduction to quantum chemistry
Approximations in quantum chemistry
Electronic structure of atoms
Hydrogen atom
Angular momentum
Electron spin
Many-electron atoms
Periodic system of elements
Pauli principle
Two electrons in two orbitals
Electronic terms
Symmetry
Ligand field theory
Some qualitative considerations
Symmetry in ligand field theory
Some quantitative considerations
Tanabe–Sugano diagrams
Introduction to computational chemistry
The wave function–based methods
The Hartree–Fock method
Post-Hartree–Fock methods
Density functional theory
Density functional approximations
Density functionals and spin states
Computational methods for excited states
Computational methods for biological systems containing transition metal
Concluding remarks
Acknowledgments
References
3 Molecular magnetochemistry
Introduction
Origin of magnetism
Contributions to angular momentum in free atoms and ions
Term symbols for free atoms and ions with one electron outside a closed inner shell
Spin–orbit coupling
Term symbols for free atoms and ions with more than one electron outside a closed inner shell
Units and definitions
Magnetic moment and the Bohr magneton
Magnetic field and magnetization
Zeeman effect
Normal Zeeman effect
Anomalous Zeeman effect
Magnetic susceptibility, effective magnetic moment and magnetization saturation
Curie law for noninteracting paramagnets
Boltzmann treatment of magnetization
Langevin paramagnetism
Brillouin function
Van Vleck equation
Curie constant and spin-only effective magnetic moment
Temperature-independent paramagnetism and the second-order Zeeman effect
Contributions to magnetism in biologically relevant ions
Orbital splitting of transition metal ions in crystal field
Effect of crystal field on magnetic properties of 3d compounds
Dimeric sites: exchange mechanisms and J values
Curie–Weiss law
Superexchange
Spin Hamiltonian
Bleaney–Bowers equation
Diamagnetism
Experimental methods
Magnetometry
Force methods
SQUID—super conducting quantum interference device
What is measured in the SQUID?
Evans NMR method
Magnetic circular dichroism
Conclusion
Problems
Answers
References
Further reading
4 EPR spectroscopy
Why electron paramagnetic resonance spectroscopy?
What is electron paramagnetic resonance spectroscopy?
Anisotropy
A comparison of electron paramagnetic resonance versus NMR
Electron paramagnetic resonance spectrometer
What (bio)molecules give electron paramagnetic resonance?
Basic theory and simulation of electron paramagnetic resonance
Saturation
Concentration determination
Hyperfine interactions
High-spin systems
Applications overview
Test questions
Answers to test questions
References
5 Introduction to biomolecular nuclear magnetic resonance and metals
Introduction
Properties of the matter relevant to nuclear magnetic resonance
Energy of nuclear magnetic resonance transitions
Macroscopic magnetization
Acting on magnetization
Pulses
The rotating frame
Relaxation
What are the physical mechanisms of relaxation?
An nuclear magnetic resonance experiment
The chemical shift
Carrier frequency
Sampling bandwidth and the Nyquist theorem
Measuring T1
Coupling: the interaction between magnetic nuclei
Decoupling
The nuclear Overhauser effect
DOSY: sizing up molecules
Chemical exchange
Multidimensional nuclear magnetic resonance
How do the correlations arise and how are cross-peaks generated?
The COSY
The NOESY
The HSQC
Metals in biomolecular nuclear magnetic resonance spectra
Transition metals and interaction with the unpaired electron(s)
Hyperfine scalar coupling
Dipolar coupling
Relaxation
Contact relaxation
Dipolar relaxation
Curie relaxation
Residual dipolar couplings
Nuclear magnetic resonance of (semi-)solid samples
Direct observation of metals by nuclear magnetic resonance
In-cell nuclear magnetic resonance
An nuclear magnetic resonance spectrometer: measuring macroscopic magnetization and relaxation
Care in obtaining nuclear magnetic resonance spectra of paramagnetic samples
Water eliminated Fourier transform and super-water eliminated Fourier transform sequences: catching up with fast relaxing s...
Evan’s method: measuring magnetic susceptibility
Conclusions
Further reading
Useful physical constants
Exercises
Answers
6 57Fe-Mössbauer spectroscopy and basic interpretation of Mössbauer parameters
Introduction
Principles
The Mössbauer light source
γ-Emission and absorption—recoil is a problem
Recoilless emission and absorption—the Mössbauer effect
The Mössbauer experiment
The Mössbauer spectrometer
57Fe hyperfine interactions
Isomer shift as informative hyperfine interaction
Electric quadrupole splitting
Magnetic hyperfine splitting
Combined hyperfine splitting
Applications—selected examples
Oxidation and spin states in a nonheme diiron center
Reaction intermediates and low- and high-valent iron complexes
The heme enzyme horseradish peroxidase
Nonheme model compounds
Synthetic iron(III) complexes with the macrocyclic ligand cyclam
Iron(II) complexes
Mixed-valence iron(III)–iron(IV) dimers and iron(IV) monomers
Iron(V) complexes
Four-coordinated iron(IV) and iron(V) compounds
The first molecular iron(VI) compound
Perspectives
Exercises
References
7 X-ray absorption and emission spectroscopy in biology
Outline of the X-ray absorption and emission spectroscopy in biology
An introductory biological X-ray absorption spectroscopy example: Mo, Cu, and Se in CO-dehydrogenase from Oligotropha carbo...
X-ray absorption (near-)edge structure
X-ray emission spectroscopy in biology
Time-resolved X-ray absorption spectroscopy
X-ray absorption spectroscopy: X-ray–induced electron diffraction
Phase shifts and effect of atom type
Plane wave and muffin-tin approximation
Multiple scattering in biological systems
Strategy for the interpretation of EXAFS
Validation and Automation of EXAFS data analysis
X-ray absorption near-edge structure simulations with three-dimensional models
Metal–metal distances in metal clusters
Nonmetal trace elements: halogens
Summary: strengths and limitations
Conclusions: relations with other techniques
Exercises
Hints and answers to exercises
References
8 Resonance Raman spectroscopy and its application in bioinorganic chemistry
Introduction
The fundamentals of vibrational spectroscopy
The classical oscillator, Hooke’s law, the force constant, and quantization
Quantization and the nature of a quantum excitation
Permanent, induced, and transition electric dipole moments
Electric dipole moments
Transition dipole moments
Polarizability, induced dipole moments, and scattering
What is ∫φvf*(x)φvidx?
Relative intensities of Stokes and anti-Stokes Raman scattering
What is a virtual state?
The (resonance) Raman experiment
Raman cross-section and the intensity of Raman bands
Raman scattering is a weak effect; but how weak?
Resonance enhancement of Raman scattering
The Raman spectroscopy of carrots and parrots
Classical description of Rayleigh and Raman scattering
The Kramer–Heisenberg–Dirac (KHD) equation
A-, B-, C-term enhancement mechanisms, overtones, and combination bands
Assigning electronic absorption spectra
Heller’s time-dependent approach
SERS and SERRS spectroscopy
Experimental and instrumental considerations
Isotope labeling and band assignment
Resolution and natural linewidth
Confocality, the inner filter effect, and quartz
Applications of resonance Raman spectroscopy
Resonance enhanced Raman spectroscopy in the characterization of artificial metalloenzymes based on the LmrR protein
Reaction monitoring with resonance Raman spectroscopy
Transient and time-resolved resonance Raman spectroscopy
Conclusions
Questions
Answers to Questions
References
Further reading
9 An introduction to electrochemical methods for the functional analysis of metalloproteins
Introduction
Basics
Redox thermodynamics: the Nernst equation
Reference potential and reference electrodes
The biological redox scale
Influence of coupled reactions (e.g., protonation or ligand binding) on reduction potentials
Electron transfer kinetics
Kinetics of proton-coupled electron transfer: stepwise versus concerted mechanisms
Electrochemistry under equilibrium conditions: potentiometric titrations
Dynamic electrochemistry
Distinction between equilibrium and dynamic electrochemistry
Electrodes for electron transfer to/from proteins
Electrochemical equipment
Vocab and conventions
The capacitive current
Diffusion-controlled voltammetry
Diffusion-controlled voltammetry at stationary electrodes
Diffusion-controlled voltammetry at rotating electrodes
Voltammetry of adsorbed proteins: protein film voltammetry
Noncatalytic voltammetry at slow scan rates to measure reduction potentials
Fast-scan voltammetry to determine the rates of coupled reactions
Catalytic protein film voltammetry and chronoamperometry
Principle and general comments
Mass-transport controlled catalytic voltammetry
Chronoamperometry to measure Michaelis and inhibition constants
Chronoamperometry to resolve rapid changes in activity
Determining the reduction potentials of an active site bound to substrate
The effect of slow intramolecular electron transfer
Slow interfacial electron transfer
Slow substrate binding
Slow, redox-driven (in)activation
Exercises
Appendices
Notations and abbreviations
Derivation of Eq. (9.9)
References
10 Structural biology techniques: X-ray crystallography, cryo-electron microscopy, and small-angle X-ray scattering
Questions and purposes
Preamble
X-ray crystallography
Protein crystallization
Protein production and sample preparation
Protein quality assessment
Protein concentration
Crystallization techniques and initial screens
Analysis of crystallization trials
Salt or protein crystals?
Crystal optimization and seeding
Cocrystallization and soaking
Membrane proteins
Harvesting and mounting of crystals
Data collection
Phase determination
Molecular replacement
Isomorphous replacement
Anomalous scattering
Direct methods
Heavy-atom derivatization
Model building and refinement
Structure analysis and model quality
Content of crystallographic models
Validation
X-ray free electron lasers
Cryo-electron microscopy
Small-angle X-ray scattering
General conclusion
References
11 Genetic and molecular biological approaches for the study of metals in biology
Introduction and aims
Basic genetics and molecular genetics: origins and definitions
The origins, evolution, and speciation
Grouping the species: classification, taxonomy, phylogeny
The fundamental molecular biological information molecules: deoxyribonucleic acid and ribonucleic acid
The central dogma
The genetic code
What is a gene?
How big are genes and genomes?
Replicons
Gene organization
Insertion elements, transposons, and repetitive deoxyribonucleic acid
How deoxyribonucleic acid moves and can be moved around between organisms: transformation, transduction, conjugation
Homologous recombination
Promoters, transcription initiation, and transcriptional regulation
Translation initiation
Setting up: regulations, equipment, methods, and resources
Regulation and approvals
Approaches and systems
Model systems
Molecular biology tools and methods
Preparation of deoxyribonucleic acid
Agarose gel electrophoresis
Pulse-field/orthoganol electrophoresis
Blotting techniques
Molecular cloning/recombinant deoxyribonucleic acid technology
The polymerase chain reaction
Deoxyribonucleic acid sequencing
Genetic and molecular genetic methods
Cloning vectors and hosts
Gene libraries
Libraries intended for genome deoxyribonucleic acid sequencing
Cosmid libraries
Mobilizable and broad-host range vectors and cosmids
Bacterial artificial chromosomes
Yeast artificial chromosomes
Deoxyribonucleic acid copy (cDNA) libraries
Protein overexpression and purification
The T7 ribonucleic acid polymerase-T7 promoter system in Escherichia coli
The Pichia pastoris system
Tags for protein purification, correct folding, improved stability
Mutagenesis
Mutants: general considerations
Chemical and physical mutagenesis
Transposable elements and their use in mutagenesis
Site-directed mutagenesis
Site-directed point mutants
CRISPR/CAS9 mutagenesis (“gene-editing/engineering”)
Bioinformatics
General bioinformatics websites
Sequence searching sites
Multiple sequence alignment
Comparative gene organization
Identification of potential domains in proteins
Genome sites
Cross-relational databases for genomes and metabolic and other pathways
Molecular phylogenies and tree drawing programs
Visualization of molecular structures
The OMICS revolution
Genomics
Transcriptomics
Proteomics
Structural genomics
Omniomics
Metabolomics
Economics
Illustrative examples in the genetics and molecular biology of N2-fixation
References
Further reading
Index
Back Cover

Citation preview

PRACTICAL APPROACHES TO BIOLOGICAL INORGANIC CHEMISTRY

PRACTICAL APPROACHES TO BIOLOGICAL INORGANIC CHEMISTRY SECOND EDITION Edited by

Robert R. Crichton Catholic University of Louvain, Louvain-la-Neuve, Belgium

Ricardo O. Louro Instituto de Tecnologia Quı´mica e Biolo´gica Anto´nio Xavier da Universidade Nova de Lisboa, Oeiras, Portugal

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 2020 Elsevier B.V. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-444-64225-7 For Information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Susan Dennis Acquisition Editor: Emily McCloskey Editorial Project Manager: Kelsey Connors Production Project Manager: Prem Kumar Kaliamoorthi Cover Designer: Christian Bilbow Typeset by MPS Limited, Chennai, India

List of Contributors W.R. Hagen Department of Biotechnology, Delft University of Technology, Delft, The Netherlands

Margarida Archer Instituto de Tecnologia Quı´mica e Biolo´gica Anto´nio Xavier (ITQB NOVA), Universidade Nova de Lisboa, Oeiras, Portugal

Irina A. Ku¨hne School of Chemistry, University College Dublin, Dublin, Ireland

Eckhard Bill Max-Planck Institute for Chemical Energy Conversion, Mu¨lheim an der Ruhr, Germany

Christophe Le´ger CNRS, Aix Marseille University, BIP, Marseille, France

Jose´ A. Brito Instituto de Tecnologia Quı´mica e Biolo´gica Anto´nio Xavier (ITQB NOVA), Universidade Nova de Lisboa, Oeiras, Portugal

Ricardo O. Louro Instituto de Tecnologia Quı´mica e Biolo´gica Anto´nio Xavier da Universidade Nova de Lisboa, Oeiras, Portugal

Wesley R. Browne Molecular Inorganic Chemistry, Stratingh Institute for Chemistry, Faculty of Science and Engineering, University of Groningen, Groningen, The Netherlands

Wolfram Meyer-Klaucke Deutsches Elektronen Synchrotron DESY, Hamburg, Germany Grace G. Morgan School of Chemistry, University College Dublin, Dublin, Ireland Robert L. Robson School of Biological Sciences, University of Reading, Berkshire, United Kingdom

Robert R. Crichton Catholic University of Louvain, Louvain-la-Neuve, Belgium Martin C. Feiters Department of Synthetic Organic Chemistry, Institute for Molecules and Materials, Faculty of Science, Radboud University, AJ Nijmegen, The Netherlands

Ineˆs B. Trindade Instituto de Tecnologia Quı´mica e Biolo´gica Anto´nio Xavier da Universidade Nova de Lisboa, Oeiras, Portugal

Vincent Fourmond CNRS, Aix Marseille University, BIP, Marseille, France

Matija Zlatar Department of Chemistry, Institute of Chemistry, Technology and Metallurgy, University of Belgrade, Belgrade, Republic of Serbia

Maja Gruden Faculty of Chemistry, University of Belgrade, Belgrade, Republic of Serbia

ix

C H A P T E R

1

An overview of the role of metals in biology Robert R. Crichton Catholic University of Louvain, Louvain-la-Neuve, Belgium O U T L I N E Introduction

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Metals in diagnosis and therapeutics

13

Essential metal ions and their functions

2

References

15

Further reading

16

Toxic metals

10

Introduction Metals play many different roles in the biological world, whether by their participation in essential biological processes, as toxic constituents of our environment, or as indispensable diagnostic and therapeutic agents in human medicine. Only a limited number of metal ions are essential for most living organisms (Fig. 1.1), and this short introduction begins by illustrating the biological importance of metals, not only in vital processes such as intermediary metabolism, electron transfer, respiration, and photosynthesis, but also in neurotransmission, cell signaling, apoptosis, and fertilization. While many essential metals can be toxic, particularly when they are in excess, in our modern environment there are a number of nonessential metals, such as cadmium, lead, mercury, and aluminum, which are themselves highly toxic. Finally, metals have assumed an extraordinary number of roles in medicine, not only therapeutically as drugs, but also as noninvasive contrast agents and radiopharmaceuticals. More detailed accounts of these aspects of metal ions are presented in the companion volume to this second edition (Crichton, 2018).

Practical Approaches to Biological Inorganic Chemistry DOI: https://doi.org/10.1016/B978-0-444-64225-7.00001-8

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© 2020 Elsevier B.V. All rights reserved.

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1. An overview of the role of metals in biology

FIGURE 1.1 A biological periodic table of the elements indicating the essential elements. The essential elements for most forms of life are shown in black with the exception of chromium (Cr), which is shown with an upward diagonal pattern, and essential elements that are more restricted for some forms of life shown in gray. Source: Reproduced from Maret, W., 2016. The metals in the biological periodic system of the elements: concepts and conjectures. Int. J. Mol. Sci. 17, pii:E66. doi:10.3390/ijms17010066. This is an open access article distributed under the Creative Commons Attribution License (CC BY) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Essential metal ions and their functions Most living organisms require some 25 elements (Maret, 2016; Chellan and Sadler, 2015) including between 10 and 14 metal ions (Fig. 1.1). In the case of Homo sapiens, there are 10 essential metal ions (sodium, potassium, calcium, magnesium, manganese, iron, cobalt, copper, zinc, and molybdenum). Of these, the first four are considered as “bulk elements” (Na1, K1, Ca21, and Mg21), representing 112 g, 160 g, 1.1 kg, and 25 g, respectively, in an “average” person of body weight 80 kg (The data on the abundance of elements in the 80 kg human body are those given in WebElements: http://www.webelements.com/.). Together, they constitute some 99% of the metal ion content of the human body. The others, manganese, iron, cobalt, copper, zinc, and molybdenum, designated “trace elements,” are present in much lower amounts than the bulk elements (respectively, 16 mg, 4.8 g, 1.6 mg, 80 mg, 2.6 g, and 8 mg in an 80-kg person). The essential alkali metal ions Na1 and K1 only weakly bind organic ligands, rendering them extremely mobile, as with H1 and Cl2. This enables them to generate ionic gradients across biological membranes. The distribution of Na1 and K1 in mammals is quite different; Na1, together with Cl2, is the major electrolyte in the extracellular fluid, whereas K1 is retained within the cells. The concentration of Na1 in the plasma is maintained within narrow limits at about 145 mmol/L, and its intracellular concentration is only about 12 mmol/L, whereas the intracellular concentration of K1 is 150 mmol/L, and typically

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Essential metal ions and their functions

3

only 4
5 mmol/L in the extracellular fluids. This concentration differential, maintained by the (Na1
K1)-ATPase of the plasma membrane, ensures a number of major biological processes, such as cellular osmotic balance, signal transduction, and neurotransmission. (Na1
K1)-ATPase transports three Na1 ions to the outside of the cell and two K1 ions to the inside (Figs. 1.2 and 1.3), contributing to the action potential involved in transmission of nerve impulses along neuronal axons. Action potentials can be generated by presynaptic neurons at the rate of about 250 per second, accounting for between one-half and two-thirds of their total ATP consumption. The repetitive G-rich sequences found in the telomeres at the ends of eukaryotic chromosomes are stabilized by K1 and Na1 ions. The retention of Na1 (hypernatremia) when Na1 intake exceeds renal clearance is one of the most common electrolyte disorders in clinical medicine. Hyperkalemia has become more common in cardiovascular practice due to the growing population of patients with chronic kidney disease and the broad application of drugs that modulate renal elimination of potassium by reducing the production of angiotensin II. The alkaline earth metal ions, Mg21 and Ca21, have greater binding strengths to organic ligands than Na1 and K1, and therefore are less mobile. Both play important structural and catalytic roles, with 99% of the body’s Ca21 found in bone and teeth. Although Mg21 is the least abundant of the “bulk elements,” the intracellular concentration of free Mg21 is around 0.5 mM, making it the most abundant cation, and less than 0.5% of total body Mg21 is in the plasma. Half of cytosolic Mg21 is bound to ATP and most of the rest, along with K1, is bound to ribosomes. Unlike the other three bulk cations, Mg21 has a much slower water exchange rate, allowing it to play a structural role, for example, participating in ATP binding in many enzymes involved in phosphoryl transfer reactions—6 of the 10 reactions of glycolysis are phosphoryl transfers. Ca21 serves as a messenger in virtually all of the important functions of cells. Why Ca21 has ended up in this position is probably due to its unique coordination chemistry, which enables it to bind to sites of irregular geometry even in the presence of large excesses of other cations such as Mg21 (Carafoli and Krebs, 2016). While the total Ca21 concentration inside cells is micromolar, in the cytosol the concentration of free Ca21 is about 10,000 times lower. This nanometer concentration is achieved by ligation of Ca21 by two broad classes of specific proteins. (1) Those which buffer Ca21 in the nanometer range, and in some cases, also process its information, by increasing, or less frequently decreasing, their biological activity upon Ca21 binding by a change in conformation, illustrated for calmodulin in Fig. 1.4—“Ca21 is not an active site metal, it is the allosteric metal par excellence” (Carafoli and Krebs, 2016). (2) Intrinsic membrane proteins which transport Ca21 in or out of cells, or between the cytosol and the lumen of cellular organelles. Apoptosis (programmed cell death) plays a major role in the maintenance of tissue homeostasis. Ca21, in addition to its role in the regulation of cellular processes, may act as a proapoptotic agent, and both intracellular Ca21 depletion or overload may trigger apoptosis (Brini et al., 2013). Hypercalcemia is a common metabolic perturbation and the increase in over-the-counter purchase of Ca21 and vitamin D supplements, notably to combat osteoporosis in the aging population, is a contributory factor. Of the six essential trace metal ions, Zn has ligand-binding constants intermediate between those of Mg21 and Ca21 and the other five. Manganese, iron, cobalt, copper, and molybdenum all have much stronger binding to organic ligands and are therefore only

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1. An overview of the role of metals in biology

FIGURE 1.2 Architecture of Na1,K1-ATPase from shark rectal gland with bound MgF422 and K1, a stable analog of the E2  Pi  2K1 state. A ribbon diagram of NKA with ouabain (shown in space fill) bound at low affinity (PDB ID: 3A3Y). Color changes gradually from the Nterminal (blue) to the Cterminal (red). ATP is taken from the E2(TG)  ATP crystal structure of Ca21-ATPase (SERCA1a) (PDB ID: 3AR4) and docked in the corresponding position. Bound K1 ions are marked (I, II, and C) and circled. Inset shows a simplified diagram of the post-Albers scheme. CLR, cholesterol; OBN, ouabain. Source: From Toyoshima et al. (2011). Copyright 2011. With permission from Elsevier.

poorly mobile. In addition, they have access to at least two oxidation states, and therefore can participate in electron transfer and redox catalysis, whereas zinc has access only to the Zn21 state. Manganese can occur in biological systems in three oxidation states, Mn(II), Mn(III), and Mn(IV). In humans, manganese is essential for development, metabolism, and the antioxidant system through its involvement in a number of enzymes, including arginase,

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FIGURE 1.3 Crystal structure of Na1, K1-ATPase in the transition state analog E1BP  ADP  3Na1. (A and B) Ribbon diagrams viewed in two orthogonal directions. Color changes gradually between the N terminus (blue) and C terminus (red) for the α- and β-subunits. Purple spheres show bound Na1 ions [three (I
III) in the transmembrane region and one (C) in the cytoplasmic region]. Sugars attached to the β-subunit are shown as ball and stick. OLA, oligomycin A. Source: From Kanai, R., Ogawa, H., Vilsen, B., Cornelius, F., Toyoshima, C., 2013. Crystal structure of a Na1-bound Na1, K1-ATPase preceding the E1P state. Nature 502, 201
206. Copyright 2013. With permission from Elsevier.

the enzyme responsible for urea production, mitochondrial superoxide dismutase, and glutamine synthetase, which plays an important role in the brain. Nevertheless, excessive exposure or intake may lead to a condition known as manganism, a neurodegenerative

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1. An overview of the role of metals in biology

FIGURE 1.4 Ribbon representation showing how target binding induces changes in the quaternary structure of calmodulin. The conformation of the two domains of calmodulin is unaffected by target binding, but the orientation of the domains with respect to each other changes drastically, bringing the two previously independent domains into contact. Calcium-loaded calmodulin (PDB code 1CLL) is shown at the top and calcium-loaded calmodulin complexed with a peptide derived from smooth-muscle myosin light-chain kinase (PDB code 1CDL) is shown at the bottom. The N-terminal domain of calmodulin is medium blue, the C-terminal domain is dark blue, and the linker loop between the domains is light blue. The peptide is red and the calcium ions are represented as yellow balls. The tint indicates the Connolly surfaces of the molecules. Source: From Johnson, C.N., Damo, S.M., Chazin, W.J., 2014. EF-hand calcium-binding proteins. In: Encyclopedia of Life Sciences. John Wiley & Sons Ltd., https://doi.org/10.1002/9780470015902. a0003056.pub3. Copyright 2014. With permission from John Wiley and Sons.

disorder that causes dopaminergic neuronal death and parkinsonian-like symptoms (Avila et al., 2013). Clearly the most important role of manganese in biology is its involvement in the oxygen evolving complex of photosystem II in cyanobacteria, algae, and green plants, which oxidizes water into dioxygen, protons, and electrons (Eq. 1.1). 2 H2 O-4H1 1 4e2 1 O2

ð1:1Þ

The determination of the structure of the Mn4CaO5 cluster (Fig. 1.5) at the center of PSII (Suga et al., 2015) has provoked an intensive flurry of biomimetic chemistry, with the aim of generating “green energy” using our unlimited access to solar power (Najafpour et al., 2015). Iron is the most abundant of the transition metal ions in humans, with the bulk present in the oxygen-binding heme proteins, hemoglobin and myoglobin. These both contain iron within the protoporphyrin IX nucleus, requiring a number of genes for biosynthesis of the porphyrin, insertion of iron, and subsequent heme transport (Crichton, 2016; Andreini et al., 2009). The remaining much smaller proportion of body iron is present in other ironcontaining proteins (heme proteins, Fe
S proteins, and nonheme, non-Fe
S proteins) with a wide variety of functions, encoded by the human genome (Crichton, 2016). A recent bioinformatics approach indicates that about 2% of human genes encode an iron protein (48% heme-binding proteins, 17% Fe
S proteins, and 35% which bind individual iron ions).

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FIGURE 1.5 (A) Schematic representation of the cofactor arrangement in the core of the reaction center (the view is along the membrane plane). Organic cofactors (for the sake of simplicity, the heme group of cytochrome b559 is omitted) are colored green (Chl), yellow (Pheo), magenta (plastoquinones QA and QB), and red (carotenoids). Ca (yellow), Fe (blue), and Mn (red) are shown as spheres; the figure was generated using PyMOL (http://www. pymol.org). The coordinating protein subunits D1 and D2 are indicated by dotted lines. (B) Structural arrangement of the Mn4CaOx cluster and Mn
O, Ca
O, Mn
water, and Ca
water distances in the oxygen evolving ˚ ) Mn1, Mn2, Mn3, and Mn4 denote the different Mn ions of the OEC. Source: (A) Reprinted complex (OEC) (in A with permission from Reger, G., 2012. Mechanism of light induced water splitting in photosystem II of oxygen evolving photosynthetic organisms. Biochim. Biophys. Acta 1817, 1164
1176. Copyright 2012 Elsevier. (B) Reprinted with permission from Suga, M., Akita, F., Hirata, K., Ueno, G., Murakami, H., et al., 2015. Native structure of photosystem II at 1.95 A˚ resolution viewed by femtosecond X-ray pulses. Nature 517, 99
103. Copyright 2015. Nature Publications.

More than half of the human iron proteins have a catalytic function, and the authors estimate that 6.5% of all human enzymes are iron-dependent (Andreini et al., 2018). Fe is a constituent of a large number of proteins involved in electron transfer chains in humans, notably the respiratory chain in the inner membrane of the mitochondria, involving cytochromes, Fe
S proteins, and quinines, channeling electrons to the terminal component, the Cu
Fe-dependent cytochrome c oxidase (COX) which mediates the reduction of O2 (Eq. 1.2). 4H1 1 4e2 1 O2 -2 H2 O

ð1:2Þ

Mammalian COX is composed of 13 subunits, three catalytic subunits I
III encoded by mitochondrial DNA, and 10 nuclear-coded subunits encoded by nuclear DNA (Fig. 1.6). Electrons from cytochrome c are transferred to the dimetallic CuA site, which rapidly ˚ away. Heme a then transfers electrons to the active site reduces the heme a, some 19 A heme a3 and CuB, where O2 binds. Copper is the third most abundant essential transition metal ion in the human body, involved, for example, in respiration, angiogenesis, and neuromodulation, yet Cu proteins represent less than 1% of the total proteome in both eukaryotes and prokaryotes

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1. An overview of the role of metals in biology

FIGURE 1.6 Crystal structure of dimeric cytochrome c oxidase from bovine heart (Tsukihara et al., 1996). The nuclear-coded subunits are in color, the mitochondrial-coded subunits I, II, and III are in yellow. Indicated schematically on the left monomer are the electron transport pathways from cytochrome c (Cyt. c) to oxygen accompanied by uptake of protons from the matrix for water formation and pumped protons (nH1). On the right monomer binding sites for 3,5-diiodothyronine (T2) and ATP or ADP are indicated. Source: From Kadenbach, B., Hu¨ttemann, M., 2015. The subunit composition and function of mammalian cytochrome c oxidase. Mitochondrion 24, 64
76. Copyright 2015. With permission from Elsevier.

(Andreini et al., 2009). Copper sites in proteins can be classified as belonging to one of three classes. Type 1 (blue Cu proteins) function in single electron transfer, type II are catalytic sites which bind directly to substrates, while type III sites are dinuclear and are involved in the activation and transport of oxygen. The copper chaperones are a specific class of proteins which ensure the safe and specific delivery of potentially harmful copper ions to a variety of essential copper proteins (Palumaa, 2013). Cu is also involved as the catalytic component in detoxification (Cu/Zn superoxide dismutase). Both iron and copper are characterized by genetic disorders associated with the accumulation of these metals in particular tissues, with toxic consequences. Wilson’s disease is a chronic disease of the brain and liver due to a disturbance of copper metabolism,

Practical Approaches to Biological Inorganic Chemistry

Essential metal ions and their functions

9

FIGURE 1.7 Structure of vitamin B12 shown as the Co31

corrin complex, where R 5 50 deoxyadenosyl, Me, OH2, or CN2 (cyanocobalamin). Source: With permission from Wikipedia (released into the public domain by its author Ymwang42 at the Wikipedia project).

accompanied by progressive neurological dysfunction, with progressive accumulation of copper in the brain, liver, kidneys, and the cornea of the eye. Iron overload can result from genetic defects in iron absorption from the gastrointestinal tract (hereditary hemochromatosis), but can also result from genetic dysfunction of erythropoiesis, as in thalassemia, necessitating regular blood transfusions (secondary hemochromatosis). Although only 1.6 mg is present in the human body, cobalt remains an essential trace element, and is required in the human diet in the form of cobalamin (vitamin B12), a product of microbial biosynthesis, where the Co is tightly bound in a corrin ring (Fig. 1.7). Vitamin B12 uptake from the gut requires a specific protein, intrinsic factor, which is secreted by the gastric mucosa and is essential for efficient absorption of the vitamin. Lack of intrinsic factor causes pernicious anemia, and vitamin B12 was identified in 1925 as the antipernicious anemia factor. This is due to B12 being an essential cofactor for a number of B12-dependent isomerases and methyltransferases (Banerjee et al., 2009) involved in DNA synthesis, amino acid and fatty acid metabolism, in the synthesis of myelin by oligodendrocytes wrapped around the axons of motor neurons, and in the maturation of developing red blood cells. Cobalt is acutely toxic in large doses and this was dramatically observed in the 1960s among heavy beer drinkers (15
30 pints/day), when Co21 salts were added as foam stabilizers, resulting in severe and often lethal cardiomyopathy (Kesteloot et al., 1968). Bioinformatics analysis of the human genome indicates that one protein in 10 (about 3000 in total) is a zinc metalloprotein (Andreini et al., 2006, 2009). Zn21 is represented in all six classes of enzymes (as defined by the International Union of Biochemistry), where it can play both a structural as well as a catalytic role, often functioning like Mg21 as a Lewis acid. It can also fulfill a very important regulatory function in the structural motifs

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FIGURE 1.8 The structure of the pyranopterin cofactor common to all of these enzymes is given at the top. Active site structures for two families of mononuclear molybdenum enzymes (xanthine oxidase and sulfite oxidase). Source: Reprinted with permission from Hille, R., Hall, J., Basu, P., 2014. The mononuclear molybdenum enzymes. Chem. Rev. 114, 3963
4038. Copyright 2014. American Chemical Society.

known as “zinc fingers” involved in the regulation of transcription and translation by its binding to DNA and RNA. Zn21 is the second most abundant of the trace metals (after iron), and is extensively involved in brain function, with mM concentrations in synaptic vesicles in which Zn is stored and from which it is released in a controlled manner (Bitanihirwe and Cunningham, 2009). In the course of meiotic maturation, oocytes take up over 2 3 109 zinc atoms, and when a sperm cell enters and fertilizes the oocyte, this triggers the coordinated release of zinc into the extracellular space in a prominent “zinc spark,” detectable by fluorescence (Duncan et al., 2016). This loss of zinc is necessary to mediate the egg-to-embryo transition. Although Mo is relatively rare in the Earth’s crust, it is the most abundant transition metal in seawater, and since the oceans are as close as we get to the primordial soup in which life first evolved, it is no surprise that Mo has been widely used in biology. While Mo is well known as an important component of the FeMo cofactor in nitrogenase, the key enzyme of nitrogen-fixing organisms, there are a number of Mo-dependent enzymes in humans, which all contain Mo in the form of a molybdenum pyranopteridindithiolate cofactor (Fig. 1.8). These include xanthine oxidase, involved in the catabolism of purine bases, sulfite oxidase involved in sulfur metabolism, and aldehyde oxidase, involved in the metabolism of many drugs (Hille et al., 2014). Ni, V, and Cr appear to be beneficial, and have been proposed to be essential for man. Although the human body contains around 8 mg of nickel, no Ni-dependent enzymes are known, however, it may be that Ni is essential for microorganisms that colonize the human gut (Zambelli et al., 2016).

Toxic metals Essential metals can be toxic if excessive concentrations of the metal ions accumulate, often in specific tissues or organs—a number of examples of which are given above. However, as a consequence of environmental exposure, a number of nonessential metal

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Toxic metals

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ions can accumulate within the body with toxic consequences. In what follows, we briefly describe some of the more common toxic metals. While little credence is given today to the theory that Roman civilization collapsed as a result of chronic lead poisoning (saturnism), a recent study has shown that “tap water” from ancient Rome contained 100 times more Pb than local spring waters (Delile et al., 2014). We are now acutely aware that saturnism is a major environmental concern, and Pb exposure remains a widespread problem, particularly in the developing world. Pb toxicity affects several organ systems including the nervous, hematopoietic, renal, endocrine, and skeletal systems. It also causes behavioral and cognitive deficits during brain development in infants and young children. Pb appears to target proteins that naturally bind calcium and zinc, and examples include synaptotagmin, which acts as a calcium sensor in neurotransmission, and δ-aminolevulinate synthase (ALAD), the second enzyme in the heme biosynthetic pathway. Human ALAD is activated by Zn21 with a Km of 1.6 pM and inhibited by Pb21 with a Ki of 0.07 pM. Pb21 and Zn21 appear to compete for a single metalbinding site (Simons, 1995). The toxicity of cadmium manifested itself among the inhabitants of the Jinzu river basin in Japan in the 1950s due to environmental Cd pollution originating in effluent from a zinc mine located in the upper reaches of the river. In the Cd-polluted areas, 50%
70% of the Cd ingested orally was derived from rice. It remains the most severe example of chronic Cd poisoning caused by prolonged oral Cd ingestion. Cd21 is a soft Lewis acid with a preference for easily oxidized soft ligands, particularly sulfur. It can displace Zn21 from proteins where the Zn coordination environment is sulfur dominated, and given the similarity of its ionic radius with that of Ca21 it can exchange with Ca21 in calcium-binding proteins. Cadmium occurs in the environment naturally and as a pollutant emanating from industrial and agricultural sources. Exposure to cadmium in the nonsmoking population (there is a high concentration of Cd in cigarettes) occurs primarily through food, and chronic exposure results in respiratory disease, emphysema, renal failure, bone disorders, and immunosuppression. The brutal reality of mercury toxicity was highlighted in 1956 by an environmental disaster which struck the population of Minamata, Japan, and its surroundings. Methylmercury was released in the industrial wastewater from a chemical factory and bioaccumulated in aquatic food chains, reaching its highest concentrations in shellfish and fish in Minamata Bay and the Shiranui Sea, which when eaten by the local population resulted in mercury poisoning. Of the 2265 victims officially recognized, 1784 died. The symptoms include ataxia, numbness in the hands and feet, general muscle weakness, narrowing of the field of vision, and damage to hearing and speech. The brain is the principal target tissue of MeHg and its major toxic effects are on the central nervous system, accumulating particularly in astrocytes. The biochemical target of Hg is the selenocysteine residues in selenoenzymes, as Hg has an affinity for Se B1 million times greater than its affinity for sulfur (Ralston and Raymond, 2010, 2018). The selenoenzymes glutathione peroxidase, thioredoxin reductase, and thioredoxin glutathione reductase are required to prevent and reverse oxidative damage to the brain and neuroendocrine system, and they undergo irreversible inhibition by methylmercury (MeHg). Selenoenzyme inhibition appears likely to cause most if not all of the pathological effects of mercury toxicity, as outlined in Fig. 1.9.

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1. An overview of the role of metals in biology

FIGURE 1.9 Se sequestration mechanism of Hg toxicity. A simplified portrayal of the normal cycle of selenoprotein synthesis is depicted on the left. Disruption of this cycle by exposure to toxic quantities of Hg (MeHg) is depicted on the right. Selenide freed during selenoprotein breakdown becomes bound to Hg, forming HgSe that accumulates in cellular lysosomes. If Hg is present in stoichiometric excess, formation of insoluble Hg selenides abolishes the bioavailability of Se for protein synthesis (indicated by gray text) and results in loss of normal physiological functions that require selenoenzyme activities. Source: From Ralston, N.V., Raymond, L.J., 2018. Mercury’s neurotoxicity is characterized by its disruption of selenium biochemistry. Biochim. Biophys. Acta Gen. Subj. 1862, 2405
2416.

Despite comprising 8% of the Earth’s crust, the most abundant metal and the third most abundant element after oxygen and silicon, aluminum is not used in biology; however, several factors have increased its access to the biosphere. First, an increase in anthropogenic acidification of soils, due to acid rain generated by emissions of sulfur dioxide and nitrogen oxides in the atmosphere, has resulted in elevated concentrations of Al31 in ground waters. On account of its lightness and corrosion resistance, aluminum is widely used for industrial purposes, from the aerospace industry to construction, from food packaging to pharmaceuticals. Aluminum salts are extensively utilized as a flocculent in water treatment. This enhanced bioavailability has resulted in the accumulation of this metal in living organisms including humans, particularly in the skeletal system, the liver, and the brain. The toxicity of Al31 is associated with anemia, osteomalacia, hepatic disorders, and certain neurological disorders. The molecular targets of Al toxicity involve disruption to the homeostasis of essential metal ions, notably Fe, Ca, and Mg. Al can replace Ca in bone and interfere with Ca-based signaling events, and can compete with Mg21 binding to phosphate groups on cell membranes, ATP, and DNA. However, it is likely that the main targets of Al toxicity are Fe-dependent biological processes. Al31 has coordination geometry similar to Fe31, which should enable Al31 to subvert the plasma iron transport pathway. In the cytosol Al31 is unlikely to be incorporated into ferritin, which requires redox cycling between Fe21 and Fe31, and it seems likely that most aluminum accumulates in mitochondria, where it can interfere with Ca21 homeostasis.

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Metals in diagnosis and therapeutics

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Metals in diagnosis and therapeutics In addition to the essential metal ions, a large number of other metals, including some that are toxic, are routinely used in clinical medicine both as diagnostic and therapeutic agents, including gadolinium and technetium complexes used in their millions every year for diagnosis. The most widely used drugs for cancer chemotherapy are platinum complexes (Chellan and Sadler, 2015). We begin by reviewing some of the growing number of metallo-organic complexes used for the noninvasive imaging techniques which have revolutionized modern medicine, allowing better diagnosis and greater patient comfort. We begin with one of the oldest noninvasive diagnostic applications, the use of the relatively insoluble barium sulfate, BaSO4, as a radiopaque contrast agent for X-ray imaging of the gastrointestinal tract. In contrast, the metastable isotope of technetium, 99mTc, was first created in 1937 by cyclotron bombardment of molybdenum, It is a γ-emitter with a half-life of 6 hours, and is used in tens of millions of nuclear medicine imaging procedures every year. The 3Dscanning technique, single-photon emission computed tomography, uses a γ-camera which usually performs a full 360 degree rotation around the patient to obtain an optimal reconstruction. 99mTc is predominantly used for bone and brain scans. For the former, pertechnetate ions are used directly, as they are taken up by osteoblasts, while for brain scans 99m Tc is chelated to hexamethylpropyleneamine oxime, which localizes in the brain according to regional blood flow. 99mTc scintigraphy can also be combined with computed tomography for more refined resolution. Millions of doses of gadolinium are administered every year as MRI contrast agents. Gd (III) has high paramagnetism (seven unpaired electrons) and a favorable slow electronic relaxation time, which makes it effective in relaxing water protons which can generate contrast in MR images. Gd as the free ion is highly toxic even at low doses (10
20 μmol/ kg), and for this reason it is necessary to use ligands that form stable complexes with the lanthanide ion. The high affinity of Gd toward polyaminocarboxylic acids has been exploited to form very stable complexes (up to log KML . 20). The first contrast agent to be approved for clinical use was [(Gd-diethylene triamine penta-acetic acid (DTPA))(H2O)]22 (Fig. 1.10) in 1988, which was administered to more than 20 million patients in the first 10 years of clinical experimentation. Turning next to metals as drugs, pride of place must go to platinum. Currently around 50% of all cancer treatments involve Pt complexes, essentially the original drug cisplatin cis-[PtCl2(NH3)2], discovered in the 1960s and approved for clinical use in the late 1970s, and subsequently joined by its analogs carboplatin cis-[Pt(1,1-dicarboxycyclobutane) (NH3)2] and oxaliplatin [Pt(1R,2R-1,2-diaminocyclohexane)(oxalate)] (Fig. 1.11). Ruthenium complexes have also shown promise as anticancer drugs (Bergamo et al., 2012). Lithium is the simplest therapeutic agent for the treatment of depression and has been used for more than 100 years. Lithium carbonate can reverse the symptoms of patients with bipolar disorder (manic-depression), a chronic disorder which affects between 1% and 2% of the population. This disease is characterized by episodic periods of elevated or depressed mood, severely reduces the patient’s quality of life, and dramatically increases

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1. An overview of the role of metals in biology

FIGURE 1.10 The structure of the MRI contrast agent Gd-DTPA (diethylene triamine penta-acetic acid). Source: From Wikipedia www.LookForDiagnosis.

FIGURE 1.11 Platinum anticancer drugs approved for clinical use: (A) cisplatin; (B) carboplatin; and (C) oxaliplatin. Source: From Wikipedia.

FIGURE 1.12 The orally active antirheumatoid arthritis drug Auranofin. Source: From Wikipedia, by Ben Mills—own work, public domain, https://commons. wikimedia.org/w/index.php?curid 5 5934319.

their likelihood of committing suicide. Today, it is the standard treatment, often combined with other drugs, for bipolar disorder and is prescribed to over 50% of bipolar disorder patients. The molecular basis of mood disorder diseases and their relationship to the effects of lithium remain unknown. Silver is well known for its potent antibacterial properties, and a combination of silver nitrate and a sulfonamide antibiotic as a topical antibacterial agent for burn management is still in use today, as are silver-containing wound dressings in lieu of antibiotics. Gold has also been used as a drug since antiquity (Raubenheimer and Schmidbauer, 2014), with the Au(I) complex auranofin (Fig. 1.12) being developed for the treatment of rheumatoid arthritis as a substitute for the injectable gold compounds aurothiomalate and aurothioglucose. Despite efficacy in the treatment of both rheumatoid arthritis and

Practical Approaches to Biological Inorganic Chemistry

References

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psoriasis, currently auranofin is seldom used as a treatment for patients with rheumatoid arthritis as more novel antirheumatic medications have become available. However, potential new applications of auranofin, notably for cancer therapy and treating infections including HIV, based on its dual inhibition of inflammatory pathways and thiol redox enzymes like mitochondrial thioredoxin reductase, have emerged (Madeira et al., 2012). Gold nanoparticles have great potential for controlled drug delivery, cancer treatment, biomedical imaging and diagnosis, and photothermal therapy (Elahi et al., 2018). While there are many more examples of metals used as diagnostics or drugs, it seems appropriate to end this section by briefly mentioning the exciting development of theranostic agents (Terreno et al., 2012). These integrate diagnosis and therapy, allowing imaging-guided drug delivery, imaging of drug release, monitoring therapy by imaging, using theranostic agents for radiation-based therapies, and the use of imaging probes in imaging-guided surgery.

References Andreini, C., Banci, L., Bertini, I., Rosato, A., 2006. Counting the zinc-proteins encoded in the human genome. J. Proteome Res. 5, 196
201. Andreini, C., Bertini, I., Rosato, A., 2009. Metalloproteomes: a bioinformatic approach. Acc. Chem. Res. 42, 1471
1479. Andreini, C., Putignano, V., Rosato, A., Banci, L., 2018. The human iron-proteome. Metallomics 10, 1223
1231. Avila, D.S., Puntel, R.L., Aschner, M., 2013. Manganese in health and disease. Met. Ions Life Sci. 13, 199
227. Banerjee, R., Gherasim, C., Padovani, D., 2009. The tinker, tailor, soldier in intracellular B12 trafficking. Curr. Opin. Chem. Biol. 13, 484
491. Bergamo, A., Gaiddon, C., Schellens, J.H., Beijnen, J.H., Sava, G., 2012. Approaching tumour therapy beyond platinum drugs: status of the art and perspectives of ruthenium drug candidates. J. Inorg. Biochem. 106, 90
99. Bitanihirwe, B.K., Cunningham, M.G., 2009. Zinc: the brain’s dark horse. Synapse 63, 1029
1049. Brini, M., Calı`, T., Ottolini, D., Carafoli, E., 2013. Intracellular calcium homeostasis and signaling. In: Banci, L. (Ed.), Metallomics and the Cell, Metal Ions in Life Sciences, 12. Springer, Dordrecht, pp. 119
168. Carafoli, E., Krebs, J., 2016. Why calcium? how calcium became the best communicator. J. Biol. Chem. 291, 20849
20857. Chellan, P., Sadler, P.J., 2015. The elements of life and medicines. Philos. Trans. A Math. Phys. Eng. Sci. 373, Available from: https://doi.org/10.1098/rsta.2014.0182, pii:20140182. Crichton, R.R., 2016. Iron Metabolism: From Molecular Mechanisms to Clinical Consequences, fourth ed. John Wiley and Sons, Chichester, UK, 556 pp. Crichton, R.R., 2018. Biological Inorganic Chemistry. A New Introduction to Molecular Structure and Function, third ed. Academic Press, London, 669 pp. Delile, H., Blichert-Toft, J., Goiran, J.P., Keay, S., Albare`de, F., 2014. Lead in ancient Rome’s city waters. Proc. Natl. Acad. Sci. U.S.A. 111, 6594
6599. Duncan, F.E., Que, E.L., Zhang, N., et al., 2016. The zinc spark is an inorganic signature of human egg activation. Sci. Rep. 6, 24737. Available from: https://doi.org/10.1038/srep24737. Elahi, N., Kamali, M., Baghersad, M.H., 2018. Recent biomedical applications of gold nanoparticles: a review. Talanta 184, 537
556. Hille, R., Hall, J., Basu, P., 2014. The mononuclear molybdenum enzymes. Chem. Rev. 114, 3963
4038. Johnson, C.N., Damo, S.M., Chazin, W.J., 2014. EF-hand calcium-binding proteins. Encyclopedia of Life Sciences. John Wiley & Sons, Ltd. Available from: https://doi.org/10.1002/9780470015902.a0003056.pub3. Kadenbach, B., Hu¨ttemann, M., 2015. The subunit composition and function of mammalian cytochrome c oxidase. Mitochondrion 24, 64
76. Kanai, R., Ogawa, H., Vilsen, B., Cornelius, F., Toyoshima, C., 2013. Crystal structure of a Na 1 -bound Na 1 , K 1 -ATPase preceding the E1P state. Nature 502, 201
206.

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Kesteloot, H., Roelandt, J., Willems, J., Claes, J.H., Joossens, J.V., 1968. An enquiry into the role of cobalt in the heart disease of chronic beer drinkers. Circulation 37, 854
864. Maret, W., 2016. The metals in the biological periodic system of the elements: concepts and conjectures. Int. J. Mol. Sci. 17, Available from: https://doi.org/10.3390/ijms17010066, pii:E66. Madeira, J.M., Gibson, D.L., Kean, W.F., Klegeris, A., 2012. The biological activity of auranofin: implications for novel treatment of diseases. Inflammopharmacology 20, 297
306. ´ Najafpour, M.M., Renger, G., Hołynska, M., Moghaddam, A.N., Aro, E.M., et al., 2015. Manganese compounds as water-oxidizing catalysts: from the natural water-oxidizing complex to nanosized manganese oxide structures. Chem. Rev. 116, 2886
2936. Palumaa, P., 2013. Copper chaperones. The concept of conformational control in the metabolism of copper. FEBS Lett 587, 1902
1910. Ralston, N.V.C., Raymond, L.J., 2010. Dietary selenium’s protective effects against methylmercury toxicity. Toxicology 278, 112
123. Ralston, N.V., Raymond, L.J., 2018. Mercury’s neurotoxicity is characterized by its disruption of selenium biochemistry. Biochim. Biophys. Acta Gen. Subj. 1862, 2405
2416. Raubenheimer, H.G., Schmidbauer, H., 2014. The late start and amazing upswing in gold chemistry. J. Chem. Educ. 91, 2024
2036. Reger, G., 2012. Mechanism of light induced water splitting in photosystem II of oxygen evolving photosynthetic organisms. Biochim. Biophys. Acta 1817, 1164
1176. Simons, T.J., 1995. The affinity of human erythrocyte porphobilinogen synthase for Zn2 1 and Pb2 1 . Eur. J. Biochem. 234, 178
183. ˚ Suga, M., Akita, F., Hirata, K., Ueno, G., Murakami, H., et al., 2015. Native structure of photosystem II at 1.95 A resolution viewed by femtosecond X-ray pulses. Nature 517, 99
103. Terreno, E., Uggeri, F., Aime, S., 2012. Image guided therapy: the advent of theranostic agents. J. Control. Release 161, 328
337. Toyoshima, C., Kanai, R., Cornelius, F., 2011. First crystal structures of Na 1 , K 1 -ATPase: new light on the oldest ion pump. Structure. 19, 1732
1738. Tsukihara, T., Aoyama, H., Yamashita, E., Tomizaki, T., Yamaguchi, H., et al., 1996. The whole structure of the 13subunit oxidized cytochrome c oxidase at 2.8 A. Science 272, 1136
1144. Zambelli, B., Uversky, V.N., Ciurli, S., 2016. Nickel impact on human health: an intrinsic disorder perspective. Biochim. Biophys. Acta 1864, 1714
1731.

Further reading Miller, A., Korem, M., Almog, R., Galboiz, Y., 2005. Vitamin B12, demyelination, remyelination and repair in multiple sclerosis. J. Neurol. Sci. 233 (1
2), 93
97. Orrenius, S., Zhivotovsky, B., Nicotera, P., 2003. Regulation of cell death: the calcium-apoptosis link. Nat. Rev. Mol. Cell Biol. 4, 552
565.

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C H A P T E R

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Introduction to ligand field theory and computational chemistry Matija Zlatar1 and Maja Gruden2 1

Department of Chemistry, Institute of Chemistry, Technology and Metallurgy, University of Belgrade, Belgrade, Republic of Serbia 2Faculty of Chemistry, University of Belgrade, Belgrade, Republic of Serbia O U T L I N E

Introduction

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Some quantitative considerations

Introduction to quantum chemistry Approximations in quantum chemistry

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Electronic structure of atoms Hydrogen atom Many-electron atoms Pauli principle Two electrons in two orbitals Electronic terms

25 25 28 32 34 37

Introduction to computational chemistry The wave function
based methods Density functional theory Computational methods for excited states Computational methods for biological systems containing transition metal

Symmetry

38

Ligand field theory Some qualitative considerations Symmetry in ligand field theory

41 44 46

Practical Approaches to Biological Inorganic Chemistry DOI: https://doi.org/10.1016/B978-0-444-64225-7.00002-X

17

48 53 55 58 62 63

Concluding remarks

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Acknowledgments

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References

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© 2020 Elsevier B.V. All rights reserved.

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2. Introduction to ligand field theory and computational chemistry

Introduction Chemists, biochemists, and biologists usually do not like mathematics. However, to understand basic concepts of chemical bonding in transition metal (TM) compounds, their spectral, magnetic, and other properties, ligand field theory (LFT), computational chemistry, and quantum mechanics (QM) are absolutely necessary. This chapter starts with an introduction to quantum chemistry, followed by a description of an electronic structure of atoms, and brief symmetry considerations. This gives the basis for the LFT which is then described. In the second part of the chapter, an overview of the techniques of computational chemistry is given, with an emphasis on the practical computation of the electronic structure and properties of TM compounds. The authors have tried to reduce the number of mathematical equations to the minimum, and strongly encourage readers not to be demoralized reading these topics, because all basic concepts necessary to understand LFT, the beauty of coordination, and computational chemistry are presented.

Introduction to quantum chemistry Understanding the properties of matter at the molecular level relies on the laws of QM. This branch of physics was developed in the first half of the 20th century to explain experimental observations unaccountable by classical physics. The clarification of blackbody radiation, photoelectric effect, the Compton effect, and the line spectrum of the hydrogen atom led to the key concepts in QM: quantization of energy and momentum, wave
particle duality, and the uncertainty principle. Quantization of physical quantities implies that they can have only discrete values and are not continuous variables. Wave
particle duality indicates the relation between momentum (property of particles) and wavelength (wave property). For example, electrons which are “normally” considered as negatively charged particles have wave-like properties and, on the other hand, light waves have particle-like properties. As particles become smaller, it is less valid to consider them as hard spheres because they are more wave-like. An electron does not move along the definite path, it is a wave distributed through space. Consequently, QM dictates that the position and momentum of a particle cannot be measured at the same time. This is nothing to do with the precision or quality of the measurements, but is intrinsic to the QM description of phenomena. It is important to emphasize that QM is made to explain the observed facts even though our common sense may be puzzled with it. Our experience is built upon macroscopic, everyday physics that may not work in the world of atoms and molecules. We will not delve here into the details of QM, but will mainly focus on concepts essential for understanding coordination compounds and the basics of computational chemistry. The interested reader is referred to more specialized textbooks on the topic (see References section at the end of this chapter; Szabo and Ostlund, 1996; Atkins and Friedman, 2005; Levine, 2017; Atkins et al., 2018). A central quantity in QM is the wave function, Ψ. It is a function, often complicated, in general complex, of the coordinates of all the particles in the system under consideration, as well as of time. It contains all the information that can be determined experimentally. However, wave function itself cannot be observed. This is one of the QM quirks—a Practical Approaches to Biological Inorganic Chemistry

Introduction to quantum chemistry

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quantity that describes everything in the system is just a mathematical construction. But the square of the wave function, Ψ2 5 Ψ Ψ (Ψ is the complex conjugate of Ψ) is a real quantity, interpreted as a probability distribution of a particle. In other words, the probability of finding a particle in a differential volume element, dV is proportional to Ψ2 dV. The measure of the probability of finding an electron at a specific location is the electron density, ρ. Electron density is determined, from its definition, by Ψ, but also, the other way around, electron density determines the wave function (apart from its phase). This is the formal foundation of density functional theory (DFT) (see below). Electron density, unlike the electron’s wave function, is observable—for example, X-ray crystallography and scanning tunneling microscopy are experimental techniques that can measure the electron density. The wave functions are solutions of the Schrodinger’s equation (SE). It is a fundamental equation of QM and applies to all kinds of systems, atoms, molecules, macromolecules, materials. In its nonrelativistic, time-independent form SE reads ^ 5 EΨ HΨ

ð2:1Þ

where E is the energy and H^ is the Hamiltonian. Eq. (2.1) hides all the complexity in its elegance. We must note that the right side of the equation is not a simple multiplication, thus we cannot eliminate Ψ from both sides to simply get that H^ and E are equal. This is because H^ is not a simple coefficient multiplying Ψ, it is an operator, and this is indicated by the funny “hat” symbol on top of the symbol H. An operator represents a set of mathematical rules that “act” on the wave function. An operator can be a simple multiplication by a number, or by a function, or a differentiation. In QM, every physical quantity is described by an operator. We have operators for the momentum, position, dipole moment, etc. Energy, E, is described by the Hamiltonian, the operator of energy. Therefore although we cannot make the equivalence of H^ and E in a strict sense, SE tells us that once we act on a wave function with a set of rules described by the Hamiltonian, we will get the energy of a system (multiplying the original wave function). The Hamiltonian can be constructed for any system, and both the energy and the wave function are obtained as a solution of SE simultaneously. The SE is an eigenvalue equation, and there are two unknowns (Ψ and E), unlike typical equations we are used to. The wave function satisfying SE for a system is the eigenfunction of the Hamiltonian for such a system, and energy is its eigenvalue. SE does not have a single, specific solution. There will be a different solution for every different distribution of particles, that is, for different values of energy. Although, there will be an infinite number of solutions, energy can have only discrete values. This quantization of energy is a direct consequence of the SE. The wave function with the lowest energy is called the ground state wave function, or simply the ground state. All the others are excited states. If two (or more) wave functions have the same E, these wave functions are degenerate. Degeneracy of the solutions of SE is a consequence of the symmetry of a system. Solving SE is not the only way of getting energies, difference in energies between the states can be determined, for example, by spectroscopic measurements. Therefore there is always a one-to-one correspondence between the QM description of the system and experiment. Initially, we said that Ψ has all the information about any experimental observation. So far, we did not specify where is this information. For every physical property of the Practical Approaches to Biological Inorganic Chemistry

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^ the average value of that property is given system O, described by a matching operator O,   Ð ^ dt 5 hΨ O^ Ψi. The wave function by the expectation value of the operator: hOi 5 Ψ  OΨ does not need to beÐ an eigenfunction of the operator, but the wave function must be normalized, that is, Ψ  Ψ dt 5 hΨ jΨi 5 1. The last expression is just stating that the probability of finding a particle anywhere in the space is equal to 1, that is, a particle exists in the system. Typically, when talking about Ψ, it is considered that they are normalized. One may always normalize a wave function, simply multiplying it with a constant, normalization constant, such that the normalization condition is satisfied. Normalization is, in general, a separate problem form the SE, and it removes any ambiguity related to the magnitude of Ψ. The expressions with “ , .” are integrals written in Dirac’s notation, which is commonly used in QM. The Hamiltonian portrays our knowledge about the system. Constructing the Hamiltonian from a few fundamental formulas is straightforward. In classical physics total energy is a sum of potential and kinetic energies. Analogously the Hamiltonian, the total energy operator, is a sum of the kinetic energy operator and the potential energy ^ The potential energy operator is the same as its corresponding operator: H^ 5 T^ 1 V. classical quantity. More precisely, it is a position operator that is the same. Considering atoms and molecules, potential energy is just the Coulombic interaction between charged particles with charges qi on a distance r. Kinetic energy requires dealing with the momentum operator, which has a special form, for the momentum along x-coordinate: ¯ d=dx , where i2 5 21, and h ¯ is the constant (the Planck’s constant divided by 2π, p^ x 5 2 i h called the reduced Planck’s constant and equal to 1:054571800 3 1034 Js). This form of the momentum operator ensures that position/momentum uncertainty is satisfied. Operators corresponding to other physical observables can be constructed in a similar manner. Expressions for the kinetic energy operator and the potential energy operator in atomic and molecular problems are given in Eq. (2.2). Examples for the form of Hamiltonian expressed as the sum of kinetic and potential energy terms for H-atom, He-atom, and H2 molecule are given in Fig. 2.1. In Fig. 2.1 expressions are given in atomic units (the numerical values of electron mass, elementary charge, reduced Planck’s constant, and Coulomb’s constant are all taken to be unity), and symbol r2 (Laplacian) is used for a sum of the second derivatives along all three Cartesian coordinates. This somewhat simplifies the expressions. V^ 5

1 q1 q2 ; 4πE0 r^

p^ 2 ¯h d2 52 T^ 5 2m d2x 2m

ð2:2Þ

We see that only two fundamental interactions are necessary to describe atoms and molecules. Note that one needs to consider all possible interactions between electrons and nuclei, and among each other, as well as the kinetic energy of all electrons. Therefore the Hamiltonian quickly becomes quite complicated as the number of particles increases. Here, and in the rest of the chapter, we consider the electronic SE—from the point of view of electrons, nuclei are not moving, therefore there is no kinetic energy of nuclei in the expressions, and V^ nn is a constant for a given molecular structure. In other words, much heavier nuclei move slower than electrons, and SE can be separated into the electronic and nuclear part. For atoms and ions, the center of mass is in a good Practical Approaches to Biological Inorganic Chemistry

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FIGURE 2.1 Schematic representation of the interactions in (from left to right) hydrogen atom, helium atom, and hydrogen molecule.

approximation at the nucleus and the motions relative to the center of mass are the motions of the electron. For molecules, separation of electron and nuclear motion is the basis of the Born
Oppenheimer (BO) approximation. In the BO approximation, we solve the SE for electrons only. Still, the resulting molecular electronic energy depends on the nuclear coordinates, leading to the molecular potential energy curve for diatomics, and potential energy surfaces for a general polyatomic molecule.

Approximations in quantum chemistry The complexity of the SE prevents the exact analytical solvability of this second-order differential eigenvalue equation, apart from some simple problems. The difficulties that arise when we attempt to solve the SE for the system that has more than one electron is not some exotic QM complexity but originates from the three-body problem that does not have analytical solutions in classical mechanics either. Even “simple” problems, like the H-atom, are not what most chemists would like to solve on paper, or even read in a book chapter. In addition, Ψ is very hard to visualize, a fact that goes against a chemist’s intuition. Thus we must turn to approximations. The role of approximations is twofold. First, simplified solutions allow us to build up models of hierarchical complexity that will let us improve our knowledge of the system. Second, approximations are the only way to solve the SE equation. Even with approximations, solving SE will rely on numerical mathematics and on the power of computers. Results of such computations need to be compared with the chemist’s qualitative understanding of the problem. To do that we need models. Any discrepancy between our qualitative model and computational results will either make us learn something new by making our models better or make us increase the accuracy of the computations. Practical Approaches to Biological Inorganic Chemistry

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There are two methods that are used in almost all computational techniques (Cramer, 2004; Atkins and Friedman, 2005; Jensen, 2017). One is the perturbational method that rewrites SE as a power series and accuracy is increased by considering higher order, more complicated terms. The other method is the variational method that uses approximate Ψ, that is adjusted to improve (lower) the energy. The approximate wave function, Ψ0 , is written with adjustable parameters in it. It will not be the exact solution of the SE, that is, it is not  an  Eeigenfunction of Hamiltonian, but the average energy is assessed via D  ^ Ψ0 H Ψ0 =hΨ0 jΨ0 i. The variational principle states that the correct wave function leads to the lowest energy. In the second step, variational parameters are adjusted to get lower average energy. The whole procedure is repeated until energy cannot be lowered further. For a qualitative understanding of some phenomenon, we use a perturbational like approach. In the Hamiltonian, we will first look for a “problematic” term, and then we will simply ignore it. The full Hamiltonian is thus written as H^ 5 H^ 0 1 H^ 1 , where the first term is the Hamiltonian of the simpler system, and the second one is the problematic one. Luckily, the SE for H^ 0 we know how to solve or has already been solved beforehand. We then observe what would be the effect of adding H^ 1 . Typically, what we observe is that solutions of H^ 0 will be highly degenerate, and that this degeneracy is removed, at least partially, with H^ 1 . We then proceed by adding more terms to the Hamiltonian. This approach will not make the mathematics more plausible than brute force calculation of the energy. Energy expressions, from the average value theorem, will be tedious, and will contain rather complicated integrals. However, in each step, we can rely on computations, and either use the perturbational or variational method and have quantitative estimates of the energies of the interactions present in our system. Our division of the Hamiltonian will be, as much as possible, based on the physics and chemistry of the studied system. Often, for integrals that appear throughout the procedure, some symbols will be given, and these integrals can be treated as parameters. If the model is built up correctly, these parameters will have a clear physical meaning. Let us look into some examples. One of the analytically solvable systems is the hydrogen atom. The wave functions for the H-atom are referred to as orbitals. However, for multielectron atoms, although chemically very simple, there is no analytical solution of the SE. This is so already for the He-atom. The electron
electron interaction makes it impossible to separate two electrons in the He-atom from the Hamiltonian. But we have found our “problematic term.” Thus H^ 5 H^ 0 1 H^ ee . If we ignore H^ ee , we conclude that our simplified Hamiltonian is H^ 0 5 h^ 1 1 h^ 2 , where h^ 1 and h^ 2 are two hydrogen-like Hamiltonians for each electron, which we know how to solve. Each electron will be described by its own orbital ψi . What we have managed to do here is to simplify the two-electron problem, which we do not know how to solve, into two one-electron problems that are solvable. The total two-electron wave function is obtained from the product of two orbitals: Ψðr1 ; r2 Þ 5 ψðr1 Þψðr2 Þ, and the energy is simply the sum of the one-electron energies. The complicated two-electron wave function that is a function of the position of both electrons is simplified. The same

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principle holds for other, multielectron atoms. The electronic state of the many-electron atom is described (approximated) by its electron configuration. The ground state of the He-atom is described by 1s2 configuration, that is, by the product of two 1s hydrogenlike wave functions (see the next subsection). Usually, it is simply said that each electron is in the 1s orbital, or that the 1s orbital is occupied by two electrons. Electron configuration describes the many-electron wave function as a product of the hydrogen-like orbitals. Of course, this is all very approximate, because we ignored a very important part of the Hamiltonian to get to this stage. The wave function can be further improved by performing actual calculations, for example, within the framework of the variational principle. Unfortunately, it is not always that simple. From the electronic configuration, we do not always have enough information about the electronic structure. In the case of the ground state of He, it is satisfactory, but this is not the universal case. For example, if we have two electrons in atomic d-orbitals, as in some TM cations, there are 45 (yes you read it right!) different ways to place these two electrons. How can we differentiate these various possibilities, in other words, how can we differentiate these 45 microstates? If there is no electron
electron interaction, all these microstates would have the same energy. However, even in this example with just two d-electrons, it is obvious that it is not the same if we have two electrons in the same orbital, or in two different orbitals. Electron
electron repulsion will differentiate some of the microstates, resulting in loss of degeneracies. By taking into account V^ ee , we arrive at five spectroscopic or electronic terms each with different energies (see discussion about electronic terms). The energy levels of these states are what we can see from atomic spectroscopy. Differences in energy between these states can be written with two parameters that are related to the integrals describing the electron
electron repulsion. These integrals can be directly calculated, or estimated from the calculations, or from experimental transitions. It is important to remember that spectroscopic transitions occur between these electronic states, and not between orbitals. All observed properties arise from states and not from orbitals. Orbitals are never observable! Orbitals are used to approximately make multielectron wave function. The electronic configuration is only sometimes enough to describe the electronic states of atoms and ions. But the qualitative picture will still hold. And importantly, the bridge between QM mathematics and chemistry is made. The concept of the orbital is kept from the simplest system to more complicated ones. Furthermore, orbitals are easily visualized, since they are functions of only three space coordinates. This way, the electron configuration, a chemically important concept, has its QM definition. Let us return for a moment to our example with two d-electrons. What happens if we have a TM ion with d2 electronic configuration surrounded by six ligands in the octahedral environment? We write now the Hamiltonian as: H^ 5 H^ 0 1 H^ ee 1 H^ LF . The last term describes a so-called octahedral ligand field (LF). We have already seen that d2 electronic configuration is 45-fold degenerate in the absence of electron
electron repulsion and now in the absence of the LF. By simply ignoring the LF term we end up with five electronic states. Realizing that the symmetry of the system is lower than spherical symmetry of the free ion will further split the electronic terms. In octahedral LF, a d2 ion

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will have 11 electronic states with different energies. These energies are described with only three parameters relative to the ground state term. In addition to the two parameters describing electron
electron repulsion, there will be one describing octahedral LF. This is the so-called weak-field LF approach. We could break degeneracies even further by including more interactions in the Hamiltonian, for example, spin
orbit coupling or magnetic field, but this would go beyond the scope of the present chapter. At this point, we wanted only to illustrate how approximations are used to make useful models and some more details on this subject will be given in the Ligand field theory section. We have already mention one other approximation, BO approximation, which is important both for performing calculations and for building up chemical concepts from a QM perspective. Almost all standard calculations are performed within the BO approximation. The coupling of electron and nuclear motion is too complicated to feasibly make efficient calculations on the molecules of the size important for chemistry. But even more important, the BO approximation, albeit just an approximation, is crucial for the connection of QM and chemistry. It is the only way we can talk about molecular geometry or potential energy surfaces. There are some interesting phenomena in chemistry that occur because of the violation of BO approximation, like the Jahn
Teller (JT) effect (Bersuker, 2006; Zlatar et al., 2009). The JT effect explains why highly symmetric molecules with degenerate ground state distort into the nuclear configuration of lower symmetry to remove degeneracy. In this case our Hamiltonian is H^ 5 H^ 0 1 H^ JT . The last term describes electron
nuclear (vibronic) coupling. If we ignore it, the highly symmetric molecular structure would be the most stable one. However, once this coupling is included, degeneracy will be removed, and the distorted structure is of lower energy. Integrals that describe the JT coupling are typically considered as parameters. What is often done is to perform electronic structure calculations on the set of nuclear geometries to obtain a potential energy curve connecting high-symmetry configuration with low-symmetry minimum energy configuration, and to fit this curve to a polynomial expression. Coefficients of that expression are related to the JT integrals, which will have physical interpretation, for example, force and force constant at high-symmetry configuration. Other models are also built to account for detailed mechanistic description of a distortion (Zlatar et al., 2009; Ramanantoanina et al., 2013). However, if we were to take the full Hamiltonian from the beginning, the electron
nuclear coupling would be there from the start. We could not talk about the molecular structure, or about a distortion of the molecular structure. We could not use concepts of the force, force constant, or even electronic degeneracy. The JT effect, which arises because BO approximation is not valid at the point of electron degeneracy, would not exist if we did not apply BO approximation in the first place. We will not be able to go here into much more detail (see the Reference section; Cramer, 2004; Jensen, 2017; Atkins and Friedman, 2005), but we hope that the reader can see how powerful approximations can be. We use them not only in the computer programs, not only to make mathematics feasible for calculations, but to make models to describe chemical systems.

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Electronic structure of atoms Hydrogen atom The hydrogen atom system with one proton and one electron is the simplest chemical system. Its SE has analytical solutions called orbitals. Because of the spherical symmetry of the system, while solving the SE, the spherical coordinate system (r,θ; φ instead of x, y, z, see Fig. 2.2) is necessary. Orbitals are written as a product of a radial and angular part, Ψðr; θ; φÞ 5 Rn;l ðrÞYl;ml ðθ; φÞ. The radial part, Rn;l ðrÞ, describes the size of the system, that is, describes the behavior of the wave function in terms of the distance r between the electron and the nucleus. The angular part, Yl;ml ðθ; φÞ (spherical harmonics), describes the shape of orbitals, that is, describes the dependence on angles θ; φ. Orbitals are classified with three quantum numbers, n, l, ml—principal quantum number, azimuthal (orbital) quantum number, and magnetic quantum number, respectively. The quantum numbers are just the integer numbers. They arise from the boundary conditions wave functions need to satisfy. Wave functions need to be periodic in θ; φ, and must be zero as r goes to infinity. The actual method for solving the SE can be found in specialized books

FIGURE 2.2 Spherical coordinate system (A), radial probability distribution (B), and spherical harmonics for H-atom (C).

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FIGURE 2.2 (Continued).

(Atkins and Friedman, 2005; Atkins et al., 2018), but we need to mention that r,θ; φ dependence is not completely independent. We do need all three coordinates to describe our system. This will imply a restriction on the values of quantum numbers: n . l $ jml j, and n . 0 and l $ 0. The quantum number n describes the size of the orbital (radial part),

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l its shape, and ml the orientation of orbital relative to the fixed coordinate system. Instead of giving the exact mathematical expression of orbitals, usually, only the quantum numbers describing them are given. In chemistry, the symbolic expression is given instead—write the n value and the symbol for the l value; for example, the 3d orbital has n 5 3 and l 5 2. For orbitals with different l values letter symbols are given: s (l 5 0), p (l 5 1), d (l 5 2), f (l 5 3). The origin of these symbols is from the early spectroscopic studies of atoms (“sharp,” “principle,” “diffuse,” “fundamental”). Quantum numbers have a very important role. They quantize not only the wave functions, but also some other physical observables. The energy depends only on n, in atomic units: E 5 2 Z2 =2n2 (Z is the nuclear charge; Z 5 1 for hydrogen). Thus in H atoms, all orbitals with the same n will have the same energy. There will be n2 such orbitals. In Fig. 2.2 the radial probability densities (for orbitals with n 5 1,2,3 and l 5 0,1,2) and spherical harmonics (for orbitals with l 5 0,1,2, and different ml values) are depicted. Orbitals will have n 2 1 nodes (points where the wave function is zero). More specifically, due to the shape of spherical harmonics, there are l nodal planes and n 2 l 2 1 radial nodes. The more nodes an orbital has, the higher its energy. If there are more nodes, the greater the curvature of the wave function. The curvature of the wave function is related to its second derivative, which itself is related to the kinetic energy (see Eq. 2.2). Therefore the more nodes a wave function has, the higher the kinetic energy will be. And orbitals with the same number of nodes will have the same energy. This is a general feature of a wave function and is often a quick way of predicting qualitatively the energy order of orbitals. The radial probability densities give the probability per unit volume that the particle will be found at certain distance r. We see that distance of the electron from the nucleus is not quantized. It also does not orbit around the nucleus. Apart from nodal points, there is a finite probability of finding the electron everywhere, even far away from the nucleus. As n increases the radial probability decays slower. That means that in the excited states, the electron is on average further away from the nucleus. This is independent of the shape of orbitals. The states that have the same n belong to the nth shell. The maximum radial probability of the 1s electron, its average distance from the nucleus, corresponds to the Bohr’s radius (a0 5 0.529 A). Angular momentum It is not just the energy that is quantized. The angular momentum, L, and the z-component of the angular momentum, Lz, are also quantized—they depend on the values of l and ml quantum numbers, respectively: L2 5 h ¯ 2 lðl 1 1Þ and Lz 5 h ¯ ml . Eigenfunctions of the ^ 2 L operator are spherical harmonics. Orbital angular momentum gives rise to observable magnetic dipole moment, μ 5 LμB and μz 5 Lz μB , where μB is a constant, “Bohr magneton.” To understand angular momentum, it is useful to make a classical analogy. We can think of an electron’s motion as being circular, like an electric current in a circular wire. Such a motion of a charged particle will generate a magnetic field proportional to its angular momentum. Therefore an electron will generate the magnetic field, however only if l . 0. Apart from being a quantized quantity, QM orbital angular momentum has one more

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property that differs from the classical picture. One can know only its magnitude, length of the vector, and its projection on the z-axis. Different orbitals will have different angular momenta. Each state with an angular momentum L will have 2l 1 1 substates, with different ml values. The energy of H-orbitals depends only on n, despite having different angular momenta. The group of states that have the same values of n and l is a subshell. In the presence of the external magnetic field, these 2l 1 1 degenerate levels will split; hence the ml is named magnetic quantum number. We see that degeneracy of the H-atom states comes up from two sources. The first is different magnitudes of the orbital angular momentum and second is different projections of that angular momentum onto the z-axis. If the l 5 0, there is no angular momentum. All the kinetic energy is in a radial motion. If l . 0, part of the kinetic energy is in radial and part in angular motion. The higher l is, the more kinetic energy is in the angular component. The total kinetic energy remains the same in the states described by the same n. This is obviously related to the total number of nodes, which depends only on n. Electron spin From previous considerations, an electron in l 5 0 state is not expected to have an internal magnetic field. Interestingly, all electrons, independently of their motion around the nucleus, even free electrons, have intrinsic angular momentum—spin angular momentum. The Stern
Gerlach experiment, from the early 1920s, showed that ground state silver atoms with one unpaired electron in s-orbital (no orbital angular momentum) still exhibit an internal magnetic field. And the field appears with only two orientations. So far it should not be surprising that this is another quantity that is quantized. The electron spin is described by a spin angular momentum quantum number s 5 1/2, and its z-component, ms, that can have values, ms 5 1/2 (“spin up” or α), or ms 5 21/2 (“spin down” or β). Spin is an intrinsic property of an electron, just like its mass and charge. The electron spin comes directly from the solution of the relativistic Dirac’s equation and in the nonrelativistic SE it is added ad hoc. The name spin for this property comes from the classical picture—a particle spinning about an axis possesses an angular momentum independent of its motion through space. The spin is independent of other motions that the electron has. In other words, it is independent of which orbital the electron occupies. Therefore to describe the spin of an electron, in addition to three space coordinates (x, y, z or r, θ; φ) we include an extra “spin coordinate” (ω) that can have two values (α or β). The total wave function, spin
orbital, is written as a product of spatial wave function (orbital) and spin wave function. Ψðr; θ; φ; ωÞ 5 ψn;l;ml ðr; θ; φÞσmsðωÞ , and we will just label two possible spin functions as either α or β. Including spin as an intrinsic property of an electron, implies that in H-atom electron is completely described by four quantum numbers n, l, ml, and ms. Degeneracy of H-atom orbitals is then 2n2.

Many-electron atoms For many-electron atoms, as we have already mentioned, SE cannot be solved analytically because of the electron
electron interaction. The starting point is to ignore this part of the Hamiltonian. This leads to the orbital approximation or the independent electron model. The multielectron wave function is approximated as a product of one-electron wave Practical Approaches to Biological Inorganic Chemistry

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functions, orbitals. This product is called electron configuration. Each electron is described with its own, hydrogen-like orbital, obtained from its own SE (obviously with modified nuclear charge Z). What we have learned in previous subsections is that the lowest energy solution of the one-electron SE is 1s orbital. Would that mean that in the ground state of a multielectron atom, all electrons are going to be described by their 1s orbitals? Obviously, this is not correct. We know that independent electrons are not really independent. It is impossible that two electrons are in the same place. Remember, electrons are moving in four-dimensional space (spatial 1 spin), but their position is not known exactly. Instead of the particle’s position, we need to think of electrons as waves described completely by the set of four quantum numbers. Thus two electrons cannot have all four quantum numbers identical. This is Pauli’s exclusion principle. Another way of stating this is that each orbital can accommodate up to two electrons (with “opposite spins,” i.e., one α, the other β). The last expression is commonly used while speaking about electron configuration. It may also look convenient to draw the configurations either in boxes or as line/arrow diagrams. It is also common to use an expression like “there are two electrons in d orbital,” or “1s orbital is occupied by two electrons.” However, never forget that an orbital (spin
orbital), by definition, is a one-electron wave function—it describes one, and only one electron. And strictly speaking, one cannot “put an electron in,” and an orbital cannot “accommodate an electron.” Again, orbitals are one-electron wave functions, described by four quantum numbers, used to approximate the multielectron wave function (Autschbach, 2012). All the statements above are, of course, completely valid, but only if we are aware of their true meaning. We need to admit that the orbital approximation is very crude. Electron
electron interactions cannot be so small to be neglected. But there is a simple way our model can be improved, while still ignoring electron
electron repulsion explicitly. An electron in a multielectron atom can be considered as moving in the effective field created by the nucleus and all the other electrons. That means that the electron is “feeling” the nuclear charge, but it is reduced due to the presence of other electrons. Thus electron
electron repulsion is mimicked by the so-called electron shielding. The electrons with different n and l quantum numbers will be shielded differently, that is, will feel different effective nuclear charge, Zeff. This points to one important difference compared to the hydrogen atom—the energy of the orbitals will depend both on n (as in the H-atom case), but also on l quantum number. Thus the degeneracy of s, p, d levels with the same n is removed. However, we still use the orbital approximation. The electron configuration is symbolically written as the product of all the “occupied” subshells, with a superscript indicating the number of electrons “in” it. A maximum number of electrons in each subshell is given by its degeneracy, 2(2l 1 1). For example, the ground state of He-atom is described as 1s2, of Li-atom 1s22s1, of O-atom 1s22s22p4. Zeff can be determined by a set of semiempirical rules. More often it is a variable which can systematically improve an initial approximate wave function by the means of the variational calculations. To understand the effect of shielding we refer to Fig. 2.2, which depicts the radial distributions of H-like orbitals. The angular wave function of each orbital is still described by spherical harmonics. While the radial part in the multielectron system will be different, the qualitative picture will still hold. Because of the higher nuclear charge (Zeff . 1) nucleus will more strongly attract electrons, and the region of maximum probability density will be closer to the nucleus. The maximum radial probability of Practical Approaches to Biological Inorganic Chemistry

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orbitals with the same n is still on a similar distance from the nucleus regardless of different l values. The shell model of an atom gives rough features of the electron density of an atom. Within each shell, a finer picture is provided by the subshells. Electrons with lower n are closer to the nucleus. They are tightly bound to the nucleus because they are only slightly shielded from the full nuclear charge. However, these electrons will reduce the average charge of the nucleus experienced by the electrons in orbitals with higher n. Thus similarly to the H-atom, energy will be higher for orbitals with higher n. Let us now try to understand why the orbitals with the same n but different l will be shielded differently. Let us think about 2s and 2p orbitals. In the one-electron system, 2s and 2p orbitals are degenerate. 2s orbital will have higher kinetic energy in the radial part of the wave function (one radial node), but 2p orbital will have higher kinetic energy in angular part (one angular node), and the total energy will be the same in both cases (as it depends only on n). But what happens in a multielectron atom, for example, in configurations 1s22s1 or 1s22p1, which one would have lower energy? If we look into radial distributions of 2s and 2p orbitals (Fig. 2.2), we see that 2s electron is, on average, slightly further away from the nucleus than a 2p electron. From that, naively, we could expect that energy of 2s electron will be higher than that of the 2p electron. But at the same time, the maximum of the 2p distribution is closer to the region occupied by the 1s electrons. Therefore 1s and 2p electrons are, on average, closer together than the 1s and 2s electrons. Consequently, the 1s and 2p electrons will repel each other more than 1s and 2s electrons (Cooksy, 2014). Here, we are approximating the electron
electron repulsion by shielding, so 2p orbitals will be more shielded than 2s, and 2s orbital will have lower energy. Similar reasoning could be applied for 3s, 3p, and 3d orbitals. Within each subshell, 2(2l 1 1) degeneracy is not removed. The energy order of orbitals in the ground state of multielectron atoms is typically 1s, 2s, 2p, 3s, 3p, (3d, 4s), 4p. . . The order of 3d and 4s orbitals will depend on atoms as they lie close in energy. Constructing the electron configuration is typically based on the Aufbau principle. The orbitals are arranged by increasing energy, and electrons are added one by one, according to this energy order, subject to the Pauli exclusion principle. A maximum of two electrons can be assigned to one orbital, and if so, then they will have paired spins. So far, we have not talked too much about the spin, but there is an additional rule that is accounted for during this build-up principle. When more than one orbital is available for occupation, which happens due to the degeneracies of subshells, electrons occupy separate orbitals (with different ml values) before entering an already half-occupied orbital. Doing so, they will have parallel spins (e.g., both with α spin). This Hund’s rule of maximum multiplicity is obviously important only if we care about the occupation of orbitals with the same l value. For example, O-atom’s configuration, 1s22s22p4, can be written as 1s2 2s2 2p2x 2p1y 2p1z , where the last two electrons are both α spin, and px, py, pz indicates three electrons with ml values 21, 0, 11, respectively. Qualitatively, occupying two different degenerate orbitals gives them a greater spatial separation, hence lower energy. Maximum spin is related to the effects of so-called spin correlation, which will be discussed shortly later. However, in the orbital approximation, all different occupations of subshells, microstates, are still degenerate. Specification of subshell occupation in electron configuration imitates the real ground state of atoms, with explicit electron
electron repulsion. Completely filled shells are tightly bound to the nucleus and these electrons are called

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core electrons. Electrons in partially filled shells are valence electrons and are further away from the nucleus. As explained, core electrons shield valence ones from the nucleus. Core electrons usually do not contribute to the chemistry of atoms. Chemistry is often related mainly to the properties of valence electrons. Periodic system of elements The electron configuration of atoms explains the common form of the periodic system of elements (Fig. 2.3). Elements are classified into “blocks” according to the subshell that is being “filled” as the atomic number increases. Each period starts with the elements whose highest energy electrons are in the ns orbital. These are s-elements, with group 1 having ns1 and group 2 having ns2 configuration of outermost electrons. The first period has only H and He because with n 5 1 only s-orbitals are available. The second period will have six more elements, p-block (groups 13
18) elements that will have 2p orbitals occupied, obviously up to 2p6. The third period will also have eight elements, because of the occupation of 3s and 3p orbitals. The complications start with the fourth period. Orbital energies depend on Zeff, which is important for the relative order of, for example, 3d and 4s orbitals (Eugen Schwarz, 2010; Eugen Schwarz and Rich, 2010). For K and Ca the order is 4s , 3d, thus these two elements belong to the s-block and start the fourth period. However, from Sc on, the order is reversed, and the 3d orbital is lower in energy. The ground state configuration of Sc is however 3d14s2. Because the 3d orbital is much more localized than the 4s orbital, the much greater repulsion energy of the two electrons

FIGURE 2.3 Periodic system of elements.

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in the 3d orbital is more important than the simple energy order of the orbitals. The total energy of the atom is lower despite populating the higher energy 4s orbital. The same reasoning is generally true for atoms Sc to Zn, and first-row d-block elements in the periodic system typically have valence electron configuration 3dn4s2. Exceptions are Cr (3d54s1) and Cu (3d104s1). A similar trend is observed in the fifth period. In the sixth period, filling 4f orbitals generates the f-block, rare earth elements. The energies of the 4f, 5d, and 6s orbitals are comparable, and true electronic configuration must be deduced considering effects that go beyond the one-electron picture. The ground state of cations is obtained by removing electrons from neutral atom configuration. In 3d metal cations that means removing electrons first from the 4s orbitals. Electronic configuration of TM cations, dn, is the starting point of the LF approach for understanding the properties of coordination compounds (see details in the Ligand field theory section, as well as a discussion related to the oxidation states and using ionic configuration in complexes in the previous edition of this book; Neese, 2013). We see that electronic configuration alone is not enough to describe the ground states of atoms. The true ground state is the state of lowest energy and needs to have included other effects, most notably electron
electron repulsion. Thus in principle, we cannot use electronic configuration and we must introduce terms. And spectroscopically determined ground electronic configurations are deduced from the transitions between true electronic states (terms).

Pauli principle The Pauli exclusion principle, mentioned briefly above, is a direct result of a more general principle—electrons are indistinguishable particles. Every electron has the same mass, the same charge, and same spin as any other electron. For example, if we consider the ground state He-atom, we said it will be described by 1s2 electron configuration, that is, that both electrons will be described by a one-electron wave function with n 5 1, l 5 0, but one will have α and the other β spin. But we cannot know which electron has α spin and which β. In other words, while we label electrons as “1” and “2” (in a two-electron system), this labeling is arbitrary. Thus the correct wave function, multielectron wave function including the spin part, must describe all electrons as indistinguishable particles. Any property of an atom, or molecule, must not depend on our choice of labeling electrons. In other words, the probability density must not depend on the label 2  2 ing we choose, that is, Ψð1; 2Þ 5 Ψð2; 1Þ . This means that the wave function can either have the same (symmetric) or can change sign (antisymmetric) while permuting labels of electrons. The Pauli principle states that the wave function must be antisymmetric with respect to the interchange of any pair of electrons. In our example of the two-electron system, this results in Ψð1; 2Þ 5 2 Ψð2; 1Þ. In fact, the same statement will apply to all pairs of identical fermions (particles with half-integer spin like electrons, protons, neutrons, 13C nuclei. . .). It will not be true for the pairs of identical bosons (particles with integer spins, like photons or 12C nuclei). But how does this statement relate to the exclusion principle? Let us take again 1s2 configuration as an example. Remember, we are dealing with the total wave function, that is a product of the orbital (space) part and spin part. Each part can be either symmetric or Practical Approaches to Biological Inorganic Chemistry

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antisymmetric, but their product must be antisymmetric. This can be true only if one is symmetric and the other antisymmetric. In this example, the orbital part is a simple product 1s(1)1s(2). Because both electrons are described by the same 1s function, the orbital part is symmetric for the permutation of labels “1” and “2.” Thus the antisymmetry of the total wave function lies in the spin part. The spin part has different possibilities: both electrons α, α(1)α(2), both β, β(1)β(2) or one α and the other β. The last case can be written as either α(1)β(2) or β(1)α(2). But, again, we cannot tell which one is α and which β, so the last case is expressed as a linear combination that will have an pffiffiffi equal probability that each electron is either alpha or beta: σ1 5 1= 2ðαð1Þβ ð2Þ 1 β ð1Þαð2ÞÞ pffiffiffi pffiffiffi or σ2 5 1= 2ðαð1Þβ ð2Þ 2 β ð1Þαð2ÞÞ, where 1= 2 is the normalization factor. Of all the possibilities for the spin part, only σ is antisymmetric. Therefore the total wave function is 1s(1)1s(2) σ . Of all possibilities only this one is valid according to the Pauli principle, and this is the one having paired spins. Paired spins mean that the spin part is described by σ . The total wave function, satisfying the antisymmetrization principle can be written as a determinant, the Slater determinant (SD):   1  1sð1Þαð1Þ 1sð2Þαð2Þ  1 pffiffiffiffiffiffi  5 pffiffiffiffiffiffi ð1sð1Þαð1Þ1sð2Þβ ð2Þ 2 1sð2Þαð2Þ1sð2Þβ ð2ÞÞ 5 1sð1Þ1sð2Þσ2  ð2Þ 1sð1Þβ ð1Þ 1sð2Þβ ð2Þ ð 2Þ Every acceptable wave function for a closed shell configuration can be expressed in the form of the SD. For open shell systems, this is not always the case. The SD describing an N-electron system has N rows and N columns, with each row corresponds to a different electron and each column to a different wave function. SD always gives an antisymmetric wave function. If we have electrons described by two different orbitals we do not need the exclusion principle. Both electrons can have the same spin. And both electrons can also have paired spins. However, the total wave function must still obey the Pauli principle and needs to be antisymmetric. Let us see how we can describe the 1s12s1 configuration. The spin part is the same as before because we have two electrons with an arbitrary spin— three symmetric spin products and one antisymmetric. The orbital part is described by a product 1s(1)2s(2) or 1s(2)2s(1). Obviously, both products suggest that we distinguish electron 1 and 2. Hence, as before with the spin part, we must take the linear combination: ψ1 5 1sð1Þ2sð2Þ 1 1sð2Þ2sð1Þ and ψ2 5 1sð1Þ2sð2Þ 2 1sð2Þ2sð1Þ. The first is symmetric and the second antisymmetric. The total wave function can be now constructed as a product of orbital and spin part. ψ1 is combined with σ to lead to an antisymmetric total wave function. And spins are paired, or more precisely antiparallel. ψ2 can be combined with any of the three symmetric spin products (α(1)α(2), β(1)β(2), σ1 ). In this case, spins are parallel. Thus there will be three total wave functions, where the space part is described with ψ2 and differing in the spin part. In the orbital approximation, all four wave functions are degenerate. When including electron
electron repulsion (note it is independent on the spin), the wave function with ψ1 space part will have a different energy than those with ψ2 . The later stays triply degenerate. The number of the spin products that append the appropriate orbital product is the spin multiplicity. In other words, total spin multiplicity is a sum of individual spins of the electrons, given by 2s 1 1 (see below). In the case of the 1s12s1 configuration, we have a singlet (two electrons with

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opposite spin) and triplet (both electrons have the same spin) states. Because electrons have a different spatial part depending on their relative spin (parallel or antiparallel), their Coulombic interaction will not be the same. The triplet state will have lower energy (the already mentioned Hund’s rule). Now when we see how the wave functions are appropriately constructed, can we understand why the triplet is more stable? We can think what happens when two electrons approach each other. The ψ2 combination will be zero if the two electrons have the same spatial coordinates. Thus there is a zero probability of finding electrons with parallel spins at the same point. This is called a Fermi hole (exchange hole). On the other hand, ψ1 will not vanish. Do not worry electrons do not collide—there is a fourth, spin coordinate that is different, that is, spins are different. Because of the Fermi hole, electrons will “avoid” each other, and average electron
electron repulsion is lower for the triplet state. There are some other subtleties related to the stabilization of the triplet state, but we cannot go here more into this not simple problem (Atkins and Friedman, 2005). It should be noted that this spin correlation is only an indirect consequence of the Pauli principle. It is of Coulombic origin and not because of interactions between spins. Let us for a moment return to the spin. We know that each electron is characterized by the spin quantum number s 5 1/2, and its projection on the z-axis, ms 5 61=2. What about multielectron systems? Can we characterize the total spin in the entire system in a similar manner? In a multielectron case, individual spin angular momenta are coupled to give a total spin S and its projection Ms, with S $ jMs j, and both can be integer or halfinteger numbers. We have seen how the spin part of the wave function is constructed. Still, it is not obvious how this is related to the S and Ms values. This would require a 2 somewhat tedious evaluation of the eigenvalues of S^ and S^ z operators, which we will skip here. Singlet states will have S 5 0, Ms 5 0. Triplet states have S 5 1 and Ms 5 21, 0, 1, corresponding to the wave functions with spin part β(1)β(2), σ2 , α(1)α(2), respectively. Spin multiplicity is equal to 2s 1 1. In a similar manner, if we would have three electrons, there will be doublet states (S 5 1/2, Ms 5 21/2,1/2,2s 1 1 5 2) and quartet states (S 5 3/2, Ms 5 23/2, 21/2, 1/2, 3/2, 2s 1 1 5 4). The wave functions are constructed in an analogous way to satisfy the Pauli symmetrization principle. This subsection is maybe a little too technical for chemists, but it is important to understand what lies behind the Pauli principle. And this principle is of fundamental importance in chemistry. Because of the Pauli principle, the ground states of atoms differ greatly along the periodic table. Electrons are described with different orbitals, and ground states will have different spin multiplicities, leading to the richness of chemistry.

Two electrons in two orbitals In this subsection, we will go into a little bit more detail regarding the “problematic” electron
electron interactions. Let us consider two electrons in two nondegenerate orbitals a and b. Here it is not important what these orbitals are. They can be 1s and 2s atomic orbitals of He-atom, or some molecular orbitals (MOs). According to the Pauli exclusion principle, we can accommodate these two electrons in six different ways, Fig. 2.4. These are six SD that can be constructed to satisfy the antisymmetrization principle. In Fig. 2.4 they are represented as line diagrams with “up” and “down” arrows

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FIGURE 2.4 Six different ways of how we can accommodate two electrons in two orbitals (single determinants). Expressions for the energies with and without electron
electron repulsion; J and K are Coulomb and exchange integrals, respectively.

representing spin of the electrons, and lines representing orbitals. For chemists, this is a familiar way of discussing different electron distributions. In the previous subsection, we explained the precise meaning of SDs. In the absence of electron
electron repulsion, these six SDs belong to three electronic configurations, a2b0, a1b1, and a0b2. Their energies are given as a sum of one-electron energies (see Fig. 2.4). The configuration a1b1 is fourfold degenerate. Determinants SD2, SD3, SD4, SD5 are four microstates that belong to this configuration and in the orbital approximation have the same energy. Once we “turn on” electron repulsion, in addition to the one-electron energies, energies of SDs will depend on two types of integrals. The first one is the Coulomb integral, J, and is positive. It describes semiclassical electron
electron repulsion. In this particular system, we have integrals Jaa, Jbb, and Jab. For example, Jab describes repulsion between charges in densities ψ2a and ψ2b . This Coulomb term will exist for each pair of electrons. It will raise the energy of each SD relative to the sum of the one-electron energies. The second integral is the exchange integral, Kab. It is also positive but exists only for pairs of parallel spins. There is no classical interpretation of the exchange integral. It is a pure QM phenomenon that arises because the wave functions are antisymmetric. Exchange interaction stabilizes configurations with parallel spins. We will not go into more details here, just the result is given in Fig. 2.4. Each of the SDs has precise Ms values. Those with “up/down,” paired spins will have Ms 5 0, and those with “up/up” parallel spins Ms 5 1, and with “down/down” spins Ms 5 21. Obviously, SD1 and SD6 are describing the state with S 5 0. In the previous section, we showed that SD1 is að1Það2Þσ2 , and SD6 is bð1Þbð2Þσ2 . SD2 and SD3 are two wave functions of the triplet state, with S 5 1, Ms 5 1 and Ms 5 21. SD2 is ðað1Þbð2Þ 2 að2Þbð1ÞÞαð1Þαð2Þ and SD3 is analogous with β ð1Þβ ð2Þ spin part. However, we face a problem with SD4 and SD5. It is not obvious which one will be the third wave function of the triplet states and which one will be the singlet state. And they both have the same energy, differing from SD2 and SD3, which clearly belong to the triplet state. In 2 fact, neither of those two SDs (SD4 and SD5) is an eigenfunction of S^ . If we look more carefully, we see that SD4 and SD5 do not satisfy that electrons are indistinguishable. As they are SD, they are antisymmetric due to the nature of determinants, but clearly, we distinguish which electron is in orbital a and which in b. This is because when we build the determinant, we need to make a choice of which spin function to associate with

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which spatial function. These two SDs are artificial mixtures of the singlet and triplet states, and they do not correctly predict the energy of the states. To make proper wave functions we need to make a linear combination of these two SDs: SD4 1 SD5 and SD4
SD5. The first combination will be multiplied with σ2 spin part to yield the open shell singlet state, and the second will be multiplied with σ1 spin part to yield the last triplet wave function. The energy of the SD4
SD5 must be the same as the energy of the Ms 5 11 or Ms 5 21 that are representable by a single determinant. Open shell singlet cannot be written in terms of a single determinant, it is an example of the multideterminantal wave function. We must note that most of the time states can be represented with single determinants (for a closed shell singlet this is always the case). However, multideterminantal states are present in open shell cases. The energy of the open shell singlet is ea 1 eb 1 Jab 1 Kab . As discussed in the previous subsection the singlet state will be higher in energy than the triplet state from the same electronic configuration. Singlet
triplet separation is directly related to the exchange integral, it is 2Kab . A schematic representation of the influence of different terms on the energy of the states is given in Fig. 2.5.

FIGURE 2.5 Graphical representation of the influence of the Coulomb and exchange integrals on the energy levels.

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The energy difference between the triplet state of the configuration a1b1 and closed shell singlet (a2b0 configuration) is Δba 1 ðJab 2 Jaa 2 Kab Þ, where Δba is the difference in one-electron energies of orbitals a and b. All the parameters are positive. Coulombic repulsion of electrons in one orbital is larger than the repulsion of the electrons in different orbitals, that is, Jab , Jaa . This means that the expression in brackets is negative. If Δba is larger than this two-electron contribution to the energy, the ground state will be the one with two electrons occupying the lowest energy orbitals with paired spins, that is, the low-spin state will be the ground state. This situation is depicted in Fig. 2.5. However, interestingly, if the energy difference between orbitals is small, two-electron energy will prevail and a high-spin state becomes the ground state. In complexes of TM ions, commonly, the spin ground state depends on the surrounding ligands. A similar situation arises in TM atoms, where the ground state is typically dn22s2, as discussed previously. On the far right in Fig. 2.5, there is one more interaction, called configuration interaction (CI) (Atkins and Friedman, 2005). In our example, the ground state and the state nominally generated from a double excitation can interact in the second order. The lowest energy state is stabilized, and the higher energy state is destabilized. The energy effect is 2 approximately x 5 Hab =ΔE, where ΔE is the energy difference between the states before CI, and Hab is the coupling matrix element. The CI becomes very important when the states are close in energy. This commonly happens in TM ions, TM complexes, excited states, or when considering a molecular dissociation. The wave functions of the two states are mixed, that is, the ground state is described as Ψg 5 Ψa 2 ðHab =ΔEÞΨb , hence the name CI—the state is a superposition of two (in general several) configurations. We will not go into more detail here, but this type of interactions exists whenever the two states have the same symmetry and is a basis of the so-called noncrossing rule.

Electronic terms In the preceding subsections, we went through the electronic structure of atoms. Our starting point was the orbital approximation which neglects electron
electron repulsion. It was shown that the electron configuration is not sufficient to describe systems with more than one electron. Therefore we must introduce electronic terms. Table 2.1 summarizes the particular electronic configurations and a number of microstates that arise from that configuration. This is thoroughly explained in the previous sections: note that microstates and SDs are the same. Some of these microstates have the same energy and belong to the same electronic term. The ground state for a given electronic configuration of the free atoms or ions is always the state with maximum multiplicity (Hund’s rule). The terms are labeled as 2S11 L, where L describes their total orbital angular momentum, and 2S 1 1 is the spin multiplicity. There are rules on how to couple individual orbital angular momenta of electrons into the total orbital angular momentum, but this is a rather lengthy exercise and here we will not go through it. Again we need to point out that the observed spectroscopic transitions occur between these terms, and the parentage of electronic configurations must be deduced. Finally, we need to mention hole formalism—configurations with N electrons are equivalent to the configurations with N “holes,” for example, d1 and d9, d2 and d8, and so on, will give rise to the same electronic terms, as indicated in Table 2.1.

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TABLE 2.1 Atomic terms for sn, pn, and dn electronic configurations. The ground term is always the first one. Electronic configuration

Number of microstates

s1

2

2

S

s2 (p6, d10)

1

1

S

p1 (p5)

6

2

P

p2 (p4)

15

3

P, 1D, 1S

p3

20

4

S, 2D, 2P

d1 (d9)

10

2

D

d2 (d8)

45

3

F, 3P, 1G, 1D, 1S

d (d )

120

4

F, 4P, 2H, 2G, 2F, 2D, 2D, 2P

d4 (d6)

210

5

D, 3H, 3G, 3F, 3F, 3D, 3P, 3P, 1I, 1G, 1G, 1F, 1D, 1D, 1S, 1S

d5

256

6

A, 4G, 4F, 4D, 4P, 2I, 2H, 2G, 2G, 2F, 2F, 2D, 2D, 2D, 2P, 2S

3

7

Electronic terms

The energy of each state from the same configuration, thus with the constant one-electron term, is expressed solely in terms of the Coulomb and exchange integrals (see the previous subsection for some details). Instead of directly calculating J and K integrals, which is a quite demanding and complicated task, energy levels of the states (multiplets) are obtained from atomic spectra and are represented with Racaha’s parameters B and C. B and C do not have physical meaning, although they are mathematically related to J and K (Bersuker, 2010). The fact that all the states are uniquely described with only two parameters is a consequence of a high, spherical symmetry of atoms and ions. As an example, let us look with some more detail into a d2 configuration. State energy levels are presented in Fig. 2.6. The energy expressions in terms of Racaha’s parameters are also indicated. We can see that energy difference between 3P and 3F is 15B, thus we can directly obtain the value of B from the difference in energy of states with the same spin multiplicity from atomic spectra. Energy for transitions between states with different spin multiplicity, spin-forbidden transitions, depends both on B and C (so when we have B, we can easily obtain C). A is just an energy shift because we set the ground term to be zero, which of course it is not.

Symmetry QM description of the electronic structure of atoms and molecules is deeply connected to the concept of symmetry. The systematic (and mathematical) discussion of symmetry is called group theory. Almost the entirety of qualitative QM can be derived from the group theoretical perspective. Of course, symmetry and group theory deserve much more space than they will get in this chapter. We will just go through some of the main points that are necessary for a discussion related to the properties of TM compounds. Practical Approaches to Biological Inorganic Chemistry

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FIGURE 2.6 Terms arising from the d2 configuration and their energy expressions described with Racaha’s parameters.

In its most rudimentary usage, group theory classifies molecules according to the symmetry operations they possess. In this process, molecules are considered as hard, 3D objects with precise geometry. Symmetry operations like rotation, inversion, reflection, and improper rotation (and identity that does nothing, but this trivial operation needs to be considered because of formal mathematical reasons) are actions that make an object looking the same after the operation has been carried out. In the case of molecules, symmetry operations convert equivalent nuclei in equivalent positions into themselves. After a symmetry operation is carried out, we get an indistinguishable nuclear configuration. For example, in the H2O molecule, if we perform a rotation for 180 degrees (2pi/2 radians) around the axis depicted in Fig. 2.7, we just interchange the position of equivalent hydrogens. We say that H2O has a twofold (C2) axis with which is associated with C2 rotation. In H2O we can also recognize two reflections, in two planes—one is the plane where all three atoms lie, and the other is bisecting the H
O
H angle in half, Fig. 2.7. A water molecule, and in fact all bent molecules of Practical Approaches to Biological Inorganic Chemistry

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FIGURE 2.7 Molecules in (A) C2v (H2O), (B) C3v (NH3), and (C) D3h (PCl5) point groups.

the type AB2 belong to the C2v point group. Ammonia, Fig. 2.7 has 2 threefold rotations (rotation for 120 and 240 degrees), and three reflections. It belongs to the C3v point group. A molecule in a trigonal bipyramidal geometry has D3h symmetry. It has 2 threefold rotations, C3, around axes that pass through the central atom and two axial ones, 3 twofold rotations perpendicular to the threefold rotations (axes lie in equatorial plane, and each one is passing through one atom), one reflection in the equatorial plane (horizontal) and three vertical reflections in planes perpendicular to the horizontal one (each defined with two axial atoms and one of the equatorial ones). This nuclear configuration also has 2 threefold improper rotations—this is a composite operation that consists of the threefold rotation and the reflection in a perpendicular plane at the same time. We will not go with more examples. Finding symmetry elements and corresponding operations is nothing but a game. But one needs to play a lot, and preferably with 3D models, to become proficient at it. All the symmetry operations that exist for molecules form a point group and all molecules are classified accordingly. All molecules belonging to the same point group, irrespective of how chemically different they may be, will have certain properties that are very similar. With the aid of group theoretical classification, one can say whether a molecule will have a dipole moment, if it is chiral, how many infrared or Raman active vibrations it will have, etc. The Hamiltonian is invariant to all the symmetry operations (in any point group). This is obvious as the energy of a molecule must not depend on symmetry operation. This has an important consequence that eigenfunctions of the Hamiltonian are also eigenfunctions of any symmetry operator (symmetry operations are operators). This means that H-atom orbitals are eigenfunctions of symmetry operations of the point group of a sphere. Each orbital will be a basis for irreducible representation (irrep) of the R3 point group. In other words, s, p, d, f. . . are not only specifying angular momenta of H-atom orbitals but also their symmetry properties. Obviously, this property is related to the spherical harmonics part of the orbital. Note that in multielectron atoms spherical harmonics are the same as in the H-atom, hence their orbitals are also carrying the symmetry information. Symbols of the electronic terms describe their total orbital angular momentum but are again also their symmetry designation. In fact, any orbital or state will belong to some of the irreps of the point group. The irreps have names that indicate how a given object (orbital) behaves under various symmetry operations. For example, five d-orbitals in Oh point group (perfect octahedral coordination) transform as T2g and Practical Approaches to Biological Inorganic Chemistry

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Eg irreps. “T” indicates that the irrep is triple degenerate, “E” that it is double degenerate, “g” that they are symmetric with respect to the inversion operation. Note the convention—orbitals are labeled with a lowercase symbol and irreps in general, as well as states with uppercase letters. Orbitals of the central atom in general ABn molecule, are so important, and the irreps that they belong to in different point groups are collected in character tables (Atkins et al., 2006). A glance at the character tables can give the irreps of orbitals in various point groups. As stated at the beginning of this section, group theory and the application of symmetry in chemistry is a subject on its own, and the reader is referred to excellent books on the subject (Cotton, 1990; Vincent, 2010).

Ligand field theory The first and the simplest theory that managed to explain the electronic structure, magnetic properties, and spectra of TM compounds is crystal field theory (CFT). In CFT only the central metal ion is considered quantum mechanically. The ligands are substituted by point charges that only exert an electrostatic field on the metal ion. CFT, therefore does not consider metal
ligand bonding (covalency). The effect of the electron
electron interaction is included in atomic like way via Racaha’s parameters. The main effect on the TM ions electronic states is that the spherical symmetry is lost because of the presence of ligands. As always the descent of symmetry leads to the removal of degeneracy. The shape and orientation of d-orbitals (Fig. 2.8) are responsible for the way of removing the degeneracy. If we first introduce just the spherical electrostatic perturbation with negative “ligands,” the energies of all five d-orbitals increase. In the octahedral environment, the redistribution of negative charges occurs, and d-orbitals will not be degenerate anymore. Two orbitals have higher energy, compared to the barycenter, while three of them are below this level. As ligands approach the metal ion, the negative charge representing the ligand will be closer to dz2 and dx2 2y2 orbitals, and farther away from dxy , dxz , and dyz orbitals. This happens because ligands are placed on a coordinate axis, the first set of orbitals is oriented along the coordinate axis, while the second set is between axes (Fig. 2.8). This will cause a particular way of splitting, due to the repulsion of like charges. Fig. 2.9 illustrates the splitting of the d-orbitals in an octahedral environment. The splitting between the two energy levels is defined as Δo or 10Dq. The lower three orbitals (t2g orbitals, using group theoretical label) are stabilized by 0.4Δo relative to the barycenter. The upper (eg orbitals) are destabilized by 0.6Δo .

FIGURE 2.8 Five d-orbitals of transition metals.

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FIGURE 2.9 Splitting of d-orbitals in octahedral environment.

FIGURE 2.10

Splitting of d-orbitals for octahedral complexes that undergoes tetragonal distortions (D4h).

Taking into consideration CFT arguments, the splitting pattern of d-orbitals can be predicted for any symmetry of the environment (point group), Figs. 2.10 and 2.11. If, for example, the octahedral complex undergoes a tetragonal distortion, the d-orbital levels split again (Fig. 2.10). Elongating the octahedron, along the z-axis, will stabilize dz2 orbital relative to the dx2 2y2 , and dxz , dyz pair relative to the dxy . If we were to completely remove two ligands along the z-axis, ending up in the square planar coordination, we see the same trend of the stabilization, with dz2 lying even below dxy orbital. A compressed octahedron has a splitting pattern like the elongated one, with a reversed order of dz2 and dx2 2y2 orbitals, as well as of the dxz , dyz pair and dxy . The same reasoning can be applied to different coordination geometries, that is, a different way as to how point charges approach the metal ion. In Fig. 2.11 splitting patterns in different coordination geometries are qualitatively depicted. For example, in the tetrahedral environment, the splitting pattern will be just opposite to the octahedral one we discussed above, with a higher lying t2 set and a lower lying e set. Tetrahedral splitting, Δt will be 4/9 times smaller than octahedral one. Practical Approaches to Biological Inorganic Chemistry

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FIGURE 2.11 Splitting of d-orbitals in tetrahedral (Td), square pyramidal (C4v), trigonal bipyramidal (D3h), and pentagonal bipyramidal (D5h) environment.

The absolute value of the splitting Δ, obtained through CF, is one order of magnitude smaller than the measured one. Thus the enormous simplicity, the most significant advantage of CFT formalism is also its biggest drawback. CFT does not take into consideration bonding. It is obvious that metal
ligand interactions cannot be properly approximated simply by considering only pure d-orbitals surrounded by the point charges. However, CFT correctly predicts the qualitative orbital splitting and its order. CFT is thus used to make parameterized models. In the LFT Δ, as well as Racaha’s interelectronic repulsion parameters B and C, are parameters obtained by fitting the experimental optical transitions. More generally, LFT considers the Hamiltonian in the form H^ 5 H^ 0 1 H^ ee 1 H^ LF , where the first term is an electronic configuration of the central metal ion, the second term is electron
electron repulsion parameterized by B and C, and the third term is parameterized in terms of one-electron (LF) parameters. The electron
electron repulsion is thus treated as atom-like, preserving spherical symmetry. LF parameters take full account for the lowering of symmetry when a spherical TM ion is introduced in a complex. The angular overlap model, a revised version of the LFT, uses adjustable parameters and the angular geometry of the metal complex to specify the LF part of the Hamiltonian. Many experimental observations are explained, rationalized, and generalized in terms of LFT. Ligands are ordered in a sequence of increasing values of Δ, the so-called

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spectrochemical series: I2 , Br2 , Cl2 , F2 , H2O , NH3 , pyridine , NO22CN2 , CO. Ligands at the beginning of the series are called weak-field ligands, while those at the end are a strong field. A similar series exists for the variation of Δ with a metal ion, independently on the nature of ligands. Δ increases with the formal charge of the ion and down the periodic table. Another generalization observed in LFT is that Racaha’s parameters B and C in complexes are smaller than those for the free ions—nephelauxetic effect. A qualitative explanation is that the metal electrons are partially delocalized onto the ligands, the orbitals are larger than in the free ion, and d
d repulsion is reduced. The nephelauxetic effect increases in order F2 , H2O , NH3 , Cl2 , CN2 , Br2 , I2. This approximately corresponds to the increasing metal
ligand covalency. It is important to point out that, LFT still considers “pure d-orbitals” in the “field of ligands” but admits the existence of covalency and admits the fact that ligands are not point charges. Albeit, bonding, and covalency are not considered explicitly. Some of the experimental characteristics of TM compounds, like charge transfer spectra, are obviously out of the scope of the LFT. Values of LF parameters extracted from experiments refer to the ground state geometry, and description of the potential energy surfaces is also not dealt with LFT. However, LFT does not even try to consider these effects; it is designed to treat partially filled d-shells of coordinated TM ions. And that is that. Most of these drawbacks of LFT are resolved within the MO theory. For example, Δ is interpreted as the energy difference between the σ and π MOs with dominant d-orbital character. In this chapter we will not go into MO theory, and the reader is referred to the previous edition of this book for a discussion about the relation between the MO and LF theories, and proper consideration of covalency (Neese, 2013). Another deficiency of LFT is a large number of parameters that are necessary to describe the electronic structure of low-symmetry complexes, and it is often impossible to obtain a unique set of parameters from the experiments. However, computational chemistry comes to the rescue, often keeping the language of LFT. In recent years LFT, as an example of effective Hamiltonian theory, has been used to interpret high-level ab initio results (Atanasov et al., 2012). DFT is also commonly used for studying the microscopic origin of LFT (Atanasov et al., 2008; Daul et al., 2015). In the rest of this subsection, we will go through some of the qualitative and quantitative consequences of LFT. The mathematical background and a detailed description of the LFT can be found elsewhere (Ballhausen, 1962; Griffith, 1964; Sugano et al., 1970) while more qualitative aspects can be found in, for example, Hauser (2004), Shriver and Atkins (2006), Bersuker (2010), Housecroft and Sharpe (2012), and Neese (2013).

Some qualitative considerations The splitting pattern of d-orbitals in a ligand field is intuitively close to chemists. This splitting is often successful in qualitative interpretation and rationalization of properties of coordination compounds (Daul et al., 2015). For example, many TM complexes, with the same central metal ion (in the same oxidation state) have different spin ground states. [Fe(CN)6]42 ion is yellow without unpaired electrons, while the [Fe(H2O)6]21 ion is pale blue and paramagnetic. Both complexes contain a Fe(II), d6 central metal ion. This is commonly explained by considering how electrons can be arranged in orbitals that are Practical Approaches to Biological Inorganic Chemistry

Ligand field theory

45

split by a ligand field. Considering hexacoordinated complexes, the different spin ground state is observed with TM ions having four to seven d-electrons (Fig. 2.12). Electrons can be arranged with a maximal number of unpaired electrons, high-spin state, or in the low-spin state with no or the minimal number of unpaired electrons. Any intermediate arrangement is called the intermediate-spin state (cannot be a ground state in the perfect octahedral environment). If the energy separation between orbitals is large enough, the pairing of the electrons in the lower set will occur before occupying higher energy orbitals. This leads to the low-spin states. If the energy splitting is small enough, the energetically more favorable situation will be one with a high-spin configuration. Note that free TM ions, with no splitting of d-orbitals, will always have a high-spin ground state, as explained before. Complexes of metal ions of the second-row transition series have a low-spin ground state because Δ is about twofold larger compared to the first-row series. We can also look into spin-state preferences in different coordination environments. Tetrahedral complexes are always in the high-spin ground state, because the splitting in the tetrahedral environment is much smaller than in the octahedral one. The strong-field CN2 ligand builds low-spin octahedral [Cr(CN)6]42, high-spin tetrahedral [Mn(CN)4]22, and high-spin penta-coordinated [Cr(CN)5]32 complexes (try playing by putting electrons in trigonal bipyramidal or square pyramidal ligand field). We see that the concepts of the strong-field and low-spin are not synonyms. Further examples include rationalization of the fact that Ni21, a d8 ion, has high-spin octahedral complexes with weak-field ligands, like [Ni(H2O)6]21, while low-spin complexes are square planar, like [Ni(CN)4]22. Intermediate-spin d6 complexes, in the perfect octahedral environment, are nonexisting, while commonly observed in metalloporphyrins with two additional ligands, where the d-orbital splitting pattern is that of an elongated octahedron (see Fig. 2.10). But this is not the whole story! It is well known that in octahedral coordination Mn21 and Mn31 form almost always high-spin complexes while Co31 is in a low-spin state in all but one complex, [CoF6]32. These observations cannot be explained with the simple difference in charge of the metal ion and Δ. Also, naively, there should be a much more pronounced tendency toward the LS ground state for Fe31 complexes comparing to Fe21 complexes because of the metal ion charge. This is because Δ is not the only factor one needs to consider. Remember what we talked in the Two electrons in two orbitals section— it is a balance of one-electron splitting and two-electron repulsion which determines the

FIGURE 2.12 Schematic representation of low-spin and high-spin d4
d7 configurations in octahedral ligand field.

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2. Introduction to ligand field theory and computational chemistry

order of states. And diagrams just discussed are only one-electron picture. LFT describes the electron
electron repulsion as a pairing energy, Π. The pairing energy is defined as the difference between the energies of electron
electron interactions in low-spin and high-spin complexes, respectively, divided by the number of pairings destroyed by the low-spin to high-spin transition. Thus in LFT, the magnitude of Δ, together with Π determines what the ground spin state will be (Daul et al., 2015). Clearly, a low-spin state is preferable if Π , Δ, and high-spin if Π . Δ. Pairing energy is expressed using Racaha’s parameters, and is different for different numbers of d-electrons: Π(d4) 5 6B 1 5C, Π(d5) 5 7.5B 1 5C, Π(d6) 5 2.5B 1 4C, Π(d7) 5 4B 1 4C (Bersuker, 2010). If we assume that B is similar in different dn configurations, and C  4B (common approximation), we arrive to approximate relations: Π(d4) 5 26B, Π(d5) 5 27.5B, Π(d6) 5 18.5B, Π(d7) 5 20B, thus Π(d6) , Π(d7) , Π(d4) , Π(d5). We can see the interelectronic repulsion is almost the same for d4 and d5 but it is much smaller for d6 (which is close to d7). The preference of Co31 octahedral complexes for low-spin can be now understood from the fact that Π for the d6 configuration is the lowest, and at the same time, Δ is high because of the high oxidation state. For d6 Fe21 complexes Δ is lower, and hence Fe21 can be found often as either a high-spin or a low-spin complex, and the most common spin-crossover compounds are those of Fe21. Another recent example of using LFT concepts is the rationalization of unusual intermediate-spin state of square pyramidal Fe31 complexes of the type [FeLN2S2X] (where LN2S2 5 2,20 -(2,20 -bipyridine-6,60 -diyl)bis(1,10 -diphenylethanethiolate) and X 5 Cl, Br, and I) (Wang et al., 2018). The intermediate-spin ground state (quartet, S 5 3/2) is a consequence of strong field in the xy plane (strong-field tetradentate L), and weak field between the metal ion and the axial X2 ligand. The orbital splitting is just approximately square pyramidal. The dx2 2y2 orbital is the highest in energy and well separated from the other four d-orbitals, including the dz2 orbital. At the same time, nephelauxetic reduction (due to the covalency of L) decreases pairing energy, leading to the double occupation of lowest energy orbital, and thus to an S 5 3/2 ground state.

Symmetry in ligand field theory Different coordination geometries of TM complexes belong to different point groups. We do not want to go into too many details of the group theory, however, we need to point out that symmetry and LFT are inseparable. The way of splitting, discussed above, can also be obtained from character tables of symmetry point groups. Perfect octahedral complexes belong to the Oh point group. In the Oh point group two of the d-orbitals (dz2 and dx2 2y2 ) span Eg and the remaining three (dxy , dxz ; and dyz ) span T2g irreps. This is exactly what we have seen from simple CFT arguments at the beginning of this section. Elongated octahedral complexes, trans-ML4L0 2 and square planar complexes are all of the D4h symmetry. Hence the similar splitting pattern is observed—dxz and dyz belong to the double degenerate Eg irrep, while dz2 , dx2 2y2 , and dxy are of A1g, B1g, and B2g symmetry, respectively. Irreps of point groups are typically used to label orbitals in different ligand fields, for example, d-orbitals split in Oh point group into t2g and eg set. The convention is to label orbitals with a lowercase symbol and irreps in general, as well as states with uppercase letters. While group theory is very powerful, it cannot predict

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Ligand field theory

the order of orbitals, just the pattern. To know the energy order of orbitals in the ligand field qualitatively, it is necessary to use chemical wisdom and/or CFT arguments. For a quantitative estimate of the splitting, first principle calculations are necessary, as will be discussed later. To summarize, when going from the simplest H-atom Hamiltonian, and then multielectron atoms, toward the LF Hamiltonian, the degeneracy of levels is removed at each step. In fact whenever we include an additional term in the Hamiltonian the levels split. When going from a spherical symmetric ion to the octahedral coordination, the symmetry is reduced. And the splitting of d-orbitals is a consequence of this lowering of symmetry, from R3 to Oh. Additional splitting occurring when moving from octahedral to square planar complexes is a consequence of further lowering of symmetry, from Oh to D4h. Very useful tools of group theory are correlation tables that show how irreps of the higher symmetry point group transform in the lower symmetry subgroup. For example, from Table 2.2, one can directly see how d-orbitals split in octahedral, square planar, or trigonal ligand fields. There are similar tables for correlations between other point groups. These are used, for example, when considering effects of lowering the symmetry when going from idealized symmetry to the real one. Idealized symmetry is used for qualitative purposes to reduce the number of parameters needed to describe the real system. For example, when six different ligands surround a metal ion in the octahedral environment, none of the five d-orbitals are degenerate, however, it is often possible to group them into approximate t2g and eg sets. In such situations, one typically does LF analysis as in the Oh point group, and finer effects are deduced from the correlation tables. One should always be aware of when to use group theoretical designation of the symmetry and when to use the type of coordination. As we have seen, “octahedral” can mean that the complex belongs to the Oh point group, but more loosely it may refer to the six-coordinate complex with an “octahedral” arrangement of ligands. H-atom orbitals are classified by their quantum numbers, n, l, ml, ms. But, the designation of H-atom orbitals as s, p, d, . . . does not only imply the value of l quantum number TABLE 2.2 Correlation table for descent in symmetry from the full rotational point group. For other correlation tables consult Cotton (1990) and Atkins et al. (2006). R3

O

D4

D3

S

A1

A1

A1

P

T1

A2 1 E

A2 1 E

D

E 1 T2

A1 1 B1 1 B2 1 E

A1 1 2E

F

A2 1 T 1 1 T 2

A2 1 B1 1 B2 1 2E

A1 1 2A2 1 2E

G

A1 1 E 1 T1 1 T2

2A1 1 A2 1 B1 1 B2 1 2E

2A1 1 A2 1 3E

H

E 1 2T1 1 T2

A1 1 2A2 1 B1 1 B2 1 3E

A1 1 2A2 1 4E

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2. Introduction to ligand field theory and computational chemistry

(l 5 0,1,2. . .), but it also gives their symmetry properties in the full rotational point group. Similarly, electronic states of multielectron atoms are given as 2S11 L, where L describes their total orbital angular momentum, but is again also their symmetry designation. When considering a complex in some point group, the states are designated as 2S 1 1 Γ , where Γ is one of the irreps of the point group. Correlation tables are particularly useful for determining how electronic states split in different point groups (see the next subsection).

Some quantitative considerations Much of the machinery of the LFT has already been described in previous parts of this chapter. The Hamiltonian is described as H^ 5 H^ 0 1 H^ ee 1 H^ LF , with the last two terms being “problematic” in our divide and conquer approach. The simplest case is a d1 system in the octahedral field. Because we have only one electron, H^ ee is missing, and the influence of the LF part is to split d-orbitals into t2g and eg set, which we discussed in great detail previously. The ground state term 2D will, therefore, split into 2T2g and 2Eg terms in Oh symmetry. The energy difference between the two is Δ. Obviously, the spin part is the same in the free ion and in the coordinated one, because symmetry will not affect the spin parts of the wave functions. Thus these states are twofold spin degenerate, the ground state is triple orbital degenerate, and the first excited state is orbitally double degenerate. Because of the electron
hole equivalence, the ground state of the d9 system is also 2D. Obviously, the splitting will be the same. However, 2Eg will be the ground state. We may point out that in Td symmetry, the splitting pattern is opposite to the one in Oh symmetry. Therefore d1 in Td will behave the same as d1 in Oh, and d9 in Td as d1 in Oh. Let us now consider the d2 case in octahedral symmetry. We have two ways to go. The first one is to note that we know what the influence of the electron
electron repulsion on this configuration is. As we have described previously, the d2 configuration will give rise to the 3F, 3P, 1G, 1D, 1S terms. Each of these will split in the Oh point group according to the correlation table, Table 2.2. Again, upon descent in symmetry, the spin multiplicity does not change! Therefore the ground state 3F in octahedral symmetry becomes 3A2g 1 3T1g 1 3T2g, the 3T1g being the ground state. 3P will become 3T1g. 1G will split into 1A1g, 1Eg, 1T1g, and 1T2g. This is a weak-field approach, but, we can go another way around, starting from three distinct configurations in Oh point group t2g2eg0, t2g1eg1, and t2g0eg2, where the first one is representing the ground state. The energy difference between the two excited configurations and the ground level will be Δ and 2Δ respectively. At this point, we are still ignoring d
d repulsion, and all the microstates within one configuration are degenerate. This is exactly what we had before for atoms, even different spin states are degenerate within a configuration. Degeneracy of the three configurations is 15, 24, and 6, respectively. Now, let us see what happens when including the H^ ee part. Each of the configurations will split due to the Coulombic repulsion. For example, t2g2eg0 will give rise to the ground 3T1g state (the highest multiplicity!) as well as

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three singlet states, 1T2g, 1Eg, and 1A1g. Similarly, electron repulsion will split the other two configurations. This is a strong-field approach. There is a somewhat tedious group theoretical approach on how to derive these states, but we will not go into more detail here. Interested readers should consult more specialized books on the subject (Griffith, 1964; Sugano et al., 1970; Cotton, 1990; Stephanos and Addison, 2017). Obviously, both weakfield and strong-field concepts must be equivalent. They both must describe a situation in the ligand field of arbitrary strength. This leads to a correlation diagram, Fig. 2.13. The states with the same symmetry and the same spin state correlate to each other in the left and right side of these diagrams. Obviously, states arising from the same free ion term do not need to correspond to the same configuration in the strong-field limit. Furthermore, the noncrossing rule must be obeyed, that is, states of the same symmetry and spin multiplicity do not cross. For example, there are two 3T1g states, one derived from 3F and the other one from the 3P ionic term. They do not cross, therefore the one from the ground 3F will correlate to the state arising from the ground t2g2eg0 configuration in the strong field. 3T1g(P), the brackets designating that it originates from the 3P term, correlates therefore with the t2g1eg1 configurations. It is noteworthy to mention, that there will be a CI mixing between these two configurations.

FIGURE 2.13 Correlation diagram for d2 configuration in the weak-field and strong-field LF schemes.

