Biophysics

An Introduction
From the hydrophobic effect to protein–ligand binding, statistical physics
is relevant in almost all areas of molecular biophysics and biochemistry,
making it essential for modern students of molecular behavior. But traditional
presentations of this material are often difficult to penetrate. Statistical Physics
of Biomolecules: An Introduction brings “down to earth” some of the most
intimidating but important theories of molecular biophysics.
With an accessible writing style, the book unifies statistical, dynamic, and
thermodynamic descriptions of molecular behavior using probability ideas as
a common basis. Numerous examples illustrate how the twin perspectives of
dynamics and equilibrium deepen our understanding of essential ideas such as
entropy, free energy, and the meaning of rate constants. The author builds on the
general principles with specific discussions of water, binding phenomena, and
protein conformational changes/folding. The same probabilistic framework used
in the introductory chapters is also applied to non-equilibrium phenomena and
to computations in later chapters. The book emphasizes basic concepts rather
than cataloguing a broad range of phenomena.
Students build a foundational understanding by initially focusing on probability
theory, low-dimensional models, and the simplest molecular systems. The basics
are then directly developed for biophysical phenomena, such as water behavior,
protein binding, and conformational changes. The book’s accessible development
of equilibrium and dynamical statistical physics makes this a valuable text for
students with limited physics and chemistry backgrounds.

73788
ISBN: 978-1-4200-7378-2

90000

ww w. c rcp ress. c o m

9 781420 073782

w w w. c rc p r e s s . c o m

Statistical Physics of Biomolecules

Daniel M. Zuckerman

Zuckerman

Statistical Physics of Biomolecules

Daniel M. Zuckerman

Statistical Physics
of Biomolecules
An Introduction



Statistical Physics
of Biomolecules
AN INTRODUCTION



Statistical Physics
of Biomolecules

AN INTRODUCTION

Daniel M. Zuckerman


CRC Press
Taylor & Francis Group
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For my parents,
who let me think for myself.



Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi
Chapter 1

Proteins Don’t Know Biology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1

1.2

1.3

1.4
1.5

1.6
1.7
1.8


Prologue: Statistical Physics of Candy, Dirt, and Biology . . . . . . 1
1.1.1 Candy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Clean Your House, Statistically . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.3 More Seriously. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Guiding Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Proteins Don’t Know Biology. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Nature Has Never Heard of Equilibrium . . . . . . . . . . . . . . . . 4
1.2.3 Entropy Is Easy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.4 Three Is the Magic Number for Visualizing Data . . . . . . . 5
1.2.5 Experiments Cannot Be Separated from “Theory” . . . . . 5
About This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.1 What Is Biomolecular Statistical Physics? . . . . . . . . . . . . . . 5
1.3.2 What’s in This Book, and What’s Not . . . . . . . . . . . . . . . . . . . 6
1.3.3 Background Expected of the Student . . . . . . . . . . . . . . . . . . . . 7
Molecular Prologue: A Day in the Life of Butane . . . . . . . . . . . . . . . 7
1.4.1 Exemplary by Its Stupidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
What Does Equilibrium Mean to a Protein? . . . . . . . . . . . . . . . . . . . . . 9
1.5.1 Equilibrium among Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5.2 Internal Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5.3 Time and Population Averages . . . . . . . . . . . . . . . . . . . . . . . . . 11
A Word on Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Making Movies: Basic Molecular Dynamics Simulation . . . . . . 12
Basic Protein Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.8.1 Proteins Fold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.8.2 There Is a Hierarchy within Protein Structure. . . . . . . . . . 14
1.8.3 The Protein Geometry We Need to Know,
for Now . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.8.4 The Amino Acid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.8.5 The Peptide Plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.8.6 The Two Main Dihedral Angles Are Not

