Table Of ContentMuscles, Reflexes, and Locomotion
Thomas A. McMahon
Muscles, Reff exes, and
Locomotion
Princeton University Press
Princeton, New Jersey
Copyright© 1984 by Thomas A. McMahon
Published by Princeton University Press, 41 William Street, Princeton, New Jersey
In the United Kingdom: Princeton University Press, Chichester, West Sussex
Contents
All Rights Reserved
Library of Congress Cataloging in Publication Data
McMahon, Thomas A., 1943-
Muscles, reflex, and locomotion.
Bibliography: p.
Includes index. Preface xiii
1 . Muscle contraction-Mathematical models.
2. Reflexes-Mathematical models. 3. Animal locomo Sources and Acknowledgments xv
tion-Mathematical models. I. Title. [DNLM:
1. Muscles-Physiology. 2. Reflex-Physiology. Chapter 1: Fundamental Muscle Mechanics 3
3. Movement. 4. Locomotion. WE 500 M478m]
QP32l.M338 1984 591.1'852 82-23156 Early Ideas About Muscular Contraction 3
ISBN 0-691-08322-3 Isolated Muscle Preparation 5
ISBN 0-691-02376-X (pbk.)
Mechanical Events: Twitch and Tetanus 6
This book has been composed in Times Roman
Tension-Length Curves: Passive and Active 8
Princeton University Press books are printed on acid-free paper,
and meet the guidelines for permanence and durability of the Quick-Release Experiments 10
Committee on Production Guidelines for Book Longevity of the
Series Elastic Component 12
Council on Library Resources
Force-Velocity Curves 12
Printed in the United States of America
Active State 14
987654
Muscles Active While Lengthening 16
Time Course of Active State in a Twitch 17
Difficulties with the Active State Concept 19
Summary and Conclusions 21
Solved Problems 21
Problems 25
Chapter 2: Muscle Heat and Fuel
27
Heat Transients 29
Resting heat 30
Initial and recovery heat 30
Fine Structure of the Initial Heat 30
Activation heat . 31
Shortening heat: The Fenn effect 33
Heat Produced by Lengthening Muscles 34
Thermoelastic Effects 36
What Is the Fuel? The Lactic Acid Era 37
High-Energy Phosphates 38
Lohmann Reaction 39
Isolating the ATP Engine 40
Light Exercise 41
vi Contents Contents vii
Aerobic Recovery 43 Chapter 4: The Sliding Movement: A. F. Huxley's
Heavy Exercise 44 1957 Model 85
Anaerobic Recovery 46 Alternative Mechanisms: A Few Discarded Ideas 85
Lactic Acid Oxygen Debt 46 Electrostatic Theories 86
Interval Running 49 Folding Thin Filaments 87
Alactic Oxygen Debt 50 Evidence for Independent Force Generators
How to Train Without Tiring 50 Operating Cyclically 87
Carbohydrate Loading 51 Points of Muscle Performance a Theoretical Model
Solved Problems 51 Should Include or Predict 89
Problems 52 Formulation of the Model 90
Chapter 3: The Contractile Proteins 53 Attachment and Detachment Rate Constants 91
Crossbridge Distributions for Isotonic Shortening 93
General Organization of Muscles 53
Developed Tension 96
Parallel-Fibered and Pennate Muscles 53
Rate of Energy Liberation 99
Fibers, Fibrils, and Filaments 55
Setting the Constants: Hill's Equations 100
"Myosin" and Its Parts 57
Independent Tests of the Model: Isotonic Stretching 103
Actomyosin Threads 58
A Subsequent Difficulty: Hill's 1964 Revisions of
Glycerinated Fibers 58
the Heat Story 103
Synthetic Actomyosin 59
Resolution of the Difficulty: Reversible Detachment 105
The Myosins 60
Solved Problems 108
The Actins 62
Problems 113
Sliding Filament Model 62
Early Evidence for the Sliding Filament Model 63 Chapter 5: Force Development in the Crossbridge 115
Later Evidence from X-Ray Diffraction in Active Muscle 65 Early Transients 115
Tension-Length Curves in Single 'Fibers 66 The Difference between the Rapid Elasticity and Hill's
Synthesizing the Contractile Apparatus: Polymerized Thick Series Elastic Component 117
