Understanding the molecular mechanism of muscle contraction started with the discovery that striated muscle is composed

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*Table of contents : Front Matter ....Pages i-xv Introduction (David Aitchison Smith)....Pages 1-19 Of Sliding Filaments and Swinging Lever-Arms (David Aitchison Smith)....Pages 21-53 Actin-Myosin Biochemistry and Structure (David Aitchison Smith)....Pages 55-99 Models for Fully-Activated Muscle (David Aitchison Smith)....Pages 101-165 Transients, Stability and Oscillations (David Aitchison Smith)....Pages 167-236 Myosin Motors (David Aitchison Smith)....Pages 237-291 Models of Thin-Filament Regulation (David Aitchison Smith)....Pages 293-346 Cooperative Muscular Activation by Calcium (David Aitchison Smith)....Pages 347-373Back Matter ....Pages 375-426*

David Aitchison Smith

The SlidingFilament Theory of Muscle Contraction

The Sliding-Filament Theory of Muscle Contraction

David Aitchison Smith

The Sliding-Filament Theory of Muscle Contraction

David Aitchison Smith Department of Physiology, Anatomy and Microbiology La Trobe University Melbourne, VIC, Australia

ISBN 978-3-030-03525-9 ISBN 978-3-030-03526-6 https://doi.org/10.1007/978-3-030-03526-6

(eBook)

Library of Congress Control Number: 2019931338 © Springer Nature Switzerland AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

For Nigel, Pamela, Samuel and Audrey

Preface

Perhaps no ﬁeld of scientiﬁc endeavour has been quite like muscle contractility for bringing together people with different backgrounds and expertise and for establishing the bona ﬁdes of biophysics as a coherent synthesis of biology and physics. The somewhat mysterious mechanism of muscle action has attracted physicists for over 50 years, and many theories of muscle action have been proposed, only to fall by the wayside as they were refuted by new experimental data. Many monographs and textbooks on muscle have been published; why should yet another one see the light of day? Most monographs concentrate on experimental studies, which range widely from muscle physiology to its biochemistry and the atomic structure of the proteins. There are fewer books on the theory of muscle action, which to some extent has lagged behind the experimental advances. Why this is so forms part of the narrative of this book. It appears that a uniﬁed theory of muscle contraction at any level of activation is ﬁnally within our grasp, despite the difﬁculties of marrying the biochemical actin-myosin cycle to mechanical events at the molecular level. The theory is necessarily mathematical, and mathematical developments are presented in enough detail that they can be veriﬁed without resorting to the literature. What distinguishes a theory from a model? In the past, many models of muscle dynamics have been invented for particular purposes, primarily to interpret new experimental data and to show that the interpretation is consistent with other aspects of muscle behaviour. As the quantity and scope of new data grew, so did the variety of models constructed to accommodate them, with the result that no uniﬁed model was apparent. Any model which can accommodate the whole range of mechanochemical observations might deserve to be known as a theory, but this is not a sufﬁcient criterion. The study of muscle contraction has suffered from a surfeit of “data-driven models whose structure and parameters lack physical meaning”, to quote O’Shaughnessy and Pollard (2016). This book is an account of the search for a uniﬁed theory which embodies the true molecular mechanism of muscle action, tested against the wealth of available data.

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The presentation is suitable for biophysicists and mathematicians with a basic knowledge of graduate-level mathematics and chemical physics. No prior knowledge of muscle structure or function is assumed, but the selection of topics is tightly focussed on molecular mechanisms of contractility. This focus covers a fraction of the phenomena of interest to muscle researchers, ranging from the mechanism of excitation-contraction coupling and the regeneration of ATP by mitochondria to the growth of muscle cells, the phylogeny of muscles in mammals, vertebrates and invertebrates, cardiac muscle and the law of the heart, muscular dystrophy and cardiac myopathies, muscle fatigue and ageing, steroids, sports medicine and the treatment of injuries, and more. Melbourne, VIC, Australia June 2018

David Aitchison Smith

Acknowledgements

I am grateful to my colleagues and many people who encouraged my involvement in muscle research for the last 30 years. My interest started with the realization that mechanisms of cell motility were poorly understood and might be tested by appropriate modelling. This soon turned into a gut feeling that force generation by myosin S1 in muscle could be the driver of motile slime ﬁlaments and amoeboid cells. This insight turned out to be largely incorrect; we now know that the movements of many motile cells are driven by actin polymerization. However, I was by then committed to exploring theories of muscle action. I am very grateful to Prof. Mike Geeves for allowing me to do this during a 6 month sabbatical at the University of Bristol: my collaboration with Mike on different aspects of muscle contraction has proved very fruitful. I am also much indebted to Prof. Bob Simmons of King’s College London for inviting me to join his laboratory with John Sleep and Walter Steffen at King’s College in 1995. At that time, the group was building an optical trap system for detecting the working stroke of a single myosin molecule, and my contribution was the development of software for automatic detection of stroke events in the presence of Brownian noise. Success arrived just in time for the new millennium, and results obtained with this highly position-stabilized system showed that the tethered myosin was binding to target zones separated by 36 nm, the periodicity of the actin ﬁlament. However, optical-trap systems developed by Justin Molloy at York, David Warshaw at Vermont and Amit Mehta at Stanford all gave the same result; the working stroke of myosin S1 from a fast muscle was 5–6 nm, whereas atomic structures indicated that the length of the lever arm was 10 nm. The reason for this discrepancy is described in Chapter 6. My other areas of activity while at King’s College (1995–2002) were thinﬁlament regulation and the development of the chain model for linked tropomyosins, in collaboration with Mike Geeves and coworkers, and the perennial problem of ﬁnding a structurally-based working model of muscle contraction at the level of the half-sarcomere. All three projects were supported by project grants from the Welcome Trust. The third project continued in Australia, courtesy of an NIH-supported ix

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collaboration with Dr. Srba Mijailovich which led, inter alia, to two papers in 2008 entitled “Towards a Uniﬁed Theory of Muscle Contraction”. The title proved prophetic; in the words of A. V. Hill, “the lifetime of theories in the area of muscle contraction is generally rather short”. A major change to the 2008 model was the belated recognition that the structuralists were correct in insisting that the myosin working stroke is triggered by the release of phosphate from its active site. Doubtless, more changes will be incorporated as muscle modelling is further reﬁned. More recently, my collaboration with Prof. George Stephenson at La Trobe University, Melbourne, led to papers on the mechanism of spontaneous oscillatory contractions at low calcium and the stability of the ﬁlament lattice in striated muscle. I am grateful to Mike Geeves and John Sleep for discussions and advice over many years and for helpful discussions with Kevin Burton, Gerald Elliott, Gerald Offer, David Morgan, K. W. Ranatunga and many others at Biophysical Society meetings and elsewhere. I also wish to thank Mike Geeves, John Sleep and Gerald Offer for their comments on the manuscript. The mysterious aspect of muscular contractility is why it has taken so long to build a working model from our current understanding of the mechanism, even with the exquisitely detailed information provided by atomic structures of the proteins. The molecular mechanism (or mechanisms?) of contractility is still a subject of abiding interest, particularly now that 35 different kinds of myosin motor have been identiﬁed. Most of them have roles to play in non-muscle cells.

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Historical Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 The Sliding Filament Model . . . . . . . . . . . . . . . . . . . . 1.1.2 New Experimental Techniques . . . . . . . . . . . . . . . . . . 1.1.3 Models of Contractility . . . . . . . . . . . . . . . . . . . . . . . . 1.2 A Short Guide to Contractile Behaviour . . . . . . . . . . . . . . . . . 1.3 The Structure of Skeletal Muscle . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Muscle Ultrastructure . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 1 2 4 4 6 9 11 17

2

Of Sliding Filaments and Swinging Lever-Arms . . . . . . . . . . . . . . 2.1 Contractile Empiricism: Hill’s Equations . . . . . . . . . . . . . . . . 2.2 How Myosin Heads Find Actin Sites . . . . . . . . . . . . . . . . . . . 2.2.1 Head-Site Matching for Vernier Models . . . . . . . . . . . 2.2.2 Lattice Models: Target Zones, Layer Lines and Azimuthal Matching . . . . . . . . . . . . . . . . . . . . . . 2.3 The First Sliding-Filament Model . . . . . . . . . . . . . . . . . . . . . . 2.4 The Swinging-Lever-Arm Mechanism . . . . . . . . . . . . . . . . . . 2.4.1 Mechanokinetics of the Working Stroke . . . . . . . . . . . 2.4.2 Theory of the Rapid Length-Step Response . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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21 21 28 31

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32 33 41 44 46 52

Actin-Myosin Biochemistry and Structure . . . . . . . . . . . . . . . . . . . 3.1 How Myosin and Actin Hydrolyze ATP . . . . . . . . . . . . . . . . . 3.1.1 Myosin is an ATPase . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Actomyosin is a Better ATPase . . . . . . . . . . . . . . . . . . 3.1.3 Steady-State ATP Hydrolysis by Actin-Myosin . . . . . . 3.2 The Biochemical Contraction Cycle . . . . . . . . . . . . . . . . . . . . 3.2.1 Actin Binding Versus Nucleotide Binding . . . . . . . . . . 3.2.2 A Biochemical Cycle for Myosin-S1 . . . . . . . . . . . . . .

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55 55 56 60 64 69 69 70

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3.2.3 Evidence for Two A.M.ADP States . . . . . . . . . . . . . . . 3.2.4 Evidence for Two M.ATP States . . . . . . . . . . . . . . . . . 3.3 Coordinating Lever-Arm Movements with Biochemical Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 What Biochemical Event Triggers the Working Stroke? . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 The Location of the Repriming Stroke . . . . . . . . . . . . . 3.3.3 An Amalgated Mechanochemical Cycle . . . . . . . . . . . . 3.4 The Atomic Structure of Myosin Complexes . . . . . . . . . . . . . . 3.4.1 Actin Binding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Phosphate Release and the Working Stroke . . . . . . . . . 3.4.3 An ADP-Release Stroke . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 ATP Binding and Actin Afﬁnity . . . . . . . . . . . . . . . . . 3.4.5 The Repriming Stroke and Hydrolysis . . . . . . . . . . . . . 3.4.6 Hydrolysis on Actomyosin? . . . . . . . . . . . . . . . . . . . . 3.4.7 The Pathway of the Stroke . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

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Models for Fully-Activated Muscle . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Strain-Dependent Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Kramers’ Method for Reaction Rates . . . . . . . . . . . . . . . 4.1.2 Actin Binding: Swing, Roll and Lock . . . . . . . . . . . . . . 4.1.3 The Kinetics of the Working Stroke . . . . . . . . . . . . . . . 4.1.4 An ADP-Release Stroke . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Evolution of Contraction Models . . . . . . . . . . . . . . . . . . . . 4.2.1 A Two-State Stroking Model . . . . . . . . . . . . . . . . . . . . 4.2.2 The Search for a Simple Vernier Model . . . . . . . . . . . . . 4.2.3 Lattice Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Probabilistic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Monte-Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . 4.4 The Effects of Filament Elasticity . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 The Equivalent Lumped Filament Compliance . . . . . . . . 4.4.2 Experimental Consequences . . . . . . . . . . . . . . . . . . . . . 4.5 Target Zones, Dimeric Myosins and Buckling Rods . . . . . . . . . 4.5.1 Calculations with Target Zones and Dimeric Myosins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 An Updated 5-State Vernier Model . . . . . . . . . . . . . . . . 4.5.3 Buckling Rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Adding Phosphate, ADP or ATP . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Added Phosphate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Changing ADP or ATP . . . . . . . . . . . . . . . . . . . . . . . . 4.7 The Effects of Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101 102 103 104 108 109 111 113 120 126 127 128 131 132 133 135 138 138 141 143 146 147 149 153 159

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6

Transients, Stability and Oscillations . . . . . . . . . . . . . . . . . . . . . . 5.1 Chemical Jumps and Temperature Jumps . . . . . . . . . . . . . . . . 5.1.1 The Activation Jump . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Pi Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 ATP Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Temperature Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Length Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 The Length-Step Response . . . . . . . . . . . . . . . . . . . . . 5.2.2 Repeated Length Steps . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Sinusoidal Length Changes . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Force Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Isotonic Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 A Simple Quantitative Theory of Isotonic Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Ramp Shortening and Lengthening . . . . . . . . . . . . . . . . . . . . . 5.5.1 Ramp Shortening . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Ramp Lengthening . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Wing-Beat Oscillations in Insect Flight Muscle . . . . . . . . . . . . 5.7 The Longitudinal Stability of the Sarcomere . . . . . . . . . . . . . . 5.7.1 Tension Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.2 A-Band Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.3 Residual Force Enhancement . . . . . . . . . . . . . . . . . . . 5.8 The Stability of the Filament Lattice . . . . . . . . . . . . . . . . . . . . 5.8.1 Electrostatic Models of the Relaxed Filament Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.2 An Electromechanical Model for Long Sarcomeres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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167 167 168 169 171 173 174 174 180 182 187 189

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Myosin Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Single-Myosin Experiments with Optical Trapping . . . . . . . . . . 6.1.1 Detecting Events in the Presence of Noise . . . . . . . . . . . 6.1.2 Observing Target Zones with Optical Trapping . . . . . . . 6.2 Actomyosin Kinetics in the Optical Trap . . . . . . . . . . . . . . . . . 6.2.1 Monte-Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 The Generalized Smoluchowski Equations . . . . . . . . . . . 6.3 Rigor Bonds, Buckling Rods and Cy3-ATP . . . . . . . . . . . . . . . 6.3.1 Rigor Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Rigor Lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Buckling Rods on a Myosin Coﬁlament . . . . . . . . . . . . . 6.3.4 Coordinating Myosin Detachments with ATP Binding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Force-Clamp Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Motility Assays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 The Glass-Microneedle Experiment . . . . . . . . . . . . . . . . . . . . .

237 237 242 244 247 247 251 256 257 258 260 262 263 268 273

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6.7

Processive Myosin Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.1 The Myosin Superfamily . . . . . . . . . . . . . . . . . . . . . . 6.7.2 What Makes a Processive Motor? . . . . . . . . . . . . . . . . 6.7.3 The Mechanokinetics of Myosin-V Processivity . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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277 277 277 278 287

Models of Thin-Filament Regulation . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Steric Blocking Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 The Simplest Steric Blocking Model . . . . . . . . . . . . . . . 7.1.2 The Rate of Myosin Binding is Also Regulated . . . . . . . 7.1.3 Closed-Open Models . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.4 A Blocked-Closed-Open Model . . . . . . . . . . . . . . . . . . 7.2 How Is Thin-Filament Regulation Controlled by Calcium? . . . . 7.3 An On-Off Model with Tropomyosin Interactions . . . . . . . . . . . 7.3.1 The Grand Partition Function for a Single Cooperative Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 A Model with End-to-End Tropomyosin Interactions . . . 7.4 Tropomyosins as a Continuous Flexible Chain . . . . . . . . . . . . . 7.4.1 The Size of the Cooperative Unit . . . . . . . . . . . . . . . . . 7.4.2 Structural Evidence for Three Regulatory States . . . . . . . 7.4.3 A Continuous-Flexible-Chain Model . . . . . . . . . . . . . . . 7.5 Mathematical Formulation of the Chain Model . . . . . . . . . . . . . 7.5.1 The Energy of a Conﬁned Flexible Chain . . . . . . . . . . . 7.5.2 The Ground-State Energy . . . . . . . . . . . . . . . . . . . . . . . 7.5.3 Ground-State Energy of a Pinned Chain . . . . . . . . . . . . 7.6 The Distribution of Thermally-Activated Chain Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Energetics and Kinetics of Myosin Binding . . . . . . . . . . . . . . . 7.8 Solution Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.1 The Extent of Myosin Binding . . . . . . . . . . . . . . . . . . . 7.8.2 Kinetic Regulation of Myosin Binding . . . . . . . . . . . . . 7.8.3 Calcium Binding to TnC . . . . . . . . . . . . . . . . . . . . . . . . 7.8.4 Thin-Filament Regulation of ATPase Rates . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

293 297 297 299 300 301 303 305

Cooperative Muscular Activation by Calcium . . . . . . . . . . . . . . . . 8.1 Observations of Cooperative Regulation . . . . . . . . . . . . . . . . . 8.1.1 Steady-State Calcium Regulation . . . . . . . . . . . . . . . . 8.1.2 Kinetic Aspects of Thin-Filament Regulation . . . . . . . . 8.1.3 Regulation on the Descending Limb . . . . . . . . . . . . . . 8.2 A Muscle Model with Thin-Filament Regulation . . . . . . . . . . . 8.2.1 A Minimal Demonstration Model . . . . . . . . . . . . . . . .

347 347 348 351 354 357 359

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305 308 311 312 314 316 319 319 320 322 324 328 332 333 334 336 338 344

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8.3 Spontaneous Oscillatory Contractions . . . . . . . . . . . . . . . . . . . . 363 8.4 Direct regulation of Myosin Contractility . . . . . . . . . . . . . . . . . 368 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

Chapter 1

Introduction

There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy. William Shakespeare, Hamlet, Act I, scene 5.

1.1

Historical Perspectives

Muscular contraction is a unique method of movement and locomotion common to all higher life forms, from primitive ﬁshes to invertebrates, mammals and mankind. Although the Greek philosophers speculated at length about the nature of matter and living things, the ﬁrst systematic attempt to understand muscle action seems to be due to the Roman physician Galen, circa 300 A.D. Galen systematically dissected all kinds of muscle, and understood that muscles exert a tensile force: his treatise “De Motu Musculorum” (Goss 1968) includes a discussion of muscle tone and its relation to blood supply. In Renaissance times, Leonardo da Vinci catalogued human anatomy and attempted to push the boundaries of the human musculoskeletal system by constructing a winged ﬂying machine (the “ornithopter”). By this time it was known that muscles were composed of ﬁbres. In 1664, William Croone published a short treatise “On the Reason of the Movement of Muscles” (Maquet et al. 2000) which attempted a mechanical explanation of contraction. Croone thought that because the volume of an intact muscle stays constant as it is activated, the axial contraction could be explained in terms of inﬂatable bladders. In 1682, Leeuwenhoek wrote that the ﬂeshy ﬁbres were composed of globules; this seems to be the ﬁrst observation of the transverse striations of vertebrate muscle. In 1791, Galvani made a frog’s legs twitch by passing a current generated by an electric battery (the newly-discovered Voltaic pile), thus discovering what he thought was “animal electricity.” Volta disagreed, thinking correctly that the electricity was generated by his pile. In 1902, Veratti observed in the optical microscope that muscles contained a delicate network of longitudinal and transverse ﬁlaments, the latter being the T-tubule system (Martonosi 2000), and ideas of muscle action formed around the hypothesis that muscles contract because the ﬁlaments shrink. © Springer Nature Switzerland AG 2018 D. A. Smith, The Sliding-Filament Theory of Muscle Contraction, https://doi.org/10.1007/978-3-030-03526-6_1

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2

1 Introduction

In the ﬁrst half of the twentieth century, biochemists began to explore the chemical events accompanying contraction. Exhaustive stimulation of a muscle caused an increase in the free concentration of phosphate ions (PO4) and lactic acid, leading to the nostrum that muscle fatigue was caused by an excess of lactic acid. The muscle proteins myosin and actin which form the ﬁlaments were isolated. The co-enzyme ATP (adenosine triphosphate) was identiﬁed, and contraction was shown to involve the hydrolysis of ATP in which the terminal phosphate group is split off by an OH ion. The Lohmann reaction, by which ATP is regenerated by phosphocreatine, was discovered and shown to occur in vivo. The discovery of ‘superprecipitation’ by Szent-Gyorgi in 1943 gave a strong clue to the action of these ingredients: when solutions of myosin and actin were mixed together, an insoluble actomyosin precipitate was formed, and the addition of ATP caused the precipitate to dissolve, indicating that ATP breaks myosin-actin bonds. The monograph of Dorothy Needham (1971) gives a full account of the early history of muscle structure, function and chemistry. In 1935, a model for shortening muscle was proposed by A.V. Hill, in which a contractile element with viscoelastic properties was coupled to a series elastic component (SEC); although empirical in nature, this model had the virtue of being quantitative and therefore testable. In fact, the contractile element (myosin) works as a spring whose resting length is reset by a conﬁgurational change. It was natural to investigate the SEC by subjecting a muscle to a sudden length change and observing the subsequent behaviour of the tension. Much later on, the same length-step protocol was used to devastating effect in unmasking the mechanism of contraction, but Hill’s length steps were not as fast as the mechanism under investigation and his results were consequently misleading (Hill 1953). The hypothetical series elastic component was subsequently abandoned, and it is ironic that it resurfaced when the distributed elastic compliance of muscle ﬁlaments was observed with X-ray diffraction in 1994. Nevertheless, A.V. Hill’s lasting contribution to muscle science is his equation for the steady-state relation between tension and shortening velocity, viewed as a statement of energy conservation and supported by a combination of mechanical and heat measurements. Hill’s equation is described in the next chapter.

1.1.1

The Sliding Filament Model

The modern approach to contractility started in 1953 when the ﬁlament structure of vertebrate muscle was observed in the optical microscope, showing two arrays of interdigitating ﬁlaments which slide into each other rather than shrink (Fig. 1.1). This discovery was made independently by Huxley and Niedegerke (1954) and Huxley and Hanson (1954), and the sliding-ﬁlament model survives to this day as the template on which all models of muscle contraction in striated muscle must build. Major experimental advances since then include the following:

1.1 Historical Perspectives

3

Fig. 1.1 The longitudinal structure of vertebrate muscle, showing the striations which deﬁne sarcomeres and interdigitating ﬁlaments in each sarcomere (Huxley 2004). With permission of John Wiley and Sons

1. X-ray determinations of muscle ultrastructure by Hugh Huxley, Gerald Elliott and John Squire; the lattice structure of striated muscle, the periodicities of myosin dimers on thick ﬁlaments and actin monomers on the thin ﬁlament, which is a double helix. Huxley’s contribution is described in an obituary (Hitchcock-DeGregori and Irving 2014). See also Elliott et al. (1965), Squire (1981). 2. The tension-length curve of striated muscle and its interpretation: myosin heads are independent force generators (Gordon et al. 1966) 3. The biochemistry of the interactions between actin, myosin and ATP (Lymn and Taylor 1971), following the isolation of myosin and actin from thick and thin ﬁlaments. 4. Mechanical evidence that myosin-S1 on actin generates force and movement by making a working stroke (Huxley and Simmons 1971). 5. The atomic structures of actin (Kabsch et al. 1990) and myosin (Rayment et al. 1993), followed by conﬁrmation of the myosin working stroke from atomic structures of myosin-S1 (M), M.ATP, and M.ADP with various phosphate analogues. 6. The mechanism of thin-ﬁlament regulation by calcium ions, starting from X-ray studies leading to the steric blocking model (Haselgrove and Huxley 1972), and culminating in cryo-EM determinations of the locations of troponin, tropomyosin on ﬁlamentary actin under different conditions (Vibert et al. 1997). The atomic structure of troponin-C (Herzberg and James 1988; Sundaralingam et al. 1985) with and without calcium, and of the core of the troponin complex TnT-TnC-TnI (Takeda et al. 2003).

4

1.1.2

1 Introduction

New Experimental Techniques

Prompted by the initial breakthrough in 1953, research on the mechanism of muscle action became highly active, spawning many new experimental techniques and several technological breakthroughs. After building models of muscle action, it is necessary to test them against the outcomes of experiments made with the new techniques. Here is a sample: (a) The use of ﬂuorescent labelling, and spin labelling with electron spin resonance (Fajer et al. 1998), to probe molecular enviroments and dynamic changes in real time. FRET (ﬂuorescent exchange transfer) has also been widely used to measure distances of up to 10 nm between excimer states on labelled atoms, for example by Suzuki et al. (1998). (b) The motility assay. Following the visualization by Yanagida of single ﬂuorescent actin ﬁlaments, Kron and Spudich (1986) observed their motion over a lawn of molecules of heavymeromyosin ﬁxed to a glass surface. (c) Single-molecule mechanics, with a single myosin motor interacting with an optically-tethered actin ﬁlament (Finer et al. 1994). Subsequent variations include the use of evanescent-wave spectroscopy (Funatsu et al. 1995), the use of quantum dots as markers of local ﬁlament movements, and a cantilevered myosin walking on actin ﬁlaments (Kitamura et al. 1999). (d) Ultrastructure in X-ray diffraction; observation of fringe patterns and their interpretation as interference between diffracted beams from the two halves of the same sarcomere (Linari et al. 2000). The authors claimed that these patterns can be used to detect the myosin working stroke. (e) The atomic structure of chicken-gizzard myosin-S1 (Rayment et al. 1993) was obtained only after a chemical trick (methylation of lysine residues), amid rumours of a shortage of chickens in Wisconsin.

1.1.3

Models of Contractility

Theoretical contributions to the mechanism of contraction in striated muscle started with A.F. Huxley (1957), who proposed a highly simpliﬁed mechanochemical model based on sliding ﬁlaments and cyclic attachment of myosin-S1 heads to actin. Huxley’s model has attracted very wide acceptance from muscle physiologists, probably because of the lack of simple competitors, but the underlying mechanism is hidden. The accepted mechanism for force generation in muscle is related to the atomic structure of myosin-S1: myosin has a lever-arm which swings to make a different angle with the globular part bound to actin, creating a working stroke. The swinging-lever-arm mechanism has been challenged by models which postulate quite different mechanisms for generating force and movement. These are as follows:

1.1 Historical Perspectives

5

1. Iwazumi (1970) proposed a model in which actin ﬁlaments are drawn into the lattice of myoﬁlaments by electrostatic attraction. This model does not require that myosin-S1 heads bind to actin. 2. A water-jet model in which water expelled from myosin by hydrolysis of ATP was directed to cause contraction (Oplatka 1997). Apart from the difﬁculty of generating sufﬁcient axial momentum transfer, this model can be discounted because Lymn and Taylor showed that ATP hydrolysis takes place when myosin is not bound to actin. 3. The impulse model of Elliott and Worthington (2001), in which tension was created when ATP binds to the actomyosin complex. This model can be dismissed for the same reason; ATP rapidly dissociates myosin from actin after force was generated, a statement conﬁrmed by single-myosin optical trap data. 4. Soliton waves have been postulated as a means of contraction (Yomosa 1985). The proposed mechanism posits a coupling between ATP hydrolysis and vibrational excitation in the alpha-helical rod linking myosin heads to their myoﬁlament backbone. It is claimed that this vibration can create a solitary wave which bends the backbone, causing myosin heads to graze the actin ﬁlament. 5. Ratchet models (Astumian 1997). These alternatives are mostly of historical interest (Ingels 1979), but ratchet models are alive and well. They postulate that a motor molecule (such as myosin-S1) sees its complementary ﬁlament (actin) as a periodic sequence of asymmetric potential wells. Brownian motion in such a structure is not unidirectional, but if a conformational change in the motor (such as the hydrolysis of ATP) changes the bias of each well, then unidirectional walking motion is possible. In a sense, this mechanism is not just a fantasy created by theoretical physicists; there are many kinds of molecular motor which are processive, meaning that they make large numbers of walking steps on their coﬁlaments. Examples include kinesin on microtubules and various DNA walkers. That stepping requires a conformational change in the motor or coﬁlament is well understood, and synthetic walking motors, including light-driven motors (Credi et al. 2014), have been created. In Chap. 6, ratchet models are discussed in the context of an experimental tourde-force from the laboratory of Toshio Yanagida, in which a single myosin molecule walks for several steps along a bundle of actin ﬁlaments (Kitamura et al. 1999). It now appears that the small number of steps seen in this experiment is a consequence of the geometry of binding sites on the actin double-helix, regardless of whether myosin makes a working stroke on binding to actin. Finally, there are ‘black-box’ models of contraction, which aim to derive Hill’s equation from a macroscopic model. The model of Baker and Thomas (2000) is of this kind; muscle is treated as a single entity operating near equilibrium and working against a ﬁxed load. The argument is essentially thermodynamic, and does not provide a systematic way of including the deviations required to set up steadystate ATP hydrolysis. However, their novel approach suggests that a macroscopic description in terms of linear irreversible thermodynamics is possible.

6

1 Introduction

Unavoidably, this survey has used various technical terms and concepts which, for the sake of brevity, have not been deﬁned. These deﬁciencies are remedied as we proceed.

1.2

A Short Guide to Contractile Behaviour

Any theory of muscle contraction is motivated, and tested, by observations of the behaviour of activated muscle. There are striated muscles, which are composed of parallel ﬁbres and show repeated striations in the optical microscope, and smooth muscles, which are spindle-shaped with ﬁlaments arranged obliquely. Skeletal muscles are the striated muscles which link the bones of vertebrates through tendons and ligaments, and are under the control of the central nervous system. Cardiac muscle which drives the heart is also striated, but has a different structure and a different tension-length curve. Muscles vary in their speed of response to stimulation. Striated muscles may be fast or slow, because individual ﬁbres may have different myosin isoforms. Slowtwitch ﬁbres (type-I myosin) maintain tension and and appear red because of high myoglobin content, while fast-twitch ﬁbres are of type-IIA, which appear pink and have an oxidative metabolism, or type-IIB, which appear white, have a glycolytic metabolism and fatigue easily. Slow ﬁbres hold tension, while fast ﬁbres are required for movement and locomotion. The fastest ﬁbres are found in the swimbladder muscle of the toadﬁsh (Rome 2006), the throat muscles of two songbirds (the European starling and the zebra ﬁnch), and the wing-beat muscles which propel ﬂying insects such as the gnat. Slow muscles are better at holding structures in place, and there is a trade-off between efﬁciency and the speed of unloaded contraction. Smooth muscles are there to hold tension, and are present in the walls of blood vessels, where they serve to regulate blood ﬂow. In invertebrates, slow muscles are the norm, but they are not universal. In bivalves such as clams, the adductor muscle is composed of smooth muscle tissue which holds the valves shut (‘catch muscle’) and striated muscle which can open the valves. For more details, see Carlson and Wilkie (1974), Squire (1981), MacIntosh et al. (2006) or Ruegg (2017). In this book, only striated muscles are considered. How does a fast striated muscle behave when it is stimulated by nerve impulses? This question can be answered in different ways, depending on what is observed and at what scale. One can begin by treating a muscle as a black box supplied with nerves and blood vessels. In the next section, we look at what lies inside the box. Consider a fast striated muscle innervated by nerve axons. When axons deliver a single action potential (which lasts for a few milliseconds), the muscle is observed to twitch; it shortens momentarily unless it is held in place by an external load which prevents shortening. In that case the tension rises over about 50 ms before decaying back to zero; the time course of muscle tension can be measured with a transducer which senses the small length change induced within. The twitch lasts about 0.2 s.

1.2 A Short Guide to Contractile Behaviour

7

As this book is primarily concerned with the mechanism of contraction, the way in which an intact muscle responds to nervous stimulation is peripheral to the main inquiry. However, it is worth noting that individual ﬁbres within the muscle are generally not all activated at the same time. The muscle is innervated at multiple points along its length, each of which is under the control of efferent nerves from the central nervous system. The region of ﬁbre that can be activated by just one such innervation deﬁnes a motor unit. Efferent nerve axons innervate different parts of the muscle, and more motor units will be recruited as the number of active nerve axons transmitting action potentials, or the frequency of action potentials in each axon, is increased, so the net tension is a smoothly graded function of the overall frequency of nervous stimulation. On top of that, afferent nerve axons sense the level of activation and relay these signals back to the central nervous system, creating a feedback loop for exquisite control of tension and movement (Katz 1966). It is desirable to study muscular contraction in isolation from its nervous inputs and outputs, and this is often done by extracting individual ﬁbers. If a string of action potentials is delivered with a frequency above 100 Hz, the potential spikes merge to give a tetanus, and the unloaded muscle remains in a state of full activation while shortening at a speed which depends on the type of muscle. For fast muscles, the unloaded shortening speed v0 is typically about 2 muscle lengths/second; for slow muscles the speed of shortening may be 4–5 times smaller. With a transducer attached, the tension rises to a steady level which is the isometric tension T0. For undamaged frog sartorius muscle at 3 C or rabbit psoas muscle at 12–15 C, isometric tension is typically 2–3 105 Pascals (Newtons/m2). However, the cross-section is not uniform; a muscle is generally thinner at the ends where it is attached to tendons. Measurements were made on intact single ﬁbres which were electrically stimulated, or skinned ﬁbres activated by calcium ions in Ringer’s solution. If a muscle is tetanised and forced to shorten at a ﬁxed velocity v, the tension quickly adjusts to a steady value T(v) < T0 until shortening stops, for structural reasons if no other. Conversely, if the muscle is attached to an external load F < T0, it will shorten at a speed v(F) which should be the solution of the equation T(v) ¼ F. However, muscle stiffness, measured by applying a small high-frequency length oscillation during shortening, also falls with shortening velocity but remains ﬁnite when the tension has dropped to zero. For frog muscle, this behaviour is illustrated in Fig. 1.2. The rate of ATP consumption increases with the speed of shortening, and can rise to a factor of 3–6 times the isometric rate. Somewhat opposite behaviour is observed in stretching. When striated muscle is stretched at constant speed, tension rises and the rate of ATP turnover falls to a very low level. A very limited amount of stretch (about 5%) is possible before the ﬁlaments move out of overlap, but by then the tension may have reached a steadystate value in excess of T0. Thus stretching muscle acts a brake. The steady-state tension rises with the speed of stretching and saturates at somewhere between 1.2T0 and 2T0, depending on conditions (Lombardi and Piazzesi 1992). If the tensile load is increased above this saturation value, the muscle cannot hold and stretches rapidly. Figure 1.3 shows steady-state tensions for shortening and stretching.

8

1 Introduction

Fig. 1.2 The shortening behaviour of frog muscle at 0–3 C: (a) Shortening velocity as a function of load (Edman 1979). With permission of John Wiley and Sons Inc. and K.A.P. Edman. (b) The high-frequency stiffness as a function of load (Piazzesi et al. 2007), normalised to isometric tension T0 and isometric stiffness S0. In their experiments, the averaged values were T0 ¼ 240 kPa and S0 ¼ 50.3 kPa/nm. Note that 1 N/mm2 ¼ 1000 kPa. With permission of Elsevier Press and G. Piazzesi Fig. 1.3 Steady-state tension versus velocity for shortening and lengthening frog muscle on the plateau of the tension-length curve (Lombardi and Piazzesi 1992). Note the apparent change in slope at zero velocity. With permission, Cambridge University Press

This discussion shows that activated muscle is almost never in thermodynamic equilibrium, even when no mechanical work is generated. The maintenance of isometric tension is accompanied by steady heat production, which is a signature of the consumption of chemical energy. In fact the energy comes from the hydrolysis of adenosine triphosphate (ATP), which in turn is generated by glycogen from

1.3 The Structure of Skeletal Muscle

9

carbohydrates. Viscous damping ensures that there is mechanical equilibrium of internal forces, but there is no chemical equilibrium. Even in an isometric tetanus, ATP hydrolysis dissipates energy at a constant rate, staying close to a steady state for a period of about 1 s, when the products of hydrolysis (ADP and inorganic phosphate) start to build up. Many mechanisms, including a build-up of lactic acid, have been proposed to explain muscle fatigue after repeated stimulation, but it now seems that the main culprit is inorganic phosphate rather than the reduction in pH (Westerblad et al. 2002). Equilibrium conditions are approached during rapid stretching, when ATP consumption is minimal, and when the ATP supply is cut off, after which rigor mortis sets in. Some tension remains in rigor, which is best likened to a glassy state in which some non-equilibrium conditions are frozen in.

1.3

The Structure of Skeletal Muscle

Skeletal muscles in vertebrates are a variety of striated muscle under the control of the nervous system. The most important structural feature is the presence of periodically repeating striations, spaced by about 2 μm, which span the cross-section. The region between adjacent striations deﬁnes the sarcomere, which can (somewhat simplistically) be considered as a one-dimensional unit cell for contractile behaviour. However, the three-dimensional structure of muscle has several layers of complexity, including the transverse tubules which distribute the action potentials arriving at nerve terminals and convert them into calcium ions, through a mechanism known as excitation-contraction coupling. Calcium ions are responsible for switching on the molecular mechanism of contraction, and the way in which this occurs is the subject of Chap. 7. A top-down description of muscle structure will be familiar to many readers, so the following presentation is intended as an outline of what has been studied in depth. Skeletal muscle is made up of bundles of cylindrical ﬁbres (Fig. 1.4), enclosed in a sheath (fascia) of connective tissue (collagen). Biologists identify each ﬁbre as a multi-nucleated cell, called a myocyte, and the membranous wall of the cell is called the sarcolemma. The ﬁbres are mainly aligned parallel to each other, although not necessarily along the axis of the muscle. However, spaces must be left for other structures, notably mitochondrial cells, fed by internal blood vessels, which maintain the supply of ATP. An individual ﬁbre may be as small as 0.2 mm in diameter, and can with a little practice be dissected from the muscle. At each striation the transverse-tubule system invaginates the membrane and conducts depolarising axon potentials to terminal cisternae, where they are converted to calcium ions by the ryanodine receptor, which implements excitation-contraction coupling. Calcium ions are then fed to the sarcoplasmic reticulum, a network of longitudinal tubes within each striation (Peachey 1965; Ruegg 2017). The whole structure is designed to convert action potentials efﬁciently and quickly to calcium ions and distribute them throughout the interior of the ﬁbre (the sarcoplasm), without having to diffuse

10

1 Introduction

Fig. 1.4 Ultrastructure of a striated muscle ﬁbre (diameter ~ 1 mm) and its sarcolemma membrane, with myoﬁbrils (diameter ~ 1 μm), and the transverse tubule network. (Blausen 0801 Skeletal Muscle.png, from www.teachmeanatomy.info)

over long distances. After stimulation ceases, an ionic pump removes calcium ions to the reticulum so that the muscle can relax. Quid pro quo, each ﬁbre is composed of parallel bundles of myoﬁbrils. The diameter of a myoﬁbril is typically about 1 μm, and individual myoﬁbrils can be isolated with a mechanical blender. Finally, each myoﬁbril is made up of parallel myoﬁlaments, arranged over the cross-section in a regular two-dimensional lattice. The lattice spacing is typically between 40 and 50 nm. Looking along a single ﬁbre reveals the periodic striations and the structure of each sarcomere (Fig. 1.1). Each striation appears as a dark region, known as the Z-line. Each sarcomere is seen to contain bundles of thick ﬁlaments in the central region, called the A-band, and interdigitating thin ﬁlaments centred on the Z-lines. Thick ﬁlaments do not extend to the Z-line, and the intervening spaces which contain only thin ﬁlaments are the I-bands, which appear much lighter in an optical microscope with phase contrast. There is also a small lighter region, the bare zone, in the centre of each sarcomere. Extraction and analysis revealed that the thick ﬁlaments are made of myosin and the thin ﬁlaments are of actin. In each half of every sarcomere, thin ﬁlaments are attached to their Z-line, a polymeric trellis which locates them in the cross-section of each myoﬁbril and deﬁnes the lattice structure.

1.3 The Structure of Skeletal Muscle

11

Over the cross-section of the whole ﬁbre, the Z-lines may be staggered, but the Z-line is normally continuous over each myoﬁbril unless the ﬁbre has been damaged. The pivotal discovery made simultaneously by Hugh Huxley and Andrew Huxley in 1953 was that the thick and thin ﬁlaments slide against each other as the length of the muscle, and hence the length of each sarcomere, is changed. As the muscle was shortened, the width of the I-bands in every sarcomere diminished but the width of each central region (the A-band) stayed constant. At a stroke, this ‘Eureka moment’ put paid to a century of speculation based on the hypothesis of shrinking ﬁlaments, not to mention centuries-old notions based on blood ﬂow and the swelling of imaginary bladders. From then on, it looked as if the ﬁlaments could be regarded as rigid structures. A new era of muscle research then began in earnest, with X-ray diffraction employed by Hugh Huxley, Gerald Elliott, John Squire and others to establish the structures of the myosin and actin ﬁlament lattices. The information required was the crystal structure of the two-dimensional lattices, the lattice spacings, and the axial periodicities and structures of the ﬁlaments. The history of this period is well documented by Hitchcock-DeGregori and Irving (2014). Meanwhile, a decade passed before the connection between sliding ﬁlaments and the isometric tension-length curve was made by Gordon et al. (1966). For a single ﬁbre of frog sartorius muscle, the isometric tension stayed constant on a plateau for sarcomere lengths of 2.05–2.25 μm, dropping linearly as length was increased from 2.20 to zero tension at 3.65 μm (Fig. 1.5). Tension also decreased at lengths below 2.05 μm, slowly at ﬁrst and then more steeply for lengths below 1.65 μm. As the ﬁlaments slide against each other, this information enables us to deduce the lengths of the thick and thin ﬁlaments (1.60 μm and 2.05 μm) and the width of the bare zone (0.2 μm). Thus maximum tension was achieved when thin ﬁlaments completely overlapped the two non-bare sections of each thick ﬁlament, and tension on the descending limb was proportional to overlap length despite the concomitant changes in lattice spacing. As is now well-known, sarcomeric tension is created by individual force-generating units (myosin crossbridges) periodically distributed along the non-bare portions of thick ﬁlaments, so that tension is proportional to the number of force-generating units in overlap with the thin ﬁament. Figure 1.5 also shows that the decrease in tension at sarcomere lengths below 2.05 μm is due to actin ﬁlaments poking into the other half of the sarcomere.

1.3.1

Muscle Ultrastructure

Looking inside the sarcomeres of striated muscle shows a rich structure associated with myosin and actin ﬁlaments. Some key results of two decades of X-ray and EM studies, up to but not including atomic-structure determinations of the 1990’s, can be summarized as follows: 1. In many skeletal muscles, the cross-sectional lattice of myosin ﬁlaments is hexagonal, having a rhombohedral unit cell with myosin ﬁlaments at each vertex

12

1 Introduction

Fig. 1.5 The shape of isometric tension as a function of sarcomere length is explained by interdigitating ﬁlaments (Gordon et al. 1966). If myosin and actin ﬁlaments in each half-sarcomere have lengths LM, LA and the halfwidth of the bare zone is LB, then L0 ¼ 2(LA LB), L1 ¼ 2LA, L2 ¼ 2(LA + LB) and L3 ¼ 2(LA + LM)

(Fig. 1.6). In the A-band where myosin and actin ﬁlaments overlap, each actin ﬁlament is located in the centroid of its three neighbouring myoﬁlaments, so that there are two actin ﬁlaments per unit cell. Equatorial X-ray reﬂections showed that, in skinned ﬁbres of frog semitendinosus muscle, the myosin-myosin lattice spacing dM of the relaxed ﬁbre was 48 nm, reducing to 44 nm when activated (Matsubara et al. 1984). However, an intact muscle maintains constant volume when activated because the sarcolemma is relatively impermeable and inelastic. The density of ﬁlaments in a myoﬁbril follows from the lattice spacing, which supplants older estimates based on muscle mass and molecular weights. For pﬃﬃﬃ dM ¼ 48 nm, the area of the rhomboid unit cell is 3d2M =2 ¼ 2.0 1015 m2. As there is one myoﬁlament per unit cell, the density of myoﬁlaments is 0.5 1015 per m2. The density of actin ﬁlaments is twice that number. In a

1.3 The Structure of Skeletal Muscle

13

Fig. 1.6 The unit cell of muscle lattices in cross-section, with crowns of three myosin dimers on each myoﬁlament. (a) The rhomboidal unit cell of the lattice of bony ﬁsh muscle, with one myoﬁlament and two actin ﬁlaments per cell. In X-ray diffraction patterns, the (1,0) and (0,1) reﬂections detect myoﬁlaments while the (1,1) reﬂection also includes contributions from actins. (b) For tetrapods, some myosins in that unit cell are oppositely oriented, giving the supercell proposed by Luther and Squire (1980)

ﬁbre, about 85% of the cross-section is ﬁlled with myoﬁbrils, the rest being taken up by mitochondria and other vesicles. 2. The actin ﬁlament (F-actin) is a double helix of repeating monomers of G-actin (Fig. 1.7). This double helix has a half-repeat of 36.0 nm, so that each strand has a periodicity of 72 nm. These repeats show up as layer lines on the meridian of X-ray diffraction patterns. There are 13 monomers per 72 nm repeat, so that the monomer spacing is 72/13 ¼ 5.54 nm and the right-handed rotation per monomer is 360/13 ¼ 27.7 . Comparing the two strands, adjacent monomers are spaced by half that distance, so the strands appear staggered by 2.77 nm. The rotation required from one monomer to its neighbour on the other strand is 166.15 , because two such rotations bring us back to 27.7 . Thus the actin double helix is actually left-handed (Fig. 1.7b). After the atomic structure of G-actin was discovered, Holmes et al. (1990) showed how these monomers could bind to generate the structure of F-actin. The outer radius is about 4.5 nm. 3. The thick ﬁlament is composed of a backbone woven with myoﬁlaments, which project radially from the axis at intervals of 14.3 nm (Fig. 1.7c, d), giving a prominent layer-line reﬂection on the meridian in X-ray observations. The projecting part of each myoﬁlament consists of a rod structure, designated S2, which is a double coiled-coil with a length of 60–100 nm. Each terminal of this double helix is ﬂexibly joined to the C-terminal of one molecule of myosin-S1 (Fig. 1.8). Myosin-S1 (or just S1) is a large molecule with a molecular weight of 120 KiloDaltons. It has a globular motor domain of roughly 5 nm in diameter, coupled to a 10 nm-long neck made from a heavy chain of amino-acid residues. It

14

1 Introduction

Fig. 1.7 (a and b) The double-helical structure of the actin ﬁlament, with a period of 72 nm and thirteen monomers per period on each strand. (c) myosin-S1 heads on the thick ﬁlament. With crowns of three dimeric myosins per layer and layers spaced by 14.3 nm (Hudson et al. 1997, with permission of Elsevier Press). (d) Dimers on adjacent layers are rotated by 40 , deﬁning three helices with periods of 3 42.9 nm

is the distal (C-terminal) end of this neck that is joined to one terminal of S2, while the motor domain is free to bind to F-actin. In this book we refer to myosinS1 as a head, on the understanding that this includes the motor domain and the neck.

1.3 The Structure of Skeletal Muscle

15

Fig. 1.8 The myosin ﬁlament (molecular weight 520,000), and its sections determined by tryptic digestion. The tail is woven into the myoﬁlament backbone. The projecting part is lightmeromyosin (M.W. 160,000), hinge-jointed to heavymeromyosin (M.W. 340,000) which consists of the S2 rod (M.W. 100,000) and two myosin-S1 heads of M.W. 120,000, each joined to one strand of S2. All parts except S1 are α-helical coiled coils

How myosin-S1 interacts with actin to generate force and movement is the central business of theories of muscle contraction; the now-dominant explanation is that the neck acts a lever-arm. However, early studies of this interaction focussed on the radial spacing between F-actin and myosin heads. In relaxed muscle, the heads of each dimer fold up against the thick ﬁlament, either as an intimate pair (Hudson et al. 1997) or in opposite directions (Offer et al. 2000). In rigor muscle, all heads are bound to actin (Cooke and Franks 1980) and there is a jumble of orientations as all heads locate actin sites on a double helix, some with favourable axial positions or azimuthal orientations, some not. A myosin head bound to actin is often called a crossbridge. In activated muscle, not all heads are bound, and the (1,0) and (1,1) equatorial X-ray reﬂections provide a convenient way of monitoring how heads move radially towards actin in the presence of calcium. On going from relaxed to rigor muscle, the intensity of the (1,0) reﬂection, which is generated by myoﬁlaments only, is unchanged, while that of the (1,1) reﬂection, which includes myosin and actin ﬁlaments, increases (Fig. 1.6a). 4. In the sarcomere, the myosin ﬁlament backbone is bipolar, and so are the actin ﬁlaments. There is a mirror symmetry between the two halves of every sarcomere, ﬁrst seen by H.E. Huxley (1963) from electron-microscope (EM) studies of isolated myosin ﬁlaments as arrowheads pointing away from the central M-band in each half-sarcomere. Isolated actin ﬁlaments decorated with myosinS1 show the same arrowhead struture (Moore et al. 1970), conﬁrming that F-actin is also a polar ﬁlament. Actin ﬁlaments have a a barbed end (the ‘plus’ end) and a pointy end (or ‘minus’ end), and uncapped ﬁlaments polymerize more rapidly at the +end (Howard 2001). In striated muscle, the +ends are attached to Z-lines so the -ends point towards the M-band. As a contracting muscle shortens, myosin heads detach from actin and rebind to actin sites closer to the Z-line, pulling it towards the centre. Thus myosin-S1 is a +end motor. Here we are talking about muscle myosin, namely myosin-II in the phylogenetic tree. Most, but not all, myosins created by evolution are +end motors. This bipolarity enables us to make sense of the tension-length characteristic of isometric muscle at sarcomere lengths below the plateau (Fig. 1.5) When actin ﬁlaments are pushed into the ‘wrong’ half of the sarcomere, myosin heads must do an ‘about-turn’, so to speak, to be able to bind to actin ﬁlaments of the wrong

16

1 Introduction

polarity for their preferred orientation. Consequently, their afﬁnity for actin is reduced and this is reﬂected as reduced isometric tension relative to the plateau, where all heads can bind to actin sites of the right polarity. However, these heads retain the ability to hydrolyse ATP as before (Stephenson et al. 1989). 5. The azimuthal orientations of myosin heads on the thick ﬁlament has been established by a combination of X-ray diffraction and electron microscopy of thin sections (Squire 1981). A crown of three myosin dimers, attached to three S2 rods, project out from the thick ﬁlament every 14.3 nm, generating a prominent layer-line X-ray reﬂection. The azimuthal orientations of the three dimers on the crown are separated by 120 . Adjacent crowns along the backbone are rotated by 40 , so that the orientations of crowns are repeated with a period of 3 14.3 ¼ 42.9 nm (Fig. 1.7). In one half-sarcomere, the overlap section of a myoﬁlament of frog sartorius muscle has a length of 700 nm, which means that it holds 3 700/14.3 ¼ 147 dimers. The starting orientations of the dimers on the ﬁrst crown after the bare zone are not necessarily the same on all myoﬁlaments. This has been studied by Luther and Squire (1980) by looking at myoﬁlament orientations in the bare zone. In muscles of bony ﬁsh, sharks, rays and sturgeons, all myoﬁlaments have the same starting orientation, However, in tetrapods (mammals, reptiles, birds and amphibians), the starting orientations of adjacent myoﬁlaments in the lattice may be equal or opposite, according to the following rules: the orientations of three adjacent myoﬁlaments on a triangle cannot be the same, and neither can the orientations of three adjacent colinear myoﬁlaments (Luther 2004). This has the effect of creating a superlattice whose unit cell is three times the area of the rhombohedral cell (Fig. 1.6b). Actin ﬁlaments also have starting orientations. Considering that the orientations of myosin heads at the pointy end of F-actin varies with sarcomere length, it probably doesn’t matter very much what those orientations are. Nevertheless, Hirose and Wakabayashi (1988) have shown that they are the same for all F-actins in the lattice. 6. Myoﬁlaments and F-actins are not the only ﬁlaments in the sarcomere. In each half-sarcomere there are extensible ﬁlaments that connect the M-band to the Z-line and stop the sarcomeres from disassembling when stretched beyond the point of zero overlap. These are titin ﬁlaments; titin is a giant polymeric molecule made of repeating units (either PVEK or ﬁbronectin) connected by elastic links. Their tension-length curve shows hysteresis; under tension these links may unwind and then wind up again, so that a ﬁbril stretched beyond overlap will contract in discrete steps (Miklos et al. 1997; Rief et al. 1997; Tskhovrebova et al. 1997). Titin ﬁlaments are responsible for the tension-length characteristic of relaxed muscle, which in skeletal muscle appears as an exponential rise starting near the bottom of the descending limb. While titin ﬁlaments, and perhaps nebulin ﬁlaments also, are responsible for maintaining the integrity of sarcomeres pulled beyond zero ﬁlament overlap, the M-band and Z-line structures play a role in stabilising the lattice spacing of relaxed muscle in the zero-overlap region. However, under physiological

References

17

conditions the thick and thin ﬁlaments do overlap, and then the stability of the lattice is thought to be controlled by electrostatic forces, primarily a balance of screened electrostatic repulsion between negatively charged ﬁlaments and a weak attractive force from van der Waals interactions. This subject is explored in Chap. 5.

References Astumian RD (1997) Thermodynamics and kinetics of a Brownian motor. Science 276:917–922 Baker J, Thomas DD (2000) A thermodynamic muscle model and a chemical basis for A.V. Hill’s muscle equation. J Muscle Res Cell Motil 21:335–344 Carlson FD, Wilkie DR (1974) Muscle physiology. Prentice-Hall, Englewood Cliffs Cooke R, Franks K (1980) All myosin heads form bonds with actin in rigor rabbit skeletal muscle. Biochemist 19:2265–2269 Credi A, Silvi S, Venturi M (2014) Molecular machines and motors, Topics in Current Chemistry, vol 354. Springer, Berlin Elliott GF, Worthington CR (2001) Muscle contraction: viscous-like frictional forces and the impulsive model. Int J Biol Macromol 29:213–218 Elliott GF, Lowy J, Millman BM (1965) X-ray diffraction from living striated muscle during contraction. Nature 206:1357–1358 Fajer PG, Fajer EA, Thomas DD (1998) Myosin heads have a broad orientational distribution during isometric muscle contraction: time-resolved EPR studies using caged ATP. Proc Natl Acad Sci USA 87:5538–5542 Finer JT, Simmons RM, Spudich JA (1994) Single myosin molecule mechanics: piconewton forces and nanometre steps. Nature 368:113–119 Funatsu T, Harada Y, Tokunaga M, Saito K, Yanagida T (1995) Imaging of single ﬂuorescent molecules and individual ATP turnovers by single myosin molecules in aqueous solution. Nature 374:555–559 Gordon AM, Huxley AF, Julian F (1966) The variation in isometric tension with sarcomere length in vertebrate muscle ﬁbres. J Physiol (London) 184:170–192 Goss CM (1968) On movement of muscles by Galen of Pergamon. Am J Anat 123:24–25 Haselgrove JC, Huxley HE (1972) X-ray evidence for a conformational change in the actincontaining ﬁlaments of vertebrate striated muscle. Cold Spring Harb Symp Quant Biol 37:341–352 Herzberg O, James MNG (1988) Reﬁned crystal structure of troponin C from turkey skeletal muscle at 2.0Å resolution. J Mol Biol 203:761–779 Hill AV (1953) The mechanics of active muscle. Proc R Soc B141:104–117 Hirose K, Wakabayashi T (1988) Thin ﬁlaments of rabbit skeletal muscle are in helical register. J Mol Biol 204:797–801 Hitchcock-DeGregori S, Irving T (2014) Hugh E. Huxley: the compleat biophysicist. Biophys J 107:1493–1501 Holmes KC, Popp D, Gebhard W, Kabsch W (1990) Atomic model of the actin ﬁlament. Nature 347:44–49 Howard J (2001) Mechanics of motor proteins and the cytoskeleton. Sinauer Assoc. Inc., Sunderland Hudson L, Harford JJ, Denny RC, Squire JM (1997) Myosin head conﬁguration in relaxed ﬁsh muscle: resting state myosin heads must swing axially by up to 150Å or turn upside down to reach rigor. J Mol Biol 273:440–455

18

1 Introduction

Huxley AF (1957) Muscle structure and theories of contraction. Prog Biophys Biophys Chem 7:257–318 Huxley HE (1963) Electron microscope studies on the structure of natural and synthetic protein ﬁlaments from striated muscle. J Mol Biol 7:281–308 Huxley HE (2004) Fifty years of muscle and the sliding ﬁlament hypothesis. Eur J Biochem 271:1403–1415 Huxley HE, Hanson J (1954) Changes in the cross-striations of muscle during contraction and stretch and their structural interpretation. Nature 173:973–976 Huxley AF, Niedegerke R (1954) Structural changes in muscle during contraction: interference microscopy of living muscle ﬁbres. Nature 173:971–973 Huxley AF, Simmons RM (1971) Proposed mechanism of force generation in striated muscle. Nature 233:533–538 Ingels NB (1979) The molecular basis of force development in muscle. Palo Alto Medical Research Foundation, Palo Alto, pp 147–162 Iwazumi T (1970) A new ﬁeld theory of muscle contraction. Ph.D. thesis, University of Pennsylvania Kabsch W, Mannherz HG, Suck D, Pai EF, Holmes KC (1990) Atomic structure of the actin: DNase I complex. Nature 347:37–44 Edman KAP (1979) The velocity of unloaded shortening and its relation to sarcomere length and isometric force in vertebrate muscle ﬁbres. J Physiol (London) 291:143–159 Katz B (1966) Nerve, muscle and synapse. McGraw Hill, Inc, New York Kitamura K, Tokunaga M, Iwane AH, Yanagida T (1999) A single myosin head moves along an actin ﬁlament with regular steps of 5.3 nanometres. Nature 397:120–134 Kron SJ, Spudich JA (1986) Fluorescent actin ﬁlaments move on myosin ﬁxed to a glass surface. Proc Natl Acad Sci USA 83:6272–6276 Linari M, Piazessi G, Dobbie I, Koubassova N, Reconditi M, Narayanan T, Diat O, Irving M, Lombardi V (2000) Interference ﬁne structure and sarcomeric length dependence of the axial x-ray pattern from active single muscle ﬁbers. Proc Natl Acad Sci USA 97:7226–7231 Lombardi V, Piazzesi G (1992) Force response in steady lengthening of active single muscle ﬁbres. In: Simmons RM (ed) Muscular contraction. Cambridge University Press, Cambridge Luther PK (2004) Evolution of the muscle lattice in the vertebrate kingdom. Microsc Anal, March: 9–11 Luther PK, Squire J (1980) Three-dimensional structure of the vertebrate muscle A-band. J Mol Biol 141:409–439 Lymn RW, Taylor EW (1971) Mechanism of adenosine triphosphate hydrolysis by actomyosin. Biochemist 10:4617–4624 MacIntosh BR, Gardner PF, McComas AJ (2006) Skeletal muscle, form and function. Human Kinetics, Champaign Maquet P, Nayler M, Ziggelaar A, Croone W (2000) William Croone: on the reason of the movement of the muscles. Trans Am Philos Soc 90(1):130 Martonosi A (2000) Animal electricity, Ca2+ and muscle contraction. A brief history of muscle research. Acta Chim Pol 47:493–516 Matsubara I, Goldman YE, Simmons RM (1984) Changes in the lateral ﬁlament spacing of skinned muscle ﬁbres when cross-bridges attach. J Mol Biol 173:15–33 Miklos S, Kellermayer Z, Smith SB, Granzier HL, Bustamente C (1997) Folding-unfolding transitions in single titin molecules characterised with laser tweezers. Science 276:1112–1116 Moore PB, Huxley HE, DeRosier DJ (1970) Three-dimensional reconstruction of F-actin, thin ﬁlaments and decorated thin ﬁlaments. J Mol Biol 50:279–288 Needham DM (1971) Machina Carnis: the biochemistry of muscular contraction in its historical development. C.U.P., Cambridge Offer G, Knight PJ, Burgess SA, Alamo L, Padron R (2000) A new model for the surface arrangement of myosin molecules in tarantula thick ﬁlaments. J Mol Biol 298:239–260

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Oplatka A (1997) Critical review of the swinging crossbridge theory and of the cardinal active role of water in muscle contraction. Crit Rev Biochem Mol Biol 32:307–360 Peachey LD (1965) The sarcoplasmic reticulum and transverse tubules of the frog’s sartorius. J Cell Biol 25:209–231 Piazzesi G, Reconditi M, Linari M, Lucii L, Bianco P, Brunello E, Decostre V, Stewart A, Gore DB, Irving TC, Irving M, Lombardi V (2007) Skeletal muscle performance determined by modulation of number of myosin motors rather than motor force or stroke size. Cell 131:784–795 Rayment I, Rypniewski WR, Schmidt-Base K, Smith R, Tomchick DR, Benning MM, Winkelmann DA, Wesenberg G, Holden HM (1993) Three-dimensional structure of a myosin subfragment-1: a molecular motor. Science 261:50–65 Rief M, Gautel M, Oesterhelt F, Fernandez JM, Gaub HE (1997) Reversible unfolding of individual titin immunoglobulin domains by AFM. Science 276:1109–1112 Rome LC (2006) Design and function of superfast muscles: new insights into the physiology of skeletal muscle. Annu Rev Physiol 68:193–221 Ruegg JC (2017) Calcium in muscle activation; a comparative approach. Springer, Berlin Squire J (1981) The structural basis of muscle contraction. Plenum, New York Stephenson DG, Stewart AW, Wilson GJ (1989) Dissociation of force from myoﬁbrillar MgATPase and stiffness at short sarcomere lengths in rat and toad skeletal muscle. J Physiol (London) 410:351–366 Sundaralingam M, Bergstrom R, Strasburg G, Rao ST, Rowchowdhury P, Greaser M, Wang BC (1985) Molecular structure of troponin C from chicken skeletal muscle at 3-angstrom resolution. Science 227:945–948 Suzuki Y, Yasunaga T, Ohkura R, Wakabayashi T, Sutoh K (1998) Swing of the lever arm of a myosin motor at the isomerization and phosphate-release steps. Nature 396:380–383 Takeda S, Yamashita A, Maeda K, Maeda Y (2003) Structure of the core domain of human cardiac troponin in the Ca2+-saturated form. Nature 424:35–41 Tskhovrebova L, Trinick J, Sleep J, Simmons RM (1997) Elasticity and unfolding of single molecules of the giant muscle protein titin. Nature 387:308–312 Vibert P, Craig R, Lehman W (1997) Steric model for activation of muscle thin ﬁlaments. J Mol Biol 266:8–14 Westerblad H, Allen DG, Lannergren J (2002) Muscle fatigue: lactic acid or inorganic phosphate the major cause? News Physiol Soc 17:17–21 Yomosa S (1985) Solitary excitations in muscle proteins. Phys Rev A 32:1752–1758

Chapter 2

Of Sliding Filaments and Swinging Lever-Arms

Bananas in pyjamas are coming down the stairs Playschool, ABC television

2.1

Contractile Empiricism: Hill’s Equations

Before sliding ﬁlaments and swinging lever arms were even dreamt of, A.V. Hill (1938) proposed simple formulae for the way in which muscle tension and energy consumption vary with shortening velocity. Both equations are empirical and based on experimental data. Hill’s equation for the steady-state tension-velocity function is widely used to this day; it is best written in the form ðT þ aÞv ¼ bðT o T Þ:

ð2:1Þ

Solving for the tension gives the hyperbolic function T ðvÞ αð1 v=v0 Þ ¼ T0 α þ v=v0

ðα a=T 0 ; v0 bT 0 =aÞ

ð2:2Þ

where T0 is isometric tension (set v ¼ 0) and v0 is the speed of unloaded shortening (set T ¼ 0). As in Fig. 1.2a, Hill’s equation can provide an excellent ﬁt to shortening behaviour, except at very low speeds. Values of the Hill ratio α vary considerably, for example 0.25 for frog sartorius muscle and 0.17 for rabbit psoas muscle (Woledge et al. 1985). Hill was led to postulate his equation by combining tension-velocity data with heat measurements. During an isometric tetanus, heat is generated at a constant rate Q_ 0 , which is the maintenance heat (rate). While the muscle is shortening, the rate of heat generation increases and mechanical work is done in moving a load. Conservation of energy means that the excess rate of energy loss during shortening is

© Springer Nature Switzerland AG 2018 D. A. Smith, The Sliding-Filament Theory of Muscle Contraction, https://doi.org/10.1007/978-3-030-03526-6_2

21

22

2 Of Sliding Filaments and Swinging Lever-Arms

ΔE_ ðvÞ ¼j W_ ðvÞ j þ j ΔQ_ ðvÞ j

ð2:3Þ

where j W_ ðvÞ j¼ Tv is the mechanical power output. In 1963, he showed that the shortening heat (rate) j ΔQ_ ðvÞ j was proportional to shortening velocity (Hill 1953, 1964), and the rate of energy loss was proportional to the fall in tension T0T below the isometric level, except at very low loads. When fed into Eq. 2.3 these results produce Eq. 2.1. Thus Hill’s tension-velocity equation has an energetic interpretation, expressed by the conditions j ΔQ_ ðvÞ j¼ av

ð2:4aÞ

ΔE_ ðvÞ ¼ bðT 0 T ðvÞÞ:

ð2:4bÞ

and

These constraints can be checked against heat-rate measurements provided that the tension-velocity characteristic is measured simultaneously. In passing, note that tensions are measured per unit area of cross-section of muscle, and velocities are generally quoted in units of muscle length per second, so the quantities in Eqs. 2.4 would then be normalised to unit volume. The connection between Hill’s tension-velocity equation and energetics is not as tight as might be supposed. Experimental data suggest that Hill’s tension-velocity function can be obeyed even when the heat-rate and energy-rate equations are not. If that is the case, then Eqs. 2.4a and 2.4b are not independent postulates and it is sufﬁcient to test only one of them. To this end, we now reanalyse Hill’s heat-rate data on the basis that the constants a and b in Eqs. 2.4 are identical with those in his tension-velocity equation. Figure 2.1 shows the results of applying this logic to the data in Table 2 of Hill (1964), taking tension as the independent variable. The velocity-tension curve was ﬁtted by Eq. 2.2, ﬁrstly by allowing v0 and α to vary, which yielded v0 ¼ 3.5 muscle lengths/second, and then by constraining v0 to a more realistic value of 3.0 L/s as in Fig. 1.2a. Then the re-normalised velocity-tension curve (Fig. 2.1a) is well ﬁtted with α ¼ 0.271 0.013. Hill’s equations for the normalised velocity u v/v0 and energy rate e_ E_ =v0 T 0 are uð τ Þ ¼

αð1 τÞ , αþτ

e_ ðτÞ ¼ q_ 0 þ αð1 τÞ

ð2:5Þ

where τ T/T0 and q_ 0 is the normalised maintenance heat. (Incidentally, u(τ) is the same function of τ that τ(u)is of u, which is a useful aide memoire). So the energy rate should fall linearly with τ, and the slope should be equal to the value of α which ﬁts the velocity-tension curve. Figure 2.1b shows the normalised versions of Hill’s data for power output w_ vT=v0 T 0 , heat rate q_ j Q_ j =v0 T 0 and their sum e_ . The energy points lie close to a straight line except at low tensions, where they fall below

2.1 Contractile Empiricism: Hill’s Equations

23

Fig. 2.1 The energetics of shortening frog muscle, and a test of A.V. Hill’s equation from experimental data in Table 2 of Hill (1964). (a) The normalised tension-velocity curve, obtained by choosing an appropriate value of v0 (see main text), is well ﬁtted with α ¼ a/T0 ¼ 0.25. (b) Plots of power output W_ ¼ vT, the rate Q_ of heat production (maintenance heat + shortening heat) due to activation, and the rate of energy production E_ ¼ W_ þ Q_ : All three functions were normalised to v0T0. If the two lowest-tension points on the energy graph can be ignored, the energy rate is reasonably well-ﬁtted by a straight line whose slope is also 0.25, which conﬁrms the assumptions leading to Hill’s equation

the extrapolated line. Ignoring the two left-hand points allows a linear ﬁt with α ¼ 0.227 0.011, which is lower than the value for the velocity-tension curve. However, we started by assuming that both values of α must be the same. In support of this assumption, good ﬁts to both curves are obtained by setting α ¼ 0.25, and it is these curves that are shown in Fig. 2.1. To conclude, the tension-velocity curve is well ﬁtted by Hill’s equation over almost the whole range of tensions, but the energy-rate equation is valid only for tensions above 0.4T0. In fact, heat measurements on different muscles show a variety of behaviour, none of which offers convincing support for Hill’s energetic equations. For example, Barclay et al. (1993) measured tension and shortening heat (rate) as a function of velocity for the extensor digitoris longum (EDL) and soleus muscles of the mouse, which are fast and slow twitch muscles respectively. Their data, reproduced in Fig. 2.2, shows that the shortening heat was indeed linear in velocity, but the slopes were not as predicted by Eqs. 2.4. To appreciate the consequences, let Eq. 2.4a be replaced by ΔQ_ ðvÞ ¼ a0 v

ð2:6Þ

and compare the dimensionless quantity α0 ¼ a0 /T0 for each muscle with its mechanical equivalent α from the tension-velocity curves, which are well ﬁtted by Eq. 2.2. Barclay’s data gives α ¼ 0.34 and α0 ¼ 0.20 for EDL ﬁbres, and α ¼ 0.14 and α0 ¼ 0.28 for soleus ﬁbres. Thus α0 < α for the fast muscle and α0 > α for the slow muscle. Using Eq. 2.6 in

24

2 Of Sliding Filaments and Swinging Lever-Arms

Fig. 2.2 Tension and heat-rate data of Barclay et al. (1993) for mouse EDL and soleus ﬁbres versus shortening velocity. With permission of John Wiley and Sons Inc.

conjunction with Hill’s tension-velocity equation gives the following formulae for the extra energy rate from shortening, written in normalised form: Δe_ ðτÞ

ΔE_ ðvÞ ðT þ a0 Þv αð1 τÞ ¼ ¼ ðα 0 þ τ Þ T 0 v0 T 0 v0 αþτ

ð2:7aÞ

or Δe_ ðuÞ ¼

αð1 þ αÞ 0 þ α α u: αþu

ð2:7bÞ

For the EDL and soleus ﬁbres, these functions are also plotted in Fig. 2.2. As a function of tension, the energy production rate is linear for α0 ¼ α, falls below that line for α0 < α, and rises above it for α0 > α. As a function of velocity, the energy rate is hyperbolic forα0 ¼ α, approaching a plateau when u 1/α (a region of negative tension). For α0 < α, the rate is lowered and reaches a maximum at a ﬁnite value of u, whereas for α0 > α the rate is raised and continues to increase with u. As a function of shortening speed, the energetics of these fast and slow mouse ﬁbres have qualitatively different characteristics. It is tempting to suppose that α0 < α for all fast ﬁbres and α0 > α for all slow ﬁbres, but this hypothesis would be limited to muscles for which Eq. 2.6 is obeyed. Woledge et al. (1985) survey many measurements of heat production as a function of shortening velocity, which show that for some muscles the shortening heat is linear in the velocity whereas in others it is a convex or hyperbolic function. Nevertheless, the results of Barclay et al. from the mouse do provide a clear-cut

2.1 Contractile Empiricism: Hill’s Equations

25

distinction between its fast and slow ﬁbres, which needs to be explained in terms of models of contractility. A.V. Hill’s interpretation of measurements of shortening heat was always open to the challenge that there are many sources of heat generation during a tetanus. There is an extensive literature, reviewed by Smith et al. (2005), devoted to unravelling different contributions, for example the initial heat, labile heat and recovery heat, as well as the maintenance heat Q_ 0 from an isometric tetanus and the shortening heat ΔQ_ . The decrease in shortening heat observed by Hill at low tensions (Fig. 2.1b) may reﬂect the difﬁculty of measuring heat output during rapid shortening over the small time interval available on the plateau of the tension-length curve, where all heads overlap the actin ﬁlament. Alternatively, the ﬁbre might become partially deactivated during rapid shortening, but this would reduce tension as well as the rate of heat production, in which case Hill’s equation would need to be reinterpreted. Muscle is a machine which can do mechanical work in pulling a load. As with any other motor, its behaviour can be characterised by its power output and efﬁciency as a function of speed. The power output is the rate of working Tv. Thus Hill’s mechanical equation predicts that the power output is w_ ðvÞ

vT ðvÞ αuð1 uÞ : ¼ v0 T 0 αþu

ð2:8Þ

when normalised as shown. This function is zero at u ¼ 0 and u ¼ 1, where no work can be done. Maximum power output occurs when n pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃo1 u ¼ umax ¼ 1 þ 1 þ 1=α , and the maximum normalised power is just u2max : For frog sartorius muscle at 3 C, α ¼ 0.25 so umax ¼ 0.31. With T0 ¼ 3 105 Pa and v0 ¼ 2.5 ML/s, the absolute maximum power output j W_ max j¼ T 0 v0 u2max per unit volume of ﬁbre would be 7.2 104 W/m3. The mechanical conversion efﬁciency of shortening muscle is here deﬁned as the ratio of work to (work + heat), namely η ð vÞ

vT ðvÞ : E_ ðvÞ

ð2:9Þ

This is the customary engineering deﬁnition of efﬁciency for heat engines. Using Hill equations in the form of Eqs. 2.5 and 2.7b gives ηðuÞ ¼

αuð1 uÞ : q_ 0 α þ fq_ 0 þ αð1 þ α0 Þgu þ ðα0 αÞu2

ð2:10Þ

which is similar to Eq. 2.8 for the normalised power output, but with a maximum at a lower value of u. For the data of Barclay et alia, these quantities are plotted in Fig. 2.3. For direct calculations from experimental data, Woledge et al. (1985) and Smith et al. (2005) should be consulted.

26

2 Of Sliding Filaments and Swinging Lever-Arms

Fig. 2.3 Derived plots for the heat-rate measurements of Barclay et al. (1993) on EDL and soleus ﬁbres from the mouse. (a and b) normalised shortening heat rate as a function of tension and velocity, from Eqs. 2.7, and (c and d) normalised power output and efﬁciency as a function of velocity, from Eqs. 2.8 and 2.10. Here q_ 0 ¼ 0.0137 (EDL) and 0.0561 (soleus). Values of α and α0 are in the main text, and α0 < α for EDL, whereas α0 > α for soleus. For comparison, the dashed lines in (a and b) were constructed with α0 ¼ α

The next stage in the evolution of muscle energetics arrived in the form of measurements of the rate of ATP hydrolysis during shortening (Kushmerick et al. 1969). Without delving into the history of muscle biochemistry (Needham 1971, Rall 2016), it is sufﬁcient to say that muscle contraction is driven by the hydrolysis of ATP, and this was ﬁrst demonstrated by W.O. Fenn in 1924. The hydrolysis reaction ATP þ H2 O Ð ADP þ Pi þ Hþ dissociates the terminal phosphate ion, leaving the adenine-ribose structure with two PO4 groups. Energy conservation for active shortening should now be restated in

2.1 Contractile Empiricism: Hill’s Equations

27

terms of the enthalpy change associated with ATP hydrolysis, since enthalpy H ¼ U + PV is the appropriate thermodynamic potential for changes at constant pressure. This follows from the ﬁrst law of thermodynamics for the internal energy U which includes elastic strain energy. For a muscle of length L carrying load F, dU ¼ dQ þ dW, dW ¼ PdV þ FdL, ðdH ÞP ¼ dU þ PdV ¼ dQ þ FdL:

ð2:11Þ

Following the usual convention, Q and W are heat and work inputs (hence the modulus signs in Eq. 2.3). If ATP is being hydrolysed at a rate R, then the rate of energy production is E_ ¼ H_ ¼j ΔH j R

ð2:12Þ

where ΔH < 0 is the molar enthalpy of hydrolysis. The value of |ΔH| has been hotly disputed, but is now agreed to be 34 KJ/mole at 0 C and somewhat less (about 30 KJ/mole) at 12 C (Woledge et al. 1985). Measurements of ATPase rates provide an attractive alternative to heat-rate measurements for validating Hill’s energy equation. Such measurements have been made with a linked assay system, whereby ATP breakdown is regenerated by the oxidation of NADH, ﬁrstly by Kushmerick and Davies (1969) on frog sartorius muscle, and later by Potma et al. (1994) and Sun et al. (2001) on rabbit psoas muscle (frog sartorius and rabbit psoas are both fast muscles). ATPase measurements on rabbit psoas have also been made with a ﬂuorescent version of the phosphatebinding protein (He et al. 1999). The results conﬁrm the variety of behaviour revealed by heat-rate experiments. ATPase measurements of Potma et al. (1994) conﬁrm Hill’s heat-rate data which show that Eq. 2.4b is not obeyed at low tension. For rabbit psoas muscle, Sun et al. found that at 10 C the hydrolysis rate R(v) increased linearly with shortening velocity out to v0, whereas at a very similar temperature (12 C) He et al. found that R(v) was a hyperbolic function of v, asymptoting to a plateau of 3.7 times the isometric rate R0. In principle, ATPase measurements can also be used to test Hill’s energy-rate equations, but the different results obtained for rabbit psoas muscle with the linked NADH assay and the phosphate-binding assay may reﬂect the way in which these assays operate in practice. For the record, Hill’s Eqs. (2.2) and (2.4b) predict that the extra ATPase rate due to shortening would decrease linearly with tension, and increase hyperbolically with velocity, as ΔRðvÞ ¼ where

bð T 0 T ð v Þ Þ u ¼ ΔRmax j ΔH j uþα

ð2:13Þ

28

2 Of Sliding Filaments and Swinging Lever-Arms

ΔRmax ¼

αð1 þ αÞv0 T 0 : j ΔH j

ð2:14Þ

and u ¼ v/v0, α ¼ a/T0 ¼ b/v0 as before. The predicted rate is linear in tension, and hyperbolic in shortening velocity with a half-maximal rise when u ¼ α. As the molar enthalpy of ATP hydrolysis provides the link between ATPase rate and the rate of energy production, so the efﬁciency of shortening muscle is measured by combining tension and ATPase data. Thus ηðvÞ ¼

vT ðvÞ j ΔH j RðvÞ

ð2:15Þ

where R(v) ¼ R0+ΔR(v) includes the isometric rate. Although this formula follows from the thermodynamic deﬁnition for heat engines (Eq. 2.9), other deﬁnitions appear in the literature. ΔH is sometimes replaced by ΔG, the change in Gibbs energy, presumably because maximum work output, equal to ΔG, is achieved when the change is reversible (Zemansky and Dittman 1996). Wilkie (1960) has also argued that Eq. 2.15 should not be used because muscle is not a heat engine working between temperature reservoirs. While a deﬁnition can only be useful rather than right or wrong, the objection seems spurious because enthalpy is the thermodynamic potential for heat and work at constant pressure. Furthermore, muscle is a chemical engine, analogous to a heat engine because it operates between reservoirs with different chemical potentials (for ATP and ADP + Pi). In conclusion, Hill’s analysis is useful as a semi-quantitative description of shortening muscle, but on the evidence available it falls short of giving quantitative connections between tension and ATPase data. Also, it is not relevant to what happens when a muscle is stretched. Ramp stretching raises the steady-state tension (Fig. 1.3) and the rate of ATP hydrolysis drops to very low levels, meaning that the muscle acts a passive brake, albeit with a maximum holding tension. Hill and Howarth (1959) claimed that ATP hydrolysis could even be reversed by stretching, but to my knowledge this result has not been repeated.

2.2

How Myosin Heads Find Actin Sites

Before building a dynamical model of striated muscle, it is desirable to know how myosin heads tethered to thick ﬁlaments interact with binding sites on an actin ﬁlament, which sits in the centre of three neighbouring myoﬁlaments. One can start from the structure of the decorated actin ﬁlament revealed by cryomicroscopy of acto-S1 in the absence of ATP (Fig. 2.4a). All sites are occupied by crossbridges (bound myosin heads), which are shaped like bananas, and this proﬁle is clearly seen in a reconstruction of the cross-section of rigor actomyosin (Fig. 2.4b). There is very little space between the ﬁlaments for crossbridges; the myosin-actin spacing is 25 nm in frog sartorius muscle and 27 nm in rabbit psoas muscle

2.2 How Myosin Heads Find Actin Sites

29

Fig. 2.4 (a) Cryomicroscopic reconstruction of the decorated actin ﬁlament in rigor, showing myosin-S1 heads bound to all actin sites (Whittaker et al. 1995). (b) A cross-sectional view of decorated actin, shown as a density map (Hawkins and Bennett 1995). With permission of Elsevier and Springer Nature

(Millman 1991), myosin-S1 has a 10 nm lever-arm and a 5 nm motor domain, but the myosin and actin ﬁlament radii are 8.0 nm and 4.5 nm respectively. It seems that myosin’s ‘banana’ proﬁle is essential for rigor packing. In the words of Fred Julian (1979), “Perhaps anyone who looks at a banana tree and thinks about crossbridges and thick ﬁlaments is going bananas.” With these structures in mind, Fig. 2.5 has been constructed to summarise the ﬁlament geometry of the half-sarcomere. Crowns of three dimers of myosin heads are spaced at intervals of 14.3 nm along their myoﬁlament, but their azimuthal orientations rotate by 40 and therefore repeat themselves at intervals of 42.9 nm. As F-actin is a double helix, not all actin sites have azimuthal orientations available for myosin heads to bind to, and clusters of available sites (or target zones) repeat themeselves every 36 nm, albeit on opposite strands of the double helix. Myosin heads generally do not line up with actin sites, either longitudinally along the ﬁlament axis, or transversely in the plane of cross-section. How do these mismatches affect their ability to bind to actin?

30

2 Of Sliding Filaments and Swinging Lever-Arms

Fig. 2.5 (a) The axial geometry of dimeric myosins on one myoﬁlament addressing binding sites on the actin double-helix. Only one head per dimer is shown, and actin sites are also addressed by dimers on the other two nearest-neighbour myoﬁlaments, as shown below. (b) A cross-section of the ﬁlament lattice looking towards the Z-line, showing myosin dimers on myoﬁlaments in the unit cell for the rhombohedral lattice of bony ﬁsh muscle, as described by Squire et al. (1990). There are three dimers per crown on each 14.3 nm layer line (m ¼ 1, 2, 3) in a 42.9 nm period. k ¼ 1, 2, 3 indexes the lines of centres (ladders of interaction) from the three myoﬁlaments surrounding each F-actin (α ¼ 1,2). Crowns on adjacent layer lines are rotated by 40 , repeating over three layer lines to give the 42.9 nm myoﬁlament periodicity. In any given layer line, crowns on all myoﬁlaments have the same orientation

To answer this question, it might seem that some kind of atomic model for myosin-S1 and monomeric actin is required, so that electrostatic forces between them can be quantiﬁed as a function of their separation, both axially and radially. However, this approach could be unnecessary as the interactions are quite shortranged because of electrostatic screening. The ionic strength of the sarcoplasm is typically 0.15–0.2 M, which implies Debye screening lengths of 0.7–0.8 nm. Under these conditions, the thick-thin ﬁlament spacing is typically 27 nm, or 15 nm when measured from the surfaces. To span this distance, an S1 head needs to swing out from the myoﬁlament backbone, and the S1-S2 junction may also need to move away by a few nanometres. This must be accomplished by Brownian forces, but binding to actin can only take place if head and actin site both have favourable orientations in the plane of cross-section. Fortunately, the junction between two S1 heads and their S2 rod is very ﬂexible. In relaxed muscle, the heads fold up against the myoﬁlament backbone, and there is a calciumactivated interaction with calmodulin on their light chains which causes them to swing out

2.2 How Myosin Heads Find Actin Sites

31

towards the actin ﬁlament. This ﬂexible joint enables heads to rotate about an axis perpendicular to the ﬁlaments. In this way, Brownian motion can cause the motor domain of each head to swing over a range of longitudinal positions on the actin ﬁlament, which is a potent mechanism for locating actin sites of the correct orientation. Van der Waals forces, or electrostatic attraction between charges on the motor domain and the actin interface, provide the mechanism to capture a head as it swings past. What happens after that is a matter of biochemistry; electrostatic interactions are augmented by hydrophobic bonds as the head aligns with an extended actin interface. A quantitative version of this mechanism is explored in Chap. 4.

2.2.1

Head-Site Matching for Vernier Models

The longitudinal mismatch between myosin-S1 heads and actin sites is such that only a fraction of heads can bind to actin (Fig. 2.5a), even with Brownian-driven rotations about the S1-S2 junction. And not all heads are favourably oriented in the plane of cross-section (Fig. 2.5b). Suppose for the moment that we disregard azimuthally skewed heads and consider only those on every third layer line, which have the same favourable orientation. These heads repeat every 42.9 nm whereas the target zones on F-actin repeat every 36 nm, so on moving to the next head and the next site towards the Z-line, the axial mismatch changes by 6.9 nm, creating mismatches of multiples of 6.9 nm until the nearest sites appears on the positive side. In this way the head-site spacing obeys the vernier law X m ¼ ðX 0 6:9mÞ modulo 36 ðm ¼ 1; . . . ; MÞ

ð2:16aÞ

where m indexed myosin heads according to their longitudinal position, and in this context ‘modulo 36’ means add or subtract a multiple of 36 to put X in the range 18 nm to +18 nm. When compared with the range of electrostatic interactions, these mismatches are so severe that very few heads could bind if they were rigidly attached to the thick ﬁlament. The vernier model proposed by A.F. Huxley (1957), as part of his pioneering contraction model, also employs only heads and sites of favourable orientation, as described above. However, Huxley assumed that each head can bind independently, so that binding is controlled only by the head-side spacing X and not the locations of heads along their myoﬁlaments. Hence the M heads can be reindexed by a new integer variable m0 (m) such that (m0 (1),. . .. . . m0 (M)) is a permutation of (1,. . . .,M) and the head-site spacings are a monotonic increasing function of m0 . As the permuted spacings span the 36 nm actin period (several times), there is no need to keep the true values, and it is convenient to replace them by a linear law, namely X m0 ¼ 18 þ 36m0 =M,

ðm0 ¼ 1; ; . . . ; MÞ

ð2:16bÞ

32

2 Of Sliding Filaments and Swinging Lever-Arms

for M heads mapped to one F-actin in the half-sarcomere. On each myoﬁlament there are 147 dimers equating to 294 heads. Only one in three on each crown is eligible to address a given F-actin, but that actin has three nearest-neighbour myoﬁlaments. If all eligible heads were able to bind to one F-actin, then M ¼ 294. However, the true value of M must be smaller, even when longitudinal mismatches are discounted, because some eligible heads have unfavourable orientations in the plane of cross-section. For a given ﬁlament lattice, an appropriate value of M can be determined by imposing cut-offs for azimuthal mismatching. The method described in Appendix A uses angular cut-offs of below 30 for misalignment of heads and actin sites with their line of centres. For vernier models, the axial selection rule can be ignored because that aspect is already built into the vernier head-site spacings. Moreover, there are always actin sites on the double helix which are azimuthally aligned with the line of centres to a given myoﬁlament. Hence the fraction of heads that are azimuthally matched to actin sites is just the fraction of heads aligned within 30 to their line of centres. For the rhomboidal 2:1 unit cell of bony ﬁsh, Fig. 2.5b shows that this fraction is 2/3 and 1/3 respectively for the two F-actins, giving M ¼ 147 for the whole unit cell. For tetrapods with the supercell of Fig. 1.6b, nature has almost doubled this number. There are six F-actins per unit cell, and the net fraction of azimuthally mapped heads rises to 0.9, giving M ¼ 265.

2.2.2

Lattice Models: Target Zones, Layer Lines and Azimuthal Matching

The structural assumptions behind single-head single-site vernier models are rather questionable. For a start, the two heads of dimeric myosin cannot both bind to the same actin site. However, they can share target zones of adjacent sites on the same strand of the double helix; a target zone of three such sites seems feasible, and has been conﬁrmed by X-ray studies on insect ﬂight muscle (Tregear et al. 2004). In that case, the choice of sites on each zone must be recorded; when one head is bound the other head has only two sites to choose from. Secondly, allowing only one head per 42.9 nm repeat to interact with actin is unnecessarily restrictive, especially when azimuthal mismatching is included in the mix. With vernier models, it is sometimes difﬁcult to account for observed isometric tensions, even for the tetrapod lattice where 265 heads per half-sarcomere are azimuthally aligned to one F-actin. Matters can be improved by employing a combination of two-headed myosins and multiple actin sites on each actin half-period. Then isometric tensions of up to 300 pN per F-actin can be accounted for, and the fraction of heads able to interact with F-actin rises to an acceptable level, generally estimated at 0.3–0.4 for skeletal muscle. For true lattice models, these problems can also be addressed using the selection rules in Appendix A, including an axial cut-off distance dco. Choosing dco ¼ 7.0 nm is sufﬁcient for strain-dependent binding, but small enough to prevent heads in adjacent layer lines from accessing the same site. For actin-1 of the rhomboidal unit cell, the number of dimers

2.3 The First Sliding-Filament Model

33

accessing each actin strand is now of order 30, varying from 26 to 33 with sarcomere length over the plateau of the tension-length curve (Fig. A1c). These numbers virtually demand that dimers be able to access three-site target zones, especially when it comes to devising theories of thin-ﬁlament regulation. In this book, tension and other extensive quantities are generally normalised to one actin ﬁlament per half-sarcomere. However, experimental results are often normalised to one myosin head, using molar S1 concentrations of 0.15–0.20 mM; this carries an implicit assumption (intended or not) that all heads are participating in the contraction process. Where does this number come from? Two methods are described by He et al. (1997). The molar concentration of S1 heads in the ﬁbre can always be determined by chemical analysis. However, when the lattice spacing has been determined by X-ray diffraction, it is instructive to make the calculation on a structural basis, with a known number Mtot of heads per myoﬁlament. If d10 is thep spacing between the (1,0) ﬃﬃﬃﬃ planes, the area A of the rhombohedral unit cell is 2d 210 = 3: Because there is one myoﬁlament per unit cell, the areal density of myoﬁlaments is 1/A. An oft-quoted estimate is 1/A 0.5 1015 m2, which implies that d10 41.6 nm. Somewhat smaller values apply for intact relaxed muscles of frog and rabbit (Table I of Millman 1991), but measurements are often made on skinned ﬁbres, which swell slightly even in standard Ringer’s solution. Starting from unit area of a half-sarcomere of length L, the number of heads per unit volume of the muscle ﬁbre is fMtot/AL molecules/m3 after allowing for the volume fraction f of ﬁlament lattice in the ﬁbre, the remainder being occupied by mitochondria and other vesicles. In SI units, the molar concentration is [S1] ¼ ( fMtot/AL)/(6.02 1026) moles/m3, to be converted to moles/litre. With f ¼ 0.83, Mtot ¼ 294, L ¼ 1.3 nm (for rabbit psoas muscle) and the standard value for A, [S1] ¼ 0.156 mM.

2.3

The First Sliding-Filament Model

The ﬁrst quantitative model of muscle contraction based on sliding ﬁlaments was formulated by Sir Andrew Huxley (Huxley 1957). While this ground-breaking paper made many simplifying assumptions, it also deﬁned the form of a mathematical model of contraction which has stood the test of time and inspired many elaborations and improvements. The basic idea is that myosin heads go through cycles of binding and detachment, and tension is created when they are bound to F-actin. If muscle length is held constant, net tension is developed; if the muscle is unloaded, the tension in each head is reduced as the actin ﬁlament moves towards the M-band, and the detached head seeks another actin site nearer to the Z-line. A generic form of Huxley’s model can be deﬁned as follows: 1. The interdigitating myosin and actin ﬁlaments are rigid and slide against each other. 2. Myosin-S1 heads make independent attachments to F-actin.

34

2 Of Sliding Filaments and Swinging Lever-Arms

3. The vernier assumption discussed in Sect. 2.2.2; the axial spacings X between myosin heads and their nearest actin sites are uniformly distributed over the range (b/2, b/2), where b ¼ 36 nm is the F-actin half-period. 4. The rates f(X) and g(X) of binding and detachment vary with the head-site spacing X to deﬁne a binding range lying within (b/2, b/2). 5. The elastic strain in a bound head is equal to the head-site spacing X, and the tension t(X) obeys Hooke’s law. 6. The cycle of myosin-actin attachment is not in thermodynamic equilibrium; binding and dissociation are irreversible transitions driven by the hydrolysis of one molecule of ATP per cycle. 7. The above behaviour is replicated in the other half-sarcomere and in all other sarcomeres along the muscle. While these features of the model can be challenged, they can be considered as generic because they point the way to future developments. The more speculative aspects of the model have to do with the kinetics of myosin-actin attachments and how they are affected by head-site spacings. Assumption (iii) implies that each myosin head sees only one actin site available for binding. Assumption (iv) implies just two states per head (free or bound), despite the changes caused by ATP binding, hydrolysis and product release. Hence the kinetics of the mth head are described by a single rate equation for the probability pm(t) of the bound state at time t: p_ m ðt Þ ¼ f ðX m ðt ÞÞð1 pm Þ gðX m ðt ÞÞpm :

ð2:17Þ

Here {Xm, m ¼ 1,. . . .,M} are the head-site spacings for M heads addressing one actin ﬁlament in the half-sarcomere. Under isometric conditions, the spacings are constant. If the half-sarcomere length L is changing, then X m ðt Þ ¼ X m ð0Þ þ ΔLðt Þ:

ð2:18Þ

This formulation may be thought of as Lagrangian, because it identiﬁes speciﬁc myosin heads and tracks their dynamics in time. Throughout this book, extensive variables such as tension are calculated in one half-sarcomere and normalised to one F-actin ﬁlament. Thus the net tension per F-actin is T ðt Þ ¼

M X

tðX m ðt ÞÞpm ðt Þ

ð2:19Þ

m¼1

where t(X) is tension in the bound head (or crossbridge), which is transmitted to both ﬁlaments. The high-frequency stiffness of the muscle, measured by applying a small oscillatory length perturbation, samples the derivative of t(X):

2.3 The First Sliding-Filament Model

Sð t Þ ¼

35 M X

t0 ðX m ðt ÞÞpm ðt Þ

ð2:20Þ

gðX m ðt ÞÞpm ðt Þ:

ð2:21Þ

m¼1

and the net rate of detachment is Rðt Þ ¼

M X m¼1

Under steady-state conditions, R is also the rate of binding, which under assumption (vi) is also the rate of hydrolysis of ATP. For analytical developments, it is convenient to convert these formulae to integrals over a continuous distribution of X. With discrete heads, the vernier assumption implies that the head-site spacings advance by δX ¼ b/M over the range (b/2, b/2). A continuum version of the vernier model is deﬁned by taking the limit M ! 1 and δX b/M ! 0. At this point, the formalism is the same as that proposed by Euler, which is almost universally used in ﬂuid mechanics; the Eulerian description does not identify speciﬁc heads but works with the probability p(X,t) that whatever head has head-site spacing X at time t is bound. The corresponding rate equation for p(X,t) follows by treating X and t as independent variables, and the derivative following the motion is DpðX; t Þ ∂pðX; t Þ ∂X ∂pðX; t Þ ¼ þ Dt ∂t ∂t ∂X

ð2:22Þ

where L(t) is half-sarcomere length and dX/dt ¼ dL/dt for all heads in overlap with F-actin. Hence ∂p ∂p vð t Þ ¼ f ðX Þð1 pÞ gðX Þp ∂t ∂X

ð2:23Þ

where v(t) ¼ dL/dt is the instantaneous shortening velocity. Which head sees a site at distance X depends upon whether muscle length is stationary or changing. However, there is some irony in noting that a numerical solution to this partial d.e. can be found with Cauchy’s method of characteristics (Webster 1955), which brings us right back to Eq. 2.17 and the Lagrangian formulation. In the Eulerian picture, the constitutive equations are T ðt Þ ¼ and similarly for S(t), R(t).

M b

Z

b=2

b=2

tðX ÞpðX; t ÞdX

ð2:24Þ

36

2 Of Sliding Filaments and Swinging Lever-Arms

The model is simple and analytic solutions are available. Under isometric conditions, v ¼ 0 and p(X,t) ¼ p0(X) f(X)/( f(X) + g(X)), generating isometric tension T0, stiffness S0 and so on. For steady-state shortening, Eq. 2.23 has the general solution pðX Þ ¼ pðb=2Þe

λðX Þ

Z

b=2

þ

0

f ðX 0 ÞeλðX ÞλðX Þ dX 0 =v

ð2:25Þ

X

where Z rðXÞ ¼ f ðXÞ þ gðXÞ, λðXÞ ¼

b=2

0

0

rðX ÞdX =v:

ð2:26Þ

X

What are the appropriate boundary conditions? During shortening, the X value of a given head is decreasing at velocity v. During the time taken for X to traverse one half-period b of the actin double-helix, that head will bind to actin and then detach, ready for binding when it enters the next actin half-period nearer the Z-line. Thus the head that leaves its half-period at X ¼ b/2 enters the next half-period with X ¼ b/2 (Fig. 2.6). Here the analogy between myosin heads and rowers in a boat is very much to the point. If one rower still has his/her oar in the water when the boat is ready for the next stroke, smooth rowing will cease and the boat will slow or stop. Unlike a good rowing crew, myosin rowers are not synchronised; because of the vernier effect their head-site spacings traverse the binding range at different times. Nevertheless they must all ‘have their oars out of the water’ before actin sites on the next halfperiod come into view. This is another deﬁning constraint for Huxley’s model: (viii) The model must allow continuous shortening under a reduced load. Let X+ and X be strict upper and lower limits of the myosin binding range, such that p (X+) ¼ p(X) ¼ 0 and jX j < b/2. From the myosin with head-site spacing X, the distance to the actin site in the next half-period towards the Z-line is X + b (Fig. 2.6a). Any myosin bound to that site must have detached before the actin ﬁlament carries it within range of the ﬁrst myosin; for this to occur there must be a zone of detachment in the range (X+, X+b). Such a zone exists only if X þ X < b:

ð2:27Þ

This condition is certainly satisﬁed if all heads are detached at each end of the vernier range: pðb=2Þ ¼ 0,

pðb=2Þ ¼ 0:

ð2:28Þ

The ﬁrst equation is the boundary condition for solving Eq. 2.23. The second equation should be regarded as a constraint on model parameters, which ensures that the detached head at X ¼ b/2 can be remapped to X ¼ b/2 as the actin site on the next half-period becomes the nearest one. If p(b/2) > 0, then zero binding at each end of the range of X can sometimes be restored by shifting the vernier to

2.3 The First Sliding-Filament Model

37

Fig. 2.6 Continued shortening requires a zone of detachment outside the myosin binding range X < X < X+ in each actin half-period (|X| < b/ 2 ¼ 18 nm). Two heads seeing adjacent half-periods, with X2 ¼ X16.9 nm, are shown at different stages of shortening. In the top diagram the second head is about to detach (X2 ¼ X). After more shortening, the middle diagram shows the ﬁrst head about to detach (X1 ¼ X). Finally, the bottom diagram shows the ﬁrst head about to bind to the site made available by the second head (X1 + b ¼ X+). Thus binding of head 1 to site 2 follows detachment of head 2 from site 2 if X > X+b

(Xmaxb, Xmax) where Xmax < b/2. If the domain of p(X) is still too wide, a window of detachment may still exist but some heads entering the vernier at X ¼ Xmax may be bound to the old site with strain Xmaxb, with a much lower tension. This is not illegal, but it does create an awkward accounting problem in calculating net tension, which is best avoided. Finally, if the binding range under shortening really does exceed the half-period b, then the simple vernier model fails, and a larger vernier over two actin half-periods, with multiple actin sites in the vernier range, will be required. An example appears in Chap. 4. These equations complete the generic description of A.F. Huxley’s 1957 model. For quantitative predictions, explicit forms are required for the rate functions f(X), g (X) and tension t(X) per head. Several forms were investigated, but Huxley’s preferred functions are f(X) ¼ f1X/d in 0 < X < d and zero otherwise, g(X) ¼ g1X/d for X > 0 and g(X) ¼ g2 for X < 0, t(X) ¼ κX (Fig. 2.7). There are ﬁve disposable constants f1, g1, g2, d and κ. The use of Hooke’s law for head tension t(X) is an obvious starting-point, but net tension is created by forbidding heads with X < 0 to bind. How this might happen is not explained.