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2. Introduction to ligand field theory and computational chemistry

In d8 complexes, the energy order of the configurations is t2g6eg2, t2g5eg3, and t2g4eg4, or if we would write configurations in terms of holes instead of electrons t2g0eg2, t2g1eg1, and t2g2eg0. A d8 system, therefore behaves as if, relative to d2, ordering of the t2g and eg levels is inverted. This means, that the correlation diagram for d8 is like that for d2 but with the ordering of the strong-field states inverted. The free ion terms stay the same, but the states arising from the single free ion term are inverted also, to correlate them with the strong-field ones. Thus the ground state of d8 octahedral complexes is 3A2g. For the same reason, d8 tetrahedral complexes have the same correlation diagram as octahedral d2, and tetrahedral d2 the same as octahedral d8. There is a general rule that d102n (Oh) is like dn (Td) and dn (Oh) like d102n (Td). Also the relation between dn(Oh) and d102n(Oh) is analogous to the d1/d9 and d2/d8 cases, just described. The principle of equivalent complementary configurations applies also for partially filled t2g and eg levels. That means that, for example, configurations eg3 and eg1 will give rise to the same states. We will not go here into correlation diagrams for other configurations but will just summarize how the ground state free ion terms split in the octahedral field, and how each ground state strong-field configuration is split due to the electron
electron repulsion, Table 2.3. In the weak-field limit, all the states arising from the same free ion term have the same spin multiplicity, because lowering the symmetry due to the ligand field does not affect spin. The ground configuration in the strong-field limit is always the one with a maximum number of paired electrons because in this limit the electron
electron interaction is ignored. The real ground state of the complex ion obviously corresponds to the intermediate situation and will depend on the nature of both central metal ion and ligands. In the d4
d7 cases, weak-field and strong-field limits give rise to different ground TABLE 2.3 Splitting of the ground electronic states in the weak-field scheme and splitting of the ground electronic configurations in the strong-field scheme. The ground state in the Oh symmetry is given in bold. Weak-field limit

Strong-field limit

Free ion

Electronic states

d1

2

D

2

d2

3

F

3

d3

4

F

4

d4

5

D

5

6

d6

5

d

7

4

d

8

3

d9

2

d

Configuration

Electronic states

T2g 1 2Eg

t2g1

2

T2g 1 2Eg

A2g 1 3T1g 1 3T2g

t2g2

3

T1g 1 1T2g 1 1Eg 1 1A1g

A2g 1 4T1g 1 4T2g

t2g3

4

A2g 1 2Eg 1 2T1g 1 2T2g

5

T2g 1 5Eg

t2g4

3

T1g 1 1T2g 1 1Eg 1 1A1g

A

6

A1g

t2g5

2

T2g 1 2Eg

D

5

T2g 1 5Eg

t2g6

1

A1g

F

4

A2g 1 T1g 1 T2g

t2g6eg1

2

Eg

F

3

A2g 1 T1g 1 T2g

t2g6eg2

3

A2g 1 1A1g 1 1Eg

D

2

T2g 1 2Eg

t2g6eg3

2

T2g 1 2Eg

4 3

4

3

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states. Moreover, as qualitatively discussed in previous subsections, they will have different spin multiplicity. The ground state of a complex under study in these d-configurations will be either high-spin or low-spin. This depends on the balance of electron
electron repulsion and LF splitting, or, in other words, on the balance of pairing energy and LF splitting. If the central metal ion in a complex is only slightly perturbed from its free ion state, we have the high-spin state. Otherwise, we have a low-spin situation. We see now that the orbital diagrams presented in Fig. 2.12, are an oversimplification of the real situation. These diagrams represent just one microstate, or more precisely just one SD of a given strong-field configuration. And in the high-spin case this configuration is not the ground configuration in the strong field limit, but the configuration that would correlate with the weak-field ground electronic term. However, we must admit that from a chemist’s point of view, orbital diagrams are very attractive and intuitive. And as we have seen, lots of chemistry can be explained with them. But one should be aware of the real, physical situation. And there is a danger that a chemist’s preoccupation of where to place the electrons in diagrams masks the fact that only states exist, and can be observed, and not the orbitals. There is one more misconception related to the orbital diagram description of the electronic structure of TM complexes. It is often thought that the first transition in electronic spectra is equal to Δ, that is, to the transition of the electron from the lower t2g to the higher eg set. The first transition is equal to Δ just in the following cases: obviously d1/d9, d3 (4A2g to 4T2g), and d8 (3A2g to 3T2g). In all the other cases the transition energy will depend both on Δ and on B. In the case of “spin-forbidden” transitions, energy will depend on C as well. Thus the equivalence of the spectroscopic transitions and Δ is at best just a rough estimate. Tanabe
Sugano diagrams So far, we have presented how rich the electronic structure of coordinated TM ions is. Admittedly, it is not simple, even with the approximations described. However, the beauty of the LF approach is that the energy of all the states arising from dn ionic configuration in the ligand field are expressed in parameterized form. Electron repulsion is always described with two parameters (B and C). Thus in the octahedral ligand field, the energy of all the states will be given in terms of only three parameters—B, C, and Δ. These parameters are optimized to reproduce available experimental data. Actual LF calculation (yes, LFT is not just a model), is done first by forming a matrix, in the basis of all SDs (microstates) of a given dn configuration, where all the matrix elements are expressed in parameterized form. Even CI matrix elements are parameterized in the same manner. Diagonalizing such a matrix yields the energies of all the states, as well as all the wave functions (states themselves). States are in general represented as a linear combination of the microstates. They are often called multiplets. With this procedure, we obtain information about all the states not just those that are observed experimentally and that are used to deduce the actual values of the parameters. We will not go into more details of such LF calculations, but it is important to mention that qualitative diagrams like those as in Fig. 2.13 can be constructed in a quantitative manner. The famous Tanabe
Sugano diagrams portray the parameterized form of the LFT. In Fig. 2.14 we present two such

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FIGURE 2.14

2. Introduction to ligand field theory and computational chemistry

Tanabe
Sugano diagrams for d2 (A) and d6 (B) octahedral complexes.

diagrams (for d2 and d6 cases) as examples. Each dn case has its own diagram. Herein they are both constructed with B 5 700 cm21 and C 5 3000 cm21 and varying Δ. In these diagrams, the energy of each state (expressed as E/B), relative to the ground state is plotted against ligand field strength (expressed as Δ/B). Relative units are used because the same diagrams can be used for different metal ions with the same dn configurations. In situations where there is a change of the ground state, as the LF strength is increased, there will be an apparent discontinuity in the diagrams, see Fig. 2.14 for the d6 case. We see that this splits diagrams in two, separated with a demarcation line, where the left part corresponds to the high-spin ground state, and the left part to the low-spin ground state. These diagrams are used for assignment, but also for detailed and quantitative analysis of electronic spectra of TM compounds. As already mentioned, if the symmetry of the studied complex is lower than Oh (or Td) the analysis based on these diagrams will give the main features, and further reduction of symmetry can be applied using correlation tables. This is particularly important for the ground states that are subject to the JT distortion (all the degenerate states; Table 2.3). The strong distortion is characteristic of Eg states.

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Introduction to computational chemistry

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FIGURE 2.14 (Continued).

Introduction to computational chemistry LFT is handy for understanding and visualizing a splitting pattern of d-orbitals. However, for a quantitative consideration of the splitting, first principle calculations are necessary. In the previous sections, we explained the QM description of atoms and ions as well as how LFT uses empirically derived parameters to describe the electronic structure of TM complexes. With the advancement in computer technology, in recent decades the electronic structure of TM complexes can be obtained by a variety of QM methods (Cramer, 2004; Jensen, 2017). The quantum mechanical methods that do not utilize any system dependent empirical parameters are often referred to as ab initio methods. These methods can be categorized into two main realms: (1) wave function
based methods and (2) density functional methods. In theory, both approaches should give the same exact energy and any observables we are interested in. Unfortunately, the fundamental equations of QM are not exactly solvable for anything but a few simple model systems, and both methodologies are essentially trying to find the best approximate approach for QM descriptions of the system under study. The wave function
based approaches have the

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advantages that they are systematically improvable, and consequently, they are considered to be highly accurate and very reliable. Their disadvantages are that they are very timeconsuming and limited by the system size. Moreover, for complicated electronic structures, none of the methods are straightforward. Instead of calculating a wave function, in DFT, the electronic energy can be expressed solely in terms of the density. In the next subsections simplified and very brief treatments of these methods will be given with an emphasis on problems related to TM compounds. It is noteworthy to mention that all computational methods were developed first for the organic molecules. There is still no universal method for the complicated electronic structure of TM compounds. Therefore lots of benchmarking and testing alongside experiments are needed. All herein computational methods concentrate on the electronic SE and use BO approximation to separate the nuclear and electronic motions. Irrespective of which computational strategy will be employed to predict properties of molecules, we need first to know the molecular geometry. Sometimes it is possible to use experimental geometry, for example, from X-ray or electron diffraction studies. However, this is usually not possible. Therefore a typical computational study starts by adjusting the positions of the atoms to find the minimum energy on the potential energy surface. This procedure is called geometry optimization (Cramer, 2004; Jensen, 2017). Note that sometimes it is possible to use one method for geometry optimization, and another one for the calculations of properties. One often needs to use the most economical strategy, to have a balance between the precision and time of calculations. The geometry optimization involves a series of computational steps. First, the electronic energy is calculated at the judiciously chosen starting nuclear geometry. The electronic energy is added to the nuclear potential energy to find the effective potential energy at starting geometry. In the second step, the derivative of the effective potential energy with respect to each geometrical coordinate is calculated. In the third step, these forces are used to estimate how the geometry should change to reach a minimum value of the energy. In the next step, atoms are moved according to the estimate from the previous step. A new geometry is obtained, and the procedure is repeated until the forces are all essentially zero, and there is no change in the energy. This indicates that a stationary point on the potential energy surface has been found. Sometimes this series of steps may arrive at a stationary point that is not a minimum, but a saddle point. To know in what type of stationary point the geometry optimization ended up, all the second derivatives of potential energy need to be calculated. This is the most time-consuming part. If we are at the minimum, we will have only positive second derivatives. Different algorithms exist for geometry optimization, including transition-state searches, where the aim is to find a saddle point on the potential energy surface. Symmetry constrained optimizations are also possible. In cases where the surface is complicated, one needs to repeat the geometry optimization from different starting geometries, to ensure that the global minimum is found and not the local one. These methods are typically used for the ground electronic state of a molecule. Sometimes, these methods may also be used to find geometries and properties of excited states. This is especially true if the considered excited state is the lowest energy state of a given symmetry and given total electron spin. The choice of the basis set isvital for all computational methods (Cramer, 2004; Jensen, 2017). For all molecules, the trial wave function is constructed as an SD consisting of MOs. These MOs are in turn most often expressed as a variationally optimized linear

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combination of functions centered at atoms. This is a very reasonable approach since, at large internuclear separations, MOs became atom-like orbitals. The set of these functions used to create MOs is the basis set. In practical computational work, the most often used basis sets utilize ether Gaussian-type orbitals, or Slater-type orbitals (STOs). In physics and material chemistry plane-wave basis sets are commonly used. STOs have the cusp at r 5 0, that is, they show correct behavior expected from atomic orbital. They also demonstrate correct behavior at large values of r (Gaussians do not capture the exponential decay (e2r ) 2 naturally since they have e2r form). Thus Slater basis functions are closer to the actual solution, and therefore fewer of them are needed for accurate results. Linear combination of Gaussian basis functions can be used to reproduce correct behavior by curve fitting to a Slater orbital, but any orbital made from Gaussians have a slope of zero at the origin. The most significant advantage of Gaussian functions is that their integrals can be evaluated analytically, and, even more importantly, that a product of Gaussians at different centers can be expressed as one Gaussian function. This enables us to reduce all multicentered integrals to two-centered integrals, which then can be evaluated analytically. Depending on the number of functions in the basis set we distinguish minimal and expanded basis sets. Minimal basis sets have one function per atomic orbital, and therefore are the least accurate. Double-zeta has two basis functions per orbital and allows treatment of spatially different bonds at the same atom. Triple zeta analogously has three functions, quadrupole four, etc. Having different sizes in exponentials allows the orbital to change size with perturbations from approaching atoms. A split valence basis set has one basis function for core atomic orbitals and more for the valence ones. In some basis sets polarization and diffuse functions are included. The choice of the basis set in practical computations of TM compounds strongly depends on the system at hand.

The wave function
based methods The Hartree
Fock method The simplest and first useful ab initio method is the Hartree
Fock (HF), which approximately solves the electronic SE. It is based on a simple approximation we have already dealt with, where the multielectron wave function is represented by a single SD. This assumption is equivalent to the mean-field approximation. Each electron moves independently of all the others but feels the average Coulombic repulsion of all the other electrons. In addition, the electron also feels “weird” exchange interaction, because of the antisymmetrization characteristics of the SD. When the wave function in the form of SD is combined with the variational principle, the optimum wave function, related to the lowest total energy, must satisfy a set of equations similar to the original SE we want to solve approximately. These, HF equations are a set of one-electron equations of the form: F^ i ψi 5 Ei ψi , where F^ i is the so-called Fock operator, ψi is one-electron wave function (orbital), and Ei is the corresponding orbital energy. The Fock operator consists of terms that express the kinetic energy of the electron in the orbital ψi , the potential energy of interaction between the electron in that orbital and the nuclei, repulsive interactions between the electron in ψi and all the other electrons, and, finally, the effects of the spincorrelation between electrons (remember electrons are indistinguishable). Formally,

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PP P F^ 5 i hi 1 i i ðJij 2 Kij Þ where hi is the core Hamiltonian, and J and K are Coulomb and exchange operators. In this expression, Coulomb and exchange parts are not constrained with i 6¼ j, so the terms describing the interaction of the electron with itself emerge. This term in HF is however exactly zero because Jii 5 Kii . Because the Fock operator depends on J and K, orbital energies are not the one-electron energies. They will contain electron
electron interaction. The total energy is not the simple sum of orbital energies, because the electron
electron interactions would be double counted, that is, the interaction between electrons i and j would be accounted both in the energy of the orbital with electron i and in the energy of the orbital with electron j. There is one more twist. The Fock operator includes the effects of all the other electrons on the electron we look into. Thus the Fock operator depends on the wave functions of those other electrons. In other words, the solution of HF equation for each electron enters the equations for each of the other electrons. To solve this problem, it is necessary to follow a self-consistent field (SCF) procedure. We start from a set of an initial, guess orbitals which are used to build initial Fock operator, and HF equations are solved. The solution is used to build the revised Fock operator in the next iteration. The loop is maintained until self-consistency is found, that is, each cycle gives the energies and wave functions unchanged to within a chosen criterion. In molecular calculations, the MOs are expressed as the linear combination of P atomic orbitals ψ 5 N i ci φ, where N is the size of the basis set. For a given basis set, solving HF means determining the coefficients in the expansion. This formulation leads to a set of equations, Roothaan equations that are represented in matrix form. Solving these matrix equations is the problem of linear algebra, with known and well-developed techniques (Szabo and Ostlund, 1996). There is one more subtle issue. Depending on the treatment of the spin
orbitals in the HF calculations, two different variants of the HF exist. In this restricted Hartree
Fock (RHF) method, the spatial part of the orbital is forced to be the same for both spins. In closed shell systems, this assumption is always true. In open shell systems, it is introduced ad hoc. The second variant is the unrestricted Hartree
Fock (UHF). The spatial part for both spins is treated separately and spin polarization appears. It is worth noting that UHF and RHF will give exactly the same results for a closed shell calculation. The HF method, by using a single determinant form of the wave function, neglects correlation between the electrons, thus making it not suitable for TM compounds. The correlation energy in HF is defined as the difference between the HF energy and the exact energy of the system. There are different types of correlations. The first one is a dynamical correlation. Dynamical correlation is related to the short-range electron interactions. There should exist instantaneous repulsion between two electrons. This is not considered in the mean-field HF. This physically wrong assumption is thus at the heart of HF theory: since the electron is interacting with the average density distribution of the other electrons, there is the same probability for the two electrons to be next to each other and at very distant points. By introducing this correlation, the Coulomb hole, not present in the HF, appears. Dynamical correlation is usually corrected using perturbation theory methods. The spincorrelation has been already discussed. The exchange hole is captured with the asymmetry of the starting wave function by the construction of the HF method. The absence of any kind of correlation between the electrons with the opposite spin, as opposed to some

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degree of correlation for the same spin electrons, leads to the artificial stabilization of the configurations with more unpaired electrons at HF level of theory. Lastly, nondynamical correlation or static correlation is important when there are electronic configurations very close in energy to the HF reference. Static correlation is important, for example, in TM ions or excited states. It is corrected using multiconfigurational calculations. If we want to improve the precision of the HF calculation and to make it suitable for TM compounds, we have to include part of the electron correlation into the wave function, as done in postHF methods. The basis set expansion is an additional limitation of the method. In practice, the convergence of the basis set must be studied to verify its completeness. It is not all so bad with HF. Since it is a variational method, the SD with the lowest energy, is as close as possible to the true wave function, for the presumed single SD form of the wave function. With HF we get spin
orbitals which minimize the energy, and often it is stated that they give “the best” SD. A single-electron picture of the electronic structure remains in HF and for example concepts such as labeling of electrons by angular momenta are still valid. Finally, approximations integrated into the HF are “well controlled.” That means that usually, we are aware of their accuracy or inaccuracy. And we usually know how to improve them. We need to mention that HF theory is the starting point for two types of calculations (Cramer, 2004; Jensen, 2017). The first one is ab initio, including HF calculations and post-HF calculations. They calculate all the integrals that appear in the HF formulation. The second one is a semiempirical approach that introduces further approximations. In the semiempirical methods, some of the integrals are set to zero and the other ones are treated as parameters fitted to some experimental data, like geometries or energies. These methods are computationally very efficient, but obviously of limited applicability, especially for TM compounds. They can be used only for systems for which parameters exist. Post-Hartree
Fock methods All post-HF methods have the goal of capturing the part of electron correlation missing in the original HF formulation. This can be done principally in two ways. One way tries to correct the single determinant approximation and the other tries to introduce correlation energy through perturbation theory. The most popular post-HF methods that are utilized in modern quantum chemical program packages are based on various flavors of the CI approach, or Møller
Plesset (MP) perturbation theory. All forms of CI are good at retrieving static correlation but differ by the amount of dynamical correlation that is corrected. MP methods are often dealing only with dynamical correlation. All these methods are characterized by systematic improvement of the results. In the MP approach, the Hamiltonian is expressed in a perturbative form, where the perturbation is a difference between real electron
electron repulsion and an average one. The zeroth order energy and first perturbative correction are equal to the HF energy. The first contribution to the electron correlation is the second-order correction (MP2). MP2 captures a considerable amount of dynamical correlation. It is not significantly more computationally expensive than HF. Since MP is perturbative and not a variational approach, it is not known if the energy is lower or higher than the exact one. The most important sources of problems are the systems where the electron correlation is too large, and the

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perturbative approach becomes questionable. This can lead to the divergent behavior of the MP expansion. Thus MP methods tend to produce poor results for spin-state energetics in TM compounds. In CI methods the multielectron wave function is written as a linear combination of different electron configurations, using the HF wave functions for each configuration. If the expansion is complete, the best possible wave function for a given set of orbitals is constructed. During the SCF procedure, both the coefficients in the expansion and orbitals are optimized. This is called a full CI. It includes all possible correlations, both dynamical and nondynamical. This is the exact solution of the SE for a given basis set. However, it is too computationally demanding and cannot be used in practice, except for systems with a very small number of electrons. If the CI method is limited to only single and double excitations, we get CISD (CI singles and doubles). The computational cost of the CI methods is greatly reduced if only the most important SDs in the CI procedure are used. The complete active space self-consistent field (CASSCF) consists of performing a full CI calculation, but only within a given set of orbitals, active orbitals. They are previously selected in a judicious way, so that they play a major role in the chemistry of the studied system. A variation of this method is the state-averagedCASSCF (SA-CASSCF) in which the final orbitals are a weighted average of the orbitals for both ground and several excited states. The major drawback of the CASSCF method is that it is necessary to select a different set of active orbitals for every different situation. It also requires a previous understanding of the problem. Correlations coming from the inactive orbitals are missing. In fact CAS is bringing mainly nondynamical correlation within the active space. Enlarging the active space improves the results. But obviously at the cost that computations become more demanding. There is another way of improving the CAS method—by combining it with perturbative treatment to include dynamical correlation e.g., CASPT2 or n-electron valence state perturbation theory (NEVPT2). All post-HF methods share a common limitation of a strong basis set dependence. It is commonly found that the use of improved methods requires a larger basis set, further increasing the cost of calculation. Another disadvantage of these methods is poor scaling with the system size. This restricts the usage of these methods to systems of relatively modest size. In recent years the density matrix renormalization group (Reiher, 2009), which considerably reduces computational demand, has been put forward for studies in TM chemistry. Without going into more detail, we have just listed some of the post-HF methods, and for more details, the reader is referred to the textbooks on the subject (Cramer, 2004; Jensen, 2017).

Density functional theory DFT is an exact reformulation of many-body QM in terms of the probability density rather than the complicated multielectron wave function (Parr and Yang, 1989). The biggest advantage is that the electron density only depends on three spatial variables while the wave function depends on 3N variables. Besides, density is a physical observable with a clear meaning that can be related to several important properties, such as polarizability, electronegativity, etc. Its advantages include less computer time, and, particularly

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for TM complexes, better precision than the HF. The essence of the DFT is that all the properties of the system, every observable quantity, can be calculated from the ground state electron density of the system alone. In mathematics, a function of a function is called a functional. “Functional” in the name DFT comes from the fact that the energy is a function of the electron density, E[ρ], and the electron density is itself a function of a position. Here only a brief overview of DFT and its applicability will be given, and a reader more interested in the subject is referred to excellent literature on the topic, for example (Koch and Holthausen, 2001; Cramer, 2004; Cramer and Truhlar, 2009; Neese, 2009; Perdew and Ruzsinszky, 2010; Burke and Wagner, 2013; Becke, 2014; Yu et al., 2016; Jensen, 2017). The first Hohenberg
Kohn theorem states that the energy of the ground state of a system is exclusively determined by the electron density. And this density is uniquely defined by the external potential. In DFT terminology external potential is the electron
nuclear potential energy operator. Thus it actually represents the potential energy, and generally is representing the field generated by nuclei (nuclei are “external” to the electrons). In other words, electron density determines external potential. From this, it follows that the electron density uniquely determines the Hamiltonian. This is because the Hamiltonian is specified by external potential and the total number of electrons, which can be computed from the electron density by integration over space. Note that ρ determines not only the ground state wave function, but the complete Hamiltonian, and thus all excited states as well. This theorem can be summarized as ρðrÞ2vext ðrÞ2ΨðrÞ. It implies that all properties of a system are functionals of the ground state density. The second Hohenberg
Kohn theorem states that it is possible to find the ground state electronic density by minimizing the energy of the system. This is equivalent to the variational principle in wave functions methods and allows us to develop similar procedures to HF but using the density. From the form of the SE, we can see that the energy functional contains three terms: the kinetic energy, the interaction with the external potential (nuclear
electron interaction), and the electron
electron interaction. The kinetic energy and the electron
electron interaction are described by the universal operators as they are the same for any system. Whether the system under consideration is an atom, small molecule, macromolecule, or solid depends only on the external potential. Ð The total energy can be rewritten as: E½ρ 5 T½ρ 1 Vne ½ρ 1 Vee ½ρ 5 F½ρ 1 Vne ½ρ 5 F½ρ 1 ρðrÞvext ðrÞdr. F½ρ is a universal functional of the density, valid for any system, whose precise form is unknown. It contains both the kinetic energy and the electron
electron interaction but is independent of vext . Thus providing the universal functional is known, the best density can be obtained using the variational procedure. These two theorems provide a conceptual framework for the utilization of a first principle QM description of system dynamics as a way of bypassing wave functions. Unfortunately, although these theorems are exact, they do not provide any prescription on how to obtain ground state energy from ρ, nor do they tell us how to find ρ if we first do not have a wave function. Most of today’s applications of DFT in quantum chemistry use the Kohn
Sham one-electron formulation of DFT. Kohn and Sham introduced a fictitious system of noninteracting electrons, moving in an external potential vs (“s” indicating this is not the real system) and having the same density as the real system. This fictitious system is

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described by a single determinant wave function. KS Hamiltonian is just a sum of one-electron Hamiltonians. Since we have the expression for the external potential, and there is no electron
electron interaction (“noninteracting electrons”), we only have to solve one-electron equations (exactly solvable) and generate so-called KS spin
orbitals. The KS equations are solved in the same way, with a smaller computational cost, as the HF equations. Orbitals are used to determine the kinetic energy of the fictitious system, Ts ½ρ. If we also note that a significant component of the electron
electron interaction is the classical Coulomb interaction, J ½ρ, the energy functional can be rearranged as E½ρ 5 Ts ½ρ 1 Vne ½ρ 1 J ½ρ 1 EXC ½ρ. The exchange-correlation (XC) functional, EXC ½ρ, has been introduced as EXC ½ρ 5 ðT ½ρ 2 Ts ½ρÞ 1 ðVee ½ρ 2 J ½ρÞ. It consists of all unknown termsthe residual part of the true kinetic energy and nonclassical electrostatic interactions. It is simply the sum of the error made in using noninteracting kinetic energy and the error made in treating the electron
electron interaction classically. It does not contain only exchange and correlation energy but also the correction to the kinetic energy. Both terms in the definition of EXC, are expected to be small. The success of the KS method depends on finding a good expression capable of reproducing these two small quantities. Let us look into the above energy expression in some detail. The first term, representing kinetic energy, is challenging to calculate directly from the density accurately, and that is the main reason that the abovementioned KS orbitals are introduced (i.e., instead of varying ρ, we are varying the KS orbitals which determine ρ). Admittedly, this represents a clear deviation in attempts to represent the energy in terms of the density only. This approach strongly resembles the one incorporated in HF theory, and the variationally obtained occupied KS orbitals resemble MOs calculated by the HF method, and they can be utilized in qualitative MO considerations. Of course, the obtained kinetic energy is by no means exact, since it originates from the fictitious KS system of noninteracting electrons, and it requires additional corrections. The term, describing the interaction of the electron density distribution with the external potential (nuclei) is exact. The third term is the classical expression for the electrostatic interelectronic repulsion energy if the electrons were smeared out into a continuous distribution of charge with electron density ρ. It is essentially identical to the Coulomb interaction term in HF theory. The final term, or XC functional, should, in principle, contain the correction to all the previous contributions. Although the exact XC functional should incorporate the effect of electronic interactions on kinetic energy, in practice, such a term is not explicitly present in most approximate functionals used. It is common practice to further separate this XC corrective term in a part corresponding to “exchange,” and the other corresponding to “correlation.” Naturally, the purpose of the exchange term is to correct neglect of the exchange in DFT and the correlation part should consider the existence of the Coulomb hole. The different forms of the Columbic and exchange expressions in DFT, as opposed to HF, prevent the perfect cancelation of the fictitious self-interaction contributions, and thus require that the exchange contributions consider this problem as well. Density functional approximations The exact form of EXC is unknown. Several types of EXC have been proposed and studied with the goal of giving a better description of the systems. Unfortunately, there are no straightforward and systematic ways of improving this functional. Since the exact form of

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the XC term is unknown, in practice the used EXC are called the XC approximations, and various approximations to the Hamiltonian in DFT are called density functional approximations (DFAs). In this way the distinction between the exact theory and approximations used in practice is clear. Although some of the DFAs are developed on the basis of constraints emerging from the physical considerations, the more successful ones do contain empirical parameters that must be determined by some means. This is usually done by fitting a functional to a few ab initio or experimental atomic properties, although it is preferable to include some molecular observables and reactivity patterns. For example, the low-quality performance of many early functionals for the spin-state energetics of TM complexes is partially a consequence of the fact that in the fitting procedure during their development, the spin-state energetics was never used in a data set. The important consequence of the fitted parameters besides the unwanted premise of semiempirical character is that these DFAs cannot be improved in a systematic manner. This is a major drawback compared to the wave function
based approaches. DFAs are commonly described by using the acronyms from the initials of the quantum chemists who have proposed them. DFAs are arranged into several classes with varying levels of complexity, the so-called Jacob’s ladder of increasing accuracy and sophistication in DFAs design and construction. The ladder begins at the simplest local density approximation (LDA) and should end up in the hypothetical exact functional. LDA depends solely upon the value of the electron density at each point in space. Since density in molecular systems is nonlocal, and a purely local description is obviously insufficient, the further developments include functions of the gradient, so-called generalized gradient approximations (GGAs). Meta-GGA DFAs incorporate the Laplacian and/or the kinetic energy density. Qualitatively, the incorporation of the derivatives is justified by the fact that energy is different in regions where density varies rapidly (close to the nuclei) compared to those where there are no abrupt changes (far away from the nuclei). The hybrid DFAs are obtained by a linear combination of the exact exchange interaction calculated from the HF theory, and XC from standard DFAs. HF exchange can also be included in a range-separated fashion, where long-range interactions are treated with HF while short-range interactions are modeled with GGA DFA. One of the most popular flavors of the hybrid DFAs is B3LYP (Becke, 3-parameter, Lee-Yang-Parr). Hybrid DFAs, and especially B3LYP, are widely accepted as a DFT “gold standards” for accurate property calculation. Consequently, they are by far the most cited and utilized DFAs by the chemist community. However, different researchers have their own “favorite” DFA, reflected in an annual online DFT popularity poll organized by Swart, Duran, and Bickelhaupt (http://www.marcelswart. eu/dft-poll/). One of the weak points of early DFAs was the description of dispersion interactions that led to the developments of dispersion (empirically) corrected and some other specialized functionals. It should be mentioned that B3LYP is not a “golden standard” for systems that contain TMs, but also that a “golden standard” for the complicated electronic structure of TM compounds does NOT exist (yet!). Density functionals and spin states The problem in the application of DFT for spin states was first noted in 2001 (Paulsen et al., 2001). Early GGAs favor low-spin states, while hybrid DFAs favor high-spin states. The larger the amount of HF exchange, the more biased toward HS states. This was used Practical Approaches to Biological Inorganic Chemistry

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by, for example, Reiher and coworkers to construct the B3LYP* functional that contains only 15% HF exchange (instead of 20% as in the original B3LYP). In the following years, many DFAs showed partial success, like TPSSh (meta-hybrid functional with 10% HF exchange, by Tao-Perdew-Staroverov-Scuseria), and B2PLYP (double hybrid that combine exact HF exchange with an MP2-like correlation to a DFA), but mainly failures in an attempt to tackle the problem of close-lying spin states in TM complexes. The combination of OPTX (Handy 2 Cohen Optimized Exchange) functional, and PBE (Perdew-BurkeErnzerhof) correlation parts, that is, OPBE, gave excellent results for the spin states of iron complexes. The initial success and some failures of OPBE led to the SSB-D (Swart-Sola`Bickelhaupt) DFA in which there is Grimme’s second generation (D2) dispersion contribution. Further refinements to make the DFA numerically more stable and to include Grimme’s third generation (D3) dispersion energy led to the improved SSB-D successor, S12g DFA. Demand for simple and accurate treatment of complicated electronic states of TM systems led to many validation studies of different DFAs, on various systems that proved to be challenging for spin state energetics. The three abovementioned DFAs that are specifically designed for spin states (e.g., OPBE, SSB-D, and S12g) have been shown to be excellent starting points for the vast diversity of interesting coordination compounds (Swart and Gruden, 2016). Just to repeat: the choice of the basis set is important also in DFT calculations. Because of the complicated electronic structure of TM a triple zeta (or even higher) basis set is required.

Computational methods for excited states As we have seen, depending on the nature of ligands, TM complexes with the same metal ion in the same oxidation state may exhibit different spin ground states. The consequence of being in either a high-spin or low-spin state is structurally usually accompanied only with changes of metal
ligand bond distances, but spectroscopic, magnetic, and other properties are drastically different. A complete understanding of the electronic structure of TM compounds, therefore requires explorations that go beyond solely that of the ground state. For experimentalists, it is well known that knowledge of electronic transitions in TM complexes is essential for understanding their chemistry and physics. Coordination compounds are so colorful, due to d
d transitions, and computational simulations are very useful tools for the deep understanding and even prediction of the excitation energies of various systems. Computational methods for excited states again belong to the two realms: (1) wave functional
based methods and (2) DFT-based methods. When we talk about wave functional
based methods for excited states (Neese et al., 2007) we again must mention CASSCF, CASPT2, or NEVPT2. Also, we can use multireference CI (MRCI) or spectroscopically orientated CI (SORCI). In addition, there are highly correlated schemes like equation-of-motion-couple-cluster (EOMCC) or algebraic diagrammatic construction (ADC). However, all these simulations are computationally very demanding, and when concerning TM systems, true experts in the field are required. Also, they suffer from the same problems when dealing with open shell TM complexes. For an accurate description of the excitations, it has been shown that results strongly depend on the selection of the active space and basis set used.

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Again, alternatives are provided by methods governed by DFT. The most popular DFT method for the calculation of the excited states is time-dependent density functional theory (TDDFT). The time-dependent Kohn
Sham equation is the time-dependent analog of the standard time-independent Kohn
Sham equation used to compute the ground state properties, as described previously. Description of the multiplets by TDDFT is given by a linear combination of single excitation (Vlahovi´c et al., 2015). TDDFT suffers from the same problem as DFT, as the exact time-dependent exchange-correlation potential functional is not known, although it is the most popular method for the excited states. Concerning d
d transitions, the most serious problem of TDDFT is related to the lack of orbital relaxation (of empty orbitals) in the transitions that depend only on Δ. It also has problems with states with the substantial character of double excitation that cannot be accessed by TDDFT. One of the most notorious examples where TDDFT failed are transitions in [Ni(H2O)6]21 where both effects are present (Vlahovi´c et al., 2015). The reader is directed elsewhere for a more detailed description of TDDFT and its success and drawbacks (Dreuw and Head-Gordon, 2004; Casida and Huix-Rotllant, 2012; Zlatar et al., 2016). Another method rooted in DFT is ligand field DFT (LFDFT) specially designed to treat multiplet levels of TM compounds (Atanasov et al., 2004; Daul et al., 2015). It allows the Racaha’s and LF parameters describing the excited state of a complex to be obtained by calculating the energies of all the microstates of a dn configuration by means of constrained DFT. The LFDFT thus combines both statistical and dynamical correlations, but it is limited only to d
d transitions.

Computational methods for biological systems containing transition metal Enzymes are highly developed molecular catalyst, essential for living organisms. Roughly speaking, one-third of enzymes have TM in their active sites. To understand and establish the mechanisms of their catalyzed reactions, molecular simulations and modeling are increasingly important, alongside experimental techniques. In this section, we only briefly outline some of the most widely used methods for modeling the structure and dynamics of metalloenzymes, and the reader is directed elsewhere for a more detailed description (Cramer, 2004; Cui, 2016; Warshel and Bora, 2016; Jensen, 2017). Bearing in mind all that has already been said about the complicated electronic structure of TM compounds, especially concerning spin which is an entirely electronic property, it is clear that for treatment of active sites of TM containing enzymes the QM level of theory is absolutely necessary. A common approach to investigate enzyme reactions, the so-called cluster approach, is to cut out the small model from the active site of the protein and to employ a continuum solvation model to model (mimic) the surrounding protein in an approximate manner. Another approach is to use a hybrid QM/MM (molecular mechanics) method, introduced in the 1976 paper of Warshel and Levitt (1976). They, along with Martin Karplus, won the 2013 Nobel Prize in Chemistry for “the development of multiscale models for complex chemical systems.” The active site of TM containing enzymes is treated with a QM method, while the rest of the enzyme is treated with an MMs force field. In MM energy functions (force fields), for example, harmonic terms

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represent the energy of bond stretching and valence angle bending, and simple periodic terms describe torsional angles. Van der Waals interactions are included by a simple Lennard
Jones function. Electrostatic interactions in MM force fields are usually described simply by using fixed atom-centered point charges. All these functions are parameterized together to represent the force field. The biggest advantage of MM methods is their computational cost, but they cannot model the bond breaking and bond making, and electronic reorganization, involved in a chemical reaction. Developed parameters are also usually transferable only to similar chemical compounds. In many (but not all) QM/MM methodologies the two regions, QM and MM interact with each other. Both approaches (cluster and QM/MM) have advantages and disadvantages, especially in terms of convergence. In QM/MM simulations additional problems are a selection of the region, the polarizability of QM atoms by MM region, and vice versa, the interaction between QM and MM regions, etc. Concerning the level of theory for QM region, all methods mentioned previously can be in principle used, but for the large biologically relevant systems only DFT is computationally tractable. Note that the choice of DFA here is also of the utmost importance. However, even hybrid QM/MM calculations using a modest DFT or ab initio region are usually limited to a minimum energy path type of calculation, or free energy simulations with a minimal amount of sampling. In some applications the biological system of interest is highly flexible and therefore a meaningful study would require an extensive degree of sampling; examples include metalloenzymes that feature a high degree of catalytic promiscuity, metal ion transporters/transcription factors, proteins involved in the assembly of catalytic metal cofactors, and peptides/proteins whose (mis)folding and aggregation behaviors are influenced by TM binding. For these problems, it is important to have a QM method that allows adequate sampling while treating the TM ions at an adequate level of accuracy. In this context, the traditional semiempirical quantum chemical methods, like MNDO (modified neglect of diatomic overlap), AM (Austin model), PMX (Parametric models, X 5 3, 5, 6), although being two to three orders of magnitude faster than DFT and ab initio QM methods, face difficulties since they are usually not accurate or transferable enough to deal with the complexity of TM ions. This is due in part to the fact that the basis of these semiempirical methods is HF, in which electron correlation is neglected entirely. Density functional tight binding (DFTB) emerged in recent years as an alternative semiempirical approach (Christensen et al., 2016). Since it is derived from DFT, the effects of electron correlation are included explicitly, albeit in an approximate fashion. The first set of DFTB2 parameters (mio) was developed for several first-row transition elements (i.e., Sc, Ti, Fe, Co, and Ni) by Morokuma and coworkers. They found reasonable accuracy in predicting geometrical properties of organometallic compounds and small metal clusters, although accuracy in energetics was lacking. Cui, Elstner et al. developed parameters for Cu and Ni in a DFTB3 model (3ob). Structures and ligand binding energies were found to be generally in good agreement with reference B3LYP calculations. However, parameters for other TM are still missing in the 3ob set, mainly because the average treatment of intra-d electron interaction is not enough to deal with strongly correlated d-electrons of TM ions.

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Concluding remarks This chapter has provided an introduction to LFT and computational chemistry, with the latter focused on TM compounds. All topics covered by this chapter deserve much more detailed explanations, however, the authors wanted to give the readers just an overview and refer readers to further reading in the particular subject. Computational chemistry of TM compounds is a challenging task, as there is no universal method so far for their complicated electronic structure. Thus this area is still wide open for new improvements.

Acknowledgments The authors thank the Ministry of Education, Science and Technological Development of the Republic of Serbia, Grant No. 172035, for the financial support.

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5337. Available from: https://doi.org/10.1021/acs.chemrev.5b00584. Cooksy, A., 2014. Physical Chemistry—Quantum Chemistry and Molecular Interactions. Pearson Education, Inc. Cotton, F.A., 1990. Chemical Applications of Group Theory, third ed. Willey. Cramer, C.J., 2004. Essentials of Computational Chemistry: Theories and Models, second ed. John Wiley & Sons Ltd, Chichester. Cramer, C.J., Truhlar, D.G., 2009. Density functional theory for transition metals and transition metal chemistry. Phys. Chem. Chem. Phys. 11 (46), 10757
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Cui, Q., 2016. Perspective: quantum mechanical methods in biochemistry and biophysics. J. Chem. Phys. 145 (14), 140901. Available from: https://doi.org/10.1063/1.4964410. Daul, C., Zlatar, M., Gruden-Pavlovi´c, M., Swart, M., 2015. Application of density functional and density functional based ligand field theory to spin states. Spin States in Biochemistry and Inorganic Chemistry. John Wiley & Sons, Ltd, Oxford, pp. 7
34. Available from: https://doi.org/10.1002/9781118898277.ch2. Dreuw, A., Head-Gordon, M., 2004. Failure of time-dependent density functional theory for long-range chargetransfer excited states: the zincbacteriochlorin 2 bacteriochlorin and bacteriochlorophyll 2 spheroidene complexes. J. Am. Chem. Soc. 126 (12), 4007
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Vincent, A., 2010. Molecular Symmetry and Group Theory, second ed. John Wiley & Sons. Vlahovi´c, F., Peri´c, M., Gruden-Pavlovi´c, M., Zlatar, M., 2015. Assessment of TD-DFT and LF-DFT for study of dd transitions in first row transition metal hexaaqua complexes. J. Chem. Phys. 142 (21), 214111. Available from: https://doi.org/10.1063/1.4922111. Wang, L., Zlatar, M., Vlahovi´c, F., Demeshko, S., Philouze, C., Molton, F., et al., 2018. Experimental and theoretical identification of the origin of magnetic anisotropy in intermediate spin iron(III) complexes. Chem. A Eur. J. 24 (20), 5091
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10674. Available from: https://doi.org/10.1021/acs.jpcc.6b02660. Zlatar, M., Schlaepfer, C.-W., Daul, C., 2009. A new method to describe the multimode Jahn-Teller effect using density functional theory. The Jahn-Teller Effect—Springer Series in Chemical Physics 131
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C H A P T E R

3

Molecular magnetochemistry Grace G. Morgan and Irina A. Ku¨hne School of Chemistry, University College Dublin, Dublin, Ireland O U T L I N E Introduction Origin of magnetism Contributions to angular momentum in free atoms and ions

69 70

Units and definitions Magnetic moment and the Bohr magneton Magnetic field and magnetization Zeeman effect Magnetic susceptibility, effective magnetic moment and magnetization saturation Curie law for noninteracting paramagnets Boltzmann treatment of magnetization

77

Contributions to magnetism in biologically relevant ions Orbital splitting of transition metal ions in crystal field Effect of crystal field on magnetic properties of 3d compounds

71

78 80 82 83 85 86 92 93 95

Dimeric sites: exchange mechanisms and J values Curie
Weiss law Superexchange Spin Hamiltonian Bleaney
Bowers equation

99 101 101 103 104

Diamagnetism

105

Experimental methods Magnetometry Evans NMR method Magnetic circular dichroism

108 108 113 114

Conclusion

115

Problems

115

Answers

116

Acknowledgments

119

References

119

Further reading

119

Introduction Magnetism is a fascinating property of both biological and nonbiological forms of matter, and has been intensively studied within different disciplines including chemistry, Practical Approaches to Biological Inorganic Chemistry DOI: https://doi.org/10.1016/B978-0-444-64225-7.00003-1

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© 2020 Elsevier B.V. All rights reserved.

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physics, and engineering. Modern theories of magnetism have therefore evolved from different scientific standpoints and this had led to parallel terminology and unit systems. Broadly speaking magnetic materials can be divided into molecular systems, including metalloproteins which contain magnetic ions, and solid-state materials, such as metals and metal oxides, which often comprise highly cooperative interacting magnetic sites. We are broadly interested in paramagnetic compounds, that is those which have unpaired electrons and hence a net magnetic moment. Paramagnetic properties are highly dependent on the periodic configuration of the magnetic ion—d-block or f-block—and the degree of interaction between neighboring magnetic ions. We will consider only molecular systems and from the perspective of bioinorganic chemistry we can focus our discussion on the biologically relevant magnetic ions, that is those of the 3d series, which are typically found as mononuclear or dinuclear sites in metalloproteins, such as the copper proteins. Sites of higher nuclearity are also known, the tetranuclear manganese site of photosystem II being a well-known example, and here the multiple interactions between many magnetic sites requires more complex treatments. At the end of this chapter you should be able to interpret the magnetic data for a simple metalloprotein, including quantifying the degree of interaction between adjacent sites in a dimeric system.

Origin of magnetism All matter is either paramagnetic or diamagnetic. Paramagnetism is typically associated with the presence of unpaired electrons and is an intrinsic property. In any paramagnetic material the elementary quantity is the magnetic moment, μ, which has both quantum and classical contributions. The quantum mechanical contribution arises from the intrinsic angular momentum associated with the electron spin. Separately, as the charged electron moves around the nucleus the orbital angular momentum associated with the motion through space generates a current loop which has an associated magnetic moment, as predicted by classical theories of electromagnetism. Both spin and orbital contributions are important and can be modeled as vector quantities. In a free atom or ion there is an additional contribution arising from how the spin and orbital angular momentum vectors combine with one another, this is termed spin
orbit coupling. However for most transition metal ions which are coordinated by ligands, including those coordinated by amino acid residues in metalloproteins, the change in energy of the d-orbitals which accompanies ligand-field formation reduces the degree of orbital motion and therefore also the possibility of spin
orbit coupling. For such systems the spin contribution is the most dominant and the most important in modeling the magnetic properties in metalloenzymes. Each of the contributions, spin, orbit, and spin
orbit coupling will be treated in turn in the following sections. Finally, we must also consider the weak negative magnetic effect termed diamagnetism, which arises in all matter when it is placed in an applied magnetic field. Diamagnetism is an induced effect which does not require the presence of unpaired electrons and which arises due to the change in orbital motion which results on application of the external field. Diamagnetism is also quantum mechanical in origin but is typically treated as a classical effect. It is important to be able to quantify the diamagnetic contribution of the closed shell components of a metalloprotein in order to accurately determine the paramagnetic contribution to the magnetic moment in open shell metal ions. This is normally achieved by adding the published values for specific atoms and functional groups in the coordination sphere. Practical Approaches to Biological Inorganic Chemistry

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Contributions to angular momentum in free atoms and ions In order to understand the contributions of spin and orbital angular momenta to the magnetic properties of transition elements in coordination environments we should consider first the internal electron
electron interactions in the free ion. Later we will examine how these are affected when the free ion is coordinated, for example, by amino acids in a metalloprotein. Atomic terms and states are a method of treating the electron
electron interactions in all atoms and ions, not just transition metal ions. We will first consider the electron
electron interactions in some free transition metal ions with unpaired electrons, and then consider how these are affected by a crystal field. Term symbols for free atoms and ions with one electron outside a closed inner shell In a simplified model we can consider that there is a momentum associated with both the spin and orbital motion of the single electron (Fig. 3.1). FIGURE 3.1 Depiction of spin (blue) and orbital (red) motion of a single electron beyond a closed shell of electrons in a free (uncomplexed) atom or metal ion.

In fact the spin angular momentum is not associated with motion as spin is a quantum mechanical property. It is however easier to visualize that the electron is spinning on its axis in a classical sense so this is the analogy that we will use. Orbital motion of the electron is classical so the model will fit well. Spin angular momentum (sam) of a single electron is quantified by the spin quantum number lower case s. Orbital angular momentum (oam) of a single electron is quantified by the orbital quantum number lower case l. Angular momentum is a vector quantity; therefore it has both magnitude and direction: s. We will consider now how both are quantified. 1. Magnitude The magnitude of the angular momentum vectors for spin and orbital motion is given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jsamj 5 sðs 1 1Þ ħ J s and joamj 5

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi lðl 1 1Þ ħ J s

where ħ 5 h=2π: For a single electron s 5 1/2 and the magnitude of the spin angular momentum vector is given by

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jsamj 5

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi 1=2 1=2 1 2=2 ħ J s pffiffiffi 3 ħ J s 5 2

The orbital angular moment can however take different values depending on the orbital quantum number l: Orbital

s

p

d

f

l

0

1

2

3

Consider now the specific case for a single electron in a d-orbital, for example, the valence electron in the Ti31 free ion. The orbital angular momentum value is l 5 2 so pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi joamj 5 2ð2 1 1Þ ħ J s pffiffiffi 5 6ħJs 2. Direction Quantum mechanics requires that the components of spin angular momentum along an arbitrary reference axis are half-integer and the components of orbital angular momentum are integer. For example, for a single electron in a d subshell the two directions for the spin angular momentum vector are 11/2 and 21/2. In a real sample of ions the arbitrary reference axis for spin or orbital angular momentum vectors will span all directions but in the aligning force of an applied magnetic field it will tend toward vertical (Fig. 3.2). The number of orientations (projections) for the sam of a single electron is given by ms where ms 5 2s 1 1 For a d ion ms 5 2 3 /2 1 1 5 2, that is, two directions, spin up or spin down, and the state is hence termed a spin doublet (Fig. 3.2). The number of orientations (projections) or directions for the oam of a single electron in a d-subshell is given by ml where 1

1

FIGURE 3.2 Left: Orientations of quantum mechanically allowed electron spin orientations with respect to a reference axis oriented in the vertical direction, for example, by application of an external magnetic field. Right: Depiction of random orientations of reference axes in a sample of paramagnetic ions in the absence of an aligning force.

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ml 5 2l 1 1 52321155 The quantum mechanically allowed orientations along an arbitrary axis are shown in Fig. 3.3 with the reference axis shown aligned in the vertical direction as would be the case on application of a magnetic field. Overall five orientations are possible for the orbital precession of a single electron in a d-orbital, each vector with the same magnitude of pffiffiffi 6 ħ J s. Spin
orbit coupling

The spin and orbital angular momenta of the single electron in the d-subshell are not independent of each another due to spin
orbit coupling. Spin
orbit coupling arises due to the interaction of the magnetic fields associated with sam and oam leading to j values for total angular momentum (tam). This is the vector sum of sam and oam and in quantum mechanics this can take values of         j 5 ðl 1 sÞ; ðl 1 sÞ 2 1; ðl 1 sÞ 2 2; . . . ; ðl 2 sÞ Therefore for a particular configuration there may be several values of total angular momentum j. For example, for the d1 configuration where s 5 1=2 and l 5 2; j can take values:      4 1   4 1      j5 1 2 ... 2 2   2 2  5

5 3 and 2 2

FIGURE 3.3 Projections of quantum mechanically allowed orbital orientations for an electron in a d-subshell with respect to a reference axis oriented in the vertical direction.

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3. Molecular magnetochemistry

What this reveals is that the electron configuration of 3d1 is just a simple address for the electron, but within that address there is more than one way to organize the spin and orbital degrees of freedom of that single electron and these represent states with different energies. The degeneracy of the resultant energy states is summed up in a term symbol which has the general form: 2S11 LJ where S is the total spin angular momentum for all electrons; L is the total orbital angular momentum for all electrons; and J is the total angular momentum for all electrons. A term symbol captures the degeneracy of the spin and orbital degrees of freedom via the 2S 1 1 and L parts of the symbol. The overall degeneracy of a free ion is ð2S 1 1Þ 3 ð2L 1 1Þ and this can be lifted by spin
orbit coupling into a subset of degenerate arrangements as illustrated below. The spin degeneracy is explicitly stated as 2S 1 1. The orbital degeneracy of the state is given as a letter with an associated 2L 1 1 multiplicity: Orbital state

S

P

D

F

G

H

I

L

0

1

2

3

4

5

6

The degeneracy of the orbital component of a state is analogous to the degeneracy of an atomic orbital with the same lower case label, for example, a D-state is fivefold degenerate like a d-orbital. The orbital component of the term symbol may have a high degeneracy in a multielectron ion so the labeling continues alphabetically beyond F, labels only up to a value of L 5 6 are shown in the table. For a single electron in a d-subshell the S and L values are s 5 S 5 1/2 l5L52 leading to an overall ð2S 1 1Þ 3 ð2L 1 1Þ 5 10-fold degeneracy written as 2D, that is, 2 3 5. The 10-fold degeneracy is lifted by spin
orbit coupling yielding two values for J, each with associated 2J 1 1 degeneracy: j 5 J 5 3=2 or 5=2 These are separated on an energy scale such that the state with J 5 3/2 lies lowest (Fig. 3.4).

FIGURE 3.4 Energy states arising from different combinations of the spin and orbital angular momentum vectors for the d1 configuration. The (5 3 2) 5 10-fold degeneracy is lifted resulting in two states with fourfold (J 5 3/2) and sixfold (J 5 5/2) degeneracy.

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FIGURE 3.5 Representation of spin (blue) and orbital (red) motion of two electrons beyond a closed shell of electrons in a free (uncomplexed) atom or metal ion.

We now consider how to combine spin and orbital angular momenta when more than one electron is present. Term symbols for free atoms and ions with more than one electron outside a closed inner shell We now need to consider electron
electron interactions as well as spin
orbit coupling. We will start with the strongest interactions and work down through weaker interactions in order. In our simplified pictorial model we can visualize two unpaired electrons outside a closed shell or shells of paired electrons, each orbiting on its axis and each orbiting through space (Fig. 3.5). If spin
orbit coupling is weak, as it usually is for light atoms and ions, including first row transition metal ions, the relative energetic order of the interactions between the two electrons is: 1. spin
spin coupling, 2. orbit
orbit coupling, and 3. spin
orbit coupling. This order forms the basis of Hund’s three empirical rules for predicting the ground state electronic arrangement: (1) the lowest energy arrangement is that with the maximum number of parallel spins, that is, greatest spin multiplicity, this rule takes precedence; (2) for a term of given spin multiplicity the term with highest value of L lies lowest in energy; and (3) if the subshell is less than half-filled Jmin lies lowest and if the subshell is more than half-filled Jmax lies lowest. The spin
orbit coupling value J can be calculated in two ways using either (1) the Russell
Saunders coupling method or (2) the heavy atom j
j coupling model. For the majority of elements in the periodic table the Russell
Saunders model gives the most accurate match to experimental results and it is always used for 3d transition metal ions. In the Russell
Saunders model the spin contributions (s) of all the unpaired electrons are combined to give a total S value and the orbital contributions (l) of all the unpaired electrons are combined to give a total L value (Fig. 3.6, left). S and L are then combined in the Clebsch
Gordan series to yield the possible values of J:         J 5 ðL 1 SÞ; ðL 1 SÞ 2 1; ðL 1 SÞ 2 2 . . . ðL 2 SÞ

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FIGURE 3.6 Comparison of Russell
Saunders and j
j coupling models: in the Russell
Saunders scheme (left) all spin contributions and all orbital contributions are summed separately; in the j
j coupling model (right) the spin and orbital contributions of each electron are summed individually before combining for the total number of electrons.

For much heavier elements, such as the heavy transition metals, the spin
orbit coupling becomes more dominant and the j
j coupling model is used. In the j
j coupling model the si and li components of each electron are combined to yield a total angular momentum value, ji , for that single electron. The ji values for all the unpaired electrons are then combined to give a total J value (Fig. 3.6, right). In the j
j coupling model the separate spin (S) and orbital (L) angular momentum values are not specified so Hund’s rules do not apply. Using Hund’s rules for the lighter elements we can accurately predict the spin and orbital arrangements for the ground state of most atoms or ions, predicting the arrangements in the excited states is less straightforward. However in the majority of transition metal complexes the ground state is the only one which is populated at room temperature and below. Let us now consider as a simple example the V31 free ion which has the electronic configuration [Ar]3d2 . Many possible arrangements are possible for the two electrons in the d-subshell, for example, excited state arrangements with S 5 0 and L 5 4 (1G), S 5 0 and L 5 2 (1D), and that with S 5 1 and L 5 1 (3P) are depicted below.

These are allowed states but none represent the ground electronic state where the electrons will arrange so as to have maximum values for the spin and orbital multiplicities and the minimum value for J as follows.

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77

FIGURE 3.7 Energy diagram for the d2 configuration showing effects on energy levels for different combinations of spin and orbital angular momentum vectors and associated degeneracies.

The relative energies of the states arising from different combination of the spin and orbital angular momentum vectors are shown in Fig. 3.7 with indicative wavelength ranges for the energy gaps. In Section 2 we will now consider what happens to the angular momentum when the atom or ion is placed in a magnetic field. Later in Section 3 we will consider the different types of magnetic response which arise when an ion with unpaired electrons is placed in a crystal field.

Units and definitions The different theoretical perspectives on magnetism which have evolved within the disciplines of chemistry and physics have resulted in two different systems of units to describe magnetic properties. Syste`me International (SI) units are most widely used in the current literature of magnetic compounds but many chemistry reports still use centimeter
gram
second (cgs) units and much of the earlier work on bioinorganic magnetism is quantified in this way. Conversion factors will be supplied as required for the definitions and derivations which are described below. When deriving equations it is also useful to be familiar with the fundamental constituents of the SI units shown in Table 3.1:

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TABLE 3.1 Fundamental constituents of some SI units. SI unit

Constituent units

Newton (N)

kg m s22

Joule (J)

kg m2 s22

Tesla (T)

kg s22 A21

Weber (Wb)

kg m2 s22 A21

Henry (H)

kg m2 s22 A22

FIGURE 3.8 Left: Magnetic moment arising from current loop around area A. Right: The total magnetic moment is modeled as the sum of the current loops in the sample.

Magnetic moment and the Bohr magneton In classical electromagnetism the magnetic moment, μ, is modeled as the vector quantity normal to the plane of a circulating current I around a loop of area A (Fig. 3.8) so that dμ 5 I dA and the moment has units of current times area, that is, A m2. The total magnetic moment is calculated by summing the moments of all the current loops over the total area (Fig. 3.8) so that ð ð μ 5 dμ 5 I dA A clockwise current loop corresponds to the south pole of a bar magnet and anticlockwise to the north pole (Fig. 3.9). The current I is modeled as arising from the motion of massless charges (electrons). However the electron has mass and its motion results in an orbital angular momentum L. The classical theory of electromagnetism predates quantum mechanics and the discovery of spin therefore it does not take into account spin angular momentum or spin
orbit coupling. In the original theory the angular momentum was modeled as arising solely from motion through space, that is, orbital angular momentum, L. The magnetic moment is, in general, proportional to angular momentum of all types and we will return to this relationship several times, μ~L

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79

FIGURE 3.9 Top: Direction of field lines around a bar magnet from north to south pole. Bottom: Direction of current flow and resulting magnetic pole type.

FIGURE 3.10 Pathway of a single electron in a hydrogen atom orbiting around the nucleus with velocity v.

The proportionality constant is termed the gyromagnetic ratio, γ μ 5 γL The magnitude of the magnetic moment of any atom or ion with a number of unpaired electrons in a real chemical environment is usually reported in multiples of the Bohr magneton, μB , a constant of value 9.274 3 10224 A m2. The Bohr magneton is the magnetic moment associated with a single electron in an isotropic environment, that is, not influenced by chemical bonding. The Bohr magneton factors in the contribution of the angular momentum associated with electron motion by including the reduced Planck’s constant ħ 5 h=2π, first introduced by Bohr as the quantum of angular momentum for a single electron. The value for the Bohr magneton is obtained by calculating the size of the magnetic moment generated by the current loop of a single electron and including an angular momentum contribution of unity, that is, ħ. For example, if we consider the single electron in the hydrogen atom tracing a loop of area A, with radius r and velocity v (Fig. 3.10) the time, t, required for the electron to complete one complete circuit around the area ðA 5 πr2 Þ is distance/velocity, that is, 2πr/v. t5

2πr v

The current I is the charge ( 2 eÞ per time ðtÞ: I5

2e t

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and substituting for t gives: I52

ev 2πr

We have already defined the magnetic moment as the product of current times area, μ 5 IA, and substituting for I and A gives: μ52

ev 3 πr2 2πr

which reduces to μ5

2evr 2

The contribution of angular momentum can be given by factoring in the mass (me ) and velocity (v) of an electron orbiting with radius r so that ħ 5 me vr ħ 5 vr me μ5

2eħ  2μB 2me

where μB is the Bohr magneton, defined as μB 5

eħ 2me

The value of the Bohr magneton can then be determined using the values: e 5 1:6021766208 3 10219 A s ħ 5 1:054571800 3 10234 J s me 5 9:10938356 3 10231 kg This works out as 9.274 3 10224 A s2 J/kg and with unit conversions (Table 3.1) is usually given as 9.274 3 10224 J/T or 9.274 3 10224 A m2.

Magnetic field and magnetization Magnetic field can be described by the vectors B and H which in a vacuum are related linearly by B 5 μ0 H where μ0 (not to be confused with the magnetic moment which we indicate in bold font as μ) is the permeability of free space, and has the value μ0 5 4π 3 1027 H=m1

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81

FIGURE 3.11 Magnetic flux lines inside a paramagnetic sample (left) and diamagnetic sample (right).

with units of henry per meter, H/m. Most literature refers to H as the magnetic field and it has units of A/m. Strictly speaking, H is the magnetic intensity and the units of amperes per meter reflect that it is a measure of how much current is flowing through the coil of an electromagnet to produce the external field. The value of H is not affected by the material being measured. When an external magnetic field H acts on a material the response is commonly termed the magnetic induction B which is a measure of the degree to which the external field H permeates the sample and is given by B 5 μH where μ is the permeability of the sample. The magnetic induction simply describes how much field exists inside the sample once the external field is applied and is also referred to as the magnetic flux density, or simply magnetic flux. The magnetic induction can be modeled as lines of flux with a greater density of flux lines inside the sample than outside in paramagnets and a reduced density inside the sample compared with outside in diamagnets (Fig. 3.11). Paramagnets are therefore attracted to a magnetic field while diamagnetic materials are repelled. In a real sample μ differs from μ0 and the sample is said to be magnetized by the external field such that B 5 μ0 ðH 1 MÞ where M is the magnetization or magnetic moment of the sample with SI units of A/m. The SI units of both M and H are A/m and those of μ0 are henry per meter (H/m). Using the values in Table 3.1 we can show that the units of B thus correspond to tesla (T). Therefore the SI units of B and H are not the same: the SI unit of magnetic induction (B) is tesla (T) and that of magnetic field (H) is A/m. In practice in the literature the magnetic field H is often reported in units of tesla. In cgs units B and H are related by the expression: B 5 H 1 4πM Note that in the cgs system the units of B and H are also not the same: the cgs unit of magnetic induction (B) is gauss and that of magnetic field (H) is oersted. Magnetization is a measure of the magnetic moment per unit volume and in the cgs system has units of erg/gauss or emu per volume: M5

m emu V cm3

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FIGURE 3.12 Generic energy diagram for Zeeman splitting of angular momentum vectors in applied magnetic field.

The erg is a cgs unit of energy equivalent to 1027 J. The term “emu” is not a unit but is used to describe a system of units and is used when reporting M in the most common experimental setup, the SQUID magnetometer. Multiplying M by the molar mass/mass of sample yields the molar magnetization Mmol .

Zeeman effect In its most general form the Zeeman effect describes the energy difference between states with more than one direction for the angular momentum vectors when their degeneracy is lifted by an external magnetic field (Fig. 3.12). The angular momentum vector may arise from quantum mechanical spin, classical orbital motion, spin
orbit coupling, or some combination of the three types. The magnetic field applied during the experiment may be weak or strong. Therefore the manifestation of the Zeeman effect depends on both the internal electronic structure and the magnitude of the applied field. The overall goal is to measure the energy difference, ΔE between the states with different orientations of the angular momentum vector. Normal Zeeman effect The normal Zeeman effect is observed in systems with closed shells of electrons and arises due to lifting of the orbital degeneracy by an external magnetic field of moderate strength. It is manifest as equally spaced fine structure lines, for example, in absorption or emission spectra (Fig. 3.13, left) and is a property of diamagnetic materials. The energy of each state is E 5 ml μB H; where ml is the component of the magnetic quantum number, l. Anomalous Zeeman effect Paramagnetic materials have unpaired electron spins and exhibit the so-called anomalous Zeeman effect. In spectroscopy this is manifest as further hyperfine structure in absorption or emission spectra collected in an applied magnetic field. Different degrees of splitting can be observed depending on the complexity of the internal electronic structure

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FIGURE 3.13 Left: Normal Zeeman effect showing allowed transitions between s- and p-orbitals in a diamagnetic molecule and resulting absorption or emission spectra with and without an applied magnetic field. Right: Anomalous Zeeman effect showing corresponding transitions in a paramagnetic molecule.

of the magnetic ion and the strength of the magnetic field. The primary splitting is termed the first-order Zeeman effect and additional splitting may be termed second order or higher. In practice, however, effects beyond second order are not observed. Second-order effects do become significant for some transition metal compounds however, particularly for those with significant zero field splitting (ZFS). Second-order Zeeman effects correspond to changes proportional to the square of the magnetic field H2 and are opposite in sign to first-order effects. In the first-order effect the energy gap between states with different values of the angular momentum MJ factors in the contribution of spin and spin
orbit coupling by including the Lande´ g factor, a constant analogous to the gyromagnetic ratio. The energy of each level is E 5 gmJ μB H and the energy difference between mJ levels is ΔE 5 gμB H The Lande´ constant factors in the contributions from the vectors associated with spin, orbital, and spin
orbit angular momentum in the following relationship: g511

J ðJ 1 1Þ 1 SðS 1 1Þ 2 LðL 1 1Þ 2J ðJ 1 1Þ

Magnetic susceptibility, effective magnetic moment and magnetization saturation The procedure to characterize the magnetic properties of any sample is to place it in an applied magnetic field, H, and measure the magnetization, M. The magnetization is a

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FIGURE 3.14 Left: Magnetization response (M) at different magnitudes of applied field (H) recorded at constant temperature. Right: Thermal response of magnetic susceptibility at low applied fields (region A of the MvH plot) showing inverse temperature dependence.

measure of the sum of the individual magnetic moments in the sample and is temperature dependent at low magnetic fields, that is, follows Boltzmann statistics. The applied field may be weak, typically less than 1 T, or strong, 2
14 T in most experimental setups. The magnitude of the applied field guides the property to be measured because at low fields M is linearly proportional to H, Fig. 3.14, region A. M~H The constant of proportionality is termed the magnetic susceptibility, χ: M 5 χH where χ is a measure of how susceptible the unpaired spins in the sample are to the aligning force of the external field and is temperature dependent. At low temperature the system has less thermal energy to resist the aligning force and in paramagnets the susceptibility increases (Fig. 3.14, right). At high temperature thermal energy opposes the aligning force of the field and the susceptibility falls. At higher fields, typically above 2 T, the linear relationship between M and H is lost as all the unpaired spins align with the external field and the system is said to be saturated, that is, reaches magnetization saturation (Fig. 3.14 region B). In this regime it is possible to measure the magnetic moment per ion. Typically one of two measurement sequences is used: 1. sweeping the temperature at constant (low) applied magnetic field to measure χ (region A) and 2. sweeping the magnetic field at constant (low) temperature to measure Mion (region B). Typically what is reported is one of the following: 1. the susceptibility χ in units of cm3/mol (region A); 2. the effective magnetic moment μeff , in multiples of the Bohr magneton, μB (region A); and 3. the magnetic moment per ion, Mion , also in multiples of the Bohr magneton, μB (region B).

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μeff is really a measure of the susceptibility, χ, which also factors in the temperature dependence. It is seldom used in current literature although it persists in the inorganic curriculum. It has largely been replaced with the quantity χm T, where pffiffiffiffiffiffiffiffiffi μeff 5 2:828 χm T Mion and μeff are not the same quantity despite both being termed “magnetic moment.” This is because they are measured under different conditions. Mion is obtained only at high fields and low temperatures when all the spins are forced to align with the field (region B). In transition metal compounds the value of Mion should match the number of unpaired spins in the sample, for example, Mion 5 5μB for a mononuclear Mn21 complex with five unpaired spins. In the cgs system Mion is obtained using the following: Mion 5

Mmol μ NA B

with units of erg/gauss for the Bohr magneton μB . In contrast the effective magnetic moment μeff is obtained only at low fields and often at high temperature, for example, room temperature and is applicable only in region A of the MvH plot in Fig. 3.14.

Curie law for noninteracting paramagnets At low magnetic fields in noninteracting paramagnets, that is, region A of the MvH plot, there is an inverse relationship between magnetic susceptibility, χ, and temperature, T: χ~

1 T

The constant of proportionality is the Curie constant C leading to the expression for the Curie law: χ5

C T

The Curie law allows the temperature dependence of the magnetization to be expressed according to Boltzmann statistics because M 5 χH Therefore M C 5 H T The value of the Curie constant is unique to each sample and takes into account the number of individual magnetic moments aligned with or against the field. This is obtained using a Boltzmann partition function which we consider in the following section.

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Boltzmann treatment of magnetization As a general principle an increase in magnetic field tends to align the individual magnetic moments while an increase in temperature tends to disorder them. The relative populations of individual moments aligned with or against the field follow Boltzmann statistics, that is, are partitioned across the allowed orientations to different degrees depending on thermal energy. Two aspects must be considered in any theoretical treatment: first the number of ions in the sample (taken as Avogadro’s number for one mole of sample NA ), and second the number of allowed orientations (or states) for spin alignment within each ion, μn . The bulk magnetization per mole can then be estimated by summing together all orientations at a given temperature using the partition function: P 2E=kB T n μn e M 5 NA P 2E=k BT ne We can write an expression for μn if we consider that in classical mechanics the magnetization of a sample perturbed by an external magnetic field H is related to the change in energy by M52

δE δH

The microscopic magnetization μn of any state can be written as μn 5 2

δEn δH

The partition function for magnetization can now be written as P  δEn  2En =kB T 2 e M 5 NA n P δH 2E =kB T n ne Langevin paramagnetism The first approach to sum the individual moments was developed by Paul Langevin in 1905, that is, before the advent of quantum mechanics. The Langevin function considers that the magnetic moment of an individual spin can lie at any angle θ to the axis of the applied field H (Fig. 3.15), rather than starting with the more complete model of quantized orientations which is included in later treatments. The contribution of all ions lying at an angle of θ 1 dθ can be estimated by calculating the area of the shaded annulus (Fig. 3.15), factoring in the Boltzmann probability for the number of spins adopting this orientation at a given temperature. The Langevin function sums the areas for all angles with a rather complex expression (see Spaldin for complete treatment) which can be expanded as a Taylor series (which introduces the factor of 1/3) to yield the more manageable relationship: Mmol 5

NA μ2 H 3kB T

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FIGURE 3.15 Area of shaded annulus used by Langevin to map out the fraction of paramagnetic moments between θ and θ 1 dθ around an axis to sum the average contributions from each.

From this we can extract an expression for the Curie constant C: χ5

M H

5

NA μ2 3kB T

5

C T

Dividing through by T gives the Curie constant as C5

NA μ2 3kB

In older nomenclature (cgs) the moment is defined as the effective magnetic moment and is measured in Bohr magnetons so the susceptibility is given by χ5

NA μeff 2 μB 2 3kB T

kB 5 1:38064852 3 10216 erg K21 NA 5 6:022140857 3 1023 mol21 μB 5 9:274009457 3 10221 erg G21 Rearranging gives μeff 2 5

3kB χT NA μB 2

μeff 2 5 7:99χT

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which yields the expression we introduced in Section 2.4 (Effective Magnetic Moment) where μeff is given in Bohr magnetons: pffiffiffiffiffiffiffiffiffi μeff 5 2:82 χm T

Brillouin function The quantum mechanical approach to summing the individual moments introduces the constraint that only quantized orientations (J levels) for the magnetic moment are permitted. This is captured in the Brillouin function which is equivalent to the Langevin function when J -N (see Blundell or Spaldin for complete treatment). The Brillouin function can also be expanded in a Taylor series to yield the quantum mechanical expression for the susceptibility: χ5

NA g2 J ðJ 1 1ÞμB 2 C 5 T 3kB T

Dividing through by T gives the expression for C: C5 5

NA g2 J ðJ 1 1ÞμB 2 3kB NA μeff 2 3kB

where μeff 5 g

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J ð J 1 1Þ μ B

The Brillouin function is applicable over the whole MvH range, that is, in both regions A and B and every part in between. Van Vleck equation The Boltzmann treatment to sum all the individual orientations for the magnetization is accurate and general, that is, can be applied using both classical and quantum mechanical approaches.   P δEn 2En =kB T 2 e n δH P 2E =k T M 5 NA n B ne However, it is often difficult to apply since it requires knowledge about all the electronic states En in order to calculate the microscopic μn states of the magnetization. In 1932 van Vleck simplified the partition function by making two assumptions: Assumption 1: Van Vleck used the fact that in a general sense the energy En of a particular level n of an ion in a magnetic field can be expanded according to the increasing

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powers of H where Enð0Þ is the energy of level n in zero field, and Enð1Þ and Enð2Þ represent the first- and second-order Zeemann coefficients: En 5Enð0Þ 1 Enð1Þ H 1 Enð2Þ H 2 1 ? If higher Zeeman coefficients beyond second order are ignored, differentiation with respect to H gives: δEn 5 Enð1Þ 1 2Enð2Þ H δH Using the fact that μn 5 2

δEn δH

It follows that μn can be written as μn 5 2 Enð1Þ 2 2Enð2Þ H Assumption 2: Van Vleck applied the Boltzmann distribution of the magnetization only in the magnetic field range where MvH exhibits a linear slope. The van Vleck equation is therefore only valid in region A of the MvH plot, that is, where the magnetic field H is not too large and the temperature T is high. At high temperatures it is assumed that kB TcEn So 2 kB TcEð0Þ n 1 Enð1Þ H 1 Enð2Þ H 1 ?

And the exponential term in the Boltzmann distribution can be rewritten as   Enð1Þ H 2Ek nTð0Þ 2En =kB T e D 12 e B kB T

2Enð0Þ

Combining Assumptions 1 and 2, and substituting the new expressions for μn and e into the Boltzmann distribution gives: 2Enð0Þ  P E H e kB T 2 Enð1Þ 2 2Enð2Þ 1 2 knð1Þ BT MDNA 2Enð0Þ P E H e kB T 1 2 knð1Þ T B

kB T

And rearranging gives the Van Vleck formula, where it is only necessary to know the quantities Enð0Þ ; Enð1Þ , and Enð2Þ : 2Enð0Þ P Enð1Þ 2 kT n kB T 2 2Enð2Þ e M 5 NA H P 2Ek nTð0Þ B ne

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The only energy levels required are those in the absence of a magnetic field (Enð0Þ ) and those with first (Enð1Þ ) and second (Enð2Þ ) order Zeeman splitting, beyond that the energies are so small they can be ignored. Recalling that χ 5 M=H leads to the following expression for the magnetic susceptibility: P χ 5 NA

2Enð0Þ

Enð1Þ 2 n ð kB T

2 2Enð2Þ Þe P 2Ek nTð0Þ B ne

kB T

The second-order Zeeman coefficient Enð2Þ vanishes when all energies En are linear in H, that is, region A of the MvH graph and under these conditions (low field, high temperature) the susceptibility due to first-order effects is given by NA χ5 : kB T

P

n ðEnð1Þ

P

2

2Enð0Þ

Þe

kB T

2Enð0Þ

ne

kB T

Curie constant and spin-only effective magnetic moment The Langevin and Brillouin functions yield useful expressions for the Curie constant which relate the susceptibility to the effective magnetic moment (Langevin) or define the effective moment in terms of the quantized J energy levels (Brillouin). As we will see in the next section, orbital angular momentum is often reduced or quenched in complexes of transition metals in contrast to the free ions. This is because the crystal field lifts the degeneracy of the d-orbitals, thereby curtailing the number of orbital pathways within a single set of equi-energetic orbitals. For this reason the main contribution to the susceptibility (and therefore μeff ) in many first row transition metal compounds is just spin, and very often μeff is reported only for the spin contribution, the so-called spin-only effective magnetic moment. This is usually obtained from application of the Van Vleck equation to the quantized energy levels in a similar approach to Brillouin, but now applied to just the S levels and only in region A of the MvH plot. Van Vleck also introduced the simplification that only the ground state energy term would be included, that is, a large separation between the ground state and the first excited state is assumed, which would rule out any coupling between the two levels. The isolated 2S 1 1 ground state Zeeman states are degenerate in the absence of an external magnetic field, while the presence of a magnetic field splits the degeneracy evenly with the Ms levels varying in integer steps from 2 S to 1 S. En 5 Ms gμB H Choosing the energy state of the ground state as the energy origin, is a convenient choice, still with the assumption that the first excited state is too high in energy to allow coupling with the ground state: E n ð0 Þ 5 0

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If H/kB T is small, the energy levels En are proportional to the magnetic field H and the van Vleck equation can be applied yielding Enð1Þ 5 Ms gμB with Ms5 5 2 S; 2 S 1 1; . . . S 2 1; S This leads to the expression for the molar magnetic susceptibility: χ5

NA g2 μB 2 X1S MS 2 MS52S ð2S 1 1Þ kB T

Rearranging gives χ5

X1S NA g2 μB 2 1 : MS 2 MS52S ð2S 1 1Þ kB T

P1S 2 and MS52S MS can be determined using the mathematical expression for the sum of squares for first n positive integers, n in this case multiplied by a factor of 2 as the values span from 2 S to 1 S, not just from 0 to 1 S. This gives: 1S X MS 52S

M2S 5

2SðS 1 1Þð2S 1 1Þ 6

We can now write the following expression for the susceptibility: χ5

NA g2 μB 2 1 2SðS 1 1Þð2S 1 1Þ : : ð2S 1 1Þ 6 kB T

yielding the following expression for the susceptibility: χ5

NA g2 μB 2 :SðS 1 1Þ 3kB T

Recalling that χ5

C T

the Curie constant can be expressed as C5

NA g2 SðS11ÞμB 2 1 2  g SðS 1 1Þ 8 3kB C5

NA g2 J ðJ 1 1ÞμB 2 3kB

and μeff 5 g

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SðS 1 1ÞμB

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TABLE 3.2 Spin-only χT and μeff values for S states S value

χT (cm3K/mol)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μeff 5 2 SðS 1 1ÞμB

1/2

0.375

1.73

1

1.000

2.83

3/2

1.875

3.87

2

3.000

4.90

5/2

4.375

5.92

3

6.002

6.92

7/2

7.878

7.94

where C5

NA μeff 2 3kB

The Curie law is only valid, when the applied field H is small and the temperature T is relatively high, leading to H { kB T. If the applied field is large, H=kB T gets too big, and the magnetization curve will reach its saturation level, MS, where all magnetic dipoles are ordered. In the current literature paramagnetic properties at low field are usually reported as χT but μeff is used in older texts. Values for both are listed in Table 3.2 for values of S in the range 1/2
7/2. Temperature-independent paramagnetism and the second-order Zeeman effect The Van Vleck treatment also explains why some metal complexes with J 5 0 (or just S 5 0, as we will see in the next section) still show a small paramagnetic response. This is termed van Vleck paramagnetism or temperature-independent paramagnetism. It arises due to the contribution from second-order Zeeman splitting. When the magnetic field is applied the primary effect is the proportional response to H, this is the first-order Zeeman splitting that is, the symmetrical splitting of the orientation of the ion’s magnetic moment with and against the field (Fig. 3.13; Zeeman splitting). In addition there is a change in energy due to H2 , the second-order Zeeman splitting which can be regarded as a distortion of the electron distribution in the ground state by the field.

Contributions to magnetism in biologically relevant ions Biologically relevant magnetic ions include many of the first row transition elements in a variety of oxidation states. We will also compare the electronic structure of coordinated transition metal ions with those of the heavier 4f block of elements to illustrate the

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FIGURE 3.16 Crystal field energy splittings of the d-orbitals in different geometries.

difference in magnetic properties which arises between coordinated ions with a strong overlap with the ligand orbitals (3d series) and those with poor overlap (4f series).

Orbital splitting of transition metal ions in crystal field The magnetic properties of transition metal ions are profoundly affected when they are coordinated by donor atoms in a crystal field. Most notably the degree of orbital freedom is quenched in the majority of cases so the primary contribution to the magnetic properties is that of spin. However, in some cases orbital contribution and hence spin
orbit coupling may persist to a limited degree. We will look at each situation in turn and also compare with the magnetic lanthanide metal ions which, unlike their lighter transition metal analogues, retain their free ion characteristics even when bonded to a set of ligand donors thus requiring a different treatment of their magnetic properties. Spectroscopic term symbols are no longer a good description of the electronic arrangements of transition metal ions in the presence of a crystal field as relatively strong bonding interactions lift the degeneracy of the d-orbital energies. In the free ion all five d-orbitals are degenerate but in a crystal field the d-orbital energies are perturbed. The degree and arrangement of perturbation depends on the number of incoming ligands and resultant geometry yielding a different set of orbital energy diagrams depending on the shape (Fig. 3.16) In addition the crystal field may be weak or strong as shown for the octahedral case (Fig. 3.17). It should be noted that transition metal ions in an octahedral environment, which belong to one of the four electronic configurations of d4 to d7 can exist in the high-spin form where the number of unpaired electrons is maximized, or in the low-spin form where the number of unpaired spins is minimized. In rare cases, these metal complexes can undergo spin transition by application of an external stimulus. This is most commonly observed in Fe21 compounds, which can undergo a spin transition from S 5 0 to S 5 2,

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FIGURE 3.17

3. Molecular magnetochemistry

Spin populations of the d-orbitals in octahedral geometry with increasing crystal field strength.

but it is also possible in compounds containing Fe31, Co21, or Mn31. The high- and lowspin arrangements for d5 are illustrated in Fig. 3.17. In a weak
medium crystal field, Δo is small. In terms of its effect on the manifold of energies of the d-electrons, we can consider that it is treated as a further perturbation of the electronic energy levels of the free ion that arise due to spin
spin and orbit
orbit interactions. In terms of treating the magnetic properties of biologically important transition metal ions we should note that crystal field splitting is generally larger than spin
orbit coupling so the contribution of spin
orbit coupling to the magnetic response of transition metal compounds is usually discounted in a crystal field (Fig. 3.18).

FIGURE 3.18

Perturbation of the manifold of energy levels in the d1 ion by a weak
medium crystal field.

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TABLE 3.3 Symmetry terms arising from free ion terms of transition metal complexes in octahedral geometry. Free ion orbital terms

Crystal field orbital terms (Oh)

L

2L 1 1

Label

0

31

S transforms as

A1g

1

33

P transforms as

T1u

2

35

D transforms as

Eg, T2g

3

37

F transforms as

A2g , T2g , T1g

In the crystal field the spectroscopic labels S, P, D, F; etc. are no longer valid as the orbital degeneracy is lost when the d energy levels are split. Instead we must consider the symmetry of the complex and describe the new energy states with symmetry labels. Spin multiplicity is still denoted as a 2S 1 1 superscript and the orbital degeneracy is now described as A, E; or T. A 5 nondegenerate (“singly” degenerate) E 5 doubly degenerate T 5 triply degenerate The S, P, D, F . . . terms are split into components with the same (or similar) symmetry labels (A, E, or T) as components arising from orbitals of the same symmetry (Table 3.3). The subscripts in the crystal field orbital terms include g or u labels which stand for gerade or ungerade. These are the German expressions for even or odd and the label indicates whether inversion through the center results in a change of phase (u) or if the orbital sign is unchanged by the operation (g).

Effect of crystal field on magnetic properties of 3d compounds As noted in the introduction the elementary property of paramagnets is the magnetic moment, μ, which originates in the spin and orbital motion of the unpaired electrons and in spin
orbit coupling. We now consider the respective contributions of each type of angular momentum to the observed magnetic moment in metal complexes (Fig. 3.19). Complexes with open shell ions from different parts of the periodic table have different contributions from S, L, and J resulting in different methods of calculating the magnetic moment. The magnetic moment is always directly proportional to the total angular momentum of the electron. In complexes of the 4f block crystal field effects are very small (Fig. 3.20, left), so the magnetic moment of lanthanide metal compounds is modeled as being proportional to the total angular momentum J: μ~J In contrast in most coordinated transition metal ions orbital motion is halted by the loss of the degeneracy of the five 3d orbitals in a crystal field which thus constrains free

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FIGURE 3.19 Depiction of three contributions to magnetic moment: spin, orbital, and spin
orbit coupling. (A) Spin (S): this is a quantum effect as spin is inherently magnetic. (B) Orbital (L): this is a classical effect—the motion of a charged particle through space gives rise to a magnetic field. (C) Spin
orbit (J): this is essentially a relativistic effect because from the reference frame of the electron a charged nucleus appears to move through space. The electron is inherently magnetic because of its spin and therefore it senses the magnetic field arising from the relative motion of the charged nucleus.

FIGURE 3.20

Comparison of the effect of complex formation in f-block and d-block metal ions.

movement through space (Fig. 3.20, right). Therefore the sole contribution to the angular momentum is from the electron spin as the orbital and spin
orbit contributions are quenched or considerably damped. The magnetic moment of transition metal compounds is therefore modeled as being primarily proportional only to spin angular momentum S: μ~S The magnetic moment is also defined differently under different experimental conditions as we noted in Section 2. We can consider either the total magnetic moment Mion obtained when all spins are forced to align at high magnetic fields, this is also termed the saturation magnetization. Alternatively we can consider the so-called effective magnetic moment μeff which is obtained experimentally at much lower magnetic fields and usually much higher temperatures. Mion reflects the proportional relationship between moment and angular momentum and the expression includes a unique proportionality constant g for each of the lanthanides, as this varies depending on the degree of spin
orbit coupling

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TABLE 3.4 Contribution of the angular momentum to the magnetic moment. 4f Complex

3d Complex

Angular momentum contribution

μ~J

μ~S

Moment in high magnetic field

Mion 5 g  JμB pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μeff 5 g JðJ 1 1ÞμB

Mion 5 2:SμB pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μeff 5 2 SðS 1 1ÞμB

Moment in low magnetic field

TABLE 3.5 Energy states and typical magnetic moments for first row transition complexes. Ground state term symbol in Oh geometry

Ion

dn

Ti31

[Ar]3d1

2

2

3

3

4

4

5

5

6

6

5

7

4

8

3

9

2

V

41

V

31 31

Cr V

T

Mn

[Ar]3d [Ar]3d

T A

[Ar]3d

E

[Ar]3d

A

21

Fe

21

Co

21

Ni

21

Cu

g

μeff (μB ) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SðS 1 1ÞμB

μeff (μB ) observed

154

1

1

1.55

1.73

1.7
1.8

209

2

2

1.63

2.83

2.8
3.1

273

3

3

0.77

3.87

3.7
3.9

4

4

0

4.90

4.8
4.9

5

5

5.92

5.92

5.7
6.0

410

4

4

6.70

4.90

5.0
5.6

535

3

3

6.63

3.87

4.3
5.2

649

2

2

5.59

2.83

2.9
3.0

829

1

1

3.55

1.73

1.9
2.1

352

460 347

Mn

21

μeff (μB ) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g J ðJ 1 1ÞμB

230

Cr Fe

Mion (μB )a

167

21

31

Number of UPE




21 31

ζ (cm21)

[Ar]3d [Ar]3d [Ar]3d [Ar]3d

T T A E

a

The saturation magnetization values of TM complexes are more likely to depend solely on spin contribution but the degree of spin
orbit contribution varies from sample to sample. UPE, unpaired electrons.

(Table 3.4). The expression to calculate g factors in angular momentum contributions from S, L, and J is given by the Lande´ formula: g511

J ðJ 1 1Þ 1 SðS 1 1Þ 2 LðL 1 1Þ 2J ðJ 1 1Þ

In contrast the proportionality constant assumes a single value of 2 for transition metal complexes when orbital angular momentum and hence spin
orbit coupling are modeled as being quenched (Tables 3.4 and 3.5). Using these expressions it can be seen that the effective magnetic moment calculated using the Lande´ formula is a good match for the experimental values obtained for Ln31 complexes (Table 3.6), which retain their free ion character due to the nonbonding nature

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FIGURE 3.21 Transformation of one orbital to another within the interaxial t2g orbital set is possible by 90-degree rotation (top) but not within the axial eg* set due to the different shapes of the orbitals.

of the magnetic 4f orbitals (Fig. 3.20). For compounds of the transition series an improved match is obtained by use of the spin-only formula, but it is not an accurate model for all ions (Table 3.5). This is because in some transition metal complexes a degree of orbital motion is possible if the t2g set (in octahedral complexes) has an asymmetric population. The condition for an electron to have orbital angular momentum is that it must be in an orbital set where it is possible to transform one orbital into another by a 90-degree rotation. For example, the three t2g orbitals in an Oh complex can be interconverted by 90-degree rotations (Fig. 3.21, top). In contrast the eg* orbitals cannot be interconverted as the shapes are different (Fig. 3.21, bottom). Therefore electrons in the eg* set do not have orbital angular momentum. In most cases orbital angular momentum is quenched for a complexed d metal ion—unless it has an asymmetric population in t2g. This always corresponds to an orbital triplet, that is, a T term. In Co21, for example, spin
orbit coupling is quite large for the 4T state, explaining why the observed value of μeff is significantly larger than that predicted by the spin-only formula (Table 3.5). A third formula for μeff can be used which takes into account only spin and orbital contributions but not spin
orbit coupling: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μeff 5 4SðS 1 1Þ 1 LðL 1 1ÞμB In practice this is not much used as it does not yield a good match with experimental data. The experimental values of μeff in Bohr magnetons for d1
d9 configurations (Table 3.5) are a closer match to those obtained by inclusion of just S rather than those obtained by inclusion of the S and L or S, L, and J terms. This is in contrast to the corresponding values of μeff for the f 1
f 13 configurations, where comparison of experimental values with those calculated using the full expression for magnetic moment are in close agreement for most of the tripositive lanthanide ions (Table 3.6). Notable exceptions are Sm31 and Eu31 which both have low-lying excited states which are populated over a wide temperature range and which contribute to the observed magnetic moment. In Sm31 the

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Dimeric sites: exchange mechanisms and J values

TABLE 3.6 Energy states and typical magnetic moments for lanthanide complexes. g

ζ (cm21)

Number of UPE

Mion (μB )

μeff (μB ) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g J ðJ 1 1ÞμB

Observed μeff (μB )

χm T

2

6/7

644

1

2.14

2.54

2.3
2.5

0.80

[Xe]4f2

3

4/5

730

2

3.20

3.58

3.4
3.6

1.60

Nd31

[Xe]4f3

4

8/11

844

3

3.27

3.62

3.5
3.6

1.64

Pm31

[Xe]4f4

4

3/5

1070

4

2.40

2.68

2.7

0.90

Sm31

[Xe]4f5

6

2/7

1200

5

0.71

0.84

1.5
1.6

0.09

Eu31

[Xe]4f6

7

5

1320

6

0.00

0

3.4
3.6

0.00

31

[Xe]4f

7

8

2

1583

7

7

7.94

7.8
8.0

7.88

[Xe]4f

8

7

3/2

1705

6

9

9.72

9.4
9.6

11.82

[Xe]4f

9

6

4/3

1900

5

10

10.63

10.4
10.5

14.17

[Xe]4f

10

5

5/4

2163

4

10

10.60

10.3
10.5

14.07

[Xe]4f

11

4

6/5

2393

3

9

9.58

9.4
9.6

11.48

[Xe]4f

12

3

7/6

2656

2

7

7.56

7.1
7.4

7.15

[Xe]4f

13

2

8/7

2883

1

4

4.54

4.4
4.9

2.57

Ground state term symbol

Ion

fn

Ce31

[Xe]4f1

Pr31

Gd

31

Tb

31

Dy

31

Ho Er

31 31

Tm

31

Yb

F5/2 H4 I9/2 I4 H5/2 F0 S7/2 F6 H15/2 I8 I15/2 H6 F7/2

moment has contributions from both the 6H5/2 ground state and 6H 7/2 first excited state leading to an observed moment of 1.5
1.6μB . In Eu31 the moment has contributions from the 7F0 ground state and the first two excited states, 7F1 and 7F2, leading to an observed moment of 1.5
1.6μB . The large spin
orbit coupling in the lanthanide series can be quantified by the single electron spin
orbit coupling constant ζ which increases on moving toward the heavier elements to values approaching 3000 cm21 for the heaviest lanthanides. Typically the values are much smaller in the first row transition metal ions (Table 3.5), but the general trend of increasing spin
orbit coupling on moving to heavier elements is clear. An important consequence of spin
orbit coupling is the opening of a gap between MS or MJ levels in the absence of a magnetic field, this is hence termed zero field splitting (ZFS). The main consequence of ZFS is that the magnetic moment is often not temperature independent as predicted by Curie law.

Dimeric sites: exchange mechanisms and J values Up to now we have considered only isolated paramagnetic centers where there is no collective interaction between the atomic magnetic moments. However there is usually some interaction between paramagnetic centers, either between magnetic sites within a cluster compound, or between adjacent compounds in a crystalline lattice. In some cases

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3. Molecular magnetochemistry

FIGURE 3.22

Spin orientations for different types of magnetic order: paramagnetic (blue), ferromagnetic (green), antiferromagnetic (red), ferrimagnetic (purple).

FIGURE 3.23

Comparison of the thermal dependence of χ, χT; and 1=χ for paramagnets, ferro- (and ferri-) magnets, and antiferromagnets.

this can lead to very strong interaction between the moments on neighboring atoms. Paramagnets can therefore be classified according to the type of ordering that is manifest as the temperature is lowered and the different types are depicted in Fig. 3.22. Paramagnetism: In a perfect paramagnet, the individual magnetic moments are well separated and therefore do not interact magnetically with each other. In the absence of an externally applied magnetic field, the spins are randomly oriented due to their thermal motion. Ferromagnetism: Ferromagnetism occurs when spins of neighboring paramagnetic centers exhibit magnetic interactions with each other in a way that the individual moments are aligned parallel to each other, which leads to an increase in the magnetic moment. Due to this possibility of parallel alignment, ferromagnetic materials are usually divided into domains to minimize their total free energy, the so-called Weiss magnetic domains. When a typical ferromagnet is cooled down, the material spontaneously divides its aligned magnetic moments into many small magnetic domains where each domain points in a uniform direction. A spontaneous magnetization arises in each domain even in the absence of an externally applied magnetic field. The saturation of the magnetization under an applied magnetic field corresponds to the complete alignments of all magnetic domains. Ferromagnetic materials show large and positive values in their magnetic susceptibility. However, above the so-called Curie temperature (TC) (Fig. 3.23) the interactions are no longer strong enough to maintain the alignment of all individual moments, and the substance divides into magnetic domains which then behave as a simple paramagnet. Antiferromagnetism: Antiferromagnetism occurs when the spins of neighboring paramagnetic centers orient in an antiparallel fashion, resulting in a complete compensation of the magnetic moments. Orientation of the spins is thermally dependent with ordering occurring below the Ne´el temperature (TN) (Fig. 3.23).

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Ferrimagnetism: Ferrimagnetism is a special case of antiferromagnetism, where the magnetic moments of adjacent sites have different magnitudes. This form of cooperative magnetism can be found in either compounds containing two different paramagnetic species, such as magnetite, which contains Fe(II) and Fe(III) ions, or when for example, a transition metal is antiferromagnetically coupled to a lanthanide ion. Alternatively, it can be found in materials where the spins are oriented in the two directions. Ferrimagnets exhibit spontaneous magnetization as in the case of ferromagnets.

Curie
Weiss law Pierre Curie showed that for isolated paramagnetic substances the magnetic susceptibility depends inversely on the temperature, χ 5 C/T, and the magnetization of paramagnets follows Curie’s law to a good approximation. When the paramagnetic ions or atoms are magnetically coupled to a neighboring paramagnetic center, this law is no longer valid, and the magnetic exchange between spin carriers then needs to be included in the model. This leads to a modification of the Curie law that includes the Weiss constant θ and is therefore named Curie
Weiss law: χ5

C T2θ

In ferromagnets and antiferromagnets the Curie
Weiss law can be applied to the magnetic susceptibility and leads to positive θ values for ferromagnetic coupling, and negative values for θ for antiferromagnetic coupling, respectively. There are three ways to plot the temperature dependence of the magnetic susceptibility, and these are shown in Fig. 3.23. These plots give information about the kind of magnetic interactions present in a compound. Plotting 1/χ against T, which is often referred to as the Curie
Weiss plot, usually gives a straight line, although it may start to curve at lower temperatures. The Curie constant C can be obtained from the gradient, and the Weiss constant θ from the extrapolated intercept on the temperature axis. Paramagnets should have this intercept at the origin, while for ferromagnets and antiferromagnets the intercept is positive or negative, respectively. For all three classes of substance, at higher temperatures χ shows a steady increase as the temperature decreases. For an antiferromagnet, χ starts to decrease again as the temperature goes below TN, while for a ferromagnet (or a ferrimagnet) χ starts to increase very sharply as the temperature passes below TC. For a paramagnet, χT is independent of temperature, and the plot shows a horizontal straight line. For antiferromagnets, the plot shows a decrease of χT with decreasing temperature, extrapolating to zero if the ground state has S 5 0. For a ferromagnet, χT increases as the temperature is lowered, while for a ferrimagnet, χT first decreases to a minimum and then increases sharply as the temperature is further decreased.

Superexchange The mechanism of communication between neighboring ions which leads to one of the types of behavior described above is exchange interaction. In the broadest sense an exchange interaction is just that: an exchange of electrons between orbitals. The simplest

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3. Molecular magnetochemistry

type of exchange is that represented by Hund’s first rule of maximum spin multiplicity. The electrons on a single atom or ion keep their spins oriented in the same direction so that they can freely exchange between the available orbitals. In polynuclear systems exchange mechanisms are either direct or indirect. Direct exchange occurs in bulk metals where the electrons on neighboring atoms change places. In polynuclear metal clusters including dimeric and polynuclear metalloproteins the magnetic ions are not directly bonded to one another so they need a bridging group to facilitate the exchange interaction. Several types of indirect exchange are possible but we will only consider superexchange where the orientation of the electron spins on two paramagnetic ions is communicated through a bridging group, often an oxo group. In linear dimers with a 180-degree angle between paramagnetic centers the exchange is said to be kinetic (Fig. 3.24). We assume here that the metal ion carries just one spin, but the value of S can range from 1/2 to 5/2 in transition metal systems. The bridging O22 oxo group hosts two p electrons and if the four electrons can exchange across the three participating orbitals in an antiferromagnetic arrangement as illustrated in Fig. 3.24 (top) then the energy is minimized. The arrangements in Fig. 3.24 middle and bottom represent excited states but the overall interaction is always antiferromagnetic. Deviation from 180-degree geometry can facilitate ferromagnetic exchange if more than two orbitals on the bridging group are involved (Fig. 3.25). This type of exchange is less

FIGURE 3.24

A superexchange pathway to antiferromagnetic interaction.

FIGURE 3.25

A superexchange pathway to ferromagnetic interaction.

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103

common but still observed in many dinuclear and polynuclear sites and is termed potential rather than kinetic.

Spin Hamiltonian Magnetic interactions between paramagnetic centers that are either isotropic or weakly anisotropic are generally characterized by the Heisenberg
Dirac
Van Vleck (HDVV) or “spin-only” Hamiltonian: H 5 2 2JðS1 US2 Þ where S1 and S2 are the spins on the two metal centers, and J is the exchange integral characterizing the coupling between them, which is usually given in units of cm21 or K, with 1 cm21 5 1.44 K. For antiferromagnetic coupling J is negative while for ferromagnetic coupling it is positive. The type of magnetic coupling within a molecule can be evaluated from the plot of the magnetic susceptibility versus T. The 1/χ versus T plot will already determine the sign of J: the intercept is positive for ferromagnets and negative for antiferromagnets, leading to J . 0 for ferromagnets and J , 0 for antiferromagnets, respectively. It is more common to use χ or χT (or μeff) to determine the temperature dependence and the challenge is to find an expression to model the magnetic susceptibility which includes the magnetic coupling, J, and the temperature, T. Intramolecular exchange coupling was first and well studied on the dimeric complex copper acetate [Cu2(OAc)4(H2O)], where μeff for two independent CuII sites was ffi expected to show a value of 2.58 μB using pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the expression μeff 5 g S1 ðS1 1 1Þ 1 S2 ðS2 1 1Þ and g 5 2.11. However the measured temperature-dependent susceptibility plot (Fig. 3.26) clearly shows that below 10 K the magnetic moment is almost zero. This suggests that the ground state needs to be diamagnetic, which can be achieved if the two copper sites are aniferromagnetically coupled.

FIGURE 3.26

Plots of magnetic susceptibility (left) and μeff (right) versus T for a dimeric CuII complex without magnetic interactions (J 5 0), with antiferromagnetic interactions (J , 0) and ferromagnetic interactions (J . 0).

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FIGURE 3.27

3. Molecular magnetochemistry

Heisenberg
Dirac
van
Vleck coupling between the Zeeman split terms (left) and their ener-

gies (right).

Bleaney
Bowers equation This challenge was met in 1952 when Bleaney and Bowers used the simplest type of binuclear system, consisting of two magnetically coupled S 5 1/2 centers, to find a fit for the magnetic interactions. The general isotropic exchange Hamiltonian contains one J value and two spin quantum numbers: H 5 22J(S1  S2) Bleaney and Bowers could further simplify this by including the assumption that the dimer has an inversion symmetry, leading to S1 5 S2 and g1 5 g2. Using the Hamiltonian with the S 5 1/2 basis set, the energetic ordering of allowed states was assumed to be that shown in Fig. 3.27. Bleaney and Bowers initially used the Van Vleck equation to describe the magnetic susceptibility  P  Enð1Þ 2 2 Enð0Þ n kB T 2 2Enð2Þ exp kB T  χ 5 NA P 2 Enð0Þ n exp kB T h  i h 2 2  i h  i h 2 2  i μB g μB g 2J 2J 2J 0 0 0  exp 2 1  exp 1  exp 1  exp kB T kB T kB T kB T kB T kB T kB T kB T     χm 5 NA  exp 2 kB0T 1 exp k2JB t 1 exp k2JB t 1 exp k2JB t h χm 5NA 

μ2B g2 kB T



 exp



2J kB T

i

1

h

1 1 3  exp



μ2B g2 kB T

2J kB T





 exp



2J kB T

i

Which was then simplified to the so-called Bleaney
Bowers equation to describe a dimeric system of pairs of S 5 1/2 ions:  2J 2 2 2exp kT NA μB g  χ5  kB T 1 1 3exp 2J kB T

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Diamagnetism

It is of course also possible to combine paramagnetic species, but the larger the spin gets, the more complicated the equation becomes. The equations for the magnetic susceptibility for binuclear complexes with combinations of various spin quantum numbers under the effect of Heisenberg’s intramolecular magnetic exchange: 2 2JS1S2 are given in Table 3.7. In order to simplify the table, C is substituted as C5

Ng2 μ2B kB T

and y includes the magnetic coupling constant J, as a function of the temperature, y 5

J kB T .

Diamagnetism Diamagnetism results from the interaction of paired electrons with an applied magnetic field, therefore all matter is diamagnetic. Diamagnetism is induced when the electron pairs in closed orbitals precess about an applied magnetic field generating a magnetic moment in the opposite direction. The effect is entirely quantum mechanical in origin but is typically simplified using a semiclassical treatment which invokes Lenz’s law. For an explanation of the quantum mechanical treatment please see Blundell. In the Langevin semiclassical treatment the precessing electrons are modeled as orbiting perpendicular to the applied field, generating a current in the opposite direction to the motion of the electrons. An electromotive force ε is induced due to the change in flux ɸ in the current loop which results as the field is applied. The rate of change of the magnetic flux is equal to the electromotive force ε which is defined as the line integral of the induced electric field E around the circumference of the current loop with radius r: dɸ dt

ε 5 E 3 2πr 5 2 So it follows that E52

dɸ 1 dt 2πr

The quantity of interest is the diamagnetic susceptibility χdia as this is required to correct for the diamagnetic contribution when measuring the paramagnetic susceptibility of a sample with unpaired electrons. χdia can be extracted by considering the rate of change in the angular momentum dL=dt which is equal to the torque exerted on the electron by the induced electric field which is 2 eEr. dL 5 2 eEr dt 52e2

dɸ 1 r dt 2πr

Practical Approaches to Biological Inorganic Chemistry

TABLE 3.7 Expressions for χ in dinuclear complexes with various combinations of spin quantum numbers. S1 5 1/2

S1 5 1

S1 5 3/2

1 2

χ5C 

2e 1 1 3e2y

S2 5 1

χ5C 

1 1 10e3y 2e2y 1 10e6y χ5C  4ð1 1 2e3y Þ 1 1 3e2y 1 5e6y

3 2

χ5C 

2 1 10e4y 3 1 5e4y

1 1 10e3y 1 35e8y 4ð1 1 2e3y 1 3e8y Þ

χ5C 

2e2y 1 10e6y 1 28e12y 1 1 3e2y 1 5e6y 1 7e12y

S2 5 2

χ5C 

10 1 35e5y 2 1 10e4y 1 28e10y χ5C  4ð2 1 3e5y Þ 3 1 5e4y 1 7e10y

χ5C 

1 1 10e3y 1 35e8y 1 84e15y 4ð1 1 2e3y 1 3e8y 1 4e15y Þ

S2 5

S2 5

S2 5

S1 5 2

S1 5 5/2

2y

5 10 1 28e6y χ5C  2 5 1 7e6y

χ5C 

χ5C 

χ5C 

2e2y 1 10e6y 1 28e12y 1 60e20y 1 1 3e2y 1 5e6y 1 7e12y 1 9e20y

5 1 35e5y 1 84e12y 2 1 10e4y 1 28e10y 1 60e18y 1 10e2y 1 35e8y 1 84e15y 1 165e24y χ5C  χ 5 C  1 4ð1 1 2e3y 1 3e8y 1 4e15y 1 5e24y Þ 4ð2 1 3e5y 1 4e12y Þ 3 1 5e2y 1 7e10y 1 9e18y

χ5C 

2e2y 1 10e6y 1 28e12y 1 60e20y 1 1 3e2y 1 5e6y 1 7e12y 1 9e20y

Diamagnetism

5

107

e dɸ 2π dt

The flux can be defined as ɸ 5 μHA where A is the area of the current loop, πr2 , μ is the permeability, and H is the applied field. We can take the permeability μ 5 μ0 for a single atom. It follows that the change in flux can be written as dɸ dH 5 μ A dt dt 0 5

dH μ πr2 dt 0

We can now rewrite dL=dt as dL e dH 5 μ πr2 dt 2π dt 0 5 ΔL 5

er2 μ0 dH 2 dt er2 μ0 H 2

Now recall from the calculation of the value of the Bohr magneton that the angular momentum for a single electron equals ħ 5 me vr. Here the angular momentum is more generally defined as L and so L 5 me vr and L 5 vr me Recall also that we defined the magnetic moment μ as μ5

2 evr 2

So e L 2 me e Δμ 5 2 ΔL 2me μ52

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3. Molecular magnetochemistry

Using our expression for ΔL derived above we obtain: Δμ 5 2

e er2 μ0 H 2me 2

Δμ 5 2

e2 r2 μ 0 H 4me

This expression for the change in moment was derived by Langevin and Pauli for the diamagnetic contribution of a single electron orbiting perpendicular to the applied magnetic field. However in the classical description all orientations are allowed so the average value of the square of the projection of r onto the field direction should be used and this reduces the magnetic moment by a factor of 2/3. Considering electrons from

 many atomic orbitals on each atom necessitates inclusion of all occupied orbital radii r2 av and multiplication by the total number of electrons Z. Taking into account all atoms in the sample requires multiplication by Avogadro’s number so:

 ZNA e2 r2 av μ0 H Δμ 5 2 6me Now substituting H for M/χ yields:

 ZNA e2 r2 av μ0 M Δμ 5 2 6me χ

Assuming that Δμ 5 M and rearranging gives the dimensionless expression for the diamagnetic susceptibility:

 ZNA e2 r2 av μ0 χ52 6me Corrections for the diamagnetic contribution of all atoms in a metal complex, including the paramagnetic ions should be made using published values. A selection is given in Table 3.8 from the data published by Bain and Berry.

Experimental methods A variety of methods are currently used to measure the magnetic susceptibility of materials including various types of magnetometry for solid-state studies, Evans’ method for NMR studies in solution and magnetic circular dichroism (MCD) for investigations in both solution and the solid state. The principles of each method are outlined below.

Magnetometry All matter is either attracted (paramagnetic compound) or repelled (diamagnetic compound) by an external magnetic field and the degree of attraction from or repulsion to the field can be measured as the magnetic susceptibility. The early magnetometry techniques

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Experimental methods

TABLE 3.8 Values for χdiamagnetism for selected anions, ligands, solvents, and cations (in 1 3 1026 emu/mol). Anions

χdi

Ligands

Solvents

χdi

Acetic acid Acetonitrile

2 10.0

C5H52

BF42

2 37.0

acac

Br2

2 34.6

bipy

Cl2

2 23.4

CO

ClO2 4

30.2

CN2

2 13.0

χdi

2

2 52.0

Cations

χdi

Cations 1

χdi

2 31.8

Sc

31

2 6.0

Li

2 1.0

2 27.8

Ti31

2 9.0

Na1

2 6.8

Acetone

2 33.8

Ti41

2 5.0

K1

2 14.9

2 65.0

Benzene

2 54.8

V31

2 10.0

NH41

2 13.3

en

2 46.5

Butanol

2 56.4

V51

2 4.0

Be21

2 0.4

ethylene

2 15.0

Chloroform

2 58.9

Cr31

2 11.0

Mg21

2 5.0

2 105

F2

2 9.1

I2

2 50.6

H2O

2 13.0

Cyclohexane

2 68.0

Mn21

2 14.0

Ca21

2 10.4

IO32

2 51.0

hydrazine

2 20.0

Dichloromethane

2 46.6

Mn31

2 10.0

B31

2 0.2

NO2 3

2 18.9

NH3

2 18.0

Diethylether

2 55.5

Mn71

Ethanol

2 33.7

Hexane

2 74.1

Methanol

2 21.4

Pentane

2 61.5

Toluene

2 65.6

Triethylamine

2 83.3

2

NCS

2 31.0

O22

2 12.0

2

OH

2 12.0

22

S

2 30.0

CO22 4

2 40.1

22

Se

2 48.0

phen

2 128

o-PBMA

2 194

phthalocyanine

2 442

PPh3

2 167

pyridine 22

salen

2 49.0 2 182

2 3.0

Al31

2 2.0

21

2 13.0

31

2 8.0

31

2 10.0

Fe Fe

21

2 12.0

31

2 10.0

Co Co

21

Ni

2 12.0

21

2 11.0

21

2 15.0

Cu Zn

Ga

41

2 0.1

31

2 4.0

C P

21

2 32.0

1

2 28.0

1

2 40.0

21

2 40.0

Pb

Ag

Au Pt

acac, acetylacetonate; bipy, 2,2’-dipyridyl; en, ethylenediamine; phen, phenanthroline; PBMA, phenylenebisdimethylarsine; salen, ethylenebis(salicylaminate).

were based on the displacement force which depends on the magnetization and the gradient of the magnetic field. The magnetic susceptibility can be determined by measuring this force, which was originally achieved using force magnetometers. Two force methods were developed: The Gouy balance and the Faraday balance. Current methods rely on superconducting quantum interference devices or SQUID magnetometers. In general solid samples are prepared by grinding to a fine powder to achieve uniformity in the sample and to guarantee a random orientation of the magnetic moments. Force methods The Gouy method is simple and may be found in many chemistry laboratories. The principle of operation uses the fact that the attraction force into the uniform magnetic field is proportional to the volume susceptibility. The Gouy balance (Fig. 3.28) measures the apparent change in the mass of the sample as it is repelled or attracted by the region of high magnetic field between the poles. The sample tube is positioned between the two magnetic poles with one end of the tube within the homogeneous field while the other end is in the region of zero field. Alternatively, the experimental procedure can also be done through two separate readings with an initial balance reading performed without a magnetic field (ma) and a subsequent balance reading taken with an applied magnetic field

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FIGURE 3.28

3. Molecular magnetochemistry

Gouy balance (left) versus Faraday balance (right).

(mb). The difference between the two masses (mb 2 ma) relates to the magnetic force on the sample. A paramagnetic sample will be pulled down toward the magnetic field and therefore will provide a positive difference in the difference of mass while a diamagnetic sample can either exhibit no apparent change in weight or a negative change as the sample is slightly repelled by the applied magnetic field. The Faraday balance (Fig. 3.28) measures the susceptibility of a sample toward a magnetic field which exhibits a field gradient where H(dH/dz) is constant over the volume of the sample. In contrast to the Gouy balance, the Faraday balance the measurement is independent of the density of the material and therefore only requires a small sample. The sample is placed in a region of the magnetic field where the field gradient is constant, which then produces force that acts on a magnetized object. In Faraday force magnetometry the force on the sample can be measured by a scale (hanging the sample from a sensitive balance), or by detecting the displacement against a spring. Commonly, a capacitive load cell or cantilever is used because of its sensitivity, size, and lack of mechanical parts. This method is approximately one order of magnitude less sensitive than a SQUID. SQUID—super conducting quantum interference device The SQUID is a sensor which can detect smallest changes in the magnetic flux of a magnetic field and is therefore a highly precise and powerful tool which can measure even extremely weak signals of any physical quantity related to the magnetic flux (e.g., magnetic field, current, voltage, and magnetic susceptibility). The SQUID is built up of a superconducting ring which is intercepted by Josephson junctions which are lightly separate the superconducting ring with normal conducting or even electric insulating material (Fig. 3.29). These Josephson junctions have to be extremely thin so that the Cooper pairs of electrons can pass from one superconductor to the other even without any applied voltage. The superconducting Cooper pairs can tunnel between closely spaced superconductors even with no potential difference. There are two types of SQUID depending on the number of junctions. On the one side consisting of only one Josephson junction, the radio

Practical Approaches to Biological Inorganic Chemistry

Experimental methods

111

FIGURE 3.29 Organization of sample and pickup coil in SQUID magnetometer.

FIGURE 3.30 Josephson junction in SQUID magnetometer.

frequency SQUID and on the other side with two positions, the direct current SQUID (DC), which is the more common version used for magnetic measurements. The greater number of Josephson junctions leads to a higher sensitivity and use of two Josephson junctions in parallel allows for enhanced sensitivity. Without a magnetic field, the input current splits equally between the branches, which maintains an externally connected tank circuit at resonance, and the electrons which are tunneling through the junctions demonstrate quantum interference (Fig. 3.30). By applying an external field, a change in the resonant frequency in the tank circuit is induced, which causes a current imbalance that leads to a voltage across the Josephson junction. The voltage is a function of the magnetic flux and can therefore be measured and used to calculate the magnetic flux. Tiny variations in a magnetic field cause resistance in

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3. Molecular magnetochemistry

FIGURE 3.31

Susceptibility (χ) versus temperature (T) (left) and magnetization (M) versus magnetic field (H)

(right).

the SQUID which in turn enables detection of such minute changes. SQUID magnetometers exhibit extreme sensitivity for measuring magnetic flux and with this high level of sensitivity comes the possibility of measuring small magnetic field changes from magnetic samples to human hearts or brains with high accuracy, while the sample itself does not need to come in contact with the SQUID. The SQUID is cooled with liquid helium to maintain superconductivity. What is measured in the SQUID?

In general, there are two types of measurements for magnetic materials that give information about the magnetic behavior of the samples. On one side the magnetic susceptibility, χ, which requires a temperature sweep under a certain applied field (usually smaller than 0.2 T), and on the other side the magnetization, M, where the magnetic field is swept while keeping the temperature stable at very low temperatures (Fig. 3.31). The magnetic susceptibility, χ, is usually quoted in cm3/mol and the magnetization is given in μB. The setup of the SQUID magnetometer uses the older unit of Oersted (Oe) for describing the strength of the magnetic field, with 1.0 T (Tesla) 5 10,000 Oe. We have already shown that the magnetic susceptibility, χ, can be described as a function of the temperature and that Curie has derived the empirical equation showing that the magnetic susceptibility is inversely proportional to the temperature (Curie law): χ 5 C/T. For an orbitally quenched ion, the Curie constant C is dependent on the number of unpaired electrons and the g value of the compound, which leads to the spin only magnetic susceptibility, which is described as χ5

NA g2 μ2B SðS 1 1Þ 3kB T

N μ2

with 3kA B B  18 and g 5 2. This leads to the following equation, which can be used to describe the magnetic behavior of transition metal ion complexes: χT 5

1 SðS 1 1Þ 2

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FIGURE 3.32 Temperature dependence of χMT for a mononuclear Mn31 complex at an applied field of 0.1 T.

The magnetic properties of samples are usually measured as well-ground polycrystalline samples or in the form of single crystals. In the following example, a Mn31 containing compound was studied between 1.8 and 300 K under an applied field of 1000 Oe (50.1 T) and the susceptibility data are shown as χMT versus temperature plot in Fig. 3.32. At room temperature, 300 K, the value is a good match for the expected spin-only value for one noninteracting Mn31 ion (S 5 2, g 5 2, χMT 5 3.0 cm3K/mol).

Evans NMR method While the solid-state magnetic properties are usually obtained using a SQUID magnetometer, in solution the Evans method can be performed utilizing 1H NMR measurements on a paramagnetic sample. The Evans method, which is named after the British chemist Dennis F. Evans, relies on the fact that the chemical shift of an NMR active nucleus (e.g., the 1H signal in TMS (tetramethylsilane) or even the solvent itself), is dependent on the magnetic environment of its environment. The susceptibility can be measured by the differences in the chemical shift of a probe molecule (usually TMS) in the presence or absence of the dissolved paramagnetic sample in solution. Experimentally, the sample preparation requires an NMR tube filled with the NMR solvent, the probe molecule (TMS) and the sample to be measured. A tube with a smaller radius is placed into this normal NMR tube where the inner tube contains the same NMR solvent and the probe molecule without the paramagnetic sample, see Fig. 3.33. This setting can also be turned the other way around, if required. The 1H NMR shift of the TMS in the outer tube differs from that in the inner one due to the extra field of the paramagnetic sample in the outer tube. The magnetic susceptibility can be determined by the change in chemical shift between the two TMS resonances, using the following equation: χm 5 χ0

MW ðsampleÞ 3000 Δυ 1 MW ðsolventÞ 4πυ0 ½CMW ðsampleÞ

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FIGURE 3.33

3. Molecular magnetochemistry

Experimental setup for Evans method using 1H NMR.

where χ0 is the molar susceptibility of the solvent, υ0 is the operating frequency of the NMR spectrometer (in Hz), Δυ is the shift in frequency (in Hz), and [C] is the concentration of the sample (in mol/L).

Magnetic circular dichroism Information about internal magnetic and electronic structure can be obtained by applying a magnetic field when recording absorption spectra using the MCD technique. MCD is the realization of the Faraday effect whereby optical activity is induced in a sample by applying a magnetic field in the same direction as a circularly polarized light source (Fig. 3.34). Circularly polarized light over a range of wavelengths from terahertz to X-ray has been used in the MCD experiment but UV
vis absorption is the most common. There is a fieldinduced differential absorption of the left and right circularly polarized (LCP and RCP) light and the difference spectrum, ΔA, gives information on the energy of the Zeeman levels. MCD 5 ½ALCP 2ARCP H applied 2 ½ALCP 2ARCP H50 The absorption is corrected for the difference spectrum in zero applied field. The MCD experiment does not require the presence of unpaired spins as the normal Zeeman effect is observed in diamagnetic compounds, that is, lifting of orbital degeneracies in systems with closed shells of electrons (Fig. 3.13). The technique also does not require the presence of molecular chirality: although both conventional circular dichroism (CD) and MCD experiments result in the differential absorption of LCP and RCP radiation the selection rules and underlying principles are different. Difference spectra are quantified by A, B; and C terms which are complex expressions describing the different probabilities for absorption of right or left circularly polarized light; these are more fully explained in the comprehensive text by Mason. Diamagnetic

FIGURE 3.34 Absorption of right and left circularly polarized (RCP, LCP) photons leads to a differential spectrum when the sample is magnetized.

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115

FIGURE 3.35 Energy levels of ground and first electronic states of paramagnetic transition metal complex and resulting MCD absorptions.

compounds are characterized by the A and B terms, paramagnetic systems by the C term. In paramagnetic compounds differential absorption arises due to the anomalous Zeeman splitting of the Ms states in the ground and excited electronic states (Fig. 3.35). It is a particularly useful technique for differentiating the magnetic environment of different metal sites in a polynuclear cluster. Due to the Curie law MCD spectra of paramagnetic compounds show a marked temperature dependence, whereas those for diamagnetic compounds are temperature independent.

Conclusion Magnetic properties of transition metal ions in living systems are a continued source of fascination and challenge in bioinorganic chemistry. Methods of analysis are varied and complex and are underpinned by complex and long-standing theories which seek to accurately model and fit the observed experimental data so as to elucidate structural and functional motifs of an enzymatic cycle. Some of the most important concepts which are important for understanding magnetic properties of metal ions have been introduced here and will serve as a platform for deeper study of the subject.

Problems 1. Write the term symbols for the following atoms and free ions C, Pb, Mn21, Mn31, Mn41.

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3. Molecular magnetochemistry

2. Determine the spin value for the two monomeric paramagnetic compounds A and B which were measured using a SQUID magnetometer to yield the following χ values at different temperatures: Temperature T (K)

χ (cm3/mol) sample A

χ (cm3/mol) sample B

300.01

0.01458

0.00127

249.95

0.01749

0.00153

199.78

0.02185

0.00191

149.83

0.02907

0.00254

100.06

0.04334

0.00378

75.048

0.05741

0.00378

50.031

0.0851

0.00743

3. Suggest the identity for the magnetic ion which is common to both samples in question 2 and explain why the S values are different. Calculate the Lande splitting factor g for the uncomplexed free ion. 4. A dimeric Mn(II) containing sample was measured using a SQUID magnetometer, and the following χT values (in cm3K/mol) at different temperatures (in K) are observed. Use the Curie
Weiss law to estimate the coupling between the two Mn(II) sites and determine whether the coupling is ferromagnetic or antiferromagnetic. T (K) χT T (K) χT

300 8.75 30 4.26

270 8.75 22 3.78

230 8.57 20 3.67

200

170

8.39 18

130

8.17 16

3.57

7.74 14

3.48

3.40

100

80

7.21 12

6.67

60 5.91

40 4.88

10

3.32

3.31

Answers 1. 2S11

LJ

C 5 ½He2s2 2p2 : S 5 2=2 5 1; L 5 1 3 1 1 1 3 0 5 1 5 P; J 5 L
S 5 1
1 5 0 : -3 P0 Pb 5 ½Xe4f14 5d10 6s2 6p2 : S 5 1; L 5 1; J 5 0-3 P0

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Answers

Mn21 5 ½Ar3d5: S 5 5=2-2S 1 1 5 6; L 5 2 3 1 1 1 3 1 1 1 3 0 1 ð
1Þ 3 1 1 ð22Þ 3 1 5 0 5 S; J 5 5=2
0-6 S5=2 Mn31 5 ½Ar3d4: S 5 4=2-2S 1 1 5 5; L 5 2 3 1 1 1 3 1 1 1 3 0 1 ð21Þ 3 1 5 2 5 D; J 5 L 2 S 5 2 2 2 5 0-5 D0 Mn41 5 ½Ar3d3: S 5 3=2-2S 1 1 5 4; L 5 2 3 1 1 1 3 1 1 1 3 0 5 3 5 F; J 5 L
S 5 3 2 3=2 5 3=2-4 F3=2 2. Calculation of χT for both samples: Sample A—300.01K: 4.374 cm3K/mol; 249.95K: 4.372 cm3K/mol, etc. Sample B—300.01K: 0.381 cm3K/mol; 249.95K: 0.382 cm3K/mol, etc. 1 8 Use χT 5  g2 SðS 1 1Þ transform to: S2 1 S 2 2 UχT 5 0 8 g pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 216 12 14U1U8:75 216 35:992 2166 55=2 5 For sample A : S 1S
2;  4:37450-S1;2 5 5 2 2 2U1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 216 12 14U1U0:76 2162 2 For sample B : S 1S
2;  0:38150-S1;2 5 51=2 5 2 2U1 2

3. Fe(III) can exist in the high-spin state with S 5 5/2 which has the maximum number of unpaired spins, but there is also the low-spin state with S 5 1/2 which has the highest number of paired spins. J ðJ 1 1Þ 1 SðS 1 1Þ 2 LðL 1 1 g511 2JðJ 1 1Þ Fe31 5 Mn21 5 ½Ar3d5 : S 5 5=2; L 5 0; J 5 5=2       5=2 ð5=2Þ11 15=2 5=211 20ð011Þ ð35=4Þ1ð35=4Þ20 35=2 511 511 511152 g Fe31 511 235=2ð5=211Þ ð35=2Þ 35=2 4. Calculation of 1/χ for both samples: T (K)

χT (cm3K/mol)

χ (cm3/mol)

1/χ (mol/cm3)

300

8.75

0.02916667

34.2857143

270

8.75

0.03240741

30.8571429

230

8.57

0.03726087

26.8378063 (Continued)

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3. Molecular magnetochemistry

(Continued) T (K)

χT (cm3K/mol)

χ (cm3/mol)

1/χ (mol/cm3)

200

8.39

0.04195

23.8379023

170

8.17

0.04805882

20.8078335

130

7.74

0.05953846

16.7958656

100

7.21

0.0721

13.8696255

80

6.67

0.083375

11.994003

60

5.91

0.0985

10.1522843

40

4.88

0.122

8.19672131

30

4.26

0.142

7.04225352

22

3.78

0.17181818

5.82010582

20

3.67

0.1835

5.44959128

18

3.57

0.19833333

5.04201681

16

3.48

0.2175

4.59770115

14

3.4

0.24285714

4.11764706

12

3.32

0.27666667

3.61445783

10

3.31

0.331

3.02114804

Linear curve fit of 1/χ in the temperature range between 300K and 40K, leading to y 5 4.0078 1 0.09972x, Using the Curie
Weiss law as following: χ1 5 T 2C Θ 5 C1 T 2 Θ C 3 3 Therefore 2 Θ C 5 4:00 mol=cm , with C 5 8.75 cm K/mol.

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mol 3 This leads to a Weiss constant of: Θ 5 2 4:0 cm 3  8:75 cm K=mol 5 2 35:0 Kantiferromagnetic coupling between the two centres.

Acknowledgments We gratefully acknowledge Conor Kelly at University College Dublin for preparing several figures in this chapter.

References Bain, G.A., Berry, J.F., 2008. Diamagnetic corrections and Pascal’s constants. J. Chem. Ed. 85, 532
536. Bleaney, B., Bowers, K.D., 1952. Anomalous paramagnetism of copper acetate. Proc. Roy. Soc. London A214, 451. Blundell, S., 2006. Magnetism in Condensed Matter. Oxford University Press. Mason, W.R., 2007. A Practical Guide to Magnetic Circular Dichroism. Wiley-Interscience. Spaldin, N.A., 2011. Magnetic Materials. Cambridge University Press.

Further reading Benelli, C., Gatteschi, D., 2015. Introduction to Molecular Magnetism—From Transition Metals to Lanthanides. Wiley-VCH. Earnshaw, A., 1968. Introduction to Magnetochemistry. Academic Press. Kahn, O., 1993. Molecular Magnetism. Wiley-VCH. O’Connor, C.J., 1982. Magnetochemistry—advances in theory and experiment. Prog. Inorg. Chem. 29, 203
283.

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C H A P T E R

4

EPR spectroscopy W.R. Hagen Department of Biotechnology, Delft University of Technology, Delft, The Netherlands O U T L I N E Why electron paramagnetic resonance spectroscopy?

121

What is electron paramagnetic resonance spectroscopy? 122 Anisotropy

125

A comparison of electron paramagnetic resonance versus NMR 126 Electron paramagnetic resonance spectrometer

128

What (bio)molecules give electron paramagnetic resonance?

132

Basic theory and simulation of electron paramagnetic resonance 133 Saturation

135

Concentration determination

137

Hyperfine interactions

139

High-spin systems

145

Applications overview

150

Test questions

152

Answers to test questions

153

References

153

Why electron paramagnetic resonance spectroscopy? Electron paramagnetic resonance (EPR) spectroscopy in biology is applicable to paramagnetic molecules with one (low spin) or more (high spin) unpaired electrons, that is, radicals and transition metal ion complexes. This chapter explains the basic phenomena whose understanding is required for a meaningful analysis and (bio)chemical interpretation of spectroscopy, namely electronic Zeeman interaction, resonance, anisotropy, saturation, hyperfine interaction, and zero-field interactions. The concept of the spectral powder pattern from randomly oriented samples is treated, and its computed simulation by

Practical Approaches to Biological Inorganic Chemistry DOI: https://doi.org/10.1016/B978-0-444-64225-7.00004-3

121

© 2020 Elsevier B.V. All rights reserved.

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4. EPR spectroscopy

unit-sphere walking is introduced. The biologically key application of spin counting or quantitative EPR is addressed also in relation to the notion of effective spins. In biochemistry spectrometers are used for two reasons: to determine a concentration or to determine a structure. Measured as a function of time, a change in concentration gives a reaction rate, and a change in structure affords information on a reaction mechanism. The reactions that we are interested in are typically conversions of a substrate into a product under the influence of a biocatalyst. By far the easiest way to measure a concentration is by UV
vis spectroscopy. If a substrate and its product do not have a measurable absorption in the spectral range from near-UV to near-IR, we turn to other detection methods. EPR (electron paramagnetic resonance) spectroscopy can only detect systems with unpaired electrons. Most metabolites in living cells are relatively stable organic molecules; these are so-called closed-shell systems, that is, they have an even number of electrons arranged in a pairwise manner (according to the Pauli exclusion principle) such that no electron remains unpaired, and no EPR is possible. Only in relatively rare cases does a substrate or product have an odd number of electrons with one electron unpaired. In the language of quantum mechanics (QM) these molecules are called spin-one-half-systems, or S 5 1/2. Some of these radicals (or paramagnets) may have a color and others don’t, but they all have an EPR spectrum. A more consequential application area of EPR spectroscopy pertains not to the substrate but to the biocatalyst, which is typically an enzyme with an active site that may encompass a radical, but much more frequently a transition metal ion, or a cluster of metal ions. By definition transition ions are open-shell systems, that is, they have partially filled d- or f-shells, and they have one (S 5 1/2), or more (S . 1/2) unpaired electrons (i.e., they carry paramagnetism) in at least one of their common oxidation states. Thus biological EPR spectroscopy is predominantly a means to study the structure and functioning of active sites of enzymes. This usually includes the quantitative measurement of concentration of spin systems for the determination of the stoichiometry of paramagnetic prosthetic groups per enzyme molecule. It sometimes also encompasses figuring out the mutual magnetic interaction of spin systems within an enzyme molecule to pin down geometrical constraints. This chapter is an introduction to the subject of continuous-wave EPR of biomolecules and their models. EPR is a quantum-mechanical phenomenon, and the theory of EPR likewise makes ample use of QM methods. Knowledge of QM is not required to read this chapter, nor to do a range of useful EPR experiments on biological systems. Key equations are given here without derivation. Those interested in a more extensive treatment of the subject, including derivation of equations, are referred to the book Biomolecular EPR spectroscopy (Hagen, 2009).

What is electron paramagnetic resonance spectroscopy? EPR spectroscopy is the absorption of microwave radiation between energy levels of molecules. In EPR one does not vary the frequency ν of the radiation (or the wavelength λ 5 c/ν in which c 5 299,792,458 m/s is the speed of light), but one uses a monochromatic source at a single fixed frequency. The most commonly used frequency is in the range

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What is electron paramagnetic resonance spectroscopy?

123

9
10 GHz (part of the X-band). The unit of energy used in EPR is the “wavenumber” or “reciprocal cm.” A frequency of ν 5 9.5 GHz means an energy hνD0.3 cm21 (so λD3 cm). This is a very small energy quantum. Compare an optical transition at 500 nm, which represents an energy quantum of (1/500)nm21 or 20,000 cm21. The small molecular energy splittings required for EPR are due to quantization of the electron spin S and require an external dipole magnet: when the paramagnetic molecule is placed in an axial magnetic field B, which is a field with a north and a south pole (i.e., a vector B⃑ ), the unpaired electron behaves like a little bar magnet but with the quantum-mechanical property that its orientation in the field can assume only two values “parallel” or “antiparallel” to the external field. These orientations correspond to a higher and a lower energy state of the molecule between which microwave energy can be absorbed. The spin quantum numbers are mS 5 11/2 and mS 5
1/2 and in the common so-called “bra-ket” notation of QM the states are denoted as |mS 5 11/2i and |mS 5
1/2i. The two states are “degenerate” (i.e., they have identical energies) when there is no external field (when the magnet is switched off). Since the frequency ν is fixed we have to vary the field B to create a spectrum. So in contrast to optical spectroscopy where we submit an invariant molecule to radiation of continuously varying energy, in EPR we throw radiation of invariant energy on a molecule whose paramagnetism is continuously varied by means of a scanning magnetic field. Absorption occurs when the splitting of the two spin energy levels, caused by the magnetic field, happens to be exactly equal to the energy of the microwave. This is called the resonance condition, hν 5 gβB, in which h 5 6.62607015 3 10234 Js is Planck’s constant and β 5 9.27400999 3 10224 J/T (T is tesla; 1 T10,000 gauss) is the Bohr magneton. The two electron spin energy levels are E 5 6gβB/2. The proportionality constant g is what is determined in an EPR experiment: the g value is specific for the molecule under study; it contains electronic information of (bio)chemical relevance. Therefore we rewrite the resonance condition in the practical form g 5 0:714477

ν ðMHzÞ B ðgaussÞ

ð4:1Þ

As an estimate of practical magnetic field values we can use the (theoretical) g value for a free electron in vacuo, ge 5 2.00232, and a microwave frequency of ν 5 9500 MHz. This requires a field of BD3390 gauss, or 0.339 T, a field that is readily produced with an electromagnet. Since radicals are molecules with a reactive, delocalized, loosely bound unpaired electron, they typically exhibit g values close to ge, say 2.00260.005, for carbon, nitrogen, and/or oxygen-based radicals (and with somewhat greater deviations from ge for heavier atoms like sulfur). This does not hold for transition metals which exhibit more pronounced deviations from ge (e.g., for Cu(II) complexes, typically, 2.0 # g # 2.5) because the unpaired electron’s movement (in QM language, its orbital angular momentum) is more strongly influenced by the metal nucleus. Furthermore, in high-spin systems (more than one unpaired electron) the measured g value, or effective g value (see below), can essentially take any value between zero and infinity as a consequence of magnetic interaction between the different unpaired electrons. The interaction between the electron spin S and a magnetic field B, which causes the splitting of otherwise degenerate spin states, is called the electronic Zeeman interaction.