Independent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.8.7 Correlations Reduce Configuration Space, but Not
Enough to Make Calculations Easy . . . . . . . . . . . . . . . . . . . . . 18
1.8.8 Another Exemplary Molecule: Alanine Dipeptide . . . . . 18

vii


Contents

viii

1.9 A Note on the Chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Chapter 2

The Heart of It All: Probability Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.1 The Monty Hall Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Basics of One-Dimensional Distributions. . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1 What Is a Distribution? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.2 Make Sure It’s a Density! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.3 There May Be More than One Peak:
Multimodality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.4 Cumulative Distribution Functions . . . . . . . . . . . . . . . . . . . . . 26
2.2.5 Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.6 Sampling and Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.7 The Distribution of a Sum of Increments:

Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.8 Physical and Mathematical Origins of Some
Common Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2.9 Change of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3 Fluctuations and Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.1 Variance and Higher “Moments” . . . . . . . . . . . . . . . . . . . . . . . 37
2.3.2 The Standard Deviation Gives the Scale of a
Unimodal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3.3 The Variance of a Sum (Convolution) . . . . . . . . . . . . . . . . . . 39
2.3.4 A Note on Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3.5 Beyond Variance: Skewed Distributions
and Higher Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3.6 Error (Not Variance). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3.7 Confidence Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4 Two+ Dimensions: Projection and Correlation . . . . . . . . . . . . . . . . . 43
2.4.1 Projection/Marginalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4.2 Correlations, in a Sentence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.4.3 Statistical Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.4.4 Linear Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.4.5 More Complex Correlation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.4.6 Physical Origins of Correlations . . . . . . . . . . . . . . . . . . . . . . . . 50
2.4.7 Joint Probability and Conditional Probability . . . . . . . . . . 51
2.4.8 Correlations in Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.5 Simple Statistics Help Reveal a Motor Protein’s
Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.6 Additional Problems: Trajectory Analysis . . . . . . . . . . . . . . . . . . . . . . 54
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55


Contents


Chapter 3

ix

Big Lessons from Simple Systems: Equilibrium Statistical
Mechanics in One Dimension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1.1 Looking Ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2 Energy Landscapes Are Probability Distributions . . . . . . . . . . . . . . 58
3.2.1 Translating Probability Concepts into the
Language of Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . 60
3.2.2 Physical Ensembles and the Connection with
Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2.3 Simple States and the Harmonic Approximation . . . . . . 61
3.2.4 A Hint of Fluctuations: Average Does Not Mean
Most Probable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3 States, Not Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.3.1 Relative Populations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.4 Free Energy: It’s Just Common Sense. . . If You Believe in
Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.4.1 Getting Ready: Relative Populations . . . . . . . . . . . . . . . . . . . 67
3.4.2 Finally, the Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.4.3 More General Harmonic Wells . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.5 Entropy: It’s Just a Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.5.1 Entropy as (the Log of) Width: Double
Square Wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.5.2 Entropy as Width in Harmonic Wells . . . . . . . . . . . . . . . . . . 73

3.5.3 That Awful p ln p Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.6 Summing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.6.1 States Get the Fancy Names because They’re Most
Important . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.6.2 It’s the Differences That Matter. . . . . . . . . . . . . . . . . . . . . . . . . 77
3.7 Molecular Intuition from Simple Systems . . . . . . . . . . . . . . . . . . . . . . 78
3.7.1 Temperature Dependence: A One-Dimensional
Model of Protein Folding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.7.2 Discrete Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.7.3 A Note on 1D Multi-Particle Systems . . . . . . . . . . . . . . . . . . 81
3.8 Loose Ends: Proper Dimensions, Kinetic Energy . . . . . . . . . . . . . . 81
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Chapter 4

Nature Doesn’t Calculate Partition Functions: Elementary
Dynamics and Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.1

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.1.1 Equivalence of Time and Configurational Averages. . . 86
4.1.2 An Aside: Does Equilibrium Exist? . . . . . . . . . . . . . . . . . . . . 86