38

2 Of Sliding Filaments and Swinging Lever-Arms

Fig. 2.7 The strain-dependent kinetics of A.F. Huxley’s model, and its predictions. (a) The rates f (X) and g(X) of actin binding and detachment. (b) The probability of binding when the ﬁlaments are stationary, and shortening at a ﬁxed velocity (Eq. 2.30). (c) Tension, stiffness and ATP-ase rate as a function of shortening velocity v (Eqs. 2.31), normalised to the unloaded velocity v0 ¼ 946 nm/s predicted by the model with the original parameter values f1 ¼ 42.7 s1, g1 ¼ 9.85 s1, g2 ¼ 205.9 s1 and d ¼ 9 nm. (d) The rise of tension and stiffness after the start of tetanic stimulation (Eqs. 2.32), as a function of dimensionless time ( f1 + g1)t

Analytic solutions of this model are as follows. For isometric conditions, p0(X) ¼ f1/( f1 + g1) for 0 < X < d and zero elsewhere, giving isometric tension, stiffness and ATP rate as T0 ¼

Mκd 2 p0 , 2b

S0 ¼

Mκdp0 , b

R0 ¼

Mg1 dp0 : 2b

ð2:29Þ

2.3 The First Sliding-Filament Model

39

For steady shortening at velocity v, f1 r1 d g X 1 exp exp 2 2v v r1 f r 1 ðX 2 d 2 Þ pðXÞ ¼ 1 1 exp 2dv r1

pðXÞ ¼

pðXÞ ¼

ðX < 0Þ, ð0 < X < dÞ

ð2:30Þ

ðX > dÞ:

0

where r1 ¼ f1 + g1. The exponential decay for negative X is generally sufﬁcient to reduce p(b/2) to zero (Fig. 2.7). At velocity v, net tension, stiffness and ATP rate satisfy ( ) T ð vÞ v v2 þ ¼12 f1 expðr 1 d=2vÞg T0 r 1 d ðg2 d Þ2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Sð v Þ v 1 expðr 1 d=2vÞ 2v=r 1 dF ¼1þ r 1 d=2v S0 g2 d R ð vÞ 2f v ¼ 1 þ 1 ð1 expðr 1 d=2vÞÞ R0 g1 r 1 d

ð2:31aÞ ð2:31bÞ ð2:31cÞ

where Z F ð xÞ ¼

x

exp t 2 x2 dt

0

is Dawson’s integral (Abramowitz and Stegun 1965). By now you may have noticed that Huxley’s model can be cast in dimensionless form by choosing d, κd and r1d as units of distance, force and velocity, leaving just two dimensionless parameters, say f1/g1 and g2/g1. This model was very successful in predicting tension and efﬁciency as a function of shortening velocity v. By setting f1/g1 ¼ 13/3 and g2/g1 ¼ 20.9, the tension-velocity curve can be ﬁtted to Hill’s equation with a/T0 ¼ 0.25 for frog muscle (Fig. 2.1a) by normalising the velocity to the unloaded value predicted by Eq. 2.31a. It predicts that the number of bound heads, and hence muscle stiffness, decreases with velocity but remains ﬁnite at v ¼ v0, indicating an equal mixture of positively and negatively strained bound heads. The predicted ATPase rate rises by a factor of ﬁve as the speed of shortening is increased. The model can also be tested against experimental values of isometric tension, stiffness and ATPase. For frog muscle at 3 C, T0 ¼ 290 pN, T0/S0 ¼ 4.5 nm (Ford et al. (1977) and R0 150 s1 (Kushmerick and Davies 1969) per actin ﬁlament per half-sarcomere. In Huxley’s model, T0/S0 ¼ d/2 (Eq. 2.29), so d ¼ 9 nm. By using a known value of head stiffness κ, the formula for T0 allows us to

40

2 Of Sliding Filaments and Swinging Lever-Arms

estimate the number of available heads M per F-actin. Estimates of κ have had an interesting history, having risen consistently from 0.25 pN/nm (Huxley and Simmons 1971) to about 3 pN/nm over the last four decades. Based on new experimental techniques, current values for frog and rabbit muscle are 3.2 and 2.7 pN/nm respectively (Piazzesi et al. 2007, Kaya and Higuchi 2010). If κ ¼ 3.0 pN/nm for frog muscle, then M ¼ 2bT0/κd2p0 ¼ 106. If M is reset to 265 as in Fig. 1.6b, it is clear that Huxley’s model is very good at generating tension. The isometric ATP-ase rate can be ﬁtted if g1 ¼ 14 s1. Analytical expressions for the rise of tension, stiffness and ATPase rate after the onset of an isometric tetanus are also available. The bound-state probability p(X,t) is proportional to 1-exp(r(X)t), which gives T ðt Þ Rðt Þ 2 2 1 ¼ ¼1 þ 1 þ expðr 1 t Þ T0 R0 r1 t ðr 1 t Þ2 r 1 t

ð2:32aÞ

Sð t Þ 1 ¼1 ð1 expðr 1 t ÞÞ: S0 r1 t

ð2:32bÞ

As a function of r1t there are no free parameters, and the mixture of exponentials and power laws is a consequence of X-averaging. Experimentally, the key observation is that stiffness rises faster than tension, for example with a half-rise time of 20 ms compared with 40 ms (Cecchi et al. 1991). However, Eqs. 2.32 show contrary behaviour; the predicted tension rises faster than stiffness (Fig. 2.7d), presumably because heads with large X bind more rapidly. These difﬁculties suggest that the binding rate f(X) should not increase with X or that the detachment rate g1(X) should increase faster than linearly. It seems likely that heads with X < 0 should also be allowed to bind to actin. In conclusion, it must be said that such difﬁculties should not be allowed to detract from the great utility of Huxley’s mathematical model. Moreover, there is really no ground for supposing that his speciﬁc assumptions about the rate functions f(X), g(X) are a priori unreasonable. Here are two ways in which heads could bind to distant sites. Firstly, they should be able to swing about their S1-S2 junction. If this junction were rigid, current values of head stiffness κ would make it impossible for Brownian forces to stretch the neck region by 9 nm; with κ ¼ 3.0 pN/nm the expected r.m.s. thermal displacement (kBT/κ)1/2 is (4/3)1/2 ¼ 1.15 nm. Perhaps the S1-S2 junction is ﬂexible only for swings in the forward direction. Secondly, the S2 rod can buckle in compression (Kaya and Higuchi 2010), which will allow binding at more negative X values. Non-symmetric binding kinetics might also be due to axially non-symmetric actin monomers; imagine a distribution of ﬁxed or induced charges which would bias myosin binding to positively-displaced sites, analogous to the induction principle for magnetically-levitated trains. The more fundamental question raised by A.F. Huxley’s model is whether net isometric tension in muscle is developed as a result of strain-based kinetics or whether there is a purely mechanical mechanism for tension generation. Such

2.4 The Swinging-Lever-Arm Mechanism

41

questions were answered deﬁnitively in 1971 by the discovery of just such a mechanism: the myosin-S1 motor is a ‘rower’ because its neck region acts as a swinging-lever arm. This is the subject of the next section.

2.4

The Swinging-Lever-Arm Mechanism

In 1971, Huxley and Simmons performed a key experiment which suggested that myosin-S1 makes a unitary force-generating event after binding to actin. The activated ﬁbre was subjected to a rapid shortening step of 0.2–1%, equivalent to 2–10 nm per half-sarcomere. This caused an immediate fall in tension, followed over a few milliseconds by a recovery phase which varied in a characteristic way with the amount of shortening. They interpreted this recovery as the signature of a ‘rowingstroke’ action of myosin-S1, now known as the working stroke. This mechanism was the trigger for many experimental studies, culminating in the determination of the atomic structure of myosin-S1 by Ivan Rayment and colleagues in 1993. Myosin-S1 has a 10 nm lever arm attached to a globular motor domain which binds to actin, and the working stroke is generated by the lever arm swinging against its motor domain, while the distal tip of the lever arm is ﬂexibly attached to the myosin-ﬁlament backbone. For this process to pull an actin ﬁlament, the motor domain must be rigidly bound to actin (Fig. 2.8). The phenomenology of this classic length-step experiment, summarised in Fig. 2.9, is widely known. The immediate tension response from isometric tension T0 to tension T1(ΔL ) (phase 1) is elastic and essentially linear in the size of the length step ΔL. At the end of rapid recovery (phase 2), the tension T2(ΔL ) is closer to T0

Fig. 2.8 Cartoons showing the working stroke of myosin-S1 bound to actin, where the neck region acts as a lever arm. (a) The pre-stroke (A-state) head bound to a site at distance x from the detached head (dotted outline), and bearing tension tA(x) ¼ κx. (b) The R-state head after a working stroke h, bearing tension tR(x) ¼ κ(x+h), showing the detached post-stroke head. Under isometric conditions, the ﬁlaments are ﬁxed, and the lever arm bends as its junction with the motor domain swings through the working stroke. For clarity, the neck region is shown as uniformly bent, but the elastically compliant region may be close to the motor domain (Kohler et al. 2002). With unloaded ﬁlaments, the working stroke is revealed as a displacement of the actin ﬁlament towards the M-band

42

2 Of Sliding Filaments and Swinging Lever-Arms

Fig. 2.9 Observations of Ford et al. (1977) for rapid tension recovery following length steps of various sizes. (a) The four-exponential curves, constructed from their Eq. 2.5, which ﬁt the tension transients. (b) The tension T2 after rapid recovery, and initial tension T1 after the length step. (c) The rate of phase-2 recovery, as the reciprocal of the half-recovery time τ0.5, is a very asymmetric function of step size. With permission of John Wiley and Sons Inc. and R.M. Simmons for graphs b, c

and varies in a characteristic way with ΔL. For release steps, the rate of recovery r2(ΔL ) increases dramatically with the size of the release, and for stretches the recovery rate drops with the size of the stretch. The interpretation of these experiments is that a quick release causes some bound myosins to make a working stroke, thereby increasing the tension, and a quick stretch causes post-stroke heads to make a reverse stroke, with tension decreasing. The ﬁnal tension and the rate are very asymmetric functions of step size. To ﬁt their data, Huxley and Simmons devised a semi-empirical theory in which all myosin heads see the same reaction-energy proﬁle for the working stroke, namely

2.4 The Swinging-Lever-Arm Mechanism

43

a quadratic variation in lever-arm angle θ with sharp potential wells at each end to deﬁne the pre-stroke and post-stroke states. This model predicts that r 2 ðΔLÞ ¼ k f1 þ expðαΔLÞg, h αΔL T 2 ðΔLÞ ¼ T 0 þ κ ΔL tanh 2 2

ð2:33aÞ ð2:33bÞ

where α κh/kBT and k is the rate of the backward stroke. Here κ is the axial stiffness of the head, just as in the previous section. The rate function ﬁts their data if α ¼ 0.5 nm1. If h ¼ 8 nm, this implies that κ ¼ 0.25 pN/nm, which is an order of magnitude smaller than later values from direct measurements. Nevertheless, this was the ﬁrst estimate of myosin stiffness. The T2 function also ﬁts their data, which requires that the slope for small steps is small and positive. In this region T2(ΔL) T0 ¼ κ(1 αh/4)ΔL+O(ΔL3), and with the above values the slope at the origin is zero. As the authors point out, higher values of κ are not permitted because the T2 curve would show a region of negative slope. The authors also suggested that the actual working stroke might occur as two consecutive 4 nm sub-strokes, a piece of blue-sky thinking that became mandatory when a more comprehensive model was developed. To understand the length-step response from a modern perspective, the argument of Huxley and Simmons should be reconstructed from a different starting point. Instead of assuming that all myosin heads react in the same way to a length-step perturbation, it is better to suppose that bound myosin heads initially have a distribution of elastic strains, as in Huxley’s 1957 model. Even before a length step is applied, some of them may have already made a working stroke, and will be in a rigor-like (R-state) conﬁguration. Others will have been unable to do this and will be sitting in an attached state (A-state). A bound head will make a stroke when it is energetically favourable to do so, meaning that its Gibbs energy is lower in the ﬁnal state. However, that Gibbs energy must include the potential energy of elastic strain. The result is that the pre-stroke strain should be more negative than the poststroke strain is positive. Here the working stroke h is deﬁned as the increase in axial strain X in the head when it makes a stroke under isometric conditions, as in Fig. 2.8. The origin of X can be chosen so that its pre-stroke and post-stroke strains when bound to actin are X and X + h respectively. Assuming that Hooke’s law applied, the tensions and strain energies in these two states are tA ðX Þ ¼ κX,

t R ð X Þ ¼ κ ð X þ hÞ

1 1 vA ðX Þ ¼ κX 2 , vR ðX Þ ¼ κðX þ hÞ2 : 2 2 Thus

ð2:34aÞ ð2:34bÞ

44

2 Of Sliding Filaments and Swinging Lever-Arms

GA ðX Þ ¼ GA þ vA ðX Þ,

GR ðX Þ ¼ GR þ vR ðX Þ

ð2:35Þ

are the Gibbs energies including elastic strain energy. For a head to make a working stroke, ΔGS(X) GR(X) GA(X) < 0, or ΔGS þ vR ðX Þ vA ðX Þ < 0

ð2:36Þ

where ΔGS is the energy change for an untethered head, and ΔGS < 0. Now vR(X) vA(X) ¼ κh(X+h/2), so the condition for a stroke is that h ΔGS : X < X∗ 2 κh

ð2:37Þ

R-state heads are stable for X < X*, whereas A-state heads are stable for X > X* because the strain-energy cost of the working stroke exceeds the drop in chemical energy. It follows that the isometric state will have a mixture of A-state and R-state heads provided that X* lies within the binding range. Then a quick release generates tension recovery by triggering working strokes in A-state heads whose strain is decreased from X to X|ΔL| < X*. Similarly, a quick stretch causes tension to fall by triggering reverse strokes in R-state heads whose strain is increased from X+h to X+ΔL +h > X∗+h. That phase-2 recovery is observed for releases and stretches proves that the isometric state contains a mixture of pre-stroke and post-stroke heads. Moreover, a length step triggers synchronized strokes only for those heads in a strain band of width ΔL that cross the threshold strain X*.

2.4.1

Mechanokinetics of the Working Stroke

What can be said about the rate of tension recovery? At the level of a single myosin head, how is the basal rate of the stroke reaction modiﬁed by elastic strain? Mechanokinetics is a subject which lies at the core of the molecular mechanism of contractility, and it will appear time and again throughout this book. The kinetics of the working stroke provides the simplest example which can be worked through in detail. As always in chemical kinetics, the starting point for modelling is the Gibbsenergy landscape of the reaction, but in this case with the addition of elastic strain energy. What might this landscape look like if the bound head were not tethered to the myoﬁlament backbone? In a one-dimensional model of the stroke reaction, the reaction coordinate is the angle θ that the lever arm makes with F-actin. Huxley and Simmons (1971) proposed that this landscape is ﬂat apart from two narrow potential wells which deﬁne the A-state and R-state conﬁgurations. To stop the head from detaching, the energy of the plateau must be below zero. Let the depths of the wells be BA and BR respectively. Then a thermally-activated working stroke would proceed at a rate kS ¼ AA exp (βBA), and the backward stroke at rate

2.4 The Swinging-Lever-Arm Mechanism

45

kS ¼ AR exp (βBR), where AA and AR are attempt frequencies and β ¼ 1/ kBT. Moreover, Gibbs’ thermodynamic identity gives the equilibrium constant KS kS/kSas exp(βΔGS). However, in the muscle ﬁbre, the bound head is tethered and elastically strained in a way which varies with the reaction coordinate. Under Hooke’s law, the elastic energy of the head with lever-arm angle θ is of the form 1 vðθ; X Þ ¼ κ ðX þ Rð cos θ cos θA ÞÞ2 2

ð2:38Þ

where R is the length of the lever-arm and the A-state head is at angle θA. If θR is the angle of the R-state head, then this equation agrees with Eq. 2.34b if h ¼ Rð cos θR cos θA Þ:

ð2:39Þ

Setting θA ¼ 90 and θR ¼ 45 used to be a popular choice for modelling. In fact, X-ray measurements coupled with length steps suggest that θR < 90 < θA (Irving et al. 1995), for example θA ¼ 110 and θR ¼ 60 , which with R ¼ 10 nm gives h ¼ 8 nm for the working stroke. The reaction energy landscape is now a function of θ and X, as in Fig. 2.10, and the heights of the potential wells are BA + vA(X), BR + vR(X) respectively. The theory of reaction kinetics is highly developed (Weiss 1986), but an unsophisticated approach which goes back to Arrhenius is to assume that the rate of reaction is determined by the highest barrier in the reaction-energy landscape. This approximation can be improved on, but it does yield simple analytic formulae for the forward and backward stroke rates. Figure 2.10 shows that the location of the higher energy barrier varies with elastic strain, according to whether the A-state or R-state has the higher strain energy. With vR(X) vA(X) ¼ κh(X+h/2), we have the following scenarios: (i) For X > h/2 the higher barrier lies above the R-state well; the thermally-activated A ! R transition requires this extra energy to complete the stroke instead of falling back into the A-state well. On the other hand, the reverse transition proceeds by escaping from the R-state well which is already at higher strain energy, so there is no strain-energy cost to this reaction. (ii) When X < h/2, the higher barrier lies above the A-state well, so the A ! R transition proceeds with no cost in strain energy, whereas its reversal is inhibited by the extra strain energy of the A-state. Thus k S ðXÞ ¼ kS exp βκhðX þ h=2Þ , ðX > h=2Þ k S ðXÞ ¼ kS , ðX < h=2Þ kS ðXÞ ¼ kS ,

ðX > h=2Þ kS ðXÞ ¼ kS exp βκhðX þ h=2Þ , ðX < h=2Þ

ð2:40aÞ ð2:40bÞ

46

2 Of Sliding Filaments and Swinging Lever-Arms

Fig. 2.10 (a) A reaction-energy landscape for the myosin working stroke as a function of its leverarm angle θ, assuming sharp potential wells for the pre-stroke and post-stroke states (A and R) and an elastic lever arm with longitudinal stiffness κ for displacements of the distal end relative to the motor domain. When the head is untethered, the energy between the wells is assumed to be constant. For a tethered head, the shape of the reaction energy between wells reﬂects elastic strain in the lever arm, and the point of highest energy is determined by the pre-stroke strain X, which is also the axial displacement of the actin site from the unbound head. (b) rate constants for the forward and backward stroke transitions as a function of X

Figure 2.10 also shows that, for some values of X, the strain energy is zero in the middle of the stroke reaction, creating a shallow minimum in reaction energy; the above results require that the lever-arm does not dwell in this position. These formulae differ from those proposed by Huxley and Simmons (1971), who assumed that all heads have the same reaction rates, with only the forward rate being exponentially dependent on the size of the length step. This assumption was justiﬁed because the rate of tension response to a quick release increased exponentially with step size up to 6 nm, beyond which the response was too rapid to follow. However, Eq. 2.40a says that the forward rate is strain-sensitive only for X > h/2, which implies that the observed rate of recovery should ﬂatten out for larger releases.

2.4.2

Theory of the Rapid Length-Step Response

Now that a distributed-strain description of pre-stroke and post-stroke myosin heads and their stroke rates is available, this information can be assembled to model the tension response to a length step ΔL. To do this, it is not necessary to make assumptions about how myosin heads attach to actin; all that is required is the probability p0(X) that the head with axial offset X is bound. Whether it is bound in an A-state or R-state is determined by the equilibrium constant KS(X) of the stroke transition, which follows from Eqs. 2.40 or directly from Gibbs’ identity KS(X) ¼ exp (βΔGS(X)):

2.4 The Swinging-Lever-Arm Mechanism

47

k S ðX Þ ¼ K S expðβκhðX þ h=2ÞÞ k S ðX Þ

K S ðX Þ

ð2:41Þ

where KS ¼ exp (βΔGS). Hence, from Eq. 2.37, h kB T ln K S X∗ ¼ þ 2 κh

ð2:42Þ

which ties this quantity more closely to kinetics. With the stroke transition in reaction equilibrium, KS(X)/(1 + KS(X)) is the fraction of bound heads in the R-state. For this model, the net tension per F-actin is TðtÞ ¼

Mκ b

Z

b=2

b=2

fXpA ðX, tÞ þ ðX þ hÞpR ðX, tÞgdX

ð2:43Þ

in terms of the occupation probabilities of A and R states. Before the step, the A and R states are in equilibrium, so pA ð X Þ ¼

p0 ð X Þ , 1 þ K S ðX Þ

p R ðX Þ ¼

K S ðX Þp0 ðX Þ 1 þ K S ðX Þ

ð2:44Þ

where p0(X) is the net bound-state probability, so isometric tension is Mκ T0 ¼ b

Z

b=2

b=2

X þ K S ðX ÞðX þ hÞ p0 ðX ÞdX: 1 þ K S ðX Þ

ð2:45Þ

Equation 2.43 gives the tension after the step, provided we remember that X is now A-state strain after the step. The state probabilities satisfy p_ A ðX; t Þ ¼ kS ðX ÞpA þ k S ðX ÞpR , p_ R ðX; t Þ ¼ kS ðX ÞpA k S ðX ÞpR :

ð2:46Þ

which reduce to a single d.e. because pA ðX; t Þ þ pR ðX; t Þ ¼ p0 ðX ΔLÞ is a conserved quantity, and X–ΔL is the strain before the step. Hence

ð2:47Þ

48

2 Of Sliding Filaments and Swinging Lever-Arms

pA ðX; t Þ erS ðX Þt 1 erS ðX Þt ¼ þ , p0 ðX ΔLÞ 1 þ K S ðX ΔLÞ 1 þ K S ðXÞ

pR ðX; t Þ K S ðX ΔLÞerS ðX Þt K S ðX Þ 1 erS ðX Þt ¼ þ p0 ðX ΔLÞ 1 þ K S ðX ΔLÞ 1 þ K S ðX Þ

ð2:48Þ

where rS(X) ¼ kS(X)+kS(X). From Eq. 2.43, the tension response is Z Mκ b=2 X þ K S ðX ÞðX þ hÞ p0 ðX ΔLÞ 1 erS ðX Þt dX b b=2 1 þ K S ðX Þ Z b=2 Mκ X þ K S ðX ΔLÞðX þ hÞ p0 ðX ΔLÞerS ðX Þt dX: þ b b=2 1 þ K S ðX ΔLÞ

T ðt Þ ¼

ð2:49Þ

In particular, tensions T1 and T2 follow by setting t ¼ 0 and 1 respectively: T 1 ¼ T 0 þ S0 ΔL, T2 ¼

Mκ b

Z

b=2

b=2

S0 ¼

Mκ b

Z

b=2

p0 ðX ÞdX

ð2:50Þ

X þ K S ðX ÞðX þ hÞ p0 ðX ΔLÞdX 1 þ K S ðX Þ

ð2:51Þ

b=2

and the tension transient can be calculated conveniently from T ðt ÞT 2 ¼

Mκh b

Z

b=2 b=2

K S ðX ΔLÞ K S ðX Þ p0 ðX ΔLÞerS ðX Þt dX 1 þ K S ðX ΔLÞ 1 þ K S ðX Þ ð2:52Þ

which follows from Eqs. 2.49 and 2.51. These formulae becomes transparent if we note that the stroke constant KS(X) is a rapidly varying function of X. As KS(X) ¼ exp(βκh(X X∗)), hence K S ðX Þ 1 ¼ 1 þ K S ðX Þ expðβκhðX X ∗ ÞÞ þ 1

ð2:53Þ

is a Fermi-Dirac function, switching from 1 to 0 as X increases above X*. The switching range is kBT/κh, which with κ ¼ 3.0 pN/nm and h ¼ 8 nm is a tiny 0.17 nm compared to the binding range, which could be as high as 10 nm. Thus the quantity in brackets in Eq. 2.52 can be set equal to sgn(ΔL ) for X 2 (X∗, X∗+ΔL ) and 0 elsewhere, so

2.4 The Swinging-Lever-Arm Mechanism

T ðt Þ T 2 ﬃ

Mκh b

Z

49 X ∗ þΔL

p0 ðX ΔLÞerS ðX Þt dX:

ð2:54Þ

X∗

Thus the approach to the ﬁnal phase-2 tension is generated only by those heads that are driven past the stroking threshold X ¼ X* by the length step. Closed formulae for phase-2 recovery can be obtained if p0(X) is approximated by a box function, constant within its binding range (XR, XA) where XR < X* < XA. Then Mκp0 2 X A X 2R þ 2hðX ∗ X R Þ 2b Mκp0 T1 ¼ T0 þ ðX A X R ÞΔL b 0 1 X XA, ΔL < X ∗ X A Mκhp0 @ ∗ ΔL, X ∗ X A < ΔL < X ∗ X R A T1 T2 ¼ b X∗ XR, ΔL > X ∗ X R T0 ¼

ð2:55aÞ ð2:55bÞ ð2:55cÞ

Thus T1(ΔL ) T2(ΔL) is linear in ΔL for small steps, but constant below the critical value X* XA < 0 for which all A-state heads make a stroke, and constant above the critical value X* XR > 0 for which all R-state heads make a reverse stroke. These formula combine to give tension T2(ΔL ), and the length step for which T2 ¼ 0 is XA þ XR ΔL2 ¼ h þ 2

ð2:56Þ

on using the ﬁrst entry in Eq. 2.55c. It has been claimed that |ΔL2| is equal to the working stroke h, but this is not exactly true when the distributed strains of bound heads are accounted for. This equation can be understood as follows: a release of this size forces all heads to make a stroke, and the centre of their X-distribution, at (XA+XR)/2 |ΔL2|, must be equal to –h for zero net tension. In general, XR XA, so |ΔL2| does provide a rough estimate of the stroke distance. However, Eq. 2.56 is only as good as the box approximation for p0(X). The time course of recovery follows from Eq. 2.54, which can be evaluated in terms of the exponential integral E1(x) (Press et al. 1992). The normalised recovery function is F ðτ; δLÞ where

T ðt Þ T 2 T1 T2

ð2:57Þ

50

2 Of Sliding Filaments and Swinging Lever-Arms

eτ δL

E 1 ðτ Þ , E1 τ e δL > δL∗ , ð2:58aÞ δL eK S τ eτ Fðτ, δLÞ ¼ ðδL δL∗ Þ, δL < δL∗ ð2:58bÞ E 1 ðτeδL∗ Þ E1 ðτÞ þ δL δL

F ðτ; δLÞ ¼

in terms of dimensionless coordinates τ ¼ kSt and δL ¼ βκhΔL. Also, δL∗ ¼ ln KS is the dimensionless form of the release step which moves heads from X* to –h/2 where the forward stroke rate becomes ﬂat. Equations 2.58 hold for moderate length steps such that the region of integration lies inside the box, and the second term in (Eq. 2.58b) is an approximation valid when KS 1. As a demonstration of plausibility, Fig. 2.11 shows how the box approximation performs for the amplitude and rate of recovery as a function of step size, taking XA ¼ XR ¼ 4.5 nm and KS ¼ 100. In Fig. 2.11a are the predicted forms of T1/T0 and T2/T0 for κ ¼ 3.0 pN/nm, h ¼ 8 nm, while Fig. 2.11c shows the time course of the normalised recovery function, plotted on a logarithmic scale to test how many exponential terms might be involved. Comparisons with Fig. 2.9 reveal serious differences with the experimental data. The predicted plots of T1 and T2 are too steep. The predicted time course of recovery shows fast and slow exponential components, as seen by Ford et al. (1977), but the rates of both components do not increase with the size of the release step. What has gone wrong here? There are at least two answers to this problem, which introduce new ideas. Firstly, the combination of a 3 pN/nm myosin stiffness and an 8 nm working stroke cannot quite generate enough isometric tension; with the above values of κ and h, X∗ ¼ 3.23 nm which barely lies inside a typical binding range of 4.5 nm, and Eq. 2.55a gives T0 ¼ 180 pN with p0 ¼ 0.8 and M ¼ 265. Modellers generally avoid these problems by supposing that the working stroke occurs in two steps of 4 nm, but there is little evidence for the existence of the intermediate state. Secondly, the ﬁlaments are not completely rigid (Huxley et al. 1994, Wakabayashi et al. 1994). Under isometric tension, actin ﬁlaments stretch by about 3 nm, which allows about half of any change in sarcomere length to be absorbed in F-actin. Thirdly, the lack of variation of recovery rate with step size may be due to the highest-energy-barrier approximation. For quick releases, Eqs. 2.58 predicts that the fast rate increases only when |ΔL| < |ΔL∗|. With κ ¼ 3.0 pN/nm, h ¼ 8 nm and KS ¼ 100, jΔL∗j ¼ 0.77 nm, which does not allow increased recovery rates out to 6 nm. A rough quick ﬁx can be made by considering the response to two 4 nm strokes, where the time course of recovery is calculated by considering just one 4 nm step. Let us assume that two 4 nm strokes in series will double the isometric tension obtainable from one 8 nm stroke, while the time course of phase-2 recovery can still be calculated from Eqs. 2.55 with h ¼ 4 nm. In addition, ΔL should be reinterpreted as half of the imposed step size per half-sarcomere as a result of elastically compliant ﬁlaments (mainly F-actin). Then the results are similar to observations on frog muscle. In Fig. 2.11b, the T1 line now intercepts the axis at ΔL1 ¼ 4.5 nm as observed by Ford et al. (1977), and the T2 curve, apart from kinks generated by the box approximation, is also much closer to the experimental curve.

2.4 The Swinging-Lever-Arm Mechanism

51

Fig. 2.11 Predictions of an approximate analytic theory of the phase-2 length-step response (Eqs. 2.55, and 2.57), showing tensions T1/T0, T2/T0 before and after rapid recovery (a and b) and the normalised time course of recovery (c and d) from shortening steps (6, 4.5, 3, 1.5 nm) as a function of dimensionless time τ. The logarithmic ordinate reveals the presence of at least two exponential components. (a and c) were generated for κ ¼ 3.0 pN/nm and a single 8 nm working stroke, with binding limits XA ¼ XR ¼ 4.5 nm and KS ¼ 100. (b and d) were generated by a 4 nm working stroke, but isometric tension was doubled to simulate the effects of two such strokes. Also, length steps per half-sarcomere were halved to simulate the effects of elastically compliant F-actin

Moreover, for release steps Fig. 2.11d shows that the faster recovery rate now increases with the size of release. A further improvement can be made by replacing the highest-energy-barrier approximation by a more sophisticated theory of the stroke reaction. A better theory is available, namely the Kramers-Smoluchowski theory of overdamped unimolecular reactions. It yields an analytic expression for the stroke rate kS(X) (Smith and Sleep 2006), but the integration in Eq. 2.52 has to be performed numerically. There is a slight improvement in the expected increase of recovery rate with release size, because the unaveraged rate kS(X) is now a decreasing function over the whole range of X.

52

2 Of Sliding Filaments and Swinging Lever-Arms

Although this theory of the rapid length-step response is simple and exact, at the same time it is incomplete because it cannot specify the true form of the probability p0(X) that a head with site offset X is bound to actin. Huxley and Tideswell (1996), on the basis of a more complete model in which actin binding and detachment reactions were speciﬁcally included, arrived at the same conclusion, that two strokes and ﬁlament compliance are both necessary to explain the observed length-step response. Although heads are not expected to detach from actin during the phase-2 process, both formulations are based on the vernier approximation with a single actin site rather than a three-site target zone per myosin head. These and other challenges to simple theories are considered in Chap. 4. Having got this far, the next step is to formulate comprehensive models of contractility which include all the reaction steps in which myosin-S1 binds to actin, makes a working stroke, and then detaches. How do these reactions dovetail with the hydrolysis of ATP and the release of the product of hydrolysis? Now it is time to turn to the biochemistry of the actin-myosin-ATPase, which for the most part has been explored with the proteins in solution rather than the muscle ﬁbre.

References Abramowitz M, Stegun IA (1965) Handbook of mathematical functions. Dover Pubs. Inc., New York Barclay CJ, Constable JK, Gibbs CL (1993) Energetics of fast- and slow-twitch muscles of the mouse. J Physiol (London) 472:61–80 Cecchi G, Grifﬁths PJ, Bagni MA, Ashley CC, Maeda Y (1991) Time-resolved changes in equatorial x-ray diffraction and stiffness during rise of tetanic tension in intact length-clamped single muscle ﬁbres. Biophys J 59:1273–1283 Ford L, Huxley AF, Simmons RM (1977) Tension responses to sudden length change in stimulated frog muscle ﬁbres near slack length. J Physiol (London) 269:441–515 Hawkins CJ, Bennett PM (1995) Evaluation of freeze substitution in rabbit skeletal muscle. Comparison of electron microscopy to X-ray diffraction. J Muscle Res Cell Motil 16:303–318 He ZH, Chillingworth RK, Brune M, Corrie JET, Trentham DR, Webb MR, Ferenczi MA (1997) ATPase kinetics on activation of rabbit and frog permeabilised isometric muscle ﬁbres: a realtime phosphate assay. J Physiol (London) 501(1):125–148 He ZH, Chillingworth RK, Brune M, Corrie JET, Webb MR, Ferenczi MA (1999) The efﬁciency of contraction in rabbit skeletal muscle ﬁbres, determined from the rate of release of inorganic phosphate. J Physiol (London) 517(3):839–854 Hill AV (1938) The heat of shortening and the dynamic constants of muscle. Proc R Soc B126:136–195 Hill AV (1953) The mechanics of active muscle. Proc R Soc B 141:104–117 Hill AV (1964) The effect of load on the heat of shortening of muscle. Proc R Soc B 159:297–318 Hill AV, Howarth JV (1959) The reversal of chemical reactions in contracting muscle during an applied stretch. Proc R Soc B 151:169–193 Huxley AF (1957) Muscle structure and theories of contraction. Prog Biophys Biophys Chem 7:257–318 Huxley AF, Simmons RM (1971) Proposed mechanism of force generation in striated muscle. Nature 233:533–538

References

53

Huxley AF, Tideswell S (1996) Filament compliance and tension transients in muscle. J Muscle Res Cell Motil 17:507–511 Huxley HE, Stewart A, Sosa H, Irving M (1994) X-ray diffraction measurements of the extensibility of actin and myosin ﬁlaments in contracting muscle. Biophys J 67:2411–2421 Irving M, Allen TS-C, Sabido-David C, Craik JS, Brandmeier B, Kendrick-Jones J, Corrie JET, Trentham DR, Goldman YE (1995) Tilting of the light-chain region of myosin during step length changes and active force generation in skeletal muscle. Nature 375:688–691 Julian FJ (1979) In: Ingels NB (ed) The molecular basis of force development in muscle. Palo Alto Medical Research Foundation, Palo Alto, pp 37–58 Kaya M, Higuchi H (2010) Nonlinear elasticity and an 8-nm working stroke of single myosin molecules in myoﬁlaments. Science 329:686–689 Kohler J, Winkler G, Schulte I, Scholz T, McKenna W, Brenner B, Kraft T (2002) Mutation of the myosin converter domain alters crossbridge elasticity. Proc Natl Acad Sci USA 99:3557–3562 Kushmerick MJ, Davies RE (1969) The chemical energetics of muscle contraction. II. The chemistry, efﬁciency and power of maximally working sartorius muscles. Proc R Soc B 174:315–353 Kushmerick MJ, Larson RE, Davies RE (1969) The chemical energetics of muscle contraction. I. Activation heat, heat of shortening and ATP utilization for activation-relaxation processes. Proc R Soc B 174:293–314 Millman B (1991) The ﬁlament lattice of striated muscle. Phys Rev 78:359–384 Needham DM (1971) Machina Carnis: the biochemistry of muscular contraction in its historical development. Cambridge University Press, Cambridge Piazzesi G, Reconditi M, Linari M, Lucii L, Bianco P, Brunello E, Decostre V, Stewart A, Gore DB, Irving TC, Irving M, Lombardi V (2007) Skeletal muscle performance determined by modulation of number of myosin motors rather than motor force or stroke size. Cell 131:784–795 Potma EJ, Steinen GJM, Barends JPF, Elzinga G (1994) Myoﬁbrillar ATPase activity and mechanical performance of skinned ﬁbres from rabbit psoas muscle. J Physiol (London) 474 (2):303–317 Press WH, Teukolsky SA, Vetterling WT, Flannery BR (1992) Numerical recipes in Fortran, 2nd edn. Cambridge University Press, Cambridge, pp 729–731 Rall JA (2016) Mechanism of muscular contraction. Springer, New York Smith D, Sleep J (2006) Strain-dependent kinetics of the myosin working stroke, and how they could be probed with optical-trap experiments. Biophys J 91:3359–3369 Smith NP, Barclay CJ, Loiselle DS (2005) The efﬁciency of muscle contraction. Prog Biophys Mol Biol 88:1–58 Squire JM, Luther PK, Morris EP (1990) Organisation and properties of the striated muscle sarcomere, in “Molecular mechanisms of muscle contraction,” Macmillan, London pp1–48 Sun Y-B, Hilber K, Irving M (2001) Effect of active shortening on the rate of ATP utilisation by rabbit psoas muscle ﬁbres. J Physiol (London) 531(3):781–791 Tregear RT, Reedy MC, Goldman YE, Taylor KA, Winkler H, Franzini-Armstrong C, Sasaki H, Lucaveche C, Reedy MK (2004) Cross-bridge number, position and angle in target zones of cryoﬁxed isometrically active insect ﬂight muscle. Biophys J 86:3009–3019 Wakabayashi K, Sugimoto Y, Tanaka H, Ueno Y, Takezawa Y, Amemiya Y (1994) X-ray evidence for the extensibility of actin and myosin ﬁlaments during muscle contraction. Biophys J 67:2422–2435 Webster AG (1955) Partial differential equations of mathematical physics. Dover Inc., New York Weiss GH (1986) Overview of theoretical models of reaction rates. J Stat Phys 42:1–36 Whittaker M, Carragher BO, Milligan RA (1995) PHOELIX: a package for semi-automated helical reconstruction. Ultramicroscopy 58:245–259 Wilkie DR (1960) Thermodynamics and interpretations of biological heat measurements. Prog Biophys Biophys Chem 10:259–298 Woledge RC, Curtin NA, Homsher E (1985) Energetic aspects of muscle contraction, Monographs of the Physiological Society no. 41. Academic, London Zemansky MW, Dittman RH (1996) Heat and thermodynamics. McGraw Hill, New York

Chapter 3

Actin-Myosin Biochemistry and Structure

One can only admire those scientists who have chosen to dissect in almost painful detail the components of muscle and the interactions of these, examining the objects of their research over and over again, with a will to learn and modify. On the other hand, one can understand this kind of religiosity. In the heaven of muscle scientists the rewards shall be great, for at the core of their research lies one of the most central issues in biology, the way in which chemical energy translates to coordinated rhythmic movement of structural elements. S. Seifter and P.M. Gallop, “The Proteins: Composition, Structure and Function,” 1966, Elsevier

As a result of biochemical and mechanical investigations on a wide front, deﬁnite answers to these questions have been obtained over the decades from 1970 to 2000 (Cooke 1997), although the issue of tight coupling continues to be argued in some quarters. Then a new era dawned with the discovery of the atomic structures of the proteins (Kabsch et al. 1990; Rayment et al. 1993). This chapter offers a summary of these advances.

3.1

How Myosin and Actin Hydrolyze ATP

The vast majority of biochemical investigations relevant to contraction have been carried out on ﬁlaments and isolated proteins in solution (Taylor 1981; Hibberd and Trentham 1986; Geeves 1991). After rapid mixing of reactants, the time course of the reaction can be observed by optical monitoring (of absorption, scattering, ﬂuorescence) after stopping the ﬂow in a reaction line. If no optical signal is available, the reaction can be terminated by chemical analysis after quenching in acid at different time intervals. Standard methods for analysing the time course of chemical reactions are described in textbooks of biochemistry (e.g. Segel 1978), and summarized by Woledge et al. (1985). These experiments are generally carried out at lower concentrations than in muscle ﬁbres, to be able to measure © Springer Nature Switzerland AG 2018 D. A. Smith, The Sliding-Filament Theory of Muscle Contraction, https://doi.org/10.1007/978-3-030-03526-6_3

55

56

3 Actin-Myosin Biochemistry and Structure

transient actin-myosin kinetics with saturating concentrations of actin. Such experiments are often made at 100 mM KCl (ionic strength 0.12 M), 20 C and pH 7, used in this chapter as a standard for quoting reaction rates in the literature. Such parameters need to be extrapolated to ﬁbre conditions, where the ionic strength is typically 0.2 M. Incidentally, 1 M ¼ 1 mole/litre, and one mole of a pure substance is its molecular weight in grams.

3.1.1

Myosin is an ATPase

~ ¼ 34 KJ/mole when To begin: ATP hydrolysis is an exothermic reaction, with ΔH ATP is regenerated by phosphocreatine (Woledge et al. 1985), and the Gibbs energy of hydrolysis is about the same (Alberty 1968). ATP in solution hydrolyses very slowly, which in muscle is not of practical interest. ATP binds to actin and myosin, but its hydrolysis is catalysed only by myosin, so myosin is an ATPase. The kinetics of hydrolysis is described by the cyclic reaction shown in Scheme 3.1, Scheme 3.1 Kinetic scheme for myosincatalysed hydrolysis of ATP

M + ATP

~ kT k −T

M.ATP

kH

k−H

M.ADP.Pi

kP

M+ADP+Pi,

which is an example of an enzyme-substrate reaction. Myosin (or, more precisely, myosin-S1) is an enzyme: ATP as a substrate on myosin is converted to the bound products of hydrolysis, which cleaves the terminal phosphate group of ATP. For these reactions, the molar concentrations satisfy the rate equations d ½M ¼ k~T ½M½ATP þ k T ½M:ATP þ kP ½M:ADP:Pi, dt d½M:ATP ~ ¼ k T ½M½ATP ðk T þ kH Þ½M:ATP þ kH ½M:ADP:Pi, dt d½M:ADP:Pi ¼ k H ½M:ATP ðkH þ k P Þ½M:ADP:Pi: dt

ð3:1aÞ

ð3:1bÞ

ð3:1cÞ

Reaction rates with tildes are for a 2nd-order reaction with two reactants. Summing these equations shows that the total mole numbers of myosin (as M + M.ATP + M. ADP.Pi) and nucleotide (as ATP + M.ATP + M.ADP.Pi) are conserved. Thus there are only two independent rate equations, and their solution is presented in Appendix B. With kP > 0 the transient solution tends to a steady state in which ATP is

3.1 How Myosin and Actin Hydrolyze ATP

57

continuously hydrolysed, but if kP ¼ 0 the preceding reactions would eventually go to equilibrium. In equilibrium, the rate equations satisfy the law of mass action ½M:ATP k~T ¼ K~ T , ½M½ATP kT

½M:ADP:Pi kH ¼ KH ½M:ATP k H

ð3:2Þ

in terms of the equilibrium constants K~ T and KH. These enzymatic reactions have been studied intensively (Trentham et al. 1976; Johnson and Taylor 1978). Although the release of hydrolysis products is reversible, especially by ADP, the release rate kP ( 0.06 s1) is very slow compared with the preceding steps. Thus the net ATPase rate per myosin is equal to kP multiplied by the equilibrated fraction of myosins in the M.ADP.Pi state, namely R¼

K~ T ½ATPK H kP 1 þ K~ T ½ATPð1 þ K H Þ

ð3:3Þ

This hyperbolic function of [ATP] has a maximum rate of kPKH/(1 + KH), achieved 1 when ½ATP >> K~ T ð1 þ K H Þ : As KH 4, this maximum is 0.8kP. The time course of the myosin-ATPase was studied by Lymn and Taylor (1970, 1971), using the quenched ﬂow method with γ-32ATP, in which the radioactive isotope 32Pi was incorporated into the terminal (γ) PO4 group of ATP. Binding to myosin and the subsequent hydrolysis liberated the labelled phosphate group from adenosine, but left it bound to myosin until released very slowly. Both free and bound 32Pi were detected after quenching in acid, which denatures M.ADP.Pi. The results, reproduced in Fig. 3.1a, show a burst of liberated Pi, generated by hydrolysis of M.ATP at a rate of 55 s1, much faster than the steady-state rate. Thus myosin is good at catalysing the phosphate-cleavage step, but poor at releasing the products of hydrolysis. The phosphate burst can be ﬁtted to the analytic solution of Eqs. 3.1a, 3.1b, 3.1c. As ATP was in excess, its concentration can be regarded as constant, giving a pseudo-ﬁrst-order reaction for ATP binding with kT k~T ½ATP: However, be warned: in fact ATP binds in two steps, so the dependence on [ATP] is not linear but hyperbolic (Appendix C). The amount of labelled inorganic phosphate liberated from γ-32ATP after quenching follows from a separate rate equation d½Pilib d½Pi d½M:ADP:Pi þ ¼ ¼ dt dt dt

d þ kP ½M:ADP:Pi dt

so Z ½Pilib ðt Þ ¼ ½M:ADP:Piðt Þ þ kP

0

t

½M:ADP:Piðt 0 Þdt 0 :

ð3:4Þ

58

3 Actin-Myosin Biochemistry and Structure

Fig. 3.1 Results of Lymn and Taylor (1971) for the burst of inorganic phosphate from hydrolysis of ATP on myosin and actomyosin. (a) The time course of liberated phosphate after mixing 25 μM of labelled ATP with HMM (ﬁlled circles) or actoHMM (open circles). The burst size B is estimated by extrapolating the linear region of steady-state hydrolysis back to zero time. That the two bursts are nearly identical suggests that in the second experiment ATP binding breaks the actomyosin bond, leaving M.ATP to hydrolyse as in the ﬁrst experiment. (b) The burst rate λ as a function of ATP concentration, obtained by ﬁtting the transients to the function B(1-exp(λt)) + Rt. In the units of the ordinate, B ¼ 1641 14, λ ¼ 68 5 s1, R 70 s1 for HMM and B ¼ 1635 58, λ ¼ 44.8 5.6 s1, R ¼ 760 300 s1 for actoHMM. If the burst size per myosin was 0.9, as expected if KH ¼ 10, then the ordinate scales by 0.9/1641 and 0.9/1635, giving R/ [HMM]0 ¼ 0.04 s1 and 0.42 s1 respectively. With permission of the American Chemical Society

Starting from the time of mixing myosin and ATP, the general solution of Eqs. 3.1a, 3.1b, 3.1c is ½M:ADP:Piðt Þ X ¼ Cα ð1 expðλa t ÞÞ ½Mtot α¼

ð3:5Þ

(Appendix B) so the amount of liberated phosphate per myosin obeys the equation

3.1 How Myosin and Actin Hydrolyze ATP

plib ðt Þ

59

½Pilib ðt Þ X kP ¼ Cα 1 ð1 expðλa t ÞÞ þ CkP t ½Mtot λa α¼

ð3:6Þ

where C C++C. Hence the burst size and steady-state hydrolysis rate are B¼

X

Cα

α¼

kP 1 , λa

R ¼ Ck P

ð3:7Þ

and the coefﬁcients Cα, λα are given in Appendix B. In practice, the time course of liberated phosphate is usually well-ﬁtted by a single exponential rise prior to steady ATP turnover, so plib ðt Þ ﬃ Bð1 expðλt ÞÞ þ Rt

ð3:8Þ

Why is a single exponential ﬁt possible when the exact solution contains two exponential components? This happens when the products are released very slowly (kP 0), and become very large when kT kH+kH where the eigenvalues λα are almost equal (Eq. B9). With kP 3000 M

T

k~TA ¼ 3 106 M1s1 k~PA ¼ 3 106 M1s1

K~ TA < 600 M1 K~ PA ¼ 3000 M1

Although ATP binds far more tightly to myosin than to actomyosin, ATP binding to the latter is almost irreversible because of rapid dissociation from actin; the population of A.M.ATP is so low that this state can be omitted, giving the cycle in Scheme 3.3, Scheme 3.3 A four-state reduction of the LymnTaylor cycle

M.ADP.Pi A k

P A

A.M.ADP.Pi

kH

M.ATP k

A T

A.M

A ATP

k PA Pi,ADP

which is very useful for later developments. The rate kA T for ATP binding includes very rapid dissociation from actin. At this point we see that the theory of the myosin-only phosphate burst can be taken over intact, with kT replaced by kA T , k-T ¼ 0 and kP reinterpreted as the net rate for actin binding and irreversible product release. For example kP ¼

kPA k A P P kA þ k P A

ð3:10Þ

if AM.ADP.Pi is treated as a steady-state intermediate (Appendix C). We will later argue that product release is always faster than actin binding, so that it is sufﬁcient to set kP ¼ k PA , but for the moment this question can be left open. The Lymn-Taylor cycle solves the problem of how the myosin ATPase rate is enhanced by actin binding, but it is worth noting that the conditions of Lymn and Taylor’s 1971 experiment were quite special. The initial concentrations of actin

62

3 Actin-Myosin Biochemistry and Structure

(14 μM) and actoHMM (2.9 μM) were very small, so that steady-state hydrolysis was limited by the actin-binding rate of M.ADP.Pi. However, for actomyosin the apparent burst rate is a smooth function of ATP, as in Fig. 3.1b. How can this be if the same theory applies to both experiments? Presumably the [ATP] dependence of the rate of ATP binding to actomyosin is described by Michaelis-Menten kinetics (Appendix C), which indicates that binding is a two-step reaction via a collision complex. Then kA T ¼

k~A T ½ATP 1 þ ½ATP=C A T

ð3:11Þ

where CA T , the Michaelis-Menten constant, is responsible for the hyperbolic rise. What happens at higher actin concentrations? Rosenfeld and Taylor (1984) measured the burst size B and steady-state hydrolysis rate R for various actin concentrations, up to a value in which R had saturated and 70% of myosin-S1 was bound to actin. B decreased with actin level, reaching 0.1 at maximum actin. This is as expected; at saturating actin concentrations, there should be no phosphate burst if actin binding and product release are much faster than hydrolysis off actin. Their experiment can be modelled from Eq. 3.7, which gives the exact solution of the burst equations, after making the above replacements for kT and kP and setting k–T ¼ 0. In the original notation, R¼

kT kH kP ðkH þ k H ÞkT þ ðk T þ kH Þk P

ð3:12aÞ

so 1 1 1 1 1 ¼ þ þ 1þ R kT kH K H kP which sums the lifetimes of the three states, and the burst size is B¼

kT r H kH kP k2P fkT r H þ ðk T þ kH Þk P g2

kT kH :

ð3:12bÞ

where rH ¼ kH+kH. Moreover, in this 3-state reduction of Scheme 3.3, the fraction of myosin-S1 bound to actin is just p ¼ R=k T :

ð3:12cÞ

These quantities are plotted in Fig. 3.2, which shows that the burst size is a decreasing function of kP and eventually becomes negative, turning the burst into a lag. The fraction of myosin-S1 bound to actin stays low even at high actin levels, unless the ATP concentration is low enough for kT kH.

3.1 How Myosin and Actin Hydrolyze ATP

63

Fig. 3.2 Predictions for (a) the steady-state hydrolysis rate, (b) the burst of liberated phosphate per myosin and (c): the fraction of myosins bound to actin, from Eqs. 3.12a, 3.12b, 3.12c for the mixing of actomyosin and ATP. Here kT is the rate of ATP binding and kP the combined rate for actin binding and release of hydrolysis products (Pi and ADP). All rates are normalised to kH, the rate of the Pi-cleavage step on myosin. As kP/kH is increased, the burst size decreases and is replaced by a slight lag (B < 0), but the fraction of bound myosins remains low except at low ATP where kT/ kH < 1. This behaviour addresses experimental results of Stein et al. (1981) and Rosenfeld and Taylor (1984)

The transient responses predicted by this model behave in a slightly bizarre fashion which does not seem to have been explored in the laboratory. When kP > kT,kH, the burst is replaced by a lag, which is understandable. In between, bursts occur at high ATP and lags at low ATP (Fig. B3). This behaviour hides a damped oscillatory response, obvious only in the ﬁrst exceptional case mentioned above. Bursts, lags and ringing behaviour are all consequences of this three-step kinetic scheme, which is the classical example of enzyme-substrate kinetics. Apparently there is more to enzyme kinetics than meets the eye.

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3 Actin-Myosin Biochemistry and Structure

3.1.3

Steady-State ATP Hydrolysis by Actin-Myosin

Long ago, steady-state hydrolysis rates as a function of [ATP] and [actin] were measured by Eisenberg and Moos (1970), using very low concentrations to slow the turnover to manageable levels. Their results set an interesting test of the LymnTaylor cycle. At the same time, a quite different cyclic reaction scheme was proposed by Eisenberg and collaborators (Stein et al. 1981), in which myosin and every myosin-nucleotide complex (M.ATP, M.ADP.Pi, M.ADP, M and M.ATP) was in rapid equilibrium with actin. In their view, hydrolysis of ATP could occur equally well on actomyosin as myosin alone. This controversy is still alive, but the evidence does favour actin dissociation rather than hydrolysis as the dominant pathway from A.M.ATP unless all myosins are cross-linked to actin (Stein et al. 1985). It might seem outlandish to argue that anything as simple as Eq. 3.12a, which came from a 3-state reduction of the Lymn-Taylor cycle (Scheme 3.3), can provide an adequate description of ATP hydrolysis rates for actin-myosin in solution. Nevertheless, this equation works when rates of phosphate release within the cycle, and of ADP release, are faster than the actin binding and cleavage steps. To test this hypothesis, it is useful to have a general expression for the hydrolysis rate of the 4-state cycle, as an explicit function of ATP and actin concentrations. The net forward ﬂux in a non-branching system of cyclic reactions can easily be calculated if one reaction is irreversible. Consider the four-state version of the LymnTaylor cycle (Scheme 3.3). Labelling the states numerically gives the scheme k1

k2

k3

k1

k2

k 3

k4

1Ð2Ð3Ð4!1

ð3:13Þ

where 1 ¼ M.ATP, 2 ¼ M.ADP.Pi, 3 ¼ A.M.ADP.Pi and 4 ¼ A.M. In a steadystate, the net ﬂux R from left to right depends on the occupation probabilities p1,. . .p4 of the states as follows: R ¼ k1 p1 k1 p2 ¼ k 2 p2 k 2 p3 ¼ k3 p3 k3 p4 ¼ k 4 p4 :

ð3:14Þ

Hence R R k3 , p3 ¼ 1þ , k4 k3 k 4 R k2 k3 1þ 1þ , p2 ¼ k2 k3 k4 R k1 k2 k3 1þ 1þ 1þ : p1 ¼ k1 k2 k3 k4 p4 ¼

ð3:15Þ

and the state probabilities sum to unity, which determines R. Collecting terms by rate constants gives the desired result

3.1 How Myosin and Actin Hydrolyze ATP

1 ¼ R

65

1 1 1 1 1 1 þ 1þ þ 1þ þ k1 K 1 k2 K 2 K 1 K 2 k3 1 1 1 1 þ 1þ þ þ K 3 K 2 K 3 K 1 K 2 K 3 k4

ð3:16Þ

where Kj kj/kj. The form of Eqs. 3.15 can be understood as follows. If τj is the lifetime of state j under steady-state conditions, then τ τ1 + τ2 + τ3 + τ4 is the cycling time and R ¼ 1/τ. Hence pj ¼ Rτj ¼ τj/τ,which expresses the fact that a particle travelling through state j at rate R spends a fraction pj of its cycling time in that state. Before moving on to data-ﬁtting, note that a 3-state version of Eq. 3.16 reproduces the ATPase rate in Eq. 3.12a, if states 1,2,3 are M.ATP, M.ADP.Pi and A.M respectively. Hence k1 ¼ kH, k2 ¼ kP and k3 ¼ kT, with K2 ¼ 1 for irreversible product release and K3 ¼ 1 for ATP binding and dissociation from actin. To ﬁt Eq. 3.16 to ATPase measurements, we need to group terms which depend on the concentrations of actin and ATP. To do this, two-step binding kinetics and their associated Michaelis-Menten constants must be included. For Scheme 3.3, k PA ~½A , 1 þ ½A=C PA ~ kA T ½ATP k4 ¼ kA T ¼ 1 þ ½ATP=C A T

k 1 ¼ k H , k 2 ¼ k PA ¼

k1 ¼ kH , k2 ¼ kPA ,

k3 ¼ kA P,

k3 ¼ kA P :

ð3:17aÞ

ð3:17bÞ

with Ki ¼ ki/k-i. In passing, note that k3 will vary with the solution concentrations of ADP and Pi. Then Eq. 3.16 translates to 1 1 1 1 1 ¼ þ þ þ R R1 R2 ½A R3 ½ATP R4 ½A½ATP

ð3:18Þ

1 1 1 1 ¼ þ þ P , R1 kH k A R 2 CA P ! 1 1 1 1 1 ¼ 1þ þ , þ A R2 KH R k~PA K~ PA kA 4 CT P 1 1 1 1 ¼ 1þ A Aþ , R3 K P k~T R4 C PA

ð3:19aÞ

where

ð3:19bÞ ð3:19cÞ

66

3 Actin-Myosin Biochemistry and Structure

1 ¼ R4

1 1 1þ : ~A K H K~ PA K A P kT

ð3:19dÞ

There is also the familiar hyperbolic function for the actin dependence of R, namely Rð½AÞ ¼

R0 ½A , ½A þ C

ð3:20Þ

where 1 1 1 , ¼ þ R0 R1 R3 ½ATP

C 1 1 : ¼ þ R0 R2 R4 ½ATP

ð3:21Þ

Figure 3.3 shows how Eqs. 3.18 and 3.19a, 3.19b, 3.19c, 3.19d can be used to ﬁt the results of Eisenberg and Moos (1970), presented as a 3D mesh plot against the inverse concentrations. The plot shown derives from plots of 1/R versus 1/[ATP] for four actin concentrations (their Fig. 3a). Not shown is another mesh plot, derived from 1/R against 1/[A] for four ATP concentrations (their Fig. 3b). Both 3D plots lie very close to a planar surface, which suggests that 1/R4 is almost negligible. However, ﬁtting both data sets to the full four-parameter function (Eq. 3.18) gave the following results: 1 ¼ 0:0838 0:0077s, R1 1 ¼ 0:440 0:022μM:s, R3

1 ¼ 1:173 0:071μM:s, R2 1 ¼ 1:31 0:20μM2 :s R4

ð3:22aÞ

(χ2 ¼ 0.055) after weighted combination of the separate ﬁts. A coarser ﬁt (χ2 ¼ 0.012) is obtained by setting 1/R4 ¼ 0, namely 1 1 1 ¼ 0:050 0:009s, ¼ 1:540 0:065μM:s, ¼ 0:556 0:021μM:s: R1 R2 R3 ð3:22bÞ However, these ﬁts do not include errors in the experimental data, and inspection of Fig. 3.3 suggests that they would remove any preference for the ﬁrst ﬁt. There are nine kinetic constants in the right-hand side of Eqs. 3.19a, 3.19b, 3.19c, 3.19d; too many to be determined by the ﬁrst ﬁt. One way through this particular maze is described in the box below.

3.1 How Myosin and Actin Hydrolyze ATP

67

Fig. 3.3 Data of Eisenberg and Moos (1970) for the steady-state rate of ATP hydrolysis as a function of actin and ATP concentrations, plotting inverse quantities on all axes. The points are ﬁtted by the function in Eq. 3.18 of the main text, which deﬁnes a nearly planar surface. Actin concentrations (M.W. 42,000) have been scaled by a factor of 23.81 to convert from mg/ml to μM, and ATPase rates by a factor of 2.834 to convert from μmoles/min.mg.HMM (M.W. 340,000) to s1 (moles ATP per mole of myosin-S1). With permission of the American Society for Biochemistry and Molecular Biology

To determine the kinetic constants in Eqs. 3.19a, 3.19b, 3.19c, 3.19d from R1,. . .R4, four of them could be regarded as given. Setting kH ¼ 55 s1 and 1 kA P ¼ 500 s (for ADP release) allows Eq. 3.19a to ﬁx the Michaelis-Menten P constant C A for actin binding at 18.4 μM, which is acceptable (Geeves and in Eq. 3.19c determines Conibear 1995). Setting KA P >> 1 A 1 1 ~ k T ¼ 2:71 μM s , also as expected (e.g. Ma and Taylor (1994). Although Eq. 3.19d can then be used to determine K~ PA , a delicate balance with K A P must be struck to get the former up to an acceptable value; to get K~ PAup to 0.3 μM1 requires K A P ¼ 20, which is in line with the estimate of Sleep and Hutton (1980) but far smaller than the value of 104 from detailed balancing at higher ionic strengths. Finally, Eq. 3.19b provides a single constraint on the two remaining unknowns, k~PAand CA T . The latter is expected to be about 300 μM (Taylor 1991), which would give k~PA¼1.2 μM1 s1. Thus the only difﬁculty that arises is the low value of K A P , which may be ionic-strength dependent. A similar procedure can be implemented for the constrained ﬁt which gave the values of R1,R2,R3 in Eq. 3.22b. Setting R4 ¼ 0 is a consequence of the 3-state model embodied in Eq. 3.12a, which is duplicated here in the limit where K A P >> 1 and 1 A P kP >> kA : Assuming that kH ¼ 55 s and KH ¼ 4, the simpliﬁed version of (continued)

68

3 Actin-Myosin Biochemistry and Structure

Eqs. 3.19a, 3.19b, 3.19c give the following estimates for the 3-state model: 1 1 k~PA ¼ 0:81 μM1 s1 and C PA ¼ 48 μM1 : The k~A T ¼ 1:80 μM s , 2nd-order actin binding rate is lower but the Michaelis-Menten constant is higher, so that the actin-binding rate at saturating actin levels is 39 s1 instead of 22 s1. Apart from that, there is not much to choose between the two models. In view of the uncertainties in assigning values to kH, KH and k A P for the conditions of Eisenberg and Moos’ data, it seems that the four-state model and its 3-state reduction are both capable of ﬁtting ATPase data as a function of [actin] and [ATP]. What these models have in common is rapid product release relative to the actin binding and hydrolysis steps. That these models are able to work at all is because actin-binding (and ATP binding also) is a two-step process via a collision state. The post-collision isomerization is the step which limits the actin binding rate at saturating levels of actin, say [A] > 3 C PA , and the limiting rate is k~PA C PA , here estimated at 20 s1 and 40 s1 respectively for the two models, is less than the value assumed for kH, the rate of the hydrolysis step. Thus there is no need for a slow transition after the products state binds to actin; that is already built in to the binding reaction, and the values of the Michaelis-Menten constants C PA and C A T are in line with those in the literature (Geeves and Conibear 1995). Although the Lymn-Taylor cycle provides an adequate description of steady-state hydrolysis under a variety of conditions, this does not mean that alternative cycles can be disregarded. Stein et al. (1985) have shown that ATP is hydrolysed, at almost the same rate, by actomyosin when myosins are cross-linked to actin ﬁlaments. However, without cross-linking, hydrolysis on actin is not the dominant pathway; A.M.ATP formed from rigor actomyosin is dissociated at least 100 times faster than ATP can hydrolyse. The question is then how rapidly can M.ATP rebind to actin to reverse this dissociation? The actin binding rate of M.ATP has been measured in many laboratories and is about the same as for other myosin-nucleotide complexes, including non-hydrolysable nucleotides such as ATP-γS and AMP-PNP (Taylor 1991; Geeves and Conibear 1995). In this sense, Eisenberg and coworkers are technically correct in claiming that M.ATP and A.M.ATP are in rapid equilibrium, but this equilibrium is heavily biased in favour of the ﬁrst state. At saturating actin concentrations, kTA e100 s1 from Table 3.1 if C TA ¼ 30 μM, whereas kTA 5000s1 : However, in the next section, evidence is presented for the existence of two M.ATP states on the contraction pathway, and the second state has an actin afﬁnity comparable to that of M.ADP.Pi. We shall see that this state can bind to actin and then hydrolyse ATP without prejudicing the mechanical events that accompany the biochemical cycle.

3.2 The Biochemical Contraction Cycle

3.2

69

The Biochemical Contraction Cycle

The question is often asked; why should we bother with a complete reaction scheme for the interactions of actin, myosin and ATP in muscle, when the one-line scheme of Lymn and Taylor appears to be sufﬁcient? Under normal physiological conditions, models based on this scheme can be made to work. However, a complete model of contractility must also be able to explain how contractile behaviour varies with the ambient concentrations of ATP and Pi. After repeated stimulation, the ﬁbre may become fatigued and this may be due to a decreased concentration of ATP and/or an increase in inorganic phosphate. As the ATP level in a skinned ﬁbre is systematically reduced, tension ﬁrst rises as the rigor state A.M becomes more populated and then falls at concentrations below 50 μM (Ferenczi et al. 1984), presumably because A.M dissociates from actin and rebinds to different sites. There is also the thorny question of which tension-bearing states dissociate from actin when the muscle is continuously stretched.

3.2.1

Actin Binding Versus Nucleotide Binding

Studies of actin-myosin binding for myosin and the three myosin-nucleotide complexes reveal that there is a reciprocity between the strengths of actin binding and nucleotide binding, with huge variations in the stability of the different complexes (Geeves 1991). The products state M.ADP.Pi binds only weakly to actin, with an afﬁnity of 103-4 M1 under solution conditions. Actin binding becomes progressively stronger as the products of hydrolysis are released; M.ADP has an afﬁnity of 105–6 M1 and the afﬁnity of apomyosin is 107–8 M1. These increases support the legitimacy of the Lymn-Taylor scheme for ﬁbres at millimolar levels of ATP; while the actin binding of M.ADP.Pi is reversible, A.M.ADP and especially A.M will not readily dissociate from actin. When ATP binds to rigor myosin (A.M), there is a huge reversal of stability. At physiological ionic strengths, A.M.ATP is essentially unstable; dissociation from actin is so fast and the afﬁnity of M.ATP for actin is so low that the dissociation might as well be treated as totally irreversible. For reasons to be explained later, this turns out to be an important condition for the success of contraction models. Detailed balancing relates the actin afﬁnities of myosin nucleotide complexes to the equilibrium constants of the product-release and ATP binding steps in the LymnTaylor cycle and their counterparts for detached myosin. The latter are known from the work of Bagshaw and Trentham (1973, 1974), followed by Greene and Eisenberg (1980), Johnson and Taylor (1978), Mannherz et al. (1974), Marston and Weber (1975), White and Taylor (1976). For example, consider the reactions shown in Scheme 3.4:

70

3 Actin-Myosin Biochemistry and Structure

Scheme 3.4 The actinbinding and ATP-binding steps for myosin

M

~ KT

~ KA A.M

M.ATP ~ K AT

~ K TA

A.M.ATP

When all reaction steps have come to equilibrium, ½A:M:ATP ½A:M ½A:M:ATP ½M:ATP ½A:M:ATP ¼ ¼ ½M ½M ½A:M ½M ½M:ATP

ð3:23Þ

so detailed balancing in the form ~ ~T K~ A K~ A T ¼ K TK A

ð3:24Þ

follows, using the law of mass action. Despite this derivation, detailed balancing is a universal law of chemistry because it involves only the equilibrium constants and not the concentrations. It also follows from Gibbs’ thermodynamic identity K ¼ exp (ΔG/kBT) and the additivity of Gibbs reaction energies. Detailed balancing provides a useful check on experimental results, and helps to ﬁll in some of the gaps. The afﬁnity of apo-myosin for ATP is so large that it is very difﬁcult to measure, but a value K~ T 101112 M1can be estimated from the identity that the product of the equilibrium constants for the chain of myosin-nucleotide transitions is the dissociation constant for the hydrolysis of free ATP, which is nearly 106 M (Trentham et al. 1976). However, actomyosin binds ATP more weakly, with 45 1 an afﬁnity K~ A M : Then the 107-fold decrease in the strength of ATP T 10 binding would be matched by a 107-fold decrease in the actin afﬁnity of M.ATP relative to myosin.

3.2.2

A Biochemical Cycle for Myosin-S1

Scheme 3.5 is intended as a slightly simpliﬁed version of the biochemical actinmyosin-nucleotide cycle, suitable as a starting point for a comprehensive model of muscle action. It is constructed on the basis that Pi is released before ADP. The dominant pathway is anticlockwise around the perimeter, but other paths for dissociating the actomyosin bond allow deviations from so-called tight coupling to be explored within the context of known reaction rates and afﬁnities in solution. For example, the dominant pathway shifts to dissociation of A.M when [ATP] is reduced to submillimolar levels and the muscle moves into rigor mortis.

3.2 The Biochemical Contraction Cycle

M.ADP.Pi

~ KP

~ K ADP

M.ADP

71

~ KD

~ K AD

A.M.ADP.Pi

~ K PA

M

~ KA

A.M.ADP

~ K DA

A.M

~ KT

KH M.ATP ~ K AT

~ K TA

A.M.ATP

Scheme 3.5 A biochemical actin-myosin-ATP cycle with hydrolysis on detached myosin

Table 3.2 An actomyosin-ATP cycle with myosin equilibrium, rate and Michaelis-Menten constants K, k, C from solution measurements at 20 C, pH ¼ 7, adjusted to ionic strength ¼ 0.1–0.2 M. kY X is the rate constant for binding/detachment of ligand X to/from the complex MY/MXY. Secondorder quantities have a superscript tilde. Michaelis-Menten constants refer to two-step attachment reactions (Appendix C), giving forward and backward rates k as follows. For binding, k þ ¼ k~½X ~ K~ , and for detachment, k+ ¼ k and k ¼ K~ 1 k ½X=ð1 þ ½X=C Þ: =ð1 þ ½X=C Þ and k ¼ k= Equilibrium constants in brackets are derived by detailed balancing. The myosin-nucleotide sequence is consistent with the scheme described by Trentham et al. (1976). Sources by data rows and columns: Criddle et al. 1985 (R7–8,C3), Dantzig et al. 1992 (R4C1), Geeves et al. 1986 (R7–8,C4), Highsmith 1976 (R7C2), Johnson and Taylor 1978 (R1C4,R2C4), Lymn and Taylor 1971 (R5C3), Ma and Taylor 1994 (R4–6,C3), Marston and Weber 1975 (R7C3), Siemankowski et al. 1985 (R5C2), Sleep and Hutton 1978 (R4C3,R7C2,R7C4), Taylor 1991 (R7–9,C3), for AMP-PNP, Trentham et al. 1976 (R1-R3,C1-C3) and White and Taylor 1976 (R4C2,R5C3). 4 ~T ~ ~ ~ DP ~ Note that K~ ATP ¼ 3.2 105 M and K~ A T =K A ¼ K T =K A ¼ 10 as in Eq. 3.24. The product K A K 4 A DP ~ P ¼ 4 10 has been partitioned to give a ﬁrst-order afﬁnity K A ½A below unity for the muscle ﬁbre with [A] ¼ 0.15 mM X K~ X , K X

P 0.01 M

D 2 105 M

T 4 1011 M1

H 4

k~X , k X CX(M) A K~ A X, KX

0.06 s1

1.4 s1

2 106 M1s1

100 s1

0.007 (10 M)

3 104 2 104 M

2 104 4 105 M1

-------

A k~A X , kX

105 s1?

>500 s1

2-4 106 M1s1

----

CA X (M) X 1 K~ X A (M )

----

?

3 104

----

DP 4000

D 4 106

---4 107

T (40)

1 1 s ) k~X A (M

4X105?

4X105?

4X106

4X105?

CX A (M)

?

?

5

3X10

4X105

For this scheme, Table 3.2 gives a consistent set of rate constants, afﬁnities and Michaelis-Menten constants obtained from solution-kinetic measurements. There are various uncertainties involved; many experiments are conducted at low salt concentrations where ionic strength is low and electrostatic interactions less wellshielded by the electrolyte, and some reactions are strongly pH-dependent. Where possible, equilibrium constants and rate constants have been adjusted to an ionic

72

3 Actin-Myosin Biochemistry and Structure

strength of 0.1 M, approaching that of muscle sarcoplasm (~0.15–0.2 M), guided by measurements over a range of salt concentrations. For actin binding of M.ADP, different results were obtained by Geeves (1989) and Taylor (1991), partly from using different concentrations of ADP. A more fundamental difﬁculty is associated with Pi release from A.M.ADP.Pi. The apparent rate of release is slow, of order 10–50 s1, as measured by phosphate jump experiments in ﬁbres (Dantzig et al. 1992), but the relevant rate for the operation of the cycle is for Pi release from its active site, which is probably much faster; this theme is taken up in Sect. 2.4. This reaction scheme allows us to track all changes in Gibbs energy. In particular, the net drop in energy over the cycle of actin binding, product release, detachment and ATP cleavage must be equal to the energy of hydrolysis of free ATP. The corresponding ~ ATP =kB T , and values of K~ ATP range from equilibrium constant is K~ ATP exp ΔG 5 6 5 10 M to 3 10 M (Alberty 1968; Goody et al. 1977; Trentham et al. 1976), giving ~ ATP j¼ 32–36 kJ/mol ¼ 53–60 zJ/molecule. This is an important test, because it ties j ΔG together the sequence of reactions which are often studied in isolation or limited combination. In fact, detailed balancing around this cycle shows that the net equilibrium constant K~ is considerably less than K~ ATP , which means that there must be a hidden transition, or possibly several, not revealed experimentally. For Scheme 3.4, the net afﬁnity around the cycle is ~A ~A ~A ~T K~ ¼ K H K~ DP A K P K D K T =K A

ð3:25Þ

1 ~T ~A ~A where K~ DP A , K A (in M ) are actin afﬁnities and K P , K D (in M) are dissociation constants for Pi and ADP. Values for all of these quantities are not available; K~ A T and K~ TA are inaccessible experimentally because of the difﬁculty of separating ATP-toactomyosin binding from the subsequent rapid dissociation from actin. For most ~T purposes, it is only the net equilibrium constant K~ A T =K A that is required, but even this quantity is hard to estimate accurately. Here are three approaches:

1. Perhaps the best method is from detailed balancing. Equation 3.24 shows that K~ A T= 4 11 12 1 7 8 1 T ~ ~ ~ ~ ~ K A ¼ K T =K A , which is of order 10 as K T~10 –10 M and K A~10 –10 M , depending on conditions. 2. ATP binding to A.M is known to be a two-step process with an intermediate collision state (Ma and Taylor 1994). The afﬁnity of the ﬁrst step is the inverse of the Michaelis-Menten constant CA T ¼ 320 μM. If the subsequent isomerization has 1 ~T ~T ~T an equilibrium constant above unity, then K~ A T > 3000 M . Next, K A ¼ k A =k A where k~TA 106 M1s1 and k~TA > 5000 s1 (Geeves et al. 1986). So K~ TA < ~T 200 M1 and, since the inequalities can be combined, K~ A T =K A > 15. As an estimate rather than a lower bound, this value is certainly too low. 3. Sleep and Hutton (1978) found that ATP-induced dissociation from actin is reversible; they found that A + M.ATP ! A.M.ATP ! A.M + ATP ran at a rate of 2.7 104 M1 s1 at 40 mM KCl. Assuming that the ﬁrst step is faster, this rate is an estimate for the quantity K~ TA k A T : Now this information can be combined with the kinetics of ATP binding to A.M; Ma and Taylor (1994) found

3.2 The Biochemical Contraction Cycle

73

6 1 1 that k~A at 100 mM salt. A two-fold rate reduction would T ¼ 3:7X10 M s convert Sleep and Hutton’s result to 100 mM salt (Johnson and Taylor 1978), and ~ T ~A A ~ T combining these results gives K~ A T =K A ¼k T =k T K A ¼ 274. 4 ~T Taking K~ A T =K A ¼ 10 , the net afﬁnity around the cycle can now be calculated 1 ~ A 4 ~A from Eq. 3.25. With KH ¼ 4, K~ DP A ¼ 4000 M , K P ¼ 0.01 M, K D ¼ 2X10 M, the product gives K~ ¼ 320M. Hence there is a missing reaction, presumably in the midst of the product-release steps on actin, and the equilibrium constant KS of this missing step is

KS

K~ ATP 3:2X105 ¼ 1000 320 K~

ð3:26Þ

with K~ ATP ¼ 3.2 105 M as in Table 3.2. How could an almost irreversible step in the cycle have been missed? And what signiﬁcance does it hold for the contraction cycle in muscle?

3.2.3

Evidence for Two A.M.ADP States

A warning sign that something might be amiss came from an unexpected quarter in 1980, from a phosphate exchange experiment. ATP can be synthesized on myosin from ADP and Pi (Goody et al. 1977). In the actin-myosin system, radioactivelylabelled Pi can be tracked backwards through the cycle, starting when it binds to A. M.ADP and ending with the release of ATP from A.M.ATP. Needless to say, the net rate for this reverse sequence is far slower than the forward rate for ATP hydrolysis. Sleep and Hutton (1980) devised an ingenious protocol to measure the rates of ATP hydrolysis and Pi exchange simultaneously. 2 mM each of unlabelled ATP and labelled Pi were mixed with 0.5 μM myosin-S1 bound to higher concentrations of actin. Mixing initiated the hydrolysis of unlabelled ATP and the simultaneous exchange of labelled Pi, which binds to A.M.ADP formed by the ﬁrst process. Their results are reproduced in Fig. 3.4. This phosphate exchange experiment can be analysed by separating the states with labelled and unlabelled phosphate, giving coupled actomyosin-nucleotide cycles as in Fig. 3.5. Starting from rigor myosin, the myosin engine can be driven around the outer track by binding unlabelled ATP, releasing phosphate to arrive at the A.M.ADP state. At this station, labelled Pi has a chance to bind, sending the engine backwards around the inner track. If at some stage the reverse reactions 3 ! 2, 2 ! 1, 1 ! 4 with labelled Pi proceed serially, then labelled ATP has been synthesized and will be released in the last step. However, the probability of each forward step is greater (ki > k-i in each case), so many forward steps and complete cycles occur in between the intervals in which the entire reaction sequence is reversed. Myosin is a stochastic engine, condemned to shunt backwards and forwards according to the laws of chemistry.

74

3 Actin-Myosin Biochemistry and Structure

Fig. 3.4 Data of Sleep and Hutton (1980) from a phosphate exchange experiment. (a) The initial rates of hydrolysis of unlabelled ATP and generation of labelled ATP, as a function of actin concentration. (b) The time course of the appearance of labelled ATP, followed to a time when about 30% of unlabelled ATP was hydrolysed. With permission of the American Chemical Society and J. Sleep

By such analysis, the authors calculated the ratio r of exchange rate to hydrolysis rate and ﬁtted their data at various actin concentrations. The calculation is quite involved; here we quote an approximate version of their Equation 6, namely rﬃ

32 K TA kA T ½ Pi : ~A A K H K DP A K P kD

ð3:27Þ

T where K DP A and K A are pseudo-ﬁrst-order equilibrium constants for actin binding to M.ADP.Pi and M.ATP. Here k A D is the rate of ADP release from A.M.ADP.Pi. With T 4 the following values, r ~ 104 as observed. Setting KH ¼ 4, kA T K A ¼ 2:7X10 ½A 4 1 A 1 DP A s (Sleep and Hutton 1980), k D ¼ 500 s , K A ¼5X10 [A], K~ P ¼ 0:01 M and [32Pi] ¼ 2 mM gives r ¼ 2.1X104. Equation 3.27 was derived by assuming that A kA D 20 for the conditions of their experiment, or 1000 from Eq. 3.26. The functional signiﬁcance of this prediction is that the A.M.ADP state formed when ADP binds to rigor myosin is different from the state, say A.M.ADP#, which can bind Pi prior to detaching from actin. This is the ﬁrst of two ways in which the cycle in Scheme 3.4 needs to be modiﬁed.

3.2.4

Evidence for Two M.ATP States

When the interaction of myosin-S1 with ATP was ﬁrst studied intensively, the ﬂuorescence studies of Bagshaw and Trentham (1974) showed the existence of two M.ATP states. At the time, it was thought that the ﬁrst state was a collision complex. Then a much later study by Malnasi-Csizmadia et al. (2000) on dictyostelium myosin II (a non-muscle myosin from a slime-mould) showed that, although a collision complex is present, there are also two subsequent states, distinguished by different levels of tryptothan ﬂuorescence. The transition between

76

3 Actin-Myosin Biochemistry and Structure

these states is rapid (k > 350–800 s1) and the equilibrium constant between them is about 0.2, which means that a constant of 20 for the subsequent cleavage step gives an overall KH of 4. Muscle myosin II probably functions in the same way. The true signiﬁcance of this discovery is related to the atomic structure of the myosin molecule, discussed in the next section. The M.ATP state formed by the dissociation of A.M.ATP is an “open” state which cannot hydrolyse ATP, but the “closed” state which follows does permit hydrolysis. In principle, the closed M.ATP state could bind to actin and possibly hydrolyse ATP, but the low equilibrium constant of the open-closed transition means that the closed state will not be highly occupied, at least under steady-state conditions.

3.3

Coordinating Lever-Arm Movements with Biochemical Events

Biochemical studies of the myosin-actin-nucleotide system have taken us to the point where biochemistry joins mechanics. The mechanochemistry of the myosin motor is deﬁned, in the ﬁrst place, by associating the pre-stroke and post-stroke positions of its lever arm with speciﬁc biochemical states. Following the discussion of the previous section, the main conclusion can be stated quite simply. The working stroke, which generates tension and shortening in the muscle ﬁbre, occurs between the two A.M.ADP states, and the repriming stroke occurs between the two M.ATP states. Such associations are probably the raison d’etre of these isomeric states. Moreover, both of them may exist on actomyosin and detached myosin. This rather bald proposition has been challenged on many fronts. In this section we begin to examine the evidence for and against, which comes from a variety of experimental probes. In addition to biochemical techniques with ﬂuorescent markers, we need to consider results from mechanical measurements on muscle ﬁbres, EPR (Fajer et al. 1990; Ostap et al. 1995; Baker et al. 1998), FRET (Suzuki et al. 1998; Shih et al. 2000; Muretta et al. 2013), and single-molecule studies of various kinds. The interpretation of single-molecule experiments with optical trapping is taken up in Chap. 6. In the next section, the argument is continued in terms of the atomic structure of myosin. With this information at hand, a contraction cycle can be constructed which synthesizes mechanical and biochemical events, to form the basis of a rigorous theoretical description of muscle action. Whether this level of detail is necessary, or even sufﬁcient, will become clearer as we formulate mechanochemical models of muscular contraction.

3.3 Coordinating Lever-Arm Movements with Biochemical Events

3.3.1

77

What Biochemical Event Triggers the Working Stroke?

The transition between the two A.M.ADP states, which is fast and moderately irreversible, is the logical place for the myosin working stroke (Sleep and Hutton 1980). The rate constant kS for the forward transition must be at least as fast as the subsequent release of ADP, over 500 s1 for rabbit at 20 C (Siemankowski et al. 1985). The estimated equilibrium constant KS 1000 is large, but still not large enough for a single 8 nm stroke; Eq. 2.42 with κ ¼ 3.0 pN/nm shows that two 4 nm substrokes are needed to make X∗ 0. If the working stroke occurs between two A.M.ADP states, then the stroke must be triggered by the release of the phosphate ion in A.M.ADP.Pi, which is released before ADP (Bagshaw and Trentham 1974; McKillop and Geeves 1990). However, which event comes ﬁrst has always been a bone of contention, and there has been a long-standing belief that the working stroke precedes Pi release (Geeves et al. 1984; Hibberd and Trentham 1986; Dantzig et al. 1992; Homsher et al. 1997; Takagi et al. 2004; Smith and Sleep 2004). Here we have already made a simplifying assumption, that the stroke transition is distinct from reaction events such as actin binding or ligand release. This statement is hard to test, but seems plausible as short-lived intermediate states continue to be revealed by ﬂuorescence spectroscopy and, more recently, time-resolved FRET from David Thomas’ laboratory (Nesmelov et al. 2011; Muretta et al. 2015). There are three lines of evidence from ﬁbre experiments which favour Pi release before the working stroke, but there is a caveat; release of Pi is here deﬁned as release from the active site in the base of myosin’s nucleotide pocket, which is not the same as release to the surrounding medium. There is a tube-like structure through which Pi can escape to the base of the actin-binding cleft (Pate et al. 1997), and the passage of Pi down this tube will be regarded as a side-reaction to the mechanochemical contraction cycle. So the central hypothesis is: Phosphate release from the active site on actomyosin is fast and reversible and triggers the working stroke. With this in mind, consider the following observations: (a) Isometric tension in skinned ﬁbres is phosphate-dependent. The ambient level of inorganic phosphate in a fast striated muscle ﬁbre is under 1 mM. Extra phosphate can be added to a skinned ﬁbre, and the addition of 25 mM Pi produces a large decrease in isometric tension, up to 70% in some cases (Dantzig et al. 1992; Fryer et al. 1995; Tesi et al. 2000; Coupland et al. 2001; Caremani et al. 2008). Muscle stiffness decreases by a smaller fraction, typically 10%, while the isometric ATPase rate is hardly affected (Potma et al. 1995; Kawai et al. 1987; Bowater and Sleep 1988). This behaviour, summarised in Fig. 3.6, presents the clearest indication that phosphate is released from the A.M.ADP.Pi state ﬁrst formed by actin binding, preceding any working stroke (Smith 2014). The reason is simply that, in the muscle ﬁbre, this state is very weakly bound in the ﬁrst place. The pseudo-ﬁrst-order actin afﬁnity of M.ADP.Pi, here called KA(x), appears to be of order unity, for two reasons. As discussed in Sect. 2.2, the effective actin

78

3 Actin-Myosin Biochemistry and Structure

Fig. 3.6 (a) Data of Bowater and Sleep (1988) for isometric tension and ATPase rate as a function of the concentration of inorganic Pi in the muscle ﬁbre. With permission of the American Chemical Society and J. Sleep. (b) The tension is approximately linear in log[Pi] over two decades, and is linear below 0.5 mM Pi with a smaller slope (Tesi et al. (2000)). With permission of Elsevier Press and C. Tesi

concentration in the ﬁbre is 0.15–0.2 mM, and at physiological ionic strengths a second-order afﬁnity of about 4000 M1 is expected, giving KA ¼ 0.6–0.8. Secondly, myosin heads are tethered to thick ﬁlaments and there will be an energy cost, met by thermal energy, in moving radially inwards to meet the actin ﬁlament. There is also an axial energy cost, so KA also depends on X, the axial head-site spacing. If the central hypothesis is correct, pre-stroke events are summarised by the reactions K A ðxÞ

~ K P

A þ M:ADP:Pi Ð A:M:ADP:Pi Ð A:M:ADP# þ Pi

ð3:28Þ

where the two actomyosin states are lumped in rapid equilibrium. The net afﬁnity to this lumped state is K A ðxÞ 1 þ K~ P =½Pi : If there were no other reactions, then the probability of this state would be K A ðxÞ 1 þ K~ P =½Pi K A ðxÞ pA ð x Þ ¼ ~ K ð x Þ þ C A ð½PiÞ 1 þ K A ðxÞ 1 þ K P =½Pi A

ð3:29Þ

where C A ð½PiÞ ¼

½Pi : ½Pi þ K~ P

ð3:30Þ

3.3 Coordinating Lever-Arm Movements with Biochemical Events

79

Fig. 3.7 Predictions for the extent of actin binding of M. ADP.Pi at 1 mM and 25 mM [Pi], including equilibrated phosphate release and rebinding but excluding any working stroke. The pA curves are calculated from Eq. 3.29 with K~ P ¼ 5 mM. The quantity ΔpA, which is the difference of the two curves, measures the sensitivity to phosphate

Thus the afﬁnities of actin binding and Pi release are intertwined. Figure 3.7 shows how the occupation of this ﬁrst bound state varies with KA at low and high phosphate; there is a range of values near KA ¼ 0.3 which is sensitive to Pi, but this sensitivity is lost when KA > > 1, and KA < < 1 gives negligible binding anyway. To make the argument precise, the x-dependence of KA should be speciﬁed, and these reactions need to be embedded in the crossbridge cycle. Some myosin heads with negative values of x will make a working stroke, but the population of poststroke states would still be proportional to pA(x). A calculation similar to those presented in Sects. 2.3–2.4 leads to isometric tension as a function of phosphate; experimentally, To([Pi]) obeys a logarithmic law over almost two decades of [Pi] (Fig. 3.6b). There is also a rider: the lower the isometric tension generated under physiological conditions, the bigger the effect of exogenous phosphate (Caremani et al. 2008); this behaviour is also predicted by Eq. 3.29. Now it can be understood why Pi must be released before the working stroke. The equilibrium constant KS of the strain-free stroke is expected to be about 103–4 to ensure that a reasonable fraction of bound heads make a stroke under isometric conditions. If KA(x) were of order 104, then changing [Pi] from 1 mM to 25 mM would cause p(x) to decrease by only 7 parts in 105 if K~ P ¼ 0.005 M. (b) The slow kinetics of phosphate jumps. At this stage, you might well object that phosphate release does not appear to be a fast transition. “Phosphate jump” experiments, in which ﬂash-photolysis from a caged compound releases Pi into the sarcoplasm of a skinned ﬁbre in a matter of milliseconds (Hibberd and Trentham 1986; Goldman 1987; Dantzig et al. 1992) showed that isometric tension decreased exponentially to the new leve1 at a rate of 10–70 s1, varying hyperbolically with ﬁnal [Pi] (Fig. 3.8). Similar results were

80

10%

a

a b c 20 ms

b kll (s–1)

Fig. 3.8 Results of Dantzig et al. (1992) for tension after photorelease of caged Pi in rabbit psoas ﬁbres. (a) Tension transients for small amounts of released phosphate (0.2, 0.4 and 1.2 mM from top to bottom) at 20 C. (b) The rate constant of the dominant exponential process increases hyperbolically with ﬁnal [Pi], ﬁtting a formula such as Eq. 3.31 with K~ P ¼ 12.3 mM. With permission of John Wiley and Sons Inc.

3 Actin-Myosin Biochemistry and Structure

60 40 20 0 0

2

8 10 4 6 Final [Pi] (mM)

12

obtained by exchanging solutions on myoﬁbrils, whose small diameter permits a rapid ingress of added phosphate (Tesi et al. 2000). If the phosphate dependence of isometric tension requires rapid reversible Pi release before the working stroke, the slow tension transient following a Pi jump must be due to a related transition in the crossbridge cycle. Without having to invoke hidden slow transitions, one such transition is the reversal of the ﬁrst step in Eq. 3.28, namely detachment of M.ADP.Pi from actin, at a rate of 50–100 s1 in rabbit psoas muscle at 10 C. It also appears to be the only slow transition linked to Pi release by a common state. Then the net rate of equilibration from/to this lumped state would be r P ð xÞ ¼ k A ð xÞ þ

½Pi kA : ½Pi þ K~ P

ð3:31Þ

assuming that k-A is x-independent. After x-averaging, this rate constant describes the Pi-dependent rate at which tension decays exponentially to the high-Pi level, which is a reasonably accurate description of the tension transient. However, at high [Pi], there is some overshooting followed by a slower recovery phase. Crossbridge models with distributed x-averaging can reproduce both phases (Smith 2014), and the only point of departure from experimental records is a rapid tension change on the millisecond time-scale of Pi release from its cage.

3.3 Coordinating Lever-Arm Movements with Biochemical Events

81

Fig. 3.9 Bi-exponential rates of phase-2 tension recovery from length steps for different phosphate levels (Ranatunga et al. 2002). Rates as a function of step size show the separation of its component phases 2a (fast) and 2b (slow). The slow rate is markedly phosphatedependent in stretch, but otherwise all rates vary little with [Pi]. For releases, phase 2a is dominant. With permission of Elsevier Press

So far, the central hypothesis remains intact: the tension response to a phosphate jump can also be explained in terms of rapid reversible Pi release before the working stroke. (c) The rate of slow tension recovery from a quick stretch is phosphate-dependent. A revealing set of length-step experiments at different levels of phosphate was made by Ranatunga et al. (2002). To discuss them, a little terminology is required for the phases of tension recovery observed after length-stepping. Phase 2 is the rapid recovery ﬁrst observed by Huxley and Simmons, and modelled in Sect. 2.4. It is followed by a weak anti-recovery phase (phase 3), which often appears only as a plateau from 3 to 10 ms, then the ﬁnal approach to isometric tension (phase 4) with a time constant of 0.1–0.2 s. To complicate matters, the rapid recovery has two exponential sub-phases; phase 2a is fast and dominant in release, whereas phase 2b is slow and dominant in stretch. Although the apparent rate constant for phase 2 as a function of step size seems to make a smooth transition from releases to stretches, closer inspection shows that two sub-phases are present for any given length change (Burton et al. 2006). Figure 3.9 shows the results of these experiments. The rate of phase-2b recovery is markedly phosphate-sensitive, increasing by a factor of 10 from 1 to 25 mM Pi, whereas phase-2a recovery is less sensitive (by a factor of 2–3). Moreover, the Pi sensitivity of phase 2b could be measured only for stretches, where it is the dominant recovery phase. Again, the working stroke and phosphate release are intimately connected. These [Pi]-dependent recovery rates can be explained in terms of a slow Pi-release transition between bound states immediately before or after the working stroke (Smith and Sleep 2004). In that paper, arguments were presented for

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3 Actin-Myosin Biochemistry and Structure

preferring the scenario where slow Pi release comes ﬁrst. However, both possibilities have unwelcome consequences which are aspects of the same problem; a slow transition between actomyosin states on the main path of the contraction cycle will limit the unloaded shortening velocity of the muscle. If that slow transition occurs after the working stroke, it would limit the effective rate of ATP-induced dissociation, which is known to be limited by ADP release (Siemankowski et al. 1985). On the other hand, if a slow Pi-release transition occurs before the stroke, then pre-stroke A.M.ADP.Pi heads may be driven into negative strain before they can release Pi and generate tension by making a working stroke. The error in such modelling arises from the assumption that the Pi-release transition reﬂects the rate of phase-2b recovery from a quick stretch. Once this assumption is removed, the way is open to allowing very rapid Pi release from its active site in myosin, possibly at a rate of 104–5 s1, while associating the slow recovery observed in length steps and Pi jumps with a linked transition (Smith 2014). To avoid compromising the unloaded contraction velocity of the ﬁbre, this transition has to be the detachment of M.ADP.Pi from actin. Then the reaction scheme in Eq. 3.28 describes the observations of Ranatunga et al. as follows. For some heads in the pre-stroke state A.M.ADP#, a quick release triggers a working stroke which puts them out of reach for phosphate binding and actin detachment. Whereas a quick stretch induces some heads in the post-stroke state A.M.ADP to make a backward stroke, after which they can bind Pi and detach from actin. From this analysis of the three muscle-ﬁbre experiments, we can conclude that the event which triggers the working stroke is indeed Pi release, which occurs immediately after the myosin-products complex binds to actin. What other experiments bear on this conclusion? Other biochemical locations for the working stroke can be ruled out. In striated muscle, there is no stroke triggered by ADP release (Gollub et al. 1996). Adding millimolar ADP to a skinned ﬁbre from striated muscle causes a small increase in isometric tension, of order 20% (Cooke and Pate 1985), and this effect can be explained in terms of competitive inhibition of ATP. However, for smooth-muscle and non-muscle myosins, ADP release is accompanied an additional stroke of about 3 nm (Gollub et al. 1996; Whittaker et al. 1995; Jontes et al. 1995; Veigel et al. 2002), and a second stroke may also be present in human cardiac myosin (Liu et al. 2018). As with the working stroke, this second stroke acts as a strain-sensitive gate, the difference being that it can hold positive tension created by the working stroke and reduce the rate of ATP consumption. There is also clear evidence that the working stroke is not triggered when ATP binds to the rigor state A.M, although a model along these lines was developed by Worthington (1962). His “charging” model was overtaken by experimental facts such as Lymn and Taylor’s discovery that ATP triggers rapid dissociation from actin. Subsequently, single-myosin experiments in the optical trap, which reveal the working stroke (Finer et al. 1994), show that the lifetime of post-stroke states grows as [ATP] is reduced; ATP binding terminates the post-stroke actomyosin condition rather than initiating it.

3.3 Coordinating Lever-Arm Movements with Biochemical Events

3.3.2

83

The Location of the Repriming Stroke

After the working stroke, it is thought that myosin binds ATP and detaches from actin with its lever-arm in the post-stroke position, to be reprimed for a new cycle of attachment. There are structural and functional reasons why the repriming stroke should take place off actin. As this transition must take place on the main contraction cycle, the only possible location appears to be the transition between the two M.ATP states discovered by Malnasi-Csizmadia et al. (2000). The structural argument, taken up in the next section, is that the transition from the ‘open’ to the ‘closed’ form of M.ATP is coupled to the reversal of the working stroke. At the level of muscle function, there is also good reason for requiring the repriming stroke to take place after M.ATP has detached from actin. Consider again the claim of Eisenberg and coworkers that ATP can be hydrolysed on actomyosin, perhaps as rapidly as when on myosin alone. If the correlation between hydrolysis and the repriming stroke also applies to actomyosin, then such hydrolysis would not take place until the lever arm has reversed its power stroke. If this event occurred for all bound myosins, there would be dire consequences for the unloaded shortening velocity, a fact amply supported by modelling. As hydrolysis on actomyosin is observed in muscle where heads are cross-linked to actin (Stein et al. 1985), it is necessary to suppose that in native muscle hydrolysis on actin is generally avoided by the more rapid dissociation of A.M.ATP. In this way, hydrolysis of the ATP on post-stroke A.M.ATP lies off the main pathway of the contraction cycle.

3.3.3

An Amalgated Mechanochemical Cycle

Scheme 3.6 is an actin-myosin-ATP contraction cycle for muscle, built from Scheme 3.5 by including the working and repriming strokes of the myosin lever arm. The

Scheme 3.6 A mechanochemical actin-myosin-ATP cycle for striated muscle

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Table 3.3 Equilibrium constants, rate constants and Michaelis-Menten constants for the mechanochemical actin-myosin-ATP cycle of Scheme 3.6. Abbreviations: T ¼ ATP, D ¼ ADP, P ¼ Pi. Kinetic constants for transitions common to Scheme 3.5 are unchanged. Hydrolysis is possible only when the lever-arm has reprimed. The pre-stroke state M.T# is free to hydrolyse before or after actin binding, but the latter pathway is diminished by the low occupancy of the MT# state with KR ¼ 0.2 and KH ¼ 20. The M.D# state corresponds to state M*.ADP in the Bagshaw-Trentham scheme (Trentham et al. 1976) X K~ X , K X

P 0.1 M

S 0.1

D 2 105 M

T 4 1011 M1

R 0.2

H 20

k~X , k X CX(M) A K~ A X, KX

0.06 s1

104

1.4 s1

2 106 M1

400

500

0.007 0.01 M

---1000

3 10 2 104 M

2 104 4 105 M1

-------

-------

A k~A X , kX

105s1?

>104s1

>500 s1

3 106 M1s1

?

?

CA X (M)

----

----

?

