# Entropy and Free Energy in Structural Biology: From Thermodynamics to Statistical Mechanics to Computer Simulation 0367406926, 9780367406929

Computer simulation has become the main engine of development in statistical mechanics. In structural biology, computer

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English Pages 396  Year 2020

Cover
Half Title
Series Page
Title Page
Dedication
Preface
Acknowledgments
Author
Section I: Probability Theory
1: Probability and Its Applications
1.1 Introduction
1.2 Experimental Probability
1.3 The Sample Space Is Related to the Experiment
1.4 Elementary Probability Space
1.5 Basic Combinatorics
1.5.1 Permutations
1.5.2 Combinations
1.6 Product Probability Spaces
1.6.1 The Binomial Distribution
1.6.2 Poisson Theorem
1.7 Dependent and Independent Events
1.7.1 Bayes Formula
1.8 Discrete Probability—Summary
1.9 One-Dimensional Discrete Random Variables
1.9.1 The Cumulative Distribution Function
1.9.2 The Random Variable of the Poisson Distribution
1.10 Continuous Random Variables
1.10.1 The Normal Random Variable
1.10.2 The Uniform Random Variable
1.11 The Expectation Value
1.11.1 Examples
1.12 The Variance
1.12.1 The Variance of the Poisson Distribution
1.12.2 The Variance of the Normal Distribution
1.13 Independent and Uncorrelated Random Variables
1.13.1 Correlation
1.14 The Arithmetic Average
1.15 The Central Limit Theorem
1.16 Sampling
1.17 Stochastic Processes—Markov Chains
1.17.1 The Stationary Probabilities
1.18 The Ergodic Theorem
1.19 Autocorrelation Functions
1.19.1 Stationary Stochastic Processes
Homework for Students
References
Section II: Equilibrium Thermodynamics and Statistical Mechanics
2: Classical Thermodynamics
2.1 Introduction
2.2 Macroscopic Mechanical Systems versus Thermodynamic Systems
2.3 Equilibrium and Reversible Transformations
2.4 Ideal Gas Mechanical Work and Reversibility
2.5 The First Law of Thermodynamics
2.6 Joule’s Experiment
2.7 Entropy
2.8 The Second Law of Thermodynamics
2.8.1 Maximal Entropy in an Isolated System
2.8.2 Spontaneous Expansion of an Ideal Gas and Probability
2.8.3 Reversible and Irreversible Processes Including Work
2.9 The Third Law of Thermodynamics
2.10 Thermodynamic Potentials
2.10.1 The Gibbs Relation
2.10.2 The Entropy as the Main Potential
2.10.3 The Enthalpy
2.10.4 The Helmholtz Free Energy
2.10.5 The Gibbs Free Energy
2.10.6 The Free Energy, , H.(T,µ)
2.11 Maximal Work in Isothermal and Isobaric Transformations
2.12 Euler’s Theorem and Additional Relations for the Free Energies
2.12.1 Gibbs-Duhem Equation
2.13 Summary
Homework for Students
References
3: From Thermodynamics to Statistical Mechanics
3.1 Phase Space as a Probability Space
3.2 Derivation of the Boltzmann Probability
3.3 Statistical Mechanics Averages
3.3.1 The Average Energy
3.3.2 The Average Entropy
3.3.3 The Helmholtz Free Energy
3.4 Various Approaches for Calculating Thermodynamic Parameters
3.4.1 Thermodynamic Approach
3.4.2 Probabilistic Approach
3.5 The Helmholtz Free Energy of a Simple Fluid
Reference
4: Ideal Gas and the Harmonic Oscillator
4.1 From a Free Particle in a Box to an Ideal Gas
4.2 Properties of an Ideal Gas by the Thermodynamic Approach
4.3 The chemical potential of an Ideal Gas
4.4 Treating an Ideal Gas by the Probability Approach
4.5 The Macroscopic Harmonic Oscillator
4.6 The Microscopic Oscillator
4.6.1 Partition Function and Thermodynamic Properties
4.7 The Quantum Mechanical Oscillator
4.8 Entropy and Information in Statistical Mechanics