Contents

x

4.2

Newtonian Dynamics: Deterministic but Not Predictable. . . . . . 87

4.2.1 The Probabilistic (“Stochastic”) Picture of
Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.3 Barrier Crossing—Activated Processes . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.3.1 A Quick Preview of Barrier Crossing . . . . . . . . . . . . . . . . . . 89
4.3.2 Catalysts Accelerate Rates by Lowering Barriers . . . . . . 91
4.3.3 A Decomposition of the Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.3.4 More on Arrhenius Factors and Their Limitations . . . . . 92
4.4 Flux Balance: The Definition of Equilibrium . . . . . . . . . . . . . . . . . . . 92
4.4.1 “Detailed Balance” and a More Precise Definition
of Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.4.2 Dynamics Causes Equilibrium Populations . . . . . . . . . . . . 94
4.4.3 The Fundamental Differential Equation . . . . . . . . . . . . . . . . 95
4.4.4 Are Rates Constant in Time? (Advanced) . . . . . . . . . . . . . . 95
4.4.5 Equilibrium Is “Self-Healing” . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.5 Simple Diffusion, Again . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.5.1 The Diffusion Constant and the Square-Root Law
of Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.5.2 Diffusion and Binding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.6 More on Stochastic Dynamics: The Langevin Equation . . . . . 100
4.6.1 Overdamped, or “Brownian,” Motion and Its
Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.7 Key Tools: The Correlation Time and Function . . . . . . . . . . . . . . 103
4.7.1 Quantifying Time Correlations: The
Autocorrelation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.7.2 Data Analysis Guided by Time Correlation
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.7.3 The Correlation Time Helps to Connect Dynamics
and Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.8 Tying It All Together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.9 So Many Ways to ERR: Dynamics in Molecular

Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.10 Mini-Project: Double-Well Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . 108
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Chapter 5

Molecules Are Correlated! Multidimensional Statistical
Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.1

5.2

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Many Atoms in One Molecule and/or Many
Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.2 Working toward Thermodynamics. . . . . . . . . . . . . . . . . . . .
5.1.3 Toward Understanding Simulations . . . . . . . . . . . . . . . . . .
A More-than-Two-Dimensional Prelude . . . . . . . . . . . . . . . . . . . . . .
5.2.1 One “Atom” in Two Dimensions . . . . . . . . . . . . . . . . . . . . .
5.2.2 Two Ideal (Noninteracting) “Atoms” in 2D . . . . . . . . . .

111
111
112
112
112
113
114


Contents


xi

5.2.3 A Diatomic “Molecule” in 2D . . . . . . . . . . . . . . . . . . . . . . . .
5.2.4 Lessons Learned in Two Dimensions. . . . . . . . . . . . . . . . .
5.3 Coordinates and Forcefields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Internal Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.3 A Forcefield Is Just a Potential Energy
Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.4 Jacobian Factors for Internal Coordinates
(Advanced) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 The Single-Molecule Partition Function . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Three Atoms Is Too Many for an Exact
Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.2 The General Unimolecular Partition Function . . . . . . .
5.4.3 Back to Probability Theory and Correlations . . . . . . . .
5.4.4 Technical Aside: Degeneracy Number . . . . . . . . . . . . . . .
5.4.5 Some Lattice Models Can Be Solved Exactly. . . . . . . .
5.5 Multimolecular Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 Partition Functions for Systems of Identical
Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.2 Ideal Systems—Uncorrelated by Definition. . . . . . . . . .
5.5.3 Nonideal Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 The Free Energy Still Gives the Probability . . . . . . . . . . . . . . . . . .
5.6.1 The Entropy Still Embodies Width (Volume) . . . . . . . .
5.6.2 Defining States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.3 Discretization Again Implies S ∼ − p ln p . . . . . . .
5.7 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 6

115
119
119
119
120
121
123
124
125
126
127
128
129
130
131
132
132
133
134
134
135
135
135

From Complexity to Simplicity: The Potential of Mean Force . . . . . 137
6.1
6.2


6.3

6.4

Introduction: PMFs Are Everywhere . . . . . . . . . . . . . . . . . . . . . . . . . .
The Potential of Mean Force Is Like a Free Energy . . . . . . . . . .
6.2.1 The PMF Is Exactly Related to a Projection . . . . . . . . .
6.2.2 Proportionality Functions for PMFs . . . . . . . . . . . . . . . . . .
6.2.3 PMFs Are Easy to Compute from a Good
Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The PMF May Not Yield the Reaction Rate or Transition
State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Is There Such a Thing as a Reaction Coordinate? . . .
The Radial Distribution Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.1 What to Expect for g(r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.2 g(r) Is Easy to Get from a Simulation . . . . . . . . . . . . . . . .
6.4.3 The PMF Differs from the “Bare” Pair Potential . . . .
6.4.4 From g(r) to Thermodynamics in Pairwise
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.5 g(r) Is Experimentally Measurable . . . . . . . . . . . . . . . . . . .