3 104

----

----

X 1 K~ X A (M ) k~X (M1s1)

A CX A (M)

4

DP 4000

D# 400

D 4 106

---4 107

T (40)

4 105?

4 105?

4 105?

4 106

4 105?

?

?

?

5

3 10

4 10

T# ? 5

? ?

working stroke is the transition between the two A.M.ADP states, separated by an equilibrium constant K A S 1000 when the head is untethered, with a forward rate constant near 104 s1 to accomodate the phase-2 response to quick release steps. As discussed in the last section, the value of K A S is chosen to satisfy detailed balancing around the cycle under solution conditions. In the ﬁbre, K A S is hugely straindependent, as in Eq. 2.41. Scheme 3.6 also locates the repriming stroke off actin as the transition from open to closed M.ATP states, which proceeds at about 400 s1 with an equilibrium constant near 0.2. Kinetic constants for this scheme are given in Table 3.3 Scheme 3.6 also contains actin-binding transitions which are not on the main pathway of the Lymn-Taylor cycle. As in solution, the post-stroke states are strongly bound to actin; rates of dissociation are about 10 s1 for A.M.ADP, and 0.1 s1 for A.M, which is negligible over the duration of a tetanus. However, the pre-stroke state (A.M.ADP#) is less strongly bound; its actin afﬁnity is comparable with that of M. ADP.Pi. This state may be prone to detachment in its own right, which is to say that it could detach to M.ADP# instead of binding phosphate and detaching to M.ADP. Pi. However, as with M.ADP.Pi, the actin-binding kinetics of M.ADP# in solution is obscured by the subsequent working stroke. For myosin heads in muscle that sit in a stable pre-stroke state, this possibility remains open but unproven. Another example in which transitions off the main pathway become relevant occurs at low concentrations of ATP. With no ATP present, the muscle goes into rigor, where all heads in the sarcomere are bound to actin (Cooke and Franks 1980).

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85

Fig. 3.10 Isometric tension of frog muscle at 0-4 C (Ferenczi et al. 1984) as a function of the concentration of MgATP concentration (in M), relative to tension at 0.005 M. With permission of John Wiley and Sons Inc. and M. Ferenczi

However, the rigor state A.M is not strongly populated at millimolar levels of ATP, which drives cycling on the Lymn-Taylor pathway. Between these extremes, Fig. 3.10 shows that tension reaches a maximum near 50 μM ATP (Ferenczi et al. 1984). For lower concentrations, the population of the A.M state increases, because the ATP-binding and actin-dissociation transitions in the contraction cycle are replaced by the box of states in Scheme 3.4. At micromolar ATP and below, the rate of equilibration is limited by the dissociation of A.M.

3.4

The Atomic Structure of Myosin Complexes

At the molecular level, a huge advance in our understanding of muscle action was made possible by the X-ray determination of the atomic structure of myosin-S1 by Ivan Rayment and coworkers, after persuading myosins to crystallize. Electron microscopy was not quite equal to the task, although single-particle analysis (Burgess et al. 1997) did reveal the motor domain and lever-arm in conﬁgurations related by a working stroke. Cryo-EM can now resolve atomic structures with a resolution of 2 Å. The atomic structure of myosin-S1 structure consists of a 95 KD heavy chain of 834 amino-acid residues, the essential and regulatory light chains (~25 KD), plus an indeterminate number of water molecules. The sequence of residues on the chains was determined and their topology mapped to a resolution of 2 Å (Rayment et al. 1993). Orthogonal views of this structure are shown in Fig. 3.11. As the atomic structure of monomeric actin and the actin ﬁlament had previously been determined (Kabsch et al. 1990), the next step was to study how myosin-S1 could dock on to the actin ﬁlament. The Rayment structure, if bound to actin, was initially thought to correspond to rigor myosin, but it now appears to be a post-rigor state, in which the lever arm remains in the post-stroke position after ATP binds and the complex dissociates from actin (Geeves and Holmes 2005). A better approximation to the rigor state of myosin II is thought to be the non-muscle myosin V with no bound

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Fig. 3.11 Ribbon diagrams of the atomic structure of rigor myosin (Holmes et al. 2004), as viewed (a) from the pointy () end of the actin ﬁlament and (b) at right angles, with the pointy end of F-actin (to the M-band) at the top of the ﬁgure. With permission of the Royal Society

nucleotide, whose structure was determined by Coureux et al. (2005). The structure of actomyosin complexes cannot be determined by X-ray crystallography, so electron microscopy was used to dock the actin and myosin structures (Holmes et al. 2004).

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87

Fig. 3.12 Atomic structures of the four structural states of actin-myosin-nucleotide in the crossbridge cycle (Sweeney and Houdusse (2010). Proceeding clockwise from lower right are the pre-stroke states M.ATP/M.ADP.Pi, the pre-stroke A.M.ADP.Pi state, the rigor state A.M and the detached post-stroke state M.ATP formed when ATP binds to rigor and dissociates the actin-myosin bond. The second state has not been determined directly from a crystal structure. With permission of Annual Reviews Inc.

For myosin II, we have atomic structures for rigor myosin (A.M), the post-rigor state M.ATP and the products states M.ATP/M.ADP.Pi (probably a mixture). In the latter, the lever arm is in the pre-stroke position, but in rigor and post-rigor the lever arm has swung through 60 to the post-stroke position, where its distal end has been displaced by 8–10 nm along the actin axis (Fig. 3.12). The missing structure is the actomyosin-products state A.M.ADP.Pi, which is a transition state preceding phosphate release. The lever-arm of this state should be in the pre-stroke position. To check this, the terminal phosphate ion was replaced by non-dissociating analogues, namely vanadate (Smith and Rayment 1996), aluminium ﬂuoride and beryllium ﬂuoride (Fisher et al. 1995). For VO4 and AlF4 the lever-arm was in the pre-stroke position, but BeFx generated the post-stroke position, so the results do not extrapolate to A.M.ADP.Pi. Recently, the atomic structure of A.M.ADP has been determined for myosin V (Wulf et al. 2016); here the lever-arm has made a stroke, so the working stroke must be associated with phosphate release.

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Fig. 3.13 Intermediate representations of myosin II structure. (a) The subdomains and their connections (Sweeney and Houdusse 2010). (b) A 2D rendering of its topology in terms of loops, helices (circles) and β-sheets (triangles), from Geeves et al. (2005). With permissions of Annual Reviews Inc. and Springer Nature respectively

For a deeper understanding of the myosin cycle of actin binding, stroking, ATP binding and hydrolysis, we need to be familiar with the characteristic features of these atomic structures. In common with many proteins, sections of the heavy chain adopt characteristic shapes such as beta sheets and helices, joined by ﬂexible loops. A grand tour of the full 3D structure of myosin-S1 entities requires 3D visualization software such as PyMOL, RasMol or Molscript. Starting with the motor domain and the lever-arm, one can get to grips with it by identifying subdomains (Fig. 3.13a)

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89

before exploring the convolutions of the heavy-chain, indexed by its sequence of residues. A useful guide is the 2D rendering shown in Fig. 3.13b. Subdomains of myosin-S1 were initially deﬁned by tryptic digestion, which strips off the light chains and gives three heavy-chain subdomains of 20 KD, 50 KD and 25 KD. Figure 3.13a shows the general disposition of these subdomains. The 20 KD domain is the lever-arm which reaches down to the actin interface, and its COOH (carboxyl) terminal is at the distal end, which in muscle is connected to one strand of the myosin rod (a coiled coil). In the motor domain, the 50 KD subdomain binds to actin and is divided into lower and upper parts (L50, U50) by a deep cleft running from the actin interface to a hinge. These parts are also loosely connected by disordered loops, which may not appear in X-ray diffraction images. The U50 domain is also linked to a 25 KD nucleotide binding domain, which incorporates the NH2 (amide) terminal of the heavy chain. The junction between these two domains deﬁnes a pocket in which ATP can enter and be hydrolysed. The leverarm is joined to the motor domain via a converter domain, which is linked to the lower 50 KD actin-binding domain by the relay helix, and to the N-terminal domain by another helix (SH1). Movements in the L50 and U50 domains triggered by strong actin binding are transmitted to the converter by these two helices, causing the leverarm to rotate about the SH1 axis to give the working stroke. Subdomain movements also open or close the actin-binding cleft and the nucleotide pocket. Biochemically, we know that the two events are closely correlated; strong actin binding closes the cleft and opens the pocket so ADP can be released and ATP can bind. ADP in the pocket is bound when the pocket is closed by a salt bridge between sulfhydryl groups, one on U50 and one on the N-terminal domain; this bridge must be broken to let ADP out. However, the γ-Pi of A.M.ADP.Pi is released before ADP, and ADP in the pocket would block its escape. So it was proposed that the γ-Pi can escape via a backdoor route to the base of the actin-binding cleft (Yount et al. 1995), and the backdoor may be a trapdoor which opens and closes (Pate et al. 1997; Lawson et al. 2004; Cecchini et al. 2010). In all there are four classes of reversible events, namely actin binding, the working stroke, and the opening of the nucleotide pocket and the trapdoor. How are these events linked together? Potential answers to these questions have been formulated in terms of structural elements, notably two switches which determine how the terminal phosphate of A. M.ADP.Pi is bound to the site which catalysed the hydrolysis of M.ATP. The γ-Pi is surrounded by three loop motifs of the heavy chain, namely switch 1, switch 2 and a P-loop. SW1 lies in the U50 domain, and always interacts with the Mg++ ion of Mg. ATP. In the closed position, it also interacts with the γ-Pi, if present, through a hydrogen bond, but in the open position it moves away (Reubold et al. 2003). SW2 is a loop between the relay helix and the β-sheet (Fig. 3.13a), and connects the L50 and U50 domains. When SW2 is closed it also forms a H+ bond with the γ-Pi (loc. cit.). The γ-Pi also interacts with the P-loop within the N-terminal domain. This and related observations suggest the entries shown in Table 3.4.

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3 Actin-Myosin Biochemistry and Structure

Table 3.4 Closed and open states of the two switch loops for the four actin-myosin-nucleotide complexes, and their relation to the conﬁgurations of other components. Adapted from Geeves and Holmes (2005)

SW1 SW2 Lever-arm Cleft Pocket β-Sheet Relay

M.ATP# M.ADP.Pi A.M.ATP# C C Primed O C Flat Kinked

A.M.ADP.Pi

A.M.ADP

M.ATP

A.M.ADP# O C Primed C C Flat Kinked

A.M O O Stroked C O Twisted Straight

A.M.ATP C O Stroked O O Flat Straight

To focus on how this table could work in practice, consider the following questions: Q1: How does myosin make a working stroke? Q2: What triggers the stroke? Q3: How does actin-binding trigger the release of Pi? Q4: Could there be two sequential working strokes? Q5: How does the working stroke trigger the release of ADP? Q6: How does ATP binding to rigor myosin reduce its actin afﬁnity so dramatically? Q7: How does ATP binding to detached myosin trigger the repriming stroke? Q8: Can repriming take place on actin as well as off it? Q9: Why is repriming a prerequisite for myosin to catalyse the hydrolysis of ATP? Q10: Can myosin also hydrolyse ATP when bound to actin? It would be very misleading to suggest that we have all the answers, or that there is general agreement about the entries in Table 3.4. The following is intended as an introduction to a better description of the transitions between the four structural states in Fig. 3.12.

3.4.1

Actin Binding

To answer the ﬁrst question, we could begin by asking how the pre-stroke A.M. ADP.Pi state is formed from its detached counterpart. If M.ADP.Pi has the same structure as M.ATP, the actin-binding cleft is open and both switches are closed, which means that the γ-Pi group is trapped and cannot escape. However, it is generally agreed that A.M.ADP.Pi is a strongly-bound state, in which the actinbinding cleft has closed and the L50 and U50 domains are both bound to actin. Consequently, the motor domain is stereospeciﬁcally bound, and can support tension. Although the structure is not available, the current view is that SW1 must have opened.

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91

Why does the cleft close when M.ADP.Pi binds to actin? Closure must require some distortion of the L50 and U50 domains, since they are unstrained in the detached state where the cleft is open. Recent cryo-EM studies of a human rigor actomyosin complex (von der Ecken et al. 2016) explore how actin binding occurs in two stages, governed by the interaction of six loops on the interface of the motor domain, so that the L50 domain binds ﬁrst and then rotates to bring more loops into contact, including those in the U50 domain, to close the cleft. So SW1, which is rooted in the U50 domain, should open as a means of relieving some of the elastic strain generated. During this process, SW2 stays closed and traps the γ-phosphate ion, which is crucially important in stabilising the pre-stroke position of the converter-lever-arm assembly attached to the relay helix and SH1. During actin binding, the β-sheet remains unstrained, and to accommodate it the relay helix is kinked.

3.4.2

Phosphate Release and the Working Stroke

In rigor myosin, the cleft stays closed, SW2 has opened and the lever-arm is in the post-stroke position, because the L50 domain has rotated, pushing the relay helix and the converter to a new orientation which deﬁnes the working stroke of the leverarm. To do this, the beta sheet has twisted, which generates a space for the relay helix to straighten out. Now myosin in rigor is more strongly bound to actin than the products state; the net change in Gibbs energy is negative. Hence the energy required to twist the beta-sheet has been outweighed by a decrease in energy from some other source. It seems unlikely that this could come from straightening the relay helix, which is a much smaller structure than the β-sheet, so the decrease in energy must have come from stronger actin-binding, presumably from the U50 domain. What triggers the working stroke? The trigger is presumably the opening of SW2, which would release γ-Pi from its catalytic site, because SW1 is already open. Whether the opening of SW2 is caused by another localised rearrangement in the structure is not clear, but a time-resolved FRET study (Muretta et al. 2013) suggests that the relay helix straightens out before Pi is released. Contrary to expectations, this result implies that the working stroke would precede Pi release, but that depends on how the release is measured. Solution studies show that Pi is released from A.M.ADP.Pi to the medium at 75 s1 (White et al. 1997), which is much slower than the working stroke. This release rate could be limited by its passage down the backdoor tube to the actin cleft, but only if the phosphate ion is sequestered along the way; Houdusse and Sweeney (2016) have suggested that it could bind transiently to a site near the mouth of the cleft. Then rapid Pi release from its active site can still open SW2 to drive the relay helix and the working stroke, leaving Pi free to ﬂoat internally in an active site free of switch elements. Is it possible for Pi to be released without the subsequent working stroke? Such a state is not accessible to X-ray crystallography, but it could exist if the lever-arm were forcibly restrained, for example in a single-molecule load clamp. The

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3 Actin-Myosin Biochemistry and Structure

reversibility of Pi release suggests that a pre-stroke A.M.ADP state does exist in the muscle ﬁbre, even if it is short-lived in solution (Sleep and Hutton 1980). The state formed after Pi release and the working stroke is the more stable of the two A.M.ADP states proposed by Sleep and Hutton. For myosin II in solution, this state releases ADP quite rapidly (Siemankowski et al. 1985). For this to occur, the nucleotide pocket must have opened, and this process must have occurred as part of the stroke transition. Perhaps SW2 opens up the nucleotide pocket by breaking the salt bridge between the sulfhydryl groups.

3.4.3

An ADP-Release Stroke

Modelling contractile behaviour from a strain-dependent actin-myosin-ATP cycle generally requires a second lever-arm stroke associated with the release of ADP. This stroke serves to stop post-stroke heads with positive tension from releasing ADP, binding ATP and detaching from actin, all of which are rapid events. Thus an ADP-release stroke acts as a strain gate, removing negatively-strained post-stroke heads which would lower the net tension while enhancing the lifetime of positivelystrained post-stroke heads. Although there is no evidence for an ADP-release stroke in myosin II, the potential for an ADP release stroke should be present in its atomic structure. In the next chapter, we present a hypothetical mechanism for strain-gated ADP release via an excited state with no net working stroke. This mechanism could be compatible with a visible second stroke for other myosins if the excited state becomes stable. A new high-resolution cryo-EM structure of myosin-1B, with and without ADP, from Ostap’s laboratory (Mentes et al. 2018) reveals the structural basis for a second stroke. They found two A.M.ADP states separated by a 25 swing of the lever-arm, starting from a lever-arm angle above 90 , held in place by a hitherto unknown interaction with the N-terminal subdomain above the nucleotide pocket. This interaction blocks ADP release from the pocket and must be relieved by the second stroke. Thus the second stroke takes place before ADP release. If this interaction reforms after ADP is released, then no second stroke would be apparent as excess tension after adding ADP to the muscle ﬁbre. But the interaction would have to be broken again to permit ATP to enter and bind to complete the cycle of actomyosin attachment.

3.4.4

ATP Binding and Actin Afﬁnity

When ATP binds to rigor myosin II, the ﬁrst state formed is the post-rigor state M. ATP, in which SW1 has closed and SW2 stays open (Table 3.4). Hence the leverarm remains in the post-stroke position, while the closure of SW1 means that the cleft has opened and the U50 domain is no longer bound to actin. This change is

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93

quite sufﬁcient to explain the huge loss in actin afﬁnity, but it does imply that the actin afﬁnity of M.ATP is much lower than that of M.ADP.Pi, in which both parts of the 50KD domain bind to actin. From Table 3.3, the afﬁnity of M.ADP.Pi is 104 times weaker than rigor, whereas that of M.ATP is about 106 times weaker. So the connection between actin binding and cleft closure supplied by atomic structures backs up the biochemistry in this respect.

3.4.5

The Repriming Stroke and Hydrolysis

It was a solution-kinetic experiment (Malnasi-Czizmadia et al. 2000, Urbanke and Wray 2001) rather than an atomic structure determination that showed that the postrigor state M.ATP is the initial state for the lever-arm to reprime to its pre-stroke position. The transition is relatively fast and very reversible, with K 0.2. The ﬁnal state M.ATP# is the enzymatically active state from which hydrolysis proceeds, and for this to occur SW2 must have closed. SW1 had already closed to form the postrigor state, and the closure of both switches also closes the nucleotide pocket to bind Mg.ATP and prevent it from escaping. Under these conditions Mg.ATP will be hydrolysed, after which Mg.ADP in the pocket is also bound and the liberated Pi is bound to its active site at the base of the pocket.

3.4.6

Hydrolysis on Actomyosin?

Actin-binding titrations show that myosin complexed with a non-hydrolysable ATP analogue has about the same actin afﬁnity as M.ADP.Pi. Thus there seems to be no reason why the enyzmatically active state M.ATP#, which is a pre-stroke state with SW2 closed, could not bind to actin and hydrolyse ATP. In this way, the sequence of hydrolysis and actin-binding could be reversed without any mechanical penalty associated with working strokes. The importance of this pathway is reduced by the low equilibrium constant of the repriming stroke, which lowers the occupancy of the M.ATP# state; the equilibrium occupancies of states M.ATP, M.ATP# and M.ADP. Pi are in the ratio 1:0.2:4. However, this pathway is not the only way in which hydrolysis on actin could take place. As originally proposed by Eisenberg and collaborators, the other possibility is that ATP could be hydrolysed from the post-rigor state A.M.ATP. The difference is that this is a post-stroke state, although very weakly bound. Fortunately, there is no such thing as a post-stroke A.M.ADP.Pi state, which would be the product of a futile hydrolysis. Thus the lever-arm would need to reprime before hydrolysis on actin could take place. There seems to be no structural reason why repriming could not occur, particularly as SW2 would not close until repriming is complete. However, repriming when bound to actin would greatly reduce the unloaded contraction velocity of the muscle. Although this objection appears ad hoc, it does have

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3 Actin-Myosin Biochemistry and Structure

substance. To recapitulate; the rate of dissociation of A.M.ATP is about 100 times faster than the expected rate of hydrolysis, so dissociation would be the dominant pathway and stroke reversals on actin would occur only with a frequency of 1 in 100.

3.4.7

The Pathway of the Stroke

A deeper understanding of the mechanism of the working stroke, and its reversal, in myosin motors would require knowledge of the reaction pathway, which in principle is determined by molecular dynamics. In practice, the energy landscape of the stroke reaction cannot be calculated in this way. Computational molecular dynamics requires time steps of 1015 s and runs beyond 1 ns generally require parallel processing, whereas ﬁbre experiments suggest that the stroke transition occurs on a time scale of 10–100 μs. However, the reaction must proceed through one or more saddle points, where the energy is a local maximum along the reaction coordinate but a minimum with respect to all conjugate coordinates. The conjugate gradient method (Fischer and Karplus 1992) has been developed to locate saddle points in the space of atomic coordinates. If a saddle point is found, steepest-descent methods can be used to construct the minimum-energy pathway which links it to the initial and ﬁnal conﬁgurations. If there is more than one saddle-point, then multiple pathways are possible, although there may a dominant path with the fastest reaction rate. Methods for calculating reaction rates from the energy landscape are highly evolved (Hanggi et al. 1990; Weiss 1986), but the initial interest is in describing how the working stroke, which is a concerted transition in a large protein, evolves from a very localised internal transition, thought to be the opening of SW2. For myosin from Dictyostelium discoideum, a calculation of this kind has been performed by Fischer et al. (2005) to map the pathway of the repriming stroke M. ATP ! M.ATP#. This is a slightly simpler task as it occurs on detached myosin, although the time scale of the transition is longer than the working stroke. Hence they proceeded by locating saddle points to construct the minimum energy path. The result can be described as a cascade of transitions spreading through the structure, ostensibly driven by the relay helix. In the initial post-stroke state, they found that the relay helix was not kinked but bent around the residue Phe652, which acted as a fulcrum. The repriming transition occurred in two phases; ﬁrstly the relay helix straightened out, causing a 40 back-rotation of the lever-arm, and then it partially unwound, causing a further 25 rotation which completed the repriming stroke. In principle, similar calculations could be made for the working stroke itself, which occurs on actomyosin. However, the reaction pathway cannot be the same, for at least two reasons. The ligands are different; we have argued that the working stroke is the transition between the two A.M.ADP states proposed by Sleep and Hutton (1980), with an equilibrium constant K A S > > 1, whereas the repriming stroke occurs between M.ATP states with an equilibrium constant KR < 1. Secondly, once SW2 has opened, the beta-sheet twists to accommodate the stronger actin-myosin

3.4 The Atomic Structure of Myosin Complexes

95

interactions associated with the post-stroke state. Whereas in the repriming stroke, which occurs on detached myosin, the beta-sheet remains ﬂat. Hence the reaction pathways of the forward and backward strokes are different, and so are the initial and ﬁnal states. This short survey is intended to give the uninitiated reader a taste of a very intensive area of research, spanning many kinds of myosin motors. For brevity, ligand-residue interactions, such as the interactions between the phosphate ion and SW1, SW2 and the P-loop (Reubold et al. 2003), have not been speciﬁed. Many important aspects of mechanism have been omitted, for example the roles of the strut and loops in the myosin head which control different degrees of actin binding, from ﬂexible attachment to stereospeciﬁc states of increasing strength. For example, there is a cardiomyopathy loop which contributes to systolic tension in heart muscle; the mutation which deletes it causes cardiac hypertrophy, in which the heart wall becomes larger and stiffer. What take-home messages encapsulate these biochemical and structural advances? Here are four observations intended as a pithy summary of actin-myosin biochemistry: 1. Actin and myosin are yin and yang to ATP. Myosin is good at hydrolysing ATP but very bad at releasing the products of hydrolysis (Pi and ADP), whereas actin bound to myosin is very good at releasing the bound products. Actin also hydrolyses ATP, but very slowly, and myosin bound to actin is of no help in releasing the products on actin. 2. Actin and ATP are yin and yang to myosin. The sequential release of the products of hydrolysis on myosin strengthens the actin-myosin bond, while the afﬁnity of these products for actomyosin relative to myosin becomes weaker. In a mirror image of this picture, the subsequent binding of ATP dramatically weakens the actin-myosin bond while the afﬁnity of ATP for myosin relative to actomyosin is much stronger. 3. The karma of myosin’s working stroke. The working stroke is the result of a cascade of local rearrangements within the motor domain, starting from the opening of the switch-2 loop which is connected to the relay helix. Switch-2 is held in place by interactions with the terminal Pi of hydrolysed ATP, which in turn is held by a P-loop and also by switch-1 if closed. When switch-2 opens, it moves away from the γ-Pi which is freed to escape provided switch-1 has already opened. 4. The ﬂux of reactions which produces the working stroke closes back on itself to form a cycle. Post-stroke myosin with bound ATP must make a repriming stroke for the nucleotide pocket to close, and hydrolysis cannot occur until the pocket is closed.

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References Alberty RA (1968) Effect of pH and metal ion concentration on the equilibrium hydrolysis of adenosine triphosphate to adenosine diphosphate. J Biol Chem 243:1337–1348 Bagshaw CR, Trentham DR (1973) The reversibility of adenosine triphosphate cleavage by myosin. Biochem J 133:323–328 Bagshaw CR, Trentham DR (1974) The chacterization of myosin-productr complexes and of product-release steps during the magnesium-ion-dependent adenosine triphosphate reaction. Biochem J 141:331–349 Baker J, Brust-Mascher I, Ramachandran S, LaConte LEW, Thomas DD (1998) A large and distinct rotation of the myosin light chain domain occurs upon muscle contraction. Proc Natl Acad Sci USA 95:2944–2949 Bowater R, Sleep J (1988) Demembranated muscle ﬁbers catalyze a more rapid exchange between phosphate and adenosine triphosphate than actomyosin subfragment 1. Biochemistry 27:5314–5323 Burgess SA, Walker ML, White HD, Trinick J (1997) Flexibility within myosin heads revealed by negative stain and single-particle analysis. J Cell Biol 139:675–681 Burton K, Simmons RM, Sleep J (2006) Kinetics of force recovery following length changes in active skinned ﬁbers from rabbit psoas muscle: with an Appendix: Simmons RM, Burton K, Smith DA. Analysis and modelling of the late recovery phase. J Physiol (London) 573:305–328 Caremani M, Dantzig J, Goldman YE, Lombardi V, Linari M (2008) Effect of inorganic phosphate on the force and number of myosin cross-bridges during the isometric contraction of permeabilized muscle ﬁbres from rabbit psoas. Biophys J 95:5798–5808 Cecchini M, Alexeev Y, Karplus M (2010) Pi release from myosin; a simulation analysis of possible pathways. Structure 18:458–470 Cooke R (1997) The mechanism of muscle contraction. Crit Rev Biochem 21:53–118 Cooke R, Franks K (1980) All myosin heads form bonds with actin in rigor rabbit skeletal muscle. Biochemistry 19:2265–2269 Cooke R, Pate E (1985) The effects of ADP and phosphate on the contraction of muscle ﬁbres. Biophys J 48:789–798 Coupland ME, Puchert E, Ranatunga KW (2001) Temperature dependence of active tension in mammalian (rabbit psoas) ﬁbres: effect of inorganic phosphate. J Physiol (London) 536:879–891 Coureux P-D, Wells AL, Menetrey J, Yengo CM, Morris CA, Sweeney HL, Houdusse A (2005) A structural state of the myosin V motor without bound nucleotide. Nature 425:419–423 Criddle AH, Geeves MA, Jeffries T (1985) The use of actin labeled with N-(1- pyrenyl) iodoacetamide to study the interaction of actin with myosin subfragments and troponin/tropomyosin. Biochem J 232:343–349 Dantzig JA, Goldman YE, Millar NC, Lacktis J, Homsher E (1992) Reversal of the cross-bridge force-generating transition by photogeneration of phosphate in rabbit psoas muscle ﬁbers. J Physiol (London) 451:247–278 Eisenberg E, Moos C (1970) Actin activation of heavy meromyosin adenosine triphosphatase. J Biol Chem 245:2451–2456 Fajer PG, Fajer EA, Thomas DD (1990) Myosin heads have a broad orientational distribution during isometric muscle contraction: Time-resolved EPR studies using caged ATP. Proc Natl Acad Sci USA 87:5538–5542 Ferenczi MA, Goldman YE, Simmons RM (1984) The dependence of force and shortening on substrate concentration in skinned muscle ﬁbres from Rana Temporaria. J Physiol London 350:519–543 Finer JT, Simmons RM, Spudich JA (1994) Single myosin molecule mechanics: piconewton forces and nanometer steps. Nature 368:113–119 Fischer S, Karplus M (1992) Conjugate peak reﬁnement: an algorithm for ﬁnding reaction paths and accurate transition states in systems with many degrees of freedom. Chem Phys Lett 194:252–260

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Fischer S, Windschugel B, Horak D, Holmes KC, Smith JC (2005) Structural mechanism of the recovery stroke in the myosin molecular motor. Proc Nat Acad Sci USA 102:6873–6878 Fisher AJ, Smith CA, Thoden JB, Smith R, Sutoh K, Holden HM, Rayment I (1995) X-ray structures of the myosin motor domain of Dictyostelium discoideum complexed with MgADP•BeFx and MgADP•AlF4. Biochemistry 34:8960–8972 Fryer MW, Owen VJ, Lamb GD, Stephenson DG (1995) Effects of creatine phosphate and Pi on Ca2+ movements and tension development in rat skinned skeletal muscle ﬁbres. J Physiol (London) 482(1):123–140 Geeves MA (1989) Dynamic interaction between actin and myosin subfragment 1 in the presence of ADP. Biochemistry 28:5864–5871 Geeves MA (1991) The dynamics of actin and myosin association and the crossbridge model of muscle contraction. Biochem J 274:1–14 Geeves MA, Conibear PB (1995) The role of three-state docking of myosin S1 with actin in force generation. Biophys J 68:194s–201s Geeves MA, Holmes KC (2005) The molecular mechanism of muscle contraction. Adv Protein Chem 71:161–193 Geeves MA, Goody RS, Gutfreund H (1984) The kinetics of acto-S1 interaction as a guide to a model for the crossbridge cycle. J Muscle Res Cell Motil 5:351–361 Geeves MA, Jeffries TE, Millar NC (1986) ATP-induced dissociation of rabbit skeletal actomyosin subfragment. Characterization of an isomerization of the ternary acto-S1-ATP complex. Biochemistry 25:8454–8458 Geeves MA, Fedorov R, Manstein DJ (2005) Molecular mechanism of actomyosin – based motility. Cell Mol Life Sci 62:1462–1477 Goldman YE (1987) Kinetics of the actomyosin ATPase in muscle ﬁbers. Ann Rev Physiol 49:637–654 Gollub J, Cremo CR, Cooke R (1996) ADP release produces a rotation of the neck region of smooth myosin but not skeletal myosin. Nat Struct Biol 3:796–802 Goody RS, Hofmann W, Mannherz HG (1977) The binding constant of ATP to myosin S1 fragment. Eur J Biochem 78:317–324 Greene LE, Eisenberg E (1980) Dissociation of the actin•subfragment 1 complex by Adenyl-50 -yl Imidophosphate, ADP and PPi. J Biol Chem 255:543–548 Hanggi P, Talkner P, Borkovec M (1990) Reaction-rate theory: ﬁfty years after Kramers. Rev Mod Phys 61:251–341 Hibberd MG, Trentham DR (1986) Relationships between chemical and mechanical events during muscular contraction. Ann Rev Biophys Biophys Chem 15:119–161 Highsmith S (1976) Interaction of the actin and nucleotide binding sites on myosin subfragment 1. J Biol Chem 251:6170–6172 Holmes KC, Schroder RR, Sweeney HL, Houdusse A (2004) The structure of the rigor complex and its implications for the power stroke. Phil Trans Roy Soc B 359:1819–1828 Homsher E, Lacktis J, Regnier M (1997) Strain-dependent modulation of phosphate transients in rabbit skeletal muscle. Biophys J 72:1780–1791 Houdusse A, Sweeney HL (2016) How myosin generates force on actin ﬁlaments. TIBS 41:989–997 Johnson KA, Taylor EW (1978) Intermediate state of subfragment 1 and actosubfragment ATPase: reevaluation of the mechanism. Biochemistry 17:3432–3442 Jontes JD, Wilson-Kubalek EM, Milliagn RA (1995) A 32 tail swing in brush border myosin I on ADP release. Nature 378:751–753 Kabsch W, Mannherz HG, Suck D, Pai EF, Holmes KC (1990) Atomic structure of the actin: DNase I complex. Nature 347:37–44 Kawai M, Guth K, Wiinikes K, Haist C, Ruegg JC (1987) The effect of inorganic phosphate on the ATP hydrolysis rate and the tension transients in chemically skinned rabbit psoas ﬁbers. Pﬂugers Arch 408:1–8

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Lawson JD, Pate E, Rayment I, Yount RG (2004) Molecular dynamics analysis of structural factors inﬂuencing back door Pi release in myosin. Biophys J 86:3794–3803 Liu C, Kawana M, Song D, Ruppel KM, Spudich JA (2018) Controlling load-dependent kinetics of β-cardiac myosin at the single-molecule level. Nat Struct Mol Biol 25:5050–5514 Lymn RW, Taylor EW (1970) Transient state phosphate production in the hydrolysis of nucleoside triphosphates by myosin. Biochemistry 9:2975–2983 Lymn RW, Taylor EW (1971) Mechanism of adenosine triphosphate hydrolysis by actomyosin. Biochemistry 10:4617–4624 Ma Y-Z, Taylor EW (1994) Kinetic mechanism of myoﬁbril ATPase. Biophys J 66:1542–1553 Malnasi-Csizmadia A, Woolley RJ, Bagshaw CR (2000) Resolution of conformational states of Dictyostelium myosin II motor domain using tryptothan (W501) mutants: implications for the open-closed transition identiﬁed by crystallography. Biochemistry 39:16135–16146 Mannherz HG, Schenk H, Goody RS (1974) Synthesis of ATP from ADP and inorganic phosphate at the myosin-subfragment 1 active site. Eur J Biochem 48:287–295 Marston S, Weber A (1975) The dissociation constant of the actin-heavy meromyosin subfragment1 complex. Biochemistry 14:3868–3873 McKillop DFA, Geeves MA (1990) Effect of phosphate and sulphate on the interaction of actin and myosin subfragment 1. Biochem Soc Trans 18:585–586 Mentes A, Huehn A, Liu X, Zwolak A, Dominguez R, Shuman H, Ostap EM, Sindelar CV (2018) High-resolution cryo-EM structures of actin-bound myosin states reveal the mechanism of myosin force sensing. Proc Natl Acad Sci USA 115:1292–1297 Muretta JM, Petersen KJ, Thomas DD (2013) Direct real-time detection of the actin-activated power stroke within the myosin catalytic domain. Proc Natl Acad Sci USA 110:7211–7216 Muretta JM, Rohde JA, Johnsrud DO, Cornea S, Thomas DD (2015) Direct real-time detection of the structural and biochemical events in the myosin power stroke. Proc Nat Acad Sci USA 112:14272–14,277 Nesmelov YE, Agafonov RV, Negrashov IV, Blakely SE, Titus MA, Thomas DD (2011) Structural kinetics of myosin by transient time-resolved FRET. Proc Natl Acad Sci USA 108:1891–1896 Ostap EM, Barnett VA, Thomas DD (1995) Resolution of three structural states of spin-labelled myosin in contracting muscle. Biophys J 69:177–188 Pate E, Naber N, Matuska M, Franks-Skiba K, Cooke R (1997) Opening of the myosin nucleotide triphosphate binding domain during the ATPase cycle. Biochemistry 36:12115–12166 Potma EJ, van Grass A, Steinen GJM (1995) Inﬂuence of inorganic phosphate and pH on ATP utilization in fast and slow skeletal muscle ﬁbers. Biophys J 69:2580–2589 Ranatunga KW, Coupland MB, Mutungi G (2002) An asymmetry in the phosphate dependence of tension transients induced by length perturbation in skinned ﬁbres from mammalian (rabbit psoas) ﬁbres. J Physiol (London) 542(3):899–910 Rayment I, Rypniewski WR, Schmidt-Base K, Smith R, Tomchick DR, Benning MM, Winkelmann DA, Wesenberg G, Holden HM (1993) Three-dimensional structure of a myosin subfragment-1: a molecular motor. Science 261:50–65 Reubold TF, Eschenburg S, Becker A, Kull FJ, Manstein DJ (2003) A structural model for actininduced nucleotide release in myosin. Nat Struct Biol 10:826–830 Rosenfeld SS, Taylor EW (1984) The ATPase mechanism of skeletal and smooth muscle actosubfragment. J Biol Chem 259:11908–11919 Segel IH (1978) Enzyme Kinetics. Wiley Interscience, New York Shih WM, Gryczynski Z, Lakowicz JR, Spudich JA (2000) A FRET-based sensor reveals large ATP-hydrolysis-induced conformational changes and three distinct states of the myosin motor. Cell 102:683–694 Siemankowski RF, Wiseman MO, White HD (1985) ADP dissociation from actomyosin subfragment 1 is sufﬁciently slow to limit the unloaded shortening velocity in vertebrate muscle. Proc Natl Acad Sci USA 82:658–662 Sleep JA, Hutton RL (1978) Actin mediated release of ATP from a myosin-ATP complex. Biochemistry 17:5423–5430

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Sleep JA, Hutton RL (1980) Exchange between inorganic phosphate and adenosine 50 - triphosphate in the medium by actomyosin subfragment 1. Biochemistry 19:1276–1283 Smith DA (2014) A new mechanokinetic model for muscle contraction, where force and movement are triggered by phosphate release. J Muscle Res and Cell Mot 35:295–306 Smith CA, Rayment I (1996) X-ray structure of the magnesium (II)•ADP•vanadate complex of the Dictyostelium discoideum myosin motor domain to 1.9A resolution. Biochemistry 35:5404–5417 Smith DA, Sleep JA (2004) Mechanokinetics of rapid tension recovery in muscle: the myosin working stroke is followed by a slower release of phosphate. Biophys J 87:442–456 Stein LA, Chock PB, Eisenberg E (1981) Mechanism of the actomyosin ATPase: effect of actin on the ATP hydrolysis step. Proc Natl Acad Sci USA 78:1346–1350 Stein LA, Green LE, Chock PB, Eisenberg E (1985) Rate-limiting step in the actomyosin adenosinetriphosphatase cycle: studies with myosin subfragment 1 cross-linked to actin. Biochemistry 24:1357–1363 Suzuki Y, Yasunaga T, Ohkura R, Wakabayashi T, Sutoh K (1998) Swing of the lever arm of a myosin motor at the isomerization and phosphate-release steps. Nature 396:380–383 Sweeney HL, Houdusse A (2010) Structural and functional insights into the myosin motor mechanism. Ann Rev Biophys 39:539–557 Takagi Y, Shuman H, Goldman YE (2004) Coupling between phosphate release and force generation in muscle actomyosin. Phil Trans Roy Soc B 359:1913–1920 Taylor EW (1981) Mechanism of actomyosin ATPase and the problem of muscle contraction. Crit Rev Biochem 6:103–104 Taylor EW (1991) Kinetic studies on the association and dissociation of myosin subfragment 1 and actin. J Biol Chem 266:294–302 Tesi C, Colomo F, Nencini S, Piroddi N, Poggesi C (2000) The effect of inorganic phosphate on force generation in single myoﬁbrils from rabbit skeletal muscle. Biophys J 78:3081–3092 Trentham DR, Eccleston JF, Bagshaw CR (1976) Kinetic analysis of ATPase mechanisms. Quart Rev Biophys 9:217–281 Urbanke C, Wray J (2001) A ﬂuorescence temperature-jump study of conﬁgurational transitions in myosin subfragment 1. Biochem J 358:165–173 Veigel C, Wang F, Sellers JR, Molloy JE (2002) The gated gait of the processive molecular motor myosin V. Nat Cell Biol 4:59–65 Von der Ecken J, Heissler SM, Pathan-Chhatbar S, Manstein DJ, Raunser S (2016) Cryo-EM structure of a human cytoplasmic actomyosin complex at near-atomic resolution. Nature 534:724–728 Weiss GH (1986) Overview of theoretical models of reaction rates. J Stat Phys 42:1–36 White HD, Taylor EW (1976) Energetics and mechanism of actomyosin adenosine triphosphatase. Biochemistry 15:5818–5826 White HD, Belknap B, Webb MR (1997) Kinetics of nucleoside triphosphate cleavage and phosphate release steps by associated rabbit skeletal actomyosin, measured using a novel ﬂuorescent probe for phosphate. Biochemistry 36:11828–11,836 Whittaker M, Wilson-Kubalek EM, Smith JE, Faust L, Milligan RA, Sweeney HL (1995) A 35-Å movement of smooth muscle myosin on ADP release. Nature 378:748–751 Woledge RC, Curtin NA, Homsher E (1985) “Energetic aspects of muscle contraction,” Monographs of the Physiological Society, no. 41, Academic Press, London Worthington CR (1962) Conceptual model for the force-velocity relation of muscle (Hill’s equation). Nature 193:283–1284 Wulf SF, Ropars V, Fujita-Becker S, Oster M, Hofhaus G, Trabuco LG, Pylypenko O, Sweeney HL, Houdusse A, Schröder RR (2016) Force-producing ADP state of myosin bound to actin. Proc Natl Acad Sci USA 113:1844–1852 Yount RG, Lawson D, Rayemnt I (1995) Is myosin a “back-door” enzyme? Biophys J 66:44S–49S

Chapter 4

Models for Fully-Activated Muscle

“Theories should be as simple as possible, but not simpler.” Albert Einstein.

The success of A.F. Huxley’s pioneering vernier model in modelling steady-state tension and ATPase as a function of velocity has given it an almost mythical status in the minds of physiologists, with the result that modelling post 1957 has developed in a rather piecemeal way, often motivated by ﬁtting new experimental data. Here are some ways in which his model can be developed to put it on a sound physicochemical footing: (a) Ideally, the vernier model of interactions between myosin heads and actin sites should be replaced by a three-dimensional lattice model with the correct geometry of myosin heads on thick ﬁlaments and monomeric binding sites on the actin double-helix, as in Sect. 2.1. (b) A comprehensive model should include the biochemical actin-myosin-nucleotide cycle studied in solution (Chap. 3), or some simpliﬁcation of it which can be justiﬁed in terms of solution kinetics. (c) To predict contractile tension, models should specify the conﬁgurations of the myosin lever-arm associated with different biochemical states. In other words, the working stroke proposed by Huxley and Simmons (1971) to explain rapid tension recovery after a length step should be embedded in a biochemicallybased contraction cycle. In a limited way, Sect. 2.3 provides a framework to show how this can be achieved. (d) Although Huxley recognized that the kinetics of transitions between myosin states must be altered by elastic strain in actomyosin, appropriate laws of straindependent kinetics need to be derived from ﬁrst principles rather than introduced empirically. The strain-dependent transitions are attachment to actin (binding and dissociation), and bound-state transitions such as the working stroke which involve a movement of the lever arm. A start was made by T.L. Hill (1974), who reminded us that the equilibrium constants of strain-dependent transitions are coupled by Gibbs’ fundamental equation K ¼ exp(βΔG) to the change in © Springer Nature Switzerland AG 2018 D. A. Smith, The Sliding-Filament Theory of Muscle Contraction, https://doi.org/10.1007/978-3-030-03526-6_4

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Gibbs energy, including elastic strain. However, deriving the strain-dependence of reaction rates from ﬁrst principles is a more formidable task which is explored in the next section. (e) The elastic compliance of the ﬁlaments should be included. By the same token, the S2 rod to which myosins are attached buckles in compression (Kaya and Higuchi 2010), and this phenomenon has generally not been incorporated in models.

4.1

Strain-Dependent Kinetics

As the atomic structures of myosin, actin and ATP are all known, it should be possible to map the energy landscape of the strain-dependent reactions between them, where strain is introduced by holding the actin monomer and the distal end of myosin’s lever arm at ﬁxed positions. In practice, this remains a distant goal of computational biophysics, although a start has been made by Fischer et al. (2005) in mapping the reaction energy of the (strain-independent) repriming stroke, and by Takano et al. (2010) in following the actin interaction of myosin-S1 tethered to a cantilever. In the meantime, progress can be made with a simple model where myosin-S1 is regarded as a rigid motor domain which binds to actin, plus a lever arm with two stable orientations relative to the motor domain. These orientations are deﬁned at the base of the lever-arm, where it meets the converter region of the motor domain. The lever-arm is elastically compliant for bending, as demonstrated by changes in the orientation of the regulatory light chain with applied tension (Irving et al. 1995). Kohler et al. (2002) have shown that the region near the converter domain is elastically compliant, but this might also reﬂect the response of a lever-arm with distributed compliance and cantilever boundary conditions at the base. Purists may object to a model with this degree of structural reduction, but it does lead to analytic formulae for strain-dependent reaction rates if the reaction-energy landscape is reduced to its minimum-energy pathway. The geometry of the model also predicts the availability of actin sites. The theory developed by H.A. Kramers provides a rigorous approach to the calculation of reaction rates. In the simpliﬁed form valid for an overdamped system (Smoluchowski 1916), it can be applied to strain-dependent myosin-actin binding, the working stroke, and the gating stroke associated with ADP release. Smoluchowski’s method is summarized below, and its application to muscle kinetics is worked out in Appendices. A further simpliﬁcation is possible for reactions whose highest energy barrier ΔE >> kBT, giving the familiar equation k AexpðβΔEÞ

ð4:1Þ

of Eyring’s transition-state theory. Strain-dependent reaction rates in the highestenergy-barrier approximation are presented in this section of the main text.

4.1 Strain-Dependent Kinetics

4.1.1

103

Kramers’ Method for Reaction Rates

The seminal paper by Kramers (1940) provided a major advance in our understanding of unimolecular chemical reactions which proceed by a sequence of inﬁnitesimal changes, such as generated by a smooth intermolecular potential. Kramers’ theory addresses the motion of a point particle activated by Brownian noise and moving in a bistable potential V(x), with two minima separated by a single maximum (Fig. 4.1). It starts by including the effects of inertia, and the stochastic motion of the particle is described a Fokker-Planck equation for its joint probability distribution with respect to position and momentum. The key result is a now-famous formula for the reaction rate over a barrier of height EB as a function of the damping time τ, valid for ωBτ 1:

k¼ 00

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ ð2ωB τÞ2 1 ωA 2ωB τ

2π

expðβEB Þ:

ð4:2Þ

Here ωA ¼ (V (xA)/m)1/2 is the frequency of simple harmonic motion of a particle of mass m at the bottom of the potential well at point A, and ωB is similarly related to the downward curvature of the potential at its highest point B. The ﬁrst factor is commonly referred to as the transmission coefﬁcient. At high viscosities (ωBτ > 1), a separate calculation shows that the rate is proportional to viscosity. In the middle, the transmission coefﬁcient approaches unity, for which the reaction rate is just that proposed by Eyring many years ago. In Eyring’s view, the unstable maximum B was considered to be a transition state whose population is in equilibrium with that in the well, and ωA/2π is the attempt frequency

Fig. 4.1 The potentialenergy landscape of a one-dimensional unimolecular reaction, with two wells A, C separated by a barrier at B

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4 Models for Fully-Activated Muscle

deﬁned by simple harmonic motion. Kramers’ formula shows that the rate predicted by transition-state theory is an upper bound. For comprehensive reviews, see Weiss (1986) and Hanggi et al. (1990). Reaction rates for particle motion in a potential V(x) can be simply derived in the limit of high damping, where inertial forces can be neglected in favour of viscous drag. For a drag coefﬁcient μ, the drift velocity is v ¼ μ1(dV/dx), which is overlaid by diffusive motions from Brownian forces. Let ρ(x,t) be the probability distribution of particle position x at time t. It satisﬁes the continuity equation ∂ρ/ ∂t ¼ dJ/dx where the ﬂux J ¼ ρv D∂ρ/∂x is the sum of drift and diffusion terms. Thus ∂ρðx; t Þ ∂ 1dV ∂ρ ¼ ρþD , ∂t ∂x μ dx ∂x

ð4:3Þ

an equation ﬁrst formulated by Smoluchowski (1916). The mobility 1/μ and the diffusion constant D are connected by Einstein’s relation ð4:4Þ

D kB T=μ,

which can be derived by equating the steady-state solution to the Boltzmann distribution exp(V(x)/kBT). So Einstein’s identity shows that viscous drag is the net result of Brownian forces acting on a moving particle. Kramers showed that Smoluchowski’s equation generates the following general formula for the reaction rate from A to C: k AC

D=Z A ¼ R xC βV ðxÞ dx xA e

Z ZA ¼

e

βV ðxÞ

dx

ð4:5Þ

A

and the derivation is given in Appendix D. ZA is the partition function for the particle in well A, so the integration should be conﬁned to that well. For smooth potentials, this result generates Eq. 4.2 in the limit of high damping (ωBτ 0Þ

ð4:13aÞ

ðΔvðX Þ < 0Þ:

ð4:13bÞ

where Δv(X) ¼ vR(X) vA(X) is the difference of strain energies for an A-R stroke. It follows that K S ðX Þ ¼

k S ðX Þ kS ¼ expðβΔvðX ÞÞ k S ðX Þ kS

ð4:14Þ

as required by detailed balancing. These functions were ﬁrst proposed by Julian et al. (1974); they have been used extensively in muscle modelling. Slightly different results follow from an application of Smoluchowski’s equation. The main difference is that the forward rate continues to increase slightly as Δv falls below zero, and the back rate increases similarly as Δv rises above zero. The working is presented in Appendix F.

4.1.4

An ADP-Release Stroke

As foreshadowed in Sect. 3.4, there are good mechanical reasons for invoking a second stroke in the biochemical contraction cycle. In fast skeletal muscle, ADP release from actomyosin, and the subsequent binding of ATP, are both fast reactions (Table 3.3), which will deplete the post-stroke A.M.ADP state. To make this state hold about 50% of isometric tension, it must be populated at positive strains, and depleted at negative strains to allow rapid unloaded shortening. This can be achieved by postulating a small gating stroke, of order 0.5–1 nm. Evidence for such a stroke has not been found in fast skeletal muscle (Gollub et al. 1996), but in slow muscles and non-muscle myosins there is a larger stroke (2–3 nm) associated with ADP release (Jontes et al. 1995; Whittaker et al. 1995; Gollub et al. 1996; Veigel et al. 1999). The negative result for fast skeletal muscle would allow a very small gating stroke hD ( 0.5 nm) beyond the limit of detection. In that case, the strain-dependent ADP-release rate would be of the same form as Eq. 4.13a with an appropriate deﬁnition of Δv(X). The structural interpretation of this stroke is that it opens the nucleotide-binding pocket to allow ADP to escape. Once the pocket is open, ATP would be free to enter the pocket provided that it does not shut beforehand; in that case there would be no reason to suppose that ATP binding is strain-dependent. As

110

4 Models for Fully-Activated Muscle

Scheme 4.1 A mechanism for strain-dependent ADP release through an excited state

A.M.ADP

A.M

K* (X)

K*(X)

A.M.ADP*

kD*

A.M*

ATP binding to rigor is followed by very rapid dissociation from actin, an ATP gating stroke would be quite difﬁcult to detect. Scheme 4.1 shows an alternative way of introducing strain-controlled ADP release without invoking a gating stroke (Smith and Geeves 1995). ADP release may occur through an excited state, generated from the A.M.ADP ground state by a 1–2 nm gating stroke, after which the excited A.M state returns to its ground state by a reverse stroke of the same magnitude. Then the ground-state A.M.ADP and A.M states would have the same strain energy. Assuming that the excitation and de-excitation strokes are fast and reversible, the reaction rate for this process is k D ðX Þ ¼

K ∗ ðX Þ k D∗ , K ∗ ðX Þ þ 1

K ∗ ðX Þ ¼ K ∗ expðβΔvðX ÞÞ

ð4:15Þ

where ΔvðX Þ ¼ vE ðX Þ vR ðX Þ ¼ κhD ðX þ h þ hD =2Þ

ð4:16Þ

for harmonic strain energies. Here kD∗ is the rate of ADP release in the excited state. Note that the equivalent solution rate is not kD∗ but K∗kD∗/(K∗+1), which follows by setting the exponent to unity. Scheme 4.1 could also be applied to the subsequent ATP binding event. After ADP release, whether the nucleotide pocket closes before ATP can bind depends on the relative rates of these reactions, which is not known. This scheme can also be interpreted as a uniﬁed mechanism for strain-dependent ADP release in fast and slow myosins. For slow and non-muscle myosins, the stable rigor state would have to be A.M* rather than A.M. These myosins are kinetically distinguished from fast myosins by slow ADP release, which is thought to be due to a side reaction to the main path of the contraction cycle (Cremo and Geeves 1998). In summary, the functional forms of strain-dependent reaction rates in the highest-energy-barrier approximation are shown in Fig. 4.4, for actin binding, the working stroke and ADP release. They may be compared with rates calculated with Smoluchowski theory in Appendix F.

4.2 The Evolution of Contraction Models

111

105

rates (s–1)

104 103 102 101 100 –8

–6

–4

–2

0

2

4

X (nm)

Fig. 4.4 The forms of strain-dependent reaction rates calculated in the highest-energy-barrier approximation with κ ¼ 3 pN/nm. Black: Two-step actin attachment rates from the swing-capture-roll-lock mechanism where the capture step is rate-limiting at small strains (Eq. 4.8 with kA2/kA1 ¼ 10). Red: Rates for a 4 nm working stroke (Eqs. 4.13). Green: Rates for ADP release + ATP binding + detachment, either as a working stroke (full line) or via an excited state with no net stroke (dashed line, Eqs. 4.15 and 4.16 with K* ¼ 0.25, kD* ¼ 2500 s1). The black and red dashed lines are backward rates

4.2

The Evolution of Contraction Models

Models of contraction in striated muscle come in two ﬂavours, namely those that rely on Huxley’s vernier model of head-site spacings and those which use the actual geometry of heads and sites in the ﬁlament lattice. Improvements to the vernier model have proceeded on several fronts. An obvious move is to include the working stroke in a contraction cycle. A simple two-state model of this kind, akin to Huxley’s model, was explored by Duke (1999). Vernier models can be improved by using a more realistic actin-myosin-nucleotide cycle, and there have been many attempts to incorporate aspects of the cycle revealed by solution-kinetic studies (Guo and Guilford 2006; Julian et al. 1974; Lan and Sun 2005; Linari et al. 2010; Mansson 2010; Nishiyama et al. 1977; Piazzesi et al. 1995; Pate and Cooke 1989; Smith and Geeves 1995; Smith 2014; Wood and Mann 1981). The model of Pate and Cooke, which uses a ﬁve-state contraction cycle, has been inﬂuential because it tackled a wide range of contractile behaviour, despite its use of empirical strain-dependent rate functions. A third line of improvement is to use physically-motivated forms of strain-dependent kinetics, as described in the last section. Motivated by experiment, there now seems to be general agreement that strain-dependent actin-myosin kinetics is an essential ingredient of contraction models (Bickham et al. 2011; Iwamoto 1995; Mijailovich et al. 2016; Siththanandan et al. 2006; West et al. 2009). A consensus about the appropriate form of these functions has been slow to emerge, but debate now centres around which speciﬁc mechanisms are appropriate, while the functional form for a given mechanism is generally well-understood.

112

4 Models for Fully-Activated Muscle

Some progress has been made with lattice models (Chase et al. 2004; Hussan et al. 2006; Smith and Mijailovich 2008; Smith et al. 2008; Tanner et al. 2007), and the same modiﬁcations should also be used here. The structural basis of lattice models was described in Sect. 2.1 and Appendix A. A partial improvement on vernier models is to recognise that myosin heads are dimeric; two heads are attached to the S2 rod which joins them to the thick ﬁlament, and these heads must compete for actin sites of favourable orientation. In each half-period of F-actin, there is a target zone of three adjacent azimuthally-available sites on the same helical strand (Tregear et al. 2004), so the two heads have three sites to choose from if axial matching is satisﬁed. The list of desirable improvements does not stop here. Myosin and actin ﬁlaments are not truly rigid; there is enough compliance in F-actin for isometric activation to produce a 3 nm extension per half-sarcomere (Bagni et al. 1990; Huxley et al. 1994; Wakabayashi et al. 1994). This produces a slight alteration in the distribution of head-site spacings but, more importantly, it doubles the net compliance of the halfsarcomere in the isometric state (Linari et al. 1998; Piazzesi et al. 2002), and this effect prompted an upward revision of estimates of myosin stiffness (Piazzesi et al. 2007). On top of that, the nano-manipulation experiment of Kaya and Higuchi (2010) showed that the S2 rod which attaches heads to the thick ﬁlament buckles in compression. In a contraction model, S2 buckling is driven by the net compressive force (negative tension) from both heads of dimeric myosin, which increases the complexity of numerical calculations considerably. Contraction models can also be processed in different ways. The probabilistic formalism used in Huxley’s model can be used in Lagrangian or Eulerian mode. When muscle length is changing, the Lagrangian form, which tracks individual heads in space and time, is more convenient for simulating transient responses, and for testing the assumption that shortening at constant velocity/load arrives at a steady state with constant tension/velocity. This is an important issue. Tension in a fully-activated fast muscle takes 3–10 ms to reach a steady state after isovelocity shortening begins. However, models have been proposed in which a steady state is not present during isotonic shortening (Duke 1999; Vilfan and Duke 2003), and this could produce non-equivalent behaviour in different sarcomeres. For isotonic shortening on the plateau of the tension-length curve, local monitoring of sarcomere lengths with the spot-follower device does reveal velocity disorder (Edman et al. 1988), but the variations were attributed to different compositions of myosin isoforms along the ﬁbre. Multi-sarcomere models with identical myosins should not exhibit velocity disorder under these conditions. There are some general mechanical properties of muscle contraction that should be common to all models. Firstly, inertial forces are so small that they can be entirely neglected. A tensile load F applied to the ﬁbre is transmitted equally to all halfsarcomeres, with no unbalanced force on any half-sarcomere that would cause local acceleration. Consider the Newtonian equation of motion m

dv þ μv ¼ T ðt Þ F dt

ð4:17Þ

4.2 The Evolution of Contraction Models

113

for the shortening velocity of one half-sarcomere with a constant external load F. This equation can be applied to one actin ﬁlament with mass m and viscous drag coefﬁcient μ. The general solution is vðt Þ ¼ vð0Þet=τ þ

Z

t 0

T ðt 0 Þ F ðtt0 Þ=τ 0 e dt m

ðτ m=μÞ

ð4:18Þ

where τ is the time for viscosity to damp out inertial acceleration. With m ¼ 2.8 1020 Kg, μ ¼ 2.2 109 N.s/m for an actin ﬁlament of length 1100 nm, radius 4.5 nm in water (Happel and Brenner 1983), τ 1011 s, absolutely negligible compared with the time scale of tension variation. Hence. F ¼ T ðt Þ μvðt Þ

ð4:19Þ

and viscous drag is added to the contractile tension. Under steady-state conditions, T is a function of velocity, and at small velocities T(v) T0λv where Hill’s equation gives λ ¼ (1 + α1)T0/v0, about 0.2 pN/nm for frog muscle. Contractile tension can be regarded as a (non-linear) drag force, sometimes called “protein friction” although it would be more correct to call it ‘protein drag” to distinguish viscous force from frictional force. The point here is that true viscous forces are negligible; the above estimate gives μ 105λ (Ηuxley 1980). The effect of neighbouring ﬁlaments can be estimated from the formula for a ﬁlament moving near a planar surface. However, Elliott and Worthington (2001) suggested that viscous forces might be much higher because ﬁlament motion in a restricted environment on a scale of 10 nm might not be described by classical hydrodynamics. Although this issue is not resolved, it seems likely that viscous drag will still be orders of magnitude smaller than protein friction. Another mechanical property of muscle is that it generates radial as well as axial contractile forces. In an intact muscle, this phenomenon is obscured by the sarcolemma which maintains constant volume as ﬁbre length is changed, so it is better studied in skinned ﬁbres where activation generally causes radial shrinkage (Millman 1991). There is a balance of radial forces which stabilises the ﬁlament lattice, which has been studied experimentally by using high-molecular-weight solutes to lower osmotic pressure. This balance of forces can also be modelled, and the modelling tested against observations of lattice spacing as a function of osmotic pressure and sarcomere length. This aspect of contraction is investigated in Chap. 6.

4.2.1

A Two-State Stroking Model

This model can be regarded as an updated version of A.F. Huxley’s model, which incorporates a working stroke h and more realistic strain-dependent kinetics (Duke 1999). Here we consider how it behaves within the context of Huxley’s vernier

114

4 Models for Fully-Activated Muscle

approximation, in which head-site spacings are uniformly distributed within the 36 nm actin half-period. There are still just two states of the myosin head, a detached myosin state and a single actomyosin state with pre-stroke and post-stroke states lumped together. A biochemically stripped-down version of this cycle is shown in Scheme 4.2: Scheme 4.2 A two-state cycle with an equilibrated working stroke

M.ADP.Pi

M.ATP g2(X )

f (X ) g1 A.M.ADP.Pi

A.M.ADP

KS (X )

Using only one detached state implies that hydrolysis of M.ATP is fast, an assumption which can be tested within the framework of the model. The pre-stroke and post-stroke states (A and R states) are combined by assuming that the stroke transition remains in rapid equilibrium over all strain values. This assumption cannot be correct for strains where the forward/backward stroke rate becomes very small, and equilibration means that the model cannot predict the time course of phase-2 recovery from a length step, although it can predict the ﬁnal tension T2. The strain-dependent kinetics of Huxley’s model was replaced by the ‘Brownian post’ model of actin attachment, plus strain-dependent ADP release from the post-stroke A.M.ADP state. Mathematically, the model is as follows. The strain-dependent rate constants are f ðX Þ ¼ f exp βκX 2 =2

ð4:20aÞ

for A-state actin-binding, which incorporates the probability that Brownian motion drives the head to the site with displacement X. The rate g1 of A-state detachment is strain-independent, while the rate of R-state detachment is g2 ðXÞ ¼ g2 exp βκhD ðX þ h þ hD =2Þ , g2 ,

X þ h þ hD =2 > 0 X þ h þ hD =2 0

ð4:20bÞ

as proposed in the previous section. The equilibrium constant of the working stroke is K S ðX Þ ¼ K S expðβκhðX þ h=2ÞÞ

ð4:21Þ

and the tension and cycling rate of the bound head with strain X are tðX Þ ¼

X þ K S ð X Þ ð X þ hÞ , 1 þ K S ðX Þ

ð4:22aÞ

4.2 The Evolution of Contraction Models

r ðX Þ ¼

115

K S ðX Þ g ðX Þ: 1 þ K S ðX Þ 2

ð4:22bÞ

which includes the equilibrated stroke transition. With the vernier assumption and probability p(X) that the head is bound to its site at spacing X, the net tension, number of bound heads and hydrolysis rate per actin ﬁlament per half-sarcomere are T¼

M b

Z

X max

X max b

tðX ÞpðX ÞdX,

Z M X max pðX ÞdX, NB ¼ b X max b Z M X max rðX ÞpðX ÞdX: R¼ b X max b

ð4:23aÞ ð4:23bÞ ð4:23cÞ

For reasons that appear shortly, the assumption of head-site vernier spacings over b, the actin half-period, is used here with an adjustable maximum spacing Xmax b/2. For isometric conditions, p(X) ¼ f(X)/( f(X)+g(X)) where gð X Þ ¼

g1 þ K S ðX Þg2 ðX Þ 1 þ K S ðX Þ

ð4:24Þ

is the net stroke-equilibrated detachment rate. Note also the forms of the quantities X∗ (the critical stroking strain of Eq. 2.42) and X+ (the A-state binding range): ln K S , X ∗ ¼ h=2 þ βκh

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 ln ðK 1 þ 2Þ Xþ ¼ βκ

ð4:25Þ

where K1 ¼ f1/g1 is the zero-strain afﬁnity for A-state binding. The second equation follows by setting p1(X+) ¼ 0.5p1(0), where p1(X) ¼ f(X)/( f(X)+g1). For steady shortening or lengthening, p(X) satisﬁes v

dp ¼ f ðX Þð1 pÞ gðX Þp dX

ð4:26Þ

which is the steady-state version of Eq. 2.23. Exact analytic solutions are not available so, as with all models described from here on, numerical solutions are required. In this case, steady-shortening behaviour is calculated from the solution of Eq. 4.26, starting from p(X) ¼ 0 at X ¼ Xmax, where Xmax b/2 ¼ 18 nm. Apart from the scaling factor M/b, there are seven model parameters f, g1, g2, κ, h, hD, KS. Rewriting these equations in dimensionless form by using h, κh and 1/g2 as units of distance, force and time generates four independent

116

4 Models for Fully-Activated Muscle

Table 4.1 Predictions for isometric and shortening properties of the two-state stroking model, computed from Eqs. 4.20, 4.21, 4.22, 4.23, 4.24, 4.25, and 4.26 for two parameter sets. A: f ¼ 40 s1, g1 ¼ 2 s1, g2 ¼ 80 s1, κ ¼ 1.3 pN/nm, h ¼ 11 nm, hD ¼ 0.5 nm, KS ¼ e15 3.27 106 from ΔGS ¼ 15kBT (Duke 1999). B: f ¼ 40 s1, g1 ¼ 8 s1, g2 ¼ 500 s1, κ ¼ 3.0 pN/nm, h ¼ 4 nm, KS ¼ 1000, hD ¼ 1.0 nm. These calculations used M ¼ 264 heads per actin ﬁlament per half-sarcomere (Sect. 2.1), and a vernier width b ¼ 36 nm A B

T0 (pN) NB0 R0 (s1) X∗ (nm) v0 (nm/s) R(v0) (s1) W_ max (zJ) vmax (nm/s) ηmax 181 47.0 166 1.30 922 1004 4.0 104 325 0.74 137 20.1 253 0.303 1480 815 1.4 104 480 0.33

dimensionless parameters. This step is not essential but is useful in exploring the behaviour of the model, as described in the original paper. The behaviour of this model with a 36 nm head-site vernier is summarised in Table 4.1 and Fig. 4.5. With parameter set A, the results, shown in Fig. 4.5a, are qualitatively different from those in the original paper, in which the vernier was replaced by a long 1D lattice. (i) The tension-velocity curve for ramp shortening rises to a maximum value above T0 before falling at higher velocities. The origin of this behaviour is the combination of a low value of g2 (80 s1) with a large (11 nm) working stroke. Although there are few post-stroke heads in the isometric state, T(v) increases at low v, because the heads that bind make a large stroke and cannot detach fast enough to reduce the tension, although the number of bound heads decreases slightly with velocity. (ii) The low value of g2 does not allow heads to complete a cycle of binding and detachment over 36 nm of travel when shortening at nearly 1000 nm/s, which invalidates the use of a 36 nm vernier model. (iii) Isometric tension is low, because the fraction of bound heads is 0.18, less than 0.3–0.4 (Linari et al. 1998, 2007). This is a consequence of the simple vernier model with one available actin site per half-period. These discrepancies illustrate the importance of specifying the geometry of the ﬁlament lattice used for numerical calculation. The ﬁrst two problems disappear with parameter-set B, where the rate constant g2 for post-stroke ADP release is increased to 500 s1 (Siemankowski et al. 1985), κ is increased to 3.0 pN/nm, and KS is reduced to accommodate the biochemical cycle. Although these numbers are measured values for rabbit psoas muscle at 10–12 C, the lower optimal working temperature of the frog suggests similar values for frog at 0–3 . The stroke size is also reduced from 11 nm to 4 nm to increase the fraction of post-stroke heads under isometric conditions. The results of making these changes are shown in row B of Table 4.1 and Fig. 4.5b. At the unloaded contraction velocity (1480 nm/s), p(X) has a narrower domain of support, giving a 15 nm-wide window of detachment in which detached heads can remap to sites on the next actin half-period. The third problem is not solved by these modiﬁcations; a single 4 nm stroke cannot generate sufﬁcient isometric tension. The obvious conclusion is to use models with two 4 nm strokes, which will accommodate the net stroke inferred from atomic structure studies. That an 8 nm working stroke should occur in two 4 nm

4.2 The Evolution of Contraction Models

117

Fig. 4.5 Isometric and shortening behaviour of the two-state stroking model, calculated probabilistically from a 36 nm vernier of head-site spacings. Columns a and b correspond to parameter-sets A and B of Table 4.1. Upper graphs: bound-state probabilities vs. pre-stroke strain under isometric conditions, including pre-stroke (red) and post-stroke states (green). Middle graphs: tension (black), no. of bound heads (red) and ATPase rate (blue) vs. shortening velocity. Lower graphs: Bound-state probabilities for unloaded shortening. In a, the value of Xmax was lowered in an attempt to accommodate the tail of the distribution

118

4 Models for Fully-Activated Muscle

steps is a consequence of high myosin stiffness, which makes a single 8 nm stroke energetically unfavourable for all (A-state) bound heads. It was ﬁrst suggested by Huxley and Simmons (1971), even though their estimate of myosin stiffness (0.25 pN/nm) was far too low. Now that κ has been measured directly, as 2.7 pN/nm for rabbit psoas myosin (Kaya and Higuchi 2010) and 3.2 pN/nm for frog myosin (Piazzesi et al. 2007), we can speculate how muscle myosin might have evolved to optimise contractile behaviour. After the atomic structure of the myosin lever-arm and its converter evolved, an optimum myosin-II motor might have emerged by converting a single working stroke into multiple strokes. Figure 4.6 shows the behaviour of a one-stroke model of contractility as a function of myosin stiffness and stroke size. As a function of κ, isometric tension ﬁrst rises as expected, and then falls as the critical stroking strain X∗ moves below the binding range. The ATPase rate also falls as more heads are prevented from stroking. As h is increased, tension also rises and then falls for the

Fig. 4.6 The behaviour of the two-state stroking model as a function of myosin stiffness κ and working stroke h. (a) Isometric tension, (b) number of bound heads and (c) ATPase rate, calculated with parameter-set B of Table 4.1

4.2 The Evolution of Contraction Models

119

same reason; the number of heads that can make a stroke decreases faster than the increase in tension. Having two 4 nm strokes allows the equilibrium constant KS to be divided equally between them, in which case KS ¼ 1000 would gives √KS ¼ 31.6 for each substroke. In this way, the critical strain X∗ for the ﬁrst stroke is increased from 2.85 to 0.85 nm. The other side of the coin is the question of how to calculate the contractile properties of models which do not allow all heads to detach over 36 nm of shortening. At ﬁrst glance, it seems highly improbable that a bound head could support about 30 nm of compression strain without detaching from actin. This idea is not as silly as it sounds; the S2 rods which carry myosin dimers can bend, so they can buckle in compression, requiring a dimer-controlled modiﬁcation to the tension-strain function t(X). Calculations can be made with the same 1D lattice employed by A.F. Huxley, and the computational methods required are discussed in the next section. A third class of questions arise from the time-dependent behaviour of the model. Edman and Curtin (2001) observed that isotonic shortening at a high load F T0 generates damped oscillations in ﬁbre length on top of steady shortening. Similar oscillations, but more pronounced, occur in this model, and theorists have been quick to point out that they arise from synchronized working strokes. This phenomenon is addressed in the next chapter, which considers the response to force steps. With a revised choice of parameters, Duke’s model is an improvement of A.F. Huxley’s 1957 model because it uses a working stroke and more realistic strain-dependent reaction rates. However, it has the same limitations as any other two-state model. Lumping pre-stroke and post-stroke states together precludes calculations of the rapid length-step response, and the omission of the hydrolysis step off actin mean that it overestimates ATPase rates. The latter can be corrected in semi-empirical fashion by lowering the binding rate and the actin afﬁnity. Another aspect of this model that can be tested within a wider context is a mechanism for generating the double-hyperbolic tension-velocity curve observed in frog muscle (Edman 1988; Edman et al. 1997). The most likely explanation is that the steep drop in tension at small velocities arises from A-state heads that detach before their strain drops to the critical value X* which triggers a working stroke (Duke 1999). In this respect, a low value of X* (generally but not necessarily negative) will be required. Two-state models keep appearing in the literature. The model of Chen and Gao (2011) also contains an embedded working stroke, but the tension-strain relationship for a bound head was assumed to mirror the T2 curve as a function of length step ΔL. This assumption was meant to be valid for time scales of 0.1 ms or longer, during which the stroke transition would have equilibrated. However, the true equilibrated tension-strain function for a single head has a completely different character; the integrand in Eq. 2.43 switches from κX to κ(X + h) over a narrow range of X values, with constant slope outside the switching range.

120

4.2.2

4 Models for Fully-Activated Muscle

The Search for a Simple Vernier Model

What properties deﬁne the simplest acceptable vernier model for contraction in striated muscle contraction? The model should be user-friendly, which requires a certain simplicity while amenable to adaptation when required. The model should not be in conﬂict with the underlying biochemistry of the actin-myosin-nucleotide cycle, which in principle can be achieved by eliminating short-lived states and lumping states considered to be in rapid equilibrium. With the purely mechanical aspects of the model, there is more freedom of choice but, as always, Occam’s razor applies; arbitrary hypotheses not supported by observation should be kept to a minimum. The ﬁrst two criteria are often in conﬂict, so the “simplest” model is inevitably a compromise and there can be no unique solution. Models can be categorised without reference to computational methods; the stochastic nature of contraction means that numerical computation can be made in terms of state probabilities, by Monte Carlo simulation, or by more exotic methods. With these criteria in mind, consider the examples in Table 4.2 from the gallery of models generated over the last 60 years. There are common features that should be retained, and others which should be discarded because they conﬂict with later experimental discoveries or because they are unnecessarily empirical. And any model worth its salt should be able to duplicate the wide range of experimental protocols now available. Here we consider only predictions for steady-state behaviour; transient responses to various perturbations are taken up in Chap. 5.

Table 4.2 Basic properties of selected vernier models for striated muscle contraction; no. of states NS, site spacing b, myosin stiffness κ and working stroke h. HEB ¼ highest-energy-barrier approximation for strain-dependent rate constants. All models use single heads addressing single actin sites Julian et al. (1974) Hill et al. (1975) Nishiyama et al. (1977) Wood and Mann (1981) Pate and Cooke (1989) Smith (1990) Piazzesi and Lombardi (1995) Linari et al. (2010) Mansson (2010)

b (nm) ? 36

κ (pN/nm) 0.22 0.8

h (nm) 10 0

3

37

0.23

8

Comments HEB stroke rate Self-consistent version of Huxley (1957) HEB stroke rate

6

50

1.0

7.65 + 2.35

HEB stroke rates

5

20(?)

4.48

7.5

Empirical rate fns.

5 5

36 5.3

0.6 0.7

10 2 4.5

N/A

3.2 3.2 2.8

4 2.8 4 2.8 8+1

HEB rates HEB stroke rates, extra detachment pathway after 1st stroke No strain-dependent rate constants

NS 4 2

5 7 4,5

36

Velocity-dependent binding rate

4.2 The Evolution of Contraction Models

121

Table 4.2 summarises key parameter values and steady-state predictions for a representative selection of vernier models. They differ in the number of biochemical states in the contraction cycle, the number of working strokes and the form of straindependent reaction rates, but otherwise have many features in common. All of them use single myosin heads, rather than dimers, which address actin sites independently, and in most cases they use a 36 nm vernier. Explicitly or implicitly, they all address the observed behaviour of frog muscle or rabbit psoas muscle, and in most cases there is qualitative agreement with experiment apart from the magnitude of extensive quantities per F-actin. Even simple vernier models require many parameters, some of which are not tightly constrained by experimental data, so it should not be surprising that models with different structures can be persuaded to yield similar predictions. Apart from the problem of insufﬁcient isometric tension, the simplest vernier model which generates reasonable predictions appears to be a 5-state single-head model with two detached states (M.ATP and M.ADP.Pi), two 4 nm working strokes and hence three bound A.M.ADP states (attached, paused and rigor), as in Scheme 4.3. To include phosphate-dependent contractility, the ﬁrst bound state is assumed to be an equilibrium mixture of A.M.ADP.Pi and A.M.ADP, but in other respects this is the model proposed by Offer and Ranatunga (2013). It is a simpliﬁed version of the biochemical cycle because ADP release, ATP binding and dissociation from actin are lumped into a single irreversible reaction, made strain-dependent by introducing a small gating stroke. This reduction is valid if the population of the intermediate rigor state A.M remains low over all strains for which A.M.DR is populated. Now ADP release is gated at positive strains, so if the subsequent ATP binding is also gated, the population of A.M may be signiﬁcant. Hence their model assumes that ATP is present in millimolar quantities and ATP binding to rigor is not strain-gated, a possibility discussed in Sect. 4.1. It also assumes that the repriming stroke on M.T is fast compared with the subsequent ATP cleavage step to M.D.P (state 2). Mathematically, this 5-state model is deﬁned by its rate constants and tension in states 3-5 for all head-site spacings X in the 36 nm vernier. The functions listed below are those used by Offer and Ranatunga, except that k23(X) describes two-step binding (Eq. 4.8) and k51(X) describes a forward and backward ADP-release stroke via an excited state (Eq. 4.15): k23 ðX Þ ¼ k 23

Scheme 4.3 A 5-state contraction cycle with two working strokes

ð1 þ γ ÞexpðβvA ðX ÞÞ , 1 þ γexpðβvA ðX ÞÞ

ð4:27aÞ

2 M.D.P

A.M.D.PA

1 M.T

3

A.M.DA

A.M.DP 4

A.M.DR 5

122

4 Models for Fully-Activated Muscle

k 34 ðXÞ ¼ k34 exp β½vP ðXÞ vA ðXÞ ðvP ðXÞ > vA ðXÞÞ, k 34 ðvP ðXÞ < vA ðXÞÞ, ðvR ðXÞ > vP ðXÞÞ, k 45 ðXÞ ¼ k 45 exp β½vR ðXÞ vP ðXÞ k45 ðvR ðXÞ < vP ðXÞÞ, k51 ðX Þ ¼ k 51

ð1 þ K ∗ Þexpðβ½vE ðX Þ vR ðX ÞÞ : 1 þ K ∗ expðβ½vE ðX Þ vR ðX ÞÞ

ð4:27bÞ ð4:27cÞ ð4:27dÞ

The reverse rate constants follow from the strain-dependent equilibrium constants K 23 ðX Þ ¼ K 23 expðβvA ðX ÞÞ

ð4:28aÞ

K 34 ðX Þ ¼ K 34 expfβðvP ðX Þ vA ðX ÞÞg

ð4:28bÞ

K 45 ðXÞ ¼ K 45 expfβðvR ðXÞ vP ðXÞÞg

ð4:28cÞ

where vA(X), vP(X) and vR(X) are strain energies in the A,P,R states 3,4,5. Their Xderivatives are the tensions tA(X), tP(X), tR(X) on F-actin directed towards the M-band (hence the absence of the minus sign). For the linear axial model, 1 1 vA ðXÞ ¼ κX 2 , vP ðXÞ ¼ κðX þ h1 Þ2 , 2 2 1 1 vR ðXÞ ¼ κðX þ h1 þ h2 Þ2 , vE ðXÞ ¼ κðX þ h1 þ h2 þ h3 Þ2 , 2 2

ð4:29aÞ

so tA ðX Þ ¼ κX,

tP ðX Þ ¼ κðX þ h1 Þ,

tR ðX Þ ¼ κðX þ h1 þ h2 Þ:

ð4:29bÞ

Let {pim(t), i ¼ 1,. . .,5} be state probabilities for the mth head with head-site spacing Xm(t) at time t. Then the net tension per F-actin per half-sarcomere is T ðt Þ ¼

M X

tðX m ðt ÞÞ • pm ðt Þ

ð4:30Þ

m¼1

in terms of the 5-dimensional state vectors t(X) and pm. Similar equations apply to the instantaneous stiffness S(t), where t(X) is replaced by κ ¼ (0,0,κ,κ,κ ), and the ATPase rate R(t), with t(X) replaced by r(X) ¼ (0,0,0,0,k51(X)). With this choice, R (t) is the instantaneous rate of ATP binding. Under steady-state conditions, R(t) is also the rate of ADP release or of ATP cleavage off actin, but under transient conditions they may not have the same time dependencies. The performance of the Offer-Ranatunga model can be judged in various ways. The authors set the initial value of κ at 1.7 pN/nm, but allowed it and the other 16 parameters to ﬂoat to ﬁt a combination of isometric, steady-state and transient

4.2 The Evolution of Contraction Models

123

behaviour from frog muscle (the anterior tibialis muscle of Rana temporaria). Their calculations used single S1 heads and three-site target zones on actin. Here it is sufﬁcient to document the behaviour of a basic 5-state model with two working strokes and an ADP-release stroke, single heads and single actin sites per 36 nm halfperiod, so that the effects of changing κ and the kinetic parameters, as in Table 4.3, can be appreciated. Filament compliance is not included here. Its effects are explored in Sect. 4.4, but it is useful to note that compliant ﬁlaments should have no effect on the isometric properties of active muscle on the plateau of the tension-length curve. Comparisons with the behaviour of fast muscles such as frog anterior tibialis and rabbit psoas reveal the merits and deﬁciencies of this model, whose predictions are shown in Table 4.4: (a) Isometric tension per F-actin per h-s can be as high as 300 pN for frog anterior tibilias muscle at 3 C (Ford et al. 1977), or 250 pN for rabbit psoas at 10 C (Coupland et al. 2001). On this basis, trials 3 and 4 give insufﬁcient isometric tension even when the number of available heads is set at 264: single-head models with one azimuthally-matched head per 42.9 nm per myoﬁlament and one azimuthally-matched actin site per 36 nm period do not form enough crossbridges. This can be corrected by working with dimeric myosins and multi-site target zones on actin. (b) In trials 3 and 4, the fraction of heads bound to actin in the isometric state (17–19%) is smaller than estimates obtained from stiffness measurements (30–40%), although the latter are complicated by the elastic compliance of the ﬁlaments. (c) Isometric ATPase values should be from 60–300 s1 per F-actin, equivalent to 0.2–1 s1 per head (Curtin et al. 1974; Levy et al. 1976; Arata et al. 1977; Takashi and Putnam 1979; Glyn and Sleep 1985). The isometric rate is sensitive to the rate k51(X) of strain-dependent detachment at the end of the cycle, which is controlled by the ADP-release stroke h3. When κ is raised to 3 pN/nm, the isometric ATPase rate is too small because of sharper ﬂux gating by the working strokes. (d) For shortening muscle, the tension-velocity curves for trials 0 and 1 show insufﬁcient curvature (Fig. 4.7), because the actin binding rate k23 is high and the detachment rate is very low. This means that, as heads are swept through the binding range d set by k23(X) (about 4 nm), the bound fraction changes little with velocity; the relevant criterion for a linear tension-velocity curve is that k23 ðX Þ v=d > 1: Lowering k23 to 40 s1 restores the familiar hyperbolic shape of T(v). (e) In this model, the unloaded contraction velocity vo is mainly controlled by k51, the rate constant for ADP release followed by ATP-induced detachment from actin. In the muscle ﬁbre, detachment is fast only when heads are driven into the drag region X < (h1 + h2 + h3/2), where all three strokes proceed without being slowed by a strain-energy barrier. This issue has been widely studied (Cooke et al. 1994). (f) The Fenn effect is larger than observed. For rabbit psoas muscle, the increase in ATPase rate for unloaded shortening relative to isometric conditions is typically

T 1 2 3 4

k12 s1 100 100 100 100

K12 23.9 23.9 5 5

k23 s1 266 266 40 40

K23 310 310 10 10

k34 s1 2058 2058 104 104 K34 234 234 100 100

k45 s1 7672 7672 104 104 K45 173 173 100 100

k51 s1 332 332 500 800 κ pN/nm 1.7 1.7 1.7 3.0

h1 nm 5.60 5.60 4.5 4.0

h2 nm 4.55 4.55 4.5 4.0

h3 nm 0.7 0.7 0.7 0.6

0 100 100 100

γ

Table 4.3 Trial parameter values for the 5-state model of Scheme 4.2. Trial 1 used the parameter values for model 340 of Offer and Ranatunga (2013), and trial 2 shows the effects of two-step actin binding set by γ. Trials 3 and 4 show the results of reducing the actin binding rate k23 and afﬁnity K23, for two values of myosin stiffness κ. The number M of heads per F-actin was set at 264, and K* at 0.1

124 4 Models for Fully-Activated Muscle

4.2 The Evolution of Contraction Models

125

Table 4.4 Results of trials for isometric and shortening properties of the 5-state model. The isometric variables are T0 (tension), NB0 (number of heads bound) and R0 (ATPase rate) per actin ﬁlament per half-sarcomere. v0 is the unloaded shortening velocity and R(v0) the corresponding ATPase rate. Maximum power output W_ maxis the maximum value of vT(v), for which v/v0 is shown in the next column. Maximum efﬁciency was calculated from the formula η ¼ vT(v)/|ΔH|R(v) where |ΔH| ¼ 56.5 zJ is the molar enthalpy of phosphocreatine splitting (Woledge et al. 1985) 1 2 3 4

T0 (pN) 320 343 205 200

NB0 R0 (s1) 69.7 24.6 77.7 165 51.0 175 44.1 31.7

v0 (nm/s) 980 1160 1730 1760

NB(v0) 50.1 74.0 12.0 7.4

R(v0) (s1) 4300 6620 1970 1550

W_ max (zJ) 3.43 104 6.62 104 2.74 104 1.91 104

v/v0 0.510 0.466 0.347 0.284

ηmax 0.463 0.549 0.349 0.344

v/v0 0.013 0.052 0.047 0.014

Fig. 4.7 The steady-state behaviour of the 5-state model as a function of shortening velocity per h-s. The top graphs show tension (black line), stiffness (red) and ATPase rate (green), while the bottom graphs show mechanical power output (black line) and efﬁciency (blue). Graphs (a, b) were constructed from the parameters of trial 1 in Table 4.3, and graphs (c, d) from trial 3

126

4 Models for Fully-Activated Muscle

a factor of 3–5 (He et al. 1999, Sun et al. 2001), whereas these models predict a ten-fold increase or more. In unloaded shortening, all bound heads are swept into the drag region where the rate of ATP-induced detachment is fast, so the maximum turnover rate is controlled by k12 and k23, the rate constants for hydrolysis on M.T and actin binding of M.ADP.Pi. However, turnover rates during rapid shortening could be lowered by detachment of heads prior to ATP binding (Brenner 1991). (g) As a function of shortening velocity, maximum power output for the frog is typically 7 104 W per F-actin per h-s at v/v0 ~0.33 (Edman et al. 1997). The lower values appearing in Table 4.2 reﬂect low values of isometric tension, and/or excessive curvature in the tension-velocity curve (low values of the Hill ratio α ¼ a/T0). The above remarks show that curvature is controlled by the actin-binding rate k23. (h) The maximum efﬁciency ηmax of shortening muscle, where η is deﬁned as rate of working/enthalpy of ATP hydrolysed (Eqs. 2.9 and 2.12), is typically 0.3–0.4 for fast muscles. Because of the Fenn effect, maximum efﬁciency occurs at lower values of v/v0 (under 0.1) than maximum power. It is clear that this single-site version of the 5-state model fails when myosin head stiffness is upgraded to 2.7 pN/nm or higher (Kaya and Higuchi 2010; deCostre et al. 2005). But all is not lost: this model is revisited in Sect. 4.5, where it is updated to include 3-site target zones and dimeric myosins. The original model of Offer and Ranatunga did include target zones, in the sense that multiple actin sites were available to each head.

4.2.3

Lattice Models

The study of lattice models is still in its infancy. Models in which myosin dimers are correctly positioned and oriented on myoﬁlament backbones, and actin sites on the double helix, are well equipped to deal with azimuthal mismatching of heads and sites as described in Sect. 4.1, but to date such effects have only been studied with empirical mismatch functions (Smith et al. 2008; Smith and Mijailovich 2008; Mijailovich et al. 2016). They also provide the only correct way of incorporating the distributed elastic compliances of the ﬁlaments (Daniel et al. 1998), since the correct axial positioning of myosin heads is removed in vernier models. Other issues that can be explored with lattice models are (i) the variation in the fraction of available heads between muscles with different unit cell structures, (ii) the effects of torsional movements in actin ﬁlaments (Tsaturyan et al. 2005), (iii) the extent to which elastic compliance in myoﬁlaments couples strains induced by head interactions with one actin ﬁlament to head interactions with neighbouring F-actins. Whether this last effect is important is not yet clear, but extensive Monte-Carlo simulations on a model with multiple unit cells (Chase et al. 2004) are required to

4.3 Computational Methods

127

deal with it. (iv) Lattice models will also be required for models of striated muscle contraction at partial activation, because cooperative thin-ﬁlament regulation by tropomyosin-troponin means that the spatial pattern of actin sites not blocked by tropomyosin cannot be ignored. Regulation of contractility, and associated modelling, is taken up in the concluding chapters. Studies on lattice models with one unit cell and two actin ﬁlaments have revealed some unexpected behaviour. Isometric tension on the plateau of the tension-length curve becomes length-dependent, with periodic variations that are not observed experimentally. Multi-cellular lattice models, with a lattice of 6–10 unit cells, show that this behaviour is smoothed out through random variations in the spatial distribution of bound heads on different F-actins (Mijailovich et al. 2016). More can be done to clarify the extent to which vernier models can give an accurate account of the contractile behaviour of striated muscle, but both kinds of model need to proceed from the correct geometry of heads and sites on their ﬁlaments in the half-sarcomere.

4.3

Computational Methods

There is a common mathematical framework for calculating the behaviour of any contractile model which seeks to calculate the behaviour of a single half-sarcomere in terms of the mean instantaneous state occupancies of myosin-S1 heads. Nearly all models of muscle contraction, be they vernier models or lattice models, are of this kind. The only exceptions are so-called stochastic models where deviations from the mean occupancy of each state are calculated explicitly from Langevin equations or a Fokker-Planck equation (Lan and Sun 2005). A good case can be made for ignoring these ﬂuctuations, even at the level of an actin ﬁlament in its half-sarcomere. About 260 heads are available for interaction with actin; if these heads behave independently at full activation, then the r.m.s. deviation from the mean occupancy of each state should be reduced by a factor of √260 ¼ 16.1. In a myoﬁbril, there are about 1000 actin ﬁlaments in the cross-section, and approximately 106 myoﬁbrils in a 1 mm ﬁbre, so ﬂuctuations in measureable quantities such as net tension should be reduced by a factor of 105. An explicit search for ﬂuctuations has proven negative. Iwazumi (1984) found “no tension ﬂuctuations in normal and healthy myoﬁbrils,” but used this fact to support an alternative model of contractility (Iwazumi 1988). That said, contractile behaviour of a single half-sarcomere at full activation by calcium can generally be reduced to the following piece of mathematics. Let there be NS states available to each head m ¼ 1,. . . .,M, and let nim(t) (i ¼ 1,. . . .,NS) be a binary number for the occupancy of state i on head m at time t. These occupation numbers change according to the rate constants kij(Xm) of reactions in the contraction cycle, which act as transition probabilities per unit time. If Aij(X) is the rate constant for transitions from j to i, then

128

4 Models for Fully-Activated Muscle

Ai j ðXÞ ¼ kji ðXÞ, ði 6¼ jÞ NS X k i j ðXÞ: Aii ðXÞ ¼

ð4:31Þ

j¼1

assuming that kii(X) ¼ 0. The instantaneous net tension per actin ﬁlament per halfsarcomere is calculated as the sum over states and heads of tension ti(Xm) multiplied by the occupation number nim. The quantity of interest is the averaged tension Tðt Þ ¼

NS M X X

ti ðX m ðt ÞÞpim ðt Þ,

pim ðt Þ < nim ðt Þ >

ð4:32Þ

m¼1 i¼1

and analogous formulae hold for other extensive quantities per actin ﬁlament per half-sarcomere, such as net stiffness and ATPase rate. Here pim(t) is the probability that the mth head is in state i at time t. By deﬁnition, state probabilities must be normalized to unity on summing over states: NS X

pim ðt Þ 1:

ð4:33Þ

i¼1

They can be calculated in two different ways:

4.3.1

Probabilistic Methods

Here we work directly with the state probabilities pim(t). In an ensemble of many myoﬁlament+actin-ﬁlament systems, pim(t) ¼ < nim(t)>, the ensemble-averaged values of the occupation numbers. Hence they also satisfy the familiar chemical rate equations which apply to an ensemble of many molecules of each kind: NS dpim ðt Þ X ¼ Aij ðX m ðt ÞÞp jm ðt Þ dt j¼1

ð4:34Þ

This matrix equation can be solved by standard methods provided that two potentially troublesome aspects are kept in mind. Firstly, the component equations are not all linearly independent, because normalization implies that NS X dp i¼1

im

dt

¼

NS X NS X

Aij ðX m Þp jm 0:

i¼1 j¼1

This equation is identically satisﬁed because, from Eq. 4.31,

ð4:35Þ

4.3 Computational Methods

129 NS X i¼1

Aij ¼

NS X

k ji þ A jj ¼ 0

ð4:36Þ

i¼1

so the NS X NS matrix Aij actually has rank NS–1. Secondly, strain-dependent transition rates imply that the matrix elements may be ill-conditioned, making (4.34) a very stiff set of differential equations. The ﬁrst problem can be dealt with by working with linearly independent rate equations for the ﬁrst NS–1 states. Using Eq. 4.34 to eliminate pNS , m gives S 1 dpim ðt Þ NX ¼ AijR ðX m ðt ÞÞp jm ðt Þ þ bim ðt Þ, dt j¼1

ði ¼ 1; . . . ; NS 1Þ

ð4:37Þ

where AijR ðX Þ ¼ Aij ðX Þ AiNS ðX Þ bim ðt Þ ¼ Ai, NS ðX m ðt ÞÞ:

ði; j ¼ 1; : . . . ; NS 1Þ

ð4:38Þ

For example, steady-state isometric properties can be calculated from the solutions of the NS–1 dimensional matrix equation obtained from dpim/dt ¼ 0. The large variation in the size of matrix elements demands a robust matrix-inversion routine such as LU decomposition or singular-value decomposition, and packaged routines are available from the Numerical Recipes collection (Press et al. 1992) or elsewhere. For calculating the transient behaviour of muscle subject to some perturbation, Eq. 4.37 are stiff differential equations which need to be solved numerically by implicit methods to guarantee stable solutions. Double-precision arithmetic is also generally required. The simplest implicit integration routine is the ﬁrst-order method due to Euler, with time-dependent matrix elements calculated at the mid-point of each integration step (Gourlay 1970). This ‘backward Euler’ method is simply speciﬁed by some general formulae. In matrix-vector form, the equation dp ¼ Aðt Þ • p þ bðt Þ dt

ð4:39Þ

is replaced by the solution of the ﬁnite difference equation pðt þ Δt Þ pðt Þ ¼ Aðt þ Δt=2Þ • pðt þ Δt Þ þ bðt Þ Δt

ð4:40Þ

pðt þ Δt Þ ¼ fI Aðt þ Δt=2ÞΔt g1 • ðpðt Þ þ bðt ÞΔt Þ

ð4:41Þ

namely

130

4 Models for Fully-Activated Muscle

which also requires a matrix inversion routine. LU decomposition is robust enough for the purpose, but in practice the Gauss-Jordan algorithm, which is the simplest method for matrix inversion, is sufﬁciently stable. For muscle which is shortening at constant speed v, the full time-dependent solution can always be calculated from Eq. 4.34 for the state probabilities pim(t). However, experiments show that the tension reaches a steady state after several milliseconds (Lombardi and Piazzesi 1990, 1992), and it is more convenient to calculate the steady-state response to ramp shortening directly. As in Sect. 2.2, this can be done by working with the Eulerian description, where the L.H.S. of Eq. 4.34 is interpreted as the derivative following the motion. Thus NS ∂pi ðX; t Þ ∂p ðX; t Þ X v i ¼ Aij ðX Þp j ðX; t Þ ∂t ∂X j¼1

ð4:42Þ

The mth head is now the head with site displacement X ¼ Xm at t ¼ 0. As the actin ﬁlament move towards the M-band, the physical head leaves point Xm and other heads take up the value X ¼ Xm. As discussed previously, the Xm are to be distributed according to head-site geometry in the half-sarcomere, e.g. Xm ¼ b/2 + mb/M for a simple vernier model with periodicity b ¼ 36 nm. Steady-state shortening behaviour can now be calculated from Eq. 4.42 by setting ∂pi(X)/∂t ¼ 0. The resulting differential equation in X should also be solved by an implicit method, such as the backward Euler method. The integration should start from X ¼ XM ¼ b/2 where all heads are detached, and proceed to X ¼ X0 ¼ b/2. Periodic boundary conditions apply, namely pi ðX 0 Þ ¼ pi ðX M Þ,

i ¼ 1, . . . , NF ,

ð4:43aÞ

pi ðX 0 Þ ¼ 0,

i > NF :

ð4:43bÞ

for a model with NF detached states. It is important to check that bound states at the end of integration are not occupied. Equations 4.43 can often, but not always, be satisﬁed by choosing Xmax just above the binding range. If not, then the model is incompatible with a 36 nm vernier and Monte-Carlo methods on a long ﬁlament lattice should be used instead. Analogous operations apply to steady-state stretching, where the integration should start at X0 and work upwards. This probabilistic method can be generalized to dimeric myosins and heads interacting with multi-site target zones, by enlarging the state space as described in Sect. 4.5, but probabilistic methods can soon become unwieldy because of the need to invert ill-conditioned matrices of large dimensionality. Probabilistic methods also run into difﬁculties when different heads compete for the same actin site, because in that situation the heads do not satisfy independent rate equations. This would happen if heads spaced by 14.3 nm on the same myoﬁlament can overcome their azimuthal mismatch to select an intervening site, a rather unlikely event which requires an axial

4.3 Computational Methods

131

reach of at least 7.15 nm. The same difﬁculty arises when heads cannot complete a cycle of detachment and binding over 36 nm of travel. These situations can be handled by the Monte Carlo method, to which we now turn.

