Statistical Physics of Biomolecules AN INTRODUCTION

Daniel M. Zuckerman

~CRC Press ~ Taylor & Francis Group

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an lnforma business

Contents Preface .................... .. .. . .... . .. . ..... .. . . ..... .. .... .... .... ....... .. ... .. ..... xix Acknowledgments .... . ......... .. ............ ... . . ........ . ...... . .... .. .. . ..... . .... xxi

Chapter 1 Proteins Don't Know Biology ........ .... ....... ... ........... .. . ... .... I

1.1 Prologue: Statistical Physics of Candy, Dirt, and Biology ...... 1 1.1.1 Candy ............ .. ........ . .. . . .... . .... .... ...... . ....... I 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 ... . ... .. ........ . ....... II

1.6 A Word on Experiments . . . ........... .. ........... ....... ...... . II 1.7 Making Movies: Basic Molecular Dynamics Simulation ...... 12 1.8 Basic Protein Geometry .. . ... .. ...... .... .. ...... .... .. .. ........ 14

1.8. L 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

viii Contents

1.9 A Note on the Chapters ... .. . . . . .. ... . .... . .. . .. . . . ... . ....... . : . 18 Further Reading ..... . .. ... . .. . . . . ......... . .. . . . ... . ... . ................ 19

Chapter 2 The Heart of It A ll : Probability Theory ............ , . ... .. . . . . .. . . . .... 21

2. 1 Introduction .. . ..... . ... . ... .. .. . ................. ... . . . ....... . . . . 2 1 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:

2.2.4 2.2.5 2.2.6

Multimodality . . . . .......... . . .... ............. .. ... . . .. .. 25 Cumulative Distribution Functions . .. ... ... .. . .... . . ... 26 Averages . . . . ................ .. . .. .. .. ... .. ...... . . .. . . ... 28 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 Variab les . .... . . . . ... . . . . .. .... . ... . .... . .. . .. 36

2.3 F luctuations 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 Hi <>her 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 Co~relations, 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 Phys ical 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 Add itional Problems: Trajectory Analysis .. . . . . . . . . . ... . .. . . .. . 54 Further Reading ... . ... . ............. . ... .. . . ... . .......... . . .. . .. . . .... . 55

Contents

Chapter 3

3.1

3.2 Energy 3.2.1

3.2.2

3.2.3 3.2.4

3.3

3.4

3.4.3 3.5

3.5.2 3.5 .3

3.6

3.6.2 3.7

3.7.2 3.7.3

3.8 Loose Further Reading.

Chapter 4 Nature Doesn't C Dynamics and Eq

4.1 Introductio~

4.1.1 Equ 4.1.2 An J

Contents

... . . . ... . . . .. . . .. .. . . .. . ..... .... : . 18

······ ·· ·· ·········· ·· ····· · ········ 19

:ory .. . .... . ... . .. ... . . .......... . . . 21

.·· .. · ··········· .. . . .. .. ... . ... ·· .· · · · 21 lem ...... .. .. ....... . ...... ... . . .. . 21 )istributions ... ....... . ...... . . ... . 22 1? .. ....... .. . ... .. .. ...... . .... . ... 22 ;; ity! .. .... . .... ...... . ... . . .. . .. ... 25 :han One Peak: ......... .. . .. ...... ···· · .. ......... 25 on Functions ....... ... . .. .. ... ... 26 ..... .. ... ... ... . . ... ... . . ... . .. .. . . 28

:s ......... .. . . . ... .. . ........... .. . 29 Sum of Increments: . .... .............. .. . ... . . . ... ..... 31 atical Origins of Some IS . ....... . .. . ........ . . .. .....•.... 34 . . . . . . .. . . . . . . . . .... .... ... . ... . .... 36

..... . . ....... . . . . ... .... .. .. ... . .... 36 "Moments" . . . ................... . 37 on Gives the Scale of a n . .... .. ...... ........... . . . ...... . . 38 m (Convolution) ... .. .. .... . ...... 39 . .. . ... .... . . . . .. . .. .. .. .. .. ... .... .. 40 ewed Distributions ................. ..... .. ... . .. .. . .. . 41

... . ...... . ... .. .... ....... . . . ....... 4].