4.9 The Configurational Partition Function
Homework for Students
References
5: Fluctuations and the Most Probable Energy
5.1 The Variances of the Energy and the Free Energy
5.2 The Most Contributing Energy E*
5.3 Solving Problems in Statistical Mechanics
5.3.1 The Thermodynamic Approach
5.3.2 The Probabilistic Approach
5.3.3 Calculating the Most Probable Energy Term
5.3.4 The Change of Energy and Entropy with Temperature
References
6: Various Ensembles
6.1 The Microcanonical (petit) Ensemble
6.2 The Canonical (NVT) Ensemble
6.3 The Gibbs (NpT) Ensemble
6.4 The Grand Canonical (µVT) Ensemble
6.5 Averages and Variances in Different Ensembles
6.5.1 A Canonical Ensemble Solution (Maximal Term Method)
6.5.2 A Grand-Canonical Ensemble Solution
6.5.3 Fluctuations in Different Ensembles
References
7: Phase Transitions
7.1 Finite Systems versus the Thermodynamic Limit
7.2 First-Order Phase Transitions
7.3 Second-Order Phase Transitions
References
8: Ideal Polymer Chains
8.1 Models of Macromolecules
8.2 Statistical Mechanics of an Ideal Chain
8.2.1 Partition Function and Thermodynamic Averages
8.3 Entropic Forces in an One-Dimensional Ideal Chain
8.5 The Critical Exponent ν
8.6 Distribution of the End-to-End Distance
8.6.1 Entropic Forces Derived from the Gaussian Distribution
8.7 The Distribution of the End-to-End Distance Obtained from the Central Limit Theorem
8.8 Ideal Chains and the Random Walk
8.9 Ideal Chain as a Model of Reality
References
9: Chains with Excluded Volume
9.1 The Shape Exponent ν for Self-avoiding Walks
9.2 The Partition Function
9.3 Polymer Chain as a Critical System
9.4 Distribution of the End-to-End Distance
9.5 The Effect of Solvent and Temperature on the Chain Size
9.5.1 θ Chains in d = 3
9.5.2 θ Chains in d = 2
9.5.3 The Crossover Behavior Around
9.5.4 The Blob Picture
9.6 Summary
References
Section III: Topics in Non-Equilibrium Thermodynamics and Statistical Mechanics
10: Basic Simulation Techniques: Metropolis Monte Carlo and Molecular Dynamics
10.1 Introduction
10.2 Sampling the Energy and Entropy and New Notations
10.4 The Metropolis Monte Carlo Method
10.4.1 Symmetric and Asymmetric MC Procedures
10.4.2 A Grand-Canonical MC Procedure
10.5 Efficiency of Metropolis MC
10.6 Molecular Dynamics in the Microcanonical Ensemble
10.7 MD Simulations in the Canonical Ensemble
10.8 Dynamic MD Calculations
10.9 Efficiency of MD
10.9.1 Periodic Boundary Conditions and Ewald Sums
10.9.2 A Comment About MD Simulations and Entropy
References
11: Non-Equilibrium Thermodynamics—Onsager Theory
11.1 Introduction
11.2 The Local-Equilibrium Hypothesis
11.3 Entropy Production Due to Heat Flow in a Closed System
11.4 Entropy Production in an Isolated System
11.5 Extra Hypothesis: A Linear Relation Between Rates and Affinities
11.5.1 Entropy of an Ideal Linear Chain Close to Equilibrium
11.6 Fourier’s Law—A Continuum Example of Linearity
11.7 Statistical Mechanics Picture of Irreversibility
11.8 Time Reversal, Microscopic Reversibility, and the Principle of Detailed Balance
11.9 Onsager’s Reciprocal Relations
11.10 Applications
11.11 Steady States and the Principle of Minimum Entropy Production
11.12 Summary
References
12: Non-equilibrium Statistical Mechanics
12.1 Fick’s Laws for Diffusion
12.1.1 First Fick’s Law
12.1.2 Calculation of the Flux from Thermodynamic Considerations
12.1.3 The Continuity Equation
12.1.4 Second Fick’s Law—The Diffusion Equation
12.1.5 Diffusion of Particles Through a Membrane
12.1.6 Self-Diffusion