Filaments and Decorated Actin 68 The T Transient and Its Time Constant 118
2
Activation of Contraction: the Sarcoplasmic Reticulum 70
A Mathematical Model for the Tension Transients:
The On-Off Switch: Troponin and Tropomyosin 72 Two Attached States ll9
Natural History Aspects: Clues to the Origin of Motility 74 Potential Energy of the S-2 Spring 122
Stretch Activation in Insect Muscle 74 Potential Energy of Conformation: The S-1 Link 123
The Problem of Static Stability in the Sliding Filament Model 75 Total Potential Energy 124
"Permanent" Extra Tension 77 Kinetics of the State Transitions 125
Tension "Creep" in a Fixed-End Tetanus 79 The Rate Equation and Its Solution 128
Sarcomere Length Nonuniformity as a Unifying Principle 79 Tension 130
Review of the Events of a Single Contraction 81 Fixing the Constants 131
Solved Problems 82 Oscillatory Work 132
Problems 83 Review and Conclusions 133
viii Contents Contents ix
Solved Problems 135 Extensor Thrust Reflex 176
Problems 137 Spinal Reflexes Involving All Limbs 176
Chapter 6: Reflexes and Motor Control 139 Spinal Locomotion on Treadmills: Constancy of the
Swing Duration 177
Organization of the Motor Control System 139
Stopped Limb Experiments 178
Spinal cord 139
Brain stem 140 Entrainment 178
Superharmonic Entrainment 179
Sensorimotor cortex and basal ganglia 141
Cerebellum 141 The Pattern Generator Could Employ Coupled Oscillators 180
Stimulated Locomotion 180
Muscle Fiber Types and the Size Principle 143
De-afferented Spinal Walking 182
Fiber types 143
Feedforward 182
Motor units and their order of recruitment 143
Vehicles with Legs 183
The Muscle Proprioceptors 144
Solved Problems 186
Spindle organs 145
Problems 187
The Golgi tendon organ 146
Afferents and Efferents 146 Chapter 8: Mechanics of Locomotion 189
The Stretch Reflex 147 Force Plates 189
Coactivation of a and 'Y Motor Neurons 148 Force Plate Records of Walking and Running 190
Reflex Stiffness 148 Determinants of Gait 192
A Lumped, Linear Model of the Spindle 152 Compass gait 194
How Velocity Sensitivity Acts to Stabilize a Reflex Loop 154 Pelvic rotation 194
Tremor 155 Pelvic tilt 195
How Time Delay in a Negative-Feedback Loop Can Stance leg knee flexion 196
Cause Oscillations 156
Plantar flexion of the stance ankle 196
The Role of Loop Gain in Stability 158
Lateral displacement of the pelvis 196
The Role of Loop Gain in Performance 159
Ballistic Walking 198
Renshaw Cells 161
Defining the model 198
Solved Problems 162
Results of the ballistic model 199
Problems 166
Extensions to include additional gait determinants 201
Chapter 7: Neural Control of Locomotion 168
Conclusions from the ballistic walking studies 202
Gait: Comparative Distinctions 168 Locomotion in Reduced Gravity 203
Gait: Classifications 170 Elastic Storage of Energy 204
Bipedal gaits 170 Energetics of kangaroos 205
Quadrupedal gaits 170 Maintaining a resonant system in motion 205
Control of a Single Limb: Reciprocal Inhibition 172 Cost of Running 208
Placing Reactions and Reflex Reversal 173 Cost of running formula 209
A Mechanical Oscillator 174 Influence of limbs 210
x Contents
Contents xi
Up and Down Hills: Efficiency of Positive and Negative Work 211 Allometric Rules 265
Tilting treadmill 211 Body Proportions 266
Efficiency 213 Resonant Rebound 270
Running with Weights 214 Trunk 270
Oxygen consumption and load 215 Limbs · 271
Carrying load is not equivalent to increasing speed 215 Galloping Frequencies 273
Enhanced Gravity: Running in Circles 217 Height of Rise of the Center of Mass 273
Utilizing Elastic Rebound: The Tuned Track 219 Joint Excursion Angles 274
A model of the leg 219 Intrinsic Muscle Velocity 274
Ground contact time 220 Phasing of the Limbs 275
Step length 222 Metabolic Power for Running 275