··· ···· ··········· · . .... . . .. ..... . · ... 43 1 and Correlation ................ .43 ~ation . .. . ... ........... ... . .... ... . 44 1tence .... ... ........ . ...... ..... .. . 45 ice . . .. ... . ... . .. . . .... ...... . .. . .. . 46 . ..... ........ . ... ... . .. .. . . ......... 46 lation ........ . . .. ...... ....... .. .. . 48 :orrelations ..... . . . ... . . ... . .. . .... 50 Conditional Probability .. ...... . . 51 .... . ..... . .. . ....... ..... ... ... . . ... 52 I a Motor Protein 's .. .. ..... .. . .. ... .. ... . .. , . . .. . .. .... 54 ory Analysis .. .... .. .. . .. .. .. ..... 54 . ........ . ............... .. ........ . . 55

Contents ix

Chapter 3 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 LP lnp 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 .I 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 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

XII Contents Contents

6.5 PMFs Are the Typical Basis for "Knowledge-Based" 7.7 The Secon ("Statistical") Potentials ·· ···· ········ ··· ··· · · · ···· ··· · ··· · ·· ·· 150 Minimiza1

6.6 Summary: The Meaning, Uses, and Limitations of 7.7.1 A the PMF . ..... . ... ..... . ...... ... .. ... ..... . .. .......... .. ...... . 150 7.7.2 s~

Further Reading .... ...... ... . .... . ..... . ........ .. .. . . .. .. .... . . . ... .. 151 7.7 .3 Tt Er

Chapter 7 What's Free about "Free" Energy? Essential Thermodynamics . ... 153 7.7.4 p~

Pr 7.1 Introduction ....... . ................. . . . ..... ....... . ............ 153 7.7.5 Tl

7.1.1 An Apology: Thermodynamics Does Matter! ...... .. 153 A!

7.2 Statistical Thermodynamics: Can You Take a Derivative? . .. 154 7.7.6 S1

7.2.1 Quick Reference on Derivatives ··· · ··· ·· ··· ··· ·· ·· ·· · 154 7.8 Calorimel

7.2.2 Averages and Entropy, via First Derivatives ........ . 155 7.8.1 In

7.2.3 Fluctuations from Second Derivatives .. . .......... . .. 157 31

7.2.4 The Specific Heat, Energy Fluctuations, and the 7.8.2 D

(Un)folding Transition : 157 F1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.3 You Love the Ideal Gas ..... ... . . .. . .. .... .. .......... . .. . ... . . 158 7.9

7.3.1 Free Energy and Entropy of the Ideal Gas ... .. ... . .. . 159 7.10

7.3.2 The Equation of State for the Ideal Gas .............. 160 7.4 Boring but True: The First Law Describes Energy

Conservation . .. .. .. ... . ........... .. .. .... . ..... ... . .. ... . .. .. .. 160 Chapter 8

7.4.1 Applying the First Law to the Ideal Gas: Heating 8.1 at Constant Volume ......... . . ... ....... ......... ...... 161

7.4.2 Why Is It Called "Free" Energy, Anyway? The Ideal Gas Tells All .. . . ... . ............ . .......... . . .... 162 8.1.2

7.5 G vs. F: Other Free Energies and Why They (Sort of) 8.1.3 Matter .. . ....... . ....... .. ...... .. ' · ... . .. . ........ . .... ... ...... 164 8.1.4 7.5 .1 G, Constant Pressure, Fluctuating Volume- A 8.1.5

Statislical View .......... .... .. ....... ...... . ..... . . ... 164 7.5.2 When Is It Important to Use G Instead of F? ... . . ... 166 8.2 7.5.3 Enthalpy and the Thermodynamic