4.3.2

Monte-Carlo Simulation

The Monte-Carlo simulation method was invented by Fermi and Ulam, but is often credited to Metropolis and others. It simulates the real time-dependent behaviour of a stochastic Markov system by generating random numbers to build a time sequence of state transitions. For example, if transitions are possible only at multiples of some time period Δt, they can be generated by the following algorithm. Let the system at time tn ¼ nΔt be in state i. The set of non-zero reaction rates kij determine a set of states to which the system might transition in time Δt. A Monte-Carlo sequence of such states is determined by the following rule: generate a uniform random number r in the range (0,1) and compare it with the transition probability kijΔt for each possible ﬁnal state j. Then i ! j at time t nþ1 ¼ t n þ Δt if r < k ij Δt:

ð4:44Þ

If there is more than one ﬁnal state j for which kij > 0, then these states can be tested in turn. But selecting j as the state with the largest value of kij is not a good idea, because that favours more probable transitions over and above what is selected by the random number r. A sequence of states generated over a predetermined time interval constitutes a Monte-Carlo run. The states can be speciﬁed by the occupation numbers ni(tn) for all i ¼ 1,. . .,NS and each sampling time tn or, more efﬁciently, by a set of state indices i(tn). For muscle modelling, extensive quantities such as net tension per actin per half-sarcomere from a single Monte-Carlo run will still show signiﬁcant ﬂuctuations despite having been summed over M available heads, so averaging over repeated Monte Carlo runs will be required to force the averaged occupation numbers to converge towards their true mean values pim(t). At each sampled time, the amplitude of ﬂuctuations should be proportional to N1/2 where N is the number of runs. In practice, this method can be computationally time-consuming because the sampling interval Δt must be such that kijΔt < 1 for the largest value of kij. A much more attractive version of the Monte-Carlo method has been formulated by Gillespie (1976). Gillespie’s method does not integrate chemical rate equations and does not require small time steps. Instead, it calculates the waiting time τ to the next reaction event by treating τ as a stochastic variable. Let kμ be the rate constant of the μth reaction from the current state. The probability p(t) of no reactions in time t satisﬁes

132

4 Models for Fully-Activated Muscle

pðt þ dt Þ ¼ pðt Þð1 kdt Þ,

k¼

X μ

kμ

ð4:45Þ

so p(t) ¼ exp(kt). Hence the probability of a reaction in time τ is 1p(τ), which can be sampled by generating a uniform random variate r1. Setting 1 expðkτÞ ¼ r 1

ð4:46Þ

generates the waiting time τ. Which reaction then occurs can be determined by a second random variate r2 such that qμ1 ðτÞ < r 2 < qμ ðτÞ,

qμ ðτÞ ¼

μ X

kυ =k

ð4:47Þ

υ¼1

where qμ(t) is the probability that the next reaction is the one indexed by μ. When applied to the model embodied in Eq. 4.31, Gillespie’s method can be used for each head m ¼ 1,. . . ..,M in turn, and the order of access is immaterial. Given that reaction rates for the contractile cycle can vary from 0.1 s1 to 104 s1, Gillespie’s method for this problem is expected to be about 100 times faster than the classical Monte-Carlo method. In practice, Monte-Carlo simulation method with Gillespie’s algorithm is the method of choice for contractile models with a large number of states per head, whether because of a more complete biochemical cycle, or by allowing each head to bind to adjacent actin sites because myosin heads come in pairs tethered to S2 rods. A considerable increase in complexity results if these rods are allowed to buckle in compression. It is worthwhile to explore probabilistic methods for the beneﬁt of faster computation, but more complex coding is required and the demands on random-access memory are correspondingly greater.

4.4

The Effects of Filament Elasticity

The sliding ﬁlament model was quantiﬁed on the assumption that myosin and actin ﬁlaments were rigid structures, but it was only a matter of time before this assumption was tested by measuring their longitudinal compliances. X-ray determinations of the extension of both ﬁlaments by tetanic activation in frog muscles were made by Huxley et al. (1994) and Wakabayashi et al. (1994), and published simultaneously. Both groups found that isometric tension extended F-actins by about 2.5 nm per half-sarcomere, and myoﬁlaments by about 0.6 nm. Although these extensions are a small fraction of ﬁlament length, the shifts in head-site spacing are not negligible. The gross effect of elastic compliance in the ﬁlaments can be understood by lumping the distributed compliances into a single series

4.4 The Effects of Filament Elasticity

133

elastic component (SEC), say between F-actin and the Z-line. The above results indicate that the lumped series compliance CS is of the order of 3.1/ 240 ¼ 0.013 nm/pN, assuming that T0 was 240 pN. As this value is comparable with the net compliance of bound heads in the isometric state (4.5/240 ¼ 0.019 nm/ pN, from length-step data), it is clear that ﬁlament compliances cannot be neglected. This situation was anticipated in Sect. 2.3, where we saw that if the ﬁlaments were rigid structures then the same tension response would be produced by a length step of about half the size. This raises the general question of what muscle properties are affected by compliant ﬁlaments. Isometric properties on the plateau of the tensionlength curve should be unchanged provided that the ﬁlaments remain in full overlap. Length-step experiments and all protocols involving length change will be affected, but the above rule of thumb, namely halve the applied length change or shortening velocity and calculate the contractile response with rigid ﬁlaments, applies only if muscle stiffness is constant. This is almost true for length-step protocols, but in ramp shortening or stretching the net stiffness from crossbridges may change in proportion to changes in the fraction of heads bound to actin. Thus a quantitative treatment is required, starting with a reliable estimate of what lumped series compliance produces the same effects as distributed ﬁlament compliance. For those who say that what goes around comes around, it is interesting to note that a series elastic component was ﬁrst proposed by A.V. Hill (1938), albeit for quite different reasons which are now of historical interest.

4.4.1

The Equivalent Lumped Filament Compliance

Given the known elastic properties of the ﬁlaments, what value of lumped compliance is equivalent to the system of compliant ﬁlaments and myosin crossbridges? This problem can be solved analytically if all elastic elements are linear and crossbridges are distributed uniformly along the overlap region. For a spatially continuous distribution of crossbridge elements in the 2:1 lattice, this calculation was made by Thorson and White (1969) and Ford et al. (1981), giving a series elastic compliance CS LM 2LA þ 2L cA LA 2LM þ 2L þ ¼ cM 3 3 2 2

ð4:48aÞ

per myosin ﬁlament for a half-sarcomere of length L, where CS is the series compliance per F-actin. Here cM and cA are compliances per unit length for myosin and actin ﬁlaments with lengths LM and LA, and cA is halved because there are two F-actins per F-myosin in the rhomboid unit cell. This formula is more transparent if written in terms of the overlap length LO LA + LM – L, namely

134

4 Models for Fully-Activated Muscle

CS cA LO cA ¼ cM ðLM LO Þ þ cM þ þ ðL LM Þ 2 2 3 2

ð4:48bÞ

where the second term arises from distributed crossbridge stiffness in the overlap zone. These formulae are valid when μLO/2 1, where μ2 ¼

SB cA cM þ LO 2

ð4:49Þ

and SB is the net stiffness/F-myosin per h-s from crossbridges (Ford et al. 1981). The calculation can also be made with ﬁnite elastic elements (Smith 2014, Supplementary Material). Later X-ray determinations of ﬁlament compliances (Dobbie et al. 1998; Piazzesi et al. 2002) led to the following values for cM and cA in frog muscle. The authors gave strains as a fraction of ﬁlament length induced by isometric tension, namely cM(2T0) ¼ 0.0014 and cAT0 ¼ 0.0026, which are dimensionless numbers. For example, if T0 ¼ 200 pN/F-actin, then cM ¼ 0.0035 nm/pN.μm and cA ¼ 0.013 nm/pN.μm. The latter can be compared with a direct measurement of the compliance of isolated actin-tropomyosin ﬁlaments (Kojima et al. 1994), namely cA ¼ 1/(65.3 6.3) ¼ 0.015 0.001 nm/pN per micron of ﬁlament. With the former value of cA, and LM ¼ 763 nm, LA ¼ 1000 nm (Barclay et al. 2010), Eq. 4.48a predicts that CS ¼ 0.0088 nm/pN and 0.0092 nm/pN at each end of the plateau of the tension-length curve (2L ¼ 2.10 and 2.15 μm). Somewhat larger values are found by mechanical experiments. For modelling contractile behaviour, it is easy to incorporate a lumped series elastic component (SEC) between F-actin and the Z-line. That means that in activated isometric muscle, head-site displacements are reduced by the extension of the SEC, namely xm(t) ¼ Xm CST(t) where T(t) is the net tension at time t and Xm is the head-site displacement of the mth head in the relaxed muscle, where T ¼ 0. When muscle length changes in time, this equation becomes xm ðt Þ ¼ X m CS T ðt Þ þ ΔLðt Þ

ð4:50Þ

where Xm is deﬁned at t ¼ 0 and ΔL(t) ¼ L(t)L(0) is the change in half-sarcomere length. For bound heads, this equation is valid for displacements of any size. However, if both heads of the dimer are detached, actin sites outside the halfperiod range b should be replaced, meaning that a multiple of b should be added to xm(t) so that j xm ðt Þ j< b=2

ð4:51Þ

and the detached dimer addresses the nearest target zone. This remapping condition should be tested after every reaction event.

4.4 The Effects of Filament Elasticity

135

For modelling, it is convenient to place the SEC between the actin ﬁlament and the Z-line. The net tension is usually not known in advance, so an iterative procedure is required to ﬁnd the correct head-site displacements xm(t). The exception to this rule is for isotonic responses, where T(t) ¼ F, the external load. An interesting consequence of ﬁlament elasticity is that it can slow the tension response to an external perturbation, even if there is no change in sarcomere length. This effect was ﬁrst described by Luo et al. (1994) for the tension rise at the start of an isometric tetanus. When contractile tension increases, the series elastic element is stretched, and the effect on the contractile apparatus is equivalent to slow shortening, which reduces the tension and possibly the number of crossbridges also. With the aid of a suitable model, Luo et al. showed that the rise in tension could be made to lag the stiffness by 10–20 ms, even if the model predicted no lag in the absence of ﬁlament compliance. The mathematical tools required to do this will be developed in the next chapter. It should be said that compliant ﬁlaments can be meaningfully incorporated into vernier models only by using an equivalent series elastic element. Replacing headsite displacements in the ﬁlament lattice by graduated vernier spacings destroys the spatial location of myosin heads, so vernier models with distributed ﬁlament compliance carry extra spatial information which should be regarded as redundant. On the other hand, lattice models should include distributed ﬁlament compliances (Daniel et al. 1998), and can be used to test the hypothesis that isometric tension is unaffected.

4.4.2

Experimental Consequences

Early measurements of isometric tension (Gordon et al. 1966) showed that T0 is independent of sarcomere length on the plateau of the tension-length curve; when the ﬁlaments are stretched the crossbridges adjust by detaching and binding to different sites. However, other workers have observed a slightly-rounded tension-length curve on the full-overlap region, as shown in Fig. 2.6 of Woledge et al. (1985). What does change on the plateau is the stiffness of the non-overlap portion of F-actin, which decreases slightly on increasing the sarcomere length from L1 to L2 (Fig. 1.3). This effect was exploited by Linari et al. (1998) by measuring sarcomere stiffness of frog ﬁbres at 2.1 μm and 2.15 μm, using small length steps. They found that CST0 ¼ 3.9 nm for the activated ﬁber and 2.6 nm in rigor, where all heads are bound to actin. The F-actin compliance cA per unit length was measured from the length dependence of CS, giving cAT0 ¼ 2.3 nm/μm. By comparing results from activated and rigor ﬁbres, the fraction of heads bound to actin in the active state was estimated as 0.43. Later work by the same group, using a different protocol, reduced this number to 0.33 (Linari et al. 2007). Stiffness measurements have also been made during isotonic shortening with various loads (Piazzesi et al. 2007), for frog ﬁbers in the active and rigor states, with the aim of measuring the mean force per crossbridge. Crossbridge stiffness is

136

4 Models for Fully-Activated Muscle

assumed to be proportional to the number of bound heads, but the net stiffness is also lowered by the elasticity of the ﬁlaments. Comparing the two sets of results enables the ﬁlament compliance to be separated out if all heads in the rigor state remain bound to actin. Their data are shown in Fig. 4.8. As in their paper, Fig. 4.8b plots the net strain F/S, where S is h-s stiffness per F-actin, against reduced load f ¼ F/T0. The compliances of crossbridges and ﬁlaments combine to give a net stiffness S( f ) per half-sarcomere: 1 1 ¼ þ CS Sðf Þ κN B ðf Þ

ð4:52Þ

with net crossbridge stiffness proportional to the number NB( f ) of bound heads. At full load, F/S ¼ T0/S(1) ¼ 4.9 nm. As the load was decreased towards 0.5T0, the slopes of the active and rigor plots stayed constant and were almost equal, while one expects that NB( f ) falls in the active state but stays constant in rigor. The following treatment differs from the original presentation by ﬁtting these plots to the functions

Fig. 4.8 Data of Piazzesi et al. (2007) for frog muscle during isotonic shortening, as a function of load F relative to isometric tension T0. (a) The shortening velocity per h-s. (b) Instantaneous stiffness S in the active state (●) and rigor state (O), plotted as net h-s strain F/S and ﬁtted to the functions in Eqs. 4.53. With permission of Elsevier Press and G. Piazzesi. (c) The number of bound heads as a function of reduced load, calculated from these ﬁts and Eq. 4.55 (see main text)

4.4 The Effects of Filament Elasticity

G a ðf Þ

fT 0 af þ ca f , ¼ S a ðf Þ f þ b

137

Gr ðf Þ

fT 0 ¼ cr f : Sr ð f Þ

ð4:53a; bÞ

On comparing with Eq. 4.52, we see that T0 a þ ca , þ CS T 0 ¼ f þb κN Ba ðf Þ

T0 þ C S T 0 ¼ cr : κN Br

ð4:54a; bÞ

so that rigor stiffness is independent of load. Considering the limited number of data points, the data of Piazzesi et alia are well ﬁtted by these functions, with a ¼ 3.95 1.70, b ¼ 0.400 0.200, ca ¼ 2.065 0.872, cr ¼ 3.768 0.066. Here T0 ¼ 246 pN for these experiments, and κ ¼ 3.1 pN/nm for the frog (Decostre et al. 2005). From Eq. 4.54a, the number of bound heads as a function of load is given by N Ba ðf Þ ¼

T0 f þb , κ a þ ð ca C S T 0 Þ ð f þ b Þ

ð4:55Þ

in which all coefﬁcients except CS are now determined. Now CS can be extracted from Eq. 4.54b if we assume that all heads are bound in rigor, namely N Br ¼ 294 for the frog. Then Eq. 4.54b gives CST0 ¼ 3.50 nm and CS ¼ 0.0142 nm/pN, so from Eq. 4.55, N Ba ð1Þ ¼ 57: Hence the duty ratio in the active state, expressed as a fraction of all heads addressing one F-actin, is p ¼ 57/294 ¼ 0.19. This value is arguably too small. Now substituting for CST0 shows that N Ba ð1Þ is a hyperbolic function of T0/κ. Given the ﬁtted data in Fig. 4.8b, this estimate of duty ratio would be increased by raising isometric tension or reducing myosin stiffness below 3.1 pN/nm. In contrast to the normalized data, this calculation requires a precise value of isometric tension, which in practice varied from ﬁbre to ﬁbre. A key conclusion of Piazzesi et al. (2007) was that the number of bound heads in the active state drops linearly with load, so that the load per bound head is constant, at about 6 pN for the frog. Because ca 6¼ CST0, the above analysis does not quite bear this out. Figure 4.8c shows that N Ba ðf Þ is slightly concave, which belies the idea that the number of bound heads changes little at high loads (Barclay et al. 2010). At zero load, not all heads are detached; the tension-velocity curve at v ¼ v0 has a ﬁnite slope, indicating an equal mixture of positively and negatively strained crossbridges (Seo et al. 1994). Hence the origin of the graph in Fig. 4.8b is a ﬁxed data point for F/S. In conclusion, simultaneous measurements of tension and stiffness as a function of load in isotonic shortening experiments provide a powerful way of measuring net ﬁlament compliance and the number of bound heads as a function of load. A variation of this protocol is to study tension and stiffness as a function of the level of activation (Linari et al. 2007), which yielded similar results. However, one must be careful to analyse such data in an appropriate fashion. For example, if the rigor data were also ﬁtted by the hyperbolic function of Eq. 4.53a, the number of bound

138

4 Models for Fully-Activated Muscle

heads in rigor would also be highly load-dependent. This is not observed and not expected from the very low detachment rate of rigor heads in solution.

4.5

Target Zones, Dimeric Myosins and Buckling Rods

A general feature of vernier models with single-headed myosins and single actin sites per 36 nm is that they often yield insufﬁcient isometric tension, even for the tetrapod lattice where the number of azimuthally matched heads per F-actin is set at 264 (Sect. 1.4 and Appendix A). This problem is solved by allowing heads to bind to target zones of three adjacent sites (Tregear et al. 2004) every 36 nm. The difference in azimuthal angle between adjacent sites is 360/13 ¼ 27.7 nm. If the central site is azimuthally matched to the myosin dimer, then the whole zone can be accessed if the maximum mismatch angle on either side is 30 . This situation will be facilitated by the torsional ﬂexibility of F-actin (Tsaturyan et al. 2005). Three-site target zones and myosin dimers go together (Fig. 4.9a). Isometric tension is increased because the vernier effect now allows a bigger fraction of heads to bind under isometric conditions. If dco is the binding range of head-site displacements, then the fraction of heads that can bind is increased from 2dco/b to 2 (dco + c)/b where c ¼ 5.54 nm is the axial actin site spacing. If dco ¼ 5 nm, the increase is from 0.28 to 0.59. However, the two heads of the dimer compete for these sites, so there are effectively ﬁve sites available and this estimate should be reduced by a factor of 1/6. The single-head single-site model, where heads bind independently to dedicated actin sites, is incompatible with dimeric myosin, because the two heads of a dimer compete for sites on the same target zone.

4.5.1

Calculations with Target Zones and Dimeric Myosins

Monte-Carlo simulations are well adapted to cope with target zones and dimeric myosins. The number of bound states in the modelled contraction cycle should be enlarged to specify the site to which the head is bound. For a cycle with NF detached states and NB bound states, there are now OS NF þ 3NB

ð4:56Þ

state-site possibilities for each head. These ‘superstates’ will be indexed by uppercase integers I,J, etc. Dimers will be indexed by m ¼ 1,. . .,M and heads on each dimer by α ¼ 1,2, so Monte-Carlo simulations can proceed as before by tracking the occupancies of states indexed by (I,m,α). Let superstates be indexed as follows:

4.5 Target Zones, Dimeric Myosins and Buckling Rods

139

Fig. 4.9 Predictions for the shortening behaviour of the 5-state vernier model with target zones and dimeric myosins. Graphs (a, b) and (c, d) are for parameter-sets I and III respectively of Table 4.5. Upper graphs: tension (black line), stiffness (red), number of bound heads (green) and number of heads on doubly-bound dimers (yellow). Lower graphs: power output (black) and efﬁciency (blue). All quantities are expressed per actin ﬁlament per half-sarcomere

I ¼ 1, . . . , NF ðhead detachedÞ I ¼ NF þ 1, . . . , NF þ NB ðhead bound to site 1Þ I ¼ NF þ NB þ 1, . . . , NF þ 2NB ðhead bound to site 2Þ I ¼ NF þ 2NB þ 1, . . . , OS ðhead bound to site 3Þ

ð4:57Þ

where the three zonal sites are indexed towards the Z-line. If Xm is the head-site displacement to the central site is Xm, then the displacement to site s (s ¼ 1,2,3) is Xm + (s–2)c where c ¼ 72/13 ¼ 5.54 nm is the actin site periodicity on one strand. However, the two heads of a dimer cannot bind to the same site. If nI,m,1 ¼ 1 for I (NF+(s 1)NB+1, NF+sNB) (s ¼ 1,2,3), then nJ,m,2 ¼ 0 for J in the same range: if head 1 is bound to site s, head 2 is excluded from that site, and vice versa.

140

4 Models for Fully-Activated Muscle

Probabilistic methods can also be extended to cope with target zones and dimeric myosins. This may not be immediately obvious, because site-exclusion means that the two heads are not statistically independent entities. This problem is solved by working with conditional probabilities. For each head, there are four conditional probability functions to keep track of, according to whether the other head of the dimer is detached from actin, or bound to site s of the target zone. For the 1st head, let these probabilities be P0(I,m,1) when the 2nd head is detached, and {Ps(I,m,1), s ¼ 1,2,3} when the 2nd head is bound to site s. These four probabilities can be calculated from the same kinetic equations formulated in Sect. 4.3. Note that Ps(I, m,1) ¼ 0 when I indexes the same site, namely I (NF+(s 1)NB+1, NF+sNB), and there is a computational advantage in reducing them to vectors of dimension OS–NB. Additionally, let p0(m,1) be the probability that the ﬁrst head is detached, and ps(m,1) the probability that the ﬁrst head is bound to site s. They satisfy the identities p0 ðm,1Þ

¼

NF X 3 X

Pt ðI,m,1Þpt ðm,2Þ

ð4:58aÞ

I¼1 t¼0

ps ðm,1Þ

NFX þsNB

¼

3 X

Pt ðI,m,1Þpt ðm,2Þ

ð4:58bÞ

I¼NF þðs1ÞNB þ1 t¼0

But since the heads are identical, their indices 1,2 can be dropped, whereupon Eqs. 4.58 become a closed set of four coupled equations for the unconditional site probabilities. Because they are homogeneous equations, they must be solved subject to the normalization condition 3 X

ps ðmÞ ¼ 1:

ð4:59Þ

s¼0

Finally, the unconditional state probabilities for either head follow as PðI, mÞ ¼

3 X

Ps ðI, mÞps ðmÞ:

ð4:60Þ

s¼0

To complete the picture, here is the formula for net tension in a vernier model, per F-actin per half-sarcomere as always: T ¼2

OS M X X m¼1 I¼1

where

τI ðX m ÞPðI; mÞ

ð4:61Þ

4.5 Target Zones, Dimeric Myosins and Buckling Rods

τI ðXÞ ¼ 0, τI ðXÞ ¼ ti ðX cÞ, τI ðXÞ ¼ ti ðXÞ, τI ðXÞ ¼ ti ðX þ cÞ,

141

I ¼ 1, . . . , NF I ¼ i NF þ 1, NF þ NB , I ¼ i þ NB , i NF þ 1, NF þ NB , I ¼ i þ 2NB , i NF þ 1, NF þ NB ,

ð4:62Þ

and tI(X) are the single-site tension-strain functions for the model cycle. Similar equivalences apply to the net number of bound heads and the ATPase rate. The factor of two is there to remind the reader that M is now the number of available dimers.

4.5.2

An Updated 5-State Vernier Model

The next step in modelling contractile behaviour is to formulate a relatively simple and accessible model which predicts the steady-state and time-dependent behaviour of fast skeletal muscle. When updated to include three-site target zones and dimeric myosins, the Offer-Ranatunga model, with two working strokes and an ADP-release stroke, is recommended as the simplest model for predicting steady-state contractile behaviour at full activation, given that myosin stiffness is of the order of 3 pN/nm. The ingredients have already been presented, so it remains to assemble this 5-state model in a suite of computer programs for the dynamical behaviour of fullyactivated muscle, using either probabilistic or Monte-Carlo methods. Some key predictions with the former method are shown in Table 4.5, using strain-dependent stroke rates derived from Kramers’ formula, which do not saturate at negative strains (Fig. F2). The presence of three binding sites per dimer boosts isometric tension and stiffness to adequate levels, but the model has some difﬁculty in generating observed ATPase rates (150–300 s1/F-actin per h-s) under isometric conditions, because high myosin stiffness does not allow heads to bind with enough negative strain to permit full cycling. This problem is partly relieved by raising the branching ratio γ for two-step binding (Eq. 4.27a) from 1 to 10 or more, but doing this accelerates the rate of tension rise when the relaxed muscle is activated. A Table 4.5 Steady-state predictions of a 5-state vernier model with 132 myosin dimers and 3-site target zones, calculated with the probabilistic method and Kramers’ formulae for strain-dependent stroke rates (Appendix F, Eq. F12). Listed here are isometric tension, number of bound heads, ATPase rates and maximum power per F-actin per h-s, maximum efﬁciency and the Hill ratio α. The parameter values were as follows: h1 ¼ h2 ¼ 4 nm, h3 ¼ 0.5 nm, CS ¼ 0.012 nm/pN, κ ¼ 2.7 pN/ nm, k12 ¼ 50 s1, K12 ¼ 5, k23 ¼ 30 s1, K23 ¼ 0.5, k34 ¼ k45 ¼ 1 104 s1, K34 ¼ K45 ¼ 30 and k51 ¼ 340 s1. The entries in rows I-III are for different values of (γ,ε), namely (0,1), (2,10) and (2,50) respectively I II III

T0 pN 221 278 329

NB0 54.4 57.4 59.1

R0 s1 9.6 33.5 64.5

v0 nm/s 1754 1744 1771

NB(v0) 7.81 11.3 12.2

R(v0) s1 1440 2089 2261

W_ max zW 2.0 104 3.0 104 3.6 104

ηmax 0.28 0.29 0.33

α 0.23 0.22 0.19

142

4 Models for Fully-Activated Muscle

second reﬁnement is peculiar to dimeric myosin: if one head is bound, then the second head of the dimer binds more rapidly to a vacant site on the same target zone (Brunello et al. 2007; Fusi et al. 2010). The acceleration factor ε for faster binding of the second dimeric head is not well-characterised experimentally, and can probably be set anywhere between 10 and 100. Table 4.5 shows the effects of the changes in γ and ε. Neither is able raise the isometric ATPase to 300 s1 (1 s1/head), and this is a consequence of high myosin stiffness, which prevents most heads from completing all the strokes. On the other hand, shortening behaviour is reasonably well-behaved (Fig. 4.9); maximum power is about 4 104 zW per F-actin, although the ATPase rate for unloaded shortening is somewhat larger than desired. The kinetic parameters are roughly as required for rabbit psoas muscle at 10 C, and are quite conservative; the stroke equilibrium constants K34, K45 are set according to Eq. 3.26. More power is achieved by raising the binding rate k23 and/or these stroke constants, and a higher unloaded shortening velocity by raising k51. In this way, the model can also simulate the steady-state behaviour of frog muscle, which is more powerful. At 0–3 C, frog sartorius or anterior tibialis muscles have isometric tensions of up to 290 pN, a Hill ratio of 0.25 and a maximum power output of 7 104 zW/F-actin (Ford et al. 1977; Edman 1988). On the other hand, slow muscles such as rabbit soleus and tortoise muscle have similar isometric tensions to their fast cousins, but with lower v0, a low Hill ratio (α 0.1) and hence low power output, but higher efﬁciency (Woledge 1968; Woledge et al. 1985) through lower ATP usage. Tuning the model in this way can start with k51, which controls v0, and k23, which directly controls the shortening component of the ATPase rate. And the discussion of the Offer-Ranatunga model in Sect. 4.2 shows that curvature in T(v) is reduced by increasing the actin-binding rate k23(X), whether at zero strain through k23 or at high strains by raising the value of γ. There is a price to be paid for reducing the biochemical contraction cycle to just ﬁve states. The effects of changing the concentrations of ATP and ADP are quite distinct, and need to be teased out of the rate constant k51 which includes ADP release, ATP binding and dissociation from actin in one reaction step. In fact, the 5-state cycle cannot predict the transition to rigor mortis as the concentration of ATP is reduced, and extra states are required. In the same vein, one can ask if there is any fundamental reason (other than simplicity) for limiting the number of working strokes to two. At present, this question can only be answered by modelling, and the optimal number is still disputed. Piazzesi et al. (2014) have proposed a model with four equal strokes of 2.7 nm each, presumably because a larger number of small strokes makes for a smoother kinetic response to length steps of different sizes. This aspect of modelling will be explored in the next chapter. Finally, it should be said that the kind of models we have constructed do not transfer to smooth muscle or non-muscle myosins. For these entities, the biochemical contraction cycle may be different in several respects. Electron cryomicroscopy and single-molecule studies both show that slow muscles have an additional gating stroke of 2–3 nm associated with ADP release (Whittaker et al. 1995; Veigel et al. 1999). They may also make a slow transition to another A.M.ADP state off the main

4.5 Target Zones, Dimeric Myosins and Buckling Rods

143

pathway of the cycle (Cremo and Geeves 1998). It is interesting to speculate whether this second state could also be the transition state suggested in Sect. 4.1 for straindependent ADP release in fast muscle, where there is no detectable gating stroke associated with ADP release (Gollub et al. 1996).

4.5.3

Buckling Rods

All contractile models considered so far have assumed that myosin dimers are rigidly tethered to the thick ﬁlament via S2 rods, at least as far as axial positioning is concerned. This property allows heads to carry positive or negative tension, according to the initial head-site displacement X before binding and any subsequent working strokes. It was therefore somewhat shocking when the nano-manipulation experiment of Kaya and Higuchi (2010), using a synthetic myoﬁlament sparsely decorated with myosin heads, revealed that a very small compressive axial force can cause S2 to buckle. This result is conﬁrmed by values of the bending stiffness κS2 obtained from the persistence length of isolated S2 rods in solution, or from a molecular dynamics simulation (Adamovic et al. 2008). Perhaps it is a moot point that S2 rods in contracting muscle do actually buckle. The extreme ﬂexibility of the isolated rod, which is seen in solution and conﬁrmed computationally, might not be present in the electrostatically charged environment of the muscle sarcomere; charge interactions between S2 and the myoﬁlament backbone might act to stiffen it. Knupp et al. (2009) have argued that the ﬁne structure in X-ray diffraction observed by Linari et al. (2000), which they attributed to mirror lever-arm movements from heads in opposite halves of each sarcomere, is actually due to axial shifts in the motor domains. Such shifts could well be generated by heads on buckling rods. Linari et al. (2011) report that their X-ray data on the M3 reﬂection cannot be explained in this way, but it seems that the issue remains open. It should still be useful to have a working contractile model in which S2 rods buckle under compressive strain. With buckling S2 rods, a dimeric-myosin model is almost obligatory because each S2 rod is a double helix whose ends each bind one myosin-S1 (Fig. 4.10). What causes S2 to buckle is the net compressive force generated by both heads, and a lot of computational bookkeeping is required to cope with the case when both heads are bound. In Appendix I, the computational tools needed to formulate a quantitative contractile model with target zones, dimeric myosins and buckling rods are developed ab initio. When only one head is bound, its A-state tension-strain characteristic is given by t A ðX 1 Þ ¼ κX 1 κκb ðX 1 X b Þ tA ðX 1 Þ ¼ κX b þ κ þ κb

ðX 1 > X b Þ, ðX 1 < X b Þ

ð4:63Þ

144

4 Models for Fully-Activated Muscle

(Figure 4.10b) where Xb < 0 is the critical buckling strain and κb is the stiffness of the already-buckled rod. When both heads of the dimer are bound, the A-state tension depends on the site and tension-state (A,P or R) of the other head, and all the required functions can be generated from the tension function tAA(X1,X2) (Appendix I, Eq. I6) in which the second head is A-state bound to a site displaced by X2. For the three-site target zone, the site spacing X2 – X1 can be c or 2c where c ¼ 5.3 nm is the actin monomer spacing on one strand. The contractile consequences of buckling rods have been explored by MonteCarlo simulation, using a 9-state model with dimeric myosins and three-site target zones (Smith 2014). Its behaviour deviates in subtle ways from that of conventional models. Heads addressing sites with X < 0 can bind freely because S2 can buckle spontaneously under thermal agitation; the buckling energy is κX 2b =2 ¼ 0.33 zJ, well below thermal energy (kBT 4 zJ). Consequently, the strain-dependent binding rate of Eq. 4.8 is modiﬁed to the form shown in Fig. 4.10c, with a signiﬁcant tail-off at negative strains. This feature allows heads to bind for X < X+, where X+ is the binding range at positive strains (about 4 nm in the ﬁgure). Heads which bind at the negative end of this range will make working strokes, raising isometric tension, and those at the extreme negative end will complete the cycle, raising the isometric ATPase rate as well. In unloaded shortening, buckling removes the capacity for dimers to carry much negative tension, but the unloaded shortening velocity will not be affected if the two heads of a dimer can bind with opposing tensions. In rapid shortening, Monte-Carlo simulations show that buckling extends the negative tail in the X-distribution of bound states without destroying the ability to complete a contraction cycle within the 36 nm period. The most dramatic effect of S2 buckling is to raise the isometric ATPase rate to observed levels (~300 s1 per F-actin, equivalent to 1 s1 per head on counting all 300 heads per F-actin in the assay (Glyn and Sleep 1985). Buckling also raises the duty ratio above 0.4, a ﬁgure reported by Linari et al. (1998). For shortening muscle, the unloaded contraction velocity is close to that found by Cooke et al. (1988) for rabbit psoas (1750 nm/s), and the maximum ATPase as a fraction of its isometric value is close to 5 (He et al. 1999). Maximum power was 3.2 104 zW, and the maximum efﬁciency was 0.48. Buckling rods also modify the net stiffness seen during shortening. A constant stiffness κ per head applies only for dimers whose net strain is above the buckling threshold, so that net stiffness as a function of shortening velocity falls more steeply as more heads are carried by buckling rods. This can be expressed in terms of ratios. If stiffness per head were indeed constant, then net stiffness S(v) per h-s at velocity v would satisfy 1/S(v) ¼ 1/κNB(v) + CS, so SðvÞ=Sð0Þ 1 þ κCS NB ð0Þ ¼ > 1: NB ðvÞ=NB ð0Þ 1 þ κC S NB ðvÞ

ð4:64Þ

4.5 Target Zones, Dimeric Myosins and Buckling Rods

145

Fig. 4.10 Aspects of a model with dimeric S1-myosins on buckling S2 rods. (a) Buckling is driven by the net compressive force supplied by both heads of each dimer. Upper ﬁgures (X > 0): A dimer seeks a target zone displaced towards the Z-line, and a bound head stretches S2 which behaves as a rigid rod. Lower ﬁgures (X < 0): The dimer seeks a negatively-displaced site, which is more easily available if the rod has buckled spontaneously by distance y under thermal agitation. (b) The tension-strain characteristic tA(X) of a singly-bound A-state dimer, and the corresponding strain energy vA(X). (c) Strain-dependent rate constants for actin attachment, calculated with two-step kinetics from Eqs. 4.27a and 4.28a with γ ¼ 1.0

146

4 Models for Fully-Activated Muscle

However, at v ¼ v0 the model gave a value of 20/29 < 1 for the left-hand side, which can be expressed as an apparent stiffness κ(v) proportional to the fraction of heads on unbuckled rods. At this juncture, the pros and cons of rigid and buckling models can be summarized as follows. For ease of calculation, rigid-rod models win hands down; the effort involved in programming the 5-state model in terms of state probabilities increases considerably to accommodate buckling rods, since there are nine conditional states of the second head (A,P,R X 3 zonal sites) to keep track of. However, the effects of buckling rods can largely be predicted by a simpler model in which each head is supported by its very own S2 rod, and this approximation is employed in the next chapter. The great advantage of buckling models is that, under isometric conditions, they allow enough heads to cycle to account for the ATPase rates observed in vivo, while keeping high myosin stiffness for positively strained heads. In the next section, we shall see that buckling models can also explain the relative indifference of ATPase rate to added phosphate, an effect that is very hard to predict with more conventional models. Time will tell if S2 rods do actually buckle in shortening muscle as well as single-ﬁlament constructs in the optical trap.

4.6

Adding Phosphate, ADP or ATP

Muscles become fatigued after repeated stimulation, in the sense that isometric tension and the capacity for doing work are lowered. The text-book explanation of fatigue is that when the supply of ATP from aerobic metabolism cannot keep up with demand, anaerobic metabolism takes over with an accumulation of hydrogen ions and lactic acid (Carlson and Wilkie 1974), which breaks down to lactate and hydrogen ions. However, this explanation was challenged by Westerblad et al. (2002) on the basis that reduced pH has little effect on muscle tension. Instead, they suggested that fatigue occurs because of an accumulation of phosphate ions in the sarcoplasm. The rate of ATP consumption is almost unchanged by excess phosphate, so the efﬁciency of contraction is impaired. The effects of adding ADP or ATP are quite different. The resting concentration of ADP is 10–30 μM, and raising this concentration to millimolar levels increases isometric tension and reduces the ATPase rate, because ADP release from the poststroke A.M.ADP is inhibited (Cooke and Pate 1985). Adding ATP has the opposite effect. However, when [ATP] in a tetanized muscle is reduced below physiological levels (1–5 mM), tension rises to a maximum value near 50 μM ATP (Cooke and Bialek 1979; Ferenczi et al. 1984), then falls as it approaches the rigor condition where [ATP] is at nanomolar levels or entirely absent. At some stage, the simpliﬁed contraction cycle fails because the rigor state A.M becomes the dominant bound state, and tension would then be controlled by its rate of dissociation. Ligand concentrations also affect steady-state shortening behaviour and transient responses, and Table 4.6 summarizes the trends for a representative set of properties.

4.6 Adding Phosphate, ADP or ATP

147

Table 4.6 Sensitivity of the contractile properties of fast muscle (rabbit psoas) to changes in the concentration of Pi, ADP and ATP. Arrows indicate changes of more than 10% in the logarithmic derivative d(lnY )/d(lnX) (dY/Y )/(dX/X), where X is ligand concentration and Y is the property in each column. a: Millar and Homsher (1990), b: Dantzig et al. (1992), c: Tesi et al. (2000), d: Caremani et al. (2008), e: Kawai et al. (1987), f: Bowater and Sleep (1988), g: Potma et al. (1995), h: Cooke and Pate (1985), i: Potma and Steinen (1996), j: Cooke et al. (1988), k: Sleep and Glyn (1986), l: Cooke and Bialek (1979), m: Ferenczi et al. 1984, n: Glyn and Sleep (1985) [Pi] [ADP] [ATP]

4.6.1

T0 #abc "h #lm

S0 #d " #

R0 efg #k "n

v0 h #h "lm

R(v0) #i #? "

α j ? ?

W_ max # ? ?

Added Phosphate

Adding phosphate reduces tension on a logarithmic basis (Millar and Homsher 1990; Dantzig et al. 1992). For rabbit psoas muscle, isometric tension can be halved by increasing [Pi] from 1 to 25 mM (Tesi et al. 2000), as in Fig. 3.6b, and Pi release has always been regarded as closely coupled to the working stroke (Pate and Cooke 1989). Finding the correct explanation of how exogenous phosphate alters the performance of the contraction cycle has been one of the thorniest problems in muscle contraction. For a long time, it was thought that heads in the newly-bound product state A.M.ADP.Pi performed a working stroke before releasing phosphate (Dantzig et al. 1992; Takagi et al. 2004). The physiological basis for this line of thinking was that added Pi would reduce tension by reducing the net actin afﬁnity of the poststroke states A.M.ADP and beyond. This hypothesis has been tested by exploring the kinetic aspects of phosphate release. A step increase in phosphate level produces a transient decay of tension with a response time of 10–100 s1, depending hyperbolically on the ﬁnal concentration (Dantzig et al. 1992), and this can be modelled by assuming that Pi is released slowly after the working stroke (Smith and Sleep 2004). The kinetics of the phosphate-jump response have already been discussed in Sect. 3.3, where it was proposed that Pi release is not a slow step in the contraction cycle, but a rapid reversible transition which occurs before any working stroke. Experiments with the Pi-binding protein show that Pi is released from myosin-S1 in solution with actin at a rate which increases to 75 s1 at saturating actin concentrations (White et al. 1997); in the muscle ﬁbre this technique provides a straightforward way of measuring ATPase rates (He et al. 1999). What is not clear is whether this slow transition is part of the contraction cycle. If it is, then, as argued in Sect. 3.3, it is difﬁcult to understand how fast muscles can shorten at 2000 nm/s or more, regardless of whether Pi release occurs after or before a working stroke. These difﬁculties are avoided if one assumes that Pi release from the active site is rapid and reversible, and that release is the trigger for the working stroke. Then Pi release to the solvent is a side transition to the actin-myosin-nucleotide cycle, in which Pi released into the bottom of the nucleotide pocket escapes through the Pi-tube to the actin-binding cleft.

148

4 Models for Fully-Activated Muscle

Experimentally, the effects of added Pi on isometric tension appear somewhat variable. The fractional reduction in tension is greater when the starting tension under physiological conditions (with endogenous [Pi] < 1 mM) is smaller (Caremani et al. 2008). If this variability is due to changes in the actin afﬁnity of the product state M.ADP.Pi, then the net afﬁnity to an equilibrated lumped pre-stroke state A.M. ADP.Pi + A.M.ADP# is K A ðX Þ 1 þ K~ P =½Pi , where K~ P is typically 0.01 M or less. Following the discussion in Sect. 3.3, the isometric effects of added phosphate follow from Eq. 3.29 for the probability pA(X) of the lumped state. Figure 3.7 shows that the Pi dependence of pA(X) is maximized when KA(X) ¼ 0.37, whereas there is very little change when KA(X) > 1. To take full advantage of this effect, the basal afﬁnity KA at zero strain should be below unity. Then added Pi has the ability to switch the dominant pre-stroke state from A.M.ADP# to A.M.ADP.Pi. Lumping the two states together is justiﬁed if the Pi-binding transition is as fast, if not faster, than the working stroke which proceeds from the A.M.ADP# state. How does this line of thinking explain why tension falls linearly with ln [Pi]? At ﬁrst glance, this is easy to understand; tension is controlled by the critical strain X* for a stroke following rapid Pi release (Eq. 2.42), which is proportional to the logarithm of its equilibrium constant KSKP/(1+KP) where K P ¼ K~ P =½Pi < 1 at high Pi. With a simpliﬁed one-stroke model, one can derive the approximate formula T 0 ð½PiÞ ¼ T 0 ð1Þ Aln½Pi,

A¼

MkB T 2ð1 þ K A Þh

ð4:65Þ

which predicts that A ¼ 44 pN with M ¼ 264, h ¼ 8 nm and KA ¼ 0.5. Hence the fractional reduction in tension from 1 to 25 mM Pi should be 0.57 if T0 ¼ 250 pN, close to what is observed. Note especially that the reduction is greatest when KA 0 and exothermic when ΔH < 0. If ΔG < 0 and ΔH > 0, the reaction is endothermic for entropic reasons, so heating produces a more disordered structure. In the absence of a phase change, ΔH can be taken as a constant, so ln K is linear in 1/T. The v’ant Hoff equation shows that changing the equilibrium constant by a factor of 0.2 over a 6 temperature drop requires ΔH ¼ 291 zJ, or 175 KJ/mol on the chemist’s scale. This is 60% larger than values quoted for the actin binding of M. ADP.Pi (Woledge et al. 1985), but enthalpies in the highly charged and structured environment of the muscle ﬁbre could be greater than for the proteins in solution. Moreover, modelling temperature-dependent contractility from a strain-dependent actin-myosin-ATPase cycle requires separate enthalpies for equilibrium constants and rate constants, and actin-products binding is not the only reaction involved. Table 4.8 summarizes experimental data for the reaction and activation enthalpies of assorted myosins. As the table shows, not all the information required is experimentally available. Contraction models can certainly generate sigmoidal temperature curves for isometric tension. For example, Offer and Ranatunga (2015) were able to ﬁt tension data for frog Rana temporaria from their 5-state model, by treating nearly all the required enthalpies as adjustable parameters. However, this approach is only as good as the set of parameters which deﬁne contractile behaviour at the control temperature. To understand what enthalpies control the temperature-dependence of muscle contraction, it is instructive to look simultaneously at model predictions for isometric and shortening behaviour. For edl ﬁbres from the rat, Ranatunga found that changing temperature from 10 to 30 C increased the unloaded contraction velocity by a factor of 4.5, while the Hill ratio for the tension-velocity curve increased from 0.14 to 0.35 (Fig. 4.13b). To generate qualitative ﬁts to Coupland and Ranatunga’s data, the 5-state model described in Table 4.5 was used to generate control parameters, nominally for rabbit at 10 C, but with K~ P ¼ 0.005 M. Some enthalpies were held constant once their effects were established. For example, the enthalpies ΔH12 and

156

4 Models for Fully-Activated Muscle

Table 4.8 Known enthalpies ΔH and activation enthalpies Δh for the temperature-dependent kinetics of the contraction cycle. Sources: (a) Woledge et al. (1985), p141, (b) White and Taylor (1976), (c) Ford et al. (1977), (d) Siemankowski et al. (1985), (e) Millar and Geeves (1988), (f) Nyitrai et al. (2006). (g) Millar et al. (1987). For data-row 3, the activation enthalpy for each working stroke follows from Q10 ¼ 1.85 for the rate of phase-2 recovery from a quick release Reaction M.ADP.Pi + A A.M.ADP.Pi ! A.M.ADP + Pi A.M.ADP ! A.M.ADP# A.M.ADP# ! A.M + ADP A.M + ATP ! M + M.ATP M.ATP ! M.ADP.Pi

ΔH (KJ/mol) 110 (a) ? ? ? 0 (f) 64 (g)

Δh (KJ/mol) 103–126 (b) ? 40 (c) 64,76 (d) 66 (b), 52 (e), 50 (f) ?

Δh12 of the ATP cleavage step were ﬁxed conservatively at 64 KJ/mol. For the working strokes, ΔH34 ¼ ΔH45 ¼ 0 and Δh34 ¼ 40, Δh45 ¼ 60 KJ/mol. That leaves just three enthalpies ΔH23, Δh23 and Δh51 to play with. The 4.5-fold ratio of zeroload shortening velocities at 10 and 30 C can be plugged into the v’ant Hoff equation to estimate Δh51 at 53 kJ/mol. The results of this modelling study are shown in Fig. 4.14 and Table 4.9. For isometric properties, Fig. 4.14a shows how difﬁcult it is to generate an adequate variation in tension from 0 to 30 C. The tension data require about twice the actin-binding enthalpies needed for the shortening behaviour. To understand this, remember that v0 is controlled by the rate k51 of ATP-induced dissociation, so its temperature dependence is ﬁxed by Δh51. However, the Hill ratio α a/ T0 of the tension-velocity curve and its temperature dependence are controlled bv k23 and Δh23 respectively. As k23 is increased, the tension-velocity curve becomes less convex and α increases. Thus the Hill ratio is an increasing function of temperature if Δh23 > Δh51 and a decreasing function if Δh23 < Δh51. For temperature-dependent isometric tension, my tentative conclusion is that rabbit psoas ﬁbres below 10 C are deactivated even when tetanised, because the predicted number of bound heads decreases. As discussed in Chaps. 7 and 8, this possibility is always there because activation by calcium is a feedback process; the regulatory ﬁlament on F-actin is activated by bound myosins as well as calcium. A separate problem with these simulations is that tension continues to increase at high temperatures. Tension can be made to saturate at temperatures approaching 30 C by setting ΔH34 ¼ ΔH45 ¼ 100 KJ/mol, making the working strokes strongly exothermic, but I know of no evidence to support this move. For the temperature dependence of shortening muscle, observations such as Ranatunga’s can be simulated from the 5-state model by adjusting enthalpy values. The variation of v0 with temperature has been simulated by adjusting Δh51, after which Δh23 was adjusted to control the variation of the Hill ratio (Fig. 4.14b). Table 4.9 shows that α is very sensitive to the relative size of these two activation enthalpies. Figure 4.14 also shows the predicted temperature dependence of maximum power, the unloaded ATPase rate and maximum efﬁciency. Maximum power increases by a factor of 28 between 0 and 30 C, while the unloaded ATPase rate

4.7 The Effects of Temperature

157

Fig. 4.14 Temperature-dependent muscle behaviour predicted from the 5-state model of Sect. 4.5, with parameter values intended for rabbit psoas muscle at 10 C (Table 4.5). (a) Isometric tension (solid line), stiffness (long dashes) and the number of bound heads (short dashes). (b) unloaded shortening velocity in nm/s (●) and ATPase rate (O). (c) Maximum power (zW) from shortening (●), maximum efﬁciency η (Δ) and the Hill ratio α a/T0 (~) of the tension-velocity curve. All plots were generated with the following activation and reaction enthalpies: Δh12 ¼ ΔH12 ¼ 64, Δh23 ¼ 60, ΔH23 ¼ 120 , Δh34 ¼ 40, Δh 45 ¼ 60, ΔH34 ¼ ΔH45 ¼ 0, Δh51 ¼ 50 KJ/mol

Table 4.9 Trial enthalpies for temperature-dependent properties of the 5-state contraction model, showing tensions, unloaded contraction velocities and Hill ratios at 0 and 30 C. The v’ant Hoff equation was used to generate rate and equilibrium constants from Table 4.5 with K~ P ¼ 5 mM for a temperature of 10 C Δh23 (KJ/mol) 0 60 60 60 80

ΔH23 120 120 100 120 120

Δh51 50 50 50 80 50

T0(0) (pN) 169 164 176 170 162

T0(30) 335 336 323 329 336

v0(0) (nm/s) 725 737 742 495 741

v0(30) 7620 7580 7560 7460 7520

α(0) 0.67 0.32 0.30 0.54 0.24

α(30) 0.11 0.59 0.60 0.16 0.93

158

4 Models for Fully-Activated Muscle

increases by a factor of 22. Hence the maximum efﬁciency also rises with temperature. These predictions are very similar to measurements on rat edl ﬁbres (Ranatunga 1984, 1988). Finally, if the net afﬁnity for actin binding + Pi release is the key quantity which links the effects of temperature and phosphate, what is the effect of varying actin afﬁnity alone? This question was posed by Karatzaferi et al. (2004), who claimed that isometric tension is directly proportional to the decrease ΔG ¼ kBT ln KA in Gibbs energy. This proposal can be understood in terms of strain-dependent binding. If, for the sake of a simple formula, we assume that T0 is proportional to its A-state contribution, then the box approximation employed in Sect. 2.3 gives Mκ T0 ¼ b

Z

XA

Xp0 ðXÞdX

X∗

Mκ 2 X p 2b A 0

ð4:72Þ

on setting X* ¼ 0. For an equilibrium distribution, p0 ð X Þ ¼

K A exp βκX 2 =2 K A exp βκX 2 =2 þ 1

ð4:73Þ

Setting p0(XA) ¼ p0/2 gives the cut-off strain XA, which satisﬁes βκX 2A =2 ¼ ln ðK A þ 2Þ: Hence T0 ¼

Mκ 2 MkB T K A X A p0 ln ðK A þ 2Þ: 2b b KA þ 1

ð4:74Þ

This function has the property that T0(KA) ~ ln KA when KA >> 1, but also that T0(0) ¼ 0, so it falls below this logarithmic law when KA 1. In this sense, Eq. 4.74 should have a more general validity than its derivation suggests. Karatzaferi’s prediction can be tested directly from complete contraction models. With the 5-state model. Figure 4.15 shows that isometric tension is roughly proportional to ln KA but the slopes at high and low afﬁnity are different. The downward curvature at low KA is consistent with Eq. 4.74, but one should not expect an exact ﬁt. If we pause to reﬂect on how muscle contraction is modulated by temperature change, one has to admire the way in which fast muscles become stronger, faster and more powerful as the temperature is raised. The maximum power output increases, not just because isometric tension is increased, but because the velocity of unloaded shortening increases substantially and the Hill ratio increases, meaning that the tension-velocity curve becomes less convex. The combination of these two effects increases power output at intermediate shortening velocities, and keeps pace with the increased ATPase rate, in which the isometric and shortening components both increase. However, the fastest muscles in nature cause vibratory motion rather than animal locomotion; they drive the wing-beats of small ﬂying insects, the swimbladder of the toadﬁsh (Rome 2006) and the throat muscles of songbirds

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159

Fig. 4.15 Isometric tension as a function of actin afﬁnity for M.ADP.Pi, calculated from the 5-state model

(Elemans et al. 2008), typically with v0 ~2 104 nm/s per half-sarcomere. Modelling can explain why they also operate efﬁciently, and how their behaviour is generated by the temperature-dependent kinetics of speciﬁc reactions in the biochemical contraction cycle. Perhaps it is not too fanciful to speculate that muscles with improved high-temperature performance could be genetically engineered by manipulating the activation enthalpies of actin binding and ATP-induced detachment.

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Luo Y, Cooke R, Pate E (1994) Effect of series elasticity on delay in development of tension relative to stiffness during muscle activation. Am J Phys 267:C1598–C1606 Mansson A (2010) Actomyosin-ADP states, interhead cooperativity, and the force-velocity relation of skeletal muscle. Biophys J 98:1237–1246 Mijailovich SM, Kayser-Herold O, Stojanovic B, Nedic D, Irving TC, Geeves MA (2016) Threedimensional stochastic model of actin-myosin binding in the sarcomere lattice. J Gen Physiol 148:459–488 Millar N, Geeves MA (1988) Protein ﬂuorescence changes associated with ATP and adenosine 50 -[γ-thio]triphosphate binding to skeletal muscle myosin subfragment 1 and actomyosin subfragment 1. Biochem J 249:735–743 Millar N, Homsher E (1990) The effect of phosphate and calcium on force generation in glycerinated rabbit skeletal muscle ﬁbers. J Biol Chem 264:20234–20240 Millar N, Howarth V, Gutfreund H (1987) A transient kinetic study of enthalpy changes during the reaction of myosin subfragment 1 with ATP. Biochem J 248:683–690 Millman B (1991) The ﬁlament lattice of striated muscle. Physiol Rev 78:359–384 Nishiyama K, Shimizu H, Kometani K, Chaen S (1977) The three-state model for the elementary process of energy conversion in muscle. Biochem Biophys Acta 460:523–536 Nyitrai M, Rossi R, Adamek N, Pellegrino MA, Bottinelli R, Geeves MA (2006) What limits the velocity of fast-skeletal muscle contraction in mammals? J Mol Biol 355:432–442 Offer G, Ranatunga KW (2013) A cross-bridge cycle with two tension-generating steps simulates skeletal muscle mechanics. Biophys J 105:928–940 Offer G, Ranatunga KW (2015) The endothermic ATP hydrolysis and crossbridge attachment steps drive the increase of force with temperature in isometric and shortening muscle. J Physiol (London) 593(8):1997–2016 Park-Holohan SJ, West TG, Woledge RC, Ferenczi MA, Barclay CJ, Curtin NA (2010) Effect of phosphate and temperature on force exerted by white muscle ﬁbres from dogﬁsh. J Muscle Res Cell Motil 31:35–44 Pate E, Cooke R (1989) A model of crossbridge action: the effects of ATP, ADP and Pi. J Muscle Res Cell Motil 10:181–196 Piazzesi G, Lombardi V (1995) A cross-bridge model that is able to explain mechanical and energetic properties of shortening muscle. Biophys J 68:1966–1979 Piazzesi G, Reconditi M, Linari M, Lucii L, Sun YB, Narayanan T, Boesecke P, Lombardi V, Irving M (2002) Mechanism of force generation by myosin heads in skeletal muscle. Nature 415:659–662 Piazzesi G, Reconditi M, Koubassova N, Decostre V, Linari M, Lucii L, Lombardi V (2003) Temperature dependence of the force-generating process in single ﬁbres from frog skeletal muscle. J Physiol (London) 549(1):93–106 Piazzesi G, Reconditi M, Linari M, Lucii L, Bianco P, Brunello E, Decostre V, Stewart A, Gore DB, Irving TC, Irving M, Lombardi V (2007) Skeletal muscle performance determined by modulation of number of myosin motors rather than motor force or stroke size. Cell 131:784–795 Piazzesi G, Dolﬁ M, Brunello E, Fusi L, Reconditi M, Bianco P, Linari M, Lombardi V (2014) The myoﬁlament elasticity and its effect on kinetics of force generation by the myosin motor. Arch Biochem Biophys 552:108–116 Potma EJ, Steinen GJM (1996) Increase in ATP consumption during shortening in skinned ﬁbres from rabbit psoas muscle: effects of inorganic phosphate. J Physiol (London) 496(1):1–12 Potma EJ, van Grass A, Steinen GJM (1995) Inﬂuence of inorganic phosphate and pH on ATP utilization in fast and slow skeletal muscle ﬁbers. Biophys J 69:2580–2589 Press WH, Teukolsky SA, Vetterling WT, Flannery BR (1992) Numerical recipes in Fortran, 2nd edn. Cambridge University Press, Cambridge Rall JA, Woledge RC (1990) Inﬂuence of temperature on mechanics and energetics of muscle contraction. Am J Phys 259:R197–R203 Ranatunga KW (1984) The force-velocity relation of rat fast- and slow-twitch muscles examined at different temperatures. J Physiol (London) 351:517–529

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Ranatunga KW (1988) Temperature dependence of mechanical power output in mammalian (rat) skeletal muscle. Exp Physiol 83:371–376 Rees BB, Stephenson DG (1987) Thermal dependence of maximum Ca2+-activated force in skinned muscle ﬁbres of the toad Bufo marinus acclimated at different temperatures. J Exp Biol 129:309–327 Rome (2006) Design and function of superfast muscles: new insights into the physiology of skeletal muscle. Annu Rev Physiol 68:193–221 Seo JS, Krause PC, McMahon TA (1994) Negative developed tension in rapidly shortening whole frog muscles. J Muscle Res Cell Motil 15:59–68 Siemankowski RF, Wiseman MO, White HD (1985) ADP dissociation from actomyosin subfragment 1 is sufﬁciently slow to limit the unloaded shortening velocity in vertebrate muscle. Proc Natl Acad Sci USA 82:658–662 Siththanandan VB, Donnelly JI, Ferenczi MA (2006) Effect of strain on actomyosin kinetics in isometric muscle ﬁbres. Biophys J 90:3653–3665 Sleep J, Glyn H (1986) Inhibition of myoﬁbrillar and actomyosin subfragment 1 adenosinetriphosphatase by adenosine 50 -diphosphate and adenyl-50 -yl imidodiphosphate. Biochemistry 25:1149–1154 Smith DA (1990) The theory of sliding ﬁlament models for muscle contraction. III Dynamics of the ﬁve-state model. J Theor Biol 146:433–466 Smith DA (2014) A new mechanokinetic model for muscle contraction, where force and movement are triggered by phosphate release. J Muscle Res Cell Motil 35:295–306 Smith DA, Geeves MA (1995) Strain-dependent cross-bridge cycle for muscle. Biophys J 69:524–537 Smith DA, Mijailovich SM (2008) Towards a uniﬁed theory of muscle contraction II: predictions with the mean-ﬁeld approximation. Ann Biomed Eng 36:1353–1371 Smith DA, Sleep J (2004) Mechanokinetics of rapid tension recovery in muscle: the myosin working stroke is followed by a slower release of phosphate. Biophys J 87:442–456 Smith DA, Geeves MA, Sleep J, Mijailovich SM (2008) Towards a uniﬁed theory of muscle contraction. I: foundations. Ann Biomed Eng 36:1624–1640 Smoluchowski MV (1916) Uber Brownsche Molekularbewegung unter Einwirkung aubere Krafte und deren Zusammenhang mit der verallgemeinerten Diffusionsgelichung. Ann Physik 352:1102–1112 Steffen W, Smith D, Sleep J (2003) The working stroke upon myosin-nucleotide complexes binding to actin. Proc Natl Acad Sci USA 100:6434–6439 Sun Y-B, Hilber K, Irving M (2001) Effect of active shortening on the rate of ATP utilization by rabbit psoas ﬁbres. J Physiol (London) 531(3):781–791 Takagi Y, Shuman H, Goldman YE (2004) Coupling between phosphate release and force generation in muscle contraction. Proc R Soc B 359:1912–1920 Takano M, Terada TP, Sasai M (2010) Unidirectional Brownian motion observed in an in silico single molecule experiment of an actomyosin motor. Proc Natl Acad Sci USA 107:7769–7774 Takashi R, Putnam S (1979) A ﬂuorimetric method for continuously assaying ATPase: applications to small specimens of glycerol-extracted muscle ﬁbres. Anal Biochem 92:375–382 Tanner BCW, Daniel TL, Regnier M (2007) Sarcomere lattice geometry inﬂuences cooperative myosin binding in muscle. PLOS Comp Biol 3:1195–1211 Tesi C, Colomo F, Nencini S, Piroddi N, Poggesi C (2000) The effect of inorganic phosphate on force generation in single myoﬁbrils from rabbit skeletal muscle. Biophys J 78:3081–3092 Thorson J, White DCS (1969) Distributed representations for actin-myosin interaction in the oscillatory contraction of muscle. Biophys J 9:360–390 Tregear RT, Reedy MC, Goldman YE, Taylor KA, Winkler H, Franzini-Armstrong C, Sasaki H, Lucaveche C, Reedy MK (2004) Cross-bridge number, position and angle in target zones of cryoﬁxed isometrically active insect ﬂight muscle. Biophys J 86:3009–3019

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Tsaturyan AK, Bershiksty SY, Butrns R, Ferenczi MA (1999) Strutural changes in the actin-myosin crossbridges associated with force generation induced by temperature jump in permeabilised frog muscle ﬁbres. Biophys J 77:354–372 Tsaturyan AK, Koubassova N, Ferenczi MA, Narayanan T, Roessle M, Bershiksty SY (2005) Strong binding of myosin heads stretches and twists the actin helix. Biophys J 88:1902–1910 Veigel C, Coluccio LM, Jontes JD, Sparrow JC, Milligan RA, Molloy JE (1999) The motor protein myosin-I produces its working stroke in two steps. Nature 398:530–533 Vilfan A, Duke T (2003) Instabilities in the transient response of muscle. Biophys J 85:818–827 Von der Ecken J, Heissler SM, Pathan-Chhatbar S, Manstein DJ, Raunser S (2016) Cryo-EM structure of a human cytoplasmic actomyosin complex at near-atomic resolution. Nature 534:724–728 Wakabayashi K, Sugimoto Y, Tanaka H, Ueno Y, Takezawa Y, Amemiya Y (1994) X-ray evidence for the extensibility of actin and myosin ﬁlaments during muscle contraction. Biophys J 67:2422–2435 Weiss GH (1986) Overview of theoretical models of reaction rates. J Stat Phys 42:1–36 West TG, Hild G, Siththanandan VB, Webb MR, Corrie JET, Ferenczi MA (2009) Time course and strain dependence of ADP release during contraction of permeabilized skeletal muscle ﬁbers. Biophys J 96:3281–3294 Westerblad H, Allen DG, Lannergren J (2002) Muscle fatigue: lactic acid or inorganic phosphate the major cause? News Physiol Soc 17:17–21 White HD, Taylor EW (1976) Energetics and mechanism of actomyosin adenosine triphosphatase. Biochemistry 15:5818–5826 White HD, Belknap B, Webb MR (1997) Kinetics of nucleoside triphosphate cleavage and phosphate release steps by associated rabbit skeletal actomyosin, measured using a novel ﬂuorescent probe for phosphate. Biochemistry 36:11828–11836 Whittaker M, Wilson-Kubalek EM, Smith JE, Faust L, Milligan RA, Sweeney HL (1995) A 35-Ao movement of smooth muscle myosin on ADP release. Nature 378:748–751 Woledge RC (1968) The energetics of tortoise muscle. J Physiol (London) 197:685–707 Woledge RC, Curtin NA, Homsher E (1985) Energetic aspects of muscle contraction, Monographs physiological society no. 41. Academic Press, London Wood JE, Mann RW (1981) A sliding-ﬁlament cross-bridge ensemble model of muscle contraction for mechanical transients. Math Biosci 57:211–263

Chapter 5

Transients, Stability and Oscillations

Sic transit gloria

5.1

Chemical Jumps and Temperature Jumps

Making a step change in ligand concentration is a standard biochemical technique for studying kinetic responses. For muscles, perhaps the oldest example of this technique is, so to speak, built in. When a muscle is tetanised by a string of nervous impulses, calcium ions ﬂood into the sarcoplasm and switch on the ﬁlaments, causing tension to rise to a steady state. One can study the effects of making a step change in [Pi], [ATP] or [ADP]. Experimentally, the key problem is to effect a rapid change in ligand concentration, so that the transient response, in tension, stiffness, X-ray diffraction peaks, ﬂuorescent labels or whatever, can reﬂect the full range of fast and slow transitions in the contraction cycle. For some ligands, a step increase can be generated by laser-induced release from a caged compound, and this “ﬂash and smash” technique has been used to generate upward jumps in Pi (Dantzig et al. 1992; Millar and Homsher 1990, 1992), and ATP (Goldman et al. 1982, 1984a, b). Rapid exchange of the bathing solution is simpler but not feasible for ﬁbres, where the rate at which solutes are exchanged is limited by internal diffusion. However, this method can be used for myoﬁbrils, and has the advantage that the effects of downward as well as upward jumps can be studied. For Pi jumps, this method was pioneered by Tesi et al. (2000), and the results were quite surprising. The temperature jump is a scatter-gun technique which probes all transitions in the contraction cycle via their temperature-dependent kinetics. It can be implemented by infra-red laser heating (Davis and Harrington 1987; Goldman et al. 1987). Interpreting the results of these experiments has been controversial, because, as argued in the preceding chapter, we still do not have ﬁrm numbers for the enthalpies of all relevant transitions.

© Springer Nature Switzerland AG 2018 D. A. Smith, The Sliding-Filament Theory of Muscle Contraction, https://doi.org/10.1007/978-3-030-03526-6_5

167

168

5 Transients, Stability and Oscillations

Fig. 5.1 The rise of tension (●) and stiffness (○) at the start of a tetanus in frog muscle, from data of Cecchi et al. (1991). The half-rise times of 44 and 27 ms are matched by predictions of the 5-state model of Sect. 4.5 (black and red lines respectively), generated with k23 ¼ 15 s1, K23 ¼ 0.5, γ ¼ 2, ε ¼ 10, and other parameters listed in Table 4.5. With permission of Elsevier Press and C. Ashley

5.1.1

The Activation Jump

When a fast muscle in the relaxed state is tetanized, tension rises more slowly than stiffness (Ford et al. 1986; Cecchi et al. 1991). The measurements of Cecchi et al. are reproduced in Fig. 5.1. The transients cannot really be ﬁtted by a single exponential rise; there is a latency period after the onset of tetanus, and the subsequent rises in tension and stiffness are too steep to be ﬁtted by single exponentials. This behaviour shows that the regulatory system is still being switched on after the start of tetanus, which makes it difﬁcult to interpret the transients in terms of contractile behaviour at full activation. The solid lines are the predictions of the 5-state model adjusted to match the observed half-rise times as closely as possible. Despite the complexity of the model, the predicted transients are well ﬁtted by single exponentials, whereas the measured transients are not. Nevertheless, much can be learnt by simulating the transients from contractile models after a step change from the relaxed state to full activation. In a fully activated system, the time scale of activation is determined by the rate of actin-binding, primarily from heads in the detached state M.ADP.Pi. This fact alone constrains the binding rate of unstrained heads to somewhere between 20 and 40 s1, as recognised by Huxley in 1957. Tension rises more slowly than stiffness because heads at large positive strain generate more tension but bind more slowly. Filament compliance provides a second mechanism for doing this, because stiffness will saturate when the crossbridge contribution rises to a level in excess of the axial stiffness of the ﬁlaments. See also Campbell (2006). The kinetic determinant of these processes is not the rate kA(X) of actin binding but the equilibration rate kA(X) + k-A(X), which must not be allowed to increase too rapidly with X in either direction. The solid curves in Fig. 5.1 are the rises calculated from the 5-state model with target zones and dimeric myosins. The key parameters are k23 and K23, which control the magnitude of the equilibration rate, and γ and ε, which control its strain dependence through two-step binding and accelerated binding of the second heads. Increasing the value of k23 accelerates the rise of

5.1 Chemical Jumps and Temperature Jumps

169

tension and stiffness, whereas increasing K23 has almost no effect on the rate of stiffness rise at low Pi where KP ¼ 10 and (1 + KP)K23 > 1; under these conditions k-A(X) kP ¼ kTR. However, Sleep et al. (2005) reported that force redevelopment at 20 C was much faster, namely kTR ¼ 68 s1, when ATP was released from a caged complex. Thus Stehle’s criterion remains to be tested under various conditions.

5.1.3

ATP Jumps

The use of caged compounds to generate a chemical concentration jump in muscle was pioneered by Goldman et al. (1982) for releasing ATP, using skinned rabbit psoas ﬁbres to permit rapid inward diffusion. Before release, the ﬁbre was prepared in the rigor condition, using length changes to generate different initial tensions. The ﬁnal concentration of ATP was varied over a range of about 100–300 μM by changing the intensity of the laser pulse, and these experiments were made with and without Ca2+ present. Thus this protocol enabled them to study the transition from rigor to the relaxed state and the active state. In the absence of calcium, ATP jumps of 100–300 μM caused the tension to fall to zero over 50 ms (Fig. 5.3a). The tension responses in ﬁbres with different initial tensions converged after 20 ms, indicating that ATP detached rigor heads on a time scale of 10 ms or so. This aspect of the response was veriﬁed by varying the amount of ATP released, indicating that the rate of the ﬁrst phase was linear in [ATP], with a second-order rate constant of 5 105 M1 s1 (Goldman et al. 1984a), close to values observed in solution. However, the subsequent common decline to zero tension indicates that detachment from actin must have been followed by reversible rebinding, otherwise there would be no way of explaining why the tension traces converged.

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5 Transients, Stability and Oscillations

Fig. 5.3 Isometric tension responses after the release of caged ATP from rigor muscle (Goldman et al. 1982). (a) With no Ca2+, transients from rigor states incubated with different tensions converge to a single curve before reaching the relaxed state. (b) The effects of calcium. With Ca2+, tension from a high-tension rigor state falls before rising to the steady-state level. (c) A model tension response for the +Ca case (Goldman et al. 1984b); the function of Eq. 5.1, plotted with F/T0 ¼ 0.825 and kT ¼ 150 s1, kA ¼ 100 s1. With permission of Springer Nature

At this stage, one could well ask why myosin heads should be rebinding to regulated actin in the absence of calcium before the ﬁbre reaches a totally relaxed state with zero tension. This phenomenon provides an interesting insight into the cooperative nature of thin-ﬁlament regulation, which is the subject of Chaps. 7 and 8. The tropomyosin ﬁlaments on regulated actin which block actin binding in the relaxed state can be switched open by calcium, but also by bound myosins. After 10 ms, ATP will have detached some crossbridges, leaving the remainder to hold some tropomyosin ﬁlaments in an open state to allow localised reversible rebinding. This process must compete with the effects of the tropomyosins that block actin sites evacuated by ATP. Because each tropomyosin covers seven actin sites, reversible rebinding cannot keep pace with the increasing number of blocked sites, so that the system evolves to the relaxed state where no heads can bind.

5.1 Chemical Jumps and Temperature Jumps

173

Following the discussion in Sect. 4.5, we can also expect that rebinding will be faster for heads whose dimeric partner remains bound. This aspect was included by the authors in a simple model, but a comprehensive model of the tension response should include the cooperative mechanism of thin-ﬁlament regulation. In the presence of calcium, an ATP jump also produces a two-phase tension response, with a common second phase rising to the steady-state tension T0 of the active muscle (Goldman et al. 1984b). The authors show that their observations can be ﬁtted by a two-step reaction of the following form (Scheme 5.1): Scheme 5.1 A two-step scheme for an ATP-induced transition from rigor to the active state

rigor → ‘relaxed’ → active kT kA

For an initial tension F, this scheme gives the tension response T ðt Þ ¼ FexpðkT t Þ þ T 0

kA ð1 expðkT t ÞÞ k T ð1 expðk A t ÞÞ kA kT

ð5:2Þ

if state tensions are proportional to their occupancies and no tension is associated with the intermediate state. Figure 5.3 plots this function for the parameter values which ﬁt their data.

5.1.4

Temperature Jumps

Laser-induced temperature jump experiments, mainly with rabbit psoas ﬁbres, have been made in various laboratories (Davis and Harrington 1987; Goldman et al. 1987; Bershitsky and Tsaturyan 1992; Davis and Rodgers 1995; Coupland and Ranatunga 2003), with essentially the same results. For a temperature rise of 5–10 C, there is an initial thermoelastic contribution, followed by two kinetic phases of tension response with rates r2 between 60 and 400 s1 and r3 from 5 to 20 s1. r2 is relatively independent of ﬁnal temperature, whereas r3 increases with ﬁnal temperature over the range 6–40 C. Most of the kinetic response occurs in the slow phase. What temperature-dependent transitions in the contraction cycle generate these two phases? The same transitions generate the ﬁnal steady-state tension, and in Sect. 4.7 we concluded that the primary candidates were the binding of M.ADP.Pi, ATPinduced detachment of A. M, or the hydrolysis of M.ATP. The relatively temperature-independent fast transition seems too fast for actin binding, but it could be generated by detachment of negatively-strained crossbridges or hydrolysis off actin. However, it might also be generated from the working strokes, whose kinetics are weakly temperature-dependent. The rate of the ﬁnal phase is similar to isometric ATPase rates, so it is probably generated by strain-gated cycling, in which all transitions contribute.

174

5 Transients, Stability and Oscillations

Fig. 5.4 Simulations of the tension response to a 10 temperature jump, ostensibly for rabbit psoas muscle at 10 C, using the 5-state model of Sect. 4.5. The three curves illustrate the effects of the reaction enthalpy for actin binding, with Δh23 ¼ 60, 120 and 240 kJ/mole generating increasingly rapid responses. Values of the other reaction enthalpies and the equilibrium enthalpy changes ΔH were Δh12 ¼ ΔH12 ¼ 64, ΔH23 ¼ 120, Δh34 ¼ 40, ΔH34 ¼ 0, Δh45 ¼ 60, ΔH45 ¼ 0 and Δh51 ¼ 50 KJ/mole

Simulating the T-jump response from a contractile model poses the same problem as the temperature-dependence of isometric tension. It is difﬁcult to disentangle the effects of a particular choice of enthalpies from the chosen parameters of the crossbridge cycle at the starting temperature, and in this case the problem is compounded by the need to match the time-scale of the tension rise. One way of proceeding is to explore the effects of changing the reaction enthalpy of the actinbinding transition, namely Δh23 in the 5-state model. This is the one parameter that is not accessible from solution kinetics; the energy barrier to be overcome by tethered myosins in the muscle ﬁbre has nothing to do with whatever barrier exists for myosin-S1 in solution, although the equilibrium change ΔH23 is presumably the same. Figure 5.4 shows how the tension response to a 10 temperature jump speeds up as Δh23 is increased, starting from a set of parameters for rabbit psoas muscle at 10 C. The choice which most closely resembles the data of Davis and Harrington is the middle curve, constructed with Δh23 ¼ 120 KJ/mole. The acceleration with temperature is generated primarily by the actin-binding rate k23 for an endothermic reaction (Δh23 > 0), while the ﬁnal tension is controlled primarily by K23 and its enthalpy ΔH23.

5.2 5.2.1

Length Steps The Length-Step Response

A step change in sarcomere length produces the characteristic rapid tension transients described in Sect. 2.4, which constitute evidence for working strokes of the

5.2 Length Steps

175

myosin lever arm. However, the character of the complete tension response is quite complex. There are at least four phases, ending in a steady state. Phase 1 is the elastic response, which is virtually instantaneous, ending in tension T 1 ðΔLÞ ¼ T 0 þ S0 ΔL

ð5:3aÞ

where S0 ¼

κNB0 : 1 þ CS κNB0

ð5:3bÞ

S0 is isometric stiffness including the lumped ﬁlament compliance CS, and NB0 is the number of bound heads. Phase 2 is the rapid response terminating in tension T2(ΔL), deﬁned as maximum/minimum tension for releases/stretches, or as a point of inﬂection (zero curvature) for releases where no maximum exists (Ford et al. 1977). Phase 3 can be present as an antirecovery phase or a ﬂat plateau, whereas phase 4 is the ﬁnal slow recovery. If the length step keeps the sarcomere on the plateau of the tension-length curve, the ﬁnal tension should equal the initial isometric tension T0. Figure 5.5 shows a set of tension transients for frog muscle (Rana esculanta) and phase-2 analysis from Piazzesi and Lombardi (1995). T1(ΔL) is not quite linear; the deviation for large releases can be explained in terms of buckling S2 rods (Smith 2014a). The phase-2 response can usually be ﬁtted by a single exponential exp(λ2t) starting from the end of the elastic response and ﬁnishing at tension T2 as deﬁned above. For large releases, the ﬁbre remains slack in the ﬁrst stage of rapid recovery, so data-ﬁtting starts when the ﬁbre begins to generate positive tension again. T2(ΔL ) falls to zero when ΔL ¼ ΔL2, here equal to ¼ 10.5 nm. |ΔL2| is related to the net working stroke h, but the discussion in Sect. 2.4 shows that, even after allowing for ﬁlament compliance, the two quantities are not equal. Sometimes the rapid tension response cannot be ﬁtted by a single exponential. This is not surprising, because the distribution of head strains generated from vernier spacings generates a range of exponents, as demonstrated by an oversimpliﬁed model (Fig. 2.11c, d). However, there is no general formula available from modelling (Burton et al. 2006). A two-exponential ﬁt gives subphases 2a and 2b, whose amplitudes vary in a characteristic way with step size. The fast phase (2a) is dominant for releases whereas the slow phase (2b) is dominant for stretches (Ranatunga et al. 2002). Moreover, the slow rate λ2b is very sensitive to [Pi], whereas λ2a is not. This asymmetry has been modelled in terms of a slow Pi-release transition (Smith and Sleep 2004), but the same behaviour is predicted if Pi release is fast and reversible and the slow phase is produced by detachment from actin. Then the rate constant for phase 2b as a function of [Pi] is λ2b ð½PiÞ ¼ k A ðX Þ þ

½Pi kA ðX Þ ½Pi þ K~ P

ð5:4Þ

176

5 Transients, Stability and Oscillations

Fig. 5.5 The tension responses of frog muscle to length steps (Piazzesi and Lombardi 1995), for steps of 1.8, 2.5, 4.6, 6.1 and 8.4 nm respectively. (a) The tension transients (original data kindly supplied by Prof. Gabriella Piazzesi). (b) Tensions T1 (ﬁlled circles) and T2 (open circles) before and after phase-2 recovery. (c) The rate of recovery during phase 2, where the dotted line is an exponential ﬁt. With permission of Elsevier Press and G. Piazzesi

where the strain-averages are functions of ΔL. The [Pi] dependence of this rate agrees with Ranatunga’s stretch data (Fig. 5.6). However, for releases phase 2b is overshadowed by the effects of the working strokes and phase 2a is only weakly Pi-dependent. This asymmetry is best interpreted as the result of different phase-2 mechanisms for releases and stretches. For releases, Pi is released before the working stroke to give phase 2a, with virtually no downstream phase-2b component from the ﬁnal A.M.ADP state, whereas for stretches Pi binds after a reverse stroke, followed by slow detachment of A.M.ADP.Pi from actin. A consequence of fast and reversible Pi release before the working stroke is that in the limit ΔL ! 0, Eq. 5.4 is identical with Eq. 5.1. With this assumption, tension redevelopment, phosphate jumping and the phase-2b response to a quick stretch are all governed by the same kinetic process.

5.2 Length Steps

177

Fig. 5.6 Rates of fast and slow recovery from 5 nm length steps as a function of Pi concentration (Ranatunga et al. 2002). The slow stretch rate is Pi-dependent and can be ﬁtted by Eq. 5.4 with K~ P ¼ 11 mM: With permission of John Wiley and Sons Inc. and K.W. Ranatunga

Turning now to phase 3, Fig. 5.5a shows that after rapid tension recovery from a release step, phase 3 appears as a plateau for the smallest release (2.5 nm) and as a weak antirecovery phase for larger releases (4.6 and 6.1 nm). However, in rabbit psoas ﬁbres, phase 3 appears as an antirecovery phase for a release of 1.5 nm (Davis and Rodgers 1995), which is most pronounced at low temperatures. The amplitude and rate of phase 3 both increase with added inorganic phosphate (Kawai 1986). The wing muscles of some ﬂying insects, notably the giant water beetle Lethocerus, exhibit a pronounced antirecovery phase (Steiger 1977) which is crucial to their ﬂight mechanism. Finally, phase 4 is a true recovery phase in which the tension appears to asymptote to a steady state value. However, phase 4 does not always allow a complete recovery of isometric tension before the tetanus is exhausted, which suggests that some very slow transitions in the contraction cycle are operating. A minimal heuristic description of the whole tension response requires a threeexponential ﬁt, namely T ðt Þ ¼ T 1 þ

4 X

A j 1 exp λ j t

ð5:5Þ

j¼2

where T0 ¼ A0+A2+A3+A4. When A3 has the opposite sign to A2 and A4, phase 3 is an antirecovery phase. This formula has a fundamental validity for small steps where Aj / ΔL, but it is also useful for larger steps where rates and amplitudes are functions of ΔL. Table 5.1 summarizes three-exponential ﬁts to the responses in Fig. 5.5a. How can we understand the different phases of recovery in terms of kinetic events? Phase-2 recovery from a quick release is generally interpreted as the result of one or more working strokes, and the recovery from a stretch step as the result of

178

5 Transients, Stability and Oscillations

Table 5.1 Amplitudes and rate constants for a three-exponential ﬁt to the length-step data of Piazzesi and Lombardi (1995) ΔL (nm) 1.8

T1 1.466

2.5 4.6 6.1 8.4

0.445 0.123 0.076 0.108

A2 0.1086 0.3574 0.485 0.716 0.850 0.61

A3 0.03 0.03 0.40 0.30 0.25

A4

1.40 1.50 0.6

λ2 (s1) 4000 260 1540 2170 2500 2000

λ3 (s1) 20 20 20 100 340

λ4 (s1)

5 10 6

Phase-2 recovery for the stretch step needs to be ﬁtted by two exponential rises, and two exponentials also give a better ﬁt to phase-2 for releases. All amplitudes are normalized to isometric tension T0

stroke reversals. In both cases, it is assumed that the contributing heads stay bound throughout. However, large releases which trigger all the strokes should lead to rapid detachment of negatively-strained A.M.ADP heads after ADP release and ATP binding, and this can be checked from modelling. In contrast, phases 3 and 4 are generated by actin binding and detachment. For example, the phase-3 tension response to a quick release is a balance between the binding of M.ADP.Pi at moderate positive strains, to ﬁll up the hole left by the release, and the ATP-induced detachment of positively-strained A.M.ADP heads after a strain-gated ADP-release stroke. In this way, the tension may rise or fall during phase 3. The ﬁnal phase 4 is generated by reversible binding of M.ADP.Pi at large X values where binding is slowed by strain energy. Modelling the step responses veriﬁes these statements but also reveals a potential problem. Simulating a satisfactory set of tension responses for release steps is a revealing test of how many working strokes are required. There is a danger that the tension transients for different steps will cross over; a small release may trigger only one stroke, whereas a larger release may trigger two strokes. Experimentally, crossovers are not observed. They can sometimes be removed by reducing the rate constant of the second stroke, but a better solution is to use a bigger number of smaller strokes. It seems that three strokes will sufﬁce, not counting the ADP release step. Piazzesi et al. (2014) have proposed a model with four 2.7 nm working strokes, making a total stroke of 11 nm, but the measurements of Kaya and Higuchi indicate that the net working stroke is nearer 8 nm. Then the fourth stroke, if it exists, could be interpreted as a reversible ADP-release stroke via an excited state, as in Sect. 4.1. The only experimental evidence for intermediate stroke states comes from a singlemyosin optical trap experiment (Capitanio et al. 2006), who found substrokes of 3–5 nm and 1 nm. Nevertheless, Occam’s razor applies: the number of hypothetical intermediate states should be kept to a minimum. Figure 5.7 shows tension transients without crossovers, generated from the 5-state model with two 4 nm working strokes. It is worth noting that, for large releases, an anti-recovery phase can be generated simply by reducing the size of the ADP release step h3, as illustrated for h3 ¼ 1.5, 1.0 and 0.5 nm. The responses for h3 ¼ 1.0 nm are close to those seen by Piazzesi et al. (Fig. 5.5). The amplitude of the anti-recovery phase increases with the size of the release, which suggests that a hidden instability is

5.2 Length Steps

179

Fig. 5.7 Tension transients generated by the 5-state model with target zones, dimeric myosins, buckling rods and two 4 nm working strokes, for release steps of 2, 4, 6, 8, 10 nm. The phase-3 response is controlled by the size of the ADP release step h3. Stroke kinetics were calculated with the highest-energy barrier approximation and ADP release kinetics with the excited-state model (Eqs. 4.27, 4.28 and 4.29), modiﬁed for buckling with κ b ¼ 0.03 pN/nm and Xb ¼ 0.5 nm

at work. The strain distribution of bound states shows that antirecovery is generated by the detachment of positively-strained R-state heads. Reducing the size of the ﬁnal strain-gating stroke h3 increases the ATP-induced detachment rate for positivelystrained heads and reduces their population over the lifetime of phase 3, as shown by the lower graphs in Fig. 5.8. This prediction does not seem to be in the literature. The mechanism which produces the ﬁnal recovery phase can be tracked in the same way. For releases, the phase-4 response is due to the binding of A-state heads over a range of positive strains to ﬁll the gap left by the length step. There are two exponential subphases, which can be thought of as arising from each end of the gap (Simmons et al. 2006). This discussion shows that a coherent explanation of the length-step response in terms of crossbridge dynamics is available. Different ﬁbre types (IIB, ID, IIA) show the same T2 curves as a function of step size but different rates for phase-2 and phase3 recoveries (Galler et al. 1996), which implies that their working strokes have the same amplitudes but different kinetics. What follows shows that our understanding of the length-step response can be tested in ways which go beyond the original experiments.

180

5 Transients, Stability and Oscillations

Fig. 5.8 Bound-state occupancies before and after phase-3 recovery from a quick release of 8 nm, taken from the modelling in Fig. 5.7. Colour codes: black for A-state (3), red for P-state (4), green for R-state (5). Occupation probabilities are summed over the three sites of the target zone, and A-state strain X refers to the site in question. Graphs (a) and (b) correspond to Fig. 5.7a (no antirecovery), and graphs (c) and (d) to Fig. 5.7c (large antirecovery). (a) and (c) were calculated at t ¼ 2.4 ms (end of phase 2) and (b) and (d) at t ¼ 20 ms (end of phase 3). Graph (d) shows that the loss of R-state tension at positive strains (X > 8 nm) is responsible for the pronounced antirecovery of tension during phase 3 in Fig. 5.7c

5.2.2

Repeated Length Steps

An ingenious extension of the length-step protocol was demonstrated by Lombardi et al. (1992) and Irving et al. (1992). They investigated the effect of a second release step of 2 nm made at various times Δt after a 5 nm release. With Δt ¼ 2 ms, phase-2 recovery from the second step was essentially additive, giving a tension close to the value T2(7) for a 7 nm release. But when Δt was increased, rapid tension recovery

5.2 Length Steps

181

from the second step was more complete until, when Δt ¼ 15 ms, the tension after the second step was close to the tension T2(5) produced after the ﬁrst step. This behaviour continued for another 15 ms. How can this result be interpreted? During the 15 ms delay, the response to the ﬁrst step was predominantly in phase 3, where the tension barely changed. Nevertheless, the strain distribution of bound states must have been regenerated by binding at the +ve edge of the strain distribution of A-state heads, and detachment at the –ve edge of R-state heads. Modelling by Chen and Brenner (1993), and the distributions in Fig. 5.8 from the 5-state model, conﬁrm this interpretation. Moreover, the straindependent probabilities of the three bound states just before the second step and 15 ms afterwards are barely changed; phase-2 recovery from the second step is more complete than from the ﬁrst step. That phase-3 recovery involves regeneration of bound states is conﬁrmed by a second experiment with repeated length steps. A staircase of step releases spaced by a ﬁxed time interval Δt of 8–20 ms (Lombardi et al. 1992, Irving et al. 1992) generates progressively more complete tension recovery as more steps are made, so that the tension response asymptotes towards a periodic response, known as a limit cycle. When that occurs, phase-3 binding and detachment in each period produce a tension increase which exactly cancels the tension decrease after phase2 recovery. This statement and the approach to a limit cycle is conﬁrmed by modelling (Fig. 5.9); myosin-actin binding and detachment over phase 3 resets the envelope of the strain distribution of bound states, just as a pianist resets his hand to replay the same sequence of notes. This conclusion was not always accepted. Lombardi et alia suggested that a branching contraction cycle with repeated working strokes per single hydrolysis of ATP was required, because the phase-3 rate of tension regeneration was faster than the ATP turnover rate, but Chen and Brenner (1993) showed that a branching cycle is not required. However, the approach to a limit cycle requires small steps and a sufﬁciently large time interval between steps, otherwise the attachment transitions cannot keep pace with repeated stepping, even

Fig. 5.9 Modelling the tension response to a staircase of 4 nm steps with the 5-state buckling model. The parameters were those used to generate Fig. 5.7b

182

5 Transients, Stability and Oscillations

after phase-2 recovery. Further study is needed to quantify the necessary conditions for a limit cycle.

5.3

Sinusoidal Length Changes

A small sinusoidal length change ΔL(t) ¼ δL cos ωt is commonly used to measure muscle stiffness via the induced sinusoidal tension response, provided the frequency is high enough; frequencies of about 5 KHz are sufﬁcient. If the length perturbation is small enough, the tension response ΔT(t) will be linear in ΔL(t) and oscillate at the same frequency ω (the circular frequency in radians/s), so that it can be written in the form ΔT ðt Þ ¼ AðωÞδL cos ðωt þ ϕðωÞÞ

ð5:6Þ

in terms of a frequency-dependent stiffness A(ω) and phase shift ϕ(ω). At high frqequencies, the phase shift tends to zero and A(ω) ! S0, the instantaneous stiffness of the muscle. However, in striated muscle these functions show an interesting and characteristic behaviour at lower frequencies. At low frequencies, ϕ(ω) > 0 just as for any viscoelastic medium. At intermediate frequencies, of order 10 Hz, the phase becomes negative, and at higher frequencies becomes positive again before tending to zero in the kilohertz range (Kawai and Brandt 1980; Rossmanith et al. 1980). AC stiffness has also been studied as a function of Pi (Kawai 1986; Kawai and Halvorson 1991), ATP and ADP (Kawai and Halvorson 1989) and temperature (Zhao and Kawai 1994). The challenge is to relate this behaviour to the straindependent kinetics of the contraction cycle (Murase et al. 1986; Iwamoto 1995), which has been an area of some controversy (Horiuti and Sakoda 1993). To begin, it is convenient to recast the linear response in terms of complex pﬃﬃﬃﬃﬃﬃﬃ exponentials, using the identity exp(iωt) ¼ cos (ωt)+i sin (ωt) where i ¼ 1: If we write ΔL(t) ¼ δLexp(iωt) and ΔT(t) ¼ δT(ω)exp(iωt), then the complex linear response function is Sð ωÞ ¼ 0

00

δT ðωÞ δL

ð5:7Þ

where S(ω) ¼ S (ω)+iS (ω) in terms of its real and imaginary parts. Because cosωt ¼ (exp(iωt)+ exp (iωt))/2, the physical response is obtained by averaging over frequencies ω and –ω. Now a real sinusoidal perturbation must generate a real tension response, so expanding the function S(ω) exp (iωt)+S(ω) exp (iωt)into 0 00 real and imaginary parts shows that S (ω)is an even function and S (ω) is an odd function. Hence the physical response is the real part of ΔT(t), and

5.3 Sinusoidal Length Changes

Re

183

ΔT ðt Þ ¼ ReðS0 ðωÞ þ iS00 ðωÞÞð cos ωt þ i sin ωt Þ δL ¼ S0 ðωÞ cos ωt S00 ðωÞ sin ωt:

ð5:8aÞ

As expected, the right-hand side of Eq. 5.8a is an even function of ω. Hence the amplitude and phase functions in Eq. 5.6 are AðωÞ ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ S0 ðωÞ2 þ S00 ðωÞ2 ,

ϕðωÞ ¼ tan 1 ðS00 ðωÞ=S0 ðωÞÞ:

ð5:8bÞ

Kawai’s work showed that S(ω) could be represented by a simple formula such as 3 X

SðωÞ ¼

j¼1

aj

iω iω þ λ j

ð5:9Þ

in terms of three viscoelastic modes. A single-mode viscoelastic response is generated by a spring and dashpot connected in series, so that the length changes for the same tension are additive. A parametric plot of the imaginary part of this function against the real part gives three semi-circular Nyquist loops centred on the real axis, such that the imaginary component has maximum amplitude when ω ¼ λj. This result follows from a simple piece of algebra; eliminating ω from the function x +iy ¼ iω/(iω+λ) gives the equation of a circle through the origin and centred on x ¼ 0.5, y ¼ 0. The notable feature of the Nyquist plots measured for fast striated muscle is that a2 < 0 for the intermediate frequency loop, while a1, a3 > 0 for the loops at low and high-frequencies respectively (Fig. 5.10). This three-mode formula for the AC susceptibility is directly related to Eq. 5.5 for the length-step response, and the equivalence can be demonstrated with the aid of Fourier transforms. A length step can be represented as an integral of sinusoidal steps. The unit length step is the Heaviside function θ(t) ¼ 1 for t > 0 and 0 for t < 0, which is the indeﬁnite integral of the Dirac delta function δ(t), which has a Fourier representation (Lighthill 1970). Thus Z θ ðt Þ ¼

t

1

δðt Þdt ¼

1 2π

Z

t

Z

1

1

-1

eðiωþεÞt dω ¼

1 2π

Z

1

eiωt dω 1 iω þ ε

ð5:10Þ

is a weighted integral of sinusoidal functions, with ε ¼ 0+. Hence 1 δT ðt Þ ¼ 2π

Z

1

eiωt SðωÞδLdω 1 iω þ ε

ð5:11Þ

is the linearised response to a length step δL starting at t ¼ 0. Substituting Eq. 5.9 for S(ω) generates the result

184

5 Transients, Stability and Oscillations

Fig. 5.10 The frequency-dependent stiffness S(ω) of rabbit psoas muscle at 20 C (Kawai 1986), with and without 8 mM added Pi. (a) The Nyquist plot of real and imaginary parts with frequency. (b) The phase function tan1(Im κ(ω)/ Re κ(ω)). In both cases, the imaginary part is extremal at 1, 11 and 100 Hz. With permission of Springer Nature and M. Kawai. (c) The Nyquist plot from Eq. 5.9, with λ1 ¼ 6.0, λ2 ¼ 80, λ3 ¼ 628 rad/s, a1 ¼ 7.0, a2 ¼ 2.8, a3 ¼ 5.6 pN/nm per F-actin (black line), and the result of subtracting a hypothetical ﬁlament compliance of 0.022 nm/pN (red line), as suggested by measurements of Linari et al. (2007)

δT ðt Þ ¼

3 X

a j exp λ j t δL

ð5:12Þ

j¼1

using the calculus of residues and closing the contour of integration in the upper complex plane. Comparing this result with Eq. 5.5 shows that A jþ1 ¼ a j δL

ðj ¼ 1; 2; 3Þ

ð5:13Þ

Thus the three-mode representations for the length-step response for small steps and the Nyquist response to a sinusoidal step are indeed equivalent, and the characteristic frequencies λj are the same if renumbered. Moreover, the negative-amplitude Nyquist loop (j ¼ 2) corresponds to the antirecovery phase in the length-step

5.3 Sinusoidal Length Changes

185

response. This result provides a guarantee that phase-3 antirecovery, as part of a three-exponential process, is always present in the response to small length steps even if it is not obvious. The sinusoidal tension response at different frequencies can be calculated directly from models of the contraction cycle, either directly in the frequency domain or in the time domain. For the linear response, the former method is very direct, while the latter method will reproduce the harmonic responses generated by ﬁnite-amplitude perturbations. Both methods can be formulated in terms of state probabilities, and it is convenient to use the Eulerian version of the kinetic equations. A linear response theory can be formulated as follows. Let Δpj(X, t) be the change in state probability produced by a small length change ΔL(t). As before, this equation refers to whatever head has site-displacement X at time t. The linearised version of Eq. 4.42 is NS dδpi ðX; t Þ dΔLðt Þ dp0i ðX Þ X þ ¼ Aij ðX Þδp j ðX; t Þ dt dt dX j¼1

ð5:14Þ

where p0i ðX Þ are the isometric probabilities. For sinusoidal perturbations, Δpi(X, t) ¼ δpi(X, ω) exp (iωt), so we have to solve the matrix equation NS X j¼1

dp0 ðX Þ δL iωδij Aij ðX Þ δp j ðX; ωÞ ¼ iω i dX

ð5:15Þ

in the frequency domain. Then the frequency-dependent stiffness S(ω) follows from δT ðωÞ ¼ SðωÞδL ¼

M b

Z

b=2

NS X

b=2 j¼1

t j ðX Þδp j ðX; ωÞdX:

ð5:16Þ

Equation 5.15 shows that the major computational task is the solution of a complex matrix equation. For single-site models with ﬁve biochemical states, this method is viable and can give a Nyquist plot with a negative loop (Smith 1990). With three-site target zones and dimeric myosins, the state-site matrix A(X) has dimension OS ¼ 11. With this dimensionality, solving an ill-conditioned matrix equation, even with singular-value decomposition (Press et al. 1992), is more difﬁcult when the matrix is complex. This becomes clear on working with the real and imaginary parts of δp(X, ω), which satisfy the coupled equations dp0 ðX Þ ω2 I þ AðX Þ2 • ReδpðX; ωÞ ¼ ω2 dX

ð5:17aÞ

186

5 Transients, Stability and Oscillations

ImδpðX; ωÞ ¼

AðX Þ ReδpðX; ωÞ ω

ð5:17bÞ

For this problem, the effects of an ill-conditioned matrix A(X) are made much worse because the square of the matrix appears in Eq. 5.17a. Another approach, which remains to be tested, follows from writing z ¼ iω and treating z as a complex variable. If z is real, then the mathematical problem posed by Eq. 5.16 is no worse than ﬁnding the inverse of A(X). As far as I know, a numerical calculation of the corresponding stiffness function S(z) cannot be analytically continued onto the imaginary axis! But if this function can be ﬁtted by the three-mode formula (Eq. 5.10) with iω ! z, then the amplitudes and frequencies of the three Nyquist loops can be extracted. How small should length oscillations be to guarantee a truly linear tension response? Kawai’s measurements typically used driving amplitudes of 0.25% (2.5 nm per half-sarcomere), but even this low level may not avoid generating harmonics in the tension response. In an insect ﬂight muscle, Cuminetti and Rossmanith (1980) found that amplitudes above 0.05% give signiﬁcant harmonic content. A fully non-linear theory would be useful, and this approach has been explored by Iwamoto (1995), working in the time domain. This problem is taken up in Sect. 5.5, where it is developed in a different way. It remains to consider the effects of ﬁlament compliance on the sinusoidal response. In Sect. 2.4, we saw that the length-step response is scaled back because over half of the length step appears in the ﬁlaments, primarily actin. The same partition of crossbridge compliance and ﬁlament compliance affects the frequency-dependent stiffness. Approximating the latter by a series elastic compliance CS gives Sð ωÞ ¼

SX ð ωÞ 1 þ C S SX ð ωÞ

ð5:18Þ

where SX(ω) is the contribution of the crossbridges. The inverse formula, for SX(ω) in terms of S(ω), has the same form with a sign change for CS, and in this way one can work back from experimental data to reveal the crossbridge contribution if CS is measured simultaneously. Figure 5.10c shows how this procedure works for the three-mode formula (Eq. 5.9), which provides a reasonable ﬁt to the low-frequency end of Kawai’s data in Fig. 5.10a. Setting CS ¼ 0.022 nm/pN in Eq. 5.18 shows that the high-frequency loop is enhanced and moved to a higher frequency, while the other loops are less affected. This demonstration shows that ﬁlament compliance does not just reduce the size of the length perturbation seen by the crossbridges; it also reduces the characteristic frequencies of the tension response. This can be understood in terms of springs and dashpots (Luo et al. 1994); a dashpot with damping coefﬁcient η in series with a spring of stiffness κ has a characteristic damping frequency of κ/η, which is reduced if the spring is made more compliant. To conclude, it is clear that the mechanism which generates the negative loop is the same as that which generates phase-3 antirecovery after a step release, namely

5.4 Force Steps

187

the detachment of positively-strained post-stroke myosins, although constrained by the binding of detached myosins in the M.ADP.Pi state. This second aspect explains why the amplitude of the negative loop increases with added Pi (Kawai 1986; Kawai and Halvorson 1991). With Pi release as a rapid reversible transition preceding the working strokes, added Pi increases the detachment rate of pre-stroke A.M.ADP heads and limits the increase in tension from M.ADP.Pi binding that works against antirecovery from a release step. The same is true of the Fourier transform of the step response, which takes us into the frequency domain. There is no reason why Nyquist plots could not be generated directly from step-response data, by taking the inverse transform of Eq. 5.11. This idea has been extended by Rossmanith (1986), who realised that the tension response over the whole frequency spectrum could be generated without frequency sweeping by modulating the length of the ﬁbre with white noise, and Fourier-analysing both the modulation and the tension changes.