137
137
138
140
141
142
143
144
145

146
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149
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6.5

PMFs Are the Typical Basis for “Knowledge-Based”
(“Statistical”) Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.6 Summary: The Meaning, Uses, and Limitations of
the PMF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Chapter 7

What’s Free about “Free” Energy? Essential Thermodynamics . . . . 153
7.1
7.2

7.3

7.4

7.5

7.6


Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.1 An Apology: Thermodynamics Does Matter! . . . . . . . .
Statistical Thermodynamics: Can You Take a Derivative? . . .
7.2.1 Quick Reference on Derivatives . . . . . . . . . . . . . . . . . . . . . .
7.2.2 Averages and Entropy, via First Derivatives . . . . . . . . .
7.2.3 Fluctuations from Second Derivatives . . . . . . . . . . . . . . . .
7.2.4 The Specific Heat, Energy Fluctuations, and the
(Un)folding Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
You Love the Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.1 Free Energy and Entropy of the Ideal Gas. . . . . . . . . . . .
7.3.2 The Equation of State for the Ideal Gas . . . . . . . . . . . . . .
Boring but True: The First Law Describes Energy
Conservation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.1 Applying the First Law to the Ideal Gas: Heating
at Constant Volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.2 Why Is It Called “Free” Energy, Anyway? The
Ideal Gas Tells All . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
G vs. F: Other Free Energies and Why They (Sort of )
Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5.1 G, Constant Pressure, Fluctuating Volume—A
Statistical View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5.2 When Is It Important to Use G Instead of F? . . . . . . . .
7.5.3 Enthalpy and the Thermodynamic
Definition of G. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5.4 Another Derivative Connection—Getting
P from F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5.5 Summing Up: G vs. F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5.6 Chemical Potential and Fluctuating Particle
Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Overview of Free Energies and Derivatives . . . . . . . . . . . . . . . . . . .
7.6.1 The Pertinent Free Energy Depends on the
Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6.2 Free Energies Are “State Functions” . . . . . . . . . . . . . . . . .
7.6.3 First Derivatives of Free Energies Yield
Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6.4 Second Derivatives Yield
Fluctuations/Susceptibilities . . . . . . . . . . . . . . . . . . . . . . . . . .

153
153
154
154
155
157
157
158
159
160
160
161
162
164
164
166
168
169
170
171
173

173
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7.7

The Second Law and (Sometimes) Free Energy
Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.7.1 A Kinetic View Is Helpful . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.7.2 Spontaneous Heat Flow and Entropy . . . . . . . . . . . . . . . . .
7.7.3 The Second Law for Free
Energies—Minimization, Sometimes. . . . . . . . . . . . . . . . .
7.7.4 PMFs and Free Energy Minimization for
Proteins—Be Warned! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.7.5 The Second Law for Your House: Refrigerators
Are Heaters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.7.6 Summing Up: The Second Law . . . . . . . . . . . . . . . . . . . . . . .
7.8 Calorimetry: A Key Thermodynamic Technique . . . . . . . . . . . . .
7.8.1 Integrating the Specific Heat Yields Both Enthalpy
and Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.8.2 Differential Scanning Calorimetry for Protein
Folding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.9 The Bare-Bones Essentials of Thermodynamics . . . . . . . . . . . . .
7.10 Key Topics Omitted from This Chapter . . . . . . . . . . . . . . . . . . . . . . .

Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 8

175
175
175
177
179
181
181
182
182
183
183
184
184

The Most Important Molecule: Electro-Statistics of Water . . . . . . . . . 185
8.1

8.2
8.3
8.4

8.5

Basics of Water Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.1 Water Is Tetrahedral because of Its Electron
Orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.2 Hydrogen Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.1.3 Ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.4 Fluctuating H-Bonds in Water . . . . . . . . . . . . . . . . . . . . . . . .
8.1.5 Hydronium Ions, Protons, and Quantum
Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Water Molecules Are Structural Elements in Many Crystal
Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The pH of Water and Acid–Base Ideas . . . . . . . . . . . . . . . . . . . . . . . .
Hydrophobic Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.1 Hydrophobicity in Protein and Membrane
Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.2 Statistical/Entropic Explanation of the
Hydrophobic Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Water Is a Strong Dielectric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.5.1 Basics of Dielectric Behavior . . . . . . . . . . . . . . . . . . . . . . . . .
8.5.2 Dielectric Behavior Results from Polarizability . . . . .
8.5.3 Water Polarizes Primarily due to
Reorientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.5.4 Charges Prefer Water Solvation to a Nonpolar
Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.5.5 Charges on Protein in Water = Complicated!. . . . . . . .

185
185
185
186
187
187
188
188
190

190
190
192
193
194
195
196
196


Contents

xiv

Charges in Water + Salt = Screening . . . . . . . . . . . . . . . . . . . . . . . . .
8.6.1 Statistical Mechanics of Electrostatic Systems
(Technical) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.6.2 First Approximation: The Poisson–Boltzmann
Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.6.3 Second Approximation: Debye–Hückel Theory . . . . .
8.6.4 Counterion Condensation on DNA . . . . . . . . . . . . . . . . . . .
8.7 A Brief Word on Solubility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.8 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.9 Additional Problem: Understanding Differential
Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.6

Chapter 9


197
198
200
200
202
202
203
203
204

Basics of Binding and Allostery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
9.1

9.2

9.3

9.4

9.5
9.6

9.7

A Dynamical View of Binding: On- and Off-Rates . . . . . . . . . .
9.1.1 Time-Dependent Binding: The Basic Differential
Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Macroscopic Equilibrium and the Binding Constant . . . . . . . . .
9.2.1 Interpreting Kd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Is Based on a
9.2.2 The Free Energy of Binding Gbind
0
Reference State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.3 Measuring Kd by a “Generic” Titration
Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.4 Measuring Kd from Isothermal Titration
Calorimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.5 Measuring Kd by Measuring Rates. . . . . . . . . . . . . . . . . . . .
A Structural-Thermodynamic View of Binding . . . . . . . . . . . . . .
9.3.1 Pictures of Binding: “Lock and Key” vs.
“Induced Fit”. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.2 Many Factors Affect Binding . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.3 Entropy–Enthalpy Compensation . . . . . . . . . . . . . . . . . . . . .
Understanding Relative Affinities:
G and
Thermodynamic Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.1 The Sign of
G Has Physical Meaning . . . . . . . . . . . .
9.4.2 Competitive Binding Experiments . . . . . . . . . . . . . . . . . . . .
9.4.3 “Alchemical” Computations of Relative
Affinities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Energy Storage in “Fuels” Like ATP . . . . . . . . . . . . . . . . . . . . . . . . . .
Direct Statistical Mechanics Description of
Binding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.6.1 What Are the Right Partition Functions? . . . . . . . . . . . . .
Allostery and Cooperativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.7.1 Basic Ideas of Allostery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.7.2 Quantifying Cooperativity with the Hill Constant . . .