Definition of G ..... .. ....................... . .......... 168 8.3 7.5.4 Another Derivative Connection- Getting 8.4

P from F .. . . . . ........ ..... .. . ...... .. .. ........ .. ..... 169 7.5 .5 Summing Up: G vs. F .. . . . .. . ... ..... .. ... .. ..... . .... 170 7.5.6 Chemical Potential and Fluctuating Particle 8.4.2

Numbers ... .. ..... . .. .... .. . .......... . . .. ....... .. ..... 171 7.6 Overview of Free Energies and Derivatives ........ ... ........ 173 8.5 Water

7.6.1 The Pertinent Free Energy Depends on the 8.5 .1 Conditions . ............................... ...... ........ 173 8.5.2

7.6.2 Free Energies Are "State Functions" ....... . . .. .... . . 174 8.5 .3

7.6.3 First Derivatives of Free Energies Yield Averages .. ................................ . .. .. . .. . . ... 174 8.5.4

7.6.4 Second Derivatives Yield Fluctuations/Susceptibilities .......... . ....... . ...... . 174 8.5 .5

Contents

;; for "Knowledge-Based"

ses, and Limitations of 150

150 151

? Essential Thermodynamics ... . 153

153 10dynamics Does Matter! . . ... ... 153 ;: Can You Take a Derivative? ... 154 Derivatives .... .. .. .............. 154

py, via First Derivatives . . . . . . . . . 155 econd Derivatives.. . .. . . . . . . . . . . . 157 :;;nergy Fluctuations, and the ton .. . ...... .. : . ... · . . ... . ..... · . .. . . 157

158 tropy of the Ideal Gas... . .. .. . ... 159 tte for the Ideal Gas . . . . . . . . . . . . . . 160 ~aw Describes Energy

160 Jaw to the Ideal Gas: Heating

161 ree" Energy, Anyway? The

162 :sand Why They (Sort of) .. . .. ; . . ............ . .... . ...... ... . 164 e, Fluctuating Volume-A

164 tt to Use G Instead ofF? . . ... ... 166 termodynamic

168 Connection-Getting

. . ...... .. . . ············· ····· ··· ·· · 169 F ... ... ............... . ....... . ... 170

:md Fluctuating Particle 171

:md Derivatives ... . ..... . .. .. ..... 173 ~nergy Depends on the

173 State Functions" ........ . . .. . . ... 174 Free Energies Yield

174 Yield tibilities ... . . . .......... .. ... ..... 174

Contents xiii

7.7 The Second Law and (Sometimes) Free Energy Minimization ...... .......... . ... ..... .. ......... .......... ..... 175 7.7.1 A Kinetic View Is Helpful . ......... ..... . . . . . . . ...... 175 7.7.2 Spontaneous Heat Flow and Entropy .... . ......... ... 175 7.7.3 The Second Law for Free

Energies-Minimization, Sometimes ..... . .... . . ..... 177 7.7.4 PMFs and Free Energy Minimization for

Proteins-Be Warned! .... ................... .. ...... . 179 7.7.5 The Second Law for Your House: Refrigerators

Are Heaters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 7.7.6 SummingUp:TheSecondLaw .. . . . .. .. . . ... . ... . .... 181

7.8 Calorimetry: A Key Thermodynamic Technique ............. 182 7.8.1 Integrating the Specific Heat Yields Both Enthalpy

and Entropy ................... . ....... . . ... . . ... . . ... .. 182 7.8.2 Differential Scanning Calorimetry for Protein

Folding ..... . . .. . . .. ....... . .................. .......... 183 7.9 The Bare-Bones Essentials of Thermodynamics ... .... ...... 183 7.10 Key Topics Omitted from This Chapter ...... . . .. .. . ..... . .... 184 Further Read ing . ............ .. ....... . ........ . . . . .... . ... ....... . . .. . 184

Chapter 8 The Most Important Molecule: Electro-Statistics of Water ... ...... 185