12.2 Brownian Motion: Einstein’s Derivation of the Diffusion Equation
12.3 Langevin Equation
12.3.1 The Average Velocity and the Fluctuation-Dissipation Theorem
12.3.2 Correlation Functions
12.3.3 The Displacement of a Langevin Particle
12.3.4 The Probability Distributions of the Velocity and the Displacement
12.3.5 Langevin Equation with a Charge in an Electric Field
12.3.6 Langevin Equation with an External Force—The Strong Damping Velocity
12.4 Stochastic Dynamics Simulations
12.4.1 Generating Numbers from a Gaussian Distribution by CLT
12.4.2 Stochastic Dynamics versus Molecular Dynamics
12.5 The Fokker-Planck Equation
12.6 Smoluchowski Equation
12.7 The Fokker-Planck Equation for a Full Langevin Equation with a Force
12.8 Summary of Pairs of Equations
References
13: The Master Equation
13.1 Master Equation in a Microcanonical System
13.2 Master Equation in the Canonical Ensemble
13.3 An Example from Magnetic Resonance
13.3.1 Relaxation Processes Under Various Conditions
13.3.2 Steady State and the Rate of Entropy Production
13.4 The Principle of Minimum Entropy Production—Statistical Mechanics Example
References
Section IV: Advanced Simulation Methods: Polymers and Biological Macromolecules
14: Growth Simulation Methods for Polymers
14.1 Simple Sampling of Ideal Chains
14.2 Simple Sampling of SAWs
14.3 The Enrichment Method
14.4 The Rosenbluth and Rosenbluth Method
14.5 The Scanning Method
14.5.1 The Complete Scanning Method
14.5.2 The Partial Scanning Method
14.5.3 Treating SAWs with Finite Interactions
14.5.4 A Lower Bound for the Entropy
14.5.5 A Mean-Field Parameter
14.5.6 Eliminating the Bias by Schmidt’s Procedure
14.5.7 Correlations in the Accepted Sample
14.5.8 Criteria for Efficiency
14.5.9 Locating Transition Temperatures
14.5.10 The Scanning Method versus Other Techniques
14.5.11 The Stochastic Double Scanning Method
14.5.12 Future Scanning by Monte Carlo
14.5.13 The Scanning Method for the Ising Model and Bulk Systems
14.6 The Dimerization Method
References
15: The Pivot Algorithm and Hybrid Techniques
15.1 The Pivot Algorithm—Historical Notes
15.2 Ergodicity and Efficiency
15.3 Applicability
15.4 Hybrid and Grand-Canonical Simulation Methods
15.5 Concluding Remarks
References
16: Models of Proteins
16.1 Biological Macromolecules versus Polymers
16.2 Definition of a Protein Chain
16.3 The Force Field of a Protein
16.4 Implicit Solvation Models
16.5 A Protein in an Explicit Solvent
16.6 Potential Energy Surface of a Protein
16.7 The Problem of Protein Folding
16.8 Methods for a Conformational Search
16.8.1 Local Minimization—The Steepest Descents Method
16.8.2 Monte Carlo Minimization
16.8.3 Simulated Annealing
16.9 Monte Carlo and Molecular Dynamics Applied to Proteins
16.10 Microstates and Intermediate Flexibility
16.10.1 On the Practical Definition of a Microstate
References
17: Calculation of the Entropy and the Free Energy by Thermodynamic Integration
17.1 “Calorimetric” Thermodynamic Integration
17.2 The Free Energy Perturbation Formula
17.3 The Thermodynamic Integration Formula of Kirkwood
17.4 Applications
17.4.1 Absolute Entropy of a SAW Integrated from an Ideal Chain Reference State
17.4.2 Harmonic Reference State of a Peptide
17.5 Thermodynamic Cycles
17.5.1 Other Cycles
17.5.2 Problems of TI and FEP Applied to Proteins
References
18: Direct Calculation of the Absolute Entropy and Free Energy
18.1 Absolute Free Energy from
18.2 The Harmonic Approximation
18.3 The M2 Method
18.4 The Quasi-Harmonic Approximation