5.4

Force Steps

One might imagine that the length response to a force step, ﬁrst studied by Podolsky and others (Civan and Podolsky 1966; Podolsky and Nolan 1973), would contain no information not present in the tension response to a length step. This expectation is neither true nor false; the rapid force-step response can duplicate the phase-2 tension response to a length step, but a small force step can also trigger an oscillatory response which has no obvious equivalent in the tension transients produced by a length step. However, there is another good reason for studying force-step responses. The interpretation of length-step experiments is complicated by the effects of ﬁlament compliance (see Sects. 2.4 and 4.4). The axial displacement of bound myosin heads is slightly less than half of the applied length change per halfsarcomere, and there is a distribution of displacements resulting from the distributed elasticity of the ﬁlaments. Force-step experiments do not suffer from this complication. Because viscous forces are negligible, a step change in load is transmitted equally to all the sarcomeres, so they should all make the same length change unless there are inbuilt structural inhomogeneities (Edman et al. 1988). Moreover, actin ﬁlaments in all sarcomeres, and myoﬁlaments also, make the same length change, which makes it easier to isolate the length-step response produced by the crossbridges. The transient length response to a force step usually displays three kinetic phases, which are the analogues of the tension response to a length step. Figure 5.11a shows what is observed after a sudden drop in load. There is the same elastic response (phase 1) with length change ΔL1 ¼ ΔF/S0 as in Eq. 5.2, followed by a rapid but decelerating length change which is thought to be the analogue of phase 2. There follows a period of slower shortening (phase 3), followed by acceleration to a prolonged period of shortening at constant velocity. Phase 4 is also visible if that acceleration overshoots the ﬁnal shortening velocity. To begin, consider what is required to produce a theory of the rapid force-step response in isolation, assuming that it is generated by working strokes in the absence

188

5 Transients, Stability and Oscillations

Fig. 5.11 Length changes induced by force steps mirror the tension changes induced by length steps (Piazzesi et al. 2002). (a) The responses observed for various force steps, from which the length changes ΔL1, ΔL2 after phases 1 and 2 can be extracted. (b) Plotting ΔL2(F) as a function of ﬁnal force gives the same result as plotting T2(ΔL ) after a length step. With permission of John Wiley and Sons Inc.

of actin attachment transitions. The Florence group adopted an intuitive approach to the analysis of force-step experiments, as follows. If the phase-2 response is entirely due to working strokes, then sarcomere length cannot change once that response is complete. However, phase-3 shortening starts simultaneously with the force step, so the length change ΔL2 at the end of phase 2 is assumed to be the extrapolation of the phase-3 region of the length response back to the time of the step. The truth of this ansatz was amply demonstrated by comparing the results of length-step and force-step experiments on the same ﬁbres (Piazzesi et al. 2002). The length change after phase-2 recovery from a force step is equal to the length step which produces the same tension after rapid recovery (Fig. 5.11b). In mathematical language, F ¼ T 2 ðΔL2 Þ:

ð5:19Þ

This equation becomes trivial if there were no actin attachment transitions leading to steady shortening. Then the force step would trigger stroke transitions leading to a new equilibrium between bound states, which is the same equilibrium produced by the equivalent length step ΔL2. Calculating the transient phase-2 response to a force step is not so easy. A crude approach is possible by using a box approximation for strain-dependent state probabilities, as in Sect. 2.4. For small steps, a linearised approach is possible. But the complete length response to a force step can always be calculated from a contractile model, by integrating the kinetic equations for state probabilities over a small time interval for a trial length step δL, such that the tension does not deviate from the applied load F. Figure 5.12 shows the results of calculations with the 5-state model, using the working strokes which generated the length-step responses in Fig. 5.7a. With ﬁlament compliance modelled by a lumped elastic element connected in series with crossbridges and rigid ﬁlaments, head-site displacements after a force step ΔF change by ΔX(t) ¼ ΔL(t) CSΔF where ΔL(t) is the observed

5.4 Force Steps

189

Fig. 5.12 Force-step responses from the 5-state model with h1 ¼ h2 ¼ 4.0 nm, h3 ¼ 1.5 nm for ten force releases (F/T0 ¼ 0(0.1)0.9), showing the transient approach to steady shortening. There is rapid shortening after the elastic response, followed by a period of damped oscillation which is the analogue of phases 3 and 4 in the length-step response. With this large value of h3, oscillations are most pronounced at large loads (F/T0 ~ 0.9). The corresponding length-step response is shown in Fig. 5.7a Fig. 5.13 Oscillations in ﬁbre length observed by Edman and Curtin (2001) after a small force step. With permission of John Wiley and Sons Inc.

length change per h-s. Whereas after a length step ΔL, an iterative procedure is required to calculate the tension changes constrained by ΔX(t) ¼ ΔL CSΔT(t).

5.4.1

Isotonic Oscillations

A very curious aspect of muscle dynamics is the phenomenon of damped oscillations in the length response to a small force step (|ΔF|/T0 < 0.2), made during a previously isometric tetanus (Armstrong et al. 1966, Edman and Curtin 2001), as in Fig. 5.13. This result was quite unexpected and does not seem to ﬁt comfortably with the threephase kinetic response to a length step; many cycles of oscillation are observed. Length oscillations were observed in different segments of the ﬁbre, and were

190

5 Transients, Stability and Oscillations

initially synchronous but became slowly disordered. The stiffness of the ﬁbre oscillated at the same frequency, and its phase was 90 behind the oscillation in shortening velocity. That simple crossbridge models can generate length oscillations after a small force drop was demonstrated by Duke (1999). As with a length step, the initial elastic shortening triggers synchronized working strokes from those heads whose strain crosses the threshold X ¼ X*. Rapid (phase-2) shortening follows to maintain constant tension in the face of working strokes (and detachment of negativelystrained R-state heads), until positively strained R-state heads begin to detach (the onset of phase 3). The gradual loss of tension must then be eliminated by a slow stretch, which has to overcome not only the loss of positively-strained post-stroke heads but stroke reversals of heads that are driven back across the threshold. This phase-3 process is terminated by recruitment of detached heads (phase 4), leaving a distribution similar to that at the start of phase 2. And then the process starts all over again. The initial distribution is not reproduced exactly, and the oscillations are slowly damped out. The ﬁve-state model also generates damped isotonic length oscillations for small force-release steps. Figure 5.12 shows prominent oscillations for F/T0 ¼ 0.9, which can be traced back to the use of a large ADP release stroke h3 ¼ 1.5 nm. A large value of h3 reduces the rate at which positively-strained rigor heads can detach from actin. A small length release pushes some R-state heads (A.M.ADP or A.M) into negative strain where they detach rapidly, but also pushes a larger number of such heads into a state where they still carry positive strain and detach slowly, thus creating the anti-recovery phase-3 response. The oscillatory response is present only for small force steps, since stretching occurs only when phase-2 shortening produces very few negatively-strained R-state heads. Larger force steps generate more negatively-strained R-state heads, whose detachment causes an increase in tension which is removed by more shortening to maintain the isotonic condition. In this way, Edman and Curtin’s isotonic oscillations can be understood without requiring any exotic additions to existing theories of contraction. Finally, note that if h3 is decreased (as in Fig. 5.7b, c), isotonic oscillations after small force-release steps disappear, but reappear for larger releases. In fact, this behaviour is characteristic of the ﬁbrillar ﬂying muscles of small insects, as we shall see in Sect. 5.6. To summarise, this description of force-step responses is a natural extension of the theory of the length-step response in Sect. 4.2. But it is not reducible to simple formulae, even with the artiﬁces used in Sect. 2.4. How can it be tested quantitatively without resorting to numerical calculation on a strain-dependent model of the contraction cycle?

5.4.2

A Simple Quantitative Theory of Isotonic Oscillations

For a simple but empirical theory, one need look no further than the three-mode description of the linear response to sinusoidal length modulation, valid for small

5.4 Force Steps

191

amplitudes. We have already seen how this formulation can be adapted to predict the response to length steps, by treating the step as a Fourier integral of sinusoidal perturbations (Eq. 5.11). The same procedure can be applied to a force step (Smith 1990). Since δL(ω) ¼ δT(ω)/S(ω), the inverse formula to Eq. 5.11 is 1 δLðt Þ ¼ 2π

Z

1

eiωt δF dω: 1 iω þ ε SðωÞ

ð5:20Þ

This is a universal formula for the response to a small step. With the three-mode representation of S(ω) (Eq. 5.9), it yields an analytic result for the time course of the length change. A little algebraic processing shows that 1 1 ðiω þ λ1 Þðiω þ λ2 Þðiω þ λ3 Þ ¼ SðωÞ iω BðiωÞ2 þ Ciω þ D

ð5:21Þ

B ¼ a 1 þ a2 þ a3 , C ¼ a1 ðλ2 þ λ3 Þ þ a2 ðλ3 þ λ1 Þ þ a3 ðλ1 þ λ2 Þ, D ¼ a1 λ 2 λ 3 þ a2 λ 3 λ 1 þ a3 λ 1 λ 2 :

ð5:22Þ

where

The integrand of Eq. 5.20 can be expanded in partial fractions, as ðiω þ λ1 Þðiω þ λ2 Þðiω þ λ3 Þ

1 ðiωÞ

2

2

ðiωÞ þ iωc þ d

¼

P ðiωÞ

2

þ

Q Rþ R þ þ iω iω zþ iω z

ð5:23Þ

where c ¼ C/B, d ¼ D/B, pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 c c2 4d 2

ð5:24Þ

λ1 λ2 þ λ2 λ3 þ λ3 λ1 cλ1 λ2 λ3 d d2

ð5:25aÞ

λ1 λ2 λ3 d

ð5:25bÞ

z ¼ and P¼

Q¼ R ¼

ðz þ λ1 Þðz þ λ2 Þðz þ λ3 Þ : z2 ðzþ z Þ

ð5:25cÞ

Now the integral can be evaluated by the calculus of residues, closing the contour in the upper half of the complex ω plane to ensure convergence for t > 0. The single and double poles at ω ¼ 0 should be moved just above the real axis. It is easy to check that the double pole contributes minus the time derivative of the single-pole

192

5 Transients, Stability and Oscillations

contribution. The poles at ω ¼ iz contribute if x Re z < 0, in which case the length response satisﬁes δLðt Þ 1 ¼ fP þ Qt þ Rþ ezþ t þ R ez t g: δF B

ð5:26Þ

Moreover, damped oscillations occur if c2 < 4d and c > 0, which together imply that x ¼ c/2 < 0 as required. The ﬁrst inequality also implies that d > 0. Then Eq. 5.26 can be rewritten as o δLðt Þ 1 n ¼ P þ Qt þ 2jRþ ject=2 cos ðΩt þ ϕÞ δF B

ð5:27Þ

where Ω

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ d c2 =4,

tan ϕ ¼

ImRþ : ReRþ

ð5:28Þ

Equation 5.27 predicts that oscillations are superimposed on the elastic response at t ¼ 0, which with the aid of the identity P+R++R ¼1 gives δL(0)/δF ¼ 1/Bas expected, and a very slow shortening velocity v ¼ QδF/B. For reference, Eq. 5.25c can be rewritten in terms of real quantities as 1 λ1 λ2 þ λ2 λ3 þ λ3 λ1 cλ1 λ2 λ3 1 þ ReRþ ¼ , ð5:29aÞ 2 d d2 1 c cðλ1 λ2 þ λ2 λ3 þ λ3 λ1 Þ ðc2 2dÞλ1 λ2 λ3 λ1 λ2 λ3 þ ImRþ ¼ : 2Ω 2 2d 2d2 ð5:29bÞ It remains to show how these oscillations arise from realistic values of the parameters a1, a2, a3, λ1, λ2, λ3 of the three-mode representation of AC stiffness S (ω) in Eq. 5.9. The number of independent parameters can be reduced by measuring length change in units of δF/B and time in units of 1/λ2. That leaves just four dimensionless parameters b1 ¼

a1 a3 , b3 ¼ , B B

r1 ¼

λ1 λ3 , r3 ¼ λ2 λ2

ð5:30Þ

to manipulate, where r1 < 1, r3 > 1 and b2 ¼ 1b1b3. With a negative second loop in the Nyquist plot, b1,b3 > 0 and b2 < 0. The key variables c and d can now be written in terms of these parameters, as

5.4 Force Steps

193

c ¼ x y þ r1 þ r3 , λ2

d ¼ r3 x r1 y þ r1 r3 , λ22

c2 4d ¼ ðy x r 3 þ r 1 Þ2 4xðr 3 r 1 Þ λ22

ð5:31aÞ ð5:31bÞ

where the combinations x ¼ b1 ð1 r 1 Þ,

y ¼ b3 ð r 3 1Þ

ð5:32Þ

appear as a delightful simpliﬁcation. The phase diagram for the boundary of damped oscillations can now be plotted in the (x,y) plane for given values of r1 and r3 (Fig. 5.14a). Equations 5.31 shows that oscillations occur in the region x þ r3 r1

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4xðr 3 r 1 Þ < y < x þ r 3 þ r 1

ð5:33Þ

between the lower branch of c2 4d ¼ 0 and the line c ¼ 0. One would expect that a sufﬁcient condition for oscillation is b2 < 0, which implies that b1 + b3 > 1. Since x b1 but y > > b3, the phase diagram shows that larger values of b1 rather than b3 are more effective in generating oscillations. To predict the character of these oscillations, it is helpful to plot the damping constant and the frequency against b1 and b3. Figure 5.14c, d, constructed for r1 ¼ 0.1 and r3 ¼ 10, show how the region of oscillations is bounded by the curve c2 4d ¼ 0 (where Ω ¼ 0) and the straight line c ¼ 0 (no damping). By moving within this space, one can construct highly damped and lightly damped oscillatory responses over a range of frequencies centred about λ2, as in Fig. 5.14b. And the linear response function δL(t)/δF is the same for force steps in either direction, as found experimentally. Edman and Curtin also found that the oscillatory response disappeared for larger release steps |ΔF|/T0 > 0.2, when it was replaced by the three-phase kinetic response equivalent to the tension response observed for length steps. A linearised theory contains no information which can predict this changeover. Nevertheless, it is interesting to ask how a large force step might move the amplitudes of the three Nyquist modes out of the region of oscillation. In this respect, the tension response to large releases may provide the best clue. In Figs. 5.5a and 5.7c, the antirecovery phase is more prominent for larger releases; from Table 5.1 and Eq. 5.14, the equivalent Nyquist amplitude a2 ¼ A3/|ΔL| is most negative for a 4.5 nm release before phase 3 starts to disappear. It would be interesting to establish the route to the phase boundary by measuring the damping rate and oscillation frequency as a function of ΔF.

194

5 Transients, Stability and Oscillations

Fig. 5.14 Isotonic length oscillations after a small force step, calculated from the three-mode formula for frequency-dependent stiffness with r1 λ1/λ2 ¼ 0.1 and r3 λ3/λ2 ¼ 10. (a) The domain of oscillations in the (x,y) plane, where x ¼ b1(1 r1), y ¼ b3(r3 1) and bi ¼ ai/ (a1+a2+a3), i ¼ 1–3. The upper boundary y ¼ x+r3+r1 corresponds to c ¼ 0 and the lower boundary pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y ¼ x þ r 3 r 1 4xðr 3 r 1 Þ to c2 4d ¼ 0 (see main text). (b) Two examples calculated from Eq. 5.27: i: b1 ¼ 3.0, b3 ¼ 1.4, giving c/2λ2 ¼ 0.1 and Ω/λ2 ¼ 5.17, ii: b1 ¼ 2.0, b3 ¼ 1.0, for which c/2λ2 ¼ 1.45, Ω/λ2 ¼ 4.00. (c and d) The normalized damping coefﬁcient c/2λ2 and oscillation frequency Ω/λ2 as a function of b1 and b3

5.5

Ramp Shortening and Lengthening

The classical signatures of muscle contraction are its steady-state behaviour under an isovelocity length change, or at constant load. With length steps and force steps, we saw that steady-state behaviour follows after a damped transient response, usually as a sum of exponential decays. Here we consider the time response to the start of a length change at constant velocity. In performing these experiments, it is desirable to

5.5 Ramp Shortening and Lengthening

195

stay on the plateau of the tension-length curve so that the number of heads addressing F-actin stays constant. The transient tension responses to ramp shortening and stretching are completely different. Ramp shortening leads to a steady-state tension after 10 ms or so, while a rapid stretch raises the tension to 2–3 T0 before subsiding to a steady-state value still in excess of T0, and this behaviour poses an interesting challenge for models (Harry et al. 1990).

5.5.1

Ramp Shortening

It may come as a surprise to discover that the tension response to a shortening ramp does not always reach a steady state. Early measurements of the tension-velocity function were nearly always made by the after-loaded technique, where the tension developed by a tetanus was held constant after rising to a predetermined level. Under these isotonic conditions, the speed of shortening does reach a steady state after 10–50 ms (Fig. 5.11), which is conﬁrmed by modelling (Fig. 5.12). For ramp shortening, measurements of Cecchi et al. (1978, 1981) on frog ﬁbres showed steady tension after a similar interval, but for rat ﬁbres Roots et al. (2007) found a slow decline in tension after the initial rapid decrease. How the tension behaves in the ﬁrst 20 ms of ramp shortening may hold some clue to this variability. Experimentally, the initial decline in tension shows a point of inﬂection P1 after several milliseconds, depending on velocity (Ranatunga and Offer 2017 and refs. therein). This feature is reproduced by the 5-state model (Fig. 5.15a), and marks the end of the ﬁrst round of working strokes. However, the subsequent more gradual transition around point P2 to a slow decrease is not matched by modelling. For fast ramp shortening, the tension makes an oscillatory approach to a steady state, which appears to be a general feature of models at the level of a single half-sarcomere. In this example, the tension levels out after 15 ms, corresponding to a travel distance of 26 nm in which detached heads will be engaging sites on the next actin half-period. However, one cannot expect that the timing of these damped oscillations will be reproduced exactly in all half-sarcomeres, in which case a slow decline in tension could result. The bound-state probabilities of individual heads as a function of head-site spacing X also settles down to a steady state, even though all spacings change as the muscle shortens. A snapshot taken after the steady state is reached shows the same distributions, as in Fig. 5.15b. The occupation of the last bound state shows the expected tail, and the whole distribution of bound states is contained within the 36 nm actin half-period. To do this it was necessary to set Xmin ¼ 26 nm rather than 18 nm, because three zonal sites are involved. For unloaded shortening, this model predicts that very few heads are bound.

196

5 Transients, Stability and Oscillations

Fig. 5.15 (a) Transient responses to rapid shortening at 1750 nm/s, calculated from the 5-state model with Kramers’ stroke rates: tension in pN (black line), stiffness in pN/nm (red), number of heads bound (green), number of heads on doubly-bound dimers (yellow) and ATPase rate (blue), calculated as the rate of step 5 ! 1 for ATP-induced detachment. (b) Occupation probabilities of the bound states (3 ¼ black, 4 ¼ red, 5 ¼ green) after 40 ms, as a function of the head-zone spacing X. Parameter values are as listed in row II of Table 4.5

5.5.2

Ramp Lengthening

When muscle is stretched at constant velocity, it acts as a brake with limited powers. The tension rises in proportion to velocity, as you would expect from a purely passive elastic element, until the muscle ‘gives’ and the tension falls towards a steady-state value. This response has been extensively studied (Lombardi and Piazzesi 1990, 1992; Piazzesi et al. 1992; Bagni et al. 1998; Getz et al. 1998; Bickham et al. 2011; Pinniger et al. 2006; Brunello et al. 2007; Roots et al. 2007; Colombini et al. 2009; Nocella et al. 2013). In practice, the complete response can only be observed with rapid stretches, otherwise the length ramp has to be truncated to avoid running off the plateau of the tension-length curve. With rapid stretching, Colombini et al. (2009) found that the tension rises steeply to a peak value Tp(v) before subsiding towards the isometric level. The peak tension increases with velocity and approaches a maximum value of 3.49T0 for v > 2000 nm/s. The time tp(v) to the peak is such that the muscle is stretched by an approximately constant distance ΔLp ¼ vtp(v), estimated at 10–12 nm by Lombardi and Piazzesi (1990) and 11.9 nm by Colombini et al. This constancy is a strong sign that heads are forcibly detached by stretches of this size. The steady-state tension also increases with stretch velocity, as in Fig. 1.4, reaching a maximum value which can vary from 1.2T0 to 2.2T0 depending on ambient conditions (Lombardi and Piazzesi 1992). The rise in tension is almost linear at a rate proportional to stretch velocity, which would be produced by a passive elastic element in the absence of kinetic events. This is easy to test. As a function of length change ΔL(t) ¼ vt, the slope should be the same as the isometric stiffness S0, which can be measured independently by applying a small high-frequency length oscillation. If stiffness stays constant, then ΔT (t) ¼ S0ΔL(t) and the increase ΔX(t) in crossbridge strain satisﬁes

5.5 Ramp Shortening and Lengthening

ΔX ðt Þ ¼ ΔLðt Þ C S ΔT ðt Þ ¼ ð1 C S S0 ÞΔLðt Þ:

197

ð5:34Þ

If S0 ¼ 50 pN/nm and CS ¼ 0.012 nm/pN, then ΔX(t)/ΔL(t) ¼ 0.4, which suggests that crossbridges are detached by a positive strain of 0.4ΔLp ¼ 4.8 nm. This has been tested by Nocella et al. (2013), who monitored stiffness with a time resolution of 40 μs during ramp stretching. During the tension rise, stiffness increased roughly in proportion to the tension, reaching a peak value of 1.25So when the tension peaked. Two possible mechanisms come to mind; either more heads are recruited as stretching proceeds, or the tension-strain characteristic of each head is intrinsically non-linear, and distinguishing between them would not be straightforward. ATP turnover during ramp stretching has also been measured, although not as often as tension. For slow to moderate stretch velocities, the ATPase rate drops to very low levels, and Hill and Howarth (1959) claimed that ATP hydrolysis could actually be reversed by stretching. This does not seem to be the case, but the absence of signiﬁcant turnover conﬁrms that extending muscle behaves like a passive viscoelastic element, with no expenditure of chemical energy. For this to happen, heads in the M.ADP.Pi state should bind to actin at low strain, ride the ‘escalator’ and detach at high strain in the same state. This means that the heads that bind should not perform a working stroke, or if they do then that stroke would be reversed as stretching continues, a process described by Getz et al. (1998) as ‘rectiﬁcation.’ As stretching continues, some heads could be recruited to vacant sites on the same target zones, which could explain the small rise in stiffness. To model the stretch response, the critical questions are (a) what determines the peak tension, (b) the character of the subsequent recovery and (c) the magnitude of the ﬁnal steady-state tension? A mechanism is required for A-state heads to detach rapidly at large positive strains, without prejudicing the isometric and shortening behaviour of existing models. In Sect. 4.1, two such mechanisms were formulated, namely two-step actin binding (Eq. 4.8) and Arrhenius enhancement of attachment rates at large strains by lowering the energy barrier of the potential well (Fig. 4.3, Eq. 4.12). Both mechanisms work equally on actin binding and detachment rates, since the straindependent afﬁnity in the Hooke’s law region is always proportional to exp(βκx2/2). The strain dependence of these two mechanisms is quite different. Two-step binding produces a detachment rate which asymptotes to a constant value at large strains, namely (1 + γ) times the zero-strain rate. On the other hand, the Arrhenius mechanism gives an exponential increase exp(βκd|x|) in attachment rates, controlled by the Arrhenius length d which is a measure of the half-width of the binding well. As such, neither mechanism delivers realistic predictions. Both mechanisms generate a peak tension, but the subsequent tension fall is too steep and the steady-state tension can fall below the isometric level. In this respect, the two-step mechanism can be saved only by using unrealistically large values of γ and ε (50, 1000) which accelerate the rate of tension activation from the relaxed state, unacceptably. What is required is Arrhenius enhancement of strain-dependent actin attachment rates which saturates at a high level. This behaviour follows from the underlying physics; the lowering of the energy barrier with strain x can be modelled as

198

5 Transients, Stability and Oscillations

Fig. 5.16 Transient ramp-stretch responses predicted by the 5-state model for a stretch velocity of 3000 nm/s, showing tension in pN (black line), the rate of ATP-induced detachment (blue line), stiffness in pN/nm (red), and the number of bound heads (green). The model and parameter values were those used for Fig. 5.15, modiﬁed for Arrhenius enhancement of M.ADP.Pi-actin attachments at large strain with d ¼ 0.7 nm, xc ¼ 6 nm (Eq. 5.35). The spike in ATPase rate occurs when detached heads ﬁnd sites on the next actin half-period nearer to the M-band, bind at negative strain, make a working stroke and complete the contraction cycle before stretching can reverse their strokes. This event is repeated every time the travel distance increases by 36 nm

ΔEðxÞ ¼ κdjxj ΔEðxÞ ¼ κdxc

ðjxj xc Þ, ðjxj > xc Þ

ð5:35Þ

where ΔE(xc) is the height of the energy barrier at zero strain. Thus strains in excess of xc ‘empty the well’ by destroying its capacity to bind. The height of the barrier is not necessarily the same as the binding energy, so xc can be treated as an adjustable parameter. Then the rate of the roll-lock transition described in Sect. 5.4.1 will be enhanced by the Arrhenius factor exp(βΔE(x)), which can be incorporated in Eq. 4. 27a for the forward rate (but not in Eq. 4.28a for the afﬁnity). Figure 5.16 shows the result of modelling a rapid stretch with this version of the Arrhenius enhancement mechanism. With d ¼ 0.7 nm and xc ¼ 6 nm, stretching at 3000 nm/s produces a tension rise of 4.1T0, subsiding over 5 ms towards an apparent steady-state tension of approximately 1.5T0. This value of d is larger than expected from a single covalent bond (0.2–0.3 nm), but seems compatible with multiple binding loops to the actin interface. With κ ¼ 2.7 pN/nm, this value of xc implies that the M.ADP.Pi-actin bond has an energy barrier of 11.3 zJ (just under 3kBT), compatible with actin afﬁnities of order 5 to the lumped A.M.ADP.Pi-A.M.ADP# state at low Pi. This modelling works as follows. Bound pre-stroke heads ride the stretch ‘escalator’ in the face of faster and faster detachment rates, so some of them are thrown off in the rising phase, to be replaced by other heads recruited at low strain to start the same process. This is evident in the strain distribution of A-state heads. Figure 5.17 shows A-state occupancies for each site of the target zone, referenced by the distance

5.5 Ramp Shortening and Lengthening

199

Fig. 5.17 Snapshots of the fractional occupation of actin sites by A-state (pre-stroke) heads during ramp stretching, as in Fig. 5.16. In contrast to Fig. 5.8, bound heads are indexed by their distance to the centre of the target-zone, and colour-coded by site (black for the trailing site (site 1), red for the central site (2) and green for the leading site (3). When X ¼ X2, which generates maximum occupation for the central site, X ¼ X1 ¼ X2 + c gives maximum occupation for the trailing site and X ¼ X3 ¼ X2c for the leading site, where c ¼ 5.57 nm. (a) After 5 ms, during the rising phase. (b) At the time of peak tension (9.3 ms), (c) After 40 ms, where the tension is approaching a steady state

between the head and central site. Rebinding is apparent in the negative tails of these distributions, especially at the end of the rising phase (Fig. 5.17b). Recruitment does not make up for the number of heads lost by detachment, but that loss is never enough to stop the tension from rising. So the tension rise is not really the stretching of a passive elastic element; the maximum rise rate in Fig. 5.16 is 135 pN/s, or 45 pN/nm in terms of stretching distance, whereas the isometric stiffness is 58 pN/ nm. There is a shoulder near the end of the rise (point P2) in which detachment starts to bite, and also a point of inﬂection (point P1) which marks the end of stroke reversals. These subtle features are observed experimentally (Pinniger et al. 2006). When does the tension peak? Figure 5.17a, b show that the strain in A-state heads continues to increase beyond xc ¼ 7.0 nm while the tension is still rising, so point P2 appears to correspond to strains of order 7.0 nm for heads on the central site. Beyond P2, the tension rise is slowed and eventually falls, during which the rate of detachment is held constant. It is this feature which controls the ﬁnal tension and stops it from dropping below the isometric level; Fig. 5.17c shows that this tension is held by a small number of highly strained A-state heads. In fact, strains of 25 nm are unrealistic; bound heads cannot bend to accommodate that amount of stretching, but the modelling did not allow for that.

200

5 Transients, Stability and Oscillations

Fig. 5.18 Signatures of the ramp-stretch response for a stretch velocity of 3000 nm/s per halfsarcomere, showing (a) the peak tension Tp and (b) the ﬁnal steady-state tension Ts as a function of the Arrhenius length d and cut-off length xc in Eq. 5.35

Figure 5.18 shows how peak tension and steady-state tension vary with d and xc. They are decreasing functions of both variables, and these graphs show that relatively small changes can produce large variations in peak tension and the ﬁnal steady-state tension. In this way, the variations seen experimentally (Tp/ T0 ¼ 3.5 at 2000 nm/s, Ts/T0 from 1.2–2.2) can be reproduced by modelling. As a function of velocity, the steady-state and peak tensions are shown in Fig. 5.19. Tension continues to rise slowly as stretch velocity is increased beyond 1000 nm/s, a result which is observed experimentally (Lombardi and Piazzesi 1990) but hard to reproduce by modelling. Also shown in Fig. 5.19 is the stretch distance ΔLp required for peak tension. Contrary to experiment, this quantity continues to rise at high velocities, and this may be an indication that another factor is preventing heads from stretching to the large strains generated with this model. In modelling, a strain cut-off is required to stop a head from being strained beyond the limit set by bending its lever-arm; the elastic constant for stretching the lever-arm must be orders of magnitude higher. This constraint is just one aspect of the possibility of a non-linear tension-strain law for bound myosin heads, as proposed by Nocella et al. (2013). One way of achieving a non-linear law for axially strained myosin heads is purely geometrical (Appendix G). The ramp-stretch protocol has been especially fruitful at yielding puzzling tension behaviour. One is the phenomenon of permanent extra tension, otherwise known as residual force enhancement, evident in many experimental results where the ramp has been terminated before the transient tension response was completed (Edman 2012). When muscle length is held constant at the end of the ramp, tension drops slowly to a level above the original isometric tension as long as the muscle remains activated. This phenomenon occurs even when the ﬁnal length per sarcomere still lies on the plateau of the tension curve, and it is not reproduced by the model just described, where tension falls abruptly and then recovers to the isometric level. Possible solutions are discussed in Sect. 5.7.

5.6 Wing-Beat Oscillations in Insect Flight Muscle

201

Fig. 5.19 The velocity dependence of the tension response to ramp stretching, showing (a) the peak tension (●) and ﬁnal steady-state tension (○) and (b) the travel distance to the peak

To complete this survey of stretch phenomena, consider the likely behaviour of muscle subject to a step increase in load. Experimentally, a small extra load generates the sort of length response seen in Fig. 5.11, which asymptotes to continuous lengthening at velocity v such that F ¼ T(v), where T(v) is the steadystate length-ramp response. What happens when F > Tp? Ramp stretching shows that the muscle cannot support this load; heads detach and the muscle will be stretched at high speed. But the current model predicts that Tp itself is velocity-dependent, so the peak load would not be uniquely deﬁned. This problem is left as a modelling exercise for you to program.

5.6

Wing-Beat Oscillations in Insect Flight Muscle

In common with many other forms of locomotion, the wing muscles of many ﬂying insects are driven by periodically repeated trains of nerve impulses. However, the wing-beat frequencies of many small insects are too high to allow a cycle of activation and deactivation over one wing-beat. The ﬂight muscles of these insects, which include ﬂies, bees, beetles, cicads, gnats, thrips, wasps and some water bugs, are asynchronous with nerve impulses, and they can beat while in a state of continuous activation (Pringle 1949). This discovery triggered a period of systematic study of asynchronous insect ﬂight muscles, as summarised by Pringle (1977, 1978) and other contributors to the symposium organized by Richard Tregear. The most important ﬁnding was that these muscles exhibit a delayed rise in tension after a quick stretch (Fig. 5.20a). This phenomenon was called stretch activation, but the name is slightly misleading because it suggests that the regulatory system is being activated, and there is no direct evidence for that. In fact, rat and cardiac muscles also

202

5 Transients, Stability and Oscillations

Fig. 5.20 (a) Stretch activation in the ﬁbrillar dorsal wing-beat muscles of the giant water-bug Hydrocyrius colombiae (Jewell and Ruegg 1966). The tension responses to a step release and step stretch both show a pronounced anti-recovery phase (phase 3), but antirecovery after the stretch is slower. With permission of the Royal Society. (b) Modelling the stretch response with the 5-state model for stretch steps of 2, 4, 6, 8, 10 nm, with the parameters used for Fig. 5.7c

Fig. 5.21 The ﬁlament structure of a relaxed and unstretched ﬁbre from the carpenter bee ﬂight muscle (Pringle 1978). (With permission of the Royal Society)

show a delayed rise in the tension-stretch response (Steiger 1977). Thus stretch activation appears to be nothing more than the anti-recovery phase (phase 3) of the response to a quick stretch. Phase 3 is more pronounced in stretch than release, and can last for 0.25 s (Jewell and Ruegg 1966; White and Thorson 1972). Stretch activation is predicted by the 5-state model (Fig. 5.20b). Insect ﬂight muscles are structurally very different from vertebrate striated muscles (Reedy 1968; Taylor et al. 1999; Tregear et al. 2004). In the microscope, the ultrastructure of the myosin and actin ﬁlaments in the rigor state is very regular, and the I-bands are very narrow (Fig. 5.21). The ﬁlament lattice is rhomboidal but with a different basis: F-actins lie midway between two F-myosins at adjacent vertices, giving three F-actins per F-myosin. Myoﬁlaments have crowns of two pairs of oppositely-directed myosin dimers (the “ﬂared-X” conﬁguration), rotating azimuthally by 67.5 per 14.5 nm layer line, giving a helical periodicity of 8 14.5 ¼ 116 nm. The actin helix has a half-period of 38.7 nm, giving three

5.6 Wing-Beat Oscillations in Insect Flight Muscle

203

half-periods in the same distance. The narrow I-bands and the very regular appearance of the half-sarcomeres are due to short connecting ﬁlaments (such as titin) between the M-band and the Z-line, which give the relaxed muscle a far greater stiffness than its non-ﬂight counterparts (Pringle 1978). Wing-beat oscillations in asynchronous insect-ﬂight muscle arise when the muscle is coupled to a load with mass and stiffness. If the resonant frequency of the load, in this case the wing itself, is in the range in which the AC stiffness of the muscle has a negative imaginary part, then that ‘viscous’ component accelerates the motion instead of retarding it, setting up a cycle of spontaneous oscillation limited by nonlinearities. This idea was explored experimentally by Machin and Pringle (1959), using an electro-mechanical feedback system where the effects of an auxotonic load were simulated electronically. This was accomplished by driving muscle length from a tension change constructed as the sum of a virtual mass term, a damping term and a stiffness term. Oscillations, when present, were at the resonance frequency of the simple harmonic oscillator deﬁned by the combined mass and stiffness of muscle and load. It follows that the conditions for oscillation can be calculated by solving the Newtonian equation of motion for an auxotonic load (Fig. 5.22a), namely

€ðt Þ þ μΔL_ ðt Þ þ SL ΔLðt Þ : ΔT ðt Þ ¼ mΔL

ð5:36Þ

The minus sign is there because ΔL is the extension of the muscle, not the load. To make contact with formulae for muscle tension, Eq. 5.36 will be interpreted as tension per F-actin per half-sarcomere, so we need to know how to scale it up for the whole muscle. Let the muscle have n half-sarcomeres in length L, area A, N actin ﬁlaments per unit area, with tension T. Then L ¼ nL, T ¼ NAT so the mass, damping coefﬁcient and stiffness of the load are all scaled up by the factor f ¼ NA/n. For a ﬂight muscle with L ¼ 20 mm and A ¼ 20 mm2, n 2 104 and N 1.5 1015 F-actins/m2, so f 1.5 106. To see what an auxotonic load can do, consider the small-signal version of tension change, where ΔL(t) ¼ ΔL exp (iωt) and ΔT(t) ¼ S(ω)ΔL exp (iωt). Then the secular equation derived from Eq. 5.36 reduces to mω2 þ iμω þ SðωÞ þ SL ¼ 0

ð5:37aÞ

where Sð ωÞ ¼ SP þ

3 X j¼1

aj

iω iω þ λ j

ð5:37bÞ

from Eq. 5.9, plus passive muscle stiffness SP. For zero damping, Eq. 5.37a reduces to

5 Transients, Stability and Oscillations

transducer

204

SL

muscle

T

T L

M U

Fig. 5.22 The model of Sicilia and Smith (1991) for predicting spontaneous oscillations in striated muscle coupled to an auxotonic load. (a) Schematic diagram of the coupling, where the load has mass M, damping coefﬁcient U and stiffness SL. (b) Tension-length oscillations predicted by the parameters given in Table 5.2. (c and d) phase diagrams for domains of different behaviour as a function of damping coefﬁcient μ, mass m and stiffness S ¼ SP + SL of the relaxed muscle and load, expressed per F-actin. α: damped oscillations leading to a stable steady state, β: stable oscillations with a loop in the T-L plane, γ: oscillations with no power output because of a ﬁgure-of-eight loop in the T-L plane, and δ: unstable oscillations with continued shortening. All variables are expressed in the reduced units of Table 5.2. With permission of Elsevier Press

ω2 ¼

Sð ωÞ þ SL m

ð5:38Þ

This is an implicit equation for the mechanical resonant frequency of the load coupled to a frequency-dependent muscle stiffness. Its solution yields a complex frequency ω ¼ ω0 +iω00 so that the tension oscillates as exp(iωt) ¼ exp(iω0 t)exp(–ω00 t). Hence growing oscillations are predicted when ω00 < 0, which occurs when ω0 lies in the window for which Im S(ω0 ) < 0, present only when a2 < 0. Then ωA < ω0 < ωB where ωA, ωB are the frequencies for which the Nyquist plot intersects the real axis. In practice, this means that the mass of the load must be tuned to lie within the range (mA,mB) obtained by substituting the limiting frequencies in Eq. 5.38.

5.6 Wing-Beat Oscillations in Insect Flight Muscle

205

Table 5.2 Parameter values for oscillations in a wing-beat model for insect ﬂight muscle coupled to a load with mass m, damping μ and stiffness SL (Sicilia and Smith 1991) m μ SP SL λj aj bj b0j

Reduced units 0.1 0.2 5.0 5.0 1.42, 13, 80 23, 24, 21 9.0, 0, 9.0 20, 100, 2400

np units 6 104 0.012 3.0 3.0 14.2, 130, 800 13.8, 14.4, 12.6 0.108, 0, 0.108 2.4, 12, 288

cj c0j

10, 0, 0 10, 0, 0

0.024, 0, 0 0.024, 0, 0

Conv. factor 0.006 0.06 0.6 0.6 10 0.6 0.012 0.12 0.0024 0.0024

np unit g pN.s/nm pN/nm pN/nm rads/s pN/nm pN.s/nm2 pN/nm2 pN.s/nm3 pN.s/nm3

All quantities refer to one F-actin per half-sarcomere

A linear theory does not predict the ﬁnal amplitude and the waveform of growing oscillations. The simplest non-linear theory that can be constructed is an extension of the Kawai three-mode formula (Eq. 5.9) to terms of second and third order in the length change ΔL(t) (Sicilia and Smith 1991). Kawai’s formula is an approximation to the linear response function of a general vernier contraction model, so the secondand third-order responses can be deduced as approximations to general formulae for the non-linear case. As described in an appendix to that paper, they lead to the following formulae in terms of the velocities v(ω) ¼ iωΔL(ω): R1 ΔT ðωÞ ¼ ηð1Þ ðωÞvðωÞ þ 1 ηð2Þ ðω; ω0 Þvðω ω0 Þvðω0 Þdω0 R 1 R 1 ð3Þ þ 1 1 η ðω; ω0 ; ω00 Þvðω ω0 Þvðω0 ω00 Þvðω00 Þdω0 dω00

ð5:39Þ

where 3 X

aj iω þ λ j j¼1 ! 0 3 X b 1 1 j bj þ ηð2Þ ðω; ω0 Þ ¼ 0þλ iω þ λ iω iω þ λ j j j j¼1 ηð1Þ ðωÞ ¼

ηð3Þ ðω; ω0 ; ω00 Þ ¼

3 X j¼1

c0j cj 1 1 1 þ 0 iω þ λ j iω þ λ j iω þ λ j iω0 þ λ j iω00 þ λ j

ð5:40aÞ

ð5:40bÞ

ð5:40cÞ

neglecting third-order terms with triple poles and more double poles. Now these functions need to be Fourier-transformed into the time domain to give the tension response as convolution integrals. With

206

5 Transients, Stability and Oscillations

Z ΔT ðt Þ ¼

1 1

ΔT ðωÞeiωt dω

ð5:41Þ

(which puts a factor of 1/2π in the back transform), the contributions in each order are convolution integrals: ΔT ΔT

ð2Þ

ΔT ð3Þ ðt Þ ¼

Z ðt Þ ¼

Z

1

1

1 1

Z

dt 0 2π

ð1Þ

Z

dt 0 2π

1

Z

dt00 2π

1

ð5:42aÞ

dt 00 ð2Þ η ðt t 0 ; t 0 t 00 Þvðt 0 Þvðt 00 Þ 2π

ð5:42bÞ

1

1

1

Z

1

dt 0 ð1Þ η ðt t 0 Þvðt 0 Þdt 0 2π

ðtÞ ¼

1

1

dt000 ð3Þ η ðt t 0 ; t0 t 00 ; t00 t000 Þvðt0 Þvðt00 Þvðt000 Þ: 2π

ð5:42cÞ The functions in the integrands are all Fourier transforms of v(ω) and the functions in Eqs. 5.40, constructed by the same rules. Hence ð 1Þ

η ðt Þ ¼

3 X

Z aj

j¼1

3 X eiωt dω ¼ 2π a j eλ j t θðt Þ 1 iω þ λ j j¼1 1

ð5:43aÞ

where θ(t) ¼ 1 (t > 0) or 0 (t < 0) is Heaviside’s step function. Similarly, ηð2Þ ðt; t 0 Þ ¼ ð2π Þ2

3 X 0 b j þ b0j t eλ j ðtþt Þ θðt Þθðt 0 Þ

ð5:43bÞ

j¼1

ηð3Þ ðt; t 0 ; t 00 Þ ¼ ð2π Þ3

3 X

0 00 c j t þ c0j t 0 eλ j ðtþt þt Þ θðt Þθðt 0 Þθðt 00 Þ:

ð5:43cÞ

j¼1

Hence we have speciﬁc equations for the tension contributions in Eqs. 5.42, namely ΔT

ð1Þ

ðt Þ ¼

3 X j¼1

ΔT ð2Þ ðt Þ ¼

3 Z X j¼1

t 1

Z aj

t

1

0

dt 0 eλ j ðtt Þ vðt 0 Þ

Z dt 0 b j þ b0j ðt t 0 Þ

t0

1

ð5:44aÞ 00

dt 00 eλ j ðtt Þ vðt 0 Þvðt 00 Þ

ð5:44bÞ

5.6 Wing-Beat Oscillations in Insect Flight Muscle

ΔT ð3Þ ðt Þ ¼

3 Z X

t

dt 0

Z

1

j¼1

t0

dt 00

Z

1

Xe

dt 000 c j ðt t 0 Þ þ c0j ðt 0 t 00 Þ

t 00

1

λ j ðtt 000 Þ

207

0

00

ð5:44cÞ

000

vðt Þvðt Þvðt Þ

in the form of multiple integrals. Converting them to differential equations makes them much easier to solve, and this can be done by introducing auxiliary functions. In ﬁrst order, the function Z

ð1Þ

ψ j ðt Þ

t

0

1

dt 0 eλ j ðtt Þ vðt 0 Þ

ð5:45aÞ

appears in Eq. 5.44a for the ﬁrst-order tension term. Similarly, in second order, the functions ð2Þ

ψ j ðtÞ

!

ð2Þ

ϕj ðtÞ

Z ¼

t

0

dt e

λj ðtt 0 Þ

1

1 t t0

ð1Þ

vðt 0 Þψ j ðt 0 Þ

ð5:45bÞ

generate the second-order tension. In third order, the auxiliary function Z

ð3Þ

ψ j ðt Þ

t

ð2Þ

0

1

dt 0 eλ j ðtt Þ vðt 0 Þψ j ðt 0 Þ

ð5:45cÞ

does not appear in the tension term, but is required to generate those that do, namely ð3Þ

ϕ j1 ðt Þ ð3Þ

ϕ j2 ðt Þ

!

Z ¼

t

0

dt e 1

λ j ðtt 0 Þ

ð 2Þ

0

vð t Þ

ðt t 0 Þψ j ðt 0 Þ ð2Þ

ϕ j ðt 0 Þ

! :

ð5:45dÞ

These functions are all in a form where they can be converted to differential equations. Thus the change in muscle tension when muscle length is changing at velocity vðt Þ ¼ ΔL_ ðt Þ is ΔT ð1Þ ¼ SP ΔL þ

3 n X

ð1Þ

ð2Þ

ð2Þ

ð3Þ

ð3Þ

a j ψ j þ b j ψ j þ b0j φ j þ c j φ j1 þ c0j φ j2

o

ð5:46Þ

j¼1

where

d ð1Þ þ λ j ψ j ðt Þ ¼ vðt Þ dt

ð5:47aÞ

208

5 Transients, Stability and Oscillations

d ð2Þ ð1Þ þ λ j ψ j ðt Þ ¼ vðt Þψ j ðt Þ dt d ð2Þ ð2Þ þ λ j ϕ j ðt Þ ¼ ψ j ðt Þ dt d ð3Þ ð2Þ þ λ j ψ j ðt Þ ¼ vðt Þψ j ðt Þ dt d ð3Þ ð3Þ þ λ j ϕ j1 ðt Þ ¼ ψ j ðt Þ dt d ð3Þ ð2Þ þ λ j ϕ j2 ðt Þ ¼ vðt Þϕ j ðt Þ dt

ð5:47bÞ ð5:47cÞ ð5:47dÞ ð5:47eÞ ð5:47fÞ

With appropriate values for the Nyquist-loop frequencies λj and their amplitudes a j , b j , b0j , c j , c0j , j ¼ 1–3, numerical solutions to these equations will generate spontaneous oscillations. Values of λj and Aj are available from small-signal stiffness data, but the remainder must either be estimated from contractile models or adjusted empirically. To avoid working with a combination of very large and very small numbers, the calculation beneﬁts from the use of dimensionless units, The original calculations were performed with [L] ¼ 5 nm, [F] ¼ 3 pN, and [t] ¼ 0.1 s for the units of length, force and time. Hence the unit of mass was [M] ¼ [F][t]2/ [L] ¼ 0.006 gm, and the same reasoning shows that the unit of mass in the (nm. pN.s) system of units is 1gm. The following table gives parameter values similar to those used in 1991, plus their conversion to nanopico units: This model generates oscillations only under certain conditions. The mass of the load must be tuned so that it lies within the window (mA,mB) such that mA ¼

SL þ Sð ω A Þ , ω2A

mB ¼

SL þ Sð ω B Þ ω2B

ð5:48Þ

where Im S(ωA) ¼ Im S(ωB) ¼ 0. For the values in the table, mA ¼ 1.07 and mB ¼ 0.010 in reduced units, or 6.4 103 g and 6.0 105 g. Then a scaling factor of 1.5 106 gives limiting mass values of 9600 and 90 g for the combined mass of wing and muscle. Simulations with this set of parameters will generate steady oscillations after a period of growth from an equilibrium state perturbed by a small length change. When tension is plotted against muscle length, the oscillations appear as a loop traversed in the anticlockwise direction, so that power is generated during each cycle (Fig. 5.22b). The mechanical work output per cycle is the area of the loop, namely I W¼

T ðLÞdL:

ð5:49Þ

5.6 Wing-Beat Oscillations in Insect Flight Muscle

209

and the power output is P ¼ ωW/2π. With the parameters of Table 5.2, W ¼ 16.4 r. u.’s ¼ 246 zJ, ω/2π ¼ 30.7 Hz, so P ¼ 7.55 1018 W per F-actin per h-s. For the hypothetical muscle with a volume of 400 mm3, the conversion factor is NAn ¼ 6 1014, giving a power output of 4.5 mW, equivalent to 11 mW per gram of muscle. Larger values (30–60 mW/g) are observed in insect ﬁbrillar muscles (Jewell and Ruegg 1966), with peak-to-peak amplitudes oscillations of up to 4% of muscle length rather than 2% as found here. These estimates from modelling are subject to uncertainities about Nyquist plot parameters, especially the non-linear ones, and ﬁlament packing density. Nevertheless, they represent an improvement on the results reported in 1991, for two reasons. One is that the stiffness amplitudes aj in Table 5.2 have been increased to match the amplitude a2 ¼ 16 pN/nm reported by White and Thorson (1972) for Lethocerus cordofanus with no added Pi. Secondly, the amplitude of oscillation was doubled by a serendipitous change of sign in c1. In Fig. 5.22, modelling predictions are presented in reduced units to facilitate comparisons with the 1991 work. All the necessary conditions for steady oscillation, including mass tuning, are contained in the phase diagrams shown in Fig. 5.22c, d. There is a critical value of the combined stiffness SP + SL, below which the muscle shortens continuously in an oscillatory fashion; some passive stiffness is required so that energy can be exchanged between muscle and load in each cycle. At zero damping, the critical value is 8.17 r.u. or 4.90 pN/nm per F-actin, which can easily be exceeded by insect ﬁbrillar muscle, whose passive stiffness rises sharply with muscle length because of shorter connecting ﬁlaments (Pringle 1977). There is also an upper limit to the amount of stiffness in the system, above which oscillations damp out. Secondly, some damping is required to generate an oscillatory loop in the tension-length plane. If μ is too low, oscillations may still occur but the phase-plane plot gives a ﬁgure-ofeight loop for which there is no power output. Thus the load must be damped to ensure a continuous transfer of mechanical energy from muscle to load. As μ is increased, the width of the mass window for oscillations narrows, until an upper critical value is exceeded and oscillations vanish. Although this non-linear three-mode theory gives a good account of the mechanics of wing-beat oscillations, it is essentially a semi-empirical theory, particularly as values of the non-linear mode coefﬁcients b j , b0j , c j , c0j are not supported by experiment. In principle, they can be extracted by driving the muscle with a largeamplitude sinusoidal length change and extracting harmonic content in the tension, provided that the ambient conditions do not generate spontaneous oscillations. The non-linear tension response is also predicted from the three-mode model. To avoid working with complex quantities, one can start from Eqs. 5.41 and 5.42 in the time domain; this calculation is left as an exercise for the tenacious reader.

210

5 Transients, Stability and Oscillations

In its present form, the theory does not address the increase in ATPase rate which provides the chemical energy driving oscillatory power output, although conservation of energy requires that the rate of consumption of chemical energy must appear as mechanical power output plus heat generated by the damping of the load. The oscillatory component of ATPase was seen as a problem for the two-state model advanced by Thorson and White (1969), in which λ2 and the detachment rate g were equal. Measurements of ATPase activity as a function of [Ca++] found that λ2 increased by a factor of 3.5, whereas the tension cost (tension/ATPase) as a measure of g was almost unaffected (Pringle 1977). Clearly, a model is required in which λ2 and the tension cost are decoupled, as in contractile models (such as the 5-state model) where tension, ATPase and phase-3 antirecovery arise from heads with different strain windows. This is also a problem which can be addressed by applying sinusoidal length modulation. Such measurements of ATP turnover rates in insectﬂight muscle have been made by Ruegg and Tregear (1966) and Steiger and Ruegg (1969). To conclude, the successes of this relatively simple three-mode theory suggest that there is no reason why a fully-ﬂedged contractile model coupled to an auxotonic load could not be made to work. By removing the auxotonic load and applying a large-amplitude length sinusoid, the amplitudes of the non-linear mode coefﬁcients can be extracted, compared with experimental data, and used to improve the empirical theory presented here which is now 27 years old. It now seems clear that “stretch activation,” previously regarded as an oscillatory mechanism peculiar to insect ﬂight muscle (Wray 1979), is actually a generic property of striated muscle. This was demonstrated long ago by Goodall (1959) and Lorand and Moos (1956), who showed that rabbit psoas muscle could be made to undergo spontaneous oscillations at high Ca++ when stiffened by glycerol extraction. This phenomenon suggests that the main reason that striated muscles with an auxotonic load do not oscillate is because they have insufﬁcient passive stiffness.

5.7

The Longitudinal Stability of the Sarcomere

Modelling muscle action by focussing on just one actin ﬁlament in one halfsarcomere is a gross simpliﬁcation of muscle structure, but it works well if actin and myosin ﬁlaments are in full overlap, so that all sarcomeres lie on the plateau of the isometric tension-length curve (Fig. 1.5). When the average sarcomere length lies on the descending limb, a pandora’s box of new behaviour opens up. Muscle creep and A-band creep are phenomena which arise because the sarcomeres are longitudinally unstable; no longer can we assume that all sarcomeres behave identically. Creep phenomena pose challenges of a different kind for theories of muscle contraction (Telley et al. 2003; Telley and Denoth 2007; Edman 2012), and we still do not understand all the processes involved.

5.7 The Longitudinal Stability of the Sarcomere

5.7.1

211

Tension Creep

When a ﬁxed-end muscle is tetanised on the descending limb, the tension does not saturate after 20–40 ms but continues to creep upwards, reaching a maximum after 0.5–1 s (Huxley and Peachey 1961). The ‘creep’ component of tension is most pronounced when muscle length per sarcomere is near the middle of the descending limb. Huxley and Peachey explored this phenomenon by observing sarcomeres along the ﬁbre, and concluded that it was driven by shorter sarcomeres present at each end of the relaxed ﬁbre. During tension creep, these sarcomeres became shorter; those in the middle of the ﬁbre lengthened, although some of them would start to shorten as the tension approached its peak value (ter Keurs et al. 1978; Julian and Morgan 1979; Edman and Reggiani 1984; Burton et al. 1989; Granzier and Pollack 1990). Figure 5.23 shows examples of muscle creep obtained by Altringham and Bottinelli (1985). That muscle creep is driven by short end sarcomeres in the relaxed state is one example of a general principle: on the descending limb the sarcomeres in a ﬁxed-end ﬁbre are axially unstable. Consider some virtual displacements ΔL of sarcomere lengths about an initially homogeneous distribution with length L. Although all sarcomeres carry the same force F, a longer sarcomere, with length L + ΔL, will generate less isometric tension and will stretch at the speed v required to achieve force balance. Similarly, a shorter sarcomere will generate more tension and will

Fig. 5.23 Observations of muscle creep after activation on the descending limb of the tensionlength curve. Data of Altringham and Bottinelli (1985) on ﬁxed-end ﬁbres, for nominal sarcomere lengths of 2.70 (a) and 3.26 μm (b) in the centre of the ﬁbre, showing the tension response and sarcomere lengths at marked positions on the rulers. (With permission of Springer Nature)

212

5 Transients, Stability and Oscillations

shorten at the appropriate speed, so any deviation from the homogenous steady-state is unstable, as A.V. Hill observed in 1953. The peak tension achieved by this process lies above the linear tension fall on the descending limb (ter Keurs 1978; Granzier and Pollack 1990), but the ﬁbre must be held at ﬁxed length. If the length of a segment in the middle of the ﬁbre is held constant, muscle creep is largely removed and the peak tension lies much closer to the descending limb of the isometric tension-length curve. In a ﬁxed-end ﬁbre, the fate of the sarcomeres is determined by a balance between active and passive axial forces. We have seen that connecting ﬁlaments such as titin stop sarcomeres from lengthening beyond zero overlap, where there can be no active tension from crossbridges. Passive tension from titin increases exponentially with sarcomere length; at zero overlap it may be comparable with active tension on the plateau. Moreover, passive tension is much stronger in the end sarcomeres, which in relaxed frog ﬁbres can sit at 2.2 μm when those in the middle are at 3.5 μm (Huxley and Peachey 1961). Whatever the structural reasons for shorter end sarcomeres, they act in the same way as virtual displacements to trigger a redistribution of sarcomere lengths when the ﬁbre is activated. Each sarcomere is stabilized only when it shortens over the plateau to the ascending limb or lengthens towards the bottom of the descending limb where passive tension takes over (Fig. 5.24). These observations lead directly to a simple one-dimensional model of muscle creep (ter Keurs et al. 1978; Edman and Reggiani 1984; Saldana and Smith 1991), in which the rise of active tension is treated as instantaneous because creep occurs on a longer time scale. The basic assumptions are as follows. (i) All sarcomeres hold the same tension T(t) per F-actin at time t. (ii) In the nth sarcomere with length Ln(t), the tension is the sum of active and passive components. (iii) Active tension is approximated by its steady-state value for length Ln(t) and velocity vn(t), although these quantities change in time. (iv) Passive tension TP has elastic and viscous components, reﬂecting the dynamic nature of titin links which fold up after stress. (v) Passive tension is progressively stronger for sarcomeres nearer each end of the ﬁbre. Fig. 5.24 The tensionlength diagram with passive tension, in the relaxed state (----) and the active state (__). U denotes the unstable ﬁxed point at L ¼ 3.0 μm where the tension has negative slope, whereas points S of the same tension are stable. The red lines indicate the corresponding quantities for an end sarcomere in which the strength of passive tension has been increased by a factor of 5

5.7 The Longitudinal Stability of the Sarcomere

213

The deﬁning equations of this model are ðnÞ

T ðt Þ ¼ T 0 ðLn ðt ÞÞð1 þ a vn ðt ÞÞ þ T P ðLn ðt ÞÞð1 þ μvn ðt ÞÞ þ T E ðLn ðt ÞÞ ðn ¼ N; : . . . ; NÞ ð5:50Þ where active tension is modelled by a linear velocity dependence with different coefﬁcients a according to the sign of vn, and passive tension in the middle of the ﬁbre is viscoelastic. For the passive components, T P ðLÞ ¼ T P0 fexpðξðL L3 ÞÞ expðξðL∗ L3 ÞÞg ) ðnÞ ðnÞ T E ðLÞ ¼ T E0 fexpðηðL L3 ÞÞ exp ηðL∗ L3 Þ

ð5:51aÞ ð5:51bÞ

with ðnÞ

T E0 ¼ T E0 expðγ ðN jnjÞÞ

ð5:51cÞ

for the distribution of extra passive tension at each end. L* is the slack length for zero tension in the relaxed ﬁbre. Equations 5.50 are differential equations for Ln(t), namely ðnÞ

vn ð t Þ

dLn T ðt Þ T 0 ðLn ðt ÞÞ T P ðLn ðt ÞÞ ¼ ðnÞ dt a T 0 ðLn ðt ÞÞ þ μT P ðLn ðt ÞÞ

ð5:52Þ

with N X

vn ð t Þ ¼ 0

ð5:53Þ

n¼N

for ﬁxed ends. However, the tension T(t) is not known in advance, so it must be determined from Eq. 5.52 by an iterative procedure such as bracketing and bisection. To save on computation, one can model one half of the ﬁbre, say n ¼ 1,. . ., N with ﬁxed ends. Table 5.3 lists the parameter values used to generate predictions of this model for frog ﬁbres at 3 C with low passive tension and a distribution of short end sarcomeres occupying about 5% of the total length. They are mostly as used by Saldana and Smith (1991). All tensions are measured in units of isometric tension on the plateau. The velocity coefﬁcients a are such that a is the slope of the tangent of the tension-velocity curve in shortening. From Hill’s equation, a ¼ (1 + α)v0/α where α ¼ 0.25 is the Hill ratio, giving a ¼ 2 (s/μm) if v0 ¼ 2.5 μm/s. Figure 5.25 shows how this model behaves for different starting lengths L0. The time course of tension does reaches a peak after about 1 s, comparable with

214

5 Transients, Stability and Oscillations

Table 5.3 Parameters for the simple 1D model of muscle creep with N ¼ 500 L1 (μm) 2.0

L2 2.2

L3 3.6

a+ (s/μm) 40

a 2.0

μ 3.0

TP0 0.1

TE0 0.4

ξ (μm1) 4.65

η 1.0

L* (μm) 2.1

γ 0.075

experimental ﬁndings, but the rise to the maximum is much faster than observed (Fig. 5.25a). Peak tensions as a function of L0 do lie above the descending limb, but are only about half way towards the experimental points shown in Fig. 5.25b. This represents an improvement of the results of Saldana and Smith; increasing the magnitude TE0 of extra passive tension raises the peak tension and the time to reach it. Tension creep is also observed when L0 > L3, where the middle sarcomeres are beyond overlap and have no active tension; then creep is driven entirely by active tension from the end sarcomeres. The redistribution of sarcomere lengths (Fig. 5.25c) shows how an increasing number of sarcomeres near the end of the ﬁbre shorten onto the ascending limb while the majority in the middle lengthen slowly as required by a constant-length ﬁbre, until they are stabilized by passive tension. Further improvements along these lines seems problematic. The rapid initial rise in tension would be alleviated if crossbridge dynamics were included to limit the rate of activation, but this would only slow the rise by 20–40 ms. A much slower process seems to be limiting the tension rise after activation. Similarly, it does not seem to be possible to raise TE0 much further, because very short end sarcomeres require less shortening to go over the plateau, and the time to peak tension decreases considerably. Other changes to the values of model parameters can be discounted; giving the extra passive tension a different slack length makes little difference, and the value of the exponent μ was chosen to maximize the peak tension. It seems difﬁcult to avoid the impression that another mysterious process operates to double excess tension above the descending limb, and that the whole subject is, well, somewhat creepy. These uncertainties have given rise to extensive modelling (Denoth et al. 2002), and modiﬁcations of this ‘no-frills’ theory. Morgan et al. (1982) have proposed that sarcomeres interact in a way which goes beyond carrying the same tension, such that the length of each sarcomere is inﬂuenced by the lengths of its adjacent sarcomeres. This interaction will slow the rate at which sarcomeres peel off the middle distribution to join end sarcomeres on the ascending limb, and it will smooth any other source of intersarcomeric discontinuity. The authors suggest that this interaction could arise from the constant volume of the intact ﬁbre. Secondly, the sarcomeres may not be structurally homogeneous. Differences in myosin isoform composition along the ﬁbre may lead to inhomogeneities in active or passive tension for sarcomeres in the middle of the ﬁbre (Edman et al. 1988), so that it is of interest to study the effects of random variations in both quantities. Morgan (1990, 1994) has suggested that a single sarcomere with low active tension could lengthen rapidly to near zero overlap where passive tension is dominant. However, ‘popping sarcomeres’ have not been observed, and neither have abrupt discontinuities in the spatial distribution of sarcomere lengths (Burton et al. 1989), even after

5.7 The Longitudinal Stability of the Sarcomere

215

Fig. 5.25 Results of a one-dimensional model for muscle creep following activation on the descending limb, where creep is generated by shorter sarcomeres at each end. (a) Tension transients after activation for various initial sarcomere lengths L0 in the middle of the ﬁbre. (b) Peak tensions obtained during creep, as predicted (●) and data of ter Keurs et al. (1978) (○), with permission of Rockefeller University Press and H.E.D.J. ter Keurs. (c) The time course of sarcomere lengths near one end, plotting every fourth sarcomere in turn. Calculations were made with 500 sarcomere in each half of the ﬁbre

ramp stretching (Telley et al. 2006a). Intersarcomere interactions would act to smooth out discontinuities arising from such random variations, but it is not clear if they should be included in the ﬁrst place.

216

5 Transients, Stability and Oscillations

Perhaps the best way of improving the theory of muscle creep would be to construct a realistic 2D or 3D theory with inclined Z-lines. Inclined or staggered Z-lines are often seen in ﬁbres stretched beyond overlap, and this would be a logical way of generating more tension as a result of sarcomeric creep. However, nearly all the complexity of sarcomere creep in ﬁxed-end ﬁbres could be removed by observing the behaviour of ﬁbres under isotonic conditions. In the absence of intersarcomere interactions, each sarcomere would move independently in a manner determined only by its initial length, and the 1D model is just as simple. The velocity of each sarcomere is given by Eq. 5.52 with T(t) ¼ F, and short sarcomeres will shorten and long sarcomeres will lengthen as before. Studying the motions of individual sarcomeres under isotonic conditions, with the spot-follower technique or with ﬂuorescent markers (Telley et al. 2006b), might reveal more clearly whether structural inhomogenities are present.

5.7.2

A-Band Creep

As if the whole phenomenon of sarcomere-length instability were not enough, Horowitz and Podolsky (1987) discovered that, on the descending limb, the two halves of each sarcomere do not remain at the same length after activation. If the sarcomere is on the descending limb, the M-band does not remain at the centre but drifts off to one side. This shift also appears as an imbalance in the widths of the A-bands which signify overlapping ﬁlaments. Thus ﬁlament overlap and isometric tension increase in one half-sarcomere, causing it to shorten while the other half lengthens. The imbalance in active tension drives A-band creep until the edge of the A-band reaches the Z-line or the other side is stabilized by passive tension (Fig. 5.26a, b). This phenomenon is much slower than sarcomere creep, and was observed in rabbit psoas ﬁbres with a sequence of electron-microscope images. Clearly, this instability is also a consequence of partial ﬁlament overlap. Using just two half-sarcomeres, it can be modelled in the same way as intersarcomere creep (Horowitz and Podolsky 1988; Shabarchin and Tsaturyan 2010), provided that sarcomere length is held constant (Fig. 5.26c). A-band creep proceeds to completion over several minutes, while the net tension remains substantially constant. Why is this process so much slower than intersarcomere creep? Their modelling shows that the answer lies in the initial conditions. Starting from the symmetrical structure in which each half-sarcomere has the same length means that tensions in each half are balanced, but the slightest perturbation will cause them to depart from this unstable steady state. Horowitz and Podolsky used an initial displacement of 0.09 nm, and state that “a change in this initial value alters the position of the curves along the time axis without changing their shape.” As their predictions are well ﬁtted to experiment, we must presume that a perturbation of similar magnitude triggered their observed time course of A-band creep.

5.7 The Longitudinal Stability of the Sarcomere

217

Fig. 5.26 A-band creep between the two halves of a sarcomere on the descending limb. (a) The initial symmetrical sarcomere. (b) The sarcomere after the A-band has translated to one side. (c) Simulations of Horowitz and Podolsky (1988) for movement of the thick ﬁlaments. With permission of Rockefeller University Press

5.7.3

Residual Force Enhancement

Muscles and muscle ﬁbers may exhibit enhanced isometric tension after a ramp stretch, as mentioned in Sect. 5.5. The extra tension is not permanent, but remains as long as the muscle/ﬁbre is activated (see for example Abbott and Aubert 1952; Edman et al. 1978; Julian and Morgan 1979; Edman and Tsuchiya 1996). Its origin is still controversial, but perhaps the strongest clues are that it increases with stretch amplitude and is more prominent at longer starting lengths on the descending limb of the tension-length curve (Edman 2012 and references therein). When this second condition is met, tension during the stretch also shows a slow rise, rather than a fall, after the initial rapid rise to what would otherwise have been the peak tension. Various explanations have been offered. Passive tension may be partly responsible. The slow rise could be a creep phenomenon, signalling the onset of inhomogeneous sarcomere lengths. If this explanation is correct, then the slow tension rise during stretch would be generated by intersarcomere creep or popping sarcomeres and maintained over the post-stretch activation period. Electron microscopy of ﬁbres ﬁxed after active stretching shows evidence of A-band creep (Edman 2012). Campbell et al. (2011) have developed a model along these lines, but it requires the operation of intersarcomere interactions as well as the basic constraint imposed by equal tensions in all structures connected in series. The interested reader is referred to the reviews by Telley and Denoth (2007), Campbell and Campbell (2011) and Edman (2012). At the time of writing, the cause of residual force enhancement remains an open problem.

218

5.8

5 Transients, Stability and Oscillations

The Stability of the Filament Lattice

An entirely different set of questions for striated muscle arises when we ask what holds the lattice of interdigitating ﬁlaments together. In activated muscle, and especially in rigor, there is no problem; the interﬁlament spacing will be determined primarily by bound myosins. But in relaxed muscle, detached myosins stay clear of their actin ﬁlaments. Since the ﬁlaments are negatively charged, one can expect that the equilibrium lattice spacing will be determined by a balance of opposing electrostatic forces, namely Coulombic repulsion and van der Waals attraction between the ﬁlaments. However, these forces are proportional to the amount of overlap; what happens when the ﬁbre is stretched beyond zero overlap? All actin-myosin interactions are removed, leaving only myosin-myosin and actin-actin interactions, and the M-band and Z-line structures which support them. In this situation, the M-band and Z-line lattices must be able to keep F-myosin and F-actin spacings in register so they can interdigitate correctly when the ﬁbre is brought back into overlap. And a theory of lattice stability should reproduce the effects of ionic strength and pH of the bathing solution. The whole subject has been comprehensively reviewed by Millman (1991). An important tool for studying interﬁlament forces has been the use of highmolecular-weight polymer solutes such as polyvinylpyrrolidone (PVP) or dextran to compress the ﬁbre. Increasing the osmotic pressure of the bathing solution lowers the partial pressure of water, causing water to ﬂow out of the ﬁbre through its semipermeable membrane until the partial pressures are equalised. By van’t Hoff’s law, the osmotic pressure is proportional to the solute concentration n: Π ¼ nRT

ð5:54Þ

where R is the gas constant. The technique is also applicable to skinned ﬁbres, since high-MW PVP and dextran cannot penetrate the ﬁlament lattice. Measuring the lattice spacing d as a function of osmotic pressure determines the radial stiffness of the ﬁbre, here deﬁned as S ¼ d

∂Π : ∂d

ð5:55Þ

Measurements of lattice spacing and radial stiffness as a function of sarcomere length and ionic conditions provide the main tests for models of lattice stability. Here is a brief summary of the salient facts. (a) Osmotic compression of relaxed ﬁbres. X-ray measurements of the lattice spacing d10 of frog ﬁbres in the relaxed state show a logarithmic dependence on osmotic pressure (Maughan and Godt 1979; Umazume and Kasuga 1984; Millman and Irving 1988). In the last two examples, the data shows that there is a critical lattice spacing dc at which the pressure must be increased signiﬁcantly for further compression. It is convenient to invert this relationship mathematically by writing

5.8 The Stability of the Filament Lattice

219

ΠðdÞ ¼ afexpðbðd0 dÞÞ 1g

ðd > d c Þ

ð5:56aÞ

ΠðdÞ ¼ afexpðbðd0 dÞÞ 1g þ Πc

ðd < d c Þ

ð5:56bÞ

where do is the equilibrium spacing at zero pressure and dc < d0. Then the radial stiffness from Eq. 5.55 is Sðd Þ ¼ abdexpðbðd 0 dÞÞ:

ð5:57Þ

For the sake of future developments, we use the myosin-actin spacing d ¼ 2d10/3. Figure 5.27 shows examples of frog data at different ionic strengths, ﬁtted in this way. In Fig. 5.27b, d0 exceeds dc by about 2.0 nm. The signiﬁcance of this ‘pressure bump’ in osmotic compression data was investigated by Umazume et al. (1991) and Brenner et al. (1996), in the light of previous studies in which myosin-S1 heads in relaxed muscle were found to interact weakly with F-actin at low ionic strength. Evidence from rapid stretches (Brenner et al. 1982) and X-ray diffraction (Brenner et al. 1984), showed that osmotic compression encounters increased resistance when the lattice is compressed to the point where myosin heads span the surface-to-surface myosinactin ﬁlament spacing, estimated at 12 nm. This corresponds to a critical spacing dc ¼ 25 nm, taking RM ¼ 8.5 nm and RA ¼ 4.5 nm for F-myosin and F-actin respectively. Thus it appears that the pressure bump is generated by heads touching the actin ﬁlament, without having to be thermally excited to span the gap. That the size of the bump increases at low ionic strength is further evidence for weak actin-myosin interactions in relaxed ﬁbres. (b) Rigor ﬁbres. Putting a relaxed ﬁbre into rigor shrinks the lattice by 5–10%, depending on muscle type, when the ﬁlaments are in full overlap (Matsubara et al. 1984). So the crossbridges exert an inward radial force, equal to the osmotic force required to produce the same shrinkage in a relaxed ﬁbre. Equilibrium occurs when that inward force is balanced by electrostatic repulsion. In skinned frog ﬁbres, the transition to rigor produced an 11% reduction in lattice spacing, matched by an osmotic pressure of 6.0 KPa in the relaxed ﬁbre (loc. cit.). How does this pressure compare with isometric axial tension in the frog? Let V(d ) be the potential energy of electrostatic interactions per unit cell, where d is the my-ac lattice spacing. To make contact with osmotic forces, consider a cylindrical ﬁbre of radius R, containing N unit cells. Then the equation dV 2πRLΠ ¼ N : dR

ð5:58Þ

expresses the balance between osmotic forces on the periphery of the halfsarcomere and the outward electrostatic repulsion. The cross-sectional area of the pﬃﬃﬃ ﬁbre is that of N unit cells, so for the rhombohedral unit cell πR2 ¼ 3N 3d 2 =2 p ﬃﬃ ﬃ apart from a packing correction of O(d2). Hence R∂R=∂d ¼ 3 3Nd=2π and

220

5 Transients, Stability and Oscillations

Fig. 5.27 Osmotic compression data ﬁtted to Eqs. 5.56 with dc ¼ 26.8 nm (Smith and Stephenson (2011). (a) Data of Umazume and Kasuga (1984), ﬁtted with a ¼ 0.11 0.01 kPa, b ¼ 0.66 0.01 nm1. (b) Data of Millman and Irving (1988), with a ¼ 0.082 0.045 kPa, b ¼ 0.91 0.09 nm1, shows a larger pressure bump of 1.7 kPa at the critical lattice spacing dc. With permission of Elsevier Press

1 ∂V Π ¼ pﬃﬃﬃ 3 3Ld ∂d

ð5:59Þ

(Schoenberg 1980; Matsubara et al. 1984; Smith and Stephenson 2011). For the frog data of Matsubara et alia, Π ¼ 6 KPa, d ¼ 25 nm and L ¼ 1.1 μm, so the outward tension ∂V/∂d per unit cell generated when the ﬁbre goes into rigor is 857 pN, or 428 pN per triangular cell containing one F-actin, larger than the maximal axial isometric tension of 290 pN/F-actin for the frog at 3 C. Essentially the same estimate was obtained by Brenner and Yu (1991). These numbers conﬁrm that the radial component of crossbridge tension should really be included in contractile modelling. The lattice spacing in rigor ﬁbres must be such that myosin-S1 heads span the distance between the ﬁlaments. In the frog, the rigor spacing d ¼ 26 nm between ﬁlament axes allows a surface-to-surface spacing of 13.0 nm, taking RM ¼ 8.5 nm and RA ¼ 4.5 nm for the radii of myosin and actin ﬁlaments respectively. That there is only a slight increase from 12 nm in the relaxed ﬁbre is probably a consequence of strongly-bound myosins angled at the post-stroke conﬁguration. (c) Intact and skinned ﬁbres. Skinned ﬁbres are preferred for osmotic studies because the lattice spacing is primarily controlled by impermeable solutes in the bathing solution. For intact ﬁbres, the situation is more complicated, because the ﬁbre is encased by the sarcolemma membrane which keeps it at constant volume; the average cross-sectional radius is proportional to L1/2 when ﬁbre length is changed (Elliott et al. 1963). However, the sarcolemma is permeable to water, so the lattice spacing decreases in proportion to the osmolarity of the bathing solution (Millman et al. 1981; Irving and Millman 1992), and can therefore be said to act as an osmometer. Osmolarity is deﬁned as solute concentration including dissociated ions, and a hypertonic solution is one whose osmolarity exceeds that of normal Ringer’s solution (0.2 M), used to

5.8 The Stability of the Filament Lattice

221

simulate physiological conditions. The sarcolemma is also permeable to small ions, but the rate at which they are exchanged is very slow (Millman 1991), perhaps because of an existing Donnan equilibrium. Thus a hypertonic solution acts directly to shrink the lattice, and indirectly because loss of water increases ion concentrations within the ﬁbre. The constraint imposed by the sarcolemma can mean that A-bands are wider than I-bands, giving sarcomeres a barrel-like appearance (Bergman 1983). Ionic equilibrium across the membrane, when it exists, must be controlled by charged ﬁlaments as well as free ions within the ﬁbre. In principle, it should be possible to calculate these ﬁxed charges by measuring the difference in electrode potentials across the membrane, and applying the conditions of Donnan equilibrium (Plonsey and Barr 1988). Measurements were made by comparing electrode potentials with the microelectrode placed in the bathing solution, or in the A-band and I-band regions of a skinned ﬁbre (Aldoroty et al. 1985; Bartels and Elliott 1985). The disruption to the ﬁbre when a microelectrode of diameter 0.4 μm is inserted must be severe. Nevertheless, such measurements can be analysed by assuming a Donnan equilibrium across the A-band/I-band interface, which separates regions of ﬁxed charge on the ﬁlaments (Maughan and Godt 1980a). Hence ﬁlament charge densities were estimated at 6.6e/μm to 20e/μ m for F-actin and –59e/μm to 68e/μm for F-myosin, including myosin heads. Bartels and Elliott have suggested that the charge density on the myoﬁlament backbone is close to 13e/μm, which implies that the heads on one myoﬁlament have a net charge of 36e/μm to 55e/μm. Here e is the proton charge. (d) The effects of pH and ionic strength. The lattice spacing decreases with decreasing pH (Maughan and Godt 1980b). Increasing the concentration of hydrogen ions neutralizes some negative charge on the ﬁlaments, reducing the Coulombic repulsion between them. A quantitative relationship involves the proton afﬁnity KH for each ﬁlament, so that the lineal charge density satisﬁes σ ð½Hþ Þ ¼ σ ð½0Þ þ

K H ½Hþ ne 1 þ K H ½Hþ

ð5:60Þ

where n is the linear density of binding sites. An afﬁnity KH ¼ 104.5 M1 gives 50% saturation when pH ¼ 4.5. Site densities can be estimated from the isoelectric point where the charge density has been brought to zero. A further reduction in pH would make myosin and actin ﬁlaments carry positive charge, and the lattice would expand if all charge interactions became repulsive. However, model calculations suggest that myosin heads will be touching actin before the isoelectric point is reached (Smith and Stephenson 2011). Measurements of pH-dependent lattice spacing in the frog (Maughan and Godt 1980b) are consistent with these predictions, while measurements on crayﬁsh ﬁbres show a rapid decrease in lattice spacing down to pH ¼ 4.0 (April et al. 1972), which suggests that the isoelectric point in this muscle is below 4.0.

222

5 Transients, Stability and Oscillations

The effects of ionic strength on lattice spacing are complex. The starting point is that all Coulomb interactions between charged groups are reduced at high ionic strength, because the Debye shielding length is lowered. The inverse screening length λ derives from the Poisson-Boltzmann equation, and is given by the well-known formula λ2 ¼

e2 X 2 z ni εkB T i i

ð5:61Þ

for charged species i of valence zi at concentration ni in an electrolyte with permittivity ε. The ionic strength IS is deﬁned as half the sum in this formula, so λ2 ¼ (2e2/εkBT)IS. For aqueous solutions, ε/εo ¼ 80, so λ ¼ (IS/0.092)1/2 nm1 when IS is in moles. Thus electrostatic screening of Coulombic repulsion between negatively-charged ﬁlaments is more effective at high ionic strength, so the lattice spacing should decrease. On the contrary, the lattice spacing of relaxed skinned ﬁbres expands as ionic strength is increased, from 0.11 M upwards for the frog and over a very wide range for rabbit psoas ﬁbres (Maughan and Godt 1980b; Millman 1991). The measurements were made by adding KCl, and Cl anions are known to bind to speciﬁc sites on the ﬁlaments. If the increased charge density on the ﬁlaments overrides the increase in electrostatic shielding, then the lattice will expand. In addition, the ﬁlaments may swell, and myoﬁlaments disintegrate at KCl concentrations of 0.6 M or more. However, the swelling is insufﬁcient to be the sole cause of lattice expansion. (e) The lattice at long sarcomere lengths. In skinned ﬁbres and intact muscle, the lattice spacing under relaxing conditions is a decreasing function of sarcomere length. In intact muscle, this is a result of constant ﬁbre volume, so 1/d2 is proportional to L as expected (Elliott et al. 1963, Matsubara and Elliott 1972). In skinned frog ﬁbres, the decrease is less severe, typically about 7% from 2.2 to 3.5 μm, and can be attributed to the progressive loss of Coulombic actin-myosin repulsion as ﬁlament overlap is reduced to zero (Millman 1991). However, the lattice spacing continues to decrease as sarcomere length is increased beyond zero overlap (Higuchi and Umazume 1986). Obviously, this behaviour cannot be attributed to F-actin-to-F-myosin interactions, so we need to consider the effects of titin ﬁlaments and the M-band and Z-line in stabilizing the lattice under these conditions. The work of Irving et al. (2011) shows that titin ﬁlaments are controlling the lattice spacing when the ﬁlaments are in partial overlap, and in this range the lattice expands when titin is removed by trypsin treatment (Higuchi 1987). However, the roles of M-band and Z-line elasticity in stabilizing the lattice beyond zero ﬁlament overlap are still not clear, and at this point one must resort to modelling.

5.8 The Stability of the Filament Lattice

5.8.1

223

Electrostatic Models of the Relaxed Filament Lattice

There is a long history of modelling studies which aim to predict the lattice spacing of relaxed muscle under various conditions (Millman and Nickel 1980; Millman 1991). There is general agreement that the ﬁlament lattice is stabilized by electrostatic interactions, namely Coulomb repulsion between negatively charged actin and myosin ﬁlaments, balanced by van der Waals attraction. Myosin-S1 also carries a net negative charge, primarily on the regulatory light chains, but there must also be charges of opposite sign to facilitate actin binding; the distribution of charge on myosin heads will be crucial in determining the correct set of interactions. This section formulates models with purely electrostatic interactions, so predictions of lattice spacing as a function of sarcomere length can reveal the extent to which they are responsible for the spacings observed. In the next section, these models are extended to include radial mechanical forces from titin and scaffolding structures. A simple model which avoids lengthy calculations based on atomic structures can be constructed by using charged cylinders aligned in parallel. If we know the charge densities σA, σM per unit length on cylindrical actin and myosin ﬁlaments, what is the potential energy of interaction between them in the 2:1 lattice? Formulae for the Coulombic interaction potential between charged cylinders in an electrolyte have been calculated by Brenner and McQuarrie (1973), but the result is in the form of an inﬁnite series. Fortunately, only the leading term is required when the ﬁlament radii exceed the Debye shielding length. When λRA >> 1, λRM >> 1, this interaction has the form vCAM ðdÞ ¼

σAσM

expðλðd RA RM ÞÞ

ð2π 3 RA RM Þ1=2 ελ

ðλdÞ1=2

ð5:62Þ

(Smith 2011), where λd >> 1 follows because d > RA + RM. The form of the van-derWaals interaction between polarizable cylinders is also available as a series expansion in RA/d and RM/d (Brenner and McQuarrie 1973), and the leading term is proportional to d5. Debye shielding does not operate here, so it is convenient to use the approximate formula of Moisescu (1973), namely vWAM ðdÞ ¼

3πA h 8

R2A R2M d i3=2 h i3=2 d2 ðRA þ RM Þ2 d2 ðRA RM Þ2

ð5:63Þ

where A, the Hamaker constant, is related to the dielectric polarizability α of an atom by A ¼ π 2ρ2α where ρ is the atomic number density. This formula can be derived by an approximate analytic integration of the point-to-point interaction –a/r6 over both cylinders (Smith 2011). It tracks the result with reasonable accuracy over the whole range of d except at large separations where a correction factor of 1.21 is required.

224

5 Transients, Stability and Oscillations

Fig. 5.28 Cartoons for an extended electrostatic model for charged ﬁlaments of radii RA, RM in relaxed muscle. (a) The 2:1 lattice, where charges on myosin-S1 heads on each myoﬁlament are simulated by charged cylinders at distance lc from their tether points. (b) When d –RA–RM > lS1, the head is free to rotate about the S1-S2 junction. (c) When d –RA–RM < lS1, the head tilts to interact with F-actin and the charged cylinder is displaced to distance d1 from the myoﬁlament axis. From Smith and Stephenson (2011), with permission of Elsevier Press

The electrostatic potential energy V(d ) per unit cell of the 2:1 lattice can now be assembled from the various cylindrical interactions. Thus vAM ¼ vCAM+vWAM is the potential energy per unit length of the actin-myosin interaction. There are also actinactin, myosin-myosin interactions and interactions between myosin heads and both ﬁlaments. Charges on myosin-S1 can be modelled by the following device (Smith and Stephenson 2011), in which the heads are replaced by three charged cylinders of charge density σS1, radius RS1 at distance d1 from their myoﬁlament, as in Fig. 5.28a. Assembling all contributions gives the energy per unit cell as V ðdÞ ¼ V AM ðd Þ þ V AA ðdÞ þ V MM ðd Þ þ V MS1 ðd 1 Þ þ V AS1 ðd 2 Þ

ð5:64Þ

where V AM ðdÞ ¼ 6ðLA þ LM LÞvAM ðdÞ

ð5:65aÞ

V AA ðdÞ ¼ 4LA vAA ðdÞ, pﬃﬃﬃ V MM ðdÞ ¼ 2LM vMM 3d ,

ð5:65bÞ ð5:65cÞ

5.8 The Stability of the Filament Lattice

225

V MS1 ðdÞ ¼ 3ðLA þ LM LÞvMS1 ðd 1 Þ,

ð5:65dÞ

V AS1 ðdÞ ¼ 3ðLA þ LM LÞvAS1 ðd 2 Þ:

ð5:65eÞ

in terms of ﬁlament lengths and half-sarcomere length L. Actin-actin interactions provide a residual interaction independent of sarcomere length, while myosinmyosin ﬁlament interactions are almost negligible because their spacing is √3d. Here d1, d2 are the S1-myosin and S1-actin spacings, such that d1 + d2 ¼ d. Note that VMS1 includes only interactions between the S1 cylinder and the F-myosin to which it is tethered. S1 will also interact with neighbouring F-myosins; this contribution was modelled by lumping the three S1 cylinders into a single cylinder of radius 0.5 (RM + d1) and charge density σM + 3σS1. At this point, it is instructive to introduce an extra degree of freedom into what is otherwise a very straightforward calculation. The pressure bump seen in osmotic compression measurements could arise from a model in which S1 heads tilt to maintain contact as the lattice is compressed beyond the spacing dc at which they ﬁrst touch F-actin. Thus the S1-myosin cylinder separation d1 is no longer ﬁxed but decreases as the S1 head tilts. If S1 has length lS1 and the S1 charge cylinder is separated from the S1-S2 junction by distance lc, then dc ¼ RM þ RA þ lS1

ð5:66Þ

and d 1 ¼ R M þ lc d 1 ¼ RM þ

ðd d c Þ

lc ðd RM RA Þ ðd < dc Þ lS1

ð5:67aÞ ð5:67bÞ

as in Fig. 5.28b, c. This has the effect of moving S1 charge closer to the myoﬁlaments when the lattice spacing is pushed below dc. Figure 5.29 shows the form of the interaction potential and the various contributions for a sarcomere at full overlap (L ¼ 1050 nm) at an ionic strength of 0.14 M, with other parameters listed in the ﬁrst row of Table 5.4. The Coulombic repulsive potentials increase sharply at small spacings, and even more so when d < dc, here set at 26.8 nm, whereas the attractive van der Waals potentials vary slowly over the whole range of spacings. In the ﬁgure, the equilibrium spacing determined by the minimum of the potential lies at 27.3 nm. Using Eq. 5.59 and the values listed in Table 5.4, this model provided good ﬁts to the measurements of osmotic pressure versus lattice spacing shown in Fig. 5.27. The ionic strengths listed in the table are close to those used experimentally. The osmotic data in Fig. 5.27, including the pressure bump in the second panel, were also ﬁtted by the exponential functions in Eqs. 5.56. The Coulombic repulsion terms in VAM(d) and VAA(d ) contains the factor exp(λd), and if these were the only interactions, then the coefﬁcient b in the exponent of the osmotic pressure equations

226

5 Transients, Stability and Oscillations

Fig. 5.29 The potential energy V per unit cell of the Coulomb and van der Waals interactions between ﬁlaments in the 2:1 lattice, as a function of myosin-actin spacing d. The heteroﬁlament interactions A-S1 and M-S1 show a steep increase for d < dc RA + RM + LS1 < 26.8 nm, when S1 heads contact F-actin and S1 charges are brought closer to their myoﬁlaments. The homoﬁlament interactions AA and MM have weaker Coulombic repulsion and their minima lie at much smaller spacings. These plots were constructed with parameter values from the ﬁrst row of Table 5.4. From Smith and Stephenson (2011), with permission of Elsevier Press Table 5.4 Parameter values of the electrostatic model used for measurements of lattice spacing by Umazume and Kasuga (1984) and Millman and Irving (1988) as a function of osmotic pressure UK84 MI88

IS (M) 0.14 0.11

L (nm) 1350 1300

σA (e/nm) 4.6 4.0

σM 6.5 8.0

σS1 9.2 10

lc (nm) 5.05 5.0

RS1 0.41 0.65

lS1 13.8 13.8

The model used ﬁxed values RM ¼ 8 nm and RA ¼ 5 nm for myosin and actin ﬁlament radii, and LM ¼ 750 nm, LA ¼ 1050 nm for ﬁlament lengths. The Hamaker constant A was set at 5.8 zJ

(Eqs. 5.56) should be equal to λ. In practice, this is not the case. The ionic strengths listed in the table correspond to λ ¼ 1.23 and 1.09 nm1, whereas the best-ﬁt values of b were 0.66 0.01 and 0.91 0.09 nm1. These values of b are telling us that S1-M and S1-A interactions are major contributors, because they are associated with the factors exp(λd1) and exp(λd2). Thus exp(bd) is a weighted average of these exponential factors as well as exp(λd ). The model also predicts lattice spacings which decrease with ionic strength and increase with pH. For these results and more discussion, see Smith and Stephenson (2011).

5.8.2

An Electromechanical Model for Long Sarcomeres

The lattice spacing of the relaxed ﬁbre is a decreasing function of sarcomere length. Measurements of Matsubara and Elliott 1972, Matsubara et al. 1984, Kawai et al. 1993) show a decrease of about 10% from full ﬁlament overlap to near zero overlap.

5.8 The Stability of the Filament Lattice

227

Fig. 5.30 Experimental determinations of lattice spacing versus sarcomere length in relaxed muscle (ﬁlled circles), and predictions of an electrostatic model (Smith and Stephenson 2011). (a) Results of Matsubara et al. (1984) for frog muscle in the overlap region, compared with the extended electrostatic model, which predicts a precipitous collapse just below 3.6 μm where heads touch F-actin. With permission of Elsevier Press. (b) Measurements of Higuchi (1987) show a slower decrease for sarcomeres beyond overlap, which is predicted only by including titin linkages (Eqs. 5.68, 5.69, and 5.70) and Z-lines with non-linear elasticity (Eq. 5.71). The dashed line shows that the equivalent harmonic potential is unable to stop the collapse generated by the loss of S1-myosin and S1-actin interactions (Smith 2014b). With permission of John Wiley and Sons Inc.

The extended electrostatic model, in which S1 charge moves radially as S1 heads contact actin, ﬁts their data well enough (Fig. 5.30a), but it also predicts an abrupt decrease in spacing at the point of zero overlap (L ¼ L3 ¼ 1.8 μm) where actinmyosin interactions cease and control is transferred primarily to actin-actin interactions. For longer sarcomeres, there is not much published data, but measurements of Higuchi and Umazume (1986), shown in Fig. 5.30b, show that this collapse is not present and the lattice spacing continues its slow decrease beyond zero overlap. To predict their observations, the electrostatic model must be supplemented by other radial forces, and the three contenders are titin ﬁlaments, M-bands and Z-lines (Irving et al. 2011). The argument can be put most clearly for titin ﬁlaments, since we know that they can exert axial forces comparable with isometric tension in the activated muscle. That titin tension can also have a radial component is a result of the way in which they are attached to the Z-line. However, titin ﬁlaments support tension when stretched, but compressed titins simply buckle because of the ﬂexible links between their structural units. Titin ﬁlaments in the relaxed muscle will stop the lattice from expanding but will not resist radial compression. Thus the collapsing lattice predicted by electrostatic models cannot be stabilized by titin or other connecting ﬁlaments, but only by the elastic properties of the Z-line and/or M-band. That said, it is still necessary to calculate the effect of titin ﬁlaments on the lattice spacing of long sarcomeres, where titin tension comes into play. To do this, we need to know where titin ﬁlaments emanating from each myoﬁlament are connected to the

228

5 Transients, Stability and Oscillations

Z-line lattice. Determining these connections has been a non-trivial problem, because the A-band and Z-line lattices have different unit cells. The unit cell of the Z-line is tetragonal (Pringle 1968), with the same area as the triangular subcell of the A-band in the 2:1 lattice. The tetrahedral angle has to be 81.8 so that the spatial relationship between A-band and Z-line unit cells is repeated over the cross-section of the ﬁbre (Knupp et al. 2002). The unit cell of the combined A-Z lattice is a supercell of four triangular sub-cells. Each subcell contains just one actin ﬁlament, and the electrostatic potential energy is repeated in all four subcells. However, the elastic energy of stretched titin ﬁlaments is a function of their radial displacements from A-band to Z-line. There are six titins per myoﬁlament, of which three are attached to vertices of the Z-line which lie within the area deﬁned by the rhombohedral A-band cell. One way of assigning these connections is shown in Fig. 5.31. The passive elastic energy of stretched titin ﬁlaments per A-band unit cell at halfsarcomere length L can now be expressed as a function of their radial displacement vectors ΔPi(d), i ¼ 1–3, namely V P ðd; LÞ ¼ 2

3 X

vP ðLPi Þ,

LPi ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðL LM Þ2 þ ΔPi2

ð5:68Þ

i¼1

where vP(LP) is the energy of one ﬁlament of length LP, measured from the end of the myoﬁlament. For example, the function n o vP ðLP Þ ¼ CP eξðLP LP∗ Þ 1 ξðLP LP∗ Þ

ð5:69Þ

gives

Fig. 5.31 A scheme for connecting titin ﬁlaments from their A-band myoﬁlaments to the Z-line lattice (Smith 2014b). The diagram shows one A-Z supercell which consists of two rhombohedral A-band unit cells. Open circles show the position of F-actins equidistant from the three neighbouring myosins. Superimposed are the front and back faces of the tetragonal Z-line lattice, with arrows indicating the displaced F-actin positions to vertices of the Z-line lattice. Black arrows show the displacements to the front face, and red arrows to the back face, as described by Knupp et al. (2002). There are different ways of linking titins to the Z-line (Liversage et al. 2001). The scheme shown here distributes the six titins from each myoﬁlament (shown as wavy lines) so that four of them go to vertices on the front face of the Z-line and two to the back face (Smith 2014b). Titins drawn in black come from the half-sarcomere in front of the Z-line and those in red come from the half-sarcomere behind. The diagram shows only those titins connecting to Z-line vertices within the one supercell. With permission of John Wiley and Sons Inc.

5.8 The Stability of the Filament Lattice

T P ðLP Þ

229

n o dvP ¼ ξC P eξðLP LP∗ Þ 1 dLP

ð5:70Þ

for the tension, where LP* is the slack length of the ﬁlament. Equation 5.68 requires only the magnitudes of the radial displacements, and for the connection scheme in Fig. 5.31, ΔP1 ¼ 0.5728d, ΔP2 ¼ 1.2055d, ΔP3 ¼ 1.3520d (Smith 2014b). That titin tension is acting at large sarcomere lengths to compress the lattice is clear from measurements of Higuchi (1987), who showed that the lattice expands when titin ﬁlaments are removed with trypsin. Then the lattice spacing remains substantially constant as sarcomeres are stretched beyond the zero-overlap length L3. When the elastic energy of stretched titins is included in the electrostatic model, the lattice still collapses at L ¼ L3, but only slightly, and the lattice spacing decreases beyond zero overlap while the lattice spacing in the overlap range is reduced. In this way, we may conclude that Z-line and/or M-band elasticity is essential to prevent the lattice from collapsing when the relaxed ﬁbre is stretched beyond zero overlap. To ﬁt the measurements of Higuchi and Umazume, one has to postulate non-linear elastic properties for these scaffolding structures when stretched or compressed. An elastic Z-line energy function which does the job is κZ V Z ðd Þ ¼ ηZ

eηZ jdd∗ j jd d ∗ j ηZ

ð5:71Þ

which is symmetrical about the resting spacing d*. For computational convenience it is expressed inpterms ﬃﬃﬃﬃﬃ of the actin-myosin spacing d rather than dM or the vertex 21=4 d ¼ 1:1456d of the Z-line lattice. With d* ¼ 27.5 nm, spacing dZ κZ ¼ 0.54 pN/nm and ηZ ¼ 0.693 nm1, there is a satisfactory ﬁt to Higuchi’s data (Fig. 5.30b), except for the last point at 2L ¼ 5.0 μm. Only a very small amount of Z-line strain energy is required to overcome the van-der-Waals attraction between actin ﬁlaments which causes the lattice to collapse in the ﬁrst place. The collapse at 3.5 μm is still there, but its magnitude is much smaller and appears to be compatible with Higuchi’s measurements. Finally, we can ask: is it the Z-line or the M-band (or both) which stabilizes the lattice with long sarcomeres? Fortunately, these scaffolding structures are linked by titin, and there is a very cogent reason why their lattice spacings must remain locked together. The relative position of a Z-line unit cell and its M-band counterpart must remain ﬁxed over the cross-section of the ﬁbre, and this can only happen if the ratio of their lattice spacings remains constant as the scaffoldings expand or contract. Otherwise, the radial displacements required of titin, and F-actin also, will change proportionally on moving over the cross-section, and actin ﬁlaments will not be able to interdigitate with the same set of myosin ﬁlaments when the ﬁbre is returned to overlap, with concomitant structural mayhem in the A-band lattice. However, if Z-line stiffness is stronger than M-band stiffness, then the sarcomeres will take on a barrel-like appearance, which is sometimes observed (Bergman 1983). In this regard, it is interesting to calculate the effective radial stiffness of titin links as a function of sarcomere length and compare it with the value assigned to the Z-line.

230

5 Transients, Stability and Oscillations

This calculation might explain why the ratio of the Z-line and A-band spacings can only differ by 2–3%, and then under conditions of extreme shrinkage or swelling (Irving and Millman 1992).