8.1 Basics of Water Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 8.1 .1 Water Is Tetrahedral because of Its Electron

Orbitals . .............. . ... . .. . . .. . ... ........ . ..... . .... 185 8.1 .2 Hydrogen Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 8.1.3 Ice ........ .. .. .. ...... . . . .. ...... ... . ... . ....... ........ 186 8.1.4 Fluctuating H-Bonds in Water ......... . .... ... ....... 187 8.1.5 Hydronium Ions, Protons, and Quantum

Fluctuations ... ..... . ... .. . ... .......... ..... . .. ... .. ... 187 8.2 Water Molecules Are Structural Elements in Many Crystal

Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 8.3 The pH of Water and Acid-Base Ideas .. . . ............ . .... . . . 188 8.4 Hydrophobic Effect ... . . .. .. .. . . . ... . .. . .... ... . .. ..... ..... . .. 190

8.4.1 Hydrophobicity in Protein and Membrane Structure .. . .. . .. . .. ............................ ... ...... 190

8.4.2 Statistical/Entropic Explanation of the Hydrophobic Effect. . . . ........ . ............ . ... .. ..... 190

8.5 Water Is a Strong Dielectric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 8.5 .1 Basics of Dielectric Behavior ....... .. ..... ..... .. .... 193 8.5.2 Dielectric Behavior Results from Polarizability ..... 194 8.5.3 Water Polarizes Primarily due to

Reorientation .... ... . .. ... ....... . . .. ....... . ..... ..... . 195 8.5.4 Charges Prefer Water Solvation to a Nonpolar

Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 8.5 .5 Charges on Protein in Water= Complicated! ....... . 196

xiv Contents

8.6 Charges in Water+ Salt= Screening .......... .. .. . .. .. .... .. 197 8.6.1 Statistical Mechanics of Electrostatic Systems

(Technical) . . ... .. ... ... ..... ... . . . .... :. . . . . . . . . . . . . . . . 198 · 8.6.2 First Approximation: The Poisson- Boltzmann

Equation . . . .. .. .. ........... .. .. . .. ... ... . . . ... . .... . ... 200 8.6.3 Second Approximation : Debye-Hiickel Theory . .. .. 200 8.6.4 Counterion Condensation on DNA .. .. ... . . . . . . . .. . .. 202

8.7 A Brief Word on Solubili ty . ... .. .. . . .. ... .. . . .. .. . ....... ... . . 202 8.8 Summary . ... .. . .. .. . .... . .. . . . . ... ... . . . . . ........... . ... . . .... . 203 8.9 Additional Problem: Understanding Differential

Electrostatics .. .. . ... . . . . .. . .. . . ... .. .. . . .. . .... . . . ... . ...... . .. 203 Further Reading . . ... . . . . . ... . ... . . . . .. . . ... .. . .. . . ...... . ... . .... . .. .. 204

Chapter 9 Basics of Binding and Allostery .. . . . ... . ........ . . . .. .......... ..... 205

9.1 A Dynamical View of B inding: On- and Off-Rates .... . .. . . . 205 9. 1.1 Time-Dependent Binding: The Basic Differential

Equation . . . ........ .. .... . ... .. . . . . ..................... 207 9.2 Macroscopic Equilibrium and the Binding Constant ... .. .. . . 208

9.2.1 Interpreting Ku . ..... . .. . . .. ...... . ...... .. . ... . ... ... .. 209 9.2.2 The Free Energy of Binding ~G~ind Is Based on a

Reference State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 0 9.2.3 Measuring Ku by a "Generic" Titration

Experiment ... . ... . .. . ... .... . . . ... . ... .. . .. . .. .. .. . . ... 211 9.2.4 Measuring Kct from Isothermal Titration

Calorimetry .... .. .... . . ....... . . ... .. ... .. . .. . ... .. ... . 211 9.2.5 Measuring Ku by Measuring Rates ... . . .. .... . . . . . .. .. 212

9.3 A Structural-Thermodynamic View of Binding . . .. .. . . . .. . . . 212 9.3.1 Pictures of Binding: "Lock and Key" vs.