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4. EPR spectroscopy

If Eq. (4.1) were a sufficient description of EPR, then all spectra would consist of a single line, and the frequency of the microwave source would be an irrelevant choice. In practice EPR spectra can be fairly complex with many details, and they usually change with the frequency (note a change in microwave frequency typically means the use of two or more spectrometers that operate with different source frequencies). There are four main reasons why this complexity (therefore increased chemical information content) occurs. Firstly, the electronic Zeeman interaction between unpaired electrons and the magnet (S2B interaction) is essential for EPR to occur, but it is not the only magnetic interaction to determine the spectrum. Also nuclei can have a spin, I, and this nuclear magnet can interact not only with the external magnet (I2B or nuclear Zeeman interaction, which is the basis for NMR spectroscopy) but also with the electron spin (S2I or hyperfine interaction). If a paramagnet has more than one unpaired electron (i.e., a high-spin system), these electron spins can mutually interact (S2S or zero-field interaction). An equivalent effect occurs for nuclear high-spin systems (I2I or quadrupole interaction). When a system has more than one paramagnet, either within one molecule or between molecules, this can lead to dipolar interaction between spins (another form of S2S interaction). Finally, many metalloproteins contain clusters of metals (i.e., metal ions at a mutual distance of one or two chemical bonds, for example, in iron
sulfur clusters) leading to the coupling of electron spins associated with individual metal ions into a new system spin, or cluster spin (in QM language this is called exchange interaction). Since only the Zeeman interactions are linear in the field (therefore linear in the frequency) and all other interactions are independent of the field, changing the microwave frequency (and therefore the field) changes the relative weight of different interactions and thus changes the EPR spectrum. Note that the quadrupole interaction and, especially, the nuclear Zeeman interaction are very often too weak to be resolved in regular EPR, however, they are observed in more elaborate double-resonance experiments like electron-nuclear double resonance (ENDOR) or electron spin echo envelope modulation (ESEEM) spectroscopy. Secondly, as a consequence of the nonspherical structure of molecules the abovementioned interactions are all dependent on the orientation of the dipole magnet (the vector ⃑B ) with respect to the molecule (or with respect to a Cartesian coordinate system defined by the molecular structure of the paramagnet). In other words, the EPR spectrum depends on the direction of orientation of the molecule in the magnetic field, and even if the electronic Zeeman interaction is the only relevant spectral determinant, the spectrum is almost never a single line. Thirdly, if we use a microwave to excite an S 5 1/2 molecule from its lower spin energy level, or ground state, to its higher level, or excited state, then the molecule subsequently has to return to its ground state again by some mechanism if only to be ready to absorb a next microwave quantum and thus to maintain the spectrum in time. This so-called “spin relaxation” influences the shape and, particularly, the width of EPR spectra, and it makes the spectral shape dependent on the temperature T and on the intensity P (for power) of the microwaves. Fourthly, all molecules are subject to conformational distribution: the exact relative atomic coordinates slightly (or not so slightly) vary in a real sample from one molecule to the other. This means that each molecule has a slightly different g value (and values for the other interactions), which in turn leads to EPR broadening and spectral changes. This phenomenon is called “g strain.”

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Anisotropy

125

Understanding EPR means to understand the nature and the practical consequences of these phenomena, and this is the subject of the remainder of this chapter.

Anisotropy The unpaired electron of a paramagnetic molecule does not only experience the external magnetic field, but it also “sees” the molecular structure as it is part of the molecule’s electronic structure. Electrons as moving charges represent a magnetic field, and this internal field adds to, or subtracts from, the external field in the Zeeman interaction: hν 5 geβ(Bext 1 Bint). By convention, this is usually written as an observed shift in the g value of the free electron: hν 5 (ge 1 Δg)βB, or simply hν 5 gβB, where g, or rather the deviation from ge, now contains electronic (and therefore structural) information on the molecule. Axial magnetic fields are vectors in 3D space, for example, in the space defined by the coordinates of the paramagnetic molecule. The external field is the dominant one, so it determines the direction along which the EPR spectroscopist “looks at” the molecule. In other words, the orientation of the molecular structure in the external field is of the essence: rotation of the molecule in the field (or, alternatively, rotation of the magnet around the molecule) will result in a change of the EPR spectrum. Each orientation has its own spectrum, and so a single molecule can give rise to an infinite number of different spectra. If the spectra are determined by the electronic Zeeman interaction only, then they all consist of a single absorption line. Single-molecule EPR spectroscopy does not exist (yet?), because the signal of a single molecule is too weak to be detected. However, a single crystal consists of many identical molecules all with the same orientation in space. A single crystal of identical S 5 1/2 molecules gives a single-line Zeeman EPR spectrum; its g value changes when the crystal is rotated. Our biomolecular or model-compound samples will usually not be single crystals; they are homogeneous solutions, or frozen solutions, or powder samples. Each molecule has a different orientation and the EPR of such a sample is the sum of many different single-line spectra. All these spectra have different g values, because the internal field seen by the unpaired electron depends on the orientation of the molecule in the external monitoring field. The result, illustrated in Fig. 4.1, is a specific spectral shape called the “powder pattern”; it covers a defined field range (or g value range) between two extreme values. Most EPR spectrometers produce the first derivative of this powder absorption spectrum, and since the slope of the powder pattern rapidly changes around the three g values gx, gy, gz (together also called the g tensor or—by mathematics purists—the g matrix), corresponding to the molecular x, y, z-axes, one gets the impression that the EPR derivative spectrum consists of three “peaks.” This three-featured form (a peak, a derivative, and a negative peak), called the “rhombic powder pattern,” is the general fingerprint of an S 5 1/2 system without any specific symmetry properties (gx6¼gy6¼gz). Note that the labeling with (x, y, z) is arbitrary; one might just as well use (z, y, x) or (x, z, y) or (a, b, c) or (1, 2, 3), etc. Symmetric properties of the coordination complex may simplify the EPR pattern. A metal ion at the center of a perfect octahedron with six identical ligands with identical metal-to-ligand bond lengths will give

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4. EPR spectroscopy

FIGURE 4.1 The construction of an X-band (ν 5 9.5 GHz) EPR powder pattern (the black trace) by summation of single-orientation spectra from individual molecules in the XYZ molecular axis system with respect to the magnetic field vector B. Red traces are for B along one of the molecular axes, and blue traces are examples from a large number of spectra from intermediate orientations. The label ZY, for example, means that B is oriented halfway between the molecular Z- and X-axis. The magenta trace is the observable first derivative of the EPR powder absorption spectrum with g values: gx 5 1.62, gy 5 2.09, gz 5 2.95 (which could stem from a low-spin heme Fe(III) complex). Note that the spectra from individual molecules at different orientations have been given different widths to illustrate linewidth variation common in experimental spectra.

a single (derivative) line spectrum, called an isotropic pattern, with gx 5 gy 5 gz. Such a highly symmetrical structure is not likely to occur in biology. However, for metalloproteins we do frequently find (near) “axial” spectra with gz6¼gyDgx. An axial pattern can occur when a perfect octahedron is elongated (or compressed) along one of the axes (which is then defined as the z-axis). More generally, if one, or two of the ligands along the z-axis are different form the others (as in many tetrapyrrole complexes, for example in hemoproteins) the local structure and the EPR spectrum can be nearly axial. Strictly speaking axiality of the EPR spectrum reflects axiality in the electronic structure, that is, in the wave function(s) occupied by the unpaired electron, and may not necessarily be retraceable to an exact axial geometry. Several so-called blue copper proteins, such as plastocyanin with His, His, Met, and Cys ligands, exhibit axial EPR, although the copper coordination is a very strongly deformed NNSS (nitrogen-nitrogen-sulfur-sulfur) tetrahedron. In summary the basic EPR spectrum is a one-, two-, or three-featured pattern, but it is not always easy to link this to structural symmetry.

A comparison of electron paramagnetic resonance versus NMR The EPR and the NMR effect were both discovered in the mid-1940s, and their subsequent technical and theoretical developments initially took a parallel course. It is instructive to compare the two spectroscopies in terms of present-day similarities and differences. The simplest system giving EPR is one with a single unpaired electron, S 5 1/2. Most nuclear magnetic resonance spectroscopy, for example, 1H-NMR, is done on

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A comparison of electron paramagnetic resonance versus NMR

127

systems with a nuclear spin I 5 1/2. There is a strong analogy between S 5 1/2 and I 5 1/2 in that they both give rise to two quantized energy states in an external magnetic field by means of a Zeeman interaction, although in EPR, due to the negative sign of the electron charge, the orientation antiparallel to the field is the lowest in energy, while in NMR the parallel orientation is the ground state. Both the nuclear Zeeman interaction and the electron Zeeman interaction are measured in the form of a spectroscopic shift. The electron shielding-induced chemical shift away from the resonance frequency of a standard compound such as tetramethylsilane is the NMR equivalent of the internal fieldinduced shift Δg from the free-electron value ge in EPR. Of course for the same external field strength, nuclear spin energy level differences are much smaller than the separations of electron spin energy levels, and the radiation to induce nuclear resonance is in the radio frequency range (MHz instead of GHz), which is one of the reasons why NMR spectroscopy typically has a lower concentration sensitivity than EPR spectroscopy. Arguably the most important historical divergence between NMR and EPR was the emergence of commercial pulsed-NMR spectrometers starting in the late 1960s. From then on it became routine to apply a broad spectrum of frequencies (i.e., a short singlefrequency pulse turning into a frequency range by virtue of Heisenberg’s uncertainty principle) to a sample in a single-valued, invariant external magnetic field. In this approach the recording of a single NMR spectrum takes a short time, and thus extensive averaging becomes practical. Even today such an experiment is technically impossible in EPR spectroscopy because one cannot make a sufficiently short GHz pulse of sufficient intensity and homogeneity to cover all the frequencies of an EPR spectrum at constant field. Therefore EPR still uses monochromatic continuous wave (CW) radiation in combination with a varying magnetic field, and the recording of a single EPR spectrum typically takes a few minutes, mainly to allow for noise reduction by means of a low-pass filter. Pulsed versions of EPR have also been developed based on nanosecond electronics, but there is a paradigmatic difference in the application of pulsed EPR versus that of pulsed NMR. EPR studies always start with the CW experiment; pulsed EPR is an optional “next step” or “advanced” follow-up experiment providing information additional to the CW data and obtained at additional cost and effort. Because the EPR pulse cannot cover the whole spectrum, it is used to measure a small range of the spectrum at increased resolution. Technically this is typically set up in the form of a double-resonance experiment. With the CW spectrum known, the magnetic field is fixed at a value corresponding to a single point of the powder spectrum, and then a pulse of radiation is applied to probe a small frequency spectral range around the fixed powder point. Today a variety of pulsed-EPR methods are at one’s disposition such as pulsed ENDOR, ESEEM, pulsed electron
electron double resonance (ELDOR) also called PELDOR or DEER (double electron
electron resonance). These techniques are usually only available in specialized laboratories, and each one has its own possibilities and limitations. Their details are beyond the scope of this chapter. From the perspective of the biochemist there is also a major conceptual difference in the application of NMR versus EPR spectroscopy. A high-resolution solution NMR experiment affords a global picture of all protons (and/or 13C, 15N, etc.), that is, of all parts of a biomacromolecule. On the contrary, EPR only looks at paramagnets and thus typically provides a picture of a local spot of the macromolecule, for example, a metal coordination complex

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in a metalloprotein. In other words, the EPR result is spatially limited and spectroscopically more simple. On the other hand, the focus point is usually on the most important part of the molecule, the active site. And here the two methods show considerable complementarity because the NMR resonances are extensively broadened (frequently beyond detection) for the nuclei near the paramagnetic center. Both S 5 1/2 EPR and I 5 1/2 NMR exhibit anisotropy, however, in liquid-state NMR tumbling of molecules affords complete averaging of this pattern to single-line spectra for each nucleus even in large biomacromolecules, hence its “high resolution.” In EPR this only works for relatively small molecules; proteins are so big that their tumbling on the EPR timescale is too slow to average anisotropy away, and therefore, for example, metalloproteins always give anisotropic, powder pattern spectra even when in solution. In solidstate NMR one does observe powder patterns, very similar to the EPR ones, as spectra for individual nuclei. Finally, “increasing the frequency” has a rather different connotation in NMR as compared to EPR. Over the years the proton NMR frequency has been steadily increased from tens of megahertz to around 1 GHz at present, concomitant with improving technology for the generation of stable, homogeneous static magnetic fields first with electromagnets and subsequently with superconducting solenoids. Each frequency field increase has led to an increase in sensitivity and an increase in resolution in these nuclear Zeeman interaction dominated spectra. On the contrary, concentration sensitivity of the EPR detection system typically decreases with increasing frequency above X-band (9
10 GHz) due to several reasons such as increasingly noisy detection diodes or increased power losses in highfrequency microwave components. Sensitivity also decreases with decreasing frequency below X-band mainly due to a less favorable Boltzmann distribution of molecules over the spin energy levels (see below). The combined effect of these trends is that starting more than six decades ago (Bagguley and Griffiths, 1947) until this day EPR at X-band has generally been found to be clearly the optimal choice in terms of sensitivity (although not necessarily in terms of spectral resolution).

Electron paramagnetic resonance spectrometer The vast majority of EPR spectrometers operate at a frequency in the range 9
10 GHz, which is part of the X-band of c. 8
12 GHz. The most efficient, or least lossy, way to transfer microwaves at X-band frequencies is the waveguide, typically a rectangular tube made of brass filled with air. The waves move through the inner “skin” of the waveguide, that is, through a layer a few micrometers thick on the inside of the guide. The second-best choice for transport in X-band is the coaxial cable, an inner and outer conductor separated by a dielectric such as Teflon. Thus the microwave part of the spectrometer is a spaghetti of components connected by waveguide and coaxial cable, most of which is, however, invisible to the operator, because it is built in a box called “the bridge.” This design gives the spectrometer a somewhat austere look, in which only a single piece of waveguide sticks out of the bridge to terminate in the heart of the machine called the cavity, centered in between the poles of the magnet (Fig. 4.2).

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FIGURE 4.2 Drawing of a conventional continuous-wave EPR spectrometer consisting of a dipolar electromagnet, a microwave bridge (with waveguide schematic) with a reflection resonator or cavity, and a computer console whose screens exhibit an EPR spectrum and a tuning mode pattern.

The cavity is also called the resonator, but note that this name has nothing to do with the resonance (the R in EPR) of the sample. The bridge-coupled cavity can be described as an electronic resonator circuit (also tank circuit or tuned circuit) with a quality factor Q. In brief this means that the inner dimensions of the cavity and the materials properties of its inner walls are such that a unique frequency in X-band affords the sustaining of a standing-wave pattern with an energy density that is Q-times greater than in a setup in which the cavity would be absent. Typical Q factors for X-band cavities are roughly of the order of 5000, and the sensitivity of the spectrometer is approximately increased by this factor compared to that of a simple transmission or reflection instrument without a cavity.

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The “mode pattern,” or the microwave electromagnetic field lines in the cavity, are such that the magnetic component of the microwave (i.e., the field required to make EPR transitions) is maximal along a vertical line through the middle, where the electric component of the microwave (i.e., the field that only disturbs the EPR measurement by nonresonant absorption) is minimal. The shape of an EPR sample in an X-band cavity is a vertical cylinder with a length of c. 15 mm and a diameter of c. 1 mm for aqueous samples and a diameter of c. 3.5 mm for all other samples including frozen aqueous solutions. This gives approximate volumes of 15 and 175 μL held in quartz tubes; the reduced diameter for watery samples is to partially compensate for the increased nonresonant absorption of microwaves due to the high dielectric constant of water, which is some 25 times higher than that of ice. Microwave intensities are expressed in decibel attenuation with respect to a reference value in watts. The decibel scale is a logarithmic one: n ðdBÞ2100:n

ð4:2Þ

so for a source of 200 mW maximal power (which is a common value for commercial spectrometers) the attenuated power in dB versus watt is as in Table 4.1 (in which mW 5 milliwatt, μW 5 microwatt, nW 5 nanowatt). The source of X-band microwaves is either a klystron (vacuum tube technology) or a Gunn diode with a typical initial output power of c. 400 mW slowly decreasing over the source’s lifetime ( . 10,000 hours of operation). This power is “leveled” (i.e., reduced) to a fixed value of 200 mW to assure constant output over time. The “road map” of the microwaves is then as follows (cf. Fig. 4.2): a wave of 200 mW leaves the source. A small amount of this intensity (c. 1% or
20 dB) is “coupled out” to the reference arm by a device called a directional coupler. The remaining 99% intensity in the main arm can be reduced to the required level (see “Saturation” section) by means of an attenuator, with the maximal attenuation for a good spectrometer being typically 60 dB (i.e., 1,000,000 times attenuation to 200 nW). After the attenuator the main wave enters a circulator, a device that can be thought of as a right-hand roundabout, and is forced to go into the waveguide that ends with the cavity. Waves reflected back from the cavity pass the circulator to go to the third TABLE 4.1 Conversion of decibel attenuation to power in watt for a source leveled at 200 mW. Decibel

0

22

24

26

28

0

200 mW

126 mW

79.6 mW

50.2 mW

31.7 mW

210

20.0 mW

12.6 mW

7.96 mW

5.02 mW

3.17 mW

220

2.00 mW

1.26 mW

796 μW

502 μW

317 μW

230

200 μW

126 μW

79.6 μW

50.2 μW

31.7 μW

240

20.0 μW

12.6 μW

7.96 μW

5.02 μW

3.17 μW

250

2.00 μW

1.26 μW

796 nW

502 nW

317 nW

260

200 nW

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arm with the detector diode, and any wave reflected from there is almost completely caught in the dead end of the fourth arm, called “load” to be converted into heat waste. As an essential prelude to actual measurement the spectrometer has to be “tuned,” which means that we have to (1) adjust the frequency, (2) make the spectrometer reflectionless, and (3) adjust the reference arm. The microwave frequency of the source in the bridge is tunable over a small range (9
10 GHz in X-band) to make it correspond to the unique lowest “eigenfrequency” (or the frequency of the fundamental mode) of the loaded cavity with a sample tube and (if necessary) a cooling system in place. This is accomplished by inspection of an oscilloscope tracing of the power reflected from the cavity as a function of a small frequency scan, say 10 MHz, around the set microwave frequency (cf. the right-hand screen on the computer in Fig. 4.2) and tuning this mode pattern to be symmetric with the “dip” centered. Then the spectrometer is made “reflectionless,” which is brought about in practice by adjusting a Teflon screw, located at the back of the cavity, with a metal end plate in front of a little aperture called the iris between the waveguide and the cavity until the dip in the tuning pattern is maximally deepened, and no current is measured at the detection diode. Reflectionless means that a standing wave is set up in the cavity whose energy dissipates only through the cavity’s side walls with no loss by means of back-reflection into the waveguide. In the language of electronics the impedance of the cavity is now “matched” to that of the rest of the system. This delicately balanced system will be detuned when, in an actual experiment, some of the radiation is absorbed by the sample, and as a result some radiation will be reflected out to cause a voltage change over the detection diode. However, for proper operation of the diode the voltage change should not be around zero but around a finite value called the voltage bias. One can create this bias by slightly detuning the cavity with the iris screw, but the disadvantage of this approach is that the spectrometer has to be readjusted every time the power, that is, the intensity of the microwaves, is changed by adjustment of the attenuator. To avoid this complication we use the reference arm through which a constant fraction of the unattenuated microwave is directly passed to the detection diode. The reference arm has its own attenuator, usually called “bias,” to optimize its output power to the characteristics of the diode (i.e., typically a diode current output of 200 μA). The reference arm also has a device called “phase shifter” to make the phase of the microwave equal to that through the main arm, which is accomplished by adjusting the phase shifter until the tuning mode pattern is perfectly symmetric. The apparatus is now ready to run a spectrum, which means that we scan the magnetic field (the x-axis of the spectrum) over the required range over a period of typically a few minutes. The field is produced by a water-cooled electromagnet, or in high-frequency EPR (c. ν $ 90 GHz) by a superconducting solenoid. To further increase spectrometer sensitivity we use a technique called phase-sensitive detection: the very slowly varying magnetic field is modulated with a very rapidly varying (100 kHz) small sinusoidal field of the order of 61 gauss, and the EPR signal from the detection diode is measured with a lock-in amplifier, that is, a device that takes an in-phase 100 kHz “look” at the signal. The result of this technique is twofold: (1) the EPR signal-to-noise ratio increases because electronic system noise at frequencies other than 100 kHz is not amplified, and (2) we obtain, by necessity, the first derivative of the EPR absorption spectrum. Note that the choice for 100 kHz is again one of optimization: the noise in an EPR spectrometer is found to be

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mainly of the “pink” or “one-over-f” (1/f ) type, which means that low-frequency noise dominates and noise amplitude decreases with increasing noise frequency, so the fieldmodulation frequency should be as high as possible. On the other hand, modulation fields of frequencies significantly above 100 kHz have increasing difficulty in penetrating the thin side walls of the cavity and reaching the sample.

What (bio)molecules give electron paramagnetic resonance? All compounds with one or more unpaired electrons give an EPR spectrum. The very vast majority of biologically relevant radicals have one single unpaired electron: a doublet with S 5 1/2 (e.g., flavin radicals; amino acid-based radicals; nitric oxide, NO; superoxide, O2•—). In rare cases one finds two unpaired electrons: a triplet with S 5 1/2 1 1/2 5 1, the most notorious example being molecular oxygen, O2. Radicals with more than two unpaired electrons exist, but none has yet been reported for a biological system. Occasionally, it may be possible to excite a diamagnetic system (no unpaired electrons) by continuous illumination with UV
visible light into the cavity to an excited triplet state with two unpaired electrons. If the lifetime of the triplet is sufficiently long it can be detected with cw-EPR, for example, photosynthetic reaction centers (Dutton et al., 1972). Transition ions can have up to five unpaired d-electrons—a quintet state with S 5 5/2 (e.g., high-spin Fe31)—or up to seven unpaired f-electrons—a heptet state with S 5 7/2 (Gd31)—and exchange-coupled clusters of transition ions can have many unpaired electrons, for example, the P-cluster with 8Fe in the nitrogenase enzyme, although it contains only d-ions, has seven unpaired electrons and a system spin of S 5 7/2 (Hagen et al., 1987). From an EPR spectroscopist’s point of view it is practical to divide all molecular systems into four groups: diamagnets (S 5 0), doublet systems (S 5 1/2), half-integer highspin systems (S 5 n/2), and integer high-spin systems (S 5 n). Diamagnets have a ground state that is not split by a magnetic field, that is, a singlet system, and they cannot have an EPR spectrum. Doublet systems are “easy” not only because they have only a single electron Zeeman transition, but also because electron spin relaxation (see “Saturation” section) is typically slow, which means that their EPR can be measured at relatively high temperatures (i.e., ambient temperature or nitrogen-flow temperatures). However, note that relaxation is usually not slow for metal clusters with a system spin of S 5 1/2. Half-integer high-spin systems are “not so easy” because they have more than one electron Zeeman transition, and because their relaxation is usually so fast that cooling with cryogenic helium gas is required to obtain sharp lines. Integer high-spin systems are “difficult” because usually they have very broad, asymmetric, and weak EPR features from (almost) forbidden transitions. Some S 5 n/2 systems behave as “effective” S 5 1/2 systems, which means that only one of the several Zeeman transitions is detectable, but the relaxation can still be fast. Some S 5 n systems behave as effective S 5 0 systems, which means that none of the several possible Zeeman transitions is detectable, and so these systems are called “EPR silent.” An overview of these classes each with an iron
protein example is given in Table 4.2.

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TABLE 4.2 Overview of classes of spin systems. A

S50

Diamagnets

Fe(II) in oxy-myoglobin

B

Low spin

S 5 1/2

Fe(III) in ferrimyoglobin-sulfide

C

High spin

S 5 n/2

Fe(III) in rubredoxin

C’

Single-transition high spin

Effective S 5 1/2

Fe(III) in ferrimyoglobin

D

Integer spin

S5n

Fe(II) in deoxy-myoglobin

D’

EPR silent integer spin

Effective S 5 0

Fe(II) in rubredoxin

TABLE 4.3 Examples of biologically relevant spin systems. S50

Most organic molecules; Complexes of all main group elements; Low-spin Fe(II), Co(III), square planar Ni(II), Cu(I), Zn(II), Mo(VI), W(VI); Clusters, for example, [2Fe
2S]21, [4Fe
4S]21

S 5 1/2

Most organic radicals, for example, flavin radicals, quinone radicals, amino acid radicals; Most inorganic radicals, for example, nitric oxide NO, superoxide O2•2; Low-spin Fe(III), low-spin Co(II), Ni(III), Ni(I), Cu(II), Mo(V), W(V); Clusters, for example, [2Fe
2S]11, [3Fe
4S]11, [4Fe
4S]31, [4Fe
4S]11, [Fe(II)
O
Fe(III)]

S 5 n/2

Mn(II), Mn(IV), high-spin Fe(III), high-spin Co(II); Clusters, for example, linear [3Fe
4S]11, some [4Fe
4S]11

S5n

Biradicals (triplets), for example, light-excited reaction centers, molecular oxygen; Mn(III), high-spin Fe(II), Fe(IV), high-spin Ni(II); Clusters, for example, [3Fe
4S]0, [Cu(II)
heme Fe(III)] in cytochrome oxidase

Table 4.3 gives a more extensive list of biologically relevant systems and the spins of their electronic ground states.

Basic theory and simulation of electron paramagnetic resonance EPR spectroscopy of single crystals from biomolecules is rare because, for example, protein crystals are much smaller than the X-band sample size of 175 μL, and they do not give sufficient signal intensity. EPR samples are almost always (frozen) solutions of biomolecules, which means that they contain very many molecules (c. 1017 for a millimolar solution), each one with a different orientation with respect to the external axial magnetic field. To quantitatively understand EPR spectra we must carry out what is called a “walk over the unit sphere,” which means that we conceptually place the molecule of interest at the origin of an x, y, z Cartesian axes system (i.e., the molecular axes system), and then let a vector from the origin, of unit length, parallel to the magnetic field B, sample “all” possible orientations with respect to the molecule. This is done by defining a sphere around the

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FIGURE 4.3 Sketch of a walk over the unit sphere. The solid blue arrow is a unit vector along the magnetic field B with polar angles θ and φ in the Cartesian XYZ molecular axis system. The broken blue arrow is the projection of the unit vector onto the XY plane. The paramagnet is at the origin; the surface of a surrounding unit sphere is divided into fragments of equal area defined in terms of the polar angles as d cos θ 3 dφ. The unit vector along B samples each of these fragments once.

origin with a radius of unity, dividing up the surface of this sphere in a large number of little areas of equal size and shape, and letting the unit vector subsequently point toward each one of these little areas. The “walking” of the unit vector is conveniently defined in polar coordinates (1, θ, ϕ) as shown in Fig. 4.3, and we must make steps in cos θ (i.e., not in θ), and in ϕ to keep the little areas constant in size. For a rhombic spectrum as in Fig. 4.1, this means that we must solve the equation for the field position of the EPR absorption, B 5 0.714477  ν/g with g(θ, ϕ) in terms of sines and cosines: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gðθ; ϕÞ 5 g2x l2x 1 g2y l2y 1 g2z l2z ð4:3Þ in which we used the definition of the so-called direction cosines lx 5 sin θ cos ϕ; ly 5 sin θ sin ϕ; lz 5 cos θ and the equation for the intensity (or transition probability) X ðg2i 2 g22 l2i g4i Þ I ðθ; ϕÞ 5 g21

ð4:4Þ

ð4:5Þ

i 5 x;y;z

by making equidistant steps in cos θ and in ϕ. For example, to generate a simulation such as the spectrum in Fig. 4.1, we write a computer program that has three nested FOR-loops (or DO-loops), two for the walk over the unit sphere, and one for a scan through a line shape function, F, for example, a Gaussian distribution (Eq. 4.6):   2 ðln 2ÞðBr 2BÞ2 F 5 Ir exp ð4:6Þ W2 in which the subscript r stands for “resonance,” so Ir is the transition probability at resonance field Br, B is a scan through the field, and W is the half width at half height of the Gaussian line. The program is as follows:

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Saturation

Pseudo-code: a basic program to generate EPR spectra of S 5 1/2 systems. FOR

cosθ 5 1 to 0

! step from z to xy plane

ϕ 5 0 to π/2 gr 5 g(θ,ϕ) Ir 5 I(θ,ϕ)

! step from x to y ! compute g value

Br 5 0.714484 * ν / gr FOR b 5 Br- 3.16 W to Br 1 3.16 W IF Bstart , b , BendTHEN F 5 F 1 Irexp[-ln2(Br-b)2/W2]

! compute resonance field ! step in Gaussian line shape ! check spectrum limits ! compute amplitude

FOR

! compute intensity

The scan in field goes out to 3.16 times the linewidth W (half width at half height) to where the Gaussian has 0.1% of its maximal intensity. This example program is written in “pseudo code” and should be rewritten in your favorite programming language in order to work. Alternatively, you can of course use existing programs, some of which can be downloaded for free from the internet. Note also that the walk over the unit sphere in the example is actually over only one octant, because the other seven octants give identical g (θ,ϕ) and I(θ,ϕ) values. Also fortunately we do not really have to calculate 1017 different molecular orientations; typically some 100 steps in cos θ and 100 in ϕ, that is, c. 104 orientations, are enough, which means that an increase in the number of steps will not change the shape of the final simulation. For a digital spectrum of say 1024 points this means calculating c. 107 amplitudes, which your garden-variety laptop will accomplish in less than a second.

Saturation How much microwave power should we use? How far should we open the attenuator of the main arm in the bridge of Fig. 4.2? There is no danger of destroying the sample: the maximum output power is typically 200 mW, which is some 5000 times less than what one uses in a household microwave oven. In X-band an aqueous sample at room temperature subjected to full power may warm up by a few degrees, which is why most work on aqueous samples is done at reduced power of
10 dB (i.e., 20 mW) or less. The temperature of a frozen sample in a cryogenic flow of nitrogen or helium will not even noticeably change at full power. From a statistical viewpoint higher power means more transitions per unit time and therefore higher EPR amplitude. However, the phenomenon of saturation limits the maximum power that we may use. This is readily illustrated on the energy level scheme of a simple S 5 1/2 system. At resonance the two levels are separated by an energy difference ΔE 5 hν, and this means a Boltzmann population distribution n1 5 n0e2ΔE/kT, in which k 5 0.69503476 cm21/K is the Boltzmann constant. In other words, for a total number of n0 1 n1 molecules we will have n0 in the ground state and n1 in the excited state. For a sample in X-band (ν 5 9.5 GHz; λ 5 3.156 cm) at room temperature (T 5 295 K) we find n1/n0 5 0.9985, that is, a very small difference in population indeed. Inducing transitions by microwave absorption will

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further reduce this small difference, the more so at higher power levels. This reduction is counteracted by relaxation, that is, the falling back to the ground state of excited molecules by dissipation of the energy difference ΔE to the surroundings (in EPR literature these surroundings are also called “lattice” or “bath”). If the relaxation cannot keep up with the microwave power input, then there will be a net decrease in population difference. At high microwave power the difference will eventually become zero and the possibility for net microwave absorption will be abolished: the spin system is completely saturated and the EPR signal has disappeared. In summary with increasing microwave power the EPR amplitude increases linearly with low power, then levels off at higher power, then decreases at even higher power, and eventually disappears at very high power. This amplitude versus power relation is usually measured and plotted for normalized amplitudes, that is, corrected for the power. Experimentally, this is done as follows. The power expressed in decibels is a logarithmic scale; the gain, or electronic amplification, on EPR spectrometers is also expressed on a logarithmic scale either in hardware on older spectrometers or in software on newer spectrometers. In other words, one can only choose from a limited number of gain values, and these are 1.25, 1.6, 2.0, 2.5, 3.2, 4.0, 5.0, 6.3, 8.0, and 10 times a power of 10. These numbers are equal to 100.n times a power of 10, where n 5 1 through 10, that is, a logarithmic scale. In practice this means that when a signal is not saturated, increasing the power by m 3 2 dB and at the same time decreasing the gain by m steps should afford the same EPR amplitude. So we can make a saturation graph (or “power plot”) by running a spectrum at a power of, say,
30 dB and a gain of, say, 4.0 3 103; then we take the next spectrum at higher power 5
28 dB and lower gain 5 3.2 3 103, and so on until we reach a power/gain combination at which the EPR amplitude starts to decrease. We have then reached the onset of power saturation, and we should go back one step (2 dB) to the optimal measuring condition in terms of signal-tonoise ratio for the given temperature. This, then, is the setting to obtain “publicationquality” data. The experiment is schematically illustrated by the green trace in Fig. 4.4: increasing the power from the very low value of
60 dB initially leaves the normalized amplitude unaffected. When we reach a power of c.
30 dB the signal starts to decrease, and at the highest power of 0 dB we face some 90% saturation. The optimal power value for this case would be approximately
32 dB, that is, the spectrum has maximal signal-to-noise ratio without being significantly saturated. There are two good reasons why saturation is to be avoided. Firstly, under (partial) saturation the signal amplitude is no longer linear in the applied power (expressed in dB) and so determination of the spin concentration (cf. next section) versus an external standard is no longer possible. Secondly, relaxation rate, and therefore saturation, is anisotropic: its extent with increasing power is different for different parts of the spectrum. Therefore under partial saturation a spectrum will change shape in a complex manner for whose analysis no theory is available to date. Fig. 4.4 gives the power plots for one single EPR signal taken at four different sample temperatures with T1 . T2 . T3 . T4. The values of the Ts are not specified because the quantitative temperature dependence of relaxation rates in EPR is usually rather complex.

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FIGURE

4.4 Theoretical power plots showing saturation as a collapse of normalized EPR amplitude with increasing microwave power. Traces are shown for four different temperatures with T1 . T2 . T3 . T4. At lower temperature the onset of saturation occurs at lower power. At T1 the signal is only slightly saturable; at T4 the signal cannot be measured under nonsaturating conditions.

Qualitatively, however, we can take the relaxation rate to always decrease with decreasing temperature: it is easier to saturate a signal at low T than at high T. This has the very important practical implication that for a given EPR signal we have to experimentally determine the optimal microwave power for each temperature at which we want to measure. In literature one frequently finds reports of the temperature-dependence of an EPR spectrum taken at a single, intermediate power level. It should be obvious from Fig. 4.4 that such an approach is far from optimal. Suppose, for example, that we take spectra at power 5
30 dB for all four Ts. At T1 the signal is unsaturated, but the signal-to-noise ratio is suboptimal; at T2 we have a near-optimal situation with high signal-to-noise and hardly any saturation; at T3 however the signal is seriously saturated (and therefore deformed) at this power; and at T4 there is hardly any intensity left. In fact for the example of Fig. 4.4 it is impossible to measure an EPR spectrum at T4 with any available power level without serious saturation problems. T4 is simply too cold for this sample. This situation is commonly encountered, for example, for S 5 1/2 systems, such as [2Fe
2S]11 proteins, recorded in X-band at temperatures close to the boiling point of liquid helium, that is, T  4.2 K. As a final practical observation on the subject of saturation note that modern computerized EPR spectrometers may offer the possibility to produce 2D power saturation data which are stacks of spectra taken automatically at fixed intervals of power values. Unfortunately these plots usually do not conform to the format in Fig. 4.4. In contrast the spectral amplitudes are corrected for the used power, so that the spectra taken at low power have correspondingly very small amplitudes, and it is not easily possible to make a reliable estimate of the onset of saturation. Correcting the amplitudes of the individual spectra for the used power would solve the problem, and when the commercial software that comes with the spectrometer does not provide for this option, writing a little routine that just does this job may significantly save time and avoid frustration.

Concentration determination Arguably one of the most useful applications of EPR in biochemistry is “spin counting,” that is, the determination of the concentration of spin systems. Picture a complex protein

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containing several paramagnets, for example the enzyme succinate dehydrogenase that contains a [2Fe
2S] cluster, a [3Fe
4S] cluster, a [4Fe
4S] cluster, a heme, and an FAD (flavin adenine dinucleotide) radical. Determination of their stoichiometry (at a given redox potential) would be a very difficult task indeed if one did not have quantitative EPR spectroscopy available. Since the intensity of a spectrum is defined by Eq. (4.5), one can simply take the area under the EPR absorption envelope (cf. the black trace in Fig. 4.1), corrected for the intensity expression, as a measure for the concentration of a paramagnet. An absolute concentration is obtained by comparison with the area under the EPR absorption envelope from a standard compound of known concentration. A commonly used standard is a frozen solution of the hydrated Cu21 ion such as 10 mM CuSO4 1 10 mM HCl 1 2 M NaClO4 with g values 2.404, 2.076, and 2.076 (Hagen, 2006). Note that addition of the “nonligand” perchlorate is to avoid di- and polymerization of the copper ions which would otherwise lead to severely broadened spectra through dipolar interaction. And adding 10 mM HCl is not the same as titrating to pH 2. In these analytical experiments the anisotropic intensity expression in Eq. (4.5) is usually replaced with an approximating average scalar according to Aasa and Va¨nnga˚rd (1975): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ðg2x 1 g2y 1 g2z Þ ðgx 1 gx 1 gx Þ 2 ð4:7Þ 1 I5 3 9 3 In practice the procedure is as follows: run a spectrum of the paramagnet with unknown concentration C Ð ÐU; read the approximate g values from the spectrum (cf. Fig. 4.1); take the second integral U (i.e., the area under the EPR absorption spectrum); and calculate the scalar intensity factor I in Eq. (4.7) for normalization. Do the same with Ð Ð the standard compound of known concentration CK, and then correct the final number K for any difference in experimental measuring conditions between the unknown and the standard. When the two spectra are taken under identical experimental conditions, we have RR IK CU 5 RR U ð4:8Þ CK I K U Most of the experimental differences between unknown and standard require linear corrections, for example, the modulation amplitude (M), the electronic amplification (G), the sample tube diameter (d), and/or the sample temperature (T). However, if the two spectra have different x-axis dimensions, that is, different magnetic field scan widths (W), then this correction should be taken squared, because the EPR derivative spectrum is integrated once to obtain the absorption and then once more to obtain the area under this absorption. Note also that a difference in used microwave power (P) in dB requires a logarithmic correction. Altogether, the concentration CU is now obtained as RR   IK MK GK dK TU WK 2 ðPU 2PK Þ CU 5 RR U 10 20 ð4:9Þ CK K I U M U G U d U TK W U Sometimes it may be preferable to replace the experimental spectrum with its simulation, for example, when integration of the experimental spectrum is unreliable because the

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FIGURE 4.5 Spin counting in a complex EPR spectrum. The experimental X-band (ν 5 9.25 GHz) spectrum of oxidized beef heart cytochrome c oxidase is simulated as a sum of four signals with intensity ratios 1:1:0.04:0.0005. The relative intensities of the simulated spectral components show that the dinuclear mixedvalence copper-A center and the cytochrome a are stoichiometric, that the EPR-silent dimer of copper-B and cytochrome a3 is slightly uncoupled affording a small amount of high-spin heme Fe(III), and that the preparation is very slightly contaminated with “dirty” iron.

baseline is of poor quality, or because the spectrum overlaps with other spectra, or because the spectrum extends over a field range that is not covered by the used electromagnet. And internal stoichiometries (e.g., of the different paramagnets in a multicenter protein) follow directly from the weighing factors used in the simulation. An example is given in Fig. 4.5: the spectrum of oxidized bovine cytochrome oxidase (EXP) is simulated (SIM) with four different spectral components, CuA, cyt a, cyt a3, dirty Fe, with relative stoichiometries 1:1:0.04:0.0005. These ratios make biochemical sense: CuA is the mixed-valence [Cu(II)
Cu(I)] cluster with S 5 1/2, cyt a is the low-spin heme of cytochrome a with S 5 1/2, and they occur once per cytochrome oxidase molecule. The other two metal centers, CuB (S 5 1/2) and cytochrome a3 (S 5 5/2), are antiferromagnetically exchange-coupled to a cluster spin S 5 5/2
1/2 5 2, which is not detectable under the used conditions. A small fraction of the Cu
Fe cluster is uncoupled (presumably by reduction of CuB to Cu(I)), and we observe a high-spin heme a3 spectrum of low intensity. The fourth component is a contamination with non-specifically bound high-spin Fe(III) of very low intensity. Note that this analysis could not have been based on inspection of the amplitudes of the four spectral components.

Hyperfine interactions Many nuclei have a nuclear spin I6¼0 and they are experienced by unpaired electrons in EPR as extra magnets, affording S2I or hyperfine interactions. Nuclei studied with NMR are mainly limited to those with I 5 1/2 (because the complexity of NMR spectroscopy with I $ 1/2 rapidly increases with I value), with two possibilities for the nuclear spin

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quantum number mI 5 1/2 and mI 5
1/2, corresponding to two energy states in an external field written in QM language with the “bra-ket” labeling as |mI 5 1/2i and |mI 5
1/2i. This limitation to I 5 1/2 does not usually hold for EPR because the S2I hyperfine interaction is a perturbation to the electronic Zeeman interaction (S2I is much weaker than S2B), and therefore allowed transitions are limited by the selection rule ΔmI 5 0. For example, for the copper ion Cu21, with S 5 1/2 and I 5 3/2, one finds only four EPR transitions, in combining bra-ket notation: |mS(i); mI(k)i2|mS(j); mI(k)i (i.e., the mI value does not change). The copper EPR line is split by a magnitude A (the hyperfine splitting expressed in gauss) into four lines centered around the field corresponding to the g value. So for an S 5 1/2 system with a single nuclear spin the number of EPR transitions is equal to 2I 1 1. This description holds under the condition that the hyperfine interaction is indeed significantly less than the Zeeman interaction, which is a reasonable assumption at X-band and higher frequencies. At low frequencies (c. 1 GHz or less for copper) the assumption no longer holds, and the spectra become more complex and their analysis more complicated. Just like the Zeeman interaction, the hyperfine interaction also is generally anisotropic, that is, different for different molecular orientations in the external magnetic field. The magnitude of the splitting in terms of the polar angles θ and ϕ of Fig. 4.3, used to define the direction cosines of Eq. (4.4), is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   A li gi 5 g22 l2x g4x A2x 1 l2y g4y A2y 1 l2z g4z A2z ð4:10Þ in which g is defined by Eq. (4.3), and from which it can be seen that the three features in a powder pattern (cf. the black trace in Fig. 4.1), namely the low-field absorption-like peak around gz, the intermediate-field derivative-like line around gy, and the high-field negative absorption-like peak around gx, will each be split into 2I 1 1 features, but with different splitting magnitudes Ai (together also called the A-tensor or the hyperfine tensor) for the different g values. Spectra with hyperfine structure can now be analyzed by simulation using Eq. (4.3) for the anisotropic g value and Eq. (4.10) for the anisotropic hyperfine splitting with the resonance field defined as Bres 5

X hν 2 AmI gβ mI

ð4:11Þ

A simple, but practically important, example is given in Fig. 4.6. Spin traps are diamagnetic compounds that readily react with unstable radicals to from meta-stable paramagnetic adducts. The spin trap 5,5-dimethyl-pyrroline N-oxide (DMPO) reacted with superoxide radical, O2•2 (added as KO2), affords the spectrum given in the red trace (exp). The unpaired electron of the superoxide delocalizes over the DMPO molecule, and consequently has hyperfine interaction with a nitrogen (14N with I 5 1) and a hydrogen (1H with I 5 1/2) nucleus. Because the adduct is a small molecule in water, all anisotropy is averaged away by tumbling, and one observes a single g value and single hyperfine splitting A values according to the isotropic resonance field expression Bres 5

hν N H H 2 AN iso mI 2 Aiso mI giso β

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FIGURE 4.6 Analysis of isotropic hyperfine interaction. The spin trap 5,5-dimethyl-pyrroline N-oxide (DMPO) has been mixed with superoxide and the room temperature aqueous solution EPR spectrum of the adduct has been simulated as a single peak split by hyperfine interaction with one nitrogen and one proton with AN  AH.

which has been used to simulate the spectrum in the black trace (sim). To understand how such a spectrum comes about the figure also gives simulations for the theoretical cases of interaction only with a single nitrogen (blue) or only with a single proton (green). The experimental spectrum can be thought of as a single line split into three by the nitrogen, with subsequent splitting of each of these three into two by the hydrogen. In general this should result in a total of six lines, however, since the splittings from the nitrogen and from the hydrogen by chance happen to be identical (AN 5 AH 5 15 gauss) some lines overlap and the result is a four-line spectrum with 1:2:2:1 intensity pattern. A more complicated example, now involving anisotropy, of the hydrated vanadyl ion (VOSO4 dissolved in acidified water) is analyzed in Fig. 4.7. Vanadium proteins with the vanadium reduced to V(IV) exhibit similar spectra (but with more noise due to the lower concentration). The spin system is S 5 1/2 and I 5 7/2, so we expect a powder pattern whose features are split into 2 3 7/2 1 1 5 8 lines. The structure of the hydrated VO21 ion (five H2O ligands) is quasi octahedral with the molecular z-axis defined along the V 5 O bond. The x- and y-axes are equivalent, so the symmetry is axial, and this implies that gx 5 gy and also Ax 5 Ay (but this is not generally the case in vanadium proteins, since their metal binding sites are of lower symmetry). The red trace in Fig. 4.7 is the experimental X-band spectrum taken with the sample immersed in liquid nitrogen (i.e., T 5 77 K). The black trace is an axial simulation. The blue trace is a simulation of what the spectrum would look like if there were no hyperfine interaction. The green trace is a simulation of the EPR absorption spectrum for a single molecular orientation (the “parallel” orientation), namely, for the external magnetic field B along the molecular z-axis; similarly, the magenta trace is for a single molecular orientation (“perpendicular”), namely, for B along the molecular x-axis (or, for that matter, anywhere else in the xy plane). Note that the hyperfine splitting in the z-direction, Az, affords positive peaks on the low-field side but negative peaks of the high-field side of the powder pattern. A good estimate of the value Az is obtained by taking the difference in field position between the highest-field negative peak and the lowest-field positive peak divided by seven (i.e., 2 3 I). Also from the average field

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FIGURE 4.7 Analysis of anisotropic hyperfine interaction. The 9.42 GHz frozen solution spectrum (T 5 77 K) of the S 5 1/2 system V41 in vanadyl sulfate in acidic water is split by hyperfine interaction with the 51V nucleus (I 5 7/2). The hydrated vanadyl ion VO(H2O)521 has axial symmetry with g|| 5 1.927, g\ 5 1.972, A|| 5 201 gauss, A\ 5 75 gauss. The blue trace “nohyp” is a simulation with I 5 0; the green trace “parl” simulates the absorption EPR for θ 5 0 (i.e., along the z-axis), and the magenta trace “perp” is for θ 5 90 degrees (i.e., in the xy plane).

position of these eight peaks (or of the first and the last one) one can make a good estimate of the gz value. It is more difficult to estimate the gx and Ax values from the “messy” middle part of the spectrum. One of the reasons can be appreciated by inspection of the single-orientation spectrum of the magenta trace: the distance between the eight hyperfine peaks is not constant, and the width of the peaks is also not constant. The first phenomenon is called a “second-order effect,” which means that Eq. (4.9) (derived using perturbation theory) is not exactly correct for these large hyperfine interactions and requires a significant correction in X-band. The details are beyond the scope of this chapter but the required equations can be found in Hagen (2009). The variable line width is a reflection of conformational distribution of the VO(H2O)521 structure, which leads to an mI-dependence of the width of the form W ðmI Þ 5 W0 1 c1 mI 1 c2 m2I

ð4:13Þ

The bottom line is that approximate g and A values may be estimated directly from the spectrum, however, accurate analysis of spectra of the type shown in Fig. 4.7 requires computer simulation affording high-quality fits (Hagen, 2009). Even without this sophisticated analysis, inspection of hyperfine structure can be very useful because simple “line counting” provides direct information on the chemical elements involved in the spin system. For example, in the spectrum of Fig. 4.7 with two Az peaks on the low-field side resolved and three Az (negative) peaks on the high-field side,

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TABLE 4.4 Biological transition metal ions and their nuclear spin. Metal

Isotope mass number

Spin (abundance)

EPR hyperfine lines (intensity)

V

51

7/2

8

Mn

55

5/2

6

Fe

54, 56, 57, 58

0, 1/2 (2%)

1, 2(1%)

Co

59

7/2

8

Ni

58, 60, 61, 62, 64

0, 3/2 (1%)

1, 4 (0.25%)

Cu

63, 65

3/2

4

Mo

92, 94, 95, 96, 97, 98, 100

0, 5/2 (25%)

1, 6(4%)

W

180, 182, 183, 184, 186

0, 1/2 (14%)

1, 2(7%)

it is straightforwardly established that there are eight Az peaks in total, and therefore that I 5 7/2. In a biochemical setting this would define the spectrum to be either from a lowspin Co(II) complex or from a V(IV) complex, since V and Co are the only biometals with I 5 7/2. Since crystal-field theory dictates that Co(II) has g . ge and V(IV) has g , ge (ge 5 2.0023) the distinction is readily made. There are no other splittings in addition to those due to the vanadium nucleus, which is consistent with all ligands being oxygen (16O has I 5 0). The protons (1H has I 5 1/2) of the water ligands are apparently too far away from the unpaired electron to afford resolved splittings, but they could be studied with special techniques, for example, ENDOR or ESEEM. Table 4.4 lists metal isotopes that are relevant in the frame of hyperfine structure in biological EPR; the mass numbers of isotopes with a nuclear spin are underlined. Note that when the percentage natural abundance of an isotope with nuclear spin is less than 100, the remainder is made up of isotopes with I 5 0, and the EPR spectrum is the sum of hyperfine split and unsplit spectra. For example, natural molybdenum consists of the isotopes with mass number 92, 94, 95, 96, 97, 98, and 100, and the EPR spectrum is for 25.5% split into six lines and for 74.5% unsplit. Since the splitting into six reduces the amplitude by a factor of six, the amplitudes of the split spectrum and the unsplit spectrum relate as 25.5/6 versus 74.5 or in percentages 5.4% versus 94.6%. Sometimes the symmetry of (biological) complexes is so low that it is not clear how to define a Cartesian axis system, with related direction cosines (cf. Eq. 4.4), in terms of the molecular structure. Analysis of the EPR may then reveal that the axis system for the g values defined by Eq. (4.3) differs from the axis system for the hyperfine A values defined by Eq. (4.10). Mathematically this means that Eq. (4.10) has to be rotated in 3D space with respect to Eq. (4.3) (in formal wording the g tensor and the A tensor diagonalize in different axes systems), and the EPR analysis becomes very involved. In practice this so-called “axes noncolinearity” from low molecular symmetry can be qualitatively recognized in the EPR spectrum as small extra peaks and shoulders in between the hyperfine lines (Hagen, 2009).

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TABLE 4.5 Biologically relevant ligand atoms and their nuclear spin. Ligand atom (intensity)

Isotope mass number

Spin (abundance)

Superhyperfine lines (intensity)

H

1, 2

1/2, 1(0.01%)

2, 3

C

12, 13

0, 1/2 (1.1%)

1, 2 (0.5%)

N

14, 15

1, 1/2 (0.4%)

3, 2

O

16, 17

0, 5/2 (0.04%)

1, 6 (0.07%)

F

19

1/2

2

P

31

1/2

2

S

32, 33, 34

0, 3/2 (0.8%)

1, 4 (0.2%)

Cl

35, 37

3/2

4

As

75

3/2

4

Se

76, 77, 78, 80, 82

0, 1/2 (7.6%)

1, 2 (3.8%)

Br

79, 81

3/2

4

I

127

5/2

6

The donor atoms of ligands that are coordinated by a metal can also have a nuclear spin. In EPR this nuclear spin may be detected through superhyperfine interaction, that is, the interaction of the electron spin S with the ligand atom nuclear spin I. Its observation implies that the unpaired electron(s) of the metal has spin density at the ligand atom, which is a direct manifestation of the covalent character of the coordination bond. In biological EPR spectroscopy it is used as an identifier of structure (and change of structure following a reaction) of prosthetic groups. An especially relevant case for enzymology is the observation of superhyperfine splittings from atoms (possibly enriched in certain isotopes) of a substrate that binds directly to the metal ion in an active site of an enzyme. Some biologically relevant ligand atom isotopes and their nuclear spins are given in Table 4.5; again the mass numbers of nuclei with I6¼0 are underlined. Possibly the most common superhyperfine pattern found in the EPR of biological systems is the three-line pattern of the metal-coordinated nitrogen in histidine ligand. A notorious example is that of NO-reacted ferrous heme in enzymes as exemplified in Fig. 4.8 (Fraaije et al., 1996). In particular the gy derivative-shaped feature is split into three by the nitrogen from the axial NO ligand, and each of these three lines is again split (by a smaller amount) by the nitrogen from the other axial ligand which is a histidine nitrogen. The experiment does not only unequivocally establish histidine as an axial ligand, it also shows that, after binding, the unpaired electron of the NO radical has delocalized over the iron onto the histidine nitrogen where it now has some finite spin density.

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FIGURE 4.8 EPR spectrum at 9.18 GHz (top) of a heme-containing enzyme (catalase-peroxidase) reacted with NO to form an Fe(II)
NO paramagnet with S 5 1/2. The rhombic spectrum is split by a strong superhyperfine interaction with the NO nitrogen and a weaker superhyperfine interaction with a histidine nitrogen. The bottom trace is a simulation with Azyx 5 20.5, 12, 10 gauss for the NO nitrogen and Azyx 5 6.5, 6.5, 6.5 gauss for the histidine nitrogen. Reproduced with permission from Fraaije, M.W., Roubroeks, H.P., Hagen, W.R., Van Berkel, W.J.H., 1996. Purification and characterization of an intracellular catalaseperoxidase from Penicillium simplicissimum. Eur. J. Biochem. 235, 192
198.

High-spin systems High-spin systems (S . 1/2) are intrinsically more complex than S 5 1/2 systems because they have more than two (namely 2S 1 1) magnetic sublevels, and this has two important consequences: (1) the number of possible EPR transitions is greater than one, and (2) the observed spectral features define “effective” g values, that is, the real g values from the Zeeman interaction are dramatically shifted by the S2S interaction between unpaired electrons (also called zero-field interaction because it is always present even when the magnet is turned off). We use the example of S 5 5/2 (e.g., high-spin Fe31) to illustrate these matters. There are six sublevels labeled with the magnetic quantum numbers mS 5 15/2, 13/2, 11/2,
1/2,
3/2,
5/2, which could in principle afford up to 15 EPR transitions. In practice, however, this number is usually seriously restricted by the selection rule ΔmS 5 1 and by the fact that some zero-field interlevel splittings are greater than the microwave quantum hν (which makes a transition impossible). The ΔmS 5 1 selection rule is a quantum mechanical shorthand notation to state that some of the possible transitions have low probability, and are practically undetectable. In many high-spin biological systems and model compounds at X-band frequencies the zero-field S2S interaction is much stronger than the Zeeman S2B interaction, and this results in a grouping of sublevels in so-called Kramers pairs (or doublets) with mS 5 6n/2 separated by an energy Δ . hν. For S 5 5/2

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FIGURE 4.9 S 5 5/2 spin manifold for axial symmetry (E 5 0) and for maximal rhombicity (E/D 5 1/3). Effective g values are given for all three intradoublet spectra assuming a real isotropic g 5 2.00.

we find the pairs mS 5 61/2, mS 5 63/2, and mS 5 65/2, which in hemoproteins (e.g., ferrimyoglobin) are separated by Δ1 5 2D (see below) and Δ2 5 4D with typically D  10 cm21, that is, energy differences much greater than the X-band quantum of c. 0.3 cm21, cf. the left-hand panel of Fig. 4.9. Together with the ΔmS 5 1 selection rule only a single transition is possible, namely, the intradoublet transition within the mS 5 61/2 pair. This would make the system very similar to a low-spin S 5 1/2 one; indeed it is therefore called an “effective” spin 1/2 system. There are two important differences with real S 5 1/2 systems. Firstly, the real g values for a system with a half-filled outer electron shell (Fe31 is d5) are predicted to be very close to ge, however, for this effective S 5 1/2 system we find two of the effective g values close to g 5 6. Secondly, although the mS 5 63/2 and mS 5 65/2 doublets may not contribute to the EPR spectrum, they are populated by molecules, and so the temperature dependence of the EPR intensity is more complex than that of a real S 5 1/2 system. At high temperature only one-third of all molecules is in the mS 5 61/2 doublet, so when counting spins we have to multiply the double integral value by three. At very low temperature, for example, 4.2 K, only the mS 5 61/2 doublet may be populated, and so no correction is required. At intermediate temperatures a correction has to be made according to the Boltzmann distribution over the three doublets (see below). In 3D space the anisotropic Zeeman interaction S2B and the anisotropic hyperfine interaction S2I are characterized by sets of three parameters: gx, gy, gz (the g tensor), and Ax, Ay, Az (the A tensor). Similarly, the anisotropic zero-field interaction S2S between electrons is characterized by the set of parameters Dx, Dy, Dz (the D tensor). However, in this case it can be shown that the three parameters are not independent, Dx2 1 Dy2 1 Dz2 5 0, and so the set can be reduced to two independent parameters, D 

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3Dz/2 and E  (Dx 2 Dy)/2. In the case of axial symmetry we have gx 5 gy and Ax 5 Ay, but Dx 5 Dy implies that E 5 0, so an axial zero-field interaction is described by a single parameter D. In X-band the hyperfine interaction is a perturbation to the Zeeman interaction (S2I ,, S2B) leading to characteristic splittings of the EPR spectrum as described in the previous section. Similarly, if the zero-field interaction is a perturbation to the Zeeman interaction (S2S ,, S2B), it leads to splittings of the EPR spectrum. For example the S 5 1 EPR of organic biradicals is split by D/geβ and (D63E)/geβ gauss (cf. Wasserman et al., 1964). However, for metal complexes we noted above that in X-band the zero-field interaction is usually dominant (S2S .. S2B) and, therefore the spectrum of a high-spin transition ion complex does not show spectral splittings from S2S interactions. And while we have generally valid simple analytical expressions for the resonance field as a function of g and A values (cf. Eqs. 4.3 and 4.10) the situation is somewhat different for systems with dominant zero-field interaction. In the EPR literature this is also called the “weak-field” case, since the Zeeman interaction, which depends on the magnetic field, is weak compared to the field-independent zero-field interaction. For the lowest doublet (mS 5 61/2) of an S 5 5/2 system (e.g., high-spin ferric heme proteins) the following equations hold in the weak-field limit (Hagen, 1981): " #  2 gx βBx 4E eff gx 5 3gx 1 2 2 ð4:14Þ D 2D2 " #  2 gy βBy 4E eff gy 5 3gy 1 2 1 ð4:15Þ D 2D2 geff z 5 gz

ð4:16Þ

in which the superscript “eff” of the left-hand symbols stands for effective g values, or real g values shifted by the zero-field interaction. The effective g values are the observables of an effective S 5 1/2 system and their angular dependence is as in Eq. (4.3). Note that the right-hand sides of Eqs. (4.14)
(4.16) contain a total of five unknowns, so with three observables the system is underdetermined when the EPR spectrum is measured at one frequency only. For the 3d5 high-spin system Fe31 the real g values happen to be very close to ge but even this simplification does not help much since, due to the large value of D eff eff compared to the X-band microwave quantum, the effective values gx and gy are very insensitive to the value of D unless one would collect multifrequency data including spectra at high frequencies well above 100 GHz (Van Kan et al., 1998). Thus it turns out that we can only determine the quantity E/D with some accuracy, because the splitting eff eff between gx and gy is approximately equal to 48E/D (assuming gx 5 gy 5 2). This is an example of a very general observation that the effective g values of half integer high-spin (S 5 n/2) systems, in particular metalloproteins, are dominated by a single parameter η 5 E/D also known as the rhombicity parameter. We can estimate the value of D from a fit of the temperature dependence of the EPR intensity to a Boltzmann distribution over all the sublevels of the spin manifold. For S 5 5/2 the relation is

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FIGURE 4.10 Rhombogram for S 5 5/2. Effective g values as a function of rhombicity are given for all three intradoublet spectra: the red traces are for the 61/2 doublet, the black traces are for the 63/2 doublet, and the blue traces are for the 65/2 doublet. The real g tensor is assumed to be isotropic, g 5 2.00. The arrows indicate rhombicities for the real spectra in Fig. 4.11.



21 I ~ 11e22D=kT 1e26D=kT T21

ð4:17Þ

in which T is the absolute temperature and it is assumed that D .. E. An example of such a determination for an S 5 7/2 system can be found in Hagen et al. (1987). To understand high-spin EPR spectra generally requires making simulations based on energy matrix diagonalization techniques. This complex subject is beyond the scope of this chapter (it is addressed in detail, e.g., in Hagen, 2009). However, its application results in effective g values as a function of rhombicity η 5 E/D only. Combined with the theoretical result that rhombicity is limited to 0 # η # 1/3 (Troup and Hutton, 1964) this allows for the practical procedure of getting effective g values from tables or figures or fast computer programs (Hagen, 2009). An example is given in Fig. 4.10. These plots are called “rhombograms”; the rhombicity E/D is on the x-axis and it runs from zero to its theoretical maximum 1/3. On the y-axis are effective g values, and the eff eff eff graph defines gx , gy , and gz for the intradoublet spectra from the three doublets mS 5 61/2 (red), mS 5 63/2 (black), and mS 5 65/2 (blue). So a given S 5 5/2 compound will have its unique rhombicity value η 5 E/D, related to the molecular symmetry at the paramagnetic center, which defines, according to Fig. 4.10, three sets of g values for three overlapping spectra. The graph also gives an indication of the relative intensity of these three spectra: the closer the three g values are to each other the narrower the spectrum and the higher its amplitude will be. Note, however, that spectra are taken on a linear

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High-spin systems

FIGURE 4.11 Examples of S 5 5/2 spectra (red traces) and their simulations (black traces) based on effective g values from rhombograms with rhombicity η 5 E/D. The top spectrum (ν 5 9.18 GHz; T 5 15 K) is from high-spin heme Fe(III) in PAC (Penicillium simplicissimum atypical catalase), and the bottom spectrum (ν 5 9.30 GHz; T 5 90 K) is from high-spin Fe(III) in Escherichia coli iron-superoxide dismutase.

magnetic field scale, that is, on a reciprocal g value scale. So moving g values toward zero will rapidly increase the field range of the spectrum and thus rapidly decrease its amplitude. In fact, geff 5 0 means a resonance at infinite field and thus an infinitely wide scan range and an infinitely small amplitude. In other words, the transition is forbidden. This is what we noted above to be the case for ferrimyoglobin, for which the energy level of the left-hand panel of Fig. 4.9 holds: two of the three possible spectra have geff 5 0, and therefore are totally unobservable. On the other hand, the Fe(III) in the protein rubredoxin, coordinated by four cysteine sulfurs in a strongly deformed tetrahedron, has η 5 E/D  1/3, for which the right-hand panel of Fig. 4.9 holds: all three spectra are allowed, but the one from the middle doublet will have the highest amplitude by far, because it incidentally has an isoeff eff eff tropic geff value (gx 5 gy 5 gz 5 4:29). Fig. 4.11 gives two experimental example spectra for cases of intermediate rhombicity: the slightly rhombic (η 5 0.023) heme iron site in a catalase, and the rather rhombic (η 5 0.238) iron site in a superoxide dismutase. Their effective g values are indicated by arrows on the top of the rhombogram of Fig. 4.10. Hagen (2009) gives rhombograms and example spectra for S 5 3/2
9/2. All metalloproteins and models are subject to conformational distributions, which in turn lead to distributions in EPR parameters, that is, g strain. Where a distribution in g values leads to broadening and skewing of the main features in the powder pattern, and a distribution in A values leads to nuclear-orientation dependent line width (cf. Eq. 4.13), a distribution in D values leads to broadening of all features in high-spin spectra,

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however, with the broadening increasing over the field scan. In other words, the low-field features of spectra from S 5 n/2 systems with large zero-field splittings always exhibit the best resolution (Hagen, 2007). Thus far, we have limited the discussion to half-integer spin systems. Integer spin systems with S 5 1, 2, 3, etc., form a special class from the viewpoint of EPR spectroscopy. Many of these systems are “EPR silent,” which means that no signal is found in X-band EPR. This situation is very different from that of the half-integer spins, which should always afford at least one signal at any frequency. The reason is that the equivalents of Kramers pair doublets in half-integer spins are non-Kramers pair doublets in integer spin. According to QM Kramers pairs are always degenerate (i.e., they have the same energy) in zero field; non-Kramers pairs do not have this restriction, and, in fact they are always split in zero field (with the exception of axial S 5 1 systems) (Hagen, 2009). If this splitting happens to be greater than the microwave quantum hν, then no transitions are possible at frequency ν, and one has to turn to a spectrometer with (much) higher frequency to detect EPR. When the zero-field intradoublet splitting is small (Δ , hν) then EPR is possible, but the transition probability is low for normal spectrometers in which the external field B is perpendicular to the magnetic component B’ of the microwave (B\B’). One frequently has to use a special cavity, the parallel-mode resonator, in which the external field is parallel to the magnetic component of the microwave, B||B’, to get reasonably sharp spectra of reasonable intensity (Hagen, 1982). See Hagen (2006, 2009) for further theoretical and experimental details. A brief, practical summary of integer-spin X-band EPR of metalloproteins and models is as follows. S 5 2 is the most commonly observed system. S 5 1 systems are rarely detectable. S 5 2 systems are either EPR silent or give a single, broad line at low field with geff $ 8 (e.g., the [3Fe
4S]0 cluster). S . 2 systems in biology are restricted to metal clusters (e.g., the P-cluster in nitrogenase enzymes), and they may exhibit a single, relatively sharp line in parallel-mode EPR with geff  4S (Hagen, 2009).

Applications overview EPR is a “something for everyone” spectroscopy: practical and useful EPR applications on biomolecules and models can range from very simple to very involved experiments and analyses. “Wow, this protein contains a metal cofactor ‘X’!” could well be the verbal synopsis of a breakthrough result from a 5-min, first-trial EPR measurement. Such an identification does not necessarily require any knowledge of EPR theory, as it can be based on fingerprint correlation, that is, comparing the EPR spectrum (e.g., shape, peak positions, broadening with increasing temperature) with literature data on characterized systems. Rather more often, however, some qualitative or (semi-)quantitative understanding of EPR may support assignments. For example, “This signal is split into four lines, so it must be from a copper complex, and since the Az value is in the range of 30
100 gauss we must have a type-I, or ‘blue’ copper protein.” Or, “The gz . g\ so EPR combined with ligandfield theory stipulated that this Cu21 probably has an axially elongated octahedral coordination.” In other words, the element identification usually also affords a qualitative

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151

conclusion about the coordination environment. And, of course, observing a copper EPR signal from a protein also immediately defines the oxidation state of the metal to be 2 1 . Input of slightly more effort can make the experiment quantitative, and thus considerably increase its biological relevance. Establishing nonsaturating measurement conditions and then counting spins from the doubly-integrated EPR with respect to an external standard, provides a concentration of the metal, which, when compared to protein concentration, allows for important interpretational discriminations such as “The Cu/protein ratio is approximately unity, so this must be a copper protein” versus “The Cu/protein ratio is c. 0.05, so this is either a heavily demetallated copper protein, or, perhaps more likely, it is a noncopper protein that in the course of the purification procedure became contaminated with extraneous copper.” Observation and qualitative understanding of superhyperfine splittings (the “super” means from ligands, not from the central metal) can identify ligands such as nitrogens in metal coordination. For example, observation of a line that is split into five lines with an intensity ratio of 1:2:3:2:1, would be consistent with coordination by two equivalent 14Ns (i.e., with the same A value). Alternatively, a line that is split into nine lines of equal intensity, would be consistent with coordination by two nonequivalent Ns (i.e., with significantly different A values). A chemically more sophisticated form of this type of experiment is to isotopically label an enzyme’s substrate (or better a nonconvertible substrate analogue), and then, after reaction of substrate and enzyme, to look for hyperfine splittings, for example, in the ENDOR of the metal in the catalytic center of the enzyme. Identification of such splittings from, for example, 2H, 13C, or 15N, not only proves that the substrate directly binds to the metal (which is also shown by a change in the EPR upon incubation with substrate), but it may also reveal which part of the substrate is coordinated by the metal. By the way, observation of superhyperfine interaction in EPR is the most direct way a (bio)chemist has available to show that a coordination bond possesses covalent character: it proves that the unpaired electron of the metal must be delocalized and spend some of its time on the ligand. Clustering of metal ions, especially in iron
sulfur clusters, is another phenomenon that is readily identified in EPR experiments. Since iron
sulfur clusters in proteins are relatively labile entities (they contain acid-labile sulfur, so the cluster disintegrates upon acidification), iron is easily liberated from the protein and iron content is readily determined colorimetrically. Suppose a protein is found to contain c. 4 irons. If the EPR of the protein only shows a single spin system, then the straightforward conclusion is that all irons are electronically coupled by exchange interaction (more precisely superexchange, because the coupling is not directly between irons, but rather over two chemical bonds via the sulfurs). Interaction of electrons by exchange is significant only over a distance of one or two chemical bonds, but through-space interaction between electrons (i.e., dipolar interaction) is significant over longer distances. This particular form of S2S interaction gives small splittings and shoulders in X-band EPR for spin systems at mutual distances of the order ˚ . Its observation is a qualitative indicator for the fact that two centers “see each of 5
15 A ˚ , and therefore are located in the same protein other,” that is, are separated by some 10 A complex. Quantitative interpretation in terms of accurate distance and mutual orientation is possible but requires involved analyses usually based on data taken at several microwave frequencies.

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If the identification of a signal with a particular structure is unequivocal, then one can also forget about EPR theory and use the spectrometer as a black box. In these types of experiments the amplitude of a signal is usually measured as a function of some external parameter, for example, redox potential (to determine reduction potentials), concentration (to determine binding affinities), reaction time (to determine rate constants), acidity (to determine pKs), etc. The experimental challenges are now not so much in the EPR spectroscopy, but rather in the setups to chemically prepare the EPR samples, for example, anaerobicity, rapid mixing, rapid freezing, etc. On a final note, let us consider the particular situation of EPR spectroscopy on complex biological systems such as multicenter enzymes, complete respiratory chains, cell organelles such as mitochondria, or even whole prokaryotic or eukaryotic cells. We are now facing a situation of several-to-many overlapping spectra while at the same time the concentration of the individual paramagnets drops down to the micromolar range or less. Nevertheless, it is particularly on this battlefield that biological EPR has savored some of its most consequential victories such as the initiation of the ever-expanding field of iron
sulfur biochemistry that started off with the discovery of an “unusual” EPR signal with g 5 1.94 in mammalian heart mitochondria some six decades ago (Beinert and Sands, 1960). A range of special measures has been developed to support this type of experimentation: (1) procedures to maximize concentration, for example, by centrifugation of samples in EPR tubes; (2) statistics on EPR from multiple, similar preparations; (3) extensive deconvolution of spectra by thermodynamic (redox equilibrium titrations) and/or kinetic (rapid redox reactions) resolution; (4) deliberate partial power saturation and/or magnetic field overmodulation of EPR; and (5) molecular biological constructs that overexpress paramagnetic metalloproteins of interest. Some of these approaches (and their putative pitfalls) were discussed in more detail in a recent review (Hagen, 2018).

Test questions [q1] (A) The g value of a radical measured at ν 5 9 GHz is g 5 2.006; what is its g value at ν 5 250 GHz? (B) Why does an S 5 1/2 EPR spectrum generally have three “peaks”? [q2] (A) Why do EPR spectrometers have a resonator/cavity? (B) Why does the microwave bridge of an EPR spectrometer have a reference arm? [q3] (A) What is the EPR equivalent of the extinction coefficient in optical spectroscopy? (B) The detection limit for X-band EPR of a reduced [2Fe
2S] cluster is c. 5 μM. How many mg of an 11 kDa ferredoxin do you have to prepare at minimum to measure a spectrum? [q4] In a small coordination complex in water an S 5 1/2 metal ion is coordinated by four nitrogens. (A) How many EPR lines do we observe when the hyperfine interaction with all 14Ns is of the same strength? (B) And how many lines do we get if each N has a very different A value? [q5] In Fig. 4.7, why does the vanadium hyperfine pattern in the z-direction give positive peaks at low field but negative peaks at high field?

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[q6] With reference to the rhombogram in Fig. 4.10, what approximately would be the relative intensities of the three subspectra of an S 5 5/2 system with rhombicity E/D 5 0.11? [q7] Why are some integer-spin systems EPR silent?

Answers to test questions [a1] (A) A real g value is frequency-invariant: if ν goes up, then Bres goes up too, and their ratio is constant, therefore also g 5 hν/βB is constant. (B) The derivative of the EPR absorption spectrum emphasizes the turning points of the powder pattern. [a2] (A) Creating a standing microwave increases the spectrometer’s sensitivity by Q  5000. (B) Diverging a fixed, small amount of power to the detection diode makes it possible to change the power to the sample without need to retune the spectrometer. [a3] (A) The EPR extinction coefficient is unity, and the EPR intensity is given by Eq. (4.5) or is approximated by the average scalar in Eq. (4.10). (B) 1 mM is 11 mg/mL, so 5 μM in 0.2 mL is 0.011 mg. [a4] (A) 9. (B) 81. [a5] Experimental EPR spectra are the first derivative of EPR absorption spectra. The first derivative gives the slope of the absorption (the latter is positive by definition); approaching the end of the spectrum the slope must be negative in order to return to the zero level of the baseline. [a6] The anisotropy in the effective g tensor of the 61/2 and the 63/2 doublet are similar, and, therefore the two subspectra will have comparable intensity. Anisotropy for the 65/2 intradoublet transition is very extensive, so the intensity of this subspectrum will be extremely low. [a7] For integer-spin systems (also non-Kramers systems) it is possible that all splittings between the energy levels of the spin manifold are greater than the microwave quantum.

References Aasa, R., Va¨nnga˚rd, T., 1975. EPR signal intensity and powder shapes: a reexamination. J. Magn. Reson. 19, 308
315. Bagguley, D.M.S., Griffiths, J.H.E., 1947. Paramagnetic resonance and magnetic energy levels in chrom alum. Nature 160, 532
533. Beinert, H., Sands, R.H., 1960. Studies on succinic and DPNH dehydrogenase preparations by paramagnetic resonance (EPR) spectroscopy. Biochim. Biophys. Res. Commun. 3, 41
46. Dutton, P.L., Leigh, J.S., Seibert, M., 1972. Primary processes in photosynthesis: in situ ESR studies on the light induced oxidized and triplet state of reaction center bacteriochlorophyll. Biochem. Biophys. Res. Commun. 46, 406
413. Fraaije, M.W., Roubroeks, H.P., Hagen, W.R., Van Berkel, W.J.H., 1996. Purification and characterization of an intracellular catalase-peroxidase from Penicillium simplicissimum. Eur. J. Biochem. 235, 192
198. Hagen, W.R., 1981. Dislocation strain broadening as a source of anisotropic line width and asymmetrical line shape in the electron paramagnetic resonance spectrum of metalloproteins and related systems. J. Magn. Reson. 44, 447
469.

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Hagen, W.R., 1982. EPR of non-Kramers doublets in biological systems: characterization of an S 5 2 system in oxidized cytochrome c oxidase. Biochim. Biophys. Acta 708, 82
98. Hagen, W.R., 2006. EPR spectroscopy as a probe of metal centers in biological systems. Dalton Trans. 2006, 4415
4434. Hagen, W.R., 2007. Wide zero field interaction distributions in the high-spin EPR of metalloproteins. Mol. Phys. 105, 2031
2039. Hagen, W.R., 2009. Biomolecular EPR Spectroscopy. CRC Press Taylor & Francis Group, Boca Raton, FL. Hagen, W.R., 2018. EPR spectroscopy of complex biological iron-sulfur systems. J. Biol. Inorg. Chem. 23, 623
634. Hagen, W.R., Wassink, H., Eady, R.R., Smith, B.E., Haaker, H., 1987. Quantitative EPR of an S 5 7/2 system in thionine-oxidized MoFe proteins of nitrogenase; a redefinition of the P-cluster concept. Eur. J. Biochem. 169, 457
465. Troup, G.J., Hutton, D.R., 1964. Paramagnetic resonance of Fe31 in kyanite. Br. J. Appl. Phys. 15, 1493
1499. Van Kan, P.J.M., Van der Horst, E., Reijerse, E.J., Van Bentum, J.M., Hagen, W.R., 1998. Multi-frequency EPR spectroscopy of myoglobin; spectral effects for high-spin Fe(III) ion at high magnetic fields. J. Chem. Soc. Faraday Trans. 94, 2975
2978. Wasserman, E., Snyder, I.C., Yager, W.A., 1964. ESR of the triplet states of randomly oriented molecules. J. Chem. Phys. 41, 1763
1772.

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C H A P T E R

5

Introduction to biomolecular nuclear magnetic resonance and metals Ineˆs B. Trindade and Ricardo O. Louro Instituto de Tecnologia Quı´mica e Biolo´gica Anto´nio Xavier da Universidade Nova de Lisboa, Oeiras, Portugal O U T L I N E Introduction

156

Properties of the matter relevant to nuclear magnetic resonance

157

Energy of nuclear magnetic resonance transitions

Coupling: the interaction between magnetic nuclei Decoupling

170 174

The nuclear Overhauser effect

174

158

DOSY: sizing up molecules

176

Macroscopic magnetization

160

Chemical exchange

177

Acting on magnetization Pulses The rotating frame

161 161 162

Relaxation What are the physical mechanisms of relaxation?

162

Multidimensional nuclear magnetic resonance How do the correlations arise and how are cross-peaks generated? The COSY The NOESY The HSQC

An nuclear magnetic resonance experiment The chemical shift Carrier frequency Sampling bandwidth and the Nyquist theorem Measuring T1

Practical Approaches to Biological Inorganic Chemistry DOI: https://doi.org/10.1016/B978-0-444-64225-7.00005-5

163 166 167 168 168 170

179 180 181 181 182

Metals in biomolecular nuclear magnetic resonance spectra 183 Transition metals and interaction with the unpaired electron(s) 184 Hyperfine scalar coupling

184

Dipolar coupling

184

155

© 2020 Elsevier B.V. All rights reserved.

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Relaxation

186

Contact relaxation

186

Dipolar relaxation

186

Curie relaxation

188

Residual dipolar couplings

188

Nuclear magnetic resonance of (semi-)solid samples 188 Direct observation of metals by nuclear magnetic resonance 189 In-cell nuclear magnetic resonance An nuclear magnetic resonance spectrometer: measuring macroscopic magnetization and relaxation

190

191

Care in obtaining nuclear magnetic resonance spectra of paramagnetic samples 192 Water eliminated Fourier transform and super-water eliminated Fourier transform sequences: catching up with fast relaxing samples 193 Evan’s method: measuring magnetic susceptibility 195 Conclusions

197

Further reading

197

Useful physical constants

198

Exercises

198

Answers

199

Introduction Like any other kind of spectroscopy, nuclear magnetic resonance (NMR) spectroscopy uses the outcome of the interaction between electromagnetic radiation and matter to obtain information on the sample being analyzed. The sample is placed in a strong and homogeneous magnetic field and is subjected to electromagnetic radiation. Some nuclei will absorb radiation at a frequency determined by three factors: the chemical nature of nucleus, the chemical environment surrounding the nucleus, and the strength of the magnetic field. These three aspects can be manipulated experimentally giving rise to the myriad of applications of NMR. These applications, in addition to the biomolecular studies that will be the focus of the present chapter, extend to studies in materials sciences and medical imaging, two examples of contemporary applications of great societal importance. It should however, be realized that these present day applications are the product of the continuous support for nearly a century of fundamental physical research. The first suggestion that atomic nuclei could have magnetic properties was made by Pauli in the 1920s, at a time when Bohr’s atomic model was being developed. In the 1930s, Rabi (Nobel Prize in Physics in 1944) and collaborators observed absorption of energy by a beam of molecular hydrogen subject to a homogeneous magnetic field. Observation of the NMR phenomenon in “ordinary matter” was made by the end of World War II by Purcel and Bloch (Nobel Prize in Physics in 1951) using water and paraffin, respectively. Chemists began to take note of the NMR technique at the beginning of the 1950s following the observation of three signals in a sample of ethanol, therefore revealing the existence of chemical shifts. However, the low sensitivity of the technique remained a challenge for biomolecular applications. This issue was only solved in the late 1960s and early 1970s

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with the convergence of three independent developments: the increase in computing and memory storage power enabled the widespread transition to pulsed NMR techniques associated with Fourier transform; the introduction of superconducting magnets which led to a dramatic increase in the field strength; and, finally, the concomitant increase in spectral sensitivity and dispersion supported the literal explosion in the development of applications of NMR spectroscopy. The award of the 1991 Nobel Prize in Chemistry to Richard Ernst for his pioneering contributions to high-resolution NMR highlights the importance of these developments for chemistry and biology. Applications that explored strategies to obtain structural information from biological molecules using either distance-geometry constraints from paramagnetic probes or from the nuclear Overhauser effect (NOE) were the focus of great efforts. These were recognized in 2002 by the award of the Nobel Prize in Chemistry to Kurt Wu¨thrich for pioneering the strategies to determine the structure of biological macromolecules by NMR, establishing this as a bona fide method for structural biology. Furthermore, during the 1970s, Paul Lauterbur introduced the concept of projection reconstruction using field gradients to encode spatial information on the recorded data. Together with the speeding up of the data collection by the echo-planar method developed by Sir Peter Mansfield, these two methods are the bases for the magnetic resonance imaging technique for which both received the Nobel Prize of Physiology and Medicine in 2003.

Properties of the matter relevant to nuclear magnetic resonance NMR spectroscopy can be performed on nuclei that are sensitive to an external magnetic field, B0, generated by the spectrometer. Both neutrons and protons have the quantum mechanical property of spin (I), which assumes values of 1 or
1/2. Just as in the case for electrons in orbitals, nuclear particles occupy specific energy levels. Within each of these levels the most stable configuration for a pair of protons or a pair of neutrons is established when these spins are antiparallel. Therefore atomic nuclei with an even number of neutrons and protons possess zero or integer net spin. Nuclei with an odd number of neutrons and of protons have an integer spin, whereas nuclei with either an odd number of neutrons or an odd number of protons have a noninteger spin. Only nuclei with nonzero spin are sensitive to the external magnetic field. Table 5.1 shows some nuclei of biomolecular relevance that are NMR active. However, some exist in low abundance in nature, forcing the use of chemically or biochemically enriched samples. For biomolecular NMR, the most commonly used isotopes for enrichment are 13C or 15N and, alternatively, isomorphous replacement can substitute one nucleus with another with similar chemical properties and more convenient NMR characteristics. This is often performed when studying metalloproteins to facilitate the characterization of the metal-containing active site. The spin of subatomic particles, like the neutron, the proton, and the electron, is a quantum mechanical attribute intrinsic to these particles. However, some of its properties relevant for NMR can be described using analogies from classical electromagnetism. In the context of classical electromagnetism, spin arises from the rotation of a charged particle. A rotating particle has a mass and possesses an angular momentum J. This is a vectorial

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TABLE 5.1 Examples of relevant nuclei for biomolecular nuclear magnetic resonance and respective spin properties. Natural abundance (%)

Gyromagnetic ratio ( 3 106 rad/T/s)

Sensitivity (relative to proton)

Larmor frequency at 2.3488 T (MHz)

1

100.00

Nucleus

Spin

1

H

1/2

2

H

1

1.5 3 1022

41.06

1.45 3 1026

15.35

1.108

67.28

1.76 3 1024

25.15

99.63

19.34

1.01 3 1023

7.23

0.37

2 27.13

3.85 3 1026

10.14

99.98

267.52

13

C

1/2

14

N

1

15

N

1/2

23

Na

3/2

100

70.80

9.27 3 1022

26.47

31

P

1/2

100

108.39

6.65 3 1022

40.52

quantity pointing perpendicular to the direction of rotation according to the right-hand rule. This rule states that the thumb indicates the direction of the angular momentum and the other fingers curl in the direction of rotation. A classical rotating particle with charge also has a magnetic moment μ which is also a vectorial quantity. These two vectorial quantities are related to each other by a scalar quantity called the gyromagnetic ratio γ (Eq. 5.1 and Fig. 5.1). The gyromagnetic ratio is a constant that is specific for each nucleus and depends on a dimensional number called the nuclear g factor, on the charge of the nucleus, and on its mass. As can be observed in Table 5.1, the gyromagnetic ratio can assume positive or negative values which indicate whether the angular momentum and the magnetic moment are parallel or antiparallel, respectively. μ 5 γUJ

ð5:1Þ

When nuclei with nonzero spin are placed in an external magnetic field, the magnetic moment of the nuclei will not align with the direction of the magnetic field. Instead, the nuclei will precess about the direction of the field. This precessional motion occurs with a frequency ω, which is proportional to the intensity of the effective magnetic field and to the gyromagnetic ratio of the nucleus. ω 5 γB ðrad=sÞ

ð5:2Þ

Therefore in the presence of an external magnetic field, nuclei with different gyromagnetic ratios can be discriminated because their precession frequency is different. Eq. (5.2) is at the basis of the common designation of NMR magnets by the precession frequency of protons (e.g., 800 MHz) by showing that this property is directly related with the field strength.

Energy of nuclear magnetic resonance transitions The magnitude of the magnetic moment μ depends on the angular momentum which is defined with respect to the nuclear spin quantum number I. Because this is a quantum

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159

FIGURE 5.1 A classical view of spins. For particles of positive gyromagnetic ratio the angular momentum and the magnetic moment point in the same direction.

FIGURE 5.2

Nuclear Zeeman splitting for commonly used nuclei in biomolecular NMR given to scale to show in a pictorial way that the energy splitting of protons is much larger than for other nuclei.

mechanical property, a discrete number of states is possible and are given by (2I 1 1). Quantum mechanical selection rules determine that only integer transitions are allowed between the
I to the 1 I states. For example, for I 5 1/2, only one transition exists from I 5
1/2 to I 5 11/2. These states are degenerate in the absence of a magnetic field, that is, they have the same energy. It is only in the presence of a magnetic field that the degeneration is lifted and the states assume different energies. This is called the Zeeman effect (Fig. 5.2). The potential energy of a magnetic dipole in a magnetic field is given by: E 5 2 μB

ð5:3aÞ

Because μ is dependent on the angular momentum, which is quantized according to the nuclear spin quantum number (m), Eq. (5.3a) can be rewritten as: E 5 2 gI μN Bm 5 γh=2π Bm

ð5:3bÞ

Only transitions between states that lead to unitary changes in the spin quantum number are allowed. The energy associated with such transitions is: h B 2π

ð5:4Þ

h B 2π

ð5:5Þ

ΔE 5 γ And according to Planck’s equation: hν 5 γ

Eliminating h from both sides of the equation and converting the frequency from Hz to radians/s (multiplying by 2π) we obtain again Eq. (5.6), but in this case it was derived from quantum mechanical considerations: ω 5 γB

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ð5:6Þ

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5. Introduction to biomolecular nuclear magnetic resonance and metals

This is called the Larmor equation. This is the fundamental equation in NMR because it relates the quantum mechanical and the classical description of the behavior of magnetic particles in the presence of an external magnetic field. The Larmor equation defines both the frequency of the precession of the magnetic moment about the direction of the external field and the energy splitting associated with the transitions between quantized nuclear magnetic states.

Macroscopic magnetization Up to now, we have dealt with an idealized isolated spin in order to obtain the fundamental expressions that define its behavior in a magnetic field. However, a real NMR sample contains a large number of spins (of the order of 1017 for a micromole of material), which when placed in a magnetic field, will be found precessing predominantly in the Zeeman state of lower energy (Fig. 5.3). The difference in the number of spins in the various Zeeman states depends on I, and on the energy separation between the states. For two states with ΔE energy difference, the Boltzmann equation defines the ratio between the respective populations as Nβ ΔE 5 e2 kT Nα

ð5:7Þ

with Nβ the population of the high energy state. At contemporary operating field strength of NMR spectrometers this difference is tiny, with a ratio of the order of 0.999999. This small number reveals two important aspects of NMR: first, the technique is inherently insensitive and therefore it requires a considerable amount of material for good signal detection; second, from Eqs. (5.4) and (5.7) it is clear that the stronger the field, the larger the population difference. Altogether this gives rise to a more intense signal, and therein lies one of the drivers for the development of stronger fields (with some caveats further discussed in this chapter). When faced with a dilute sample and if a spectrometer with a more intense field is not available, the sensitivity can be improved by repeating the experiment several times and FIGURE 5.3 Representation of an ensemble of spins precessing in an external magnetic field showing the slight excess in the low energy state.

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161

adding up the results. The signal to noise ratio will increase with the square root of the number of repetitions. These repetitions are called scans for historical reasons. The original continuous wave (CW) NMR machines would scan a range of field strengths to detect the signals. Nowadays the field is fixed but the designation has survived the instrumental evolution. The small excess of spins precessing in the low energy state and the fact that these spins are found randomly distributed throughout the surface of the precessional cone, generates a macroscopic magnetization vector M, which aligns with the magnetic field. A good approximation of M can be given by the Curie Law: M

γ2 h ¯ 2 BIðI 1 1Þ 3kT

ð5:8Þ

This is a vectorial description of the NMR phenomenon in terms of a macroscopic magnetization that behaves in a classical fashion. Although it is applicable strictly to noninteracting spins, the pictorial representations that are derived from this description are often useful to interpret more complex situations. In this description the NMR experiment involves manipulation of the orientation of the magnetization vectors and therefore it is convenient to define an axes system. In this axes system, the B0 field is aligned with the z direction, and the x and y axes are oriented in the same way as the fingers curl around the thumb in the right hand.

Acting on magnetization Pulses The introduction of pulsed NMR methods associated with fast Fourier transform for processing the data was a tipping point in the development of NMR, opening the possibility for numerous experimental developments. In pulsed NMR the macroscopic magnetization is disturbed by applying a magnetic field B1 perpendicular to the static field generated by the magnet. This B1 field is applied as electromagnetic radiation during a short period and from there came the designation pulse. This pulse is a polychromatic emission of radiofrequency, and to a good approximation can be considered to cover a bandwidth of frequencies given by the reciprocal of the pulse length. Therefore the shorter the pulse the larger the bandwidth covered. When the B1 field matches the Larmor frequency, energy will be absorbed by the nuclei and the difference between spins up and down will be reduced, reducing the macroscopic magnetization in the z direction. However, magnetization will not be eliminated, instead it will be flipped away from the direction of the static B0 field by an angle theta (θÞ because the nuclear spins are no longer randomly distributed but will tend to point in the direction of the B1 field, that is, they develop coherence (Fig. 5.4). The flip angle achieved by the pulse depends on the nature of the nucleus, the strength of the B1 field, and on the duration of the pulse. It will become apparent below that observation of NMR signals occurs in the xy-plane, and therefore the maximum

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FIGURE 5.4 Effect of the application of a pulse about the x direction.

signal is observed for a pulse with a flip angle of 90 degrees, which is also typically called a π/2 pulse. θ 5 γB1 t

ð5:9Þ

From Eq. (5.9) it is evident that multiple combinations of pulse power and length can be used to obtain the same flip angle. Pulses also have phases. Most frequently they are applied in the x, y,
x, or
y direction. A 90 degrees pulse about the x direction will leave the magnetization in the y-axis.

The rotating frame The description of NMR often includes the implicit assumption that the axes system has been transformed from the static laboratory frame to the rotating frame. The description of the pulse represented in Fig. 5.4 follows this frame. The rotating frame is an axis system with the z-axis aligned with the B0 field and x and y axes rotating. When the frequency of the rotating frame is equal to the Larmor frequency, a magnetization vector that has been flipped by a B1 pulse will appear to be stationary, simplifying the description of the events. This is equivalent to the everyday situation of describing the trajectory of an object that is dropped as a vertical fall. Such description ignores the fact that the Earth is rotating at about 1600 km/h (at the equator) while the object is falling. The advantage of using the rotating frame, just like in the gravitational analogy, is that it simplifies the mathematical expressions used.

Relaxation After switching off the B1 field the spins gradually loose coherence, and the macroscopic magnetization returns to the direction of the static B0 field. These two phenomena are called relaxation (transverse and longitudinal relaxation, respectively) and are assumed to follow an exponential decay. This is the free induction decay or FID, and it is described by the Bloch equations: Mx ðtÞ 5 ½Mx ð0Þ cosðωtÞ 2 My ð0Þ sinðωtÞeð2t=T2 Þ

ð5:10aÞ

My ðtÞ 5 ½Mx ð0Þ sinðωtÞ 1 My ð0Þ cosðωtÞeð2t=T2 Þ

ð5:10bÞ

Mz ðtÞ 5 Meq 1 ½Mz ð0Þ 2 Meq  eð2t=T1 Þ

ð5:10cÞ

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FIGURE 5.5 Time evolution of the macroscopic magnetization during relaxation following a 90 degrees pulse: Mz, continuous line; Mx, dotted line; My, dash-dot line. 3D plot of the trajectory followed by the magnetization.

Each of the equations defines how the macroscopic magnetization evolves in the direction of each axis and combining them shows how the precessing macroscopic magnetization returns to the direction of the static field (Fig. 5.5). The Bloch equations show that the time it takes to recover the macroscopic magnetization in the direction of the static field B0 depends on the Larmor frequency and on two new parameters called T1 and T2. Only Mx and My depend on the Larmor frequency, and therefore the detection of NMR signals has to be made on the xy-plane in order to be able to distinguish different nuclei in the sample. The maximum signal is obtained when the magnetization is placed on the xy-plane. This is what is achieved by a 90 degrees pulse. The T1 and T2 parameters introduced by the Bloch equations are time constants that describe the relaxation of the macroscopic magnetization. T1 is called longitudinal relaxation because it affects solely Mz, and defines how fast the macroscopic magnetization returns to the direction of the static field. T2 is called transverse magnetization because it affects Mx and My and defines how fast the magnetization loses coherence after the B1 pulse has been switched off. It is clear from Eqs. (5.10a)
(5.10c) and Fig. 5.6 that coherence can be lost without return of the magnetization to the direction of the static field. Therefore T2 is always shorter or equal to T1.

What are the physical mechanisms of relaxation? The difference in energy levels in an NMR experiment is very small. In these conditions spontaneous emission has a low probability of occurring and for this reason is not considered as an effective relaxation mechanism. Instead collisions are also capable of promoting relaxation of nuclear dipoles because nuclei are not naked, instead they are surrounded by

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FIGURE 5.6 Representation of the magnetization vectors looking down on the xy-plane from the z-axis. The rotating frame tracks the Larmor frequency and therefore the macroscopic magnetization is stationary. In equilibrium, the magnetization is aligned with the z-axis and pointing towards the observer (left panel). After the B1 pulse all vectors are aligned (middle panel). As time goes by (right panel), the projection on the xy-plane becomes shorter due to T1 and the vectors spread evenly about the Larmor frequency (loose coherence) due to T2.

electrons. Therefore relaxation occurs because of local fluctuations on the effective magnetic field that affect the nuclei stimulating them to switch between the energy levels. These fluctuations can be: dipolar interactions with other nuclei; paramagnetic interactions with unpaired electrons; or quadrupolar interactions in the presence of nuclei with I . 1/2. For nuclei with I 5 1/2 the dominant sources of relaxation are dipolar interactions modulated by molecular motion. However, the underlying physical mechanism is different in each case: T1 arises from energy transfer to the surrounding medium, and for this reason is also called spin
lattice relaxation. It is caused by rotations and translations of a frequency similar to the Larmor frequency, so that spin transitions take place. This is the mechanism by which the nuclei dissipate the energy absorbed from the B1 pulse and it is an enthalpic process. T2 arises from energy transfer to other spins, and for this reason is called spin
spin relaxation. It is stimulated by molecular motions of frequencies similar to the Larmor frequency and also by molecular motions of very low frequency close to 0 Hz. Because this is associated with the loss of coherence of the magnetization, that is, increased disorder, it is an entropic process. As seen above, relaxation depends on molecular motions and therefore on changes with time which are described by a correlation function. For the rotation of a rigid body under Brownian motion this function is given by Eq. (5.11): 2τ

GðτÞ 5 Gð0Þeτc

ð5:11Þ

where τc is the rotation correlation time, which is the average time taken by a molecule to rotate by a radian. This is obviously a function of molecular size and shape, temperature, and medium viscosity. And it is also evident that, all else being equal, the larger the molecule the slower the typical motions and consequently, the longer the correlation time. More precisely, the spectral density function, which defines the distribution of frequencies of molecular motions, depends on τc according to: JðωÞ 5

2τc 1 1 ω2 τ2c

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ð5:12Þ

Relaxation

165

Fig. 5.7 shows that as the τc decreases, that is, the molecular size decreases, the range of motional frequencies sampled extends to higher values. The spectral density function shows that for shorter τc, slower motions are less probable. Given that T1 depends on motions of frequency similar to the Larmor frequency, this will be shorter when the molecular motions occur close to this frequency. On the other hand, because T2 depends on motions with frequencies close to the Larmor frequency and motions with frequency close to 0 Hz, T2 continues to decrease as the correlation time increases, that is, as the molecular size increases. This is seen in Fig. 5.8.

FIGURE 5.7 Normalized spectral density functions calculated for different values of τc. Yellow, orange, and blue lines were calculated for decreasing τc.

FIGURE 5.8 Dependence of T1 and T2 with correlation time. Blue, red, and green lines are calculations made respectively at 300, 600, and 900 MHz. Molecular size increases to the right.

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Because T2 is related with the way Mx and My lose coherence, that is, with the increased spreading of vectors in the xy-plane, T2 is related with the signal linewidth. Eq. (5.13) shows the relationship of T2 with the linewidth at mid-height of the NMR signal. Δν 1=2 5

1 πT2

ð5:13Þ

Therefore as the molecular size increases, lines become broader. When estimating T2 from the experimental linewidth, the value obtained will include the contributions from sample and field inhomogeneities.

An nuclear magnetic resonance experiment All NMR experiments can be described as organized in two general parts (Fig. 5.9): • the preparation period, during which the spins are manipulated in order to generate the effect of interest; and • the detection period, where the time evolution of the transverse magnetization is observed. The time dependence of the magnetization decay (the free induction decay or FID) is measured at discrete intervals and subjected to a mathematical procedure called Fourier transform to give the familiar frequency domain spectrum. This very simple NMR experiment consists of a 90 degrees pulse followed by the detection of a single signal which gives rise to two observables: • The intensity of the signal. This is dependent on the concentration of the sample, the strength of the B0 magnetic field, and also on the value of the gyromagnetic ratios of the nuclei being excited and detected (which are the same for this particular experiment). • The linewidth of the signal. As seen above, this is inversely proportional to the rate of transverse magnetization decay (T2). • Three consequences arise from these observables: • The stronger the magnetic field or the larger the gyromagnetic ratio of the nucleus, the stronger the signal. • If the transverse relaxation is too fast the signals will be too broad to be detected. Therefore very large molecules will be difficult to study. FIGURE 5.9 Diagram of a simple nuclear magnetic resonance experiment.

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• If there were no transverse relaxation the signal would be very sharp, and the linewidth would arise only from the Uncertainty principle. A spectrum such as that in Fig. 5.9, with a single signal is not of much use. Fortunately, nuclei of the same nature can have slightly different Larmor frequencies depending on the surrounding environment, therefore increasing the information content in a NMR experiment.

The chemical shift Atomic nuclei are immersed in atomic or molecular electron clouds. Electrons, being charged particles that are not static, are sensitive to the presence of the magnetic field of the spectrometer and generate a very weak local magnetic field. Therefore the nuclei sense an effective field that is different from the static field due to the influence of the surrounding electronic environment—the nuclear shielding (σ). Typically, the distribution of electrons around the nuclei is not spherical. The nuclear shielding tensor is anisotropic and thus is defined with respect to each direction of the axes system (the three principal components) giving rise to the phenomenon of chemical shift anisotropy. Fortunately, in solution, this effect averages out due to the fast and isotropic reorientation of the molecules, and thus a single value for the nuclear shielding can be used. Subsequently, the Larmor frequency can be expressed as: ω 5 2 γð1 2 σÞB0 ;

ð5:14Þ

In these conditions, a different chemical environment gives rise to a characteristic nuclear shielding. Therefore there are very typical spectral regions where the signals of nuclei with a particular chemical nature can be found. As seen from the Larmor equation, the resonance frequency is dependent on the static field strength, and therefore the frequency difference between two signals increases with field strength. On the other hand, the chemical shift (δ) is defined as the ratio between the nuclear shielding and the static magnetic field and is reported in parts per million. The very useful outcome in this definition of chemical shift is that signals measured in spectrometers with different field strength can be matched because they will have the same chemical shift (in ppm). Nonetheless, the spectral dispersion is larger in the experiment performed at the strongest field. In practice, chemical shifts are measured as the difference in parts per million between the signal of interest and the signal of a reference substance added to the sample. Different compounds are used as reference depending on the solvent, applications, and nuclei being observed. In water or deuterated water, the methyl signal of tetramethylsilane (TMS) is typically used as a marker of 0 ppm for protons. δ5

ν sample 2 ν ref :106 ν ref

ð5:15Þ

In this way, a positive δ indicates that the frequency of resonance of the nucleus under observation is higher than that of the reference. In the old CW spectrometers where the spectrum was obtained by varying the intensity of the field, observation of such signal

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required that the field was lower relatively to that required to observe the reference. Therein lies the still often used designation of “low field” for signals with positive chemical shift. The phenomenon of chemical shift has a very important operational consequence in modern FT spectrometers with fixed B0. The frequency of the B1 pulse may need to be modified depending on the chemical shift of the nuclei to be observed.

Carrier frequency Differences in nuclear shielding place NMR signals in different spectral positions. Therefore the choice of the carrier frequency (Cf) is essential for a good performance of the NMR experiment. It was seen above that when the B1 field matches the Larmor frequency there is absorption of energy. By analogy with a relay race where the runners match their speed to transfer the baton, the Cf should match as closely as possible the Larmor frequency of the nuclei being observed so that excitation occurs more effectively.

Sampling bandwidth and the Nyquist theorem In addition to correctly choosing the B1 pulse power, length, and Cf to ensure that all interesting nuclei are excited, the existence of chemical shifts also forces the need to ensure that the measurement conditions are chosen so that all signals of interest are observed. NMR spectrometers record a FID that is Fourier transformed to obtain a spectrum. Because of this arrangement, the sampling bandwidth (usually referred to as spectral width) observed is dependent on the sampling frequency during the time the FID is recorded. According to the Nyquist theorem, for any signal present in the FID, at least one point per period must be sampled in order to correctly define the frequency upon Fourier transformation. F 5 1=ð2ΔtÞ

ð5:16Þ

In Eq. (5.16) F is the highest frequency that can be correctly defined when using the sampling interval Δt. Nonetheless, in some cases where the spectra are sparsely populated with signals, researchers may deliberately opt for sampling below the requirements of the Nyquist theorem. The advantage is that for the same experimental time and data size, better spectral resolution is obtained, and signals outside of the spectral window are aliased (often referred to as “folded”) into the spectrum. A signal with a frequency equal to F 1 f will be aliased into a position which is equal to F 2 (Fig. 5.10). Therefore these aliased signals can be identified by changing the sampling bandwidth because unlike all other signals they will change position. At the current operating field strengths, the Larmor frequency is of the order of hundreds of MHz whereas in typical solution state biomolecular applications, the chemical shifts show differences, at most, in the order of kHz. Therefore the importance of correctly selecting the Cf becomes apparent (Fig. 5.11). The most favorable condition occurs when the Cf can be placed in the center of the spectrum, since (1) the excitation profile of the B1 pulse extends symmetrically from the Cf, covering a range inversely proportional to the

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The chemical shift

FIGURE 5.10 Effect of incorrectly chosen sampling frequency. The top panel shows the continuous decay of two signals of different frequency. The bottom panel shows the result of the discrete measurements at a sampling frequency that is clearly inadequate for the signal in the thick blue line, which upon digitization appears of a lower frequency than the true frequency.

FIGURE 5.11

Effect of different choices of carrier frequency (Cf) upon a spectral window with two signals of interest. While the first option (Cf 1) shows a good choice for the Cf, the option in the bottom (Cf 2) is clearly not the best choice for the two signals of interest.

pulse length. Therefore, at the center of the spectrum we have the lowest demands on pulse duration and power. (2) Also this is the condition where the minimum sampling is necessary because the difference between the Cf and any signal in the spectrum is, at most, half of the spectral width. The consequence is the minimization of the experimental time, since the total experimental time is the number of samplings made, multiplied by the sampling interval. Until now we have seen that NMR provides information on the number and relative abundance of chemically distinguishable nuclei in the sample. It also provides information on the transverse relaxation rate that relates with the molecular size. Additionally, if it is necessary to perform multiple scans to improve the signal to noise ratio of the spectrum, the T1 must be determined in order to know how long one must wait until the magnetization has returned to the z direction. T1, unlike T2, cannot be directly estimated from the spectrum, instead its measurement requires a specially designed experiment.

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FIGURE 5.12 Schematic representation of the temporal evolution of magnetization in an experiment designed to measure spin
lattice relaxation.

Measuring T1 Spin
lattice relaxation is measured using the inversion recovery experiment. In this experiment, the sample is subjected to a 180 degrees pulse to invert the magnetization. After the pulse, the magnetization will start relaxing back to the equilibrium position. Waiting a variable length of time and applying a 90 degrees pulse, snapshots of the recovery process back to equilibrium will be provided. This is modeled as a first-order process, and thus the null point occurs at a time equal to T1 3 ln 2 (Fig. 5.12). With the above information it would still be difficult to obtain the exquisitely detailed information that NMR can provide for complex molecules. However, magnetic nuclei in a molecule create a slight disturbance of the effective magnetic field felt by each other. This disturbance, called coupling, can be propagated through chemical bonds or through space, providing chemical or spatial information.

Coupling: the interaction between magnetic nuclei Two types of interactions between nuclei are known: dipolar couplings and scalar couplings. Dipolar couplings depend on the relative orientation of the vector connecting the interacting nuclei with respect to the direction of the static magnetic field. Dipolar couplings are observed in solids and in liquid crystal samples. The interaction energy is given by μ μ E 5 Iz 3 Sz ð3cos2 θ 2 1Þ ð5:17Þ r This equation applies to all kinds of dipoles and the dipolar interaction disappears for the angle 54.74 degrees (decimal degrees), called the “magic angle” for this reason (Fig. 5.13). In amorphous solid samples, this effect is used to simplify the spectrum by

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FIGURE 5.13

Schematic representation of the angle between the internuclear bond and the static field giving rise to the RDC, using an NH bond as an example (left), and angular dependence of the RDC (right). The actual values depend on characteristics of the alignment and on the internuclear distance, but the crossing point occurs always at the magic angle.

FIGURE 5.14 Diagram of two bonded nuclei A and B with spin 1/2, and the two options for the orientation of the nuclear spin B. The configuration on the left has slightly lower energy. The short arrows crossing the circles are the nuclear spins. The long arrows are the paired electron spins responsible for the bonding orbital depicted in gray. The nucleus A shows two signals, one of each of the two configurations and the same happens for nucleus B.

rotating the sample holder tilted at this angle, and therefore removing the dipolar couplings (see below). In solution, by contrast, the molecular motion is typically sufficiently fast and random for this mode of coupling to average to zero. However, it is possible to prepare samples with a preferred molecular orientation and measure the residual dipolar coupling (RDC) that is generated in these conditions. Also macromolecules with metal centers that contain unpaired electrons can spontaneously assume a slight preference for a particular orientation in solution in the presence of the static magnetic field. This is a consequence of the interaction of the unpaired electron with the magnetic field, allowing for measurement of the resulting RDCs without further manipulation. Scalar couplings are another consequence of the electron orbitals that envelope the magnetic nuclei. The scalar coupling is mediated by the Fermi contact interaction which describes the coupling between nuclear and electronic spins (Fig. 5.14). The consequence is that only electrons in orbitals with nonzero spin density in the nuclei will contribute to this effect, that is, bonds with some sigma character. Scalar couplings are mediated by the electrons that form the chemical bonds connecting the nuclei. Therefore the effect fades as the number of bonds increases. According to the

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TABLE 5.2 Multiplicity and relative intensity for the various peaks of the multiplets of a signal coupled to nuclei with spin 1/2. Number of I 5 1/2 nuclei

Multiplicity

Relative intensity

0

1 (singlet)

1

1

2 (doublet)

1:1

2

3 (triplet)

1:2:1

3

4 (quadruplet)

1:3:3:1

Pauli principle the electrons in the bonding orbital will be antiparallel. The consequence is that parallel and antiparallel nuclear spins connected by such a bond will have a slightly different energy and therefore the signal will be split 2I 1 1 times. For example, a nucleus coupled with another nucleus of spin 1/2 will have its signal split in two. When several equivalent nuclei are the source of the coupling some signals will overlap. For nuclei of spin 1/2 the multiplicity becomes N 1 1 and relative intensities are given by the Pascal triangle as seen in Table 5.2. Scalar couplings are named according to the number of intervening bonds between the coupled nuclei and designated nJ: 2 J or geminal couplings are observed between nuclei of different chemical shifts bonded to one common nucleus. The value depends on the hybridization of the molecular orbitals of the connecting atom. 3 J or vicinal couplings are observed between nuclei separated by two other nuclei. Vicinal couplings have a well-defined geometrical dependence on the dihedral angle between the coupled nuclei. This is easily visualized in diagrams called Newman projections that indicate the dihedral angle between two hydrogen atoms bonded to the front and back carbons. There are different mathematical expressions for different kinds of nuclei, however, all are denominated Karplus equation (Eq. 5.18). For vicinal protons A 5 7 Hz, B 5
1 Hz, and C 5 5 Hz. If rotation is fast, the coupling settles into an average value of the dominant configurations, but when rotation is prevented the value of the scalar coupling provides geometrical constraints for structural determination and definition of stereochemistry (Fig. 5.15). 3

J 5 A 1 B cos φ 1 C cos 2φ

ð5:18Þ

Up to now this chapter showed that NMR can provide knowledge on: • • • • •

the number of different magnetically active nuclei, from the number of signals; how many equivalent nuclei contribute to each signal, from its intensity; how many nuclei are coupled to each nucleus, from the pattern of scalar splitting; molecular motions, from the signal linewidths; the stereochemistry from the magnitude of the vicinal couplings.

However, Fig. 5.16 shows that this information is insufficient for the characterization of biological molecules even of moderate size. Fortunately, more information can be obtained from NMR experiments.

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Coupling: the interaction between magnetic nuclei

FIGURE 5.15 Newman projection and angular dependence of J according to the dihedral angle between vicinal protons.

FIGURE 5.16 1D 1H spectrum of sucrose and respective molecular structure represented with the Haworth projection.

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FIGURE 5.17 Comparison of the pulse sequences for a standard acquisition and an acquisition with a B2 decoupling pulse during acquisition. The effect on the spectrum is represented on the right side.

Decoupling We have seen that scalar coupling is of considerable use to identify the nature of neighboring groups and to define molecular geometry. However, in a complex spectrum such as that shown in Fig. 5.16 it may not be obvious which pairs of signals are coupled and therefore belong to neighboring nuclei. Saturation of the signals of one nucleus with a weak selective B2 pulse during detection causes the coupled nuclei to feel the average orientation of the irradiated nucleus and loosen the splitting, becoming a singlet. In this way, the signals of nuclei separated by a few chemical bonds can be identified, and the S/N ratio is increased, improving the experimental sensitivity (Fig. 5.17).

The nuclear Overhauser effect As explained above, dipolar couplings typically cannot be observed directly in solution due to fast and random reorientation of the molecules. However, their presence influences the spectra in ways that can be detected. The NMR signal arises because the various states of a nucleus (two in the case of nuclei with spin 1/2) have different energies. Nuclei of different chemical shifts have a different energy separation. In these conditions the Boltzmann distribution determines that the populations will also be different. When nuclei are coupled, their resonance frequencies are modified depending on the spin state of the coupled nucleus. For the simplest case of two nuclei with spin 1/2, four states exist with a population slightly different depending on their energy separation according to the Boltzmann distribution, as shown in Fig. 5.18. In equilibrium the states α1α2, β1α2, α1β2, β1β2 will be successively less populated. It is possible to equalize experimentally the

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FIGURE 5.18 Energy diagram for a pair of coupled spins. Zero quantum (W0), single quantum (W1), and double quantum (W2) transitions are represented. The height difference between the transitions provides a visual cue for the kind of transition connecting each pair of states.

population of one of these two nuclei, for example, nucleus 1, by applying a selective pulse at the Larmor frequency of that nucleus—this is called a saturating pulse. The consequence is that the states α1α2 and β1α2 will be equally populated. The same happens to the states α1β2 and β1β2. This means that the coupled system is out of equilibrium, and as soon as the saturating signal is switched off reequilibration will occur. This reequilibration can occur through any of three pathways: along the edges of the diagram (the single quantum transitions, W1), sideways across the diagram (the zero quantum transitions, W0), or longitudinal between top and bottom (the double quantum transitions, W2). As the name of the transitions implies, zero quantum transitions occur at frequencies close to 0 Hz, single quantum transitions have frequencies close to the Larmor frequency, whereas double quantum transitions have frequencies close to double the Larmor frequency. The consequence of the reequilibration is that the intensity of the signal of the coupled nucleus (in this case nucleus 2) will be modified—it is this change in intensity that reveals the NOE. η 5 ðI 2 I0 Þ=I0

ð5:19Þ

The change in intensity is designated the NOE enhancement (Eq. 5.19). It depends on the distance and on the molecular motion, and therefore is related to molecular size. η 5 fðτ c r26 Þ

ð5:20Þ

It is the dependence with distance (Eq. 5.20) that makes the NOE effect one of the most important tools for the characterization of molecular structures. The dependence with molecular motion, and therefore size, arises from the fact that zero, single, and double quantum transitions, like all NMR transitions are stimulated by fluctuating magnetic fields. For smaller molecules, molecular motion is fast and therefore transitions of higher frequency (double quantum) are enhanced. For large molecules motion is slow and transitions of low frequency (zero quantum) are enhanced. This has two consequences: (1) for small molecules the coupled signal is larger than in the absence of saturation, whereas for large molecules the coupled signal is smaller; and (2) for molecular sizes where the correlation time is similar to the reciprocal of the Larmor frequency there is no NOE enhancement (Fig. 5.19).

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FIGURE 5.19 Dependence of the NOE enhancement on correlation time calculated at 500 MHz for a ˚. pair of protons separated by 2 A

A simple rule of thumb provides a rough estimate of the correlation time for proteins in water at room temperature: the molecular weight in kDa is approximately twice the correlation time in nanoseconds. For the conditions of Fig. 5.19 at a molecular weight of approximately 700 Da there is no NOE enhancement. It also shows that the maximum enhancement for macromolecules (
1) is double and of opposite sign to what can be obtained for small molecules (0.5).

DOSY: sizing up molecules The size of molecules is clearly an important parameter for selecting the experimental approach for analysis, and indeed in a complex mixture NMR can discriminate molecules with different sizes by measuring the diffusion of molecules in the NMR tube. The name DOSY stands for diffusion ordered spectroscopy. In this experiment, the external magnetic field is deliberately forced to change along the length of the tube (a static field gradient) so that a particular field strength corresponds to a particular position in the tube. Therefore as a molecule diffuses the signals change position and the amount of change depends on the diffusion coefficient, that is, the hydrodynamic radius. The data are plotted as a bidimensional map where the signals of each molecule are reported in a line and the different lines correspond to different diffusion coefficients. If a molecule is unknown, its hydrodynamic radius can be estimated using the Stokes
Einstein equation (Eq. 5.21) and a calibration curve with known molecules (Figs. 5.20 and 5.21). r5

kT 6πηD

ð5:21Þ

where k is the Boltzmann constant, T the absolute temperature, η is the fluid viscosity, and D is the diffusion constant.

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Chemical exchange

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FIGURE 5.20 DOSY spectrum of a mixture of phospholipids. The yy-axis parses different mobilities. The arrow indicates the solvent. The ellipses indicate the presence of at least three different components in the mixture.

Chemical exchange The phenomenon of chemical exchange arises from the fact that nuclei may experience different chemical environments during an NMR experiment. For example, two molecules in a mixture may be in equilibrium between the bound and unbound forms. Another example is the study of biological macromolecules, where the conformational interconversion of a flexible region of a protein makes the chemical shift of a particular nucleus different in the various conformations. The simplest formulation for chemical exchange showing the forward and backward rate constants for two states is given by: k1



! A



B k21

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FIGURE 5.21 Calculated nuclear magnetic resonance spectra for exchange between two sites with equal occupancy.

If a nucleus experiences two chemical environments it can have signals in two different spectral positions. Furthermore, for the simple case of a two-state exchange, kinetics will be of paramount importance. Indeed, the appearance of the spectrum will be determined by the ratio between the rates of the exchange phenomenon and the frequency difference between the signals in exchange. Therefore we can modify the appearance of the spectrum by modifying the experimental conditions. This can be achieved by changing sample concentration, solvent viscosity, temperature, or pH. Also, the spectral appearance can be modified by running the experiment at different field strengths, which changes the frequency difference between the signals in exchange (because of the different B0, see Eq. 5.5) without modifying the sample. Three limiting regimes of chemical exchange can be defined as seen in Fig 5.21: • Slow exchange: the rates of chemical exchange are much slower than the difference in the Larmor frequency of the signals. For the case of two-site exchange two signals are observed, one for each chemical environment with intensities proportional to the relative occupancy of each chemical environment. • Fast exchange: the rates of chemical exchange are much faster than the difference in the Larmor frequency of the signals for the (two) chemical environments. They must also be faster than the sums of spin
spin and spin
lattice relaxation rates, that is, 1/T1 and 1/T2, for the (two) chemical environments. In this case, a single signal is obtained in a position that is the average of the extreme positions weighted by the relative occupancy of each chemical environment (Eq. 5.22). ν 5 ν A fA 1 ν B fB

ð5:22Þ

• Intermediate exchange: also known as coalescence, which occurs when the exchange rates are of the order of magnitude of the differences in Larmor frequencies. For experiments run in this regime a very broad peak between the two extreme positions of the slow exchange regime may be observed under favorable conditions. Therefore NMR experiments also allow: • The identification of nuclei separated by a short number of chemical bonds using decoupling. Practical Approaches to Biological Inorganic Chemistry

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FIGURE 5.22 1D 1H nuclear magnetic resonance spectrum of a protein.

• The determination of distance constraints between pairs of nuclei from the magnitude of the NOE effect. • The determination of molecular size and discrimination of molecules with different sizes in a mixture. • The observation of exchange phenomena. However, spectra of biological macromolecules are extremely complex with hundreds to thousands of signals, as is evident in Fig. 5.22. Spectral overlap of the signals is severe, and the techniques described up to now do not match the challenge. Two strategies can be used to address this problem: spectral editing, and expansion into multiple dimensions. Spectral editing, which is typically performed to distinguish the hybridization of carbons in biomolecules, requires sample labeling to achieve convenient sensitivity, and will not be described here.

Multidimensional nuclear magnetic resonance To obtain a 1D NMR spectrum, the nuclei are excited with a radio frequency pulse, the decay of the signal is recorded in the time domain and Fourier transformed to obtain the spectrum in the frequency domain. Multidimensional NMR requires additional frequency dimensions. This is achieved by introducing further time domains that can be Fourier transformed. Without loss of generality, we can consider the case of a two-dimensional spectrum that is obtained by recording a series of 1D spectra. The experiment is initiated by a preparation procedure that can have different degrees of complexity, depending on the specific experiment but it includes a 90 degrees pulse. Then there is a waiting period, Practical Approaches to Biological Inorganic Chemistry

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called t1 during which the magnetization evolves. The magnetization is again manipulated by procedures that include another 90 degrees pulse and, finally, signals are recorded during a period called t2. For each subsequent 1D spectrum in the series, the duration of t1 is changed so that the signal recorded is a function of the t1 period. At the end of the experiment we have a series of FIDs that depend on the t1 period and therefore can be Fourier transformed relative to this t1 time and also Fourier transformed relative to the normal acquisition time t2. The results are reported as a 2D contour map with signals located at the correct frequency for both evolution periods. Multidimensional techniques allow the dispersion of the signals in various dimensions, and the observation of correlations between different signals. Depending on how the magnetization is manipulated during the experiment, different correlations are generated.

How do the correlations arise and how are cross-peaks generated? The vector model is not suitable to describe the process in detail but an energy diagram will be helpful. Without loss of generality, consider once more the energy diagram of a pair of coupled nuclei (Fig. 5.23). For the present case it is not relevant whether the coupling is scalar or dipolar because the preparation period is designed to select which kind of coupling will be observed during the experiment. After the first B1 pulse, the macroscopic magnetizations of these two nuclei were flipped to the xy-plane and they begin to precess during t1 at their characteristic Larmor frequencies. During the second manipulation period, in particular after the 90 degrees pulse, some magnetization will be flipped to the z direction (becoming invisible), and some remains in the xy-plane to be detected. This is the trivial consequence of pulsing twice. However, because the energy of the B1 pulse is of the correct frequency to excite the nuclei, some of it will be used to promote the single quantum transitions depicted by thick lines in Fig. 5.23. Following this, the magnetization decay is detected in the t2 period. The consequence of this is that some signals will precess with a particular frequency during t1, but will also precess with a different frequency during t2. Therein lies the origin of the cross-peak. The coupling between nuclei therefore provides a path for magnetization to be transferred during the second pulse.

FIGURE 5.23 Energy level diagram for a pair of coupled spins. Single quantum transitions between pairs of states are depicted as thick lines.

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Multidimensional nuclear magnetic resonance

The enhanced discrimination provided by the multiple dimensions, together with the knowledge on the physical bases of the couplings that give rise to the cross-peaks, allows the characterization of biological macromolecules such as proteins and nucleic acids. Multidimensional NMR experiments can be homonuclear when they correlate signals of nuclei of the same nature, or heteronuclear when nuclei of different nature are correlated. Increasing the number of dimensions requires that relaxation is slow (long T1 and T2) so that there is still sufficient magnetization in the xy-plane to be observed after the various waiting periods. Despite the myriad of experiments developed over the years, a few basic types are of widespread usage and important for biomolecular NMR.

The COSY The name stands for correlation spectroscopy and this is the simplest 2D experiment with respect to implementation. It detects scalar couplings (Fig. 5.24). The first 90 degrees pulse generates transverse magnetization that evolves during t1. The second 90 degrees pulse flips part of that magnetization into the z-axis but leaves part in the xy-plane. Very importantly, it also promotes transitions among the coupled spins. The signals are detected during t2. For those spins that were stimulated to change state by the second pulse, the precessional frequency during t2 is different from that during t1 and thus a cross-peak will be obtained. The cross-peaks in this experiment indicate the presence of scalar coupling between a pair of nuclei.

The NOESY The name stands for nuclear Overhauser enhancement spectroscopy and it is the 2D implementation of the detection of transient NOE effects (Fig. 5.25). For crowded spectra it is more convenient because the selective saturation in 1D NOE experiments cannot be strictly achieved. As a consequence, some spillover of the saturation to the neighboring signals cannot be avoided, leading to uncertainty in assigning the effect observed. The NOESY experiment can also be used to observe chemical exchange. Chemical exchange provides an easier visualization when describing what happens in the NOESY experiment. The first 90 degrees pulse generates transverse magnetization that evolves during t1. The second 90 degrees pulse flips part of the magnetization into the z-axis and leaves part in the xy-plane. The longitudinal magnetization is unobservable but is evolving. In case of FIGURE 5.24 Diagram of the pulse sequence experiment.

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a

COSY

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FIGURE 5.25 pulse sequence experiment.

Diagram of the of a NOESY

chemical exchange during the mixing time (tm) the nucleus can change chemical environment. The third 90 degrees pulse brings this longitudinal magnetization back into the xyplane to be detected during t2. For those nuclei that experienced exchange during tm, the precession frequency during t1 and during t2 is different giving rise to a cross-peak.

The HSQC The name stands for heteronuclear single quantum coherence, and this experiment generates a signal for each pair of different NMR active nuclei that are bonded together. When developing heteronuclear correlation experiments two aspects must be considered: (1) the larger difference in Boltzmann population difference for protons (the nucleus with the highest gyromagnetic ratio) favors the transfer of magnetization from protons to the heteronuclei; and (2) the higher detection sensitivity of protons favors the transfer of magnetization from the heteronuclei to the protons. Since the S/N is approximately γ excited(γ detected)3/2 experiments that start and finish in protons with a passage by the heteronucleus of interest have a much better S/N. A diagram of the pulse program for an HSQC organized this way is depicted in Fig. 5.26. A further advantage is that the spin
lattice relaxation of protons is typically faster than that of the other nuclei in a biomolecular sample. This allows for a shorter acquisition time for each scan. The experiment is initiated by using the protons coupled to the heteronuclei to generate a population difference in the heteronuclei that is larger than what could be achieved by a 90 degrees pulse applied to the heteronuclei. This is achieved by introducing a delay Δ that is equal to the reciprocal of the coupling constant between protons and the heteronucleus of interest. The enhancement is approximately γ proton/γ heteronucleus. The magnetization of the heteronucleus evolves during t1. In order to avoid decoupling of the signals of the heteronucleus by the protons, a 180 degrees pulse is applied to protons in the middle of the t1 period. 90 degrees pulses applied simultaneously to the proton and heteronucleus promote magnetization transfer. Detection is made in the proton frequency during t2. If decoupling of the 1H signals by the heteronucleus is to be prevented, a broadband decoupling pulse has to be applied in the heteronucleus channel after waiting for a delay with the length of Δ. This is necessary to prevent complete loss of the coupling that would result in an empty spectrum. With the sequence represented in Fig. 5.26 a seemingly impossible feat is achieved. A spectrum is obtained that results from scalar coupling between two nuclei, and decoupling is achieved in both dimensions.

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FIGURE 5.26 Top: diagram of a 1H15N HSQC pulse sequence. Thin bars represent 90 degrees pulses and thick bars represent 180 degrees pulses. Bottom: A 1H15N HSQC of a protein showing the region of the NH groups with the good spectral dispersion characteristic of a properly folded protein. These spectral features made this experiment very popular in protein NMR studies because each signal is associated with a single amino acid backbone NH or a side chain.

Metals in biomolecular nuclear magnetic resonance spectra Metals are an essential component of the biological systems. They participate in the catalytic activity of numerous enzymes, in the capture of small molecules, intercellular signaling, in the structural organization of numerous proteins, or as extracellular structural

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support of cells and tissues. The presence of metals in an NMR sample can be used to observe the effect of the metal or its electrons on other NMR active nuclei or to directly observe the metal when it is NMR active.

Transition metals and interaction with the unpaired electron(s) Transition metals are characterized by having orbital shells that are not fully occupied. Depending on the bonding environment of the metal some orbitals may be partially filled leading to electronic spin states that can range from 1/2 to 9/2. Electrons, like protons and neutrons, are subatomic particles with the quantum mechanical property of spin. Being a spin, typically denoted S, the electron has a magnetic moment that is quantized. When placed in a magnetic field the energy of this magnetic moment is expressed in a fashion similar to that of the nuclear magnetic moments seen in Eq. (5.3a). E 5 gμB B0 S

ð5:23Þ

Given that the behavior of electrons is described by orbitals with defined shapes, g is not a number but instead a matrix. When the orbital is not spherically symmetrical the g matrix accounts for the differences in energy that arise from the orientation of the molecule relative to the magnetic field. Unpaired electrons are spins and interact with nuclear spins via the same physical mechanisms that were previously described (for a refresh, go back to the “Coupling: the interaction between magnetic nuclei” section) and the combination of such is called the hyperfine interaction.

Hyperfine scalar coupling Scalar, or contact, coupling arises from the fact that according to quantum mechanics the probability of finding the unpaired electron at the nucleus is not zero and its value is independent of the molecular orientation in the magnetic field. For transition metals, these electrons are located in the valence d orbitals that participate in bonds with the remainder of the molecule. The electron has a magnetic moment and generates a local magnetic field that modifies the effective field that is felt by the nuclei in its proximity. The Fermi contact shift generated by the unpaired electron is described by Eq. (5.24). δcon 5

A ge μB SðS 1 1Þ h 3γkT ¯

ð5:24Þ

A is the coupling constant between electron and nucleus. It has the same physical meaning as J between nuclei.

Dipolar coupling As previously described, dipolar interactions are propagated through space. In this case they depend on the distance between the unpaired electron and the nucleus. They also

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Dipolar coupling

depend on the orientation of the vector connecting the electron and the nucleus with respect to the external magnetic field. When g is isotropic and under fast rotation, the dipolar interaction energy averages to 0. In these conditions, the chemical shift of the signal is not perturbed, and the dipolar effect arising from the presence of the unpaired electron only leads to enhanced relaxation, that is, broader signals (see below). When the unpaired electron is found in an orbital of nonspherical symmetry, g is orientation dependent and the dipolar contribution does not average to zero. This gives rise to the pseudocontact shift. The dipolar coupling is very complex but when describing its effect in protons, a simplification can typically be applied. This simplification considers that the unpaired electron orbital is centered in the metal coordinates and is axially symmetrical. The axial symmetry means that the x and y directions are equivalent and cannot be distinguished. With this simplification the pseudocontact shift is defined by Eq. (5.25) δpc 5

μ0 μ2B SðS 1 1Þ 2 1 ðg! 2 g2| Þ 3 ð3cos2 θ 2 1Þ 9kT r 4π

ð5:25Þ

The g values can be measured in an EPR experiment (see Chapter 4 on EPR spectroscopy). Eq. (5.25) reveals several important aspects of the pseudocontact shift. First, it shows that isotropic electrons do not generate a pseudocontact shift because g!2 and g|2 are equal; second, the shift has a different sign depending on the position of the atom versus the magic angle; and third, for the same position the pseudocontact shift changes sign depending on whether the g! or g| is larger. Fig. 5.27 shows how the pseudocontact shift extends around a paramagnetic center of axial symmetry, clearly revealing the existence of the magic angle. It also shows that the effect extends more in the direction of the z-axis than in the direction of the xy-plane. Therefore for nuclei not bonded to the metal and its ligands, the magnitude of the pseudocontact shift provides information on the distance and geometry relative to the magnetic FIGURE 5.27 Contour of pseudocontact shifts of equal magnitude for an axial system calculated using Eq. (5.25). Solid and dotted lines represent shifts of opposite sign.

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axes defined by the unpaired electron orbital. This is the basis for using paramagnetic shift constraints to improve the structure of proteins containing unpaired electrons. Furthermore, as can be seen from Eqs. (5.24) and (5.25), both contact and pseudocontact effects are dependent on the reciprocal temperature. Therefore paramagnetic shifts decrease with increasing temperature as long as there are no changes in coordination or transitions to excited electronic states. In solution the molecule under observation is typically in conditions of fast reorientation and therefore the contact and pseudocontact terms of the hyperfine interaction are measured together.

Relaxation When paramagnetic molecules are placed in a magnetic field B0 the unpaired electron (s), which are magnetic spins 1/2, will distribute among the two Zeeman energy levels. Spontaneous transitions between the two Zeeman energy levels are accompanied by changes in the electron magnetic moment. Therefore these transitions generate local fluctuations in the magnetic field capable of stimulating nuclear relaxation.