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Chapter 6

Myosin Motors

I came here confused about actin and myosin. Now I am still confused, but at a higher level. Sir Andrew Huxley F.R.S., on summing up the symposium on Frontiers in Molecular Motors, Osaka, Japan, hosted by Toshio Yanagida (Cyranoski 2000).

6.1

Single-Myosin Experiments with Optical Trapping

A new dawn in the history of muscle research arrived with the news that a single molecule of myosin-S1 could be observed making repeated working strokes with F-actin. The most popular conﬁguration has been the “three-bead experiment” (Finer et al. 1994; Molloy et al. 1995; reviewed by Tyska and Warshaw 2002), where an actin ﬁlament is suspended between two micron-sized polystyrene beads trapped in the foci of convergent laser beams (Fig. 6.1). Extreme mechanical stability, preferably enhanced by position feedback to piezoelectric drives, is required. When this assembly is lowered onto a single molecule of myosin-S1 on a ﬁxed bead, the actinbeads assembly makes intermittent axial displacements out of its equilibrium position. As expected, the average lifetime of these displacements increased as the concentration of ATP was reduced. Figure 6.2 shows an example; the displacements of the beads show simultaneous low-noise periods with different levels, where the myosin has bound to F-actin. How can these bound levels be interpreted as evidence for a working stroke? Such displacement records show that a single myosin molecule in this environment behaves very differently from the collective behaviour of myosins in muscle. Firstly, the interaction with F-actin is a stochastic process. Binding to actin is a random event controlled by the probabilistic interpretation of kinetics; if the ﬁrstorder binding rate is kA, then kAΔt is the probability that a detached head binds during time Δt. Secondly, the beads are weakly trapped and Brownian forces drive the actin-beads ‘dumbbell’ out of the trap centres, typically with an RMS displacement of 15 nm. Thus many actin sites are presented in turn to the tethered myosin. © Springer Nature Switzerland AG 2018 D. A. Smith, The Sliding-Filament Theory of Muscle Contraction, https://doi.org/10.1007/978-3-030-03526-6_6

237

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Fig. 6.1 The three-bead experiment: An actin ﬁlament is linked to two micron-sized polystyrene beads, which are trapped in the foci of two laser beams. The actin-beads ‘dumbbell’ is lowered onto a single molecule of myosin-S1 attached to a ﬁxed bead. Intermittent displacements of the dumbbell are produced by actin-myosin attachments which are biased in one direction when the myosin makes its working stroke. The microscope stage is stabilized in three directions by piezoelectric transducers. From Steffen et al. (2001), with permission, copyright of the National Academy of Sciences, USA

Thirdly, does the dumbbell rotate about its axis through the trap centres? Finally, there is a signiﬁcant signal-to-noise problem in identifying working-stroke events in the presence of Brownian motion, and algorithms for detecting these steps in the presence of noise are required. Nevertheless, the ﬁrst observations allow an appealingly simple interpretation. A binned histogram of the displacements in low-noise periods approximates a shifted Gaussian distribution whose standard deviation σ describes the Brownian motion of the free dumbbell. If we assume that available binding sites are uniformly distributed along the actin ﬁlament, then the shift in the distribution can be equated with the myosin working stroke h. The dumbbell displacement induced by myosin binding to a site at distance X on the resting dumbbell will be u ¼ X + h,p with a frequency ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ proportional to exp(X2/2σ 2) ¼ exp ((u h)2/2σ 2) where σ ﬃ kB T=κ t and κt is the stiffness of both traps (Fig. 6.3). Data from several laboratories (Veigel et al. 1998; Smith et al. 2001; Guilford et al. 1997; Mehta et al. 1997) reproduced the result of Molloy et al. (1995) that the estimated working stroke was 5–6 nm, not 8–10 nm as expected. What is going on here? Are there, as some skeptics have suggested, extra compliances in the system which could deﬂate the displacement in the distribution of bound levels? There are two strands to the resolution of this anomaly. To get to the answer, we have to look more closely at compliant elements in the system, whether the distribution of binding sites is in fact uniform, and how actin-binding kinetics is affected by the Brownian motions of the dumbbell. It is also helpful to understand the

6.1 Single-Myosin Experiments with Optical Trapping

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Fig. 6.2 Examples of displacement records of the actin-beads dumbbell in the three-bead experiment shown in Fig. 6.1. (a and b) Displacements of the left- and right- hand beads, as a function of sampling number. (c–e) Records for running variances constructed from the above traces; autovariances for the left- and right-hand beads (c and d) and the covariance (e). The sampling frequency was 10 KHz, and running variances used a window of 100 data points. (Unpublished data, reproduced with permission of Dr. John Sleep)

different methods for detecting bound levels in a noisy record. A little dose of theory is required! Firstly, actin-bead linkages are not rigid, so bead displacements in the traps are not the same as the actin ﬁlament displacement (Fig. 6.4). Let us assume that the myosin head has bound to a site at distance X on the resting dumbbell, and made its working stroke. If the left-and right-hand beads are displaced by u1,u2 and F-actin by u, then the potential energy of the dumbbell is 1 1 1 1 V ðu1 ; u2 ; uÞ ¼ κt u21 þ u22 þ κ1 ðu1 uÞ2 þ κ 2 ðu2 uÞ2 þ κðX þ h uÞ2 4 2 2 2 ð6:1Þ where κt is the stiffness of both traps, κ1, κ2 are the elastic constants of the links and κ, as usual, is axial myosin stiffness. As in Fig. 6.4, u1, u2 and u are left-directed displacements but X is measured to the right. This potential energy is a harmonic

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Fig. 6.3 A typical histogram for the frequency of bound levels in the displacements records of early double-beam trap experiments, which obeyed a shifted Gaussian distribution. The width of the distribution is accounted for by Brownian forces on the beads of the free dumbbell, which introduces many actin sites to the ﬁxed myosin head. The shift in mean value is widely interpreted as a measure of the working stroke

Fig. 6.4 Schematic of the three-bead experiment, showing the actin-beads dumbbell with beads displaced by u1,u2 from their trap centres, F-actin displaced by u, and a ﬁxed myosin selecting an actin site at distance x from the resting position of the dumbbell. Here κ t/2 is the effective stiffness of each bead-trap restoring force, and the actin-bead linkages have stiffnesses κ 1 and κ 2 respectively

function, so its equilibrium displacements are also the average displacements in the presence of Brownian motion. Setting all partial derivatives to zero gives κ ð X þ hÞ uðX Þ ¼ κ þ ~κ 1 þ ~κ 2 u1 ðX Þ ¼

~κ 1 ¼

κ1 uðX Þ, κ 1 þ 12κt

1 2κ 1 κ t κ 1 þ 12κt

u2 ðX Þ ¼

; ~κ 2 ¼

1 2κ 2 κ t κ 2 þ 12κ t

κ2 uðX Þ κ 2 þ 12κ t

! ð6:2aÞ ð6:2bÞ

Equation 6.2a shows how the dumbbell displacement is slightly reduced by the beadtrap linkages acting in parallel with the bound myosin. Equation 6.2b shows why bead displacements may be smaller than actin displacement because of compliant actin-bead linkages. The actin ﬁlament can be regarded as inextensible, but it can be

6.1 Single-Myosin Experiments with Optical Trapping

241

tensioned in situ to stop it from going slack under Brownian compression. You might object that Eq. 6.1 does not contain the pre-tensioning potential, but the displacements in Eqs. 6.2 are exactly the same provided that the resting position of the actin ﬁlament has not changed. Why? Because the potential is harmonic. At this point, a diversion: the potential energy function also determines the amplitude of Brownian noise, which is much reduced when the myosin is bound. This property was exploited by Molloy et al. (1995) to facilitate the detection of bound levels. Detecting bound periods directly from the raw time series of the bead displacements is difﬁcult, because of the multiplicity of levels present over all displacements of the free dumbbell. It is much easier to detect a simple binary signal in the presence of noise, and a running variance of the original data is a time series of this kind. A certain number of data points, say 50, is required to construct a useful variance-time record, so there is a trade-off between ﬁdelity and time resolution, ultimately limited by the sampling rate of data collection. There is no point in increasing the sampling rate in the variance record beyond the inverse correlation time of variance noise. The statistical properties of running variance signals are also determined by the potential energy of Eq. 6.1. In the two-bead experiment, there are three variance signals, the autovariances v11, v22 and the covariance v12, where vij ðt k Þ ¼

n 1 X ðui ðt kþm Þ ui ðt k ÞÞ u j ðt kþm Þ uj ðt k Þ 2n m¼n

ð6:3aÞ

n 1 X ui ðt kþm Þ 2n m¼n

ð6:3bÞ

ui ðt k Þ ¼

over 2n + 1 sampled time points, as in Fig. 6.2c–e. What values are expected in an equilibrium ensemble? They may be calculated from a Boltzmann distribution with the potential of Eq. 6.1: RRR vij ¼

ðui ui Þðuj uj ÞexpðβVðu1 , u2 , uÞÞdu1 du2 du RRR expðβVðu1 , u2 , uÞÞdu1 du2 du

ði, j ¼ 1, 2Þ ð6:4Þ

which gives v11 ¼ κt 2

v22 ¼ κt 2

v12 ¼ κt 2

κ þ κ 1 þ ~κ 2 kB T, þ κ 1 κ þ ~κ 1 þ ~κ 2

ð6:5aÞ

κ þ ~κ 1 þ κ2 kB T, þ κ 2 κ þ ~κ 1 þ ~κ 2

ð6:5bÞ

þ κ1

κ

t

2

κ1 κ2 kB T: þ κ2 κ þ ~κ 1 þ ~κ 2

ð6:5cÞ

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By setting κ ¼ 0, these equations also give the expected running variances in the free periods. It is a moot point whether u should be treated as a degree of freedom, because the same results can be achieved by inserting the equilibrium value of u for each pair of values of u1,u2. This method looks simpler, but it has the effect of requiring some heavy algebraic manipulation sooner rather than later. For Eqs. 6.4, some typical numbers are κt ¼ 0.08 pN/nm, κ ¼ 2.0 pN/nm, κ1 ¼ κ2 ¼ 4.0 pN/nm (Smith et al. 2001), giving v11 ¼ v22 ¼ 3.95, v12 ¼ 1.89 nm2 with myosin bound and v11 ¼ 50.5, v12 ¼ 49.5 nm2 in the free periods. With these values, bound periods reduce the autovariances by a factor of 12.8 and the covariance by a factor of 26.2, so there can be a deﬁnite advantage in working with the covariance signal. The covariance record is particularly useful if both actin-bead links are weak (Mehta et al. 1997). A covariance record is shown in Fig. 6.2e. Having calculated what to expect from variance records, the experimenter still has to use a reliable method of detecting low-noise periods, so that each ‘bound level’ (the mean displacement in a bound period) can be extracted from the original displacement-time records, preferably from the bead with the stronger link. Only then can one build histograms for the distribution of bound levels.

6.1.1

Detecting Events in the Presence of Noise

There are various statistical methods for detecting a binary signal in the presence of noise. In view of the variety of applications (single-channel ion recordings, voice recognition, earthquake detection, to name just a few), there is an extensive literature (Jaswinski 1970; Bozic 1979; Colquhoun 1998; Stranneby and Walker 2004). Here we focus on a very simple example which covers the running-variance construct of the previous section. Consider a noisy time series x(t) in which a signal intermittently switches on and off between two levels x ¼ 0 and x ¼ s. We wish to estimate all the switching times and the mean value of the time series in every ‘on’ period and every ‘off’ period. Averaging over all ‘on’ periods gives an estimate for s and its standard error. The simplest method is to use the student-t statistic and impose a threshold (Weatherburn 1968). Switching times can be detected by constructing means and variances in a running window of 2n + 1 points, with separate values for the front and rear halves of the window: x ðt k Þ ¼ so

n 1X xðt km Þ, n m¼1

σ ðt k Þ2 ¼

n 1X fxðt km Þ x ðt k Þg2 , n m¼1

ð6:6Þ

6.1 Single-Myosin Experiments with Optical Trapping

tðt k Þ ¼

j xþ ðt k Þ x ðt k Þ j pﬃﬃﬃ : ðσ þ ðt k Þ þ σ ðt k ÞÞ= n

243

ð6:7Þ

tests the hypothesis that the two means belong to different populations. There is a threshold value tc(n) for t(tk), above which the probability that the two means belong to different populations exceeds 95%. tc(n) decreases slowly with n; for example, tc(20) ¼ 2.09 and tc(50) ¼ 2.01. Thus t(tk) should rise to a maximum when the running window is centred on a switching event, and the threshold determines whether that event is signiﬁcant. Once the switching times are determined, mean levels and their standard deviations can be calculated within each ‘on’ period and ‘off’ period, leading to estimates for the true levels x ¼ 0 and x ¼ s. There are variations on the student-t test (Page 1954; Smith 1998a), and Page’s test is discussed in the review by Knight et al. (2001). The student-t test has limited success in extracting signals from a high-noise environment. Better results can be obtained if the algorithm is given more information about the nature of the signal. In the context of single-motor trap recordings, the amplitude of the signal is ﬁxed (but unknown), while the switching times are generated by a stochastic process, namely actin binding and detachment of the motor protein. Although this means that there is no causal relationship between the times of these events, they can be characterized in terms of transition probabilities which connect signal values and states over one time step. These quantities can be fed into a chain of probabilities for a given sequence of states. Maximising the joint probability of this sequence relative to all possible state sequences constitutes the maximum likelihood method, otherwise known as the Hidden Markov method (Rabiner 1989), for the detection of a stochastic signal in the presence of noise. A formal exposition is given in Appendix J. The Hidden Markov method is ideally suited to detecting events in a runningvariance record, because only two states are involved, but the resolution of the method is limited. For raw displacement data sampled at intervals of Δt, the time resolution of events detected from a running-variance window of W points is WΔt/2 (Smith et al. 2001), for example 5 ms if Δt ¼ 0.1 ms and W ¼ 100. It would be preferable to detect myosin transitions directly from the displacement record. Athough working strokes occur too quickly to detect pre-stroke states, many actin sites can be chosen, and the chain of probabilities is more difﬁcult to unravel. Many elaborations of these methods can be found in the literature. It may be advantageous to ﬁlter the time series before introducing it to an event detector, and there is a wide choice of algorithms (Jaswinski 1970; Press et al. 1992; Block and Svoboda 1995; Carter et al. 2008). For example, a linear ﬁlter adopts a weighted average of all points in the running window. However, median averaging, with as many points below the mean as above, is less sensitive to outliers, and so is the forward-backward switching algorithm of Chung and Kennedy (1991). There are also methods based on spectral analysis of the time series, and in this context it is worth remembering that the Fourier transform of a signal deﬁned at discrete time intervals Δt is limited to frequencies below the aliasing frequency ωc ¼ π/Δt. Apart

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from ﬁnite sampling, there are intrinsic limits to the frequency spectrum of the time series which impose an upper limit on the rate at which data should be collected; for the three-bead experiment, the frequency spectrum of bead displacements is limited by the correlation time of Brownian dumbbell noise. In the time domain it may be instructive to construct a running auto-correlation function, although this method is generally overwhelmed when applied to raw displacement data from the three-bead experiment. To assess the quality of such data before event detection, it is helpful to plot the running variance against the running mean (Guilford et al. 1997), or as a three-dimensional histogram (Patlak 1993). One necessary modiﬁcation, which occurs in many applications, is to correct the time series for a drifting base line. In the case of trap displacement recordings, the base-line may drift from slow movements of the microscope stage relative to the laser foci, or in the associated electronics. It is desirable to use the same algorithm to detect baseline drift and noise, and this can be done by combining the HiddenMarkov method with Kalman ﬁltering (Bruno et al. 2013; Milescu et al. 2006).

6.1.2

Observing Target Zones with Optical Trapping

In the earliest single-myosin trap experiments, there were hints that the distribution of bound levels was not exactly a shifted Gaussian; some of the expected Gaussian distribution was missing (Molloy et al. 1995). This was not evident in some later work (e.g. Veigel et al. 1998). where more unitary events were detected. However, Guilford et al. (1997) observed two groups of bound levels of opposite sign, separated by 11 nm. These apparently conﬂicting results can be explained if the actin-beads dumbbell was not rotating in the traps, in which case not all actin sites would be available to the myosin. Only those sites of near-correct azimuthal alignment, in other words target zones, would be presented to myosin, so the distribution of bound levels would look like a Gaussian distribution with large sections missing. The bits of the distribution that were present could be shifted by displacing both traps along the ﬁlament axis, which suggests that a complete Gaussian distribution could have been generated by a slow drift in the x-position of the microscope stage. In this way, target zones at all displacements from myosin might have been generated in the course of data collection. Such speculations were tested with an optical-trap assembly where the position of the microscope stage was feedback-stabilised in three dimensions, to an accuracy of 0.7 nm in the x, y directions and 5 nm in the z-direction (Steffen et al. 2001). With this system, the bound levels fell into one or two groups with a width of 10 nm. The spacing between the groups was 30 nm, not too different from the 36 nm halfperiod of the actin double helix. These observations constitute direct evidence for target zones of three azimuthally-accessible sites on the actin double-helix, linked to beads which do not rotate freely in the optical traps. This hypothesis was conﬁrmed by taking multiple data sets with different x-positions of the traps; the distributions of bound levels then shifted as expected within the Gaussian envelope (Fig. 6.5).

6.1 Single-Myosin Experiments with Optical Trapping

245

Fig. 6.5 The distribution of ‘bound levels’ (averaged displacements in each bound period) is a function of the axial position of the traps, which was varied as shown in graphs (a–c). The character of the distributions is bimodal, because Brownian displacements of the actin-beads dumbbell access target zones on adjacent 36 nm half-periods of the actin double-helix. The relative amplitudes of the two Gaussian modes vary with the x-position of the traps, according to which half-period is more frequently accessed. From Steffen et al. (2001), with permission, copyright of the National Academy of Sciences, USA

A quantitative description of these results is possible by ﬁtting the distribution of bound levels U to a Gaussian function multiplied by a target function A(U ), which expresses the aziumthal accessibility of actin sites on a torsionally compliant actin ﬁlament-beads assembly. Thus the frequency of bound levels has the form ð U hÞ 2 J ðU; X Þ / AðU X Þexp 2σ 2

! ð6:8Þ

where X denotes the axial position of the traps and σ 2 ﬃ kBT/κt with strong links. The target function A(U ) is periodic, with a period equal to the actin half-repeat distance b. It can be identiﬁed with the probability exp(κ effϕ(U )2/2kBT), where ϕ(U ) ¼ πU/ b modulo 2π, that Brownian torques will activate a hindered rotation of the dumbbell to correct the angular mismatch. Here κeff is a torsional stiffness for the combination of the trapped beads, F-actin and the bound myosin. Hence

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no. binding events J(U,X)

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Fig. 6.6 The bimodal distribution of bound levels in Fig. 6.5 can be described by a Gaussian distribution of Brownian displacements of the free dumbbell modulated by a periodic ‘target’ function which expresses the azimuthal availability of actin sites to the ﬁxed myosin. This modulation shows that the dumbbell is not rotating about its axis in the optical traps, or makes hindered rotations about a preferred orientation. In either case, this data shows that a tethered myosin-S1 cannot reach all sites on an actin ﬁlament which is not freely rotating. (Steffen et al. (2001), with permission, copyright of the National Academy of Sciences, USA)

~ 2 =2 , AðU Þ ¼ exp αU

~j jU

< b=2

ð6:9Þ

~ ¼ U nb for the integer n which satisﬁes the where α ¼ π 2κ eff/2b2kBT and U inequality. A(U ) is well localized within one period, and is essentially the function used in the original paper. Figure 6.6 shows how this target function ﬁts under its Gaussian envelope, with a width sufﬁcient to accommodate a target zone of three actin sites on the same strand. An apparent working stroke can be calculated as the mean value of U from the normalized frequency function of Eq. 6.8, but this led to values which varied all the way from 0 nm to 10 nm, depending on the position of the traps (Steffen et al. 2001). The real working stroke should be determined by the Gaussian envelope of J(U,X), and the best ﬁt to the data yielded a result close to the X-average of the apparent working strokes, namely h ¼ 5.5 nm. In conclusion, this highly-stabilized trap experiment succeeded in demonstrating how myosin heads can be used to map target zones on F-actin, and even the 2.75 nm stagger between its two helical strands, but it did not close the gap between opticaltrap determinations of the working stroke and the 10–11 nm stroke deduced from crystal structures. A subsequent trap experiment with higher time resolution by Capitanio et al. (2006) uncovered evidence for two substrokes of 4.3 0.9 nm

6.2 Actomyosin Kinetics in the Optical Trap

247

and 1.15 0.15 nm, giving a net working stroke of 5.5 nm. Thus the optical-trap data on myosin II tended to give a working stroke of slightly more than half of what is expected on structural grounds. The solution to this puzzle was ﬁnally achieved by a piece of theory, presented in the next section.

6.2

Actomyosin Kinetics in the Optical Trap

The problems posed by single-myosin experiments with an optically-trapped actin ﬁlament opened up a new kind of mechanokinetics. As with myosins in muscle, there are two theoretical approaches. Mock displacement recordings can be generated by Monte-Carlo simulation and used to detect the efﬁciency of event detectors, whether they be actin attachment events or the start of a working stroke within a bound period. Apart from the fact that all such events can be marked, Monte-Carlo simulations have their limitations; they give no guiding principles or general formulae for the distribution of bound levels or their lifetimes. The more powerful approach (when computationally feasible) describes dynamical interactions in terms of the probabilities of actomyosin states and the probability distribution of dumbbell displacements in the three-bead experiment. This can be achieved by marrying Smoluchowski’s equation with the rate equations of actomyosin kinetics (Smith 1998b). It is the strain-dependence of these transitions which couples the two processes tightly together.

6.2.1

Monte-Carlo Simulations

Brownian motions of an optically-trapped actin-beads dumbbell can be simulated by solving a Langevin equation (Chandrasekhar 1943), which is Newtons’s equation of motion including viscous drag and Brownian forces. The following formulation holds for any particle subject to an elastic restoring force, when viscous drag is large enough for inertial forces to be neglected. In hydrodynamics this is the limit of low Reynolds number, which applies to many cellular and intra-cellular motions as well as the optically-trapped dumbbell that concerns us here. It is simplest to start with just one available actin site. Let the dumbbell be displaced (to the left in Fig. 6.4) by u(t) at time t from its resting position, where X is the right-directed axial displacement of the resting actin site from the myosin. Then the equation of motion in the presence of a random Brownian force FB(t) is μ

du þ κt uðt Þ ¼ κ ðX uðt ÞÞnðt Þ þ F B ðt Þ dt

ð6:10Þ

where n(t) ¼ 0 for the free dumbbell and 1 with myosin bound. The damping coefﬁcient can be estimated from Stokes’ law μ ¼ 6πηR for each bead of radius R,

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where η is the viscosity of the solution, plus an estimate for the actin ﬁlament approximated by a long ellipsoid (Happel and Brenner 1983), which gives a much smaller value. The response to any driving force will be delayed over a time which is the inverse of λ0 or λ1, where λ0 ¼

κt , μ

λ1 ¼

κt þ κ μ

ð6:11Þ

for free and bound myosin respectively. These quantities are the so-called ‘corner’ frequencies for the frequency spectrum of Brownian noise, which is derived in the accompanying text box. With η ¼ 0.001 N.s/m2 and R ¼ 0.5 μm, μ ¼ 2.0 108 N. s/m. If κ t ¼ 0.08 pN/nm and κ ¼ 2.0 pN/nm, then λ0 ¼ 4000 s1 and λ1 ¼ 1.04 105 s1. Relative to myosin kinetics, these are high frequencies; λ0 is almost three orders of magnitude above actin binding rates in the trap, and λ1 is higher than the stroke rate. The problem with Langevin’s equation is that it cannot be causally integrated in the presence of random Brownian force, modelled as white noise. However, integration over a time interval Δt 18.4/κ th, around 30 nm with κt ¼ 0.08 pN/nm and h ¼ 8 nm. The exclusion zone is admittedly not symmetric in X, but is so wide that in practice it seems safe to assume that all binding events will be followed by a working stroke. This argument needs to be made more precise by including strain-dependent myosin binding. Nevertheless, it suggests that the true reason for the deﬂated displacements of bound levels is the presence of many actin sites available for binding. To understand how this might happen, we also need to employ the whole straindependent contraction cycle under steady-state rather than equilibrium conditions. A start can be made with Monte-Carlo simulations and the Langevin equation. Figure 6.7 shows a sample of Monte-Carlo simulations with strong actin-bead links, many accessible actin sites spaced by 5.5 nm and a three-state contraction cycle with pre-stroke and post-stroke states (A and R). To guarantee stable solutions with larger time steps, it may be helpful to use implicit integration, by replacing u(tk) by u(tk+1) on the right-hand side of Eq. 6.12. The coding of this model starts with detached myosin and a DO-loop over the actin sites. Eventually one is chosen for binding, and the X-displacement of that site must be carried through to all subsequent transitions until the head detaches and another site is selected. With κ ¼ 2.0 pN/nm, λ1 ¼ 105 s1 so, with time steps below 10 μs, very long runs are required to build

Fig. 6.7 (a) Monte-Carlo simulations of bead displacements in the three-bead experiment, from step integration of the Langevin equation with Δt ¼ 106 s, sampled every 0.1 ms. (b) The derived running-variance record with a window of 100 points, sampled every 5 ms. Simulations were made with κ t ¼ 0.1 pN/nm, κ ¼ 1.0 pN/nm and h ¼ 5.5 nm. The kinetics of the simulation determined apparent binding and detachment rates of 4.6 s1 and 12.7 s1 respectively. From Smith et al. (2001), with permission of Elsevier Press

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251

up 10–100 s worth of displacements. With attachment rates deduced from experimental data, for example f ¼ 5 s1 and g ¼ 15 s1 at 5 μM ATP, runs of this length are required for good event statistics. Because the times of all reaction events are known, the results can be used to test event detectors, such as the Hidden-Markov method operating on a derived running-variance record. Event times from simulations can also be used to build up the distribution of displacements in each bound period, separating the A and R states in a way which cannot be done experimentally. In this way the fraction of bound heads that make a stroke can be calculated. In view of these computational hurdles, it is fortunate that there is a probabilistic formalism which can make general predictions about the frequency of attachment events, the occurrence of working strokes on different sites and the lifetimes of bound periods. This is the subject of the rest of this section.

6.2.2

The Generalized Smoluchowski Equations

For simplicity, consider again the case of an optically-trapped actin-beads dumbbell with stiff links and one actin site available to the myosin head. For this system, let Pi(u, t)du be the joint probability of myosin in state i and dumbbell displacements in the range (u, u+du). For a given myosin state, Pi(u, t) satisﬁes a Fokker-Planck equation for the motion of a Brownian dumbbell in the potential well of the traps, plus myosin when bound. In the limit of high damping, this equation simpliﬁes to Smoluchowski’s Eq. (4.3). With transitions between myosin states, reaction terms must be added, giving the generalized equation ∂Pi ðu; t Þ ∂ 1 ∂V i ðu; X Þ ∂Pi ðu; t Þ ¼ Pi ðu; t Þ þ D ∂t ∂u μ ∂u ∂u NS X Aij ðX uÞP j ðu; t Þ þ

ð6:20Þ

j¼1

where A(X) is the reaction matrix of Eq. 4.31 and the diffusion constant satisﬁes Einstein’s relation D ¼ kBT/μ (Eq. 4.4). The potentials are different for detached, pre-stroke and post-stroke states (D,A,R), namely 1 V i ðu; X Þ ¼ κt u2 ðD-statesÞ 2 o 1n 2 κ t u þ κ ð X uÞ 2 V i ðu; X Þ ¼ ðA-statesÞ 2 o 1n 2 κt u þ κðX þ h uÞ2 ðR-statesÞ V i ðu; X Þ ¼ 2 with similar formulae for multi-stroke models.

ð6:21Þ ð6:22Þ ð6:23Þ

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These equations become much simpler in the limit where Brownian motions are much faster than crossbridge kinetics. This is true when the head is detached and also when bound. Let kA(X) be the actin-binding rate for M.ADP.Pi, kS(X) the stroke rate and kD(X) the post-stroke detachment rate. Then the required conditions are λ0 >> k A þ kA

ðdetachedÞ

ð6:24aÞ

ðboundÞ

ð6:24bÞ

λ1 >> kS þ kS , k D

omitting the X-dependencies. All these conditions are easily satisﬁed in trap experiments. Then we can insert Boltzmann distributions into the joint probabilities: Pi ðu; t Þ pi ðt Þϕi ðuÞ,

ϕi ðuÞ / expðβV i ðu; X ÞÞ

ð6:25Þ

so rﬃﬃﬃﬃﬃﬃ βκt 1 ϕ i ð uÞ ¼ ði D Þ exp βκt u2 2 2π rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ( ) β ðk þ κ t Þ 1 κX 2 exp βðκ þ κt Þ u ϕ i ð uÞ ¼ ði A Þ 2π 2 κ þ κt rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ( ) β ðk þ κ t Þ 1 κ ð X þ hÞ 2 exp βðκ þ κ t Þ u ϕ i ð uÞ ¼ ði RÞ 2π 2 κ þ κt

ð6:26aÞ ð6:26bÞ

ð6:26cÞ

Rate equations for the state probabilities pi(t) follow by substituting in Eq. 6.20 and integrating over all u. The drift and diffusion terms disappear at inﬁnity, leaving the equations NS ∂pi ðt Þ X ¼ Aij ðX Þp j ðt Þ, ∂t j¼1

Aij ðX Þ ¼

Z

1

-1

Aij ðX uÞϕ j ðuÞdu

ð6:27Þ

with rate constants averaged over Brownian dumbbell motions in the initial state j of kji(X) ¼ Aij(X). The strain-dependence of rate constants constructed for the muscle ﬁbre is partially washed out in the single-myosin trap experiment, and this effect is most pronounced for detached states where the free dumbbell makes large Brownian excursions. Here are some examples. For the ‘Brownian post’ formula for actin binding (Eq. 4.7), and the highest-energy-barrier formula for the stroke rate and its reversal (Eqs. 4.13), Brownian-averaged rate and equilibrium constants are

6.2 Actomyosin Kinetics in the Optical Trap

kA ðX Þ ¼

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ κt kA exp βκS X 2 =2 κ þ κt

1 kS ðX Þ ¼ k S ½erfcðx1 Þ þ expfβκ S hðX þ h=2Þgerfcðx2 Þ 2 k~S ðX Þ ¼ K S expfβκS hðX þ h=2Þg: KS ðX Þ k~S ðX Þ

253

ð6:28aÞ ð6:28bÞ ð6:28cÞ

where rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ β ðκ þ κ t Þ h κt X þ x1 ¼ , 2 2 κ þ κt rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ β ð κ þ κ t Þ h κ t ð X þ hÞ x2 ¼ 2 2 κ þ κt

ð6:29Þ

and κS ¼ κκt/(κ+κt) is the stiffness of myosin and traps in series. (Remember that myosin-actin and actin-trap connections act in parallel for virtual displacements of the dumbbell but act in series for virtual displacements of the traps). Except for Eq. 6.28b, these formulae have the same X-dependence as their un-averaged counterparts with κ replaced by κS. These functions are plotted in Fig. 6.8. How does this probabilistic formalism relate to reaction events with a single myosin? The state probabilities refer to average occupancies over many periods in a single displacement-time record or, if the ergodic hypothesis applies, in an ensemble

Fig. 6.8 Strain-dependent rate constants for actin-myosin transitions in a weakly-trapped actin ﬁlament, averaged over axial Brownian motions generated by its attachment to optically-trapped beads (Fig. 6.1). Rates kA(X) of attachment, kS(X) for the working stroke and kS(X) for its reversal, were generated from Eqs. 6.26, using κ ¼ 2.0 pN/nm, κ t ¼ 0.08 pN/nm, kA ¼ 25 s1, KA ¼ 0.5, kS ¼ 2000 s1, KS ¼ 100. The rate constant kD(X) for ADP release was generated from a simpliﬁed version of Eq. 4.27d for release, namely kD(X) ¼ kD exp (βκ ShD(x+h+hD/2)) with kD ¼ 300 s1

254

6 Myosin Motors

of records sampled at the same time. Although kA ðX Þis a transition probability per second for actin binding, the lifetime of the detached head is a stochastic variable. Let p(τ) be the probability distribution of lifetime τ. It can be calculated from the probability p>(t) that the head remains detached over (0,t), which satisﬁes p> ðt þ dt Þ ¼ p> ðt Þð1 kA ðX Þdt Þ: Thus p> ðt Þ ¼ expðkA ðX Þt Þ and p(t) ¼ dp>(t)/dt, giving pðt Þ ¼ kA ðX ÞexpðkA ðX Þt Þ

ð6:30Þ

Some standard integrations show that the mean lifetime is 1=kA ðX Þ and the standard deviation is equal to the mean, which is a characteristic of Poisson distributions (Weatherburn 1968). In this way, rate constants can be extracted from the lifetimes of detached and bound periods in trap displacement records. Brownian-averaged rate constants can now be used to generate an analytic solution to the problem of deﬂated working strokes in the optical trap (Sleep et al. 2006). The preliminary calculation, in which every binding event is accompanied by a working stroke, suggests that the problem is caused by unequal access to different actin sites. Moreover, site access should be calculated on the basis of steady-state conditions rather than a hypothetical equilibrium, for example with the cycle shown in Scheme 6.1. Scheme 6.1 A simpliﬁed actin-myosin cycle with Brownian-averaged rate constants

1

~ k1 ( X ) k−1

2

~ k2 ( X ) ~ k−2 ( X )

3

k3 k −3 ADP

g 4

1 ATP

Here 1 ¼ M.ADP.Pi+M.ATP, 2 ¼ A.M.ADP.Pi+A.M.ADP, 3 ¼ A.M.ADP, 4 ¼ A.M. State 2 is the pre-stroke state, and 3 and 4 are post-stroke states. The true rigor state A.M is separated from its ADP counterpart because [ATP] in trap experiments is usually at micromolar levels to prolong the lifetime of bound states, whereas A.M is relatively short-lived at the millimolar levels of ATP in active muscle. For simplicity, the ADP release and ATP binding transitions are not assumed to be strain-dependent. With multiple actin sites, Scheme 6.1 applies in turn to each site, whereas the detached state 1 is held in common. We wish to calculate the frequency of bound levels on each site, but that depends on which bound state is detected. State 2 has displacement X but is very short-lived. States 3,4 have displacement X + h, but since g ~ 10 s1 and k3 300 s1, the myosin will be in state 4 for most of its lifetime. In fact, the time resolution of event detectors is such that all detected bound levels will be in state 4. Hence we require the forward ﬂux into state 4, namely

6.2 Actomyosin Kinetics in the Optical Trap

255

J 4 ð X k Þ ¼ k 3 p3 ð X k Þ

ð6:31Þ

from which the mean bound level can be calculated as P U¼

k

ðX k þ hÞJ 4 ðX k Þ P : J 4 ðX k Þ

ð6:32Þ

k

The calculation turns on the occupation of state 3 on site n, which follows under steady-state conditions by equating the net ﬂuxes through each transition: k1 ðX Þp1 k 1 p2 ¼ k2 ðX Þp2 k2 ðX Þp3 ¼ k 3 p3 k 3 p4 ¼ gp:4 :

ð6:33Þ

After a little manipulation, it may be veriﬁed that p3 ð X Þ k1 ðX Þ k2 ðX Þ ¼ p1 k1 þ g2 ðX Þ k2 ðX Þ þ g3

ð6:34Þ

where k2 ðX Þg3 g~2 ðX Þ ¼ , k2 ðX Þ þ g3

g3 ¼

k3 g : k3 þ g

ð6:35Þ

g3 is the effective detachment rate from state 3, and g~2 ðX Þ likewise from state 2. The two factors in Eq. 6.34 show how p3(X) can become asymmetric from the actin-binding or stroke transitions. With g ¼ 10 s1, k3 ¼ 300 s1 and K3 ¼ 7, g3 ¼ 57 s1, whereas with k2 ¼ 5000 s1 and K2 ¼ 20, k2 ¼ 250 s1. Thus the second factor in Eq. 6.34 stays close to K 2 ðX Þ except when X > 20 nm, where actin binding is very weak; over most of the domain of support, the stroke transition is close to equilibrium. So an asymmetric population in state 3 can only arise from the ﬁrst factor, which describes the effects of actin binding. Signiﬁcant asymmetry arises only when g2 ðX Þ K2 ðX Þg3 –10 nm is obtained by two straight lines, of slopes 2.7 pN/nm and 0.03 pN/nm, intersecting at X ¼ 0.35 nm. This is what is expected if the S2 rod buckles for compressive strains above this threshold (Howard 2001). The buckling strain XB ¼ 0.35 nm is

6.3 Rigor Bonds, Buckling Rods and Cy3-ATP

261

consistent with the molecular dynamics simulation of Adamovic et al. (2008). And the downturn in tension for X < 60 nm is due to S2 turning around to face the M-line. Similar experiments were conducted to determine how the apparent working stroke changes when more than one myosin is bound. The idea behind this, and other muscle ﬁbre experiments, is best conveyed by considering what happens under isotonic conditions. With a single-bead puller, this can be arranged by a feedback system which keeps the trap a ﬁxed distance in front of the bead. Suppose that there are n myosin heads bound to actin, and the trap supports a tension F ¼ T(X) where X speciﬁes the position of the actin ﬁlament. What happens when another head binds to a site at distance x and makes a working stroke h, as would happen with M.ADP.Pi heads in the presence of ATP? The bead will be displaced by an additional distance u out of the trap, which is calculated by balancing forces. Thus F ¼ T ð X Þ ¼ κ t u þ T ð X uÞ þ κ ð x þ h uÞ

ð6:41Þ

if the S1-S2 combination is supporting tension after the working stroke. Let S (X) ¼ dT/dX be the differential stiffness of the initially-bound heads. If the displacement is small, or if S(X) is a constant, then u¼

κ ð x þ hÞ κ ð x þ hÞ κ þ κ t þ Sð X Þ κ þ Sð X Þ

ð6:42Þ

If the available actin sites are uniformly distributed along the ﬁlament (and the head always makes a working stroke after binding!) then x averages to zero and the mean displacement u is reduced from h to κh/(κ+S). For example, if there was no buckling under compression, then S(X) ¼ nκ for n bound heads, and u ¼

h : 1þn

ð6:43Þ

The apparent working stroke is reduced by the load created by n bound heads. However, if all of these heads had already been put into the buckling range, then their series stiffness is reduced to almost zero and u h. The terminology associated with this topic is potentially confusing. The term ‘load-dependent working stroke’ often appears in the literature (Reconditi et al. 2004; Piazzesi et al. 2007) but in this context ‘working stroke’ is synonomous with ‘step distance’ or ‘sliding distance,’ meaning the F-actin displacement that occurs when a bound head makes a stroke. In this book, the term ‘working stroke’ denotes the transition between the two states of the lever arm, but its magnitude h is deﬁned as the axial displacement of the tip under zero load. Then h is a constant of myosin structure, as determined by X-ray crystallography.

262

6 Myosin Motors

Fig. 6.14 Actin-myosin interactions in the optical-trap combined with ATPase measurements by total internal reﬂection spectroscopy (Ishijima et al. 1998). From top to bottom, each graph shows a time record of beads-ﬁlament displacement, stiffness as the inverse of running variance, and ﬂuorescence intensity from 10 nM Cy3-ATP, which increases when bound to myosin. (a) An event in which the binding of Cy3-ATP dissociates the actomyosin bond. (b) An event in which Cy3-nucleotide (probably Cy3-ADP) was released before binding took place. About 50% of all myosin-actin bindings were delayed. With permission of Elsevier Press

6.3.4

Coordinating Myosin Detachments with ATP Binding

In the hands of Ishijima et al. (1998), the three-bead experiment was adapted to allow simultaneous monitoring of myosin force-generating events and ATP binding. This was achieved by using a ﬂuorescent analogue (Cy3-ATP), in which the ribose group is ﬂuorescently labelled. The ﬂuorescence intensity increases when Cy3-ATP binds to myosin (Funatsu et al. 1995). Nucleotide attachment events, in which Cy3-ATP binds to actomyosin or Cy3-ADP was released, were observed by trapping the actinbeads dumbbell just below the surface of the bathing solution, so that ﬂuorescence changes could be detected by total internal reﬂection in the layer of the evanescent beam. A very low concentration of Cy3-ATP (10 nM) gave periods of several seconds for the bound myosin. The key results of this experiment, shown in Fig. 6.14, can be summarized as follows. Actomyosin dissociation events, detected by running variance records as the ends of bound periods, were accompanied by an increase in Cy3-ATP ﬂuorescence as expected, signalling Cy3-ATP-induced detachment of A.M, but the two events were not always simultaneous; the ﬂuorescence signal sometimes preceded and sometimes followed detachment event, with an average time difference of 0.05 s either way. Secondly, myosin binding events with force generation, detected as the beginning of bound periods, were accompanied by a decrease in ﬂuorescence, signalling the release of ﬂuorescent nucleotide, presumably Cy3-ADP. In about

6.4 Force-Clamp Spectroscopy

263

half of their observations, these events were simultaneous but otherwise the decrease in ﬂuorescence preceded myosin binding, generally by 0.5 s but sometimes more. How can these results be explained in terms of the actin-myosin contraction cycle? The major problem is the drop in ﬂuorescence which precedes myosin binding, which suggests that Cy3-nucleotide in whatever form (triphoshate or diphosphate) is being released while the myosin is detached. It cannot be possible for M.Cy3-ATP, initially present after dissociation from actin, to release Cy3-ATP before hydrolysis; the afﬁnity of ATP for apomyosin is so high (~1010 M1) that the release rate would be of order 10610 ¼ 104 s1. However, there seem to be unusually long detached periods in this trap experiment, so Pi and Cy3-ADP might occasionally dissociate from the products complex M.Cy3-ADP.Pi formed after hydrolysis. In that case, myosin binding would precede silently, with a time lag modulated by the z-position of the F-actin dumbbell. Alternatively, Cy3-ATP may have bound to other myosins on the coﬁlament, which was constructed to hold single myosin heads rather than dimers. This would allow some ﬂuorescence changes uncoupled to attachment events for the active myosin, but it would probably not explain the mistimed ﬂuorescence increases associated with actomyosin dissociation. With such a low nucleotide concentration, rigor myosins might occasionally dissociate before binding Cy3-ATP. This would silence their detachment and the subsequent rebinding, which would occur before Cy3-ATP at 10 nM could bind (kT 1068 ¼ 0.01 s1). Either way, we are left with a partial and highly speculative explanation of their observations. To my knowledge there has been no resolution of this problem, which remains open for further investigation.

6.4

Force-Clamp Spectroscopy

In its original form, the ‘three-bead’ experiment for observing the interactions of a single myosin with an optically-trapped actin ﬁlament (Sect. 6.1) suffers from a number of limitations. The presence of large random ﬂuctuations in the position of the actin-beads dumbbell prevents the experimenter from controlling which actin site is interacting with the tethered myosin. Attachment events have to be detected by a reduction in noise amplitude, which is best performed by creating a running variance signal, and these methods limit the time resolution of detected events to 5–10 ms. Thus the study of rapid events such as the working stroke is beyond the reach of this technique. The time resolution of single-molecule interaction records in the dualbeam optical trap would be improved if one or both beads were held ﬁxed, and this can be achieved by optical feedback, where the position of a bead is relayed back to acousto-optic deﬂectors in the path of the laser beam. Thus the position of either bead can be controlled, and clamping one bead in a ﬁxed position was used in the early experiments in an attempt to reduce positional noise. A highly desirable aim of single-myosin trap experiments is to measure the lifetime of bound states under different loads. For rigor myosin, we have seen that

264

6 Myosin Motors

this can be done with a single trapped bead. For myosin in the presence of ATP, the binding event must be detected before a force can be applied. This can be done by sinusoidal modulation of the position of one bead, and detecting that signal in the position of the other bead (Veigel et al. 2005), which marks the onset of a bound period. Using 1 kHz modulation at 35 nm r.m.s, the authors claim that binding events can be detected with a resolution of 1–2 ms. The trap system used by Takagi et al. (2006) provides another example of the use of bead-trap feedback. Instead of clamping just one bead, both beads were clamped to maintain the actin ﬁlament at a constant length, which can increase the stiffness of the whole dumbbell system by stretching the links. An interaction with the ﬁxed myosin pulls the dumbbell out of the traps, and the position signal from one bead was used to monitor the force produced. That force signal was integrated and fed to the other bead to oppose the motion. In this way the ﬁlament was kept at a constant length, and the force increased in time during each bound period. Perhaps the most satisfactory way of detecting attachment events and applying a known force to the bound myosin is by a variation of the positional-feedback technique. In the previous section, we saw that moving both traps at constant velocity allows the tethered myosin to ﬁnd different sites on actin. The dumbbell is almost halted when myosin binds, and the slight change in position during a bound period provides a measure of myosin stiffness (Lewalle et al. 2008). In the hands of Capitanio et al. (2012), this method has been reﬁned to automatically apply a ﬁxed load to the myosin after each binding event, reducing the time resolution of detected binding events to 10–100 μs, a factor of 100–1000 times better than detection from a running variance record. The method also uses the dual-beam optical trap system, and it works as follows. Using beed-trap feedback on both beads, a constant force F was applied to one bead and a different force F + ΔF to the other bead (Fig. 6.15a). With myosin-S1 detached, the dumbbell moves at a constant velocity ΔF/μ determined by viscous drag. When myosin binds, the unbalanced force is transferred to the myosin molecule, and the steady motion of the dumbbell is eliminated entirely, leaving only small random Brownian displacements determined entirely by myosin stiffness. After 200 nm of travel, the unbalanced force is reversed, creating a triangular sweep across the ﬁxed myosin. During this process, the motion of the dumbbell reﬂects actin-myosin transitions in the contraction cycle at a load ΔF, which is held constant to an accuracy deﬁned by the loop gain of the feedback system and its response time. The performance of this system can be analysed by splitting the displacement response of the dumbbell into causal and stochastic parts. This can be done quite cleanly, because the system is operating isotonically rather than in an unloaded or isovelocity conﬁguration. Figure 6.15a deﬁnes the geometry of this system for analysis. Here u(t) is the leftdirected displacement of the dumbbell at time t, and the unbalanced force ΔF (rightdirected when ΔF > 0) is applied to the right-hand bead. This bead is assumed to hold the +end of F-actin, so that any working stroke increases u(t) and a positive force resists the stroke. This geometry is actually the same as used for the right-hand half-sarcomere of the muscle ﬁbre, in the sense that ΔF > 0 creates a stretch and

6.4 Force-Clamp Spectroscopy

265

Fig. 6.15 Dual-beam optical trapping with unbalanced force clamps (Capitanio et al. 2012). (a) Cartoons of the trapped beads-ﬁlament dumbbell, with the +end of F-actin attached to the righthand bead. The left- and right-hand beads are clamped to hold forces F and F + ΔF respectively. In the upper drawing, the tethered myosin is about to bind to an actin site displaced by a small distance X. In the lower drawing, about 10 μs after myosin has bound, myosin holds the unbalanced force ΔF and the dumbbell is left-displaced by XΔF/κ and stationary apart for Brownian motion. (b) The unbalanced force is reversed periodically to let myosin scan for actin sites, and the dumbbell moves under viscous forces until myosin binds. (c) The left-directed dumbbell displacement expected after binding, followed by a working stroke after a time delay determined stochastically by the rate constant kS(ΔF/κ). The rising displacement would occur over the time required for the lever-arm to swing against viscous forces (about 10 μs). (d) Ensemble-averaged displacement records obtained under resistive force (ΔF > 0) by Capitanio et al. (2012), superimposed to show binding events at the same time with the same displacement. Short-lived bound periods did not make a working stroke. With permission of Springer Nature

du/dt is the analogue of the shortening velocity. For the detached dumbbell, du/ dt ¼ ΔF/μ. Suppose that, at time t ¼ 0, the ﬁxed myosin binds to a site at distance X to the right. The Langevin equation for the subsequent dumbbell motion is

266

6 Myosin Motors

μ

du ¼ κðX uðt ÞÞ ΔF þ F B ðt Þ dt

ð6:44Þ

which should be compared with Eq. 6.10 for isometric traps. Setting κ ¼ 0 deﬁnes the stochastic motion of the free dumbbell, which separates into causal and stochastic components: uð t Þ ¼

ΔF t þ uB ð t Þ μ

ð6:45Þ

where uB(t) is the result of Brownian motion, and duB/dt ¼ FB(t)/μ. The solution of this stochastic equation is ‘solved’ by introducing the impulse function B(t) used in Sect. 4.2, leading to the famous formula uB ðt Þ2 ¼ 2Dt

ðD ¼ kB T=μÞ

ð6:46Þ

for the position variance of a Brownian particle. The derivation is straightforward and follows from Eqs. 6.12, 6.13, 6.14 and 6.15. Since the dumbbell is moving at a constant average velocity, the velocity noise is of more interest here. Over a time Δt greater than the correlation time of force noise, the change in displacement deﬁnes a coarse-grained random velocity vB ðt Þ

uB ðt þ Δt Þ uB ðt Þ Δt

ð6:47Þ

and Eq. 6.46 leads directly to the mean-square ﬂuctuation in velocity vB ð t Þ 2 ¼

2D Δt

ð6:48Þ

which grows as the sampling time is reduced. The central-limit theorem guarantees that the distribution of velocities is Gaussian. The same kind of analysis can be carried through for the dumbbell displacement after myosin binds, starting from Eq. 6.44. Again there is the same clean separation of causal and stochastic components, and it is sufﬁcient to quote the results: uð t Þ ¼

ΔF X 1 eλt þ uB ðt Þ κ

uB ð t Þ 2 ¼

kB T 1 e2λt κ

ð6:49aÞ ð6:49bÞ

where λ ¼ κ/μ and uB ðt Þ ¼ 0. Dumbbell displacements are here referred to its position at t ¼ 0 when myosin binds to a site with offset distance X (probably less than 2–3 nm in either direction). Thus the dumbbell reverts to a position of mechanical equilibrium where κ(Xu) ¼ ΔF over a time 1/λ, which is of order 10 μs. During that time, the

6.4 Force-Clamp Spectroscopy

267

noise component grows towards the value deﬁned by equipartition of energy, rising to an RMS value of 1.2 nm if κ ¼ 2.7 pN/nm. This analysis ignores elastic compliance in the bead-ﬁlament links, but their extensions are constant because they hold the same forces F, F + ΔF before and after myosin binds. If a working stroke h occurs during a bound period, the displacement should rise abruptly by the same amount, provided that mechanical equilibrium is established ﬁrst. For a single molecule, the rate constant of the stroke will be kS(Xu), asymptoting to kS(ΔF/κ) after equilibration. Here the stochastic interpretation of rate constants is all-important; kS(ΔF/κ)t is the probability of the stroke occurring after time t, so the strain- dependent stroke rate determines the distribution of delay times before the stroke occurs as a very rapid event, as depicted in Fig. 6.15b. Rarely has there been such a direct way of verifying this stochastic interpretation of chemical kinetics, and of measuring the strain-dependence of the stroke rate! However, this prediction is obscured by the common practice of averaging singlemolecule responses to improve the signal-to-noise ratio, and averages of many such events starting from the instant of binding yield a smoothly rising displacement over several milliseconds (Fig. 6.15c). In this case, a positive unbalanced force is expected to reduce the stroke rate (Figs. 4.4 or F2) and increase the delay. Studying the lifetimes of the bound states as a function of force ΔF has proven to be a very powerful technique for characterizing the biochemical contraction cycle. Results from many runs were used to generate a cumulative distribution of lifetimes p 40 nm at high myosin densities, which suggests that b is the 36 nm actin halfperiod rather than the monomer spacing. Presumably an actin ﬁlament does not rotate freely about its axis in the high-density motility assay. Given that b ¼ 36 nm, measurements of Lmin provide an estimate for the duty ratio required for the second sliding distance. A different approach to measuring d~ATP was formulated by Uyeda et al. (1990), who deliberately used low densities of ﬁxed myosins and ﬁlaments shorter than Lmin. High viscosity solutions ensured that ﬁlaments did not fall off the tracks deﬁned by a line of myosins. Under these conditions, the actin ﬁlaments are expected to move intermittently, with an average velocity reduced by the fraction of time in which at least one myosin head is bound. Thus n o v ¼ v0 1 ð1 pÞNM

ð6:52Þ

where NM is the number of myosins that can overlap the ﬁlament. The reader is referred to their paper for a description of how this approach was implemented. For the motility assay, results from different laboratories, summarized in Table 6.1 shows a huge range of values of d~ATP , from 8 nm to 155 nm. It is important to understand why this is so. The measurements were made under similar conditions, but still yield different values for dATP. Part of the problem lies in the way in which the duty ratio has been estimated. Harada et al. (1990) attempted to measure p from dwell times; τ ¼ 1/r and τon ¼ τ τhyd 1/kA where kA was estimated from separate experiments with short ﬁlaments. On the other hand, Toyoshima et al. did not measure the duty ratio as such but made a related assumption, that on average only one head per minimum-length ﬁlament (MLF) is bound at any time. Hence τon ¼ 1/RLmin (R ¼ ATPase rate per unit length of moving ﬁlaments) and d~ATP ¼ v=RLmin v=rN min

6.5 Motility Assays

271

Table 6.1 A digest of motility assays: myosin spacings, minimum lengths for moving ﬁlaments, sliding velocities, ATPase rates R per unit length of moving F-actin and r per myosin head in interaction, duty ratios and the corresponding ATP sliding distances dATP and d~ATP

Authors Harada et al. (1990) Toyoshima et al.(1990) Uyeda et al. (1990)

Temp ( C) 22 30 30

dM (nm) 11 30

Lmin (nm) 40 40 150

30

36–140

_

v (μm/ s) 5.5 11.2 4.6 7.4

R (s. nm)1 1.1 5.5 3.95

r (s1) 22 61 64

dATP (nm) 250 185 39

p 0.80 0.32 (0.2)

d~ATP (nm) 200 60 8

_

13

560

0.05

28

The mean nearest-neighbour spacing dM is calculated from the areal density ρM of randomly placed 1=2 ﬁxed myosins byd M ¼ 0:5ρM . R was calculated from the total ATP turnover rate and the total length of all moving ﬁlaments, and r ¼ RdM. The duty ratio was estimated in different ways, described in the main text

where Nmin ¼ Lmin/dM is the number of myosins addressing a MLF. According to this formula, p ¼ 1/Nmin for the duty ratio. Because of the above uncertainties in estimating the duty ratio, no conclusion from motility assays can be drawn in favour of tight or loose coupling in the contraction cycle. To vary the duty ratio, motility assays can be implemented at various concentrations of ATP. The sliding velocity increases hyperbolically with [ATP], with a typical half-maximum concentration of 200 μM (Harada et al. 1990), while the myosin-driven ATPase rate appears to saturate at about 10 μM ATP (Harada et al. 1987); thus dATP rises with [ATP] while the duty ratio would fall. The change in duty ratio could be estimated from the second-order rate constant for ATP-induced detachment, which would give d~ATP as a function of ATP level. To derive the relations between dATP, d~ATP and the working stroke from fundamentals, consider a simple motility model in which strain-dependent kinetics is set aside. For given spacings between ﬁxed myosins, and a given periodicity of binding sites, let us assume that (i) myosins in overlap can bind to one site in any 36 nm period, (ii) all bindings are accompanied by the working stroke and (iii) detachments are post-stroke, triggered by ATP binding, and followed by hydrolysis on M.ATP. Then mechanics enters only via the size of the sliding distance per binding event (Eq. 6.43), which is reduced by the number of myosins already bound. For attachment rates f and g (where f is reduced by the rate of hydrolysis off actin), the average number of bound myosins is nM ¼

f NM: f þg

ð6:53Þ

Per ﬁlament, the frequency of binding events is also the frequency of detachment events, which in this model is the ATPase rate R. For a ﬁlament with NM myosins in interaction, R ¼ f(1 p)NM ¼ gpNM where p ¼ f/( f+g) is the duty ratio. The sliding

272

6 Myosin Motors

velocity is the reduced stroke (Eq. 6.43) multiplied by its frequency, which with tight coupling is the ATPase rate per ﬁlament. So v¼

h R 1 þ pNM

ðR ¼ N M r Þ

ð6:54Þ

and the two sliding distances per ATP are v NM h hﬃ ðNM >> 1Þ dATP ¼ r 1 þ pNM p pv d~ATP ﬃ h ðNM >> 1Þ: r

ð6:55aÞ ð6:55bÞ

It is this relationship which prompted the second deﬁnition of the ATP sliding distance. Motility assays with a saturating concentration of myosins and millimolar concentrations of ATP can still yield sliding distances in excess of what this equation predicts, and there are several reasons why this could be so (Burton 1992). Apart from the different methods used for estimating the duty ratio, the ﬁlament can lift off its track and be propelled by Brownian forces. Estimates of the average sliding velocity can also be derailed in various ways. At low densities, the ﬁlament can become pinned at its leading edge, which causes it to buckle in the plane of the surface and then rotate about the pinning centre. This ‘spiral defect’ is driven by the torque created by intermittently bound motors, and can be modelled in its own right (Bourdieu et al. 1995). At high densities, there is also a variety of collective behaviours (Schaller et al. 2010), which should serve as a warning that the ﬁlaments may not move independently along separate myosin tracks. For ﬁbres, the interpretation of ATP sliding distances is much clearer. ATPase measurements give the ATP sliding distance as a function of shortening velocity under load, and there are many examples to choose from, for example Harada et al. (1987), Higuchi and Goldman (1991), Yanagida et al. (1985), plus those referenced in Sect. 2.1. Figure 6.17 shows ATPase and the ﬁrst sliding distance as a function of shortening velocity, from measurements of He et al. (1999). For unloaded shortening, the duty ratio can be estimated from AC stiffness measurements. At zero load, dATP ¼ 107 nm and p ¼ 0.1, which is compatible with h ¼ 10.7 nm or two smaller working strokes. On the modelling side, it must be admitted that the models described in Chap. 4 do not allow post-stroke A.M.ADP or A.M heads to detach under rapid shortening, followed by rapid rebinding without change of state or lever-arm conﬁguration. In the muscle ﬁbre, evidence for such events comes from measurements of ﬁbre stiffness as a function of load (Brenner 1991). In this way, tight coupling may be avoided occasionally, allowing the sliding distance per ATP hydrolysed to increase above what is predicted from Eq. 6.54 without requiring multiple strokes per ATP.

6.6 The Glass-Microneedle Experiment

273

Fig. 6.17 ATPase rate r(v) per myosin head as a function of shortening velocity (red line) in rabbit psoas ﬁbres (He et al. 1999), and the corresponding ATP sliding distance dATP ¼ v/r (black line). These plots were constructed from the hyperbolic function r(v) ¼ 5.1 + 0.02030v/(1 + 0.00149v), equivalent to that used by He et al. when the unit of velocity is converted from muscle lengths to nm/s, with a half-sarcomere length of 1300 nm

6.6

The Glass-Microneedle Experiment

In 1999, Kitamura, Tokunaga, Iwane and Yanagida announced the results of a new kind of nano-manipulation experiment, in which a single myosin head on the tip of a very ﬁne cantilever was lowered onto a ﬁxed bundle of actin ﬁlaments (Fig. 6.18a). The cantilever was repeatedly observed to make several 5.3 nm steps in the same direction before returning to rest (Kitamura et al. 1999). Is this discovery compatible with the swinging lever-arm mechanism which generates movement when myosin is hydrophobically bound to actin, or does it point to a ratchet mechanism where myosin steps along F-actin in a weak periodic potential well which changes shape when ATP binds? This experiment, and the formidable technical difﬁculties involved in repeating it, became the bete noire of muscle mechanics, prompting a now-egregious symposium organised by Toshio Yanagida in which East met West without any apparent resolution (Cyranoski 2000). A close look at the Kitamura-Yanagida experiment reveals the distinguishing features of their ﬁndings. (i) Within the limitations imposed by Brownian noise, the size of each step was always 5.3 nm or a multiple thereof (Fig. 6.18b). (ii) The number of steps varied from one to ﬁve, with one step occurring most frequently (Fig. 6.18c). (iii) During stepping, the myosin head was interacting strongly with actin; the inverse running variance κeff/kBT rose to 0.5 nm2, for which κeff ¼ 2 pN/ nm, as expected for strongly-bound myosin. (iv) Stepping nearly always occurred in the same direction, regardless of the initial displacement of the cantilever. Thus the direction of stepping is determined by the polarity of the selected actin ﬁlament. (v) The exponential distribution of dwell times between steps was substantially independent of ATP concentration; the mean lifetimes were 3.2 ms and 4.8 ms at 0.1 and 1 μM ATP, but decreased signiﬁcantly with rising temperature. (vi) The

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6 Myosin Motors

Fig. 6.18 (a) The experiment of Kitamura et al. (1999), where a single molecule of myosin-S1 was attached to a ZnO whisker on a glass microneedle, forming a delicate cantilever to probe the actin ﬁlament. (b) Displacement records of the cantilever showed repeated steps of 5.3 nm, the actin monomer spacing. (c) The pair histogram of repeated displacements reveals the 5.3 nm periodicity, especially after subtracting the Gaussian distribution of initial displacements (inset). With permission of Springer Nature

sequence of steps was terminated by detachment from actin. When the cantilever returned to its rest position, Brownian noise increased; the stiffness κc of the cantilever p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ was typically 0.03 pN/nm, for which the RMS displacement noise kB T=κ c would have been 11.5 nm. (vii) The ﬁnal detachment was probably triggered mechanically rather than by ATP binding, which would have allowed a net bound period of 0.2–0.5 s at 1 μM ATP. There seem to be two key questions. Firstly, what role, if any, does the working stroke play in generating 5.3 nm steps? A 5 nm working stroke would generate unidirectional motion, and the head presumably detaches from actin before stepping, but (vii) shows that repeated stepping could not be generated by ATP-driven cycling. Thus repeated stepping probably occurs with rigor heads, not accompanied by biochemical events once the head has bound and made an initial working stroke. This hypothesis can be tested in the same way that working strokes are detected in the three-bead experiment, by looking at the frequency of initial displacements u. However, the same problem arises: if the nearest actin site lies at distance x from the resting cantilever, the distribution would be biased about u ¼ x+h, and a Gaussian distribution biased about x ¼ h would arise only after sampling a 36 nm range of xpositions of the cantilever (Steffen et al. 2001).

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275

Secondly, is the maximum number of steps determined by the stiffness of the cantilever; would a more loosely-tethered head make more steps? This can only be answered experimentally, but it seems likely that the number of steps is determined by the helical structure of F-actin, in which the azimuthal angles of adjacent sites differ by 27 . Myosin attached to a ﬂexible cantilever will not be able to access more than a few such sites on a ﬁxed actin ﬁlament, and the maximum number of sites visited (ﬁve) would require two sites on either side of an optimally-oriented central site. The positional constrains on this myosin are probably more severe than for myosins in muscle, where the heads are tethered by ﬂexible S2 rods, a situation explored in Appendix G. Nevertheless, myosin-S1 on the cantilever was able to access three actin sites, corresponding to the target zones seen in rigor insect-ﬂight muscle (Tregear et al. 2004). As multiple working strokes can be ruled out, what does it take to persuade rigor heads to jump between actin sites? Ratchet models (Astumian 1997) postulate that the myosin motor undergoes biased Brownian motion, driven by a periodic but asymmetric potential well which changes shape in response to a chemical event such as ATP binding. However, in this experiment, repeated stepping occurs before ATP binding and is decoupled from ATP-driven cycling. Takano et al. (2010) proposed that the potential well seen by the tethered myosin oscillates with the 5.3 nm period of actin sites, but also with the 36 nm half-period of the actin double helix, decreasing to a minimum where the azimuthal head-site mismatch tends to zero (Fig. 6.19). In this potential, the Brownian stepping motion of a tethered head would precede in the same direction until stabilised at the bottom of the potential well, or until further displacements of the cantilever require too much strain energy. The direction of stepping would be determined by which site was initially chosen by the free cantilever. However, Kitamura et al. found that stepping proceeded in the same direction whether the initial displacements were positive or negative. In fact, the potential shown in Fig. 6.19 is not just hypothetical but was generated by Takano et al. from a molecular-dynamics simulation, using the atomic structures

Fig. 6.19 The potential energy of a vertically constrained myosin moving along F-actin, calculated from the molecular dynamics simulation of Takano et al. (2010). This asymmetric funnel produced unidirectional stepping towards the plus-end of F-actin for most initial displacements in a 36 nm period. With permission, copyright of the National Academy of Sciences, USA

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6 Myosin Motors

of F-actin and rigor myosin. They were able to reproduce the observed unidirectional stepping motion because the 36 nm component of the potential was asymmetric, decreasing slowly and rising rapidly as the head was moved towards the plus-end of F-actin. This asymmetry implies that the energy associated with azimuthal mismatching varies with axial position along the ﬁlament. For example, myosins sitting 5.3/2 ¼ 2.65 nm in front and 2.65 nm behind an azimuthally matched site would have different interaction potentials, because the atomic structure of myosin does not have that symmetry. However, the direction of motion is still determined by the x-position of the initial binding site. With different starting positions, Takano et al. found that six out of seven sites led to stepping in the same direction. Later work by the same group showed that stepping was sensitive to mutations affecting loop 2 on the motor domain (Nie et al. 2014). In view of this promising explanation, what are the wider implications for myosin motility and muscle action? Kitamura’s experiment differs radically from optical trap experiments with a weakly-trapped actin ﬁlament; the cantilever had a bending compliance similar to that of an optically-trapped bead, but it could not be stretched, and motion in the z-direction is virtually prohibited. The head-site interaction must be critically dependent on the height of the cantilever stage, but it must have been lowered to the point where the head made a strong rigor bond to an optimallyoriented site. In the molecular-dynamics simulation, the depth of the potential shown in Fig. 6.19 is over 60 zJ, more than expected for rigor bonds in muscle. The difference must be due to ionic conditions which control the strength of electrostatic interactions; the Kitamura-Yanagida experiment used a low-salt buffer solution with 25 mM KCl, whereas muscle operates at high salt with an ionic strength of 100–200 mM. Takano et al. showed that the strength of the potential and the amplitude of 5.3 nm-period oscillation was much reduced when the ionic strength was raised to 100 mM and, most importantly, the potential was much less asymmetric, allowing stepping in either direction. One can also compare the lifetime of the putative rigor bonds implied by this stepping behaviour with direct measurements of actin-S1 bond lifetimes under load, discussed in Sect. 6.3. Measurements of Nishizaka et al. (2000) showed that rigor-S1 bond lifetimes could be reduced to about 1 s by loads of 2–3 pN, and microneedle experiments with a stiffer cantilever (Kitamura et al. 2005) produced axial forces of that magnitude after a 5 nm step. However, that does not explain the millisecond dwell times of the original experiment, which were presumably created by vertical tension in the cantilever. To conclude, it seems that a vertically-tethered rigor myosin-S1 at low ionic strength and submicromolar ATP can make a few Brownian-driven steps along F-actin before being stopped by the tether, or by azimuthally misaligned sites on the double helix. Helical positioning of actin sites dictates that stepping is unidirectional once the starting site is selected. However, the asymmetry of the myosin-S1 motor interface creates a directional bias with respect to the starting site. All of this appears to be compatible with the swinging-lever-arm mechanism for contractility provided that myosin entered the rigor state after the ﬁrst attachment. However, physiological conditions in the muscle ﬁbre are very different; ionic strength is high

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277

and an active muscle requires millimolar ATP, which ensures that rigor heads will bind ATP and detach long before they can step between sites. Heads in rigor muscle might occasionally step in this way as part of the process of reattachment; rigor muscle is a glassy state with no deﬁned equilibrium, and slow detachments and rebinding to adjacent sites is only to be expected as the tension falls.

6.7 6.7.1

Processive Myosin Motors The Myosin Superfamily

Although the primary subject of this book is the myosin-II motor that powers muscle contraction, this chapter would not be complete without a brief look at the whole family of myosin motors that have evolved for different purposes. There are now 35 genetically distinct myosin motors. Striated muscle in all its forms is driven by members of the myosin II family (myosin IIA,IIB,IIX). Smooth muscle contains various forms of myosin I, whose job it is to hold tension without rapid shortening. The remainder are non-muscle myosins which perform a variety of cellular tasks (Sellers 2000; De La Cruz and Ostap 2004; Hartman and Spudich 2004). Some of them, notably myosin V, are two-headed processive motors which can carry vesicles over long distances on actin ﬁlaments. Other members of the superfamily have different roles again; myosin VII moves the cilia of the inner ear, and there is a mutation which causes deafness. Most myosins are plus-end motors, meaning that they migrate to the plus-end of F-actin, but myosin VI, which is involved in several intracellular processes, is a minus-end motor (Wells et al. 1999). Minus-end motors can be engineered by inverting the direction of the lever arm (Tsiavaliaris et al. 2004). For the phylogenetic tree of the myosin superfamily, see Berg et al. (2001) and Hodge and Cope (2000).

6.7.2

What Makes a Processive Motor?

A single head of myosin II in solution cannot be processive, as it will fall away from F-actin when detached by ATP. A two-headed motor such as heavymeromyosin is required so that one head can stay bound while the other head detaches to ﬁnd a new site. With heavymeromyosin, only one bound head per F-actin might be sufﬁcient for continuous motion in the motility assay. In that way, the low duty ratio of the single head (0.3–0.4 at millimolar ATP) might be mitigated. However, myosin II in the form of HMM does not walk along F-actin any more than a single head does, despite generating twice as much force as single-headed myosin and a bigger mean displacement in the optical trap (Tyska et al. 1999). This can be understood statistically; a duty ratio well above 0.5 is needed to keep a single head bound for many steps, and two-headed myosin V is such a motor (Mehta et al. 1999).

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6 Myosin Motors

The third requirement is structural, and related to the way in which a two-headed motor can move along its ﬁlament. There seem to be two possibilities. The motor may behave like an inchworm, and wriggle forwards until the leading head detaches in favour of the next site ahead, when the trailing head would be forced to detach. Alternatively, the motor makes hand-over-hand movements, so that the detached trailing head swings around the leading bound head and selects a site in front, giving an alternating sequence of leading and trailing heads. For myosin V, this ‘hand-overhand’ (or more syntactically, ‘head-over-head’) motion has been beautifully demonstrated by attaching ﬂuorophores to the neck region of both heads (Forkey et al. 2003). Dimeric myosin V has a much longer lever-arm than myosin II, which enables it to make steps equal to the 36 nm actin repeat (Purcell et al. 2002; Moore et al. 2004). It has six IQ motifs rather than two, and its lever arm is 23 nm long compared with 9.5 nm for myosin II (Purcell et al. 2002; Brenner 2006). Incidentally, an IQ motif (shorthand for isoleucine and glutamine) spans 25 amino acid residues and carries calmodulin or a calmodulin-binding light chain. As with myosin II, the lever-arms terminate in an alpha-helical S2 tail. Myosin V, in company with all non-muscle myosins, does not bundle to form a myoﬁlament, but the distal end of the tail has a linker which can bind to cargo. The atomic structure of dimeric myosin V has a lot in common with myosin II (Coureux et al. 2004; Sweeney and Houdusse 2004) but there are signiﬁcant differences which may explain its high duty ratio. Its kinetics differ sharply from myosin II by having a very low rate of ADP release, circa 14 s1, when bound to actin (De La Cruz and Ostap 2004), and this effect alone will generate a high duty ratio. Optical trap experiments show that myosin-V, in company with myosin-I and many other non-muscle myosins, has a second stroke which gates the release of ADP (Veigel et al. 1999, 2001). After this introduction, it is time to look at the range of single-molecule experiments which have elucidated the stepping mechanism of myosin V. This mechanism may well serve as a template for other kinds of processive motors.

6.7.3

The Mechanokinetics of Myosin-V Processivity

Of the various studies of myosin V motility, it was the use of single ﬂuorophores attached to the light chains of the lever arms that provided clear proof that myosin V walks on actin by a hand-on-hand mechanism (Forkey et al. 2003, using immobilized F-actins). Monitoring the polarization of their ﬂuorescence signals showed that each lever-arm swung between angles of 40 and 140 , while the two signals were always out of phase. This observation is exactly what is expected from a hand-to-hand action, whereas little change in the orientations of the lever-arms is expected from an inchworm model. This technique was then reﬁned to monitor the spatial location of each signal to an accuracy of 1.5 nm (Yildiz et al. 2003), which revealed a sequence of alternating steps exactly as expected from hand-to-hand action (Fig. 6.20). In an inchworm model, uniform steps of 36 nm would be required, which was not observed.

6.7 Processive Myosin Motors

279

Fig. 6.20 Schematic representation of myosin-V walking hand-over-hand on actin, as seen with by monitoring the polarization from ﬂuorescent probes attached to the light chains (Forkey et al. 2003). Subsequent observations of probe positions (Yildiz et al. 2003) showed a sequence of steps of various sizes, either a uniform sequence of 74 nm steps, alternating steps of 42 and 33 nm, or alternating 52 and 23 nm steps. They attributed the different sequences to populations of myosin V with ﬂuorophores on different IQ domains. With hand-over-hand motility, a ﬂuorophore attached at distance x away from the mid-point junction makes alternating steps of 37 + 2x nm and 37–2x nm, so their observations required x ¼ 0, 2 or 7 nm

The same walking motion was observed when the actin ﬁlament is not ﬁxed to a substrate but tethered in solution between two beads (Ali et al. 2002); this conﬁguration allows each motor molecule to explore the whole ﬁlament. The tagged myosin V rotated very slowly (6 per step on average) as it processed towards the barbed (+) end, performing a left-handed spiral although each actin strand constitutes a right-handed spiral. Clearly, this processive motion does not follow the sequence of actin monomers on one strand, but jumps to sites on the other strand separated by about 36 nm. The authors found that an unloaded myosin V steps to every 13th site or every 11th site, counting monomers on both strands, for which the spacings are the 36 nm half-period or the half-period minus one monomer spacing on the same strand (Fig. 1.7b). The next question is how this stepping geometry is generated by a working stroke. If the axial angles θA and θR of the pre-stroke and post-stroke states are known, then the working stroke is given by h ¼ Rð cos θR cos θA Þ:

ð6:56Þ

Optical-trap determinations of the working stroke of single-headed My5 gave h ¼ 24 nm (Veigel et al. 2001) and 21 nm (Moore et al. 2004). And electron microscopy of myosin-decorated F-actins ﬁxed in the absence of ATP (Walker et al. 2000) showed these angles directly; one of their images is shown in

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6 Myosin Motors

Fig. 6.21 (a) A cryo-EM image of rigor myosin V doubly bound to F-actin, with the barbed end to the right (from Fig. 1 of Walker et al. 2000). The two lever arms span the 36 nm half period, and the lever-arm of the trailing head is at an angle of 40 . The lever-arm of the leading head was always observed in a bent conﬁguration, suggesting that the leading head was in a pre-stroke state with an angle of 115 at the motor domain. With permission of Springer Nature. (b) A cartoon reconstruction, showing a 24 nm working stroke within the 36 nm half-repeat of the actin ﬁlament

Fig. 6.21a. Since myosin V processes towards the +end of actin, the leading head must be in a pre-stroke state and the trailing head in a post-stroke state. It is clear that θR ¼ 40 for the trailing head, but the bent appearance of the leading head suggests that θA < 140 . With R ¼ 23 nm, these angles are reconciled with a 24 nm working stroke if θA ¼ 105 , which gives h ¼ 23.6 nm. For dimeric MyV, it is generally agreed that the unitary 36 nm step is a combination of a 24 nm working stroke plus 12 nm of Brownian searching for the next actin site. This interpretation can be ﬂeshed out as follows, based on the cartoon in Fig. 6.21b. The EM images of Walker et al. (2000) imply that, during the period when both heads are bound, the leading head is positively strained by (36–24)/2 ¼ 6 nm and the trailing head is negatively strained by the same amount (Fig. 6.21b). Here ‘positive strain’ is used in the sense deﬁned for the muscle ﬁbre (Fig. 2.9). As shown in Fig. 6.22, the stepping cycle can be achieved by the following sequence of transitions. For clarity, assume that the motor is unloaded. 1. State I is the doubly-bound state with the leading head stable in the +vely strained A-state (A.M.ADP.Pi or A.M.ADP#), and the trailing head stable in the –vely strained R-state (A.M.ADP). This state is long-lived because ADP release in My5 is so slow. It terminates when ADP is released, allowing ATP to bind and detach the trailing head. 2. State II is the resulting singly-bound state, in which the detached head swings around the tail junction to ﬁnd actin sites on the next half-period of actin. Because the leading head is now strain-free, the tail junction has moved forward by a distance

6.7 Processive Myosin Motors

281

Fig. 6.22 Cartoons for the mechanical stepping cycle of myosin V walking on F-actin. In one cycle, the trailing head moves forward by one actin period (72 nm), while the junction between the lever-arms moves by 36 nm

U II ¼

b þ R cos θA ¼ 18 6 ¼ 12 nm 2

ð6:57Þ

assuming that in state I the junction is mid-way between the bound motors. This state ends when the leading head makes a working stroke. 3. In state III the leading head has made a 24 nm working stroke from its strain-free conﬁguration, a process which requires that X* > 0 for the critical stroking strain (Eq. 2.42). This condition can probably be met if the stroke is composed of two or more substrokes. The tail junction has moved forward by the stroke distance, giving a net displacement U III ¼ U II þ h ¼

b þ R cos θR ¼ 18 þ 17:6 ¼ 35:6 nm 2

ð6:58Þ

4. In state IV, the detached head that has swung forwards can now get so close to actin sites on the next half-period that it can bind with ease; Fig. 6.22 shows that it can sit at a distance 2R cos θR ¼ 35.2 nm ahead of the occupied site. However, UIV ¼ UIII. 5. State V. The stepping cycle is completed when the leading head binds to the site on the next half-period, giving a net displacement UV ¼ b ¼ 36 nm of the tail junction. However, stereospeciﬁc binding forces its lever arm to adopt the 105 angle, which introduces a positive strain of 6 nm as before, and the trailing head is negatively strained by the same amount.

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In this way, one cycle of processive motion can been completed with very minimal strain-dependent binding of the leading head, and ATP-induced detachment of the negatively-strained trailing head. The geometry of myosin V is exquisitely tuned to give steps equal to the actin half-period, or sometimes one monomer short of that. Moreover, the experiments of Uemura et al. (2004), with dimeric MyV linked to an optically-trapped bead, revealed state II, in which the tail junction moved by 12 nm, as a short-lived intermediate at the start of a new 36 nm step. Myosin stiffness is a critical parameter for the strain-dependence of transitions. Veigel et al. (2001) have measured the axial stiffness of My5S1 with optical trapping, using the three-bead conﬁguration. Bound periods observed with sinusoidal modulation of trap positions gave κ ¼ 0.2 pN/nm, after correcting for the compliance of the traps and actin-bead links. It seems that myosin V is a much more compliant beast than myosin II with κ ¼ 2.7–3.2 pN/nm, probably because of its longer lever-arm (Howard and Spudich 1996). Even with this compliant leverarm, Eq. 2.42 shows that two or more strokes will be needed to make X* > 0 for the 24 nm working stroke. The kinetics of the stepping cycle also determine how many steps myosin-5 makes on average before falling off the ﬁlament. The calculation is simple if we make some approximations. To begin, assume that actin attachment transitions are strain-independent. Secondly, some transitions can be ignored. The detachment of the pre-stroke leading head, at rate g1, can be omitted if the trailing head remains bound, because it will be held within a nanometer of the actin site and will rebind again at rate f, where f > g1. The detachment of the post-stroke trailing head, at rate g2 determined by the ATP concentration, will be irreversible if the leading head remains bound, because it will be removed from F-actin by the leading lever-arm reverting to its strain-free attachment angle of 105 . However, the singly-bound motor can still unbind at any stage. Sakamoto et al. (2003) comment that “processivity can be viewed as a race between the time needed for the unattached head to ﬁnd a suitable actin binding site and the ATP cycle time for the attached head. If the bound head dissociates before the unbound head ﬁnds a target, the myosin dissociates and the processive run terminates.” With the lever-arm geometry of myosin V, Brownian rotations of the trailing head will bring it close to the next actin site in less than 0.1 ms (Appendix E), so it would normally bind to actin long before the other head detaches. Thus f >> g2, which should be satisﬁed at micromolar levels of ATP. Secondly, the leading head can still detach at rate g1 and ﬂoat off if the trailing head has already detached. Thus the stepping cycle augmented by detachment of singly-bound heads gives the sequence of transitions shown in Scheme 6.2: Scheme 6.2 A schematic reaction scheme for processive motion of a two-headed motor

AB

g2

B

g

f

BA

g2

A

g O

f

AB

g2

B

g O

O

f

6.7 Processive Myosin Motors

283

where the singly-bound states A and B include pre-stroke and post-stroke forms, as in diagrams (ii) and (iii) of Fig. 6.22. The effective rate for detachment from the composite states A,B is g¼

g1 kS g þ g, g1 þ k S 1 g1 þ k S 2

ð6:59Þ

controlled by the stroke rate kS and the branching ratio. In practice, kS >> g1 so g g2. In fact, the duty ratio which determines processivity is p ¼ f/( f+g) even although g2 is the detachment rate for post-stroke heads. This is because doubly-bound to singly-bound transitions just couple singly-bound states from which the motor can detach completely. From these states the motor has a choice between detaching at rate g or rebinding at rate f, and the branching ratio of these rates means that f/( f+g) is the probability of the next doubly-bound state. Hence the probability of n such steps is pn, and multiplying by the probability 1p of the ﬁnal detachment gives the termination probability Pn ¼ ð1 pÞpn :

ð6:60Þ

Hence the mean number of steps before termination is n¼

1 X

nPn ¼ ð1 pÞ1 :

ð6:61Þ

n¼1

Although this hand-waving argument is correct, other formulae have appeared in the literature. Based on Scheme 6.2, a rigorous derivation of Eq. 6.60 is available (Smith 2004). Equation 6.61 predicts that a processive run of 720 nm, or 20 steps, requires p ¼ 0.95, or f/g2 ¼ 19 with g ¼ g2. At 10 μM ATP, g2 ~ 10 s1, similar to the rate of ADP release, so this duty ratio would require f ¼ 190 s1. An actin binding rate as high as this is explicable only for a head held in proximity to actin, whereas binding rates in solution are similar to those observed for myosin II (De La Cruz and Ostap 2004). Kinetics also determines the average stepping speed v in terms of the mean dwell time τ per step, which is the sum of the mean lifetimes of singly-bound and doublybound states. Thus b v¼ , τ

τ¼

1 1 þ f g2

ðb ¼ 36 nmÞ

ð6:62Þ

where 1/f includes the time for ATP cleavage off actin. This formula assumes tight coupling as well as no detachment of single-head states. If the duty ratio is close to unity, then v ¼ g2b. For myosin V, this was tested by Forkey et al. (2003), who observed stepping velocities as a function of ATP level and ﬁtted their data to the formula

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v ¼ vmax

½ATP ½ATP þ CT

ð6:63Þ

with vmax ¼ 453 nm/s and CT ¼ 11.7 μM. Their results are consistent with Michaelis-Menten kinetics for ATP binding to rigor, namely g2 ¼ kTA of Eq. 3.11, such that k~TA C T ¼ vmax =b ¼ 453/36 ¼ 12.6 s1. Hence k~TA ¼ 1.08 106 M1 s1, a reasonable value for the second-order rate constant of ATP binding and detachment from actin. However, this agreement may mask a different [ATP] dependence of g2. The detachment of post-stroke states is a two-step process where ADP is released prior to ATP binding. At high concentrations of ATP, the ADP release step should be ratelimiting for detachment, and this can be tested in several ways. Adding millimolar ADP will increase the step lifetime by reducing the net rate of ADP release. And optical-trap measurements on single-headed myosin V support a strain-gated transition prolonging the lifetime of the dominant A.M.ADP state (Veigel et al. 2005). However, a slightly different picture is revealed by optical-trap experiments from Ishiwata’s laboratory. Uemura et al. (2004) observed myosin V stepping on a ﬁxed actin ﬁlament as a function of load, using various level of ATP, ADP and BDM. In a single run, substeps were sometimes observed, with a 12 nm step followed after several milliseconds by a 24 nm step. The dwell time of each complete step increased exponentially with load, and also by lowering [ATP] or increasing [ADP] and [BDM] (Fig. 6.23). BDM acts to slow the contraction cycle in various ways (Fortune et al. 1994), including a reduction in the equilibrium constant for Pi release on actomyosin (Zhao and Kawai 1994), which reduces the lumped equilibrium constant of the subsequent working stroke. Without incorporating Pi release and the working

Fig. 6.23 The mean lifetime of a processive step of myosin V as a function of load (Uemura et al. 2004), under the various conditions listed in the ﬁgure. With permission of Springer Nature

6.7 Processive Myosin Motors

285

Table 6.2 Fitted values of Uemura et al. (2004) for the parameters in Eq. 6.64 Conditions 1 mM ATP +100 mM ADP +100 mM BDM 10 μM ATP

τ1 (ms) 5.9 5.9 28.9 5.9

d1 (nm) 2.2 2.2 2.2 2.2

τ2 (ms) 1.1 2.8 1.7 1.1

d2 (nm) 12.5 12.5 12.5 12.5

τ3 (ms) 80 137 111 547

stroke into the mix, this effect is tantamount to reducing the actin afﬁnity and the binding rate of M.ADP.Pi. Uemura et al. ﬁtted load-dependent dwell times to a sum of three terms, two of which vary exponentially with load F: τðF Þ ¼ τ1 expðβd 1 F Þ þ τ2 expðβd2 F Þ þ τ3

ð6:64Þ

Table 6.2 shows that the longest dwell time is for state 3, which is strainindependent and prolonged by raising [ADP] or reducing [ATP]. It can therefore be identiﬁed as the combination A.M.ADP + A.M of post-stroke states so that, at high ATP, 1/τ3 is the ADP release rate observed by De La Cruz and Ostap (2004). State 1 is prolonged by BDM and is modestly strain-dependent, indicating that it is the detached state terminated by actin binding, Thus state 2 must be the intermediate singly-bound state (state II in Fig. 6.22) where the leading head is yet to make a stroke. Its millisecond lifetime and extreme strain dependence are compatible with a 24 nm working stroke with two 12 nm substrokes (the transition II ! III). These assignments can now be matched to strain-dependent rate constants for MyV transitions. That requires known strains for both heads in each state of the stepping cycle. Following the notation used for the muscle sarcomere, x denotes A-state strain. In state I, the leading head is in an A-state so let its strain be x1. The trailing head is in an R-state; let its strain be x2 + h. Since the heads are distance b apart, x1 x2 ¼ b. With no load, the tensions in the lever-arms are equal and opposite, so x2+h ¼ x1. Hence x1 ¼ (b h)/2, giving x1 ¼ 6 nm when h ¼ 24 nm. These assignments change under load. For load F, force balance in the lever arms requires κx1 ¼ F κðx2 þ hÞ

ð6:65Þ

so 1 x1 ¼ ðF=κ þ b hÞ, 2

1 x2 þ h ¼ ðF=κ b þ hÞ: 2

ð6:66Þ

Thus the change Δx in x1 and x2 imposed by the load is shared equally between the two heads. However, this formula applies only to doubly-bound states. So

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6 Myosin Motors

Δx ¼ F=2κ Δx ¼ F=κ

ðdoubly-boundÞ ðsingly-boundÞ:

ð6:67Þ

In this way, strain-dependent reaction rates predict their dependence on an external load. Returning to Table 6.2, we can now use the rate functions proposed in Sect. 4.1 to relate d1 to strain-dependent binding and d2 to the working stroke h. Motivated by Fig. 6.21, a highly empirical equation for the stepping time is τ¼

1 1 1 1 þ þ þ kD kT k S ðxII Þ k A ðxIV Þ

ð6:68Þ

where kD, kT are strain-independent rate constants for ADP release and ATP binding on actomyosin, and the three terms are to be identiﬁed with τ3, τ2 and τ1 respectively. The strains involved are xII ¼ F=κ ðhead 1Þ xIV ¼ b 2R cos θR þ F=κ ðhead 2Þ:

ð6:69Þ

Here xII is initial strain for the working stroke in state II, and xIV is head-site distance for the binding of the detached head in state IV. Both are increased by the load pulling on the singly-bound head. For the working stroke, kS(x) / exp(βκhx), so the lifetime of state II is increased by the factor exp(βκhxII) ¼ exp(βFh). Hence d2 ¼ h. As d2 12 nm in Table 6.2, this indicates that the 24 nm stroke has two equal substrokes. For the actin binding of head 2, the trailing head is already bound, so one must go back to fundamentals and consider the reaction pathway. The most favourable path seems to be that the lever-arm of the trailing head continues to support load F while the leading head uses Brownian strains to ﬁnd its next site. In the Brownian-post model, the rate of actin binding is kA(x) / exp(βκx2/2) with x ¼ xIV. Then the potential energy barrier is 1 1 1 κx 2 ¼ κ ðd1 þ F=κÞ2 ¼ κd 21 þ Fd 1 þ F 2 =2κ 2 IV 2 2

ð6:70aÞ

d 1 ¼ b 2R cos θR :

ð6:70bÞ

with

If the quadratic term in F can be neglected, the lifetime of the singly-bound state IV is proportional to exp(βd1F), and Uemura’s data (Table 6.2) has d1 ¼ 2.2 nm. The zero-load lifetime τ1 implies a binding rate of 1000/5.9 ¼ 170 s1. A similar value of d1 is predicted from Eq. 6.70b; our previous estimates of myosin V geometry, with R ¼ 23 nm and θR ¼ 40 , give d1 ¼ 0.8 nm if b ¼ 36.0 nm. In fact, the subtraction of these large parameters means that the argument should be

References

287

inverted; Uemura’s value for d1 gives 2RcosθR ¼ 36.0–2.2 ¼ 33.8 nm. Thus the geometry of myosin V makes it uniquely well-adapted for making 36 nm steps on actin ﬁlaments. To conclude, this discussion shows that the data of Uemura et al. allow an interpretation of myosin V stepping that is consistent with trap data and solution kinetics. What is not conﬁrmed by the data in Table 6.2 is the dictum that ADP release is strain-dependent, although it may very well be so. Strain-gated ADP release is implemented by a lever-arm stroke preceding the release of ADP, and such a stroke may well be part of the overall 24 nm working stroke, although it would occur quickly under unloaded conditions. Once that stroke has occurred, solution kinetics shows that ADP release from actomyosin V is very slow, and Uemura’s data show that there is no subsequent working stroke to prolong the lifetime of the resulting rigor state. Much more can be said about the processivity of myosin V (for example, Kolomeisky and Fisher 2003), but this short introduction should convey something of the ﬂavour of an exciting area of research.

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Sleep J, Lewalle A, Smith DA (2006) Reconciling the working strokes of a single head of skeletal muscle myosin estimated from laser-trap experiment and crystal structures. Proc Natl Acad Sci USA 103:1278–1282 Smith DA (1998a) A quantitative method for the detection of edges in noisy time series. Philos Trans R Soc B 353:1969–1981 Smith DA (1998b) Direct tests of muscle cross-bridge theories: predictions of a Brownian dumbbell model for position-dependent cross-bridge lifetimes and step sizes with an optically trapped actin ﬁlament. Biophys J 75:2996–3007 Smith DA (2004) How processive is the myosin motor? J Muscle Res Cell Motil 25:215–217 Smith DA, Steffen W, Simmons RM, Sleep J (2001) Hidden-Markov methods for the analysis of single-molecule actomyosin displacement data: the variance-hidden-Markov method. Biophys J 81:2795–2816 Steffen W, Smith DA, Simmons RM, Sleep J (2001) Mapping the actin ﬁlament with myosin. Proc Natl Acad Sci USA 98:14949–14954 Stranneby D, Walker W (2004) Digital signal processing and applications, 2nd edn. Elsevier, Amsterdam Sweeney HL, Houdusse A (2004) The motor mechanism of myosin V: insights for muscle contraction. Philos Trans R Soc B 359:1829–1841 Takagi Y, Homsher EE, Goldman YE, Shuman H (2006) Force generation in single conventional actomyosin complexes under high dynamic load. Biophys J 90:1295–1307 Takano M, Terada TP, Sasai M (2010) Unidirectional Brownian motion observed in an in silico single molecule experiment of an actomyosin motor. Proc Natl Acad Sci USA 107:7769–7774 Thomas DD, Ramachandran S, Roopnarine O, Hayuden DW, Ostap EM (1995) The mechanism of force generation in muscle: a disorder-to-order transition, coupled to internal structural changes. Biophys J 68:135s–141s Toyoshima YY, Kron SJ, Spudich JA (1990) The myosin step size: measurement of the unit displacement per ATP hydrolyzed in an in vitro assay. Proc Natl Acad Sci USA 87:7130–7134 Tregear RT, Reedy MC, Goldman YE, Taylor KA, Winkler H, Franzini-Armstrong C, Sasaki H, Lucaveche C, Reedy MK (2004) Cross-bridge number, position and angle in target zones of cryoﬁxed isometrically active insect ﬂight muscle. Biophys J 86:3009–3019 Tsiavaliaris G, Fujita-Becker S, Manstein DJ (2004) Molecular engineering of a backwards-moving myosin motor. Nature 427:558–561 Tyska MJ, Warshaw DM (2002) The myosin power stroke. Cell Motil Cytoskeleton 51:1–15 Tyska MJ, Dupuis DFE, Guilford WH, Patlak JH, Waller GS, Trybus KM, Warshaw DM, Lowey S (1999) Two heads are better than one for generating force and motion. Proc Natl Acad Sci USA 96:4402–4407 Uemura S, Higuchi H, Olivares AO, De La Cruz EM, Ishiwata S (2004) Mechanochemical couping of two substeps in a single myosin V motor. Nat Struct Biol 11:877–883 Uyeda TQP, Kron SJ, Spudich JA (1990) Myosin step size: estimation from slow sliding movement of actin over low densities of heavy meromyosin. J Mol Biol 214:699–710 Veigel C, Bartoo ML, White DCS, Sparrow JC, Molloy JE (1998) The stiffness of rabbit skeletal actomyosin cross-bridges determined with an optical tweezers transducer. Biophys J 75:1424–1438 Veigel C, Coluccio LM, Jontes JD, Sparrow JC, Milligan RA, Molloy JE (1999) The motor protein myosin-I produces its working stroke in two steps. Nature 398:530–533 Veigel C, Wang F, Bartoo ML, Sellers JR, Molloy JE (2001) The gated gait of the processive motor, myosin V (2001) Nat Cell Biol 4:59–65 Veigel C, Schmitz S, Wang F, Sellers JR (2005) Load-dependent kinetics of myosin-V can explain its high processivity. Nat Cell Biol 7:861–869 Walker M, Zhang X-Z, Jiang W, Trinick J, White HD (1999) Observation of transient disorder during myosin subfragment-1 binding to actin by stopped-ﬂow ﬂuorescence and millisecond time resolution electron cryomicroscopy: evidence that the start of the crossbridge power stroke in muscle has variable geometry. Proc Natl Acad Sci USA 96:465–470

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Chapter 7

Models of Thin-Filament Regulation

Cooperation is the best form of regulation.

anon.

Tropomyosin (Tm) was discovered by Ebashi (1963), in much the same way that the actin-myosin interaction was revealed as ﬂocculent clumps (‘super-precipitation’) when the proteins were mixed in solution. If regulated actin ﬁlaments were mixed with myosins in solution, superprecipitation was not observed unless calcium was present, and then the results were indistinguishable from the clumps generated from non-regulated ﬁlaments. Subsequently, X-ray measurements (H.E. Huxley 1972, Haselgrove 1972) showed two possible Tm orientations on F-actin; a state in which Tm units sit in the groove between the two helical strands, and a second state in which each Tm has moved to expose the groove. This was strong evidence for the concept of ‘steric blocking,’ where Tm in the groove blocks myosin binding to the actin sites it covers. These two states of Tm will be called ‘blocked’ and ‘open.’ For an overview, see Ruegg (1988), Gordon et al. (2000), Geeves and Lehrer (2002) and, for more recent developments, Lehman et al. (2013), Geeves (2016). The geometry of regulation by tropomyosin is quite appealing; seven actin monomers cover a distance of 38.5 nm along the curve of the groove, and a 40 nm tropomyosin molecule spans that distance with a few nanometres to spare, enabling the ends of adjacent Tm units to overlap (Perry 2001). Why they should arrange themselves in this way, with minimal overlap and no gaps (Palm et al. 2003), is addressed in Sect. 7.4. Early theories of thin-ﬁlament regulation were built around the idea that each Tm protomer could move independently as a rigid body between the blocked and open positions, thus regulating myosin binding to seven actin sites (Fig. 7.1a). This regulatory mechanism is highly cooperative, so it should be very efﬁcient in speeding up the rate of tension development. The tropomyosin ‘on-off’ switch is activated by calcium, which binds to troponin (Tn), an auxiliary protein bound to tropomyosin, with one Tn per Tm protomer. Exactly how calcium regulates myosin binding via tropomyosin-troponin complexes has been the subject of many studies, culminating in the atomic structures of both

© Springer Nature Switzerland AG 2018 D. A. Smith, The Sliding-Filament Theory of Muscle Contraction, https://doi.org/10.1007/978-3-030-03526-6_7

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7 Models of Thin-Filament Regulation

Fig. 7.1 Cartoons for the regulated actin ﬁlament, showing a single chain of tropomyosins (Tm) on one strand. TnT is bound to one end of tropomyosin, and the C-terminus of TnC and the N-terminal of TnI are bound to TnT. (a) Steric blocking in the absence of calcium, where the hands of the N-terminal region of TnC are shut and the C-terminal of TnI is bound to actin, blocking myosin binding. (b) At high calcium, two Ca2+ ions are bound to TnC, allowing the hands of its N-terminal region to open and bind TnI. From Smith and Geeves (2003). With permission of Elsevier Press

proteins. Troponin has three parts, namely TnT which is attached to tropomyosin, TnC which in skeletal muscle binds two calcium ions, and TnI which can bind to actin, thereby holding tropomyosin over the groove (Fig. 7.1b). There is an interaction between TnC and TnI which is crucial to the whole business of thin-ﬁlament regulation. At submicromolar calcium levels, the binding sites on TnC are vacant, which allows TnI to bind to actin, taking tropomyosin with it and blocking myosin binding. This is the blocked or ‘off’ state. At high [Ca2+] (generally above 30 μM), the calcium-binding sites on TnC are occupied, which forces TnI to detach from actin. In the process, the whole TmTn complex moves to a new stable orientation on actin, the open or ‘on’ state, which allows myosins to bind. That is the content of the steric-blocking model. In the light of new data, the steric blocking model has undergone several revisions before arriving at a model with three distinguishable tropomyosin orientations on each tropomyosin protomer, as proposed by McKillop and Geeves (1993) on kinetic grounds. The model with independent tropomyosin units was modiﬁed to include end-to-end Tm interactions (Hill et al. 1980), and ﬁnally by reinterpreting each endto-end tropomyosin assembly as a continuous ﬂexible chain (Smith 2001; Smith et al. 2003; Smith and Geeves 2003). The three-state model and its extension to a continuous ﬂexible chain have now been validated by cryo-EM microscopy which revealed the existence of the three azimuthal states, the ‘blocked’ state present at – Ca, with TnI bound to actin, a ‘closed’ state present at +Ca in the absence of myosin, and an ‘open’ state in the vicinity of bound myosins (Vibert et al. 1997). The closed state is also observed in the absence of troponin, and may span a range of Tm orientations between the blocked and open states.