205
207
208
209
210
211
211
212
212
212
213
215
216
216
218
218
220
221
221
222
222
224


Contents

xv

9.7.3


Further Analysis of Allostery: MWC and KNF
Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.8 Elementary Enzymatic Catalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.8.1 The Steady-State Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.8.2 The Michaelis–Menten “Velocity” . . . . . . . . . . . . . . . . . . .
9.9 pH AND pKa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.9.1 pH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.9.2 pKa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.10 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 10

227
229
230
231
231
232
232
233
233

Kinetics of Conformational Change and Protein Folding . . . . . . . . . 235
10.1 Introduction: Basins, Substates, and States . . . . . . . . . . . . . . . . . .
10.1.1 Separating Timescales to Define Kinetic
Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 Kinetic Analysis of Multistate Systems. . . . . . . . . . . . . . . . . . . . . .
10.2.1 Revisiting the Two-State System . . . . . . . . . . . . . . . . . . . .
10.2.2 A Three-State System: One Intermediate . . . . . . . . . . .

10.2.3 The Effective Rate in the Presence of an
Intermediate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2.4 The Rate When There Are Parallel Pathways. . . . . . .
10.2.5 Is There Such a Thing as Nonequilibrium
Kinetics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2.6 Formalism for Systems Described by Many
States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3 Conformational and Allosteric Changes in Proteins . . . . . . . .
10.3.1 What Is the “Mechanism” of a Conformational
Change? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.2 Induced and Spontaneous Transitions . . . . . . . . . . . . . . .
10.3.3 Allosteric Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.4 Multiple Pathways. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.5 Processivity vs. Stochasticity . . . . . . . . . . . . . . . . . . . . . . . .
10.4 Protein Folding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4.1 Protein Folding in the Cell . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4.2 The Levinthal Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4.3 Just Another Type of Conformational
Change? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4.4 What Is the Unfolded State? . . . . . . . . . . . . . . . . . . . . . . . . .
10.4.5 Multiple Pathways, Multiple Intermediates . . . . . . . . .
10.4.6 Two-State Systems, Values, and Chevron
Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

235
235
238
238

242
246
250
251
252
252
252
253
254
255
255
256
257
258
258
259
260
262
264
264


xvi

Chapter 11

Contents

Ensemble Dynamics: From Trajectories to Diffusion
and Kinetics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

11.1 Introduction: Back to Trajectories and Ensembles . . . . . . . . . .
11.1.1 Why We Should Care about Trajectory
Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.1.2 Anatomy of a Transition Trajectory . . . . . . . . . . . . . . . . .
11.1.3 Three General Ways to Describe Dynamics . . . . . . . .
11.2 One-Dimensional Ensemble Dynamics . . . . . . . . . . . . . . . . . . . . . .
11.2.1 Derivation of the One-Dimensional Trajectory
Energy: The “Action” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2.2 Physical Interpretation of the Action . . . . . . . . . . . . . . . .
11.3 Four Key Trajectory Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3.1 Initialized Nonequilibrium Trajectory
Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3.2 Steady-State Nonequilibrium Trajectory
Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3.3 The Equilibrium Trajectory Ensemble . . . . . . . . . . . . . .
11.3.4 Transition Path Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.4 From Trajectory Ensembles to Observables . . . . . . . . . . . . . . . . .
11.4.1 Configuration-Space Distributions from
Trajectory Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.4.2 Finding Intermediates in the Path Ensemble . . . . . . . .
11.4.3 The Commitment Probability and a
Transition-State Definition . . . . . . . . . . . . . . . . . . . . . . . . . .
11.4.4 Probability Flow, or Current . . . . . . . . . . . . . . . . . . . . . . . . .
11.4.5 What Is the Reaction Coordinate? . . . . . . . . . . . . . . . . . . .
11.4.6 From Trajectory Ensembles to Kinetic Rates . . . . . . .
11.4.7 More General Dynamical Observables from
Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.5 Diffusion and Beyond: Evolving Probability
Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.5.1 Diffusion Derived from Trajectory Probabilities . . .

