Statistical Physics of Biomolecules AN INTRODUCTION Daniel M. Zuckerman TECHNiSCHE INFORMATIONSSIBLIOTHEK UNiVERSITATSBIBLIOTHEK HANNOVER V. J CRC Press Taylor & Francis G roup CRC Press Is an imprint of the Taylor & Francis Croup, an Inform* business
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Statistical physics of biomolecules : an introduction · Statistical Physics ofBiomolecules AN INTRODUCTION DanielM.Zuckerman TECHNiSCHE INFORMATIONSSIBLIOTHEK UNiVERSITATSBIBLIOTHEK
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Statistical Physicsof BiomoleculesAN INTRODUCTION
Daniel M. Zuckerman
TECHNiSCHE
INFORMATIONSSIBLIOTHEK
UNiVERSITATSBIBLIOTHEKHANNOVER
V. J
CRC PressTaylor & Francis Group
CRC Press Is an imprint of the
Taylor & Francis Croup, an Inform* business
Contents
Preface xix
Acknowledgments xxi
Chapter 1 Proteins Don't Know Biology 1
1.1 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
1.2 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
1.3 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
1.4 Molecular Prologue: A Day in the Life of Butane 7
1.4.1 Exemplary by Its Stupidity 9
1.5 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
1.6 A Word on Experiments 11
1.7 Making Movies: Basic Molecular Dynamics Simulation 12
1.8 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
vjjj Contents
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 ix
Chapter 3 Big Lessons from Simple Systems: Equilibrium Statistical
Mechanics in One Dimension 57
3.1 Intrpduction 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 Slatistical 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 Inp 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 ID Multi-Particle Systems 81
3.8 Loose Ends: Proper Dimensions, Kinetic Energy 81