7 Models of Thin-Filament Regulation

295

Fig. 7.2 The actin ﬁlament (green) with tropomyosin coiled coils (red, yellow), shown in (a, b) as ribbon and space-ﬁlling diagrams respectively. From Holmes and Lehman (2008), with permission of Springer Nature

As the atomic structures of tropomyosin and troponin became available, many subsequent studies have conﬁrmed this interpretation, using EM to dock the structures together (Lehman et al. 2000, 2001), Pirani et al. (2005), Li et al. (2010a, 2011). Tropomyosin is an α-helical coiled coil which sits loosely on the actin ﬁlament at a radius of 4 Å (Fig. 7.2), with a speciﬁc sequence of residues which give it an intrinsic curvature (Li et al. 2010b, c). Both the residue sequence and the curvature are required for it to maintain loose electrostatic binding with corresponding charged groups on F-actin (Dominguez 2011), a phenomenon christened as ‘gestalt binding’ by Holmes and Lehman (2008) and developed by Lehman et al. (2013). Some mutations of tropomyosin residues destroy gestalt binding and stop the protomers from forming an end-to-end assembly on F-actin (Singh and Hitchcock-DeGregori 2003). There is also evidence that the two C-terminals of Tm splay open to accommodate the N-terminal end of an adjacent Tm unit (Greenﬁeld et al. 2009; Frye et al. 2010). The atomic structure of troponin, depicted schematically in Fig. 7.3, was revealed slowly from the work of different laboratories. First came the atomic structure of TnC, obtained simultaneously by Herzberg and James (1985) and Sundaralingam et al. (1985). Skeletal TnC has four putative calcium-binding sites, two at each end of the main α-helix. Those at the N-terminal end are low-afﬁnity Ca++-binding sites which regulate contraction, but those at the C-terminal end are high-afﬁnity sites normally occupied by Mg++. Structures of TnC-TnI complexes came next

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7 Models of Thin-Filament Regulation

Fig. 7.3 A cartoon for the action of troponin (TnC + TnT + TnI) in Ca2+-regulation by the thin ﬁlament (Takeda et al. 2003). TnC (magenta) is linked to α-helix H1 of TnI, which is hinged to its second helix H2 which has a regulatory domain which binds to actin in the absence of Ca++. The hinge also incorporates the junction between the two α-helices of TnT. With permission of Springer Nature

(Vassylyev et al. 1998), and ﬁnally the core structure of troponin, namely TnC-TnITnT2, with some minor deletions (Takeda et al. 2003). The coupling between Ca++ binding to TnC and the detachment from actin is achieved when two bound Ca++ ions open up the “E-F hands” of the upper end of TnC, which captures the C-terminal end of TnI. A few more facts before turning to models. In cardiac muscle, TnC has a different structure; there is only one regulatory calcium-binding site so only one calcium ion need bind to detach TnI from actin and initiate contraction. Mutations of TnC which stop calcium binding result in familial cardiac hypertrophy (Palmiter and Solaro 1997). There is also the question of what happens to calcium ions released into the sarcolemma by the ryanodine receptor. Not all of them bind to TnC; some remain in the medium and some are sequestered by proteins such as parvalbumin or pumped back to the receptor, so that contraction does not go on indeﬁnitely. For the same reason, calcium binding to TnC is reversible and reasonably fast; the dissociation rate is about 100 s1 (Johnson et al. 1981; Dong et al. 1996). Thus a single action potential triggers a ‘twitch’ response in the intact muscle, whose duration is determined primarily by the lifetime of free Ca++ ions in the sarcolemma.

7.1 Steric Blocking Models

7.1

297

Steric Blocking Models

In this section we consider only models in which different tropomyosins move independently between one or more ﬁxed azimuthal positions on actin. Each tropomyosin is regarded as a rigid body.

7.1.1

The Simplest Steric Blocking Model

Based on the above ideas, a simple regulatory model of myosin binding can be constructed as follows. Consider seven actin sites spanned by one tropomyosin protomer with two positions, namely the blocked state in which myosins cannot bind to actin, and the open state in which they can. Let K M K~ M ½S1 be the equivalent ﬁrst-order afﬁnity for myosin-S1 binding, and K be the equilibrium constant of the Tm switch from blocked to open. All states are assumed to be in chemical equilibrium, so this model does not extend to the proteins in the presence of ATP, which sets up a dissipative steady state as shown in Scheme 7.1: blocked

○○○○○○○ K open

○○○○○○○

KM

○●○○○○○

KM

○●○●○○○

KM

….. ●●●●●●●

Scheme 7.1 The steric blocking model with two regulatory states covering seven actin sites

In this scheme, myosins bind independently to the seven sites provided that tropomyosin is in the open state. To save space, only one choice of site occupancies is shown here. Conversely, tropomyosin can move back to the blocked state covering all actin sites only when the sites are empty; K is the equilibrium constant from the closed state to the empty open state. All on-off models presume that the tropomyosin complex moves as a rigid body. State occupancies are determined from the conditions of chemical equilibrium. Let the probabilities of the open and blocked states be p and q respectively. p includes all possible occupations of the seven actin sites by myosins, so p + q ¼ 1. However, the equilibrium constant K is equal to po/q where po is the probability of the empty open state. The connection with p is simply that. po ¼ ð1 þ K M Þ7 p

ð7:1Þ

298

7 Models of Thin-Filament Regulation

Fig. 7.4 Myosin-S1 binding to regulated actin as a function of net myosin afﬁnity KM, showing data of Maytum et al. (1999) for solution measurements at high and low calcium, and ﬁts to the steric blocking model with independent 7-site regulatory units (Eq. 7.3). S1 concentrations were converted to KM using an afﬁnity of 2 107 M1 to the strong-binding state. Data supplied by Prof. Michael Geeves

since 1/(1 + KM) is the probability that any particular site is empty. Hence the normalization condition is (1 + KM)7Kq + q ¼ 1, which gives q, p0 and p¼

K ð1 þ K M Þ7 1 þ K ð1 þ K M Þ7

:

ð7:2Þ

One quantity of experimental interest is the fractional occupation θ of the actin sites, which can be measured in solution by titrating with different myosin concentrations. If Tm is in the open state, the probability that any one actin site is occupied is equal to KM/(1 + KM). Hence the fractional occupancy of the seven-site cooperative unit by myosins is. θ¼

KM KK M ð1 þ K M Þ6 p¼ : 1 þ KM 1 þ K ð1 þ K M Þ7

ð7:3Þ

Figure 7.4 shows how this overall occupation of actin sites varies with KM, which is proportional to [S1]. When K >> 1, the cooperative unit is effectively switched on and the binding curve is just proportional to KM/(1 + KM), as expected from a single site. This is the scenario expected at high levels of calcium (above 100 μM), but in practice the binding curve falls below a hyperbola at very low myosin levels (Maytum et al. 1999), which is ﬁtted if K 0.5. At lower calcium concentrations (1–10 μM), cooperative inhibition at low myosin levels is more pronounced and the data shown can be ﬁtted with K ¼ 0.011. When myosin concentration is raised above the point where K(1 + KM)7 1, the population of occupied actin sites forces tropomyosin into the open state despite the low value of K, and the binding curve rises steeply towards the full-activation curve. This S-shaped curve is a signature of cooperative myosin binding to regulated actin, because moving TmTn out of the groove exposes not just one actin site but seven sites. To switch the regulatory system fully on requires not just high calcium but a high concentration of myosin as well (Heeley et al. 2006).

7.1 Steric Blocking Models

7.1.2

299

The Rate of Myosin Binding is Also Regulated

At this juncture, we depart from the historical development of regulatory models and explore the kinetics of myosin binding; can the binding transient also be simulated with the two-state steric blocking model? When myosin and regulated actin are mixed in a stopped-ﬂow experiment, the initial binding rate at low calcium is lowered to about one-third of the rate observed at high calcium (Trybus and Taylor 1980; Geeves et al. 2011). As one might guess, the binding rate at high calcium is the same as observed with unregulated actin, although the detachment rate is reduced by tropomyosin (Maytum et al. 1999). To follow the sequence of binding reactions in Scheme 7.1, suppose that a mixing experiment is initiated at t ¼ 0. Let the states be numbered by the number of myosins bound, with pj(t) (j ¼ 0,1,. . . .,7) for the probability of state j at time t. Let kM and k-M be the myosin binding and detachment rates for a single actin site. The corresponding rates kj and k-j between states j-1 and j are different because of the numbers of sites involved. Note also that state 0 is special because it must include the blocked state of the regulatory system as well as the empty open state. We assume that these two states are in rapid equilibrium, so they can be lumped together. Then the effective binding rate k1 out of state 0 must be reduced by the probability K/ (1 + K ) that the TmTn complex is in the open state. Taking all site degeneracies into account, the transition rates are given by 7K kM , 1þK kj ¼ ½7 ðj 1Þk M ,

k1 ¼

k1 ¼ kM , kj ¼ jkM

ðj ¼ 2, :::::7Þ:

ð7:4Þ

On this basis, the rate equations for the state populations in an ensemble are. p_ 0 ðtÞ ¼ k1 p0 þ k1 p1 , p_ j ðtÞ ¼ k j pj1 ðk j þ kjþ1 Þpj þ kðjþ1Þ pjþ1 p_ 7 ðtÞ ¼ k 7 p6 k 7 p7 :

ðj ¼ 1, . . . , 6Þ,

ð7:5Þ

Then the mean fractional occupation of states by myosin-S1’s is given as. θðtÞ ¼

7 1X jp ðtÞ: 7 j¼1 j

ð7:6Þ

Also, it can be veriﬁed that the equilibrium population of state j relative to state 0 is. p j ð 1Þ ¼ p0 ð 1 Þ

K 7 ðK Þj j 1þK M

ðj ¼ 1; : . . . ; 7Þ

ð7:7Þ

where the combinatorial factor is 7!/j!(7–j)! and j! ¼ j(j–1),. . . .,1. This formula leads back to Eq. 7.3 by a different method. Equations 7.5 need to be solved numerically,

300

7 Models of Thin-Filament Regulation

Fig. 7.5 Myosin-S1 binding transients to regulated actin, showing experimental data of Geeves et al. (2011) under conditions of excess S1. Measurements at low Ca++ are roughly ﬁtted by a kinetic version of the steric blocking model with kM ¼ 60 s1 and KM ¼ 8, after scaling the ordinate by a factor of KM/(1+ KM) for the predicted asymptotic binding fraction. But the high-calcium data cannot be ﬁtted simply by changing the value of the regulatory constant K as the model requires. Data supplied by Prof. Michael Geeves

and this is best accomplished by an implicit method (Press et al. 1992), because of the wide range of rate constants in Eqs. 7.4. Figure 7.5 shows myosin-S1 binding data of Maytum et al. (1999) at low and high calcium, and the transients predicted by using the same values of K (0.011 and 0.5) that ﬁtted the titration data, KM ¼ 8 (for 5 μM S1) and a single-site binding rate of 60 s1 at both calcium levels. Apart from the initial lag, there is a reasonable ﬁt to the low-Ca data, but the high-Ca transient is about 10 times faster than observed. As there is no reason to suppose that the S1 binding rate kM to an open site should be calcium-dependent, this disagreement should be taken as a sign that the two-state steric blocking model is defective.

7.1.3

Closed-Open Models

Historically, the steric blocking model lasted for about a decade before it was severely modiﬁed, for reasons unrelated to the above kinetic discrepancy. At the time, the problem was that myosins were observed to bind weakly to actin at low calcium. At low ionic strength, myosin binds to regulated actin in the absence of calcium (Brenner et al. 1982), and ATPase activity, although inhibited, is also present (Chalovich and Eisenberg 1982). Hence the model was rejigged to allow weak myosin binding at low calcium, with a corresponding change in terminology; the ‘blocked’ state was replaced by a ‘closed’ state. This kind of model was ﬁrst proposed by Hill et al. (1980), and in a simpler form by Geeves and Halsall (1987). Both models can be described as ‘on-off’ models, since they still employ only two regulatory states. The Geeves-Halsall model is equally good at ﬁtting myosin

7.1 Steric Blocking Models

301

titration data, so in that respect it cannot be discriminated from the original steric blocking model. Models based on closed and open states also had a rather limited lifetime. The ﬁrst attempt at resolving the discrepancy between equilibrium S1 binding and S1 binding kinetics was made by McKillop and Geeves, who proposed a model with three regulatory states.

7.1.4

A Blocked-Closed-Open Model

In the model of McKillop and Geeves (1993), myosin-S1 cannot bind to actin in the blocked state, weak binding is permitted in the closed state, and in the open state weakly-bound myosins can isomerize to a strongly-bound conformation. As in closed-open models, the underlying assumption is that weakly- and stronglybound acto-S1 states coexist in the same S1-nucleotide complex. Many such complexes have been used in solution studies; for example, S1, S1.ADP, S1.ATP and S1. ADP.BDM. Experimentally, bound myosins were detected by a ﬂuorescence signal from pyrene-labelled actin, which is assumed to respond only to strongly-bound heads. Note also that the weak-to-strong isomerization need not be accompanied by a working stroke; a stroke occurs only for S1.ADP.Pi and then only after Pi is released internally from its active site. This model is summarized in Scheme 7.2, where KA is the myosin afﬁnity for weak binding and KS is the isomerization constant to the strongly-bound state. blocked

○○○○○○○ KB closed

KA

○○○○○○○ KT

open

○○○○○○○

KA

○●○○○○○ KA

KA

○●○●○○○ KA

○●○○○○○ KS

○▼○○○○○

KA

○●○●○○○ KS

○▼○▼○○○

….. ●●●●●●● ….. ●●●●●●● KS

….. ▼▼▼▼▼▼▼

Scheme 7.2 The three-state regulatory model of McKillop and Geeves

The motivation for introducing a third regulatory state is two-fold. Firstly, equilibrium S1-titration data at high calcium require only modest unblocking, namely K ¼ 0.2–0.5 in the steric blocking model; calcium alone is not enough and bound myosins must complete the job. This state of affairs seems to contradict the observation that the rate of S1 binding to A.Tm.Tn at high Ca is as fast as for

302

7 Models of Thin-Filament Regulation

unregulated actin (Trybus and Taylor 1980), which suggests that K > > 1. But experimental uncertainties may mean that K values near unity could work. A more compelling reason is as mentioned before; when low Ca is replaced by high Ca, the observed binding rate increases only by a factor of 2–4, whereas the equilibrium data are ﬁtted by K values of 0.011 and 0.5 (Fig. 7.4), which implies that the fraction of open states increases by the ratio of the values of K/(1 + K ), namely 30. Thus the connection between binding rate and the fraction of Tm units in the open state is broken. McKillop and Geeves solved this problem by assigning the calcium dependence of regulation to the blocked-closed transition, with an equilibrium constant KB of 0.3 at low Ca and large values, say 30, at high Ca. However, the detection of only strongly-bound myosins meant that the closed-open transition, with an equilibrium constant KT, also had to take place, and KT was assigned values of 0.01–0.2, essentially independent of calcium level. For titration data, the BCO model behaves in almost the same way as the steric blocking model with K ¼ KBKT. With KT ¼ 0.01, K varies from 0.003 to 0.3 with Ca++, in the right range to ﬁt S1 binding curves such as those in Fig. 7.4. However, the regulatory factor K/(1 + K ) which reduces the initial binding rate in the steric blocking model is now replaced by KBKT/ (1+ KB(1 + KT)), which varies with Ca from (0.3/1.3)KT to KT when KT 1. The time course of S1 binding has to be calculated numerically from an extended version of Eqs. 7.5. As we have seen, the middle term in the denominator, which ﬁxes the population of closed states, cannot be neglected in generating the kinetic response. The BCO model provides much better ﬁts to myosin titration data and myosin binding transients. The slight S-shape to the titration data at high calcium is reproduced correctly, and the pronounced S-curve at low calcium is accurately ﬁtted. Moreover, numerical solution of the appropriate rate equations gives a good ﬁt to the binding transients, with a ratio of 4.4 for the initial rates at low and high calcium. The original paper should be consulted for these results and values of the ﬁtting parameters. The lasting value of the BCO model lies not so much in its detailed structure but in the structural signiﬁcance of the three regulatory states. Putting all the calcium

7.2 How Is Thin-Filament Regulation Controlled by Calcium?

303

dependence into the blocked-closed transition implies that this transition corresponds to dissociation of TnI from actin, whereas the calcium-independent closedopen transition is driven entirely by bound myosins. Thus the closed state is the apo-state with all TnI’s and myosins detached from actin; it should therefore be the native state of A.Tm and pure actin ﬁlaments in the absence of myosin. These conclusions were dramatically conﬁrmed four years later by structural studies with X-ray diffraction and cryo-electron microscopy. They opened the way for a totally different model of thin-ﬁlament regulation, which is the subject of Sects. 7.4, 7.5, 7.6, 7.7, and 7.8. To complete this survey of cooperative-unit models, consider two more aspects of the problem. One is the way in which the regulatory switch is driven by calcium ions, and the other is the extension of c-u models to include interactions between the ends of adjacent tropomyosins, as proposed by Hill et al. (1980). This model needs to be introduced with a small dose of statistical mechanics.

7.2

How Is Thin-Filament Regulation Controlled by Calcium?

Calcium controls the way in which the tropomyosin-troponin complex regulates myosin binding to sites on the thin ﬁlament (regulated F-actin), and it does that by binding to troponin-C. But how does that cause troponin-I to detach from actin? It is known that when this happens, the actin-binding arm of TnI is captured by TnC, which provides the link with calcium binding to TnC. Hence it is natural to suppose that an allosteric transition occurs between two states of TnC and TnI in the troponin complex, while the third component TnT remains bound to tropomyosin. The following reaction scheme is a classical example of the general allosteric principle formulated by Monod et al. (1965). For most muscles, two calcium ions must bind to complete the allosteric switch between the blocked and open states of the Tm-Tn complex. Scheme 7.3 shows a simple version of this reaction scheme. Scheme 7.3 An allosteric reaction scheme for coupling calcium binding to the blocked-open transition

blocked

L

A.TnI + TnC

open A + TnC.TnI 2Kc / e

2K c eL

A.TnI + CaTnC 1 2

A + CaTnC.TnI

Kc

1 2

Kc / e

e L 2

A.TnI + Ca 2TnC

A + Ca2TnC.TnI

304

7 Models of Thin-Filament Regulation

Note that the calcium afﬁnities satisfy detailed balancing. The switch from the open to the blocked state is controlled by the quantity L, which looks like the afﬁnity of TnI for actin. It is truer to say that L is a transfer afﬁnity (and hence dimensionless), since the arm of TnI must ﬁrst detach from TnC before it can bind to actin. At very low calcium, the uppermost blocked state is stabilized if L >> 1, which is presumed to be the case. This afﬁnity is progessively weakened by a factor of ε ( 1) the maximum level of activation is λ/(1+λ), less than unity. p(C) is a sigmoidal function of calcium, and the calcium level for half-maximal activation and the slope at the half-way point can be calculated explicitly. The results are given in Appendix K. In the next section, the same result follows from some more powerful statisticalmechanical machinery, which allows this scheme to be included from the outset in a regulatory model with interactions between cooperative units. This is the method of the grand canonical ensemble, where the partition function is calculated by allowing the numbers of particles (here myosins and calcium ions) to vary.

7.3 An On-Off Model with Tropomyosin Interactions

7.3

305

An On-Off Model with Tropomyosin Interactions

Two lines of argument suggest that thin-ﬁlament regulation by independent TmTn complexes may not be a good model of how muscle is activated by calcium (and myosin). The ﬁrst is structural; adjacent tropomyosin units, which in fast striated muscle are 40 nm long, do have overlapping ends, so they can regulate all seven sites of each 36 nm period of each strand of F-actin. There is also evidence that the effective number of actin sites in the cooperative unit can be greater than the seven sites covered by overlapping tropomyosins. All this begs the question of why the tropomyosin ﬁlaments are arranged in this way; how do they manage to position themselves with overlaps along the actin ﬁlament with its 36 nm periodicity? Before these questions became acute, a comprehensive on-off model with end-toend tropomyosin interactions was proposed by Hill et al. (1980). This model became the template for ﬁtting various kinds of experimental data that characterise thinﬁlament activation. The model is formulated only for regulated actin ﬁlaments in solution, but there is also much experimental data for skinned ﬁbres, where the calcium ion concentration can also be varied and precisely controlled. To ease you in gently, the statistical-mechanical formulation of the HillEisenberg-Greene model, which uses the grand canonical ensemble, is ﬁrst applied to an isolated TmTn unit, as in the model of Geeves and Halsall (1987). This serves as the foundation of the HEG model. In both models there are two regulatory states, closed and open, with myosin binding weakly in the former and strongly in the latter.

7.3.1

The Grand Partition Function for a Single Cooperative Unit

In statistical mechanics, a partition function is the sum over states of a Boltzmann factor exp.(βEα), where α labels a state of energy Eα and β ¼ 1/kBT. In this canonical ensemble the total number of particles is ﬁxed. However, ‘on-off’ models for the cooperative unit (one molecular TmTn complex) do not have ﬁxed mole numbers of bound myosins and calcium ions; instead these numbers are controlled by their concentrations in solution through the corresponding afﬁnities or, equivalently, chemical potentials. This is where the grand partition function (g.p.f.) comes to the rescue; in the simplest case where the particles are of the same kind, the energy Eα in the Boltzmann factor is replaced by Eα–μNα, where the number of particles Nα is state-dependent and the chemical potential μ is the energy required to add a particle while maintaining thermodynamic equilibrium (Hill 1960). In the present context, this deﬁnition of the g.p.f. needs to be generalized. There are two kinds of particles (myosin-S1 and Ca++) that bind to the cooperative unit, which in this case is seven actin sites covered by one TmTn protomer. Secondly, the chemical potentials for adding these particles both depend on the regulatory state of the unit, on or off. To keep track of all this, a more systematic notation is desirable.

306

7 Models of Thin-Filament Regulation

Let m be the number of myosins bound to actin on the cooperative unit (c.u.), and n the number of Ca2+ ions bound to its TnC. The calcium afﬁnities were deﬁned in Scheme 7.3, and a similar notation for myosin afﬁnities is desirable. For the GeevesHalsall model, let K 0M and K 1M be the pseudo-ﬁrst-order afﬁnities, proportional to [S1]. By Gibbs’ thermodynamic identity, the corresponding chemical potentials are μσM ¼ kB T ln K σM ,

μσC ¼ kB T ln K σC

ðσ ¼ 0; 1Þ

ð7:11Þ

where σ ¼ 0 and 1 for closed and open states, and K 0C ¼ K c , K 1C ¼ K c =ε as in Sect. 7.2. The energy Eσ of the unit also depends on its regulatory state. Setting E 0 ¼ kB T ln L,

E1 ¼ 0

ð7:12Þ

accounts for the TnI-actin transfer energy relative to the open state in the absence of bound myosin and Ca2+. Hence the g.p.f. of the unit is 1 X 7 X 2 X 7 2 βðEσ mμσM nμσC Þ ξ¼ e m n σ¼0 m¼0 n¼0

ð7:13Þ

where the combinatorial factors are the degeneracies of 7 actin sites and 2 calcium sites with m and n sites occupied. In terms of afﬁnities, ξ¼

7 X 2 X 7 2 M m C n M m C n Ko þ K1 K1 L Ko m n m¼0 n¼0

and the sums (not surprisingly) are binomial expansions. Thus ξ ξ0+ξ1 from the closed and open states respectively, where 7 2 ξ0 ¼ L 1 þ K 0M 1 þ K 0C ,

7 2 ξ1 ¼ 1 þ K 1M 1 þ K 1C :

ð7:14Þ

and n of bound myosins and Next are recipes for calculating the mean numbers m calcium ions in the grand ensemble, and the fraction σ of cooperative units in the open state. From the deﬁnition in Eq. 7.13, m¼

∂lnξ , ∂ln½S1

n¼

∂lnξ , ∂ln½Ca2þ

1σ ¼

∂lnξ , ∂lnL

ð7:15Þ

where the concentrations enter through K σM ¼ K~ σM ½S1 and K σC ¼ K~ σC Ca2þ : For the present ensemble of isolated c.u.’s, it follows that

7.3 An On-Off Model with Tropomyosin Interactions

307

m K 0M ξ0 K 1M ξ1 ¼ þ 7 1 þ K 0M ξ 1 þ K 1M ξ

ð7:16aÞ

n K 0C ξ0 K 1C ξ1 ¼ þ , C 2 1 þ K 0 ξ 1 þ K 1C ξ

ð7:16bÞ

σ¼

ξ1 : ξ

ð7:16cÞ

These formulae are to be understood as fractional site occupations weighted by the fractional occupation of open and closed states. After substituting from Eq. 7.14, what can we learn from these results? of actin sites agrees with Firstly, Eq. 7.16a for the fractional occupancy θ ¼ m=7 Eq. 7.3 for steric blocking if K 0M ¼ 0, K 1M ¼ K A and K¼

1 þ K 1C 1 þ K 0C

2

1 L

ð7:17Þ

is the function deﬁned in Eq. 7.9. In this way, the grand partition function reproduces the MWC allosteric scheme for Ca-TnC binding. Secondly, Eq. 7.16b for the calcium occupancy of TnC can be analysed in the same way. Just as the extent of myosin-actin binding is regulated by calcium bound to TnC, so is the extent of calcium binding to TnC regulated by bound myosins. Here is a beautiful symmetry, but also a trap for the unwary; in these on-off models, the apparent equilibrium constant of the regulatory switch is different for myosin binding and calcium binding! The analogous formula is n K 0C 1 þ K 0C þ K TC K 1C 1 þ K 1C ¼ 2 2 2 1 þ K C þ K TC 1 þ K C 0

ð7:18Þ

1

where K TC

7 1 þ K 1M 1 1 þ K 0M L

ð7:19Þ

is a function of [S1]. Thus the myosin-actin binding curve is modulated by the Ca2+ concentration in solution, and the calcium-TnC binding curve is modulated by the concentration of myosin-S1, but more cooperatively because of seven actin sites against two calcium sites.

308

7 Models of Thin-Filament Regulation

Fig. 7.6 A cartoon of the regulatory model of Hill et al. (1980), simpliﬁed to allow the same attractive end-to-end interaction between adjacent tropomyosins in the closedclosed and open-open states

Finally, note that the truest measure of the equilibrium between closed and open states is deﬁned by a third equilibrium constant, namely KU

ξ1 ¼ ξ0

1 þ K 1M 1 þ K 0M

7 2 1 þ K 1C 1 1 þ K 0C L

ð7:20Þ

which is the ratio of the occupancies of all open and closed states in the cooperative unit.

7.3.2

A Model with End-to-End Tropomyosin Interactions

In describing the model of Hill et al. (1980), it is convenient to work with a strippeddown version with fewer parameters than the original. Figure 7.6 shows a diagrammatic version of their model, simpliﬁed to allow the same attractive interaction J between adjacent tropomyosins in the same state, open or closed. Then a grand partition function ΞN for N interacting TmTn units can be deﬁned and, because the problem is essentially one-dimensional, calculated by the transfer-matrix method. The calculation goes as follows. Let F ðσ; m; nÞ ¼ E σ mμσM nμσC be the energy function for a single cooperative unit that appears in the exponent of Eq. 7.13. For the simpliﬁed HEG model considered here, the equivalent function for N interacting units is. N X

fF ðσ α ; mα ; nα Þ J ½σ α σ α1 þ ð1 σ α Þð1 σ α1 Þg:

ð7:21Þ

α¼1

Hence, following the deﬁnition in Eq. 7.13, the grand partition function for N units is.

7.3 An On-Off Model with Tropomyosin Interactions

ΞN ¼

309

N X 1 X 7 X 2 Y 2 7 eβfFðσα ;mα ;nα ÞJ ½σα σα-1 þð1σα Þð1σα-1 Þg n m α α α¼1σ ¼0 m ¼0 n ¼0 α

¼

N X 1 Y

α¼1σ α ¼0

α

ξσ α e

α

ð7:22Þ

βJ ½σ α σ α-1 þð1σ α Þð1σ α-1 Þ

where ξ0 and ξ1 are the constrained single-unit partition functions of Eq. 7.14. To calculate ΞN by the transfer-matrix method (Hill 1960), it is necessary to work with a constrained g.p.f. in which the regulatory state of the last cooperative unit is held ﬁxed. Let ΞN,σ be the g.p.f. for which the Nth unit is in state σ. Then ΞNþ1, σNþ1 ¼

1 X σ N ¼0

ξσNþ1 eβJ ½σNþ1 σ N þð1σNþ1 Þð1σN Þ ΞN, σN

or, in matrix-vector form,

ΞNþ1, 0 ΞNþ1, 1

¼

ξ0 eβJ ξ1

ξ0 ξ1 eβJ

ΞN, 0 ΞN, 1

ð7:23Þ

which reveals the 2 2 transfer matrix T. Iterating this equation gives the desired solution ΞN + 1 ¼ TN Ξ1. The transfer matrix can be diagonalised by a bi-orthogonal transformation (Press et al. 1992). In this way one can show that, in the limit of large N, ΞN ΞN, 0 þ ΞN, 1 bλþN

ð7:24Þ

where λ+ is the larger eigenvalue of T. The constant b does not concern us, because only terms of O(N) in ln ΞN are required to calculate average values in the limit N > > 1. Averages over the N-unit system can be calculated in the same way, namely. m¼

N∂lnλþ , ∂ln½S1

n¼

N∂lnλþ , ∂ln½Ca2þ

Nð1 σÞ ¼

Nlnλþ : ∂lnL

ð7:25Þ

Now λþ ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 ðξ1 þ ξ0 ÞeβJ þ ðξ1 ξ0 Þ2 e2βJ þ 4ξ1 ξ0 2

ð7:26Þ

and this formula shows how end-to-end coupling increases cooperativity. Consider the limiting cases of small and large coupling:

310

7 Models of Thin-Filament Regulation

1. When J ¼ 0, then λ+ ¼ ξ0+ξ1 ξ and the single-unit g.p.f. is recovered. 2. When βJ > > 1, λ+ ¼ max{ξ0,ξ1}exp(βJ). The exponential factor disappears from the averages listed above, but what remains is the single-site g.p.f. of the dominant regulatory state, rather than the average g.p.f. over closed and open states. As we have seen, whether the closed or open state predominates is controlled by the concentrations of calcium and myosin. In this way, end-toend tropomyosin interactions increase the extent of cooperative behaviour already present within each regulatory unit. From Eqs. 7.25–7.26, explicit formulae for the fractional occupations of actin sites, TnC calcium sites, and the open regulatory state when N >> 1 are as follows: mN 1 K 0M ∂λþ K 1M ∂λþ ¼ ξ þ ξ , 7N λþ 1 þ K 0M 0 ∂ξ0 1 þ K 1M 1 ∂ξ1 nN 1 K 0C ∂λþ K 1C ∂λþ ¼ ξ þ ξ , θC 2N λþ 1 þ K 0C 0 ∂ξ0 1 þ K 1C 1 ∂ξ1

θM

θU

σN ξ ∂λþ ¼1 0 : N λþ ∂ξ0

ð7:27aÞ ð7:27bÞ ð7:27cÞ

How they behave as functions of [Ca2+], [S1] and J is summarized in Fig. 7.7. A footnote for users of the HEG model; the quantity L has the same meaning as that symbol in their 1980 paper. The other important parameters are L0 ¼

1 þ K 0C 1 þ K 1C

2 L

1 , K

Y ¼ e2βJ

ð7:28Þ

where K is the function in Eq. 7.17. Remarkably, the original HEG model with many more interaction parameters is ﬁxed by L’ and Y, so there is no loss of generality in using the simpliﬁed version described here. The Hill-Eisenberg-Greene model has inspired various models along the same lines, while retaining the basic restriction to rigid ﬁlaments and nearest-neighbour interactions. For example, Robinson et al. (2002) have proposed a heuristic model to describe the kinetic effects of interactions, and Zou and Phillips (1994) formulated a model based on cellular automata. The HEG model will generate myosin binding transients if more parameters are introduced to handle the effect of interactions on myosin binding rates (Chen et al. 2001), and Mijailovich et al. (2012) have concluded that the three-state BCO model offers a more straight-forward description of myosin kinetics. However, end-to-end tropomyosin interactions need to be re-evaluated if Tm protomers are not rigid units but ﬂexible structures. In the next section, we examine the evidence in favour of ﬂexible tropomyosin units with strong end-to-end interactions, which is the overture to a radically different type of model for regulated myosin binding to the thin ﬁlament.

7.4 Tropomyosins as a Continuous Flexible Chain

311

Fig. 7.7 The effect of end-to-end tropomyosin interactions predicted by the model of Hill et al. (1980). (a) the fractional occupation of actin sites by myosin-S1, (b) the fractional occupation of the regulatory sites of TnC by calcium ions and c: the fraction of regulatory units in the open state. Colour codes: black (–S1,–Ca); red (+S1,–Ca); green (–S1,+Ca); yellow (+S1,+Ca). Calculations were made from Eqs. 7.27a, 7.27b, 7.27c, and the abscissa Y ¼ exp(2βJ) is a measure of interaction strength. All variations are shown at low/high concentrations of myosin and calcium as measured by their pseudo-ﬁrst-order afﬁnities, namely K 1M ¼ 0.5 and 2, K 0C ¼0.2 and 0.5. Other parameter values were L ¼ 90 and ε ¼ 0.15. More widely separated values yield limiting binding fractions which are unaffected by the interaction. Note that all fractions are much more sensitive to end-to-end interactions at the higher level of S1, where nearly all cooperative units are in the open state

7.4

Tropomyosins as a Continuous Flexible Chain

In this chapter we present arguments, evidence from solution-kinetic experiments, and structural determinations by X-ray and electron microscopy, that all point towards a different mechanism of regulation by tropomyosin. Instead of ‘on-off’ models where tropomyosins behave as rigid switching units, tropomyosins should be regarded as ﬂexible structures which interact to form a quasi-continuous ﬂexible chain (Lehrer et al. 1997), particularly when each tropomyosin comes with troponin attached. The notion that tropomyosin is a ﬂexible structure is supported by the way in which tropomyosins attach to, or rather sit on, the actin double-helix. In practice, a single Tm protomer binds very weakly to actin with an afﬁnity of 2–5 103 M1 (Wegner 1980); the ﬁrst-order afﬁnity at 1 mM Tm would be of order 3, for which

312

7 Models of Thin-Filament Regulation

the binding energy is barely above kBT. So individually-bound Tm protomers are likely to ﬂoat off into solution. However, by raising the Tm concentration to the point where protomers are positioned along the whole length of F-actin, end-to-end interactions hold the Tm assembly together to form a chain. Then a kind of loose binding does occur, since weak Tm-actin interactions over a 1 μm actin ﬁlament (a chain of 25 units) would have a net afﬁnity of 325 (Tobacman and Butters 2000). These weak interactions allow the Tm chain to move in and out of the groove as necessary for regulated S1-actin binding. There have been various claims that the chain not only rocks about the groove, but also rolls, a truly groovy concept now supported by multiple ﬂuorescent-labelling (Bacchiocchi et al. 2004). The formation of a chain of linked tropomyosins is obviously facilitated if each tropomyosin is ﬂexible and bends under Brownian forces.

7.4.1

The Size of the Cooperative Unit

Strong evidence for interacting ﬂexible tropomyosins came when tropomyosin movements relative to S1 binding were studied in solution with pyrene-labelled tropomyosin (Geeves and Lehrer 1994). The pyrene ﬂuorescence signal monitored the time course of azimuthal tropomyosin movements, while the extent of S1 binding was measured from light scattering. The results were interpreted in terms of independent tropomyosin units, and the number n of actin sites covered by each unit was treated as a free parameter rather than ﬁxed at seven. Two solution-mixing protocols were used, which gave very similar results. In the ﬁrst, S1 was mixed with regulated actin (A.Tm or A.Tm.Tn) to follow the time course of Tm movement as well as S1-actin binding. Here the Tm signal was much faster than S1 binding (Fig. 7.8a), as each Tm unit switched on in response to the ﬁrst S1-binding event. If kM is the rate of S1 binding to a single actin site, then the rate constant for binding to n sites is nkM (Eq. 7.4), and this rate controls the Tm

Fig. 7.8 Simultaneous observation of the time courses of S1 attachments and tropomyosin movements on A.Tm ﬁlaments, monitored by light scattering and ﬂuorescent-labelled tropomyosin respectively. (a) S1 binding and (b) ATP-induced dissociation. From Geeves and Lehrer (1994), with permission of Elsevier Press

7.4 Tropomyosins as a Continuous Flexible Chain

313

movement. As S1 was in excess, with K~ M ½S1 60 at 5 μM S1, actin binding was almost irreversible. Then the probabilities of bound S1’s and the ‘on’ state of Tm are pS1 ðtÞ ¼ 1 ekM t ,

pTM ðtÞ ¼ 1 enkM t

ð7:29Þ

and the Tm ﬂuorescence signal leads by a factor of n. With A.Tm.Tn and low Ca, the lag in S1 binding would be due to the ﬁrst S1 binding in the blocked state. In the second protocol, bound myosins were already present in equilibrium, and this solution was mixed with ATP at a concentration sufﬁcient to detach them all. ATP-induced detachment of S1’s from actin is so fast that the effective rate of this process is the rate of ATP binding, proportional to the ATP concentration. Now the tropomyosins of each cooperative unit must wait until all myosins have detached before they can return to the ‘off’ state (Fig. 7.8b). If kATP is the rate of ATP binding to a single actin-bound S1, the probability that the site is vacant after time t is q(t) ¼ 1 exp (kATPt). The probability that all n sites are vacant is q(t)n and this is the probability of the ‘off’ state. If, as before, the base line of the ﬂuorescence signal was the ‘off’ state, then that signal measures the probability pTM(t) ¼ 1 q(t)n of the ‘on’ state, so pS1 ðtÞ ¼ ekATP t ,

pTM ðtÞ ¼ 1 ð1 ekATP t Þn :

ð7:30Þ

These functions provide a good ﬁt to the two experiments. Using 5 μM S1 and 1 μM actin, the authors found that the curves in Fig. 7.8a were ﬁtted with n ¼ 6.4 for actin.Tm, and n ¼ 7.6 and 10.2 for actin.Tm.Tn at low and high Ca respectively. The obvious interpretation of these results is that n is the size of the cooperative unit. If this is true, then the fact that n ¼ 10–12 for A.Tm.Tn at low Ca spells the end for all models based on independent 7-site tropomyosins. The cooperative unit is instead a measure of the size of the umbrella formed when the tropomyosin chain is pushed out of the groove by one bound myosin-S1. This claim can be backed up by calculating the transient responses from a rigid Tm model covering n actin sites. The differential equations required are an obvious extension of Eqs. 7.4, 7.5, with the ﬂuorescence signal identiﬁed with the probability pTM ðt Þ ¼ 1

p0 ð t Þ 1þK

ð7:31Þ

that the regulatory system is in the open state. (Remember that p0 is the probability of no bound myosins, so in Scheme 7.1 the probability of the blocked state is p0/(1 +K )). As a ﬁrst step, assume that n ¼ 7 and calculate the two transient responses, assuming that KM > > 1 because S1 was in excess. Figure 7.9 shows what happens when this model is tuned to ﬁt the results for A.Tm. By adjusting the value of K, the rate constants obtained by ﬁtting the two signals are in the ratio 6.4:1, as observed. K ¼ 10 is optimal, and this ratio is an increasing function of K, falling to unity when K 0, and the chain is forced to wrap itself around the far side of the bound myosin, in a narrow range of angles above ϕM. The chain is effectively pinned at this angle by a bulge in the actin interface (Lehman et al. 2013). a

b

φM2

φM2

0

0

φM1

φM1 M1

M2

M1

M2

B

B

M1

M2 M1

B

Fig. 7.11 Cartoons illustrating the continuous-ﬂexible-chain (CFC) model of thin ﬁlament regulation. (a) At high calcium, the chain is displaced to angle ϕM2 > 0 by strongly bound myosins (M2), which enables it to avoid weakly-bound myosins (M1) at a range of angles around ϕM1 < 0. (b) At low Ca, the chain is pinned at the blocking angle ϕB ϕM1 where TnI (shown as circles on the chain) is bound to actin. Strong myosin binding near TnI pinning centres is inhibited by the required distortion of the chain. From Geeves et al. (2011), with permission of Elsevier Press

7.4 Tropomyosins as a Continuous Flexible Chain

317

6. The chain ends up on the far side of any strongly-bound myosin because weak binding occurs ﬁrst, in a range of orientations below the closed-state range but above ϕB. In that case, the chain lies above the weakly-bound myosin, and is pushed towards higher angles as the myosin seeks a bigger interface on actin to bind more strongly. 7. The persistence length LP of the chain is determined by its bending stiffness κΤ (deﬁned for angular displacements on actin) and the curvature α of the azimuthal conﬁning potential. In the absence of pinning centres, the range of angles generated by thermal agitation has a standard deviation σ o. Exact formulae for these quantities, derived in the next section, are LP ¼

1=4 4κ T , α

σo ¼

1=2 kB TL3P : 8κ T

ð7:32Þ

κT can be estimated from the persistence length LPS of tropomyosin in solution, where LPS ¼ κT/R2kBT (Howard 2001), with radius R. Note that the bending stiffness as normally deﬁned is κ T/R2. A molecular-dynamics simulation which included the intrinsic curvature of the tropomyosin coiled-coil gave LPS ¼ 460 nm (Li et al. 2010a), about twice experimental values analyzed without allowing for curvature. As R ¼ 3.9 nm (Holmes and Lehman 2008), then κT ¼ 2.80 1044 N.m4. The curvature α can only be estimated from a model of electrostatic Tm-actin potentials. However, if the persistence length LP of the conﬁned chain is 20 nm (a number justiﬁed by subsequent simulations), then σ o ¼ 0.38 radians or 22 (these numbers replace incorrect values given in Geeves et al. 2011). 8. The width of the ‘umbrella’ of actin sites covered by the chain on either side of a pinning centre (either myosin-S1 or TnI) is 2LP. This umbrella does not have a well-deﬁned edge; as one looks along the chain away from the pinning site, the chain reverts smoothly to its equilibrium position ϕ ¼ 0 unless there are other pinning centres in the vicinity. The number of actin sites covered by the umbrella is n 2LP/c + 1 where c ¼ 5.5 nm is the actin monomer spacing, and this deﬁnition allows qualitative comparisons with the number of actin sites derived from models with independent regulatory units. For example, n ¼ 10 for A.Tm. Tn (Geeves and Lehrer 1994) equates to a persistence length of 25 nm. Figure 7.12 illustrates several consequences of this model, which can be understood intuitively without more mathematics. One is that not all actin sites will be covered by the chain in its blocked state, where all TnI’s are bound to actin. The distance between adjacent TnI’s on the chain is 38.5 nm, approximately two persistence lengths. In between these TnI’s, the chain relaxes partially towards the closed state and there will be some opportunity for myosins to bind under the chain. This resolves the original dilemma of having to choose between uni-angular blocked and closed states. Similarly, strongly bound myosins create umbrellas in the opposite direction, pulling the chain towards positive angles and allowing more myosins to bind weakly. To summarise, the new deﬁnitions of blocked and open states refer to local, not global, chain positions; the new deﬁnition of the closed state is indeed intermediate in angle but occupies a range of positions because of thermal agitation.

318

7 Models of Thin-Filament Regulation

Fig. 7.12 A gallery of chain conﬁgurations generated during MonteCarlo simulation with the CFC model. (a) Excess actin and high calcium, with myosins occupying 10% of actin sites. (b) Excess actin and low calcium, showing clustering of bound TnI’s, with 10% myosins bound and 14% of TnI’s. (c) Excess myosin, high calcium; here 10% myosin occupancy is sufﬁcient to open the chain over nearly all of its length, except for just one bound TnI. (d) Excess myosin, low calcium, where open segments are disrupted by clustering of bound TnI’s. From Geeves et al. (2011), with permission of Elsevier Press

Regions of the chain that lie between pinning centres may be considered as local versions of the closed state, but the range of angles involved depends on whether the nearest pinning centres on either side are both myosins, both TnI’s or a mixture of the two. The next sections provide a complete and self-contained formulation of the mathematics of the chain model, which has previously been scattered over several puiblished papers. The algebra is complicated and somewhat garrulous, since solutions of a fourth-order differential equation are required. Nevertheless, the results are essential for making predictions and testing the model against experiments. These aspects of the chain model are addressed in Sect. 7.8 and the ﬁnal chapter.

7.5 Mathematical Formulation of the Chain Model

7.5

319

Mathematical Formulation of the Chain Model

A “worm-like chain” model (Marko and Siggia 1995) is often used in biophysics to simulate the wriggling motions of a long-chain polymer in solution, provided that the external forces are properly speciﬁed (Wiggins et al. 1998). In the last section we introduced a continuous-ﬂexible-chain (CFC) model for linked protomers of tropomyosin-troponin on actin, where wriggling is restricted because the chain is conﬁned to one groove of the actin double-helix. However, conﬁnement introduces a higher level of complexity. This section starts with a general expression for the energy of the conﬁned chain, from which we calculate the minimum energy and conﬁguration of a chain pinned at one place by bound myosin or TnI, and then at two places. In between pinning sites, the chain is free to move and will exhibit thermally-driven positional ﬂuctuations. Thus the next step is to calculate the distribution of thermally-activated displacements between pinning centres; this result is far-reaching because it enables a model of regulated myosin binding which includes both the afﬁnity and the binding kinetics. For practical purposes, this step completes the mathematical development of the model in terms of closed formulae. However, these analytical results are approximate, because only the nearest pinning centres are considered. When multiple pinning centres are present, an estimate of the errors involved in making the ‘pair approximation’ is required.

7.5.1

The Energy of a Conﬁned Flexible Chain

Let the chain at distance s along its length have angle ϕ(s) relative to its position in the groove. The simplest model assumes that the chain sits in a harmonic potential well with energy proportional to ϕ(s)2. The bending strain is proportional to the inverse radius of curvature, which reduces to the second derivative for small strains, so the bending energy is proportional to (d2ϕ/ds2)2. Thus the energy of this conﬁguration of a chain of length L is. 1 E fϕð s Þ g ¼ 2

Z

L

κT ϕ00 ðsÞ2 þ αϕðsÞ2

ds

ð7:33Þ

0

which is a functional of ϕ(s). The bending stiffness κ T is a property of the chain, which is assumed to be homogenous despite its modular construction from Tm units. We also assume that the conﬁnement stiffness α is homogenous, and that the chain stays wrapped around the actin ﬁlament. Any reduction in end-to-end length as chain bending increases is small and can be neglected. To use this formula, we seek the minimum-energy conﬁguration subject to boundary conditions at each end, and then ask how this energy is raised when the chain is pinned in various places or constrained to go around obstacles. The chain is

320

7 Models of Thin-Filament Regulation

pinned where a molecule of TnI binds to actin, and is walled in when a myosin is strongly bound, the wall being treated as a one-sided constraint ϕ > ϕM. These problems can be solved with reference to the minimum-energy or ‘ground-state’ conﬁguration of the chain, to use quantum-mechanical terminology. The mathematical development is a simpliﬁed version of that published previously (Smith 2001), the difference being that the regularization procedure for calculating partition functions is omitted.

7.5.2

The Ground-State Energy

Suppose that the chain is ﬁxed at each end, where its displacement and slope are both speciﬁed. The variational condition for minimum energy is Z

L

δE ¼

ðκT ϕ00 δϕ00 þ αϕδϕÞds ¼ 0

ð7:34Þ

0

where δϕ ¼ 0 and δϕ0 ¼ 0 at each end. Integrating the ﬁrst term twice by parts produces vanishing contributions at the ends, and the integrand is reduced to (κTϕ(4)+αϕ)δϕ. So the extremal conditions are satisﬁed when d4 ϕ þ 4ξ4 ϕ ¼ 0 ds4

α 4 ξ : 4κT

ð7:35Þ

There are four linearly independent solutions of this d.e., which are the functions {uk(ξs), k ¼ 1,..,4} where u1 ðxÞ ¼ sinhx sin x, u3 ðxÞ ¼ sinhx cos x,

u2 ðxÞ ¼ coshx sin x, u4 ðxÞ ¼ coshx cos x:

ð7:36Þ

The persistence length LP is here deﬁned as 1/ξ. The general solution for a conﬁguration is ϕð s Þ ¼

4 X

ck uk ðξsÞ

ð7:37Þ

k¼1

and the coefﬁcients ck can be determined from the boundary conditions ϕð 0Þ ¼ ϕ a ,

ϕ0 ð0Þ ¼ ξψ a ,

ϕðLÞ ¼ ϕb ,

ϕ0 ðLÞ ¼ ξψ b

ð7:38Þ

at the ends a, b. It is convenient to set ϕ0 (s) ¼ ξψ(s) throughout. From this point on, algebraic manipulations (although straightforward) become somewhat tedious. The solution for the ck is given in Appendix A of Smith (2001).

7.5 Mathematical Formulation of the Chain Model

321

Integrating Eq. 7.33 by parts gives Eo in terms of the boundary conditions, because the main integral vanishes when Eq. 7.35 is satisﬁed. Thus Eo ¼

iL κT h 00 000 ϕ ðsÞϕ0 ðsÞ ϕ ðsÞϕðsÞ 0 2

ð7:39Þ

and Eq. 7.37 is required to calculate the derivatives at the ends. Using the scaled derivative ψ(s), the solution can be written in matrix form as E o ¼ κT ξ3 Φa AðX ab ÞΦa þ Φa BðX ab ÞΦb þ Φb AS ðX ab ÞΦb

ð7:40Þ

where Xab ¼ ξL, Φa ¼

ϕa , ψa

Φb ¼

ϕb , ψb

ð7:41Þ

and sinhXcoshX þ sin X cos X d ðX Þ sinh2 X þ sin 2 X A12 ðX Þ ¼ A21 ðX Þ ¼ d ðX Þ sinhXcoshX sin X cos X A22 ðX Þ ¼ d ðX Þ A11 ðX Þ ¼ 2

ð7:42Þ

sinhXcos X þ coshXsin X dðXÞ sinhXsin X B12 ðXÞ ¼ B21 ðXÞ ¼ 4 dðXÞ coshXsin X sinhXcos X B22 ðXÞ ¼ 2 dðXÞ B11 ðXÞ ¼ 4

Here d(X) ¼ sinh2X sin2X, and AS denotes the skew complement of A, whose off-diagonal elements have the opposite sign. The matrices A and B are fundamental to what follows. A simple example is the case of a long chain (much longer than the persistence length) clamped at one end (ϕa ¼ ψ a ¼ 0). The only matrix required here is Að1Þ ¼

2 1

1 1

and the ground-state energy is E 0 ¼ 2κξ3 ϕb 2 ϕb ψ b þ ψ 2b =2 : For a given displacement ϕb, this energy has a minimum value of κξ3 ϕ2b : when the slope ψ b is varied. However, E 0 ¼ 2κξ3 ϕ2b when the slope is held to zero, and this is just half the energy required to pin a longer chain to angle ϕb at a point far away from each end.

322

7 Models of Thin-Filament Regulation

7.5.3

Ground-State Energy of a Pinned Chain

By joining chains together, the above results for a chain constrained at each end can be applied to a multiply-pinned chain with different pinning angles. To begin, consider a single pinning site. Let site 1 be near the middle of a long chain, clamped at each end with zero displacement and zero slope. For a pinning angle ϕ1 and reduced slope ψ 1, Eq. 7.40 gives the energy as E o ¼ κT ξ3 Φ1 AS ð1Þ þ Að1Þ Φ1 ¼ κ T ξ3 4ϕ21 þ 2ψ 21

ð7:43Þ

which has a minimum value of 4κT ξ3 ϕ21 when ψ 1 ¼ 0, as claimed above. Next, consider two pinning sites with different angles ϕ1,ϕ2, separated by a distance of X in units of the persistence LP ¼ 1/ξ. Both sites should be far away from the ends of a long end-clamped chain. For given slopes at the pinning sites, Eq. 7.40 leads to the ground-state energy Eo ¼ κT ξ3 Φ1 AS ð1Þ þ AðX Þ Φ1 þ Φ1 BðX ÞΦ2 þ Φ2 AS ðX Þ þ Að1Þ Φ2 ð7:44Þ which can be understood by the same trick, constructing the chain from three segments a-1, 1–2 and 2-b. Now this energy must be minimized with respect to the slopes ψ 1,ψ 2 for which the matrix-vector products have to be written out in full. The answer is obviously a quadratic form in the pinning angles, namely Eo ðϕ1 ; ϕ2 jX Þ ¼ 4κT ξ3 Γ11 ðX Þϕ21 þ Γ22 ðX Þϕ22 þ 2Γ12 ðX Þϕ1 ϕ2 :

ð7:45Þ

Straightforward but lengthy algebra enables the coefﬁcients Γij(X) to be written in terms of the A and B matrices. However, full substitution leads to prohibitively lengthy equations, and symbolic manipulation software is essential to achieve a useful simpliﬁcation. I ﬁnd that Γij(X) ¼ Nij(X)/D(X) where N 11 ðX Þ ¼ N 22 ðX Þ ¼ 2e2X ð2 cos 2X þ sin 2X Þ 2e4X , N 12 ðX Þ ¼ 2eX ð cos 3X 2 cos X 3 sin X Þ þ 2e3X ð sin X þ cos X Þ, DðX Þ ¼ 5 þ 2e2X ð3 cos 2X þ 2 sin 2X Þ 2e4X þ 2ð cos 2X 3 sin 2X Þ þ cos 4X þ sin 4X: ð7:46Þ These daunting formulae are equivalent to those in Appendix B of the original paper. In the limit X >> 1, Γ11(X) ! 1 and Γ12(X) ! 0. At the origin, both are singular; Γ11(X) ~ 0.5/X2 and Γ12(X) ~ 0.5/X2. However, Γ11(X) + Γ12(X) is ﬁnite at the origin, and equal to 0.5. For a doubly-pinned chain, these functions give the following results, graphed in Fig. 7.13a:

7.5 Mathematical Formulation of the Chain Model

323

Fig. 7.13 (a) Energy E of the doubly-pinned chain as a function of reduced distance X between the pinning sites, for pinning angles ϕ1 ¼ ϕ2 ¼ 1 and ϕ1 ¼ 1, ϕ2 ¼ 1. The energy is normalised to the energy cost 4κ Τξ3 of a singly-pinned chain of unit displacement. (b) Mean and standard deviation of displacements of a thermally-excited chain as a function of reduced distance X ¼ ξx away from a pinning site. The standard deviation is in units of the free-chain value (kBT/8κ Τξ3)1/2

1. If the chain is pinned with the same displacement ϕ1 at both sites, the chain energy varies from 4κ T ξ3 ϕ21 when X ¼ 0, as expected for a single pinning site, to exactly twice that value when X > 5 and the pinning sites cease to interact. 2. If the doubly-pinned chain has different displacements at the pinning sites, the chain energy diverges as X ! 0, because the chain is forced to adopt different displacements over a very small distance. However, when X > 5, Eo ! 4κT ξ3 2 ϕ1 þ ϕ22 , which is the sum of the single-centre energies. Having got this far, we can generate an approximate expression for the groundstate energy of a chain with an arbitrary number of pinning sites, by including only interactions between neighbouring pinning sites. This “pair approximation” can be formulated by identifying single-site energies, pair interactions, triplet interactions, etc. For example, consider ﬁrst the case of two pinning sites 1,2 with separation X, and then three sites 1,3,2 separated by X and Y. The exact ground-state energies of these two examples can be written as. E o ðϕ1 ; ϕ2 jX Þ ¼ Aðϕ1 Þ þ Aðϕ2 Þ þ V ð2Þ ðϕ1 ; ϕ2 jX Þ E o ðϕ1 ; ϕ3 ; ϕ2 jX; Y Þ ¼ Aðϕ1 Þ þ Aðϕ3 Þ þ Aðϕ2 Þ þ V ð2Þ ðϕ1 ; ϕ3 jX Þ þ V ð2Þ ðϕ3 ; ϕ2 jY Þ þ V ð2Þ ðϕ1 ; ϕ2 jX þ Y Þ þ V ð3Þ ðϕ1 ; ϕ3 ; ϕ2 jX; Y Þ

ð7:47aÞ ð7:47bÞ

where A(ϕ) ¼ 4κTξ3ϕ2 is the one-site pinning energy, V(2) is the interaction energy of a pair of sites and V(3) is the residual triplet interaction. For two sites, the triplet interaction is absent, and subtracting the one-site energies from Eq. 7.45 gives the pair interaction energy V ð2Þ ðϕ1 ; ϕ2 jX Þ ¼ 4κT ξ3 ðΓ11 ðX Þ 1Þ ϕ21 þ ϕ22 þ 2Γ12 ðX Þϕ1 ϕ2 ,

ð7:48Þ

324

7 Models of Thin-Filament Regulation

using Γ22 Γ11. Why are there triplet interactions for three or more sites? Remember that the slope of the chain at each pinning site is free to adjust for minimum energy. For two sites this adjustment is calculated correctly in Eqs. 7.45 and 7.46, but when a pinning site (site 3 above) has other sites on each side, the slope adjustment cannot be made independently for each neighbouring site; the true slope will in general be different from that which minimizes the interaction energy of each pair (1–3 or 3–2). The error involved is assessed in the next section.

7.6

The Distribution of Thermally-Activated Chain Displacements

A different way of looking at the problem of pinning interactions arises from the following question: what controls myosin binding under the chain to an actin site between two pinning centres? To answer this question, it is not enough to know the mean displacement, which is associated with the ground-state chain conﬁguration. Thermally-activated conﬁgurations must also be considered, so we wish to calculate the probability distribution of displacements away from the mean value, using Boltzmann’s law. This is obviously a central issue for estimating the extent of myosin binding, and also for the rate of binding which will determine the rate of tension development in a newly-activated muscle ﬁbre. How can thermally-excited conﬁgurations be calculated if we know only the ground-state conﬁguration of a multiply-pinned chain? A frontal assault on this problem can be made in terms of the path-integral formulation of statistical mechanics (Feynman and Hibbs 1965), to which this problem is ideally suited. Distributions of displacements can be calculated from partition functions for chain segments, but only by using a ﬁnite-difference version of Eq. 7.33 and regularising the approach to the limit (Smith 2001). Fortunately, this approach can be avoided if we are interested only in probability distributions, and the argument is as follows. Suppose we seek the probability distribution p(ϕ1) of displacement ϕ1 at position 1 of a chain clamped at the ends, a and b. Although ϕ1 may not be the mean displacement, we do know the ground-state energies Ea1 and E1b of the segments a-1 and 1-b, where the slope ψ 1 has been adjusted for minimum energy. The sum of these energies must appear as a Boltzmann factor in the distribution, so pðϕ1 Þ / expðβðEa1 þ E 1b ÞÞ:

ð7:49Þ

where β ¼ 1/kBT. The same factor must appear in the partition function for the chain with a given displacement at site 1. What else is required? For the partition function, there is an additional factor W which counts the number of thermally-excited conﬁgurations, and Boltzmann’s famous formula S ¼ kB ln W connects it with entropy. For Eq. 7.49, the missing constant is determined by normalizing the distribution, and this statement applies to probability distributions within a chain segment with ﬁxed displacement and slopes at each end. Nevertheless, the following

7.6 The Distribution of Thermally-Activated Chain Displacements

325

results can be rigorously derived from partition functions which average over all values of ψ 1. In this way, the probability p(ϕ1) of displacement in the middle of a long chain is seen to be a Boltzmann distribution for the energy 4κT ξ3 ϕ21 , as if the chain were pinned at position 1. This distribution is a Gaussian centred on zero displacement:

1 pðϕ1 Þ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp ϕ21 =2σ 2o , 2πσ 2

σo ¼

kB T 8κT ξ3

1=2 :

ð7:50Þ

For the TmTn chain, the standard deviation σo is about 20 . More interesting distributions follow in exactly the same way. The conditional distribution p(ϕ2| ϕ1, X) of ϕ2, where the chain at distance X is pinned at displacement ϕ1, follows from the joint distribution pðϕ2 jϕ1 ; X Þ ¼

pðϕ1 ; ϕ2 jX Þ pð ϕ1 Þ

ð7:51Þ

where pðϕ1 ; ϕ2 jX Þ / expðβðE a1 þ E 12 ðX Þ þ E 2b ÞÞ:

ð7:52Þ

The sum of the energy terms in this Boltzmann factor is given in Eq. 7.45, so that the joint distribution is a bi-variate Gaussian, namely pðϕ1 , ϕ2 jXÞ ¼ CðXÞexpð4βκ T ξ3 ϕ12 ΓðXÞϕ12 Þ,

ð7:53Þ

where ϕ12 ¼ (ϕ1, ϕ2) and the matrix elements of Γ(X) are in Eqs. 7.46. This distribution can be normalized over both displacements by using the identity

ZZ expðxMxÞd x ¼ 2

1=2 π2 jMj

ð7:54Þ

where x is a 2-vector and M is any 2 2 matrix (Doi and Edwards 1988). Hence CðXÞ ¼ 4βκT ξ3 jΓðXÞj1=2 =π jΓðXÞj1=2 =2πσ 2o and pðϕ2 jϕ1 ; X Þ ¼

Γ11 ðX Þ 1=2 Γ11 ðX Þϕ22 þ ðΓ11 ðX Þ 1Þϕ21 þ 2Γ12 ðX Þϕ1 ϕ2 exp 2πσ 2o 2σ 2o ð7:55Þ

using the identities

326

7 Models of Thin-Filament Regulation

Γ22 ðX Þ ¼ Γ11 ðX Þ,

Γ21 ðX Þ ¼ Γ12 ðX Þ,

j ΓðX Þ j¼ Γ11 ðX Þ:

ð7:56Þ

For the determinant, the last identity was conﬁrmed by symbolic manipulation. Equation 7.55 can be put in a more useful form by identifying the mean and S.D. of ϕ2. The exponent is a perfect square, meaning that no constant term is left over (do check that this is true). So this distribution is a shifted Gaussian: 2 ! ϕ2 ϕ2 ð ϕ1 ; X Þ 1 pðϕ2 jϕ1 ; X Þ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp 2σ ðX Þ2 2πσ ðX Þ2

ð7:57aÞ

Γ12 ðX Þ ϕ , Γ11 ðX Þ 1

ð7:57bÞ

with ϕ2 ð ϕ1 ; X Þ ¼

σo σ ðX Þ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : Γ11 ðX Þ

Figure 7.13b shows how ϕ2 moves from its pinning value ϕ1 when X ¼ 0 to the mean value (zero) when X > 5 persistence lengths. The S.D. grows from zero to its freechain value rather more quickly, essentially when X > 2. Finally, we need the distribution of displacements between two pinning sites. What is the distribution of angles at a point in between nearest-neighbour pinning centres 1 and 2, one on each side, at distances X and Y? Instead of working from the joint distribution, there is a quicker way of getting to the answer, which proceeds from energy changes (Geeves et al. 2011, supplementary material). The energy required to push the chain at point 3 from its equilibrium value to a displacement ϕ3 is ΔE ðϕ3 jϕ1 ; ϕ2 ; X; Y Þ ¼ Eo ðϕ1 ; ϕ3 ; ϕ2 jX; Y Þ E o ðϕ1 ; ϕ2 jX þ Y Þ ¼ Aðϕ3 Þ þ V ð2Þ ðϕ1 ; ϕ3 jX Þ þ V ð2Þ ðϕ3 ; ϕ2 jY Þ þ V ð3Þ ðϕ1 ; ϕ2 ; ϕ3 jX; Y Þ

ð7:58Þ

in terms of the single-site, pair interaction and triplet interaction energies in Eqs. 7.47a, 7.47b. Neglecting the triplet energy gives the energy cost in the pair approximation, namely ΔE pair ðϕ3 jϕ1 ; ϕ2 ; X; Y Þ ¼ 4κT ξ3 ϕ23 þ ðΓ11 ðX Þ 1Þ ϕ21 þ ϕ23 þ 2Γ12 ðX Þϕ1 ϕ3 2 þ ðΓ11 ðY Þ 1Þ ϕ3 þ ϕ22 þ 2Γ12 ðY Þϕ3 ϕ2 : ð7:59Þ The corresponding distribution is p(ϕ3| ϕ1, ϕ2, X, Y ) / exp (βΔEpair), which can be processed by completing the square in Eq. 7.59 with respect to ϕ3. This time the answer is not a perfect square; there is a term left over. Speciﬁcally,

7.6 The Distribution of Thermally-Activated Chain Displacements

βΔE pair ðϕ3 jϕ1 ; ϕ2 ; X; Y Þ ¼

2 ϕ3 ϕ3 ðϕ1 ; ϕ2 ; X; Y Þ 2σ ðX; Y Þ2

327

þ βδE

ð7:60Þ

where the mean and standard deviation are ϕ3 ðϕ1 ; ϕ2 ; X; Y Þ ¼

Γ12 ðX Þϕ1 þ Γ12 ðY Þϕ2 , Γ11 ðX Þ þ Γ11 ðY Þ 1

σo σ ðX; Y Þ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Γ11 ðX Þ þ Γ11 ðY Þ 1 ð7:61Þ

and ðΓ11 ðX Þ 1ÞðΓ11 ðY Þ 1Þ ϕ21 þ ϕ22 2Γ12 ðX ÞΓ12 ðY Þϕ1 ϕ2 δE ¼ 4κT ξ : ð7:62Þ Γ11 ðX Þ þ Γ11 ðY Þ 1 3

As functions of X,Y, the mean chain displacement and S.D. are shown in Fig. 7.14. However, for the pair energy the δE term must be neglected on physical

Fig. 7.14 Mean (a, b) and standard deviation (c) of chain displacement between two pinning sites, as functions of the reduced distances X and Y on either side. The mean displacements in (a, b) are for pinning angles ϕ1 ¼ ϕ2 ¼ 1 and ϕ1 ¼ 1, ϕ2 ¼ 1 respectively

328

7 Models of Thin-Filament Regulation

grounds; the energy required to move the chain from its mean displacement ϕ3 to some other value ϕ3 must be zero when ϕ3 ¼ ϕ3 : Then the probability distribution follows as the Gaussian distribution 2 ! ϕ3 ϕ3 ðϕ1 ; ϕ2 ; X; Y Þ 1 pðϕ3 jϕ1 ; ϕ2 ; X; Y Þ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp : 2σ ðX; Y Þ2 2πσ ðX; Y Þ2

ð7:63Þ

This equation provides the key to formulating the energetics and kinetics of protein binding to regulated actin. Because of its central role, it is important to make a quantitative assessment of the errors incurred by making the pair approximation. It can be shown that the energy correction δE is almost negated by the neglected triplet interaction energy, leaving a subthermal correction to the pair energy. Thus the pair approximation should sufﬁce for applications of the chain model.

7.7

Energetics and Kinetics of Myosin Binding

How is myosin’s afﬁnity for actin and its rate of binding regulated by a chain of tropomyosin/troponin units? All the mathematical tools required to answer this question have been developed in the last section. We have already seen that myosin-actin binding in striated muscle is modulated by strain energy in myosins tethered to the thick ﬁlament. However, even in solution, the binding of myosins to regulated actin is modulated by the TmTn chain because of the strain energy required to displace it. Local displacements of the chain occur when myosin binds strongly to actin, pushing the chain out of the way and effectively pinning it at an angle removed from its equilibrium position, deﬁning the ‘open’ state. At low calcium, the chain can also be pinned in the opposite direction when molecules of TnI bind to actin, giving the ‘blocked’ state. The essential features of the chain model were described at the end of Sect. 7.4, which emphasises that blocked and open states are merely local conﬁgurations of the chain created by bound myosins or TnI’s. To codify these ideas, note that myosin-nucleotide complexes always bind to a weakly-bound state (state M1) which then isomerises to a more strongly-bound state (state M2). These two states are distinguished by the extent of their binding interface with actin; weakly-bound myosin forms electrostatic and covalent bonds over its lower 50 K domain, while the strongly-bound form has additional bonds from its upper 50 K domain (Fig. 7.11). This statement is supported by experimental evidence, regardless of whether the weak-to-strong isomerization is accompanied by a working stroke. Solution kinetics shows that S1 and S1.ADP bind in two steps (Geeves 1991), while single-molecule experiments with the same complexes in the optical trap (Steffen et al. 2003) show no net working stroke. With S1.ADP.Pi, the weakly-bound state in solution is not observed because the phosphate ion is released as part of the working stroke, ending up with a strongly-bound S1.ADP state; only

7.7 Energetics and Kinetics of Myosin Binding

329

the complex with vanadate (a phosphate analogue) forms a stable weakly-bound state with actin. In the strongly-bound state, the myosin lever arm is in the rotated (or ‘near-rigor’) position. Structural evidence from X-ray and cryo-EM studies on regulated actin (Lehman et al. 2000, 2001) supports the concept of two-step binding by identifying the azimuthal regions of actin covered by weakly and strongly bound myosins. For weak binding, the interface lies in a range of negative angles (ϕM1 ΔM1, ϕM1+ΔM1) relative to the equilibrium angle ϕ ¼ 0 of the free chain, so the chain probably sits at angles above ϕM1 + ΔM1 near a weakly-bound myosin. Strongly bound myosin occupies a bigger interface on actin with an angular range (ϕM1 ΔM1, ϕM2) where ϕM2 > 0, pushing the chain to angles above ϕM2 (Fig. 7.11b). It is convenient to assume that the chain passing over a strongly-bound myosin is pinned at angle ϕM2, and this constraint is vindicated by a bulge in the actin interface (Lehman et al. 2013). Thus two-step myosin binding has the effect of keeping the chain on the positive-angle side of myosin in both its weakly and strongly-bound forms. For a Tm.Tn chain, structural studies also show that TnI binds to actin at low calcium at an angle ϕB < 0 which inhibits weak myosin binding, so ϕM1 ΔM1 < ϕB < ϕM1 þ ΔM1 :

ð7:64Þ

In the absence of structural information to the contrary, it is convenient to set ϕM1 ¼ ϕB. At this stage a change of terminology is required. In the early models of regulation, TmTn was thought to have three state conﬁgurations, blocked, closed and open. The closed state is the sum of all conﬁgurations of the free chain, unencumbered by bound myosins or TnI’s; it exists in the absence of myosin and at high calcium, and also with no troponins. The blocked state exists at low calcium, where every molecule of TnI is bound to actin. However, the open state is not a property of the chain alone because it is a local conﬁguration created by each bound myosin; therefore it is better to call it a “myosin” state. The new terminology has been widely adopted, even though the hybrid syntax of “blocked, closed and myosin” is somewhat awkward. The effect of the chain on the actin afﬁnity of myosins in solution can now be written down, following the development in Geeves et al. (2011). Let KM1 be the equivalent ﬁrst-order myosin-S1 afﬁnity into the weakly bound state in the absence of chain distortion, and KM2 ¼ KM1KS be the net afﬁnity to the strongly-bound state, where KS is the isomerization constant. And KTI ¼ 1/K(C), the inverse of the function in Eq. 7.9, is the afﬁnity of TnI for actin, averaged over the allosteric transitions in Scheme 7.3. How are these quantities affected by chain distortion? In the pair approximation, the net afﬁnity to the strongly-bound state on actin is a function of chain angle (ϕM2 or ϕB) at the binding site, and its nearest pinning centres on either side. For pinning ði; jÞ centres i,j at distances X,Y, let the myosin and TnI afﬁnities be K M2 ðX; Y Þ and ði; jÞ K TI ðX; Y Þ, where i,j ¼ 1 for a blocked state (bound TnI) or + 2 for a stronglybound myosin (state M2). Gibbs’ thermodynamic identity shows that.

330

7 Models of Thin-Filament Regulation

ði; jÞ ði; jÞ K M2 ðX; Y Þ ¼ K M2 exp βΔE M2 ðX; Y Þ

ði; jÞ ði; jÞ K TI ðX; Y Þ ¼ K TI exp βΔE B ðX; Y Þ ði, jÞ

ð7:65aÞ ð7:65bÞ ði, jÞ

where ΔE M2 ðX, YÞ ΔE pair ðϕM2 jϕi , ϕj , X, YÞ, ΔE B ðX, YÞ ΔE pair ϕB jϕi ; ϕ j ; X; Y in the more explicit notation of Eq. 7.59. These quantities are the energies required to move the chain from its mean equilibrium angle to the pinning angle (ϕM2 or ϕΒ) in the presence of neighbouring pinning centres. If these centres are absent, the formulae can still be used, by setting X,Y >> LP. In principle, the energy cost in the exponent should be a Gibbs energy, which includes an entropic component, but an explicit calculation of the entropic term is beyond the scope of this book. Why, you may ask, is the weak-binding myosin state not included? In solution, all heads that bind make a working stroke, so the weak-binding state is a short-lived intermediate. In the absence of neighbouring pinning centres, the chain ﬂuctuates over a range of angles about zero, and some of these ﬂuctuations block myosin binding by covering the weak-binding interface. The actin-binding rate is known to be Ca2+-regulated. Kinetic regulation is achieved because the binding rate will be proportional to the probability P that the chain avoids the weak-binding site. Complete inhibition is not expected, even in the most inhibitory case where ϕB ¼ 0. Similar inhibition occurs in the general case with pinning centres on either side, where thedistribution is a shifted Gaussian of S.D. σ(X, Y ) about a mean angle ϕij ðX; Y Þ ϕ3 ϕi ; ϕ j ; X; Y (Eq. 7.61). Hence the binding rate is of the form ði; jÞ kM1 ¼ k M1 P ϕij ðX; Y Þ; σ ðX; Y Þ

ð7:66Þ

where kM1 is the unregulated rate and ! Z ϕM1 þΔM1 1 ðϕ ϕÞ2 exp Pðϕ, σÞ ¼ 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ dϕ 2σ 2 2πσ 2 ϕM1 ΔM1 1 ϕM1 þ ΔM1 ϕ ϕM1 ΔM1 ϕ pﬃﬃﬃ pﬃﬃﬃ erf ¼ 1 erf 2 2σ 2σ

ð7:67Þ

in terms of the error function. Examples are shown in Fig. 7.15. Excluding the chain from the weak-binding site does cost energy, but this energy will be subthermally small if the weak-binding site is itself small (ΔM1 σo). Then myosin afﬁnity to the weakly-bound state will not be regulated by the chain. In that case, the detachment rate will be kinetically regulated in the same way as the binding rate, through the factor P. However, the stroke transition to the strongly-bound state is regulated through the energy cost ΔEM2 in pushing the chain out to angle ϕM2, where it is effectively pinned. Thus the net afﬁnity to the post-stroke state is

7.7 Energetics and Kinetics of Myosin Binding

331

Fig. 7.15 Factors controlling the kinetic regulation of weak myosin-actin binding by a ﬂexible tropomyosin chain. (a) The probability po(ϕ) of angular displacement ϕ of the free chain in the parabolic well of curvature α. The binding rate is proportional to the probability that the chain does not block the weak-binding site, shown as the grey shaded area over the groove of the double-helix at angle ϕM1 ¼ ϕΒ (the blocking angle when occupied by TnI). Black shading indicates the interface occupied by strongly-bound myosin. po(ϕ) is a Gaussian distribution of mean angle ϕðϕ1 ; ϕ2 ; X; Y Þ and S.D. σ(X, Y ) (Eq. 7.61), which referencethe angles and distances of the nearest pinning sites on either side. (b) The resulting binding rate k M ϕ; σ as a function of ϕ for two values of σ, namely the free-chain S.D. σo ¼ (kBT/8κ Tξ 3)1/2 (with ϕ 0), and 0.1σo (ϕ in Fig. 7.13b) to show the effect of neighbouring pinnings. From Geeves et al. (2011), with permission of Elsevier Press

ði; jÞ ði; jÞ ði; jÞ ði; jÞ K M2 ðX; Y Þ K M1 ðX; Y ÞK S ðX; Y Þ ¼ K M1 K S exp βΔE M2 ðX; Y Þ :

ð7:68Þ

Because the stroke transition is rapid ( 104 s1), the net binding rate is limited by the initial rate (Eq. 7.66) to the weakly-bound state. Although KS > > 1, the stroke transition is still reversible on the time scale of actin binding in solution, so the two states can be considered to be in equilibrium. Hence the detachment rate from the strongly-bound state M2 is ði; jÞ

kM2 ¼

ði; jÞ

kM1 ðX; Y Þ ði; jÞ

K M2 ðX; Y Þ

:

ð7:69Þ

The actin binding of TnI is also regulated by the chain. Because TnI is tethered to tropomyosin, the binding rate is likely to be very fast, so it too can be considered to be in reaction equilibrium with actin, with an afﬁnity given by Eq. 7.65b. With these preliminaries, we are now in a position to review solution-kinetic experiments on myosin binding to regulated actin. Both the extent and the rate of binding have been studied, as a function of myosin concentration and calcium also. These experiments provide challenging tests of the validity of the chain model.

332

7.8

7 Models of Thin-Filament Regulation

Solution Experiments

In Sect. 7.1, experimental studies of myosin binding to regulated actin ﬁlaments in solution were used to test early models of cooperative binding. A key property of these models was the number n of actin sites in the cooperative unit, deﬁned by assuming that adjacent units do not interact. Experimental data analysed in this way suggested that n ¼ 6–7 for skeletal A.Tm and n ¼ 10–12 for A.Tm.Tn; these results form part of the argument for treating tropomyosin units as a single ﬂexible chain. In the chain model, the size of the cooperative unit is estimated at 2LP/c + 1 where c is the actin monomer spacing. By adjusting the persistence length, the equivalent unit size can be estimated by ﬁtting the predicted binding curve θ([S1]) to titration data, for example from Maytum et al. (1999) where S1 concentration was slowly increased while pyrene ﬂuorescence data was collected. The best way of generating binding curves from the chain model is by MonteCarlo simulation. Transfer-matrix methods are possible and have been implemented, but (in contrast to the Hill-Eisenberg-Greene model) nearest-neighbour interactions are not sufﬁcient and some truncation of the pair-interaction potential is required (Smith et al. 2003; Smith and Geeves 2003). In these papers, the pair interaction was truncated at twice the persistence length (X ¼ 2), which ignores the repulsive tail of the interaction. The results are sensitive to truncation at this length, but the transfermatrix method is still useful in detecting trends when model parameters are varied. To simulate titration data, Monte-Carlo methods require an algorithm for the approach to equilibrium. One way of doing this is to model the binding transient in real time; then the same code can be used for simulating the extent and the rate of myosin binding to regulated actin. Functional forms for the actin-binding rate constant and the equilibrium constants were discussed in the last section; the only kinetics involved is for S1 binding into the weakly-bound state (Eq. 7.66), because the rapid isomerization to the strong state can be treated as an instantaneous equilibrium. The actin binding of TnI can be treated in the same way. For the chain model, the basic Monte-Carlo method (Sect. 4.3) can be implemented is as follows. On each actin strand, let ns be the state of the binding site indexed by s, with ns ¼ 0 for the empty state, 1 for the site with weakly-bound S1, 2 for strongly-bound S1 and 1 for bound TnI. TnI can only bind to every 7th site, and S1 and TnI compete for these sites. If k is the rate constant for S1 binding, then the probability of binding in time Δt is kΔt. If a site is empty, S1 binds to it within that interval if r < kΔt, where r is a uniformly distributed random number in [0,1). A similar recipe holds for detachment when ni ¼ 1. Rapid equilibria can also be simulated, by forming equilibrium fractions. For example, if KS is the equilibrium constant for the weak-to-strong isomerization 1 ! 2, then KS/(KS + 1) is the probability of the strong state. If these are the only two actomyosin states, their occupation can be speciﬁed by ns ¼ 2 if r < KS/(KS + 1) where r is another uniform random number in [0,1), otherwise ns ¼ 1. TnI-actin binding is treated in the same way.

7.8 Solution Experiments

7.8.1

333

The Extent of Myosin Binding

Here we present results of Monte-Carlo simulations which ﬁt the titration data of Maytum et al. (1999). Separate binding transients were generated for various initial [S1] concentrations, optimising the ﬁt by adjusting the persistence length and the TnI afﬁnity KTI, which is a proxy for the calcium level. The weak myosin-actin afﬁnity K~ M1 was ﬁxed at 2 105 M1, with KS ¼ 100 for the isomerization and a maximum S1 concentration of 0.2 μM. The chain pinning angles ϕB and ϕM2 for TnI and strongly-bound myosin were set at 1.0 and 0.4 in units of the free-chain standard deviation σo (Eq. 7.32), which was used as the unit for angular chain displacements; the ratio of these angles agrees with electron-microscopic observations (Vibert et al. 1997). The equivalent binding rate was arbitrarily set at 10 s1, and Δt was generally set at 104 of the end-time. Figure 7.16 shows the quality of the best ﬁts, achieved with LP ¼ 24 nm and KTI ¼ 5 (high Ca) or 200 (low Ca). The persistence length is somewhat higher than in previous modelling studies, so stronger TnI binding at low calcium was required to stop the chain from opening up at too low a concentration of S1. The fraction of bound TnI monitors the transition from the blocked to the open state, which occurs over a very small range of S1 concentrations, indicating a high level of cooperative S1 binding under the chain as it is pushed towards the open position ϕ ¼ ϕM2. The same high value of LP is required if the pinning angles are changed to favour TnI binding, since the chain creates cooperative inhibition at low [S1] and/or low[Ca2+] and cooperative activation at high [S1]. Similar ﬁts are generated by the transfermatrix method with the pair interaction truncated as described.

Fig. 7.16 Myosin loading curves of Maytum et al. (1999) at low and high Ca, ﬁtted with the chain model, where calcium enters through the actin afﬁnity of TnI (KTI ¼ 200 and 5). For chain parameters, LP ¼ 24 nm, ϕM/σo ¼ 0.4, ϕB/σo ¼ 1.0, ΔM1/σo ¼ 2.0. S1 concentrations of the experimental data were converted to the net afﬁnity K M K~ M ½S1 using K~ M ¼ 2 107M1. The weak-binding myosin state was not included. The experimental data was supplied by Prof. Michael Geeves

334

7.8.2

7 Models of Thin-Filament Regulation

Kinetic Regulation of Myosin Binding

The rate of myosin binding to regulated actin is also calcium-dependent, but the rate increases only by a factor or 2–3 over the range of calcium activation (Geeves et al. 2011). In Sect. 7.1 we showed that the steric blocking model cannot duplicate this behaviour. While the BCO model comes close, and a forced ﬁt can be achieved with an extension of the Hill-Eisenberg-Greene model, the development of a physicochemically based kinetic model has been overdue. Can titration and transient myosin-binding data be ﬁtted by the chain model with the same set of parameters? This exercise requires measurements made under the same solution conditions, including ionic strength, because afﬁnities, binding rates and the the persistence length are all controlled by electrostatic interactions. The unregulated actin afﬁnity of myosin-S1 can be extracted from high-calcium titrations on A.Tm.Tn, and such a test becomes possible if the low-calcium afﬁnity of TnI for actin is treated as an adjustable parameter. A thorough test of the chain model requires myosin-S1 binding transients for different concentrations of S1 and calcium. The essential test is made under excessmyosin conditions, with enough myosins to saturate the actin ﬁlament once chainrelated inhibition is overcome; a pronounced lag phase is observed at low calcium, and there is also a small lag at high calcium. The subsequent rise can usually be ﬁtted by a single exponential, giving a calcium-dependent rate constant for binding under the chain. Data of this kind (Geeves et al. 2011) is shown in Fig. 7.17; from low calcium to high calcium, the rate constant kobs from a single-exponential ﬁt after the lag phase increases by a factor of three. A similar result was found with excess actin, where only one S1 might be available to bind between each TnI, and cooperative activation is not observed. Figure 7.18 shows the complete binding transients obtained by Geeves et al. (2011) under these conditions. At low calcium, S1 binding is initially inhibited by bound TnI, so the transient shows a lag which terminates as activation gets under way, with more myosins binding under chain umbrellas. With excess myosin, these transients are complicated by another phenomenon, namely the slowing of S1 binding as the bound Fig. 7.17 The calcium dependence of myosin binding rates to regulated actin, obtained by ﬁtting the transient responses observed by Geeves et al. (2011) to single exponentials. Legend: (Ο) excess actin, (●) excess myosin. From low to high calcium, the rates increase by a factor of only 2.5–4. With permission of Elsevier Press

7.8 Solution Experiments

335

Fig. 7.18 Myosin binding transients to regulated actin at low and high calcium (Geeves et al. 2011) under conditions of excess myosin (a, b) and excess actin (c, d), showing data of Geeves et al. (2011) (black lines) and ﬁts to the chain model (red lines). (a, c) are for high calcium (pCa ¼ 4.6, KTI ¼ 0.25) and (b, d) for low calcium (pCa ¼ 7.0, KTI ¼ 40 and 32 respectively). The ﬁts used kM1 ¼ 18 s1, KM1 ¼ 1.0 with excess myosin and kM1 ¼ 0.9 s1, KM1 ¼ 0.05 with excess actin. Parameters with common values were KS ¼ 100, LP ¼ 21 nm, and the pinning angles used in Fig. 7.16. Steric blocking of solution myosins was incorporated as described in the original paper. Green lines show the predicted fraction of TnI’s bound to actin. With permission of Elsevier Press

fraction increases and actin sites become obscured by neighbouring myosins. Steric slowing is a general phenomenon, observed in unregulated actin and so unrelated to regulatory mechanisms. An empirical way of dealing with this effect is to reduce the binding rate by a factor of 1–γ for bound myosins on each nearest-neighbour site, but the quality of the ﬁts suggest that this is not a totally adequate description. What is clear that the values of LP and γ should not be allowed to vary with calcium level, which was buffered without changing ionic strength. With this caveat in mind, reasonable ﬁts to experiment are obtained with LP ¼ 21 nm and γ ¼ 0.39. In the above paper, myosin binding transients were measured over a range of calcium concentrations to give the apparent binding rates shown in Fig. 7.17. However, the binding rates predicted by the chain model were calculated as a function of KTI, the actin afﬁnity of TnI. The two variables should be linked by the allosteric calcium-TnC-TnI binding scheme described in Sect. 7.2. Combining these data sets yields KTI as a function of calcium, and Fig. 7.19 shows how the

336

7 Models of Thin-Filament Regulation

Fig. 7.19 Determinations of the calcium dependence of actin-TnI afﬁnity, obtained by ﬁtting myosin binding transients at various calcium concentrations (Geeves et al. 2011). Legend: (Ο) excess actin, (●) excess myosin. The data were ﬁtted to the allosteric model of Ca-TnC-TnI binding scheme in Sect. 7.2, using Eq. 7.71. For these conditions, the ﬁtting parameters were respectively K~ C ¼ ð2:31 0:10Þ 105 M1 and (1.77 0.16) 105M1, ε ¼ 0.034 0.0015 and 0.0327 0.0019, plus KTI ¼ 86.5 and 93.5 for TnI-actin afﬁnity at zero calcium. With permission of Elsevier Press

allosteric model can be ﬁtted to the experimental data, for both excess-myosin and excess-actin conditions. In principle, more information on allosteric parameters can be obtained from measurements of calcium-TnC binding, to which we now turn.

7.8.3

Calcium Binding to TnC

Measurements of calcium-TnC binding as a function of free calcium level provide a test of the chain model combined with the allosteric scheme of Sect. 7.2. At ﬁrst glance it might appear that calcium loading does not involve chain interactions, but this is not the case. Two bound calcium ions on the TnC component of one troponin complex cause the TnI component to detach from actin, which lifts the local conﬁguration of the TmTn chain and decreases the actin afﬁnity of TnI’s on neighbouring troponins. Then the allosteric model (Scheme 7.3) predicts that the calcium afﬁnity of the neighbouring TnC’s will increase. The net effect is an indirect coupling of the calcium afﬁnities of neighbouring TnC’s. In fact, models can be formulated where the chain is removed and replaced by interaction potentials between bound myosins and bound TnI’s or bound calcium ions. Fortunately there is a simple recipe for calculating the interaction of calcium with TnI. If one molecule of TnI is known to be detached or bound to actin, then the calcium afﬁnity of the corresponding TnC is known from Scheme 7.3. In each case the fraction of occupied calcium sites on TnC is known. The overall fractional

7.8 Solution Experiments

337

occupancy θC of the two regulatory sites on TnC is the average of these fractions, weighted by the probabilities that TnI is detached or bound. Thus θC ¼

K~ C C K~ C C=ε ð1 θTI Þ θTI þ ~ 1 þ K CC 1 þ K~ C C=ε

ð7:70Þ

where C ¼ [Ca2+], K~ C is the single-site calcium afﬁnity for A.TnC.TnI and θTI is the fraction of TnI’s bound to actin. Binding is a binary choice, so θTI is also the probability that a single TnI is bound. This formula has the same structure as Eq. 7.16b, whose validity extends beyond independent-unit models: the effects of interactions, whether in the chain model or otherwise, appear in the behaviour of θTI as a function of calcium and myosin concentrations. To calculate θC, the previous Monte-Carlo method can be used to calculate θTI as a function of KTI, which is a function of calcium concentration C. The function 2 ½blocked 1 þ K~ C C ¼L : K TI ðCÞ ½open 1 þ K~ C C=ε

ð7:71Þ

(the inverse of K(C) in Eq. 7.9) describes one of many possible allosteric models for Ca-TnC-TnI interaction, but in Fig. 7.19 it was used successfully to ﬁt myosin binding data. It is important to understand that this equivalence operates at the level of a single troponin complex and is independent of chain-mediated interactions between them. Calcium loading curves are traditionally plotted against pCa –logC and characterised by two parameters. The pCa value for half-maximum loading measures the sensitivity of activation to calcium, while the steepness of the curve as measured by the Hill coefﬁcient dlog θðC Þ= 1 θðCÞ n dlogC C¼C 0:5

ð7:72Þ

is a measure of the cooperativity of activation. The form of the numerator follows from the empirical function θ(C) ¼ KCn/(1+KCn) (Hill 1913), which makes the right-hand side equal to n for all Ca levels. A quick and easy way of estimating the steepness of the loading curve is to take the difference of the pCa values for 10% and 90% saturation. If you wish to use this function, a cute little piece of algebra shows that n ¼ 2log9/ΔpCa ¼ 1.91/ΔpCa, independent of K, so a completely non-cooperative loading curve (n ¼ 1) would have ΔpCa ¼ 1.91, close to two decades of [Ca2+]. While skeletal TnC has two low-afﬁnity Ca-binding sites, allosteric schemes such as Scheme 7.3 generally give n < 2 in the absence of chain interactions, so that calcium binding curves with n > 2 can be taken as a sign of positive feedback from chain interactions between TnI’s. Figure 7.20 shows a demonstration of calcium loading curves predicted by the chain model, using ε ¼ 0.001 and λ ¼ 3.0 where λ 1/Lε2 is the dissociation

338

7 Models of Thin-Filament Regulation

Fig. 7.20 Calcium loading curves predicted by the chain model under rigor conditions at high/low [S1], showing that bound myosins increase the sensitivity and the cooperativity of calcium binding to TnC

constant of TnI at high-calcium. With no myosin present, pCa ¼ 5.66 and n ¼ 1.44. With KM ¼ 1, which corresponds to [S1] 0.1 μM, pCa ¼ 6.66 and n ¼ 2.26; calcium sensitivity has increased and the binding is more cooperative. Similar results were found experimentally by Grabarek et al. (1982), who also found that Ca2+ binding to isolated TnC and the isolated troponin complex (TnC.TnT.TnI) was non-cooperative. Their results were obtained under cycling conditions, where there was a high concentration of ATP. The model produces highly-cooperative Ca2+ binding only if ε dc Þ, ðd < dc Þ

ð8:10Þ

gives the dependence expected from a screened electrostatic actin-binding potential when d > dc, the critical spacing where myosins just touch F-actin. We also require a balance of radial forces, which can be written in terms of the radial stiffnesses SM and SE from crossbridges and electrostatic interactions at full overlap. Setting the net inward radial force to zero gives xSM ðd d c Þ þ xSE ðd d E ðxÞÞ ¼ 0

ð8:11Þ

as electrostatic myosin-actin interactions are also proportional to the degree of overlap (Sect. 5.8). Writing SM ¼ γT M0 ðdÞ dE ðxÞ ¼ dE0 þ dE1 x

ð8:12Þ ð8:13Þ

closes this set of equations. Equation 8.12 expresses the proportionality between axial and radial tensions, and Eq. 8.13 gives the decrease in lattice spacing with sarcomere length seen in the relaxed ﬁbre (Fig. 5.30). Hence Eqs. 8.9, 8.10, 8.11, 8.12, and 8.13 combine to give K ðd Þ ðd dc Þ þ εðd d E0 d E1 xÞ ¼ 0 K ðd Þ þ 1

ð8:14Þ

where ε SE/γT00. This equation determines the equilibrium spacing d(x) and the tension, and it must be solved numerically. Experimental data for radial tension as a function of lattice spacing are available for relaxed, active and rigor ﬁbres at full overlap (Brenner et al. 1996). For active rabbit psoas ﬁbres, they found that SM ¼ 75.6 pN/nm per F-actin and dc ¼ 23.5 nm after adding the sum of the ﬁlament radii. For relaxed ﬁbres, SE ¼ 15.5 pN/nm and dE0 + dE1 ¼ 25.0 nm. If T00 ¼ 240 pN, then γ ¼ SM/T00 ¼ 0.31 and ε ¼ 0.2. Thus the lattice spacings of the relaxed and active ﬁbre differ by 1.5 nm. Larger lattice spacings are found in frog muscle (Fig. 5.30). Finally, if myosin-actin afﬁnity is electrostatically screened, then λ 0.6 nm1 for the inverse screening length at an ionic strength of 0.25 M.

8.2 A Muscle Model with Thin-Filament Regulation

357

Fig. 8.4 Solutions of Eqs. 8.14 and 8.9 for a model of stretch activation at partial Ca2+-activation on the descending limb. (a) Tension and (b) lattice spacing versus the fraction x of heads in overlap with F-actin, for ﬁve values of the maximal myosin-actin afﬁnity K0. Other parameters of the model were dE0–dc ¼ 0.5 nm, dE1 ¼ 1.7 nm, λ ¼ 1.65 nm1, ε ¼ 2.0. For graph b, dc was set at 26.5 nm

Solving Eq. 8.14 for lattice spacing and tension as a function of overlap can reproduce the trends seen in Fig. 8.2, but only if ε is much larger than 0.2. The curves in Fig. 8.4 are appropriate for various activation levels from 6% to 97%, judging by the tension predicted at full overlap (x ¼ 1). This is achieved with a hundred-fold range of values of K0, even though K0 ¼ 10 at the lowest level of activation. By using K0 to mimic the effects of calcium on plateau tension, this model predicts the transition to complete activation near zero overlap (x ! 0) at any calcium level. Cardiac muscle is also Ca2+-regulated by the thin ﬁlament, although some details are different (cardiac TnC has only one Ca2+-binding site, kTR is less regulated or unregulated (Hancock et al. 1997), and tension increases with sarcomere length in the region of partial overlap because of increased tension from short connecting ﬁlaments). Enhanced activation on the descending limb is also observed in cardiac muscle, and is also attributed to a decreased lattice spacing, since the same effect can also be generated with osmotic compression (Martyn et al. 2004). Length-dependent activation has important consequences for cardiac function. The Frank-Starling law of the heart says that increased ventricular volume at the end of diastole, which translates to longer sarcomeres in the heart muscle, results in a bigger pressure during systole (the period where the heart muscle is contracting) which increases the pumping stroke. For a critical summary, see the short review of Moss and Fitzsimmons (2002).

8.2

A Muscle Model with Thin-Filament Regulation

What is the simplest way of constructing a contraction model for striated muscle as a function of calcium? The basic unit is one actin-Tm-Tn ﬁlament in one sarcomere, with tropomyosin and troponin giving cooperative regulation of myosin binding. For

358

8 Cooperative Muscular Activation by Calcium

modelling, thin-ﬁlament regulation generates a major increase in complexity; myosin heads do not bind independently to regulated actin, because each strongly-bound head moves the tropomyosin chain and opens up binding sites for neighbouring heads. A one-dimensional array of interacting units appears in many guises, such as the Ising model for magnetic interactions, and physicists have an armoury of mathematical tools to ﬁnd exact solutions. Such problems are easier to solve if the interactions are restricted to nearest-neighbour units. On the other hand, approximate solutions rely on the mean-ﬁeld approximation, which is more accurate when each unit interacts with a large number of neighbours. Regulation of striated muscle via a continuous tropomyosin chain lies between these two extremes, and existing models have sought to simplify matters by working with nearest-neighbour interactions on a periodic lattice (Shiner and Solaro 1982; Razumova et al. 2000; Robinson et al. 2002; Hussan et al. 2006). There are also models at the level of a single myosin head with calcium-regulated kinetics; these models have no cooperativity and will not be considered here. Many elaborate regulatory schemes have been devised in order to ﬁt a rich gallery of experimental data, and it is difﬁcult to devise critical tests to discriminate between these models. Now that the structural dynamics of tropomyoin-troponin on F-actin is better understood, it is possible to explore a minimal contraction model based on TmTn as a continuous ﬂexible chain. That tropomyosin protomers behave in this way is not in doubt. The model proposed by Tobacman and Butters (2000) starts from this premise, although it is formulated in terms of conformational changes in actin monomers. Models of cooperative contractility should also be based on the geometry of the muscle lattice, including the positions and orientations of myosin heads and actin sites. Vernier models are not up to the task, so we start with the ﬁlament lattice and the axial and azimuthal selection rules in Appendix A that determine which heads can access sites on unregulated actin. Adding a tropomyosin chain with calciumbinding troponin units at regular intervals gives a one-dimensional model of myosin heads per thin ﬁlament which interact indirectly via the chain, much as paramagnetic ions interact indirectly via their magnetic ﬁeld. As each strand of the actin doublehelix carriers its own Tm-Tn chain, the book-keeping must be done separately for each strand. Now the spatial structure of the model can be speciﬁed. As in Appendix A, let A X mM (m ¼ 1,. . .,NM) and X nσ (n ¼ 1,. . .NA, σ ¼ 0,1) be the axial coordinates of the mth myosin and the nth actin site on strand σ, starting from the M-band. The selection rules locate a subset of both vectors, indexed by actin sites n ¼ nsσ, s ¼ 1,. . . .., Mσ on each strand. Indexed by s, the required spatial coordinates are the axial position Xsσ of site s measured from the –end of F-actin, and the headsite spacing xsσ: A X sσ ¼ X nσ L þ LA ðn ¼ nsσ Þ A xsσ ¼ X nσ X mM n ¼ nsσ ¼ nmap ðm; k; σÞ

ð8:15aÞ ð8:15bÞ

where nsσ was constructed from the myosin-to-actin mapping nmap(m,k,σ) in Appendix A. The selection rules ensure that each selected myosin is associated with a

8.2 A Muscle Model with Thin-Filament Regulation

359

different actin site. Also required are the axial coordinates of every 7th site on each strand, which is available to TnI. Thus there are actin sites selected by myosins, sites selected by TnI and a few sites selected by both. To reference all these selected sites, a combined list can be constructed in which the type of site selected is ﬂagged by a function ksσ such that ksσ ¼ 1 (myosin only), 1 (TnI only) or 0 (myosin and TnI). This list should then be sorted in ascending order of the corresponding axial coordinate Xsσ for sites of both kinds. For each strand σ, the length of the list will be Mσ plus the number of TnI sites in the overlap region which are not shared with myosins. As stated before, the contents of each list will vary with sarcomere length, even on the plateau of the tension-length curve. For the simple rhomboid unit cell, Mσ ¼ 30–31 on the plateau, with some excursions between 27 and 33 (Fig. A1c), so that the total number of sites selected on both strands is close to 60. If the selected single sites are replaced by three-site target zones there will be some overlapping zones and more scope for myosin binding, although the 1-1 nature of the mapping will be destroyed. With these preliminaries out of the way, a contractile model with thin-ﬁlament regulation can be constructed by grafting an actomyosin contraction cycle onto each site with ksσ ¼ 1. For sites with ksσ ¼ 1, TnI must be allowed to bind reversibly, for example as in Scheme 7.3. For sites with ksσ ¼ 0, the contraction cycle must be expanded to allow a choice between myosin and TnI, which is easy to do if MonteCarlo simulations are used throughout. TnI-to-actin attachment rates are sufﬁciently fast to put this step into equilibrium.