Tuned track prototypes 227 Blood Pressure and Myocardial Stress 278
Solved Problems 229 Basal Metabolic Rate 278
Problems 232 Enzymatic Activities 281
Chapter 9: Effects of Scale
234 Surface Area and Cross-Sectional Area 282
Dimensional Analysis 234 Physiological Frequencies 283
An example: The period of a pendulum 234 Pulse-Wave Velocity and Blood Velocity 284
Fundamental quantities 236 Aortic Hydraulic Impedance 285
Dimensionless variables 236 Alveolar Diffusion 287
The air resistance of a runner 237 Time Constants 288
The pi-theorem 242 Life Span 290
Scaling by Geometric Similarity 243 Feeding Gulliver 292
Rowing: A comparative analysis based on Concluding Remarks 292
geometric similarity 243 Solved Problems 294
Animal performance: A. V. Hill's scaling rules from
Problems 296
geometric similarity 245
Answers to Selected Problems 297
Some experimental challenges to Hill's arguments 247
List of Symbols 303
A conflict within the model 252
References 311
The Role of Gravity: Variable-Density Scaling 253
Index 325
How Scale-Dependent Distortions in Dimensionless Groups
Are Accommodated 255
Distortions in Geometry 257
Deriving Elastic Similarity 258
Buckling 258
Bending 260
Generality of the rules 262
Constant Stress Similarity 264
Preface
Everyone knows that meat is really muscle. Muscle is the only known piece of
machinery which can be cooked in many ways. This knowledge, by itself, is not
good for much. One has to do experiments with living muscle, again prepared
in different ways, to have any useful knowledge about it. As with any scientific
subject, the knowledge gained from the experiments is more satisfying if it can
be fit into a conceptual scheme showing how the experimental evidence is
related.
In this book, the conceptual scheme is a mathematical model, wherever that
is practical. Mathematical models have always done a lot for physics, but they
are only just now beginning to get a good reputation in biology.
As you will see if you flip through the pages, the concerns of this book are a
bit more global than muscle alone, although muscle is always the basic issue.
The choice of subjects is just another way in which this book is idiosyncratic. I
wanted to find a rational, consistent framework within which to understand
muscles and how they work in the body, as if muscle physiology were an
engineering subject. In order to do that, I had to pick out what I thought were
some specific, related success stories, and leave behind many stories of other
worthy investigations which might have appeared in a longer book for a more
specialized readership.
This brings up the question of who is the readership. I expect this book will
be worth its price to students of biophysics and bioengineering, and also to that
broad population of biologists and medical scientists who are used to talking
about actin and myosin as if they were somehow like positive and negative
electricity, i.e., useful, necessary, and fundamental, but fairly mysterious.
The main idea behind the work presented here is that muscle (like positive
and negative electricity) is illuminated in wonderful ways by mathematical
arguments about how it works-arguments which bring a very nice harmony
to all the observations. Because I was cautioned to do so by a biologist friend, I
have not skipped any steps in the mathematical arguments. He said that
biologists would be willing to follow along, provided that there were no gaps;
that is what kills them. I have taken him at his word. I have also made an
effort to explain the experimental evidence behind every assumption, behind
every tentative conclusion. Why not? The reader should never be forced to
wonder what is a measurement and what is a guess.