Contact relaxation This mechanism arises from the delocalization of the electron onto the nucleus and is proportional to the electronic spin, and to the correlation function. In this case there is no contribution of the molecular motion to relaxation, which is restricted to electronic relaxation and (eventually) chemical exchange. It is significant for nuclei directly bonded to the metal through bonds displaying significant covalence. Although the unpaired electron(s) occupies an orbital with a defined shape, relaxation can be analyzed considering that the unpaired electron(s) is localized on the metal nuclear coordinates. This approximation ignores the contact contribution towards relaxation and simplifies the calculation of the dipolar contribution.

Dipolar relaxation Within this simplification, and in the fast motion limit, that is, the transition frequencies for the nuclei and the electron are slower than the reciprocal of the correlation time for the electron
nuclear interaction, the simplest formulation of the Solomon equation is obtained: Rð1;2Þ 5

4 γ 2 g2e B2e SðS 1 1Þ τc 3 r6

ð5:26Þ

Even in this simplified form (Eq. 5.26) it is easy to see that relaxation is proportional to the square of the gyromagnetic ratio of the nucleus under observation, and therefore much more dramatic for protons than for other nuclei. This together with the strong dependence of this effect with distance, means that the radius around the paramagnetic center where

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broadening is so excessive that signals cannot be observed is much larger for protons than for nuclei with low gyromagnetic ratios, such as 15N or 13C. This is the driving force for the development of direct heteronuclear detection (protonless) NMR experiments for the characterization of paramagnetic proteins. The dipolar relaxation of the nucleus under observation caused by the unpaired electron can be partitioned into three components: 1. transition between the Zeeman levels inverts the induced field generated by the unpaired electron that is felt by the observed nucleus; 2. rotation of the molecule changes the angle subtended by the vector connecting the unpaired electron and the observed nucleus versus the direction of the static field; and 3. chemical or conformational exchange modifies the relative position (distance and/or angle) between the unpaired electron and the observed nucleus. All these components are of a dynamic nature, and as previously described (see “What are the physical mechanisms of relaxation?” section) the dynamic effects caused by random and unpredictable changes in a system that is at equilibrium are described by a correlation function characterized by a correlation time. In the case of the electron
nuclear dipolar relaxation three mechanisms contribute towards the correlation function as previously described. Given their nature, it is reasonable to assume that they are independent and therefore the correlation function decays as the product of the three exponential components. As a consequence, the overall correlation time relates to the correlation times for each mechanism according to the following expression: 1 1 1 1 5 1 1 τC τS τR τM

ð5:27Þ

where τ S is the correlation time for electron relaxation, τ R is the correlation time for molecular tumbling, and τ M is the correlation time for chemical exchange. Electronic relaxation times range from 10213 to 1027 s. Rotational correlation times at room temperature range from 10211 s for aqueous metal complexes, to 1026 s for large macromolecules, and exchange correlation times are slower than 10210 s. It is clear from these ranges that, depending on the molecule under analysis, one of the components may dominate the overall correlation time. If the overall correlation time is dominated by electronic relaxation, a situation that occurs for metals with short τ S, nuclear relaxation is slow and relatively sharp signals can be observed. This is the case for low spin Fe(III), high spin Fe(II), and some lanthanides historically known as shift agents, such as Pr(III) and Eu(III) which typically have electronic relaxation correlation times in the range of 10212
10211 s. However, if electronic relaxation is slower and the rotational correlation time dominates the overall correlation time, nuclear relaxation is enhanced leading to broad signals that may be undetectable in the vicinity of the metal. This is the case for Cu(II), high spin Fe(III), and lanthanides such as Gd(III) which are known as relaxation agents. In the case of Gd(III) this property is used in magnetic resonance imaging to improve image contrast by administering to the patient compounds containing the metal.

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Curie relaxation In the description of dipolar relaxation made so far, it was implicitly considered that electronic spins were evenly distributed among the two states. However, in fact this is not true and there is a slight excess of electronic spins in the low energy state. As expected the difference in population increases with increasing field strength. Curie relaxation arises from this difference in electron population in the presence of the external magnetic field. It causes a small electronic magnetic moment aligned with the field, the Curie spin. Molecular motion modulates the effect of this spin on the nuclei which gives rise to a dependence with r26. As the molecular size increases the contribution of Curie relaxation towards the total nuclear spin relaxation increases and can become dominant. Curie relaxation is also proportional to B20 which means that for paramagnetic molecules, the use of a stronger field may not be the most favorable for observing the signals of interest, in particular for large paramagnetic macromolecules ( . 40 kDa). If a significant contribution of this relaxation mechanism is suspected, this can be identified by observing dramatic increases in the linewidth of the signals with increased field strength.

Residual dipolar couplings The fact that electronic spins have a preferential orientation when placed in an external magnetic field has one very important consequence. Because the electron is a much stronger spin than any nuclear spin (about 650 times stronger than the proton) in strong magnetic fields, an unpaired electron in an orbital that is not isotropic generates a measurable residual preference for the orientation of the whole molecule in solution. In these conditions, the movement of the molecule is not free and random anymore, and thus the dipolar coupling between the nuclei does not average to zero. This so-called RDC can be measured as a change in the peak splitting with field strength of coupled spins. For example in a NH pair, an HSQC experiment performed without decoupling will reveal a total splitting that is no longer just J, but J 1 D (where D is the residual dipolar contribution). It should be noted that RDCs are ambiguous, meaning that they depend on the angle but not on the direction. Therefore “H
N” and “N
H” give the same RDC. For this reason, they are most often used for structural refinement. Because the dipolar coupling decays with the third power of distance (Eq. 5.17), whereas the NOE decays with the sixth power of distance (Eq. 5.20), RDCs provide long-range structural constraints for the determination of structures. If a macromolecule does not contain a paramagnetic center or if a suitable paramagnetic tag cannot be attached, nonspherical molecules can be dissolved in jellified media that induce a preferential orientation of the molecules, leading to observable RDCs.

Nuclear magnetic resonance of (semi-)solid samples As molecular motion becomes more restricted, for example in very large biological structures, such as protein complexes in biological membranes or fibrils, dipolar couplings Practical Approaches to Biological Inorganic Chemistry

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and chemical shift anisotropy become prevalent. Furthermore, T1 becomes longer since there is less motion to modulate dipole
dipole relaxation, and T2 becomes short since there is a flip between fixed spin pairs. All these factors broaden the spectrum in such a way that liquid NMR methods cannot be applied to obtain significant information on the sample, and thus solid-state NMR comes to the rescue. Biological samples for solid-state NMR are not normally prepared in single crystals and thus a spectrum with a shape called the “powder-spectrum” is obtained (see also the chapter on EPR spectroscopy). The shape of this spectrum derives from the dipolar coupling between nuclear spins and from the chemical shift anisotropy of the signals. Both effects can be eliminated by rotating the sample at the magic angle relative to the external magnetic field, because both phenomena contain the 3cos2θ21 term which is null for this angle. The spinning rate must be faster (in Hz) than the linewidth of the powder-spectrum to be observed (in Hz). If the spinning rate is less than this value, the so-called spinning side-bands become visible and occur at regular intervals (in Hz) equal to the rate of spinning. The linewidth of the powderspectrum is dominated by dipolar coupling between nuclei. Between protons and carbons ˚ distance this is of the order of 30 kHz, and for this reason high-resolution magic at 1 A angle spinning requires NMR probes with rotors that can go beyond this rate of rotation. However, given the higher sensitivity of protons, and their predominance in biomolecules, it is convenient to be able to observe these nuclei. This is now possible but requires an engineering feat of rotating the samples faster than 100 kHz. This is a consequence of the fact that for a proton
proton pair, the dipolar coupling may be larger than 100 kHz. As expected these fast rotation rates lead to the centrifugation of the sample into the rotor wall which may result in phase separation or solvent exclusion. When instrumentation to observe protons is not available, or it is desirable to observe other nuclei bonded to protons, the presence of protons in the sample can still be useful by applying an experimental strategy similar in concept to the basis of HSQC experiments. This is called crosspolarization where polarization is transferred from the abundant 1H nuclei to 13C or 15N nuclei, increasing the sensitivity of the experiment by a maximum of γ H/γ X (X is any heteronucleus of less abundance than proton). The more abundant nuclei relax faster due to a more efficient longitudinal relaxation, and altogether both effects contribute to a decrease in experimental time by reducing the necessary number of scans and the recycling delay (the time between successive scans).

Direct observation of metals by nuclear magnetic resonance Of those metals that can be directly observed in biological samples and tissues, Na1 and K1 are sufficiently abundant and display an effective sensitivity that is compatible with NMR studies. These monovalent cations play key roles in cellular transport, maintenance of transmembrane electrical potential, and cell signaling. However, both nuclei present I 5 3/2. This means that these nuclei have three allowed single quantum transitions. Nuclei of I . 1/2 have an electric quadrupole moment, which can be interpreted as arising from a nonspherical charge distribution on the nucleus. The presence of the electric quadrupole moment gives rise to greater spectral complexity. In a uniform electric field all three transitions have the same energy and a single signal is observed. However, when the nucleus is in an electric field gradient the three transitions can have different energies, Practical Approaches to Biological Inorganic Chemistry

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with the splitting of the three signals proportional to the electric field gradient. This can be observed in single crystals or in liquid crystalline samples. An important consequence is that when analyzing unoriented samples or samples where the nuclei are subject to rapidly fluctuating electric fields only the central resonance may be visible. This leads to an apparent reduction of the signal intensity to approximately 40% relative to the expected value, a phenomenon known as “invisibility.”

In-cell nuclear magnetic resonance Although the vast majority of biomolecular NMR experiments are performed with samples that have somehow been manipulated in ways that are not compatible with the survival of the original biological source, all of the methodologies can be applied to samples containing living cells in order to explore the molecular phenomena inside those cells. The observation of biological processes in living cells by NMR has a long history. Two mature examples are the monitoring of cellular metabolic state by following 31P signals of phosphorylated metabolites such as adenosine triphosphate (ATP) and adenosine diphosphate (ADP), and also the monitoring of 23Na or 39K transport across cellular envelopes. However, the study of the structure, dynamics, and interactions of biological macromolecules in cells only became feasible in the beginning of the 21st century. Nowadays, numerous biological systems exist for in vivo NMR with the bacterium Escherichia coli the most explored, but also, yeast, insect, and mammalian cells. To study biological macromolecules by in-cell NMR, the macromolecules need to be labeled, typically with 15N or 13C, to discriminate them from the background of the other cellular components. Labeling can be performed endogenously by overexpression of the desired macromolecule or exogenously followed by purification and microinjection or transfection into the cells. Furthermore, to further remove the effect of background in the spectra, two main methods can be applied: one, where the cells are washed and resuspended in a “NMR-friendly” media; and the other, where the spectra of the other cellular components without the molecule of interest is acquired and then used for subtraction. Using 1H13C or 1H15N HSQC experiments, the molecule of interest is observed provided that the cellular concentration reaches a value of approximately 5% of the total soluble protein content, and that it tumbles freely. In the crowded environment of the cell, molecular tumbling is typically slowed down by two to three times versus what is observed for aqueous solutions, leading to increased line broadening that may render the signals undetectable. One class of proteins that has attracted considerable interest for study by in-cell NMR due to their implications for health are the intrinsically disordered proteins (IDPs). IDPs and partially unfolded proteins give sharp in-cell spectra due to their transient interactions with other cellular components and higher internal flexibility. In the case of stably folded proteins, very often the backbone signals are not detectable due to slow tumbling, most likely caused by interactions with other cellular components. In these cases, specific labeling of methyls in the side chain of amino acids provides an observation window into the behavior of these proteins due to their short relaxation times. Another alternative is the use of 19F labeled amino acids such as the aromatics phenylalanine, tryptophan, and tyrosine. If these strategies are not sufficient to obtain workable spectra, solid-state NMR methods exist for observation in-cell.

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These methods often rely on the detection of 13C or 15N instead of proton detection, since these provide less background noise from the other cellular components. However, the lower gyromagnetic ratios require longer acquisition times.

An nuclear magnetic resonance spectrometer: measuring macroscopic magnetization and relaxation Considering the separation between the energy levels of magnetic nuclei in contemporary magnetic fields, the electromagnetic radiation that is absorbed during an NMR experiment has a wavelength of the order of a meter. This is the range of radio waves and therefore apart from the magnet, the NMR spectrometer can be viewed as a very sophisticated radio that generates “music” for the nuclei and then observes the resulting choreography. An NMR spectrometer is composed of: • • • • • •

a magnet; probeheads; the lock system; the shim system; signal generators; and analogue to digital converter (ADC).

All of these are now operated via a computer interface that also serves to store and process the experimental data. The magnet is the core of the spectrometer and defines it. Nowadays it is composed of a coil of wire that becomes superconducting when immersed in a liquid helium bath. This is isolated from room temperature by a liquid nitrogen bath and vacuum seals. The coil is superconductive so that a strong electrical current can flow through it indefinitely without generating heat that would boil off the helium bath. It is this electrical current flowing in the coil that generates the static field. Among the NMR community, the strength of the magnetic field is typically indicated by the Larmor frequency that the field imposes on hydrogen nuclei instead of the SI unit Tesla. Therefore a spectrometer of 21.2 T is referred to as 900 MHz. In the center of the coil and of the liquid helium and nitrogen baths there is a hole called a bore, where the probehead is inserted. The probeheads serve to hold the sample in the correct place to run the experiment. These can be compared to radio antennae that emit the electromagnetic pulses and detect the magnetization decay of the sample by containing coils of wire of specific design to perform both tasks. An RF pulse is generated by passing current in the coils, and subsequently the signal is detected by measuring the current induced in the coils by the macroscopic magnetization of the sample. When the sample contains very strong signals, the current induced in the detecting coils interacts with the nuclei in the sample. This phenomenon, called radiation damping, causes extra broadening of the signals and this is relevant for biomolecular samples, given that protons in water can have a concentration of up to 110 M.

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The probes must have at least two coils. One for generating and detecting the signals of interest, and the other for locking the sample (see below). Because many modern NMR experiments used in biomolecular NMR excite and detect more than one kind of nucleus, the probes to execute such experiments must have more coils and are called multichannel. For MAS experiments with semisolid samples, the probe construction is different and includes a rotor bearing that keeps the sample holder (commonly referred to as the rotor) with the sample in the magic angle and spins it by using a flow of air. The larger the sample holder diameter, the slower the maximum rotation frequency that can be achieved. As a consequence, there is an inverse correlation between the sample size and the maximum rotation frequency. The lock system is used to track the natural drift in the static magnetic field with time. Because field stability is essential for NMR experiments, the lock system measures the position of deuterium nuclei added to the sample (for instance, the deuterated water in biomolecular samples) and makes sure that it does not change by applying an electrical current in the coil inside the magnet. Given the need to follow the deuterium signal, all probes have a dedicated coil to perform this task. The shim system is used to correct small imperfections in the static magnetic field. This is achieved by passing electrical current in coils of specific shapes. The shims can be on-axis (in the zz direction) or off-axis (other directions). Different shim names represent different effects on the static field. Signal generators produce the radiation that is used to manipulate the nuclei such as the B1 pulse. They produce sinusoidal waves of the desired frequency, phase, and shape. Typically, there is more than one signal generator, and each is connected to a channel dedicated to a specific nucleus or a frequency range. The ADC measures the analogue signal induced in the detection coil by the macroscopic magnetization of the sample at discrete time points. The consecutive measurements are stored in a digital form that can subsequently be Fourier transformed. The period separating consecutive measurements is called the sampling interval. Contemporary ADCs contain 16 bits, which means that the dynamic range is 216. The consequence is that the smallest signal that can be distinguished from noise needs to be larger than 1/65,536 the size of the largest signal. A computer is used for controlling all components, and for recording and processing the experimental data. As seen above, the NMR signal is a digitized periodic variation of electrical current induced in a coil that decays with time. For a simple case, inspection of the signal may reveal the frequency. When several nuclei with different chemical shifts exist, the signal becomes too complex for direct inspection of the FID. In these conditions a Fourier transform converts the time domain data into the frequency domain spectrum.

Care in obtaining nuclear magnetic resonance spectra of paramagnetic samples NMR samples of biomolecules are in the liquid-state for the vast majority of cases. Therefore the target molecules should be soluble at concentrations preferably above 1 mM due to the low sensitivity of the NMR phenomenon. For diamagnetic samples, molecular oxygen must be removed because it is paramagnetic and enhances the relaxation of the nuclei.

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Very often paramagnetic samples are oxygen sensitive and therefore oxygen also has to be excluded to avoid degradation or changes in the redox state. This is typically achieved by subjecting the sample to repeated cycles of vacuum and inert gas. Given the dependence of the paramagnetic shift with temperature, a very good control of the sample temperature is necessary to obtain good quality spectra. Furthermore, careful temperature calibration is necessary to allow the matching of signals measured in different experiments, for instance, for the structure determination of paramagnetic proteins by NMR. Temperature control is also important since often the addition of standards for calibration is not convenient due to spectral overlap, and the residual HOD signal can be used to calibrate the spectra in samples prepared in deuterated water as is typical in biomolecular NMR. δ 5 5:051 2 0:0111 T

ð5:28Þ

Eq. (5.28) shows that the chemical shift of HOD is temperature dependent and this dependence is approximately linear in the range of 0 C
50 C, for which Eq. (5.28) is valid. Because of the enhanced spectral dispersion caused by the paramagnetic shifts, a larger spectral bandwidth has to be excited and this requires short strong pulses to achieve the required flip angles. NMR pulses are made of electromagnetic radiation and therefore samples should contain the minimum amount of salt. This minimizes sample heating resulting from the movement of ions stimulated by the radiation. When analyzing really demanding paramagnetic samples that contain for instance signals with very large spectral dispersion, sampling of the FID may need to be so fast that special digitizers are required to handle data collection if aliased peaks are to be avoided. Alternatively, a weaker field strength can be used, where the spectral window in frequency units is narrower, therefore reducing the demands on the digitizer. This is also advantageous for large paramagnetic proteins due to the reduced contribution of Curie relaxation. Paramagnetically relaxed signals can be very broad and require a large number of scans to be observed. In order to optimize the signal strength for the available experimental time, the recycle delay, that is, the time between successive scans, may need to be reduced from the typically recommended value of 5T1. With recycle delays between T1 and 1.5T1 a better sensitivity per unit time is obtained but the B1 pulse should not aim to achieve a 90 degrees flip angle. For each recycle time and T1, there is an optimum flip angle known as the Ernst angle:
2T  αE 5 arccos eT1 ð5:29Þ

Water eliminated Fourier transform and super-water eliminated Fourier transform sequences: catching up with fast relaxing samples Taking advantage of the fast relaxation rates of the paramagnetic system, paramagnetic samples can be analyzed using the water eliminated Fourier transform (WEFT) and super-WEFT pulse sequences (Fig. 5.28A). These consist of an inversion pulse (180 degree
τ
90 degree) with interpulse delays to selectively enhance fast relaxing signals (paramagnetic) while saturating slow relaxing signals (diamagnetic) (Fig. 5.28B). The recycle time in these experiments needs to be short in comparison to the T1 value of the

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signals to be suppressed (diamagnetic) but long in comparison to the T1 values of the signals to be enhanced (paramagnetic). This way, when the spectrum is acquired, signals with long T1 values will have minimal intensity, whereas signals with short T1 values will be detected (Fig. 5.28B). The super-WEFT pulse sequence (180 degree
τ A
90 degree
τ B) is an optimized version of WEFT (T-180 degree
τ n
90 degree) whereby shorter τ A and τ B delays are used. A series of “dummy” sequences (the experiment is performed but data are not recorded) is performed prior to data collection to allow the system to reach a steady state in the magnetization pulsing and evolution. Thus super-WEFT performs better than WEFT in suppressing H2O signals and it is the most commonly used for metalloproteins. These pulse sequences can be used in 2D experiments by simply adding the 180π
τ module in front of any 2D sequence to use in paramagnetic samples. The general drawback of these pulse sequences is that the decrease in the delays used does not necessarily decrease the number of points to be sampled, given that paramagnetic systems often have signals that span over large spectral widths, and thus require a more frequent data point sampling. This is usually attenuated by oversampling techniques, where the data are acquired at a faster rate than necessary for the desired spectral window. To cover large spectral widths, pulses of less than 90π can also be used and an adequate ADC will be required (currently the most commonly used are 16
32 bits) with dwell times (sampling interval) of 1 μs or less. With very large spectral widths baseline distortion problems will arise resulting from imperfections in instrumentation and sample preparation. To correct these baseline distortions, other pulse sequences have been developed such as SHWEFT-PASE pulse sequences that employ Polychromatic and Adiabatic-Shaped pulses instead of the conventional rectangular-shaped pulse. These have been shown to produce uniform excitation over large spectral widths (75,000 Hz) providing flat baselines while maintaining peak shape fidelity (Fig. 5.28).

FIGURE 5.28 Diagram of WEFT and super-WEFT pulse sequences (A) and schematic representation of the evolution of magnetization in the WEFT and super-WEFT experiments designed to filter signal intensity based T1 relaxation differences (B). Blue arrows represent diamagnetic species and red arrows represent paramagnetic species in solution.

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Evan’s method: measuring magnetic susceptibility Often the spin state of a sample is an important information and NMR can provide that information in a convenient and simple experiment. When a sample is placed in a magnetic field it will respond by acquiring magnetization. This magnetization (M) will be proportional to the external magnetic field (H) and it will depend on the bulk magnetic susceptibility of the material (χ), in other words, the ability of the material to acquire a magnetic moment. M 5 χH

ð5:30Þ

The bulk magnetic susceptibility of materials is mainly derived from the interaction of electrons with the external magnetic field. For diamagnetic molecules, the bulk magnetic susceptibility is small and negative (χ , 0) (Table 5.3). This occurs due to the filled electron shells in diamagnetic molecules that shield the nuclei when magnetization is induced, and thus very weak magnetization is experienced. Additionally, as described by Lenz’ law, the induced magnetization will oppose the magnetic field. Diamagnetic susceptibility is temperature independent. On the other hand, if electrons are not coupled in spin-up spin-down pairs, the way these unpaired electrons interact will determine the kind of paramagnetism of the molecule (Table 5.3) and will also cause an extra contribution to the chemical shift. Based on this observation, the molar magnetic susceptibility can be determined using the Evan’s

TABLE 5.3 Most common types of magnetism and respective properties. Magnetic Type of magnetism susceptibility

Properties

Diamagnetism

χ , 0 (small values)

No electronic spin
spin interaction. Spins align antiparallel to the external magnetic field. Magnetization is lost when external magnetic field is removed. Temperature independent.

Paramagnetism

χ . 0 (small values)

No electronic spin
spin interaction Spins align parallel to the external magnetic field. Magnetization is lost when external magnetic field is removed. Temperature dependency—follows Curie law.

Ferromagnetism

χ . 0 (large values)

Electronic spin
spin ordering is parallel. Magnetization is kept after removal of external magnetic field. Temperature dependency—follows Curie
Weiss law (θ . 0).

Antiferromagnetism χ . 0 (large values) Ferrimagnetism

χ . 0 (large values)

Electronic spin
spin ordering is antiparallel. No net magnetization. Temperature dependency—follows Curie
Weiss law (θ , 0). Type of antiferromagnetism. Electronic spin
spin ordering is antiparallel; however the magnitude of spins is unequal.

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method. The Evan’s method takes advantage of the fact that the magnetic susceptibility of the medium, in which the resonating nuclei are immersed, affects greatly resonance line positioning. Thus when a paramagnetic species is added to a diamagnetic solvent, changes in chemical shift will occur, and this relationship can be described by: p

Δδ 5 CM χM =3;

ð5:31Þ p

where CM represents the millimolar concentration of the paramagnetic species and χM is the paramagnetic contribution of the paramagnetic species added to the molar magnetic susceptibility of the medium. In this equation, the diamagnetic effects of the solvent and the differences of solvent density between reference sample and paramagnetic sample are not considered. p From the paramagnetic contribution, χM , the magnetic moment of the paramagnetic species μeff can be determined using: p

μ2eff 5 χM 3kT=NA μ0 ;

ð5:32Þ

where k is the Boltzman constant, T is the absolute temperature (K), NA is the Avogadro’s number, and μ0 is the vacuum permeability. From the magnetic moment, μeff , the spin state of the paramagnetic species in solution can be determined since the magnetic moment is given by: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μs 5 g SðS 1 1Þ; ð5:33Þ where g is the gyromagnetic ratio of the free electron (Lande constant) and S is the spin state of the compound (Table 5.4). The experimental setup for the Evan’s method requires a coaxial or capillary NMR tube. The diamagnetic sample (solvent) is placed in the inner capillary tube and the paramagnetic sample will be placed in the outer tube (solvent 1 paramagnetic species). For work on biomolecules, the most commonly used inert reference solvent contains tert-butyl alcohol and 1,4-dioxane. TABLE 5.4 Magnetic moment values depending on electronic spin state. S

μs (units of Bohr magnetons)

1/2

1.73

1

2.83

3/2

3.87

2

4.90

5/2

5.92

3

6.93

7/2

7.94

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The Evan’s method can also be used to identify the type of paramagnetism by exploring how the magnetic properties of the species in solution change with temperature, in other words, by verifying if the Curie or Curie
Weiss laws apply. The system is paramagnetic when the system is composed by noninteracting electronic spins and thus the Curie law p applies, with the magnetic susceptibility χM varying with the temperature according to: p

χM 5 C=T

ð5:34Þ

where C is the Curie constant and T is the absolute temperature (K). If the Curie law does not apply, the system is composed of localized interacting electronic spins and thus magnetic susceptibility varies with the Curie
Weiss law: p

χM 5 C=ðT 2 θÞ

ð5:35Þ

where C is the Curie constant, T is the absolute temperature and θ is the Weiss constant. If the Weiss constant is negative the electronic spins are antiferromagnetically coupled, whereas if the Weiss constant is positive the electronic spins are ferromagnetically coupled. When using the Evan’s method for temperature dependence studies of magnetic susceptibility the variation of solvent density with changes in temperature should be considered.

Conclusions The contents of this chapter were the result of a choice of subjects that touched various key aspects of NMR relevant for biomolecular samples and, in particular, those containing metals. Given the intended target audience, and the available space, depth was traded for wider coverage. Considering the introductory nature of this chapter, an effort was made to the greatest possible extent to describe NMR spectroscopy in ways that are pictorial rather than mathematical. However, it should be clear that true mastery of NMR will require a good command of the mathematical tools that support the underlying physical phenomena. The interested reader should further explore more detailed literature, and some examples are listed below.

Further reading Bertini, I., Luchinat, C., Parigi, G., 2001. Solution NMR of Paramagnetic Molecules. Elsevier, Amsterdam. Cavanagh, J., Fairbrother, W.J., Palmer III, A.G., Skelton, N.J., Rance, M., 2007. Protein NMR spectroscopy, Principles and Practice, 2nd ed. Academic Press, San Diego. Derome, A.E., 1993. Modern NMR Techniques for Chemistry Research. Pergamon Press, Oxford. Freeman, R., 1997. Spin Choreography, Basic Steps in High Resolution NMR. Spektrum Academic Publishers, Oxford. Gil, V.S.M., Geraldes, C.F.G.C., 1987. Ressonaˆncia Magne´tica Nuclear, Fundamentos Me´todos e Aplicac¸o˜es, Fundac¸a˜o Calouste Gulbenkian, Lisboa (in Portuguese). Levitt, M.H., 2008. Spin Dynamics, Basics of Nuclear Magnetic Resonance, 2nd ed. John Wiley & Sons.

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Useful physical constants Name

Symbol

Value

Free-spin electron g factor

ge

2.0023

Electron Bohr magneton

μB

9.2740 3 10224 J/T

Nuclear Bohr magneton

μN

5.0508 3 10227 J/T

Plank’s constant

H

6.6261 3 10234 Js

Boltzmann constant

K

1.3807 3 10223 J/K

Vacuum permeability

μ0

1.2566 3 1026 Tm/A

Exercises 1. Calculate the resonance frequency of protons in a spectrometer that has a magnet of 18.79 T. 2. Calculate the energy difference for nuclear spin transitions of proton, 13C and 15N at 9.39 T. 3. Calculate the ratio for the spin-up and spin-down populations for the nuclei used in the previous exercise at 25 C. 4. Calculate the ratio between the macroscopic magnetization of protons and 15N at 9.39 T. 5. Calculate the strength for the B1 field necessary to generate a 90 degrees pulse in proton and 13C with a pulse length of 8 μs. 6. Calculate the difference in frequencies between two signals separated by 0.1 ppm when measured in a 500 and a 900 MHz spectrometer. 7. A spectrometer has the choice of a 16 bit ADC with a minimum sampling interval 7 μs, or a 12 bit ADC with a sampling interval of 1 μs. Determine the maximum frequency that can be sampled by each ADC. 8. Calculate the vicinal coupling for a pair of protons when the dihedral angle is 0 and 60 degrees. 9. Assuming that the coupling constant between the unpaired electrons and the nucleus does not change, what is the relationship between the contact shift for a low spin paramagnetic molecule (S 5 1/2) and a high spin molecule (S 5 5/2)? 10. A proton in a paramagnetic protein exchanges between two positions that have the same angle with respect to the orbital of the unpaired electron. One of these positions ˚ from the center of coordinates and gives a pseudocontact shift of (A) is located 2 A ˚ away from 1 ppm. What is the pseudocontact shift for the other position (B) that is 4 A the center? 11. Considering the approximation that the electron is found in the coordinates of the metal nucleus that gives rise to Eq. (5.26), how is the relaxation between the two positions related, considering that the metal nucleus is located at the center of coordinates of the previous exercise?

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12. Copper II has an electronic relaxation correlation time of 1 3 1029 s. Considering that there are no exchange phenomena, in a small molecule of 0.5 kDa and a protein of 40 kDa. Estimate the rotational correlation time for each case and indicate what mechanism dominates the correlation function for the electron nuclear dipolar interaction in both cases. Do you expect sharp or broad lines for protons? 13. Considering that you have a paramagnetic sample with T1 equal to 3.7 ms, calculate the best flip angle to maximize the signal to noise ratio of the experiment if the recycle time is 16 ms. 14. Consider that you have a sample of 320 μM of a paramagnetic protein that upon analysis using the Evan’s method gives for the peak of 1-4-dioxane a delta shift of 0.0019 ppm in a 500 MHz spectrometer. Determine the spin state of your protein.

Answers 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

800 MHz. 2.65 3 10225 J, 0.67 3 10225 J, 0.27 3 10225 J. 0.999936, 0.999984, 0.999993. 97.23. 0.734 3 1023 T, 2.92 3 1023 T. 50 and 90 Hz. 71.4 and 500 kHz. 11 and 4 Hz. The magnitude of the contact shift for the low-spin case is 8.6% of that observed for the high-spin case. 0.125 ppm. Relaxation in position A is 64 times the relaxation in position B. The correlation times are approximately 0.25 and 20 ns. For the small molecule it is the rotational relaxation time that dominates whereas for the macromolecule it is the electronic relaxation time that dominates. 89.24 degrees. S 5 1/2.

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Fe-Mo¨ssbauer spectroscopy and basic interpretation of Mo¨ssbauer parameters Eckhard Bill

Max-Planck Institute for Chemical Energy Conversion, Mu¨lheim an der Ruhr, Germany O U T L I N E Introduction

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Principles The Mo¨ssbauer light source γ-Emission and absorption—recoil is a problem Recoilless emission and absorption—the Mo¨ssbauer effect The Mo¨ssbauer experiment The Mo¨ssbauer spectrometer

202 203

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Fe hyperfine interactions

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Isomer shift as informative hyperfine interaction

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Electric quadrupole splitting

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Magnetic hyperfine splitting

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Combined hyperfine splitting

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Applications—selected examples

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Practical Approaches to Biological Inorganic Chemistry DOI: https://doi.org/10.1016/B978-0-444-64225-7.00006-7

Oxidation and spin states in a nonheme diiron center Reaction intermediates and lowand high-valent iron complexes The heme enzyme horseradish peroxidase Nonheme model compounds Iron(II) complexes Mixed-valence iron(III)
iron(IV) dimers and iron(IV) monomers Iron(V) complexes Four-coordinated iron(IV) and iron(V) compounds The first molecular iron(VI) compound

214 216 216 217 218 219 220 221 222

Perspectives

224

Exercises

224

References

225

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© 2020 Elsevier B.V. All rights reserved.

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Introduction More than five decades after the discovery of nuclear gamma resonance absorption by R.L. Mo¨ssbauer in 1958 (Mo¨ssbauer, 1958a,b), the spectroscopy based on this so-called Mo¨ssbauer effect is an established tool in bioinorganic spectroscopy. It is useful for the investigation of all types of iron-containing materials including metalloenzymes. The exotic nuclear method owes its success to the fact that the 57Fe nucleus has the most favorable physical properties among the family of Mo¨ssbauer isotopes, and iron in general is one of the most relevant transition metal ions in chemistry. Mo¨ssbauer spectroscopy detects iron in all oxidation and spin states, provided that the samples are solids, powders, or frozen solutions. The spectra can be readily assigned to local properties of the iron sites and effects from other constituents in the sample cannot obscure the comprised chemical information. Moreover, the isotope is not particularly expensive, and small molecules up to molar masses of a few kDa can be studied with naturally abundant 57Fe (2.2%). Introductions into Mo¨ssbauer spectroscopy on various levels of sophistications can be found in a number of reviews and books (Debrunner, 1989, 1993; Mu¨nck, 1978, 2000; Trautwein et al., 1991; Martinho and Mu¨nck, 2010; Schu¨nemann and Winkler, 2000; Paulsen et al., 2005; Gu¨tlich et al., 2011; Gu¨tlich and Schro¨der, 2010; Gu¨tlich and Enssling, 2012; Pandelia et al., 2015), some of which address particularly applications in transition metal chemistry (Gu¨tlich and Schro¨der, 2010; Greenwood and Gibb, 1971; Long and Grandjean, 1989; Gu¨tlich et al., 2011), and also applications in biochemistry are covered in great broadness and depth by a number of the articles. The purpose of this contribution is to provide a concise overview of what Mo¨ssbauer spectroscopy can do for the solution of scientific problems in (bio)inorganic chemistry and how to look at Mo¨ssbauer data published in literature. The chapter is mostly focused on the interpretation of isomer shifts and electric quadrupole splittings obtained from zero-field measurements, but magnetically perturbed spectra and paramagnetic systems are also mentioned.

Principles Mo¨ssbauer spectroscopy is based on resonant absorption of γ-radiation by atomic nuclei. Chemists are familiar with resonant absorption of electromagnetic radiation from the phenomena of light-induced electronic transitions. Visible light from a white incident beam is absorbed by coordination compounds at exactly the energies of d-electron transitions or metal-to-ligand charge transfer transitions; this is the main cause of the colors of inorganic complexes. Only when the quantum energy of the light (photons) matches the energy gap between the electronic states does such resonant absorption occurs. An analogous process is possible for γ-radiation, for which nuclear states are involved as emitters and absorbers (Fig. 6.1). Observation of this resonance is interesting because nuclear levels are extremely sharp, such that hyperfine interaction between the nucleus and the electrons in the atomic shell can be observed. This is the foundation of Mo¨ssbauer spectroscopy. Electric and magnetic fields from the electron shell affect the energies of the nuclear levels, which can be seen as shifts and splittings of the resonance lines in the

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Principles

FIGURE 6.1 Emission of γ-rays from the excited state of 57Fe in the Mo¨ssbauer source and resonant absorption in the absorber.

FIGURE 6.2 The light source for 57Fe Mo¨ssbauer spectroscopy. Metastable 57Fe nuclei are generated by K-capture decay of radioactive 57Co. The nuclear reaction yields energy to populate the excited state at 136 keV, the decay γ-cascade of which passes through the Mo¨ssbauer level at 14.4 keV.

spectra. Since the properties of the nucleus are known and invariant, the data can be readily interpreted in terms of electronic parameters that are related to the valence state and the bonds of the Mo¨ssbauer atom.

The Mo¨ssbauer light source The γ-photons used in the experiments usually originate from nuclear decay of a radioactive mother isotope (Fig. 6.2). The decay cascade in the Mo¨ssbauer source passes through the excited nuclear state of the corresponding Mo¨ssbauer isotope. In the case of 57 Fe spectroscopy, the source isotope is 57Co, which decays by so-called K-capture with a lifetime of 270 days. In these events an electron from the K-shell reacts with a proton of the nucleus, yielding 57Fe in a metastable excited state. The decay of this state yields a cascade of γ-transitions, the lowest of which has quantum energy E0 5 14.4 keV. For reasons given below the radiation coming from this transition is selected for the Mo¨ssbauer experiment. The lifetime τ of the Mo¨ssbauer level is about 100 ns. According to Heisenberg’s uncertainty principle the lifetime of 100 ns is equivalent to a natural line width ΔE of the Mo¨ssbauer emission line of only 4.55 3 1029 eV, which corresponds to an energy resolution ΔE/E0 of the order 3 3 10213. This is about 100,000 times higher than what is known for typical optical transitions in the electronic shell. For instance, the D-line of sodium atoms has an energy resolution of ΔE/E 5 2.1 3 1028. The narrow width of the γ-transition leads to complete selectivity for resonant γ-spectroscopy; photons originating from a nuclear transition of a certain Mo¨ssbauer isotope cannot be resonantly reabsorbed by nuclei other than those of the same isotope. This means that a Mo¨ssbauer spectrometer equipped with 57Co as a “light source” can and will detected only 57Fe in the Mo¨ssbauer sample, which is not radioactive since 57Fe is a stable isotope.

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γ-Emission and absorption—recoil is a problem Unfortunately, the narrow width of γ-lines in principle may prevent nuclear γ-resonance, because photons have a momentum and γ-transitions are afflicted with loss of nuclear energy due to recoil imparted to the nucleus during emission and absorption. This is a general effect that occurs also for optical transitions, but there it is usually insignificant because the recoil energy ER is negligible compared to the experimental line widths. The recoil energy depends on the square of the photon energy, ER ~ E20 , and for the 14 keV radiation of 57Fe, ER exceeds the width of the Mo¨ssbauer line by a factor of 3,000,000! As a consequence, nuclear γ-resonance absorption is not possible when the Mo¨ssbauer atoms are free to move (Fig. 6.3).

Recoilless emission and absorption—the Mo¨ssbauer effect Rudolf L. Mo¨ssbauer discovered in his PhD work (Mo¨ssbauer, 1958a,b) that in solids, photon emission and absorption is not necessarily affected by recoil energy loss. Instead, there is a certain probability f that a γ-transition does not pass energy to the “crystal lattice.” This means that in this case no vibration of the Mo¨ssbauer atom (a so-called phonon) is excited during photon emission and absorption; the elastic bonds prevent translational motions anyhow. Such zero-phonon processes yield the sharp and unaffected lines of natural line width in the spectrum of a γ-source, since recoil is taken by whole solid particles and therefore negligible because of the relatively huge mass. Resonant absorption of the corresponding photons is possible by nuclei of the same kind. The discovery and understanding of the effect (Mo¨ssbauer, 1958a,b, 1959) was worth a Nobel Prize. Since then nuclear resonance absorption of γ-rays observed in solid materials, powders, and frozen solutions is called the Mo¨ssbauer effect. FIGURE 6.3 The natural line width of the excited state of the Mo¨ssbauer nucleus is many orders of magnitude lower than the loss of the photon energy Ephoton due to recoil energy ER imparted to nuclei when it is free to move during the nuclear emission and absorption. Since the resulting energy separation of the Lorentzian emission and absorption profiles by 2ER prevents any significant overlap of the two lines, resonant nuclear absorption of γ-photons is not possible for free atoms.

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FIGURE 6.4 Table of Mo¨ssbauer active elements. The most suitable isotope for applications and most easy to

use is 57Fe. Source: Courtesy Prof. J.G. Stevens, Mo¨ssbauer Effect Data Center, previously UNC North Carolina, since 2010 Dalian Institute of Chemical Physics within Chinese Academy of Sciences. For online database of Mo¨ssbauer isotopes and more, see: http://www.medc.dicp.ac.cn/Resources.php.

The probability factor f for zero-phonon processes is obtained from a quantum mechanical treatment of the nucleus and its vibrational environment. It is found to be inversely proportional to the recoil energy, or in terms of the γ-energy: f ~ 1=E20 . Sufficiently high f factors of source and absorber are essential for the intensity of the Mo¨ssbauer spectrum. 57 Fe is a good Mo¨ssbauer isotope, because it has comparably high f factors due to the lowenergy transition at 14.4 keV (but the radiation is still “hard” enough to penetrate real samples). The Mo¨ssbauer f factor, also known as Lamb
Mo¨ssbauer factor, is not quite the same, but practically equivalent to the Debye
Waller factor for elastic X-ray scattering by atoms. Characteristic f-values for the 14.4 keV transition of 57Fe can be as low as 0.1 when the Mo¨ssbauer atom is in a “soft” environment like a protein sample at 270 K, but the values can be also 0.6
0.9 for bulk solids, or upon cooling to 77 K or lower. Therefore freezing of solution samples is essential, and cooling to cryogenic temperature may significantly enhance the absorption effect. An overview of the isotopes for which Mo¨ssbauer effect has ever been demonstrated is given in Fig. 6.4. In summary the above arguments about nuclear states, vibrations, and γ-rays make 57Fe virtually the best Mo¨ssbauer isotope, not only because it has an appropriate low-lying excited nuclear state, affording high f factors, but also the lifetime is suitable, yielding a (not too) sharp γ-line. Moreover, a convenient mother isotope is available, 57 Co, and last-not-least, the nuclear spin and hyperfine interactions are reasonable and sensitive, providing valuable chemical information (which will be described below).

The Mo¨ssbauer experiment According to the decay scheme of Fig. 6.2, the radioactive Mo¨ssbauer source emits monochromatic γ-rays that are perfectly tuned for the observation of the Mo¨ssbauer effect when the Mo¨ssbauer isotope has the same chemical environment as in the source and thus

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FIGURE 6.5 Recording the line shape of the nuclear energy level(s) in an absorption experiment requires energy variation of the radiation emitted by the Mo¨ssbauer source. To this end, the source is being moved relative to the absorber with different velocities v, which results in Doppler-modulated quantum energies, Eγ 5 E0(1 1 v/c), where E0 is energy of the nuclear transitions and c is the speed of light. Different absorption probability yields a “dip” in the transmission spectrum, at the Doppler velocity for which resonant absorption occurs. The shape of the absorption profile, which is given by the counts of photons passed through the absorber per measuring time and per velocity increment, reveals the Lorentzian line shape and natural line widths of the nuclear energy levels in source and absorber (more detail in Fig. 6.6).

has exactly the same nuclear energies. However, for recording the absorption spectrum of a general Mo¨ssbauer absorber, eventually having several absorption lines, the energy of the incident photons arriving at the absorber has to be varied to probe the absorption probability as a function of photon energy—just like in any spectroscopy. Since the nuclear levels in the Mo¨ssbauer source cannot be tuned experimentally, the energy modulation has to be done during γ-emission by moving the source relative to the absorber, so that the Doppler effect will add or subtract energy increments to the energy of the photons in the laboratory system. The Doppler modulation depends on the ratio of the source velocity v and the speed of light, v/c (Fig. 6.5). Nevertheless, the observation of hyperfine splitting in 57Fe-Mo¨ssbauer spectra usually does not require higher Doppler velocities than c.610 mm/s; for 57Fe, 1 mm/s corresponds to 4.8 3 1028 eV.

The Mo¨ssbauer spectrometer Most Mo¨ssbauer spectra are recorded with a transmission setup as shown in Fig. 6.6. The light comes from the Mo¨ssbauer source and passes through the absorber, which is the sample under investigation. The absorber is not radioactive and rests in a cryostat or another compartment. The source is mounted on the moving part of a Mo¨ssbauer drive system. For recording the spectrum, the number of γ-rays passing the absorber is counted with a γ-counter and an electronic detection system that is tuned to discriminate the 14.4 keV Mo¨ssbauer photons from other radiation. During the measurement, the radioactive source is periodically moved with controlled Doppler velocities, 1 v towards and 2 v away from the absorber. The counting system is synchronized with the velocity control of the driving system such that the full velocity range 6v is divided into a number of discrete velocity increments, vi, for storing the detected data (so-called channels). The number of Practical Approaches to Biological Inorganic Chemistry

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Fe hyperfine interactions

FIGURE 6.6 Mo¨ssbauer transmission experiment in five steps. The absorption probability depends on the energy shift of the emission line due to Doppler modulation [Eγ 5 E0(1 1 v/c)] and the resulting “overlap” of emission and absorption lines. The transmission spectrum T(v) is usually normalized to the counts C(N) at high velocity, v-N. In the so-called thin absorber approximation, the absorption spectrum of a single-line absorber is a Lorentzian line given by CðNÞ 2 CðvÞ Γ2 5 fs 2t , where fS is CðNÞ ½E0 ðv=cÞ2ΔE2 1 Γ2 the Lamb
Mo¨ssbauer factor for the source, and t is the so-called effective thickness of the absorber (Gu¨tlich et al., 2011). Γ is the natural line width of the Mo¨ssbauer transition, and ΔE denotes a hyperfine shift of the absorber line. Note that the experimental line width, 2Γ, is twice the natural width because an emission line is used to scan an absorption line of the same width.

counts, C(v), arriving at the detector for each velocity (increment) v, being stored in the array of channel, represents the Mo¨ssbauer spectrum. In this experiment, the probability for resonant nuclear γ-absorption at any velocity v is determined by the amount of overlap of the Doppler-shifted Lorentzian emission line and the Lorentzian absorption line, as indicated on the left side of Fig. 6.6. Stronger overlap yields higher absorption probability, and thus weaker transmission. Maximum resonance occurs at complete overlap when the energies of emission and absorption lines match exactly. 57

Fe hyperfine interactions

The electrons in the vicinity of the Mo¨ssbauer nucleus affect the nuclear energy levels in the Mo¨ssbauer experiment due to the electric and magnetic fields that they generate. There are essentially three different types of such hyperfine interactions that shift and split the nuclear states. The effects are seen in the Mo¨ssbauer spectra as isomer shift, electric Practical Approaches to Biological Inorganic Chemistry

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FIGURE 6.7 Overview of

57

Fe hyperfine interactions. Electric monopole interaction in source and absorber causes the isomer shift δ of the spectra (left spectra panel). Electric quadrupole interaction splits the excited state in the absorber and yields a symmetric quadrupole doublet in the spectrum with quadrupole splitting ΔEQ (middle panel). The center of the quadrupole doublet may be additionally subjected to isomer shift δ. Magnetic hyperfine interaction totally lifts the degeneracy of the nuclear spin states, such that six transition lines are observed due to the selection rule ΔmI 5 0, 61 (right panel). The center of the magnetic spectrum also may be displaced by an isomer shift δ.

quadrupole splitting, and nuclear magnetic Zeeman splitting (Fig. 6.7). Hyperfine interactions can be formally described in terms of the nuclear spin of the Mo¨ssbauer isotope, which is I 5 1/2 for the ground state of 57Fe and I 5 3/2 for the excited state. The radioactive source in a Mo¨ssbauer experiment is selected such that the chemical matrix in which 57Co is embedded does not induce any splitting of the nuclear levels of the Mo¨ssbauer isotope, obviously in order to get emission from a nonsplit single-line source (Fig. 6.7, top left). In the Mo¨ssbauer sample (absorber), magnetic dipole interaction may completely lift the degeneracy of ground and excited state (Fig. 6.7, top right). However, only six transitions are allowed in this case, because of the selection rules ΔmI 5 0, 61 for the Mo¨ssbauer radiation. As a consequence, a characteristic six-line pattern is observed. In contrast, electric quadrupole interaction affords partial splitting of the excited state levels into a doublet, whereas the ground state remains nonsplit (middle panel) because that state with I 5 1/2 does not have a quadrupole moment. A line doublet is observed in the spectrum, the splitting of which is called electric quadrupole splitting, ΔEQ. For powders and frozen solutions the intensity distribution of the quadrupole doublet is symmetric. In addition to magnetic Zeeman and electric quadrupole interaction the nuclear levels may be subjected to common energy shifts, arising from electric monopole (Coulomb) interaction of the nucleus with the surrounding electrons. Since the effect is different for ground and excited state of the Mo¨ssbauer isotope, and it is different for the sample and the source (Fig. 6.7, top left and middle), this gives rise to a uniform isomer shift of all Mo¨ssbauer lines in the spectra. It should be noted that isomer shifts can be quoted only relative to a reference, which can be the material of the Mo¨ssbauer source (e.g., 57Fe in rhodium metal) or a reference absorber. The usual reference nowadays is metallic (α-)iron, which is mostly used also as a standard absorber for velocity calibration of Mo¨ssbauer spectrometers. In old literature other reference materials, like nitroprusside, may be found.

Isomer shift as informative hyperfine interaction The phenomenon of isomer shift is related to the fact that the nucleus has a finite size, which changes upon the Mo¨ssbauer transition. Since the nucleus is immersed in a cloud of Practical Approaches to Biological Inorganic Chemistry

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FIGURE 6.8 Isomer shifts observed for 57Fe compounds relative to metallic iron at room temperature. Filled boxes indicate high-spin compounds, hashed boxes mark intermediate spin, and empty boxes are used for low-spin complexes. aOnly examples of unambiguous monovalent character are mentioned; these are structurally characterized molecular Fe(I)-diketiminate and tris(phosphino)borate complexes with three-coordination, which explains the low δ values as resulting from short bonds. bThe low-spin state of the 3d4 configuration for Fe(IV) in quasioctahedral or tetrahedral symmetry has S 5 1, but the “low-low-spin” state with S 5 0 is found for distorted trigonal-prismatic sites with strong ligands. cDiamagnetic Fe(VI) with spin S 5 0 occurs only in ferrates, and there is only one example of a molecular iron(VI) complex, which is six-coordinate and has spin S 5 0 (Berry et al., 2006b).

negative electron charge density that depends on the chemical properties and environment of the Mo¨ssbauer atom, the Coulomb energy of this system apparently depends on the chemical environment. The resulting isomer shift δ is proportional to the
 electron density e|ψ(0)|2 at the Mo¨ssbauer nucleus in the absorber (A), δ 5 α jψð0Þj2A 2 C , where ψ(0) is the of the electron wave function at distance r 5 0, and α is an isomer shift calibration constant. The constant C summarizes the relevant properties of the source material, or a reference absorber (Gu¨tlich et al., 2011). The electron density at the nucleus, e|ψ(0)|2, originates primarily from the finite density of s-electrons at the nucleus. However, in a simple ligand field picture these do not participate in chemical bonds, whereas p- and d-valence electrons essentially do not contribute to |ψ(0)|2 because their orbitals have nodes at r 5 0. Nevertheless, experimental isomer shifts depend on the electronic structure and bonds of the Mo¨ssbauer atom, particularly the oxidation state (Fig. 6.8). In an (over)simplified approach, derived for hypothetically free atoms and ions, this effect is often explained by the expansion of inner s-orbitals due to shielding of the nuclear potential by electrons in the valence shell. For instance, six d-electrons of iron(II) ions afford more shielding and thus allow more s-expansion than six d-electrons. As consequence, |ψ(0)|2 should be lower for iron(II) than for iron(III), and the isomer shift should be higher, because the coupling parameter α is negative. The explanation is qualitatively correct and in accord with experiments, but in reality, it describes only a minor contribution to the isomer shift. Practical Approaches to Biological Inorganic Chemistry

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In quantum chemical descriptions of chemical bonds, a direct contribution to the isomer shift arises in addition to the indirect shielding effect of d-electrons, due to partial population of valence 4s-orbitals and their participation in molecular orbitals (MOs). Calculations show that this effect plays a decisive role and that the 4s-contribution to |ψ(0)|2 varies most significantly in relation with iron
ligand bond lengths and coordination numbers. Chemical coordination of the Mo¨ssbauer atom causes compression of the radial distribution of the 3s- and 4s-iron orbitals and thus alters their density at the nucleus. This effect depends on the bonding situations and determines the major part of the variations of the isomer shift found for iron complexes. Based on the theoretical correlation it was suggested that average iron
ligand bond length is the key parameter for understanding isomer shifts (Neese, 2002; Neese and Petrenko, 2011). The most obvious empirical correlations of the isomer shift with chemical parameters, and the most useful for practical applications, are the general relationship with the formal oxidation state and with the spin state of iron, as shown in Fig. 6.8 (both can be readily explained by effects of the bond lengths involved!). The trends in the diagram are rather clear for high-spin complexes, but for low-spin and low-valent states, the dependence of δ on the oxidation state increasingly fades. Anyhow, the overall comparison for arbitrary types and numbers of compounds is hardly conclusive in detail. Only isomer shifts above c.1 mm/s reveal unambiguously a distinct electron configuration, namely Fe(II) high-spin; in all other cases various interpretations are possible. The ambiguity is due to the fact that isomer shifts, among other factors, depend on the number of ligands, and via the bond length relation on the σ-donor and the π-acceptor strengths of the ligands. Therefore rather conclusive isomer shift correlation diagrams are found for systematic series of related compounds with similar ligands. An example for synthetic iron centers is presented in Fig. 6.14, which allowed the assigning of distinct high-valent Fe(IV), Fe(V), and Fe(VI) compounds that model intermediates of the reaction cycle of oxygen- and nitrogen-activating enzymes. A macrocyclic ligand, cyclam, with four nitrogen donor atoms was used throughout the series of compounds and the number of valence electrons was “titrated” only within the range of t2g orbitals, that is, all compounds have been low-spin iron centers. The minimal number of chemical variations in that series yields a remarkably straight correspondence. Tight isomer shift correlations have been established also for iron
sulfur clusters, most of which are oligonuclear and include one to four tetrahedral high-spin [FeS4]-sites. The iron centers are redox-active and show a variety of charge-delocalized mixed-valence states, as has been demonstrated by Mo¨ssbauer spectroscopy (Schu¨nemann and Winkler, 2000; Pandelia et al., 2015; Beinert et al., 1997; Mouesca and Lamotte, 1998; Holm et al., 1996; Solomon et al., 2008; Rao and Holm, 2003). Isomer shifts have been empirically related to the average oxidation number x of the iron centers according to the expression δ(x) 5 [1.43
0.40x] mm/s, with x ranging from 2.0 to 3.0, depending on how many iron sites and valence electrons are available (Hoggins and Steinfink, 1976). In summary isomer shifts are experimentally correlated to bond lengths of the Mo¨ssbauer atom as follows: • Higher oxidation states show shorter bond lengths and lower (more negative) isomer shifts. • High-spin states have longer bond lengths than low-spin states and higher isomer shifts.

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• Four-coordination affords shorter bond lengths than six-coordination and lower isomer shifts. • More covalent bonds are shorter and induce lower isomer shifts. Complexes with “soft” ligands like sulfur have lower isomer shifts than complexes with “hard” ligands like oxygen or nitrogen.

Electric quadrupole splitting Electric quadrupole interaction is related to the orientation of nonspherical nuclei in the inhomogeneous electric field generated by an asymmetric charge distribution of the surrounding electrons. The resulting hyperfine coupling energy depends on the nuclear quadrupole moment and the electric field gradient (efg) generated by the electrons. Only nuclei with spin I . 1/2 are subjected to quadrupole interaction, because others do not have a quadrupole moment. Thus the ground state of 57Fe is not affected, but the excited state is split into two Kramers doublets with magnetic quantum numbers mI 5 61/2 and 63/2. (Fig. 6.7, top middle; the reader may note the similarity of nuclear quadrupole splitting and zero-field splitting in the electronic shell for S 5 3/2 systems). Therefore two γ-transitions are possible for 57Fe with electric quadrupole interaction, the energy difference of which is seen as quadrupole splitting ΔEQ of the resulting spectrum. The efg is a traceless tensor (ΣVii 5 0, see Fig. 6.9) with only two independent components in the principal axes system. Usually the so-called main component Vzz and the asymmetry parameter η 5 (Vxx
Vyy)/Vzz are selected as independent parameters. The range of η can be restricted to 0 # η # 1, since the tensor axes are selected such that Vzz is 57 the largest component. For Fe, the measured quadrupole splitting is given by the relation pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 2 ΔEQ 5 2 eQVzz 1 1 η =3, where eQVzz is called the quadrupole coupling constant.

FIGURE 6.9 Nuclei with I . 1/2 (like 57Fe in the excited state) have an electric quadrupole moment Q, which

has rotational energy in an electric field gradient (efg) rE (left). The efg is mathematically described by a 3 3 3 matrix Vij, with i,j 5 x,y,z. Precession of the electric quadrupole moment Q in the efg rE is quantitized, which gives rise to discrete quadrupole coupling energies of the nucleus and the electron shell. This is seen as quadrupole splitting ΔEQ of the Mo¨ssbauer spectrum (middle). Oblate axial electronic charge distributions give rise to a positive efg at the nucleus (Vzz . 0, η 5 0), whereas elongated, cigar-shaped charge distributions cause a negative efg (Vzz , 0, η 5 0) (right). The sign of Vzz can be determined only from magnetically perturbed spectra discussed below, not from pure quadrupole doublets as shown here.

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The main component of the efg may be positive or negative, depending on the type of asymmetry of the electronic charge distribution that causes the efg (see Fig. 6.9, right). Next we will discuss some archetypical charge distributions in the valence shell that cause an efg at the Mo¨ssbauer nucleus. The quadrupole splitting |ΔEQ| of 57Fe in bioinorganic compounds is usually found in the range 0 to about 4 mm/s; whereas some particular synthetic nitrido complexes related to nitrogen fixation reactions show up to 6.23 mm/s (Hendrich et al., 2006; Vogel et al., 2008; Scepaniak et al., 2011). A finite efg indicates the presence of a nonspherical or noncubic distribution of electronic charge around the nucleus. Axial elongated distributions cause an efg with negative Vzz (negative ΔEQ), and axial compressed or oblate charge distributions cause positive Vzz. Due to a 1/r3-dependency of the efg-components on the distance of the nucleus to the originating charges, the electrons close to the nucleus have the most important effect, that is, the valence electrons of the Mo¨ssbauer atom. The corresponding valence contributions to the efg parameters, (Vzz)val and ηval, are determined by the iron-centered MOs formed between the central Mo¨ssbauer atom and the ligands. Strong values for (efg)val arise when the oxidation state of the metal ion and the chemical bonds afford significant differences in the population of the local atomic valence orbitals. This is often the case for open-shell complexes. As long as covalence effects are not too strong, first “hand-waving” interpretations of iron quadrupole splittings can be derived from estimates of (efg)val for the known or putative 3dn electron configuration, given in a simple crystal-field model. In this approximation, (efg)val is incrementally built up by summarizing the contribution of all d-electrons, using the expectation values given in Table 6.1. (One may note the different signs of Vzz for the different “elongate” and “oblate” orbitals dz2 or dx22y2 for instance.) Addition of the tensor elements for, for example, the five d-orbitals in the 3d5 configuration of Fe(III) high-spin, yields zero (one may focus on Vzz and sum up the values in that column). The result corresponds to the experimental observation of small |ΔEQ| in the range 0
0.5 mm/s found for most “ionic” ferric high-spin compounds. In contrast, TABLE 6.1 Expectation values of (efg)val tensor elements for d-electrons in quasi-octahedral symmetry. Orbital

ðV xx Þval   e r23

ðV yy Þval   e r23

ðV zz Þval   e r23

dxy

22/7

22/7

14/7

dxz


2/7

14/7


2/7

dyz

14/7


2/7


2/7

dx22y2


2/7


2/7

14/7

12/7


4/7

12/7

dz2 23

23

The values are given in units of e hr i, where e is the proton charge and hr i is the expectation value for the cube orbital radius r23. The expectation values can be converted into quadrupole coupling energies (in mm/s) by inserting the actual values for the quadrupole moment and for hr23i of the orbitals into the appropriate equations (Gu¨tlich et al., 2011). An estimate of (ΔEQ)val may be obtained with the conversion factor 4.5 mm/s per 4/7e hr23i (obtained for hr23i 5 5a023 and Q 5 0.16b). Accordingly, a single electron in a hypothetically pure dx22y2-orbital, for example, is expected to yield a quadrupole splitting of 14.5 mm/s when covalence effects and lattice contributions are neglected.

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the contribution of six electrons for the 3d6 configuration of Fe(II) high-spin yields a finite value for (Vzz)val. If we assume that the sixth electron is found in a low-lying, doubly occupied dz2 orbital the expectation value for (Vzz)val is 24/7ehr23i. This can be converted into a quadrupole splitting of about 24 mm/s by using the parameters given in the caption of Table 6.1. This value marks a large quadrupole splitting. Alternatively, if dxz or dyz would be the ground state orbital and doubly occupied, the expectation for (Vzz)val would be only half the value, 22/7ehr23i. Correspondingly, most ferrous high-spin compounds show in fact large quadrupole splitting of the order 2
4 mm/s. In similar manner one can predict large quadrupole splitting for ferric low-spin compounds with e0g t52g configuration, having a “hole” in the t2g orbital set, or vanishing ΔEQ for ferrous low-spin compounds due to the filled t2g orbital set. More practical examples will be presented below. Crystal-field theory dealing with pure metal d-orbitals often can provide a reasonable insight into the origin of “large” or “small” quadrupole splitting, but quantitative estimates of efgs are not possible, even for ionic compounds. In reality ligand orbitals participate substantially in the valence orbitals forming covalent chemical bonds so that the electron configuration cannot be realistically described by integer numbers 3dn. Resulting nonequivalences of the d-population numbers reflects charge asymmetries around the nucleus that may cause substantial covalence contributions to the efg. In addition, ligand orbitals can also contribute to this effect. Therefore quantum chemical calculations are necessary for quantitative interpretation of Mo¨ssbauer quadrupole splitting (Neese and Petrenko, 2011). The most prominent examples of large covalence contribution to the efg in bioinorganic chemistry are found for ferric high-spin ions in μ-oxo dimers with Fe(III)
O
Fe(III) core. The centers are typically encountered in nonheme iron proteins like ribonucleotide reductase. They show quadrupole splitting as large as 22.45 mm/s (δ 5 0.53 mm/s) (Atkin et al., 1973; Vincent et al., 1990), in spite of the fact that iron has formally a halffilled d-shell, which in a crystal-field model affords vanishing efg, (ΔEQ)val 5 0. However, there is considerable charge anisotropy as the five or six covalent bonds around the iron centers are very different. In particular, the bridging oxo group contributes a uniquely ˚ (Vincent et al., 1990)) whereas the other O- and short Fe
O bond (d  1.64
1.68 A N-bonds are “normal” and less covalent. Thus one (or two) of the iron d-electrons will be much more affected by the covalence of the Fe
O bond than the others, namely those that have the right orientation for strong σ- and π-interactions. Their population numbers deviate significantly from those for a half-filled shell in a purely ionic compound, signifying distinct charge anisotropy. The (negative) sign of the resulting efg, indicating an elongate electron distribution, is consistent with strong σ-donation from the oxygen atom into the half-filled dz2 orbital, if the iron-oxo bond is in z-direction.

Magnetic hyperfine splitting The energy of the magnetic Zeeman interaction is given by the product of the nuclear magnetic moment μN and the field B at the nucleus, E 5 2μNB. Large magnetic hyperfine splitting of the nuclear levels as shown in the right panel of Fig. 6.7 is encountered in bioinorganic chemistry, for example, for iron(III) high-spin paramagnetic compounds at liquid helium temperatures with external fields applied to polarize the spin system. Then strong

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spin magnetic moments from unpaired valence electrons can yield strong internal fields of more than 50 T. The resulting magnetic splitting of the Mo¨ssbauer spectra are 16 mm/s or more. For a first interpretation of magnetic six-line spectra, the overall magnetic splitting between the outermost lines can be converted into units of the internal field at the nucleus by using the factor 3.1 T per 1 mm/s. Although applied-field Mo¨ssbauer spectroscopy of paramagnetic iron centers is an interesting topic, an introduction would lead too far here and the reader is referred to other reviews and book articles (Debrunner, 1989; Mu¨nck, 1978, 2000; Trautwein et al., 1991; Martinho and Mu¨nck, 2010; Schu¨nemann and Winkler, 2000; Paulsen et al., 2005; Gu¨tlich et al., 2011).

Combined hyperfine splitting Competing magnetic Zeeman interaction and electric quadrupole interaction are often encountered for paramagnetic iron compounds when magnetic fields are applied to the Mo¨ssbauer samples, because most iron sites have a significant efg, affording quadrupole interaction. The general description of combined hyperfine interaction is difficult and requires numerical procedures, particularly when both interactions are of comparable strength (Gu¨tlich et al., 2011). However, if one of the two interactions is weaker than the other, the resulting nuclear level splitting can be rather easily predicted from perturbation treatments (Fig. 6.10). In both cases of combined hyperfine interaction, μNB .. eQVzz as well as eQVzz .. μNB, a perturbed spectrum is observed, from the asymmetry of which the sign of the efg can be determined. The interpretation of magnetically perturbed quadrupole spectra will be demonstrated below in more detail for some applications.

Applications—selected examples Oxidation and spin states in a nonheme diiron center Most of the enzymes involved in the activation of molecular oxygen have one or more iron atoms in their active site. Our first example for the characterization of a biological iron center deals with the enzyme component called BoxB, which catalyzes the dearomatization key step in a particular pathway for metabolic degradation of aromatic compounds (Rather et al., 2011). The enzyme has a two-iron center, the structure of which is shown on the left side of Fig. 6.11. The enzyme occurs in the redox states diferric, diferrous, and semireduced, as was shown by Mo¨ssbauer and Electron Paramagnetic Resonance (EPR) spectroscopy (Rather et al., 2011). Productive O2 binding starts from the semireduced state in presence of benzoyl-CoA. Oxidized BoxB shows a quadrupole doublet with isomer shift δ 5 0.49 mm/s, and electric quadrupole splitting ΔEQ 5 0.69 mm/s (Fig. 6.11, right top). The values are typical of high-spin Fe(III) with five or six “hard” oxygen or nitrogen ligands, similar to what was found for other nonheme diiron enzymes in oxidized state (Kurtz, 1990). The values reveal in particular that there cannot be a bridging oxo-ligand, since that would induce significantly higher quadrupole splitting, as mentioned already above in the introduction to

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FIGURE 6.10 Combined magnetic Zeeman and electric quadrupole interaction for the cases of weak quadrupole perturbation of Zeeman-split levels (left), and weak magnetic perturbation of a quadrupole spectrum (right). In the first case (μNB .. eQVzz), the inner four lines of the six-line pattern are shifted relative to the outer two. The direction of the shifts depends on the sign of efg component Vij along the magnetic field (here the field B is taken parallel to Vzz . 0). The corresponding quadrupole splitting in a nonmagnetic situation would be twice the quadrupole shift EQ seen in the perturbed magnetic spectrum, ΔEQ 5 2EQ 5 0.5[(L6
L5) 2 (L2
L1)], where Li indicates the Doppler energies of lines i. In the second case (eQVzz .. μNB), the manifold of transitions from the ground state levels to the excited state levels underlying an unperturbed quadrupole doublet is split in characteristic manner, providing a 3:2 splitting pattern of the generic quadrupole lines. The asymmetry of the spectrum reveals the sign of the efg. Here the situation is shown for Vzz . 0; for negative values the perturbed quadrupole spectrum would be inverted on the velocity axis.

FIGURE 6.11 Structure of the diiron center of the enzyme BoxB (Rather et al., 2011) in oxidized state and, superimposed in gray, a so-called half-reduced state (left). The carboxylate oxygen atoms of Glu150 bridge the two-iron atoms in the oxidized but essentially not in the semireduced state. Right: zero-field Mo¨ssbauer spectra of oxidized (top) and reduced (bottom) BoxB recorded at 80 K. The red lines represent fits with one or two Lorentzian quadrupole doublets.

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quadrupole splitting. A recent example of an enzyme with a diferric center occasionally having either an oxo- or an hydroxo bridging group, is found in stearoyl-acyl carrier protein Δ9D desaturase. Mo¨ssbauer studies of the resting enzyme revealed ΔEQ 5 1.53 mm/s, δ 5 0.54 mm/s for the oxo-bridged dimer, and ΔEQ 5 0.72 mm/s, δ 5 0.49 mm/s for the hydroxo-bridged center (Fox et al., 2004). The difference owes its origin to the pronounced charge asymmetry that is induced by the short iron-oxo bond, in contrast to the “normally” elongated bonds expected for hydroxo groups. Upon reduction of BoxB the Mo¨ssbauer spectrum changes significantly and shows a new, slightly asymmetric quadrupole doublet (Fig. 6.11, right-bottom). The spectrum could be simulated with two Lorentzian doublets (in green) with parameters δ(1) 5 1.20 mm/s, ΔEQ(1) 5 2.39 mm/s, and δ(2) 5 1.31 mm/s, ΔEQ(2) 5 2.77 mm/s, using equal intensities but different line width Γ(1) 5 0.32 mm/s and Γ(2) 5 0.50 mm/s. The simulation reveals the presence of two high-spin ferrous ions (SFe 5 2) in nonequivalent sites. In particular, the high isomer shifts, which are similar to those of other diferrous centers in bioinorganic chemistry (Merkx et al., 2001), are typical for an octahedral coordination shell with hard ligands such as oxygen from carboxylates or water.

Reaction intermediates and low- and high-valent iron complexes The most common oxidation states of iron in metalloproteins are Fe(II) and Fe(III). Higher oxidation states, Fe(IV) and Fe(V), are also known or have been proposed for short-lived intermediates in the reaction cycle of oxygen-activating enzymes and model systems (Merkx et al., 2001; Costas et al., 2004; Groves, 2006; Bell and Groves, 2009; Tinberg and Lippard, 2011). This situation has fueled an interest in the chemistry of heme and nonheme iron complexes supporting high valence states, for which Mo¨ssbauer spectroscopy proved to be most valuable for the study of the electronic structure of the reaction intermediates captured by freeze-quench procedures (Krebs and Bollinger, 2009), as well as of stable model compounds.

The heme enzyme horseradish peroxidase An interesting property of heme complexes in high oxidations states is that not only iron but also the porphyrin ligand can be oxidized, forming a π-cation radical. This was first demonstrated for horseradish peroxidase (HRP), a heme enzyme that catalyzes the degradation of peroxides. In the first step of the reaction cycle, a mononuclear iron-oxo intermediate called “Compound I” (HRP-I) is formed, which formally is two equivalents more oxidized than the ferric starting compound. Moreover, a second intermediate (HRPII) could also be trapped after subsequent one-electron reduction. Mo¨ssbauer spectroscopy revealed “a fundamental change in iron configuration in going from HRP to HRP-II, and it is compatible with a net loss of electrons from the d shell. However, the Fe(V) state postulated for compound I probably does not exist: the electronic configurations of the iron in HRP-I and HRP-II appear essentially the same,” as Moss stated in 1969 (Moss et al., 1969). The conclusion was inferred from a comparison of the isomer shifts and quadrupole splittings seen in the Mo¨ssbauer spectra of HRP and its peroxide derivative (Fig. 6.12).

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FIGURE 6.12 Reaction cycle of horseradish peroxidase (HRP) (left), and zero-field Mo¨ssbauer spectra of HRP (Moss et al., 1969) and its peroxide reaction derivatives (right).

The moderately low isomer shift δ 5 0.25 mm/s found for native HRP at 77 K, in conjunction with large quadrupole, ΔEQ 5 1.96 mm/s (Moss et al., 1969), is typical of ferric low-spin heme iron complexes (Debrunner, 1989). The quadrupole splitting arises mainly from valence electrons due to the asymmetry in valence charge distribution expected for the 3d(t2g)5 configuration. After oxidation, the distinctly lower isomer shift for the second intermediate HRP-II (δ 5 0.03 mm/s at 77 K, ΔEQ 5 1.36 mm/s) reveals a loss of d-charge density due to a metal-based oxidation, which means that a ferryl species, Fe(IV) 5 O was formed. However, the second oxidation equivalent that distinguishes HRP-I from HRP-II must be ligand-centered, since HRP-I has essentially the same Mo¨ssbauer parameters (δ 5 0.0 mm/s at 77 K, ΔEQ 5 1.20 mm/s) as HRP-II. Later, applied-field Mo¨ssbauer studies (Schulz et al., 1979, 1984) yielded detailed insight into the electronic structure of the ferryl heme iron(IV) oxo complex (low-spin 3d4 configuration, S 5 1 with large zero-field splitting) and the interaction and spin coupling with the porphyrin π-cation radical species (S0 5 1/2) (Groves, 2006; Jayaraj et al., 1997; Mandon et al., 1992; Rittle and Green, 2010; Shaik et al., 2009).

Nonheme model compounds Synthetic iron(III) complexes with the macrocyclic ligand cyclam In nonheme iron systems the ligands bound to iron are generally considered to be redox-innocent, and high-valent intermediates containing Fe(IV) or Fe(V) have been found and studied by Mo¨ssbauer spectroscopy (Price et al., 2003; Pestovsky et al., 2005; de Oliveira et al., 2007; De Hont et al., 2010). A robust supporting ligand for the synthesis of high-valent iron model complexes is the macrocycle 1,4,8,11-tetraazacyclotetradecane (cyclam), which coordinates iron with four equatorial nitrogen donors akin to the coordination mode of a porphyrin. In contrast to the latter, cyclam is “redox-innocent,” that is, it cannot be oxidized or reduced. There are several cyclam derivatives, including some with a pendant acetate arm that can bind as a fifth ligand (see the following scheme).

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Cyclam and its methylated and pendant arm derivatives have been successfully used to generate high-valent transitions metal complex, particularly mononuclear complexes with high-valent Fe(IV), Fe(V), and Fe(IV) (Berry et al., 2006b; Meyer et al., 1999; Grapperhaus et al., 2000; Rohde et al., 2003; Aliaga-Alcalde et al., 2005; Petrenko et al., 2007). While the reactive high-valent iron species in biological systems are mostly stabilized by a strong electron-donating oxo group, also iron-nitrido systems are interesting and have been used in synthetic systems, not only because of their relevance for enzymatic reactions in the biogeochemical nitrogen cycle, but also for theoretical and technical reasons. An interesting preparation method for high-valent complexes is based on photolytic cleavage of coordinated ferric azide complexes (with coordinated N2 3 ions), yielding dinitrogen (N2) and high-valent Fe 5 N species (Berry et al., 2006b; Meyer et al., 1999; Grapperhaus et al., 2000; Rohde et al., 2003; Aliaga-Alcalde et al., 2005; Petrenko et al., 2007; Betley and Peters, 2004; Brown and Peters, 2005). In the first example, the starting material was a bis-azide iron(III) cyclam complex, [(cyclam)FeIII(N3)2]. This can exist in two different conformations with the two azide ligands either in cis- or in trans-positions (see insets of Fig. 6.13). The difference has major impact on the electronic structure, as can be inferred from the Mo¨ssbauer spectra (Fig. 6.13A and B). The trans-complex is a typical ferric low-spin compound with low isomer shift and large quadrupole splitting (δ 5 0.29 mm/s, ΔEQ 5 2.26 mm/s), whereas the more open arrangement of the macrocyclic ligand upon the cis-arrangement of azides allows high-spin configuration with moderately large isomer shift and small quadrupole splitting, due to the quasi cubic symmetry of the half-filled d-shell (δ 5 0.46 mm/s, ΔEQ 5 0.29 mm/s) (Meyer et al., 1999).

Iron(II) complexes Electrochemical reduction of the ferric compounds yields different ferrous products (Fig. 6.13C and D) (Meyer et al., 1999). The Mo¨ssbauer parameters of the reduced cis-complex unambiguously reveal formation of high-spin Fe(II) (δ 5 1.11 mm/s, ΔEQ 5 2.84 mm/s); whereas the major part of the reduced trans-complex is in a low-spin Fe(II) state which has low isomer shift and low quadrupole splitting, due to the high symmetry of the filled t62g subshell (subspectrum I, δ 5 0.55 mm/s, ΔEQ 5 0.72 mm/s). However, the trans-conformation is not stable for the reduced compound and tends to convert in solution to the cis-arrangement, as can be seen from the presence of the same high-spin component as found for the cis-complex (subspectrum II in panel C). The intensity of this component is found to increase with incubation time in solution.

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FIGURE 6.13 Zero-field Mo¨ssbauer spectra of trans- and cis-[(cyclam)FeIII(N3)2]1 complexes (35%

57 Fe-enriched, 1.5 mM in CH3CN, panels A and B) (Meyer et al., 1999) and their electrochemically one-electron reduced CH3CN solutions (C and D) recorded at 80 K. The spectra are simulated with Lorentzian doublets. The oxidized complex with trans-orientation of the azide (N2 3 ) ligands is a typical low-spin Fe(III) compound (A), whereas the chemically identical cis-arrangement shown in (B) affords high-spin Fe(III). After coulometric reduction at 230 C the reduced trans-complex (C) shows a superposition low-spin Fe(II) (subspectrum I) and high-spin Fe(II) (subspectrum IL); prolonged incubation in solution at 230 C provides strong enhancement of subspectrum II and concomitant attenuation of I. Reduction of the cis-complex shown in (D) yields only high-spin Fe(II). Subspectrum III in D, having parameters δ 5 0.49 mm/s, ΔEQ 5 0.90 mm/s, is most probably from a contamination with a [FeIII
O
FeIII]41 μ-oxo dimer species due to oxidation with traces of air. This spectrum was exclusively obtained when a reduced sample was deliberately oxidized with air.

Similar Mo¨ssbauer spectra and parameters as for trans-[(cyclam)FeIII(N3)2]1 (Fig. 6.13A and C) have been observed for [(cyclam-acetato)FeIIIN3]1 (not shown, δ 5 0.27 mm/s, ΔEQ 5 2.53 mm/s). Coordination of the tethered acetate arm allows only trans-configuration for this compound. Apparently, the replacement of an (N2 3 ) ligand by the acetato-oxygen group did not have much influence on the electronic structure of the metal ion. Similar trends were also observed for the corresponding reduction and photooxidation products, as discussed in the following.

Mixed-valence iron(III)
iron(IV) dimers and iron(IV) monomers Iron(III)cyclam-azide complexes are light sensitive, and illumination of trans- and cis[(cyclam)FeIII(N3)2]1 complexes in chilled fluid solution at 235 C with light of about 420 nm wavelength yields two new compounds which could be identified as mixedvalence dimers with μ-nitrodo-bridged diiron cores of the type [Fe(III)
N
Fe(IV)]41. One has cis-symmetry of nitrido- and terminal azide ligands at the Fe(IV) site, the other transsymmetry (Meyer et al., 1999). The formation of these products indicates that photooxidation as well as photoreduction must have occurred as primary reactions, such that the corresponding Fe(IV) and Fe(II) products could form the observed Fe(III)/Fe(IV) mixedvalent dimers via bimolecular reactions. Here only the properties of the high-valent Fe(IV)

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sites of these dimers shall be considered. As could be revealed by applied-field Mo¨ssbauer and EPR measurements, they are hexa-coordinated Fe(IV) low-spin centers with spin S 5 1, according to the 3d(t2g)4 electron configuration (four electrons in three orbitals). The electric Mo¨ssbauer parameters are δ 5 0.14 mm/s, ΔEQ 5 0.79 mm/s for the cis-compound, and δ 5 0.11 mm/s, ΔEQ 5 0.97 mm/s for the corresponding trans-complex (at 80 K in frozen solution). The values are similar to what is known for the Fe(IV) 5 O species with porphyrin ligands (Debrunner, 1989; Rittle and Green, 2010; Shaik et al., 2009; Jung, 2011). Monomeric iron(IV) complexes without terminal or bridging oxo- or nitrido-groups could be obtained from electrochemical oxidation of a series of halide and azide complexes of the type [(Me3cyclam-acetate)FeX]PF6, where X was Cl2, F2, and N2 3 , and Me3cyclamacetate is the methylated cyclam derivative with a pendant acetate arm shown in the scheme above (Berry et al., 2006a). The Mo¨ssbauer parameters for these compounds are rather similar to those of the Fe(IV) sites in the dimer molecules: δ 5 0.08/0.02/0.11 mm/s and ΔEQ 5 2.40/2.43/1.92 mm/s for the Cl2, F2, and N2 ligands, respectively 3 (Berry et al., 2006a).

Iron(V) complexes Genuine iron(V) is a very rare oxidation state; only three authenticated iron(V)-oxo compounds have been reported in (bio)inorganic chemistry. These were synthesized with the redox-innocent tetranionic tetra amido macrocyclic ligand (TAML) (de Oliveira et al., 2007), which provides four exceptionally strong amide-N σ-donor groups, and its biuretamide derivative (Ghosh et al., 2014; Mills et al., 2016). These are capable of stabilizing iron(V) when an iron(III) precursor complex is treated with an oxygen-transfer agent. The presence of the expected 3d3 configuration was verified by Mo¨ssbauer spectroscopy (δ 5 20.42 mm/s at 4.2 K, ΔEQ 5 4.25 mm/s), EPR spectroscopy (S 5 1/2, g 5 [1.99, 1.97, 1.74]), and X-ray absorption spectroscopy (XAS) and extended X-ray absorption fine structure spectroscopy (EXAFS), (all values for the TAML complex) (de Oliveira et al., 2007). Genuine iron(V) nitrido complexes could be generated by low-temperature photolysis of the cyclam iron(III) complexes described above (Berry et al., 2006b; Meyer et al., 1999; Grapperhaus et al., 2000; Aliaga-Alcalde et al., 2005; Petrenko et al., 2007). In frozen solution, where bimolecular reactions are prevented, the corresponding monomeric Fe(V)N species could be generated in high yield (Fig. 6.14, left). Apparently, photoreduction of the immobilized molecules virtually does not contribute to the result, presumably because in contrast to photooxidation it is reversible, and azide radicals eventually generated by photoreduction can rebind to the immobilized Fe(II) product molecules, restoring the starting material. Comprehensive studies on [(cyclam-acetato)FeVN]1 using XAS and EXAFS measurements, magnetic susceptibility, Mo¨ssbauer spectroscopy, nuclear inelastic scattering (NIS), and recently EPR in conjunction with density functional theory (DFT) calculations (Meyer et al., 1999; Grapperhaus et al., 2000; Aliaga-Alcalde et al., 2005; Petrenko et al., 2007; Berry, 2009; Chang et al., 2019) characterized the compound as a low-spin (dxy)2(dxz,dyz)1 system. The Mo¨ssbauer parameters of [(cyclam-acetato)FeVN]1 (δ 5 20.02 mm/s, ΔEQ 5 1.60 mm/s) are consistent with those found for [(cyclam)FeVN]1 in CH3CN solution, which presumably has a sixth ligand at the iron (δ 5 20.04 mm/s, ΔEQ 5 1.67 mm/s).

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FIGURE 6.14 Left: Mo¨ssbauer spectra of [(cyclam-acetato)FeIII(N3)]1 (in red) and its low-temperature iron(V) photolysis product (in green) (Grapperhaus et al., 2000), measured at 80 K. Right: Isomer shift correlation diagram for low-spin iron complexes with cyclam and cyclam-acetato supporting ligands. The iron valences range from Fe (II) to Fe(V). The blue line is shown to guide the eye; it is not based on a theoretical concept beyond the general arguments about dn-electron configurations and isomer shifts. Sometimes also straight lines have been used here.

The conclusions about the correct electronic structure have been substantially based on the close correlation of the isomer shifts of the cyclam complexes with the formal oxidation state of the iron, as shown in the right panel of Fig. 6.14. The variation of δ appears to follow the expected decrease in bond lengths of the apical ligands with increasing oxidation state. A tight, conclusive correlation is obtained here because in contrast to the general scheme shown in Fig. 6.8, a consistent series of quasi isomorphous complexes with invariant equatorial ligands has been compared. The compounds are all six-coordinate and variations in the valence electron configuration are restricted to the t2g subshell (low-spin) with increasing covalency of the axial ligand bonds. A similar correlation may be found for related complexes with axial oxygen or oxo ligands, but the slope of the corresponding plot for isomer shifts versus oxidation state will be different.

Four-coordinated iron(IV) and iron(V) compounds An amazing four-coordinate iron(V) nitrido complex with a tripodal N-heterocyclic carbene ligand was also reported (Fig. 6.15, left). The high-valent compound was generated electrochemically by oxidation of the corresponding iron(IV)-nitrodo precursor, which had been synthesized previously by photolysis of a corresponding iron(II)-azide starting complex (Scepaniak et al., 2008). The iron(IV) precursor is a diamagnetic molecule (S 5 0) with low Mo¨ssbauer isomer shift, δ 5 20.28 mm/s. The low-spin state is most unusual for four-coordination; the strong ligand field results here from short bonds and strong trigonalaxial distortion of the pseudotetrahedral symmetry. The quadrupole splitting is ΔEQ 5 6.23 mm/s, which is the largest value ever observed for an iron compound. The value slightly exceeds the splittings of 6.01 and 6.04 mm/s found for two other, electronically similar four-coordinate compounds (Vogel et al., 2008), [PhBPiPr3]FeN and [(TIMENmes)FeN]1, respectively (isomer shifts of 20.31 mm/s at 140 K and 20.27 mm/s at 77 K).

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FIGURE 6.15 Molecular structure of the cation [PhB(tBuIm)3FeVαN]1, where PhB (tBuIm)32 is the phenyltris (3-tert-butylimidazol-2-ylidene) borato ligand (Scepaniak et al., 2011) (left) and zero-field Mo¨ssbauer spectrum of a microcrystalline sample of the iron(IV) precursor (Scepaniak et al., 2008) PhB(tBuIm)3FeIVN recorded at 77 K (right). The red line represents a fit with a Lorentzian doublet and δ 5 20.28 mm/s, |ΔEQ| 5 6.23 mm/s.

The strong (trigonal) distortion of the pseudotetrahedral symmetry of these complexes affords unusually strong orbital splitting with a “three-over-two” pattern. Two nonbonding e-orbitals are low lying, rendering linear combinations of dxy and dx22y2 orbitals. Since they are energetically well isolated, the four valence electrons are accommodated in the two orbitals (Scepaniak et al., 2008), affording diamagnetism (S 5 0). The state may be called the “low-low-spin” state of the d4 configuration, since the usual (t2g)4 low-spin configuration in octahedral symmetry has S 5 1, arising from four electrons in three orbitals. Moreover, the quasi planar charge distribution expected for the discussed (dxy, dx22y2)4 configuration of the pseudotetrahedral nitrido complexes explains the unprecedentedly large efg at the Mo¨ssbauer nucleus (see Vzz expectation values for dxy and dx22y2 in Table 6.1). Although not determined, one may presume a positive sign for Vzz because of the huge (positive) valence contribution to the efg. The oxidized cation [PhB(tBuIm)3FeVN]1 is also low-spin, S 5 1/2, according to the corresponding (dxy, dx22y2)3 configuration (Scepaniak et al., 2011). Its quadrupole splitting, ΔEQ 5 4.73 mm/s, is less than that of the Fe(IV) precursor, but it is still remarkably large and higher than the values usually reported for octahedral or tetrahedral iron compounds. The isomer shift, δ 5 20.49 mm/s, is below the range observed for the cyclam complexes with Fe(V)-nitrido groups, due to the overall shorter metal
ligands bonds. This demonstrates again that a single isomer shift value may be not conclusive if it cannot be compared with reference systems.

The first molecular iron(VI) compound Basic ligand field considerations predict relatively high stability for highly oxidized iron complexes with d2 or d1 configuration, due to the reduced number of antibonding d-electrons compared to 3d3 configuration. Nevertheless, iron(VII) with 3d1 configuration is elusive and only one molecular iron(VI) compound with 3d2 configuration could be synthesized so far. That is an Fe(VI)-nitrido complex (Berry et al., 2006b) with the bulky ligand (Me3cyclam-acetate)2. The high-valent [(Me3cyclam-acetate)FeVIN]21 complex (inset of

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FIGURE 6.16 Left: Mo¨ssbauer spectra of a photolyzed acetonitrile solution of [(Me3cyclam-acetate)FeIVN3] (PF6)2 measured at 80 K (top) and at 4.2 K with an applied magnetic field of 7 T (bottom) (Berry et al., 2006b, 2008). The major subspectrum marked in red represents the high-valent photoproduct [(Me3cyclam-acetate) FeVIN]21. The red lines results from a fit with Lorentzian doublets for the 80 K measurement and from a spinHamiltonian simulation with S 5 0 for the applied-field spectrum (bottom). The subspectrum shown in green was detectable only at 80 K and presumably represents a ferric contamination, which was split at 4.2 K/7 T and broadened beyond recognition due to intrinsic magnetic moments. The inset on the left panel shows the molecular structure predicted by density functional theory (DFT). Right: orbital scheme for [(Me3cyclam-acetate)FeVIN]21 from DFT.

Fig. 6.16) was electrochemically generated from the iron(IV)-azide precursor [(Me3cyclamacetate)FeIVN3]21 by low-temperature photolysis in the sample cup of the Mo¨ssbauer spectrometer. The product is diamagnetic, S 5 0, and the FeN bond (157 pm obtained from EXAFS) is significantly shorter than that for the corresponding iron(V) compound (161 pm) (Aliaga-Alcalde et al., 2005). The isomer shift, δ 5 20.29 mm/s, is 0.40 mm/s lower than that of the corresponding iron(III)-azide starting material and 0.19 mm/s lower than that of the corresponding iron(V)-nitrido compound. The value fits perfectly to the isomer shift correlations shown in Fig. 6.14. This feature together with the short FeN bond, and the spin S 5 0 obtained from an applied-field Mo¨ssbauer measurement (Fig. 6.16, left) supports the presence of genuine iron(VI) with an electronic structure that is well described as 3d2 configuration (Berry et al., 2006b) (Fig. 6.16, right). The quadrupole splitting of the iron(VI) compound is ΔEQ 5 11.53 mm/s, with asymmetry parameter η 5 0.3. The sign of Vzz and η have been obtained from the simulation of the magnetically perturbed spectrum with spin S 5 0 shown in Fig. 6.16, left bottom. Apparently, the efg is significantly weaker than what would be expected for an ionic d2 lowspin configuration with two valence electrons in one orbital (compare Table 6.1). This reveals competition of valence and covalence contributions attenuating the actual values. The various spectroscopic and structural parameters are consistent with the overall geometry and the electronic structure predicted by quantum chemical calculations on [(Me3cyclam-acetate)FeVIN]21 (calculated Mo¨ssbauer parameters: δ 5 20.31 mm/s, ΔEQ 5 10.75 mm/s, η 5 0.7, which is excellent for δ and satisfactory for ΔEQ). The 3d orbital scheme obtained from DFT calculations as shown on the right side of Fig. 6.16 nicely

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supports the interpretation as 3d2 configuration. The highest occupied MO is best described as a doubly occupied dxy orbital. The xz and yz orbitals are strongly destabilized in energy because of considerable contributions (45%) from p orbitals of the terminal nitrido ligand and are best considered to be Fe
N p* antibonding orbitals. Occupancy of these orbitals reduces the Fe
N bond order, and is a major source of the efg (Berry et al., 2006b, 2008).

Perspectives The most spectacular chemical application of Mo¨ssbauer spectroscopy in recent years has been the search for water-dependent or water-carrying minerals on Mars by the Mo¨ssbauer spectrometers of the NASA twin Mars Exploration Rovers (MER), landed in 2004 and going on until recently (Gu¨tlich and Schro¨der, 2010; Klingelho¨fer et al., 2003; Klingelho¨fer and Fleischer, 2011). The tremendous amount of data harvested during this most successful mission has provided a world of information on extraterrestrial chemistry, which certainly prompt further extraterrestrial research missions. The miniaturized spectrometers are now also used for terrestrial applications, in geosciences for instance (Klingelho¨fer and Fleischer, 2011). Although less extravagant, but also very innovative are Mo¨ssbauer experiments with synchrotron radiation (SR) (an introduction of which is given in Chapter 9 of Gu¨tlich et al., 2011). It is easy to predict that applications of SR-based techniques in bioinorganic chemistry will further develop. The aim might be measurements of hyperfine interactions with several noniron isotopes, with microscopically small samples with minimal volume. NIS of SR is a most interesting variant of Mo¨ssbauer spectroscopy, yielding highly selective vibrational data (also called Nuclear Vibrational Spectroscopy, NRVS). But there will be also continuing applications for conventional Mo¨ssbauer spectroscopy with radioactive sources, although since long time it is in its “mature” middle age. For instance, there is a revival of research on iron
sulfur proteins due to their role in catalysis, for example, in hydrogenases (Pandelia et al., 2015, 2011), or the role of Fe:S cluster formation and enzyme maturation for gene expression and genetic disorder has captured much attention, not least due to the medical implications (Lill, 2009; Sheftel et al., 2010). In such fields should be a promising future for interesting Mo¨ssbauer spectroscopy.

Exercises 1. A synthetic iron complex was obtained in two different molecular structures (I) and (II), which both have the same type and number (6) of ligands and the same oxidation state. The zero-field Mo¨ssbauer spectra showed both a quadrupole doublet, but with very different parameters: I. δ 5 0.27 mm/s; ΔEQ 5 2.25 mm/s II. δ 5 0.48 mm/s; ΔEQ 5 0.29 mm/s Use the isomer shift correlation diagram of Fig. 6.8 and the expectation values of Vzz (Table 6.1) for the valence contribution to the efg to discuss the different isomer shifts and quadrupole splittings and try to assign oxidation and spin states of the iron sites.

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(Assume approximately octahedral symmetry, and neglect extreme oxidation states below Fe(II) an above Fe(III).) Answer: Compound I is clearly low-spin, due to the low isomer shift. The large quadrupole splitting renders it Fe(III) low-spin (add up Vzz,val for five electrons in the t2g orbitals dxz, dyz, dxy). The isomer shift of Compound II would be consistent with high-spin Fe(III) or low-spin Fe(II), but the small quadrupole splitting reveals a half-filled shell, that is, Fe(III) highspin. 2. What should be the spin state of six-valent iron (Fe(VI)) in tetrahedral coordination (as found for solid-state ferrates)? Discuss the valence contribution to efg (for a high-spin 3d2 configuration) and estimate the expected quadrupole splitting (small to medium large). Answer: Fe(IV) has a 3d2 configuration, and tetrahedral symmetry renders the eg orbital dz2 and dx22y2 to be low-lying. Two electrons in the eg subshell must adopt high-spin configuration. Since the Vzz,val for these orbital is equal but of opposite sign, the quadrupole splitting should be vanishing. 3. Try to sketch a magnetic Mo¨ssbauer of an octahedral Fe(II) high-spin compound, recorded with a field of c.7 T for positive and for negative Vzz,val (δ 5 1 mm/s; |ΔEQ| 5 4 mm/s; assume high temperature and internal field averaged to 0). What can be the reason for opposite efgs. Answer: See Fig. 6.10, right side, for positive efg, and spectrum inverted for negative efg. The sixth electron of the 3d6 configuration decides the sign of Vzz,val. This can be in dx22y2 or dz2, which have opposite efgval.

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57

Fe-Mo¨ssbauer spectroscopy and basic interpretation of Mo¨ssbauer parameters

Scepaniak, J.J., Vogel, C.S., Khusniyarov, M.M., Heinemann, F.W., Meyer, K., Smith, J.M., 2011. Synthesis, structure, and reactivity of an iron(V) nitride. Science 331, 1049
1052. Schulz, C.E., Devaney, P.W., Winkler, H., Debrunner, P.G., Doan, N., Chiang, R., et al., 1979. Horseradish peroxidase compound I: evidence for spin coupling between the heme iron and ‘free’ radical. FEBS Lett. 103 (1), 102
105. Schulz, C.E., Rutter, R., Sage, J.T., Debrunner, P.G., Hager, L.P., 1984. Mo¨ssbauer and electron paramagnetic resonance studies of horseradish peroxidase and its catalytic intermediate. Biochemistry 23 (20), 4743
4754. Schu¨nemann, V., Winkler, H., 2000. Structure and dynamics of biomolecules studied by Mo¨ssbauer spectroscopy. Rep. Prog. Phys. 63 (3), 263
353. Shaik, S., Cohen, S., Wang, Y., Chen, H., Kumar, D., Thiel, W., 2009. P450 enzymes: their structure, reactivity, and selectivity—modeled by QM/MM calculations. Chem. Rev. 110 (2), 949
1017. Sheftel, A., Stehling, O., Lill, R., 2010. Iron-sulfur proteins in health and disease. Trends Endocrinol. Metabol. 21 (5), 302
314. Solomon, E.I., Xie, X., Dey, A., 2008. Mixed valent sites in biological electron transfer. Chem. Soc. Rev. 37 (4), 623
638. Tinberg, C.E., Lippard, S.J., 2011. Dioxygen activation in soluble methane monooxygenase. Acc. Chem. Res. 44 (4), 280
288. Trautwein, A.X., Bill, E., Bominaar, E.L., Winkler, H., 1991. Iron-containing proteins and related analogs— complementary Mossbauer, EPR and magnetic susceptibility studies. Struct. Bond. 78, 1
95. Vincent, J.B., Olivier-Lilley, G.L., Averill, B.A., 1990. Proteins containing oxo-bridged dinuclear iron centers: a bioinorganic perspective. Chem. Rev. 90 (8), 1447
1467. Vogel, C., Heinemann, F.W., Sutter, J., Anthon, C., Meyer, K., 2008. An iron nitride complex. Angew. Chem. Int. Ed. 47 (14), 2681
2684.

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C H A P T E R

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X-ray absorption and emission spectroscopy in biology Martin C. Feiters1 and Wolfram Meyer-Klaucke2 1

Department of Synthetic Organic Chemistry, Institute for Molecules and Materials, Faculty of Science, Radboud University, AJ Nijmegen, The Netherlands 2Deutsches Elektronen Synchrotron DESY, Hamburg, Germany O U T L I N E

Outline of the X-ray absorption and emission spectroscopy in biology

230

An introductory biological X-ray absorption spectroscopy example: Mo, Cu, and Se in CO-dehydrogenase from Oligotropha carboxidovorans 231

Strategy for the interpretation of EXAFS

257

Validation and Automation of EXAFS data analysis

258

X-ray absorption near-edge structure simulations with three-dimensional models

260

X-ray absorption (near-)edge structure

236

X-ray emission spectroscopy in biology

239

Time-resolved X-ray absorption spectroscopy

Metal
metal distances in metal clusters

261

244

Nonmetal trace elements: halogens

262

X-ray absorption spectroscopy: X-ray
induced electron diffraction

245

Summary: strengths and limitations

264

Phase shifts and effect of atom type

248

Conclusions: relations with other techniques

266

Plane wave and muffin-tin approximation

252

Exercises

266

Multiple scattering in biological systems

Hints and answers to exercises

269

254

References

272

Practical Approaches to Biological Inorganic Chemistry DOI: https://doi.org/10.1016/B978-0-444-64225-7.00007-9

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© 2020 Elsevier B.V. All rights reserved.

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7. X-ray absorption and emission spectroscopy in biology

Outline of the X-ray absorption and emission spectroscopy in biology X-rays are electromagnetic radiation in the energy range 100
100 keV (corresponding to a wavelength range of 0.01
10 nm or a frequency range of 3.1016
3.1019 Hz). They can interact with matter by photoelectric absorption or by elastic or inelastic scattering. The interactions that do not involve exchange of energy are studied in experiments based on diffraction in the case of crystalline solids, such as the X-ray crystallography discussed in another chapter in this book, or by scattering in the case of amorphous materials. In this chapter we will discuss the application of experiments that do involve exchange of energy, viz., X-ray absorption and emission spectroscopy to biological inorganic chemistry. X-ray spectroscopy can be applied regardless of the physical state of a sample and is also used for the characterization of other noncrystalline systems such as metal ions in solution or on catalyst supports (van Bokhoven and Lamberti, 2016). Following the advent of synchrotrons as sources of intense white X-rays in the second half of the 20th century, X-ray absorption spectroscopy (XAS) has become established as one of an armory of physical techniques that life scientists apply to the study of metal ions or other trace elements in whole biological systems or isolated components thereof. Many metal ions are now recognized as essential to life in addition to the “bulk” elements (C, H, N, O, S, and P). The majority of biological systems studied by XAS are metalloproteins, but there are an increasing number of studies on other biomolecules and/or other trace elements. The so-called absorption edge of an element of choice is element specific (see Fig. 7.1 for the energies of the edges of the elements). Thus the XAS shows fine structures only for one absorber element at a time. Both the absorption edge itself as well as the fine structure above the edge are related, by physical processes that will be described below, to the chemical environment of the element that has been selected. From the so-called X-ray absorption near-edge structure (XANES) at and near the absorption edge, information on the valence state and coordination geometry of the element involved can be derived; the extended X-ray absorption structure (EXAFS) beyond the edge is interpreted to give information on the type, distance, and number of the surrounding ligands. In recent years, X-ray emission spectroscopy (XES) has also been applied to biological problems, and found to give information on electronic structure and ligand identity that overlaps with or is complementary to the results of XANES and EXAFS. Even more recently, a more intense source of X-rays has become available with the X-ray free electron laser (XFEL), of which the applications to biological spectroscopy promise to give further insight into the dynamic of reactions. In this chapter, we will introduce the most important aspects of X-ray spectroscopy that are relevant to biological studies. The main line of the text will discuss most principles and examples, while the reader is referred to specific Boxes with details on certain aspects: Box 7.1 on the BioXAS experiment, Box 7.2 on L-edge and ligand-edge XAS, Box 7.3 on XES techniques, Box 7.4 on data reduction, Box 7.5 on phase shift calculations, and Box 7.6 on the Debye
Waller factor. The first section discusses how XANES can be interpreted in terms of electronic spectroscopy, in order to extract information on oxidation state and ligand geometry. We proceed with XES because the transitions involved are related to those in XANES, and it also provides an improved approach to XANES measurements.

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An introductory biological X-ray absorption spectroscopy example

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FIGURE 7.1 Name, symbol, atomic number, K (blue) and L3 (red)-edges in eV of biologically important elements (black) and some others (gray) in the Periodic Table.

Subsequently we introduce the X-ray
induced electron diffraction that underlies EXAFS. We then treat the theory in terms of a single scattering approximation, and discuss the phase shifts that need to be calculated in order to be able to extract information on the distance and approximate atom type and number of the ligands by simulation. The next topic is multiple scattering, which is found to be particularly important in biological systems. We then discuss a strategy for the interpretation of the EXAFS and how it can be automated, strengths and limitations of the technique, and the relation with other spectroscopic techniques and crystallography. The chapter concludes with a number of special topics and conclusions.

An introductory biological X-ray absorption spectroscopy example: Mo, Cu, and Se in CO-dehydrogenase from Oligotropha carboxidovorans Using monochromatized synchrotron radiation (see Box 7.1 for experimental aspects), it is possible to select an element and scan the X-ray energy around the so-called edge, that is around the energy that is required to liberate an electron from that atom. Fig. 7.1 gives an overview of the element specificity of these electron binding energies; it summarizes the edge energies for the most important elements in biology. For most elements, in particular the 3d transition metals, the K-edges (blue numbers in Fig. 7.1), where the 1s electron is excited, are readily accessible; for elements higher up in the Periodic Table, it is more convenient to use the L-edges where 2s and 2p electrons are excited (red numbers give L3-edges in Fig. 7.1). Absorption is described by the Beer
Lambert law, with the transmitted intensity It depending on the incident intensity I0, the sample thickness x, and the energy-dependent absorption coefficient μ(E). It 5 I0 expð2 μxÞ

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ð7:1Þ

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Thus the X-ray absorption spectrum is represented by the dimensionless X-ray absorption coefficient μ:
 Io ln 5 μx ð7:2Þ It with I0 and It the X-ray intensities before and after the sample, respectively; it can also be measured, as in most examples discussed here, in fluorescence as μf (see Box 7.1). Fig. 7.4 shows XANES and EXAFS spectra of examples that were selected to demonstrate the power of XAS to probe chemical information that is relevant to biological structure
function relationships. In the top panel we follow molybdenum uptake and utilization by an enzyme and show the XANES of the K-edge of molybdenum (absorption of X-rays by the 1s electron of Mo) of bioavailable, aqueous molybdate (MoO22 4 ), molybdate in one of the Mo transport proteins, here ModG, and finally Mo in the enzyme CO-dehydrogenase in its oxidized and reduced forms. The reduction of Mo in CO-dehydrogenase (from Mo61 to Mo41) leads to a shift of the absorption edge (the part of the spectrum where the absorption increases most, 20,000
20,030 eV) to lower energy; this is a general observation in XANES. It is in agreement with the intuitive notion that it should require a little more energy to excite an electron from a metal ion in a relatively high oxidation state, when it bears more positive charge, than in a low oxidation state. There is also an interesting preedge structure at

BOX 7.1

The BioXAS experiment XAS of samples as dilute in the element of interest as biological systems requires an intense source of X-rays continuous in wavelengths around the absorption edge. This requirement was fulfilled in the 1970s at synchrotrons. In principle the spectrometer consists of the following (Fig. 7.2): X-rays of all wavelengths (white beam) originating from the bending magnet (or insertion device like wiggler or undulator) in the synchrotron enter the spectrometer from the left and pass through an entrance slit into a monochromator. According to Bragg’s law nλ 5 2d sin θ

ð7:3Þ

where n is a natural number (1, 2, 3, etc.), λ the wavelength, d the lattice parameter of

the Si, and θ the angle between monochromator and beam, X-rays of a certain wavelength are diffracted by two parallel Si crystals into the ionization chambers. The wavelength of the monochromatized beam is varied throughout the experiment by changing the monochromator angle with respect to the incident beam. Diffraction of the X-rays occurs for n 5 1 and for the harmonic contaminations (n . 1), which have shorter wavelengths, corresponding to higher energies. These harmonics are selectively rejected by an order-sorting monochromator in which the positions of the two crystals are controlled so that they are slightly nonparallel. This selection is based on the acceptance angles that are by far smaller for higher harmonics.

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An introductory biological X-ray absorption spectroscopy example

BOX 7.1

233

(cont’d)

FIGURE 7.2 Schematic representation of an X-ray absorption spectrometer.

FIGURE 7.3 (Left) In the X-ray absorption process a hole in one of the orbitals (e.g., 1s for K-edge) is created, and the excited electron leaves the atom as a photoelectron and takes the remaining energy (5photon energy
binding energy) with it. X-ray fluorescence arises when electrons from higher orbitals fill up the resulting hole, giving off the excess energy as fluorescence, which has an energy also in the X-ray range. (Right) Dependence of K- and L3-edge fluorescence yields on the atomic number Z. The probability for filling the hole with an electron from an orbital with higher energy is called fluorescence yield. Note that this is rather high for K-edges, whereas for L-edges it requires rather high concentrations to detect a sufficient signal. In order to determine the absorption of the sample according to the Beer
Lambert law (Eq. (7.1)), the intensities of the X-rays before and after passage through the sample are measured by ionization chambers, in which the X-rays ionize gas molecules; the resulting ions give a current that is proportional to the X-ray intensity. Absorption of X-rays causes fluorescence radiation

(Fig. 7.3, left) which has lower energies than incident and scattered radiation. Using detectors of the solid-state multielement type, the fluorescence (If) can be separated from scattered radiation, which represents the background, by its energy; provided that the sample is thick, that is, sufficiently absorbing the X-rays used in the experiment, but dilute in the element of interest,

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BOX 7.1 the energy-dependent absorption coefficient is given as μf 5 If/I0, which ultimately results in an extracted fine structure (see Box 7.4) equivalent to that measured in transmission mode. Biological systems are usually weakly absorbing, except for the trace elements that they are relatively dilute in, resulting in hardly any edge step against a large background in the transmission experiment; therefore fluorescence is the preferred mode of detection for these systems. The fluorescence yield depends on both the atomic number and the absorption edge (Fig. 7.3, right). Even if fluorescence detection is applied, it is still important to accumulate multiple scans of a sample with a concentration in the millimolar range in the element of interest. Because of the susceptibility of the biological matrix to radiation damage, measurements are usually carried out on frozen samples at cryogenic temperatures in order to reduce the mobility of any radicals generated by X-ray irradiation; this has also the added beneficial effect of decreasing the thermal component of the so-called Debye
Waller factor (see Box 7.6), enhancing the sensitivity to weak long-range contributions. Cryoprotectants such as glycerol may be added if it is suspected that freezing damages the biological sample, but it is necessary to check that they do not interfere with the biological activity of the biomolecule. A special kind of radiation damage is that

(cont’d) resulting in a change of the valence state of the element under investigation, for example, photoreduction, as its progress can be probed by comparing the edge structures (the so-called XANES part of the spectrum, see below) of consecutive scans during irradiation and data accumulation. Typically the requirement to have efficient data collection on a sample as small and dilute as possible using a high flux beamline (third generation source with insertion device) needs to be balanced against the risks of general and more specific radiation damage (Ascone et al., 2003). A typical solution is to spread the photon flux over a rather large area and/or replace the sample as soon as photoreduction is detected. Besides the high intensity and the energy range, synchrotron radiation has another property that is of interest for biological samples, viz., it is polarized in the plane of the synchrotron. This means that if one wants to measure typical anisotropic XAS of a crystalline sample it is advisable to use a slurry of crystals rather than a single crystal, in order to avoid effects of preferential orientation with respect to the plane of the synchrotron beam. On the other hand, advantage may be taken of the polarized beam to study the linear dichroism in anisotropic biological samples, such as protein single crystals, or the metal centers in proteins in stacked layers of membranes, such as the organelles responsible for photosynthesis in plants, the thylakoids.

approx. 20,005 eV, which is very strong in the highly symmetric (tetrahedral) molybdate and much weaker in the low-symmetry site of the enzyme. Its intensity appears to be proportional to the number of oxo (O22) ligands around Mo, which is four for molybdate in aqueous solution and in the transport protein ModG versus two for the enzyme. When the enzyme CO-dehydrogenase was initially characterized selenium was identified in various protein preparations. This led people to believe that in these preparations a

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FIGURE 7.4 Top panel, Mo K-edge XANES of (top to bottom) aqueous molybdate (blue), the Mo transport protein ModG (pink; Duhme et al., 1999), and oxidized CO-dehydrogenase (red; Gnida et al., 2003); inset, putative active site of CO-dehydrogenase. Bottom panel, experimental (black) and simulated EXAFS (left) and k3-weighted phase shift-corrected Fourier transform (right) at (top to bottom) Mo (red), Cu (blue) and Se (green) K-edge; metal
metal interactions highlighted with arrows.

dinuclear Mo
Se center catalyzes the dehydrogenation, possibly with the Se in the position of the question mark in the structure in the inset of Fig. 7.4. This was inconsistent with the Mo K-edge EXAFS analysis, however. The other panels of Fig. 7.4 highlight the element specificity of XAS and show experimental EXAFS spectra (middle panel) and their Fourier transforms (right panel), respectively, of the Mo, Cu, and Se edge of the oxidized form of the enzyme. As will be discussed in more detail below, the Fourier transforms give a radial distribution of atoms around the element at the edge of which the EXAFS was measured, and the phase relationship between Fourier transform and EXAFS, together with the characteristic backscattering in the latter, allow the ligand atom types to be identified. The analysis of the EXAFS of enzyme preparations that were fully catalytically competent revealed the presence of a dinuclear MoCu cluster, bridged by a sulfur ligand.

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7. X-ray absorption and emission spectroscopy in biology

In the final model, Mo is directly surrounded by S and O ligands, the Cu by S ligands. ˚, The Fourier transform of the Mo and Cu EXAFS both show a small peak just below 4 A which represents the distance between Mo and Cu, which are connected by a bridging ligand. The Fourier transform of the Se EXAFS reveals the presence of the C atoms of the amino acid methionine, but no metal contribution. Thus Se is not a constituent of the active site.

X-ray absorption (near-)edge structure

OxyHc 1.0

Deoxy Hc

C Cu

Linear

T-shaped

L

L

z

Cu

Cu

x

O2 Cu

0.0

O Cu O

L

4s

9000

Energy (eV)

Cu

z

L

L

L

y

y

Cu

x

9030

z y

L

Cu

x

L

x

L

pz

4px,y,x

3d 8970

L

Tetrahedral

L

Continuum

1s

Trigonal

z

y

Energy

Normalized X-ray -ray fluorescence

The near-edge region of the K-edge XAS of Cu in the oxygen-binding protein hemocyanin of the arthropod Panulirus interruptus in its deoxygenated and oxygenated forms is shown in Fig. 7.5. Upon oxygenation of hemocyanin, the oxygen molecule is bound to a pair of Cu11 ions, and both Cu ions transfer an electron to the molecular oxygen, so that they become Cu21, resulting in a shift of the edge to higher energy. Intuitively one would expect on the basis of electrostatic arguments alone that it should be easier to liberate an electron from a metal in a low oxidation state (in this case Cu11) than in a high oxidation state (Cu21). The position of the edge, taken as the excitation energy corresponding to 50% of the maximum edge intensity, depends on (1) valence state, so that the edge appears at higher energy for a higher valence, but also on (2) ligand type, and (3) the average distance to the nearest ligand atoms R, so that it appears at higher energy when R is shorter (“Natoli’s rule”). Valence state and average R play a role in the small (a few eV) edge shift observed in the direct comparison of the deoxygenated and oxygenated forms of the oxygen-binding protein hemocyanin shown in Fig. 7.5. In cases where the energy of the X-rays falls in the range 8980
8995 eV, where it is just not enough to liberate the 1s electron of Cu into the so-called continuum, it is probably still enough to excite the electron into one of the unoccupied higher orbitals of the absorber atom; the approximate energies of these transitions are schematically indicated by the arrows in the bottom of Fig. 7.5, left. Such electronic transitions are governed by the

L

pz py,z py

px,y

px

px,y,z

px

FIGURE 7.5 (Left, top) X-ray absorption near-edge spectra (XANES) of the Cu K-edge of oxygenated (blue) and deoxygenated (green) hemocyanin from the spiny lobster P. interruptus; (left, bottom) approximate energies of excitations of the Cu 1s electron to nonoccupied orbitals and to the continuum, with a color code (red, formally forbidden; green, at least dipole- or spin-allowed) indicating the probability; (inset) oxygenation in the hemocyanin active site. (Right) Effect of ligand geometry on the relative energies of the 4p orbitals of the Cu11 ion.

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same selection rules as those that apply to the UV
vis spectra of coordination compounds, and the possibilities for Cu are schematically indicated in Fig. 7.5, left, with a color code indicating the expected intensity. One rule is that they should be spin-allowed, that is, they should not be accompanied by a change in the total spin S (ΔS 5 0), or, in other words, the spin multiplicity should be maintained. This means that some mechanism of relaxation of this rule must exist to allow transitions from a filled 1s orbital to an empty higher orbital, as it is spin-forbidden and its intensity is therefore at best relatively weak. Another rule, the Laporte rule, requires that the transitions should be dipole-allowed, by a change in symmetry between initial and final state, which is equivalent to the necessity of a change in secondary quantum number; s-p and p-d transitions are allowed (this is the most important reason that K- and L3-edges look different, as they involve excitation of 1s and 2p electrons, respectively, see Box 7.2), but s-d transitions are not. Like for the d-d transitions in the UV
vis range of the electromagnetic spectrum, the transitions in systems with tetrahedral ligand geometry are more intense than for octahedral, as the rules are more likely to be relaxed in a system of lower symmetry. Cu21 ions have a vacancy in their 3d shell (d9 system) which makes a 1s-3d transition possible at 8979 eV (Kau et al., 1987); this is spin-allowed but not dipole-allowed, and therefore so weak that it is not visible in Fig. 7.5. For some other transition metal ions, for example, the d6 and d5 systems of Fe21 and Fe31, respectively, there are various vacancies at different energies in the set of d orbitals, and the pattern of transitions is found to be sensitive to the spin state (high or low) of the ion (Westre et al., 1997). The energies and probabilities of the edge transitions are also influenced by the degree of overlap between the orbitals of metals and ligands, or in other words by the covalency of the bond between metal and ligand; this covalency can also be probed by measuring and interpreting the ligand XANES (see Box 7.2). For Cu21 complexes, the K-edge is at higher energy when the ligands are more electronegative, viz., increasing from approximately 8985
8988 eV going from S via Cl and N to O ligands. The intuitive explanation is analogous to that of the effect of a higher oxidation state, viz., that a more electronegative ligand would cause a

BOX 7.2

Ligand-edge and L-edge X-ray absorption spectroscopy Ligand-edge. An alternative for probing the covalency of metal
ligand bonds in proteins by metal K- or L-edge studies is to measure the ligand K-edge XANES, for example, for sulfur. The predominant transition in S K-edge XANES is 1s-4p, but the excitation of the S 1s to the empty metal 3d level can be stimulated by mixing of the latter with the S 3p level (Fig. 7.6). Thus the intensity of this preedge transition in the S

K-edge XANES is proportional to the degree of mixing, and thereby to the covalency. In this way the covalency of metal-S bonds in a number of electron transfer proteins have been probed, viz., for Cu
S bonds in the so-called blue copper proteins (Shadle et al., 1993), and for Fe
S in the [4Fe
4S] clusters in high-potential iron
sulfur protein and ferredoxins (Dey et al., 2007).

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BOX 7.2

(cont’d) FIGURE 7.6 Illustration of probing mixed 3p orbitals of Cl or S ligands with a transition metal’s 3d orbital, in this case the singly occupied 3dx22y2 orbital of Cu21 (other Cu 3d orbitals grouped together for clarity), by either K-edge or L3-edge XAS.

L-edge. As mentioned in the text, the Xray
induced transitions involving the 3d (or 4d, 5d) valence orbitals of transition metals can give information on the energies of these orbitals and thereby about the spin state, for example, of Fe21 and Fe31 (Westre et al., 1997). On the one hand it is easier to probe the energy levels of the d orbitals, which give information on the metal ion’s spin state, from the transition metal’s L3-edge, since the 2p-3d transition induced at this edge is dipole-allowed, whereas the 1s-3d probed at the K-edge is not; moreover, the L-edge features are also sharper, and therefore better resolved, than the K-edge features. From an experimental viewpoint the measurement at the softer Ledge (see Fig. 7.1 for typical excitation energies) is more demanding, since all parts of

the spectrometer which can be in air in the instrument depicted in Fig. 7.2 (Box 7.1) such as space between ion chambers, sample holder, and fluorescence detector, have to be in vacuum to avoid loss of intensity due to the X-ray absorption of the atmospheric gases; moreover, the fluorescence yield at the L-edge is lower (Fig. 7.3, right). L-edge spectroscopy has been applied to a number of Fe and Ni proteins (Cramer et al., 1997). The higher resolution of the L-edge features compared to the K-edge makes the measurement with circularly polarized X-rays in the presence of a magnetic field (X-ray magnetic circular dichroism) very powerful, as shown in Fe L-edge studies of model compounds of nitrogenase cofactors (Kowalska et al., 2017).

decrease in electron density on the absorber atom, and X-rays of higher energy would be required to liberate an electron. An analogous covalency shift is observed for octahedral Ni21 complexes with varying amounts of S and N ligands (Colpas et al., 1991). Cu11 ions have a filled 3d shell (d10 system), but because of the lower coordination numbers they feature coordination geometries in which some or most of the 4p orbitals

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have no interaction with the ligands. For example, Cu11 with linear two-coordinate geometry has two ligands along the z-axis, which raises the energy of the 4pz orbital above that of 4px,y; this allows for “pure” 1s-4px,y transitions, which are spin-forbidden but dipole-allowed, at relatively low energy (8983
8984 eV). The effects of coordination geometry for Cu11 are illustrated in the right part of Fig. 7.5, together with ligand-field orbital descriptions derived from the analysis of relevant model compounds (Kau et al., 1987). Another example of a pure transition is the 1s-4pz transition in square planar Ni21 (d8 system with ligands in the xy plane; Colpas et al., 1991). In summary, the XANES of hemocyanin indicates that Cu11 ions with a low (3) coordination number (various 1s-4p transitions observed) with low symmetry (no strong 1s-4px,y absorption indicative of linear geometry) are oxidized (edge shift) to a Cu21 ions with some additional electronegative (O) ligands, consistent with the change in the chemical diagram in Fig. 7.5, left (inset). We conclude that the XANES is very sensitive to biologically relevant changes in the metal environment; it can also be used to probe the stability of the metal ion toward nonbiological processes like photoreduction and/or or radiation damage in other studies using X-rays, such as X-ray crystallography. Compared to EXAFS, which gives much more accurate information on ligand distances, it is much easier to record a XANES spectrum with high signal-to-noise ratio on a relatively small sample in a relatively short time. It is the only part of the spectrum that one can expect to measure satisfactorily in many time-resolved studies, and is much more suited than EXAFS for spatially resolved (imaging) studies where one aims at getting some more information (oxidation state, symmetry of coordination) on the trace element than just its distribution over the image. As will be discussed later, it has recently become possible to acquire XANES with high-energy resolution using an X-ray emission spectroscopic approach.

X-ray emission spectroscopy in biology As discussed in Box 7.1, fluorescence is the preferred mode of acquiring X-ray absorption spectra for systems as dilute as trace elements in biological systems. The solid-state detectors used for such systems can be set to discriminate fluorescence from scattered X-rays, but they record the total fluorescence. As will be discussed below, analysis of the energies and intensities of the fluorescence, the X-ray emission spectrum, can yield interesting information that is overlapping and complementary with XAS, and provides an improved means of recording XANES. The experimental approach to such a system is described in Box 7.3. In the X-ray emission spectrum of a first row transition metal (such as the Mn21 represented in Fig. 7.9), the Kα1 and Kα2 lines (Fig. 7.3, left, and Fig. 7.9, left) are well resolved and more intense, by an order of magnitude, than the Kβ1 and Kβ3 lines, which are not resolved; the Kβ1,3 line is in turn more intense than the Kβ satellite lines Kβ2,5 and Kβv. Not indicated in the scheme of Fig. 7.9 (right), but usually present for transition metals which have a total electron spin S6¼0 (such as Mn21, Fig. 7.9, left), is the Kβ0 line at slightly lower energy than the Kβ1,3 line. This results from emission from the metal 3p level combined with a spin flip of a 3d electron and is therefore sensitive to the spin state of the metal ion.

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BOX 7.3

The X-ray emission spectroscopy (XES) experiment As an example of a photon-in
photon experiment with incident energy ωin and emitted energy ωout, the analysis of the 2p to 1s emission stimulated by 1s-3d absorption of a 3d transition metal is given in Fig. 7.7, left. The accurate measurement of ωout in X-ray emission spectra requires element-specific analyzer crystals that have a high-energy resolution, close to the natural linewidth. In the Johann approach, this is achieved by point-to-point scanning using a crystal bent with radius 2R, which is placed with the sample and detector on a Rowland circle with radius R to ensure that radiation from the sample that is selected by the crystal hits the detector (Fig. 7.7, right). In a wavelength dispersive arrangement, X-rays emitted by the sample are diffracted

and the photons of different energies detected separately with a position-sensitive detector. When the experiment is set up such that the energy of the incident radiation is close to that of the absorption, the experiment is called resonant. The result of a resonant inelastic X-ray scattering (RIXS, Fig. 7.8) experiment is usually given as a twodimensional plot of the excitation energy ωin in the K-edge energy range in one dimension, and the energy transfer (ωin
ωout) in the other (Glatzel and Bergmann, 2005). This experiment gives considerably more detail of the transitions in the edge region than the XANES mentioned above, even when this is recorded as total fluorescence.

FIGURE 7.7 (Left) Principle of a hard X-ray photon-in/photon-out experiment. (Right) Experimental setup at a storage ring beamline, with the sample, analyzer crystals, and detector in Rowland circle geometry. The arrows indicate the motion of the components when a spectrum of the emitted X-rays is taken. Source: Adapted from Glatzel, P., Sikora, M., Ferna´ndez-Garcı´a, M., 2009. Eur. Phys. J. Special Top. 169, 207
214.

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X-ray emission spectroscopy in biology

BOX 7.3

241

(cont’d)

FIGURE 7.8 1s-2p RIXS spectra for Mn
O (Glatzel et al., 2004). (C) RIXS plane, with ΓK and ΓL the lifetime broadenings for respectively the intermediate and final states; (B) diagonal of (C) (constant emission energy): HERFD-XANES; (A) horizontal cut (constant energy transfer); (D) vertical cut (constant incident energy). The final state of the combined excitation and emission is identical to that of L-edge excitation (Fig. 7.7, left), but it has the experimental advantage of being probed by hard X-rays at the K-edge rather than the soft L-edge, and giving additional information. When the RIXS plane (Fig. 7.8C) is

scanned along the constant emission energy diagonal, the spectrum obtained (Fig. 7.8B) is equivalent to the XANES spectrum, but because the XES spectrometer has much higher energy resolution, the spectral features are much better resolved. XAS or XANES recorded in this way is therefore

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7. X-ray absorption and emission spectroscopy in biology

BOX 7.3

Kβ satellite Lines

Kβ main lines

Kβ1,3

Kβ’’

(x 500)

Kβ’

L2,3 2p L 1 2s

M 4,5 3d M2,3 3p

Kβ 2,5

M1 3s

Kα1 Kα2

good penetration depth) even though the K-edges of these elements are at soft X-ray energies (Table 7.1). The theory to support the interpretation of XES involves calculations of the “ligand-field multiplets” to interpret Kα and Kβ main lines, and molecular orbital theory for the Kβ satellites (Glatzel and Bergmann, 2005); the treatment of these theories is outside the scope of this chapter.

continuum

Kβ2,5

Energy

Intensity (arbitrary units)

referred to as partial fluorescence yield (PFY) or high-energy resolution fluorescence detected (HERFD). The same experimental setup can also be used for nonresonant experiments, such as X-ray Raman spectroscopy. This hard photon-in, hard photon-out experiment can be used to probe XAS of low-Z (C, N, O) elements with all the advantages of hard X-rays (no vacuum required,

(cont’d)

Kβ''

L 1 2s

Kα1

K edge excit.

(x 8) 5900

5920

6480

6520

Fluorescence energy (eV)

6560

4–

Mn2+

Mn2+

L3 edge excit.

Kα 1 fluor.

Kα2

K1 1s

5880

K1 1s

Kβ 1,3 L 3 2p 3/2 L 2 2p 1/2

3–

C ,N , 2– – O ,F

Mn2+

Mn2+

FIGURE 7.9 (Left panel) K shell emission lines in Mn
O, with Kβ main lines (magnification 8 3 ) and Kβ satellite lines (500 3 ). (Right panel) Effect of “low-Z” ligands (C, N, O, F) on the X-ray emission of a Mn21 complex. K-edge excitation (blue) leads to a 1s core hole intermediate state (central green oval), which can emit X-ray fluorescence at various wavelengths. The final state obtained with Kα1 fluorescence (red box) is identical to that obtained by direct L3-edge excitation (red). Source: Left panel, adapted from Glatzel, P., Bergmann, U., 2005. Coord. Chem. Rev. 249, 65
95, with permission.

Of the Kβ satellite lines, the crossover emission line Kβv is extremely sensitive to the nature of the coordinating ligands, because it involves emission from the ligand’s 2s level to the metal’s 1s core hole, and is therefore called the valence-to-core (“VtC” or “V2C”) transition. Because the 2s level of the ligand donor atom is characteristic of the ligand donor atom element, the energy of this line allows one to distinguish O from N and C ligands. This ligand identification is of interest as it gives information complimentary to that obtained from EXAFS, which (as discussed in the “Phase shift and effect of atom type” section) can typically not discriminate between coordination by ligands from the same row of the Periodic Table. In the case of the Mn4CaO5 cluster of the oxygen-evolving center in Photosystem II, the Kβv or V2C line has been used to probe the variation in the number of O ligands to Mn in the cycle that the cluster passes through in its light-catalyzed evolution of O2 from H2O, the so-called Kok cycle in Photosystem II (Pushkar et al., 2010).

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FIGURE 7.10 Structures of clusters in metalloproteins discussed in this chapter; charges not explicitly given. (A) Mn4CaO5 cluster in the oxygen-evolving center in photosynthetic system II. (B) MoFe7S9C cluster in the nitrogen binding site of nitrogenase (Einsle et al., 2002; Lancaster et al., 2011), with the tetraanionic nonproteinaceous coordinating cofactor homocitrate in gray. Source: (A) Adapted from Umena, Y., Kawakami, K., Shen, J.-R., Kamiya, N., 2011. Nature 473, 55
60.

This Mn4CaO5 cluster (Fig. 7.10) has been studied by a number of X-ray diffraction and spectroscopic techniques (Yano and Yachandra, 2014). EXAFS studies at the Mn and Sr K-edges of the cluster which had been modified by replacing the Ca ion, whose X-ray edges are in the soft X-ray region, by the element directly below it in the Periodic Table, strontium (Sr, Table 7.1), gave Mn
O, Mn
Mn, and Mn
Ca/Sr distances, which all appeared to be slightly shorter than those found in the crystallographic structure determination (Umena et al., 2011). A tentative explanation for the difference was that although both X-ray techniques used synchrotron radiation, the exposure in the case of the diffraction study was so much more intense than in the spectroscopic study that radiation damage had occurred. The exact position and intensity of the Kβ1,3 emission line in Mn K-edge XES spectra has turned out to be a reliable marker not only of the various states that the Mn4CaO5 cluster passes through in its light-catalyzed evolution of O2 from H2O, but also of the extent of radiation damage, in particular photoreduction (of Mn31 and Mn41 to Mn21). In a combined X-ray diffraction/spectroscopic experiment using the intense femtosecond pulses of an XFEL this feature of the XES was used to establish that the crystals of the photosystem did not suffer radiation damage in the brief exposure to XFEL X-rays needed to obtain a diffraction pattern (Kern et al., 2013). The Kβv or VtC emission line observed upon excitation at the Fe K-edges has also provided decisive evidence that the low-Z central atom in the iron
molybdenum and iron
vanadium cofactors (FeMoco, Fig. 7.10B, resp. FeVco) in nitrogenase, whose identity could not be unambiguously established by X-ray crystallography (Einsle et al., 2002), is not N, but C (Lancaster et al., 2011; Rees et al., 2015). By recording the Mo K-edge XANES in the PFY or HERFD mode (Box 7.3), it was possible to improve the energy resolution to 3.5 eV compared to 6
8 eV in the conventional Mo XAS experiment (Bjornsson et al., 2014). On the basis of comparison to model compounds and time-dependent density functional theory (TD-DFT) it was possible to conclude that the oxidation state of Mo in FeMoco is 3 1 .