"Induced Fit" .. . ... . ........... . . . .. . . . . . .... . .. .. . .. . .. 212 9.3.2 Many Factors Affect Binding ..... .. . ... . . . . .. .. . . . . . . 213 9 .3.3 Entropy-Enthalpy Compensation ....... ... . . .. ... . . . . 215

9.4 Understanding Relative Affinities: ~~G and Thermodynamic Cycles . . . . . . . . . ... . . . . . .. .. . . . .. . . .. ..... .... . 216 9.4.1 The Sign of ~~G Has Physical Meaning ....... . .... 216 9.4.2 Competitive Binding Experiments ................... . 218 9.4.3 "Alchemical" Computations of Relative

Affinities . . . . .. . . .. .... .. . .. ...................... . ... .. 218 9.5 Energy Storage in "Fuels" Like ATP . ............ . ... . .. . .... . 220 9.6 Direct Statistical Mechanics Description of

Binding . ... ... .. . ............ .. .. .. ....... . ........... .. .. . .. . . . 221 9.6.1 What Are the Right Partition Functions? .... . ... . .. . . 221

9.7 Allostery and Cooperativity . .... .. ................... . ...... . . 222 9.7.1 Basic Ideas of Allostery .. ... .. . ... . ............. .. . .. . 222 9.7.2 Quantifying Cooperativity with the Hill Constant ... 224

Contents

Chapter 10

9.7.3

9.8

9.9

9.10

10.1

10.2

10.3

10.4

10.5

10.2.3

10.3.2 10.3.3 10.3.4 10.3.5

Contents

)creening . ..................... . . . 197 s of Electrostatic Systems

198 : The Poisson- Boltzmann

······· · ··············· · ············ 200 ion : Debye-Hlickel Theory .... . 200 :ation on DNA ..... ...... ... .... . 202

. ········ · · ···· ··· ·· · ··· · · .. ··· · ···· 202 ······ · · ·· · ························· 203 :tanding Differential

··················· · ········ ·· ·· · ··· 203 ..... . . . .. . ... . .. . .... . . .. .. . . . ..... 204

······· · · · ···· ········ ·········· ···· 205

ng: On- and Off-Rates .. . ..... .. 205 nding: The Basic Differential

. .. ···· ···· ·· · ·· · ·········· ......... 207 1d the Binding Constant ......... 208 ..... . ... . ....................... . .. 209 Binding t..qind Is Based on a ......... . .......................... 210 'Generic" Titration

························· · · ·· ···· · ·· 211 Isothermal Titration

· · ··················· · ······· ·· . . ... 211 easuri ng Rates .................... 212 ic View of Binding . . .. . .. . ...... 212 "Lock and Key" vs.

. ..... . ......... . ................. .. 212 tBinding ..... . ... . ... . .... . ...... 213 :ompensation ..................... 215 inities: 1'3./':!,.G and ................. .. .... ... . .. . ...... 216 . -las Physical Meaning ... . .... .... 216 g Experiments .. ........ . . ........ 218 >Utations of Relative . . . .. . ..................... . ........ 218 ~ike ATP ..... . .............. ... .. . 220 ;; Description of

················· · · ·· ·· ·· ··· · · ······ 221 Partition Functions? . . ........... 221

. ······························· · ···· 222 ;tery ............ .. . . ............... 222 ·ativity with the Hill Constant ... 224

Contents XV

9.7.3 Further Analysis of Allostery: MWC and KNF Models ................................................. 227

9.8 Elementary Enzymatic Catalysis ...... . .. . ..... ... . .. ......... 229 9.8.1 The Steady-State Concept . .... ... ..................... 230 9.8.2 The Michael is-Men ten "Velocity" ......... . .. .. .... . 231