For the reader's convenience, a list of symbols is given at the end of the
book. This list should prove especially valuable for following the more
mathematical chapters, including Chapters 4, 5, and 9.
xiv Preface
Finally, I want to tuck in a small caution. The reader who skips the worked
problems at the ends of the chapters will be skipping quite a large fraction of
Sources and Acknowledgments
what I want him or her to see.
I am grateful to the many colleagues and students who have read all or parts
of the manuscript, and who have helped me understand aspects of this subject
I did not know. Those include: Yoram Ariel, Robert Banzette, Daniel Bogen,
Bradley Buchbinder, Shelley Copley, Chris Damm, Lincoln Ford, Peter
The following figures have been reprinted with permission of the copyright
Greene, Andrew Huxley, Gideon Inbar, David Leith, David Morgan, Mark
holder.
Patterson, James Propp, Philip Rough-Loux, Lee Sweeney, C. Richard
Taylor, and Andrew Ward. I am indebted for editorial help to Lisa Betteridge 1.5 Reprinted with permission from J. Biomech., Vol. 6, Pinto, J. G., and
and Sharon McDevitt. The manuscript was typed by Renate D'Arcangelo, Fung, Y. C., Mechanical properties of the heart muscle in the passive state,
Frances Korson, and Sharon McDevitt. The figures were drawn by William Copyright 1973, Pergamon Press Ltd.
Minty and Margo Burrelo. The indexer was Nicholas Humez. 3.1 Part (a) adapted from B. Pansky, Review of Gross Anatomy, Copyright
I wish to thank the Systems Development Foundation, Palo Alto, Califor Macmillan Publishing Co. (1979). Reproduced by permission.
nia, for support assisting the completion of this book. 3.11 Copyright 1975 by the American Association for the Advancement of
I am grateful to the many authors of previously published material who Science.
permitted me to reprint or adapt from their work many of the figures in this 3.12(c) Reproduced with permission from Wakabayashi, T., Huxley, H. E.,
book. Amos, L.A., and Klug, A. (1975). Three-dimensional image reconstruction
of actin-tropomyosin complex and actin-tropomyosin-troponin 1-troponin T
Thomas A. McMahon
complex. J. Mo/. Biol. 93:477-497. Copyright: Academic Press (London)
Cambridge, Massachusetts
Ltd.
April, 1982
4.3, 4.4, 4.5, 4.6 Reprinted with permission from Prog. Biophys. Biophys.
Chem., Vol. 7, A. F. Huxley, Muscle structure and theories of contraction,
Copyright 1957, Pergamon Press, Ltd.
5.3, 5.8 Reprinted by permission from Nature, Vol. 233, pp. 533-538,
Copyright © 1971 Macmillan Journals Limited.
6.1 Reproduced with permission from Eyzaguirre, C., and Fidone, S. J.:
Physiology of the Nervous System, 2nd edition. Copyright© 1975 by Year
Book Medical Publishers, Inc., Chicago. (Slightly modified from Mount
castle, V. B.: Medical Physiology [12th ed.; St. Louis: The C. V. Mosby
Co., 1968].)
6.2 Reproduced with permission from Eyzaguirre, C., and Fidone, S. J.:
Physiology of the Nervous System, 2nd edition. Copyright© 1975 by Year
Book Medical Publishers, Inc., Chicago. (Slightly modified from Penfield,
W., and Rasmussen, T.: The Cerebral Cortex of Man [New York: The
Macmillan Company, 1950].)
6.3 Reproduced with permission from Eyzaguirre, C., and Fidone, S. J.:
Physiology of the Nervous System, 2nd edition. Copyright © 197 5 by Year
Book Medical Publishers, Inc., Chicago. (Adapted from Barker, D., in
Barker, D, ed.: Muscle Receptors [Hong Kong, China: Hong Kong
University Press, 1962], p. 227.)