11.5.2 Diffusion on a Linear Landscape . . . . . . . . . . . . . . . . . . . .
11.5.3 The Diffusion (Differential) Equation . . . . . . . . . . . . . . .
11.5.4 Fokker–Planck/Smoluchowski Picture for
Arbitrary Landscapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.5.5 The Issue of History Dependence . . . . . . . . . . . . . . . . . . .
11.6 The Jarzynski Relation and Single-Molecule
Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.6.1 Revisiting the Second Law of Thermodynamics . . .
11.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

265
265
266
267
271
272
274
275
275
275
276
276
278
279
280
280
281
281
282

283
283
284
285
287
289
291
293
294
294
295


Contents

Chapter 12

xvii

A Statistical Perspective on Biomolecular Simulation . . . . . . . . . . . . 297
12.1 Introduction: Ideas, Not Recipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.1.1 Do Simulations Matter in Biophysics? . . . . . . . . . . . . . .
12.2 First, Choose Your Model: Detailed or Simplified . . . . . . . . . .
12.2.1 Atomistic and “Detailed” Models . . . . . . . . . . . . . . . . . . .
12.2.2 Coarse Graining and Related Ideas . . . . . . . . . . . . . . . . . .
12.3 “Basic” Simulations Emulate Dynamics. . . . . . . . . . . . . . . . . . . . .
12.3.1 Timescale Problems, Sampling Problems . . . . . . . . . . .
12.3.2 Energy Minimization vs. Dynamics/Sampling . . . . .
12.4 Metropolis Monte Carlo: A Basic Method and
Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12.4.1 Simple Monte Carlo Can Be Quasi-Dynamic. . . . . . .
12.4.2 The General Metropolis–Hastings Algorithm . . . . . .
12.4.3 MC Variations: Replica Exchange and Beyond . . . .
12.5 Another Basic Method: Reweighting and Its Variations . . . .
12.5.1 Reweighting and Annealing . . . . . . . . . . . . . . . . . . . . . . . . .
12.5.2 Polymer-Growth Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.5.3 Removing Weights by “Resampling” Methods . . . . .
12.5.4 Correlations Can Arise Even without Dynamics . . .
12.6 Discrete-State Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.7 How to Judge Equilibrium Simulation Quality . . . . . . . . . . . . . .
12.7.1 Visiting All Important States . . . . . . . . . . . . . . . . . . . . . . . .
12.7.2 Ideal Sampling as a Key Conceptual Reference . . . .
12.7.3 Uncertainty in Observables and Averages . . . . . . . . . .
12.7.4 Overall Sampling Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.8 Free Energy and PMF Calculations . . . . . . . . . . . . . . . . . . . . . . . . . .
12.8.1 PMF and Configurational Free Energy
Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.8.2 Thermodynamic Free Energy Differences Include
All Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.8.3 Approximate Methods for Drug Design. . . . . . . . . . . . .
12.9 Path Ensembles: Sampling Trajectories . . . . . . . . . . . . . . . . . . . . .
12.9.1 Three Strategies for Sampling Paths . . . . . . . . . . . . . . . .
12.10 Protein Folding: Dynamics and Structure Prediction . . . . . . .
12.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

297
297
298
299

299
300
301
304
305
305
306
307
309
310
311
312
313
313
313
314
314
314
315
316
317
318
320
321
321
322
323
323

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325




Preface
The central goal of this book is to answer “Yes” to the question, “Is there statistical
mechanics for the rest of us?” I believe the essentials of statistical physics can be made
comprehensible to the new generation of interdisciplinary students of biomolecular
behavior. In other words, most of us can understand most of what’s important. This
“less is more” approach is not an invitation to sloppiness, however. The laws of
physics and chemistry do matter, and we should know them well. The goal of this
book it to explain, in plain English, the classical statistical mechanics and physical
chemistry underlying biomolecular phenomena.
The book is aimed at students with only an indirect background in biophysics.
Some undergraduate physics, chemistry, and calculus should be sufficient. Nevertheless, I believe more advanced students can benefit from some of the less traditional,
and hopefully more intuitive, presentations of conventional topics.
The heart of the book is the statistical meaning of equilibrium and how it results
from dynamical processes. Particular attention is paid to the way averaging of statistical ensembles leads to the free energy and entropy descriptions that are a stumbling
block to many students. The book, by choice, is far from comprehensive in its coverage of either statistical mechanics or molecular biophysics. The focus is on the main
lines of thought, along with key examples. However, the book does attempt to show
how basic statistical ideas are applied in a variety of seemingly complex biophysical
“applications” (e.g., allostery and binding)—in addition to showing how an ensemble
view of dynamics fits naturally with more familiar topics, such as diffusion.
I have taught most of the first nine chapters of the book in about half a semester
to first-year graduate students from a wide range of backgrounds. I have always
felt rushed in doing so, however, and believe the book could be used for most of a
semester’s course. Such a course could be supplemented by material on computational
and/or experimental methods. This book addresses simulation methodology only
briefly, and is definitely not a “manual.”