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7. X-ray absorption and emission spectroscopy in biology

Time-resolved X-ray absorption spectroscopy XAS is highly sensitive to small changes in the vicinity of the absorber atom. Thus XANES and to a smaller extent EXAFS are perfectly suited to monitor structural changes during reactions. Modern X-ray sources such as synchrotrons produce pulses of some picoseconds. Free electron lasers that recently became available even produce pulses of some 10 fs. Both sources can be coupled to an excitation laser allowing for a pump-probe setup (Fig. 7.11). These measurements provide insights into dynamics and structural changes of optically triggered chemical reactions (Chergui, 2016). Pump-probe XAS studies target systems such as Photosystem II or copper charge transfer complexes and can be based on different detection mechanism, such as XES (Kern et al., 2013), L-edge XAS (Kubin et al., 2017), or K-edge XAS (Naumova et al., 2018). Recently, nitric oxide (NO)-heme recombination kinetics has been probed by timeresolved iron K-edge XAS. By 532-nm photoexcitation NO was released in order to study the recombination after photolysis (Silatani et al., 2015): upon photolysis deoxy Mb with domed heme geometry was formed. This was maintained in about one-fifth of the molecules, where NO escaped to the solvent. In all other cases NO rebonded to the iron heme complex after less than 200 ps. Interestingly, upon recombination the domed heme geometry initially has been conserved. After another 30 ps the planar heme geometry of nitrosylmyoglobin (MbNO) has been reestablished. This method demonstrates the flexibility of molecules and provides detailed insights into individual steps of reaction mechanisms. Thus time-resolved XAS opens a new window into the understanding of catalysis, spin-transitions, and other dynamic processes in biological as well as in biomimetic systems. The experimental setup is similar to the one given in Fig. 7.2. An additional laser has to be aligned to the sample and both laser and X-ray source have to be electronically coupled in such a way that the time difference between laser pulse (pump) and X-ray pulse (probe) is aligned to the dynamics of the process under study (Fig. 7.11). In contrast to the simplified setup in Fig. 7.11 often liquid samples are studied. Then, a high-speed liquid jet allows to continuously renew the sample and thereby to overcome photoreduction by the intense synchrotron or free electron laser beam (Milne et al., 2014).

FIGURE 7.11 Major components of a pump-probe setup used for measuring recovery dynamics. Synchronized X-ray pulses generated by either a synchrotron radiation source or an free electron laser (red) and optical LASER pulses (blue) allow for a defined recovery period after which the sample is analyzed. The time difference can be adjusted by increasing or decreasing the optical LASER pulse pathway.

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X-ray absorption spectroscopy: X-ray
induced electron diffraction

245

With the same approach intermediate states are identified and structurally characterized. A drawback of such studies is the high dynamic disorder at room temperatures, that further limits the sphere around the absorber atom structurally elucidated in the experiment compared to measurements at cryogenic temperatures (see below).

X-ray absorption spectroscopy: X-ray
induced electron diffraction The EXAFS extracted as χ(E) from the normalized X-ray absorption spectrum (Box 7.4 and Fig. 7.12) is usually presented as χ(k) with the energy axis converted from ˚ 21), using energy (E, eV) to wave vector (k, A rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2me ðE 2 E0 Þ k5 ð7:4Þ ħ2 in which me is the mass of the electron, E0 is the threshold energy (7120 eV for the Fe example in Box 7.4), and ħ is h/2π. This choice of x-axis has the advantage of showing the EXAFS oscillations as sinusoids. To offset the damping of the oscillations, the fine structure is also k3-weighted, so that the oscillations now have more or less constant amplitude over the k-range shown. In Fig. 7.13 various situations above the edge, where the X-ray energy is more than enough to liberate the electron to the continuum, and the excess energy goes into the kinetic energy of the photoelectron wave, are illustrated for an Fe ion surrounded by four nitrogen ligands (as for Fe in hemin, the example of Box 7.4). The wavelength of the photoelectron wave depends on the kinetic energy, that is, on the difference between the energy supplied by the X-rays and that required to liberate the electron. In Fig. 7.13, left, we have a situation where an integer number of wavelengths exactly fits the Fe
N distance; this means that the electron wave going out from the absorber atom and that backscattered by the backscatterer atom are exactly in phase, as indicated by the lines depicting the wave fronts. As a result, the electron waves interfere constructively at the absorber atom, resulting in a high electron density and hence a high absorption coefficient for X-rays. In Fig. 7.13, right, we have a situation with higher X-ray energy, with a higher kinetic energy and a larger electron wavelength as a result. Now the wavelength does not fit the Fe
N distance so well, and the interference of the outgoing and backscattered photoelectron waves at the central atom is destructive, resulting in a relatively low electron density, and a low absorption coefficient for the X-rays. It can be seen that when the X-ray energy is increased, going from the situation on the left in Fig. 7.13 to that at the right, the X-ray absorption coefficient goes from a minimum to a maximum. Indeed when the energy is scanned over a longer range, we will go through various situations where the electron wavelength varies such that the X-ray absorption coefficient goes through minima and maxima. This results in the extended X-ray absorption fine structure (EXAFS) which can now be seen to depend on the distance of the absorber atom to the closest backscatterers. This observed interference between the electron waves is in fact a very convincing demonstration of the proposed (de Broglie) wave character of electrons, in addition to the diffraction patterns that can be observed in an electron microscope. So although X-rays are

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7. X-ray absorption and emission spectroscopy in biology

BOX 7.4

Data reduction The initial data reduction, including checks of the individual detector contributions to the individual scans and the alignment of the scans by energy calibration, involves some computation which can in principle be automated. In order for the EXAFS to be interpreted by simulation, it

has to be extracted as χ by the procedure shown for the K-edge of iron in aqueous hemin chloride (containing the protoporphyrin IX ligand, the Fe-coordinating cofactor present in heme proteins and enzymes) at around 7120 eV in Fig. 7.12. The experiment has been set up in such a way that a

0.7 60

Fe

χ(k)

Fluorescence

0.5

μ0

N

C C C C

0.0

μ /FT/

–0.5

μ′

10

χ(k) * k3

0.0

χ(E)

0.4

0

0 0.0

–0.4 6920

–40

–10 7220

7520

Energy (eV)

7820

2

5

8

11

0

k (Å–1)

2

4 R (Å)

6

8

FIGURE 7.12 Data reduction for aqueous hemin chloride. Left panel, top: iron K-edge X-ray absorption spectrum (μf which is for thick samples independent of its thickness x) with μ0 (blue), extrapolated preedge, representing the theoretical spectrum of the sample without the Fe absorber; μ0 (red), polynomial resulting from a smoothened fine structure, representing the theoretical X-ray absorption of an Fe atom without neighboring atoms. Left panel, bottom: extracted fine structure (χ), with the green zero line corresponding to μ0 (red) in the top panel. Middle panel, extracted fine structure χ in k space, without (top) and ˚ from with k3-weighting (bottom); the blue line corresponds to a simulation representing 4 N ligands at 2 A Fe. Right panel, phase shift-corrected Fourier transform of the k3-weighted χ; inset: structure of coordinated pyrrole moiety; black and purple lines represent the modulus and the imaginary parts, respectively.

Practical Approaches to Biological Inorganic Chemistry

X-ray absorption spectroscopy: X-ray
induced electron diffraction

BOX 7.4 sufficiently long energy range in the preedge region is recorded so that it can be extrapolated to give the background absorption of the other elements in the absence of the central absorber atom Fe, μ0 (blue line in Fig. 7.12, top left). Subtraction of this background μ0 from the experimental μ gives the background-corrected spectrum μA. The next step is to construct the hypothetical X-ray absorption spectrum of Fe in the absence of surrounding atoms (atomic absorption, XAS of Fe as if it were a monoatomic noble gas), μ0. This is done by instructing the computer program for background subtraction to find a polynomial through the fine structure (red line in Fig. 7.12), which is more or less an extreme smoothing. The fine structure χ is then calculated relative to this polynomial as μ
μ0, using the value of 7120 eV (Fe K-edge) for the threshold energy (E0), and normalized relative to the edge step (μ0
μ0 ). This yields the fine structure χ(E) as oscillations around the zero level represented by the green line in the bottom left panel of Fig. 7.12, which is equivalent to the red line in the top left panel. As shown in the middle panel of Fig. 7.12, the EXAFS is usually presented as χ(k) with the energy axis converted from energy (E, eV) to wave vector (k, ˚ 21), using Eq. (7.4) in the text, which has A the advantage of showing the EXAFS oscillations as sinusoids (Fig. 7.12, middle, top). To offset the damping of the oscillations, the fine structure is also k3weighted, so that the oscillations now

247

(cont’d) have more or less constant amplitude over the k-range shown (Fig. 7.12, middle, bottom). With this weighting the experimental noise in the high-energy range is also amplified, and it is advisable to set up the experiment in such a way that some extra time is spent on recording this part of the spectrum, and to choose the energy distance between the points such that they are equidistant in k space after the conversion. In Fig. 7.12, right, we have not only plotted the real part or modulus of the phase shift-corrected Fourier transform of the EXAFS, as is customary in most literature, but also the imaginary part, which contains information on the phase relation between EXAFS and FT. Good agreement between experiment and simulation in both the real and imaginary parts of the Fourier transform inevitably implies good agreement with the EXAFS. When we compare the positions of the peaks in this radial distribution function to the known structure of hemin (see inset) we note that no contribu˚ tions of atoms in the structure beyond 4.3 A from the central Fe atom are observed. Indeed it is exceptional to detect a contribution beyond the first shell of atoms at such a distance as observed for the specific case of the pyrrole unit here; the factors that determine whether a long-range contribution will be detected or not are the electron mean free path, the presence of a rigid chemical system, and the Debye
Waller factors, which will be discussed in more detail in Box 7.6.

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7. X-ray absorption and emission spectroscopy in biology

FIGURE 7.13 Examples of interference patterns of electron waves between absorber and backscatterer atoms, leading to (left) constructive interference and maximum electron density at the absorber, and a maximum in the X-ray absorption coefficient μ and the EXAFS χ, or (right) destructive interference and minimum electron density at the absorber, and a minimum in μ and χ.

used in XAS, and it is clearly a spectroscopy that yields structural information that is comparable to or complementary with X-ray crystallography, it does not involve X-ray diffraction, but X-ray
induced electron diffraction. The distance between absorber and backscatterer determines the frequency of the EXAFS oscillation; if the neighbor atom is close to the absorber (trace element), the frequency is low, and if the neighbor atom is remote, the frequency is high. The mathematical tool of choice to analyze oscillations is the so-called Fourier transformation; applied to the EXAFS with its energy axis in the wave vector dimension (reciprocal length unit), it gives a radial distribution function with maxima at the distances from the central atoms where shells of atoms occur (Sayers et al., 1971). For a system with two shells of atoms at close and remote distances we observe a fine structure that is an interference pattern of oscillations with low and high frequency, respectively. These appear in the Fourier transform at their respective distances, provided that the phase shift, which will be discussed next, is taken into account.

Phase shifts and effect of atom type What we have not mentioned so far is that the photoelectron wave undergoes a phase shift while traveling through the atomic potentials of absorber and backscatterer; there is a phase shift when the wave leaves the absorber atom, another one when it goes back and forth through the potential of the backscatterer atom, and then yet another one when it returns to the absorber atom. Such phase shifts occur as well in refraction, when photons enter a medium of different optical density. In EXAFS this causes the nonphase shiftcorrected R0 (also indicated as R 1 φ) found in the radial distribution function immediately after Fourier transformation to be an underestimation of the true (phase shift-corrected) R ˚ depending on the type of backscatterer. It follows that for the EXAFS to be by 0.2
0.5 A interpreted in terms of absorber
backscatterer distances, knowledge of the phase shifts of the absorber
backscatterer pair is required, either from model compounds or from calculations, as discussed in Box 7.5.

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249

BOX 7.5

Phase shift calculations As mentioned before, knowledge of the phase shifts of the absorber
backscatterer pair is required for the EXAFS to be interpreted in terms of absorber
backscatterer distances. The same is true for the backscatterer amplitudes with respect to number of atoms (coordination number, occupancy), and the variation of the backscatterer amplitude with k to identify the atom type. In early studies EXAFS was simulated using phase shifts and backscattering amplitudes that were extracted from model compounds, which were shown to be transferable to unknown systems. Nowadays, theoretical calculations such as by the programs EXCURVE (Gurman et al., 1984, 1986), FEFF (Rehr et al., 1991), and GNXAS (Filipponi et al., 1995) are accurate and accessible enough for practical use, but of course it is still good practice to validate the results on model compounds of known

structure. Such calculations of phase shifts and backscattering amplitudes, usually collectively known as “phase shift calculations,” require knowledge of the potential at every place in the system of absorber and backscatterers; this is approximated by considering the individual atoms and the surrounding cloud of electrons as potential wells in an area of constant interstitial potential, in the so-called muffin-tin approximation (Fig. 7.14). The advantage of a constant interstitial potential is that in simulations the distance between atoms can be varied in order to find one which optimally reproduces the observed EXAFS, with no need to repeat the phase shift calculation for every distance. At the current level of theory, it is not necessary any more to empirically adjust or refine the amplitude reduction factor Sj nor the electron mean free path λj (see Eq. 7.6).

FIGURE 7.14 “Muffin-tin” illustrating the type of potential approximation used for phase shift calculations; VCu, etc., individual atom potentials; Vint, interstitial potential.

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7. X-ray absorption and emission spectroscopy in biology

The type of backscatterer has an effect on its phase shift, because this is directly affected by the shape of its potential. In addition, there is an effect on the intensity of the backscattered photoelectron wave. This intensity depends on the photoelectron energy and thus the X-ray energy. The envelope (variation of the backscattering amplitude, independent of the oscillatory variation due to the distance to the absorber) describing this dependency varies with the type of backscatterer, as is illustrated in Fig. 7.15, left panel. First of all, it is obvious that the contribution of H to the backscattering is very weak compared to that of other backscatterers for most of the k-range, so it is usually neglected in EXAFS simulations. The similarity of the envelope for elements that are in the same row of the Periodic Table, such as the examples of C, N, O, and F, or S and Cl, in Fig. 7.15, makes it difficult to discriminate between them; the light biological backscatters of the first row in the Periodic Table, C, N, and O, are therefore often collectively referred to as “low-Z” ligands. The reason is that the backscattering amplitudes and the phase shifts are comparable, due to the similarity in the sizes of the nuclei and the surrounding electron clouds, respectively. It is found that the backscattering amplitude envelope is similar for C, N, O, and F (a decay with increasing k for most of the k-range), and that the differences in backscattering amplitude and phase shift are too subtle to be of diagnostic value in routine cases.

100

Backscattering amplitude envelope

W

I

EXAFS * k3

10

0

Mo –10 2

3

4

5

6

7

8

9

Br

10 11 12 13 14 15 16

k (Å–1) 100

I

Zn F O N C

Cr Cl

/FT/

Br Cl

S

F H

0

0 2

3

4

5

6

7

8

9 10 11 12 13 14 15 16

0

1

2

k (Å–1)

FIGURE 7.15

3

4

5

6

R (Å)

Effect of atom type; left, backscattering amplitudes of selected atoms; right, EXAFS and phase shift-corrected Fourier transforms of elements from the same (second) row and column (halogens) of the Periodic ˚ from Zn. Table at 2.0 A

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251

The envelopes in Fig. 7.15 (left) also indicate that it should be possible to discriminate between atoms from different rows of the Periodic Table, for example, for the halogens (F, Cl, Br, and I) which are elements from the same column (see Fig. 7.1). Fig. 7.15 (right) shows the EXAFS (top) and phase shift-corrected Fourier transform (bottom) of ˚ (not realistic for all elements) from a Zn ion. The these elements when placed at 2.0 A EXAFS panel confirms that the maximum in the backscattering amplitude envelope at ˚ 21) k for F shifts to higher values for Cl (6), Br (10), and I (11
12 A ˚ 21), with low (4
5 A 21 ˚ an additional maximum at low k for I (5
6 A ). Interestingly, when the phase relationship between the EXAFS and the Fourier transform is inspected, it turns out that the EXAFS of F is approximately π out of phase (has opposite phase) with that of Cl, which in turn has opposite phase to that of Br, which is again out of phase with I. As a result, the contribution of F is approximately in phase with that of Br, and that of Cl with that of I. As will be discussed below, the phase relationship offers another possibility to identify contributions of elements from different columns, in addition to the inspection of the backscattering amplitude envelope, as will be discussed below for a biological example. Backscatterers with very high Z have multiple maxima in their backscattering envelope. This can be seen when going from Br to I in Fig. 7.15, left; the larger back˚ 21. scatterer, I, has actually a weaker backscattering amplitude in the k-range 7
9 A When the group 6 transition metals are compared, it can be observed that Cr has a ˚ 21, larger backscatterer amplitude than Mo and W in the k-ranges 5
8 and 4
10 A respectively. The backscatterer with the largest Z included in Fig. 7.15, W (Z 5 74) has the strongest backscattering amplitude only at high k; therefore substitution of Mo by W as the other metal X in mixed [FeXS2] and [Fe3XS4] clusters paradoxically leads to a loss of backscattering power in the Fe EXAFS when investigated over a short k-range (Antonio et al., 1985). Upon going from nitrogen to an element from the next row, sulfur, the amplitude of EXAFS and Fourier transform becomes bigger (as expected due to the larger nucleus and number of electrons), but just as in the example of fluorine and chlorine discussed above, the EXAFS is also π out of phase when placed at the same distance; in addition the maximum in the backscattering amplitude envelope shifts toward higher k. Fig. 7.16A illustrates that the combination of two simulated contributions of equal size but opposite ˚ ) from a Zn phase, for example, 2 N (blue) and 1 S (orange) at the same distance (2.0 A absorber, can interfere destructively and lead to a very weak total EXAFS signal (black). In reality this rarely happens, not only because of the subtle shift in maximum in the backscattering amplitude envelope, but also because due to the difference in ionic radius, N and S are never at exactly the same distance from, for example, Zn. This is illustrated in ˚ and 1 S at 2.3 A ˚ from Zn, which Fig. 7.16B for the simulated example of 2 N at 2.0 A predicts that the amplitude envelope of the total EXAFS is not as weak as in the case of the atoms at the same distance, and will go through minima and maxima over the experimental k-range. This is confirmed in Fig. 7.16C for a biological system, the coordination of Zn21 ion in a protein (HIV-2 integrase; Feiters et al., 2003) by 2 N(imidazole) ligands at ˚ and 2S at 2.3 A ˚. 2.0 A ˚ 21) in Fig. 7.16C includes some relatively noisy data The experimental k-range (2
16 A 3 at high k (also because of the k -weighting) but is long enough to resolve the N and S

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7. X-ray absorption and emission spectroscopy in biology

EXAFS * k3

8

(A)

(B)

(C)

0 –8 2

4

6

/FT/

60

8 10 12 14 16 2 4 k (Å–1)

0

6

8 10 12 14 16 2 4 k (Å–1)

2N+1S at 2.0 Å 1

2

3 R (Å)

4

8

2N 1 S at 2.3 Å

–60 0

6

5 0

1

2

3 R (Å)

4

10 12 14 16 k (Å–1)

2 Im at 2.0 Å 2 S at 2.3 Å

5 0

1

2

3 R (Å)

4

5

FIGURE 7.16 Dependence of phase shift on atom type. Top panels, k3-weighted EXAFS taken at the Zn Kedge; bottom panels, phase shift-corrected Fourier transform of (A) sum (black) of 2 N atoms (blue) and 1 S atom ˚ , (B) sum (black) of 2 N atoms (blue) at 2.0 A ˚ and 1 S atom (orange) at 2.3 A ˚ , (C) Zn K-edge experi(orange) at 2.0 A mental EXAFS of HIV-2 integrase (Feiters et al., 2003) (black) with theory (pink) calculated for two imidazole ˚ (blue) and two sulfur ligands at 2.3 A ˚ (orange). ligands at 2.0 A

contributions to the major shell. Compared to other techniques where Fourier transformation is used, such as pulse nuclear magnetic resonance (NMR) and cyclotron mass spectrometry, the range over which the Fourier transform is taken is relatively small in EXAFS, because the experimental k-range is usually limited to a few (,10) oscillations. This means that the resolution, ΔR, of EXAFS is rather limited, according to its dependence on the k-range: ˚ ΔR 5 π=2 Δk ðAÞ

ð7:5Þ

˚ 21, the distance between two shells that can be which means that for a k-range, Δk, of 14 A ˚ . So the paradoxical situation exists that the resolved ΔR is equal to π/2 3 14 5 0.11 A ˚ is limited by the fact power of EXAFS to measure distances with an accuracy of 60.02 A ˚ , that is, an order that in order to be resolved in EXAFS, two shells have to be .0.1
0.2 A of magnitude more, apart.

Plane wave and muffin-tin approximation As discussed before, the EXAFS can be conceived as the sum of oscillations that are resolved in the Fourier transform and represent a number of shells (j) of backscatterer atoms of a certain type at certain distances from the absorber. This is summarized in the expression for the EXAFS in the so-called plane wave approximation which is given below as Eq. (7.6), which gives us the opportunity to discuss a number of parameters and their effects on EXAFS simulations and the accuracy of the results obtained. In this approximation the curvature of the electron wave is neglected; its derivation from the accurate description of the process given by Fermi’s Golden Rule can be found

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253

Plane wave and muffin-tin approximation

elsewhere (Burge, 1993). It can be seen that the formula for the plane wave approximation for EXAFS: X 22k2 σ2 22rj =λ 2 χðkÞ 5 j Nj USi ðkÞUFj ðkÞU e |fflfflfflfflfflfflfflfflj fflUe ð7:6Þ j 1 φj ðkÞÞ=kr j {zfflfflfflfflfflfflfflfflffl} U sin ð2kr |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} damping

amplitude

oscillatory

contains an oscillatory part (the sine term) and an amplitude part, which in turn contains pure amplitude parts and damping factors. In the amplitude part, Fj is the backscattering amplitude of each of the Nj (coordination number, occupancy) backscattering atoms of type i in shell j. Each shell also has a damping part characterized by the Debye
Waller factor 2σ2j (discussed in Box 7.6) and an oscillatory part determined by the distance rj and the total phase shift φj (two times that of the absorber 1 once that of the backscatterer). Because of the definition of k, the choice of the threshold energy (Δ) E0 has its largest effect in the oscillatory part; this is a refinable parameter in the simulations as ΔE0. There remain in the amplitude part the amplitude reduction factor

BOX 7.6

The Debye
Waller factor As mentioned when introducing the formula for the resolution (Eq. (7.5)), the power of EXAFS to measure distances with ˚ is limited by the an accuracy of 60.02 A fact that in order to be resolved in EXAFS, ˚ , that is, two shells have to be .0.1
0.2 A an order of magnitude more, apart. In cases where shells are not resolved, the distance found in the simulation for the combination of unresolved shells will be an average distance. The increased disorder in a shell that is composed of nonresolved contributions (static disorder) is noted as a more rapid decline of the EXAFS amplitude at higher energy, and a broadening of the peak in the Fourier transform; in the refined simulation, this is reflected in a larger value for the Debye
Waller factor, which was introduced in the formula of the planar wave approximation (Eq. (7.6)) as 2σ2j . In crystallography a displacement factor is used to describe deviations of an atom’s position from its lattice point. The

Debye
Waller factor or 2σj2 used in EXAFS simulations is related to this crystallographic parameter, but it always relates to at least a pair of atoms, that is, an absorber and backscatterer, or an absorber and more backscatterers. The Debye
Waller factor describes effects of static and thermal disorder on the EXAFS spectrum. A high value for the Debye
Waller factor can be caused by a variance in the ligand distances (static disorder), as in the example of the unresolved shells discussed above. It can also be caused by disorder due to thermal effects, that is, oscillations in the absorber
backscatterer distance. Whether the origin of the disorder is static or thermal can be probed by temperature variation; upon lowering the temperature, the value that 2σj2 refines to in the simulation should go down in case of thermal disorder, because the oscillations that cause this disorder become weaker, whereas for static disorder it stays the same (Scherk et al., 2001).

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7. X-ray absorption and emission spectroscopy in biology

BOX 7.6 It should be noted that a high σj for a single absorber
backscatterer pair can be due to uncorrelated motion because of a weak chemical bond. As mentioned above, a high value for 2σj2 is reflected in a rapid decrease (damping) of the EXAFS signal in k space, and a broadening/decrease in amplitude for the peak in the Fourier transforms. The bond between an absorber and backscatterer can be so weak that its contribution to the EXAFS is virtually wiped out because of the high value of 2σj2. In fact the observation of shells beyond the first shell of ligand donor atoms, leading to the characteristic patterns for the hemin in Fig. 7.12 and for the imidazole ligand to Zn in the integrase in Fig. 7.16, is exceptional, and reserved for rigid systems like ligands with strong bonds (CO, CN) or rings (porphyrin, imidazole) only. The reasons why EXAFS dies out after one or a few shells can be summarized as follows: • The electron mean free path λ is limited. • Disorder of a static or thermal origin.

(cont’d) • Destructive interference of contributions of opposite phase. • A weak bond between absorber and backscatterer, or none at all. As the outcome of the EXAFS simulations, the (bio)chemist is interested in the type of atom (this determines the backscattering amplitude Fj; it must be chosen, and reconsidered if not adequate for the simulation) and its number Nj and distance rj (which are set at reasonable starting values, and then iteratively refined in simulation). Unfortunately, refinement of (and correlation with!) physical parameters which are chemically less interesting, such as the Debye
Waller factor and ΔE0, cannot be avoided. Please note that the coordination number N and the exponential Debye
Waller factor are both amplitude factors. They are correlated in the analysis and the degree of correlation depends on the length of the energy range, because only the exponential function including the Debye
Waller factor depends on the wave vector.

Si, which corrects for X-ray absorption processes not contributing to the EXAFS, such as multiple excitation effects, and in the additional damping factor the electron mean free path λ (see Box 7.5).

Multiple scattering in biological systems It is of interest to look in a little more detail at the relation between the EXAFS and its Fourier transform. When the Fourier transforms of a number of imidazole complexes are compared the pattern of imidazole coordination is always the same, independent of metal ions. It is worth noting that the FT patterns for Zn(im)4 and Zn(im)6 are comparable ˚ , because the imidazoles are forced to be further (although shifted by approximately 0.2 A away from the metal ions due to the stronger steric hindrance in 6-coordinated

Practical Approaches to Biological Inorganic Chemistry

EXAFS * k3

Multiple scattering in biological systems

15

(A)

(B)

0

–15 2

4

6

120

8 10 12 14 2 k (Å–1)

4

6

8 10 12 14 k (Å–1) CC

/FT/

Zn

C

Zn

–120 2

k3-Weighted Zn K EXAFS (top) and phase shift-corrected Fourier transform (bottom) of Zn imidazole complexes. (A) Zn(imidazole)4 diperchlorate (blue gray) with simulation (red) based on the crystal structure; (B) comparison of Zn(imidazole)4 diperchlorate experimental with imi˚ (blue gray) with simulation dazoles at 2.0 A ˚. (violet) of six imidazoles at 2.2 A

CC

N

N C

0

FIGURE 7.17

NN CC

0

NN

255

4

C

6 R (Å)

8

10 0

2

4 6 R (Å)

8

10

complexes), whereas the appearance of the EXAFS is different (Fig. 7.17B); the so-called ˚ 21 that is characteristic of 4-coordinate metal
imidazole camel back feature at 4
5 A ˚ ) is absent in the EXAFS of the complexes (typical metal
imidazole N distance 2 A 6-coordinate complex. This underlines the diagnostic value of the Fourier transform. Fig. 7.17A shows the experimental spectrum of Zn(imidazole)4 diperchlorate together with a simulation based on the crystal structure. We have not only plotted the real part or modulus of the Fourier transform, as is customary in the literature, but also the imaginary part, which contains information on the phase relation between EXAFS and FT. There is good agreement between experiment and simulation in both the real and imaginary parts of the Fourier transform, and this inevitably means that there is also good agreement with the EXAFS. Like the aqueous hemin chloride of Fig. 7.12, the metal
imidazole complex in Fig. 7.17 are examples of systems with rigid (heteroaromatic) ligand systems which typically allow a characteristic pattern of shells to be observed. In such systems, besides the sum of single scattering pathways of the photoelectron wave A(bsorber)
B(ackscatterer)
A(bsorber) and A(bsorber)
R(emote backscatterer)
A(bsorber) such as represented in Fig. 7.13, and represented by the equation for the plane wave approximation given above, multiple scattering pathways of the kind A(bsorber)
B (ackscatterer)
R(emote backscatterer)
A(bsorber) may exist. For many systems, these are only important at low k, that is, in the XANES region (0
50 eV above edge); this is why this part of the XAS is much more affected by the two- and three-dimensional order in the molecular structure than the EXAFS, which basically depends on the radial distribution function. However, multiple scattering pathways are important for the whole k-range of the EXAFS of complexes with rigid ligand systems where the angle A
B
R
A approaches 180 degrees ( . 140 degrees). Examples are coordinating cyanide, isocyanide, or carbon monoxide ligands (Korbas et al., 2006), or coordinating rigid heteroatomic ligands, such as pyridine, imidazole (Binsted et al., 1992), pyrrole, and porphyrin, as shown in Table 7.1. The absorber atom can also be at the center of a multiple scattering unit itself, for example, in a coordination geometry with perfect octahedral symmetry, such as the tungstate bound

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TABLE 7.1 Geometries of biologically relevant ligands with their multiple scattering pathways. Example

Single scattering Multiple scattering

CO (X 5 O) and CN (X 5 N) ligands to Fe in hydrogenase CODH I (Korbas et al., 2006)

M

C

X

Pyrrole (Y 5 C) part of porphyrin; imidazole (Y 5 N) (Binsted et al., 1992)

C

M

N

X

Y M

Tungstate or molybdate bound to protein ligands (Hollenstein et al., 2009), square planar Ni in dithionite-reduced CO-dehydrogenase II (Ha et al., 2007)

M

O

N

M

Y

O

O

M

S S

Ni

S

S

O

S S

Ni

S

S

by two monodentate carboxylate ligands in the bacterial tungstate-binding protein WtpA (Hollenstein et al., 2009), or a square planar geometry, such as Ni in dithionite-reduced CO-dehydrogenase II (Ha et al., 2007). Synthetic coordination complexes, which are model complexes in the structural and not necessarily in the functional sense, in which one or more aspects of the (expected) ligand environment in the protein, such as type and/or geometry of the ligands, are mimicked, are important as reference compounds in order to extract phase shift information, or to test the validity of ab initio approaches with single and/or multiple scattering. The important difference between the single scattering pathway A
R and the multiple scattering pathways A
B
R is the presence of the extra atom B in the pathway, whose electrons give an extra contribution to the phase shift. Attempts to simulate the multiple scattering system A
B
R as the sum of two single scattering systems A
B and A
R ignore the effect of the position of the atom B between A and R and therefore lead to anomalous amplitude and phase effects, and unrealistic results for distances and occupancies (Strange et al., 1987). The imaginary part of the Fourier transform of the Zn imidazole model compound in Fig. 7.17A shows that the ampli˚ is anomalously low compared to that of the major tude of the shell at approx. 3 A ˚ shell at 2 A (considering that it represents 2C atoms per imidazole ring), but that its ˚ has an anomalously large ampliphase is similar, whereas the shell at approx. 4 A tude, and a different phase relationship with the EXAFS. In order to calculate the multiple scattering, geometric 2-dimensional information of the imidazole unit has to be used in the simulation, in particular the Zn
N
C and Zn
N
N angles, which are ˚ is reproduced derived from the crystal structures. Only if the shell at approx. 4 A ˚ 21 in the EXAFS, with its correct phase amplitude is the “camel back” feature at 4
5 A ˚ , correctly which we know to be characteristic of imidazole coordination at 2.0 A reproduced.

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Strategy for the interpretation of EXAFS

257

Strategy for the interpretation of EXAFS Having established the most important aspects of EXAFS simulations, it is time to discuss the strategy. After elementary data reduction, the k3-weighted EXAFS will be simulated. A numerical indication of the quality of the simulations is the so-called fit index, which is a measure of the difference between experimental and simulated spectrum over the whole data range. Based on the chemical information already available on the system, atom types will be chosen for a calculation of the phase shift and backscattering amplitude. It is possible but not necessary to isolate the shells that contribute to the EXAFS and are resolved in the Fourier transform by a process called Fourier filtering, that is, Fourier transformation, selection of the R0 range of the shell, and back transformation. For every shell, a reasonable choice of atom type and of the four most important parameters: ΔE0, threshold energy; R, distance absorber
scatterer; N, occupancy; and a (52σ2), Debye
Waller factor, is made and the EXAFS is calculated to see if there is a reasonable agreement with the experimental EXAFS and FT. If such an agreement cannot be obtained, even by adjusting the parameters, it is time to reconsider the atom type. Once a full starting model is available, it is time to instruct the simulation program on the computer to start looking for the best possible fit by a process called iterative refinement. It means that the computer program will start moving the parameters in small steps and calculate the spectrum and, most importantly, compare it to the experimental by the fit index to see if this has improved. If the fit index has decreased, the computer program for EXAFS simulations will continue to move the parameter in the same direction; if not, it will try the other direction. This loop continues until no further decrease in the fit index is detected. It is expected that the computer program will then have reached an absolute minimum in the multidimensional parameter/fit index space. It is wise to check whether the computer program reaches the same minimum from a different set of starting parameters. The minimum should correspond to a simulation with good agreement to the experimental data. As in any simulation of experimental data, the resulting parameters should be critically evaluated. In case a satisfactory answer cannot be obtained at this stage, it is time to consider the atom types again. In single scattering simulations, the parameters that are allowed to float freely (that are refined) are ΔE0 for the complete simulation, and R as well as a for every shell. For multiple scattering simulations, the parameters are ΔE0 for the complete simulation, and R as well as a and the angle M
A
B for every shell. For unknown systems, in addition the occupancy of each shell might be refined. Generally, the number of parameters that is refined should be kept as low as possible; the refinement of too many parameters might lead to an overinterpretation of the data. In EXAFS data analysis the number of parameters that can be refined should always be less than the number of independent data points N(ind), which depends on the k and R range that are fitted according to: NðindÞ 5

ð2UΔkUΔRÞ 12 π

ð7:7Þ

In the case of heteroaromatic ligands, such as the porphyrin (pyrrole) and imidazole examples from Table 7.1, this means that one should take as much advantage as one

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7. X-ray absorption and emission spectroscopy in biology

possibly can from knowledge about the geometry of the multiple scattering unit (the aromatic ring) that is defined for the multiple scattering calculations. The atom-to-atom distances within the ring are not expected to change much with coordination to metals or with different orientations with respect to the metal
donor atom (typically N) vector, and it is more important to put emphasis on an independent measurement of metal
donor atom distance and metal
donor atom
other atom angles. In constrained refinement, the distances within the unit are fixed, and the parameters refined are ΔE0 for the complete simulation, one occupancy and angle for any unit, and one R and one a for every shell. This is too rigid for most simulations, and restrained refinement (imported from protein crystallography) is often preferred. In this approach, idealized values (restraints) for the distances in the unit are given, the metal
unit atom distances are allowed to vary freely in the refinement, but if this variation results in deviations from the restraints, a penalty (which can be weighed) is added to the fit index, to discourage the computer from looking further in the direction where the unit is distorted (Binsted et al., 1992). The parameters refined are ΔE0 for the complete simulation, one occupancy for every unit, and one R, one angle, and one a for every shell. Independent information is obtained for the number of units, their distance to the metal, and (if angles are refined) the orientation with respect to the metal
donor atom vector. In all simulations it is necessary to make reasonable choices for the values of the Debye
Waller factor a before refinement. In multiple scattering units this means that it increases with the distance of the shell to the metal ion. To apply the same value of the Debye
Waller factor to shells at similar distances is a useful way to limit the number of parameters in all refinements.

Validation and Automation of EXAFS data analysis The structural model resulting from EXAFS data analysis or other methods such as protein crystallography should always be compared to prior knowledge on the system under study, its chemistry and established knowledge. Ignoring the scientific skepticism might lead to the publication of crystal structures traced backward, unreasonable metal
ligand distances caused by low occupancy of the metal binding site, or in the case of EXAFS to wrong metal binding motifs due to problems with sample quality or wrong metal
ligand distances caused by photoreduction. The challenge in data analysis is to avoid such pitfalls. Therefore chemical knowledge and criteria indicating potential problems are applied. In automation such criteria become part of the routines applied searching for the structural model representing the data best. One approach in automation is based on modeling of Debye
Waller factors or the introduction of boundary conditions, for example, by the bond valence sum method (see below), and to use these values as boundary conditions in the refinement. In contrast, the second approach favors a shotgun strategy: one selects all potential ligands and their occupancy range; on this basis potential starting models for the EXAFS refinement are designed; all models are compared to the EXAFS data by refining distances, Debye
Waller factors, and ΔE0, but no occupancies; the parameters are analyzed with the help of criteria, that one can use as quality indicators:

Practical Approaches to Biological Inorganic Chemistry

Validation and Automation of EXAFS data analysis

259

1. Do the obtained distances fall into the interval of reported metal
ligand distances? Note that these values depend on the metal oxidation state (and the ligand’s chemistry). Reference data for frequently used metal ions are critically summarized by several authors (Harding, 2004) or can be extracted from small molecule databases (e.g., Cambridge structural data base). 2. Bond valence sum analysis (BVSA). This formalism which is described in detail elsewhere (Thorp, 1998) takes advantage of established empirical correlations between (1) the oxidation state of a metal, (2) its metal
ligand distances, and (3) its coordination number. Its application to results of EXAFS studies is particularly appropriate, because it can be a way to get a better indication for the coordination number (which is relatively inaccurate from EXAFS) based on the distance information (which is relatively accurate from EXAFS). 3. Debye
Waller factor criterion. Disorder in biological systems is typically larger than in small molecules, which is reflected by the Debye
Waller factors resulting from EXAFS refinement of metalloproteins. But too large Debye
Waller factors artificially decrease the contribution by the corresponding shell, whereas too small Debye
Waller factors artificially enlarge it. For automation this criterion is based on experience reflecting published values for similar systems. 4. The shift of the energy threshold (ΔE0) should be similar for similar samples—typically less than 1 eV. Note that samples differing in spin or oxidation state are not considered as being similar in this context. In automated refinement (see Fig. 7.18 for a schematic overview) for each criterion a parameter is defined, which can vary in value between 1 (entirely fulfilled) and 0 (complete failure). These criteria are weighted and multiplied; thus failure in one criterion results in the rejection of the structural model and only models with a good fit index that also fulfill the criteria will be ranked high. Those models ranked highly will have many details in common. This information is extracted by a meta-analysis, resulting, for example, in a well-defined number of low-Z ligands without claiming to differentiate their similar backscattering potentials (Wellenreuther et al., 2010).

FIGURE 7.18 Typical criteria helpful to computer programs and novice EXAFS groups comprise bond valence sum analysis, the comparison of oxidation state and coordination number-dependent metal
ligand distances taken from Cambridge structural database (CSD), Debye
Waller criterion, and ΔE0 shift. On a computer easily up to 1000 structural models can be compared to the data. A meta-analysis can extract common features of the good models (Wellenreuther et al., 2010).

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7. X-ray absorption and emission spectroscopy in biology

X-ray absorption near-edge structure simulations with three-dimensional models As explained in the introduction, most theoretical approaches for XANES (the low-energy range of XAS) are based on orbital theory, whereas those for EXAFS (the high-energy range) are based on electron diffraction. This raises the question up to what energy the orbital theory should be applied, and from which energy the electron diffraction approach is valid, and this is not easy to answer. In the preceding sections we have already discussed EXAFS as a sum of oscillations due to electron diffraction phenomena involving shells of backscatterers in a one-dimensional radial distribution function, and the single scattering theory that describes the electron diffraction phenomena at the higher energies (high k). We have also seen that for low energy (low k) the multiple scattering pathways, involving scattering of the electron wave from one atom to another, before returning to the absorber atom, are more important. In order to describe the low energy spectra accurately with the electron diffraction theory, two- and even three-dimensional information will have to be taken into account; this is equivalent to (but more complicated than) discussing the spectra in terms of orbital theory. As an example of the limitations and opportunities, simulations for the Mo K-edge XANES of molybdate are shown in Fig. 7.19; an analysis of the corresponding EXAFS just gives the distance of the Mo to the four oxo ligands, without any twoor three-dimensional information. The well-known tetrahedral alignment of the four oxo ligands yields the dotted red curve, whereas a square planar alignment of these atoms results in the dashed green curve. Although the red curve reproduces the main features of the measured XANES by approximation, it is not a perfect fit to the data. But in this ab initio simulation no parameter has been optimized and thus the fact that the green curve shows no similarity to the molybdate XANES allows a square planar coordination to be ruled out. In this manner XANES serves as a fingerprint in both the qualitative and quantitative analysis, and XANES simulations and their refinement are of increasing importance. In any case, one should keep in mind that the number of independent data points is more limited than for EXAFS and thus the result might depend on the assumptions used in the refinement; compared to EXAFS one tries in fact to derive more (two- and three-dimensional) information from a shorter range FIGURE 7.19 Comparison of the experimental XANES of aqueous molybdate (turquoise, cf. Fig. 7.4) with simulations with 4O in a tetrahedral (red dots) and square planar (green dashed) arrangement.

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261

Metal
metal distances in metal clusters

of data, without the guidance provided by the Fourier transformation for the EXAFS. An advantage of the shorter data range is that one does not have to take the dampening due to the Debye
Waller factor (see Box 7.6) into account.

Metal
metal distances in metal clusters In EXAFS contributions at higher distances frequently originate from multiple scattering or backstattering from a metal ion. In our introductory example on oxidized CO-dehydrogenase from Oligotropha carboxidovorans, metal
metal contributions are present in both the Cu edge EXAFS as well as in the Mo edge EXAFS. In Fig. 7.20 the individual contributions to the EXAFS are shown. For the Mo edge EXAFS (top panel) the ˚ . The second nearest shell correspectrum is dominated by the two oxo ligands at 1.74 A ˚ , which based on comparison to other Mo sponds to a single sulfur ligand at about 2.28 A enzymes is identified by its bond length as the ligand bridging both metal ions. The two ˚ belong to the pterin cofactor, and the last identifiable subsequent sulfur ions at 2.50 A ˚ . The Cu EXAFS (bottom panel) is contribution is refined as Cu backscattering at 3.70 A ˚ . Again, metal
metal backscattering can be dominated by the sulfur contributions at 2.18 A ˚ , this time with Cu as the absorber and Mo as the backscatterer. The identified at 3.70 A analysis of the individual traces is a good example for the general features discussed above: (1) the shorter the bond length, the lower the frequency of the oscillations; (2) the heavier the backscatterer, the more the maximum of the EXAFS intensity shifts to a higher value of the wavevector k; (3) the closer a ligand the more intense the resulting EXAFS contribution will be; (4) the more backscatterer atoms in a shell, for example, 2 versus 1 S, FIGURE 7.20 EXAFS (left) and phase-corrected Fourier transform (right) of experimental (black) and simulated (colors) Mo (top) and Cu (bottom) data of oxidized COdehydrogenase from O. carboxidovorans. The complete simulation is shown along with the experimental data; other traces represent the contributions of individual components to the model.

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7. X-ray absorption and emission spectroscopy in biology

the stronger the contribution to the EXAFS, and (5) metal
metal contributions show a high frequency in the EXAFS and require a reasonably long energy range for unambiguous identification. They are sufficiently specific to rule out the presence of Se (in case of the Mo EXAFS); the backscattering amplitudes of Cu and Se resemble those given for Zn and Br, respectively, in Fig. 7.15, left, and there are also diagnostic differences in the phase relationship between EXAFS and Fourier transform such as discussed for the halogens in Fig. 7.15, right. The long energy range also helps to distinguish metal backscatterers from contributions of a unit with multiple scattering in the same region, such as those observed ˚ in the Fourier transform for porphyrins (Fig. 7.12) and imidazole at just above 4 A (Fig. 7.17); multiple scattering contributions of systems with low-Z backscatterer atoms are very intense in the lower k-range. In this example, both spectra have been refined simultaneously. Thus Debye
Waller factor as well as the distance of the metal ions were fitted together increasing robustness and consistency of the resulting structural model. In case of homodinuclear metal sites a similar strategy can be applied, but here it is even more important to determine independently the metal content in the sample under study, because this will be reflected by the occupancy of individual binding sites (Svetlitchnyi et al., 2004).

Nonmetal trace elements: halogens Although most XAS studies have focused on the chemical environment of transition metal cations, increasing attention has been paid in recent years to metalloids (such as selenium and arsenic), halogens, and oxyanions. We discuss here features common to the studies of cations and anions, as well as differences. Transition metal cations interact in a dynamic way with a number of electron-donating ligands in order to achieve a favorable coordination number (or secondary valence); halogens are subject to solvation and H-bonding in their anionic form, and can form single covalent bonds (primary valence) in systems where they can still be H-bond acceptors or themselves be part of a halogen bond. As illustrated in Fig. 7.21, where the effect of six-membered aromatic rings is compared, the Fourier transform of the EXAFS of a coordination complex of Cu with pyridine (Feiters et al., 1999) resembles that of a compound where I or Br is covalently attached to a phenyl ring. Like for the five-membered ring systems highlighted above, the EXAFS can only be simulated satisfactorily with a multiple scattering approach in both cases. The most important difference between metal and halogens is the decrease in amplitude of the EXAFS because the metal ion has four identical ligands, whereas the halogens each have only a single covalent bond. There are also subtle differences in the relation between EXAFS and Fourier transforms due to the subtle but significant differences in the distances between the absorber and the first backscatterer. Interestingly, the accurate determination of the halogen
carbon distance by EXAFS allows one to discriminate between halogens bound to sp2- and sp3-hybridized carbons (such as occur in aromatic/olefinic and aliphatic halocarbons, respectively), because the carbon
halogen bond becomes shorter with increasing s-character of the bonding orbital on C (Feiters et al., 2005a). Brown algae such as Laminaria digitata (oarweed) accumulate iodine to concentrations 106 times that of surrounding seawater, and XAS is ideal as a noninvasive technique to study this system.

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Nonmetal trace elements: halogens

263

FIGURE 7.21

Left and middle panels: Experimental (thin red lines) and simulated (solid lines) EXAFS (left panel) and corresponding Fourier transforms (middle panel) of Cu K-edge of [Cu(pyridine)4](NO3)2 (top trace), I K-edge of 3-iodotyrosine (middle trace), and Br K-edge of 4-bromophenylalanine (bottom trace). Insets, structures ˚ ) of the respective absorbers to the nearest atom. on which the simulations are based, including distances (in A For the Cu spectrum, two oxygen atoms of the weakly coordinating nitrate anion were also included in the simu˚ (Feiters et al., 1999); for the Br spectrum, the R group was not included, whereas for the I spectrum lation at 2.5 A none of the other ring substituents was included (Feiters et al., 2005a). Right panel: Br K-edge EXAFS (top panel) and Fourier transform (bottom panel) of native bromoperoxidase from Ascophyllum nodosum (Feiters et al., 2005b). Top traces: experimental (thin red line) and complete simulation (solid line); bottom traces: contributions of aromatic ring component (blue), nearest C (green), and bromine (orange line). Inset: dibromotyrosine structure highlighting the aromatic ring; substituents other than Br were not included in the simulation.

Iodine in Laminaria gave a weak EXAFS even when compared to NaI in water, and was shown to represent iodide ions with their solvation shell displaced by H-bonding to bio˚ ) (Ku¨pper et al., 2008). On this basis a physimolecules at a comparable distance (3.5
3.6 A ological role for accumulated iodide as an inorganic oxidant is proposed. The enzymes responsible for incorporation of the accumulated halide ions into biomolecules are the so-called haloperoxidases. Fig. 7.21 illustrates one of the ways by which the bromoperoxidase of Ascophyllum nodosum (knotted wrack), can be involved in the halogen chemistry of the algae. The Br K-edge EXAFS of the native enzyme (without added Br) reveals a spectrum that is characteristic of a 1,3-dibrominated aromatic ring, such as present in 3,5-dibromotyrosine, with contributions of the ring at ˚ (Feiters et al., 2005b). The spec1.90 (nearest C atom) and a remote bromine at 5.70 A trum is almost entirely accounted for by the ring contribution, except for the Br atom which becomes stronger at high k; in addition, the calculated contribution of the Br is much stronger when it is part of the ring, because of the forward scattering and focusing effects of the ring carbons, than when it is on its own. The identification of this posttranslational amino acid modification implied an important addition to the protein crystal structure, in which the electron density had been modeled with tyrosine singly substituted with I.

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Summary: strengths and limitations X-ray absorption spectra arise because of the excitations of electrons from the inner shells of atoms with X-rays of the right energy, and can be recorded with monochromatized synchrotron radiation. The interesting part of an X-ray absorption spectrum can be divided into two regions: the XANES and the extended EXAFS. The information to be obtained from XAS is summarized in Table 7.2. In the typical BioXAS experiment, the fine structure is measured by collecting the X-ray fluorescence, which is dominated by the Kα TABLE 7.2 Information to be obtained from certain features of the XANES and EXAFS regions of XAS as well as XES. Spectral feature

Information

Accuracy and correlations

Other XAS

Other technique

Edge position

Oxidation state

Relate to model compounds; be aware of correlation with average R

ΔR: EXAFS

UV
visible spectra, electron paramagnetic resonance (EPR)

Preedge features

Ligand geometry

Relate to wellcharacterized model compounds

N from EXAFS

UV
visible spectra, EPR, crystallography

Covalency of metal
ligand bond




XANES of other EPR hyperfine metal- and structure ligand edges

Kβ0 line

Metal ion spin state







EPR, magnetic susceptibility

Kβv line

Ligand identity




EXAFS (Z61 accuracy)

Crystallography

20% (61), correlation with Debye
Waller factor

Ligand geometry from XANES

Crystallography

XANES

XES

EXAFS Amplitude Coordination number, N

Decay of amplitude with k

σ.a: static or thermal disorder, distinguish by T variation

Correlated with/spoils accuracy of N

Periodicity Distance R of scatterers

˚ if shell resolved, 60.02 A correlation with threshold energy ΔE0

Edge shift from XANES

Crystallography

Phase

Different to distinguish atom types adjacent in Periodic Table

Unambiguous from XES Kβv line

Crystallography

a

Backscatterer atom type: C, N, O (“low-Z”) versus S

Debye
Waller factor.

Practical Approaches to Biological Inorganic Chemistry

Summary: strengths and limitations

265

lines, in a solid-state detector. Detailed investigations of the Kβ main and satellite lines by XES give information on the metal ion’s spin state and the type of ligand(s), respectively. The position of the edge in the XANES goes to higher energy with higher oxidation state and with decreasing average metal-to-ligand distance. The fine structure can be interpreted in terms of transitions to empty orbitals of the metal and the transition probabilities depend on the ligand geometry. When transitions to molecular orbitals of the complex are involved, the covalency of the metal
ligand bond also has an influence; this can also be probed by metal L-edge or ligand K-edge XANES. An interpretation of the XANES in terms of a three-dimensional model extrapolated from the electron diffraction theory for the EXAFS is also possible. As the XANES energy range is short and the fine structure is relatively strong compared to that in the EXAFS region, this is the region of choice to explore in time- (reaction kinetics) and space- (element chemical state imaging) resolved studies. The fine structure in the EXAFS is caused by X-ray
induced electron diffraction phenomena. Backscattering amplitudes and phase shifts for absorber and backscatterer atoms can be derived from model compounds, or calculated using the muffin-tin approximation (and validated on model compounds). A strength of EXAFS is the accurate determination of the distance R. Limitations in this ˚ depending on the k-range) and aspect are the poor resolution (ΔR approximately 0.15 A the unreliable observation of weakly bonded atoms. The best data collection strategy is to collect as long a range of data as possible (to improve the resolution) at as low a temperature as possible (to reduce thermal vibrations and disorder). Another strength of EXAFS is the identification of atom type of the backscatterer. A limitation is that atom types that are close in Z are difficult to distinguish, for example, “lowZ atoms,” C, N, or O. We have seen (Box 7.2) that this is a particular strength of the newly emerging X-ray emission techniques. The determination of coordination numbers is not a particular strength of EXAFS, in view of the limitation that it is strongly correlated with the Debye
Waller factor. This parameter needs to be incorporated in the EXAFS theory and refined in order to account for disorder of static and/or thermal origin. It is possible to use other features of the XAS spectrum to get a more accurate idea of the coordination number. One is to interpret the geometrical information from XANES: tetrahedral geometry means 4-, octahedral 6-coordination. The other is to use the accurate distance information to take advantage of established correlations between ligand distance and coordination number by the so-called bond valence sum analysis (BVSA). The Debye
Waller factor and other amplitude effects limit the accuracy of the determination of coordination numbers by EXAFS to approx. 620%. It should be kept in mind, however, that coordination numbers may be inferred from XANES based on ligand symmetry arguments, as discussed in the XANES section, or from the aforementioned bond valence sum analysis formalism. EXAFS gives an indication of the presence and orientation (with respect to the metal
N bond) of heteroatomic ligands (imidazoles, porphyrins). A limitation is that in addition to the parameters normally refined in single scattering simulations, the angles will also have to be refined. In such cases the procedure of restrained refinement can be used. The strength and limitations of the two regions of the XAS spectrum (EXAFS and XANES) and their relations with each other and other techniques are summarized in Table 7.2.

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7. X-ray absorption and emission spectroscopy in biology

Conclusions: relations with other techniques Metals in proteins would appear to be minor components, but are usually at the active site which makes a study of their close environment by XAS particularly relevant. Even when a structure of the whole protein is already known from protein X-ray crystallography (PX) or NMR, it is usually worthwhile to study it by XAS for a number of reasons. (1) Typically the error in the determination of metal
ligand distances is much larger for macromolecular (protein) crystallography than it is for small molecules crystallography ˚ ), whereas for EXAFS it is the same (60.02 A ˚ ); this implies that (estimate 60.1 vs ,0.01 A the accuracy of the geometry of the metal environments in crystallographic studies can be improved with input from EXAFS. By using XAS results in the refinement of the crystal structure it is possible to resolve the metal environment at higher resolution, and the judicious combination of PX and XAS can even lead to structures with subatomic resolution (Strange et al., 2005). (2) With XAS it is much easier to study the effect of other reactants/circumstances on the metal environment which might be difficult to achieve in the crystalline state, or result in magnetism of the metal site interfering with the structure elucidation by NMR. Thus the important advantage of XAS over other structural and spectroscopic techniques for biological as well as other applications is that the environment of any trace element can be studied irrespective of the physical state (crystalline, solid, liquid, frozen liquid) of the sample, or the chemical state (oxidation state, spin state, magnetism) of the element. A number of other techniques whose results overlap or are complementary with those from XAS studies, such as UV
vis spectroscopy, electron paramagnetic resonance, and magnetic susceptibility measurements, are given in the appropriate places in Table 7.2. An important feature of XAS analysis that was inspired by crystallographic examples is the use of restraints (Binsted et al., 1992) for distances of ligands of well-known geometry, such as imidazoles or porphyrins, in order to reduce the number of independent parameters to be refined in an EXAFS simulation. High-resolution protein crystal structures have been surveyed to identify typical metal
ligand distances in proteins (Harding, 2004). EXAFS is also the method of choice to experimentally validate the geometrical results of computational studies on metalloproteins.

Exercises 1. What are the differences in origin and application of K- and L-edges in X-ray absorption spectroscopy? 2. How is the wavelength selected in the X-ray absorption experiment? Which fundamental law determines the energy of the photons on the sample? 3. a. Estimate the errors for signals collected in an X-ray absorption experiment in transmission mode. Typically, a small signal is measured on a large background: background: 100,000 counts/s; signal: 1000 counts/s. b. For how many seconds per data point should the data collection be set up in order to obtain a signal which at the edge will be 10 times the background’s error.

Practical Approaches to Biological Inorganic Chemistry

Exercises

4.

5.

6.

7.

8.

267

Hint: For large numbers use Poisson statistics (square root of the number estimates the error margin). An unknown “blue copper” protein is suspected to have Cu
N and Cu
S bond ˚ , respectively. lengths of 1.95 and 2.10 A ˚ 21) and energy (E, eV), is required a. What kind of energy range, in wave vector (k, A for the EXAFS to be able to resolve these shells? b. Are there problems to be expected if the protein is contaminated with Zn to an amount of 1% compared with the Cu? The XAS studies of the Mn4Ca cluster in the oxygen-evolving center of the Photosystem II are hampered by the presence of Fe, which has its K absorption edge 573 eV above the Mn K-edge. How might a long ( . 800 eV) range of Mn EXAFS be recorded without interference of the Fe? Synchrotron radiation from a bending magnet is “white”—a continuous spectrum ranging from UV
vis to hard X-rays—and needs to be monochromatized for most techniques, including XAFS. The most used monochromator material is single crystalline Si; the most used reflection is (111). a. Using Bragg’s law, calculate the angle of incidence needed to generate a ˚ . The spacing of the (111) planes in monochromatic beam with a wavelength of 0.8 A ˚ Si is 3.14 A. b. Usually a so-called double crystal monochromator is used (see Fig. 7.2). Why is that convenient? c. The first crystal of a double crystal monochromator requires cooling, the second does not. Why? In Fig. 7.22, the k3-weighted Fe K-edge EXAFS (top panel) and its Fourier transform (bottom panel) of solid hemin chloride (1) and frozen aqueous hemin (dotted lines) are ˚ , solid line) and chlorine given, along with simulations for the nitrogen (4N at 2.06 A ˚ (1Cl at 2.23 A, dashed line) contributions to the hemin chloride EXAFS. a. What problems are to be expected in the analysis of the close Fe
N and Fe
Cl ˚? contributions in the Fourier transform range 2.0
2.3 A b. Considering the geometry of the pyrrole units of the porphyrin, would an analysis in which only the single scattering is simulated be appropriate? c. What happens upon dissolution of hemin chloride in water? The binding of tungstate by the bacterial tungstate (WO422) binding protein from Archaeoglobus fulgidus has been studied by EXAFS (Fig. 7.23). The tungstate oxyanion in ˚ ) W 5 O bonds (traces B). The solution is a tetrahedron with four extremely short (1.77 A protein provides two additional ligands which results in the lengthening of one W 5 O ˚ (traces A). bond, and the formation of two W
O bonds of 2.24 A a. What experimental precaution has to be taken in order to be able to resolve the ˚ in a spectrum like A? contributions at 1.77 and 2.24 A ˚ 21)k. The position of b. Spectrum A shows a high-frequency oscillation at low (3
6 A ˚ the corresponding shell in the Fourier transform close to 4 A corresponds to the ˚ ; 1.77 1 2.24 A ˚ ). What sums of some of the W 5 O and W
O distances (1.77 1 1.77 A assumptions about the geometry of the unit and the nature of the scattering

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7. X-ray absorption and emission spectroscopy in biology

N Cl N Fe 1

N

N

HOOC

COOH

EXAFS × k 3

15

0

// 0

–15 2

3

4

5

6

7

8

9

10

11

12

13

8

9

10

k (Å–1)

100

/FT/

50

0

// 0 0

1

2

3

4

5 6 R (Å)

7

FIGURE 7.22

(Exercise 7). k3-Weighted Fe K EXAFS (top panel) and phase-corrected Fourier transform (bottom panel) of experimental (dotted lines) of solid hemin chloride (1, top traces) and aqueous hemin (bottom ˚ , solid line) and chlorine (1Cl at 2.23 A ˚, traces) are given, along with simulations for the nitrogen (4N at 2.06 A dashed line) contributions to the hemin chloride EXAFS. Source: Feiters, M.C., de Val, N., Toussaint, L., Cuypers, M., Crichton, R.R., Meyer-Klaucke, W., 2005. Hasylab Annual Report II. 505
506. Practical Approaches to Biological Inorganic Chemistry

269

Hints and answers to exercises

EXAFS × k3

8

A

0

B 0

–8 2

4

6

8 k (Å–1)

10

12

14

FT magnitude

60

30

A 0

FIGURE

7.23 (Exercise

8). k3 Weighted extended X-ray absorption fine structure (EXAFS) spectra (upper panel) and corresponding phasecorrected (assuming oxygen in main shell) Fourier transforms (FT; lower panel) at the tungsten L3-edge of tungstate in the binding protein from A. fulgidus (traces A) and buffer (traces B). Red line experimental data, black line simulation with the following parameters (distances in angstroms, Debye
Waller-type factors as 2σ2 in square angstroms in parentheses, fit index (FI) with k3-weighting): trace A, threshold energy shift:Δ ˚ (0.008), E0 5 214.619 eV, 3O at 1.786 A ˚ (0.002), 2O at 2.241 A ˚ 1O at 2.060 A

(0.003), fit index 0.29989 3 1023 ; trace ˚ B: ΔE0 5 213.54 eV, 3.7O at 1.770 A (0.005), fit index 0.14069 3 1023 . Source: Hollenstein, K., Comellas-Bigler, M., Bevers, L.E., Feiters, M.C., Meyer- Klaucke, W., Hagedoorn, P.-L., Locher, K.P., 2009. J. Biol. Inorg. Chem. 14, 663
672.

B

0 0

2

4

6

8

10

R (Å)

phenomena within it have to be made in order to explain this part of the spectrum with just the [WO6] unit?

Hints and answers to exercises 1. Hint: Check Figs. 7.6 and 7.9 as well as the L-edge paragraph in Box 7.2. Answer: For dilute samples the K-edge is preferred because of its fluorescence yield is higher than that of L-edges (Fig. 7.3). K and L3-edges look different, as the spinallowed transitions (s-p, resp. p-d) are different, and the transitions involving the d orbitals can give information on the spin state of the metal. The L-edges are experimentally more difficult, but in XES advantage can be taken from the fact that K excitation followed by Kα1 fluorescence results in the same final state as direct L3 excitation (Fig. 7.9, right).

Practical Approaches to Biological Inorganic Chemistry

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7. X-ray absorption and emission spectroscopy in biology

2. Hint: Check Box 7.1 and Fig. 7.2 on the instrumentation, and Equation 7.3. Answer: The wavelength is selected by diffraction of the white beam by two parallel Si crystals (Fig. 7.2). The wavelength of the diffracted beam is determined by Bragg’s law: nλ 5 2dsin θ (Equation 7.3) where n is a natural number (1, 2, 3, etc.), λ the wavelength, d the lattice parameter of the Si crystal, and θ the angle between monochromator and beam. 3. Hint: For large numbers use Poisson statistics (square root of the number estimates the error margin). (Check Eq. (7.1).) Answer: For a measurement of n seconds and signal 5 (background 1 signal) 2 background, one has 1000n 5 101,000n 2 100,000n, and (signal error)2 5 (error in background 1 signal)2 1 (background error)2. Following the Poisson statistics and taking the errors as the square roots of the actual numbers, one finds (signal error)2 5 101,000n 1 100,000n, so signal error 5 O201,000n 5 448On. The error in the background is O100,000n 5 316On. For a signal of 1000n to be 10 times the error in the background requires that 1000n 5 10.316On, so On 5 3.16, and n 5 10 s. ˚ 21) from Eq. (7.5), relate this to the 4. Hint: Calculate the required k-range (Δk, A maximum energy (eV) by Eq. (7.5), and compare this to the difference in K-edge energies between Cu and Zn (Fig. 7.1). ˚ ) and starting with the required Answer: Using Eq. (7.5): ΔR 5 π/2Δk (A ˚ ˚ 21. Eq. (7.4), ΔR 5 2.10
1.95 5 0.15 A, one finds for the required k-range Δk 10.5 A rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2me ðE 2 E0 Þ k5 ħ2 gives k in (m21) with Planck’s constant h 5 6.6260755 3 10234 J s, rest mass electron me 5 9.1093897 3 10231 kg, if the energy is entered in J. Energies entered in eV can be cor˚ 21 rected for by the elementary charge (electron) of e 5 1.60217733 3 10219 C. Thus k in A can be calculated from E in eV using k 5 O0.262(ΔE). It follows that the maximum k-range that can be measured at the Cu K-edge of a sample with a significant Zn contamination ˚ 21, which should be (ΔE 5 9659 (Zn K-edge) 2 8979 (Cu K-edge) 5 680 eV) is 13.35 A enough to achieve the required resolution. 5. Hint: Select the energy of the fluorescence photons to be counted with the solid-state multielement detector (Box 7.1) to reject not only scattering by the sample, but also Fe fluorescence (hint 5 answer). 6. a. Hint: Use Equation A. Answer: From Bragg’s law, 2d sin θ 5 λ (Equation A) we find:

 λ 0:8 θ 5 a sin 5 a sin 5 7:323 2d 2 3 3:14 b. Hint: Check Fig. 7.2 and the text around it in Box 7.1. Answer: In many experiments, and certainly spectroscopic experiments, the wavelength is scanned. With a single crystal monochromator, the angle of the beam

Practical Approaches to Biological Inorganic Chemistry

Hints and answers to exercises

c.

7. a.

b.

c.

8. a.

b.

271

changes with the wavelength and thus the sample and associated equipment need to be moved. With a double crystal monochromator, the second crystal brings the beam back to its original direction, irrespective of the wavelength. (Note: there is a small vertical offset difference as a function of wavelength.) If the second crystal can be oriented slightly nonparallel with respect to the first, as in the so-called order-sorting monochromator, there is also the possibility to selectively reject the higher harmonics, because their acceptance angles are much smaller. Hint: Check Fig. 7.2, and think of the difference in intensity between white and monochromatic beam. Answer: The first crystal receives “white” radiation, which can contain a lot of total X-ray power (often several hundred watts). From this, a single wavelength is selected, which corresponds to only a small amount of energy. (The bandwidth of a typical monochromator is roughly 0.01%, leaving a monochromatic energy in the mW range.) Hint: Check Fig. 7.16 and text around it. ˚ ) so care should be taken to Answer: The distances are fairly close (ΔR 5 0.3 A record a sufficiently long range of data (Δk) to be able to resolve it. Moreover, the Fe
N and Fe
Cl contributions are out of phase (as N and Cl are from subsequent rows of the Periodic Table) when placed at the same distance, so they are likely to cancel each other to some extent. Hint: Check Table 7.1, second example. Answer: The porphyrin is a rigid heteroaromatic ligand system with pyrrole units in which multiple scattering is important, as some of the angles A(bsorber)
B (ackscatterer)
R(emote Backscatterer)
A(bsorber) approach 180 degrees ( . 140 degrees). Hint: Look at the N and Cl contributions to the first shell of the FT of hemin chloride, and see what remains of the EXAFS and FT upon dissolution in water. Answer: The spectrum of aqueous hemin still has the characteristic signature of the porphyrin, and its first shell resembles the contribution calculated for the 4N in solid hemin chloride, whereas the chloride has disappeared. Apparently, Cl is no longer coordinated to Fe in aqueous hemin chloride, which implies that hemin chloride has dissociated in water into an Fe-porphyrin cation and a chlorine anion. Hint: Check Fig. 7.11 and text around it. ˚ ) and starting with the required Answer: Using Eq. (7.7): ΔR 5 π/2Δk (A ˚ ˚ 21). 12 A ˚ 21 are ΔR 5 2.2 2 1.7 5 0.5 A, one finds for the required k-range Δk π (3.14 A used for the simulation which is more than enough. Hint: See Table 7.1, last example. Answer: The [WO6] unit is distorted with some longer and shorter W
O distances. There are multiple scattering pathways that go back and forth through the central W absorber and involve the O atoms that are trans with respect to each other and the ˚ W. So the position of the corresponding shell in the Fourier transform close to 4 A ˚ corresponds to the sums of the W 5 O and W
O distances (1.77 1 1.77 A; ˚ ) involving O atoms that are trans with respect to each other and the W. 1.77 1 2.24 A

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C H A P T E R

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Resonance Raman spectroscopy and its application in bioinorganic chemistry Wesley R. Browne Molecular Inorganic Chemistry, Stratingh Institute for Chemistry, Faculty of Science and Engineering, University of Groningen, Groningen, The Netherlands O U T L I N E Introduction

276

The fundamentals of vibrational spectroscopy 277 The classical oscillator, Hooke’s law, the force constant, and quantization 279 Quantization and the nature of a quantum excitation 281 Permanent, induced, and transition electric dipole moments Electric dipole moments Transition dipole moments Polarizability, induced dipole moments, and scattering Relative intensities of Stokes and anti-Stokes Raman scattering What is a virtual state?

Practical Approaches to Biological Inorganic Chemistry DOI: https://doi.org/10.1016/B978-0-444-64225-7.00008-0

283 284 284 286 291 292

The (resonance) Raman experiment Raman cross-section and the intensity of Raman bands Raman scattering is a weak effect; but how weak? Resonance enhancement of Raman scattering The Raman spectroscopy of carrots and parrots Classical description of Rayleigh and Raman scattering The Kramer
Heisenberg
Dirac (KHD) equation A-, B-, C-term enhancement mechanisms, overtones, and combination bands

275

293 295 296 298 299 302 303 305

© 2020 Elsevier B.V. All rights reserved.

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8. Resonance Raman spectroscopy and its application in bioinorganic chemistry

Assigning electronic absorption spectra Heller’s time-dependent approach SERS and SERRS spectroscopy Experimental and instrumental considerations Isotope labeling and band assignment Resolution and natural linewidth Confocality, the inner filter effect, and quartz

305 308 310 311 311 313 314

Applications of resonance Raman spectroscopy 317 Resonance enhanced Raman spectroscopy in the characterization of artificial

metalloenzymes based on the LmrR protein Reaction monitoring with resonance Raman spectroscopy Transient and time-resolved resonance Raman spectroscopy

317 318 319

Conclusions

321

Questions

322

Answers to Questions

322

Acknowledgements

323

References

324

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Introduction The inelastic scattering of light was predicted by Adolf Smekal in 1923 and the experimental observation of this phenomenon, by C.V. Raman, was reported in the journal Nature in 1929. The award of the Nobel Prize in Physics in 1930 recognized the immediate importance of the discovery, however, it was not until the early-1990s that the instrumentation necessary to record Raman spectra routinely became available; especially sensitive detectors, steep Rayleigh scattering rejection filters, and high-quality monochromatic lasers. These technical developments moved the technique from “home-built” systems in physical laboratories out to the wider world of chemistry laboratories, and more recently first responders, factories, hospitals, and airports, as a widespread analytical tool. Despite the rapid advances in laser and detector technology, and the increasing affordablility (affordable hand held devices are now available),Raman spectroscopy is still often perceived as being a somewhat exotic or specialist tool. This perception is in part due to the long delay between discovery and routine application that is in sharp contrast to the development of the related technique, Fourier transform infrared (FTIR) spectroscopy. Raman spectroscopy is a highly adaptable spectroscopic technique requiring no or at least minimal sample preparation. In comparison with FTIR spectroscopy, the spectra have well resolved bands and are simpler because overtones and combination bands are generally weak or absent. Furthermore, the instrumentation used for Raman spectroscopy is relatively uncomplicated—indeed a Raman spectrometer is essentially a spectrofluorimeter on which one records emission spectra but with very high wavelength resolution! Indeed, it is of note that the intensity and signal-to-noise (S/N) ratio of the Raman band of water at 3500 cm21 is typically used as a figure of merit for spectrofluorimeters.

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Raman scattering is an incredibly improbable phenomenon. However, when the wavelength of the laser used coincides with the wavelength of an electronic absorption band of a molecule, the intensity of the Raman scattering can increase by many orders of magnitude. Spectra obtained under this condition are referred to as resonance Raman spectra and can allow compounds present at as low as micromolar concentrations to be observed and studied. The phenomenon of resonance enhancement has made Raman spectroscopy a key technique in bioinorganic chemistry especially. Although it may be perceived as a highly specialized technique, as we will see, in practice recording a “resonance Raman” spectrum is no different to recording a nonresonant “normal” Raman spectrum. In fact, the only practical distinction between a Raman spectrum and a resonance Raman spectrum is the choice of wavelength of laser used and consideration of inner filter effects. Both of these aspects will be discussed below. In this chapter we will first introduce the basis for vibrational spectroscopy and instrumentation used to record a (resonance) Raman spectrum, and thereafter, superficially, the theory of (resonance) Raman spectroscopy. We will then consider the phenomenon of resonance enhancement of Raman scattering. A brief discussion of the various sampling aspects that are relevant to (bio)inorganic chemistry will be given, before finally, a number of examples will be described to illustrate both the simplicity and possibilities this technique presents to (bio)inorganic chemistry.

The fundamentals of vibrational spectroscopy Before discussing resonance enhancement of Raman scattering (known colloquially as resonance Raman spectroscopy or rR), it is necessary to consider first what a “vibration” is, and the nature of the interaction of electromagnetic radiation (EM) with matter. When we think of electromagnetic radiation, we usually think of photons; packets of energy that fly through space. However, they are much more than quanta of energy. Photons carry information in the form of angular momentum and specifically one quantum of angular momentum each. The concept that photons have momentum (p) related to their wavelength (λ) arises from the de Broglie relation (p 5 h/λ). Furthermore, a photon has an electric and magnetic field that, to the observer from the side, appears to oscillate (Et 5 Eocos(ωt), where Eo is the amplitude and ω 5 2πν is the angular frequency) in the direction of propagation. The electric and magnetic components rotate about the axis of propagation. In addition to the interaction of the oscillating electric field with the electrons and nuclei in the substance, the transfer of angular momentum of the photon needs to be taken into account when considering the mechanisms by which a photon interacts with matter. In routine spectroscopy, we need also to consider the flux (number per unit time) of photons, as it is the total electric field (combination of the fields generated by each photon) that interacts with dipoles and charged particles. In the case of electrons and nuclei the response is dependent on the frequency of the light. With mid-infrared (IR) light the frequency of oscillation of the photons is close to that of nuclear oscillation (motion) while the frequency of visible and near infrared (NIR) photons is much higher and corresponds to the frequency at which electrons in molecules will oscillate.

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When UV/visible and NIR light impinges on matter, the electric field varies over time and the electrons move resonantly (oscillating with the same frequency), while the nuclei remain at their positions, thereby creating an induced dipole moment through the formation of regions of excess negative and positive charges (Fig. 8.1). In this simple model, we assume that the Born
Oppenheimer approximation holds; nuclei are essentially static on the timescale of these processes and only the electrons are affected by the interaction with light. We will see later with the Heller formulism that this is not exactly true. The oscillating motion of the electrons (i.e., accelerating and decelerating charge) in turn generates electromagnetic radiation with precisely the same frequency of the light used to excite them, which is termed scattered radiation. Rayleigh scattering is the dominant process and its wavelength dependence (1/λ4, discussed later in the chapter) gives rise to the blue color of the sky—blue light is scattered more easily than red light. It is important to note that the scattering refers to the fact that the generated Rayleigh scattered light travels in all directions and should not be confused with Mie scattering (i.e., scattering due to particles with dimensions close to those of the wavelengths of the light used). In IR spectroscopy, the energy of a photon is absorbed, resulting in excitation from the lowest vibrational state (ϕi ) to a higher energy state (ϕf ). The energy of the photon (E 5 hν) must equal the difference in energy between the two vibrational states, that is, it must be resonant. For the transition to occur, there must be a change in dipole moment between the two states, which is represented by the transition dipole moment (~ μ ) and the transition ^ The transition takes place because the nuclei oscillate at dipole moment operator (μ).

FIGURE 8.1 (A) Interaction of electromagnetic radiation (E 5 hν 5 hc/λ) with nuclei and with electrons. For photons at lower energy (frequency, IR region), the nuclei oscillate under the influence of light and the electrons respond instantaneously to the change in nuclear position (upper part). In contrast, light in the UV/vis frequency range will induce oscillation of the electrons only, and the nuclei will remain at their equilibrium positions (lower part). (B) The Born
Oppenheimer approximation; electrons move on a timescale much shorter than nuclei and hence an electric field created by light in the UV/vis region will distort the electron “cloud” around nuclei raising the energy of the molecule but without changing the coordinates of the nuclei and hence the nuclear wave function (wavepacket). (C) Oscillating dipole created by a molecular vibration.

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frequencies in the same range as of the light impinging on them (i.e., in the infrared region, c. 1012 Hz). The interaction of the oscillating electric field of light with matter at much higher frequencies (UV, visible, and NIR light) either: • induces movement of electrons, which in turn delays propagation of the wave (the real component of the refractive index/dielectric constant of a material) • results in the absorption of the energy of the photon (absorbance is the imaginary component of the refractive index) • changes the direction of propagation (Rayleigh scattering) • or occasionally results in the photon losing or gaining energy (Raman scattering) Although the interaction of a photon with matter is easiest thought of as the molecule absorbing the energy of the photon, this can only happen if a number of conditions are satisfied—the most important of which are resonance and the conservation of angular momentum. However, if the electrons in a molecule interact transiently with the electric field created by the photons without absorbing their energy then the molecule is excited to a virtual state, from which it generates a photon of the same frequency as the incident light as the electrons move back to their original positions. Note that an oscillating charged species will generate electromagnetic radiation. We will discuss the details of this in depth later in the chapter.

The classical oscillator, Hooke’s law, the force constant, and quantization The basis of vibrational spectroscopy has its roots in the study by Robert Hooke in the 17th century of oscillators, such as the motion of pendulums, and masses connected by springs. From a classical perspective, we can think of a simple molecule such as dihydrogen (H
H) as two point masses (the nuclei) that are subject to forces: • The first force is due to the mutual repulsion of positively charged nuclei. The repulsion between positively charged nuclei means that energy is needed to bring the nuclei together. The energy is stored as potential energy (PE). • The second force is the attraction of the nuclei to the shared electrons and the closer the nuclei are, the closer they are to the electrons , and hence the lower their potential energy is (i.e., separating the electrons and nuclei takes energy). Together these two interactions give rise to the London dispersion forces. If at some distances the net result of the two interactions is a decrease in potential energy (i.e., the electron nuclear attraction is greater than the internuclear repulsion) then a stable situation is realized (Fig. 8.2D). The range of distances over which the attraction outweighs the repulsion is relatively narrow and the bottom of the well has approximately the shape of a parabolic curve; a feature that is made good use of in the mathematical descriptions below. In cartoon fashion we can think of a bond as being two atoms connected by a spring (Fig. 8.2A). In the classical (Newtonian) world two properties of oscillators make immediate sense. The oscillator can be at rest (i.e., the pendulum is not swinging, balls connected by a spring are not moving) or the oscillator is in motion (oscillating). At any momentary

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FIGURE 8.2 (A) The pendulum. Epot is potential energy and Ek is kinetic energy. (B) Hooke’s law and the spring connecting two balls model of a chemical bond. (C) Local nature of vibrations. (D) Morse potential well formed by competition between internuclear repulsion and electron nuclear attraction. If the net result is a minimum (bonding interaction) then the minimum is at the equilibrium bond length and the region around it approximates a parabolic curve.

observation in time, it is most likely (maximum probability) that it will be at the extreme points of the oscillation since it is moving the slowest, and has the lowest kinetic energy/ highest potential energy, at these points. If the angle through which the oscillation takes place is small, then pendulums (oscillators) always swing (oscillate) at a set frequency. The oscillation frequency is dependent only on their length (Fig. 8.2A). For masses connected by a spring, however, both the stiffness of the spring and the effective mass determine the frequency of

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oscillation (Fig. 8.2B). These observations led Robert Hooke to establish what became known as Hooke’s law: sffiffiffiffiffiffiffiffi 1 k υosc 5 2π meff where υosc is the natural frequency of the oscillator (for molecules usually expressed in wavenumbers, cm21) and is proportional to the force constant k (a measure of the strength of a bond) and inversely proportional to the effectivemass (meff ), which for a two atom oscillator (Fig. 8.2C) is equivalent to the reduced mass μ 5

m1 m2 m1 1 m2

.

The restoring force K exerted on the atoms that are oscillating depends on the point in the oscillation that the system is at, that is, the restoring force is greatest when |ΔR| is at a maximum and zero when |ΔR| 5 0 (Fig. 8.2B). Indeed K 5 2kΔR and the potential energy is Ek 5 12 kx2 (where x 5 jΔRj), which happens to be a formula for an upward opening parabola (Fig. 8.2D). The stiffness of the bond is reflected in the magnitude of k (the force constant); the greater k is, the stiffer (stronger) the bond is and the lower the amplitude and greater the frequency of oscillation.

Quantization and the nature of a quantum excitation A particle, such as a molecule, is in a particular energy state and at a particular position, which can be described by a wave function. Changing the molecule to a different state (i.e., different energy and/or position) requires that an operation is carried out. Although macroscopically we think of moving position or changing energy as a continuous process, at the microscopic level the changes are quantized and changing the energy of an entity (be it electronic, nuclear, vibrational, or rotational—but not translational!) can only be done in discrete steps. It should not be forgotten that quantization, in quantum mechanics, is enforced by the imposition of a boundary. Only wavefunctions that satisfy the boundary conditions are allowed. In the case of translational motion, the boundaries are so far apart compared to the de Broglie wavelength of the particle that changes in energy can be considered continuous. The vibrational energy in molecules is quantized, however, due to the confinement imposed by the potential well. In mathematical terms we describe the transition between states using an integral as follows; ð ^ i dτ ψf μψ where ψf is the wave function describing the final state, ψi the initial state, and μ^ is the transition dipole moment operator (the mechanism by which a molecule changes from one state to another). It is the dipole moment operator that describes (reading left to right) how the final state is arrived at from the initial state. The integral is over all space since the wave functions describing each state are continuous over all space τ. However, typically for vibrational spectroscopy we usually consider the wave function along a single coordinate, especially where a bond elongation and contraction is all that is of concern to us (Fig. 8.3).

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FIGURE 8.3 Interaction of plane polarized (along y-axis) light with a Cl
Cl bond.

A quantum mechanical oscillator can only exist in any of a series of discrete states with energies: sffiffiffiffiffiffiffiffi    1 h k 1 Evib 5 hυosc ν 1 5 ν1 2 2π meff 2 where v 5 0, 1, 2, etc. The time-independent Schrodinger equation: 2

h ¯ 2 d2 ψ 1 2 1 kf x ψ 5 Eψ 2 2meff dx2

allows us to calculate the total energy of an oscillator ðEvib Þ as a sum of its kinetic and potential energy (Fig. 8.3). The potential depends on how much (jΔRj 5 x) longer or shorter the bond is compared with the equilibrium length of the bond (the length that corresponds to the minimum of the potential well, x 5 0). The potential varies with separation smoothly (continuous curved function) and can be modeled using a Taylor series. The potential (Vx) at any distance from the minimum can be estimated by:     dV 1 d2 V x1 x2 1 . . . Vx 5 Vx50 1 dx x50 2 dx2 x50 The first term Vðx50Þ is the potential at the equilibrium bond length (and the bottom of the well) and can be arbitrarily set to zero since it is the minimum. The potential well’s curvature approximates a parabolic curve near the minimum and hence the second term

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FIGURE 8.4 The formal definition of the force constant (kf). The stronger a bond is, smaller the amplitude of the oscillations and the narrower the potential well will be. The rate of change of the slope of the curve  2   d V increases as kf increases. dx2 x50

dV

dx x50 x

is also zero since the slope of the tangent at the minimum of an upward opening parabolic curve is by definition zero when x 5 0. The third term is the first nonzero term:   1 d2 V x2 Vx 5 2 dx2 x50 and it corresponds to the steepness of the potential well. For small displacements from the equilibrium bond length, the higher terms are negligible and hence the force constant that appears in Hooke’s law (kf) can be replaced by   d2 V : dx2 x50

indeed this is the formal definition of the force constant (Fig. 8.4).

Although the vibration of a bond in a molecule can be modeled as a harmonic oscillator, in reality the potential well is anharmonic. The anharmonicity arises becuase the bond must always be a positive distance (i.e., the separation between atoms must be greater than zero) and the bond cannot be infinitely long, as beyond a certain length the bonding interaction and the restoring force are negligible. However, at ambient temperatures the amplitude of the oscillations is small and the behavior of two bound atoms approximates, albeit not exactly, that of a harmonic oscillator.

Permanent, induced, and transition electric dipole moments The following sections are somewhat of an aside from our discussion of vibrational spectroscopy, however, for a detailed discussion of the phenomenon of resonance enhanced Raman scattering, it is pertinent to first remind ourselves of some basic concepts

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related to the interaction of electromagnetic radiation and matter. We begin with the concept of permanent electric dipole, transition dipole, and then induced dipole moments to build a basis for understanding Raman spectroscopy.

Electric dipole moments When two monopoles of opposite charge are in proximity they will generate a moment (a force) called an electric dipole moment (μ) between them. The strength of this dipole moment is macroscopically given by the charge multiplied by the distance between the two charges (μ 5 qR). The electric field (E) generated by the two charges is given by: ~5 E

1 ~ μ 4πεo R2

where R is the distance between the two charges and εo is the vacuum permittivity. Since the potential well for the oscillator is parabolic, we can calculate the dipole moment μ using a Taylor expansion as we did for potential above:     dμ 1 d2 μ μ 5 μo 1 x1 x2 1 . . . dx x50 2 dx2 x50 Where   μo is the dipole moment at the equilibrium bond length and is a constant, and dμ is the rate of change of dipole moment at the equilibrium bond length, etc. dx x50

Since we can measure the electric dipole moment (it is an observable), then in quantum ^ which describes the change in mechanics we can assign a dipole moment operator μ, dipole moment between the initial and final states, and:     dμ 1 d2 μ x1 x2 1 . . . μ^ 5 μo 1 dx x50 2 dx2 x50 (We will use this relation in the next section.) The transition dipole moment operator changes also as bond length changes with ^ where x^ is the position operator. Since ‘x^ 5 x 3 ’ then μ^ 5 qx: the dipole moment μ^ 5 qx, operator is dependent on both the charge and the difference in distance between the charges, that is, the atoms in the bond, compared to the equilibrium bond length. Permanent dipole moments and especially the strength of the dipole moments are often related directly to the strength of electronic and vibrational (IR) absorption bands. However, it is the change in dipole moment during the excitation (absorption of a quantum of energy) that determines absorptivity rather than the strength of the permanent dipole moment.

Transition dipole moments Absorbing the energy of a photon results in a change in energy of a system (molecule) and, by definition, means a change in state, specifically from the initial state (ψi ) to the

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final state (ψf ), and this can either proceed directly (i.e., as in IR absorption spectroscopy) or via intermediate states (as we will see with Raman spectroscopy). In both cases the transition involves the loss or gain of energy and angular momentum through interaction of matter with photons. Unlike in the macroscopic world, the transition between the two states does not proceed as a continuous process but involves the mixing of states; there needs to be some mechanism (an operator) that describes the change (mixing) from the initial to the final state. In the quantum mechanical approach, the transition is achieved ^ Its name suggests through an operator called the “transition dipole moment” operator (μ). the concept of dipoles changing during the change in state and for good reason. The change in dipole moment that accompanies energy loss or gain by a molecule is quantified as μfi , the transition dipole moment. It is the change in dipole moment that occurs during the transition from the initial state (ψi ) to the final state (ψf ), that is, the change in charge distribution. Of course a change in state (energy) of a molecule can mean a change in the electronic, vibrational, and rotational energy. However, because electron motion, nuclear motion, and molecular rotation occur at different timescales (frequencies), we can separate the total wave function that describe each state into three parts: ð ð  ^ i dτ 5 ^ ei ϕvi YJi dτ μfi 5 ψf  μψ ϕef ϕvf YJf μϕ electronic ðϕe Þ, vibrational ðϕv Þ, and rotational ðYj Þ. We can assume that the electronic and rotational states do not change during a vibrational excitation and since we consider only one axis (x), the equation simplifies to: ð ^ vi dx μfi 5 ϕvf  μϕ ^ reflects the redistribution of charge that The transition dipole moment operator (μ) occurs during the transition position (distance from the equilibrium bond length, see previous section). We can replace μ^ with the expansion discussed in the previous section and then separate out the terms.       ð dμ 1 d2 μ 2 μfi 5 ϕvf  μo 1 x1 x 1 . . . ϕvi dx dx x50 2 dx2 x50    ð ð   dμ 5 ϕvf  μ0 ϕvi dx 1 ϕvf  x ϕvi dx dx x50   2   ð 1 dμ x2 ϕvi dx 1 . . . 1 ϕvf  2 dx2 x50   ð ð dμ  ϕvf  ðxÞϕvi dx 5 μo ϕvf ϕvi dx 1 dx x50   ð   1 d2 μ ϕvf  x2 ϕvi dx 1 . . . 1 2 2 dx x50

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FIGURE 8.5 Variation of dipole moment with bond length.

Only the first two terms in the equation need to be considered, as third and higher terms are minor under normal conditions and can be disregarded: Ð • The first term μo ϕvf  ϕvi dx Ðis only nonzero if the initial and final states are the same; if the states are different then ϕvf  ϕvi dx 5 0. Remember that the vibrational states are orthogonal to each  other.  Ð • The second term dμ ϕvf  ðxÞϕvi dx is zero also unless there is a change in dipole dx x50

moment along the x-axis during the transition the rate of change (Fig.   8.5). Specifically dμ of dipole moment at x 5 0 must not be zero dx x50 6¼ 0 . Further analysis below Ð  (Section: What is ϕvf ðxÞϕvi dx?) shows that the selection rule is Δv 5 1/ 2 1.

Polarizability, induced dipole moments, and scattering The extent to which a material can respond to an oscillating electric field is expressed by a frequency-dependent property, polarizability (α). In the context of Raman spectroscopy, α is the ease with which the electron cloud can be disturbed by an electric field. An oscillating electric field (i.e., electromagnetic radiation) induces an oscillation in the position of the electrons in a molecule, which creates a transient (induced) dipole moment (μind). The transient-induced dipole moment differs from the permanent (static) dipole moment due to its time dependence; μind changes continuously. The light-induced

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movement of electrons away from their equilibrium positions raises the energy of the molecule by the energy of the photon (hν) and the molecule is transiently excited to a higher energy state, referred to as a virtual state. We shall return to the concept of the virtual state later. The strength of the induced dipole moment depends on how easily the electron cloud ~t ). can be distorted and is directly proportional to the strength of the electric field (~ μ ind ~ E The polarizability is dependent on bond length, and hence on the nuclear coordinates (Qij), and varies during an oscillation (vibration) (Fig. 8.6); increasing as the nuclei move further apart and decreasing as the nuclei approach each other. Hence, the polarization induced depends also on the bond length and the constant of proportionality, which varies with bond length, μind 5 αx Et . This dependence opens a potential mechanism to induce a change in state in a molecule as described by the induced dipole moment operator, ~t . μ^ ind 5 αx E The magnitude of the polarizability at position x (αx ) can be defined with reference to the polarizability when the nuclei are at their equilibrium positions (x 5 0). If we look again at the equation derived above for the transition dipole moment (Section: Transition dipole moments) but this time consider the induced dipole moment rather than a perma~t to obtain the equation: nent dipole moment, then we can substitute μ^ ind with αx E ð ð ð     ~ ~ μind fi 5 ϕvf μ^ ind ϕvi dx 5 ϕvf ðαx E t Þϕvi dx 5 E t ϕvf  ðαx Þϕvi dx The polarizability depends on position (distance from the minimum) since as the nuclei move further apart or closer together their electrostatic interaction with the electrons changes. We can calculate αx using a Taylor expansion:     dα 1 d2 α αx 5 α0 1 x1 x2 1 . . . dx x50 2 dx2 x50

FIGURE 8.6 An electric field oscillating in the 1015 Hz range will induce a transient dipole moment by moving the electrons in a molecule. The ease with which this can occur depends on the deviation in bond length with respect to the equilibrium bond length—when elongated the electrons are more loosely bound and are easier to displace and vice-versa.

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Since the bond length changes little during a vibration and the third (hyperpolarizability) and higher terms are essentially zero, we need only consider the first two terms:     ð ð ð    - ϕ  α 1 dα - ϕ  ðα Þϕ dx ϕ ðα Þϕ dx 5 E x ϕ dx 5 E μind fi 5 Ex vi 0 0 vi vf vf vi vf t t t dx x50      ð ð ð dα dα ϕvf  ϕvf  ϕvi dx 1 1 Ex ϕvi dx 5 α0 EE- ϕvf  ðxÞϕvi dx t t dx x50 dx x50 t Ð ~t ϕ  ϕ dx is zero unless the initial and final states are the same, The first term α0 E vf vi that is, ð ϕvf  ϕvi dx 5 0 if ϕvf 6¼ ϕvi and

ð

ϕvf  ϕvi dx 5 1

if

ϕvf 5 ϕvi

Hence, if the initial and final states are the same then the induced transition dipole moment is large, which is consistent with the intensity of Rayleigh scattering compared with Raman (Stokes and anti-Stokes) scattering. For Raman scattering to take place, the initial and final states cannot be the same and therefore the first term is zero and the second term has to be nonzero. The second term is nonzero if the initial and final states differ by one quantum (i.e., Δν 5 61, see the next section) and if the rate of change in polarizability at the minimum of the potential well is non  zero, that is, dα 6 0. Hence the polarizability has to change during the oscillation. dx x50 ¼ Ð What is ϕvf  ðxÞϕvi dx? The wave functions for vibrational states are based on a, the Hermite polynomials,  2 which is a set of orthogonal functions weighted by the Gaussian function e2z , that satisfies the boundary conditions for oscillation in a potential well (so-called eigenstates). Their precise description can be found elsewhere, but a brief discussion is warranted here to explain the gross selection rule for Raman and IR spectroscopy and rationalize the origin of Rayleigh scattering, and Stokes and anti-Stokes Raman scattering. In the following discussion:  2 14 h • a 5 m¯eff kf  12 • Nvi and Nvf are normalization constants with Nv 5 αpffiffiπ12v v! • Hv is a Hermite polynomial for a vibrational level (Fig. 8.7). • xa 5 y; and dx 5 a dy y2 • The wave functions ϕν are Gaussian (e2 2 ) modified by the Hermite polynomial ðHv Þ: y2

ϕν 5 Hv e2 2 . The second term in the equation for μind above can be expanded using the Hermite polynomials corresponding to each level; in this case.

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FIGURE 8.7 Potential well for a harmonic oscillator with allowed quantum states equally spaced in energy and shape of Hermitian-based wave functions. The total energy is the sum of the kinetic energy and potential energy (Evib 5 Ek 1 Epot). The shape of the stationary wave functions that describe each vibrational level are given by the Hermitian polynomials (see below) and as the energy increases, the oscillations begin to resemble the behavior of a macroscopic oscillator, with the oscillator spending most its time at the extremes of the oscillation.

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ð ϕvf



   ðN  ðN y2 y2 x 2 22 22 Hvi e ðxÞϕvi dx 5 Nvf Nvi Hvf e Hvf yHvi e2y dx dx 5 Nvf Nvi a 2N 2N ðN 2 5 a2 Nvf Nvi Hvf yHvi e2y dy 2N

We make use of the recurrence relation, yHvi 5 vHvi
1 1 12 Hvi11 , to obtain (Fig. 8.8):   ð ðN 1 2 ϕvf  ðxÞϕvi dx 5 a2 Nvf Nvi Hvf vi Hvi21 1 Hvi11 e2y dy 2 2N

ð ðN 1 N 2 2 Hvf Hvi21 e2y dy 1 Hvf Hvi11 e2y dy 5 a2 Nvf Nvi vi 2 2N 2N

FIGURE 8.8 Rayleigh, Stokes, and anti-Stokes Raman scattering. Note in the case of anti-Stokes Raman scattering the virtual state (see Section: What is a virtual state?) reached is different, but the gap between the initial and virtual states is the same as for Stokes and Rayleigh scattering. The process is allowed only if the rate of  Raman  change in polarizability at the equilibrium bond length is not zero: dα dx x50 6¼ 0.

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ÐN

291

• The first term 2N Hvf yHvi
1 e2y dy describes anti-Stokes Raman scattering and is zero unless vf 5 vi21 . ÐN 2 • The second term 12 2N Hvf yHvi11 e2y dy describes Stokes Raman scattering and is zero unless vf 5 vi11 . 2

In summary, only transitions from υi to υi11 and from υi to υi21 are allowed and hence we have the selection rule Δυ 5 61. In all other circumstances the integrals evaluate as zero and hence overtones are forbidden in Raman spectroscopy, except under specific circumstances (see below). It should be noted that for a harmonic oscillator the separation between each vibrational level is constant. However, in molecules the spacing between the energy levels decreases with increasing vibrational level number due to anharmonicity. Hence, although the probability of transitions where Δυ ¼ 6 61 should be zero, the anharmonicity provides some allowedness for direct excitation by a resonant photon, that is, infrared absorption, in addition to the fundamental transition between the 0th and 1st vibrational levels, transitions between the 0th and 2nd (1st overtone) and 0th and 3rd (2nd overtone) vibrational levels can occur, but with an order of magnitude decrease in molar absorptivity with each additional step. As a final remark in this section, in addition to fundamental transitions, other terms that appear frequently in a discussion of vibrational spectroscopy are: Normal modes: A normal mode is an independent synchronous motion of atoms for which excitation does not lead to excitation of other modes and does not lead to translation or rotation of the molecule. This term appears frequently in the discussion of vibrational spectra. Overtones: (ν 2 ’ν 0 ; ν 3 ’ν 0 . . . etc:Þ are transitions where Δν 6¼ 61 and are formally forbidden. Combination bands are bands due to the simultaneous excitation of two normal modes, in and out of phase.

Relative intensities of Stokes and anti-Stokes Raman scattering The ratio of the Stokes and anti-Stokes scattering depends primarily on the relative number of molecules that are in the lowest or first vibrational state under the conditions used, and is governed by the Boltzmann distribution:   Ni gi 2ΔE k T 5 e B Nj gj where Ni and Nj the number of molecules in the lower (i) and upper (j) vibrational states, gi and gj the degeneracy of the states, and ΔE is the energy gap between the vibrational states. The number of molecules that are in vibrational states above the lowest vibrational state is dependent on temperature. At room temperature about 200 cm21 energy is available. The larger the energy gap, or in other words, the higher the wavenumber or frequency of a mode, the less likely that molecules will be present in the 1st or 2nd vibrationally excited state, and hence in the anti-Stokes Raman spectrum the lowest wavenumber bands are generally the most intense.

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What is a virtual state? Raman spectra recorded with a excitation wavelength much longer (lower energy) than the lowest electronic transition are referred to as non-resonant. Under these conditions, the interaction of electromagnetic radiation (i.e., an oscillating electric field) with the electrons in molecules results in displacement from their lowest energy arrangement. This displacement occurs on a timescale much shorter than required for the nuclei to respond and move. The change in energy is represented as a vertical transition in the normal coordinate diagram shown below. The name “virtual state” reflects the fact that although the nuclei are still at the same position as they were before interaction with the electric field, the electrons are now disturbed from their stable distribution and as a result the energy of the molecule has increased by the energy of the photon. Hence, the “virtual” state is real but does not represent a true electronically excited (or formally a stationary) state and the wave packet (φ) has the same initial “shape” as that of the ground vibrational state (the Franck
Condon principle, Fig. 8.9A). The term “Frank
Condon factors” appears frequently in the discussion of vibrational and electronic spectroscopy and it is worth taking a moment to consider its meaning (Fig. 8.9B). Using the Born
Oppenheimer approximation, the nuclei are static on the timescale that electrons move then for a transition to take place, the overlap of the wave functions of the initial and final vibrational states (i.e., the distribution of the nuclei) has to be nonÐ zero: ϕvf  ϕvi dx 6¼ 0.

FIGURE 8.9 (A) Illustration of excitation to a virtual state “wavepacket” (φ blue dotted line), which relaxes back to (φ(i)) before the propagation of the wavepacket (φ(t), movement of nuclei) creates a significant nonzero Frank
Condon overlap with a vibrational state (φ(f)) other than the initial vibrational state (φ(i)). (B) Example of large and small overlap of wave functions and hence high and low overlap integrals (Franck
Condon factors).

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Later in our discussion of the description of the Raman effect proposed by Heller, we will see that the nuclei begin to move immediately (wavepacket propagation, φ(t)). The movement (displacement) is limited, as the electrons return to their original distribution (χ(i)) by generation of a photon of the same frequency as the incident photon (Rayleigh scattering) before significant changes in nuclear position (compared to φ(i)) can take place. Occasionally, and with a probability of the order of 1 in 10
100 million, the wavepacket will have propagated sufficiently (i.e., nuclei actually move slightly) for relaxation to a different vibrational level (φ(f)) of the ground state to occur, since the overlap integral (between the two vibrational wave functions in the virtual and ground states) is no longer zero, giving rise to Stokes or anti-Stokes Raman scattering.

The (resonance) Raman experiment From an experimental perspective, Raman spectroscopy is relatively straightforward. A laser is focused onto a sample and some of the light scattered is collected by lenses, focused into a spectrograph and dispersed on a charge coupled device (CCD) array to generate a spectrum of intensity of scattered light versus wavenumber shift (Fig. 8.10A). The vast majority of the light scattered is the same wavelength as that of the laser (Rayleigh scattering) and will swamp the weak Raman shifted scattering if it enters the spectrograph. Hence a steep long pass filter is used to allow only light of longer wavelength (or shorter wavelength in the case of anti-Stokes Raman spectroscopy) than the laser wavelength to enter the spectrograph.

FIGURE 8.10 (A) A basic experimental arrangement for Raman spectroscopy. (B) Comparison of Raman and FTIR spectra of cyclohexane.

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The Raman spectrum generated is analogous to an FTIR spectrum with the key difference that the selection rules and hence relative intensity of bands are different (Fig. 8.10B). However, the expected positions for particular bands in the FTIR spectrum are the same in the corresponding Raman spectrum, and tables of expected band positions for IR spectroscopy are equally applicable in the interpretation of Raman spectra. An important technical difference, however, is that although Raman spectra are reported with the abscissa in wavenumbers (cm21), corresponding to the far and midinfrared parts of the electromagnetic spectrum, the photons (light) that are received by the detector in Raman spectroscopy are in fact in the visible or near-infrared region. Indeed the abscissa in Raman spectra should be labeled as Raman shift (Δcm21), where Δ refers

FIGURE 8.11 (A) Relation between laser wavelength and wavelength of photons responsible for Stokes and antiStokes Raman scattering. Note that the Raman shift is the difference (in wavenumbers) between the excitation laser and the wavelengths of the Raman photons. (B) UV/Vis absorption spectrum of [Fe(bipy)3]21 (c. 0.1 mM) overlaid with Raman spectra recorded at 532 nm and 785 nm (c. 0.5 M) showing the energy of the Raman photons in nm rather than the more usual wavenumber shift. Note that the Raman scattering at 532 nm overlaps with an absorption band of the complex but that at 785 nm does not (see Section: Confocality, the inner filter effect, and quartz).

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to the difference in energy between the excitation laser and the Raman scattering, as illustrated in Fig. 8.11. Furthermore, the laser used to record Raman spectra must have a narrow spectral linewidth; that is to say that the emission spectrum of the laser must have a full width at half maximum (FWHM) that is much less than the natural linewidth of the Raman bands (i.e., ,0.3 cm21, or 1 GHz). As with IR spectroscopy, the Raman spectrum is a manifestation of both the shape (nuclear coordinates) and distribution of electron density of the molecule. Symmetry and group theory are therefore key mathematical tools in the prediction and interpretation of spectra, a discussion of which is beyond the scope of this chapter. An example of the importance of symmetry (molecular shape) to the appearance of vibrational spectra is shown in Fig. 8.12. The well-known ligand 2,20 -bipyridine adopts a conformation in which the nitrogen atoms are trans to one another to reduce steric hindrance between the H5/H50 hydrogens. When bound to a metal ion, the structure is reversed and while the spectrum is similar, in the sense that bands are present in the same regions of each spectrum, the positions and relative intensity of the bands are different, reflecting the change in shape.

Raman cross-section and the intensity of Raman bands There are two aspects of vibrational spectroscopy with which we are concerned: band position and band intensity. In both IR and Raman spectroscopy, the band position (resonant frequency) depends on the strength of the bond (kf ) and the effective mass of the oscillators. In IR spectroscopy, transmittance depends nonlinearly on concentration and the transmission pathlength and hence for quantitative work absorbance and molar absorptivity are used. Since we use the ratio of light transmitted with and without the sample we give little, if any,

FIGURE 8.12 Raman spectra (λexc 785 nm) of solid samples of 2,20 -bipyridine and its ruthenium(II) complex

[Ru(bipy)3]21.

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TABLE 8.1 1/λ4 dependence of Raman scattering on laser wavelength. Wavelength (nm) of lasers used commonly

Raman intensity relative to 1064 nm

266

256

355

81

532

16

632.8

8

785

3

1064

1

thought usually to instrumental aspects (intensity of the lamp, detector response, etc.). In contrast, band intensity in Raman spectroscopy is dependent on a number of factors including practical/physical parameters such as the intensity of excitation light (laser, Iexc), the frequency of the laser (v), the solid angle (Ω) over which the Raman scattering is collected, concentration (number of molecules in the confocal volume, N), and the square of the rate of change of polarizability (α) at the equilibrium nuclear coordinates (Q):  2 dα Intensity ~ υ4 Iexc ΩN dQ From an experimental perspective, the first four terms “υ4 Iexc ΩN” are of practical importance and responsible for the increase in Raman intensity at shorter wavelengths. All things being equal, with regard to detector sensitivity, the same Raman intensity can be achieved at 266 nm using a laser power of c. 1.2 mW as at 785 nm with 100 mW, or at 1064 nm with 300 mW (Table 8.1). However, detector sensitivity and the performance of optics and gratings can have a large influence on the overall intensity and hence the enhancement in Raman scattering is not a major consideration in the choice of wavelength of the laser. From a practical perspective the intensity of a Raman band scales linearly with concentration (Fig. 8.13) and the solvent itself, or another compound can act as an internal reference to correct for variations in other parameters between measurements.

Raman scattering is a weak effect; but how weak? As a point of reference, high-performance Raman spectrometers can be expected to achieve limits of detection between 10 and 100 mM. However, this is dependent on the nature of the analyte and its polarizability and in our own experience the limit of detection for nonresonant Raman scattering can be as low as 5 mM For compounds such as H2O2, however, even under ideal situations the limit is at best 20 mM, and more usually 100 mM. In contrast, in many biological systems the concentration of the chromophore of

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FIGURE 8.13 Variation in Raman (λexc 785 nm) band intensity with concentration for cyclooctene in CH3CN with 1,2-dichlorobenzene as internal reference. Inset: calibration curve generated from data showing linearity and limit of detection.

the species under examination can be as low as in the low micromolar concentration range. These concentrations pose a challenge to the limits of detection of many techniques, not least Raman spectroscopy. A feeling for how weak the Raman effect is can be gained by considering Fig. 8.14. The emission spectrum of the well-known luminophore [Ru(bipy)3]21 in water and acetonitrile is shown with excitation at 355 nm in this particular case. The emission spectrum, in terms of shape, does not depend on excitation wavelength and hence by choosing 355 nm we can easily see the Raman scattering from the solvent also. This demonstrates that the only difference between a spectrofluorimeter and a Raman spectrometer is performance (i.e., narrow linewidth laser and high resolution spectrograph). However, it should be noted that the emission spectra have been scaled up by several orders of magnitude, so that the Raman scattering is noticeable. If we note that the transmission is 90% at 355 nm and therefore [Ru(bipy)3]21 absorbs 10% of the 10 mW of light impinging on it and reemits at most 0.02% of that light, then the total area of the emission spectrum represents 2 μW, or 1 in 5000 photons. The area of the Raman scattering is c. 0.01% of the area of the total emission and hence only 1 in 50 million of the photons hitting the sample are scattered as Raman scattering! Furthermore, the concentration of most analytes of interest are typically in the 10
100 mM range and hence we reach the 1 in 25 billion photons range. This of course is a “back of the envelope” calculation and is therefore imprecise, however, it serves to highlight the challenge faced in recording Raman spectra. It should also be noted that, in this example, the wavelength of the laser used was chosen so that the Raman scattering would appear in a region of the spectrum where it

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FIGURE 8.14 Comparison of the intensity of the emission from the complex [Ru(bipy)3]21(10 μM) in water (blue) and acetonitrile (orange) with the Raman scattering from the solvent with excitation at 355 nm (10 mW). The spectra are expanded to reveal the Raman scattering between 350 and 410 nm.

does not overlap with the emission spectrum of the complex. If the laser used was at 532 nm, in principle, the narrow Raman bands would sit “on top” of the very broad emission band and a simple baseline correction could be applied to see only the Raman spectrum. However, this will not work due to “shot noise.” Shot noise is the major source of noise with CCD cameras used in Raman systems and the noise (N) scales with pffiffiffiffiffiffiffiffiffiffiffiffiffiffi the total intensity (Shot noise ~ Signal) and hence the noise is due to the total signal (Emission 1 Raman scattering) and not just the Raman scattering. In short, fluorescence or phosphorescence present in the wavelength range where Raman scattering is generated will often swamp the weak Raman signal and make it essentially impossible to record a useful spectrum. With non-resonant Raman the solution is of course to use a different laser wavelength, however, for resonance Raman spectroscopy, which is the subject of this chapter, this can mean “moving out of resonance”.

Resonance enhancement of Raman scattering In the study of bioinorganic systems, a major challenge is that of the intrinsically low concentration of the species under examination, for example a metalloenzyme. Indeed, even in its pure form, the weight percentage of the active site within the protein is often relatively low. Hence, it is not obvious that Raman spectroscopy, a phenomenon that is

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improbable and with detection limits typically in the 10
100 mM range, would become a leading technique in the field. The success of Raman spectroscopy lies in the effect referred to as resonance enhancement, in which the Raman scattering by a chromophore can be enhanced by orders of magnitude and resulting in limits of detection for such a species as low as the micromolar range. In this section, the phenomenon of resonance enhancement and the technique known as resonance Raman spectroscopy will be discussed. However, we will start with a simple but essential point: There is no such thing as a resonance Raman spectrometer; and there is no such thing as running a resonance Raman spectrum. The experiment, referred to colloquially as resonance Raman spectroscopy, is in all cases with all samples simply the recording of a Raman spectrum on a standard spectrometer equipped with a laser that is chosen to be “resonant” with, that is, has the same wavelength as, an optical absorption band of the chromophore of interest.

The Raman spectroscopy of carrots and parrots We will first consider the expected Raman spectrum of a carrot to illustrate the effect of resonance enhancement. The Raman spectrum of a root vegetable, for example a parsnip or a carrot, would be expected to be composed of scattering due to the major components primarily, in order of concentration, water, polysaccharides (sugars), lipids, and proteins. Trace components, that is, those with concentrations less than 10 mM, will not contribute significantly as the Raman scattering they produce is likely to be below the signal-to-noise threshold. Water and carbohydrates are by far the major components but generally give weak Raman scattering. We would expect bands due to water stretching and bending at c. 3500 and 1600 cm21, and, in addition, bands at just below 3000 cm21 due to aliphatic C
H stretching and at 1400 and 1250 cm21 due to C
H wagging and C
C stretching modes. Bands at essentially the same positions will be observed for lipids and proteins present in addition to weak contributions from esters

FIGURE 8.15 UV/vis absorption spectrum (of ethanol extract) of a carrot and Raman spectrum of untreated carrot (λexc 632.8 nm). Although the wavelength of the laser is 150 nm longer than the lowest absorption band of the carrot, preresonance enhancement is substantial.

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and amides. The spectrum of a carrot recorded with a laser at 632.8 nm is shown in Fig. 8.15. The spectrum is far from that expected. Two intense bands are present, at 1250 and 1550 cm21, that are consistent with a polyene (β-carotene) rather than any of the expected contributions from lipids and carbohydrates. The dominance of these bands are due to resonance enhancement. A more subtle example of resonance enhancement can be seen in the Raman spectra of parrot feathers. Color is of substantial importance in the natural world, seen most vividly in the striking color combinations presented by members of the parrot family. Color can arise from either of two sources; structural and pigmentation. In the case of structural color, the thickness of layers of biomaterials or periodicity of photonic crystals are close to those of the wavelength of light so that interference phenomena create color. These approaches are used especially to achieve blue/green colors. Black is typically due to cross-linked polyphenolics (i.e., melanin produced from polymerization of oxidized tyrosine). The red and yellow colors are typically produced by organic chromophores, however, their identity is difficult to determine, due to the extremely low concentrations at which they are present in biological matrices, and extraction methods can lead to reactions that alter chemical structure. Taking a parrot feather as an example, we would predict a Raman spectrum based on the composition of the feather, which is essentially collagen. Indeed when a white feather is focused on with a red laser (e.g., 632.8 nm) the spectrum, although weak, shows all expected bands of the amino acids present in collagen Fig. 8.16. Focusing on a black region results in burning the feather due to heating caused by the absorption of the light and little information can be extracted. By contrast, if we examine a red or yellow region, we see the expected weak Raman scattering of the protein matrix, but also several very intense bands at c. 1100 and 1550 cm21. These bands are due to Raman scattering from a chromophore, which, although present in micromolar or lower concentrations, shows strong Raman scattering. Comparison of the spectrum with that of the carrot above leads to the immediate conclusion that the chromophore is a carotenoid, however, the differences in wavenumber positions indicate that they are not identical structures. When the

FIGURE 8.16 Raman spectra (λexc 632.8 nm) of a white (black line) and red (red line) region of a feather from a Moluccan cockatoo (A) full scale and (B) expansion to show weak collagen bands. (C) Raman spectra of yellow (yellow spectrum) and red (red spectrum) regions showing the 6
10 cm21 shift in the main bands.

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focus point is moved to a yellow region of the feather the general shape of the spectrum obtained is largely the same, however, there are two noticeable differences. The peak positions are shifted and the signal-to-noise ratio has decreased considerably. The latter effect is consistent with a difference in the absorption spectra of the chromophores with the compound providing the yellow color absorbing less at 632.8 nm and hence the resonance enhancement expected is less (there is an increase in the denominator in the Kramers
Heisenberg
Dirac (KHD) equation for polarizability, see below). The difference in wavenumber shift of the bands is not due to optical effects as in the region between yellow and red, it is clear that the spectrum is a sum of the two contributing spectra. The change in wavenumber indicates a change in structure, but it must be remembered that this change can be due to conformational effects induced by a surrounding protein environment (as is the case with retinal), rather than a change in molecular structure. Recall that the vibrational spectrum reflects the structure in three dimensions and hence a change in conformation is sufficient to change the vibrational, as well as visible absorption, spectrum. These examples, although somewhat trivial, highlight a number of important aspects of resonance Raman spectroscopy. Firstly, in all cases the spectrum is simply a Raman spectrum in terms of how it is recorded. The wavelength of the laser used has a large effect on the absolute intensity of the Raman scattering due to the ν4 dependence on the intensity of Raman scattering on wavelength. If the laser is in resonance with an electronic transition in a compound, then the Raman scattering for that compound can be increased by many orders of magnitude. In this case, the Raman spectrum is dominated by the carotenes, despite being present in concentrations typically of 0.1 mg/g or 0.01 wt%. In the next section, we will consider the origin of this increase in intensity, but first we will look at how Raman scattering from the solvent on its own and a dissolved colored compound changes as the wavelength of the laser is changed from 785 nm to 244 nm. As the laser wavelength approaches the visible absorption bands, the Raman scattering from the compound is expected to increase substantially due to resonance enhancement. However, as we will see, this does not necessarily occur. The complex [Cu(dmbipy) (H2O)2]21 absorbs strongly in the UV region (ππ* transitions) and only weakly in the visible and NIR region (metal centered absorption bands) (Fig. 8.17). When we compare the spectra from which Raman scattering from the solvent has been removed by spectral subtraction, it becomes apparent that the band positions match exactly, that is, the Raman bands appear with exactly the same wavenumber in spectra obtained both on and off resonance. However, as resonance enhancement becomes more pronounced it becomes clear that the relative intensity of bands, even those that are of similar wavenumber, varies dramatically. This is an important feature of resonance Raman spectroscopy and these differences in relative intensity of bands allows us to probe or to verify the electronic structure responsible for the absorption bands, and in particular to identify the type of transition it is (e.g., metal-centered, ligand to metal (LMCT) and metal to ligand (MLCT) charge transfer, etc.) and which parts of the molecule undergo changes in electron density as a result of the excitation. In summary, in addition to an

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FIGURE 8.17 UV/Vis absorption and Raman spectra of [Cu(dmbipy)(H2O)2]21 at various nonresonant and resonant wavelengths (Draksharapu, et al. 2015a). The concentration used was 100 mM at 785 nm, 10 mM at 473 nm and 0.3 mM at all other wavelengths.

overall increase in Raman scattering intensity, the phenomenon of resonance enhancement is characterized by the observation that: • the Raman scattering from only a limited number of modes undergo significant enhancement, that is, not all bands observed in the Raman spectrum of a compound will be observed under conditions of resonance enhancement; • the relative intensity of bands will often be different at different resonance wavelengths; and • the position (in wavenumber, cm21) of the enhanced bands are identical to those observed in the Raman spectrum recorded under nonresonant conditions, that is, at a wavelength far removed from an electronic absorption band of the compound.

Classical description of Rayleigh and Raman scattering There are a number of approaches that can be taken to explain the interaction of light with molecules (Long 1977). The classical approach makes use of the relation between polarization P and the strength of the electric field, which microscopically is expressed ~t . In the case of electromagnetic radiation E ~t 5 E ~0 cos2πν 0 t, where E ~0 is the as ~ μ ind 5 αE amplitude of the oscillating electric field and ν 0 is the frequency of the electromagnetic radiation (in the visible region 1015
1016 Hz). The nuclei also move in an oscillatory manner and the nuclear coordinates (Qj ) vary with frequency ν j such that   Qj 5 Q0j cos 2πν j t , where Q0j are the coordinates of nucleus j at the equilibrium position and ν j is the frequency of the oscillation. The polarizability is dependent on the position

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Resonance enhancement of Raman scattering

of all the nuclei as we found above (α 5 αo 1



dα dQj

 Qj ). Hence, we can make a series of

substitutions:



μind 5 αEt 5 αE0 cosð2πν 0 tÞ 5 αo E0 cosð2πν 0 tÞ 1



 dα Qj E0 cosð2πν 0 tÞ dQj

   dα Q0j cos 2πν j t E0 cosð2πν 0 tÞ dQj     0 - dα 5 αo E0 cosð2πν 0 tÞ 1 Qj E0 cos 2πν j t cosð2πν 0 tÞ dQj 5 α o E0 cosð2πν 0 tÞ 1

remembering the identity: cos A cos B 5 cosðA 1 BÞ 12 cosðA 2 BÞ       1 0 - dα   - cosð2πν tÞ 1 1 Q0 E- dα Q μ5 α E 1 ν Þt 1 E cos 2πðν cos 2πðν0 2 νj Þt o 0 0 0 j j 0 j 0 ind 2 dQj 2 dQj Rayleigh anti
Stokes Stokes The first term corresponds to elastic or Rayleigh scattering, while the second and third terms correspond to the anti-Stokes and Stokes scattering, respectively. This formulation is wholly classical and misses many of the details of Raman scattering but already we can   see that the rate of change of polarizability at the equilibrium nuclear coordinates dα dx x50 must not be equal to zero. Furthermore, the induced dipole moment (polarization) is linearly dependent on the strength of the electric field.

The Kramer
Heisenberg
Dirac (KHD) equation The quantum mechanical treatment of polarizability (α) can be carried out using the KHD equation, and differs from the classical description in the previous section in that it rationalizes the factors that influence its magnitude, in particular the dependence of polarizability on the wavelength of excitation and resonance enhancement. The wavelength dependence of the polarizability is derived from the KHD equation: ! X hΨF jμ^ρ jΨI ihΨI jμ^σ jΨG i X hΨI jμ^ jΨG ihΨF jμ^ jΨI i   σ σ αρσ GF 5 k 1 ωGI 2 ωhυ 1 iΓI ωGI 1 ωhυ 1 iΓI I I which describes the polarizability (α) of a molecule, where ρ and σ are the polarization of the incident and scattered light and summed over all vibronic (vibrational-electronic) states of the molecule, with the subscripts G indicating the ground state, I the intermediate state, F the final state, ΨG ; ΨI ; and ΨF are the ground, intermediate, and final vibronic states, and μ^ρ ; μ^σ are the dipole moment operators going from the ground to intermediate and intermediate to final states, respectively (Rousseau et al., 1979). The first term essentially mixes the ground, final, and excited states, starting with the ground state. The second term is almost the same but in the opposite direction. The sign in the denominators is important. In the first term, as the wavelength of excitation approaches the gap between ground and excited electronic states, ωGI and ωhυ cancel each other and the term becomes very large. In contrast for the second term the denominator increases as the excitation

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wavelength shortens and the term decreases in importance. (see Section: Heller’s time dependent approach). The damping factor iΓ is related to the “lifetime” of the electronically excited state. It represents the dephasing of the excited state wavepacket that occurs due to other degrees of freedom. The value of 1/Γ is typically a few vibrational periods. Importantly it prevents the denominator from becoming 0 (which would result in infinite polarizability!). The wave functions can be simplified by applying the Born
Oppenheimer approximation, which assumes that nuclei do not move on the timescale that electrons move. The nuclear and rotational components can then be separated (and the rotational part neglected for simplicity). The electronic component is described with the term MIG(Rx50) with MIG(Rx50)0 as a correction factor. Rε describes the distance from the equilibrium position. The three terms are referred to as the A-, B-, and C- (not shown) terms: X ΦF jΦI ΦI jΦG   αρσ GF 5 kM2IG ðRx50 Þ ωGI 2 ωhυ 1 iΓI I A
Term 0

1 kMIG ðRx50 ÞM IG ðRx50 Þ

X ΦF jΦI ΦI jRε jΦG 1 ΦF jRε jΦI ΦI jΦG I

ωGI 1 ωhυ 1 iΓI B
Term

The first (A) term has as numerator the Frank
Condon factors, the overlap between the vibrational wave functions in the ground and excited states, hΦF jΦI ihΦI jΦG i. If the potential wells of the ground and excited states are identical, that is, there is no difference in bond length (mean nuclear coordinates, ΔQ . 0) or force constant (bonds involved in the mode get either stronger or weaker, Δk . 0) between the ground and excited electronic states, then this term is equal to zero (no overlap) unless the ground and final states are the same. Hence, the A-term contributes to polarizability primarily in the case of Rayleigh scattering. The importance of this term to Raman scattering increases, however, as the wavelength of excitation approaches resonance with the electronic transition. The importance is apparent from consideration of the denominator ωGI 2 ωhυ , which makes the whole term increase rapidly as ωhυ approaches ωGI . If the equilibrium bond lengths and/or force constant are different for the ground state and the electronically excited state that comes into resonance with the excitation laser, then the overlap between the ground and final states that are not the same becomes nonzero also, which is the origin of the intensity of Raman scattering only observed for those modes that are distorted in the excited state. D   E2 Finally, the first term contains M2IG ðRx50 Þ (which is ψe μψg ), and hence the magnitude of the A-term depends on the square of the electric dipole transition moment (manifested in the molar absorptivity). Taken together the two terms indicate that the increase in polarizability and hence resonant enhancement depends on the “allowedness” of the electronic transition. The second (B) term includes contributions to the polarizability due to vibronic coupling between excited electronic states (i.e., ψe and ψs ). This term is important only when the two electronic states are close in energy (ΔE in the dominator) and the transition dipole moments between the ground state and each of the excited states is nonzero

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(i.e., both transitions  components in the numerator are also important  are allowed).

The to consider, since vQj m and vQj n connect the ground (ψg ) and excited (ψe ) states that differ by one vibrational quantum.  Importantly when they are multiplied by the Franck
Condon factors (e.g., ΦRF ΦRI ) for vibrational states that have the same quantum numbers, they do not vanish (go to zero) even if the excited state displacement is equal to zero (ΔQ 5 0). Hence both totally and nontotally symmetric modes can be enhanced. The C-term is omitted above and for a detailed description the interested reader is referred to the original paper by Albrecht (Albrecht 1961). In summary, under non-resonant conditions the A-term provides for Rayleigh scattering and not Raman scattering since it is zero if the initial and final vibronic states are not the same. The B-term contributes most to polarizability and hence Raman scattering. When the laser used is resonant with an electronic absorption band, both A- and B-terms contribute to the intensity with the A-term dominating.

A-, B-, C-term enhancement mechanisms, overtones, and combination bands The mathematical basis for the theory of resonance enhancement of Raman scattering (Czernuszewicz and Spiro 1999) is beyond the scope of this chapter. However, the conclusions reached by Albrecht (Albrecht 1961), using the Herzberg
Teller formalism to treat the quantum mechanical description of dispersion, are important in regard to the excitation wavelength dependence of resonance Raman spectroscopy and its use in probing the electronic structure of chromophores. The enhancement of Raman scattering from a compound near or at resonance conditions is typically due to the so-called A-term enhancement. The primary consideration is the difference in mean nuclear coordinates (Q, i.e., bond lengths/dihedral angles) between the ground electronic state and the electronically excited state that is in resonance with the wavelength of the laser (ΔE 5 hν exc). The enhancement is also proportional to the electric dipole transition moment for the electronic transition, and hence the more allowed the electronic transition, the greater the expected enhancement. This is in part why porphyrins give intense resonantly enhanced Raman spectra when excitation is into their allowed π
π* transitions, and why the d-d bands of metal complexes do not provide significant enhancement (Fig. 8.18).

Assigning electronic absorption spectra The assignment of localization of electronic transitions in UV/Vis absorption spectra is an important application of resonance Raman spectroscopy. An example of this can be seen in the heteroleptic Ru(II) polypyridyl complex shown in Fig. 8.19. The visible absorption spectrum shows two main transitions at c. 420 and 550 nm. These visible absorption bands are metal to ligand charge transfer in nature (1MLCT), however, the excitations are relatively localized. Specifically the electron is excited from the t2g orbitals of the Ru(II) ion to a ligand π* orbital (π*’t2g). The complex [Ru(bipy)2(Me-Phpztr)]21 in this case contains two equivalent 2,20 -bipyridine ligands (bipy, shown in black) and a third ligand comprises a methylated cationic pyrazine (shown in red) and an anionic 1,2,4-triazole moiety (shown in blue). Raman spectra recorded between 355 and 561 nm are shown in Fig. 8.19. Practical Approaches to Biological Inorganic Chemistry

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FIGURE 8.18 A change in equilibrium bond length and/or force constant between the ground and excited state is required for the band associated with that normal mode to undergo resonance enhancement.

FIGURE 8.19 UV/Vis absorption spectrum of a hetereoleptic (not all ligands are the same) ruthenium(II) polypyridyl complex (shown as inset) and Raman spectra recorded at the indicated wavelengths (note the gap in the Raman spectra at ca. 1350 cm-1 is minimise interference from imperfect subtraction of solvent bands) (Draksharapu, et al. 2012).

The spectra obtained at 355, 400.8, 449, and 473 nm are essentially identical to those expected for the homoleptic (i.e., all ligands are the same) complex [Ru(bipy)3]21 (Fig. 8.12). In all four spectra, the relative intensities of the bands change but their positions do not. The assignment of these bands as due to enhancement of Raman

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scattering from the bipy ligands was confirmed from the shifts induced by deuteration of the ligands (i.e., with d 8-bipy all bands underwent a shift to lower wavenumbers). At 532 nm intense bands at c. 1630 and 1240 cm21 are observed also and at 561 nm these are the only bands that are enhanced significantly. These additional bands are assigned to the methylated pyrazine moiety again based on shifts induced by isotope labeling. These data allowed us to assign the low energy absorption as being predominantly 1MLCT (pz’t2g) and the absorption at 420 nm as 1MLCT (bipy’t2g). The wavelength dependence of the resonance Raman spectrum in this case highlights an essential point in resonance enhancement. Charge transfer transition involves formal one electron reduction of a ligand or ligand moiety. The reduction involves population of an antibonding orbital and hence there are substantial changes to bond order and equilibrium bond lengths, that is, ΔQ . 0. A nonzero value of ΔQ is an essential requirement for A-term enhancement. For the other ligands, the change in oxidation state of the metal center has relatively little effect since the electron is taken formally from a nonbonding orbital. Hence in the electronically excited state, the other ligands are unchanged in terms of bond lengths and strengths and hence ΔQ 5 0. As a result there is no mechanism for their Raman scattering to be enhanced. A second mechanism for enhancement of Raman scattering is through vibronic coupling between two electronically excited states. Only modes that are responsible for the coupling are enhanced, which are often nontotally symmetric modes. Bands that are only weakly enhanced at some wavelengths can become intense at other wavelengths depending on the electronic excited state structure. The enhancement of nontotally symmetric modes (e.g., A2g, B1g, B2g in porphyrins) is consistent with vibronic bands in the absorption spectrum. Nontotally symmetric modes are identifiable by the “anomalous polarization”; that is, if the laser used is plane polarized, then Raman scattering under resonance conditions should be polarized as well. B-term enhancement is characterized by some of the Raman bands not retaining the laser’s polarization. The third mechanism for enhancement is referred to as C-term enhancement (this term is not usually included in the form of the KHD equation above), and is much less commonly encountered. It becomes important when the excitation wavelength used is resonant with a vibronic side band of a forbidden or weakly allowed electronic absorption band such as the Q-bands of porphyrins, which although forbidden, “borrow” intensity from the allowed Soret transition (Fig. 8.20b). Without going into detail, the C-term is proportional to two integrals that depend on Q (nuclear coordinates) and connect vibrational levels in the ground jgi and electronically jei excited states that differ by one quantum of angular momentum—hence overtones are enhanced. C ~ hfjQjei  hejQjgi Raman spectroscopy of porphyrins and related compounds is a well-established field in itself due primarily to the strong resonant enhancement observed upon excitation across their electronic absorption spectra. At certain wavelengths the spectra become much more complex than expected due to enhancement of overtones (Paulat, et al. 2006). A characteristic of C-term enhancement is the enhancement of overtones and combination bands which is seen in spectra of porphyrins when excited into the Qv region of their

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absorption spectra (Fig. 8.20b). Such enhancement is notable because, in contrast to FTIR spectroscopy, overtones and combination bands are generally absent or weak in Raman spectra, and especially resonantly enhanced Raman spectra. The observation of overtones and combination bands is most likely for modes that undergo the largest displacements in the electronically excited states and when Γ is large. An example of this effect is observed in the Raman spectra of an Fe(IV) 5 O complex reported by Que and coworkers where the first overtone of the 798 cm21 stretching mode appears at 1587 cm21, and, importantly, shows the expected sensitivity to 18O labeling (see Van Heuvelen et al. 2012).

Heller’s time-dependent approach Polarizability can, in principle, be calculated using the KHD equation, in which all vibronic (vibrational-electronic) states are included. In practice these calculations are prohibitively large and expensive in computational time, even for small molecules. However, the time-independent approach taken in the KHD equation is a useful starting point for the time-dependent approach in which the calculation of polarizability is simplified considerably by taking a semiclassical approach as proposed by Eric Heller (Heller, 1981). In the Heller approach, the problem of the breakdown in the Born
Oppenheimer approximation is reduced by considering how the nuclei move during the excitation. As discussed above, the Frank
Condon principle states that electrons and nuclei move at different speeds and therefore all electronic transitions are vertical with respect to nuclear coordinates—that is, the nuclei do not move to an appreciable extent on the timescale over which the electronic distribution in the molecule is changed by an oscillating electric field (Fig. 8.20a).

FIGURE 8.20

(A) C-term enhancement allows for significant intensity for overtones, when excitation is in resonance with the second vibrational level of an excited state. (B) UV/vis absorption spectrum of zinc tetraphenylporphyrin in dichloromethane.

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After excitation, the nuclei feel a new force field and as a result begin to move to reach a new set of nuclear coordinates that minimizes the energy of the system. The distribution of the nuclei once the electronically excited state is reached is essentially that of the ground state, in other words the molecules are in an excited “virtual” state called a “wavepacket” (Fig. 8.21). Electronic transitions can be described by the overlap of the time-independent vibrational wavefunctions (eigenfunctions) to form a wavepacket. The wavepacket begins to “move” along the excited state surface—or rather the nuclei in each molecule begin to move towards their new equilibrium positions with a speed corresponding to the frequency of the normal modes that bring them to these new nuclear coordinates. Only modes associated with bonds that differ in length between the ground and excited state are involved in this process. If we consider the ground and excited states involved in an electronic transition, the initially formed vibrational wave function in the excited state is the displaced wavepacket φð0Þ, which is equal to the ground state vibrational wave function multiplied by the transition dipole operator. For simplicity we will disregard direction and orientation aspects, that is, the polarization of the incoming photons with respect to molecular axes.