8.2.1

A Minimal Demonstration Model

As a simple demonstration of how thin-ﬁlament regulation could work, consider the following model, in which Duke’s strain-dependent contraction cycle (Sect. 4.2) is coupled with the continuous-ﬂexible-chain model for thin-ﬁlament regulation. This cycle has one lumped myosin state (M.ATP + M.ADP.Pi), and one lumped actomyosin state in which the pre-stroke and post-stroke contributions are in straindependent equilibrium. The parameters of the cycle are head stiffness κ, a working stroke h and ADP-release stroke hD, plus attachment rates kA to the pre-stroke state and the rate kD of irreversible detachment from the post-stroke state. To simplify the model, only one head of each dimer will be allowed to bind; the triple restriction to single heads, single sites and a single working stroke means that the predicted isometric tension will be down by a factor of about eight unless actin binding is boosted at large strains. The transitions of this contraction cycle must be regulated by the Tm-Tn chain and, following the discussion in Sect. 7.8, this can be done as follows. Let the chain be pinned to angles ϕM1, ϕM2 > 0 by myosin bound in states M1,M2, or angle ϕB < 0 by bound TnI, which removes the assumption in chap. 7 that state M1 was not regulated by the chain. M1 and M2 are pre-stroke and post-stroke chain states, which

360

8 Cooperative Muscular Activation by Calcium

for myosin are the A and R states (here only one of each). Then the actin afﬁnities of myosin and TnI will be regulated by the energy cost of chain distortion, as follows: ði; jÞ ði; jÞ K A ðX; YjxÞ ¼ K A ðxÞexp βΔEM1 ðX; Y Þ ði; jÞ ði; jÞ K TI ðX; Y Þ ¼ K TI exp βΔE B ðX; Y Þ

ð8:16aÞ ð8:16bÞ

for pinning sites of type i,j at distances X,Y from the site in question. The chain distortion energies are given by the pinning angles and the mean and S.D. of chain angle between pinning centres (Eq. 7.61), for example ði; jÞ ΔEM1 ðX; Y Þ

¼ kB T

2 φM1 φij ðX; Y Þ 2σ ðX; Y Þ2

ð8:17Þ

for the pre-stroke state, using a condensed notation for the mean angle. The equilibrium constant of the working stroke is also regulated: h i ði; jÞ ði; jÞ ði; jÞ K S ðX; YjxÞ ¼ K S ðxÞexp β ΔE M2 ðX; Y Þ ΔEM1 ðX; Y Þ :

ð8:18Þ

For myosin kinetics, we assumed in Sect. 7.7 that the binding rate was regulated kinetically, by the probability P(i, j)(X, Y ) that the chain avoids the weak-binding site M1. This assumption must now be updated to include energetic regulation as well, where a weakly-bound myosin pins the chain at angle ϕM1. The kinetic constraint puts an energy barrier in the way of myosin binding, and the energetic constraint raises the Gibbs energy of the ﬁnal bound state. Which energy is higher; the barrier or the ﬁnal energy? The form of the binding rate depends on the answer to this question. In the highest-energy-barrier approximation, we have ði; jÞ

k A ðX; Yjx Þ ¼ k A ðxÞPði; jÞ ðX; YÞ

ði; jÞ Pði; jÞ ðX; Y Þ < exp βΔE M1 ðX; Y Þ , ði; jÞ ði; jÞ k A ðX; YjxÞ ¼ k A ðxÞexp βΔE M1 ðX; Y Þ ði; jÞ Pði; jÞ ðX; Y Þ > exp βΔE M1 ðX; Y Þ

ð8:19Þ

with P upgraded from Eq. 7.67 to P

ði; jÞ

( ! !) φM1 φij ðX; Y Þ φB ΔM1 φij ðX; Y Þ 1 pﬃﬃﬃ pﬃﬃﬃ ðX; Y Þ ¼ 1 erf erf 2 2σ ðX; Y Þ 2σ ðX; Y Þ ð8:20Þ

so that the weak-binding site covers the angular range (φB ΔM1, φM1).

8.2 A Muscle Model with Thin-Filament Regulation Scheme 8.1 Chaindependent kinetics for the two-state stroking model

361

k A(i, j) ( X , Y | x) K S(i, j) ( X , Y | x) k D ( x)

D A k −(i,Aj) ( X , Y | x)

R

D

Scheme 8.1 shows the regulated cycle. For single myosin heads and sites chosen by the selection rules, the transient approach to a steady state has been calculated by Monte-Carlo simulation, starting from the relaxed state with all heads detached and all TnI’s bound to actin. Calculations with a persistence length of 20 nm gave very little cooperativity in the tension-pCa curve, because the spacing between sites selected by myosins is too large. The average spacing is four monomers, with gaps of up to 7–8 monomers, and this degree of separation will prevent myosins from opening up the chain as required. This defect could be overcome by working with three-site target zones and dimeric myosins. With the present model, cooperative binding is partially restored if LP is set to 40 nm, and calculations were made on this basis. The behaviour of this model is illustrated in Fig. 8.5. To avoid introducing the allosteric parameters of Scheme 7.3, tension and ATPase in Fig. 8.5a are shown as functions of K ¼ 1/KTI, rather than calcium concentration. To interpret such plots, the simple steric blocking model sufﬁces; Eq. 7.3 shows that we can expect a hyperbolic curve with a half-maximal afﬁnity K 0:5 ¼ ð1 þ K M Þn

ð8:21Þ

where KM is the ﬁrst-order afﬁnity of myosin binding and n is the size of the cooperative unit. These results indicate that KM 0, the ﬁxed point is a node, which is stable if A < 0 and unstable if A > 0. If B > 0, then λ+ > 0, λ < 0 and the ﬁxed point is a saddle, with one outward and one inward trajectory

366

8 Cooperative Muscular Activation by Calcium

A quantitative theory of SPOC can be constructed along these lines. The simplest boundary condition is the isotonic one, since the tension in a ﬁbre held at constant force is transmitted equally to all sarcomeres, and a model at the level of a single half-sarcomere can be constructed. The model of Smith and Stephenson (2009) is of this kind, and passive tension was used to generate the viscoelastic response necessary for oscillations rather than popping. The theory of tension creep presented in Sect. 5.7 can be extended to a theory of SPOC by adding an empirical lengthdependent myosin afﬁnity K(L ), chosen to give the observed tension-length curve at low calcium, and an auxotonic boundary condition in which constant force is replaced by a viscoelastic load. In this theory, passive tension provides the restoring force which prevents sarcomeres from popping out beyond zero overlap, and the waveform of sarcomere length oscillation is not as asymmetric as observed. In fact, SPOC is observed in ﬁbres with little or no passive tension (Ishiwata et al. 1996; Shimamoto et al. 2008). This, and the difﬁculty of predicting the sawtooth waveform for sarcomere lengths, means that a different mechanism is required for creating oscillations about the unstable ﬁxed point. The mechanism proposed by Sato et al. (2011) gives a two-dimensional theory of SPOC by treating the lattice spacing as a dynamical variable, coupled to sarcomere length by making radial tension proportional to axial crossbridge tension. In Sect. 8.1, we saw that this coupling provides a quantitative prediction of the tension-length curve, when myosin afﬁnity is treated as a function K(d ) of lattice spacing d. Sato’s theory of SPOC, which is a logical extension of these ideas, can be rephrased as follows. The key dynamical equation is for the unitary myosin tension TM(t) introduced in Sect. 8.1, so that x(t)TM(t) is the contractile tension at partial overlap, with x ¼ (L3 L )/(L3 L2). For TM(t), the macroscopic differential equation T_ M ðt Þ ¼ ðK ðdÞ þ 1ÞgfT M T M0 ðd Þð1 þ v=v1 Þg

ð8:22Þ

gives a single-exponential approach to steady-state tension as a function of velocity v, arising from a detachment rate g and binding rate K(d)g. The linear velocity dependence on v will match the start of Hill’s tension-velocity equation if v1 ¼ v0/(1 +1/α) where α is the Hill ratio. As in Eq. 8.9, T M0 ðd Þ ¼

K ðd Þ T 00 K ðd Þ þ 1

TðtÞ ¼ xðtÞT M ðtÞ

ð8:23Þ ð8:24Þ

for the unitary steady-state tension and the net tension T(t), omitting any passive contribution. Various forms for K(d) are possible. The new ingredient is the dynamical form of radial force balance, namely Eq. 8.11 in the form γT M ðd d c Þ þ SE ðd d E0 d E1 xÞ ¼ 0

ð8:25Þ

8.3 Spontaneous Oscillatory Contractions

367

which couples TM(t) and d(t) to sarcomere length via the overlap fraction x(t). Finally, the auxotonic boundary condition T ðt Þ ¼ F κS ΔLðt Þ μS vðt Þ

ð8:26Þ

in which an applied force is augmented by an external viscoelastic load, provides for all possibilities, including the isotonic condition (κS ¼ μS ¼ 0). By choosing suitable values of κS and μS, it may be possible to mimic the effects of including this sarcomere in a ﬁxed-end ﬁbre. By formally eliminating the lattice spacing, Eqs. 8.22, 8.23, 8.24, 8.25, and 8.26 are reduced to coupled differential equations for TM(t) and x(t). Using these equations to search for SPOC solutions is akin to ﬁnding the proverbial needle in the haystack, unless the stability of the ﬁxed points is determined in advance. For isotonic boundary conditions, the tension-length curve at low calcium has three ﬁxed points P,Q,R for a certain range of loads (Fig. 8.7a). Since the slope at point P is positive, it appears that P is stable, but this is not so. Let x(d) be the overlap fraction coupled to lattice spacing d, so that x(dP)TM0(dP) is the tension at P. A small increase in length forcibly reduces tension by decreasing the number of heads in overlap, giving tension x(d )TM0(dP) (d > dP), but more heads must be recruited to bring the tension to its hypothetical steady-state value x(d)TM0(d ). When SPOC occurs, the sarcomere moves between these two extremes without asymptoting towards either. A standard stability analysis, summarized in Fig. 8.7b, gives the conditions required for sustained oscillations. Although these equations are based on the ideas proposed by Sato, there are subtle differences with Sato’s dynamical equations. Equations 8.22, 8.26 and 8.25 are roughly analogous to Eqs. 1, 2, 3 respectively of Sato et al. (2011), where P is the number of bound heads per unit length (in overlap). Thus P plays the role of unitary tension TM, but Eq. 1 does not contain a velocity term, which appears instead in Eq. 2 for axial force-balance. The status of Sato’s equations needs to be clariﬁed, but they do provide a convincing description of SPOC oscillations, including the characteristic saw-tooth waveform for sarcomere length changes. What is beyond doubt is that spontaneous oscillatory contractions are driven by stretch activation at low calcium and partial ﬁlament overlap, which is itself a consequence of the decrease in lattice spacing. A systematic modelling study of the SPOC phenomenon should include the various conditions under which SPOC is observed in the laboratory. Slow ﬁbres exhibit SPOC more readily than fast ﬁbres, and have a lower frequency of oscillation (Stephenson and Williams 1981). There is a ﬁnite range of calcium concentrations, bounded below by the relaxed state and above by the active steady state. ADP acts as an activator, as might be expected from the increased tension at full activation from adding ADP. At pCa ¼ 6.5, or even in the absence of calcium, millimolar ADP can cause oscillations or even steady contraction in the relaxed ﬁbre (Shimizu et al. 1992; Ishiwata et al. 1993). Added Pi inhibits tension, and has the reverse effect. Modelling synchronous oscillations under isotonic conditions is also the prelude to predicting the spatial organization of sarcomeric oscillations, including wave-like

368

8 Cooperative Muscular Activation by Calcium

propagation, seen under ﬁxed-end and auxotonic conditions. In all cases, it is reasonable to assume that the tension is the same in all sarcomeres; unbalanced tensions would propagate at the speed of sound and dissipate long before a SPOC wave can progress. Some progress along these lines has been made by introducing random variations in isometric tension along the ﬁbre (Smith and Stephenson 1994), which can induce phase discontinuities and quiescent regions; this may be an aspect of disorganized SPOC which could result from spatial variations in myosin isoforms. SPOC has also been studied under weakly auxotonic conditions, where the ends of the ﬁbre were attached to compliant microneedles (Anazawa et al. 1992). In this case, tension oscillations could be observed in conjunction with SPOC waves propagating repeatedly from one end to the other. Applying a ﬁxed load to the microneedles caused the period to increase proportionally. The fascinating phenomena of SPOC waves and the various degrees of phase coherence between oscillating sarcomeres in a ﬁxed-end ﬁbre deserves further study, as for example in Sato et al. (2013). Further investigation of this specialized area of muscle contraction is beyond the scope of this book.

8.4

Direct regulation of Myosin Contractility

It is ﬁtting to end this book by surveying what is now a highly active area of muscle research, namely the regulation of contractility by phosphorylation of myosin’s regulatory light chains (Irving 2017). Long ago, it was known that phosphorylating the two regulatory light chains on the neck of myosin-S1 led to increased tension and ATPase rate, in mammalian striated muscle, molluscan muscles and smooth muscles (Ruegg 2012). In striated muscle, this was seen as a secondary mechanism to thinﬁlament regulation via tropomyosin-troponin, but in molluscan muscles and smooth muscles the thin ﬁlaments lack troponin, so they cannot regulate contraction in a Ca-sensitive manner. To explain the origin of these new developments, it is necessary to take a detour to see how heavymeromyosin (dimeric myosin) can regulate itself, by a route which involves calcium only as a cofactor to regulatory-light-chain (RLC) phosphorylation. The implicit assumption is that, in striated muscle, this mechanism operates just as it does in muscles where troponin is absent. Molluscan muscles such as scallop muscle have a collinear array of myosin and actin ﬁlaments, whereas the smooth muscles in our arteries and veins have a spindle structure with inclined ﬁlaments. Both of them lack troponin, and contractility is activated by RLC phosphorylation via myosin light chain kinase (MLCK) when bound to calmodulin with four bound calcium ions. In this way calcium regulates contractility indirectly, just as it does via troponin in striated muscle. MLCK is phosphorylated reversibly by the hydrolysis of cyclic AMP, which seems a waste of energy to activate a muscle whose contractile apparatus has a very low ATPase rate. In smooth muscle, this is the primary mechanism of regulation, but there may also be thin-ﬁlament regulation via caldesmon, which binds to actin in the absence of calcium (Hodgkinson 2000).

8.4 Direct regulation of Myosin Contractility

369

Exactly how a phosphorylated RLC activates myosin-actin interactions became clear when Wendt et al. (2001) discovered the inhibitory state of heavymeromyosin. When its regulatory light chains are unphosphorylated, the actin-binding interface of one head locks on to a region of the other head which includes the nucleotide pocket, the converter domain and one of its essential light chains. In the process, the S2 rod is bent and both heads are folded back on the myoﬁlament backbone, so neither head can bind to actin and their ability to hydrolyse ATP is compromised (Fig. 8.8). Phosphorylating the regulatory light chains removes this head-head interaction and enables each head to interact with regulated F-actin (Woodhead et al. 2005; Jung et al. 2008). When dimeric myosin is activated in this way, the heads can move closer to F-actin and adopt a nearly perpendicular orientation. There is evidence that this form of activation is also cooperative. In cardiac muscle, Kampourakis et al. (2016) found that calcium-activated tension was very little affected after replacing most of the regulatory light chains with an isoform which could not be phosphorylated. Phosphorylating 23% of the locked dimers was able to persuade the remainder to unlock and swing out towards F-actin. So thick ﬁlament activation is also cooperative; how does it work? Previously, Linari et al. (2015) had addressed this question via X-ray measurements of axial myosin reﬂections before and after calcium activation. The intensities of the M1 and M2 reﬂections from the 42.9 nm periodicity, and the M3 reﬂection from the 14.34 nm layer-line spacing, all decreased on activation, and the layer-line spacing increased to 14.57 nm in response to myoﬁlament tension. Most of these Fig. 8.8 Atomic model of dimeric myosin in the superrelaxed state, where the actin-binding interface of one head is locked onto the catalytic site and converter region of the other head (Woodhead et al. 2005). With permission of Springer Nature

370

8 Cooperative Muscular Activation by Calcium

changes were reversed when the activated muscle was allowed to shorten under zero load, as expected if only 5% of heads were bound at any instant. However, imposing a small load F ¼ 0.1 T0 caused an increase in ﬁlament periodicity and a partial recovery of the intensities of the diffraction peaks. The kinetics of activation were also tested by monitoring these reﬂections during force redevelopment after unloaded shortening. The X-ray intensities recovered over a period of 20 ms, similar to that of isometric tension. This suggests that, although the initial rise in tension would be controlled by the few myosin dimers that are switched on by RLC-phosphorylation, more dimers are switched on as tension rises and the myoﬁlament is extended. This is the gist of the “mechanosensing” hypothesis for cooperative regulation of contractility by the thick ﬁlament; although myosins can be switched on by light-chain phosphorylation, they can also be switched on by other switched-on myosins via myoﬁlament strain from the tension that they generate. If the thick ﬁlament can be cooperatively activated by RLC phosphorylation, it should be possible to demonstrate this process in isolation, without activating the thin ﬁlament. By using ﬂuorescent probes on the regulatory light chains, Fusi et al. (2016) were able to show that stressing the thick ﬁlament provided the necessary interaction to unlock neighbouring dimers cooperatively. Fibres from rabbit psoas muscle were stressed under relaxing conditions by stretching them incrementally to sarcomere lengths of 3.60 μm, with passive tensions of up to 0.8T0 supplied by titin linkages. In this way, the authors were able to conﬁrm that the myosin lever-arms moved towards a perpendicular orientation as myoﬁlament tension was increased, and that this effect was reversible. It remains to explain how a small (1.6%) elastic strain in the myoﬁlament can trigger a cascade of activation by releasing locked dimers. In the totally switched-off or “super-relaxed” state, where each locked dimer is folded back on the backbone, dimers are held in place by their S2 rods, and perhaps by interactions with titin and myosinbinding-protein C as well, so that a small increase in spacing is sufﬁcient to unlock the asymmetric head-head conﬁguration of each dimer. Exactly how this happens is not clear, and further developments in this unfolding saga can be expected.

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Razumova MV, Bukatina AE, Campbell KB (2000) Different myoﬁlament nearest-neighhbour interactions have distinctive effects on contractile behaviour. Biophys J 78:3120–3137 Regnier M, Morris C, Homsher M (1995) Regulation of the cross-bridge transition from a weakly to strongly bound state in skinned rabbit muscle ﬁbers. Am J Phys 269:C1532–C1539 Robinson JM, Wang Y, Kerrick WGL, Kawai R, Cheung HC (2002) Activation of striated muscle: nearest-neighbour regulatory units and cross-bridge inﬂuence on myoﬁlament kinetics. J Mol Biol 322:1065–1088 Ruegg JC (2012) Calcium in muscle activation; a comparative approach. Springer, Berlin Sato K, Ohtaki M, Shimamoto Y, Ishiwata S (2011) A theory on auto-oscillation and contraction in striated muscle. Prog Biophys Mol Biol 105:199–207 Sato K, Kuramoto Y, Ohtaki M, Shimamoto Y, Ishiwata S (2013) Local and globally coupled oscillators in muscle. Phys Rev Lett 111(5):108104 Shimamoto Y, Suzuki M, Ishiwata S (2008) Length-dependent activation and auto-oscillations in skeletal myoﬁbrils at partial activation by Ca2+. Biochem Biophys Res Commun 366:233–238 Shimizu H, Fukita T, Ishiwata S (1992) Regulation of tension development by MgADP and pi without Ca2+: role in spontaneous tension oscillation of skeletal muscle. Biophys J 61:1087–1098 Shiner JS, Solaro RJ (1982) Activation of thin-ﬁlament-regulated muscle by calcium ion: considerations based on nearest-neighbour lattice statistics. Proc Natl Acad Sci USA 79:4637–4641 Smith DA, Stephenson DG (1994) Theory and observation of spontaneous oscillatory contractions in skeletal myoﬁbrils. J Muscle Res Cell Motil 15:369–389 Smith DA, Stephenson DG (2009) The mechanism of spontaneous oscillatory contractions in skeletal muscle. Biophys J 96:3682–3691 Stephenson DG, Wendt IR (1984) Length dependence of changes in sarcoplasmic calcium concentration and myoﬁbrillar calcium sensitivity in striated muscle ﬁbres. J Muscle Res Cell Motil 5:243–272 Stephenson DG, Williams DA (1981) Calcium-activated force responses in fast and slow-twitch skinned muscle ﬁbres from the rat. J Physiol (London) 333:637–653 Tanner BCW, Daniel TL, Regnier M (2007) Sarcomere lattice geometry inﬂuences cooperative myosin binding in muscle. PLOS Comp Biol 3:1195–1211 Tesi C, Piroddi N, Colomo F, Poggesi C (2002) Relaxation kinetics following sudden Ca2+ reduction in single myoﬁbrils from rabbit skeletal muscle. Biophys J 83:2142–2151 Tobacman LS, Butters CA (2000) A new model of cooperative myosin-thin ﬁlament binding. J Biol Chem 2756:27587–27593 Vandenboom R, Clalin DR, Julian FJ (1998) Effects of rapid shortening on rate of force regeneration and myoplasmic [Ca2+] in intact frog skeletal muscle ﬁbres. J Physiol (London) 511(1):171–180 Vandenboom R, Hannon JD, Sieck GC (2002) Isotonic force modulates force redevelopment rate of intact frog muscle ﬁbres: evidence for cross-bridge induced thin ﬁlament activation. J Physiol (London) 543(2):555–566 Von der Ecken J, Muller M, Lehman W, Manstein DJ, Penczek PA, Raunser S (2015) Structure of the F-actin-tropomyosin complex. Nature 519:114–117 Wahr PA, Metzger JM (1999) Role of Ca2+ and cross-bridges in skeletal muscle thin ﬁlament activation probed with Ca2+ sensitizers. Biophys J 76:2166–2176 Wendt T, Taylor D, Trybus KM, Taylor K (2001) Three-dimensional image reconstruction of dephosphorylated smooth muscle heavymeromyosin reveals asymmetry in the interaction between myosin head and placement of subfragment 2. Proc Natl Acad Sci USA 98:4361–4366 Woodhead JL, Zhao FQ, Craig R, Egelman EH, Alamo L, Padron R (2005) Atomic model of a myosin ﬁlament in the relaxed state. Nature 436:1195–1199 Yasuda K, Shindo Y, Ishiwata S (1996) Synchronous behaviour of spontaneous oscillations of sarcomeres in skeletal myoﬁbrils under isotonic conditions. Biophys J 70:1823–1829

Appendices

Appendix A: Axial and Azimuthal Selection Rules for HeadSite Interactions Axial and azimuthal constraints on the interaction between a tethered myosin head and an actin site can be modelled on a ‘yes-no’ basis, by deﬁning selection rules based on the structures in Fig. 1.6 (Smith et al. 2008). To illustrate the method, the following treatment is based on just one actin ﬁlament, as shown in Fig. A1a. Suppose there is a cut-off distance dco for the magnitude of axial head-site displacements ΔX, and an angular cut-off ϕco for azimuthal misalignments in the plane of cross-section. The latter can be speciﬁed in terms of angular deviations from the line of centres between myoﬁlament and F-actin. There may also be angular mismatches ΔϕM for the head and ΔϕA for an actin site, relative to the lines of centres from F-actin to myoﬁlaments. These lines of centres can be thought of as the rungs of a ladder of head-ﬁlament interactions. Then there are three conditions for matching myosins (dimers or single heads) to actin sites: Axial selection :

j ΔX j< dco

ðA1Þ

Azimuthal headladder selection :

j ΔϕM j< ϕco

ðA2Þ

Azimuthal laddersite selection :

j Δϕ j< ϕco

ðA3Þ

A

While more elaborate criteria for angular selection can be devised, these empirical equations do have the virtue of simplicity. Let a ¼ 42.9 nm and b ¼ 36 nm for the F-myosin period and the half-period of F-actin. As in Fig. 2.5, heads in adjacent layer lines on the same myoﬁlament cannot select the same actin site if dco < a/6 ¼ 7.15 nm. In each layer line there is a crown of three myosin dimers, but only one of them can select an actin site if ϕco < 30 . Structurally, these constraints seem quite realistic.

© Springer Nature Switzerland AG 2018 D. A. Smith, The Sliding-Filament Theory of Muscle Contraction, https://doi.org/10.1007/978-3-030-03526-6

375

376

Appendices

Fig. A1 (a) A triangular subcell of the 2:1 ﬁlament lattice, showing a central actin ﬁlament and three myosin heads on the three surrounding myoﬁlaments (k ¼ 1,2,3) in three adjacent layer lines (m ¼ 1,2,3) of the half-sarcomere. The shaded heads are those most closely oriented to the actin ﬁlament. (b) Head-site displacements, referred to selected actin sites on the two strands, as a function of site position, calculated at L ¼ 1150 nm for the numbers in the main text. (c) The numbers of dimers selected on each strand varies with half-sarcomere length even at full ﬁlament overlap, as determined by a combination of myosin and actin periodicities

Implementing these selection rules for all dimers addressing the same actin ﬁlament is now a matter of bookkeeping. Anticipating the application in Chap. 8 to regulated actin, these rules can be applied to each actin strand in turn, since the two strands are separately regulated by a chain of tropomyosin-troponin units. Here are the rules, formulated for sarcomeres with any degree of ﬁlament overlap. To specify the geometry of the ﬁlaments, let there be NM myosin dimers on each myoﬁlament k ¼ 1,2,3 addressing an F-actin with NA sites per strand. Measured from the M-band, the axial coordinates of myosins, sites and their axial mismatches are X mM ¼ LB þ ðm 1Þ

a 3

ðm ¼ 1; : : . . . ; NM Þ

ðA4Þ

Appendices

377

σ 2b A X nσ ¼ L þ n NA þ 2 13

ðn ¼ 1; : . . . ; NA Þ

A M ΔX nσ m ¼ X nσ X m :

ðA5Þ ðA6Þ

Here L is half-sarcomere length, LB is the half-width of the bare zone and σ ¼ 0,1 labels the two actin strands. Thus 2b/13 ¼ 5.54 nm is the site spacing on each strand, and the two strands are staggered by 2.77 nm. The ﬁlament lengths LM,LA follow from Eqs. A4 and A5. For example, if NM ¼ 50 and LB 50 nm, then LM ¼ 765 nm. For F-actin, setting LA ¼ (2NA+1)b/13 gives LA ¼ 1100 nm for NA ¼ 200. With reference to Fig. A1, azimuthal angles (in degrees) will be deﬁned clockwise from the vertical in the range j0,360), signifying 0 ϕ < 360. For myosins on myoﬁlament k, the angle of the mth head (in degrees) most closely oriented towards F-actin is ϕM mk ¼ 360

k m0 1 þ modulo 360 3 9

ðm0 1 ¼ m 1 modulo 3Þ

ðA7Þ

and these angles repeat on every third layer line. Let ψ k be the angle of the kth ladder of interaction as seen from actin: ψ k ¼ 120ðk 1Þ

ðk ¼ 1; 2; 3Þ

ðA8Þ

However the ladder as seen from myoﬁlament k is at angle 180þ ψ k, so the azimuthal mismatch between head m on myoﬁlament k is M ΔϕM mk ¼ ϕmk ð180 þ ψ k Þ

ðA9Þ

which should be reduced modulo 360 to [180,180). (Be careful in programming; modulo and mod may have different deﬁnitions). We also need the angular site-ladder mismatch. On setting the angle of the ﬁrst actin site to zero, the azimuthal angle of the nth site on strand σ will be ϕA nσ ¼ 360

n 6σ modulo 360 13

ðA10Þ

which repeats on every 13th site to give the 72 nm axial period. Note also that the stagger angle between the strands is –166 , which makes the double helix lefthanded although each strand is a right-handed screw. So the ladder-site mismatch angle is A ΔϕA knσ ¼ ϕnσ ψ k

ðA11Þ

378

Appendices

reduced to [180,180). This completes the recipe for mapping myosins to actin sites. If conditions Eqs. A1, A2, and A3 are met, the mapping is one-to-one or one-to-none. If axial matching is removed by setting dco ¼ b/2, an azimuthally matched site on the double helix can always be found. The orientations of actin sites on one strand advance by 360/13 ¼ 27.7 , so there will always be a mapped site unless ϕco < 13.85o, which is unduly restrictive. If ϕco ¼ 30o, then up to three sites may be selected. So the optimum site n, which deﬁnes the centre of a target zone, is the one which minimizes jΔϕA knσ j. Combining all the selection rules gives the mapping function n ¼ nmap ðm; k; σ Þ

ðm ¼ 1; : . . . ; NM ; k ¼ 1 3; σ ¼ 1; 2Þ:

ðA12Þ

This function can be used in various ways. Of prime importance are the numbers Mσ of dimers mapped to F-actin strand σ, and the net number M ¼ M0 þ M1, which determine the magnitudes of tension and other extensive quantities per F-actin rather than per head. For thin-ﬁlament regulation, separate mappings are required for each actin strand, and the axial positions of mapped heads on one strand determine how they are regulated by tropomyosin. For this purpose, the mapping function needs to be inverted as follows. Mappings from the three myoﬁlaments can be combined to give the locations {nsσ, s ¼ 1, . . . . ,Mσ) of mapped sites on strand σ, the axial coordinates {Xsσ} of these sites, and the head-site displacements {xsσ}. These operations are best described by hybrid computer code, omitting the deﬁnitions already given. Here sort2(A,B) is a piece of code which sorts array A into ascending order, using the same permutations to sort array B. The discrete axial positions of myosin heads implies that Mσ varies with sarcomere length, even when all heads overlap F-actin, and an example is shown in Fig. A1b. The head-site displacements produced by these mappings are plotted in Fig. A1c; they provide a window from which to view the behaviour of lattice models. DO σ ¼ 1,2 s¼0 DO k ¼ 1,3 DO m ¼ 1,NM IF nmap(m,k,σ) > 0 THEN s¼sþ1 n ¼ nmap(m,k,σ) nsσ ¼ n xsσ ¼ ΔX nσ m END IF END DO END DO Mσ ¼ s Call sort2([nsσ, s ¼ 1,Mσ], [xsσ, s ¼ 1, Mσ]) END DO

Appendices

379

The axial locations Xsσ of the actin sites indexed by s, namely A X sσ ¼ X nσ

ðn ¼ nsσ Þ

ðΑ13Þ

are also required for thin-ﬁlament regulation. For the triangular sub-cell of the ﬁlament lattice shown in Fig. A1, the functions nsσ , Xsσ and xsσ provide all the spatial information needed to set up a theory of thin-ﬁlament regulation in striated muscle, at the level of a single actin ﬁlament in the half-sarcomere. For lattice models, all three selection rules apply. Here are the results of implementing them with dco ¼ 7.1 nm, ϕco ¼ 28.8 , and NM ¼ 50, NA ¼ 200. At L ¼ 1140 nm, there are 30 and 31 dimers selected for each strand, and in Fig. A1b their head-site displacements x(n) are plotted against their axial positions on the actin ﬁlament. Hence M ¼ 61 for the number of dimers addressing both strands. In terms of single heads, out of the 300 possible heads on the three myoﬁlaments, only 122 satisfy the selection rules. Moreover, Fig. A1c shows that these numbers change with half-sarcomere length, even on the plateau of the tension-length curve, reﬂecting the sharp cut-offs inherent in the selection rules. For axial selection, a sharp cut-off of 7.1 nm easily accommodates the binding range deﬁned by myosin stiffness. For azimuthal selection, a sharp cut-off could be replaced by graded selection where actin afﬁnity is a smooth function of the mismatch angles ΔϕM and ΔϕA. However, such selection will be determined by the bending and torsional stiffnesses of tethered myosins, and possibly the torsional stiffness of F-actin as well. These results need to be extended to deal with multiple F-actins in the rhomboidal unit cell, and also for the supercell in Fig. 1.6b. But for vernier models, axial selection is built-in, and the number M of dimeric myosins addressing one F-actin ﬁlament is determined only by azimuthal head-site matching. If there are 150 dimers (300 heads) per F-actin, then inspection of Fig. 1.6 and its counterparts in adjacent layer lines shows that M ¼ 75 for the simple unit cell and M ¼ 135 for the supercell.

Reference Smith DA, Geeves MA, Sleep J, Mijailovich SM (2008) Towards a uniﬁed theory of muscle contraction. I: foundations. Ann Biomed Eng 36:1624–1640

Appendix B: Solving the Rate Equations for the MyosinATPase Reaction It should be no surprise that the rate equations (Eq. 3.1) for the hydrolysis of ATP on myosin can be solved analytically. On using the conservation law M0 ¼ [M] þ [M. ATP] þ [M.ADP.Pi] to eliminate one reactant, say apomyosin (M), they reduce to

380

Appendices

two linearly independent coupled d.e.’s. In matrix-vector notation, the general form of these equations is dxðt Þ þ Ax ¼ b dt

ðB1Þ

where x(t) is a 2-vector and A a constant 2 2 matrix. For the problem in hand, let us use fractional concentrations x1(t) ¼ [M. ATP](t)/M0, x2(t) ¼ [M. ADP. Pi](t)/M0, so A¼

kT þ kT þ k H , k H ,

k T kH , k P þ k H

b¼

kT : 0

ðB2Þ

where kT ¼ k~T ½ATP: When the exact solution is applied to the experiment of Lymn and Taylor (1970), it explains why the results could be ﬁtted to a single exponential, but the rate constant of that exponential was a piece-wise function of ATP concentration, increasing linearly with [ATP] below 100 μM and constant at higher concentrations. The initial condition of this experiment was x(0) ¼ 0, so the solution is of the form xð t Þ ¼

X

cα ð1 expðλα t ÞÞxα

ðB3Þ

α¼

where Axα ¼λxα for the eigenvalues and eigenvectors of A. In terms of its elements, λ ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 a11 þ a22 ða11 a22 Þ2 þ 4a12 a21 2 x2α λα a11 a21 ¼ ¼ : x1α a12 λα a22

ðB4Þ ðB5Þ

It is sufﬁcient to choose unnormalised eigenvectors, for example x1α ¼ a12 ,

x2α ¼ λα a11

ðB6Þ

although this choice is not unique. Then the coefﬁcients cα are determined by substitution in Eq. B1, which reduces to X α¼

and hence

c α λ α xα ¼ b

ðB7Þ

Appendices

381

c ¼

ðλþ a11 Þb1 a12 b2 , λ ðλþ λ Þa12

cþ ¼

ðλ a11 Þb1 a12 b2 : λþ ðλþ λ Þa12

ðB8Þ

For the problem in hand, a little manipulation enables the eigenvalues to be written as λ ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 r T þ r H þ k P ðr T r H kP Þ2 þ 4k T kH 4k H k P 2

ðB9Þ

where rT ¼ kT+k–T, rH ¼ kH+k–H. The occupation of the hydrolysed state M.ADP.Pi is x2 ð t Þ ¼

X

C α ð1 expðλα t ÞÞ

ðB10Þ

α¼

with C ¼ (λ – a11)c. After some more algebraic manipulation, C α ¼ α

kH kT , λα ðλþ λ Þ

ðα ¼ 1Þ

ðB11Þ

which completes the solution for the time course of hydrolysis. For example, the steady-state occupancy of the M.ADP.Pi state at large times is x2 ð 1 Þ ¼ C C þ þ C ¼

kH kT kH kT ¼ : λþ λ k T r H þ k T kH þ ðr T þ kH Þk P

ðB12Þ

The solution is complicated by the unwelcome discovery that the eigenvalues are complex for a certain range of values of kP. Hence it is best to ﬁrst consider the case when the products of hydrolysis are released very slowly.

Slow Product Release; kP 0, C+ < 0, and |C+| < C–, the predicted time course of liberated phosphate shows a slight lag arising from the faster exponential, but otherwise remains well-behaved even when the individual

Fig. B1 The burst of bound phosphate liberated on M.ADP.Pi, as predicted from ATP binding and hydrolysis using Eqs. B8, B9, B10, and B11 with k~T ¼ 106 M1 s1 : (a) The burst transient for ﬁve different ATP concentrations as shown. (b) The burst rate, approximated by the smaller eigenvalue λ, as a function of [ATP]. (c) The fraction C+/C– of the coefﬁcients of the two exponential rises in Eq. B10

Appendices

383

terms in Eq. B10 become much larger than unity. In practice, this behaviour justiﬁes the use of a single exponential to ﬁt burst observations unless high accuracy is required.

Faster Product Release (kP arbitrary) As kP is increased beyond kH and/or kT, the rate of hydrolysis increases and the burst size diminishes because the net ﬂux through the sequence of reactions stops the population of M.ADP.Pi from building up. The burst size and the steady-state rate are B¼

X

Cα

α¼

kP 1 , λa

R ¼ Ck P

(Eq. 3.7) and B is a decreasing funtion of kP. There is nothing in these results to stop the burst size from becoming negative, meaning a lag in the rise of liberated phosphate. However, these formulae must be manipulated to allow for complex eigenvalues, where the transient solution becomes a damped oscillation. Firstly, when are oscillations expected? In the (kT,kP) plane, there is a boundary curve for the domain of complex eigenvalues, obtained by setting the discriminant in Eq. B9 to zero. This boundary is determined only by the ratios x ¼ k T =kH ,

y ¼ kP =k H

ðB14Þ

and is given by the equation pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y ¼ x þ 1 ε 2 x ε,

ε 1=K H :

ðB15Þ

which is displayed in Fig. B2. Thus an exponential response occurs at small kP and large kP but, except at very small ATP levels (x < ε), there is an intermediate domain in which the response is a damped oscillation. Within this domain, the time course of liberated phosphate per myosin is

x2 ðt Þ ¼ C 1 1 eλ1 t cos λ2 t þ C 2 eλ1 t sin λ2 t þ CkP t

ðB16Þ

where the complex eigenvalues λα ¼ λ1+iαλ2 follow from Eq. B9, and C1 ¼

2λ1 k P 1 2 C, λ1 þ λ22

C2 ¼

λ1 λ21 λ22 kP C: λ2 λ21 þ λ22 λ2

ðB17Þ

384

Appendices

Fig. B2 The boundary between damped and damped-oscillatory Pi-burst responses, as a function of the rate ratios kT/kH and kP/kH (Eq. B15), and plotted for ε ¼ 0.1

Examples of this ‘ringing’ behaviour are shown in Fig. B3. For small values of kP, there may be a range of values of kT which lie just above the lower boundary in Fig. B2, which give a pronounced oscillatory response. As kP is increased, the range of kT for damped oscillations becomes much wider, although the responses are very heavily damped. At higher values of kP, low values of kT lie to the left of the boundary, giving a non-oscillatory response with a lag, which ﬁnally becomes the norm.

References Lymn RW, Taylor EW (1970) Transient state phosphate production in the hydrolysis of nucleoside triphosphates by myosin. Biochemistry 9:2975–2983 Woledge RC, Curtin NA, Homsher E (1985) Energetic aspects of muscle contraction, Monographs of the physiological society no. 41. Academic Press, London

Appendix C: The Kinetics of Two-Step Reactions It is often desirable, if not always justiﬁable, to reduce a sequence of two consecutive reactions to just one reaction step. There are two situations in which this is appropriate. The ﬁrst is when the ﬁrst step is rapid and reversible, so that it can come to equilibrium before the second step proceeds. The second case is that where the intermediate state is short-lived and its occupancy can be neglected. Both permit the derivation of a simple formula for the rate of the net reaction.

Appendices

385

Fig. B3 Examples of damped and damped-oscillatory Pi-burst transients predicted from Eqs. B10 or B16 for various values of kP (1, 50, 150, 250 s1 in graphs (a–d) respectively), and ATP concentrations as listed. Oscillatory responses are shown in red. Other ﬁxed parameters were k~T ¼ 106 M1 s1 , kH ¼ 50 s1 and KH ¼ 10

Binding via a Collision Complex When two molecules bind, they ﬁrst collide to form a collision complex which is weakly held together by electrostatic forces (screened Coulomb forces between polar groups of like sign, or van der Waals attraction through induced dipole moments). In the second step, the complex becomes more tightly bound as covalent bonds are formed. The collision step is reversible, very rapid and limited by diffusion; the second-order rate constant is typically ~108 M1s1 (Segel 1978), so this step

386

Appendices

Scheme C.1 Kinetics of enzyme-substrate binding via a collision complex

E+S

~ K1

ESX

k2 k-2

ES

equilibrates long before the subsequent isomerization. This rapid equilibrium can be formalised by lumping the states together, so that the kinetics of the lumped state does not include very rapid processes. Let E be an enzyme (such as myosin) to which a substrate S (actin, nucleotide, any ligand) binds, forming a collision complex ESX and then the stable form ES. The assumption of rapid equilibrium, as shown in Scheme C.1, is an approximation, which is justiﬁed when k1+k– 1 >> k2+k–2. Furthermore, when the substrate is in excess, the traditional approach is to treat its concentration [S] as constant. This approximation is often used in an uncontrolled manner, but if no further reactions ensue then it is easy to estimate how much substrate is consumed as the reaction goes to equilibrium. To begin, let Enet ¼ EþESX and Snet ¼ SþESX be lumped states of enzyme and substrate which include the collision complex. The reactants and the collision complex are assumed to be in rapid equilibrium, with X ES =½E½S ¼ K~ 1

ðC1Þ

so

ESX K~ 1 ½S ¼ , ½Enet 1 þ K~ 1 ½S

½E 1 ¼ : ½Enet 1 þ K~ 1 ½S

ðC2Þ

Rate equations can now be set up so that rates of transitions out of the box are reduced by the fractional occupation of ESX, while transition rates into the box are unaffected: K~ 1 ½S d ½ES ¼ k2 ½Enet k2 ½ES dt 1 þ K~ 1 ½S

ðC3aÞ

K~ 1 ½E d ½ES ¼ k 2 ½Snet k2 ½ES: dt 1 þ K~ 1 ½E

ðC3bÞ

Let E0 and S0 be the total concentrations of enzyme and substrate, so E0 ¼ [Enet]þ [ES] and S0 ¼ [Snet]þ[ES]. Hence K~ 1 ½S d½ES ¼ k2 ðE 0 ½ESÞ k2 ½ES, dt 1 þ K~ 1 ½S

ðC4aÞ

Appendices

387

K~ 1 ½E d½ES ¼ k2 ðS0 ½ESÞ k 2 ½ES: dt 1 þ K~ 1 ½E

ðC4bÞ

In fact, these two equations for [ES] are identical, and the proof is straightforward. If the substrate is in excess, the traditional approach is to treat [S] as constant in time, so that Eq. C4a gives the pseudo-ﬁrst-order rate constants k¼

K~ 1 k 2 ½S , 1 þ K~ 1 ½S

k ¼ k2 :

ðC5Þ

For applications, it is desirable to hide sufﬁces 1 and 2 and use a generic notation. k is a hyperbolic function of [S], which is better written in the form k¼

k~½S , 1 þ ½S=Cm

k~ K~ 1 k2 ;

C m 1=K~ 1

ðC6Þ

where k~ is the second-order rate constant in the limit of small [S], and Cm is a Michaelis-Menten constant. In textbooks of biochemistry, the Michaelis-Menten equa~ m ¼ k 2 , which tion is usually written in terms of the maximum rate constant V max kC is approached when [S] >> Cm. Here this notation is avoided to reserve the symbol V for potential energy in mechanics. What has happened to detailed balancing? The overall pseudo-ﬁrst-order equilibrium constant for substrate binding is K ¼ K~ 1 ½SK 2 K 1 K 2 where K2 ¼ k2/k–2, but the ratio of the transition rates is K~ 1 ½SK 2 k ¼ : k 1 þ K~ 1 ½S

ðC7Þ

These quantities are not equal because they are not ratios of equilibrium populations between the same states; K1K2 ¼ [ES]/[E] whereas k/k– ¼ [ES]/[Enet]. If two-step binding is to be treated as a single reaction by ignoring the intermediate state, then the price to be paid is that the rate of the back reaction is not reduced by the fraction of the lumped state present as free enzyme. What is reduced by the collision complex is the amount of free enzyme created by the back reaction, not the rate.

The Steady-State Approximation for Short-Lived Intermediate States Quite often it happens that the initial binding of enzyme and substrate is not fast enough for the ﬁrst bound state to qualify as a collision complex. This can be tested if the rate constant k–1 of the back reaction can be measured, for example from turbidity

388

Appendices

as seen by light scattering. The forward rate k1 can then be extracted from the Michaelis-Menten constant C m ¼ 1=K~ 1 , which requires measurements of the second-order rate K~ 1 k2 (with low substrate levels) and the maximum rate k2 at high levels. To deal with this case, Briggs and Haldane proposed that approximate rate constants for the net reaction could be derived by assuming that the concentration [ESX] of the intermediate state rises to a steady value well before the reaction is completed. This can never be exact. For example, if the ﬁrst step is more rapid then [ESX] will decrease as the second step gets underway. The steady-state condition for the intermediate state is expressed by equal forward ﬂuxes for the two reaction steps: K~ 1 ½E½S k 1 ½ESX ¼ k2 ½ESX k2 ½ES

ðC8Þ

X k~1 ½E½S þ k2 ½ES ES ¼ : k2 þ k1

ðC9Þ

d ½ES ¼ k2 ESX k2 ½ES k~þ ½E½S k ½ES dt

ðC10Þ

so

Hence

where k~þ ¼

k~1 k2 , k2 þ k1

k ¼

k2 k1 : k 2 þ k1

ðC11Þ

are effective rate constants for the net reaction, which express the “steady-state condition” employed by Briggs and Haldane. Naturally, detailed balancing is satisﬁed; k~þ =k ¼ K~ 1 K 2 : The formulae are easy to remember: the numerators are products of the two forward and two backward rate constants respectively, while the denominator is the net rate constant out of the intermediate state. The concentration of the intermediate state can be neglected if it is short-lived. Now Eq. C9 shows that [ESX] > k2

ðC12Þ

which is less restrictive than the condition for rapid equilibrium. How accurate is the Briggs-Haldane approximation? Segel (1978) expresses it thus: “As the ratio [S]0/[E] increases, the time interval before d[ESX]/dt 0 decreases and the extent of the reaction during which d[ESX]/dt 0 increases.”

Appendices

389

Reference Segel I (1978) Enzyme kinetics. Wiley-Interscience, Hoboken

Appendix D: Smoluchowki’s Equation and Kramers’ Theory To understand the foundations and limitations of Smoluchowski’s equation, it is convenient to start from the Newtonian equation of motion of a particle in the presence of viscous drag and a ﬂuctuating force f(t) due to molecular impacts, as proposed by Langevin: mv_ þ μv þ

dV ðxÞ ¼ f ðt Þ, dx

v ¼ x_ :

ðD1Þ

This Langevin equation serves to illustrate several key points. For the damping coefﬁcient μ, Stokes’ law gives μ ¼ 6πηR for a sphere of radius R in a medium of viscosity η. It is important to remember that viscous drag itself arises from molecular impacts. A formal ﬁrst integration gives vðt Þ ¼ vð0Þe

t=τ

1 þ m

Z 0

t

dV ðxðt 0 ÞÞ 0 0 þ f ðt Þ eðtt Þ=τ dt 0 dxðt 0 Þ

ðD2Þ

where τ m=μ

ðD3Þ

is the time over which inertial accelerations are damped out. On any greater time scale, the integral can be approximated by replacing the bracketed function by its value at time t, giving 1 dV ðxðt ÞÞ f ðt Þ vð t Þ μ dxðt Þ

ðD4Þ

where the velocity of the particle is always proportional to the net force on it. In hydrodynamics this is the regime of low Reynolds number (Happel and Brenner 1965). However, a rigorous derivation of Eq. D4 requires Ito’s calculus of stochastic integration (Gardner 2003). For a free particle, this equation shows that the mean-square velocity is proportional to the correlation function of the Brownian forcef(t). Moreover, statistical mechanics in the form of the law of equipartition of energy gives v2 ¼ kBT/m. Hence

390

Appendices

f ðt Þ2 ¼

μ 2 kB T m

ðD5Þ

is the explicit connection between viscous drag and Brownian forces. The second point is that a second integration of Eq. D1 leads to the well-known law of diffusion for a free Brownian particle, namely xðt Þ2 ¼ 2Dt

ðD6Þ

where D is the diffusion constant and D kB T=μ

ðD7Þ

is Einstein’s relation which links the diffusion constant to the viscous drag coefﬁcient. Einstein’s proof proceeded from a Fokker-Planck equation, but Langevin was pleased to provide a simpler proof, by using Eq. D2 to generate a d.e. for the meansquare displacement. The details are given in Gardner’s monograph. The ﬁnal point is that, in the presence of a potential V(x), Eq. D4 holds provided that V(x) is slowly varying over the range of displacements vτ generated by Brownian noise, where v ¼ (kBT/m)1/2. At this point, Langevin’s theory connects with Smoluchowski’s equation. When the distribution of momenta comes to thermal equilibrium in every part of the potential, it can be factored out of Kramers’ FokkerPlanck equation for the joint distribution of position and momentum (Kramers 1940, Hänggi et al. 1990). This requires that particle motion at the top of the potential well be highly damped, so that ωB τ > 1. The scaled variable xA is not so very different from X in nanometres, since xA/X ¼ (βκ/2)1/2 ¼ 0.61 nm1 if κ ¼ 3 pN/nm. The full X-dependence of the roll-lock reaction rate is shown in Fig. F1a. What can be deduced from this result? Because the X-dependent rate of the rolllock reaction varies from inﬁnity to zero, it cannot be the rate-limiting step at all X. At small |X| values, the capture step from the collision state to the ﬂoppy state must be rate-limiting, while the Boltzmann factor exp(βκX2/2) ensures that the roll-lock transition is rate-limiting at sufﬁciently large values of |X|. Thus the overall actinbinding rate should be of the form k c AðX ÞeβκX =2 k A ðX Þ ¼ 2 kc þ AðX ÞeβκX =2 2

ðF8Þ

as shown in Fig. F1b. Incidentally, the dimensions of A(X) are s1, despite appearances to the contrary in Eq. F7, because the diffusion constant in Smoluchowki’s equation is deﬁned relative to the reaction coordinate, which in this case is dimensionless. Here the units of D are s1.

398

Appendices

The Working Stroke Kramers-Smoluchowski theory gives a slightly different strain-dependent stroke rate from that obtained from transition-state theory with the highest energy barrier. For X < X* the rate is not constant but continues to increase as X is reduced. Using axial strains only, the Gibbs energy GS(θ,X) is as shown in Fig. 2.10a: GS ðθ, XÞ ¼ νðθ, XÞ þ gA ðθÞ þ gR ðθÞ

ðF9Þ

and v(θ, X) is the function of Eq. 4.10 with LS1 replaced by R, the chord length of the lever-arm. The functions gA(θ), gR(θ) denote sharp potential wells of depths BA, BR and width Δθ at θ ¼ θA and θR respectively. They refer to small rotational displacements of the lever-arm with respect to the motor domain, whereas G(ψ, X) in Eq. 4.9 refers to rotations of the motor domain with respect to F-actin. Equation 4.5 requires the quantities Z A ¼ Δθ eβðκX =2BA Þ , sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Z uR o 2 n x2 2 βvðθ;X Þ R F ð x Þ exA F ð x Þ e du ¼ e R A βκR2 uA 2

ðF10aÞ ðF10bÞ

where u ¼ cos θ. Here xA ¼ (βκ/2)1/2X, xR ¼ (βκ/2)1/2(X+h) and F(x) is Dawson’s integral (Eq. F4). Hence the rate constant of the working stroke is

1=2 βκR2 =2 DeβBA

k S ðX; κ Þ ¼ Δθ exp x2R x2A F ðxR Þ F ðxA Þ

ðF11Þ

and the rate normalized to its (strain-independent) value at zero stiffness is k S ðX, κÞ h~ ¼ ~ ~ 2 hðx þ h=2Þ kS ðX, 0Þ e A FðxR Þ FðxA Þ

ðF12Þ

where h~ ¼ ðβκ=2Þ1=2 h (Smith and Sleep 2006). The backward rate is obtained by combining Eq. F12 with the strain-dependent equilibrium constant ~ A þ h=2ÞÞ ~ K S ðX, κÞ ¼ K S expð2hðx

ðF13Þ

which is equivalent to Eq. 2.41. The general form of these rate functions, shown in Fig. F2, should be compared with those in Fig. 4.4 for the highest-barrier approximation. The pre-exponential factor causes the forward rate to rise as X decreases below –h/2, and the backward rate to rise as X moves in the opposite direction.

Appendices

399

Fig. F2 Predictions of Kramers-Smoluchowski theory for strain-dependent rate constants of the myosin working stroke ((a) forward, (b) backward). Rates are normalized to their strainindependent values obtained by setting κ ¼ 0, and plotted against x ¼ (βκ/2)1/2X for ten values of the dimensionless working stroke h~ ðβκ=2Þ1=2 h ¼ 0.5(0.5)5.0 as indicated by the arrows

References Abramowitz M, Stegun IA (1965) Handbook of mathematical functions. Dover Pubications Inc, New York Smith DA, Sleep J (2006) Strain-dependent kinetics of the myosin working stroke, and how they could be probed with optical-trap experiments. Biophys J 91:3359– 3369

Appendix G: A Two-Dimensional Model for S1-Actin Binding and Tension In Sect. 4.1, tension in bound heads was assumed to be proportional to the axial displacement X of the actin site from the detached head. This linear axial model ignores any radial component of tension, which can be calculated from a two-dimensional model. The results are quite far-reaching; not only do they predict the maximum values of X obtainable in the ﬂoppy state and the locked state, but the X-dependence of axial tension is no longer linear, increasing more rapidly with positive strain and decreasing less rapidly with negative strain. To date, this reﬁnement has not been used in modelling at the level of a single head. The 2D geometry of actin binding starts from the swinging head, depicted in Fig. 4.2 as a spherical motor domain of radius r with a lever-arm of length R tethered at its centre C. The distal end B is ﬂexibly attached to the S2 rod AB, which is free to bend (Adamovic et al. 2008). The presence of loops on the actin-binding surface of

400

Appendices

the motor domain suggests that binding to a ﬂoppy state occurs when one loop binds to a conjugate site on actin. There are six such loops (von der Ecken et al. 2016), which deﬁne a binding patch on the motor domain, so the roll-lock reaction must involve the binding of more loops, perhaps all of them, to provide a strongly-bound stereospeciﬁc A-state.

The Axial Reach of the Heads In this idealized picture, the bound states of the head are shown in Fig. G1. Figure G1a depicts a ﬂoppy state in which loop Q has bound to its receptor on a displaced actin site, and this displacement determines the angle θF of the lever-arm. The geometry of the ﬁgure gives the coordinates of the S1-S2 junction (point B) as xB ¼ xP þ R cos θF r ðθF θA Þ

ðG1aÞ

zB ¼ Z r R sin θF

ðG1bÞ

where θA is the ﬁxed lever-arm angle of the A-state. Point B is constrained by the elastic behaviour of S2, which can bend but not stretch. The no-stretch condition is j rB rA j¼ LS2

ðG2Þ

where rA ¼ (–LS2,0). Strictly speaking, it is the arc length rather than the chord length AB which is invariant, but this can be simulated by increasing the value of LS2; for small bending displacements it can be shown that a 20% increase is required. Hence qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ xB ¼ LS2 þ LS22 z2B :

ðG3Þ

For a given head-site distance xP, (G1) and (G3) are three equations for the three unknowns θF, xB, zB. To a very good approximation, the axial retraction xB can be set to zero, which provides a ﬁrst iterate for more accurate estimates. After the ﬂoppy state has formed, the motor domain rolls on actin until point P is in contact. Then more loops bind to lock the motor domain into its A-state conﬁguration (Fig. G1b) with lever-arm angle θA. During this process, the lever-arm bends by angle ψ A as its distal end is joined to S2 at B. In the locked state, its coordinates are different: xB ¼ xP þ R cos ðθA þ ψ A Þ

ðG4aÞ

zB ¼ Z r R sin ðθA þ ψ A Þ:

ðG4bÞ

Appendices

401

Fig. G1 Cartoons for the initial and ﬁnal states of the roll-lock reaction for S1-actin binding. The point of contact represents a loop on S1 bound to actin. (a) The ﬂoppy state, in which the lever-arm is tension-free and a loop at Q is bound. (b) The locked state, where the contact point has rolled from Q to P, and the lever-arm has bent by angle ψ A from its no-strain position CD, which has rotated anticlockwise from θF to θA by rolling. For the head-site distance, X ¼ xD. (c) Axial displacement of an actin site accessed by a swinging head in the ﬂoppy state, as a function of position ΔX of the loop in the binding patch which contacts actin. (d) The bending displacement s2 zB (Eq. G1b) of the S2 rod to which the head is tethered

As before, ψ A, xB, zB are determined in combination with Eq. G3. Because P is the point of contact during rolling, xP in Eqs. G1a and G4a are the same. The locked A-state permits a precise deﬁnition of the head-site axial distance X, so that X is also the axial strain in that state. The quantity xP does not satisfy this deﬁnition, because the lever-arm angle θA is not 90 , but closer to 110 (Irving et al. 1995). If the lever-arm of the bound head were disconnected from S2, it would straighten out with its distal end at point D. So X should be deﬁned as the x-coordinate of D, measured from origin O where the S1-S2 junction sits when the myosin is relaxed. Thus

402

Appendices

X ¼ xP þ R cos θA :

ðG5Þ

In terms of X, the axial retraction of B due to S2 bending becomes xB ¼ X þ Rð cos θF cos θA Þ r ðθF θA Þ

ðfloppy stateÞ

ðG6Þ

xB ¼ X þ Rð cos ðθA þ ψ A Þ cos θA Þ

ðlocked stateÞ

ðG7Þ

To a good approximation, xB ¼ 0. Then these equations determine the angles θF and ψ A in terms of X. However, these functions are not linear. Now the axial reach of the head is determined by the size of the binding patch. The arc length of the patch involved in rolling from angle θF to θA is ΔX ¼ rΔθ r ðθF θA Þ

ðG8Þ

X ¼ F ðΔX Þ Rf cos θA cos ðθA þ ΔX=r Þg þ ΔX

ðG9Þ

so

follows from Eq. G6 with xB¼ 0. If ξ is the half-width of the patch, then Xmax ¼ F(ξ) and Xmin ¼ F(–ξ). Figure G1c shows that the function F(ΔX) is close to linear with a slope of R/r, which is of order 4–5. If ξ ¼ 1.5 nm, then Xmax 7 nm, a comfortable value which protects myosins on adjacent layer lines from binding to the same site if they could overcome their azimuthal mismatch. Also, F(ΔX) is not an odd function when θA 6¼ 90 . In this example the axial retraction of S2 is quite small, so the axial reach of the ﬂoppy state is determined primarily by the width of the binding patches. However, an axially displaced site pulls the S1-S2 junction radially towards actin, and there may be a limit to the associated bending of S2. Figure G1d shows the amount of bending as a function of ΔX, which can be interpreted as patch half-width or position on the patch. The plot is sensitive to ﬁlament separation; with Z ¼ 15 nm, an 8 nm axial offset requires S2 to bend by 5 nm. More bending of S2 could drive it into a non-linear domain where the bending force could rise exponentially with displacement, further limiting the axial reach of a head to its collision state.

Axial and Radial Tensions in the Locked State Treating the lever-arm as an object in the x-z plane with bending compliance but no stretch compliance changes the customary linear axial tension-strain relationship tx(X) ¼ κX. The reason is simply that net tension is tangential to the arc described by the tip of the lever-arm. As the tip of the arm bends towards 180 , the tension vector rotates towards the z axis and the x-component becomes small. Hence for distant

Appendices

403

actin sites a much larger axial tension must be applied to produce the same tangential tension. Consequently, tension as a function of X in the locked state rises much faster than linearly at large strains. The appearance of non-linear elastic behaviour is created by changing from an axially linear stress-strain function to a linear stressstrain function deﬁned on a circle. Formally, axial and radial tensions in the locked A-state can be calculated from virtual displacements of the A-state bending-strain energy VA(ψ, rB) of S1 and S2. To simplify matters, the motor domain is assumed to be rigidly locked to actin, so that the elastic strain energy of S1 is due to the bent lever-arm. Then 1 1 V A ðψ A , rB Þ ¼ κS1 ðRψ A Þ2 þ κS2 ðx2B þ z2B Þ 2 2

ðG10Þ

for small strains. The values of X, Z and ψ A are ﬁxed, so the strain energy is ﬁxed by Eq. G10 in combination with Eqs. G7, G4b and G3. As κS2 0 and towards it (ΔSmin > 0) if ϕM < 0, as expected from inspection of Fig. H1. (iv) A misaligned actin site with ϕA > 0, as shown in Fig. H2, has the opposite effect, moving the loop contact towards the line of centres (ΔSmin > 0). (v) Azimuthally misaligned heads require more bending of S2, but misaligned actin sites require little or no extra bending, because RM > RA. How is the axial reach affected if S2 bending is limited by a threshold? All the conﬁgurational changes which produce the ﬂoppy state are driven by Brownian

Fig. H2 Examples of the effects of azimuthal mismatching, speciﬁed by the angles ϕM, ϕA on S1-actin binding in the ﬂoppy state (Fig. H1): a selection of axial head-site displacements (X) and S2 bending (s2) as a function of the local coordinates ΔX, ΔS of loop Q on the binding patches. (a) ϕM ¼ ϕA ¼ 0, (b) ϕM ¼ 40o, ϕA ¼ 0, (c) ϕM ¼ –40o, ϕA ¼ 0, (d) ϕM ¼ 0, ϕA ¼ –28o

Appendices

409

Fig. H3 (a, b) Maximum head-site displacements in the ﬂoppy state, as a function of the azimuthal angles ϕM, ϕA are constrained by imposing a threshold on the amount of S2 bending (6 nm in a and 10 nm in b). (c, d) The amount of S2 bending required in the locked state increases with the degree of azimuthal head-site mismatching. Calculations were made for two head-site displacements, namely X ¼ 0 in c and X ¼ 5 nm in d

motion, so the critical value s2c would be reached when the bending energy exceeds thermal energy kBT. In this case, Xmax can be calculated as a function of misalignment by the following procedure. If s2(ξ, ΔSmin) > s2c, S2 bending can be reduced to the critical level by lowering the value of ΔX, which moves the loop towards the centre of the patch. ΔSmin is not affected because the valley of the s2 function runs parallel to the ΔX axis. Figure H3a, b shows the axial reach Xmax(ϕM, ϕA) for two threshold levels of S2 bending, namely s2c ¼ 6 and 10 nm. In the ﬁrst case, the threshold is exceeded for all levels of mismatch, and the axial reach is reduced to zero when ϕM > 40o or ϕM < –20o. In the second case, the threshold is exceeded only for values of |ϕM| above 20 , and the reduction in Xmax is not so severe. In both cases, the axial reach is again relatively independent of the actin mismatch angle ϕA.

410

Appendices

However, there is no guarantee that this behaviour is duplicated in the locked state. A separate calculation is required, and the results are qualitatively different.

The Locked State Here we address a different problem. In the locked state, the head-site displacement, and hence the value of X, is given, so the question is how much S2 bending is required when head and/or site are azimuthally misaligned. The axial view of this state is shown in Fig. G1b. All loops in the patch of the motor domain have bound to their conjugate sites on the actin patch. Consequently the central S1 axis CP lies in the x-z plane, and the coordinates of the point D, where the distal end of the lever-arm would be if not tethered to S2, are obtained from Eq. H3 by setting θ ¼ θA, ϕ ¼ ϕ0 ¼ ϕA. Also required are the coordinates of C and A, where A is the back-end of S2 (Fig. 4.2): 0

1 Rcos θA rD rC ¼ @ Rsin θA sin ϕA A Rsin θA cos ϕA 1 X Rcos θA A rC ¼ @ ðRA þ rÞsin ϕA Z r þ ðRA þ rÞð1 cos ϕA Þ

ðH8aÞ

0

ðH8bÞ

and 0

1 LS2 A rA ¼ @ RM sin ϕM RM ð1 cos ϕM Þ

ðH8cÞ

are required in what follows. The lever-arm must ﬂex by angle ψ ¼ /BCD to join S2 at point B, so the ﬁxed lengths of the lever-arm and S2 are two constraints on B: j rB rC j¼ R,

j rB rA j¼ LS2 :

ðH9Þ

In three dimensions, these constraints conﬁne B to a circle in a plane perpendicular to the line AC, which is insufﬁcient to determine its coordinates uniquely. So some other criterion is needed. As B moves on this circle, the ﬂexing energy

Appendices

411

1 V S1 ðψÞ ¼ κS1 ðRψÞ2 2

ðH10Þ

of the lever-arm will vary, so the third condition is that this energy be minimized. To solve this minimization problem, begin by expressing ψ in terms of point coordinates. Let 0 1 0 1 x xB X þ Rcos θA A: rB rC ¼ @ y A @ yB ðRA þ rÞsin ϕA z zB Z þ r ðRA þ rÞð1 cos ϕA Þ

ðH11Þ

xcos θA þ ysin θA sin ϕA zsin θA cos ϕA pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : x2 þ y2 þ z 2

ðH12Þ

Hence cos ψ ^rCB ∙ ^rCD ¼

Now the ﬁrst constraint of Eq. H9 is that auxiliary variables ρ and ε, where ρ¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ x2 þ y2 þ z2 ¼ R: It is useful to introduce

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y2 þ z2 ¼ R2 x2 ,

y z sin ε ¼ , cos ε ¼ ρ ρ

ðH13Þ

so cos ψ ¼

xcos θA ρsin θA cosðϕA þ εÞ : R

ðH14Þ

Because the strain energy (Eq. H9) is quadratic in ψ, minimizing that energy with respect to ψ is equivalent to maximizing cos ψ w.r.t. ε, which is achieved by setting cos(ϕAþε) ¼ –1, or ε ¼ π ϕA :

ðH15Þ

y ¼ ρsin ϕA , z ¼ ρcos ϕA , qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ðH16Þ

Hence

ρ¼

R2 ðxB X þ Rcos θA Þ2 :

The second constraint in Eq. H10 can be temporarily removed by setting xB ¼ 0, and this gives an approximate solution to the minimization problem, in the sense that y and z, and hence yB and zB, are determined. The constant-length constraint for S2 (as a chord length) is

412

Appendices

xB ¼ LS2 þ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ LS22 y2B z2B :

ðH17Þ

With numerical computation, it can be included by making an iterative solution of Eqs. H16 and H17 for xB, where the necessary connections are supplied by Eq. H11. The desired outcome of this calculation is s2, the amount of S2 bending in the locked state, where 2 s22 ¼ xB 2 þ ðyB RM sin ϕM Þ2 þ zB þ RM ð1 cos ϕM Þ :

ðH18Þ

Computed values of s2 as a function of mismatch angles are shown in Figs. H3c,d. They can be used to deﬁne the limits of azimuthal mismatching if one accepts the idea that there is an intrinsic limit to the bending of S2. In contrast to the ﬂoppy state, for the locked state this limitation imposes much stricter limits on ϕA. For example, even when X ¼ 0 (Fig. H3c), setting s2c ¼ 8 nm leads to |ϕM| < 50 and ϕA < 25o. There is also another limitation, which is built into the mathematical structure of this solution; the argument of the square root in Eq. H17 cannot become negative, so X < xB þ Rð1 þ cos θA Þ:

ðH19Þ

The upper limit is simply the value of X for which the lever-arm lies back parallel to the actin axis, including the small negative contribution from the axial retraction of S2. For X ¼ 6 nm, this condition is violated at large values of ϕA, as in Fig. H3d, where s2 has been set to zero to signify the inaccessible range of mismatch angles.

Appendix I: Models with Buckling Rods Only a very weak compressive force (negative tension) from a bound head is needed to buckle the S2 rod which carries it. Elasticity theory gives the critical buckling force (Howard 2001) as f b ¼ π 2 LS2 κS2 =3

ðI1Þ

and a slight increase in force will compress the buckled rod by distance y, so that f b ð yÞ ¼ f b þ κ b y

ðI2Þ

except for very large compressions. When that force is supplied by myosin-S1 with head-site distance x, the balance of forces is expressed by κ ðx þ yÞ ¼ f b ðyÞ

ðI3Þ

Appendices

413

so xb ¼ –fb/κ is the critical (negative) myosin strain for S2 buckling. For x < xb, y ¼ κ(xb–x)/(κ+κ b), so myosin tension in the buckling range is tA ðxÞ ¼ κxb þ

κκb ð x xb Þ κ þ κb

ð x < xb Þ

ðI4Þ

as in Eq. 4.63. Kaya and Higuchi (2010) found that fb ¼ 1.35 pN, κ ¼ 2.7 pN/nm and κb ¼ 0.03 pN/nm, so xb ¼ –0.5 nm. If LS2 ¼ 60 nm, a similar value for fb(2.0 pN) comes from Eq. I1 and κS2 0.01 pN/nm (Adamovic et al. 2008). For strain-dependent kinetics, the corresponding strain energy of the head and buckled rod are required. Integrating Eq. I4 ensures that the buckling energy of S2 is included, because the tension is shared by S1 and S2. Thus 1 vA ðxÞ ¼ κx2 2 1 1 κκb ðx xb Þ2 vA ðxÞ ¼ κx2b þ κxb ðx xb Þ þ 2 2 κ þ κb

ðx > xb Þ, ðx < xb Þ:

ðI5Þ

In a model with dimeric myosins and target zones, both heads of the dimer can bind to actin. Hence the compressive strain which drives the buckling of S2 can be generated by one bound head or both heads. Here is a brief summary of what is required to keep track of all doubly-bound conﬁgurations in a model with two working strokes. If the two bound heads are both in the A-state, the force balance condition is fb+κby+κ(x1+y+x2+y) ¼ 0, so y ¼ κ(xb–x1–x2)/(2κ+κb). Then tAA ðx1 , x2 Þ ¼ κx1

κ2 tAA ðx1 , x2 Þ ¼ κx1 þ ðxb x1 x2 Þ 2κ þ κb

ðx1 þ x2 xb Þ ðx1 þ x2 < xb Þ:

ðI6Þ

is the tension-strain function for the ﬁrst head. For model cycles with two working strokes, there are three bound-state conﬁgurations (A-state, P-state and R-state) for each head, giving nine tension-strain functions in all. They can all be expressed in terms of tAA: tPA ðx1 , x2 Þ ¼ tAA ðx1 þ h1 , x2 Þ, tAP ðx1 , x2 Þ ¼ tAA ðx1 , x2 þ h1 Þ, tRP ðx1 , x2 Þ ¼ tAA ðx1 þ h, x2 þ h1 Þ, tPR ðx1 , x2 Þ ¼ tAA ðx1 þ h1 , x2 þ hÞ,

tRA ðx1 , x2 Þ ¼ tAA ðx1 þ h, x2 Þ, tPP ðx1 , x2 Þ ¼ tAA ðx1 þ h1 , x2 þ h1 Þ, tAR ðx1 , x2 Þ ¼ tAA ðx1 , x2 þ hÞ, tRR ðx1 , x2 Þ ¼ tAA ðx1 þ h, x2 þ hÞ:

ðI7Þ

where h ¼ h1+h2 and the two subscripts give the states of the ﬁrst and second heads. In each case, Eq. I6 shows that the critical value of x1 is different. However, for binding to sites of an actin target zone, the spacing d ¼ x2–x1 is restricted to 2c, c, c, 2c.

414

Appendices

From these formulae, stiffness and potential energy functions follow. The stiffness function sAA(x1, x2) should be deﬁned as the derivative of tAA at constant d, because that is the result of displacing the ﬁlaments. For the elastic strain energies, vAA(x, d ) is the change in strain energy of the dimer when the ﬁrst head binds, the second head being already bound, so vAA ðx1 ; x2 Þ ¼ V AA ðx1 ; x2 Þ vA ðx2 Þ

ðI8Þ

where 1 V AA ¼ κðx21 þ x22 Þ 2 1 1 κ2 ðxb x1 x2 Þ2 V AA ¼ κðx21 þ x22 Þ 2 2 2κ þ κb

ðx1 þ x2 > xb Þ, ðx1 þ x2 < xb Þ:

ðI9Þ

You can check that tAA(x1,x2) is the partial derivative of VAA with respect to x1 at constant x2. The other eight strain energies can all be written in terms of vAA by formulae strictly analogous to Eq. I7. The effects of buckling rods in contraction models are quite subtle. Heads can bind to more negatively displaced sites because S2 can buckle under thermal agitation alone. More heads can bind and stroke, generating more isometric tension than possible with rigid rods and high myosin stiffness. Some negatively strained heads will complete all working strokes and cycle, raising the isometric ATPase rate to values not attainable with rigid S2 rods. More details and modelling results are described in Smith (2014).

References Adamovic I, Mijailovich SM, Karplus M (2008) The elastic properties of the structurally characterized myosin II S2 subdomain: a molecular dynamics and normal mode analysis. Biophys J 94:3779–3789 Howard J (2001) Mechanics of motor proteins and the cytoskeleton. Sinauer Associates Inc., Massachusetts, pp 103–104 Kaya M, Higuchi H (2010) Nonlinear elasticity and an 8-nm working stroke of single myosin molecules in myoﬁlaments. Science 329:686–689 Smith DA (2014) A new mechanokinetic model for muscle contraction, where force and movement are triggered by phosphate release. J Muscle Res Cell Motil 35:295–306

Appendices

415

Appendix J: The Hidden-Markov Method for Running Variance Records The Hidden-Markov method is an algorithm for analysing a time series (x1,...,xN) of data points at equally spaced times tk ¼ kΔ (k ¼ 1,. . .,N) to uncover the parameters of the underlying stochastic process, which is assumed to be a Markov chain. The process generates random changes in a hidden set of state variables i1,..,iN within a given range, say (1,NS), and it also generates the observed changes in the time series. A Markov process is one in which the probability of the stochastic variables at time kþ1 depends only on their values at time k and not on values at any earlier time. So in this case, the chain of Markov transitions is two-dimensional. The following presentation is adapted from that of Rabiner (1989). The Markov chain is speciﬁed by the 2D matrix of transition probabilities Qij(x, x0 )dx0 which is the joint probability of state j and data in the range (x0 ,x0 +dx0 ), given state i and data-point x at the previous time-point. This joint probability cum probability distribution is normalized to unity: NS Z X

Qi j ðx, x0 Þdx0 ¼ 1:

ðJ1Þ

j¼1

If the transition probabilities are known, they can be used to construct the probability P that the entire time series of x-values was (x1,...,xN) as observed, namely P¼

NS X i1 ¼1

......

NS X iN ¼1

Pi1 ðx1 ÞQi1 , i2 ðx1 , x2 Þ . . . . . . QiN1 , iN ðxN1 , xN Þ

ðJ2Þ

with the starting probability distribution Pi(x) for state and x-value. This is a quantity independent of the Qij since Qij(x, x0 ) does not determine the probability of its initial state. The basic idea is that the parameters which deﬁne the transition probabilities can be determined by maximizing the value of P. This quantity can be very small, so it is convenient to quote values of the likelihood, deﬁned as log P. However, for a long time series this formula cannot be calculated directly, and an iterative method is required. Such a method is Baum’s forward-backward algorithm (Rabiner 1989 and references therein). The forward algorithm deﬁnes αkþ1, j ¼

X i

so that

αk, i Qij ðxk , xkþ1 Þ

ðJ3Þ

416

Appendices

α1, i ¼ Pi ðx1 Þ,

P¼

X

αN, j :

ðJ4Þ

j

Although this procedure calculates the likelihood, one still has to maximize it with respect to the transition probabilities. To this end, Baum’s backward algorithm will be required. Let βk, i ¼

X

Qi j ðxk , xkþ1 Þβkþ1, j

ðJ5Þ

j

with βN, i ¼ 1,

P¼

X

Pi ðx1 Þβ1, i

ðJ6Þ

i

Hence αk,iβk,i/P is the probability of state i at time tk. Moreover, the joint probability of states i,j at times tk and tk+1 is αk,iQij(xk, xk+1)βkþ1,j/P, so that the expected number of state transitions from i to j throughout the time series is ξi j ¼

N1 1X αk, i Qi j ðxk , xkþ1 Þβkþ1, j : P k¼1

ðJ7Þ

But the rate constants for state transitions are also deﬁned ab initio by the transition matrix Z Z qij ¼

ϕi ðxÞQij ðx; x0 Þdxdx0

ðJ8Þ

Pi ðxÞ Pi ðxÞdx

ðJ9Þ

where ϕi ð xÞ ¼ R

is the probability distribution of x in state i. Now the transition frequencies in Eq. J7 can be used to re-estimate the ‘states-only’ transition probabilities qij. Baum’s re-estimation algorithm is that ξi j q~i j P : ξi j

ðJ10Þ

j

It remains to prove that this re-estimation procedure actually maximizes the likelihood function. On varying the transition probabilities and the state probabilities about the supposed maximum, we have

Appendices

417

( ) N1 X X δQik , ikþ1 ðxk , xkþ1 Þ δPi1 ðx1 Þ þ δPðXÞ ¼ PI ðXÞ Pi1 ðx1 Þ Qik , ikþ1 ðxk , xkþ1 Þ I k¼1

ðJ11Þ

where X (x1,....,xN), I (i1, . . . . , iN) and PI(X) is the summand in Eq. J2. Since it is only the states-only transition probabilities that are being varied, an equivalent variation is ( ) N1 X X δqik , ikþ1 δpi1 þ δPðXÞ ¼ PI ðXÞ qik , ikþ1 pi1 I k¼1

ðJ12Þ

where Z pi ¼

Pi ðxÞdx

ðJ13Þ

is the probability of state i. These variations are constrained by normalization: X

qij ¼ 1,

X

j

pi ¼ 1:

ðJ14Þ

i

Now Eq. J12 can be processed, using the identities X

PI ðXÞ ¼ αk, i Qi j ðxk , xkþ1 Þβkþ1, j ðk ¼ 1, :: . . . , N 1Þ

I6¼i, j

X

PI ðX Þ ¼ α1, i β1, i

ðJ15aÞ ðJ15bÞ

I6¼i

where the sums in (J15a) are over (i1,....,ik–1,ik+2,....,iN) with ik ¼ i, ik+1 ¼ j, and the sums in (J15b) are over (i2,. . .,iN) with i1 ¼ i. Using undetermined multipliers to impose normalization, the extremal condition is ( N1 X X i, j

) αk, i Qij ðxk , xkþ1 Þβkþ1, j λi qij

k¼1

δqij X δp þ ðα1, i β1, i μpi Þ i qij pi i

¼0

ðJ16Þ

from which λi qij ¼

N1 X k¼1

αk, i Qij ðxk , xkþ1 Þβkþ1, j ,

μpi ¼ α1, i β1, i :

ðJ17Þ

418

Appendices

The multipliers follow by applying the normalization conditions. Hence the re-estimated states-only transition probabilities and the state probabilities which make the likelihood extremal are N1 P

q~ij ¼

αk, i Qij ðxk , xkþ1 Þβkþ1, j

k¼1 P N1 P j

, αk, i Qij ðxk , xkþ1 Þβkþ1, j

α1, i β p~i ¼ P 1, i α1, i β1, i

ðJ18Þ

i

k¼1

It can also be proven that these results deliver maximum rather than minimum likelihood, and proofs have been given by Baum (1972) and Levinson et al. (1983).

Running Variance Records for Optical Trap Data A particularly straightforward application of the Hidden Markov method is to the detection of state transitions in variance records {vk} constructed from optical-trap displacement data (Smith et al. 2001). There are only two states, the ‘off’ and ‘on’ states (i ¼ 1,2) for myosin detached and bound to actin. Moreover, the compound transition matrix Qij(v, v0 ) will be independent of the initial variance v provided that the variance record is not oversampled. Correlations in the variance record are undesirable; maximum information and minimal correlations can be achieved in a running variance window of W displacement points if the variance is sampled at every W/2th point, so that the sampling time is WΔ/2. Then Qij ðv; v0 Þ ¼ qij ϕ j ðv0 Þ

ði; j ¼ 1; 2Þ

ðJ19Þ

which is required to generate the ξij. The state transition probabilities follow from rate equations for qij(t), which is the probability of state j at time t given state i at time zero: dq12 ðt Þ ¼ fq11 ðt Þ gq12 ðt Þ, dt

dq21 ðt Þ ¼ gq22 ðt Þ fq21 ðt Þ dt

ðJ20Þ

where q11+q12¼1 and q21+q22¼1. Over one sampling period Δ, we have q12 ¼

f 1 eðf þgÞΔ , f þg

q21 ¼

g 1 eðf þgÞΔ : f þg

ðJ21Þ

We also need the variance distributions ϕi(v), which are determined by the standard deviations σi of the displacement record in free and bound periods. This noise is assumed to be due to Brownian forces. For example,

Appendices

419

rﬃﬃﬃﬃﬃﬃﬃﬃ kB T σ1 ¼ , κt

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ kB T σ2 ¼ κ þ κt

ðJ22Þ

for a dumbbell with strong links. The effects of averaging over a running window of W displacements uk,...,uk+W can be calculated if we assume that they are uncorrelated random variables with a Gaussian distribution of width σi. The answer is a chi-squared distribution with W1 degrees of freedom (Weatherburn 1968), namely ðW3Þ=2

W y eyi ϕi ðvÞ ¼ 2 i , 2σ i Γ ðW 1Þ=2Þ

Wv yi ¼ 2 2σ i

ðJ23Þ

in which the gamma function appears. The mean and variance of this distribution are

V i ¼ 1 W 1 σ 2i ,

2σ 4i , ðJ24Þ S2i ¼ 1 W 1 W pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ so the signal-to-noise ratio in state i is V i =Si W=2 for W >> 1. Overall, the variance record switches between two noisy levels; the signal switches between V1 and V2 while the RMS noise amplitudes switch between S1 and S2. So a signal-tonoise ratio which measures the visibility of the two levels can be deﬁned as V1 V2 R S1 þ S2

rﬃﬃﬃﬃﬃ W V1 V2 : 2 V1 þ V2

ðJ25Þ

With two-bead displacement records, a running covariance record can also be constructed, and its signal-to-noise ratio can also be calculated (Smith et al. 2001). To detect the switching times between the two states of a running variance record, Baum’s forward and backward method can be applied to generate the probabilities p~k, 2 ¼

αk, 2 βk, 2 αk, 1 βk, 1 þ αk, 2 βk, 2

ðJ26Þ

of state 2, in which the myosin head is bound, at each time-point k. These probabilities can be used to assign a deﬁnite state at every sampled time, either state 2 if p~k, 2 > 0:5, otherwise state 1. In that way, bound periods are identiﬁed throughout the variance record, and the displacement record from which it was generated. And Eq. J4 gives the value of the likelihood function log P, which is a measure of the quality of detection.

420

Appendices

References Baum LE (1972) An inequality and associated maximization technique in statistical estimation for probabilistic functions of a Markov process. Inequalities 3:1–8 Levinson SE, Rabiner LR, Sondhi MM (1983) An introduction to the application of the theory of probabilistic functions of a Markov process to automatic speech recogntion. Bell System Tech J 62:1035–1074 Rabiner LR (1989) A tutorial on hidden Markov models and selected applications in speech recognition. Proc IEEE 77:257–285 Smith DA, Steffen W, Simmons RM, Sleep J (2001) Hidden-Markov methods for the analysis of single-molecule actomyosin displacement data: the variancehidden-Markov method. Biophys J 81:2795–2816 Weatherburn CE (1968) A ﬁrst course in mathematical statistics, 2nd edn. Cambridge University Press, Cambridge

Appendix K: Calcium Activation with the Allosteric Model In Sect. 7.2, the MWCallosteric scheme for calcium activation by troponin gave the calcium-dependent activation constant K ðC Þ ¼ λ

ðε þ K c Þ2 ð1 þ K c Þ

2

K c K~ c C

ðK1Þ

where C ¼ [Ca2+]. The associated activation fraction pð C Þ ¼

K ðC Þ K ðC Þ þ 1

ðK2Þ

serves as a template for Ca-activation of tension and related properties; it predicts what would be observed in the absence of cooperative regulation. The sigmoidal tension-pCa curve observed in striated muscle is characterized by its calcium sensitivity (the calcium level for half-maximal activation) and the steepness of the curve, as measured by the Hill coefﬁcient (Eq. 7.72). Here we derive closed formulae for these quantities from Eqs. K1 and K2, using Kc as a dimensionless measure of the calcium level. As the maximal level of activation is not unity but λ/(λ þ 1), half-maximal activation occurs when K(C) ¼ λ/(2þλ). On substituting from Eq. K1, ðε þ K c Þ2 ð1 þ K c Þ so

2

¼

1 2þλ

ðK3Þ

Appendices

421

Fig. K1 Properties of the MWC allosteric scheme of Sect. 7.2. (a) The dimensionless calcium sensitivity K ðc0:5Þ and (b) the Hill coefﬁcient n, as functions of λ for ﬁve values of ε

K ðc0:5Þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1ε 2þλ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2þλ1

ðK4Þ

is the half-activation level, commonly measured in terms of pCa ¼ logC. The Hill coefﬁcient n was deﬁned by assuming that K(C) / Cn at half-maximal activation. Hence n¼

d ln K ðCÞ K c dK

d ln C K dK c K ðc0:5Þ

ðK5Þ

so n¼

ð0:5Þ

2ð1 εÞK c : ð0:5Þ ð0:5Þ ε þ Kc 1 þ Kc

ðK6Þ

This is a function of λ and ε, plotted in Fig. K1. The quoted value n ¼ 2 is never quite realised, but is approached when ε 1. These conditions need to satisfy the requirement that L 1/ε2λ > 1 for the –Ca afﬁnity of TnI for actin, to inhibit activation at low calcium.

Index

A Actin binding ﬂoppy, 395, 401, 408 stereospeciﬁc, 95, 281 strong, 89, 91, 148 two-step, 108, 124, 141, 197 weak, 301, 311, 316, 343 double helix, 5, 13, 15, 29, 30, 32, 36, 101, 126, 244, 245, 275, 276, 311, 314, 316, 319, 358 F-actin, 13–16, 29–35, 40, 47, 50, 51, 86, 105, 112, 121–124, 126, 127, 132–138, 140–142, 144, 156, 184, 195, 202–204, 209, 212, 218–222, 224–229, 237, 239, 240, 245, 246, 256, 259, 261, 263–265, 271, 273, 275–282, 293, 295, 303, 305, 312, 315, 351, 356–358, 361–363, 369 monomer, 3, 13, 40, 85, 102, 144, 274, 279, 293, 315, 317, 332, 339, 358 target zone, 29, 31, 52, 108, 112, 123, 144, 152, 245, 246, 256, 275 Actomyosin, 2, 5, 28, 58, 60–64, 68, 70–73, 75–78, 82, 83, 86, 87, 91–95, 101, 109, 114, 149–151, 247–256, 262, 263, 267, 284, 286, 287, 332, 352, 359, 420 Adenosine diphosphate (ADP) ADP release stroke, 92, 109–110, 121, 123, 141, 178, 190, 359 Adenosine triphosphate (ATP) ATP-induced detachment, 123, 126, 159, 173, 178, 179, 196, 198, 262, 267, 271, 282, 313, 350

Afﬁnity second order, 78 pseudo-ﬁrst-order, 306, 311 Arrhenius length, 197, 200, 267 ATP sliding distance, 269–273 Axial selection, 32 Azimuthal matching, 32–33

B Backward Euler method, 129, 130 Brownian forces, 30, 40, 103, 104, 108, 237, 240, 247, 248, 268, 272, 312 Brownian motion, 5, 31, 114, 238, 240, 247, 252, 253, 265, 266, 275

C Caged ATP release, 172 Calcium, 369 activation, 334, 369 allosteric scheme, 307, 336–338 release, 296, 347 Calmodulin, 30, 278, 368 Cleavage terminal phosphate, 56 Contraction, 1, 2, 4–7, 9, 15, 26, 31, 33, 55, 60, 69, 73, 82, 93, 111–127, 146, 148, 150–152, 155, 157, 158, 190, 194, 205, 257, 277, 295, 296, 347, 350, 357, 358, 362–368, 384, 414 Contraction cycle biochemical cycle, 68, 132 5-state cycle, 142

© Springer Nature Switzerland AG 2018 D. A. Smith, The Sliding-Filament Theory of Muscle Contraction, https://doi.org/10.1007/978-3-030-03526-6

423

424 Contraction cycle (cont.) strain-dependent, 111, 121, 182, 190, 250, 359 tight coupling, 269 2-state stroking cycle, 113–119 Converter, 89, 91, 102, 118, 369 Crossbridge, 11, 15, 28, 29, 34, 79, 80, 87, 133–137, 168, 172–174, 179, 186–188, 190, 196, 197, 212, 214, 219, 220, 249, 252, 338, 343, 350, 353, 355, 356, 366 Cy3-ATP, 256–263

D Debye shielding length, 223 Descending limb tension-length curve, 210–212, 217, 365 instability, 216

E Efﬁciency, 6, 25, 26, 28, 39, 125, 126, 139, 141, 142, 144, 146, 157, 158, 247 Enthalpy, 27, 28, 125, 126, 155–157, 159, 167, 174 Excitation-contraction coupling, 9, 347 Exothermic reaction, 56

F Filament actin ﬁlament (F-actin), 13, 34, 112, 229, 259, 363 compliance, 2, 52, 102, 123, 126, 132–137, 168, 175, 184, 186–188, 353 myoﬁlament, 5, 10, 12, 13, 15, 16, 28–33, 44, 105, 123, 126, 128, 130, 132, 143, 187, 202, 221, 222, 224–228, 278, 363, 369, 370 Force steps, 119, 187–194 Force-clamp method, 263–268

G Gestalt binding, 295, 315 Gibbs energy, 28, 43, 44, 56, 72, 91, 101, 106, 107, 158, 250, 330, 360 Gibbs’ fundamental identity, 101

Index H Heat rate maintenance heat, 21–23, 25 shortening heat, 22–26 Heavymeromyosin (HMM), 4, 15, 58, 67, 257–260, 268, 269, 277, 368, 369 Highest-energy-barrier approximation, 50, 51, 102, 110, 111, 120, 179, 360 Hill equation, 25 Hill ratio, 21, 126, 153, 155–158, 213, 366 Hydrolysis actomyosin.ATP (A.M.ATP), 61, 64, 68, 69, 73, 76, 83, 93, 150, 349 enthalpy, 27, 28, 126 myosin.ATP (M.ATP), 5, 8, 27, 52, 56, 57, 60, 62, 64, 69, 73, 76, 83, 88, 89, 93, 114, 173

I Insect ﬂight muscle, 32, 186, 201–210, 275, 355 Ionic strength, 30, 56, 67, 69, 71, 78, 218, 219, 221, 222, 225, 226, 276, 334, 335, 349, 350, 356

K Kramers’ formula, 104, 141

L Lactic acid, 2, 9, 146 Lattice lattice spacing, 10–12, 16, 33, 113, 218–223, 225–227, 229, 347, 354–357, 365–367 length-dependent spacings, 355 stability, 17, 218, 367 Length step response phase 2, 41, 51, 81, 178 phase 2b, 81 phase 3, 81, 181, 188 phase 4, 81, 175

M Michaelis-Menten constant, 62, 65, 67, 68, 71, 72, 84, 150

Index Minimum-length ﬁlament (MLF), 270, 271 Monte-Carlo simulation, 120, 126, 131–132, 138, 144, 170, 247–251, 318, 332, 333, 341, 359, 361, 362 Motility assay, 4, 268–272, 277, 350 Motor domain actin-binding cleft, 90, 268 L50, 89–91 lever-arm, 41, 85, 88, 89, 102, 105, 107, 108, 280 nucleotide pocket, 369 U50, 89–92 Muscle contraction activation, 127 creep, 210–212, 214–216 isometric, 15, 112, 134 isotonic shortening, 135, 136, 352 ramp lengthening, 196–201 ramp shortening, 130, 194–201, 350, 351 relaxation, 354 Myosin II fast, 110, 151 slow, 110 Myosin states A-state, 43, 44, 179–181, 198, 280 P-state, 180 R-state, 43–47, 114, 179–181, 280, 285 Myosin V, 85, 87, 277–284, 287 hand-over-hand mechanism, 278 processivity, 268, 277–287 working stroke, 87 Myosin VII, 277 Myosin-S1 actin-binding loops, 198 ATP-induced detachment, 123, 159, 173, 178, 179, 196, 198, 262, 267, 271, 282, 313, 350 forced detachment, 262 light chains, 30, 85, 89, 102, 223 motor domain, 13, 41, 88, 102, 105 substrokes, 77, 246, 281 working stroke, 41, 46, 77, 95, 237, 238 Myosin-S2, 14–16, 30, 40, 102, 104, 108, 112, 119, 132, 143–145, 148, 257, 260, 278, 369, 370

N Nyquist plot, 183–185, 187, 192, 204, 209

425 O Optical trapping three-bead experiment, 282 Oscillations isotonic, 189–194 SPOC, 367 wing-beat, 201–210

P Passive tension, 212–214, 216, 217, 365, 366, 370 pH isoelectric point, 221 Phosphate phosphate burst, 57, 59–62 phosphate jump, 72, 79–81, 147, 169–171, 176, 352 Phosphate ion, 2, 26, 60, 77, 87, 91, 95, 146, 328 Phosphorylation, 347, 368–370 Potentiation, 339, 342 Power output fast muscle, 23 insect ﬂight muscle, 210 slow muscle, 23, 142 Probabilistic methods, 128, 129, 131

R Reaction rates strain-dependent, 102, 104, 105, 110, 111, 119, 286 Regulation Ca2+, 296 direct, 368–370 thin-ﬁlament, 3, 33, 127, 172, 173, 293–343, 347, 350–352, 354, 355, 357–363, 368 Regulatory models BCO model, 302, 343 chain model, 339, 343 Hill-Eisenberg-Greene model, 305, 310, 332, 334 steric blocking model, 297–303 Residual force enhancement, 200, 217 Rigor bond lifetimes, 258–260, 276 bonds, 87, 256–263, 276

426 S Sarcomere A-band, 10, 11, 210, 216, 217, 221, 228, 229, 354, 355 I-band, 10, 11, 202, 221 inhomogeneities, 171, 187, 214, 354 M-band, 15, 16, 33, 130, 203, 216, 218, 222, 227, 229 Z-line, 10, 15, 16, 31, 33, 216, 218, 222, 227, 229 Series elastic component (SEC), 2, 132–135 Single molecule experiments isotonic (force clamp), 76 single bead, 257, 261 three bead, 263 Sinusoidal response linear, 186 non-linear, 186 Smoluchowski equation, 251–256 Spontaneous oscillatory contractions (SPOC), 363–368 Stability analysis, 367 Steric hindrance, 3, 293, 294, 297–303, 307, 334, 335, 343, 361 Stiffness (bending) F-actin, 39, 112, 122, 124, 134–137, 141, 142, 203, 229, 356 muscle, 7, 39, 77, 133, 153, 182, 203, 204 myosin lever-arm, 118 S2, 40, 112, 143, 146, 149, 261 titin, 203 Strain A-state, 45, 47, 105, 106, 114, 145, 199, 285 myosin, 41, 43, 46, 102, 110, 328 Strain-dependent reaction rates, 102, 110, 111, 119, 121, 286 Switch I, 89, 95 Switch II, 89, 95

T Tension isometric, 7, 8, 11, 12, 16, 21, 32, 36, 38–41, 47, 50, 51, 77–82, 85, 109, 116, 118, 121, 123, 126, 127, 132, 134–138, 141, 142, 144, 146–149, 153–159, 172, 174, 175, 177, 178, 210–213, 216, 217, 219, 220, 227, 349, 354, 355, 359, 363, 368, 370

Index length curve, 3, 6, 8, 11, 16, 25, 33, 112, 123, 127, 133–135, 175, 196, 210–212, 217, 354, 359, 363, 365–367 myosin, 366 pCa curve, 348, 361 temperature-dependence, 156, 174 tension-velocity curve, 23, 39, 116, 119, 123, 126, 137, 155–158, 213 transients, 42, 48, 80, 170, 174–176, 178, 179, 187, 195, 200, 215, 267, 351 Tetanus, 7, 9, 21, 25, 40, 84, 135, 168, 177, 189, 195, 350, 354 Titin, 16, 203, 212, 222, 223, 227–229, 350, 370 Transverse tubules, 9, 10, 347 Tropomyosin chain, 313, 331, 349, 358 protomer, 294, 297, 315, 358 states blocked, 294, 297, 299–301, 304, 313, 328, 329 closed, 76, 84, 90, 300, 301, 329 open (myosin), 76, 84, 90, 297, 299, 301, 304, 328, 329 Troponin troponin C, 3, 303, 347 troponin I, 303, 349 troponin T, 3, 294, 296, 303, 338

V Variance variance-Hidden-Markov method, 243, 251

W Wing-beat oscillations, 201–210 Working strokes, 3–5, 41–46, 51, 76–85, 89–95, 101, 102, 108–111, 113, 114, 116, 118–121, 123, 142–144, 147, 148, 150, 156, 173–179, 181, 187, 190, 195, 197, 198, 237–240, 243, 246, 247, 249–251, 253–255, 259, 261, 263–265, 267–272, 274, 275, 279–282, 284–287, 301, 328, 330, 339, 343, 352, 353, 359, 360