9.9 pH AND pK •...... . ........... . ...... ...... .. . ..... . .. . .... .. .. 231 9.9.1 pH .......... . .. . .. .. .. . ... . .. . . ... ........... . ....... .. . 232 9.9.2 pK ........................ . ... ...... .... . ........ .. ....... 232

9.10 Summary .. . . . ...... .... ... . .. ....... . .. . ...... . .. .. ......... . ... 233 Further Reading .................. . .... . ........... .. .. .. ... .... ....... 233

Chapter 10 Kinetics of Conformational Change and Protein Folding ... .. .. .. 235

I 0.1 Introduction: Basins, Substates, and States ........... .. ..... 235 I 0.1.1 Separating Timescales to Define Kinetic

Models ... . ................. . ........ . ......... . ....... 235 10.2 Kinetic Analysis ofMultistate Systems .... ........ . ......... 238

I 0.2.1 Revisiting the Two-State System .. .. . . ... . .......... 238 I 0.2.2 A Three-State System: One Intermediate .. . . .... ... 242 I 0.2.3 The Effective Rate in the Presence of an

Intermediate .. ....... ............ . .. .. .. ........ . ..... 246 I 0.2.4 The Rate When There Are Parallel Pathways ....... 250 I 0.2.5 Is There Such a Thing as Nonequilibrium

Kinetics? ..................... .... . .................... 251 I 0.2.6 Formalism for Systems Described by Many

States ......................... ....... .................. 252 I 0.3 Conformational and Allosteric Changes in Proteins ........ 252

I 0.3.1 What Is the "Mechanism" of a Conformational Change? .. . . .. . .......... ... .. .. .. .... ... . .. ..... .. . ... 252

10.3.2 Induced and Spontaneous Transitions . ........ .. . ... 253 10.3.3 Allosteric Mechanisms .......... ..... ................ 254 10.3.4 Multiple Pathways ..... . .......... . . ...... . ... .... .... 255 10.3.5 Processivity vs. Stochasticity . . ... .. .. ... ...... .. .. .. 255

I 0.4 Protein Folding .... ....... . . . ............. . ... . ....... . ........ 256 I 0.4.1 Protein Folding in the Cell .... ... . . ............ .. .. .. 257 10.4.2 The Levinthal Paradox .......... . ......... ..... .. .... 258 10.4.3 Just Another Type of Conformational

Change? .................... .... ................. .. .... 258 10.4.4 What Is the Unfolded State? ........ . ............... . 259 I 0.4.5 Multiple Pathways, Multiple Intermediates ... ...... 260 I 0.4.6 Two-State Systems, <I> Values, and Chevron

Plots ................................ .. ....... .......... 262 10.5 Summary ... ..... . . ... . . ... . .. . ........ .... . . .. . ....... . . .. ... . 264 Further Reading .................... . ... . ......... ..... . .. . . .. ....... 264

xv1 Contents

Chapter 11 Ensemble Dynamics: From Trajectories to Diffusion and Kinetics .. . . ... . ....... . . . .. .... ... . .. . . . . .. ........ . .. . ..... . .... 265

11 .1 Introduction: Back to Trajectories and Ensembles . . .. ... ... 265 11 .1.1 Why We Should Care about Trajectory

Ensembles ... .. . . ..... ........... . ... . ... . ... .. . . . .. . . 265 11.1 .2 Anatomy of a Transition Trajectory .... . ... . . . . . . . .. 266 11 . 1.3 Three General Ways to Describe Dynamics . . . . . . .. 267

11 .2 One-Dimensional Ensemble Dynam ics . .. . .. .. . ... . .... . . .. . 271 11.2.1 Derivation of the One-Dimensional Trajectory

Energy : The "Action" .. . . . . . . .... . .. ...... ... .. . .. ... 272 11.2.2 Physical Interpretation of the Action .. . . .. .. . . . . .... 274

11.3 Four Key Trajectory Ensembles . .. . . . ............. . .. .. .. .. .. 275 11 .3.1 Initiali zed Nonequilibrium Trajectory