FIGURE 8.21 Excitation from the ground state leads to a virtual state which has the same vibrational wave function (called a wavepacket) as the ground state. The wavepacket changes due to nuclear motion, and would ultimately evolve to a stationary state of the excited state, were it not for the instantaneous relaxation to the ground state with generation of a scattered photon.

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The frequency dependence of the polarizability (α(ω)), with a single excited state involved, is given by the KHD equation, which can be expressed when only one excited state is considered (Heller et al., 1982), by two terms (resonant and non-resonant):         X ψn μψk ψk μψo X ψn μψk ψk μψo 1 α0-n ðωÞ 5 ðEk 2E0 Þ 2 h ¯ ω 1 iΓ ðEk 2E0 Þ 1 h ¯ ω 1 iΓ k k ðresonant termÞ 1 ðnonresonant termÞ where ψo is the ground vibrational state, ψn is the nth vibrational state, ω is the frequency of the incoming photons, ψk is the kth vibrational eigenstate (wave function) in the excited state, and Ek is the energy of that state. Note that this equation is essentially a repeat of the KHD equation mentioned at the beginning of Section “The Kramer
Heisenberg
Dirac (KHD) equation”. For large molecules only a small part of the excited state potential surface is of relevance, since only a few of the nuclei in the molecule are displaced in the excited state. Hence an alternative equation, proposed by Heller and coworkers, can be applied that limits the number of states involved and considers the evaluation of the wavepacket with time: The polarizability with respect to a specific frequency is given by: ðN α0-n ðωÞ 5 eiωt2Γt hφn jφðtÞidt 1 ðnon resonant term Þ 0

where jφi 5 μjψð0Þi and jφn i 5 μjψðnÞi and φðtÞ are the same wavepacket at any point in time. The Raman scattering into a given final state n is proportional to the square of the amplitude of the Electric field. Even though the integration is from 0 to infinity, the actual integration is only necessary over short time periods due to dephasing and, less obviously, if the excitation frequency is moved away from resonance then the uncertainty Δωτ is approximately equal to 1 where Δω is the deviation from resonance and τ is lifetime of the wavepacket in the excited state, that is, the time before relaxation to the ground electronic surface takes place.

SERS and SERRS spectroscopy In surface-enhanced Raman scattering (SERS) and surface-enhanced resonance Raman scattering (SERRS) spectroscopy, nanostructured noble metals (principally silver and gold) are used to enhance Raman scattering by several orders of magnitude. SERS spectroscopy is relatively simple to implement and is used extensively in analytical applications and especially in sensor applications. Surface-enhanced spectra of compounds are obtained upon aggregation of aqueous gold or silver colloids by addition of anions (Cl2, SO422, etc.) or using nanostructured surfaces. Analyte that is present between the contacting (aggregating) nanoparticles benefits from a so-called antenna effect, arising from the intense electric field (hot spot) formed between the particles. As for resonance enhancement due to the coincidence of the laser wavelength with an electronic absorption band of the analyte, in SERS spectroscopy the aggregation is

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FIGURE 8.22 UV/Vis absorption spectrum of a gold colloid before and shortly after addition of KCl(aq) to induce aggregation.

optimized to shift the surface plasmon resonance of the colloid to be coincident with the laser used (typically 632.8 nm or 785 nm). If the analyte itself has an electronic absorption band at the wavelength of excitation, then enhancement of Raman scattering is expected due both to the surface and resonance enhancement (Fig. 8.22). A key challenge to the application of SERS in (bio)inorganic chemistry is the noninnocence of the gold and silver colloids, and especially the gold and silver ions present unavoidably in solution. As an example, the SERS spectra of two copper(II) complexes obtained using gold colloid are shown in Fig. 8.23. It is clear from comparison with the ligand only that the SERS spectra are in fact of the gold(I) complexes and not of the original copper(II) complexes. Indeed, any analyte that is a metal complex in which the ligand is labile, or which bear functional groups that can bind to gold or silver (e.g., N, S, alkynes etc.), will likely provide a spectrum of a gold or silver complex rather than the original form of the analyte. The interaction of analytes with the colloid can enhance their fluorescence, which can swamp the weak Raman signal. Often the colloid quenches the excited states of bound molecules and reduces interference from fluorescence, while providing enhancement of the Raman spectrum.

Experimental and instrumental considerations Isotope labeling and band assignment The assignment of Raman spectra is relatively straightforward, given that the band positions for particular types of normal modes (vibrations) are the same as those found in FTIR spectra. Indeed the tables available for IR spectroscopy can be used equally well for Raman spectroscopy. The only real caveat to this is that the relative intensities of Raman bands do not correspond with those in IR spectra. Recently DFT methods have developed to a sufficiently sophisticated level that we can accurately predict vibrational spectra.

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FIGURE 8.23 SERS spectra of both copper(II) complexes (black) and of the ligands alone (red) on aggregated gold colloid. The spectra are identical and are distinct from the solid state spectra of either the complexes or ligands in terms of band positions, confirming that the spectra are of the gold complexes of the ligands.

In the studies in which resonant enhancement is of interest, we generally know the structure of the compound under study already. Often we want to confirm that a particular band is assigned correctly and in this isotope labeling is especially useful, given that a primary use of Raman spectroscopy in bioinorganic chemistry is in determining the frequency and hence force constant for key bond classes (e.g., O
O, Fe
O stretching, etc.). Indeed, isotope labeling is a key tool in the definitive assignment of bands in species that are otherwise not sufficiently stable to be isolated. An example of the use of resonance enhanced Raman spectroscopy in bioinorganic and biomimetic chemistry, and the role played by isotope labeling, is found in the study of species formed upon the reaction of iron complexes with oxidants such as oxygen and hydrogen peroxide. The species of interest are typically Fe(III)
OOH, Fe(III)O2 and Fe (IV) 5 O intermediates, for example an Fe(III)
OOH species is the last detectable intermediate in the oxidative cycle of the DNA cleaving anticancer drug Bleomycin. Until relatively recently, these species had not been isolated and characterized structurally. Their near-infrared absorption bands around 600
850 nm have allowed us to use resonance enhancement to determine the strength of the Fe(IV) 5 O and O
O bonds, which can be related to the reactivity of the complexes. In the example in Fig 8.24, an Fe(III)
OCl complex is formed by reaction of an Fe(II) complex with sodium hypochlorite. Four bands are enhanced resonantly at the wavelength of excitation used, with the band at c. 845 cm21 undergoing the expected shift for an Fe(IV) 5 O species formed under these conditions (Fig. 8.24). The other three bands are due to the Fe(III)
OCl species formed Exchange of 16O with 18O shows that only two

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FIGURE 8.24 Raman spectrum of an Fe(III)-OCl species generated in solution by addition of NaOCl to the Fe

(II) complex. Replacement of H216O with H218O results in shifts of some but not all bands (Draksharapu et al. 2015b).

bands undergo a shift in wavenumber. The band at c. 670 cm21 is not sensitive to oxygen labeling as it is due to a Fe
N stretching mode. The other two bands undergo a shift consistent with their assignment as Fe
O and O
Cl stretching modes. DFT methods allow for the prediction of both their wavenumber shift and the isotope shift. This example illustrates that vibrational modes are essentially localized over a few atoms (two or three typically) and in this case the excitation is into a ligand to metal charge transfer band of the complex, in which the iron formally changes from Fe(III) to an Fe(II) oxidation state. The iron
ligand bond lengths differ in the two oxidation states and as a result there is a change in nuclear coordinate (ΔQj 6¼ 0). The resonance enhancement, however, is modest (approximately 10
50 times) which reflects the forbidden nature of the transition.

Resolution and natural linewidth In many cases Raman spectroscopy is employed to monitor changes in conditions (e.g., binding or release of a ligand, change in oxidation state, etc.). The change in the spectrum

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must be beyond the resolution limit of the spectrometer to begin with but often when changes are minor it falls to the experience of the spectroscopist to discern whether the changes are due to the species under study or simply due to instrumental/optical properties. Furthermore, although modern instrumentation can achieve resolutions as low as 0.1 cm21, this is only useful for samples in the gas phase and the real limit to resolution in studies in condensed phases (liquids and solids) is determined by natural linewidths (6
8 cm21).

Confocality, the inner filter effect, and quartz A key aspect of Raman spectroscopy is that it is inherently a confocal technique, that is, the Raman spectrum measured is due to scattering from a small volume of space called the confocal volume (Fig. 8.25). Resonance Raman spectroscopy presents an experimental dichotomy that can limit application in some cases. An optical absorption at the wavelength of excitation is required in order to achieve resonance enhancement. However, if the confocal volume of the Raman system is at the center of a 1 cm path length cuvette then the laser undergoes attenuation (absorption) before it reaches the confocal volume. Furthermore the Raman scattering has to pass through 5 mm of the sample along the path to the collection optics. Hence, although an increase in analyte concentration increases the (resonance) Raman intensity relative to the solvent bands, the concomitant increase in absorbance leads to an overall reduction in the intensity of the spectrum. In short, sometimes less is more in Raman spectroscopy. An example of this effect is shown for a manganese complex in Fig. 8.26. A major consideration in resonance Raman spectroscopy is the inner filter effect; the reabsorption of scattered light before it leaves the cuvette (see also Fig. 8.11B). As the absorbance of a sample increases there is a tendency to focus at or near the edge of the sample holder (quartz cuvette or glass capillary, etc.) and since the penetration of light can be quite limited (e.g., for total absorbance of 2, the laser is only 10% its initial power by the time it reaches the center of the 1 cm cuvette). Increasing the absorbance to 3 in a 1 cm path length cuvette means that within 1 mm, 50% of the light has been absorbed. In such situations positioning the cuvette so that the confocal volume contains only the first millimeters of solution can reduce the impact of the inner filter effect. However, this also means that the quartz or glass walls of the cuvette are within the confocal volume and Raman scattering from these materials will appear in the spectrum also. A key difficulty is that these bands are broad and are difficult to remove by subtraction, and are especially a problem in studies where isotope shifts in the ,1000 cm21 region are concerned (Fig. 8.27). As a final point of warning that is especially relevant when operating under resonance conditions, it should not be overlooked that the intense highly focused lasers used in Raman spectroscopy can be quite good at inducing photochemical reactions. In many cases it is not possible to obtain resonance Raman spectra simply because the sample has been destroyed before a spectrum can be acquired. In these cases cooling, continuous flow systems, and sensitive detectors that allow for low laser powers to be used, are essential tools.

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FIGURE 8.25 The light scattered during the passage of a laser through a sample is largely lost except for a small volume (the confocal volume in red). Light emanating from this region is collected by a lens and refocused into the spectrograph (through the slits).

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FIGURE 8.26 (A) UV/vis absorption spectrum of manganese complex 1 (1 mM in CH3CN, 1 mm pathlength) showing positions of selected Raman bands with excitation at 355 nm. (B) Raman spectrum at λexc 785 nm of CH3CN (blue), of 1 (0.1 M in CH3CN, yellow) and at λexc 355 nm of 1 (0.001 M in CH3CN, green). The two resonantly enhanced bands of 1 are indicated with arrows—the enhancement factor is estimated as 16 for the band at 700 cm21 and 50 for the band at 800 cm21. (C) Calculated relative intensity of the laser at the center of a 1 cm pathlength cuvette and Raman bands at selected Raman shifts for excitation at 355 nm as a function of concentration. The overall loss in signal intensity in this example is due to the inability of the laser to penetrate the solution, and maximum absolute Raman signal is expected at c. 4
6 mM.

FIGURE 8.27 Raman spectrum of an empty NMR tube (blue) and an empty quartz cuvette (orange) in which the wall of the cuvette is positioned with the confocal volume.

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Applications of resonance Raman spectroscopy In this last section a number of examples of the application of resonance enhancement of Raman scattering (so-called resonance Raman spectroscopy) are discussed. The goal is not to be comprehensive but instead to give an impression of both the ease with which the technique can be applied and its potential to help solve problems.

Resonance enhanced Raman spectroscopy in the characterization of artificial metalloenzymes based on the LmrR protein Resonance Raman spectroscopy, as it is often termed, is a useful asset in studies under biologically relevant conditions, that is, in aqueous buffers, when the compound or moiety of interest is present only at low concentrations. Provided that the compound absorbs light at a convenient wavelength (and by convenient we mean a wavelength close to the wavelengths of the lasers available in one’s own or a friend’s laboratory!), and that absorbance is appreciable (greater than 0.2), it may be possible to study the compound or moiety under conditions relevant to its application. The LmrR protein, a protein that lends multidrug resistance to bacteria, has been applied by the Roelfes group over recent years as a basis for artificial metalloenzymes (Drienovska´, et al. 2015). An artificial metalloenzyme is one in which a protein is modified with, for example, a nonnatural amino acid that allows it to bind a metal ion. In the case of the LmrR protein, the Roelfes group replaced an amino acid (phenylalanine) with an amino acid that bore a 2,20 -bipyridyl unit instead of the phenyl of phenylalanine. The 2,20 bipyridyl unit is positioned by the amino acid sequence within the pocket formed by the symmetric dimeric LmrR protein, and should in principle be able to bind two Cu(II) ions. The copper(II) ions then act as Lewis acid catalysts within the chiral hydrophobic pocket of the protein to engage in enantioselective catalytic reactions. Proving that binding to the 2,20 -bipyridyl takes place selectively, however, is challenging since the Cu(II) ions can bind to other residues around the protein both inside and outside of the pocket. Direct spectroscopic evidence under reaction conditions is ideal, however, the absorption spectrum of the copper(II)-bound LmrR protein shows overlap of the absorption bands of the protein (which contains tryptophan residues) and the expected absorption of the Cu(II) 2,20 -bipyridyl complex. Furthermore, the concentration of the protein available was c. 60 μM. The optimum wavelength for achieving resonance enhancement of Raman scattering of the Cu(II) 2,20 -bipyridyl complexes is 355 nm, however, the use of pulsed laser sources (that is, Nd-YAG laser) with proteins leads to laser-induced aggregation and precipitation. Continuous wave (CW) lasers are available, and in this case the limits of detection for the Cu(II) 2,2’-bipyridyl complex were determined to be c. 20 mM using NO32 (which does not show resonance enhancement) as well as the water Raman band as internal reference. The weak Raman scattering cross-section of water in this case is an enormous advantage, as the strong scattering from organic solvents would have overwhelmed the weak signals of the complex. Comparison with the Raman spectrum obtained for the LmrR Cu(II) complex was hampered by fluorescence due to residual contaminants

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FIGURE 8.28 (A) The limit of detection for the complex in buffer solution is determined by dilution to be c. 20 μM. (B) Using the NO32 anion as an internal reference allows for accurate estimation of the concentration of copper(II) complex formed in the protein by comparison.

remaining after protein purification (see Section: Raman scattering is a weak effect; but how weak?). Although the fluorescence signal is broad and can be removed by a simple pffiffiffiffiffiffiffiffiffiffiffiffiffi background correction, the associated is shot noise ð ~ signalÞ is detrimental to the limit of detection. Nevertheless, comparison of the spectrum of the in situ prepared Cu(II) 2,20 -bipyridyl complex in MOPs buffer with that of the spectra of the native and the 2,20 bipyridyl-modified LmrR protein in the presence and, absence of Cu(II) ions, shows that the Cu(II) ions bind predominantly to the 2,20 -bipyridyl residues. In this example, the resonance enhancement of Raman scattering is applied to confirm the formation of a species under reaction conditions rather than to assign spectra or probe electronic structure. Importantly it has to be emphasized that the experiment was carried out as any Raman spectrum would be recorded, apart from selection of the laser wavelength to coincide with an optical absorption band of the chromophore of interest (Fig. 8.28).

Reaction monitoring with resonance Raman spectroscopy The ability to obtain Raman spectra of compounds present at low concentration is a key advantage of the effect of resonance enhancement in the study of reaction mechanisms. As an example, we will take a manganese catalyzed oxidation of alkenes with H2O2 (Fig. 8.29). The concentration of the catalyst, [Mn(IV)2O3(tmtacn)2]21, used in this example is 1 mM and, as it is in a high oxidation state, its UV/Vis absorption spectrum allows for the observation of the catalyst’s conversion to [Mn(III)2O (μ-O2CR)3(tmtacn)2]21 under reaction conditions. Despite the extensive visible absorption

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FIGURE 8.29 (A) Conversion of 1 to 2 with H2O2 and (B) UV/vis absorption spectra of 1 before (red) and after conversion to 2 (black, 1 mM in CH3CN with 10 mM CCl3CO2H, 1 mm pathlength) with 50 mM H2O2. (C) Raman spectrum (λexc 355 nm) before (black) showing resonantly enhanced bands of 1 at 700 cm21 and at 800 cm21 and after addition of H2O2 (500 mM, purple) with the nonresonantly enhanced O
O stretch of H2O2 at 870 cm21. The bands of 1 and H2O2 decrease concomitant with the appearance of the resonantly enhanced band of 2 at 765 cm21, reaching a maximum (light green) before decreasing also until all H2O2 has been consumed (green). (D) Time dependence of Raman intensity for each of the bands indicated and the intensity of the solvent (CH3CN) band at 920 cm21 which decreases and increases concomitant with the appearance and loss of 2 due to inner filter effects. The changes in the intensity of this band correspond exactly to the changes in the UV/Vis absorption spectrum.

of both species, the Raman scattering of either complex is not enhanced when the laser wavelength is between 457 and 785 nm. However, this is consistent with the assignment of these bands as being largely metal centered (MC) transitions and therefore excitation does not affect the Mn
ligand or ligand bond lengths or strengths significantly. By contrast, excitation at 355 nm results in enhancement of some of the Mn
O
Mn modes for both complexes. The enhancement is consistent with the ligand (oxo) to metal charge transfer character of the bands since the charge transfer in the excited state involves a change in Mn
O bond strength and hence bond length. In this example, the enhancement due to resonance is approximately 50
100 times only and illustrates that even a small increase in signal can be sufficient to obtain useful data. The ability to determine the concentration of both complexes with good accuracy, as well as that of H2O2, allows for direct comparison between them over time.

Transient and time-resolved resonance Raman spectroscopy The recent increase in interest in photoredox catalysis has seen the widespread application of transition metal complexes and organic chromophores that can undergo photoexcitation and then electron transfer to or from other species in solution. When

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FIGURE 8.30 (A) Transient and time resolved Raman spectroscopy are based on exciting a significant number of molecules into their excited electronic state. The Raman spectrum acquired when these excited molecules are present will be a mixture of the Raman spectrum of the molecules in the ground state and the molecules in the excited state. (B) The ratio of the species can be changed by increasing the power of the pump pulse. The molecules relax back to their ground electronic state rapidly after the pump pulse has passed. (C) Example of resonance Raman spectra of a ruthenium(II) polypyridyl complex using a continuous wave laser (where the laser power is constant over time and very few molecules are present in their excited state) and a pulsed laser where all the light arrives within a few nanoseconds in a “packet” (and hence a large proportion of the molecules are excited into their electronically excited state). The pulsed laser also generates Raman scattering from the sample but since most molecules are in the excited state the Raman spectrum is dominated by these. In this case the excited state is a 3MLCT state where formally one ligand has undergone a one-electron reduction and the bands at 1212 and 1285 cm21 are characteristic of such an anion radical.

Raman spectroscopy is carried out using a pulsed laser, that is, a laser which delivers a packet of photons to the sample all within a few nanoseconds, then photoexcitation (excitation to metastable excited states rather than virtual states) becomes significant (see Browne et al. 2005; Browne and McGarvey, 2006, 2007). If we consider the power (number of photons) as a function of time (see Fig. 8.30), then we note that the first photons arriving at the sample will give some Raman scattering, but will excite the molecules into electronically excited states also. The extent to which the molecules in the sample are excited depends on the intensity of the laser pulse (number of photons per unit time). If a significant proportion are in an electronically excited state, then the photons which arrive later (i.e., within a nanosecond or so) will generate Raman scattering from these excited molecules as well as from molecules in the ground state. The relative contributions of Raman scattering from the molecules in the ground state and molecules in the excited state depend on the total number of photons arriving at the sample per unit time. Hence, at low power, relatively few molecules are excited electronically and the Raman scattering is weak (since the laser power is low). As a result, the Raman spectrum is close to that expected with a continuous wave (CW) laser used normally in Raman spectroscopy. If the energy per pulse is increased then the proportion of the sample that is excited will increase also and the Raman spectrum will become more intense overall (high laser power) and the contributions from Raman scattering from the molecules in

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the excited state will increase relative to molecules in the ground state. Changing the relative contributions from molecules in their electronic ground state and excited molecules by changing power is a relatively simple way to gain information on the electronic nature of the lowest excited states and the experiment is referred to as transient resonance Raman spectroscopy (TR2). The technique allows us to probe the nature of the lowest thermally equilibrated excited (THEXI) states and, for example, in ruthenium(II) polypyridyl complexes, to determine whether they are metal to ligand charge transfer (3MLCT) states localized on one or other ligand. This information is important for understanding excited state behavior such as electron transfer, photodissociation, etc. The technique relies the molecule in the excited state also absorbing (is resonant) at the same wavelength as the laser used. If this is not the case then the excited molecule will be “invisible” and the only effect increasing laser power will have is to cause a decrease in the intensity of the Raman scattering of the molecules in the ground state (relative to the solvent bands). Since some compounds have relatively stable excited states that persist for several nano- or even microseconds, and for these a two pulse experiment can give timedependent information. The first pulse arrives at the sample and promotes a part of the sample into its electronically excited state. A second pulse (of either the same or a different wavelength) that is weaker (so as not to disturb significantly the ratio of molecules in the ground and excited states) arrives at a set time after the first pulse and the Raman scattering generated by this pulse gives the spectrum. As the delay between the pulses increases the molecules have more and more time to relax back to the ground state and hence the spectrum will return eventually to that of the ground state only. In this way it is possible to build up kinetic information as to the relaxation of the excited state.

Conclusions Raman spectroscopy is fundamentally a vibrational spectroscopy and provides information as to molecular structure, functional groups, etc. in a manner complementary to FTIR spectroscopy. More than this though, Raman spectroscopy has proven itself over nearly a century of practical and theoretical development to be a versatile and informative tool in the study of molecular systems and materials. The simplicity with which samples can be analyzed and the various mechanisms which allow us to enhance signals from particular species selectively make it unique. It is therefore understandable that the technique has something to offer in every field of chemical science. In this chapter, the phenomenon of resonance enhancement is addressed, including an admittedly cursory consideration of the fundamental theory that underpins the technique. However, the main aim of the chapter is to impress upon the reader two key aspects. The first is that there is little by way of practical difference in recording resonance Raman spectra compared with recording a Raman spectrum. In fact, there is no such thing as a “resonance Raman” spectrum and it is, in my view, unfortunate albeit unavoidable, that the term has crept into the lexicon of this field. What we in fact mean by the term is that the laser wavelength used coincides with an electronic absorption band of a component of a sample under study and as a result its Raman scattering is possibly orders of magnitude greater than would otherwise

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be expected for its concentration in the sample. For the rest it is just a Raman spectrum. The second and final point regards the interpretation of Raman spectra. At its most basic, we interpret a Raman spectrum in essentially the same manner as we interpret an FTIR spectrum (wavenumber, intensity, band shape, etc.), but the technique offers much more than structural analysis and determination of concentration. It allows us to probe the behavior and interactions of molecules in real environments, since sample preparation is often unnecessary, and extract information as to electronic structure in detail. Combined with theoretical methods therefore we have, through resonance enhancement, a unique window to the world of molecules in complex environments and it is perhaps biology that presents us with the most complex of bioinorganic environments.

Questions [q1] Sketch the Raman spectrum of air with excitation at 532 nm (consider Fig. 8.27) showing both Stokes and anti-Stokes at 150 K and at 400 K. [q2] What factors determine the extent of resonance enhancement that can be achieved by excitation into an electronic absorption band? [q3] Why does the so-called A-term in the KHD equation not contribute significantly to Raman intensity when the wavelength of excitation is not in resonance with an electronic transition? [q4] In Fig. 8.24, why do only some of the bands shift when Na18OCl is used instead of Na16OCl? [q5] Why are these Raman bands of the complexes the only bands observed in the spectra in Fig. 8.24, whereas the spectrum of the complex in the Fe(II) oxidation state (Fig. 8.31) shows many more bands in the range 200
1600 cm21?

Answers to Questions [a1] The Raman spectrum of air primarily comprises the single stretching mode of O2 and of N2 and 1553 cm21 and 2330 cm21, respectively. The ratio of Stokes and anti-Stokes band intensities depends on temperature and difference in energy of the first and   second vibrational states. n1 2 1553cm21 At 298 K the ratio for oxygen is 5 exp 5 3:4 3 1027 and n0 0:695cm21 K21 3 150K   n1 2 2330cm21 for nitrogen the ratio is 5 exp 5 2 3 10210 n0 0:695cm21 K21 3 150K Hence the anti-Stokes scattering is 7
10 orders of magnitude weaker than the Stokes scattering. Notably the ratio of the intensity of the N2 band to the O2 band is 6m that observed in the Stokes Raman spectrum. At 400 K the ratio between the N2 and O2 anti-Stokes bands is 0.06 that of the Stokes bands.

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FIGURE 8.31 The effect of deuteration of the benzylic positions of an iron binding ligand on the resonance Raman spectrum of its low spin Fe(II) complex. Note that the bands at 1600 cm21 are essentially unaffected but the aliphatic in and out of plane modes at c. 1250 cm21 are changed dramatically by the deuteration (Draksharapu, et al. 2012). *Bands shifted by deuteration.

[a2] The primary factor is the transition dipole moment for the electronic transition giving rise to enhancement. Hence d-d transitions give low enhancement while π 2 π* transitions give large enhancements. [a3] The A-term has the Frank
Condon factors in the numerator, the overlap 

between

the vibrational wave functions in the ground and excited states, ΦRF ΦRI ΦRI ΦRG . If the potential wells of the ground and excited states are identical,that is, there is no difference in bond length (mean nuclear coordinates, ΔQ . 0) or force constant (bonds involved in the mode get either stronger or weaker, Δk . 0) between the ground and excited electronic states, then this term is equal to zero (no-overlap) unless the ground and final states are the same. Hence, the A-term contributes to polarizability primarily in the case of Rayleigh scattering. [a4] Only bands which involve displacement of the oxygen atom will be shifted and hence the band at 673 cm21 is likely to be a mode involving for example Fe
N stretching. [a5] The absorption bands correspond to charge transfer to oxygen or metal-centered transitions and hence only the bonds between Fe/Cl and oxygen and the nitrogen atoms are affected as they are the only bonds to change in equilibrium bond length and/or force constant (bond strength). These are prerequisites for resonance enhancement.

Acknowledgements The assistance of Doekele Stavenga, Tjalling Canrinus, Davide Angelone, Ruben Feringa, Andy Sardjan, Apparao Draksharapu, and Juan Chen in the collection of data used to prepare some of the figures and Ronald Hage,

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Hanneke Siebe, Linda Eijsink, Hans Kasper, Andy Sardjan, and Jorn Steen for comments and suggestions is greatly appreciated and acknowledged. Steven Bell and John McGarvey (1939
2017) are acknowledged for general discussion, and their boundless enthusiasm for Raman spectroscopy.

References Albrecht, A.C., 1961. On the theory of raman intensities. J. Chem. Phys. 34, 1476. Browne, W.R., McGarvey, J.J., 2006. Raman scattering and photophysics in spin-state-labile d6 metal complexes. Coord. Chem. Rev. 250, 1696. Browne, W.R., McGarvey, J.J., 2007. The Raman effect and its application to electronic spectroscopies in metalcentered species: techniques and investigations in ground and excited states. Coord. Chem. Rev. 251, 454. Browne, W.R., O’Boyle, N.M., McGarvey, J.J., Vos, J.G., 2005. Elucidating excited state electronic structure and intercomponent interactions in multicomponent and supramolecular systems. Chem. Soc. Rev. 34, 641. Czernuszewicz, R.S., Spiro, T.G., 1999. In: Solomon, E.I., Lever, A.B.P. (Eds.), In Inorganic Electronic Structure and Spectroscopy, I. John Wiley & Sons, New York, pp. 353
441. Draksharapu, A., Li, Q., Logtenberg, H., van den Berg, T.A., Meetsma, A., Killeen, J.S., et al., 2012. Ligand exchange and spin state equilibria of FeII(N4Py) and related complexes in aqueous media. Inorg. Chem. 51, 900. Draksharapu, A., Boersma, A.J., Leising, M., Meetsma, A., Browne, W.R., Roelfes, G., 2015a. Binding of copper(II) polypyridyl complexes to DNA and consequences for DNA-based asymmetric catalysis. Dalton Trans. 44, 3647. Draksharapu, A., Angelone, D., Quesne, M.G., Padamati, S.K., Go´mez, L., Hage, R., et al., 2015b. Identification and spectroscopic characterization of nonheme iron(III) hypochlorite intermediates. Angew. Chem. Int. Ed. 54, 4357. Drienovska´, I., Martı´nez, A.R.-, Draksharapu, A., Roelfes, G., 2015. Novel artificial metalloenzymes by in vivo incorporation of metal-binding unnatural amino acids. Chem. Sci. 6, 770. Heller, E.J., 1981. The semiclassical way to molecular spectroscopy. Acc. Chem. Res. 14, 368. Long, D., 1977. The Raman Effect; A Unified Treatment of the Theory of Raman Scattering by Molecules. John Wiley & Sons. Paulat, F., Praneeth, V.K.K., Na¨ther, C., Lehnert, N., 2006. Quantum chemistry-based analysis of the vibrational spectra of five-coordinate metalloporphyrins [M(TPP)Cl]. Inorg. Chem. 45, 2835. Rousseau, D.L., Friedman, J.M., Williams, P.F., 1979. Topics in Current Physics 2, 203. Van Heuvelen, K.M., Fiedler, A.T., Shan, X., De Hont, R.F., Meier, K.K., Bominaar, E.L., et al., 2012. One-electron oxidation of an oxoiron(IV) complex to form an [O~FeV~NR]1 center. Proc. Natl. Acad. Sci. 109, 11933.

Further reading Larkin, P.J., 2011. IR and Raman Spectroscopy
Principles and Spectral Interpretation. Elsevier, Oxford, UK. McCreery, R.M., 2000. Raman Spectroscopy for Chemical Analysis. John Wiley & Sons Ltd, New York, USA. Smith, E., Dent, G., 2005. Modern Raman Spectroscopy
A Practical Approach. John Wiley & Sons Ltd, Chichester, UK.

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C H A P T E R

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An introduction to electrochemical methods for the functional analysis of metalloproteins Vincent Fourmond and Christophe Le´ger CNRS, Aix Marseille University, BIP, Marseille, France O U T L I N E Introduction

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Basics Redox thermodynamics: the Nernst equation Reference potential and reference electrodes The biological redox scale Influence of coupled reactions (e.g., protonation or ligand binding) on reduction potentials Electron transfer kinetics Kinetics of proton-coupled electron transfer: stepwise versus concerted mechanisms

327

Electrochemistry under equilibrium conditions: potentiometric titrations Dynamic electrochemistry Distinction between equilibrium and dynamic electrochemistry

Practical Approaches to Biological Inorganic Chemistry DOI: https://doi.org/10.1016/B978-0-444-64225-7.00009-2

Electrodes for electron transfer to/from proteins Electrochemical equipment Vocab and conventions The capacitive current

327 329 329 330 332 334 335 336 336

Diffusion-controlled voltammetry Diffusion-controlled voltammetry at stationary electrodes Diffusion-controlled voltammetry at rotating electrodes Voltammetry of adsorbed proteins: protein film voltammetry Noncatalytic voltammetry at slow scan rates to measure reduction potentials Fast-scan voltammetry to determine the rates of coupled reactions

337 338 338 339 339 339 341 343 343 347

Catalytic protein film voltammetry and chronoamperometry 351 Principle and general comments 351

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Mass-transport controlled catalytic voltammetry Chronoamperometry to measure Michaelis and inhibition constants Chronoamperometry to resolve rapid changes in activity Determining the reduction potentials of an active site bound to substrate The effect of slow intramolecular electron transfer Slow interfacial electron transfer

353 355 358 360 362 362

Slow substrate binding Slow, redox-driven (in)activation

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Exercises

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Acknowledgements

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Appendices Notations and abbreviations Derivation of Eq. (9.9)

366 366 366

References

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Introduction Direct electron transfer (ET) to proteins (without the need for mediators) was first reported in the 1970s, opening the way for detailed studies of biological reactions, and electrochemical investigations of large redox enzymes are now common. Determining reduction potentials is only one application of the method; in studies of redox proteins or small molecules, electrochemical techniques are used for learning not only about the thermodynamics but also the kinetics of chemical reactions that immediately precede or follow ET (e.g., protonation or substrate binding). Using direct electrochemistry the turnover rate of enzymes can also be measured with very high temporal resolution and potential control. This greatly broadens the possibilities of enzyme kinetics. This technique has indeed been used to study all aspects of catalysis: interfacial and intramolecular ET, substrate diffusion along substrate channels, active site chemistry, mechanism of reaction with inhibitors, redox-driven (in)activation processes, etc. The proteins or enzymes that can be studied using direct electrochemistry have at least one surface-exposed redox center, which is the entry point for electrons from the electrode. The chance of success is greater when the protein of interest is small and hydrophilic, or, if it is an enzyme, when it has a large turnover rate. The amount of protein required depends on which method is used, but it can be as small as a few pmol for a series of experiments carried out with the same “film” of adsorbed proteins. The electrochemical equipment is particularly cheap (compared to many biophysical techniques) and available in most chemistry labs. An extensive description of most of the electrochemical techniques will be found in Compton and Banks (2011) and Bard and Faulkner (2004). The physical aspects of electrochemistry, including hydrodynamics, are discussed in Newman and Thomas-Alyea (2004) and Levich (1962). Save´ant (2006) provided a comprehensive discussion of voltammetric wave shapes under various conditions, with an emphasis on the case where the electrocatalyst diffuses in solution. Cornish-Bowden (2004) and Fersht (1999) are insightful textbooks on enzyme kinetics. There are many comprehensive reviews on the use of direct

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electrochemistry to probe the mechanism of redox proteins and enzymes (e.g., Le´ger and Bertrand, 2008; Jeuken, 2009; Gates et al., 2011; Lojou, 2011; Flanagan et al., 2016; Armstrong et al., 2016; Fourmond and Le´ger, 2016, 2017; Reeve et al., 2017; Sensi et al., 2017a; del Barrio et al., 2018b). Sensi et al. (2017a), Armstrong et al. (2016, 2009), Vincent et al. (2007), and Mazurenko et al. (2017) focus on hydrogenases. Fourmond (2018) focuses on direct electrochemistry of Mo and W enzymes. Various authors have reviewed the strategies for connecting metalloenzymes, including membrane-bound enzymes, to electrodes (Jeuken, 2009, 2016; Lojou, 2011; Mazurenko et al., 2017; Yates et al., 2018).

Basics Redox thermodynamics: the Nernst equation Consider a reaction mixture containing the oxidized and reduced forms of two different species (1 and 2): Ox1 1 Red2 -Red1 1 Ox2

ð9:1Þ

The free energy of the reaction (ΔrG, in units of J/mol) is given by Δr G 5 Δr G0 1 RT ln

½Red2 ½Ox2  ½Ox1 ½Red2 

ð9:2Þ

where R is the gas constant, T the absolute temperature, and ΔrG0 is the (tabulated) standard free energy of the reaction (“standard conditions” means that the activity of all constituents is unity, and the pressure equals one bar). If equilibrium between the different species is reached, ΔrG 5 0, and the ratio of concentrations (the reaction quotient) is linked to ΔrG0 by the relation:
 ½Red1 eq ½Ox2 eq Δr G0 5 exp  Keq 5 ð9:3Þ ½Ox1 eq ½Red2 eq RT Initially upon mixing Ox1 and Red2, the concentrations of various species in the solution do not satisfy Eq. (9.3), and ΔrG is nonzero: the system is not at equilibrium. Thermodynamics predicts that reaction (9.1) will spontaneously proceed in the direction given by dG 5 ΔrGdξ , 0, where dξ is the change in the extent of reaction, until the reaction quotient equals Keq. During the reaction, Red2 is oxidized and gives electrons to Ox1. The overall reaction can be written as the sum of two “half-reactions”: Ox1 1 ne -Red1

ð9:4aÞ



ð9:4bÞ

Red2 -Ox2 1 ne

It is possible to measure the flux of electrons from Red2 to Ox1, and therefore the rate dξ/dt of the overall reaction, by placing the species in a two-compartment cell (Ox1 in one compartment and Red2 in the other). Place in each side an electrode at which the species can interact. When the two electrodes are connected together, a current flows, while the

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system evolves toward equilibrium. This current results from a potential difference between the two electrodes, V 5 E2 2 E1, the value of which can be predicted applying the Nernst equation to each of the two electrodes. E 5 E0 1

RT ½Ox ln nF ½Red

ð9:5Þ

E0 is the standard reduction potential of the redox couple Ox/Red. F 5 96,500 C is the Faraday constant. (Note that the usage is to call E the “electrode potential,” but strictly speaking, it is the difference between the electrical potential of the metal electrode and the electrical potential of the solution adjacent to the metal, ΦM 2 ΦS (Compton and Banks, 2011). The term ΦS cancels in an expression of the potential difference between two electrodes, as in Eq. (9.6)). The electrode potential given by Eq. (9.5) cannot be measured; only the difference between the potentials of two electrodes can. The potential difference between the electrodes in compartments 1 and 2 is: V 5 E2 2 E1 5 E02 2 E01 1

RT ½Ox2 ½Red1  ln nF ½Red2 ½Ox1 

ð9:6Þ

The electrons are going to flow from the cell whose electrode potential is the lowest to the other until V and ΔrG are zero and the concentrations satisfy
 ½Red1 eq ½Ox2 eq  nF  0 0 E 2 E1 Keq 5 5 exp 2 ð9:7Þ RT 2 ½Ox1 eq ½Red2 eq This is equivalent to Eq. (9.3), since reduction potentials and free energies are linked by Δr G0 5 2 nFðE01 2 E02 Þ

ð9:8Þ

The Nernst equation can therefore be used to determine the direction in which a redox reaction will proceed spontaneously [Eq. (9.6)]. Reaction (9.1) will proceed forward significantly (Keq will be large) only if E01 . E02 . If E01 is “high,” Ox1 is called a strong oxidant, and Red1 is a weak reductant. But it may not be enough to compare standard reduction potentials, since the sign of ln([Ox2][Red1]/[Ox1][Red2]) in Eq. (9.6) can change the sign of V (and ΔrG) and thus the direction of the current flow. Also remember that thermodynamics predicts the direction but not the rate of the reaction: a “spontaneous” reaction might not happen because its rate is very small, in which case the equilibrium cannot be reached. It makes more sense, in our opinion, to call E0 a standard reduction potential than a standard redox potential: the sign of ΔrG0 relates to a reaction proceeding in a certain direction, and the sign of a standard reduction potential [as defined in Eq. (9.8)] is what it is because the associated half-reaction is a reduction. The reader should make sure he/she distinguishes between an electrode potential (the difference between two electrode potentials is measured using a voltmeter, it has units of Volts, and relates to the free energy of the reaction, in units of J/mol) and a standard reduction potential (a thermodynamic property which is related by Eq. (9.8) to a standard free energy of reaction). This should not be confused with the electrochemical potential, in units of J/mol, which is the equivalent of the partial free energy (or chemical potential) of a charged species with an additional term that accounts for the effect of the electric potential. Practical Approaches to Biological Inorganic Chemistry

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Reference potential and reference electrodes To study only one half-reaction, it is convenient to make sure that the potential difference between the two electrodes reflects the potential of the electrode one is interested in, called the “working electrode.” This can be done by using in the other compartment an electrode designed to have a constant potential: this is called a reference electrode. The standard hydrogen electrode (SHE) is one of these. It consists of a platinum electrode immersed in a pH 5 0 electrolyte under 1 bar of H2. One can sometimes see references to the “normal hydrogen electrode” — this corresponds to a slightly different version of the SHE, with slightly different potential; its use should be avoided. A real SHE is rarely practical. Instead one uses convenient reference electrodes such as: • Hg/Hg2 Cl2/KCl (standard calomel electrode, SCE, E(SCE) 5 241 mV vs SHE.) This is the most commonly used reference electrode • Hg/Hg2 SO4/K2SO4, used when chloride ions must be avoided. E(SCE) 5 615 mV versus SHE • Ag/AgCl (E 5 200 mV vs SHE) Since only potential differences can be measured, reduction potentials can only be reported against a certain reference electrode. The International Union of Pure and Applied Chemistry requires that this primary reference electrode be the SHE. All reduction potentials tabulated in the literature are (or should be) quoted versus the SHE. Therefore, by convention, the potential of the SHE is zero.

The biological redox scale Because standard conditions (pH 0!) are not suitable for biological reactions, reduction 0 potentials are usually stated for pH 7, and termed E0 or Em,7. FIGURE 9.1 The biological redox scale at pH 7. “OEC” stands for the “oxygen evolving center” of Photosystem II.

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Fig. 9.1 gives an idea of the range of reduction potentials spanned by biologically important redox cofactors and redox couples. It is important to realize that biological processes occur in water, and that the oxidation and reduction of water define the limits of a physiological potential window (although the physiological range extends slightly beyond these two limits).

Influence of coupled reactions (e.g., protonation or ligand binding) on reduction potentials Redox reactions can be coupled to other chemical equilibria such as ligand binding (e.g., protons, substrate, inhibitor) or conformational changes. A very common (and physiologically important) coupled reaction is protonation, as represented in the square scheme in Fig. 9.2. KOx and KRed are the acidity constants for Ox and Red. Utilizing the principle of thermodynamic cycles (the sum of ΔrG0 values round the square is zero), these acidity constants can be linked to the reduction potentials of the protonated and unprotonated redox couples. Note that potentials alone cannot be summed; they must be scaled by n. For an n-electron, one-proton process, the whole pH-dependence of the reduction potential is given by:  ! 1 1 ½H1 =KRed 2:3RT 00 1 0   log10 E ð½H Þ 5 Ealk 1 ð9:9Þ nF 1 1 ½H1 =KOx This equation is demonstrated in the appendix. Usually Red is a better base than Ox, so it has a higher pKa, that is, pKOx , pKRed For pH , pKOx, both Ox and Red are protonated. Ox:H 1 ne2 "Red:H

ð9:10Þ E0acid

The reduction potential is pH independent, and equals (Fig. 9.2). For pH between pKOx and pKRed, Ox is not protonated but Red is Ox 1 ne2 1 H1 "Red:H

ð9:11Þ

The reduction potential is pH dependent: it decreases by 2.3RT/nF Volts per pH unit   [2 59=n mV=pH at 25 C]. This is simply understood as follows: the reduction potential increases as the concentration of protons increases because protonation makes reduction easier. FIGURE 9.2 Square scheme for a protonation reaction coupled to a redox process.

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For pH . pKRed, neither Ox nor Red are protonated, the redox process is Ox 1 ne2 "Red

ð9:12Þ E0alk

The reduction potential is pH independent, and equals (Fig. 9.2). Redox-linked protonations are conveniently represented by a Pourbaix diagram, a plot 0 of E0 as a function of pH, as schematized in Fig. 9.3A. In general, for a redox process involving n electrons and m protons, the maximal pHdependence is 2

2:3RT m V=pH unit F n

ð9:13Þ

As an example, Fig. 9.3B (Jeuken et al., 2002b) shows the dependence on pH of the reduction potential of the type-I copper site of amicyanin, determined using protein film voltammetry (PFV) (see below, e.g., Fig. 9.10). The data can be analyzed in terms of ET being coupled to the protonation of a single group with pKRed 5 6.3 and pKOx # 3.2. Protonation occurs on one of the two histidine ligands of the copper ion.

FIGURE 9.3 (A) Schematic Pourbaix diagram for the 1e2:1H1 reaction of amicyanin, a type-I blue copper protein. (B) Reduction potential versus pH for amicyanin from Paracoccus versatus at 2 C. The values of pKRed and pKOx can easily be measured from the fit of Eq. (9.9), with n 5 1 (line). Source: Data from Jeuken, L.C., Camba, R., Armstrong, F.A., Canters, G. W., 2002b. The pH-dependent redox inactivation of amicyanin from Paracoccus versutus as studied by rapid protein-film voltammetry. J. Biol. Inorg. Chem. 7 (12), 94100. URL: Available from: http://dx.doi.org/10.1007/ s007750100269.

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FIGURE 9.4 Pourbaix-like diagram for the change in the reduction potential of the flavin cofactor of Escherichia coli fumarate reductase as a function of succinate concentration (increasing from right to left) at 20 C, pH 7. Line is best fit to Eq. (9.9), with n 5 2, but with [succinate] instead of [H1]. Source: Data from Le´ger, C., Heffron, K., Pershad, H.R., Maklashina, E., Luna-Chavez, C., Cecchini, G., et al., 2001. Enzyme electrokinetics: energetics of succinate oxidation by fumarate reductase and succinate dehydrogenase. Biochemistry 40 (37), 1123411245. URL: Available from: http://view.ncbi. nlm.nih.gov/pubmed/11551223.

It is essential to understand that these considerations apply for any ligand which binds a redox center: provided that the dissociation constants from the reduced and oxidized forms are different, the reduction potential depends on the concentration of ligand, and the dissociation constants can be measured by determining how the reduction potential depends on the concentration of ligand. The latter may be the proton, the substrate/ product or inhibitor of an enzyme, or even the apo-protein. Fig. 9.4 illustrates the succinate concentration-dependence of the reduction potential of the active site flavin in Escherichia coli fumarate reductase, determined using experiments such as those in Fig. 9.22A. In this case, the reduction potential decreases as the concentration of succinate is raised (from right to left in this figure) because binding of succinate to the oxidized form of the enzyme is tighter than to the reduced form (pKOx . pKRed). Zu et al. (2001), Butt et al. (1993, 1997), Fawcett et al. (1998), and Bak and Elliott (2013) provide examples of the effect of protonation or binding of metal or exogenous thiolate on the reduction potential of FeS clusters.

Electron transfer kinetics Electrons are transferred between the redox centers of redox proteins and enzymes involved in respiration and photosynthesis, and along chains of redox centers within certain enzymes. We first define the parameters that determine the rates of these homogeneous ET events before discussing interfacial ET. Let us consider a certain ET step, from a donor D to an acceptor A. kD-A

Dred 1 Aox " Dox 1 Ared kA-D

ð9:14Þ

Thermodynamics predicts the ratio kD-A/kA-D (Eq. 9.7) but not the values of the two rate constants. The rate of ET from D to A depends on the reduction potentials of D and A [Eq. (9.7)], but also on other parameters. This was established in the 1960s by Rudolph Marcus who developed a model based on a molecular description of ET between small

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molecules in solution and received the 1992 Nobel Prize in Chemistry for his theory. He showed that this process requires the formation of a transient complex, in which the kinetics of the ET step can be described by an equation of the form
 ðΔr G0 1λÞ2 kD-A 5 C exp ð9:15Þ 4λRT • ΔrG0 is the standard free energy of the reaction, which is related to the standard reduction potentials of the donor D and the acceptor A according to Δr G0 5 FðE0D 2 E0A Þ [cf. Eq. (9.8)]. • The parameter λ, called “reorganization energy,” is all the greater that large molecular rearrangements are coupled to the transfer (both the geometries of the molecules that are oxidized or reduced and the polarization of the surrounding solvent are considered). Biological ETs usually take place in the “normal region” (|ΔrG0| , λ), where the rate constant is increased when ΔE0 increases and λ decreases, but the “inverted region” plays a very important role in photosynthetic ETs. • The expression of the preexponential factor C depends on the strength of the electronic coupling between the acceptor and the donor. If it is strong enough (“adiabatic” transfer), C simply equates kT/h, as given by the classical transition state theory. When it is weak (this is so for long-distance, “nonadiabatic” ET) C depends on the overlap of the molecular wave functions of D and A, and therefore on the nature of the redox centers, on their distance and on the intervening medium. An exponential decrease of C with distance is observed (Gray and Winkler, 2005). In the literature, nonadiabatic transfers are often referred to as electron tunneling processes. The relation between thermodynamics and kinetics is understood by calculating an equilibrium constant from the ratio kD-A/kA-D [then compare with Eq. (9.3)]:    C exp 2 ðΔr G0 1λÞ2 =4λRT kD-A 0    5 exp2Δr G =RT 5 ð9:16Þ kA-D C exp 2 ð2Δr G0 1λÞ2 =4λRT Now consider the ET between a molecule and a metallic electrode (the case of semiconducting electrodes is treated e.g., in Bard and Faulkner, 2004). kOX

Red " Ox 1 e2 ðelectrodeÞ kred

ð9:17Þ

The equation that gives the rate of ET is a complex function of the reduction potential 0 of the molecule E0 , the electrode potential E, and the reorganization energy λ (Save´ant, 2006; Le´ger and Bertrand, 2008; Zeng et al., 2014), but a simplified rate equation, known as the “Butler Volmer equation,” predicts that the rates of oxidation and reduction are independent of λ and exponentially increase and decrease (respectively) as the electrode potential increases: 00

kOX 5 k0 expðF=2RTÞðE2E

Þ

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kred 5 k0 exp2ðF=2RTÞðE2E

00

Þ

ð9:18bÞ

The preexponential factor k0 depends on the coupling between the electrode and the redox molecule. The greater k0 the faster the electron exchange between the electrode and the redox molecule; of course, this parameter has no physiological relevance. k0 is often called the rate of ET, but it is important to understand that rates of ET depend on electrode potential, and the rate of ET equates to k0 only when there is no driving force, at E 5 E0 (in which case kox 5 kred 5 k0). The relation between thermodynamics and kinetics is understood by calculating the ratio kOx/kRed [then compare with Eq. (9.5)]: 00

kox k0 expðF=2RTÞðE2E Þ 00 5 5 expðF=RTÞðE2E Þ 00 Þ 2 F=2RT ðE2E ð Þ kred k0 exp

ð9:19Þ

The more sophisticated MarcusHush model of interfacial ET takes into account the distribution of energy states in the metal; it gives Eqs. (9.18a) and (9.18b) justification when the overpotential is small (at E  E0), but in contrast it predicts that the rate asymptotically approaches a plateau value kmax when the overpotential is greater than λ. An inverted region does not occur for an electrode reaction at a metal electrode. For further elaboration, see Save´ant (2006, Section 1.4.2.), or Newman and Thomas-Alyea (2004, Section 3.6.2). Since according to the MarcusHush model, the rate of interfacial ET depends on E and λ, the dependence of the ET rate on E could be used to determine λ, which is a biologically relevant parameter. However, a complication arises from the fact that interfacial ET may be limited (gated) by a process that is distinct from ET. In that case, the plateau of the apparent ET rate may be incorrectly interpreted as the ET rate reaching the maximal value predicted by MarcusHush theory, and the reorganization energy may be greatly underestimated [for a discussion, see Jeuken et al. (2002a) and refs. therein].

Kinetics of proton-coupled electron transfer: stepwise versus concerted mechanisms When a redox process is coupled to proton transfer (judging from the pH dependence of the corresponding reduction potential), one can discern between a stepwise mechanism (whereby ET either precedes or follows proton transfer), and a concerted mechanism (whereby both transfers occur simultaneously). The mechanisms can sometimes be discriminated on the basis of kinetic isotope effects, or from the dependence of rates on T, buffer, pH, and/or ΔG (or E). Electrochemical investigations can be highly informative in this respect (Costentin et al., 2010; Anxolabe´he`re-Mallart et al., 2011). We will discuss in the “Fast-scan voltammetry to determine the rates of coupled reactions” section the simple electrochemical experiments that establish the stepwise mechanism of proton/ET to a buried FeS cluster. We refer the reader to Costentin et al. (2010) and Save´ant (2014) for recent experimental reviews, and to Hammes-Schiffer (2012) for a thorough theoretical introduction to the kinetics of concerted protonETs.

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Electrochemistry under equilibrium conditions: potentiometric titrations The reduction potential of a redox couple is determined by recording the ratio [Ox]/[Red] observed after allowing the system to equilibrate with the electrode at different potentials. The equilibrium potential is usually varied by adding titrants of an oxidant or a reductant. Unlike small molecules, protein redox centers do not generally react rapidly with the measuring electrode and equilibrium is not established quickly. To overcome this problem, small redox agents called mediators are added to the solution to transport electrons between the active site and the electrode. For best results, these should have reduction potentials close to that of the active site being studied; mixtures of mediators are often employed to cover a wide range. A short list of mediators and their reduction potentials is given in Table 9.1. See also Wardman (1989). The ratios [Ox]/[Red] are typically determined by examination of the optical or EPR spectra. Very often the concentration of Ox or Red is plotted as a function of the equilibrium electrode potential, and the data are fitted with: ½Red 1    5 ½Red 1 ½Ox 1 1 exp nF=RT ðE 2 E00 Þ   0  exp nF=RT ðE 2 E0 Þ ½Ox    5 ½Red 1 ½Ox 1 1 exp nF=RT ðE 2 E00 Þ

ð9:20aÞ ð9:20bÞ

Fig. 9.5 is the result of a potentiometric titration of the “blue” (type-I) copper site in an azurin mutant, followed by UVvis spectroscopy. The oxidized copper site absorbs at  600 nm due to a Cys-S to Cu ligand-to-metal charge-transfer, and the intensity of this band is therefore proportional to the concentration of oxidized copper site. Performing such an experiment under fully anaerobic conditions requires a glove-box or skills. Dutton described the glassware that can be used on the bench for the anaerobic potentiometric preparation of samples to be examined by EPR or UVvis spectroscopy (Dutton, 1971; Leslie Dutton, 1978). In the case of species that can exist under three redox states, “Ox,” “Int,” and “Red,” the concentrations obey the following equations: ½Red 1       5 0 0 ½Red 1 ½Int 1 ½Ox 1 1 exp nF=RT ðE 2 E02 Þ 1 1 exp nF=RT ðE 2 E01 Þ   0  exp nF=RT ðE 2 E02 Þ ½Int       5 0 0 ½Red 1 ½Int 1 ½Ox 1 1 exp nF=RT ðE 2 E02 Þ 1 1 exp nF=RT ðE 2 E01 Þ     0  0  exp nF=RT ðE 2 E01 Þ exp nF=RT ðE 2 E02 Þ ½Ox       5 0 0 ½Red 1 ½Int 1 ½Ox 1 1 exp nF=RT ðE 2 E02 Þ 1 1 exp nF=RT ðE 2 E01 Þ 0

ð9:21aÞ ð9:21bÞ ð9:21cÞ 0

In which we denote E01 the reduction potential of the “Ox”/”Int” couple and E02 that of the “Int”/”Red” couple. The concentration of “Int” follows a bell-shaped dependence on 0 0 potential, and it is maximal at E 5 ðE01 1 E02 Þ=2.

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TABLE 9.1 Standard reduction potentials at pH 7, 25 C, of common mediators. 0

Mediator

E0 (mV versus SHE)

Benzyl viologen

2360

Lapachol

2172

Methylene blue

11

Ferricyanide [K3 Fe(CN)6]

360

Hexachloroiridate (Na2 IrCl6)

870

SHE, Standard hydrogen electrode.

FIGURE 9.5 Potentiometric titration of Pseudomonas aeruginosa His117Gly azurin, a type-I blue copper protein (0.1 mM protein, 20 mM MES, pH 6, 33 mM Na2SO4, 1 M NaCl, 20 C.) The different symbols correspond to stepwise reduction and stepwise reoxidation. Plain line is best fit to Eq. (9.20b) with n 5 1. The adsorption at 638 nm is characteristic of the oxidized (blue) form of the copper center. Mediators: K3Fe(CN)6, Na2 IrCl6, 1,2-ferrocene dicarboxyl acid. Source: Data from Jeuken, L.J.C., van Vliet, P., Verbeet, M.P., Camba, R., McEvoy, J.P., Armstrong, F.A., et al., 2000. Role of the surface-exposed and copper-coordinating histidine in blue copper proteins: the electron transfer and redox-coupled ligand binding properties of a mutant of azurin. J. Am. Chem. Soc. 122 (49), 1218612194.

Dynamic electrochemistry Distinction between equilibrium and dynamic electrochemistry In potentiometry experiments (previous section), the measurement of the electrode potential is carried out under equilibrium conditions: the stepwise addition of titrant makes the concentrations change, but when equilibrium is reached the rates of oxidation and reduction exactly cancel each other and there is no net transformation. The concentration of oxidised or reduced species is measured by spectroscopy, and therefore this approach requires that the redox center has a distinct spectroscopic signature.

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A completely different approach consists in forcing the electrode potential to take a value that is different from the equilibrium potential. In that case the system may evolve toward equilibrium by taking electrons from (or giving electrons to) the electrode. This reduction (resp. oxidation) is detected as a current (in units of ampere, that is coulombs/ second) which measures the amount of electric charge passing the electrode per unit time. Therefore it is proportional to the rate of reduction or oxidation.

Electrodes for electron transfer to/from proteins To be successful, electrodes must exchange electrons quickly with the proteins, and preserve their native properties. These electrodes may resemble natural environments or reaction partners for the protein. ET can be achieved if the redox center is exposed at the protein surface (or not too deeply buried). In the case of enzymes whose active site is buried in the protein, a favorable situation occurs when it is “wired” to the surface by a chain of redox cofactors; having one of these centers exchanging electrons with the electrode is enough for achieving an electric connection of the active site. Direct ET to enzymes was first reported in the late 1970s (Tarasevich et al., 1979; Yaropolov et al., 1984) and is now very common, but not all redox enzymes can be electrically connected to electrodes. Protein/electrode interactions may be tailored to be weak or strong. Weak interactions might ideally give rise to diffusion-controlled electrochemistry (“Diffusion-controlled voltammetry” section), whereas with strong interactions the experiment may address just a small sample (“film”) of protein molecules on the electrode (“Voltammetry of adsorbed proteins: protein film voltammetry” and “Catalytic protein film voltammetry and chronoamperometry” sections). Electrode surfaces for which protein electrochemistry is commonly observed are listed below: • Metal (Au, Pt, Ag) surfaces on which a monolayer of adsorbate is self-assembled [selfassembled monolayers (Love et al., 2005)]. The adsorbate is a bifunctional molecule of the type X-(CH2)n-Y, where X is a substituent that anchors the molecule on the metal electrode surface (e.g., a thiol) and Y is a functional group that interacts with the protein (typically carboxyl for cytochromes c, or amino for acidic proteins such as plastocyanin or ferredoxins). • Pyrolytic graphite edge (Blanford and Armstrong, 2006) or basal plane electrodes provide hydrophilic or hydrophobic interactions, respectively. The former is sometimes used with coadsorbates (aminocyclitols, polymyxin, polylysine) which probably form cross-linkages between the protein and the electrode surface. • Graphite or carbon nanotubes can be functionalized by reducing a diazonium salt (Allongue et al., 1997) to expose aromatic functionalities (Blanford et al., 2008) or amino groups (Rudiger et al., 2005; Alonso-Lomillo et al., 2007; Baffert et al., 2012; Abou Hamdan et al., 2012a) that interact favorably with hydrophobic or carboxylate-rich patches on the protein surface. Carbodiimide coupling can then be used to form an amide bond between amino groups on the electrode surface and protein carboxylates (or vice versa).

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• Various electrode materials which cover either a film of nonbiological surfactant (e.g., DDAB) or layers of polyions have been used to incorporate large membrane-bound proteins, but catalytic activity was generally greatly impaired. Suitable electrodes for integral membrane enzymes are described in Jeuken (2009) and Gutie´rrez-Sa´nchez et al. (2011). When the protein that is studied is an enzyme, proof that it is not denatured on the electrode is that it can still catalyze the transformation of its physiological substrate at a reasonable rate and in a range of electrode potential that is consistent with what we know about the catalytic cycle and the redox properties of the cofactors. Checking the effects of known specific inhibitors can also be useful. Upon starting a new project, and before embarking on detailed electrochemical studies, it is essential to make sure that the catalytic properties of the adsorbed enzyme bear some resemblance to those determined in biochemical experiments. In this chapter we illustrate this by also discussing the results of conventional experiments which confirm certain unexpected results obtained in electrochemical investigations of redox enzymes.

Electrochemical equipment The experiment is carried out using a potentiostat in conjunction with the cell. The cell consists of three electrodes. The reference electrode is often contained in a side arm linked to the main compartment by a capillary tip called a Luggin (after the glassblower who invented it). The tip is positioned close to the working electrode. To avoid passing current through the reference electrode (this would change its potential and also damage it), a third electrode, called an auxiliary or counter electrode, is used. The working electrode can also be rotated (“Diffusion-controlled voltammetry at rotating electrodes” and “Catalytic protein film voltammetry and chronoamperometry” sections) to control mass transport of solution species. The potentiostat measures the current registered in response to the potential that is applied. In general, the potential of the working electrode (versus the reference electrode) is modulated (e.g., in a linear sweep) and the current flowing between the working electrode and the counter electrode is recorded. Since the electrode potential is swept forward and back, the technique is called “cyclic voltammetry.” The scan rate ν (in units of V/s) is a very important parameter which determines the timescale of the experiment and therefore the time constant of the processes which can be resolved (Save´ant, 2006). A voltammogram is a plot of current against electrode potential. Alternatively, in an experiment called “chronoamperometry,” the electrode potential is held at a fixed value (or sometimes stepwise changed) and the current is recorded as a function of time.

Vocab and conventions A cathodic process is a reduction, the cathode is the electrode onto which the reduction occurs. A anodic process is an oxidation, occurring at the anode. In Europe a cathodic current is counted as negative and an oxidation results in a positive current. American people and software often use the opposite convention for the sign of the current, and plot the potential from right to left (see e.g., Fig. 9.7).

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The capacitive current The measured current is usually the sum of a Faradaic current (which reveals the redox transformations of molecules that come sufficiently close to the electrode) and a capacitive current ic, which does not involve the passage of electrons across the electrodesolution interface. The capacitive current (or “charging current”) arises as a consequence of the variation of the electrode potential and it is proportional to the electrode surface A: ic 5 C

dE 5 Cv dt

ð9:22Þ

where C, the capacitance of the electrode/electrolyte interface, is proportional to A. In a voltammetric experiment, the magnitude of the capacitive current is therefore proportional to scan rate ν. It is positive if E is increasing and negative if the electrode potential is swept down. There is no capacitive contribution to the current if the potential is constant, but potential steps (in chronoamperometry experiments) result in current transients that are approximately exponential (ic , 0 if the potential is stepped down, ic . 0 if the potential is stepped up). This current usually decays in less than a few seconds. The capacitive current must be subtracted from the total current to obtain the Faradaic contribution. It can sometimes be determined from a control experiment where there is no Faradaic current, or extrapolated from the part of the signal where there is no Faradaic contribution (see e.g., Fig. 9.10 and Box 9.1) (Fourmond et al., 2009a).

Diffusion-controlled voltammetry Diffusion-controlled voltammetry at stationary electrodes We consider a solution containing only the reduced form of a soluble electroactive species, and the potential is swept linearly in time, as shown in Fig. 9.6A, starting from a low potential. The current response as a function of time is plotted in Fig. 9.6B, and the cyclic voltammogram (current against potential) in Fig. 9.6C. 0 While the potential is lower than E0 , no oxidation occurs and no current 0is measured [see 0 (a) in Fig. 9.6]. This is because the rate of oxidation0 kox 5 k0 expðF=2RTÞðE2E Þ is much lower 0 than the rate of reduction kred 5 k0 expð2F=2RTÞðE2E Þ (“Electron transfer kinetics” section). 0 When the electrode potential approaches E0 , Red starts being oxidized into Ox, giving electrons to the electrode. This is measured as a (positive) current which increases as E and the rate of oxidation increase (b). However, the electrode oxidizes only species adjacent to it and the interface is soon depleted. The current reaches a maximum before it starts to decrease (c). It tends to zero like t21/2 (noting that E changes in proportion to time t). This decrease in current reveals that the size of the diffusion layer (i.e., the zone of the solution adjacent to the electrode where the concentration of species differ from that in the bulk) increases by diffusion like the square root of time, and the concentration gradient that drives the diffusion of Red from the bulk of the solution to the interface therefore decreases like t21/2. While a positive current is being measured, Ox produced by the reaction accumulates near the electrode and diffuses slowly toward the bulk.

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FIGURE 9.6 Cyclic voltammetry for a redox species free to diffuse in solution, at a stationary electrode. In a typical voltammetric experiment, the electrode potential is swept linearly in time (A), and the current recorded as a function of time (B). A convenient and usual way of displaying the results is to plot the current against potential (C). The labels (af) are referred to in the text. “arb. u.” means “arbitrary units.” See also the video that shows the evolution of the concentration profiles, posted on our YouTube channel at https://youtu.be/YnmQdOnyfi8.

After the scan is reversed, (d), the current is still positive and decreasing: Red species are still being oxidized since the electrode potential is above the reduction potential. Near the reduction potential, the Ox species which have accumulated are now being reduced and a negative current is observed (e) until the concentration of Ox near the interface drops down (f), and so does the magnitude of the current. This results in a peak-like response in both directions. The modeling of voltammograms is complex because the current response depends on the free diffusion of the species in solution, and the differential equations coupling reaction and diffusion are in general difficult to solve. For an n-electron reaction, the ideal separation between cathodic (reduction) and anodic (oxidation) peaks is given by ΔEp 5 Ep;a 2 Ep;c 5 2:218

RT 57 5 mV at 298K nF n

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ð9:23Þ

Diffusion-controlled voltammetry

FIGURE 9.7 Data for diffusion limited voltam-

2

metry of a redox protein, the cytochrome c2 from Rps. palustris. C 5 0.2 mM, cell volume 0.5 mL. From Battistuzzi et al. (1997). Mind the axis! The voltammogram is plotted with the American convention: the high electrode potentials are on the left, and the current is positive for a reduction. Source: Reprinted with permission from Battistuzzi, G., Borsari, M., Sola, M., Francia, F., 1997. Redox thermodynamics of the native and alkaline forms of eukaryotic and bacterial class I cytochromes c. Biochemistry 36, 16247. Copyright (1997) American Chemical Society.

I (µA)

1

0

–1 0.7

341

0.6

0.5

0.4

0.3

0.2

0.1

E (V) vs. SHE

The peak current is proportional to the bulk concentration of species C, and to the square root of the potential scan rate ν: rffiffiffiffiffiffiffi Fv 3=2 1=2 ip 5 0:446n AD F 3 C ð9:24Þ RT This is the RandlesSevcik equation, where pffiffiffi A is the electrode surface and D is a diffusion coefficient. A linear plot of ip against v is the criterion used to identify when the redox species are diffusing from the bulk to the electrode. A system that conforms to these criteria (peak separation and dependence of the peak current on scan rate) is said to be reversible and diffusion-controlled, and the reduction potential is obtained from the average of the cathodic and anodic peak potentials. 0

E0 

Ep;a 1 Ep;c 2

ð9:25Þ

Fig. 9.7 shows a voltammogram for the reversible oxidation and reduction of a cytochrome. Note that the concentration of the protein sample must be high. For example the experiment in Fig. 9.7 used 100 nmol of cytochrome. Deviation from this ideal behavior might arise when interfacial ET is slow (in which case the peaks broaden and tend to separate), or when one of the redox species is irreversibly transformed on the voltammetric timescale (in which case the signal might become asymmetrical) [see Save´ant (2006) for the effect of follow-up reactions on the voltammetry of diffusive species, and “Fast-scan voltammetry to determine the rates of coupled reactions” section herein for an example with an adsorbed protein].

Diffusion-controlled voltammetry at rotating electrodes The peak shape of the diffusion-limited voltammogram at a macroelectrode is due to the depletion of electroactive species near the electrode surface as they are consumed by the redox reaction.

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FIGURE 9.8 Sigmoidal wave obtained in a cyclic voltammetry of an electroactive species in solution at a rotating disc electrode.

There are many electrochemical techniques in which the solution moves with respect to the electrode in order to minimise depletion. In the most popular configuration, the electrode (called a “rotating disc electrode”) is rotated along its axis in the solution. This introduces a convective movement of the solution which increases the efficiency of the transport of species from the bulk toward the electrode. Because the depletion layer can no longer spread in the solution, the current reaches a limiting value ilim at high driving force (Fig. 9.8). The Levich equation predicts that the limiting current is proportional to the concentration of electroactive species C and to the square root of the electrode rotation rate ω: pffiffiffiffi ilim 5 0:620nFAv21=6 D2=3 3 C ω ð9:26Þ s In this equation, ν s is the kinematic viscosity of the solution. (The kinematic viscosity is the ratio of the viscosity over the density, e.g., the viscosity of pure water at 20 C is 103 μPa s, and its density  1 g/cm3; this gives a kinematic viscosity νs  1022 cm2/s.) A 0 21/2 plot of i21 is called a KouteckyLevich plot. The reduction potential E0 is lim against ω simply given by the half-wave potential E1/2, the potential at which the current reaches half its limiting value. 0

E0 5 E1=2

ð9:27Þ

The scan rate ν and direction do not enter the measurement if ν is small. When the current depends on electrode potential but is independent of time, the voltammogram is said to be at steady state. This configuration is not used for measuring the reduction potential of redox proteins, because rotating the electrode in the solution requires that the volume of the electrochemical cell (and thus the amount of protein) be large, but it is important to understand the difference between this experiment and that shown in Fig. 9.22A, since they give similar electrochemical responses for completely different reasons. The above considerations only apply to “macro”-electrodes (i.e., when the diameter of the electrode is larger than the typical size of the diffusion layer). With a “micro”-electrode,

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whose typical size is of the order of a few micrometers, the voltammogram may have a sigmoidal shape even in the absence of convection [Bard and Faulkner (2004)]. Reference Kudera et al. (2000) shows cyclic voltammograms of amicyanin from Paracoccus denitrificans at a 3 μm gold microelectrode. Microelectrodes have also been used in the context of cell biology; their size make them suitable to detect electroactive species released by a single cell. See, for example, Amatore et al. (2008) for a review.

Voltammetry of adsorbed proteins: protein film voltammetry When the protein is immobilized on the electrode surface, diffusion is eliminated and much greater thermodynamic and kinetic resolution can be obtained with extremely small sample quantities. This approach was developed by F. Armstrong (now in Oxford) in the 1970s with small redox proteins, and since the beginning of the 1980s with large redox enzymes. Over the last 20 years, we and others have used this technique to study all sorts of aspects of the mechanism of redox proteins and enzymes (Le´ger and Bertrand, 2008): proton transfer (Chen et al., 2000) (cf. “Fast-scan voltammetry to determine the rates of coupled reactions” section), inter- and intramolecular ET (Le´ger et al., 2006; Fourmond et al., 2013), diffusion along substrate channels (Leroux et al., 2008; Liebgott et al., 2009) (“Diffusion-controlled voltammetry” section), effect of light (Sensi et al., 2016, 2017b), conformational changes (Zeng et al., 2017), aerobic (Abou Hamdan et al., 2012a; Kubas et al., 2017) and anaerobic (Hamdan et al., 2012; Fourmond et al., 2014; del Barrio et al., 2018a) inactivation of various metalloenzymes, catalytic bias (Abou Hamdan et al., 2012b; Caserta et al., 2018), etc. The rest of this chapter will focus on the principle and applications of PFV.

Noncatalytic voltammetry at slow scan rates to measure reduction potentials Fig. 9.9 shows the ideal shape of a cyclic voltammogram when the redox species is adsorbed onto an electrode (Bard and Faulkner, 2004). Starting again at an electrode potential lower than E0, the redox centers are fully reduced [see (a) in Fig. 9.9]. Sweeping the potential toward high values, the protein starts being oxidized when the electrode potential approaches the reduction potential (b); a positive current is then measured, which drops down to zero at high potential when all the adsorbed protein molecules have been oxidized (c). On the reverse scan, a reductive (negative) current is observed when the electrode potential matches the reduction potential of the protein (d) until the entire sample has been reduced and the current vanishes. For an ideal, reversible system, the signal consists of symmetrical oxidation and reduc0 tion peaks centered at the reduction potential E0 0

E0 5 Ep;c 5 Ep;a

ð9:28Þ

The area under the peak (A, in units of VA) gives the charge passed for that redox couple (nF electrons per mole of adsorbed centers, ΓA, A is the electrode surface). It should be the same for the oxidative and for the reductive peaks.

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FIGURE 9.9 Cyclic voltammetry for a redox species adsorbed on an electrode surface. Rotation of the electrode should make no difference. In a typical voltammetric experiment, the electrode potential is swept linearly in time (A), and the current recorded as a function of time (B). Panel C shows the cyclic voltammogram.

A 5 nFAΓv

ð9:29Þ

The peak current is proportional to the scan rate ν, to the surface concentration of electroactive species Γ and to the square of n. Therefore, the electroactive coverage must be high enough for a current to be observed. Typically, a coverage higher than a few pmol/ cm2 will suffice. ip 5

n2 F2 AΓv 4RT

ð9:30Þ

A linear plot of ip against ν proves that the redox species are adsorbed onto the electrode. The peak width at half height, δ, is: δ  3:53

RT nF

ð9:31Þ

(91/n mV at 25 C). Note that the expected dependence of δ on temperature is not observed in experiments (McEvoy and Armstrong, 1999; Chobot et al., 2007).

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FIGURE

9.10 Cyclic voltammogram for Pseudomonas aeruginosa azurin adsorbed at a pyrolytic graphite electrode. The dashed line is the baseline and the inset shows the baseline subtracted current (the Faradaic current). 0 C, pH 8.5. ν 5 20 mV/s. ΓA  5.5 pmol.

FIGURE 9.11 Cyclic voltammogram for Sulfolobus acidocaldarius 7Fe ferredoxin adsorbed at a pyrolytic graphite electrode (McEvoy and Armstrong, 1999). The dashed line is the baseline. 0 C, pH 8.5, ν 5 20 mV/s. ΓA  3.5 pmol.

One-electron peaks are often broader than this 90 mV theoretical limit and observing a peak that is narrower than 90 mV is a clear indication that two electrons are transferred cooperatively. Indeed, fully cooperative two-ETs give signals with up to four times the height and half the width of one-ETs; they are therefore more easily distinguished. Fig. 9.10 shows a noncatalytic voltammogram that makes it possible to determine the redox potential of the copper site of azurin. In experiments, there is a capacitive contribution resulting from “electrode charging” (“The capacitive current” section). The dashed line shows the interpolated capacitive current which has to be subtracted to obtain the Faradaic current alone (Box 9.1). Fig. 9.11 shows the voltammetry of the 7Fe ferredoxin of Sulfolobus acidocaldarius, which contains one [3Fe4S] cluster, with redox transitions [3Fe4S]1/0 and [3Fe4S]0/22, and one [4Fe4S] cluster with a 2 1 / 1 redox transition. When different centers are present in a protein, the different redox transitions appear as multiple peaks, the areas of which reveal the stoichiometry of the redox processes. For example, the area under the low potential [3Fe4S]0/22 peak is twice as much as that of the high potential [3Fe4S]1/0 peak. This figure also illustrates the fact that two-electron redox processes give prominent signals (cf. Eqs. (9.30) and (9.31)].

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BOX 9.1

A note about the interpolation of baselines The interpolation of baselines as is performed in Figs. 9.109.12 is only possible in the case of adsorbed species. This is because interpolation makes the implicit assumption that, sufficiently far away from the peak on both sides, the current is only the baseline. This is true for adsorbed species for which the signal has a finite size, but wrong for diffusive species for which the oxidation or reduction peaks have a very long “tail”

pffiffiffi (that decreases as 1= E as was discussed above in the “Diffusion-controlled voltammetry at stationary electrodes” section). Thus, in the case of diffusive species, past the peak, the current never reaches the level of the baseline again, which makes it impossible to interpolate the basline. It must be extrapolated instead from the region on either side of the peak where the faradaic current is zero.

The fumarate reductase from E. coli contains 3 FeS clusters and one flavin cofactor. The two-electron signal associated with the flavin is easily distinguished from the three oneelectron peaks due to the FeS clusters (Fig. 9.12). The data clearly show that the reduction potential of the FAD is strongly pH dependent, as expected for a reduction process coupled to protonation [cf. “Influence of coupled reactions (e.g., protonation or ligand binding) on reduction potentials” section]. Voltammetry is now becoming a routine technique to measure reduction potentials, and offers several advantages over potentiometric titrations described in the “Electrochemistry under equilibrium conditions: potentiometric titrations” section: • No need for a spectroscopic “handle.” There is no requirement for a distinctive and unambiguous change in some spectroscopic parameter. The counterpart of this advantage is obvious: voltammetry provides no structural information. • Sample economy. A few pmol of protein/enzyme are adsorbed onto the electrode (although making a film sometimes requires a larger amount than that, typically 10100 times as much, this is still much less than required for bulk titration). • Quick measurements. Recording a cyclic voltammogram usually takes a few seconds to a few minutes. • Instantaneous dialysis. The electrode can be transferred in a solution of different composition/pH and the measurement repeated with the same sample. The protein might even survive long enough in a hostile environment for measurements to be performed before the electrode is taken back into a more gentle solution [see, e.g., Zu et al. (2001) for the measurement of the reduction potential of a [2Fe2S] cluster in the pH range from 2 to 14]. • Easier analysis and modeling of data. The baseline (the charging current) is easily removed (Box 9.1), and interpretation of noncatalytic data does not require

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FIGURE 9.12

Cyclic voltammograms (raw data out of scale, base-line subtracted and deconvoluted signals) for Escherichia coli fumarate reductase (FrdAB) adsorbed at a pyrolytic graphite edge electrode. This enzyme contains 3 FeS clusters and a flavin cofactor. 20 C, pH 7 (top panel) and 9 (bottom panel), ν 5 10 mV/s. At pH 7, FADox/FADred 250 mV versus SHE, [2Fe2S]21/1 240 mV, [4Fe4S]21/1 2305 mV, [3Fe4S]1/0 265 mV ΓA  0.4 pmol. Note the strong pH dependence of the FAD signal (60 mV for two pH units). Source: Data from Le´ger, C., Heffron, K., Pershad, H.R., Maklashina, E., Luna-Chavez, C., Cecchini, G., et al., 2001. Enzyme electrokinetics: energetics of succinate oxidation by fumarate reductase and succinate dehydrogenase. Biochemistry 40 (37), 1123411245. URL: Available from: http://view.ncbi. nlm.nih.gov/pubmed/11551223.

solving any transport (diffusion/convection) equations. Free softwares are available to model the data (Box 9.2). • In situ measurements. Reduction potentials can be measured as a function of temperature (McEvoy and Armstrong, 1999; Hagedoorn et al., 1998; Park et al., 1991; Brereton et al., 1998) or even pressure (Gilles de Pelichy and Smith, 1999).

Fast-scan voltammetry to determine the rates of coupled reactions The measurement of a reduction potential (an equilibrium property) theoretically requires that the system is at equilibrium. This appears to be in contradiction to measuring a current: indeed the flow of electrons when the protein is oxidized or reduced results from the fact that the system is driven out of equilibrium when the electrode potential is changed around the (equilibrium) reduction potential. In practice this does not matter too much if the scan rate (and therefore the current) is slow enough that the system is nearly at equilibrium (Laviron, 1979). When the scan rate is raised, however, departure from equilibrium can be observed, and a great deal of information about the kinetics of redox processes can be gained by looking at the scan rate dependence of the voltammograms. Theoretical voltammograms for an uncoupled one-electron redox process are plotted in 0 Fig. 9.13A. At slow scan rate the oxidation peak occurs at E0 . If the scan rate is high,

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FIGURE 9.13 Effect of scan rate on the voltammetry of a redox species undergoing a oneelectron, no-proton redox process, with k0 5 0.1 s21. A: calculated cyclic voltammograms at different scan rates (from 1000 V/s to 10 mV/s). The difference between Ep,a and Ep,c increases as the scan rate is raised. B: “Trumpet plot”: Ep,a (filled squares) and Ep,c (empty squares) as a function of the log of the scan rate. The slower electron transfer rate is, the peak will separate at a slower scan rate and the trumpet plot will shift to lower scan rates.

because it takes time for the ET between the electrode and the redox center to occur, the maximal current is observed after the redox potential has been reached, that is, at higher electrode potential. The same reasoning applied to the reductive process predicts that Ep,c 0 is lower than E0 . When measurements are performed over a large range of scan rates, the results can be displayed by plotting Ep,a and Ep,c as a function of scan rate, on a log scale, as shown in Fig. 9.13B (Laviron, 1979). This has been called a trumpet plot. The more efficient the ET between the electrode and the active site [the greater k0 in Eqs. (9.18a) and (9.18b)], the faster the scan rate at which the oxidative and reductive peaks start to separate (peak separation occurs when the scan rate is greater than about k0RT/F, Laviron, 1979). For adsorbed redox proteins, reported values of k0 vary greatly, from a few s21 to 15,000 s21, in which case the peaks remain visible at scan rates as high as 3000 V/s (Baymann et al., 2003; Hirst et al., 1998b). Needless to say, the situation of fast ET is more desirable if the focus is on studying biologically relevant processes rather than interfacial electrochemistry. Fast-scan voltammetry also gives information about the rates of the reactions that are coupled to ET. If these coupled reactions (e.g., (de)protonation) are fast on the voltammetric timescale, their effect is to shift the reduction potential of the redox couple (“Influence of coupled reactions (e.g., protonation or ligand binding) on reduction potentials” section), and also to decrease the apparent rate of ET, the parameter noted k0 in Eqs. (9.18a) and (9.18b) [this effect is discussed in a series of papers written by Etienne

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Laviron in the 1980s (Laviron, 1980; Meunier-Prest and Laviron, 1992; see also Anxolabe´he`re-Mallart et al., 2011; Le´ger and Bertrand, 2008, Section 2.1.2.1). In that case the voltammogram remains reversible at very low scan rate, and symmetrical but with greater peak separation when the scan rate is increased (Fig. 9.13). Recording voltammograms at increasing scan rates can be used for determining the value of k0, but also to determine the rates of chemical processes that are coupled to ET. From a biological point of view, proton transfer is certainly the most important reaction coupled to ET, because the synthesis of ATP in most organisms is coupled to long-range proton transfers across biological membranes (Nicholls and Ferguson, 2013), but the kinetics and mechanism of proton transfer are difficult to study using conventional techniques. It is remarkable that electrochemistry proved very useful in this context (Chen et al., 2000; Hirst et al., 1998a), as described below. Fig. 9.15 illustrates the voltammetric study of the [3Fe4S]1/0 one-electron one-proton reaction (Fig. 9.14), for a mutant of Azodobacter vinelandii ferredoxin I. High-resolution crystal structures reveal that the [3Fe4S] is buried with no access to water molecules, and that a carboxylate group from an aspartate (D15) is located close to the cluster on the protein surface. It was suggested that a movement of this position 15 side chain may transfer a proton from the solvent to the cluster. The experiments depicted in Fig. 9.15 illustrate the use of fast-scan voltammetry to determine the kinetics of protonation of the [3Fe4S] cluster in a mutant where D15 is replaced with a glutamate. They were performed at pH 5.4, greater than pKOx and 1.3 pH unit lower than the pKRed 5 6.7 of the [3Fe4S] cluster, the reduced form of which is therefore protonated at equilibrium. The scans were started from the high potential limit, and only the first scan is considered. • At slow scan rates (panels A and B in Fig. 9.15) oxidation and reduction peaks for the [3Fe4S]1/0 appear at the same electrode potential (  2350 mV). Under these “close-toequilibrium” conditions, the reduction is followed by protonation, and oxidation proceeds along the reverse route (Fig. 9.14). The peak potential is given by Eq. (9.9). • When the scan rate is increased, oxidation and reduction peaks start to separate (Fig. 9.15C). • At scan rates between 1 and 10 V/s, the reductive peak is still clearly visible, but the oxidation peak vanishes (Fig. 9.15DF), because the cluster is trapped in the protonated form: it is quickly protonated upon reduction, but the rate of deprotonation, koff in Fig. 9.14, is too small for the cluster to be deprotonated during the fast oxidative scan. The deprotonation of the cluster “gates” its reoxidation.

FIGURE 9.14 L-shape scheme used to interpret the fast-scan voltammetry of a [3Fe4S] cluster. Electrochemists call this an “EC” mechanism (Compton and Banks, 2011; Saveant, 2006).

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FIGURE 9.15 Effect of scan rate on the voltammetry of a redox species undergoing a one-electron, one-proton redox process. The data are for the [3Fe4S]1/0 redox couple of a slow proton-transfer mutant of Azodobacter vinelandii ferredoxin I (D15E) at low pH (Chen et al., 2000; Hirst et al., 1998). You can “watch” these voltammograms in real time on our YouTube channel at https://goo.gl/ TDhz7f.

• At very high scan rates, 20 V/s and above (not shown), both peaks are observed, but the average reduction potential is lower than at low scan rate, and matches the alkaline limit: this is because the scan is reversed before the reduced cluster is protonated, and ET is therefore not coupled to protonation. These very simple experiments determine the rates of (de)protonation. Electrochemistry can achieve this because the timescale of potential modulation can be changed over orders of magnitude (1 min at 10 mV/s to 1 ms at 1000 V/s) to match that of the chemical events. In conjunction with site-directed mutagenesis, crystallography, and molecular dynamics

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simulations, this has made it possible to obtain very original information about the molecular mechanism of protonation (Chen et al., 2000; Hirst et al., 1998a). Several examples of voltammetric studies of coupled reactions, involving cytochromes and FeS clusters, have been reported (see, e.g., Jeuken et al., 2000, 2002a; Armstrong et al., 2001).

Catalytic protein film voltammetry and chronoamperometry Principle and general comments In the absence of substrate and at sufficiently high coverage, a redox enzyme immobilized onto an electrode gives peak-like signals resulting from the reversible transformation of its redox centers (Fig. 9.12). Upon adding substrate, the nonturnover peaks are transformed to sizeable “catalytic waves” (Limoges and Save´ant, 2004): reaction with substrate transforms the active site, which is regenerated by electron exchange with the electrode in a succession of catalytic cycles. The magnitude of the current is proportional to electroactive coverage and to turnover rate, and so the relationship between driving force (potential) and catalytic activity is traced in a single voltammetric experiment. Note that catalysis may be observed even if coverage is too low to observe noncatalytic signals (as is unfortunately often the case). We shall discuss qualitatively a series of voltammograms and chronoamperograms obtained with three different enzymes and selected because they illustrate the variety of experiments that can be carried out and the variety of information that can be obtained. We shall emphasize what makes each result particularly relevant in the context of mechanistic studies. These three enzymes are • The soluble fraction of E. coli fumarate reductase (FrdAB), a flavoenzyme that has an ET chain consisting of three FeS clusters (Iverson et al., 1999). Its physiological function is to reduce fumarate to succinate, but it can also oxidize succinate both in vitro and in vivo, when it replaces succinate dehydrogenase (complex II). • Various hydrogenases (from Allochromatium vinosum, Desulfovibrio fructosovorans, and Aquifex aeolicus) in which the NiFe active site is connected to the protein surface by a chain of three FeS clusters and a gas channel (Leroux et al., 2008; Fontecilla-Camps et al., 2007). These enzymes reversibly catalyze the oxidation of H2. • The periplasmic nitrate reductase from Rhodobacter sphaeroides (NapAB), which irreversibly reduces nitrate into nitrite. This enzyme houses a buried molybdenum active site, one 4Fe4S cluster and two surface exposed hemes (Arnoux et al., 2003). The noncatalytic voltammetry of E. coli fumarate reductase has been discussed above (Fig. 9.12). Fig. 9.16 shows again the noncatalytic voltammetry of this enzyme in panel A and the catalytic signal that is observed when the solution contains both fumarate and succinate and the electrode is rotated at a high rate to avoid mass-transport control (panel B). The flavin cofactor is oxidized at high electrode potential, giving two electrons to the electrode. The oxidized enzyme can bind succinate, and the oxidation of succinate results in the reduction of the flavin. The reduced FAD can be reoxidized giving electrons to the electrode and so on. This results in a steady-state flux of electrons from succinate in

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FIGURE 9.16 (A) Noncatalytic voltammogram obtained for Escherichia coli fumarate reductase (FrdAB) adsorbed at a pyrolytic graphite edge electrode in the absence of substrate. The raw voltammogram (outer dash-dot line) is not to scale. Inset: background corrected current (small dots) and deconvoluted data (dashed lines). ν 5 10 mV/s, 20 C, pH 7. See also Fig. 9.12. (B) Catalytic wave showing reversible succinate oxidation and fumarate reduction by adsorbed FrdAB in a solution containing succinate and fumarate, with the electrode rotating at a high rate. ν 5 1 mV/s, 20 C, pH 7, ω 5 3000 rpm. Source: Reproduced with permission from Le´ger, C., Heffron, K., Pershad, H.R., Maklashina, E., LunaChavez, C., Cecchini, G., et al., 2001. Enzyme electrokinetics: energetics of succinate oxidation by fumarate reductase and succinate dehydrogenase. Biochemistry 40 (37), 1123411245. URL: Available from: http:// view.ncbi.nlm.nih.gov/pubmed/11551223. Copyright (2001) American Chemical Society.

solution to the electrode, via the adsorbed enzyme, which is measured as a steady-state positive current. At low electrode potential a reductive (negative) current is observed which is proportional to the rate of fumarate reduction. The filled circle in Fig. 9.16B indicates the potential of zero net current (or “open circuit potential,” OCP) at 251 mV. This corresponds to the reduction potential of the fumarate/ succinate couple which can be calculated for any concentration ratio using the Nernst 0 0 equation [Eq. (9.5)] and the published value E0  120 mV at 25 C, pH 7 (E0 1 RT/2 F ln([F]/[S]) 5 246 mV). As a general way of things, provided the solution contains both the oxidized and reduced substrates, and that the adsorbed enzyme is able to catalyze both the oxidation and reduction reactions at significant rates, the value of the OCP is given by the Nernst equation and does not tell us anything about the enzyme. It only characterizes the substrate/product redox couple. In contrast, the value of the current on either side of the OCP and its dependence on electrode potential reveal the enzyme’s intrinsic properties. Nevertheless, the determination of the OCP can sometimes prove useful to determine thermodynamic parameters that would be difficult to obtain through other means (Bassegoda et al., 2014; Kurth et al., 2015). Catalytic electrochemistry offers several advantages with respect to conventional solution assays of the enzyme’s activity. • The temporal resolution of the activity is very high (the current can easily be sampled every 0.1 seconds or faster), which is useful if the activity of the enzyme is evolving. For reasons which will become clear below, this makes it possible to study very easily the reaction of certain enzymes with their gaseous substrates or inhibitors (CO, H2, etc.).

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• In contrast to certain solution assays, anaerobicity is not required (at least on condition that the electrode potential is high enough that oxygen is not reduced on the electrode), which makes it possible to study the reaction of the enzyme with oxygen; this has proved very useful recently in the field of hydrogenase research (Vincent et al., 2007; Liebgott et al., 2009). • Last, and most importantly, the electrode potential is a very useful control parameter, which influences the redox state of the enzyme and determines the driving force for the catalytic reaction and for certain redox-driven (in)activation processes. The dependence of activity on electrode potential can be very complex, and in some cases, efforts are still being made to make sense of the data in relation to the catalytic mechanism. The magnitude and shape of the catalytic signal depend on a number of factors because catalysis involves many consecutive reactions or processes: transport by convection and diffusion of the substrate in solution toward the adsorbed enzyme, substrate diffusion within the enzyme, its binding and transformation at the active site, product release and diffusion away from the enzyme, regeneration of the redox state of the active site upon intramolecular ET (assuming there is a redox chain in the enzyme), and interfacial ET between the electrode and a redox center that is exposed at the protein surface. The overall turnover rate depends on the slowest of these steps, noting that changing the electrode potential changes the rates of the redox processes. This means that if one is interested in studying the catalytic mechanism, one should try to reach a situation where interfacial ET and mass transport in solution do not limit the current. A good kinetic model will not include all steps (this would lead to indetermination), but only those which influence the turnover rate. Often the examination of the catalytic signal gives very useful information about which steps matter, as discussed below. Although kinetic models exist and are adapted to many different situations, they will not be described below and we shall only refer the reader to the primary literature. However, it is important to acknowledge that enzyme kinetics and electrochemistry are quantitative sciences, and that by looking at the data too superficially or qualitatively, one may miss important information or misinterpret the data.

Mass-transport controlled catalytic voltammetry Fig. 9.17A illustrates the voltammetry for hydrogen oxidation by A. vinosum NiFe hydrogenase adsorbed at a rotating disc electrode. In this experiment the positive current at high potential is proportional to the rate of catalytic H2 oxidation and the negative current results from proton reduction (H2 evolution). The current tends to a limiting value at high potential, which increases dramatically as the electrode rotation rate ω is raised (Pershad et al., 1999). This is because during turnover the concentration of hydrogen near the electrode decreases, the enzyme is able to consume H2 faster than it is brought to the electrode by the convective motion of the solution. The greater the rotation rate, the more efficient the transport of hydrogen from the bulk solution toward the enzyme, and the greater the current (“Diffusion-controlled voltammetry at rotating electrodes” section). At infinite rotation rate the catalytic current is finite: mass transport is no longer rate limiting, and the extrapolated current reveals the intrinsic efficiency of the enzyme.

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FIGURE 9.17 (A) Influence of electrode rotation rate (ω, given in units revolution per minute) on the catalytic current measured for hydrogen oxidation by Allochromatium vinosum NiFe hydrogenase adsorbed at a rotating graphite electrode. (B) KouteckyLevich plot [Eq. (9.32)] showing small but nonzero intercept at infinite rotation rate, allowing one to estimate kcat 0.1 bar H2, ν 5 100 mV/s, T 5 45 C, pH 6.5. Source: Data from Pershad, H.R., Duff, J.L.C., Heering, H.A., Duin, E.C., Albracht, S.P.J., Armstrong, F.A., 1999. Catalytic electron transport in Chromatium vinosum [NiFe]-hydrogenase: application of voltammetry in detecting redox-active centers and establishing that hydrogen oxidation is very fast even at potentials close to the reversible H 1 /H2 value. Biochemistry 38 (28), 89928999.

The KouteckyLevich plot in Fig. 9.17B appears to follow: 1 ilim



1 constant 1 pffiffiffiffi nFAΓ 3 ðturnover rateÞ ω

ð9:32Þ

The equation above emphasizes departure from mass-transport control at high ω [compare to Eq. (9.26)], but it is not rigorous (see Heering et al., 1997; Merrouch et al., 2017; Le´ger and Bertrand, 2008, Section 2.3.1). This limitation by mass transport is all the more influential when the enzyme has high activity, when the electrode coverage is high, and when the bulk concentration of substrate is small compared to the Michaelis constant (indeed under saturating conditions a small decrease of interfacial substrate concentration should have no effect on turnover rate). The magnitude of the current is proportional to AΓ 3 kcat, and can give an estimate (sometimes only a lower estimate) of kcat Interestingly a study of hydrogenase showed that the turnover frequency of the enzyme is significantly higher than that observed in solution assays, using oxidizing dyes (Pershad et al., 1999); in the latter case it becomes evident that turnover in solution assays is limited by ET to the soluble electron partner, and that the electrode is a much faster electron acceptor than the soluble dye.

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Chronoamperometry to measure Michaelis and inhibition constants In conventional biochemistry experiments, the rate of turnover is measured as a function of substrate concentration to determine the MichaelisMenten parameters (kcat, the maximal turnover rate, and Km, the Michaelis constant): turnover rate 5

k  cat  1 1 Km =½S

ð9:33Þ

The same parameters can be determined from chronoamperometric experiments looking at the substrate-concentration dependence of the current recorded at a fixed potential i 5 nFAΓ

k  cat  1 1 Km =½S

ð9:34Þ

In practice this kind of measurement might be far from easy. (1) The limiting current is proportional to AΓ, the total amount of enzyme adsorbed, which can be determined (with a very relative accuracy) only when the electrode coverage is high enough for noncatalytic signals to be measured in the absence of substrate [Eq. (9.29)]. (2) The measurement of Km can be performed without knowing the exact electroactive coverage. This requires however that the adsorbed film is stable enough as a function of time for the coverage to be constant when currents are measured with the same film in solutions of different substrate concentrations. Fourmond et al. (2009b) describes methods for correcting the effect of film desorption. (3) Last, Eq. (9.34) does not take into account mass transport of substrate in solution; this is correct only if there is no depletion of substrate near the electrode. This should be checked for by looking at the rotation rate dependence of the current, or using the rotation rate dependence to extrapolate the current at infinite rotation rate. Models explicitly taking into account mass transport are available in Merrouch et al. (2017) and Reda and Hirst (2006). Fig. 9.18 shows the result of this simple chronoamperometric experiment, in a case where it is particularly informative: nitrate reduction by the molybdoenzyme R. sphaeroides periplasmic nitrate reductase. Here the effect of film desorption was corrected using a method proposed in Fourmond et al. (2009b). The nitrate reduction rates (negative current) shown in the middle row were measured at two different electrode potentials. The top panels show the concentration of nitrate against time (each step corresponds to the injection in the electrochemical cell of a small amount of a concentrated stock solution of nitrate). The bottom panels show the steady state current at the end of each step plotted against nitrate concentration. At low potential (left column), the change in current simply follows MichaelisMenten kinetics, whereas under less reductive conditions (right column) high concentrations of nitrate inhibit the enzyme. This is particularly relevant because experiments aimed at trapping catalytic intermediates before they are characterized in spectroscopy are often carried out with very high concentrations of substrates; in the case of nitrate reductase moderately reducing conditions are needed to detect a molybdenum(V) intermediate by EPR (the fully reduced Mo(IV) state is EPR-silent), Fig. 9.18 demonstrates that these conditions favor the formation of an inactive enzyme, rather than a catalytic intermediate (Fourmond et al., 2010c). Solution assays with two electron donors having different reduction potentials fully supported these electrochemical results (Fourmond et al., 2010c).

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FIGURE 9.18 Dependence of the rate of nitrate reduction on nitrate concentration, with nitrate reductase adsorbed at a rotating disc electrode spun at a high rate. The left- and right-hand sides correspond to a redox poise at 2460 and 140 mV versus SHE, respectively. Conditions: pH 6; 25 C; 5 krpm. Panels (A) and (D) show the evolution of nitrate concentration against time, when the concentration is stepwise increased by adding aliquots of a stock solution of potassium nitrate (note the logarithmic Y scale). Panels (B) and (E) show the resulting change in catalytic current. Panels (C) and (F) show the catalytic current reached at the end of each step as a function of nitrate concentration. The fit of the data in C to the MichaelisMenten returns the value of Km  85 μM. The inset shows a EadieHofstee plot (Cornish-Bowden, 2004). The red line in panel F is the best fit to an equation accounting for substrate inhibition, with Km 5 10 μM and Ki  4 mM. Source: Reprinted with permission from Fourmond, V., Sabaty, M., Arnoux, P., Bertrand, P., Pignol, D., Le´ger, C., 2010c. Reassessing the strategies for trapping catalytic intermediates during nitrate reductase turnover. J. Phys. Chem. B 114 (9), 33413347. URL: Available from: http://dx.doi.org/10.1021/jp911443y. Copyright (2010) American Chemical Society.

The high temporal resolution of the activity measurement is also useful for probing the reaction of redox enzymes with gaseous substrates and inhibitors. Many redox enzymes use, consume, or are inhibited by small molecules like O2, CO, N2, H2, NO, etc. The fact that these molecules tend to escape the solution and equilibrate with the gas phase above it makes certain experiments difficult. For example, to establish the competitive character of the inhibition of hydrogenase by CO by carrying out normal (solution) assays, one must measure the H2oxidation turnover rate as a function of the concentrations of H2 and CO dissolved in solution, but setting their concentrations precisely and maintaining them constant may be tricky and time-consuming. This problem is easily solved using electrochemistry, because it is possible to change the concentrations of dissolved gas in a controlled manner, and to simultaneously monitor the change in activity, as illustrated in Fig. 9.19 (Le´ger et al., 2004). Fig. 9.19A shows in black the change in concentration of dissolved H2 against time that is obtained when, after having maintained an atmosphere of 1 bar of H2 until t 5 0, a small

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FIGURE 9.19 Hydrogen oxidation by WT Desulfovibrio fructosovorans NiFe hydrogenase adsorbed at a rotating disc electrode: measurement of the Michaelis constant relative to H2 and of the inhibition constant relative to CO. Panel A shows the change in hydrogen (black) and CO (blue) concentrations against time. The red line in panel B shows the change in current against time in the experiment where H2 is flushed away at t . 0. The blue line is the result of a similar experiment, but a solution saturated with CO was injected at t  30 s while the concentration of H2 was decreasing. The fits to Eqs. (9.35), (9.36) are shown as dashed lines. Source: Reprinted with permission from Le´ger, C., Bertrand, P., 2008. Direct electrochemistry of redox enzymes as a tool for mechanistic studies. Chem. Rev. 108 (7), 23792438. URL: Available from: http://dx.doi.org/10.1021/cr0680742. Copyright (2008) American Chemical Society.

tube is suddenly plunged into the buffer and used to bubble argon. This flushes hydrogen away from the cell and its concentration decreases exponentially with time, with a timeconstant which we denote by τ H2 (Le´ger et al., 2004). The red curve in panel B shows the sigmoidal change in current for hydrogen oxidation by D. fructosovorans NiFe hydrogenase in the same experiment. The equation used to fit this data (black dashed line) is simply obtained by inserting into the MichaelisMenten equation a time-dependent substrate concentration: ½H2 ðtÞ 5 ½H2 0 expð2 t=τ H2 Þ: iðtÞ 5

imax  1 1 Km =½H2 0 expðt=τ H2 Þ 

ð9:35Þ

In the second experiment, shown as a blue curve in Fig. 9.19B, an aliquot of COsaturated solution is injected at t  30 seconds while the hydrogen concentration is decreasing (blue line in panel A). In this experiment, the concentrations of both CO and H2 decrease with time, but the equation for the transient current is again simply obtained by using the rate equation that considers competitive inhibition by CO, in which we insert exponential decays of both H2 and CO: iðtÞ 5

11

imax 11

Km ½H2 ðtÞ

½COðtÞ Ki



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ð9:36Þ

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The fit is shown as a dashed curve, and simultaneously determines Km for H2 and Ki relative to CO, from an experiment that lasts no longer than a few minutes! However, note that the sort of experiments shown in Fig. 9.19 cannot be very accurate; indeed with the concentrations varying exponentially with time, only the log (i.e., the order of magnitude) of the values of Km and Ki can be determined.

Chronoamperometry to resolve rapid changes in activity Nitrate reductase is reversibly inactivated at a high concentration of nitrate and moderate potential (Figs. 9.18 and 9.22F); in addition to that, it also irreversibly activates the first time it is reduced. This is clear from the chronoamperogram in Fig. 9.20, which shows the response of a fresh film of periplasmic nitrate reductase immersed in a solution of nitrate when the potential is stepped as indicated in the upper panel. When the potential is poised at E 5 2 160 mV (at t 5 40 seconds), the current is essentially constant. On the step to 2460 mV, at t 5 80 seconds, the activity first instantly decreases (the current becomes less negative) and then slowly increases before it stabilizes. This slow change in current demonstrates that the enzyme activates at low potential. The magnitude of the activation phase accounts for 20% of the current reached after activation. The instant decrease in activity when the potential is first stepped from 2160 to 2460 mV is not surprising considering the steady state profile in Fig. 9.22D. When this sequence of potential steps is repeated with the same film of enzyme (from t 5 480 seconds), the activity detected at E 5 2 160 mV is greater than at the same potential in the first experiment, as indicated by the downward arrow, and no further activation occurs on the second step to 2460 mV, at t  560 seconds. Therefore the activation proceeds only once, on the first step to 2460 mV, and it is not reversed by taking back the enzyme to oxidizing conditions (1240 mV): it is irreversible. This irreversible reductive activation cannot be detected in solution assays of the enzyme, because assaying the enzyme requires that it is reduced, and this reduction activates the sample. Therefore the enzyme molecules that are already active and those that

FIGURE 9.20 Chronoamperometric experiments demonstrating the irreversible reductive activation of periplasmic nitrate reductase (NapAB). The top panel shows the sequence of potential steps which was applied to a film of as-prepared WT NapAB. The horizontal dotted lines is for i 5 0. ω 5 5krpm, pH 6, 25 C. Reprinted with permission from Fourmond, V., Sabaty, M., Arnoux, P., Bertrand, P., Pignol, D., Le´ger, C., 2010c. Reassessing the strategies for trapping catalytic intermediates during nitrate reductase turnover. J. Phys. Chem. B 114 (9), 33413347. URL: Available from: http://dx.doi.org/10.1021/jp911443y. Copyright (2008) American Chemical Society.

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require reductive activation are indistinguishable in solution assays. In contrast the temporal resolution of the activity measurement in PFV is high enough that the activity can be measured while the enzyme is activating (Fig. 9.18). This is reminiscent of the “redoxcycling” experiment with Rhodobacter capsulatus dimethyl sulfoxide (DMSO) reductase: the enzyme has greater activity for dimethyl sulfide oxidation after it has been reduced (Bray et al., 2000). Furthermore, the reductive activation of nitrate reductase is very relevant because its amplitude correlates with the concentration of a certain Mo(V) EPR signal (the so-called “high-g” signal) present in the sample before activation, demonstrating that, unlike all expectations, this signature arises from an inactive state which activates the first time the enzyme is reduced (Fourmond et al., 2010c). On a different note, chronoampetrometry experiments with hydrogenase make it easy to characterize quantitatively the kinetics of inhibition by CO (and also O2). The experiment in Fig. 9.21 consists of monitoring the H2-oxidation current after the concentration of CO first suddenly increases when an aliquot of solution saturated with inhibitor is injected in the electrochemical cell, and then slowly returns to zero as the buffer is flushed by a stream of H2 (Le´ger et al., 2004; Le´ger and Bertrand, 2008, Section 2.4.2). The concentration of H2 is nearly constant, and the inhibitor concentration follows an exponential decay (panel A). The concentration of the inhibitor need not be independently measured because its change against time is defined by the amount of inhibitor that is injected and by the time constant of the decay, which is determined by fitting the change in current (Le´ger et al., 2004). If binding and/or release of the inhibitor is slow, the change in activity is delayed from the time of injection and the rates of inhibition can be measured (Leroux et al., 2008). In the case of the wild type (WT) enzyme from D. fructosovorans (at 40 C), the decrease in activity after CO injection is too fast to be resolved (black dots in Fig. 9.21B). Also, the

FIGURE 9.21 Inhibition by CO of H2 oxidation by WT Desulfovibrio fructosovorans NiFe hydrogenase and a variant where a double mutation narrows the gas channel (Leroux et al., 2008; Liebgott et al., 2009). Panel A: CO concentration against time. Panel B: the change in current against time for the WT enzyme (black) and the mutant (red). The modeling of these data directly gives the rates of diffusion along the channel (Leroux et al., 2008; Liebgott et al., 2009; Almeida et al., 2007). The aliquot of solution saturated with CO was injected at t 5 0. 1 bar H2, E 5 2 160 mV, pH 7, 40 C, 3krpm. Source: Data from Le´ger, C., Bertrand, P., 2008. Direct electrochemistry of redox enzymes as a tool for mechanistic studies. Chem. Rev. 108 (7), 23792438. URL: Available from: http://dx.doi.org/10.1021/cr0680742. Copyright (2008) American Chemical Society.

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recovery of activity follows exactly the decrease in CO concentration as the latter is flushed away from the cell. In contrast the red signal was obtained with a mutant of the enzyme, where amino acids whose side chains point inside the substrate channel connecting the active site to the solvent have been substituted (Leroux et al., 2008). Clearly, the binding and release of CO are much slower, suggesting that CO diffuses slowly within the mutant, to and from the active site. The data can be fit to the model in Almeida et al. (2007) to measure the rate constants corresponding to diffusion in the channel in either direction. Independent experiments based on solution assays of the isotope-exchange reaction confirmed the effects of the mutations on the rates of intramolecular diffusion, and proved that the difference between the two signals in Fig. 9.21 is not an artifact resulting from the presence of the electrode (Leroux et al., 2008). Similar electrochemical experiments could be carried out to determine how the mutations of amino acids whose side chains point into the channel of hydrogenase affect the rates of binding and release of CO, O2, and H2, and PFV is now a unique tool for studying how the structure of gas channels affects intramolecular diffusion rates (Leroux et al., 2008; Liebgott et al., 2009; Kubas et al., 2017).

Determining the reduction potentials of an active site bound to substrate Chronoamperometry experiments are usually easy to analyze, whereas understanding the profile of activity against potential sometimes demands effort. Fig. 9.22 shows various catalytic voltammograms obtained with FrdAB, NapAB, and hydrogenases adsorbed at rotating disk electrodes. We shall now discuss their shape (as opposed to merely their magnitude) and explain what can be learned about the enzyme from each signal. We examine first succinate oxidation by E. coli fumarate reductase, FrdAB (Fig. 9.22A). Like all other data in this figure, this signal was obtained with the electrode rotating at a very high rate, so that mass transport was not influential. The background (capacitive) current is shown as a dotted line. At low electrode potential, the enzyme and its active site are reduced and unable to oxidize succinate, there is no activity and no Faradaic current. At very high potential the current tends to a limit (a plateau), which is independent of electrode rotation rate, and represents the maximal turnover rate when the enzyme is oxidized at a very high rate and therefore remains fully oxidized in the steady-state. In between the two, the activity increases when the electrode potential becomes high enough that the active site flavin becomes oxidized. The position of the inflection point of the main catalytic wave is a phenomenological parameter often called the “catalytic potential.” By using a very simple kinetic model that uses the Nernst equation to relate the electrode potential to the redox state of the flavin, the position and precise shape of the voltammogram can be interpreted to determine the two reduction potentials of the active site flavin, under turnover conditions and in the presence of substrate. In contrast, in equilibrium titrations the reduction potential of the active site flavin can only be measured in the absence of substrate, otherwise the enzyme would turnover and equilibrium could not be reached. In Le´ger et al. (2001), voltammograms such as that in Panel A were used for determining the dependence of the reduction potentials of the active site flavin on succinate concentration (shown in Fig. 9.4) and pH, from which the affinity for succinate and

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FIGURE 9.22 A small collection of cyclic voltammograms, obtained with various enzymes adsorbed at a rotating electrode spun at a high rate (so that substrate transport toward the electrode is fast). In most panels the voltammogram plotted with a dotted line is a blank, recorded with no adsorbed enzyme: the contribution of the enzyme is obtained by subtracting this capacitive current. (A) succinate oxidation by E. coli fumarate reductase ν 5 1 mV/s, ω 5 3 krpm, 20 C, 16 mM succinate, pH 7.5 (Le´ger et al., 2001). (B) fumarate reduction by E. coli fumarate reductase (Hudson et al., 2005). (C) H2-oxidation by A. vinosum NiFe hydrogenase, at high temperature (60 C) and fast scan rate (ν 5 1 V/s), ω 5 2.5 krpm, 1 bar H2, pH 7 (Le´ger et al., 2002). (D) Nitrate reduction by R. sphaeroides periplasmic nitrate reductase, at very low nitrate concentration (10 μM) ν 5 20 mV/s, 25 C, pH 6 (Bertrand et al., 2007). (E) H2-oxidation by A. aeolicus NiFe hydrogenase, at slow scan rate (ν 5 0.3 mV/s) (Fourmond et al., 2010b); compare with the signal in panel C. (F) Nitrate reduction by R. sphaeroides periplasmic nitrate reductase, at very high nitrate concentration (24 mM) ν 5 20 mV/s, 25 C, pH 6 (Fourmond et al., 2010c); compare with the signal in panel D. Data from Le´ger et al. (2001, 2002), Fourmond et al. (2010b, 2010c), and Bertrand et al. (2007).

pKa of the flavin in its different redox states could be determined using equations such as Eq. (9.9). These thermodynamic properties of the active site are relevant to the catalytic cycle, regarding the protonation and binding states of the FAD intermediates. Note that the shape and position of the catalytic wave may not (and probably in most cases do not) match the thermodynamic reduction potential of the active site. One of the reasons is that chemical and intramolecular electron-transfer steps that are coupled to the oxidation/reduction of the active site in the catalytic cycle shift the catalytic potentials, as discussed in refs. Fourmond and Le´ger (2017), Sensi et al. (2017a), Le´ger et al. (2006), and Fourmond et al. (2013) and below.

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BOX 9.2

Softwares An open-source program called QSoas has been developed in our group, with the aim of analyzing one-dimensional signals (Fourmond, 2016). It offers a large set of commands for handling voltammetric and chronoamperometric data: filtering, baseline subtracting, and fitting various kinetic

models, including to catalytic wave shapes as described in Fourmond and Le´ger (2017). It is free of charge and can be compiled on computers running any operating system (including Windows and Mac OS X). Binary installers are available. See installation instructions on our web page at qsoas.org.

The effect of slow intramolecular electron transfer Consider the case where the enzyme has a chain of redox cofactors that relay electrons internally. If intramolecular ET is very fast on the timescale of turnover, then the shape and position of the catalytic wave depends only on the active site. Otherwise the wave may be shifted from the potential of the active site (Elliott et al., 2002). Models that explicitly take into account intramolecular ET predict that the shape and position of the wave then depend on the properties of the active site but also on the reduction potentials of the relays and the kinetics of intramolecular ET (Fourmond and Le´ger, 2017; Le´ger et al., 2006; Fourmond et al., 2013). How the catalytic potential compares to the reduction potentials of the relays may in some cases inform on the rate of intramolecular ET (Le´ger et al., 2006). We have identified one particular FeFe hydrogenase where slow intramolecular ET defines the rate of H2 production and oxidation in both directions (Caserta et al., 2018). In NiFe hydrogenases, modifying the ET chain also has a significant effect on the catalytic wave shape (Dementin et al., 2006; Adamson et al., 2017). For reasons that are not entirely clear yet, the shape of the catalytic wave may also have a particular feature at a potential that corresponds to one of the redox relays. For example the fumarate reductase discussed above has a series of three FeS clusters that connect the active site flavin to the solvent. The medial relay is a low potential [4Fe4S] cluster. The catalytic signal for fumarate reduction in Fig. 9.22B shows a first wave at the potential of the flavin, and a “boost” of activity at the potential of the [4Fe4S] cluster (Hudson et al., 2005).

Slow interfacial electron transfer In Fig. 9.22A and B it also appears that the current does not reach a plateau at low potential, but instead it keeps increasing linearly. In some cases, this linear increase in current hides the underlying sigmoidal wave: this is so for example in Fig. 9.22C, which shows H2 oxidation by a NiFe hydrogenase (Le´ger et al., 2002). This is observed when (1) the rate of interfacial ET is not very fast compared to the enzyme’s turnover, and (2) not all enzymes are adsorbed in exactly the same orientation, which results in a distribution of interfacial ET rates.

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The catalytic current corresponding to the assembly of enzymes molecules can be derived by averaging over the distribution of interfacial ET rate constants, and a linear change in current as a function of electrode potential at high driving force can indeed be predicted (Fourmond and Le´ger, 2017; Fourmond et al., 2013; Le´ger et al., 2002). This results from the contribution of enzyme molecules having low k0 values which contribute only at high driving force. This effect, which blurs the signal and hides its interesting features, is all the more pronounced when the dispersion of orientation is wide and when interfacial ET (k0) is slow compared to the intrinsic turnover rate of the enzyme.

Slow substrate binding Many redox enzymes exhibit complex activity profiles as a function of the electrode potential (Elliott et al., 2002). As an example, Fig. 9.22D shows a catalytic voltammogram corresponding to the reduction of nitrate by R. sphaeroides nitrate reductase (NapAB) at very low nitrate concentration. The activity of the enzyme drops down (the current becomes less negative) when the electrode potential is decreased below 2400 mV, and it is recovered on the reverse scan. One may first wonder whether this could be an artifact resulting, e.g., from the reorientation of the enzyme on the electrode at low potential. However, the decrease in activity under very reducing conditions is also observed in solution assays carried out with reduced methyl viologen (MV): under conditions where the concentration of reduced MV greatly decreases during the assay, so that the driving force for the reduction of nitrate decreases as a function of time, the activity measured in solution increases before MV is completely exhausted (Fourmond et al., 2010a) (see also Sucheta et al., 1992 for a similar experiment with complex II). This mirrors the observation in Fig. 9.22D: starting from the low potential limit, the activity increases when the electrode potential is increased. This acceleration of turnover upon consumption of reduced MV was also made with the periplasmic nitrate reductase from Paracoccus pantotrophus, which exhibits the same voltammetry as in Fig. 9.22D (Gates et al., 2008) and R. capsulatus periplasmic DMSO reductase (Adams et al., 1999), which belongs to the same family of molybdoenzymes. The membrane-bound DMSO reductase from E. coli was the first molybdoenzyme for which a signature like that in Fig. 9.22D was observed (Heffron et al., 2001). Therefore, this peculiar relation between driving force and activity appears to be a property typical of reductases from the DMSO reductase family. Although the concept of pH optima for enzyme activity is well established (CornishBowden, 2004; Laidler and Bunting, 1973), the possibility that activity of redox enzymes might be optimized within a narrow range of potential has been explored only recently. While pH optima give mechanistic information on protonation equilibria during catalysis, potential optima provide information on the roles played by different oxidation states of redox-active sites. Certain redox transitions may regulate electron flow to or from the active site. Or substrate binding or atom transfer may occur only when the active site is in a certain oxidation state. These relationships are difficult to observe by conventional techniques but can be revealed by PFV, due to its ability to measure subtle changes in catalytic activity as the electrode potential is varied. Considering the signal in Fig. 9.22D, we now wonder what makes the activity drop under very reducing conditions. It has been proposed that the “switch off” results from

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the reduction of the [4Fe4S] cluster that relays electrons toward the active site (Anderson et al., 2001). However, there appears to be no correlation between the reduction potential of the [4Fe4S] cluster and the position of the low potential feature in the voltammogram (Fourmond et al., 2010a). It has also been proposed that the complex shape may result from the fact that catalysis can follow distinct routes, as substrate binding either precedes or follows the reduction of the active site, and the relative rates of the two reactions determine which track is used (Heffron et al., 2001). Indeed changing the electrode potential changes the rate of reduction of the active site, which may make ET faster or slower than substrate binding. When substrate binding to any redox state of the active site is taken into account in kinetic models, all sorts of complex wave shapes can indeed be predicted (see Fig. 9.3 in Bertrand et al., 2007), and the signal in Fig. 9.22D can indeed be accurately predicted (Bertrand et al., 2007; Frangioni et al., 2004).

Slow, redox-driven (in)activation So far, we have only considered steady-state voltammograms, where the current depends on electrode potential but not on time, and the catalytic signal is therefore independent of scan rate and sweep direction. For example the switch on and off observed in Fig. 9.22D is observed at the same potential irrespective of whether the potential is scanned toward positive or negative potential, suggesting that whichever reaction triggers this modulation of activity is fast on the timescale of the voltammetric experiment (not necessarily so on the timescale of turnover). Sometimes, the enzyme undergoes slow activation or inactivation processes as the potential is varied, and this results in a strong hysteresis: the magnitude of the current depends on time and sample history, and the voltammogram depends on scan rate and direction (but this is not caused by the diffusion of the substrate, contrary to the situation discussed in the “Diffusion-controlled voltammetry”section). For example, NiFe hydrogenases reversibly deactivate under oxidizing conditions due to the overoxidation of the active site. The enzyme has to be reduced (e.g., incubated with hydrogen or reduced MV) for the activity to be recovered. Fig. 9.22E is a catalytic voltammogram showing H2 oxidation with A. aeolicus NiFe hydrogenase. It is recorded essentially in the same conditions as in Fig. 9.22C but for a slow scan rate (recording a whole catalytic voltammogram at this scan rate takes 1 hour). The activity of the enzyme, measured as a positive current, decreases above 2200 mV, and the activity is recovered as the potential is taken down. However, because the inactivation and reactivation are slow on the timescale of the voltammetry, the (in)activation reactions lag behind the change in electrode potential, resulting in a characteristic hysteresis. This inactivation was not observed in the electrochemical experiments depicted in Fig. 9.22C because the scan rate was too fast: the enzyme was taken to high potential and back before the inactivation reaction could proceed (Limoges and Save´ant, 2004). One can also design experiments in which the electrode potential is held, and the slow change in current resulting from the (in)activation measured, to study the kinetics and mechanism of (in)activation (Fourmond et al., 2010b; Jones et al., 2003). The shape of the signal in Fig. 9.22E is now fully understood, and a very important conclusion from the theoretical study is that the potential where the activity is recovered on

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the scan toward low potential is not the reduction potential of the inactive state (Hamdan et al., 2012; Fourmond et al., 2010b) (a clear simple observation suggesting that it is not a thermodynamic quantity is indeed that its value is greatly dependent of scan rate over the entire range of accessible scan rates). This illustrates a pitfall of the qualitative interpretation of the voltammograms: an inflection point in a voltammogram cannot always be equated to a reduction potential. FeFe hydrogenases also reversibly inactivate under anaerobic oxidative conditions (Fourmond et al., 2014; del Barrio et al., 2018a). This inactivation is dependent on the presence of chloride or bromide in the buffer—anions that are usually considered innocent. The catalytic voltammogram in Fig. 9.22F may be the most complex reported to date. It shows how the nitrate reductase signal in Fig. 9.22D is deformed by a reversible (in)activation process that decreases the activity at high potential, and increases it under reducing conditions. The resulting strong hysteresis at high potential is only detected at a high concentration of nitrate (24 mM in panel F, compared to 10 μM in panel D). This is another manifestation of the inhibition of the enzyme by its substrate that we discussed in relation to Fig. 9.18, right column (Fourmond et al., 2010c). The shape of this signal is also fully understood, and results from the fact that the nitrate-inhibited state exists under three distinct redox states which bind nitrate at different rates (Jacques et al., 2014).

Exercises • Fig. 9.12 shows that the two-electron reduction potential of the FAD in E. coli FrdAB shifts 260 mV between pH 7 and 9. Explain why the reduction potential decreases as the pH increases. Use Eq. (9.13) to determine the number of protons involved in the reduction of the flavin. • Recall the criteria (peak shape, dependence of peak current on scan rate, peak separation) that discriminate the voltammetry of a species that diffuses to/from the electrode or that is adsorbed onto an electrode. • From the data in Fig. 9.15, estimate the order of magnitude of the rate of deprotonation (in units of s21) of the [3Fe4S] cluster of D15E Av FdI. • In Fig. 9.23 we have plotted two voltammograms recorded with a film of E. coli fumarate reductase in contact with a solution containing only succinate at a concentration [S] 5 50 mM (approximately 200 times the Michaelis constant). The plain line corresponds to a stationary (nonrotating) electrode, whereas the dashed voltammogram was recorded with the electrode rotating at 3000 rpm. (1) Why is the current at high potential rotation-rate independent in this case? (2) Why are the shapes of the voltammograms different?

Acknowledgements CL is very grateful to F. Armstrong (Oxford University, UK) for introducing me to this research field. Some of the data shown herein were obtained in his lab. We also thank him and A.K. Jones (formerly in Oxford, now at Arizona State University, USA) for proofreading an early version of this document. We thank Anne K. Jones, Harsh Pershad, Raoul Camba, James McEvoy, and Lars J.C. Jeuken for kindly providing some of the data shown herein.

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FIGURE 9.23 Effect of electrode rotation on the catalytic voltammetry of Escherichia coli fumarate reductase (FrdAB) in a solution initially free succinate of fumarate. ½S 5 50 mM  200 3 Km .

Our work in Marseilles is funded by the CNRS (mainly), Aix-Marseille Universite´, the Agence Nationale de la Recherche, Excellence Initiative of Aix-Marseille University  A*MIDEX, a French “Investissements d’Avenir” programme, the Region Provence-Alpes-Cote d’Azur, and the city of Marseilles.

Appendices Notations and abbreviations A, area under a pic (in units of VA); A, electrode surface; C, concentration of species, or capacitance [Eq. (9.22)], or coupling prefactor [Eq. (9.15)]; D, diffusion coefficient; δ, peak width at half height; η, overpotential; E, electrode potential; E0, reduction potential; Ep, peak potential; ET, electron transfer; F, Faraday constant; Γ, electroactive coverage; i, current; ic, capacitive current; Km, Michaelis constant; Ki, inhibition constant; k, rate constant; λ, reorganization energy; m, number of protons; MV, methyl viologen; n, number of electrons; ν, scan rate; ν s, kinematic viscosity [Eq; (9.26)]; OCP, open circuit potential; ω, electrode rotation rate; R, gas constant; RDE, rotating disc electrode; T, temperature (in K); t, time; V, potential difference; ξ, extent of reaction.

Derivation of Eq. (9.9) We write the Nernst equation first for the alkaline couple Ox/Red, and then for both forms (protonated and deprotonated) of the redox couple:
 RT ½Ox ln E 5 E0alk 1 ð9:36aÞ nF ½Red
 RT ½Ox 1 ½OxH 0 ln E 5 E0 ð½H1 Þ 1 ð9:36bÞ nF ½Red 1 ½RedH

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We rewrite Eq. (9.36b) as follows:

   ½Ox 1 1 ½H1 =KOx RT    ln E 5 E ð½H Þ 1 nF ½Red 1 1 ½H1 =KRed   1 1 ½H1 =KOx RT ½Ox RT 00 1   ln 1 ln E 5 E ð½H Þ 1 nF ½Red nF 1 1 ½H1 =KRed 00

1

ð9:37aÞ ð9:37bÞ

Equating Eqs. (9.37b) and (9.36a) gives Eq. (9.9): 00

E ð½H

1

Þ 5 E0alk

 ! 1 1 ½H1 =KRed 2:3RT   log10 1 nF 1 1 ½H1 =KOx

ð9:38Þ

0

Check that E0 ([H1]) tends to E0alk when [H1] is small. Using E 5 E0acid 1 instead of Eq. (9.36a) gives 00

1

E ð½H

Þ 5 E0acid

RT ½OxH ln nF ½RedH

 ! 1 1 KRed =½H1  2:3RT   log10 1 nF 1 1 KOx =½H1 

ð9:39Þ

ð9:40Þ

0

Check that E0 ð½H1 Þ tends to E0acid when [H1] is large. The relation between E0alk and E0acid is simply obtained by equating Eqs. (9.37a) and (9.40): E0acid 5 E0alk 1

RT KOx ln nF KRed

ð9:41Þ

Check that with pKOx , pKRed, E0acid . E0alk .

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Structural biology techniques: X-ray crystallography, cryo-electron microscopy, and small-angle X-ray scattering Jose´ A. Brito and Margarida Archer Instituto de Tecnologia Quı´mica e Biolo´gica Anto´nio Xavier (ITQB NOVA), Universidade Nova de Lisboa, Oeiras, Portugal O U T L I N E Questions and purposes

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Heavy-atom derivatization

396

Preamble

376

Model building and refinement

397

X-ray crystallography

377

Protein crystallization Protein production and sample preparation Protein quality assessment

380

Structure analysis and model quality Content of crystallographic models Validation

400 400 401

X-ray free electron lasers

405

Data collection

391

Cryo-electron microscopy

406

Phase determination Molecular replacement Isomorphous replacement Anomalous scattering Direct methods

392 393 394 394 396

Small-angle X-ray scattering

409

General conclusion

411

Acknowledgments

412

References

412

Practical Approaches to Biological Inorganic Chemistry DOI: https://doi.org/10.1016/B978-0-444-64225-7.00010-9

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© 2020 Elsevier B.V. All rights reserved.

376

10. Structural biology techniques: X-ray crystallography, cryo-electron microscopy, and small-angle X-ray scattering

Questions and purposes • Acknowledge the existence of different structural biology techniques. • Understand the basic principles and applications of each of the presented structural biology techniques. • Comment on the major advantages and limitations of macromolecular crystallography, X-ray free electron lasers (X-FELs), cryo-electron microscopy (cryo-EM), and small-angle X-ray scattering (SAXS). • Describe briefly the main steps involved in the three-dimensional (3D) structure determination of macromolecules by X-ray crystallography, X-FELs, and cryo-EM. • Describe briefly the methodology to determine the overall shape and envelope of macromolecules by SAXS. • Understand the advantages of using X-FELs for the assessment of 3D models of macromolecules and time-resolved experiments. • Enumerate the main factors that contributed to the development of crystallography and the recent technological breakthroughs in cryo-EM. • Understand the rationale of the crystallization process and the parameters that may influence it. • Distinguish between cocrystallization and soaking techniques, and when one is preferable over the other. • Compare the different methods used to solve the phase problem in X-ray crystallography. • Specify the reasons why membrane proteins are more difficult to manipulate than soluble proteins and why crystals generally diffract poorly. Comment on their 3D structure determination by single-particle cryo-EM.

Preamble X-ray crystallography is one of the most commonly used techniques to characterize the 3D structure of biological macromolecules. A detailed description of this method is beyond the scope of this chapter; more exhaustive information can be found in some of the referenced books (Blundell and Johnson, 1976; Drenth, 1999; Rupp, 2010) and reviews (Hickman and Davies, 2001). We aim to provide graduate students or researchers working with metalloproteins with an overview on what needs to be done in order to determine the 3D structure of proteins, and what information can be extracted from the crystallographic models. We should also point out that throughout this chapter some generalizations are made (where exceptions may exist) and simplifications that may lead to omissions due to space limitation. A critical step in the process of 3D structure determination of macromolecules by X-ray crystallography is the production of well-ordered, diffraction quality crystals. We do not know a priori if a certain macromolecule will crystallize. Indeed, some of them might never be crystallized due to their intrinsic nature (e.g., naturally unfolded proteins, membrane proteins, and complexes) or simply because they are not stable or homogeneous enough. Other techniques do not present this limitation but rather study the biomolecules in solution, such as single-particle cryo-EM and SAXS.

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The recent technological breakthroughs in electron microscopy, such as the commercialization of direct detector devices (DDDs), freeze plunging of biological samples, development of sophisticated software for imaging and data processing, and increasing computer power has led to a “resolution revolution” in single-particle cryo-EM. It only needs low amounts of noncrystalline material, two major advantages when compared to X-ray crystallography. 3D structures solved by this technique are being reported at an unprecedented pace, in particular membrane proteins and large macromolecular assemblies that are resilient to crystallization, leading to a new era in structural biology. We have therefore added a brief summary on cryo-EM. SAXS study the overall shape and structural transitions of biological macromolecules in solution, from extreme (e.g., high pressure or cryo-frozen) to nearly native. SAXS provides low-resolution information on the shape, conformation, and assembly state of proteins, nucleic acids, and various macromolecular complexes. The technique also offers powerful means for the quantitative analysis of flexible systems, including intrinsically disordered proteins. Some basic notions on the technique along with its applications will be presented here. Some insights are also given to X-FELs. These revolutionary X-ray sources changed the paradigm of crystallization and data collection by providing X-ray pulses of a few tens of femtoseconds in duration facilitating entirely new and different structural approaches. The concept “diffraction-before-destruction” that X-FELs introduce paves the way for “radiation-free 3D models” and “time-resolved experiments,” very important concepts for metalloproteins, where some of the ligands and/or cofactors might be labile or prone to photoreduction, and to gain insights into enzymatic mechanisms and structural changes, respectively.