xix




Acknowledgments
I am grateful to the following students and colleagues who directly offered comments on and corrections to the manuscript: Divesh Bhatt, Lillian Chong, Ying Ding,
Steven Lettieri, Edward Lyman, Artem Mamonov, Lidio Meireles, Adrian Roitberg,
Jonathan Sachs, Kui Shen, Robert Swendsen, Michael Thorpe, Marty Ytreberg,
Bin Zhang, Xin Zhang, and David Zuckerman. Artem Mamonov graciously provided
several of the molecular graphics figures, Divesh Bhatt provided radial distribution
data and Bin Zhang provided transition path data. Others helped me obtain the understanding embodied herein (i.e., I would have had it wrong without them) including
Carlos Camacho, Rob Coalson, David Jasnow, and David Wales. Of course, I have
learned most of all from my own mentors: Robijn Bruinsma, Michael Fisher, and
Thomas Woolf. Lance Wobus, my editor from Taylor & Francis, was always insightful and helpful. Ivet Bahar encouraged me throughout the project. I deeply regret if I
have forgotten to acknowledge someone. The National Science Foundation provided
support for this project through a Career award (overseen by Dr. Kamal Shukla),
and much of my understanding of the field developed via research supported by the
National Institutes of Health.
I would very much appreciate hearing about errors and ways to improve the book.
Daniel M. Zuckerman
Pittsburgh, Pennsylvania

xxi



Don’t Know
1 Proteins
Biology
1.1 PROLOGUE: STATISTICAL PHYSICS OF CANDY, DIRT, AND
BIOLOGY

By the time you finish this book, hopefully you will look at the world around you
in a new way. Beyond biomolecules, you will see that statistical phenomena are at
work almost everywhere. Plus, you will be able to wield some impressive jargon and
equations.

1.1.1 CANDY
Have you ever eaten trail mix? A classic variety is simply a mix of peanuts, dried
fruit, and chocolate candies. If you eat the stuff, you’ll notice that you usually get a
bit of each ingredient in every handful. That is, unsurprisingly, trail mix tends to be
well mixed. No advanced technology is required to achieve this. All you have to do
is shake.
To understand what’s going on, let’s follow a classic physics strategy. We’ll
simplify to the essence of the problem—the candy. I’m thinking of my favorite
discoidally shaped chocolate candy, but you are free to imagine your own. To adopt
another physics strategy, we’ll perform a thought experiment. Imagine filling a clear
plastic bag with two different colors of candies: first blue, then red, creating two
layers. Then, we’ll imagine holding the bag upright and shaking it (yes, we’ve sealed
it) repeatedly. See Figure 1.1.
What happens? Clearly the two colors will mix, and after a short time, we’ll have
a fairly uniform mixture of red and blue candies.
If we continue shaking, not much happens—the well-mixed “state” is stable or
“equilibrated.” But how do the red candies know to move down and the blue to move
up? And if the two colors are really moving in different directions, why don’t they
switch places after a long time?
Well, candy clearly doesn’t think about what it is doing. The pieces can only move
randomly in response to our shaking. Yet somehow, blind, random (nondirected)
motion leads to a net flow of red candy in one direction and blue in the other.
This is nothing other than the power of diffusion, which biomolecules also “use”
to accomplish the needs of living cells. Biomolecules, such as proteins, are just as
dumb as candy—yet they do what they need to do and get where they need to go.

Candy mixing is just a simple example of a random process, which must be described
statistically like many biomolecular processes.

1