Ensembles . .. .... .... . . .. . . ............. .. . ... .. . . . ... 275 11 .3.2 Steady-State Nonequilibrium Trajectory

Ensembles . . . ..... ..... ... ... . .. ...... .. .. . ... . .... .. . 275 I 1.3.3 The Equilibrium Trajectory Ensemble ....... . ... . . . 276 11.3.4 Transition Path En~embles .. . . . . .. . . . . . ..... . ... . .... 276

11.4 From Trajectory Ensembles to Observables .............. . .. 278 I 1.4.1 Configuration-Space D istributions from

Trajectory Ensembles .. . . ... .. .. .... . . . . . . ... .... . . .. 279 11.4.2 Finding Intermediates in the Path Ensemble .. . .. . . . 280 11.4.3 The Commitment Probability and a

Transition-State Definition ... .. . .... .. . .... . .... ... . 280 11.4.4 Probability Flow, or Cun·ent. .. .. . . . . .. . . . . . . .. . .. . . . 281 11.4.5 What Is the Reaction Coordinate? . ... . .. . ..... . . . . .. 281 11.4.6 From Trajectory Ensembles to Kinetic Rates ...... . 282 11.4.7 More General Dynamical Observables from

Trajectories .. . .. . .. .. . . . . .. . . . .. . ... .. . .. . ... . .... . ... 283 11.5 Diffusion and Beyond : Evolving Probability

Distributions . . ... . . ....... . .... . . . ... ... . ... . . ..... . .. . ... . .... 283 11.5 .1 Diffusion Derived from Trajectory Probabilities . .. 284 11 .5.2 Diffusion on a Linear Landscape ... . . . . ..... . ... . ... 285 I 1.5.3 The Diffusion (Differential) Equation . .. .. . . . ... . ... 287 11 .5.4 Fokker-Pianck/Smoluchowski Picture for

Arbitrary Landscapes .... . .... .. .... . .. .. .. ... . .. .. .. 289 11.5.5 The Issue of History Dependence . ... . . . ... . ... . .. . . 291

11.6 The Jarzynski Relation and Single-Molecule Phenomena . . ... .. .. . .. . . ... . .. .... . . . ....... . .... . ........ . ... 293 11 .6.1 Revisiting the Second Law of Thermodynamics ... 294

11.7 Summary ........ . ... . . . ... . .. .... . . . ... .. .. . .. .. .. . . .. . . . ... . . 294 Further Reading ... .. . .. . . . . . . . .. . .. ... . . ......... . . . ... . ........ . .. . 295

Conte nts

Chapter 12 A Statistical Pe

12.1 Introduct: 12.1.1 D

12.2 First, Ch< 12.2.1 A 12.2.2 c

12.3 "Basic" S 12.3. 1 T 12.3.2 E

12.4 Metropol Variation: 12.4.1 Si 12.4.2 T 12.4.3 N

12.5 Another I 12.5.1 R 12.5 .2 PI 12.5.3 R 12.5.4 c

12.6 Discrete-: 12.7 How to J1

12.7.1 v 12.7.2 Id 12.7.3 u 12.7.4 0

12.8 Free Enet 12.8.1 pj

c 12.8.2 Tl

A 12.8.3 A

12.9 Path Ens~ 12.9.1 Tl

12. 10 Protein F1 12. 11 Summary Further Readin1

Index .. .. .. .. ...... .......... .