X-ray crystallography X-ray crystallography is the most used technique to determine the 3D structure of biomolecules, such as proteins, nucleic acids, or viral particles. Structure determination by X-ray crystallography begins with growing a single crystal of the macromolecule whose structure is to be determined. An X-ray beam is then passed through the crystal. The X-rays interact with the electron clouds of the atoms in the crystal and their regular and repeating atomic arrangement gives rise to a complex pattern of diffracted beams which are recorded by a detector as spots (the diffraction pattern). Encoded in this pattern is information about the positions of all the atoms in the crystal. The nature of the diffraction phenomenon is such that the information available from the diffraction pattern is not enough to recover this information. Additional experiments and a considerable amount of mathematical computation are needed to obtain a map of electron density, displayed as contour maps. Ideally, the peaks in the electron density map correspond to the atomic positions in the molecule. This map is interpreted by building an atomic model of the molecule into it, and this model is then refined against the experimental data until a final model of good quality is obtained. Crystallography can give reliable answers to many structure-related questions, from global folds to atomic details of bonding. The 3D structure provides detailed information

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on the atom positions, specific atomic interactions (intra- and intermolecular hydrogen bonds, salt bridges, hydrophobic pockets, etc.), as well as hints about the flexibility or mobility of the molecule. It can also give insights into the active site centers and reaction mechanism of enzymes, conformational changes occurring upon ligand binding, effects of point mutations in the protein fold, and their repercussion on its function. The knowledge of accurate molecular structures is an important prerequisite for structure-aided drug design and for structure-based reaction mechanisms. Structural studies in the crystal (by X-ray crystallography) and in solution [by nuclear magnetic resonance (NMR)] for the same macromolecule, as well as studies of the same macromolecule crystallized in different conditions, have revealed very similar models. Although conformational changes may occur in some flexible regions of the protein due to crystal packing, X-ray crystallography has proved to be an excellent technique to characterize the 3D structure of macromolecules. Moreover, many proteins were shown to be still active while in the crystal form, allowing further experiments to be performed on the crystalline state including time-resolved crystallography, UV
visible and Raman spectroscopy. An analogy can be made between X-ray diffraction analysis of crystal structures and a very powerful microscope used to visualize the shape of small objects. In optical microscopy, a beam of visible light strikes the object and is scattered in various directions. A lens then collects the scattered rays and reassembles them to form an image. In turn the use of electromagnetic radiation to visualize objects requires a wavelength comparable to the features one desires to resolve. Since atoms are separated by distances in the order of ˚ 5 10210 m), X-rays are in the right wavelength range for the resolution the angstrom (A of atomic features. X-ray diffraction may be compared with optical microscopy, except that there is no lens to focus the X-rays scattered by the electrons within the crystal, so no X-ray microscope can be built that allows direct visualization of the protein atoms. Instead, the crystallographer measures the intensity of the scattered X-rays from the crystal and combines them with additional phasing information to compute an electron density map from which a model can be built. Each diffracted beam is a wave, characterized by an amplitude and a phase angle. However, only the amplitude of each wave can be derived from the experimentally measured diffraction intensities and the phase information is lost and must be recovered. This is the so-called phase problem in X-ray crystallography. Since the signal from a single molecule would be too weak to be detected, a crystal is required because it comprises a very large number of molecules ( . 1015) in the same orientation acting as an amplifier (Hickman and Davies, 2001). Also, the scattering of X-rays from a single molecule is a continuous function. However, when one or more molecules (objects) are arranged in a 3D periodic arrangement (i.e., in a crystal), the scattering of X-rays can only occur in specific directions determined by the crystal lattice, whereas the intensity of these scattered beams is determined by the molecules in the crystal. A brief journey through the history of crystallography with some relevant discoveries and related achievements will be highlighted herein, although many others could also be added. Over the past century, several scientists made important contributions to the crystallography and/or structural biology fields and were awarded Nobel Prizes, mostly in Physics and Chemistry. First, the discovery of radiation called X-rays

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(“X” meaning “unknown”) carried out by Wilhelm Ro¨ntgen in 1895 earned him the Nobel Prize in Physics in 1901, the year the Nobel Prizes were first attributed. Max von Laue (Nobel Prize in 1914) then discovered that X-rays are diffracted by crystals, which opened doors that were tightly closed until then and a new age began where the world of atoms was no longer out of reach. One year later, William H. Bragg and his son William L. Bragg won the Nobel Prize for their work on the crystal structure determination of NaCl. At that time, scientists irradiated all kinds of materials with X-rays (from glasses to fibrous substances, from polycrystalline metals to beeswax). The first X-ray diffraction image of a hydrated protein was taken in 1934 by J.D. Bernal, a doctoral student of W.H. Bragg. Interestingly, two of Bernal’s doctoral students won Nobel Prizes due to their work in crystallography: Max Perutz (Chemistry Nobel Prize in 1962), who together with John Kendrew determined the first protein structures (myoglobin and hemoglobin); and Dorothy Hodgkin (Nobel Prize in 1964), for the determination of several important structures (penicillin, insulin, and vitamin B12). The Nobel Prize in Physiology and Medicine in 1962 was attributed to Francis Crick, James Watson, and Maurice Wilkins for their discoveries concerning the molecular structure of nucleic acids (DNA double-helix) and its significance for information transfer in living organisms. In the last 20 years there has been a considerable increase in the number of structures determined by macromolecular crystallography. As of January 2019, there were over 131,000 structures in the Protein Data Bank (PDB), however only c.2% correspond to membrane protein structures. The first X-ray structure of a membrane protein was a significant breakthrough in macromolecular crystallography. The structure of the bacterial photosynthetic reaction center granted Johan Deisenhofer, Robert Huber, and Hartman Michel the Nobel Prize in 1988. Their work pioneered the use of detergent molecules for the solubilization and crystallization of membrane proteins. Until then, attempts to extract proteins from membranes were unsuccessful due to the fact that proteins precipitated/denaturated in aqueous solutions because of the hydrophobic nature of their transmembrane domains. The knowledge obtained by Deisenhofer, Huber, and Michel was the basis for all subsequent work in the field, including the determination of the 3D structures of ATP synthase (Nobel Prize to John Walker in 1997), potassium channels (Nobel Prize to Roderick MacKinnon in 2003), and G protein
coupled receptors (GPCRs) (Nobel Prize to Robert Lefkowitz and Brian Kobilka, 2012). There is almost no limit to the size and complexity of the structures currently characterized by X-ray diffraction, provided suitable crystals are available. These achievements have resulted from: developments in molecular biology and protein engineering, which made it possible for a wide variety of proteins to be produced in sufficient amounts for structural studies; advances in computer technology, both in computational power and crystallographic software; access to increasingly more powerful X-ray sources such as synchrotrons, which provide very intense and tunable X-ray beams; the development of single-photon counting detectors (also known as “pixel detectors”), like PILATUS and EIGER detectors, with photon counting capabilities of up to 107 photons/s/pixel and dead times as low as 3.8 μs allowing for “shutterless” (continuous), data collection; and crystal cryo-cooling techniques that minimize crystal decay due to radiation damage during data collection, and in many cases allow a full data set to be recorded from one single macromolecule crystal.

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X-ray structures of several protein:protein and protein:nucleic acid complexes were also characterized which, a few decades ago, would be unthinkable. The work of Roger Kornberg on the structure determination of RNA polymerase (Nobel Prize in 2006), and the studies on the structure and function of the ribosome (pioneered by Ada Yonath in the late 1970s which, together with Venkatraman Ramakrishnan and Thomas Steitz, won the Nobel Prize in 2009), are good examples. The ribosome, a molecular machine that translates the genetic information encoded in the mRNA into proteins, has a very large ( . 2.5 MDa) and complex macromolecular structure composed of both RNA and proteins molecules (with approximate proportions of two-thirds and one-third, respectively).

Protein crystallization Protein production and sample preparation Procedures on protein purification will not be included in this section since they are protein-specific and beyond the scope of this chapter. It is noteworthy that in the case of proteins with affinity tags attached to one of their termini (or even both), protocols include a selective purification step by affinity chromatography. Short peptide tags such as a stretch of histidine residues (usually 6
10 histidine residues), or a STREP-tag with eight amino acids (Trp-Ser-His-Pro-Gln-Phe-Glu-Lys) are commonly used. Other tags are also available which are composed of proteins, such as glutathione-S-transferase, maltosebinding protein, or green fluorescent protein (GFP). These tag proteins are coexpressed with the target protein and used for purification with specific affinity chromatography resins. If tags do not interfere with the folding and bioactivity of the target protein, and suitable crystals are obtained, there is no need to remove them. If not, an additional step of tag removal has to be performed provided a protease cleavage site is present between the tag and the target protein. However, the protein of interest may also suffer proteolysis at internal sites, and sample heterogeneity may occur as a result of incomplete tag cleavage and removal. Other types of tags, like thioredoxin (Corsini et al., 2008), and coexpression systems, like split-GFP (Kamiyama et al., 2016), have been developed. These tags serve the main purpose of being used for affinity-chromatography purification, but can also be used as crystallization helpers or as labeling tools for colocalization studies.

Protein quality assessment Regardless of the protein source (e.g., bacterial, yeast, or mammalian) and production methods (native or recombinant protein), the quality of the protein sample needs to be assessed prior to crystallization experiments. The protein should have a high degree of purity, stability, and homogeneity. Denaturing gel electrophoresis such as sodium dodecyl sulfate polyacrylamide gel electrophoresis (SDS
PAGE) is usually a first indicator of the protein sample purity. It is important to examine the protein over a range of concentrations: more concentrated

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samples to visualize contaminating proteins present in low amounts, and more dilute to confirm the existence of only one band concerning the target protein, otherwise two close bands resulting from proteolysis may appear as a single band. Heterogeneity can arise from different sources: posttranslational modifications, deamidation, oxidation of amino acid side chains, multiple conformations, or oligomeric forms (often due to the formation of disulfide bridges between “free” cysteines). Addition of dithiothreitol (DTT) or 2-mercaptoethanol prevents oxidation of the cysteine residues. Multiple conformations of the protein in solution may result from flexible domains. Sizeexclusion chromatography (SEC) is frequently used during protein purification or as a last “polishing” step, where a sharp and symmetrical peak is a good indication of homogeneity. Furthermore, any aggregated material, if present in the sample, will be eluted in the column void volume. If a calibration curve is performed, the molecular weight (MW) of the protein can be estimated as well as its oligomeric form in solution. Dynamic light scattering is another method that can yield information about the sample homogeneity/ monodispersity. Protein samples can be further analyzed by native gel electrophoresis, isoelectric focusing, N-terminal sequencing, and mass spectrometry. A stable, monodisperse, and pure protein sample is the best starting point for crystallization trials. Protein concentration Another important parameter to consider is the protein quantity needed in order to proceed to crystallization trials. As a rule-of-thumb, 10 mg of protein should be available to carry out the initial screens and to perform optimization around those first hits in case crystalline material appears. Nowadays, with the advent of nanoliter crystallization robots, the amount of protein sample required to conduct screenings may be 10
20 times lower compared to manual screenings. This is particularly important for native or recombinant proteins with low production yields and for membrane proteins. Usually, the protein is at a concentration of 10
20 mg/mL, although proteins are known to have crystallized from samples as low as 2
3 mg/mL and as high as 275 mg/mL. Protein sample manipulations are usually performed at 4 C to prevent sample degradation and denaturation and to retard bacterial growth. Several methods exist to exchange the sample buffer and to concentrate the protein solution. One of the most commonly used is filtration through a membrane using ultraconcentrators under gas pressure, or smaller concentrators that operate by centrifugation. The selected membrane cutoff (e.g., 10, 30, 100 kDa) depends on the MW of the protein. It should be noted that some buffer components such as glycerol and detergents tend to concentrate above the membranes. Moreover, some proteins may stick to, or aggregate on the membrane. In these cases, buffer exchange through dialysis or SEC might be more suited. Sometimes it is also not possible to concentrate the protein above a certain threshold, so a search for the best buffer composition should be carried out. In this case, a fluorescence-based thermal stability assay (also known as “Thermofluor”; Moreau et al., 2012) can be used to quickly screen buffer, pH, salt concentration, and ligand conditions in order to assess some factors affecting protein stability by monitoring its thermal melting curves. The fluorescence signal as a function of temperature can be followed using a real-time PCR device and a fluorophore [e.g., commercially available dyes

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or intrinsic probes such as flavin adenine dinucleotide (FAD)] bound to the protein. An increase in the melting temperature (Tm) values will identify stabilizing conditions for the protein sample. The basic assumption is that a more stable protein is more amenable to crystallization than a less stable one. As crystallization trials involve changing the protein environment in a controlled manner, the protein buffer composition should be kept to a minimum, for example, 10
20 mM of a buffer adjusted to a specific pH. Eventually addition of DTT, 2-mercaptoethanol, ethylenediaminetetraacetic acid, salts, glycerol, or other agents (such as detergents for membrane proteins) may be required to further stabilize the protein. Quite often, the addition of cofactors, substrates (or substrate analogues), or reaction products (e.g., for enzymes), can also be good strategies to improve the stability of the protein. If the protein tolerates it, it is convenient to freeze small aliquots at 280 C or in liquid nitrogen followed by storage at 280 C, so that the samples can be thawed as needed. A cryo-protecting agent, like glycerol, may need to be included in the protein buffer prior to freezing. Some of the analytical methods described above can be used to confirm whether the protein is still in “good shape” after thawing; and if possible check its activity. Some cases have been reported in the literature, where protein crystals were obtained only if the crystallization drops were set up immediately after purification. Crystallization techniques and initial screens The first step in the 3D structure determination of a protein involves the preparation of crystals of suitable quality for diffraction data measurements from a highly pure and monodisperse protein sample. Crystallization is the first and most unpredictable step in protein crystallography, a consequence of the thermodynamic and kinetic variables involved in the formation of a unique crystalline 3D packing. For crystallization to occur, protein molecules must separate from solution and self-assemble into a periodic crystal lattice. Precipitants are added to the protein solution to reduce its solubility until supersaturation is achieved. In the supersaturated, thermodynamically metastable state, nucleation can occur and crystals form. Although a vast amount of knowledge has been accumulated over the last few decades, it is still not possible to know a priori in which conditions a specific protein may crystallize. Some efforts have been made in recent years to develop bioinformatic tools to assess “protein crystallizability” (http://ffas.burnham.org/XtalPred-cgi/xtal.pl), or “crystallization pH prediction” (http://www.ruppweb.org/cryspred/default.html). However, these web interfaces basically use predictive models that output “crystallization propensity” based on previous crystallized proteins. It cannot always be assumed that a protein will behave the same as others (similar or not) and crystallize in the same (or similar) conditions. It is known that single point mutations may alter the crystallization conditions of a protein (precipitant and/or buffer and pH) (Brito et al., 2015). Hence, crystallization trials very often start by exploring many different conditions before crystals can be grown. Usually a systematic search is performed by varying conjugated manner parameters such as the protein concentration, precipitating agents, temperature, pH, buffers, and ionic strength. A wide variety of commercial kits are available to set up the initial crystallization screens, such as the sparse-matrix screen first compiled by Jancarik and Kim (1991). Many laboratories are now equipped with crystallization robots which can dispense nanoliter volume drops into

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FIGURE 10.1 Schematic representation of crystallization techniques: vapor diffusion by hanging and sitting drop (A), microdialysis (B), microbatch (C), and free-interface diffusion (D). Solutions are depicted in blue for reservoir, pale yellow for oil, yellow for protein, and green for mixture of protein and reservoir; crystals are represented by a yellow cubic shape.

96-well plates. The miniaturization of crystallization experiments not only reduces the demand on the amount of purified protein required for screening but also enables extensive exploration of the chemical “space” to identify optimal conditions for crystallization. Crystallization methods commonly used include vapor diffusion (hanging or sitting drops), batch, dialysis, and capillary diffusion. The most frequent one is the vapor diffusion technique (Fig. 10.1A). A basic experiment setup is illustrated in Fig. 10.1A. Here, a small droplet of protein solution (typically 1
5 μL) is mixed with a similar volume of the crystallizing solution and is placed on a glass cover (silanized cover slip for hanging drop setup). The reservoir or well, with a much larger volume (usually 500 μL) of precipitating solution, has a greased rim and is sealed with the flipped-over cover slide. It should be noted that the precipitant in the protein
precipitant mixture in the hanging drop was diluted and so is less concentrated than the reservoir solution. The difference in the concentration of the precipitating agent between the drop and the reservoir drives the system toward equilibrium by diffusion through the vapor phase. Water will evaporate from the drop into the reservoir until the concentration of the precipitant in the drop equals the one in the well. Thus the concentrations of both protein and precipitant slowly increase in the drop. The solution in the drop should become supersaturated, that is, the protein solubility limit is exceeded, so eventual nucleation can occur and crystals grow (Fig. 10.2). Dialysis methods (Fig. 10.1B) permit the variation of many parameters that can influence the crystallization process. The protein is separated from a large volume of solvent by a semipermeable membrane that allows the passage of water and small molecules but blocks protein macromolecules [and also high MW polyethylene glycols (PEGs)]. Equilibration kinetics depends on the membrane MW exclusion size, ratio of precipitant concentration inside and outside the chamber, temperature and geometry of the dialysis cell. Microdialysis buttons exist in various sizes and forms.

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FIGURE 10.2 Solubility phase diagram. In a crystallization experiment, the protein sample should be in a supersaturated zone (metastable zone) where nucleation may occur and crystals grow.

Another crystallization method employed is the batch technique (Fig. 10.1C), where precipitant and protein solutions are simply mixed and allow kinetics to take their course. Supersaturation is achieved directly, rather than by diffusion. If very small volumes are used, this technique is called microbatch and the droplets are covered with oil, such as paraffin, to prevent evaporation. Sometimes, silicone oils, which are more water permeable than paraffin, are used to cover the microbatch wells, allowing for a partial exchange of solvent vapors. In the free-interface diffusion, the protein and precipitant solutions are brought into contact in a narrow vessel (e.g., capillaries) without premixing and the components are allowed to equilibrate by diffusion only. Due to the small size of the capillary, the precipitant usually diffuses directly into the protein solution. Analysis of crystallization trials Regardless of the chosen methods, the crystallization setups have to be monitored regularly, and depending on those observations, a decision has to be made on how to proceed next. Vapor diffusion is the most commonly used technique for the initial screenings. Visualization of the drops under a stereo microscope should be done just after they are set up, repeated every other day during the first weeks, and then at longer time intervals. If most of the drops remain clear, then the precipitant and/or protein concentrations should be increased (undersaturation zone, Fig. 10.2). On the contrary, if most drops show heavy precipitation, then the concentration of precipitant and/or protein should be decreased (supersaturation zone, Fig. 10.2). It is also important to observe whether the protein tends to denature under a specific pH range or in the presence of certain precipitants (e.g., alcohols, PEGs, salts). If no positive hits are obtained (no crystalline material observed), the search field should be enlarged: other screenings can be tried, the ratio of protein to precipitant modified (e.g., 1:1, 2:1, or 1:2) or the temperature varied (usually room

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temperature B20 C
25 C and 4 C, and if possible try intermediate and higher temperatures, e.g., 15 C and 30 C). Moreover, the addition of various compounds (e.g., additive screenings), formation of complexes with ligands, cofactors, or substrate analogues in the case of enzymes (which may decrease protein flexibility and/or increase its stability) can also influence the crystallization process. At this stage, prior knowledge of the protein and its biochemical/functional characterization can be very informative, providing additional clues or strategy options for further screening experiments. Since the number of possible combinations is unlimited, crystallization can be an ad eternum process and a tricky question is to decide whether to keep going or give up. As a final resort, other strategies such as limited proteolysis can still be explored, which can be useful for isolating protein fragments that can fold autonomously and thus behave as protein domains which are often more prone to crystallize in the absence of flexible loops. Site-specific mutagenesis, protein engineering (i.e., different constructs), and protein modification (e.g., lysine methylation) (Walter et al., 2006) can also be useful. Salt or protein crystals? If crystalline material is observed, it is important to confirm that the crystal is made of protein and not salt (or detergent in the case of membrane proteins). One simple method is to stain the drop with a dye such as methylene blue or IZIT (Hampton Research). These small-molecule dyes diffuse into the solvent channels of protein crystals, coloring the crystals blue with an intensity over and above the background drop color (Fig. 9.3E). Salt crystals do not possess these large solvent channels and generally stay colorless. Another test is to crush the suspect crystal: protein crystals are usually very sensitive and fragile and easily break with a needle or a stiff fiber, in contrast with salt crystals that are much harder and resistant, although there can be exceptions. Intrinsic ultraviolet fluorescence is another method that may be used to distinguish between salt and protein crystals, or to visualize protein crystals hidden under amorphous precipitates. Pusey et al. (Forsythe et al., 2006; Pusey et al., 2008) described a method for the general identification of protein crystals in crystallization experiments using noncovalent fluorescent dyes which might be of interest when the protein has a low content in aromatic residues, particularly tryptophan residues, which might hinder UV analysis. In both cases, the simplest setup requires a UV light source coupled to a microscope, or a more sophisticated system consisting of a UV
fluorescence microscope with a CCD (charge-coupled device) camera. Furthermore, several crystals can also be collected out of the drop with a loop, thoroughly washed and dissolved so that a sample can be run on SDS
PAGE to see whether a band corresponding to the MW of the expected protein is observed. Even so, the ultimate test will be the X-ray diffraction pattern of the crystals, where protein crystals should have many “spots” very close to one another, in contrast to only a few spots far away from one another for salt crystals; this observation relates to their different unit cell dimensions. Protein samples that are colored due to the presence of chromophore(s), such as metalloproteins or flavin-containing proteins, are easily recognized as protein crystals because of their color (e.g., red/brown for iron, blue for copper, and yellow for flavin; a few examples are shown in Fig. 10.3). Interestingly, some crystals can turn colorless upon protein reduction or loss of the “colored” cofactor or metal.

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FIGURE 10.3 Images of crystallization drops: (A) a clear drop (undersaturation zone); (B) drop with amorphous precipitate (precipitation zone); (C) drop with phase separation (metastable zone); and crystals of (D) lysozyme; (E) lysozyme after IZIT staining; (F) Rieske ferredoxin (with [2Fe
2S] center); (G) sulfide:quinone oxidoreductase (Brito et al., 2006) (FAD cofactor); (H) papain from Carica papaya visualized under polarized light (crystals were originally colorless); (I) laccase CotA (Bento et al., 2005) (Cu centers); (J) dissimilatory sulfite reductase (Oliveira et al., 2008) (sirohemes and [4Fe
4S] clusters); (K) aldehyde oxido:reductase (Romao et al., 1995) (molybdopterin and [2Fe
2S] clusters); (L) CbiK chelatase (Romao et al., 2011) (heme b); (M) RNaseII (McVey et al., 2006); (N) neelaredoxin (Pinho et al., 2010) (Fe center); (O) Hollidayjunction resolvase mutant. FAD, Flavin adenine dinucleotide.

Crystal optimization and seeding The positive hits from the initial screenings reveal conditions in which protein crystals can be obtained. Afterwards, upscaling (from the nanoliter-robot scale to the microliterhandmade scale to obtain bigger crystals), and a fine-tuning of the experimental parameters is generally necessary to improve the crystal quality. Usually, narrower grids are tried around the initial conditions, including small variations in the precipitant/protein concentrations, pH, temperature, and drop ratio, accompanied by trial of additives and/or ligands. The typical size of a protein crystal is in the order of 0.1
0.2 mm, although even smaller crystals can now be measured at synchrotron facilities using microfocus beamlines (i.e., with a very narrow X-ray beam which can be as small as 5 3 5 μm2).

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Protein crystallization

For crystal optimization, micro- and macroseeding techniques are often used. Nucleation is a key factor to crystallization and only occurs at supersaturated levels, which often leads to poor results, as too many small crystals in the drop. Seeding permits the uncoupling of nucleation and crystal growth. Seeds from the initial or previous trials are placed in low supersaturated crystallization solutions (so that there is no spontaneous nucleation) to allow for their controlled and slow growth. In microseeding a stock solution is made containing a few crushed crystal fragments or microcrystals and a series of dilutions are prepared from the initial stock solution. A small aliquot from each solution is then added to new crystallization drops of reduced supersaturation (less precipitant and/ or less protein) and the best dilution will yield a few larger crystals in the droplet. A popular variation of this technique is the streakseeding, where a thin whisker or fiber is streaked across a seed crystal and swiped through the new drop(s); sometimes several consecutive passes through various drops are necessary until only a small number of crystals are obtained. In contrast, macroseeding involves the transfer of a single well-formed crystal into a new crystallization drop (of similar condition where it was originally grown) after washing it several times. The idea is to remove imperfections or other micronuclei from the seed crystal surface and have more “fresh” protein available for further growth. Cocrystallization and soaking Once native crystals are obtained, it may be of interest to obtain the structure of the protein in complex with other molecules such as ligands, cofactors, nonhydrolyzable substrate analogues, drug lead compounds, DNA/RNA oligomers, peptides, or even with other proteins. Obtaining crystals of protein
ligand complexes can be achieved by either soaking ligands into pregrown protein crystals or cocrystallization, where the protein and its binding partner are incubated prior to crystallization setups (Fig. 10.4). FIGURE 10.4 Schematic representation of (A) soaking, where the ligand solution is added into a drop containing pregrown protein crystals and (B) cocrystallization techniques, where the protein and ligand solutions are incubated prior to crystallization experiments. Solutions are colored in blue for reservoir, yellow for protein, red for ligand, and orange for protein
ligand complex. The yellow cubic-shaped crystal becomes orange upon ligand binding.

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Protein crystals contain large solvent channels between the protein molecules which are filled with solvent (crystal solvent content is often around 50%, but can vary from 20% to 80%). The soaked ligand molecules are usually small enough to diffuse through these sol˚ wide) and bind to the protein. In soaking experiments, a vent channels (around 20
100 A solution containing the ligand is added directly to a drop with crystal(s) (Fig. 10.4A). However, some ligands can induce protein conformational changes upon binding, leading to crystal destruction. Changes in the mother buffer (crystallization solution) can also lead to crystal cracking or a decrease of their X-ray diffraction power. Stabilization of the protein crystals by cross-linking with glutaraldehyde can reduce these problems (Hofbauer et al., 2015). The soaking time and ligand concentration usually require optimization. In cocrystallization experiments (Fig. 10.4B), the ligand is added to the protein to form a complex that is subsequently used for crystallization trials. This is often the method of choice when the compounds are quite insoluble or the protein aggregates easily. It should be stressed that the crystallization conditions of the complexed protein may be different from those of the protein by itself. Membrane proteins It is estimated that about 30% of the human genome encodes for membrane proteins, and over 60% of all current drug targets are membrane receptors. Membrane proteins are key players in many cellular functions, such as energy production, transport, signal transduction, and intercellular communication. Despite the importance of membrane proteins, data analysis of the PDB reveals that very little structural information is available on membrane proteins, representing less than 2% of all known protein structures. The reason behind this huge deficit is due to the generally low abundance of membrane proteins and difficulty in obtaining crystals suitable for X-ray diffraction. Additional complications arise if membrane proteins suffer posttranslational modifications, for example, glycosylation of GPCRs. Integral membrane proteins span across the hydrophobic lipid bilayer (as transmembrane α-helices or forming β-barrels) and are insoluble in aqueous media. During overexpression, only a small portion of the expressed membrane proteins can be transported through the cell and properly incorporated into the host membrane before the cells are broken down. Extraction from the cell membranes and solubilization usually requires the use of detergents (water-soluble amphipathic molecules, commonly with a hydrophobic tail and polar head). Detergent molecules will associate with the membrane protein forming a micelle complex with their hydrophobic parts facing inwards and the polar groups in contact with the aqueous solution (Fig. 10.5). The detergent concentration has to be kept above its critical micelle concentration (CMC), otherwise the micelle complex will be disrupted with the subsequent precipitation of the protein. However, if the detergent concentration is too high above the CMC or if salts are used as precipitants with ionic detergents, a detergent-rich phase tends to form, into which the protein can get partitioned and denatured. PEGs are therefore the first choice to be used as precipitants for membrane protein crystallization. In the case of peripheral membrane proteins, which are weakly associated with the membrane, the extraction can be accomplished by the addition of high concentrations of salt, such as NaBr (Fig. 10.5).

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FIGURE 10.5 Solubilization of membrane proteins. Peripheral membrane proteins (C) can usually be extracted from the membrane with high salt concentrations (e.g., 3 M NaBr), and are soluble without detergents, while integral (B) and associated (A) membrane proteins require detergents for their extraction and solubilization forming protein
detergent micelles.

Detergents tend to inhibit crystallization and, even if crystals are obtained, their diffraction quality is usually very poor. Among the most important factors contributing to the very weak diffraction of membrane protein crystals are the occlusion of polar amino acid residues by detergent molecules, thus hindering their participation in intermolecular contacts; together with a high solvent content and internal disorder. The primary solubilization detergent often needs to be exchanged against a milder detergent more suitable for crystallization, such as sugar-based detergents of the maltoside or glucopyranoside family (e.g., n-octyl-β-D-glucopyranoside, n-dodecyl-β-D-maltoside) or zwitterionic phosphocholine detergents. The size, structure, and type (ionic vs neutral) of detergents, along with their CMC, are critical parameters for the crystallization of membrane proteins. Addition of other detergents, lipid molecules, or small amphiphiles can help to improve crystallization. Detergent kits are commercially available that can be tried as additives. Although the solubilized membrane protein can be regarded as a “soluble protein,” the presence of detergent and the delicate equilibrium of the detergent
membrane protein micelle add another level of complexity to the process. Another technique that has been developed for membrane proteins is crystallization in lipid cubic phases (Caffrey and Cherezov, 2009).

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The lipid monolein forms a complex phase system with water. One of the phases is a bilayered cubic phase containing 50%
80% lipid and interconnected solvent channels. This setup tries to mimic the native lipid bilayer environment. Harvesting and mounting of crystals The ultimate test of crystal quality is achieved when X-ray diffraction images are obtained so that its diffraction pattern and resolution limit can be analyzed. In order to do this, crystals need to be harvested from the crystallization drop and mounted on the diffractometer. Protein crystals need to remain surrounded by their mother liquor during crystal mounting and data collection otherwise they will dry out, lose long-range 3D order, and stop diffracting. Crystals can be mounted in quartz capillaries and data collected at room temperature (Fig. 10.6A). However, crystals are most frequently pooled from the crystallization drop using a tiny loop (made of, e.g., nylon fiber, Fig. 10.6B) attached to a steel pin which is in turn mounted on a steel base, and flash-cooled in either liquid nitrogen (boiling point of 77K or 2196.15 C), liquid propane (B150K or B 2 123.15 C, more rarely used), or under a cryogenic nitrogen gas stream (around 100K or 2173.15 C). The crystal-containing pin is then mounted on a diffractometer equipped with a cryostat to keep the low temperature (c.100K) during data acquisition. Alternatively, the flash-cooled crystal can be inserted into a plastic vial filled with liquid N2 and stored in a cooled Dewar for later measurements. The main benefit of low-temperature diffraction measurements is the reduction of X-radiation damage to the crystals. As a consequence, complete data sets can be recorded from a single crystal. The dynamic disorder of the crystals may also be decreased (reflected in the thermal motion parameters, see below). However, to prevent the detrimental formation of ice within the crystals and/or mother liquor, a cryoprotectant is required. Therefore crystals are either grown in solutions already containing a cryoprotectant, such as glycerol, PEG, sucrose, or salts in the appropriate concentration, or are briefly swept through a buffer, where such a compound was added prior to freezing. It is important that this addition is done by replacing water in the buffer, so that the concentration of all the other substances remains unchanged.

FIGURE 10.6 Crystal mounted in a sealed quartz capillary (A) with reservoir (usually for room temperature data collection), and in a loop (B) surround by a cryoprotectant solution (for cryogenic temperature data collection).

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Data collection Once crystallization is successful, the next step is to expose a crystal to an X-ray beam and measure the diffracted signal. A crystal is a periodic, finite assembly of unit cells, the smallest repeating unit that can generate the crystal with only translation operations (Fig. 10.7). Each unit cell contains the same number of identically arranged molecules. The asymmetric unit is the smallest fraction of the unit cell that can be rotated and translated using only the symmetry operators allowed by the crystallographic symmetry to generate one unit cell. It may contain the whole macromolecule or one (or more) subunit(s) of a multimeric protein. If two or more identical molecules are present in the asymmetric unit, they are usually related by noncrystallographic symmetry (NCS) (Fig. 10.7). The crystal acts as a 3D grating that diffracts the incident beam of X-rays only in certain directions. These directions depend on the orientation of the crystal and its unit cell dimensions. The diffraction pattern is a record of the directions and intensities of all the diffracted X-ray beams by a crystal. Scattered waves in phase add up, augmenting the signal to a measurable level in some directions, and cancel one another out in all other directions. That is why the diffraction pattern from a crystal consists of a discrete number of diffracted beams. These beams can be recorded on a detector as diffraction spots. The crystal is rotated relative to the X-ray beam so a complete diffraction pattern can be recorded. A single diffraction data set may comprise tens to hundreds of thousands of reflections, and sometimes more than one data set is needed. The diffraction data is the experimental “raw material,” from which the protein structure is derived. A good quality data set is characterized by a low background noise, data extending to high values of the reflection angle θ with no overlaps, a wide range of intensities between the weakest and strongest reflections (which should not be overloaded), a good agreement between the intensities of symmetry-related reflections. The resolution d

FIGURE 10.7 The asymmetric unit is the smallest portion of a crystal structure to which symmetry operations can be applied in order to generate the complete unit cell (i.e., the crystal repeating unit). In this case, the asymmetric unit also displays NCS, which relates the two molecules by a twofold NCS operator (180-degree rotation). Two asymmetric units form the unit cell, which is then translationally repeated in three directions to make a three-dimensional crystal. NCS, Noncrystallographic symmetry.

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depends on the angular extent to which the intensity data have been measured (θ is the diffraction angle) and on the wavelength λ of the radiation, according to Bragg’s law: λ 5 2d sin θ ˚ ) in a protein structure is usually defined as the value of dmin correspondResolution (A ing to the θmax of the data set. This parameter corresponds to the level of detail which can be seen in the resulting electron density map, and ultimately determines the quality of the structural model that can be derived from the map. Usually, in-house X-ray equipment comprises a rotating anode X-ray generator with a ˚ , a two-dimensional (2D) copper target which emits a peak of CuKα radiation at 1.5418 A X-ray detector, a CCD, and a cryo-cooling system to keep crystals at cryogenic temperatures. CCD detectors record diffraction data at a faster rate with higher sensitivity and a better dynamic range (ratio between the strongest and the weakest measurable intensities) compared to the former imaging-plate detectors. Furthermore, synchrotron beamlines are now being equipped with PILATUS and EIGER detectors, consisting of 2D hybrid pixel array detectors, which operate in single-photon counting mode. These detectors feature several advantages compared to current CCDs, namely no readout noise, superior signalto-noise ratio (SNR), shorter readout time, and higher dynamic range. The short readout and fast framing times allow for the collection of diffraction data in continuous mode without opening and closing the shutter for each frame. Some of these detectors have also been developed for in-house application, but due to their high cost, they are mostly present at synchrotron facilities. The access to synchrotron facilities is still very important in the development of X-ray crystallography. Synchrotron radiation sources provide X-ray beams with high intensity, low divergence, and a wide range of energy. The high intensity of X-rays increases the rate of data acquisition and allows data to be recorded faster and from smaller crystals than when using a home source. The low divergence reduces the overlap between adjacent reflections enabling data collection from crystals with larger unit cells (e.g., virus particles, large assemblies, or complexes). Furthermore, a beam with tunable energy offers the possibility to select a specific wavelength, which is essential for anomalous dispersion experiments (see the “Phase determination” section). On the other hand, a white X-radiation can yield a complete Laue diffraction pattern for a protein crystal on a millisecond scale. Laue diffraction is most often used in time-resolved crystallographic experiments.

Phase determination As described in the previous section, the diffraction pattern is a record of the directions and intensities of all the diffracted X-ray beams by a crystal. Each diffracted beam corresponds to the vector sum of the individual contributions from all the atoms in the unit cell, and is characterized by an amplitude |Fhkl| (related to the number of electrons in each contributing atom) and a phase αhkl (derived from the atom positions within the unit cell). Just as the atomic positions in a crystal structure can be defined using a 3D coordinate system (x,y,z), each individual diffracted beam (or diffraction spot) can be defined via three integer values (h,k,l), termed Miller (or Bragg) indices.

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Fhkl is called the structure factor (depends on the structure of the scattering atoms) and is represented by: XXX Fh k l 5 jFh k l j exp ðiα h k l Þ h

k

l

The structure factor amplitudes |Fhkl| are proportional to the square root of the intensity of the corresponding diffracted beams and are experimentally measured, whereas the phase angles are not. An estimate of these “lost” phases αhkl has to be obtained by “indirect methods,” in order to calculate an electron density map of the crystal structure. Once both the amplitudes and reasonable estimates for the phases of all the diffracted X-ray beams are known, it is possible to calculate the electron density (ρ) at any point xyz within the unit cell, represented by the following equation:    1 XXX ρðxyzÞ 5 jFh k l j expðiαh k l Þ exp 22πi hx 1 ky 1 lz V h k l where V is the volume of the unit cell (the repeating unit of the crystal) and αhkl is the phase associated with the structure factor amplitude |Fhkl|. It is important to remember that each Fhkl results from the diffraction of all atoms within the crystal lattice. Therefore the final 3D structure will be a time- and space-averaged picture of the entire volume that was irradiated by the X-rays. The so-called phase problem is the last bottleneck that lies between the crystallographer and the 3D structure of a macromolecule. There are three major methods for obtaining phase information: molecular replacement, anomalous scattering, and isomorphic replacement. Direct methods may also be a possibility when atomic resolution diffraction data is available.

Molecular replacement When the 3D structure of a macromolecule homologous to the one of interest is available, phase determination by molecular replacement is frequently the method of choice. In this method, the known structure (search model) is used to estimate the phases for the target unknown structure. First, a rotation function is computed to orient the search model in the new unit cell, followed by a translation function to determine the position of the correctly oriented model in the target cell. Thus the 3D structure of the known macromolecule is used as the starting model to provide phase angles for the observed structure factor amplitudes from the unknown structure. This method is applicable in the case of mutated proteins, proteins complexed with ligands that crystallize in different crystal forms, or structurally related proteins. However, since the structural identity is not a priori known, the sequence identity is used as a guide. A rule-of-thumb is that above 25% of sequence identity, a molecular replacement solution may be possible, since these two proteins should share structural homology. This correlation between 3D structure and amino acid sequence was first presented by Chothia and Lesk (1986), and since then other authors have contributed to this topic making molecular replacement more accessible and robust. However, it should be noted that sequence identity is not necessarily a guarantee of sufficient structural homology to successfully apply this method, and vice versa. If various

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homologous structures are available, an “average structure” can also be used as a search model highlighting the conserved structural motifs and downweighting the most dissimilar regions. Related to this approach, recent efforts have been made for the development of automated software pipelines for molecular replacement protein structure solution based on domains/secondary-structure fragments databases rather than protein structures (Long et al., 2008; Vagin and Lebedev, 2015). It should be noted that the main drawback of the molecular replacement method is the model bias, which results from using an atomic model to calculate crystallographic phases. The resulting electron density map will tend to have features in the model even if they are not actually present in the structure, so refinement procedures are used to reduce model bias. Many structures are now solved by the molecular replacement method, alone or in combination with other methods.

Isomorphous replacement The isomorphous replacement method was first applied by Max Perutz and John Kendrew back in the 1960s. It requires the attachment of a heavy atom (i.e., with high atomic number, e.g., Hg, Pt, Au, Pb, Ag) to the protein in the crystal. Usually, the protein crystals are soaked in a solution containing salts of the heavy atom at low concentrations or, alternatively, the protein can be cocrystallized with the heavy-atom compound. The heavyatom addition/substitution should neither affect the protein structure nor the crystal unit cell dimensions (isomorphism), and so soaking is usually preferred. These heavy-atom derivatized crystals should show measurable changes in the intensities of some reflections in relation to the “native” crystals, which can be used to deduce the positions of the heavy atoms. In this method, the intensity differences between the native crystal and the derivatized one (due to interference effects on the intensities of the diffracted beams caused by the addition of heavy atoms to the protein) provide the estimates of the protein phase angles. A native crystal and a good single isomorphous derivative may be sufficient for structure determination [single isomorphous replacement (SIR)], but usually multiple derivatives are necessary to derive phase information [multiple isomorphous replacement (MIR)].

Anomalous scattering Phase determination by anomalous scattering is based on the fact that heavy atoms at wavelengths near their absorption edges scatter X-rays anomalously with a change in amplitude and phase. The resulting changes in the diffracted intensities can be used to determine the heavy-atom substructure, from which phase estimates for all structure factor amplitudes can be computed. This method overcomes problems of nonisomorphism (present in SIR/MIR phasing), because all data are collected from the same crystal, but requires an accurate measurement of the diffracted intensities; however, crystal decay due to radiation damage is sometimes a problem. Hendrickson and Ogata (1997) developed a procedure making use of engineered proteins containing selenomethionine instead of methionine residues to provide a suitable anomalous scatterer. As a general rule-ofthumb, for a successful multiple wavelength anomalous dispersion (MAD) experiment on

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selenomethionine-containing proteins, at least one selenomethionine per 100 amino acids is required. However, under favorable conditions, it may be possible to succeed with a lower ratio. This method is the obvious choice for metalloproteins, which contain one or more endogenous metals (e.g., Fe, Cu, Mo, Zn, Ni). However, the presence of the metal(s) in the protein is not per se a guarantee of success toward phasing using this inherent anomalous scatterer(s). This will depend on the size of the protein, type, number and occupancy (occ) of the metals (atomic number and anomalous scattering properties), diffraction quality of the protein crystals, and accuracy of X-ray data measurements. Eventually, it may be necessary to search for additional heavy-atom derivatives where a combination with any of the aforementioned methods may contribute to solve the phase problem. In recent years, ˚ ) beamlines were built at synchrotrons, so anomalous scattering long-wavelength ( . 1.8 A of sulfur (naturally present in the amino acids cysteine and methionine), and phosphorous (present in nucleic acids and some nucleotides like ADP, ATP, FMN, or FAD), can be used to provide initial phases for proteins. A few crystallographic basic concepts need to be introduced at this stage to explain the rationale behind anomalous scattering. The atomic scattering factor has three components: a normal scattering term that is dependent on the Bragg angle and two terms that are not dependent on the scattering angle, but on wavelength: f ðθ; λÞ 5 f0 ðθÞ 1 f 0 ðλÞ 1 ifvðλÞ The latter two terms represent the anomalous scattering that occurs near the absorption edge when the X-ray energy is sufficient to promote an electron from an inner shell. The dispersive term [f0 (λ)], reduces the normal scattering factor, whereas the anomalous term, [ifv(λ)], is 90 degrees advanced in phase. This leads to a breakdown in Friedel’s law (stating that intensities of the h, k, l and 2 h, 2 k, 2 l reflections are equal) giving rise to anomalous differences that can be used to locate the anomalous scatterers. It should be noted that these anomalous scattering effects are significant for atoms such as Fe, Cd, and Se, much smaller for S and P, and virtually nonexistent for C, N, O, and H. A typical MAD experiment collects diffraction data sets at three wavelengths: at the peak of the absorption curve (highest fv), at the point of inflection of the absorption curve (minimum f0 ), and at a remote wavelength, above the absorption edge, where f0 is near zero and fv may still be relatively large. The values for f0 and fv depend on the scatterer and the theoretical values can be plotted, as illustrated in Fig. 10.8, for three elements: Fe, Hg, and Li. However, since they are also dependent on the experimental setup, the actual shape of the absorption curve must be experimentally determined by a fluorescence scan of the crystal at the synchrotron beamline prior to data collection. For example, in the crystals of cytochrome c nitrate reductase complex the Fe peak and inflection point wave˚ , respectively. As can be observed in Fig. 10.8, and in lengths were 1.7393 and 1.7408 A contrast with Fe and Hg, the absorption edge of Li lies outside the represented energy range, which is that most commonly accessible in a tunable beamline at a synchrotron X-ray source. This is also true for many other elements. In recent years, single wavelength anomalous dispersion (SAD) method has become increasingly popular. This method makes use of data collected at just one wavelength, typically at the absorption peak or high-energy remote. It minimizes problems of radiation

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FIGURE 10.8 Plot of theoretical f0 and fv values for Fe, Hg, and Li, in the energy range 5000
20,000 eV, to assess the anomalous scattering properties of the elements. Source: Adapted from http://skuld.bmsc.washington.edu/ scatter/AS_form.html.

damage and nonisomorphism, but requires very accurate measurements. The SAD method relies on density-modification protocols, such as solvent flattening, to break the phase ambiguity problem introduced by using data from only one wavelength, and provides interpretable maps.

Direct methods Direct methods are based on the positivity and atomicity of electron density which leads to phase relationships between normalized structure factor amplitudes. If the phases of some reflections are known, or can be assumed as a set of starting values, then the phases of other reflections can be deduced leading to a bootstrapping of phase values for all reflections. This methodology, also called ab initio phase determination, has limited ˚ ), but it is used routinely applicability since it requires atomic resolution (better than 1.2 A to find the heavy-atom substructures in other methods.

Heavy-atom derivatization The essential foundations of techniques for heavy-atom crystal derivatization and its theoretical basis can be found in several books (Blundell and Johnson, 1976; Drenth, 1999) and review articles (Boggon and Shapiro, 2000; Garman and Murray, 2003) and will not be extensively referred here. The International Tables for crystallography, Volume F (Rossmann and Arnold, 2006) also presents valuable information on the subject. A very useful tool prior to any derivatization experiment is the heavy-atom database (Islam et al., 1998), a website that contains information on the preparation and characterization of heavy-atom derivatives of protein crystals.

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Heavy-atom derivatives available for phasing can be divided into seven categories: endogenous metals which can be used directly for SAD/MAD (e.g., heme Fe for cytochromes), or substituted by heavier metals with similar valency (e.g., substitution of Ca for Sr which has higher anomalous signal); single metal ions which can be bound electrostatically to the protein; Se in selenomethionine; metal compounds which require a chemical reaction to take place; multimetal cluster complexes used for large proteins and multiprotein assemblies (e.g., the Ta6Br12 cluster; Knablein et al., 1997), the noble gases Xe and Kr; and halides and triiodide, including brominated nucleic acids (Garman and Murray, 2003). Historically, heavy metal salts used to successfully make derivatives are the so-called magic seven compounds (Boggon and Shapiro, 2000), namely K2PtCl4 (platinum potassium chloride), KAu(CN)2 (aurous potassium cyanide), K2HgI4 (mercuric potassium iodide), UO2(C2H3O2)2 [uranium (VI) oxyacetate], HgCl2 (mercuric chloride), K3UO2F5 (potassium uranyl fluoride), and para-chloromercurybenzoic sulfate. However, the success of the “magic seven” as derivatives may be consequential to the fact that they have been utilized more often than other compounds. Careful analysis of the protein primary sequence can give important clues as to which heavy atoms to try initially: mercurial compounds tend to bind to cysteines (free sulfhydryls and not on disulfide bridges) or histidines, while platinum compounds are more prone to bind to cysteine, histidine, and methionine residues. The usual starting conditions for trying heavy-atom derivatization are 0.1, 1.0, and 10 mM concentrations of soaking solution for 10 minutes to several days at the highest concentration that the crystal will tolerate. It is generally accepted that concentration of the heavy atom is a more useful variable than soaking time. Cocrystallization of the protein with heavy atoms can also be performed, although it may change the crystal unit cell dimensions (problems of nonisomorphism). Moreover, the composition and pH of the mother liquor can also affect the metal compound used for derivatization and should be taken into consideration when designing the derivatization experiment.

Model building and refinement Once phases are known, they can be combined with the structure factor amplitudes derived from the X-ray diffraction data to calculate an electron density map. Computer graphics are required to build a model of the structure of interest into the electron density map. This model is then refined to optimize the agreement between the observed and calculated values of the structure factor amplitudes. The quality of the electron density maps depends on the quality of the X-ray diffraction data and phase estimates. Of particular importance is the resolution of the data (Fig. 10.9). ˚ resolution may allow the identification of helices and eventually of Maps at 5
6 A ˚ the topology folding can be established, the polypeptide chain can be β-sheets; at 2
3 A ˚ or traced, and side chains assigned providing the amino acid sequence is known. At 1.5 A better, the individual atoms are almost resolved with a well-defined network of water molecules and hydrogen atoms may become visible. In contrast to well-ordered small ˚ , not many macromolecule molecules that often diffract to resolutions better than 0.5 A

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FIGURE 10.9

Electron density maps displayed for an α-helix at different resolutions. The level of detail fades ˚ , the electron density perfectly “wraps” the built model and even away with the decrease of the resolution: at 1.1 A ˚ the aromatic residues display a doughnut shape with a hole in the middle of the aromatic ring; in contrast at 6 A only an oblong shaped electron density is visible for the whole α-helix and no side chains are clearly defined.

structure crystals diffract to atomic resolution. This may result from packing defects (not all protein molecules are identically packed within the crystal), weak intermolecular interactions, protein flexibility, and high solvent content. It is important to emphasize that the diffraction pattern does not abruptly stop at high diffraction angles, but gradually fades away. This notion is important because the crystallographer will be called to analyze the statistics of data reduction and decide where to cut the resolution (see below). In the case of molecular replacement, a preliminary model (based on the similar known structure) is already available, and one has to adjust this model (by moving, adding, or removing atoms or groups of atoms, namely side chains that differ from the search model and the model to be built) into the computed electron density maps. Model building is facilitated but care must be taken to get rid of model bias arising from the initial phases. If phases were obtained experimentally (by anomalous scattering or isomorphous replacement methods) the initial map will be “empty” and the model needs to be built from scratch. The crystallographer can start by inspecting the maps and try to recognize continuous segments with a shape consistent with the geometry of secondary-structure elements such as α-helices or β-sheets. It may be possible to build some stretches of polyalanine chains and to identify residues with distinctive side chains, such as tryptophan, phenylalanine, histidine, or tyrosine. Selenium (in the case of a SeMet derivative), metal ions, or heavy atoms make good anchor points in sequence assignment. While some years ago, model building had to be done entirely by hand, at present automated modelbuilding programs are available to help with this task. Despite the best efforts of crystallographers the model will always contain errors, namely geometry and stereochemistry errors (deviation of bond lengths and angles from expected values, implausible torsion angles, and some highly correlated small errors). Therefore the model needs to be refined against the experimental diffraction data (observed structure factor amplitudes). While earlier methods used an iterative nonlinear least-squares procedure, modern refinement programs use more robust maximumlikelihood algorithms to minimize the difference between the observed and calculated

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structure factor amplitudes. The model parameters being refined are one or more overall scale factors, the atomic positions, B-factors (describing the atomic displacements), and occupancies. The high solvent content of protein crystals also needs to be taken into account. While the ordered solvent molecules can be included in the model being refined, a large fraction is disordered but still contributes to the Bragg scattering, in particular at low resolution. For this reason, parameters describing this contribution are also included in the refinement. The ratio of observations to variables is usually too small for unrestrained positional refinement due to the typical resolution we obtain for macromolecules. Hence, additional “observations” are included in the refinement procedure in the form of geometrical restraints for the atomic positional parameters, based on dictionaries of standard geometrical data [e.g., that compiled by Engh and Huber (1991) on geometrical data derived from small-molecule structures]. If there are several copies of the same molecule in the asymmetric unit, then NCS (see Fig. 10.7) restraints among them can also be included in the refinement, making the copies geometrically similar. As the model improves, so do the phases calculated from it, and so do the electron density maps. Thus the overall procedure consists of alternating cycles of restrained refinement with model correction against the improved electron density maps, until convergence is achieved (no more significant improvements are possible) and a suitable 3D structure of our protein model is ready. A schematic representation of the various steps involved in structure determination of proteins by X-ray crystallography is presented in Fig. 10.10 and herein described.

FIGURE 10.10 Schematic diagram of the steps involved in the three-dimensional structure determination of proteins by X-ray crystallography.

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Structure analysis and model quality Content of crystallographic models A refined model contains the atomic positions (coordinates x, y, z) of each element present in the asymmetric unit, its occ and B-factor. Hydrogen atoms can also be included in idealized calculated positions to help during the refinement; usually they are not included in the final model unless in ultrahigh resolution structures where sometimes they are visible in the electron density maps. At present, most scientific journals require the deposition in the PDB of the coordinates and experimental data (i.e., the structure factors) of the structure to be published. Aside from the coordinates, each entry also includes the names of molecules present in the PDB file, primary and secondarystructure information, sequence database references, ligand(s) and biological assembly information, details about data collection and structure solution, and bibliographic citations. A few lines are listed corresponding to the PDB entry (2J7A) of cytochrome c nitrite reductase complex: HEADER TITLE TITLE ... ATOM ATOM ... HETATM54766 HETATM54767 HETATM54768 ... HETATM58415 ... HETATM58624 HETATM58625 TERM

OXIDOREDUCTASE 06-OCT-06 2J7A CRYSTAL STRUCTURE OF CYTOCHROME C NITRITE REDUCTASE NRFHA 2 COMPLEX FROM DESULFOVIBRIO VULGARIS 1 2

N CA

GLY A GLY A

26 26

6.532 5.841

78.390 77.273

35.719 34.998

1.00 49.72 1.00 49.81

N C

FE CHA CHB

HEM A1001 HEM A1001 HEM A1001

-6.896 -3.984 -7.680

30.107 29.659 26.759

70.628 68.883 70.277

1.00 11.26 1.00 6.68 1.00 8.96

FE C C

CA

CA

-72.372

1.00 17.39

CA

O O

HOH A2001 HOH A2002

37.108 38.101

1.00 14.08 1.00 20.04

O O

Q1006

52.958 -9.260 0.602 -3.834

62.423 62.200

where “6.532, 78.390, 35.719” are the coordinates of the first atom of the model, corresponding to main-chain nitrogen atom (N) of glycine (GLY) 26 from chain A, show˚ 2. ATOM is used for amino acid residues ing a full occ (1.00) and a B-factor of 49.72 A and HETATM for heteroatoms (e.g., metals, ions, ligands, or solvent molecules). Notice that ATOM 2 represents the Cα of Gly26, whereas HETATM58415 CA corresponds to a calcium ion. The last column lists the atom type, in this case, C (carbon) or CA (calcium). The occ parameter measures the fraction of molecules in the crystal in which a certain atom actually occupies the position specified in the model, it can vary between 0 and 1. At ˚ , some side chains may show alternate conformations, that is, a resolution better than 2.0 A the same side chain is observed in two different confirmations; if so the side chain will be split into parts A and B and their respective occupancies will be adjusted by the refinement software (e.g., 0.3 and 0.7). Some other side chains, such as surface (solvent exposed)

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residues, or even whole amino acid residues(s), as part of disordered loops, may not be well defined in the electron density maps. Here, opinions diverge as to whether crystallographers should try to fit and include them in the model (occ of 1) and allow the B-factor to refine to high values or set the occ to 0. Sometimes no electron density is visible at all for some parts of the protein, and if so the corresponding part of the protein chain should be omitted from the model, and breaks in the polypeptide chain will be visible while displaying the model on a graphics program. This is not dependent on the resolution nor the quality of the data, rather to the intrinsic flexibility of the protein and/or the crystal packing. Another important parameter is the atomic displacement parameter, often referred to as the temperature factor or B-factor. It describes the degree of displacement of an atom within the unit cells of the measured crystal, that is, how a certain atom oscillates or vibrates around its mean position specified in the model. The lower the B-factor, the more static an atom is; the higher the B-factor, the more mobile an atom is. Very high B-factors ˚ 2) are often found in both termini of the model as well as in solvent exposed resi( . 80 A dues with long side chains (e.g., arginines, lysines, glutamates). A picture of the crystallographic model colored according to temperature factors (commonly from low B values in blue to high B values in red) provides insights into which parts of the protein have higher motion/disorder. If atomic resolution is available, B-factors may be refined anisotropically, with six parameters instead of one, and the atomic motion can be displayed as triaxial ellipsoids rather than spheres.

Validation A concern among the crystallographic community was the development of validation tools to assess the quality of the experimental diffraction data, the refined model, and the agreement between them. All publications involving X-ray structure determinations should report the relevant statistical parameters for data collection and processing, and model refinement. The parameters commonly used to assess the quality of the diffraction data include resolution, completeness, multiplicity, SNR, and the merging R-factor of the data set. ˚ ) is calculated from the highest Bragg angle q where significant difResolution (dmin, A fraction data were measured, and roughly corresponds to the distance range at which two features can be resolved in the corresponding electron density. As a matter of fact, resolution itself is not a validation parameter per se but rather a quantitative value for the analysis of the data quality. In practical terms, the resolution determines the level of detail of electron density maps (see Fig. 10.9); hence the higher the resolution (the lower its nominal value), the higher the detail. Completeness is defined as the percentage of the measured crystallographic reflections in a data set over the total number of theoretically possible unique reflections, at specified resolution limits. Completeness should be as high as possible in all resolution bins of the data set (missing reflections lead to the deterioration of the model parameters). At present, with the introduction of novel quality assessment parameters like CC1/2 (see below) and

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alternative scaling protocols (e.g., STARANISO server that scales the data anisotropically; Tickle et al., 2018), the overall completeness may be lower (30%
50%), with values as low as 5%
10% in the highest resolution shell. Redundancy (or multiplicity) is calculated as the ratio between the number of measured reflections and the number of unique reflections present at a given resolution range. This calculation gives the average number of independent measurements of each reflection in a crystallographic data set, so the more individual measurements are done, the more precise the average measurement becomes. SNR, (I/σ(I)), in practical terms, indicates how many times above noise level were the intensities measured. , I/σ(I) . , the mean value of I/σ(I), is usually listed for the whole data set and for the highest resolution shell. The diffraction limit of a given crystal was usually set at a resolution where this ratio remained above 2, but with the introduction of novel quality indicators and scaling protocols this value is gradually decreasing with data sets being now reduced to resolutions where the I/σ(I)is as low as 1. Rmerge also known as Rsym, measures the agreement between the multiple independent observations of the same reflection and used to be one of the most used parameters to decide where to cutoff the data in the highest resolution shell. Because this value tends to increase as the data multiplicity increases, two more appropriate statistics were devised, named redundancy-independent R-factor (Rrim or Rmeas) and precision-independent R-factor (Rpim) (Diederichs and Karplus, 1997). A good data set used to have low Rmerge values, with high values indicating suboptimal data quality or problems with the data processing. Rmerge values as high as 30%
40% in the highest resolution shell, or even of 60%
70% for high-symmetry space groups, were considered reasonable. Nowadays it is quite normal to find data sets where the overall Rmerge is 30%
40% with 900%
1000% in the highest resolution shell (see below) (Akey et al., 2014; Lariviere et al., 2012). In recent years, the CC1/2 parameter was introduced by Karplus and Diederichs (2012). This correlation coefficient compares two random sets of data, each one containing half of all the measurements made for each unique reflection in a complete data set. As emphasized by the authors, CC1/2 estimates the correlation of an observed data set with the underlying “true signal” in the data, thus better determining the high-resolution cutoff, where Rmerge and other parameters fail. In 2015 the same authors claimed that this parameter should be the only one used to determine the high-resolution cutoff of the data (Karplus and Diederichs, 2015). At present, this is the most used criteria to select where to cut the data pinpointing a limit up to which data are still useful. The utilization of this parameter led to other ones being overlooked (like Rmerge), although most specialty journals still request them. In addition to assessing the quality of the diffraction data, model validation against the experimental data, and examination of stereochemistry during the model building and refinement processes are also of critical importance. This is usually performed using the residual R-factors and by comparison of the model geometry against the standard values. Validation programs currently available include PROCHECK (Laskowski et al., 1993), WHATCHECK (Hooft et al., 1996), and MolProbity (Chen et al., 2010), with the latter being most used, namely as part of the refinement software pipelines like BUSTER-TNT (Bricogne et al., 2017) and PHENIX (Adams et al., 2002). “MolProbity Score” is a very useful feature that provides a single number representing the crystallographic model quality

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statistics. It is a log-weighted combination of several validation parameters (e.g., clash core, percentage Ramachandran not favored, and percentage bad side chain rotamers), giving one number that reflects the crystallographic resolution at which those values would be expected, that is, a structure with a numerically lower MolProbity score than its actual crystallographic resolution is, quality-wise, better than the average structures in the PDB at that resolution (Chen et al., 2010). Rwork or Rcryst, measures the agreement between the experimentally measured structure factor amplitudes, Fobs, and those calculated from the model, Fcalc. Rfree, or free R-factor, is computed in the same manner as Rwork, but using a small set of structure factor amplitudes, the so-called test set, which are set aside in the beginning of the refinement and are never used throughout refinement (Brunger, 1997). These structure factor amplitudes never “see” the model and they are used only for cross-validation. Rfree measures how well the current atomic model compares with that subset of the measured structure factor amplitudes that were not present in the refinement calculations, whereas Rwork measures how well the current model compares with the entire measured data set. These residual R-factors are used to evaluate the progress of structure refinement, and their final values are important criteria of model quality. The Rmerge and resolution of the data will influence the value of Rwork. The higher the resolution, the lower the Rwork is expected to be. A desirable target Rwork for a protein crystal structure refined with data to ˚ is around 20%; structures near-atomic resolution should have lower Rwork, around 2.5 A 10%
15%. The value of Rfree also depends on the data resolution. At lower resolutions it may be up to 5%
10% higher than Rwork, whereas at near-atomic resolution the difference should be 1%
2% at most. Validation of model geometry compares model properties such as stereochemistry, local chemistry environments, and packing propensity against their expected target values based on prior knowledge. In particular, properties such as bond lengths, bond angles, ϕ (phi)
ψ (psi) torsion angles, side chain torsion angles, peptide flips, clashes, Cβ deviations, Asn/His/Gln side chain flips, and local environment profiles can be checked. As introduced in the refinement section, a restraint is a subsidiary condition imposed on the atomic parameters during crystallographic refinement, based on prior knowledge. For example, in a protein structure, the bond lengths and angles are expected to be within a specified range of values. These values are standard for all structures and are extracted from high-resolution data sets and small-molecule structure databases like the Cambridge Structural Database. During refinement, the stereochemical deviations from ideal values are minimized along with the deviations between observed and calculated structure factor amplitudes. In order to ensure a well-behaved refinement providing a chemically meaningful structure, different weights are assigned to each term of the mini˚ ), a higher weight is given to the stemized function. At low resolution (below e.g., 2.5 A ˚ ) the crystallographic term reochemistry, whereas at high resolution (above e.g., 1.5 A becomes more important and the electron density is allowed to be the major driving force in the refinement. Deviations from the standard values indicate how much a model differs from geometrical parameters that are considered typical or represent chemical ˚ for bond knowledge. Reasonable root mean square deviation values are around 0.02 A lengths and 1.5 for bond angles. Higher values may indicate errors in the model or problems with the refinement protocol.

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10. Structural biology techniques: X-ray crystallography, cryo-electron microscopy, and small-angle X-ray scattering

The Ramachandran diagram plots the ϕ
ψ torsion angles of a polypeptide (Fig. 10.11). The rotations of main-chain N
Cα and Cα
C bonds, represented by ϕ and ψ, respectively, are free to rotate, but many combinations are not possible because of steric repulsion. The Ramachandran plot shows regions where conformations are allowed and others that are sterically disallowed for all amino acids except GLY which is unique in that it lacks a side chain. In fact GLYs are often found in loop regions where the polypeptide chain makes a tight turn. Most if not all amino acid residues should lie in the favored regions, although proteins can accommodate a few outliers. Structure publications often include the results of the plot, with an explanation about non-GLY residues that lie in high-energy (not allowed) areas, such as structural constraints that overcome the energetic cost of an unusual backbone conformation, and which may have functional significance. Once the validation step is accomplished, one can move on to structure
function analysis, that is, interpreting the final model based on the available experimental data (physicochemical and biochemical information). What can we get out of the 3D structure? Does it

FIGURE 10.11

Ramachandran plot calculated with the program RAMPAGE (Lovell et al., 2003). Different shades of blue refer to favored (dark blue), and allowed (light blue) regions of the Ramachandran for all residues except glycine. Dark brown and light brown represent favored and allowed regions of the Ramachandran for glycines, respectively. Squares represent all residues except prolines (triangles) and glycines (crosses).

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correlate with the function? Does it explain reported biochemical data? Are there any binding sites? Can we identify interacting partners? Are there any structural homologues? Can we propose a structure-based mechanism?

X-ray free electron lasers X-FELs are linear accelerator X-ray sources that deliver X-ray pulses with very short intervals (in the femtosecond range). Not only the geometry of an X-FEL is different from that of a synchrotron (linear vs circular), but its brilliance (the number of photons delivered per second), is nine to ten orders of magnitude higher than common synchrotrons (Neutze et al., 2015). These new features of this technology open new possibilities in the structural biology field, while posing new challenges. The first X-FEL to start operation was the Linac Coherent Light Source in Stanford (United States) (Emma et al., 2010). It began user operation in 2009 and soon after started ˚ (a wavelength most comto provide experimental end stations at a wavelength of B1 A mon to “traditional” synchrotron beamlines) (Se´bastien and Garth, 2010). The Spring-8 Angstrom Coherent Laser in Harima (Japan) was the second X-FEL to begin operation (in 2011), with PAL-FEL (Pohang, South Korea), and European XFEL (Hamburg, Germany), starting in 2017, followed by SwissFEL (Villigen, Switzerland) in 2018. As highlighted above, one of the problems in macromolecular crystallography is radiation-induced damage. Although cryocrystallography techniques can mitigate this problem, a limit of 30 MGy (1 Gy corresponds to one joule deposited per kilogram), at 100K has been reported as the limit for which useful structural information can be derived. If X-FEL delivers to a sample an X-ray pulse that is 9
10 orders of magnitude higher than a synchrotron (i.e., 1012 X-ray photons onto the sample), within a few femtoseconds, we are delivering a dose of tens up to hundreds of MGy. This dosage creates a thermal jump of c.10,000K leading the sample to literally explode. This represents the first main advantage of X-FEL data: while irradiating a sample with a dose that leads to its explosion, but still managing to retrieve structural information before that explosion, we are virtually obtaining radiation-damage-free data. This “diffraction-before-explosion” principle was first demonstrated by Chapman et al. (2006) and is one of the most important features of X-FEL, namely for metalloproteins. The second main advantage of using X-FEL radiation is that we do not need a single, diffraction quality, micrometer-sized crystal for data collection. Since the sample will be irradiated and then burst by the X-ray beam, what we really need is a stream of microcrystals continuously flowing through the beam producing diffraction patterns being collected at high rates by the detector. This technique is called serial femtoseconds crystallography (SFX). Moreover, since there is a new crystal being irradiated each time the laser hits the stream of microcrystals (hence the radiation-induced free structures), this opens possibilities for time-resolved serial crystallography experiments, also called pump-probe experiments (Spence, 2017). Here, an optical pulse (pump) triggers a reaction followed by an X-FEL pulse (probe) (Johansson et al., 2017). This time delay may vary between femtoseconds and minutes enabling the assessment of spatial and temporal variations within a structure.

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It is noteworthy that the developments achieved for X-FEL technology bring along new problems and challenges to be addressed, which are mostly solved or mitigated at the present time. An obvious issue is the development of novel crystallization strategies to obtain huge amounts of microcrystals to be used in a fluid stream, which imply rather different techniques than the ones used for “traditional” macromolecular crystallography aiming for single diffraction quality crystals. We will not address these technicalities here; for details, check Johansson et al. (2017) where the authors describe the most used methods to produce homogeneous microcrystal slurry with optimal crystal size and density. For each X-FEL experiment, tens of thousands of diffraction images are usually collected, because crystals are randomly oriented in the stream and one must collect a complete data set as highlighted above. Due to the way the sample is delivered onto the X-FEL beam, many of the generated images will be blank, that is, no crystal was hit by the beam. These issues raise obviously two problems: the amount of data generated (also present in cryo-EM—see later), and the way data-processing software handles these data. Several programs exist nowadays for the initial triage of blank/diffraction data containing images and most traditional data-processing software were adapted to handle X-FEL data. X-FEL experiments also have to overcome the so-called phase problem as traditional crystallography (as opposed to cryo-EM). Until 2014 most X-FEL structures were determined by molecular replacement. However, Barends et al. (2014) showed that experimental phasing is also possible by using a gadolinium-derivative crystal on a SAD experiment to phase lysozyme crystals. Thus X-FEL and SFX go beyond “traditional” crystallography experiments and elicit new possibilities in the study of proteins, namely metalloproteins. ˚ (for proteinase K), structures of soluble and membrane proResolutions as high as 1.20 A teins, especially GPCRs, and the light-activated proton pump bacteriorhodopsin are also on the list of achievements for X-FELs, SFX, and time-resolved experiments (Liu et al., 2013; Masuda et al., 2017; Nogly et al., 2016; Zhang et al., 2015).

Cryo-electron microscopy Single-particle cryo-EM has recently undergone a quantum leap progression in its achievable resolution and has triggered a revolution in structural biology. It became a major technique to study challenging biological systems, such as large macromolecular complexes and membrane proteins that are recalcitrant to crystallization. A very brief overview will focus on the recent technological accomplishments responsible for this revolution and remaining challenges. Electrons can be used to “look” at protein structures, as proteins scatter electrons approximately four orders of magnitude stronger than X-rays, and electrons can be accelerated in high-voltage electric fields (B300 kV) to wavelengths (picomolar range) that are much smaller than the distances between atoms in protein structures. Moreover, electrons can be focused with electromagnetic lenses, so electron microscopes can be built to make images with atomic-level details (Fernandez-Leiro and Scheres, 2016). Although the development of cryo-EM technique began in the 1970s, the technological breakthroughs in recent years have contributed to major advances in the field.

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Cryo-electron microscopy

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Since electrons are scattered by molecules in the air, electron microscopes need to be operated at high vacuum. However, the vacuum causes dehydration of biological specimens in aqueous environment and exposure to the high-energy electron beam induces radiation damage on the biological sample. These problems were overcome with quick plunge freezing (vitrification). Usually, a few microliters of purified protein solution is applied to a metal grid (commonly copper) covered with a thin layer of carbon holey film. After blotting away excess liquid with filter paper, the grid is rapidly plunged into liquid ethane cooled with liquid nitrogen. Semiautomated plungers with tunable parameters, such as Vitrobot (FEI) and Cryoplunge (Gatan), can be useful in this step. This should result in a thin layer of amorphous (noncrystalline) ice where the protein is embedded in a wide range of orientations. The frozen grid is then transferred into an electron microscope and kept at cryogenic temperature during data acquisition. Pictures taken through the holes in the carbon film contain 2D projections of individual biological macromolecules, which are called particles. Multiple 2D projections can be combined into a single 3D reconstruction, provided their relative orientations are known. However, this information is lost in the experiment, since the individual particles tumble around randomly in solution before the sample is vitrified, and it needs to be computationally determined by processing the experimental 2D projection images a posteriori. Sometimes, one cryo-EM sample can contain macromolecules in different conformations, 2D and 3D classification schemes can be used to computationally deconvolute this heterogeneity, allowing multiple 3D structures to be resolved from a single sample. The electron dose used to image biological samples is limited by the effect of radiation damage in the sample so high-resolution detail in the molecular structures can be preserved. Therefore images are recorded with a low electron dose and have very poor SNRs. Cooling the sample in liquid nitrogen helps to ameliorate the radiation damage effects, so the sample can tolerate a higher electron dose, but still far from being able to directly visualize high-resolution details from raw micrographs (Cheng, 2018). Henderson and Unwin addressed this problem using a crystallographic approach consisting of averaging images of many identical proteins packed as 2D crystals, a method known as electron crystallography (Unwin and Henderson, 1975). The difficulty of growing well-ordered 2D crystals impedes the broad application of the method. In parallel, Frank proposed to determine macromolecular structures by computationally aligning and combining images of many individual assemblies (particles) of the same type (Frank, 1975). A 3D map of these particles is calculated by combining information from images showing different views. This approach with the plunge freezing sample preparation became what we now call single-particle cryo-EM. The dose rate also depends on the type of detector used for imaging. High dose rate (high beam intensity) is typically used for short exposures (B1 second or less), which minimizes the extent of specimen drift, but longer exposures can be used when operating in movie mode. Here, the total electron dose is fractionated into a series of image frames that can be aligned to compensate for specimen drift and beam-induced movement, thus reducing image blurring (Campbell et al., 2012). After alignment, the frames are averaged, and the resulting image is used for subsequent structure determination (Cheng et al., 2015). When movie frames are averaged, a relative weighting can be applied that optimizes the signal in the final average (Campbell et al., 2012; Scheres, 2014).

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The remarkable progress in single-particle cryo-EM in the last years relates with the availability of DDD cameras that replaced the traditional film or scintillator-based CCD. The high contrast (better SNR) together with the fast readout made it possible to collect dose-fractionated image stacks, where the blurring is minimized and images can be computationally treated as movies rather than individual exposures. The development of sophisticated image processing algorithms for the correction of beam-induced motion and the classification of distinct structural states are producing images of unprecedented quality. Together with automation of data acquisition and increasing computer power they are revolutionizing cryo-EM structure determination by single-particle analysis, so it is no longer dismissed as “blob-ology.” The development of consistent validation tools is still a critical issue, for example, an objective quality criterion to assess the accuracy and define the resolution of the density map, as illustrated by the recent controversy around the structure of HIV-1 glycoprotein trimer (Henderson, 2013; Mao et al., 2013). Once an initial map is obtained, the structure has to be refined. Data going into image analysis software will invariably produce a 3D map, so the big question is to make sure the output map really corresponds to the structure of the specimen that was imaged (low SNR), let alone the true resolution of the map (Smith and Rubinstein, 2014). EM packages use a 3D projection matching procedure that modifies the orientation parameters of single-particle images (projections) to achieve a better match with reprojections computed from the current approximation of the structure. Progress of the refinement can be monitored by several parameters, in particular the Fourier shell correlation (FSC) curve, which provides information on the level of SNR as a function of spatial frequency (Penczek, 2010), and the resolution of the map. “Resolution” in single-particle EM is a somewhat arbitrarily chosen cutoff level of SNR or FSC curve (more details in Cheng, 2018, and references therein). The value of 0.143 for FSC is selected to relate EM results with those in X-ray crystallography (Rosenthal and Henderson, 2003). The elevated prices of the microscopes (e.g., 200
300 kV) used to collect high-quality data that run into the millions of euros, along with the operation and maintenance costs, are also a matter of concern. National and international facilities have been created to facilitate broad access to high-end microscopes. Nevertheless, sample optimization still requires access to an electron microscope on a regular basis. Another problem concerns storing and processing large volumes of data. Automated data acquisition on high-end microscopes can yield several terabytes of images every day and processing them demands high-performance computing infrastructure, so cloud computing may be an option. There is still ample scope for addressing these challenges and improving the available technology, namely in sample preparation, improved image acquisition and processing algorithms, better detectors, reducing beam-induced motions, more widespread use of energy filters and phase plates. The number of 3D structures in the PDB is reaching 147,000 (as of January 2019), of which around 89% were solved by X-ray crystallography, 8% by NMR spectroscopy, and only 2% by cryo-EM. Remarkably, the number of structures delivered by this last emerging technique is growing exponentially and will likely follow this trend in the future, in contrast with the other two, which have remained stable or slightly decreased (NMR) over

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Small-angle X-ray scattering

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the last 10 years. At the present time cryo-EM is still limited to the size of the sample ( . 100 kDa), but this value may tend to go lower as further developments come along. X-ray crystallography is limited by the ability to grow well-ordered 3D crystals, whereas NMR spectroscopy requires highly concentrated samples with nonoverlapping spectral peaks. Many macromolecular assemblies are too large, scarce, or unstable to be analyzed by these techniques. Single-particle electron microscopy allows high-resolution structures to be determined from small quantities of noncrystalline material. It also offers the unique opportunity to visualize various conformational states in a single experiment providing insights into their dynamics. Indeed, it had a great impact on large molecular complexes and membrane proteins. Membrane proteins can be imaged directly when solubilized in detergents or amphipols, or stabilized in nanodiscs (scaffold of an amphipathic protein belt) or saposin-lipoprotein system. The first membrane protein structure solved by singleparticle cryo-EM to near-atomic resolution was that of the transient receptor potential cation channel subfamily V member 1 (Liao et al., 2013). A panoply of 3D structures can already be enumerated to demonstrate the contribution of this technique in a new era of structural biology, for example, the mitochondrial ribo˚ (Amunts et al., 2014), the 20S proteasome at 3.3 A ˚ (Li et al., 2013), some subunit at 3.2 A mitochondrial respiratory complex I (Liao et al., 2013), and the spliceosome in different functional states (Fica and Nagai, 2017; Shi, 2017).

Small-angle X-ray scattering SAXS is a powerful method to characterize the structure of ordered and disordered proteins in solution. It provides information about the sizes and shapes of proteins and complexes in a broad range of molecular sizes ranging from kDa to GDa. SAXS is a rapid technique at synchrotron sources and time-resolved studies can yield unique information about kinetics of processes and interactions (Kikhney and Svergun, 2015). However, SAXS ˚ , where “resolution” relates to the smallest is a low-resolution technique limited to B5 A angle measured in the scattering pattern, leading to the longest distance that can by characterized by the data (Jacques and Trewhella, 2010). SAXS allows the study of macromolecules in solution, in their native conformations, and in conditions resembling their physiological environment (Dmitri and Michel, 2003). More interestingly, SAXS allows one to analyze structural changes induced by variation of external conditions (e.g., pH, temperature, presence of ligands and cofactors) and has the ability to provide structural information on completely (or partially) disordered systems, an obvious advantage over X-ray crystallography. The intrinsic relation between SAXS and the basic shape/structural information of (macro)molecules in solution dates back to the early work of Andre´ Guinier on fine particles and colloidal suspensions (Guinier, 1938), and metal alloys (Guinier, 1939). With his quintessential work, and in less than 20 years, this French physicist showed the potential of SAXS for the study of proteins, like hemoglobin and ovalbumin, and other biological macromolecules setting the foundations of modern small-angle scattering (Guinier and Fournet, 1955). In this later work, Guinier and Fournet (1955) showed that

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this method can provide not only structural information, like the size and shape of particles, but also information on the internal structure of disordered (or partially ordered) systems. In the 1960s, the method was already quite popular, yielding low-resolution structural information on the overall shape and internal structure of several biological macromolecules in solution, overcoming the obvious problem of producing crystals for X-ray structural analysis. In the 1970s, with the advent of synchrotron radiation, the technique had a tremendous boost providing important structural information on noncrystalline biochemical systems (Dmitri and Michel, 2003). In the 1990s, together with third-generation synchrotrons, the main breakthrough was in data analysis methods, where reliable ab initio shape determination and detailed modeling of macromolecular complexes could be performed using rigid body refinement. Nowadays, modern SAXS experiments utilize state-of-the-art synchrotron-based technology and computing resources yielding valuable information on the analysis of macromolecular shapes, quaternary structure of complexes, oligomeric mixtures and equilibrium systems, intermolecular interactions (also very useful for the analysis of protein crystallization), polyelectrolyte (e.g., polysaccharides), solutions and gels, time-resolved studies [e.g., assembly and (un)folding], coherence, and single-molecule scattering (Dmitri and Michel, 2003). In recent years, SAXS has even gained a new impetus, being often used in combination with other structural techniques like cryo-EM and X-ray crystallography, to correct Fourier amplitudes (Gabashvili et al., 2000; Saad et al., 2001; Thuman-Commike et al., 1999), and to generate ab initio envelopes for experimental phasing (Hao, 2001; Hao et al., 1999), respectively, being employed in conjunction with those techniques to generate consistent biochemical/structural models (Dmitri and Michel, 2003). In recent years, SEC-SAXS has emerged as a powerful technique to study samples with homogeneity issues, such as aggregation-prone multidomain proteins (Han et al., 2007) and heterogeneous protein complexes in dynamic equilibria (Ali and Imperiali, 2005; Berger et al., 2011; Knowles et al., 2014; Marsh and Teichmann, 2015; Vestergaard, 2016). SEC-SAXS is possible at most synchrotron SAXS beamlines. Here, the sample is loaded onto an analytical SEC column and, once eluted, directly flowed to the SAXS sample cell in the beamline, so data is acquired on-the-fly and on specific points of the eluting profile. A typical SAXS experiment makes a monochromatic and highly collimated X-ray beam irradiate macromolecules which are randomly oriented in solution. Their scattering pattern is then recorded by an X-ray detector (e.g., EIGER and PILATUS pixel detectors). Biological samples are measured along with those of the blank solvent, so the latter can be subtracted from the measured sample intensities. The sample solution should be homogeneous and well characterized prior to the experiment, because any sample, independent of its quality, will yield small-angle scattering. Knowing accurately the concentration and MW of the macromolecule under study is critical, as well as a good match between the sample and solvent (blank) buffers. Radiation damage may be an issue that can be minimized by beam attenuation or the use of radical scavengers, such as ascorbate, DTT, or glycerol (Bucciarelli et al., 2018; Jacques and Trewhella, 2010).

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The scattering profile of a monodisperse solution of noninteracting identical particles harbors information about the major spatial parameters of the particle, namely, its MW and radius of gyration (Rg). An increase of Rg is associated with protein aggregation, whereas a decrease relates with interparticle interference due to Coulombic repulsion forces (Chen and Bendedouch, 1986; Jacques and Trewhella, 2010). In addition, values of the hydrated particle volume (V), its specific surface (S), and maximum distance within the particle (Dmax) can also be obtained (Porod, 1982). V and MW can estimate the possible oligomeric assembly of the macromolecule. Besides these overall parameters, one-dimensional solution scattering profiles allow one to confidently analyze 3D structures, with low-resolution particle shapes being reconstructed ab initio from the scattering patterns. If a high-resolution model is available, it can be validated against the experimental SAXS data and assessed for possible conformations or oligomeric states, thus rapidly screening the identification of biologically relevant structures. Moreover, in the case of mixtures and flexible systems, solution scattering provides a means for quantitative description of the sample composition, and, for multisubunit particles, the quaternary structure of the complex can be reconstructed based on the models of individual subunits by applying rigid body analysis. Overinterpretation of data should be avoided (no crystallographic Rfree nor FSC like for EM refinement is available), so one must use extra caution when modeling SAXS 3D information. Additional biochemical and structural information can be quite helpful to validate the model. Furthermore, SAXS can be combined with other structural biology methodologies, such as NMR, X-ray crystallography, and EM, providing complementary data, as illustrated in recent publications (Hu et al., 2018; Steiner et al., 2018; Upla et al., 2017).

General conclusion Structural biology incorporates techniques and principles of molecular biology, biochemistry, and biophysics as a means of elucidating the molecular structure and dynamics of biologically relevant molecules. The three main research techniques for structural biology are X-ray crystallography, NMR, and cryo-EM, along with relevant information that can be provided by SAXS. These methods are different in many respects, from sample preparation to structure determination. It is important to appreciate the complementary nature of each technique and understand their unique advantages as well as limitations (see Table 10.1). This analysis is essential to select the best approach to tackle a biological system, so one method might be used extensively in some cases but rarely in others. Problems in structural biology are often not solved by one technique alone, but require a combination of methods including those mentioned in this chapter/book. A critical step in all techniques relates to sample preparation and the need for high-quality purified proteins or objects to be observed, so the resolution of the observations can be enhanced. The 3D structure of a protein is not per se a “finish line” but rather a “starting line” for new questions gathered from the insights that the structure provides.

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TABLE 10.1 Advantages and limitations of major methods for structure analysis of biological samples. Method

Samples

Advantages

Limitations

X-ray Crystals crystallography

Very high resolution (as high as 0.48 Ǻ), • Crystals required • Flexible domains not visible revealing fine details on atomic • Structure might be influenced structure (e.g., active sites and metal by crystal packing centers)

NMR

Solutions (5
10 mg/mL)

High resolution (as high as 2 Ǻ), of macromolecules in solution

• Hardly applicable for MW higher than B200 kDa (depends on labeling scheme)

X-ray free electron lasers

Solutions/matrices (depends on the experiment)

• Very high resolution (depending solely on crystal quality), revealing fine details on atomic structure (e.g., active sites and metal centers) • Radiation-damage-free structures • No need for crystal fishing and cryo optimization • Allows for time-resolved experiments

• Few sources available worldwide with very constrained beamtime access • New methods for crystallization • Computer demanding

Cryo-EM

Frozen solutions (,1 mg/mL)

• Low amount of material • Expensive and technically demanding • Direct visualization of particle shape and symmetry • Hardly applicable for MW lower than 100 kDa

SAXS

Solutions (1
10 mg/mL)

• Analysis of structure, kinetics, and interactions in nearly native conditions • Study of mixtures and nonequilibrium systems • Wide MW range (few kDa to hundreds MDa)

• Low resolution (not higher than 5 Ǻ) • Requires additional information to resolve ambiguity in model building

DLS

Solutions (,1 mg/mL)

• Nondestructive • Low amount of material • Simplicity of the experiments

• Yields overall parameters only

Cryo-EM, Cryo-electron microscopy; DLS, dynamic light scattering; MW, molecular weight; NMR, nuclear magnetic resonance; SAXS, small-angle X-ray scattering. Adapted from Dmitri, I.S., Michel, H.J.K., 2003. Rep. Prog. Phys. 66, 1735.

Acknowledgments We would like to thank Yong Zi Tan from Columbia University and New York Structural Biology Center (New York, United States), for revision of the cryo-EM section of the text, and all the members of the Macromolecular Crystallography Unit at ITQB NOVA, which provided the crystal photos in Fig. 3 of Ricardo Coelho, Tiago M. Bandeiras, Isabel Bento, Taˆnia F. Oliveira, Ce´lia V. Roma˜o, Colin E. McVey, and Ma´rio Correia.

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Genetic and molecular biological approaches for the study of metals in biology Robert L. Robson School of Biological Sciences, University of Reading, Berkshire, United Kingdom O U T L I N E Introduction and aims Basic genetics and molecular genetics: origins and definitions The origins, evolution, and speciation Grouping the species: classification, taxonomy, phylogeny The fundamental molecular biological information molecules: deoxyribonucleic acid and ribonucleic acid The central dogma The genetic code What is a gene? How big are genes and genomes? Replicons Gene organization Insertion elements, transposons, and repetitive deoxyribonucleic acid How deoxyribonucleic acid moves and can be moved around between organisms: transformation, transduction, conjugation

Practical Approaches to Biological Inorganic Chemistry DOI: https://doi.org/10.1016/B978-0-444-64225-7.00011-0

Homologous recombination Promoters, transcription initiation, and transcriptional regulation Translation initiation

418 419 419 419 420 422 423 424 424 425 427 428

429

430 431 433

Setting up: regulations, equipment, methods, and resources Regulation and approvals

434 434

Approaches and systems Model systems

435 436

Molecular biology tools and methods Preparation of deoxyribonucleic acid Agarose gel electrophoresis Pulse-field/orthoganol electrophoresis Blotting techniques Molecular cloning/recombinant deoxyribonucleic acid technology The polymerase chain reaction Deoxyribonucleic acid sequencing

417

437 437 438 439 440 443 444 448

© 2020 Elsevier B.V. All rights reserved.

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Genetic and molecular genetic methods 452 Cloning vectors and hosts 452 Gene libraries 454 Libraries intended for genome deoxyribonucleic acid sequencing 455 Cosmid libraries 455 Mobilizable and broad-host range vectors and cosmids 456 Bacterial artificial chromosomes 456 Yeast artificial chromosomes 456 Deoxyribonucleic acid copy (cDNA) libraries 457 Protein overexpression and purification 457 The T7 ribonucleic acid polymerase-T7 promoter system in Escherichia coli 457 The Pichia pastoris system 459 Tags for protein purification, correct folding, improved stability 459 Mutagenesis 460 Bioinformatics General bioinformatics websites Sequence searching sites Multiple sequence alignment

466 466 467 467

Comparative gene organization Identification of potential domains in proteins Genome sites Cross-relational databases for genomes and metabolic and other pathways Molecular phylogenies and tree drawing programs Visualization of molecular structures The OMICS revolution Genomics Transcriptomics Proteomics Structural genomics Omniomics Metabolomics Economics

467 468 468 468 469 469 469 470 470 470 470 471 471 471

Illustrative examples in the genetics and molecular biology of N2-fixation 471 References

472

Further reading

476

Introduction and aims 1. The closely interrelated fields of genetics and molecular biology provide a wealth of elegant and powerful approaches and techniques which can provide definitive answers to biological questions and which are highly complementary to other techniques covered in this book. The range of tools and resources is expanding rapidly, driven by the need to respond to the grand scientific challenges posed by human health, the biosphere, the environment, and diminishing natural resources and man’s insatiable curiosity to understand more about the world. 2. This chapter does not set out to provide a cookbook of recipes: there are quite extensive manuals referenced herein that fulfill that need. Rather this chapter lays out some key considerations and approaches or strategies to particular types of questions, and introduces some basic concepts and methodologies in genetics and molecular biology, especially for those whose primary areas of expertise and interest is not in biology. The chapter closes by illustrating the way that these techniques have been applied over the years to one system, nitrogen fixation, a complex and important metalloenzyme system.

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419

Basic genetics and molecular genetics: origins and definitions The origins, evolution, and speciation 1. Most biologists believe that life on Earth today evolved from a single life form which came into existence as the Earth’s crust formed about 3.4 billion years ago. This view has gained acceptance because of the unity observed in all life forms, for example, the commonality of basic macromolecular components [deoxyribonucleic acid (DNA), ribonucleic acid (RNA), polypeptides, etc.] and the underlying universal genetic code. Since Haeckel drew the first “tree” of life in 1866, biologists have visualized the relationships between organisms in the form of a tree based on comparisons of form and function. Now, molecular tools (comparison of the sequences of stable RNA molecules (16S and 18S rRNAs)) are commonly used to deduce phylogenetic relationships and construct pathways of evolution and a very approximate clock of when major developments may have occurred. This has led to a number of interesting hypotheses, for example, that the earliest organisms may have been thermophiles and hyperthermophiles consistent with Darwin’s idea that life may have “emerged in warm little ponds.” 2. The basic unit of life is the species. However, the term is interpreted in different ways by different groups of biologists. For some groups of organisms it is relatively easy to define a species as “an organism which is capable of sexual reproduction and producing fertile offspring.” However, in organisms that do not reproduce sexually such a definition is not useful and other criteria need to be applied. For example, in bacteria members of the same species have B95% DNA sequence identity. 3. The total number of recorded species is B1.5 million. At least half of these are insects, with ants being the dominant group. Surprisingly, we only know B5000 species of bacteria but we might have expected there to have been far more given they are the oldest extant forms of life. However, whilst we can be reasonably sure that there are no more large animals to be discovered, it has been estimated from studies of environmental samples that we may know only B1% of all existing bacteria and that the total sum of species on Earth may be in the order of B10 million.

Grouping the species: classification, taxonomy, phylogeny 1. Early classifications grouped organisms into two kingdoms: plants and animals. When Van Leeuwenhoek first reported his “little animalcules” the new kingdom of the Protista was born. Fungi belonged to the Plant kingdom for many years but they were proposed as a separate kingdom in 1944. The Protozoans or protists, and the Bacteria were reclassified later as separate kingdoms. In 1937 Edouard Chatton proposed a major distinction between organisms which contained a nuclear membrane (the so-called Eukaryotes consisting of animals, plants, fungi, and protozoans) and those organisms which do not (the prokaryotes: the bacteria) (see in Sapp, 2005). This is a fundamental division in biology with many implications (Fig. 11.1 and see below). However, a new classification scheme which overarches the kingdoms and which resolves life into three domains was proposed by Woese and colleagues in 1990 on the basis of the comparison of sequences of the 16S ribosomal RNAs (rRNAs) (Fig. 11.2). One domain contains all the eukaryotes (or Eukarya), the other two domains are prokaryotes. For a long time we had known about a group of prokaryotes, for example, the CH4-forming methanogens, with unusual properties including Practical Approaches to Biological Inorganic Chemistry

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FIGURE 11.1

Prokaryotes and Eukaryotes. Source: commons.wikimedia.org/wiki/File:Celltypes.svg

FIGURE 11.2

The Three Domains of Life. Source: en.wikipedia.org/wiki/Three-domain_system.

ether-linked and often branched membrane lipids and insensitivity to many antibiotics. This group is so distinctive that it formed the basis of a separate domain called the Archaea. The other prokaryotes which include most of the well-known bacteria, pathogens, etc., form the eubacteria domain. Several features of the Archaea more closely resemble the properties of the eukarya than the eubacteria. It seems likely that eukarya are more likely to have evolved from the archaea rather than the eubacteria.

The fundamental molecular biological information molecules: deoxyribonucleic acid and ribonucleic acid 1. DNA is a polymer in which the bases adenine, cytosine, guanine, and thymine are attached to a deoxyribose-phosphate backbone (Fig. 11.3). Its commonly adopted Practical Approaches to Biological Inorganic Chemistry

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FIGURE 11.3 DNA and RNA: structures and building blocks. Source: commons.wikimedia.org/wiki/File: Difference_DNA_RNA-EN.svg

structure is that of an antiparallel double helix held together by hydrogen bonds between the inward facing bases (A
T and G
C) creating major and minor grooves in the helix. However, DNA can adopt other structures under special circumstance such as cruciforms, triple-stranded forms, and bends and kinks produced by runs of particular bases (e.g., A
T-rich regions). In situ DNA is also supercoiled (negatively or positively) either through the action of specific enzymes (topoisomerases) or by being “wound” onto protein scaffold formed by histones or histone-like proteins. In cells, DNA exists in a complex with proteins (histones in eukaryotes) and there are a number of DNA-binding proteins which can create sharp bends and loops which in some cases allow interactions between proteins bound to other loci in the vicinity. Also histones and bases can be modified, often by methylation or acetylation, and such modifications are biologically highly significant, for example, in producing resistance to cleavage by restriction endonucleases in bacteria, or by controlling the expression of genes or large groups of genes in development and cell differentiation in Eukarya. Moreover, there are Practical Approaches to Biological Inorganic Chemistry

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a growing number of examples of these modifications being inherited. This rapidly growing field of study is called epigenetics and has been greatly facilitated by modern DNA sequencing techniques which can identify modified DNA bases. 2. RNA is generally a single-stranded polymer in which the bases adenine, cytosine, guanine, and thymine (in place of thymine) attached to a ribose-phosphate backbone (Fig. 11.3). However, RNA has the propensity to form complex intramolecular structures, for example, in transfer RNAs (tRNAs) or in ribosomes and spliceosomes. Messenger RNAs can also form secondary structures which can be of considerable functional significance. Again RNA contains modified bases, for example, the several modifications found in tRNAs. RNA is generally divided into two broad groups: the relatively unstable messenger RNAs and the stable RNAs involved in the protein synthesis machinery, for example, tRNAs and rRNA and RNA splicing by the RNA components of spliceosome assembly in eukarya. It is generally accepted that the larger rRNA genes are good molecules to use for identification, for constructing phylogenetic trees, and for reconstructing evolutionary pathways. 3. All organisms have genomes composed of DNA except some viruses where the genomes are solely RNA, for example, the animal pathogens such as polio, influenza, and of course the retroviruses such as HIV.

The central dogma 1. The path of information flow in biology was initially thought to be unidirectional from DNA to DNA to RNA to protein. But the discovery of retroviruses and the enzyme they contain called reverse transcriptase which synthesizes DNA from an RNA template showed that information flows back from RNA to DNA. The processes by which the information flows and the basic enzyme systems which are involved are illustrated in Fig. 11.4. FIGURE 11.4 The central dogma of biology. Source: en-wikipedia.org.

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2. All mechanisms of DNA synthesis (both replication and reverse transcription) require appropriate primers. For example in the case of the retroviruses such as HIV, in which the enzyme, reverse transcriptase, acts as both an RNA-dependent and DNAdependent DNA polymerase, the primer for DNA synthesis is a tRNA molecule hijacked from the previous host cell and encapsulated into the virus upon assembly. By contrast transcription does not require a primer.

The genetic code 1. The genetic code (Fig. 11.5) with its triplet basis is organized into a number of family groups which broadly relate to amino acid side chain functionality and/or space filling capacity such that on a statistical basis mutations tend to cause “functional” substitutions. Recent analysis suggests that the code is exquisitely evolved toward conserving functionality in proteins. ATG encodes methionine but with appropriate translation initiation signals upstream it is used as the translation initiation codon, although CTG and GTG are also used quite frequently in prokaryotes. Three codons, TAA, TAG, and TGA, signal the termination of translation. Crick’s wobble hypothesis (Crick, 1966) proposed that the triplet code is organized in order of importance: position 2 . position 1 . position 3. 2. The integrity of the genetic code is as much a function of the cognate tRNAs (often there are several tRNA genes for a single codon) and the modified bases in the vicinity of the anticodon which influence the codon
anticodon pairing specificity. The fidelity of translation is not only controlled by the codon/anticodon pair but also the aminoacyl tRNA synthetases which couple the amino acid residue to the cognate tRNA. This is FIGURE 11.5 The universal genetic code. Source: From Griffiths, Anthony J. F. et al Publisher: W.H. Freeman and Co, Modern Genetic Analysis: Integrating Genes and Genomes 2nd edition, 2002.

Second letter U

U

UUU UUC UUA UUG

First letter A

AUU AUC AUA AUG

G

Phe Leu

Leu

UCU UCC UCA

Ser

UAU UAC UAA

G

Stop

UGU UGC UGA

Stop

UGG

Tyr

UCG

UAG

CCU CCC CCA CCG

Pro

CAU His CAC CAA Gln CAG

CGU CGC CGA CGG

Thr

AAU AAC AAA AAG

AGU AGC AGA AGG

Met

ACU ACC ACA ACG

GUU GUC Val GUA GUG

GCU GCC GCA GCG

Ile

A

Ala

Asn Lys

GAU Asp GAC GAA Glu GAG

Cys Stop Trp

Arg

Ser Arg

GGU GGC Gly GGA GGG

U C A G U C A G U C A G

Third letter

C

CUU CUC CUA CUG

C

U C A G

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well illustrated by the incorporation of novel amino acids into proteins during protein synthesis in vitro by preparing synthetically aminoacylated tRNA species (Noren et al., 1989). 3. Different organisms have different “codon biases,” that is, some codons are used in preference to others. This is particularly true in the prokaryotes. Sometimes the bias is extreme especially where the G 1 C% contents of the genome are either very high or very low. Codon bias can be a reason why a particular gene does not express well in a foreign host. Codon bias is an important consideration in designing oligonucleotide primers to identify/amplify genes from genomic DNA, especially from organisms with AT- or GC-rich genomes. 4. The genetic code is not completely universal. Slight variant genetic codes are found in the small genomes present in a few animal mitochondria and in some bacteria which also have small genomes, for example, mycoplasmas. These variants may have evolved first through certain codons becoming redundant and therefore available for reassignment. The nonstandard codes mean that expression of genes from such backgrounds in other organisms will lead to aberrant proteins.

What is a gene? 1. A gene is region of DNA, or RNA in RNA viruses, which codes for a gene product. In the case of DNA this will be an RNA molecule (a transcript) which may or may not be translated into a polypeptide. However, a gene is not simply the region which is ultimately transcribed and/or translated but it also includes all the necessary signals which make it functional within its host, that is, transcription/translational initiation and termination signals, and posttranscriptional processing signals (e.g., splicing sites) and regulatory loci.

How big are genes and genomes? 1. The average polypeptide has a molecular mass (Mr) ofB33 kDal. Since the average molecular mass of amino acid residues is 108, then the average polypeptide is about 300 residues long. Given the triplet code the average gene for the average gene product is therefore about 900 base pairs (bp) or B1 kilobase (kb) long. So if you know that the genome of a bacterium is 1000 kb or 1 megabase (Mb) then it will encode approximately 1000 genes. 2. A genome is an organism’s complete set of DNA and contains all of the information needed to build and maintain that organism. In animal cells the genome includes the nuclear-borne chromosomes and the mitochondrial DNA and in plant cells it also includes DNA present in the plastids (chloroplasts). Mitochondria and plastids have evolved from prokaryotes which have become internal symbionts of eukaryotic cells. 3. Genome sizes vary greatly. Some viruses may have genomes of just a few kb. The smallest bacterial genome characterized is that of the highly specialized intracellular parasite Mycoplasma genitalium (580 kb). Escherichia coli has a genome of 4.3 Mb, yeast’s is 13.5 Mb. Amongst the multicellular organisms, the nematode, Caenorhabditis elegans,

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has a genome of 100 Mb, the higher plant, Arabidopsis thaliana, has a genome of 100 Mb, and the human genome is B3000 Mb 4. Gene density is that fraction of a genome which encodes gene products. This can be very high ( . 95%) in viruses, a little less so in bacteria ( . 80%), but is often far lower especially in higher eukaryotes.

Replicons 1. A replicon is an autonomously replicating DNA molecule. A chromosome can be defined as a replicon which contributes genes essential for the organism to grow and survive under all conditions, for example, genes required for the major metabolic, biosynthetic, and macromolecular processes, for example, ribosome formation, replication, transcription, etc. 2. Usually prokaryotes have a single circular chromosome which may be present in several copies but exceptions include the metabolically versatile autotroph Rhodobacter capsulatus which has two circular chromosomes and the causative agent of Lyme disease, Borrelia burgdorferi, which is remarkable in comprising a single linear chromosome of 910 kb and 21 other linear and circular replicons that add an additional 533 kb base pairs of DNA. 3. Most prokaryotes also contain additional and usually much smaller replicons called plasmids. Sometimes (e.g., in the case of several species of Rhizobium) these additional replicons are very large ( . 1 Mb) and are known as megaplasmids. Plasmids are generally thought to be not essential for survival of the organism in the laboratory and can be eliminated (“cured”). However they are required for survival and competitiveness of the organism in its natural environment. Examples would include the megaplasmids in rhizobia which contain genes for the formation of nitrogen fixing nodules of leguminous plants. 4. However, plasmids are usually smaller (3
200 kb). There are a great many different plasmids. Important and very well-characterized examples include drug-resistance plasmids (R(esistance)-factors) such as those found in Pseudomonas aeruginosa, which carry resistance genes for a number of antibiotics (tetracycline, ampicillin, streptomycin, kanamycin); plasmids which confer the ability to metabolize particular substrates, (e.g., toluene-, naphthalene-, and camphor-degrading pathways in pseudomonads) and also plasmids which confer the ability to kill other cells, for example, the small colicinencoding plasmids (e.g., the ColE1-plasmid) in E. coli from which some of the first recombinant cloning vectors were constructed. 5. Incompatibility. Two different plasmids with similar or identical replication mechanisms usually cannot coexist in the same cell. They are said to be incompatible and to belong to the same incompatibility group. However plasmids of different incompatibility groups can coexist. 6. Host range. Plasmids cannot replicate in all hosts. “Narrow host range” plasmids are specific for one host or closely related organisms because they depend heavily on the host replication machinery. Often this means that the origin of replication (ori) is only recognized by that host cell’s replication machinery. An example is the ColE1 group of

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plasmids which are only capable of replicating in E. coli and closely related members of the Enterobacteriaceaea, such as Salmonella pneumoniae or Klebsiella aerogenes. However, broad-host range plasmids present greater potential risks for genetic containment and these will need to be taken into account when designing research strategies. 7. Conjugation/mobilization. Conjugative plasmids are those capable of transferring themselves to other members of the same or other species. These plasmids carry several tra (transfer) genes, and usually a small locus (mob) on which those genes act. This is a useful property where one is cloning, for example, constructing a gene library for the organism of interest first in E. coli but want to transfer the recombinant plasmids in the library to the organism under study where those genes can potentially function. Some cloning systems have been developed to separate tra and mob loci onto compatible plasmids to produce what is known as a binary plasmid system. A classical example is the plasmid pRK2013. This plasmid carries an antibiotic resistance and also R1 tra genes but not a mob site therefore is nonself-transmissible. However, if a cloning vector carrying that mob site is introduced into a strain carrying pRK2013, then the tra genes will operate to transfer that vector to a new host. The tra genes are then said to function in trans. This means that the “mobilizable” cloning vector does not need to carry the often large number of tra genes and so minimizing its size. Also since the recombinant DNA carried by the cloning vector is not self-transmissible it minimizes the risk of unwanted genetic transfer events. 8. Copy number. Plasmids vary considerably in their replication capacity measured as the number of copies in the cell (copy number). Some high copy number plasmids, for example, the common ColE1-derived plasmids, can exist in .200 copies per cell. They are said to show relaxed replication. This means that they are easy to isolate in bulk. Also the presence of many copies of genes of interest in a host can be useful for overexpression work, and for finding genes in libraries, etc. High copy number recombinant plasmids can prove unstable where genes or segments of DNA they carry are toxic. For example, DNA that is A 1 T rich can often be difficult to clone into high copy number vectors in E. coli. Other plasmids have much more stringent replication and may be present in just a few copies per cell. Plasmids with low copy numbers can be useful for constructing vectors where genes or their products are toxic or cause unwanted effects, for example, titration of activators, etc. 9. Chromosomes in eukaryotes. Nuclear encoded genes in eukaryotes are organized into pairs (diploidy) of linear chromosomes which have a very complex structures (see Fig. 11.6). Ploidy is the term given to the number of copies of chromosomes in an organism. Human cells except for germ cells (sperm and egg) are diploid and contain two sets of nuclear autosomes (not the sex-determining X and Y chromosomes). Germ cells have half the chromosome number and though they are strictly monoploid, they are actually called “haploid” plus either the X or the Y chromosome. Some higher organisms are polyploids and contain multiple copies of chromosomes. Polyploid plants are often larger and more vigorous. A fascinating example is the wheat used for making bread, Triticum aestivum, which has 42 chromosomes and was derived some 8000 years ago.

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FIGURE 11.6 Eukaryotic chromosomes. Source: commons.wikimedia.org/wiki/File:Chromo some-DNA-gene.png.

DNA

Exon

Intron

Exon

Gene

Gene organization 1. Gene organization is markedly different in viruses and prokaryotes as compared to eukaryotes. In viruses, prokaryotes, mitochondria, and plastids, the genes are organized into operons which contain often functionally related genes which are cotranscribed from a single promoter (Fig. 11.7). The gaps between genes are often short and some genes even overlap so that the start codon of one gene overlaps with the stop codon of the preceding gene (e.g., ----ATGA----). Not only does this maximize gene density but it ensures translational coupling where ribosomes terminating translation of the first gene do not disengage as normal but move on to translate the second cistron so preserving equivalence in the levels of the polypeptides produced. Not all operons contain more than one gene, some contain a single gene (monocistronic) but multigene operons (polycistronic) predominate. 2. An unavoidable consequence of polycistronic operons is that mutations in the proximal genes may affect distal genes. This is especially the case where “foreign” DNA, for example, a transposon is inserted in the proximal genes so that transcription is often terminated at that site causing truncation of the mRNA and loss of the translation products of the distal genes. In such cases the phenotype of such mutants is not simply a consequence of the loss of function of the gene in which the insertion occurs. It is for this reason that “in-frame deletion” mutants, where the reading frame of the mutated gene is not disturbed, are helpful in the precise analysis of individual gene function. But even such mutants may lead to destabilization of the message or the loss of secondary and hitherto unsuspected promoters. 3. In eukaryotes each gene has its own promoter. This is true even in simple unicellular organisms such as yeast in which the genome is not substantially larger than those found in some bacteria. 4. A number of reasons have been proposed as to why gene organizations in viruses, prokaryotes, and eukaryotes is so different. These include the need for an operon structure and a much higher degree of coding efficiency in microscopic life forms

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FIGURE 11.7

11. Genetic and molecular biological approaches for the study of metals in biology

The lac operon in Escherichia coli. Source: commons.wikimedia.org/wiki/File:Lac_operon.pdf.

coupled with the selective advantage that a greater mobility or promiscuity of whole functional units can offer. Against that, placing each gene under its own promoter as in the eukarya provides greater potential for subtlety in expression of each individual gene, and the capacity for the reassortment and recombination within or between different alleles via the sexual process provides huge potential for generating variation in the populations of organisms.

Insertion elements, transposons, and repetitive deoxyribonucleic acid 1. Transposons. Transposons were first described in maize by Barbara McLintock (1950) and subsequently they have been found in almost every organism. They are so-called “mobile” or “promiscuous” genetic elements which can effectively “cut and paste” or Practical Approaches to Biological Inorganic Chemistry

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Basic genetics and molecular genetics: origins and definitions

FIGURE 11.8 Transposon Tn3. Source: flylib.com.

“copy and paste” themselves independently of the replication of the whole genome of an organism either within the same genome (cis) or into another genome in the cell (trans). There are two basic classes of transposons: class 1 or retrotransposons replicate via an RNA intermediate, whilst class 2 or DNA transposons replicate without an RNA intermediate. Most encode the transposition apparatus which in the simplest case is a transposase which usually recognizes specific sequences in the genome as insertion targets. Some elements are however dependent on a transpose gene at a separate locus. Transposons are also usually bordered by short repeat sequences at either end which are a consequence of the transposition mechanism. Some transposons carry one or more additional genes. Notable examples in bacteria being those which carry one or more antibiotic resistance genes, for example, Tn3 which is nearly 5 kb and carries a bla gene for β-lactamase (ampicillin resistance) (Fig. 11.8), Tn5 which carries genes for resistance to streptomycin and kanamycin, and Tn7 which is a large complex transposon of 14 kb encoding genes for trimethoprim and streptomycin/spectinomycin resistance. These and similar transposons have been responsible in part for the rapid spread of antibiotic resistance genes between bacteria but they have also been very useful for genetic analysis. 2. Insertion sequences (IS elements). These are the simplest forms of transposons. Many do not carry a transposase gene but can be acted on in trans by transposases located elsewhere in the genome of the organism. Examples include copia, one of 30 different elements in Drosophila which make up an astonishing 30% of its genome, Alu in humans, and the Ty elements in yeast. 3. Repetitive DNA. DNA which is present in many copies in excess of the replicon number. Repetitive DNA includes insertion sequences, satellite and microsatellite DNA. It is these sequences which can be used for genetic fingerprinting analysis in humans and other animals and also plants (so-called “bar coding”).

How deoxyribonucleic acid moves and can be moved around between organisms: transformation, transduction, conjugation 1. Transformation. Transformation is the natural or forced uptake of naked DNA into cells. A few organisms contain natural DNA uptake mechanisms. Examples include the eubacteria in the genera Haemophilus, Bacillus, Azotobacter, and Thermus. These Practical Approaches to Biological Inorganic Chemistry

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FIGURE 11.9

Electroporation cell. Source: en-wikipedia.org.

organisms develop the capacity to take up DNA (known as “competence”) at a particular stage of growth or in a particular medium. Other cells can be “forced” to take up DNA in a number of ways. For E. coli, incubation in high concentrations of divalent cations (e.g., Ca21) followed by a heat shock in the presence of the DNA (e.g., a plasmid) induces the DNA to enter the cell. A more general and often highly efficient method is electroporation which involves suspending cells in very low ionic strength water and then passing a high voltage pulse through the suspension in the presence of the DNA (Fig. 11.9). The ballistic method involves blasting cells with tungsten bullets (ballotini) coated with the DNA. This method can even succeed in delivering DNA through plant cells and into chloroplasts. 2. Transduction. Transduction is the process by which DNA is acquired by a virus (bacteriophage in bacteria) and carried to another cell and is then integrated into the host genome. In “generalized transduction” the viral particle picks up “by mistake” any virus sized piece of DNA. In specialized transduction the virus genome integrates into the host genome and when it excises for independent replication it does so imprecisely and carries with it a piece of the host genome next to the insertion site. Bacteriophage λ in E. coli is a very well understood and well exploited example of the latter. 3. Conjugation. Conjugation is seen in many organisms, bacteria, fungi, and protozoans and it is a sexual process. In terms of this discussion, conjugation in bacteria involves the natural transfer of one plasmid to another organism (Fig. 11.10). One of the earliest systems to have been identified was in E. coli where Hfr strains carry the F plasmid and are capable of mating with a “female” strain during which the whole chromosome can be transferred. It takes about 60 minutes for the whole process and that is why the position of genes on the chromosome of E. coli are mapped in minutes (relative to the start of conjugation) from a fixed point of the chromosome where the transfer originates.

Homologous recombination 1. Homologous recombination is a type of genetic recombination in which nucleotide sequences are exchanged between two similar or identical molecules of DNA. It is most widely used by cells to accurately repair harmful breaks that occur on both strands of DNA, known as double-strand breaks. Homologous recombination also produces new combinations of DNA sequences during meiosis, the process by which eukaryotes make gamete cells. These new combinations of DNA produce genetic variation in offspring,

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FIGURE 11.10

Bacterial conjugation. Source:

simple-wikipedia.org.

which in turn enables populations to adapt during the course of evolution. Homologous recombination is also used in horizontal gene transfer to exchange genetic material between different strains and species of bacteria and viruses. 2. Although homologous recombination varies widely among different organisms and cell types, most forms involve the same basic steps (Fig. 11.11). After a double-strand break occurs, sections of DNA around the 50 ends of the break are cut away in a process called resection. In the strand invasion step that follows, an overhanging 30 end of the broken DNA molecule then “invades” a similar or identical DNA molecule that is not broken. After strand invasion, one or two cross-shaped structures called Holliday junctions connect the two DNA molecules. Depending on how the two junctions are cut by enzymes, the type of homologous recombination that occurs in meiosis results in either chromosomal crossover or noncrossover. Homologous recombination that occurs during DNA repair tends to result in noncrossover products, in effect restoring the damaged DNA molecule as it existed before the double-strand break. 3. Homologous recombination is conserved across all three domains of life as well as viruses, suggesting that it is a nearly universal biological mechanism. Homologous recombination is crucially important in molecular biology for introducing genetic changes into the genomes of target organisms. The transfer of a particular mutation from an engineered recombinant construct into the genome of the organism under study involves recombination events on either side of the target location (so-called “double recombination”).

Promoters, transcription initiation, and transcriptional regulation 1. What is a promoter? A promoter is simply a region of DNA at which RNA polymerase binds and at which transcription by RNA polymerase is initiated. However, that simple definition belies the complexity of the process of transcription initiation and the

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FIGURE 11.11 Homologous recombination. Source: en.wikipedia.org/wiki/Homologous_recombi nation.

complexity of many promoter regions and the interplay between transcription initiation and transcriptional regulation. 2. Transcription initiation. The process of transcription initiation is complex and involves several steps including reversible binding of the RNA polymerase to the promoter to form a complex, isomerization, and DNA strand separation (open complex formation). Binding of RNA polymerase and other molecules to DNA can be demonstrated by DNA protection and footprinting techniques. The “simplest” system is seen in prokaryotes and especially bacteria where there is a single RNA polymerase core enzyme and interchangeable subunits (σ subunits) which help to define the specificity of the RNA polymerase binding. For example, in E. coli σ70 (so defined because of its molecular mass) is the “housekeeping” σ, the form which transcribes the basic gene set required for normal growth and division. In archaea RNA polymerases generally contain more subunits and other initiation factors are required for transcription. In eukaryotes the situation is more complex still with three types of RNA polymerase for nuclear encoded genes (I for the larger rRNAs, II for mRNAs, and III for tRNAs and 5S rRNA). Many more initiation factors are involved in the formation of the transcription initiation complex. Also mitochondria and plastids contain a fourth RNA polymerase which resembles the bacterial enzyme. 3. Transcriptional regulation. This is probably the most frequently encountered level at which gene expression is controlled and it represents a huge area to summarize briefly.

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Transcriptional regulation involves a great diversity of regulatory proteins which often have conserved fold patterns (helix-turn-helix, leucine zippers, zinc fingers) that recognize major or minor grooves of the DNA onto which sequence-specific recognition is imposed. 4. Activators are those proteins required for the expression of certain genes. Often they bind to the DNA in the vicinity of the promoter (classically called operator regions) and interact with the RNA polymerase to alter the kinetics of the transcription initiation events, for example, stimulate open complex formation. It is not necessary that they bind immediately adjacent to the promoter, or even that the activator binds upstream of the promoter because DNA easily forms loops which bring the two DNA-bound proteins together. Also it is possible to have complex arrangements where several activators can bind to the “operator” regions. In eukaryotes these regulatory regions can be quite extensive regions of DNA. Activators either have sensor domains or interact with one or more partner protein components which have sensor domains. Two component regulator systems are common in prokaryotes and other organisms. Mutations to activators or to the region of DNA at which they bind usually lead to loss of expression of the genes under their control. 5. Repressors generally bind to DNA in the vicinity of the promoter and block RNA polymerase binding. Repressor-binding sites often overlap or are close to the promoter. Loss of a repressor gene function leads to constitutive expression.

Translation initiation 1. In order for an mRNA to be read it must be recognized by the ribosome. The signals are markedly different in prokaryotes and eukaryotes. 2. In prokaryotes a sequence in the mRNA B10 bases upstream of the translation initiation codon (usually ATG) is important for translation initiation. The so-called Shine and Delgarno sequence is a purine-rich motif with the consensus of 50 GAGGAGA0 -30 which is complementary to the 30 end of 15S rRNA and postulated to be important for recognition between the ribosome and the mRNA (Shine and Delgarno, 1975). A few instances of mRNAs which start directly at the translation initiation codon and yet are still translated are known which suggests that sequences 30 to the translation initiation codon may be important too. 3. In eukaryotes mRNAs have to undergo processing of three types before they are translated: capping, tailing, and splicing (Fig. 11.12). Capping involves the addition of 7-methylguanylate at the 50 end of the message in an unusual 50
50 phosphotriester bond. It appears to be essential for translation and also for message stability. Tailing involves cleavage by a sequence-specific ribonuclease at the 30 end of the mRNA and then the addition of a polyadenylate tract at the end catalyzed by the enzyme polyadenylate polymerase. Tailing is also important for message stability. In addition, mRNA undergoes splicing out of untranslated regions (introns: so-called because they interrupt the gene) and the remaining sequences (exons: so-called because they are the part of the message which is expressed) are rejoined to create a coherent mature message. This highly complex organization provides some flexibility for the production

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FIGURE 11.12 Processing DNA_exons_introns.gif.

of

mRNA

in

eukaryotes.

Source:

wikimedia.org/wikipedia/commons/1/12/

of variant proteins (“splice” variants) through different splicing pathways in different tissues. The spliceosome is the remarkably complex protein/RNA assembly that carries out the recognition of the specific sequences surrounding the splicing sites, and the cleavage and the rejoining of the exons. Other splicing events occur in the processing of rRNAs which are not spliceosome dependent. These are the so-called type I and type II self-splicing reactions. 4. The processing of transcripts in eukarya adds additional complexity to the process of cloning specific genes. Some genes especially in the higher eukarya contain many introns and can be arranged over very long distances in the genome. Usually cloning the expressed component of the gene involves starting with the isolation of the mature mRNA and its reverse transcription into DNA (so-called copy DNA or cDNA). However, the polyadenylate tails of the mature mRNAs prove very useful because mature mRNAs can be isolated using poly-T affinity columns and a poly-T primer can be used as a universal primer for synthesis of the copy DNA by reverse transcriptase.

Setting up: regulations, equipment, methods, and resources Regulation and approvals 1. If you are considering doing recombinant DNA work for the first time in your laboratory or even handling genetically manipulated (GM) organisms obtained from another lab you will need to meet local genetic manipulation regulations and requirements. These will vary from country to country but almost certainly you will need to consider whether your lab is suitably designed and equipped or whether it might require refurbishment potentially on a significant scale. Generally the requirements are determined on the basis of the risk of release and spread of organisms/genes which are potentially harmful to man, the biosphere, and the environment more generally. Usually there are three categories of risk: category I requiring basic facilities; category II requiring a higher level of containment; and category III being required for pathogens and toxin genes. For each of these categories, laboratory fit-out requirements are specified in all countries. The basic presumption is one of over- rather than underprovision. Even in the lowest risk category it must be possible to contain and effectively decontaminate any lab spills. Therefore lab benches

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and surfaces must be sealed so that they can be decontaminated and the flooring must be chemically resistant and will contain no unsealed joins and should overlap up the walls and benches to contain potential liquid leaks. The lab should also have negative air pressure so that the air flows into, but not out of, the lab in order to avoid airborne contamination of corridors or the external environment. Waste disposals sinks will have to release aqueous waste into approved ducts or sewers. Detailed records will need to be kept of the disposal or decontamination of live recombinant organisms or DNA. If working with known pathogens then there will be a need to work in a special containment laboratory (e.g., a category III lab) which will have very stringent design specifications for air flow, access, security, storage, and record keeping. If you will need to use recombinant technologies rarely in your research, these are very powerful reasons for going to work in a lab which has the required standard and approvals from the appropriate regulatory bodies. 2. Approvals/authorizations from your institution or from a host institution for each project will almost certainly be needed before you even start to do recombinant DNA work. Most institutions have GM safety officers who will be able to advise you and take you through the process of obtaining approval. Obtaining approval is not necessarily onerous but does require appropriate forethought and planning and could mean that you will not be given permission to start the project. Also, unauthorized working with GM organisms or on recombinant DNA projects could potentially put at risk the institution-wide license of your institution or that of a host institution. 3. Once the lab/project is approved, just some of the basic equipment required or required access to include: fume hood, autoclave, orbital shakers,
20 freezer,
70 freezer with storage trays and boxes, fridges, laminar flow cabinet, ice machine, microcentrifuge, refrigerated bench centrifuge (1 L scale), thermocycler, agarose gel electrophoresis boxes, electrophoresis power packs, electroblotter, UV illuminator and camera, bacteriological incubator, electroporator, pipettors (a range), phase contrast microscope, water baths and heat blocks, timers, solvent cupboard, microwave oven, still or equivalent for the production of high-quality water, glass bottles for autoclaving reagents and media, tube racks, pipettors, glass pipettes and pipette cans, autoclave and reagent bottles, disposable pipettes, vacuum pump for aspiration, computers with broadband internet access. Consumable needs will depend on the project but include molecular biology enzymes (e.g., restriction enzymes, DNA polymerases, ribonucleases), antibiotics, nucleotides, solvents, agarose, DNA and plasmid purification kits, a range of plasticware, for example, flip cap and screw cap microfuge tubes, a range of pipettor tips, petri dishes, 50 mL tubes, and bacteriological/genetic manipulation disposal bags.

Approaches and systems 1. Defining the goals and devising appropriate and effective strategies to achieve them is crucial for any scientific project. This is particularly true in molecular biology because everything looks straightforward in principle but rushing headlong into a project without careful forethought and planning is likely to be expensive and frustrating. This

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is true whether the aims of a project are simple or ambitious. For example, the cloning of a single gene in order to express, purify, and study its gene product may look easy on paper. The cloning part can be achieved in just a few days when all resources and expertise are available. However, whether the protein will express to the desired level and be correctly folded will be a matter of fortune. This is especially true for metalloproteins where there may be a requirement to manufacture and insert specific metal centers. Even if one does succeed in overexpressing the protein in an apparently native and active state, whether it is an artefact of the system used to produce it can only be judged by reference to the native protein. Laboratories “practiced in this art” will have alternative cloning strategies available involving different host strains, expression vectors, and/or cultivation conditions etc. At the other end of the scale a more complex project might involve the identification and analysis of genes for a novel process in an organism not previously studied. Work on such projects will probably take years and require significant human resources and laboratory facilities to yield results. A project of this type will involve some or all of the following: the isolation and characterization of mutants; isolation and characterization of proteins involved, determining protein sequences for DNA primer design; raising antibodies to the protein (s); construction of gene libraries to isolate the genes and flanking DNA, and DNA sequencing and bioinformatics analysis; development of a genetic system for the organism; and site-directed mutagenesis to explore gene functions and possibly regulatory analysis. In projects of this complexity it is probably becoming cost-effective to start by having the complete genome of the organism determined, especially in the cases of prokaryotes.

Model systems 1. Genetic research has tended to focus on particular organisms with important properties (e.g., pathogens; producers of commercially important substances, for example, antibiotics, solvents; genetic tractability; and/or they are relatively easy to cultivate). This results in a critical mass of researchers working on the organism with the attendant accumulation of resources and know how. The sharing of expertise and resources and the exchange of researchers between laboratories is very much accepted practice and funding agents around the world have been very forward thinking and supported massive national/international collaborative projects, resource centers (e.g., stock collections; databases, and facilitating web-based software). Some research funders will often look more favorably on research proposals involving model organisms rather than relatively little-studied systems especially where they offer little novelty. 2. Wherever possible it is advisable to work with a model organism, or closely related species. Of course, there is often no alternative but to work on a system which has unique properties but one must recognize the challenges. These can often be lessened by carefully defining the objectives and good project planning. It is also advisable not to start a genetic and molecular genetic project without prior experience. It will save a lot of time and expense to acquire the “art” from experienced practitioners or indeed if the project is of a limited scope (e.g., cloning and expressing a single protein/enzyme) then

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probably that part of the project is best done in a lab where the experience, equipment, and materials reside. It could save a lot of time and expense in buying basic equipment, consumables, and molecular biology kits which you may use rarely. 3. The classic model organism is the Gram
ve bacterium E. coli for which there are thousands of strains (both commensal and pathogenic), strain collections, a knockout mutation collection (The Keio Collection: Baba et al., 2006), the complete genome sequences of different strains, very well studied and exploited viruses (bacteriophage), a host for gene libraries, cloning and expression vectors, a number of genetic systems, and expertise that can be found almost all over the world. There are many other model systems including the bacterium Bacillus subtilis (a Gram 1 ve bacterium); the unicellular eukaryotes, Chlamydomonas reinhardtii (a unicellular green alga), Saccharomyces cerevisiae (the budding yeast used in brewing, fermentation and the production of vitamins and biochemicals), Schizomyces pombei (the fission yeast); and in multicellular organisms Drosophila melanogaster (the fruit fly), C. elegans (a nematode), Danio rerio (zebra fish), and among the mammals (Musca domestica, the mouse). These model organisms have been effectively “domesticated” in a manner akin to man’s early domestication and breeding of animals and crops.

Molecular biology tools and methods 1. This chapter is not intended to be a “cookbook.” Instead it provides a summary of various methods and techniques, something of their history and their application. There are however a number of excellent manuals available, two of the most comprehensive being: (a) Molecular Cloning: A Laboratory Manual (Sambrook and Russell, 2000). (b) Short Protocols in Molecular Biology (Ausubel et al., 2002).

Preparation of deoxyribonucleic acid 1. There are basically two different methods for preparing DNA depending on whether the DNA is chromosomal (large linear or circular molecules with little or moderate supercoiling) or plasmid (usually small circular molecules often with a high degree of negative supercoiling). Furthermore, if it is necessary to have pure DNA it is important to recognize that it exists in cells complexed with specific proteins (histones or histonelike proteins) and these must be removed, usually by protease digestion. Also methods that purify DNA will also purify RNA too and therefore the RNA must also be removed, usually by ribonuclease digestion. Clearly any proteases or ribonucleases used must be DNAase free. For many polymerase chain reaction (PCR) amplifications the DNA need not be pure and it may be only necessary to boil up a small volume of a cell suspension to release sufficient DNA to serve as a template. 2. Chromosomal DNA. DNA is a relatively fragile molecule. The aim of methods is to avoid as little breakage (or shearing) of the DNA as possible. Key factors which assist the recovery of long DNA molecules are the ease with which the cell can be disrupted and

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the absence of endogenous endonucleases released during the cell breakage. All solutions and glassware or plasticware must be sterile to ensure there are no contaminating nucleases present and no contaminating DNA either. Some cells break easily, for example, animal cells and some bacterial cells, but others are much more robust. For example most plant cells are very difficult to break and need extreme techniques, such as first freezing the plant tissue in liquid nitrogen and/or grinding up in a pestle and mortar and sometimes with pure sand or glass particles (ballotini). The basic method derives from that published by Marmur (1961). The cells are ruptured (lysed) as gently as possible sometimes with a viscoprotectant such as sucrose present. A mixture of high salt (sodium acetate) and isopropanol is layered over the top of the lysate. The nucleic acid/protein complex and RNA will precipitate at the interface and gradually rise up in strands through the upper layer. These strands can then be hooked or spooled out on a very fine glass hook made from a disposable glass pipette. The precipitate containing the DNA is then redissolved by gentle rolling in a buffer containing DNAase-free protease and ribonuclease. Once the DNA has come into solution, it is then rolled very gently with a phenol/chloroform mixture into which the peptide fragments partition. The tube is then left to stand to allow the phases to separate, the upper aqueous phase is removed and the DNA reprecipitated from it with the salt and isopropanol mixture and collected by spooling again. By now, pure DNA should then redissolve easily in dilute buffer and any traces of phenol/chloroform are removed by ether extraction. Purity of the DNA can be checked by measuring the ratio of absorption at both 260 and 280 nm (after Warburg and Christian, 1942). For longterm storage, it can be freeze-dried or more usually kept in aliquots as alcohol precipitates. Both storage methods preserve the DNA as long molecules. Such molecules can then can be easily further fragmented by endonuclease activity or physical breakage. 3. Plasmid DNA. Methods for the purification of plasmids from prokaryotes generally exploit the relative resistance to denaturation (DNA strand separation) in the presence of mild alkali of highly supercoiled DNA molecules as compared to relaxed chromosomal molecules. In the standard method (Birnboim and Doly, 1979) which is available commercially in kit form, organisms are lysed in detergent and mild alkali to which DNAase-free RNAase can be added. RNA is hydrolyzed by ribonuclease and the alkali denatures the released chromosomal DNA and protein which can then be precipitated by the addition of high concentrations of sodium acetate. Following centrifugation, the plasmid DNA is precipitated from the supernatant by the addition of an equal volume of isopropanol. Very large plasmids (megaplasmids) can also be isolated in this way but require controlled lysis in very mild alkaline conditions.

Agarose gel electrophoresis 1. Horizontal submerged agarose gel electrophoresis is the standard and simple technique used for resolving DNA and RNA molecules of different lengths (Southern, 1975). Agarose gels are prepared from high-quality molten agarose solution, usually in a tris (hydroxymethyl)aminomethane (TRIS)-based buffer poured into the former in an

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FIGURE 11.13 Agarose gel electrophoresis equipment. Source: en.wikipedia.org/wiki/Agarose_ gel_electrophoresis.

electrophoresis box with a sample comb (Fig. 11.13). After the agarose is set, the box is flooded with buffer and the comb is removed carefully leaving behind wells in the gel into which the DNA samples mixed with a buffer containing bromophenol blue dye and sucrose to increase the density are loaded. The gel box is coupled up to a power pack and a voltage of about 10 V/cm is applied across the electrodes for between 30 minutes and 2 hours depending on the size of the gel and the degree of resolution required. The nucleic acids can be visualized after staining with ethidium bromide (which is usually placed in the agarose gel mix at the start) and illuminated with a short-wave UV transilluminator. Ethidium bromide is a carcinogen and appropriate precautions must be taken. Alternative proprietary stains are available from commercial sources. DNA fragments can be sized using DNA marker sets (ladders) containing linear DNA fragments of known different lengths, most usually between B150 bp and B25 kb depending on the agarose concentration and buffer used (Fig. 11.14). Supercoiled and nicked plasmid DNAs have different electrophoretic properties compared to linear DNA but linear DNA ladders can be used to size circular DNA molecules just as long as they have been calibrated against intact plasmid DNAs. Often plasmids and especially high copy number examples can give rise to multiple bands which can be confusing. This is because they form concatamers of interlocked circular molecules due to the failure of the molecules to separate (resolve) after replication. Digestion with a restriction enzyme should simplify the picture. 2. RNA molecules such as mRNAs can also be separated in agarose gels but require denaturing conditions produced by the inclusion of formaldehyde in the gel and running buffer so that the molecules do not form secondary structures.