6.4 Reproduced with permission, from the Annual Review of Physiology,
Volume 41. © 1979 by Annual Reviews Inc.
xvi Acknowledgments
6.6 Reprinted with permission fromJ. Biomech., Vol. 12, Greene, P.R., and
McMahon, T. A., Reflex stiffness of man's anti-gravity muscles during
kneebends while carrying extra weights, Copyright 1979, Pergamon Press,
Ltd. Muscles, Reflexes, and Locomotion
7.9 Photograph courtesy Life Magazine© 1969 Time Inc.
8.4-8.9 Inman, V. T., Ralston, H. J., and Todd, F. Human Walking.
Copyright 1981, The Williams & Wilkins Company. Reproduced by
permission.
8.10, 8.11, 8.12 Reprinted with permission from J. Biomech., Vol. 13,
Mochon, S., and McMahon, T. A., Ballistic walking, Copyright 1980,
Pergamon Press, Ltd.
8.14 Reprinted by permission from Nature, Vol. 246, pp. 313-314. Copy
right© 1973 Macmillan Journals Limited.
8.24, 8.28-8.31 Reprinted with permission from J. Biomech., Vol. 12,
McMahon, T. A., and Greene, P. R., The influence of track compliance on
running, Copyright 1979, Pergamon Press, Ltd.
8.25-8.27, 8.32 From "Fast running tracks" by T. A. McMahon and P.R.
Greene. Copyright © 1978 by Scientific American, Inc. All rights reserved.
9.5, 9.6 Copyright 1971 by the American Association for the Advancement
of Science.
9.8 Copyright 1974 by the American Association for the Advancement of
Science.
9.13, 9.16 Copyright© 1977 McGraw-Hill Yearbook of Science & Techno
logy. Used with the permission of McGraw-Hill Book Company.
9.22 Copyright 1973 by the American Association for the Advancement of
Science.
9.23 Reprinted with permission from J. Biomech., Vol. 4, Murthy, V. S.,
McMahon, T. A., Jaffrin, M. Y., and Shapiro, A. H., The intra-aortic
balloon for left heart assistance: an analytic model. Copyright 1971,
Pergamon Press, Ltd.
Chapter 1
Fundamental Muscle Mechanics
This book is about muscles and how they work in the bodies of vertebrate
animals. Its special concern is locomotion on land. It is written for a reader
who has some interest in the physical as well as the biological sciences, and
who suspects that mechanics, thermodynamics, and engineering control
theory might be useful, as well as biochemistry and histology, in understand
ing how muscles operate.
The present chapter takes up those aspects of muscle function which have to
do with force, length, and shortening velocity. Later chapters explore the way
in which muscle utilizes fuel, the way it produces heat, the nature of the
proteins which generate the force, and the control of muscles by the nervous
system. In the final chapters, the aims broaden to include the mechanics and
energetics of locomotion, and discussion of how the effects of body size
determine aspects of muscular performance. We will see that although a great
deal is known about muscle and its control, one may still hope to see many fine
new contributions to this subject in the years to come, because such essential
matters as what causes the force remain somewhat.mysterious even today.
Early Ideas About Muscular Contraction
Human imagination has been at work on animal movement for a long time.
Hippocrates and his followers thought that the tendons caused the body to
move. They confused tendons with nerves, and in fact used the same word,
neuron, for both. Aristotle compared the movements of animals to the
movements of puppets, and said that the sinewy tendons played the role of the
puppet strings, bringing about motion as they were tightened and released.
The muscles themselves were not credited with the ability to contract until
the third century B.c., when Erasistratus suggested that an animal spirit flows
from the head through the nerves to the muscles. He understood the nerves to
be hollow tubes, through which the muscles could be filled with pneuma,
causing them to expand in breadth but contract in length, thus moving the
joints.
I have a friend who earns his living by producing, in his basement, a
patented pneumatic actuator for industrial applications. This is a melon-