18.5 The Mutual Information Expansion
18.6 The Nearest Neighbor Technique
18.7 The MIE-NN Method
18.8 Hybrid Approaches
References
19: Calculation of the Absolute Entropy from a Single Monte Carlo Sample
19.1 The Hypothetical Scanning (HS) Method for SAWs
19.1.1 An Exact HS Method
19.1.2 Approximate HS Method
19.2 The HS Monte Carlo (HSMC) Method
19.3 Upper Bounds and Exact Functionals for the Free Energy
19.3.1 The Upper Bound FB
19.3.2 FB Calculated by the Reversed Schmidt Procedure
19.3.3 A Gaussian Estimation of FB
19.3.4 Exact Expression for the Free Energy
19.3.5 The Correlation Between sA and FA
19.3.6 Entropy Results for SAWs on a Square Lattice
19.4 HS and HSMC Applied to the Ising Model
19.5 The HS and HSMC Methods for a Continuum Fluid
19.5.1 The HS Method
19.5.2 The HSMC Method
19.5.3 Results for Argon and Water
19.5.3.1 Results for Argon
19.5.3.2 Results for Water
19.6 HSMD Applied to a Peptide
19.6.1 Applications
19.7 The HSMD-TI Method
19.8 The LS Method
19.8.1 The LS Method Applied to the Ising Model
19.8.2 The LS Method Applied to a Peptide
References
20: The Potential of Mean Force, Umbrella Sampling, and Related Techniques
20.1 Umbrella Sampling
20.2 Bennett’s Acceptance Ratio
20.3 The Potential of Mean Force
20.3.1 Applications
20.4 The Self-Consistent Histogram Method
20.4.1 Free Energy from a Single Simulation
20.4.2 Multiple Simulations and The Self-Consistent Procedure
20.5 The Weighted Histogram Analysis Method
20.5.1 The Single Histogram Equations
20.5.2 The WHAM Equations
20.5.3 Enhancements of WHAM
20.5.4 The Basic MBAR Equation
20.5.5 ST-WHAM and UIM
20.5.6 Summary
References
21: Advanced Simulation Methods and Free Energy Techniques
21.1 Replica-Exchange
21.1.1 Temperature-Based REM
21.1.2 Hamiltonian-Dependent Replica Exchange
21.2 The Multicanonical Method
21.2.1 Applications
21.2.2 MUCA-Summary
21.3 The Method of Wang and Landau
21.3.1 The Wang and Landau Method-Applications
21.4 The Method of Expanded Ensembles
21.4.1 The Method of Expanded Ensembles-Applications
21.6 Methods Based on Jarzynski’s Identity
21.6.1 Jarzynski’s Identity versus Other Methods for Calculating ΔF
21.7 Summary
References
22: Simulation of the Chemical Potential
22.1 The Widom Insertion Method
22.2 The Deletion Procedure
22.3 Personage’s Method for Treating Deletion
22.4 Introduction of a Hard Sphere
22.5 The Ideal Gas Gauge Method
22.6 Calculation of the Chemical Potential of a Polymer by the Scanning Method
22.7 The Incremental Chemical Potential Method for Polymers
22.8 Calculation of µ by Thermodynamic Integration
References
23: The Absolute Free Energy of Binding
23.1 The Law of Mass Action
23.2 Chemical Potential, Fugacity, and Activity of an Ideal Gas
23.2.1 Thermodynamics
23.2.2 Canonical Ensemble
23.2.3 NpT Ensemble
23.3 Chemical Potential in Ideal Solutions: Raoult’s and Henry’s Laws
23.3.1 Raoult’s Law
23.3.2 Henry’s Law
23.4 Chemical Potential in Non-ideal Solutions
23.4.1 Solvent
23.4.2 Solute
23.5 Thermodynamic Treatment of Chemical Equilibrium
23.6 Chemical Equilibrium in Ideal Gas Mixtures: Statistical Mechanics
23.7 Pressure-Dependent Equilibrium Constant of Ideal Gas Mixtures
23.8 Protein-Ligand Binding
23.8.1 Standard Methods for Calculating .A0
23.8.2 Calculating .A0 by HSMD-TI
23.8.3 HSMD-TI Applied to the FKBP12-FK506 Complex: Equilibration
23.8.4 The Internal and External Entropies
23.8.5 TI Results for FKBP12-FK506
23.8.6 .A0 Results for FKBP12-FK506
23.9 Summary
References
Appendix
Index