Pulse-field/orthoganol electrophoresis 1. Generally very large DNA molecules do not resolve well in a linear field in agarose gels even if the gel concentration is lowered. This is because the long molecules tend to get

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FIGURE 11.14 Agarose gel: visualizing DNA. Source: en.wikipedia. org/wiki/Agarose_gel_electrophoresis.

trapped in the carbohydrate network. This problem is resolved by applying either a pulsed field, where the current is reversed briefly or better still where the current is applied in several different directions but with a net forward direction. This allows the trapped molecules to wriggle through the agarose. In this way it is possible to resolve very long molecules including large megaplasmids and even bacterial chromosomes several megabases long (Schwartz and Cantor, 1984). This requires of course that the DNA is not broken on extraction from the cell and this can be achieved in some cases by embedding the cells in small agarose blocks, lysing them in situ so that the DNA is released and unfolds gently, and treating them with protease and if necessary restriction enzymes in situ. The blocks are then placed into the wells and a pulse field is applied. Separation can take as long as 10
24 hours and it is usually necessary to run such gels in a cold room.

Blotting techniques 1. Blotting techniques are common throughout molecular biology. In principle they involve the capillary or electrophoretic transfer (blotting) of the DNA, RNA, or proteins that have been separated in a gel to a porous cellulose nitrate or nylon membrane filter. 2. Southern blotting. Southern blotting is used to transfer DNA from an agarose gel onto a filter (Southern, 1975). The membrane captures the pattern of DNA molecules produced during electrophoresis and after drying it can then be probed with DNA or RNA probes to detect the presence and location of specific sequences. The technique involves

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Molecular biology tools and methods

Weight Southern blot tray lid

Absorbent paper

Filter paper Nylon membrane

Agarose gel

Filter paper bridge

Plastic support

Buffer

Buffer

Southern blot tray

FIGURE 11.15

Southern Blotting. Source: From Colin Heath, PhD, http://www.GBiosciences.com.

placing the gel on to the filter cut to exactly the same size as the gel which is then overlayed with a stack of 3MM chromatography paper rectangles and a wad of paper towels the same size as the gel. The whole assembly is then placed gel side down on a much longer piece of 3MM chromatography paper the ends of which are submerged in buffer in an electrophoresis tank. A weight such as a flat glass bottle is placed on top of the stack (Fig. 11.15). Over a period of hours, buffer is drawn up by a wicking action from the gel box tank through the gel and through the filter to the stack of filter paper and towels above. The denatured single-stranded DNA is carried along with the flow of buffer but binds to the filter. The technique requires very even wicking of buffer through the gel so that the pattern of DNA fragments in the gel is perfectly reproduced on the membrane. Also care must be taken to avoid any bubbles between any layers of the sandwich. Nowadays electroblotting is generally used for this process because it is quicker and generally more reproducible. In this technique the filter paper
agarose gelmembrane filter
filter paper sandwich is clamped between two plastic mesh plates and the whole assembly placed in an electroblotter tank full of electrophoresis buffer (Fig. 11.16). When a voltage is applied across the terminals, DNA is electrophoresed out of the gel and binds onto the membrane. 3. Northern blotting. Northern blotting is used to transfer RNA from agarose gels onto a nylon or cellulose nitrate membrane. This is essentially the same process as for Southern blotting but with the crucial difference that the agarose gel is prepared with formaldehyde to limit secondary structures forming in the RNA (Alwine et al., 1977). Sizing of RNA molecules will require an RNA ladder. It is particularly important in this technique to ensure that all reagents are not contaminated with RNAase and it is advisable to reserve gel boxes solely for running RNA gels.

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FIGURE 11.16

Electroblotting. Source: http://en.wikipedia.org/wiki/Electroblotting.

4. Detection of specific DNA or RNA sequences by hybridization. Hybridization in molecular biology involves the use of a specific labeled single-stranded DNA or RNA molecule/ fragment of between B50 and 2 kb (probe) to detect homologous polynucleotide sequences usually on Southern or Northern blots or in situ, for example, on arrayed chromosomes. Originally 32P-labeled probes were made from the template DNAs by end-labeling or nick translation using DNA polymerase I with 32P-labeled deoxynucleotide triphosphates (dNTPs). More recently nonradioactive methods are recommended because the technologies are safe, no hazardous radioactive waste storage is required, the probes are long lasting and hybridization solutions can be reused, and the exposure times are generally shorter. The two main nonradioactive methods are the biotin
streptavidin and the digoxigenin (DIG)
antidigoxigenin system. In the biotin system, biotin is incorporated into the probe during its synthesis using biotinylated dNTPs (e.g., biotin-7-dATP). The incorporated biotin is detected directly by avidin or streptavidin or an anti-biotin antibody conjugated to a fluorochrome or an enzyme such as alkaline phosphatase or horseradish peroxidase. The DIG
antidigoxigenin system uses DIG. In both systems the probe is detected with either chromogenic (colorimetric) substrates by fluorescence or chemiluminescence. DIG-labeled probes can be made using DIG-11-dUTP by a variety of methods including nick translation using DNA polymerase I, random priming using Klenow polymerase, and hexa-oligonucleotide primers. 5. Hybridization is usually carried out in a sealed bag which will contain the Southern or Northern blot, and hybridization fluid containing the labeled probe and other components including unlabeled DNA, which should be highly unlikely to bind the probe or the target on the filter (salmon sperm DNA is commonly used), and a mix of components and conditions which can be adjusted to provide for hybridization only where there is a real identity between the probe and the target sequence (stringent conditions) or even between similar but nonidentical sequences (nonstringent). Duplex DNA normally melts at around B94 C depending on the base composition. Incubation conditions just below the melting temperature provide high specificity. The lower the temperature the less stringent is the experiment. A number of factors govern the hybridization reaction, these include ionic strength, base composition, and temperature. Formamide is often added to the

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Detection in Western blots Detection signal (colorimetric or chemiluminescent)

Enzyme-conjugated Secondary antibody

Enzyme substrate

Primary antibody Target protein

Membrane containing transferred protein

FIGURE 11.17

Antibody detection of specific proteins on Western blots. Source: From Leinco Technologies, Inc.,

leinco.com.

hybridization fluid to lower the melting temperature in a controlled way and to avoid the use of hot ovens or baths whilst the reaction is taking place. After hybridization which usually takes several hours, the filter is washed, blocked to avoid nonspecific binding of the detection system, and then developed using the appropriate detection system. 6. Western blots. This is a standard technique analogous to Southern blotting but for transferring polypeptides which have been separated by SDS-polyacrylamide gel electrophoresis (SDS-PAGE: Laemmli, 1970) to a nylon membrane whilst preserving their spatial relationships. Usually this is done using unstained gels with the transfer using an electroblotter as for Southern blotting but with appropriate buffers and transfer conditions (Towbin et al., 1979; Burnette, 1981). The presence of specific proteins can be detected usually with specific antibodies (Fig. 11.17).

Molecular cloning/recombinant deoxyribonucleic acid technology 1. The origins of recombinant DNA technology or molecular cloning trace back to the early 1970s (Jackson et al., 1972; Lobban and Kaiser, 1973; Cohen et al., 1973) and the first patent was awarded in 1980 (in Hughes, 2001). In essence it was a simple but elegant and revolutionary idea: the use of purified DNA ligase to join DNA molecules end-to end in vitro. Just prior to that, DNA ligase had been purified for the first time from bacteriophage T4 (T4 DNA ligase) (Weiss and Richardson, 1967), and restriction endonucleases (restriction enzymes) were becoming available (Arber and Linn, 1969). These allowed DNA to be cleaved at specific short sequences to create 50 or 30 overhangs (so-called “sticky ends”) and then joined (ligated) using DNA ligase to any other DNA molecule with complementary sticky ends. A whole range of restriction enzymes with different target sequences have been purified from different prokaryotes. The use of plasmids or viruses as vectors into which the DNA could be ligated, and their introduction by transformation into bacteria meant that DNA from any source could be “farmed” in a bacterial host. The use of antibiotic resistance genes in the

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FIGURE 11.18

11. Genetic and molecular biological approaches for the study of metals in biology

The general cloning vector pBR322. Source: en-wikipedia.org.

recombinant plasmid allowed those bacteria that had acquired the plasmid by transformation to be selected on a medium containing the antibiotic. Early cloning vectors such as pBR322 (Fig. 11.18) (Bolivar et al., 1977) and variants thereof used the inactivation of antibiotic resistance to detect cloning events and were the workhorses of molecular cloning from 1977 until more sophisticated vectors appeared which significantly cut down the work involved in identifying recombinant molecules. 2. A now widely used method for the detection of inserts in plasmids is “blue-white” screening which allows for identification of successful cloning events through the color of the resultant bacterial colony produced by the hydrolysis of the substrate X-gal. The method is based on the principle of α-complementation of a mutant lacZ gene (lacZΔM15) for β-galactosidase deleted for residues 11
41 by a peptide comprising residues 3
90 of β-galactosidase (the α-complementing fragment; Langley et al., 1975). The widely used pUC series of cloning vectors (Vieira and Messing, 1982) were the first plasmid vectors to use this screening method (Fig. 11.19). They contain a series of adjacent restriction endonuclease sites (a multiple cloning site or MCS) in the gene for the α-complementing fragment. The system requires the correct host E. coli strain (JM109, DH5α, XL1-Blue) which carries the lacZΔM15 mutation. Transformants are plated onto nutrient agar containing X-Gal and IPTG a nonmetabolizable inducer of lacZ: most (but not all!) recombinant plasmids will give rise to white colonies (Fig. 11.20).

The polymerase chain reaction 1. The PCR has been one of the many methodologies to have revolutionized molecular biology. Invented and first patented by Kary Mullis and colleagues at Cetus in 1985 it enabled specific DNA sequences to be synthesized and amplified exponentially in vitro using the following components: the template DNA, a pair of specific oligonucleotide

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FIGURE 11.19

The general cloning vectors pUC19. Source: en.wikipedia.org/wiki/PUC19.

FIGURE 11.20

Blue white selection. Source: Reproduced with permission from Sigma-Aldrich Co. LLC.

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primers designed from the sequence of the DNA to be amplified, purified DNA polymerase, and the four deoxynucleotide substrates. The mixture is subjected to a threestep sequence (cycle) of heating at .90 C to denature the DNA strands, a short annealing step at a much lower temperature designed to preferentially allow the oligonucleotide primers (added in considerable excess) to bind to their target sequences on the template but to minimize reannealing of the template, and then an extension/synthesis step at a higher temperature in which the DNA polymerase extends the primers (Fig. 11.21). Initially the technique used a DNA polymerase preparation (the Klenow fragment) from E. coli (Klenow and Hennsingsen, 1970) but this could not survive the denaturation step so had to be added each cycle. But in 1985 use of the thermostable DNA polymerase from the thermophilic bacterium Thermus aquaticus (Brock and Freeze, 1969) overcame the enzyme stability problem (Saiki et al., 1985) and made way for the explosion in the use of this technique. More recently DNA polymerases from hyperthermophilic prokaryotes are used

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11. Genetic and molecular biological approaches for the study of metals in biology 5′

3′

Target sequence Genomic DNA 1 Denaturation: Heat briefly to separate DNA strands

2 Annealing: Cool to allow primers to form hydrogen bond with ends of target sequence

Cycle 1 yields 2 molecules

3 Extension: DNA polymerase adds nucleotides to the 3′ end of each primer

3′

5′

5′

3′

3′

5′

Primers

New nucleotides

Cycle 2 yields 4 molecules

Cycle 3 yields 8 molecules; 2 molecules (in white boxes) match target sequence

FIGURE 11.21 The polymerase chain reaction. Source: From https://en.wikipedia.org/wiki/DNA_sequencing#/ media/File:Radioactive_Fluorescent_Seq.jpg.

which are still more stable and have other desirable attributes such as greater processivity allowing longer sequences to be amplified and in some cases proofreading capacity to reduce errors introduced during DNA synthesis. The PCR has had a somewhat controversial history in terms of patents and litigation (e.g., Bartlett and Stirling, 2003). 2. The PCR has many applications from the precision cloning of a single gene or part of a gene, to the construction of gene fusions and use in diagnosis and forensics. The maximum length of DNA molecules that can be synthesized is B10 kb. For some

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purposes, for example, cloning, it is important to check the sequence of the DNA that has been amplified and cloned. Errors can be introduced into the sequences depending on the DNA polymerase used. 3. The design of the primers is an important consideration. The primer pair is designed according to the target sequence to be amplified. These can be highly specific where the sequence is known or could have a degree of redundancy introduced during primer synthesis which might allow the cloning of genes from an organism where there is no sequence available but gene sequences from closely related organisms can be used. The primers can be synthesized to create additional sequences engineered at either end of the amplified product. A common addition is a restriction enzyme site. The use of different sites at either ends of the molecule ensures directional cloning of the target sequence into the vector. Where protein expression is the goal, it is common that the 50 end primer will carry an NdeI site which allows the amplified product to be cloned in to the commonly used pT7-5/7 and pET series of protein expression vectors (see Protein Expression section). 4. There are a large number of variants of PCR and even a scientific journal solely devoted to PCR technologies and methodologies. Some of the more common variants are: a. Nested PCR is intended to mitigate against contamination in products due to the amplification of unexpected primer binding sites. It increases the specificity of DNA amplification, by reducing background due to nonspecific amplification of DNA. Two different sets of primers are used in two successive PCRs. In the first reaction, one pair of primers is used to generate DNA products, which besides the intended target, may still consist of nonspecifically amplified DNA fragments. The product(s) are then used in a second PCR with a set of primers whose binding sites are completely or partially different from and located 30 to each of the primers used in the first reaction. Nested PCR is often more successful in specifically amplifying long DNA fragments than conventional PCR, but it requires more detailed knowledge of the target sequences (Sing et al., 1999). b. Touchdown PCR (step-down PCR) like nested PCR, aims to reduce nonspecific background but by gradually lowering the annealing temperature as PCR cycling progresses. The annealing temperature during the initial cycles is usually a few degrees (3 C
5 C) above the Tm of the primers used, while in the later cycles, it is a few degrees (3 C
5 C) below the primer Tm. The higher temperatures give greater specificity for primer binding, and the lower temperatures permit more efficient amplification from the specific products formed during the initial cycles. c. Inverse PCR is commonly used to identify the flanking sequences around genomic inserts. It involves a series of DNA digestions and self-ligation, resulting in known sequences at either end of the unknown sequence (Ochman et al., 1988). d. Overlap-extension PCR or splicing by overlap extension is used to splice together two or more DNA fragments that contain complementary sequences. To splice two DNA molecules, special primers are used at the ends to be joined. For each molecule, the primer at the end to be joined is constructed such that it has a 50 overhang complementary to the end of the other molecule. Following annealing when replication occurs, the DNA is extended by a new sequence that is complementary to the molecule it is to be joined to. Once both DNA molecules are extended in such a

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manner, they are mixed and a PCR is carried out with only the primers for the far ends. The overlapping complementary sequences introduced will serve as primers and the two sequences will be fused. This method has an advantage over other gene splicing techniques in not requiring restriction sites. (Higuchi et al., 1988). e. Quantitative PCR (Q-PCR) Q-PCR quantitatively measures starting amounts of DNA, cDNA, or RNA. Q-PCR is commonly used to determine whether a DNA sequence is present in a sample and the number of copies. Q-PCR has a very high degree of precision. The methods use fluorescent dyes, such as SYBR Green, EvaGreen, or fluorophore-containing DNA probes to measure the amount of amplified product in real time. It is also sometimes abbreviated to RT-PCR (real time PCR) or RQ-PCR. f. Reverse transcription PCR (RT-PCR) is used to make DNA copies (cDNA) of RNA. Reverse transcriptase reverse transcribes RNA into a DNA copy, which is then amplified by PCR. RT-PCR is widely used in expression profiling, to determine the expression of a gene or to identify the sequence of an RNA transcript, including transcription start and termination sites. If the genomic DNA sequence of a gene is known, RT-PCR can be used to map the location of exons and introns in the gene. The 50 end of a gene (corresponding to the transcription start site) is typically identified by RACE-PCR (rapid amplification of cDNA ends) (Tan et al., 1994; VanGuilder et al., 2008).

Deoxyribonucleic acid sequencing 1. DNA sequencing is hugely important in biology and has many application including medicine, forensics, agriculture, and environmental biology. The development of different methods has been almost explosive since the first methods emerged in 1977. There are now at least eight commercially established methods of DNA sequencing exploiting a number of different technologies and yet others in development. Each has its own characteristics of accuracy, read length, speed, cost, and portability. The “evolution” of the techniques was driven by the hunger for ever more DNA sequence data and aided by valuable competition prizes. The intention in this section is only to provide an overview of the methods. However, it is important to recognize that the acquisition of DNA sequence is only the first step in the process of DNA analysis which continues in most cases (e.g., genome sequencing) with the assembly of individual reads into ever longer contiguous sequences (“contigs”) until the whole genome is assembled. There then follows a now much longer process called “annotation” of interpreting the sequence for coding, noncoding regions, regulatory elements, repetitive sequences, etc. This process will be discussed in the bioinformatics section below. With the rapidly emerging field of epigenetics it became essential to develop methods which could identify the modified bases in the sequence. 2. Originally two quite different but universal methods for DNA sequencing were both published in 1977. Both relied on radioactive labeling and high-resolution polyacrylamide gel electrophoresis to separate and detect single-stranded DNA molecules differing by a single base. The Maxam and Gilbert method (or “chemical sequencing”) (Maxam and Gilbert, 1977) depends on radioactively end-labeling one of the two DNA strands and

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4.

5.

6.

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subjecting the molecule to several different chemical treatments to cleave the DNA molecules between individual bases. This method is no longer used for general DNA sequencing but still has specialist applications, for example, the methylation interference assay used to map DNA-binding sites for DNA-binding proteins. The second method introduced in 1977 (the Sanger method; Sanger et al., 1977) used dideoxyNTPs (ddNTPS) which terminated the elongation process. This could be described as the “synthesis method.” For each template four different reactions were run each with one or other of the four different base ddNTPs. The reactions included a DNA polymerase lacking a proofreading function and an excess of all four dNTPs one of which was radioactively labeled. The reactions produced a family of molecules of different lengths each terminated at a base complementary to that in the template. The molecules were separated in a denaturing thin-layer acrylamide electrophoresis and detected by autoradiography of the dried gel and the base calling done manually. It was time-consuming and could produce read lengths of up to 400 bp at best. The majority of methods in use today can be described as “synthetic” because they involve a DNA polymerase to elongate the DNA chain along the template starting from a primer The Sanger technique rapidly evolved to using fluorescently-labeled ddNTPs each base carrying a fluor with different emission spectra. Now instead of four reactions a single one could be run and all the molecules resolved by high pressure liquid chromatography (HPLC) and detected by florescence detector. This sped up the process significantly and led to more accurate and longer reads. In each case the template DNAs had to be prepared separately and have a 50 terminus on one strand which contained a sequence complementary to a universal primer (Fig. 11.22). However, there was a demand for much higher throughput and often “shotgun” sequencing from a library of fragments of a genome or even DNA extracted from environments. This started the establishment of commercial sequencing services as in most cases it was no longer time and cost-effective for individual labs to sequence “in-house.” The great advance as exemplified by the much employed Illumina sequencing (Bentley et al., 2008) was the exploitation of array technologies and microfluidics using slides or micro- or nanowells to carry a large number of different templates each trapped by one means or another at a single location in the array (Fig. 11.23). The methods also use a cyclical base addition process which could be automated. In the main the methods still used fluorescent chain-terminating NTPs but unlike the Sanger method the chain elongation-blocking and fluorescent moieties could be chemically removed after each new single base addition had been detected by a sensitive camera and read into the computer. The cycle then begins again with removal and washing away the fluors and blocking moieties from the arrays which are then flooded again with new enzyme, NTPs and other reagents. Increased sensitivity was produced by first amplifying each initial template molecule (e.g., a single DNA fragment) to produce a large number of identical copies at each location of the array. This served to increase the signal output from each location after each base addition. A novel way of amplifying the template employed in nanoball sequencing (Drmanac et al., 2009) was to convert the individual template DNA molecules into circles and to use rolling circle replication to create a much larger molecule comprising concatenated templates. These form a “knot” or nanoball of

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FIGURE 11.22

11. Genetic and molecular biological approaches for the study of metals in biology

DNA sequencing: chain termination and dye termination outputs. Source: dna-rna.net.

negatively charged DNA which binds to arrays of positively charged locations on a slide prior to synthesis process using fluorescent dNTPs. 7. Variants of this general approach do not involve fluors and therefore fluorescent detectors. Pyrosequencing (Nyre´n, 2007) (e.g., the commercial 454 pyrosequencing) exploits the release of pyrophosphate in the dNTP addition step. In addition to DNA polymerase, the reaction mix contains ATP sulfurylase, luciferase, and the substrates adenosine 5’ phosphosulfate (APS) and luciferin. Through these coupled reactions, the production of pyrophosate leads to light emission which is detected by a camera. The intensity of the light produced in each well indicates the number of base additions made in that one cycle. Residue nucleotides are then broken down by apyrase and the cycle starts again. In this technique each cycle uses only one of the four bases in rotation. One of the limitations of the technique is that read lengths are generally quite short (300
400 bp) and long single base runs may not be accurately called. 8. Ion torrent sequencing employs nonoptical detection methods. Here protons (H1) produced on base additions are detected. The template clones are trapped in a microwell array backed by a semiconductor strip which contains an ion-sensitive field effect transistor (ISFET) under each well. Read lengths of B400 base can be achieved but as with pyrosequencing reading long single base runs can be problematic. 9. All the above sequencing methods rely on multiple identical templates produced by amplification of a single starting DNA fragment. However a significant advance has been

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FIGURE 11.23 High-throughput DNA Sequencing: Dye termination versus second generation sequencing. Source: Shendure and Hanlee, 2008 Nature Biotechnology.

the development and commercialization of methods for sequencing single molecules of as-isolated “native” DNA. Importantly these methods can identify modified bases as they exist in the native DNA. The modified bases are detected by subtle differences in the kinetics with which the bases are incorporated by the DNA polymerase. In singlemolecule real-time sequencing the DNA is synthesized from the template DNA fragments in zero-mode waveguides in small well-like containers with an unmodified DNA polymerase anchored to bottom of the well. Fluorescently-labeled deoxy nucleotides flow freely in the solution. The wells are constructed in a way that only the fluorescence occurring by the bottom of the well is detected. The fluorescent label is detached from the nucleotide upon its incorporation into the DNA strand, leaving an unmodified DNA strand. This approach also allows reads of 20,000 bases or more, greatly assisting contig and genome assembly, and of course modified bases are read too.

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FIGURE 11.24 Nanosequencing. Single-stranded DNA threads through protein pore affects ion current. Source: Oxford Nanosciences.

10. A recently developed method which sequences DNA (and RNA) directly without need for enzymes or substrate nucleotides is nanosequencing as exemplified by the method commercialized by Oxford Nanosciences (Clarke et al., 2009). It has huge potential especially for environmental studies but also has many applications including in education and even for “hobbyists.” The initial concept for nanopore sequencing involves threading individual single-stranded DNA molecules through microarrays of modified staphylococcal a-hemolysin protein pores set in an electrically-resistant polymer membrane. When the membrane is subjected to an applied potential, the DNA molecule threads through the pore and modulations of the ionic current passing through the pore are measured. Each base, including modified bases, are registered, in sequence, by a characteristic decrease in current amplitude at each pore. The data is acquired in real time. Sample template DNA preparation is also very simple. The method will also sequence RNAs. As developed by Oxford Nanosciences the most basic apparatus, the MinION, is easily held in the palm and is therefore highly portable and can be taken into the “field” (Figs. 11.24 and 11.25). Units containing multiple cells are available for laboratory use.

Genetic and molecular genetic methods Cloning vectors and hosts 1. General purpose cloning vectors. General purpose cloning vectors are usually small circular DNA molecules (B 3
5 kb in length) which replicate independently to high copy number when introduced into a suitable host, are easy to purify, and into which fragments of DNA can be inserted in vitro, for example, pBR322 and pUC18/19. These plasmids have many uses including shotgun cloning of small (B600 bp) random fragments in large-scale and genome sequencing projects. They usually comprise the following: an origin of replication derived from a naturally occurring colicin-producing

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FIGURE 11.25

453 MinION cell. Source:

Oxford Nanosciences.

2.

3.

4.

5.

or antibiotic-resistant plasmid isolated from a member of the enterobacteriaceae such as E. coli and which usually will allow the molecule to be replicated to a “high copy number” in E. coli. It will also contain an antibiotic resistance gene (or marker) which will enable the detection of the presence of the vector in the host organism; a “multiple cloning site” where several restriction enzyme sites are bunched together, and usually at the 50 -end of a gene, which is inactivated when a fragment is cloned into the site. Broad-host range vectors. These are composed of similar elements to the general cloning vectors but can replicate in a relatively wide range of bacteria. However, they are usually larger DNA molecules because they carry all genes necessary for their own for replication. Examples include derivatives of RP4 which can replicate in both E. coli and a range of other Gram
ve bacteria, for example, pseudomonads, rhizobia, rhodobacters, alcaligenes. For cloning in other bacteria, for example, Gram 1 ve bacteria different vectors are required based on compatible replication systems. Expression vectors. These vectors are commonly used in the overexpression of proteins. In principle they are based on general cloning vectors described above but in addition they contain an insert which contains a strong inducible promoter and translation initiation site behind which the gene of interest is inserted in-frame. The most common expression vectors are those based on a system originally described by Tabor and Richardson (1985) which can yield a strikingly high level of protein production. Suicide vectors. These specialized vectors are used to deliver specific mutations into the genomes of organisms, for example, in the production of site-directed knockout mutants. Usually these are binary systems useful in Gram
ve eubacteria. The DNA fragment carrying the desired mutation is constructed in vitro in a vector which carries the mob site. They are first propagated in an E. coli host and then introduced into the target organism by conjugation in tripartite mating with a second E. coli strain carrying a tra1 plasmid which will mobilize any plasmid containing a mob site. Shuttle vectors. Shuttle vectors are specialized cloning vectors which have more than one origin or replication that enable them to replicate in two quite different organisms: for

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example, a Gram
ve organism such as E. coli and a Gram 1 ve organism such as B. subtilis, or a prokaryote such as E. coli and very commonly a eukaryote such as yeast. In this way advantage can be made of performing cloning and other manipulations in E. coli but transferring the final construct to the organism of interest.

Gene libraries 1. The aim of gene libraries is to capture fragments representing the whole genome of an organism in a collection of recombinant plasmids carried individually in members of a host organism, usually either E. coli or yeast. Different types of gene libraries can be constructed depending on need. They include libraries of short fragments in E. coli as a host which are mainly used for genome sequencing projects and libraries containing very large DNA fragments hundreds of kb long in either E. coli or in yeast used for long-range physical mapping and sources of segments of the genome where there may be gaps in the sequencing data and they provide a very efficient way of storing a whole genome. Gene libraries can also be constructed in highly restricted host range vectors which may be important for genetic containment or in mobilizable broad-host range plasmid vectors which are very useful for genetic analysis in some Gram
ve bacteria. 2. Whatever the starting genome or vector to be used, constructing a gene library involves: (1) preparation of high molecular weight DNA from the target organism; (2) its fragmentation to a greater or lesser extent depending on need; (3) purification of fragments of the desired size rage; (4) cloning of the fragments into a vector in such a way as to minimize multiple genome fragments being cloned into a single vector molecule; (5) transformation into the host organism, and the isolation and storage of individual clones (the library); (6) validation of the library as being representative of the genome of the target organism with a high order of redundancy; and (7) long-term storage of the library. 3. High molecular genomic DNA used to be fragmented using restriction enzymes which recognize 4 bp sequences and which, statistically, will occur very frequently in the genome. This method produces fragments which are easily cloned into a compatible cloning site in a vector but are nonrandom by definition. Nowadays it is more usual to prepare DNA fragments by physical shearing where the breakages occur at random— greater shearing produces shorter fragments. 4. Physical breakage produces fragments with “ragged” or short single-stranded overhangs. These are end-repaired or filled-in enzymically with DNA polymerase to produce blunt ends. The inclusion of a fragment-sizing step at this stage ensures that the fragments to be cloned will be relatively uniform in length. Usually this involves separating the fragments by regular agarose gel electrophoresis for small fragments or by pulse field electrophoresis for much larger fragments. Special agaroses (e.g., low melting point agarose) are used in this separation so that the DNA can be recovered from the agarose. The fragments are then blunt-end ligated into the desired vector. Where feasible a common procedure is to use the double adaptor method (Andersson et al., 1996) in which the end-repaired fragments are ligated to oligonucleotide adaptors creating long 12-base overhangs. The use of nonphosphorylated oligonucleotides at this

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step prevents formation of adaptor dimers and ensures efficient ligation of the insert to the adaptor. The vector is digested with appropriate restriction enzymes so that they produce ends which are complementary to the overhangs created in the fragment digest. Following the annealing of insert to vector, the DNA is directly used for transformation without a ligation step. This protocol produces no chimeric clones and a high proportion (B99%) of clones contain an insert.

Libraries intended for genome deoxyribonucleic acid sequencing 1. For the greatest efficiency in genome sequencing and analysis projects it is usual to prepare two or three types of gene libraries. One library will contain very small fragments of up to 1
2 kb to be used primarily for initial DNA shotgun sequencing from one end of the fragment insert or from both ends as in the so-called “double shotgun” method which offers significant advantages in terms of efficiency and sequence assembly to produce “contigs.” A second library will contain longer fragments of say 5
10 kb and is used to confirm sequences assembled from the shotgun results (contigs) and to hunt for and fill in gaps. A further library containing even larger fragments may be required where there is significant repetition in the genome under study or large gaps to be completed. Sequences are all produced by commercial automated high-throughput sequencing techniques on different platforms. Sequences are assembled and annotated using software packages.

Cosmid libraries 1. Cosmids (Collins and Hohn, 1978) are cloning vectors used to construct libraries of intermediate fragment length (e.g., up to B50 kb) and which are characterized by containing the B200 bp cosN sequence of phage λ. This is the target site for the linearization of the circular phage λ genome by a specific λ-encoded terminase. The formation of infective phage capsids requires that the DNA be linear and of a relatively specific length of B49 kb to be encapsulated. The terminase cuts the genome within the cosN site to produce 12 bp sticky or “cohesive” ends. In addition to the cosN site, these vectors contain an origin of replication (ori) either for bacterial or mammalian cells and some selectable marker (e.g., antibiotic/drug resistance). Given that there are no other specific DNA requirements for capsid assembly apart from the DNA fragment length and the cos site, any double-stranded DNA molecule can be packaged into infective phage particles. The phage particle simply serves as a highly efficient DNA delivery vehicle. The cloning capacity of a cosmid vector is inversely related to the size of the vector itself. In the cloning protocol two vector arms are generated and these are ligated to genomic DNA fragments of the required length. The assembly of the capsids is carried out in vitro and started by adding the ligation products to a mix of λ packaging extracts prepared from two E. coli strains carrying different mutant λs: one defective in head assembly and the other defective in tail assembly. After the packaging reaction the mix is used to infect E. coli and the recombinant DNA bearing clones are selected on the appropriate drug or antibiotic for which the cosmid carries resistance.

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Mobilizable and broad-host range vectors and cosmids 1. Broad-host range and mobilizable vectors including cosmid vectors are extremely useful in identifying genes through genetic complementation including interspecific complementation and also for cloning regions around the sites of insertion of transposons. Early examples include the cosmid pLAFR1 for use in Gram
ve bacteria (Friedman et al., 1982), which is a relatively large vector of 21.6 kb constructed by inserting the cosN site and a mob site into the broad-host range P1 incompatibility group vector pRK290. Cosmid libraries constructed with this vector will accommodate DNA inserts of 20
30 kb with E. coli as a host. When mixed with a culture of a second E. coli strain containing the Tra1 helper plasmid pRK2013, the library can be mass mated into a recipient organism. The helper plasmid first transfers to the library clones and will work in trans on any plasmid containing the mob locus such as pLAFR1 to mobilize it into the intended recipient. In this way it is possible to identify specific clones from the library which will restore functions to mutants of the recipient strain. In the initial use recombinant cosmids carrying Rhizobium meliloti genome fragments were identified which would restore function to (complement) auxotrophic mutants of R. meliloti and therefore carry the relevant genes. The individual cosmids could then be isolated and their inserts studied further. A functionally similar cosmid but of only 13 kb which can carry larger fragments was described by Selvaraj and Iyer (1985). 2. Fosmid libraries. Fosmids are similar to cosmids in containing the cosN site but contain the bacterial F-plasmid origin of replication which provides low copy number control and therefore offers greater stability compared to high copy gene libraries constructed in copy number vectors. They are particularly useful for constructing stable libraries from complex genomes (Kim et al., 1992).

Bacterial artificial chromosomes 1. Bacterial artificial chromosome (BAC) libraries are based on E. coli and its single copy plasmid F factor. A BAC vector is capable of maintaining very large genomic fragments of .300 kb and even up to 1 megabase (Shizuya et al., 1992). BAC libraries have proved to be very useful for preparing stable libraries of very complex genomes and this has facilitated further physical and genetic analysis. As discussed above, high molecular weight genomic DNA is cut with a restriction enzyme and then fractionated using pulse field gel electrophoresis and extracted from the agarose. The BAC vector is digested with the same restriction enzyme which cuts at the single cloning site and the vector is then treated with phosphatase to prevent self-ligation. DNA fragments of the target organism are then ligated to the prepared vector and the ligation mix electroporated into the E. coli host with a surprisingly high frequency.

Yeast artificial chromosomes 1. The construction of yeast artificial chromosomes (YACs) was first described by Murray and Szostak in 1983. These vectors comprise the sequences for the telomeres, centromeres, and replication origins of chromosomes that will replicate and be stably

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maintained in yeast. They are used for cloning and physical mapping of large DNA fragments of between 100 and 3000 kb and are particularly valuable for cloning large genes from eukaryotes which can extend over large regions. Moreover, they are particularly effective for the cloning and expression of genes which require complex posttranscriptional processing and/or which encode proteins which require posttranslation modification.

Deoxyribonucleic acid copy (cDNA) libraries 1. Typically, many genes in eukaryotes have complex structures consisting of segments of the DNA which are not expressed in the mature message (the introns) and those segments that are expressed (the exons). Some genes can contain a number of introns which can be long and extend the gene over considerable distance in the chromosome. The introns are spliced out by the spliceosomes resulting in a mature message which is also capped and tailed with poly AAAA tail. “Copy” or “complementary” DNA (cDNA) libraries are essential in order to isolate and study genes and their products from Eukarya. A cDNA library is a collection of recombinant plasmids which contain copies of the mature mRNAs or coding regions of all the expressed genes. They are prepared by isolation of the mRNA from the organism/tissue and the copying of the RNA into a double-stranded DNA molecule by reverse transcriptase. The doublestranded DNA products are then cloned into a suitable vector. Clearly a cDNA library only represents those genes that are expressed in that organism or tissue at the time the mRNA is isolated. The composition of the library is also biased in favor of abundant mRNAs. The preparation of a cDNA libraries is an essential starting point for the overexpression of proteins from eukaryotes.

Protein overexpression and purification 1. The overexpression of proteins is frequently used in research projects where the goal is to study protein structure and function. This section will compare and contrast two of the most frequently used systems.

The T7 ribonucleic acid polymerase-T7 promoter system in Escherichia coli 1. T7 is a bacteriophage which infects E. coli. On infection it injects its DNA genome into the host and proceeds to hijack its macromolecular synthesis systems to produce new phage particles which are eventually released when the host bursts. The virus encodes its own RNA-polymerase (T7-polymerase) which is a single polypeptide which recognizes promoters present only in the phage genome and which exhibits a remarkably high degree of processivity. Exploitation of this system for overexpression of foreign proteins was first described by Tabor and Richardson (1985). Their elegant system comprised two compatible plasmid constructs: pGP1-2 carries the T7polymerase gene under the control of the phage λPL promoter and the gene for temperature-sensitive phage λ repressor (cI857) placed under the control of the lacZ

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Pst l Pvu l Sca l

AlwN l

FIGURE 11.26 Expression vector pET 5a. Source: Reproduced with permission from Promega Corporation. Copyright 2019. All Rights Reserved.

Amp Afl lll

ori pET-5a Vector (4131bp)

Tth111 l Pvu ll Acc lll BspM l Bal l Ava l

Bsm l Sty l

Ssp l Aat ll EcoR l BamH l T7 transcription control region Nhe l Nde l Xba l Bgl ll Sph l EcoN l Sal l Xma lll Nru l

promoter; pT7-1 contains the strong Ø10 T7 RNA polymerase promoter, just upstream of a multiple cloning site into which a target gene can be cloned. In the presence of the lactose inducer IPTG, the temperature-sensitive λ repressor is produced which at 30 C is active and prevents expression of the T7-polymerase However, when the culture temperature is raised to 42 C the λ repressor becomes inactive. This switches on expression of the T7 polymerase which in turn drives transcription from the Ø10 promoter and expression of the target gene. Transcriptional selectivity can be further enhanced by adding rifampicin to the culture to shut down the host’s own RNA polymerase. 2. Studier and Moffatt (1986) described a similar system that was later developed and patented and now widely used and available from several biotech companies (Studier et al., 1990). This system comprises the pET family of expression vectors which contain a T7 promoter and a means of providing directional cloning for the target gene (Fig. 11.26). This cloning site comprises an NdeI restriction site just a few bases downstream of a ribosome-binding site and the Ø10 promoter and contains a 30 -ATG-50 translation initiation codon so that the open reading frame (ORF) of the target gene can be inserted into a closely coupled transcription/translation arrangement. This optimizes high levels of protein production. The distal cloning site is a BamH1 site which can accommodate the sticky ends produced by other restriction enzymes such as BglII. The ORF of the target gene is first amplified by PCR from the source DNA using a pair of primers, one of which engineers an NdeI site at the 30 end of ORF and a BamHI site at the 50 end. A notable feature of this system is that it provides several options for ensuring no expression of a target gene until required as some proteins can be toxic even if only a few molecules of the protein were to be produced inadvertently. Therefore, the first cloning stage is usually conducted in an E. coli strain lacking the T7-polymerase gene. For overexpression, the recombinant plasmid is transformed into a special E. coli host strain [BL21 (DE3)]. This strain carries a λ DE3 lysogen (i.e., an insert into the chromosome) that has the λ phage 21 immunity region, the lacI gene, and the lacUV5-driven T7-RNA polymerase cassette. In this system, when the expression

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plasmid is present in the host and the lac inducer IPTG is added to cultures, the lacUV5 promoter is derepressed allowing overexpression of the T7-polymerase. This in turn induces expression of the target gene cloned into the pET expression plasmid. The E. coli BL21 host also lacks the lon-encoded protease that can degrade proteins during subsequent purification. The still more sophisticated host BL21-Gold (DE3) carries the plasmid pLysS CamR which expresses T7-lysozyme which binds to and inhibits transcription by T7 polymerase. This effectively silences expression from any T7 polymerase-dependent promoter in the host until addition of IPTG drives up polymerase expression levels. The intracellular T7-lysozyme aids gentle lysis which is useful when the overexpressed protein is potentially susceptible to more robust cell rupture methods. Where extremely toxic genes are being expressed, the system also allows the T7-polymerase gene to be introduced into the producer strain on a λ phage.

The Pichia pastoris system 1. Whilst E. coli is by far and away the most frequently used host for protein expression it has some limitations. These include the inability to produce disulfide bonds and the inability to glycosylate proteins (Cregg et al., 2009). The methylotrophic yeast Pichia pastoris has these capabilities and can be grown easily to a very high cell density on simple defined medium with methanol as sole C source. This is especially useful for isotopic labeling for NMR experiments. Pichia expression systems exploit one of the two alcohol oxidase gene promoters to drive expression of the target gene. As in the E. coli T7-polymerase system, commercial kits are available, one of which allows a translational fusion to be created between the secretion signal of the α-mating factor of S. cerevisiae and the ORF of the target gene. This elegant system causes the expressed protein product to be secreted into the medium which is a potential aid to purification. Protein production is not necessarily so consistent in Pichia and several clones may need to be tested for effective expression. One drawback of Pichia is that induction of expression may take several days of growth of the host compared to a matter of hours in E. coli.

Tags for protein purification, correct folding, improved stability 1. It is now straightforward to engineer gene fusions that produce “tagged” target proteins where the “tag” aids correct folding, greater stability, and purification. One of the most common methodologies is to create “his-tagged” proteins, where a polyhistidine linker peptide is engineered at the N-terminus of the target protein. Given the high affinity of clusters of histidinyl residues for divalent cations and especially Ni21, purification of his-tagged proteins can sometimes be achieved in a one-step process using Ni21 affinity chromatography. Whereas this can be very useful for many proteins it is perhaps less useful when working with metalloproteins unless the tag is removed by specific proteolytic digestion following the purification step. Even so, the exposure to relatively high levels of Ni21 means that the proteins are often

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contaminated with Ni21. Other affinity tags may prove more useful in metalloprotein work such as the chitin binding protein, maltose binding protein, and glutathione-Stransferase. The latter two tags can overcome the common problem of insolubility of overexpressed proteins. Other tags such as thioredoxin and poly(NANP) are also used to overcome insolubility problems which may have many causes including nonspecific aggregation and misfolding which may be due to a lack of a correct chaperonin complex in the host. Use of polyanionic amino acid tags, for example, FLAG-tag have been developed to alter the chromatographic properties of proteins so that they can be more easily resolved during purification steps. Tags which have proved useful for immunoprecipitation experiments and Western blotting use protein sequences which encode highly immunogenic epitopes often derived from virus sequences. These include the HA-, the V5-, and the cmyc-tags. Fluorescent tags, especially green fluorescent protein, have proven to be remarkably useful in protein expression and localization studies. 2. Whilst different tags are extremely useful, the obvious caution is that the tag might alter the properties of the target protein so that it no longer behaves like the native protein. Therefore experimental findings need to be interpreted with caution.

Mutagenesis Mutants: general considerations 1. It is crucial to understand the potential biological impacts and experimental value of different types of mutations especially when planning a mutagenesis strategy. The many types of mutations include point mutants, deletions, insertions, polar/nonpolar, lethal, conditional lethals, etc. Point mutations are those which affect a single nucleotide pair. Naturally occurring point mutations found in different alleles in human populations are known as SNPs (single nucleotide polymorphisms). Such mutations occur spontaneously but can also be induced by a range of chemical and physical treatments (see below). Point mutations which have no discernible impact on the cell function are known as silent mutations. However, point mutations can often cause subtle phenotypic change, for example, affecting the catalytic and/or regulatory properties of the protein. For this reason point mutations are very useful for fine mapping of structure/function relationships in regulatory sequences (e.g., promoters) or in proteins. Deletion or insertion mutations tend to have major impacts on gene function and in the case of prokaryotes can affect not only the gene in which the mutation has occurred through polarity effects. Many genes are essential and their disruption is lethal. However, it is often possible to isolate mutations in such genes that affect the function only under particular growth conditions such as temperature: where the mutation is silent under the permissive growth conditions but expressed under the restrictive growth conditions. This approach has been essential in the study of very complex processes such as cell division and the cell cycle more generally. Of course, diploid organisms carry two copies of most genes except those borne on the sex chromosomes. Therefore it is possible to maintain such lethal mutations in a heterozygous state in the cell and to expose them in a homozygous state only after

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sexual reproduction. Even in haploid organisms such as prokaryotes, it is possible to provide a wild-type copy of the gene on a plasmid in what is known as a partial diploid (merodiploid). Chemical and physical mutagenesis 1. Mutations occur naturally sometimes due to slight errors in DNA replication and/or repair processes or as a result of environmental factors or even intracellular activities that produce reactive species that damage DNA. The induction of mutations in organisms in the laboratory by physical and chemical means stretches back to the work of Hermann Muller with his work on the effects of X-rays on Drosophila in 1927. Lewis Stadler and colleagues demonstrated the mutagenic effects of X-rays and UV-light on cereal plants (Stadler, 1928; Stadler and Sprague, 1936) and Auerbach, Robson, and Carr showed that mustard gas induced mutations in Drosophila (Auerbach et al., 1947). Now we know that chemicals such as polycyclic aromatic hydrocarbons, alkylating agents such as N-nitrosamines, intercalators, and many other types of agents damage DNA in a wide variety of different ways producing characteristic kinds of mutations. 2. From a research point of view the use of chemical and physical mutagenesis has largely been replaced by more targeted (site-directed) mutagenesis protocols that aim to generate specific mutations. Nevertheless, much has been learned from mutants produced by chemical and physical mutagenesis. Many of the thousands of mutants of common host organisms such as E. coli, yeast and Drosophila were produced by these means. Indeed genome sequencing on lab-trained strains of E. coli such as K12 reveal the battering that they have received from successive rounds of mutagenesis over the years. However, the randomness of mutations created by such techniques often throw up the unexpected and open up new areas for study. Moreover chemical and physical mutagenesis is probably the only recourse where no system exists to engineer the desired mutant strains, for example, through site-directed mutagenesis approaches. 3. The usual protocol for producing mutants using chemical or physical agents is to expose the organism to the mutagen at a level that statistically induces only one mutation in the genome of only some individuals in the population. This mitigates against the induction of multiple mutations in an individual organism which can produce complex and potentially misleading phenotypes. In any event it is essential to check for multiple mutations if feasible by performing complementation analysis involving introducing the wild-type gene into the mutant to look for restoration of the fully wild-type phenotype or to cross the mutation back into a wild-type organism and to check the phenotype. 4. To produce controlled mutagenesis the usual approach is first to determine a dose
response curve for the mutagen measuring the levels of kill produced with increasing exposure to the agent. Once this is determined then mutagenesis should be carried out at levels of exposure to the agent which cause death of a minority of the population (e.g., 10%
25%). Once the mutagenesis step has been carried out, it is essential to “outgrow” the population for several generations in nonselective conditions to allow the mutation to segregate from the wild-type copies through cell division. This enables the mutation to express biochemically or physiologically which is essential before searching for, or attempting to select, the desired mutants.

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Transposable elements and their use in mutagenesis 1. Transposons were described in an earlier section. For the purpose of this section it is important to know only that transposons create mutations wherever they insert or where they delete (Fig. 11.27) and if the transposon carries a selectable marker, for example, an antibiotic/drug-resistance gene, then individuals carrying such mutations can be easily selected and the location of the insertion can be easily identified because it has become physically “tagged.” The DNA flanking the site can be easily isolated and characterized (Kleckner et al., 1977). 2. In bacterial systems, the usual protocol for transposon mutagenesis is to introduce the transposon into the target organism via a vector (a suicide vector) which has a limited capacity to replicate, sometimes under particular culture conditions. The marker gene carried by the transposon will be lost unless the transposon copies itself into the genome of the organism. Colonies of the organism which survive on antibioticcontaining agar are those which must have acquired the transposon in their genomes (e.g., see Morales and Sequira, 1985). 3. More recently a number of sophisticated systems have been developed which allow the transposition into the target DNA to be carried out in vitro. One system which is of potentially wide application in bacteria is the GAMBIT method (genomic analysis and mapping by in vitro transposition) (Akerly et al., 1998). In this system, originally applied to Haemophilus and Streptococcus, the transposition event is performed in vitro FIGURE 11.27 Transposition: cut and paste mode. Source: chrisdellovedova.com.

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with the transposase from the mariner-family transposon himar1 from the horn fly. This enzyme mediates transposition in vitro without other cellular factors and has very little insertion site specificity. The target DNA can be fragments of the whole genome or the insertion can be targeted at a specific region of the genome by using large fragments carrying that region (B10 kb) synthesized by extended PCR. The transposons used in the original work were artificial minitransposons carrying antibiotic resistance genes. In this method, the mutated DNA was transformed into the host using its natural competence systems and the mutated organisms selected on agar plates with the appropriate antibiotics. An essential requirement of this protocol and many other types of similar mutational strategies is the need for the transposon to recombine into the genome through double homologous recombination. Site-directed mutagenesis 1. Site-directed mutagenesis is the targeting of mutations to specific loci. It applies the knockout of single gene or clusters of genes (gene knockouts or deletions) or the mutation of a single base (known as point mutations). These techniques are extremely powerful for analysis of gene function and protein structure/function relationships. Many different approaches are available. Site-directed mutations can be created in a number of ways. Nowadays, starting with the DNA sequence of the target gene and its flanking regions, the mutation is created using the PCR to amplify two fragments, one being the flanking sequence upstream of the desired target and the other being the downstream flanking region. For the purposes of knocking out the gene in the chromosome of the organism the fragments need to be sufficiently long (B200 bp at least) to allow efficient gene exchange via homologous recombination on either side of the desired mutation. The two arms are also synthesized with linker extensions containing restriction enzyme sites so that they can be cloned sequentially into a suitable vector, and at the same time allowing the creation of a restriction site at the point of deletion into which an antibiotic resistance gene (or some other marker) can be inserted. The construct needs to be introduced into the target organisms by transformation or electroporation or using a suicide vector. Presumptive mutants are selected on a medium containing the appropriate antibiotic. Ideally the construct needs to be linearized (except in the case of a suicide vector) to avoid a single recombination into the genome which would result in integration of the whole construct into the chromosome and produce a potentially misleading genotype and phenotype. Several counterselection systems (e.g., the sacB system) have been developed to “force” the recombination and ensure that the vector is eliminated (Reyrat et al., 1998). Failure to confirm the mutation could lead to spurious results and conclusions about the possible function of the target gene. This can be done either by hybridization with a suitable probe or more usually using the PCR with primers designed to distinguish unequivocally between the mutant and wild-type genotypes. Deletions can also be constructed without the need to construct two separate flanking fragments by using overlap extension PCR (see the PCR section). 2. Delitto perfetto mutagenesis. An elegant method with wide applicability for producing markerless (sometimes called “scarless”) mutations in the genome of a target organism was first reported for yeast by Storici and Resnick (2003). The first step in this two-step

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technique involves using a gene cassette consisting of both a selectable marker (e.g., a drug or antibiotic resistance gene), and a counterselectable marker, for example, in yeast the KlURA3 or GAL1/10-p53 genes which when present in yeast prevent growth in media containing 5-fluoroorotic acid or galactose, respectively. In a yet more sophisticated variant, the cassette carries a recombinant GAL1-I-SceI construct in which the SceI gene encodes the so-called homing endonuclease under the control of the GAL1 promoter. For this technique it is essential to check that a SceI target sequence is not present in the genome of the organism under study. Using overlap extension PCR, the cassette is extended at either end with sequences amplified from the genomic DNA of the organism which flank the intended site of the mutation. The construct is then electroporated into yeast and antibiotic/drug-resistant transformants are selected. These will have the entire cassette integrated into the yeast chromosome at the required locus. In the second step, starting again with the yeast genomic DNA, the same flanking regions are amplified and fused by overlap PCR to create a single linear molecule which is then transformed into the strain constructed in the first step. Now the transformants are grown in the counterselective conditions and the survivors will be those in which the cassette has been lost through double homologous recombination. This technique, suitably modified, has been applied to several prokaryotic systems (e.g., Kristich et al., 2008). Site-directed point mutants 1. There are a number of methods of producing mutations which affect only one base or a few bases in a gene. In 1987, Kunkel introduced a very elegant and effective technique which reduces the need to select for the mutants. The vector DNA into which the target gene to be mutated was first cloned into the phage-based replicon M13mp2 from which single-stranded DNA was produced. The recombinant vector was then propagated in a dut and ung strain of E. coli resulting in DNA which contains some uracil residues. The single-stranded DNA was isolated and used as the template for mutagenesis. An oligonucleotide containing the desired mutation was used for primer extension in vitro and the heteroduplex DNA formed contained the template strand unmutated and containing uracil instead of thymine, and the newly synthesized strand mutated but containing no uracil. The DNA was then treated with uracil deglycosidase which removes the uracil from the template and then with alkali which specifically degrades the strand that contained the uracil. The surviving mutated strand was then transformed back into E. coli. Various elaborations of this technique have been developed including the use of plasmid-based systems, some of which are available commercially as kits containing DNA polymerases with greater processivity, and the use of two oligonucleotide primers, one designed by the experimenter to create the desired mutation and the other one provided in the kit which corrects a mutation in an antibiotic resistance gene in the starting vector. This allows the synthesized strand carrying both mutations to be selected for when transformed into the host. 2. One of the most efficient systems for studying protein structure and function is where the gene can be expressed in an active form in an expression vector and the sitedirected mutagenesis can be carried out in the same vector. This allows mutants to be created very rapidly and one can move straight to overexpression and characterization

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Genetic and molecular genetic methods

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of the mutant protein. This is an ideal situation but where complex metalloproteins are concerned it is rare that such an approach will be feasible. CRISPR/CAS9 mutagenesis (“gene-editing/engineering”) 1. The revolutionary CRISPR/CAS “gene-editing” methodology has wide applications in the site-directed mutagenesis (so-called “gene-editing” or “genome engineering”) of prokaryotes and eukaryotes including higher organisms and even human cell lines. 2. The methodology arose from the remarkable discovery in prokaryotes of widely occurring systems providing heritable immunity to viral attack and plasmid ingress. Several different classes have been identified but all comprise a series of clustered regularly interspersed palindromic repeats (CRISPR) in the genome often with flanking CRISPR-associated (Cas) gene clusters. The palindromic sequences are typically between 28 and 37 bp long depending on the system and can number as many as 50. They are separated by “spacers” with different sequences of between typically 32 and 38 bp. The spacers are copies of short sequences usually derived from viruses or plasmids which have previously attempted to infect the organism. Several of the casencoded proteins are involved in inactivating the infecting DNA, cutting it into short sections and inserting it together with a new CRISPR repeat into the CRISPR array. The whole CRISPR/spacer array is transcribed into a long RNA transcript which other CAS proteins complex with and cut into smaller segments (crRNAs). If the CAS-crRNAs form complexes with any DNA of infecting viral or plasmids with which the spacer sequence is homologous to, then the endonuclease in the complex cuts the doublestranded DNA so arresting the infection. 3. Whilst the whole CRISPR/CAS system is very elegant and sophisticated, the final steps of recognition and inactivation of foreign (for example, viral) DNA require few components allowing it to be developed as an effective and very efficient site-directed mutagenesis tool. The first demonstrated application used the Cas9 endonuclease complex from Streptococcus pyogenes which produces blunt-end cuts. The complex involves a guide crRNA and a transactivating CRISPR RNA (tracRNA). The two RNA molecules were fused to form a single guide RNA which could locate and cut DNA recognized by the guide RNA. The guide RNA could then be designed to recognize and cut any sequence (Jinek et al., 2012; Mojica and Montoliu, 2016). 4. The basic method first involves the construction in vitro of a plasmid which can express the Cas9 protein in the recipient cell and into which the required guide crRNA can be inserted. Delivery of the plasmid into cells can be by electroporation or chemical transfection. For the technique to be effective and not to cause potentially lasting and lethal double-strand breaks in the genome, the DNA strands must be rejoined either by an endogenous DNA repair system or by recombination with a “repair” or “accessory” DNA which can be introduced with and expressed from the plasmid. In fact the system is made highly efficient because only “repaired” cells will usually survive. The accessory or template DNA can be engineered to match the sequence of and span both sides of the cut so that homologous recombination can take place. In this way it can be employed to repair mutations and introduce “knockdown” mutations and also foreign genes. (Fig. 11.28) It is now used extensively and examples range from the yeast S. cerevisiae (DiCarlo et al., 2013) to human embryos (Baltimore et al., 2015). This technique

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11. Genetic and molecular biological approaches for the study of metals in biology

FIGURE 11.28 CRISPR/CAS 9 gene-editing (Cas9 protein illustrated as pink shading). Source: From Eurofins Genomics Europe Shared Services GmbH, https://eurofinsgenomics.eu. gRNA

Genomic DNA PAM

Donor DNA

NHEJ HDR

Knock-out by insertion

Knock-in by donor DNA

already has many variants (e.g., the use of photoactivation of the Cas9 activity) and has already found and will continue to find many applications, some of which (e.g., the production of GM crops or correction of genetic disease in humans) are subject to close ethical and moral scrutiny.

Bioinformatics 1. Bioinformatics is the application of computing and informatics to biology. The explosive growth in this area of science has been driven by a number of important factors. These include amazing developments in the technologies and the introduction of highly robotic and industrial-scale operations for genomics, transcriptomics, proteomics, and metabolomics (see below). These developments have been supported by governments and commercial concerns in North America, Europe, and Asia. The convention requiring academic and other workers to deposit and release sequence and other data upon or before publication to open source databases has also been crucial as has the ease of access to databanks, web-based search engines, and other bioinformatics softwares. The databases and other resources that we see today stem from the foresight and philosophies of the pioneers in this field who first applied the emerging developments in information technology and the worldwide web to biology. The following section provides some of the more commonly used and immensely useful web-based resources.

General bioinformatics websites 1. Websites which provide a huge range of information and links include those of the National Center for Biotechnology Information (NCBI) site (http://www.ncbi.nlm.nih.

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Bioinformatics

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gov/) and the EMBL European Bioinformatics Institute (EBI) which are gateways to vast resources of biomedical and genomic information, software, and publications. The EXPASY Life Science Directory (http://www.expasy.ch/alinks.html) contains a huge list of sites listed by category. Pedro’s tools (http://www.biophys.uni-duesseldorf.de/ BioNet/Pedro/research_tools.html) is also a huge resource for links to other databases, search engines, and methods.

Sequence searching sites 1. The NCBI site listed above gives access to the BLAST (basic local alignment tool) search sites for proteins, DNA, and RNA sequences originally developed by Altschul et al. (1990). The EBI also offers another powerful suite of sequence searches tools known as FASTA (standing for FAST-ALL) (http://www2.ebi.ac.uk/fasta3/). In both the BLAST and FASTA sites, sequences are easily copied and pasted into dialogue boxes on the webpage. The query is submitted and the software searches all sequences deposited in current databases and matches are returned rapidly in descending order listing the closest matches first. The default parameters for the search are entirely adequate but it is possible to adjust these should the need arise.

Multiple sequence alignment 1. The ability to create multiple alignments of polypeptide or nucleic acid sequences is immensely useful. For example, in proteins it allows the identification of conserved residues or domains. One site for creating multiple alignments that has proved immensely powerful is INRA’s MULTALIN software (http://bioinfo.genotoul.fr/ multalin/) (Corpet, 1988). To use this site, the first operation is to make a “file of files” containing those sequences of interest. Each sequence should be in a predetermined text file format. This should be in the PIR/FASTA format in which the first line starts with the . symbol which is immediately followed by a short unique identifier of no more than eight characters immediately following. The following lines contain the amino acid sequence in single character nomenclature. The next sequence should then start on the next line and in this way it is possible to stack up many sequences. These can be pasted into the dialogue box on the webpage and submitted to the server. The alignment is normally returned in a few minutes. As with sequence searches it is possible to adjust the parameters of the search but the default option is sufficient for most needs. 2. ClustalW (Chenna et al., 2003) is another powerful set of software tools for making multiple alignments (http://www.ebi.ac.uk/Tools/clustalw2/index.html). This program is more sophisticated. For example, it enables you to adjust alignments by eye. Also the alignment output from ClustalW can be used directly to input into phylogeny programs such as the Felsenstein package called PHYLIP (see below).

Comparative gene organization 1. The following site is exclusively dedicated to the comparison of the genetic context in which any specified gene is located within the genomes of prokaryotes for which the

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genome sequence has been determined and annotated. http://www.microbesonline. org/ (Dehal et al., 2010). This site shows the extent to which the organization of genes that encode similar or related functions has been conserved in many prokaryotes but also reveals interesting differences.

Identification of potential domains in proteins 1. ProDom is based on a comprehensive set of protein domain families automatically generated from the UniProt Knowledge Database (http://prodom.prabi.fr) (Servant et al., 2002). The software allows the search for a sequence of interest you want to explore (for example by entering a gene name) and this will display the domains within that protein and other proteins which contain similar domains but often with quite different overall activities and functions. This software also allows you to submit a new protein sequence, for example, a new gene product you have discovered but have no idea what it does. ProDom analysis can provide clues as to the possible activities of unknown proteins and therefore ideas for further experiments.

Genome sites 1. There are numerous sites that relate to total genome sequencing projects. Some are specific to major genome projects, for example, human, mouse, drosophila, etc. Some have been developed by major labs which carry out many projects. All sites are interlinked to varying degrees. One of the most productive organizations has been the J Craig Venter Institute (JCVI) (http://www.jcvi.org/). Other sites well worth exploring are those of the Department of Energy Joint Genome Institute (http://www.jgi.doe. gov/) and the Wellcome Trust Sanger Institute (http://www.sanger.ac.uk/).

Cross-relational databases for genomes and metabolic and other pathways 1. For a truly amazing cross-relational database which links genomes and metabolic, regulatory, and many other pathways, see the Kyoto Encyclopedia of Genes and Genomes (http://www.genome.ad.jp/kegg/) and in particular the search engine for biochemical and other pathways (http://www.genome.jp/kegg/pathway. html#metabolism) for all organisms for which the genome has been sequenced. As an example of how to use this software, on this page, scroll down to “energy metabolism” and click on “photosynthesis.” The page opens to reveal a reference photosynthesis system. At the top of the page on the left find a pull-down menu box containing “REFERENCE PATHWAY.” Pull down the menu which lists all the organisms for which genome sequences have been determined (totally or partially). From the list, select ANABAENA and then click on the grey EXEC button to the right. When the page loads up, you will see what has been inferred to be present in this cyanobacterium. The genes now lit up in green on the bars below have all been found in Anabaena and passing the cursor over each reveals its function and clicking on it will allow you to link

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to pages which will give you much more detail even down to the level of protein structures if determined.

Molecular phylogenies and tree drawing programs 1. The construction of phylogenetic and evolutionary relationships between organisms based on the alignment and comparison of macromolecular sequences (DNA, RNA, protein) is firmly established as the basis for constructing evolutionary trees stemming back to the work of Lane et al. (1985). Nowadays, rRNAs are the molecules most commonly used for this purpose. In an earlier section the ClustalW web software (http://www2.ebi.ac.uk/clustalw/) was highlighted as a tool for generating multiple sequence alignments in a format which could be input into a powerful commonly used software package for generating such trees known as PHYLIP (for phylogeny inference package). For example it allows you to choose many different parameters and different methods of making alignments. Pull down the menu under “Tree type” and find there several alternative methods of alignments, for example, nj, neighbor joining; dist, distance matrix. The multiple sequence alignment can then be uploaded into web-based PHYLIP software. This is a much used and sophisticated molecular phylogenetics package and you should click on “documentation” to learn more about it. PHYLIP software is accessible at the following URL: http://bioweb.pasteur.fr/seqanal/ phylogeny/phylip-uk.html. The PHYLIP software returns outputs which contains data with file extensions of dnd (the tree) and aln (the alignment). These can be uploaded into a number of different tree drawing softwares. To construct trees using the data with dnd from ClustalW and many other file extensions an effective program called TREEVIEW can be downloaded from the following website: http://taxonomy.zoology. gla.ac.uk/rod/treeview.html 2. Many similar and related web software sites for phylogenetic and evolutionary analysis are listed here: (http://evolution.genetics.washington.edu/phylip/software.html).

Visualization of molecular structures 1. There are a number of web-based softwares which support the visualization of macromolecular structures. A commonly used example is the user-sponsored molecular visualization system on an open source foundation at http://www.pymol.org/.

The OMICS revolution 1. In recent years sophisticated technologies have been developed that allow the, almost industrial-scale, sequencing and annotation of complete genomes and the analysis of large numbers of genes, RNAs, and proteins. Sequencing is becoming so cheap now that it is probably the stepping off point of choice for many molecular biological studies of organisms. The handling and analysis of such data is underpinned by bioinformatics or biocomputing.

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Genomics 1. Genomics has largely been covered by the paragraphs on DNA sequencing above.

Transcriptomics 1. Transcriptomics is one of the elements of “postgenomics.” It involves the analysis of gene expression by measuring and comparing the abundance of mRNAs for individual genes. One way in which this is done is to produce gene microarrays on glass slides which consist of fixed microspots of DNAs for each gene in the genome produced by the PCR. Total RNA can be isolated from cells under study and then fluorescently labeled and hybridized to the microarrays. Those genes which are highly expressed will bind higher amounts of mRNA and this can be measured by fluorescent detectors which read across the complete microarray. In this way it is possible to compare mRNA abundances for the complete set of genes for cells exposed to different conditions or healthy or diseased cells. Usually statistical analysis is vital because there are many steps in the process where errors could occur. Again this is a methodology which is greatly enhanced by robotics.

Proteomics 1. Proteomics is the second postgenomics technique but one which attempts to look at gene expression at the protein or polypeptide level. In this technique total polypeptides are isolated from cells and then subjected to 2D electrophoresis in a slab gel. This technique displays the polypeptides when stained as individual spots which can then by subjected to image analyzers to record and measure relative abundances. Spots of polypeptides of interest, for example, those which are upregulated under a particular condition, are cut out of the gel, digested with specific proteases and then subjected to matrix-assisted laser desorbtion/ionization-time of flight (MALDI-TOF) mass spectrometry. From this, total molecular masses can be determined for component peptides and these can be compared with a database deduced from all the genes identified in the genome sequencing project. In this way the polypeptide can be ascribed to a particular gene. Again through the use of array techniques, high-speed readers, and computing it is possible to analyze a highly complex mixture of proteins, to identify all the genes which encode them and to move to a functional analysis of those genes based on the rapidly growing database of known gene functions.

Structural genomics 1. This is a highly ambitious concept which attempts to provide high-throughput structural determinations of proteins. For example, one objective might be to determine the structures for all human proteins. In practice there are many reasons why this may be impossible. Also we may not need to know the structures for all proteins. The idea is to overexpress whole sets of genes from a specific genome. Proteins produced in great abundance can be purified easily and then high-throughput crystallization trials established. Once crystals are obtained, the diffraction qualities can be assessed at high

Practical Approaches to Biological Inorganic Chemistry

Illustrative examples in the genetics and molecular biology of N2-fixation

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speed and then further structural analysis and solving can proceed. Once the structure is obtained, it may be possible to deduce function from structure. Even if the function is not known, if the protein is a potential drug target, chemists can then try to design possible ligands to potential active sites.

Omniomics 1. The more biological sequences, structures, functions that are determined it becomes increasingly possible to carry out automated comparisons of the genetic potential of entire organisms or groups of organisms. Omniomics attempts to link all the databases into automated intelligent networks that can carry out high-speed analysis and comparison. That is, as genome sequences emerge from the sequencers the whole process of analysis down to the level of gene function and expression can be produced automatically.

Metabolomics 1. Metabolomics attempts to analyze and identify metabolites within cells and interrogate changes in them. The use of NMR of complex cell metabolite mixtures together with neural networks and other artificial intelligence systems is providing ways to probe complex mixtures and the changes occurring within them.

Economics 1. None of the above technologies is cheap. The equipment and the consumables costs are high. However, there are research institutes around the world and biotech and pharmaceutical companies which are equipped with large numbers of capillary sequencers, robotics, arrayers, etc. It is difficult to predict what the next quantum leaps will be but obviously much of the work is driven one way or another by the need to find cures for diseases.

Illustrative examples in the genetics and molecular biology of N2-fixation 1. The table in this section is intended to illustrate the application of many of the techniques described in this chapter to a complex metalloenzyme system as exemplified by selected studies which over 40 years have exploited different technological developments as they have emerged leading to our current state of understanding of the process of biological N2 fixation catalyzed by the nitrogenase enzyme family. It is not the intention here to provide a detailed account of the several nitrogenase enzyme systems now known, their biochemistry, nor the nif, vnf, and anf genes which encode them. These are described in considerable detail in the references provided which represent only a small fraction of the studies published on this important and highly complex biological process.

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Development

Illustrative publications

Groundbreaking paper on nitrogenase assay, purification, and characterization

Bulen and LeComte (1966)

Early mutants affecting nitrogen fixation

Streicher et al. (1971)Shah et al. (1973)

Mobilization of functional nif genes to Klebsiella and Salmonella

Dixon and Postgate (1972) Dixon et al. (1976) Cannon et al. (1974)

Earliest application of recombinant DNA technology to nif genes

Riedel et al. (1977)

Physical and genetic maps of K. pneumoniae nif genes

Riedel et al. (1979)

Early mapping of nif mutants of A. vinelandii

Bishop and Brill (1977)

Detection of nif structural genes in different N2-fixers using interspecific DNA hybridization

Ruvkun and Ausubel (1980)

Protein sequencing of nitrogenase proteins

Tanaka et al. (1977) Hausinger and Howard (1982)

DNA sequencing as applied to nif genes

Mevarech et al. (1980) Sunderesan and Ausubel (1981) Scott et al. (1981) Mazur and Chui (1982)

Crystal structures of MoFe nitrogenase components. Crystal structure of the VFe nitrogenase

Georgiadis et al. (1992) Kim et al. (1993) Sippel and Einsle (2017)

Knockout mutants dissect gene function and reveal new nitrogenase systems

Eady et al. (1987) Robson et al. (1987) Chisnell et al. (1988)

Protein overexpression reveals functional analysis

Evans et al. (1991) Zheng et al. (1998) Curatti et al. (2007)

Mechanistic overview of nitrogenase

Howard and Rees (2006)

Engineering nif genes into higher plants

Allen et al. (2017)

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4579.

Practical Approaches to Biological Inorganic Chemistry

Index Note: Page numbers followed by “f,” “t,” and “b” refer to figures, tables, and boxes, respectively.

A Ab initio methods, 53 54 Ab initio phase determination, 396 Absorption, 231 232 Activators, 433 ADC. See Algebraic diagrammatic construction (ADC); Analogue to digital converter (ADC) Adenine, 420 422 Adenosine 5’ phosphosulfate (APS), 450 Adenosine diphosphate (ADP), 190 191 Adenosine triphosphate (ATP), 190 191 ADP. See Adenosine diphosphate (ADP) Adsorbate, 337 Agarose gels, 438 439, 440f electrophoresis, 438 439, 439f ALAD. See δ-aminolevulinate synthase (ALAD) Algebraic diagrammatic construction (ADC), 62 Alkaline earth metal ions, 3 Allochromatium vinosum, 351 A. vinosum NiFe hydrogenase, 353 Aluminum (Al), 12 AM. See Austin model (AM) Analogue to digital converter (ADC), 191 192 Angular momentum, 27 28 contributions to, 71 77 term symbols for free atoms and ions, 71 75 spin orbit coupling, 73 75 Angular overlap model, 43 Anisotropy, 125 126 Annotation, 448 Anodic process, 338 Anomalous polarization, 307 Anomalous scattering, 394 396 Anomalous Zeeman effect, 82 83 Antennae effect, 311 Anti-Stokes Raman scattering, relative intensities of, 291 292 Antiferromagnetism, 100 Apoptosis, 3 APS. See Adenosine 5’ phosphosulfate (APS) Aquifex aeolicus, 351 A. aeolicus NiFe hydrogenase, 364

Arabidopsis thaliana, 424 425 Archaea, 419 420 Artificial metalloenzymes, Resonance enhanced Raman spectroscopy in characterization of, 317 319 Ascophyllum nodosum, 263 Asymmetric unit, 391, 391f A-term enhancement mechanism, 305 306 Atomic displacement parameter, 401 Atomic nuclei, 156 157, 167 Atomic scattering factor, 395 ATP. See Adenosine triphosphate (ATP) Aufbau principle, 29 31 Auranofin, 14 15, 14f Aurous potassium cyanide(KAu(CN)2), 397 Austin model (AM), 64 Auxiliary electrode, 338 Avogadro’s number, 108 Axes noncolinearity, 143 Azimuthal quantum number, 25 27 Azotobacter, 429 430 Azodobacter vinelandii ferredoxin I, 349 350

B BAC. See Bacterial artificial chromosome (BAC) Bacillus, 429 430 Bacillus subtilis, 437 Backscatterers, 251 Bacterial artificial chromosome (BAC), 456 Bacterial conjugation, 430, 431f Bacteriophage λ in E. coli, 430 BamH1 site, 458 459 Band assignment, 312 314 intensity, 296 position, 296 Basal plane electrodes, 337 Basic local alignment tool (BLAST), 467 Batch technique, 384 Beer Lambert law, 231 232 B-factor. See Atomic displacement parameter Binary plasmid system, 426 Biochemistry spectrometers, 122

477

478 Bioinformatics, 6 7, 466 469 analysis of human genome, 9 10 comparative gene organization, 467 468 cross-relational databases for genomes and metabolic pathways, 468 469 genome sites, 468 identification of potential domains in proteins, 468 molecular phylogenies and tree drawing programs, 469 multiple sequence alignment, 467 sequence searching sites, 467 visualization of molecular structures, 469 websites, 466 467 Bioinorganic chemistry, 69 70 Biological EPR spectroscopy, 144 processes, 2 3 redox scale, 329 330, 329f systems containing transition metal, 62 63 X-ray absorption spectroscopy example, 231 236 Biomolecular nuclear magnetic resonance spectra acting on magnetization, 161 162 care in obtaining NMR spectra of paramagnetic samples, 192 197 Evan’s method, 195 197 WEFT and super-water eliminated Fourier transform sequences, 193 195, 194f chemical exchange, 177 179 chemical shift, 167 170 contact relaxation, 186 correlations arising and cross-peaks generation, 180 181 COSY, 181 coupling, 170 174 Curie relaxation, 188 dipolar coupling, 184 186 dipolar relaxation, 186 187 DOSY, 176 energy of NMR transitions, 158 160 HSQC, 182 hyperfine scalar coupling, 184 in-cell nuclear magnetic resonance, 190 191 macroscopic magnetization, 160 161 measuring macroscopic magnetization and relaxation, 191 192 metals in, 183 184 transition metals and interaction with unpaired electron(s), 184 multidimensional nuclear magnetic resonance, 179 182 NMR experiment, 166 167 NOE, 174 176

Index

nuclear magnetic resonance of (semi-)solid samples, 188 190 properties of matter relevant to NMR, 157 158 relaxation, 162 166 residual dipolar couplings, 188 useful physical constants, 198 BioXAS experiment, 232b 2,2’-Bipyridine, 295 296 Bla gene, 428 429 BLAST. See Basic local alignment tool (BLAST) Bleaney Bowers equation, 104 105 Bleomycin, 313 Bloch equations, 162 163 Blotting techniques, 440 443 Blue copper proteins, 125 126, 150 151, 237b “Blue-white” screening, 444, 445f BO approximation. See Born Oppenheimer approximation (BO approximation) Bohr magneton, 27 28, 78 80 Bohr’s radius, 27 Boltzmann constant, 198 Boltzmann distribution, 174 175 Boltzmann equation, 160 Boltzmann treatment of magnetization, 86 92 Bond length, 210 Bond valence sum analysis (BVSA), 259, 265 Born Oppenheimer approximation (BO approximation), 20 21, 24, 278, 292 293, 304 Borrelia burgdorferi, 425 BoxB enzyme, 214, 216 Bragg indices, 392 Bragg’s law, 391 392 Brillouin function, 88 Broad-host range vectors, 453 B-term enhancement mechanism, 305 306 “Bulk elements”, 2 Butler Volmer equation, 333 334 BVSA. See Bond valence sum analysis (BVSA)

C Cadmium (Cd), 11 Caenorhabditis elegans, 424 425 Calcium ion (Ca21), 3 “Camel back” feature, 256 Capacitive current, 339 Capping, 433 434 Carbodiimide coupling, 337 Carbon nanotubes, 337 Carboplatin, 13, 14f Carrier frequency (Cf), 168 Cartesian axis system, 143 Cas gene. See CRISPR-associated gene (Cas gene)

Index

CASSCF. See Complete active space self-consistent field (CASSCF) Catalytic electrochemistry, 352 353 Catalytic potential, 360 361 Catalytic protein film voltammetry, 351 365. See also Protein film voltammetry (PFV) mass-transport controlled catalytic voltammetry, 353 354 principle and general comments, 351 353 Cathodic process, 338 CD. See Circular dichroism (CD) cDNA. See Copy deoxyribonucleic acid (cDNA) Centimeter gram second units (cgs units), 77 Cf. See Carrier frequency (Cf) CFT. See Crystal field theory (CFT) cgs units. See Centimeter gram second units (cgs units) Channels, 206 207 Character tables, 40 41 Charge-coupled device (CCD) array, 294 295 camera, 385 Charging current. See Capacitive current Chemical exchange, 177 179 Chemical potential, 328 Chemical sequencing, 448 449 Chemical shift, 167 170 anisotropy, 167 carrier frequency, 168 measuring T1, 170 sampling bandwidth and Nyquist theorem, 168 169 Chlamydomonas reinhardtii, 437 Chromium (Cr), 10 Chromophore(s), 301, 385 Chromosomal DNA, 437 438 Chromosome, 425 in eukaryotes, 426, 427f Chronoamperometry, 338, 351 365 determining reduction potentials of active site bound to substrate, 360 361 mass-transport controlled catalytic voltammetry, 353 354 to measure Michaelis and inhibition constants, 355 358 principle and general comments, 351 353 to resolve rapid changes in activity, 358 360 CI. See Configuration interaction (CI) Circular dichroism (CD), 114 cis-[Pt(1,1-dicarboxycyclobutane) (NH3)2]. See Carboplatin cis-[PtCl2(NH3)2]. See Cisplatin CISD. See Configuration interaction singles and doubles (CISD)

479

Cisplatin, 13, 14f Classical oscillator, 279 281 Classical physics, 18 Classical theory of electromagnetism, 78 Cloning specific genes, 434 vectors and hosts, 452 454 Closed-shell systems, 122 ClustalW, 467 Clustered regularly interspersed palindromic repeats (CRISPR), 465 CMC. See Critical micelle concentration (CMC) 57 Co isotope, 203 CO-dehydrogenase enzyme, 234 236 Coalescence. See Intermediate exchange Cobalamin. See Vitamin B12 Cobalt (Co), 2, 9 Cocrystallization, 387 388 Codon biases, 424 Coexpression systems, 380 Combination bands, 291, 305 306 Combined hyperfine splitting, 214 Comparative gene organization, 467 468 Competence, 429 430 Complete active space self-consistent field (CASSCF), 58 Completeness, 401 402 Computational chemistry, 18, 53 64 computational methods for biological systems containing transition metal, 62 63 for excited states, 62 63 DFT, 58 62 wave function based methods, 55 58 Computer, 192 graphics, 397 Concerted mechanism, 334 Configuration interaction (CI), 37, 58 Configuration interaction singles and doubles (CISD), 58 Confocal volume, 314 Confocality, 314 317 Conjugation, 426, 430 Conjugative plasmids, 426 Constrained refinement, 257 258 Contact relaxation, 186 Continuous wave (CW), 127 lasers, 318 319, 321 NMR, 160 161 Convective movement of solution, 342 Conventions, 338 Copper (Cu), 2, 7 8 charge transfer complexes, 244

480 Copper (Cu) (Continued) in CO-dehydrogenase, 231 236 Cu21 ions, 237 238 Copy deoxyribonucleic acid (cDNA), 434 libraries, 457 Copy number, 426 Core electrons, 29 31 Correlation spectroscopy (COSY), 181, 181f Cosmid libraries, 455 COSY. See Correlation spectroscopy (COSY) Coulomb hole, 56 57 Coulombic repulsion of electrons, 37 Counter electrode, 338 Counterselection systems, 463 Coupled reactions on reduction potentials, 330 332 Coupling, 170 174 dipolar, 184 186 residual dipolar, 188 Covalence contribution, 213 Covalency of bond, 237 238 COX. See Cytochrome c oxidase (COX) CRISPR. See Clustered regularly interspersed palindromic repeats (CRISPR) CRISPR-associated gene (Cas gene), 465 CRISPR/CAS9 mutagenesis, 465 466 Critical micelle concentration (CMC), 388 Cross-polarization, 188 189 Cross-relational databases for genomes and metabolic and other pathways, 468 469 Cryo-electron microscopy (Cryo-EM), 376, 406 409 Cryoplunge, 407 Crystal, 391 crystal-field theory, 213 optimization and seeding, 386 387 Crystal field theory (CFT), 41, 43 Crystallization content of crystallographic models, 400 401 techniques and initial screens, 382 384 C-term enhancement mechanism, 305 306 Curie constant, 90 92 Curie law, 161, 197 for noninteracting paramagnets, 85 Curie relaxation, 188 Curie temperature (TC), 100 Curie Weiss law, 101 CW. See Continuous wave (CW) Cyclic voltammetry, 338 Cyclotron mass spectrometry, 251 252 Cytochrome c nitrite reductase complex, 400 Cytochrome c oxidase (COX), 7, 8f Cytosine, 420 422

Index

D Damping factor, 303 304 Danio rerio (Zebra fish), 437 Data reduction, 246b DC SQUID. See Direct current SQUID (DC SQUID) DDDs. See Direct detector devices (DDDs) ddNTPS. See DideoxyNTPs (ddNTPS) de Broglie relation, 277 278 Debye Waller factor, 232b, 253b, 259 Decibels, 136 scale, 130 Decoupling, 174 Deletion mutations, 460 461 Delitto perfetto mutagenesis, 463 464 δ-aminolevulinate synthase (ALAD), 11 Denaturing gel electrophoresis, 380 381 Density functional approximations (DFAs), 60 61 Density functional theory (DFT), 18 19, 44, 53 54, 58 62, 220 density functionals and spin states, 61 62 DFAs, 60 61 Density functional tight binding (DFTB), 64 Deoxyribonucleic acid (DNA), 419 422, 429 430 preparation of, 437 438 sequencing, 448 452, 450f structures and building blocks, 421f Desulfovibrio fructosovorans, 351 D. fructosovorans NiFe hydrogenase, 356 357 Detergents, 389 390 DFAs. See Density functional approximations (DFAs) DFT. See Density functional theory (DFT) DFTB. See Density functional tight binding (DFTB) Dialysis methods, 383 Diamagnetism, 70, 105 108 Diamagnets, 132 DideoxyNTPs (ddNTPS), 449 Diferric redox state, 214 Diferrous redox state, 214 Diffraction data, 391 392 Diffraction pattern, 392 Diffraction-before-destruction, 377 Diffraction-before-explosion, 405 Diffusion ordered spectroscopy (DOSY), 176 Diffusion-controlled criteria, 341 Diffusion-controlled voltammetry, 339 343 at rotating electrodes, 341 343 at stationary electrodes, 339 341 DIG. See Digoxigenin (DIG) Digoxigenin (DIG), 442 DIG antidigoxigenin system, 442 Dimeric sites, 99 105 Bleaney Bowers equation, 104 105 Curie Weiss law, 101

Index

spin Hamiltonian, 103 superexchange, 101 103 Dimethyl sulfoxide (DMSO), 358 359 5,5-Dimethyl-pyrroline N-oxide (DMPO), 140 141 Dipolar couplings, 170, 174 175, 184 186 Dipolar relaxation, 186 187 Dipole moment, 284 Dipole-allowed transitions, 236 237 Dirac’s notation, 19 20 Direct current SQUID (DC SQUID), 110 111 Direct detector devices (DDDs), 377 Direct electron transfer, 326 Direct methods, 396 Dithiothreitol (DTT), 381 DMPO. See 5,5-Dimethyl-pyrroline N-oxide (DMPO) DMSO. See Dimethyl sulfoxide (DMSO) DNA. See Deoxyribonucleic acid (DNA) dNTPs. See 32P-labeled deoxynucleotide triphosphates (dNTPs) Doppler effect, 206 DOSY. See Diffusion ordered spectroscopy (DOSY) Double electron electron resonance (DEER). See Pulsed electron ELDOR Double recombination, 431 “Double shotgun” method, 455 Doublet systems, 132 Drosophila, 461 Drosophila melanogaster (Fruit fly), 437 Drug-resistance plasmids, 425 DTPA. See Gd-diethylene triamine penta-acetic acid (DTPA) DTT. See Dithiothreitol (DTT) Dynamic electrochemistry, 336 339 capacitive current, 339 distinction between equilibrium and, 336 337 electrochemical equipment, 338 electrodes for electron transfer to/from proteins, 337 338 Vocab and conventions, 338 Dynamical correlation, 56 57

E EBI. See European Bioinformatics Institute (EBI) Economics, 471 Edge, 231 Effective Hamiltonian theory, 44 Effective magnetic moment, 83 85, 96 97 efg. See Electric field gradient (efg) Eigenstates, 288 ELDOR. See Electron double resonance (ELDOR) Electric dipole moments, 284 285 Electric field gradient (efg), 211

481

Electric quadrupole interaction, 208, 214 splitting, 208, 211 213 Electroblotting, 440 441, 442f Electrochemical equipment, 338 Electrochemical potential, 328 Electrochemistry under equilibrium conditions, 335 Electrode for electron transfer to/from proteins, 337 338 potential, 328, 353 Electromagnetic radiation (EM), 277 Electron Bohr magneton, 198 Electron double resonance (ELDOR), 127 Electron paramagnetic resonance spectroscopy (EPR spectroscopy), 121 128, 214 anisotropy, 125 126 applications, 150 152 basic theory and simulation of, 133 135 biomolecules gives EPR, 132 133 concentration determination, 137 139 high-spin systems, 145 150 hyperfine interactions, 139 144 pseudo-code, 135b saturation, 135 137 spectrometer, 128 132 conventional continuous-wave, 129f Electron spin echo envelope modulation spectroscopy (ESEEM spectroscopy), 124 Electron transfer (ET), 326 kinetics, 332 334 Electron-nuclear double resonance spectroscopy (ENDOR spectroscopy), 124 Electron(s), 328, 406 407 configuration, 28 29 crystallography, 407 density, 18 19 at nucleus, 209 electron electron repulsion, 22 23 microscopy, 377 repulsion, 51 52 shielding, 29 spin, 28 of hydrogen atom, 28 tunneling processes, 333 Electronic absorption spectra, 307 309 Electronic energy, 54 Electronic structure of atoms electronic terms, 37 38 hydrogen atom, 25 28 many-electron atoms, 28 32 periodic system of elements, 31 32 Pauli principle, 32 34 two electrons in two orbitals, 34 37

482

Index

Electronic terms, 37 38 Electronic Zeeman interaction, 123 124, 126 127 Electron nuclear dipolar relaxation, 187 Electroporation cell, 429 430, 430f EM. See Electromagnetic radiation (EM) ENDOR spectroscopy. See Electron-nuclear double resonance spectroscopy (ENDOR spectroscopy) Energy of magnetic Zeeman interaction, 213 214 of NMR transitions, 158 160 of photon, 279 Energy effect, 37 Enterobacteriaceaea, 425 426 Enzymes, 63 EOMCC. See Equation-of-motion-couple-cluster (EOMCC) Epigenetics, 420 422 EPR spectroscopy. See Electron paramagnetic resonance spectroscopy (EPR spectroscopy) Equation-of-motion-couple-cluster (EOMCC), 62 Equilibrium potential, 335 Ernst angle, 193 Escherichia coli, 190 191, 332, 424 425, 454 fumarate reductase from, 346 T7 ribonucleic acid polymerase-T7 promoter system in, 457 459 ESEEM spectroscopy. See Electron spin echo envelope modulation spectroscopy (ESEEM spectroscopy) Essential alkali metal ions, 2 3 Essential metal(s), 1, 10 11 ions, 13 and functions, 2 10 ET. See Electron transfer (ET) Eukaryotes, 419 420, 420f European Bioinformatics Institute (EBI), 466 467 Evan’s method, 195 197 NMR method, 113 114 EXAFS. See Extended X-ray absorption fine structure (EXAFS) Exchange hole. See Fermi hole Exchange-correlation functional (XC functional), 59 60 Exclusion principle, 32 33 Expectation value of operator, 19 20 Expression vectors, 453 Extended X-ray absorption fine structure (EXAFS), 220, 245 248, 251 strategy for interpretation, 257 258 validation and automation of EXAFS data analysis, 258 259 Extended X-ray absorption structure, 230, 234 236, 264t. See also X-ray absorption near-edge structure (XANES)

F

19 F labeled amino acids, 190 191 FAD. See Flavin adenine dinucleotide (FAD) Faraday balance, 108 110, 110f Fast chemical exchange, 178 FAST-ALL (FASTA), 467 Fast-scan voltammetry, 347 351 Fermi hole, 33 34 Fermi’s Golden Rule, 252 254 Ferrimagnetism, 101 Ferromagnetism, 100 Ferryl species, 217 FID. See Free induction decay (FID) Fit index, 257 Flavin adenine dinucleotide (FAD), 137 138, 381 382 Fluorescence, 239 Fluorescence-based thermal stability assay. See Thermofluor Fock operator, 55 56 Force constant, 279 281 Force methods, 109 110 Fosmid libraries, 456 Four-coordinated iron(IV) and iron(V) compounds, 221 222 Fourier filtering, 257 Fourier shell correlation (FSC) curve, 408 Fourier transform infrared (FTIR) spectroscopy, 276 Fourier transformation, 248 Frank Condon factors, 292 293, 304, 309 310 FrdAB. See Fumarate reductase (FrdAB) Free electron lasers, 244 Free energy, 328 Free energy of reaction, 327 Free induction decay (FID), 162 163, 166 Free-spin electron g factor, 198 Frequency dependence of polarizability, 310 Friedel’s law, 395 Fruit fly. See Drosophila melanogaster (Fruit fly) FSC curve. See Fourier shell correlation (FSC) curve FTIR spectroscopy. See Fourier transform infrared (FTIR) spectroscopy Full Hamiltonian, 22 Full width at half maximum (FWHM), 295 Fumarate reductase (FrdAB), 351, 360 361 from E. coli, 346 FWHM. See Full width at half maximum (FWHM)

G G protein coupled receptors (GPCRs), 379 “g strain”, 124 Gadolinium, 13 GAL1-I-SceI construct, 463 464

Index

GAL1/10-p53 gene, 463 464 γ-emission and absorption, 204 γ-photons, 203 γ-radiation, 202 203 Gaussian distribution, 134 135 Gaussian-type orbitals, 54 55 Gd-diethylene triamine penta-acetic acid (DTPA), 13 Geminal couplings, 172 Gene, 424 425 Gene libraries, 454 455 Gene organization, 427 428 Gene-editing/engineering, 465 466 General purpose cloning vectors, 452 453 Generalized gradient approximations (GGAs), 61 Genetic code, 423 424, 423f Genetics and molecular genetics approaches and systems, 435 437 BAC, 456 bioinformatics, 466 469 cDNA libraries, 457 central dogma, 422 423, 422f classification, taxonomy, phylogeny, 419 420 cloning vectors and hosts, 452 454 cosmid libraries, 455 deoxyribonucleic acid, 429 430 fundamental molecular biological information molecules, 420 422 genetic/gene, 424 code, 423 424, 423f libraries, 454 455 and molecular biology of N2-fixation, 471 472 organization, 427 428 genomes, 424 425 homologous recombination, 430 431, 432f insertion elements, transposons, and repetitive DNA, 428 429 libraries intended for genome DNA sequencing, 455 mobilizable and broad-host range vectors and cosmids, 456 model systems, 436 437 molecular biology tools and methods, 437 452 mutagenesis, 460 466 OMICS revolution, 469 471 origins, evolution, and speciation, 419 Pichia pastoris system, 459 promoters, transcription initiation, and transcriptional regulation, 431 433 protein overexpression and purification, 457 regulation and approvals, 434 435 replicons, 425 426 T7 ribonucleic acid polymerase-T7 promoter system in E. coli, 457 459

483

tags for protein purification, correct folding, improved stability, 459 460 translation initiation, 433 434 YACs, 456 457 Genomes, 424 425 sites, 468 Genomics, 470 Geometry optimization, 54 Germ cells, 426 GFP. See Green fluorescent protein (GFP) GGAs. See Generalized gradient approximations (GGAs) Glycerol, 382 Gold (Au), 14 15 nanoparticles, 14 15 Gouy balance, 108 109, 110f GPCRs. See G protein coupled receptors (GPCRs) Gram ve bacteria, 456 Gram ve bacterium E. coli, 437 Graphite, 337 Green energy, 6 Green fluorescent protein (GFP), 380 Ground state wave function, 19 Group theory, 38 40 Guanine, 420 422 Gyromagnetic ratio, 79, 157 158

H Haemophilus, 429 430, 462 463 Half-integer high-spin systems, 132 Halogens, 262 263 Haloperoxidases, 263 Hamiltonian, 40 41 formulas, 19 20, 22, 24 Handy2Cohen Optimized Exchange, 61 “Haploid”, 426 Harmonic oscillator, 283 284, 289f Hartree Fock method (HF method), 55 57 Harvesting and mounting of crystals, 390 H-atom orbitals, 47 48 HDVV. See Heisenberg Dirac Van Vleck (HDVV) Heavy atom j j coupling model, 75 76 Heavy metal salts, 397 Heavy-atom derivatization, 396 397 Heisenberg Dirac Van Vleck (HDVV), 103 Heller formulism, 278 Heller’s time-dependent approach, 309 311 Heme enzyme HRP, 216 217 HERFD. See High-energy resolution fluorescence detected (HERFD) Hermite polynomials, 288 Herzberg Teller formalism, 305 Hetereoleptic Ru(II) polypyridyl complex, 307

484 Heteronuclear single quantum coherence (HSQC), 182 HF method. See Hartree Fock method (HF method) High dose rate, 407 High molecular genomic DNA, 454 High pressure liquid chromatography (HPLC), 449 High-energy resolution fluorescence detected (HERFD), 240b High-spin systems, 145 150, 146f High-throughput DNA sequencing, 449 450, 451f High-valent iron complexes, 216 Himar1, 462 463 “His-tagged” proteins, 459 460 Hole formalism, 37 Holliday junctions, 431 “Home-built” systems, 276 Homing endonuclease, 463 464 Homo sapiens, 2 Homologous recombination, 430 431, 432f Hooke’s law, 279 281, 283 Horizontal submerged agarose gel electrophoresis, 438 439 Horseradish peroxidase (HRP), 216 HRP-I and HRP-II, 216 HPLC. See High pressure liquid chromatography (HPLC) HRP. See Horseradish peroxidase (HRP) HSQC. See Heteronuclear single quantum coherence (HSQC) Hund’s rule, 37, 76 of maximum multiplicity, 29 31 Hybrid DFAs, 61 Hybridization, specific DNA or RNA sequence detection by, 442 Hydrogen atom, 25 28, 400 angular momentum, 27 28 electron spin, 28 spherical coordinate system, 26f Hydrogenases, 351 Hypercalcemia, 3 Hyperfine interactions, 139 144 scalar coupling, 184 Hyperkalemia, 2 3 Hypernatremia, 2 3 I mutations, 460 461

I IDPs. See Intrinsically disordered proteins (IDPs) Illumina sequencing, 449 450 Imidazole, 257 258 In situ DNA, 420 422 In-cell nuclear magnetic resonance, 190 191 “In-frame deletion” mutants, 427

Index

In-house X-ray equipment, 392 Independent electron model, 28 29 Induced dipole moments, 287 291 Inelastic scattering of light, 276 Infrared light (IR light), 277 278 Inner filter effect, 314 317 INRA’s MULTALIN software, 467 Insertion elements, 428 429 Insertion sequence (IS) elements, 429 Integral membrane proteins, 388 Intermediate exchange, 178 Intermediate-spin state, 44 45 Interpolation of baselines, 346b Intrinsic ultraviolet fluorescence, 385 Intrinsically disordered proteins (IDPs), 190 191 Inverse PCR, 447 Invisibility, 189 190 Ion torrent sequencing, 450 Ion-sensitive field effect transistor (ISFET), 450 IR light. See Infrared light (IR light) Iron (Fe), 2, 6 7 Fe(II), 216 complexes, 218 219 Fe(III), 216 Fe(III) OCl complex, 314 Fe(III) OOH species, 313 Fe(IV), 216 Iron(IV)-nitrodo precursor, 221 precursor, 221 Fe(V), 216 complexes, 220 221 FeFe hydrogenases, 365 iron-57 (57Fe) hyperfine interactions, 207 208, 208f spectroscopy, 203 iron-containing proteins, 6 7 Iron(VI) compound, 222 224 iron sulfur clusters, 210 Irreducible representation (irrep), 40 41 IS elements. See Insertion sequence (IS) elements ISFET. See Ion-sensitive field effect transistor (ISFET) Isomer shift, 208 as informative hyperfine interaction, 208 211 Isomorphous replacement, 394 Isotope labeling, 312 314 Isotropic pattern, 125 126 Iterative refinement, 257 IZIT, 385

J J Craig Venter Institute (JCVI), 468 Jahn Teller effect (JT effect), 24 Josephson junctions, 110, 111f

Index

K K-capture, 203 KHD equation. See Kramers Heisenberg Dirac equation (KHD equation) Kinetics of proton-coupled electron transfer, 334 Klebsiella aerogenes, 425 426 Klenow fragment, 444 446 KlURA3 gene, 463 464 Kok cycle in Photosystem II, 242 Koutecky Levich plot, 342, 354 Kramers Heisenberg Dirac equation (KHD equation), 301, 303 305, 310 KS spin orbitals, 59 60 Kunkel method, 464

L Lac operon in Escherichia coli, 427, 428f LacZ gene, 444 LacZΔM15 mutation, 444 Lamb Mo¨ssbauer factor. See Mo¨ssbauer f factor Laminaria digitata, 262 263 Lande´ constant factors, 83 Lande´ formula, 96 97 Langevin paramagnetism, 86 88 Laporte rule, 236 237 Larmor equation, 160, 167 Larmor frequency, 167 Laue diffraction, 392 LDA. See Local density approximation (LDA) L-edge X-ray absorption spectroscopy, 237b Levich equation, 342 LF. See Ligand field (LF) LFDFT. See Ligand field DFT (LFDFT) LFT. See Ligand field theory (LFT) Libraries intended for genome deoxyribonucleic acid sequencing, 455 Ligand field (LF), 23 24 Ligand field DFT (LFDFT), 63 Ligand field theory (LFT), 18, 41 54 five d-orbitals of transition metals, 41f qualitative considerations, 44 46 quantitative considerations, 48 52 Tanabe Sugano diagrams, 51 52 splitting of d-orbitals in octahedral environment, 42f symmetry in, 46 48 Ligand to metal transfer (LMCT), 302 303 Ligands, 43 identification, 242 ligand-edge, 237b Lithium, 13 14 Lithium carbonate, 13 14 LMCT. See Ligand to metal transfer (LMCT) LmrR protein, 317 319 Local density approximation (LDA), 61

485

Lock system, 192 London dispersion forces, 279 280 Low-valent iron complexes, 216 “Low-Z” ligands, 250 Luciferin, 450 Luggin, 338 Luminophore, 297 299

M Macro-electrodes, 342 343 Macrocycle 1,4,8,11-tetraazacyclotetradecane, 217 218 Macromolecular crystallography, 379 Macroscopic magnetization, 160 161 Macroseeding techniques, 387 MAD. See Multiple wavelength anomalous dispersion (MAD) “Magic angle”, 170 171 Magnesium ion (Mg21), 3 Magnet, 191 Magnetic circular dichroism (MCD), 108, 114 115 Magnetic/magnetism, 69 70 in biologically relevant ions, 92 99 crystal field effect on magnetic properties of 3d compounds, 95 99 orbital splitting of transition metal ions, 93 95 dipole interaction, 208 field, 80 82 hyperfine splitting, 213 214 intensity, 80 81 moment, 78 80, 107 origin of, 70 quantum number, 25 27 and respective properties, 195, 195t susceptibility, 83 85 measurement, 195 197 Zeeman interaction, 214 Magnetization, 80 82, 195 acting on pulses, 161 162 on rotating frame, 162 Boltzmann treatment, 86 92 Brillouin function, 88 Curie constant and spin-only effective magnetic moment, 90 92 Langevin paramagnetism, 86 88 second-order Zeeman effect, 92 temperature-independent paramagnetism, 92 van Vleck equation, 88 90 macroscopic, 160 161 saturation, 83 85 Magnetometry, 108 113 Force methods, 109 110 SQUID, 110 113 Magnitude, 71 72

486 MALDI-TOF. See Matrix-assisted laser desorbtion/ionization-time of flight (MALDI-TOF) Mammalian COX, 7 Manganese, 2, 4 6 role in biology, 4 6 Manganism, 4 6 Marcus Hush model, 334 Mars Exploration Rovers (MER), 224 Mass-transport controlled catalytic voltammetry, 353 354 Matrix-assisted laser desorbtion/ionization-time of flight (MALDI-TOF), 470 Maxam and Gilbert method, 448 449 MbNO. See Nitrosylmyoglobin (MbNO) MC transitions. See Metal centered transitions (MC transitions) MCD. See Magnetic circular dichroism (MCD) MCS. See Multiple cloning site (MCS) (Me3cyclam-acetate)2, 222 223 Mediators, 335 Megaplasmids, 425 Membrane proteins, 388 390 MER. See Mars Exploration Rovers (MER) 2-Mercaptoethanol, 381 Mercuric chloride (HgCl2), 397 Mercuric potassium iodide (K2HgI4), 397 Messenger RNA (mRNA), 422 in eukaryotes, 433 434, 434f Metabolomics, 471 Metal centered transitions (MC transitions), 319 320 Metal to ligand charge transfer (MLCT), 302 303, 307, 321 Metal-based oxidation, 217 Metal imidazole complex, 255 Metallic (α-) iron, 208 Metallic electrode, 333 Metalloproteins, 69 70, 395 biological redox scale, 329 330, 329f catalytic PFV and chronoamperometry, 351 365 diffusion-controlled voltammetry, 339 343 dynamic electrochemistry, 336 339 electrochemistry under equilibrium conditions, 335 electron transfer kinetics, 332 334 exercises, 365 influence of coupled reactions on reduction potentials, 330 332 kinetics of proton-coupled electron transfer, 334 redox thermodynamics, 327 328 reference potential and reference electrodes, 329 voltammetry of adsorbed proteins, 343 351 Metal metal distances in metal clusters, 261 262

Index

Metals, 1 biological periodic table of elements indicating essential elements, 2f in biomolecular nuclear magnetic resonance spectra, 183 184 in diagnosis and therapeutics, 13 15 platinum anticancer drugs, 14f structure of MRI contrast agent Gd-DTPA, 14f essential metal ions and functions, 2 10 toxic metals, 10 12 Methyl viologen (MV), 363 Methylene blue, 385 Methylmercury (MeHg), 11 Michaelis and inhibition constants, chronoamperometry to measuring, 355 358 Michaelis Menten parameters, 355 Micro-electrodes, 342 343 Micro-seeding techniques, 387 Microbatch, 384 Microstates, 29 31 Microwave intensities, 130 Mie scattering, 278 Miller indices, 392 MinION cell, 452, 453f MIR. See Multiple isomorphous replacement (MIR) Mixed-valence iron(III) iron(IV) dimers and iron(IV) monomers, 219 220 MLCT. See Metal to ligand charge transfer (MLCT) Mn4CaO5 cluster, 243 mob gene, 426 Mobilizable and broad-host range vectors and cosmids, 456 Mobilization, 426 “Mode pattern”, 130 Model bias, 394 Modern theories of magnetism, 69 70 Modern X-ray sources, 244 Molecular biology tools and methods, 437 452 agarose gel electrophoresis, 438 439, 439f blotting techniques, 440 443 deoxyribonucleic acid sequencing, 448 452 PCR, 444 448, 446f preparation of DNA, 437 438 pulse-field/orthoganol electrophoresis, 439 440 recombinant DNA technology, 443 444 Molecular cloning, 443 444 Molecular phylogenies, 469 Molecular magnetochemistry Bohr magneton, 78 80 Boltzmann treatment of magnetization, 86 92 contributions to angular momentum, 71 77 to magnetism in biologically relevant ions, 92 99

Index

Curie law for noninteracting paramagnets, 85 diamagnetism, 105 108 dimeric sites, 99 105 effective magnetic moment, 83 85 experimental methods, 108 115 Evans NMR method, 113 114 magnetometry, 108 113 MCD, 114 115 magnetic field, 80 82 moment, 78 80 susceptibility, 83 85 magnetization, 80 82 saturation, 83 85 origin of magnetism, 70 Zeeman effect, 82 83 Molecular orbitals (MOs), 34 35, 210 Molecular replacement, 393 394 Molecular systems, 69 70 Molecular weight (MW), 381 Møller Plesset perturbation theory (MP perturbation theory), 57 MolProbity, 402 403 MolProbity Score, 402 403 Molybdenum (Mo), 2 in CO-dehydrogenase, 231 236 Moment, 87 MOs. See Molecular orbitals (MOs) Mo¨ssbauer effect, 202, 204 205 Mo¨ssbauer experiment, 205 206, 207f Mo¨ssbauer f factor, 205 Mo¨ssbauer light source, 203 Mo¨ssbauer spectroscopy/spectrometer, 202, 206 207 perspectives, 224 principles, 202 207 Mouse. See Musca domestica (Mouse) MP perturbation theory. See Møller Plesset perturbation theory (MP perturbation theory) MRCI. See Multireference CI (MRCI) mRNA. See Messenger RNA (mRNA) Muffin-tin approximation, 249b, 252 254 Multichannel, 192 Multidimensional NMR, 179 182 Multielectron wave function, 28 29 Multiple 2D projections, 407 Multiple cloning site (MCS), 444 Multiple isomorphous replacement (MIR), 394 Multiple scattering, 230 231 in biological systems, 254 256 Multiple sequence alignment, 467 Multiple wavelength anomalous dispersion (MAD), 394 395 Multiplets, 51 52

487

Multireference CI (MRCI), 62 Musca domestica (Mouse), 437 Mutagenesis, 460 466 chemical and physical, 461 CRISPR/CAS9, 465 466 site-directed, 463 464 transposable elements and, 462 463 Mutants, 460 461 MV. See Methyl viologen (MV) MW. See Molecular weight (MW) Mycoplasma genitalium, 424 425

N N2-fixation, 471 472 Nanosequencing, 452, 452f “Narrow host range” plasmids, 425 426 National Center for Biotechnology Information (NCBI), 466 467 Natoli’s rule, 236 Natural linewidth, 314 NCBI. See National Center for Biotechnology Information (NCBI) NCS. See Noncrystallographic symmetry (NCS) NdeI site, 458 459 Near infrared photons (NIR photons), 277 278 Ne´el temperature (TN), 100 n-electron valence state perturbation theory (NEVPT2), 58 Nephelauxetic effect, 43 44 Nernst equation, 327 328 Nested PCR, 447 NEVPT2. See n-electron valence state perturbation theory (NEVPT2) Newman projections, 172, 173f Nickel (Ni), 10 NIR photons. See Near infrared photons (NIR photons) NIS. See Nuclear inelastic scattering (NIS) Nitrate reductase, 358 Nitric oxide-heme recombination kinetics (NO-heme recombination kinetics), 244 Nitrogen-nitrogen-sulfur-sulfur (NNSS), 125 126 Nitrosylmyoglobin (MbNO), 244 NMR spectroscopy. See Nuclear magnetic resonance (NMR) spectroscopy NNSS. See Nitrogen-nitrogen-sulfur-sulfur (NNSS) NO-heme recombination kinetics. See Nitric oxideheme recombination kinetics (NO-heme recombination kinetics) NOE. See Nuclear Overhauser effect (NOE) NOESY. See Nuclear Overhauser enhancement spectroscopy (NOESY) Noncatalytic voltammetry at slow scan rates, 343 347

488

Index

Noncrossing rule, 37 Noncrystallographic symmetry (NCS), 391 Nondynamical correlation, 56 57 Nonessential metals, 1 Nonheme diiron center, oxidation and spin states in, 214 216 Nonheme model compounds, 217 218 synthetic iron(III) complexes with macrocyclic ligand cyclam, 217 218 Noninvasive imaging techniques, 13 Nonmetal trace elements, 262 263 Nonphosphorylated oligonucleotides, 454 455 Nonresonant transition, 292 293 Nontotally symmetric modes, 307 Normal modes, 291 Normal Zeeman effect, 82 Normalization constant, 19 20 Northern blotting, 441 NRVS. See Nuclear Vibrational Spectroscopy (NRVS) Nuclear Bohr magneton, 198 Nuclear gamma resonance absorption, 202 Nuclear inelastic scattering (NIS), 220 Nuclear magnetic resonance (NMR) spectroscopy, 126 128, 156, 251 252, 378. See also Electron paramagnetic resonance spectroscopy (EPR spectroscopy) energy of NMR transitions, 158 160 experiment, 166 167 in-cell, 190 191 measuring macroscopic magnetization and relaxation, 191 192 multidimensional NMR, 179 182 properties of matter relevant to, 157 158 of (semi-)solid samples, 188 190 direct observation of metals, 189 190 signal, 192 Nuclear Overhauser effect (NOE), 156 157, 174 176 Nuclear Overhauser enhancement spectroscopy (NOESY), 181 182, 182f Nuclear shielding tensor, 167 Nuclear Vibrational Spectroscopy (NRVS), 224 Nucleation, 387 Nyquist theorem, 168 169

O OCP. See Open circuit potential (OCP) Octahedral LF, 23 24 Oligotropha carboxidovorans, 231 236, 261 262 Omics revolution, 469 471 economics, 471 genomics, 470 metabolomics, 471 omniomics, 471

proteomics, 470 structural genomics, 470 471 transcriptomics, 470 Omniomics, 471 One-electron peaks, 345 “One-over-f” type (1/f type), 131 132 One-proton process, 330 Open circuit potential (OCP), 352 Open reading frame (ORF), 458 459 Operator regions, 433 Operons, 427 Orbital angular momentum, 27 28, 71 72 Orbital approximation, 28 29 Orbital quantum number. See Azimuthal quantum number Orbital splitting of transition metal ions, 93 95 Orbitals, 22 27 ORF. See Open reading frame (ORF) Oscillating motion of electrons, 278 Overlap-extension PCR or splicing by overlap extension, 447 448 Overtones, 291, 305 306 [Ox]/[Red] ratios, 335 Oxaliplatin, 13, 14f Oxidation in nonheme diiron center, 214 216 Oxidized BoxB, 214 216

P 32

P-labeled deoxynucleotide triphosphates (dNTPs), 442 Paired spins, 32 33 Pairing energy, 45 46 Panulirus interruptus, 236 Para-chloromercurybenzoic sulfate, 397 Paracoccus denitrificans, 342 343 Paracoccus pantotrophus, 363 Paramagnetic shift, 185 186 Paramagnetism, 69 70, 100 Parametric models (PMX), 64 Partial fluorescence yield (PFY), 240b Particles, 281, 407 Pauli principle. See Pauli’s exclusion principle Pauli’s exclusion principle, 28 34, 171 172 pBR322, 443 444, 444f, 452 453 PCR. See Polymerase chain reaction (PCR) PDB. See Protein Data Bank (PDB) PE. See Potential energy (PE) Peak current, 341 Peak width at half height, 344 PEGs. See Polyethylene glycols (PEGs) PELDOR. See Pulsed electron ELDOR (PELDOR) Peripheral membrane proteins, 388 Periplasmic nitrate reductase, 351

Index

Permanent electric dipole moments, 284 293 Perturbational method, 22 pET 5a, 458 459, 458f PFV. See Protein film voltammetry (PFV) PFY. See Partial fluorescence yield (PFY) pH optima for enzyme activity, 363 Phase determination, 392 397 anomalous scattering, 394 396 direct methods, 396 heavy-atom derivatization, 396 397 isomorphous replacement, 394 molecular replacement, 393 394 Phase problem in X-ray crystallography, 378 Phase shifter, 131 Phase shifts calculations, 249b and effect of atom type, 248 252 Phase-sensitive detection, 131 132 Phonon, 204, 277 278 Photoredox catalysis, 321 Photosystem II complexes, 244 Phylogeny inference package (PHYLIP), 469 Physical breakage, 454 455 Physical mutagenesis, 461 Pichia pastoris system, 459 Pixel detectors. See Single-photon counting detectors Planck’s equation, 159 Plane wave, 252 254 Plank’s constant, 198 Plasmids, 425 DNA, 438 pRK2013, 426 Platinum complexes, 13 Platinum potassium chloride (K2PtCl4), 397 Ploidy, 426 PMX. See Parametric models (PMX) Point mutations, 460 461, 463 Polarizability, 287 291, 309 Poly(NANP), 459 460 Polycistronic operons, 427 Polyethylene glycols (PEGs), 383 Polymerase chain reaction (PCR), 437, 444 448, 446f Porphyrin, 257 258 Post-Hartree Fock methods, 57 58 Potassium uranyl fluoride (K3UO2F5), 397 Potential energy (PE), 279 operator, 20 Potentiometric titrations, 335 Potentiostat, 338 Pourbaix diagram, 331, 331f, 332f Powder pattern, 125 Powder-spectrum, 188 189 Preexponential factor, 334

489

Principal quantum number, 25 27 Probability factor for zero-phonon processes, 205 Probeheads, 191 PROCHECK program, 402 403 ProDom, 468 Projection reconstruction concept, 156 157 Prokaryotes, 419 420, 420f, 425 Promoters, 431 433 Protein crystallization, 380 390 protein production and sample preparation, 380 protein quality assessment, 380 390 analysis of crystallization trials, 384 385 cocrystallization and soaking, 387 388 crystal optimization and seeding, 386 387 crystallization techniques and initial screens, 382 384, 383f harvesting and mounting of crystals, 390 membrane proteins, 388 390 protein concentration, 381 382 salt or protein crystals, 385 Protein crystals, 385 Protein Data Bank (PDB), 379 Protein film voltammetry (PFV), 331, 343 351 fast-scan voltammetry, 347 351 noncatalytic voltammetry at slow scan rates, 343 347 Protein overexpression and purification, 457 Proteins, identification of potential domains in, 468 Proteomics, 470 Proton-coupled electron transfer, kinetics of, 334 Protonation, 330 Pseudocontact shift, 184 185 Pseudomonas aeruginosa, 425 [Pt(1R,2R-1,2-diaminocyclohexane)(oxalate)]. See Oxaliplatin pUC19, 444, 445f Pulse-field/orthoganol electrophoresis, 439 440 Pulsed electron ELDOR (PELDOR), 127 Pulsed EPR, 127 Pulsed NMR techniques, 156 157, 161 Pulses, 161 162 Pump-probe experiments. See Time-resolved serial crystallography experiments Pump-probe XAS studies target systems, 244 Pyranopterin cofactor, 10, 10f Pyrolytic graphite edge, 337 Pyrosequencing, 450

Q Q-bands of porphyrins, 308 QM. See Quantum mechanical/mechanics (QM) Q-PCR. See Quantitative PCR (Q-PCR)

490 QSoas, 362b Quadrupole coupling constant, 211 212 Quantitative PCR (Q-PCR), 448 Quantization, 279 284 Quantum chemistry, 18 24. See also Computational chemistry approximations in, 21 24 Quantum excitation, nature of, 281 284 Quantum mechanical/mechanics (QM), 18, 38 40, 72, 122 oscillator, 282 selection rules, 158 159 treatment of polarizability, 303 304 Quantum numbers, 25 27 Quartz, 314 317

R Racaha’s parameters, 38, 39f, 41 Radiation damping, 191 Radiation-induced damage, 405 Ramachandran diagram, 404 Raman cross-section and intensity of Raman bands, 296 297 Raman effect, 293 Raman scattering, 277, 297 299 classical description of, 303 resonance enhancement, 299 311 Raman spectroscopy, 276 277, 295 296, 314 of carrots and parrots, 300 303 of porphyrins and related compounds, 308 Randles Sevcik equation, 341 Rayleigh scattering, 278, 303 RDC. See Residual dipolar coupling (RDC) Reaction intermediates, 216 Reaction monitoring with resonance Raman spectroscopy, 319 320 Recoil energy, 204 Recoilless emission and absorption, 204 205 Recombinant deoxyribonucleic acid technology, 443 444 Recording voltammograms, 349 Recurrence relation, 291 Redox reactions, 330 Redox thermodynamics, 327 328 Redox-cycling experiment, 358 359 Redox-innocent cyclam, 217 218 Redox-linked protonations, 331 Reduction potentials, 328 329 coupled reactions on, 330 332 noncatalytic voltammetry at slow scan rates to measuring, 343 347 of redox couple, 335 Redundancy, 402

Index

Reference electrodes, 329 Refined model, 400 Relative intensities of Stokes and anti-Stokes Raman scattering, 291 292 Relaxation, 162 166, 186 agents, 187 contact, 186 dipolar, 186 187 physical mechanisms, 163 166 Reorganization energy, 333 Repetitive deoxyribonucleic acid, 428 429 Repetitive DNA, 429 Replicons, 425 426 Repressors, 433 Resection, 431 Residual dipolar coupling (RDC), 170 171, 188 Resolution, 314, 392, 397 398, 401 Resonance condition, 123 Resonance enhanced Raman scattering, 284 Resonance Raman spectroscopy, 277 applications, 317 322 in characterization of artificial metalloenzymes, 317 319 classical oscillator, Hooke’s law, force constant, and quantization, 279 281 experiment, 294 299 experimental and instrumental considerations, 312 317 fundamentals of vibrational spectroscopy, 277 284 permanent, induced, and transition electric dipole moments, 284 293 quantization and nature of quantum excitation, 281 284 Raman cross-section and intensity of Raman bands, 296 297 Raman scattering, 297 299 relative intensities of Stokes and anti-Stokes Raman scattering, 291 292 resonance enhancement of Raman scattering, 299 311 SERRS spectroscopy, 311 312 SERS, 311 312 virtual state, 292 293 Resonant experiment, 240b Restrained refinement, 258 Restricted Hartree Fock method (RHF method), 56 Reverse transcriptase, 422 Reverse transcription PCR (RT-PCR), 448 Reversible criteria, 341 RHF method. See Restricted Hartree Fock method (RHF method) Rhizobium, 425 Rhizobium meliloti, 456

Index

Rhodobacter capsulatus, 358 359, 425 Rhodobacter capsulatus periplasmic DMSO reductase, 363 Rhodobacter sphaeroides, 351 nitrate reductase, 363 periplasmic nitrate reductase, 355 Rhombic powder pattern, 125 126 Rhombograms, 148 149 Ribonucleic acid (RNA), 419 422 structures and building blocks, 421f Ribosomal RNAs (rRNAs), 419 420 Ribosome, 380 RNA. See Ribonucleic acid (RNA) Roothaan equations, 55 56 Rotating disc electrode, 342 Rotating electrodes, diffusion-controlled voltammetry at, 341 343 Rotating frame, 162 rRNAs. See Ribosomal RNAs (rRNAs) RT-PCR. See Reverse transcription PCR (RT-PCR) Rule-of-thumb, 393 395 Russell Saunders coupling method, 75 76 Ruthenium complexes, 13

S SA-CASSCF. See State-averaged-CASSCF (SA-CASSCF) Saccharomyces cerevisiae, 437 SAD method. See Single wavelength anomalous dispersion method (SAD method) Salmonella pneumoniae, 425 426 Salt crystals, 385 Sampling bandwidth, 168 169 Sampling interval, 192 Sanger method, 449 Saturnism, 11 SAXS. See Small-angle X-ray scattering (SAXS) Scalar couplings, 171 Scan rate, 338 “Scarless” mutations, 463 464 Scattered radiation, 278 Scattering, 287 291 SCF. See Self-consistent field (SCF) Schizomyces pombei, 437 Schrodinger’s equation (SE), 19 SD. See Slater determinant (SD) SDS PAGE. See Sodium dodecyl sulfate polyacrylamide gel electrophoresis (SDS PAGE) SE. See Schrodinger’s equation (SE) SEC. See Size-exclusion chromatography (SEC) SEC-SAXS. See Size-exclusion chromatography-smallangle X-ray scattering (SEC-SAXS) Second Hohenberg Kohn theorem, 59

491

Second-order effect, 142 Second-order Zeeman coefficient, 90 effects, 82 83, 92 Selenium (Se), 398 in CO-dehydrogenase, 231 236 Selenoenzyme inhibition, 11 Self-consistent field (SCF), 55 56 Semiautomated plungers, 407 Semiempirical approach, 57 Semireduced redox state, 214 Sequence searching sites, 467 Serial femtoseconds crystallography (SFX), 405 SERRS spectroscopy. See Surface-enhanced resonance Raman scattering (SERRS) spectroscopy SERS. See Surface-enhanced Raman scattering (SERS) SFX. See Serial femtoseconds crystallography (SFX) SHE. See Standard hydrogen electrode (SHE) Shift agents, 187 Shim system, 192 Shine and Delgarno sequence, 433 Short peptide tags, 380 Shot noise, 297 299 Shuttle vectors, 453 454 SHWEFT-PASE pulse sequences, 193 195 SI units. See Syste`me International units (SI units) Signal generators, 192 Signal-to-noise ratio (SNR), 276 277, 392, 402 Silicone oils, 384 Silver (Ag), 14 Single isomorphous replacement (SIR), 394 Single nucleotide polymorphisms (SNPs), 460 461 Single wavelength anomalous dispersion method (SAD method), 395 396 Single-particle cryo-EM, 406 Single-photon counting detectors, 379 Singlet state, 33 34 SIR. See Single isomorphous replacement (SIR) Site-directed mutagenesis, 463 464 Site-directed point mutants, 464 465 Size-exclusion chromatography (SEC), 381 Size-exclusion chromatography-small-angle X-ray scattering (SEC-SAXS), 410 Slater determinant (SD), 32 35 Slater-type orbitals (STOs), 54 55 Slow, redox-driven (in)activation, 364 365 Slow chemical exchange, 178 Slow interfacial electron transfer, 362 363 Slow intramolecular electron transfer, effect of, 362 Slow substrate binding, 363 364 Small-angle X-ray scattering (SAXS), 376 377, 409 411 SNPs. See Single nucleotide polymorphisms (SNPs) SNR. See Signal-to-noise ratio (SNR)

492

Index

Soaking, 387 388 Sodium dodecyl sulfate polyacrylamide gel electrophoresis (SDS PAGE), 380 381, 443 Solid-state materials, 69 70 Solomon equation, 186 Solubility phase diagram, 383, 384f Solubilization of membrane proteins, 388, 389f SORCI. See Spectroscopically orientated CI (SORCI) Southern blotting, 440 441, 441f Spatial wave function, 28 Spectral density function, 164 165 Spectral editing, 179 Spectral width. See Sampling bandwidth Spectrochemical series, 43 44 Spectroscopically orientated CI (SORCI), 62 Spherical harmonics, 25 27 Spin angular momentum, 71 72 quantum number, 28 Spin correlation, 29 31 Spin counting, 137 138 Spin Hamiltonian, 103 Spin multiplicity, 33 34, 95, 236 237 Spin relaxation, 124 Spin states density functionals and, 61 62 in nonheme diiron center, 214 216 Spin wave function, 28 Spin-allowed rule, 236 237 Spin-only effective magnetic moment, 90 92 Spin-only Hamiltonian, 103 Spin lattice relaxation, 170 Spin orbit coupling, 70, 73 75, 99 Spliceosome, 433 434 Splicing, 433 434 Split-GFP, 380 SQUID. See Super conducting quantum interference device (SQUID) SR. See Synchrotron radiation (SR) Standard hydrogen electrode (SHE), 329 State-averaged-CASSCF (SA-CASSCF), 58 Static correlation, 56 57 Stationary electrodes, diffusion-controlled voltammetry at, 339 341 Steady state voltammogram, 342 Step-down PCR. See Touchdown PCR Stepwise mechanism, 334 Stereochemistry, 172 Stern Gerlach experiment, 28 Sticky ends, 443 444, 458 459 Stokes Raman scattering, relative intensities of, 291 292 Stokes Einstein equation, 176 STOs. See Slater-type orbitals (STOs) Streakseeding, 387

Streptococcus, 462 463 Streptococcus pyogenes, 465 Strong-field approach, 48 49 Structural biology techniques cryo-electron microscopy, 406 409 data collection, 391 392 model building and refinement, 397 399 phase determination, 392 397 protein crystallization, 380 390 SAXS, 409 411 structure analysis and model quality, 400 405 X-FELs, 405 406 X-ray crystallography, 377 380 Structural genomics, 470 471 Structure factor, 393 Suicide vectors, 453 Sulfolobus acidocaldarius, 345 Super conducting quantum interference device (SQUID), 110 113 measurements in, 112 113 Super-WEFT pulse sequences, 193 195, 194f Superexchange, 101 103 Supersaturation, 384 Surface-enhanced Raman scattering (SERS), 311 312 Surface-enhanced resonance Raman scattering (SERRS) spectroscopy, 311 312 Surface-exposed redox center, 326 Symmetry, 38 41 in LFT, 46 48 Synchrotron radiation (SR), 224 sources, 392 Synchrotrons, 244, 379 beamlines, 392 Synthesis method, 449 Synthetic coordination complexes, 255 256 Synthetic iron(III) complexes with macrocyclic ligand cyclam, 217 218 Syste`me International units (SI units), 77, 78t

T T7 ribonucleic acid polymerase-T7 promoter system, 457 459 Tags for protein purification, correct folding, improved stability, 459 460 Tailing, 433 434 TAML. See Tetra amido macrocyclic ligand (TAML) Tanabe Sugano diagrams, 51 52 Taylor expansion, 288 TDDFT. See Time-dependent density functional theory (TDDFT) Technetium complexes, 13 Technetium isotope (99mTc), 13 scintigraphy, 13

Index

Temperature control, 190 191 Temperature factor. See Atomic displacement parameter Temperature-independent paramagnetism, 92 Temporal resolution of activity, 352 Test set, 403 Tetra amido macrocyclic ligand (TAML), 220 Tetramethylsilane (TMS), 114, 126 127, 167 Therapeutics, metals in, 13 15 Thermally equilibrated excited states (THEXI states), 321 Thermodynamics, 332 333 Thermofluor, 381 382 Thermus, 429 430 Thermus aquaticus, 444 446 THEXI states. See Thermally equilibrated excited states (THEXI states) Thioredoxin, 380, 459 460 Three-dimensional models, XANES simulations with, 260 261 Thymine, 420 422 Tight isomer shift correlations, 210 Time-dependent density functional theory (TDDFT), 63, 243 Time-independent Kohn Sham equation, 63 Time-independent Schrodinger equation, 282 283 Time-resolved resonance Raman spectroscopy, 321 322 Time-resolved serial crystallography experiments, 405 Time-resolved X-ray absorption spectroscopy, 244 245 TMS. See Tetramethylsilane (TMS) TMs. See Transition metals (TMs) Touchdown PCR, 447 TR2 spectroscopy. See Transient resonance Raman spectroscopy (TR2 spectroscopy) tra gene, 426 Trace elements, 2 Transactivating CRISPR RNA (tracRNA), 465 Transcription initiation, 431 434 Transcriptional regulation, 431 433 Transcriptomics, 470 Transduction, 430 Transfer RNAs (tRNAs), 422 Transformants, 444 Transformation, 429 430 Transient complex, 332 333 Transient resonance Raman spectroscopy (TR2 spectroscopy), 321 322 Transition dipole moments, 285 287 elements, 64 Transition metals (TMs), 18, 184 Transposable elements, 462 463

493

Transposons, 428 429, 462 Tn3, 5 and 7 gene, 428 429, 429f Tree drawing programs, 469 Trial wave function, 54 55 Triplet state, 33 34 Tris(hydroxymethyl)aminomethane (TRIS), 438 439 Triticum aestivum, 426 tRNAs. See Transfer RNAs (tRNAs) Trumpet plot, 348 Two-dimensional X-ray detector (2D X-ray detector), 392 Two-electron wave function, 22 23 Tyndall effect, 300 301 Type I self-splicing reaction, 433 434 Type II self-splicing reaction, 433 434

U UniProt Knowledge Database, 468 Unpaired electron(s), 184 interaction with, 184 Uranium (VI) oxyacetate (UO2(C2H3O2)2), 397 UV/visible (UV/Vis) absorption spectra, 307 light, 278

V V2C. See Valence-to-core (“VtC”) Vacuum permeability, 198 Valence contributions, 212 Valence-to-core (“VtC”), 242 Validation, 401 405 Van der Waals interactions, 63 64 Van Vleck equation, 88 90 Vanadium proteins, 140 Vapor diffusion, 383 385 Variational principle, 22 Vectorial quantity, 157 158 Vibrational spectroscopy, fundamentals of, 277 284 Vicinal couplings, 172 Virtual state, 287, 292 293 Visualization of molecular structures, 469 Vitamin B12, 9, 9f Vitrobot, 407 Vocab, 338 Voltage, 111 112 Voltammetry of adsorbed proteins, 343 351 Voltammogram, 338

W Water eliminated Fourier transform (WEFT), 193 195, 194f Wave function(s), 18 19, 25 27, 281 282, 304

494

Index

Wave function(s) (Continued) for vibrational states, 288 wave function based approaches, 53 54, 62 HF method, 55 57 post-Hartree Fock methods, 57 58 Wavepacket, 310 Wave particle duality, 18 Weak-field approach, 48 49 LF approach, 23 24 WEFT. See Water eliminated Fourier transform (WEFT) Weiss magnetic domains, 100 Western blots, 443 WHATCHECK program, 402 403 White X-radiation, 392 Wild type enzyme (WT enzyme), 359 360 Wilson’s disease, 8 9 Working electrode, 329 WT enzyme. See Wild type enzyme (WT enzyme)

X XANES. See X-ray absorption near-edge structure (XANES) XAS. See X-ray absorption spectroscopy (XAS) XC functional. See Exchange-correlation functional (XC functional) XES. See X-ray emission spectroscopy (XES) X-FELs. See X-ray free electron lasers (X-FELs) X-ray absorption near-edge structure (XANES), 230, 236 239, 264t simulations with three-dimensional models, 260 261

strengths and limitations, 264 265 X-ray absorption spectroscopy (XAS), 220, 230, 245 248 biological X-ray absorption spectroscopy example, 231 236 in biology, 230 231 time-resolved X-ray absorption spectroscopy, 244 245 X-ray emission spectroscopy (XES), 230 in biology, 230 231, 239 243 experiment, 240b X-ray free electron lasers (X-FELs), 230, 376, 405 406 X-ray(s), 378 379 beam, 377 crystallography, 376 380 x-ray induced electron diffraction, 245 248

Y Yeast artificial chromosomes (YACs), 456 457

Z Zebra fish. See Danio rerio (Zebra fish) Zeeman effect, 82 83, 158 159 anomalous, 82 83 normal, 82 Zero field splitting (ZFS), 82 83 Zero-phonon processes, 204 Zinc (Zn), 2 4, 9 10 fingers, 9 10