Contents

ctories to Diffusion ···· · ············ ..... .. .. . ... . ... . 265

ctories and Ensembles .... .. .... 265 1re about Trajectory

·· · ··· · ··· · · · ····· ··· ·· ·· .. ... . . . .. 265 >ition Trajectory .. . .... ....... .. . 266 . s to Describe Dynamics ........ 267 e Dynamics . . .......... .......... 271 ne-Dimensional Trajectory n" . ............................... 272 ion of the Action .. . .. ... . . . ... . . 274 1bles ............. . ................ 275 librium Trajectory . .. ...... ... .. . . .. . ................. 275 JUilibrium Trajectory ·········· · ····· . . . ... . . ... ... ..... . 275 ·ajectory Ensemble .... . ........ . 276 iembles . .. . .............. . . . .. .. . . 276 s to Observables ............... .. 278 :e Distributions from les . . ...... . ............ ....... ... . 279 .tes in the Path Ensemble .... . ... 280 )robability and a !tinition ... .. . .............. . ..... 280 or Cun·ent .. . ....... . ... .... ... .... 28 1 Jn Coordinate? .............. . .... 281 nsembles to Kinetic Rates . .. . . . . 282 arnica!" Observables from

······· · ············ · ···· · ·········· 283 Jiving Probabi lity

··· ······ ··· ··· ··············· · ····· 283 from Trajectory Probabil ities ... 284 ~ar Landscape . .. . . .. ... . ... ...... 285 ferential) Equation ..... ...... .... 287 toluchowski Picture for Jes .. . ....... .... .... ... ........... 289 ry Dependence ................... 29 I j Single-Molecule ···· ········· · ...... . ... ..... . . .. . .. 293 Jnd Law of Thermodynamics ... 294

········· · · ····· ·· ······ ··· · ·· ·· · ··· 294 .. . . . ...... . . . . . .. .... ....... ... .... 295

Contents xvii

Chapter 12 A Statistical Perspective on Biomolecular Simulation .... .. .. . .. . 297

12.1 Introduction: Ideas, Not Recipes ............ .............. ... 297 12.1 .1 Do Simulations Matter in Biophysics? ....... .. .... . 297

12.2 First, Choose Your Model: Detailed or Simplified ......... . 298 12.2.1 Atomistic and "Detailed" Models ...... ... .... . ..... 299 12.2.2 Coarse Graining and Related Ideas .... ... ... ... .. . . . 299

12.3 "Basic" Simulations Emulate Dynamics . . ... . ... . .. .. ... . ... 300 12.3.1 Timescale Problems, Sampling Problems ........... 30 I 12.3.2 Energy Minimization vs. Dynamics/Sampling .. .. . 304

12.4 Metropolis Monte Carlo: A Basic Method and Variations ... .. ... ................ . ... . .. ... .... . ........... .... 305 12.4.1 Simple Monte Carlo Can Be Quasi-Dynamic . .... .. 305 12.4.2 The General Metropolis-Hastings Algorithm ...... 306 12.4.3 MC Variations: Replica Exchange and Beyond .... 307

12.5 Another Basic Method: Reweighting and Its Variations .. . . 309 12.5.1 Reweighting and Annealing .............. .. ......... 310 12.5.2 Polymer-Growth Ideas ...... .. ...... .. .. .. ........ ... 311 12.5 .3 Removing Weights by "Resampling" Methods ..... 312 12.5.4 Correlations Can Arise Even without Dynamics . . . 313

12.6 Discrete-State Simulations .......... .. ............ .. ......... 313 12.7 How to Judge Equilibrium Simulation Quality ... . . . .. . .... . 313

12.7.1 Visiting All Important States .... .................... 314 12.7.2 Ideal Sampling as a Key Conceptual Reference .... 314 12.7.3 Uncertainty in Observables and Averages .. .... .... 314 12.7.4 Overall Sampling Quality . . ..... ... . .. ...... . ... . ... . 315

12.8 Free Energy and PMF Calculations . ... .. . . .... .. .. .......... 316 12.8. 1 PMF and Configurational Free Energy

Calculations .... .. .................. .. .... .. .... . ... . . 317 12.8.2 Thermodynamic Free Energy Differences Include

All Space ........... .. .......... . ... ................. .. 318 12.8.3 Approximate Methods for Drug Design ............. 320

12.9 Path Ensembles: Sampling Trajectories ...... .. .. .. . . .. .. ... 321 12.9.1 Three Strategies for Sampling Paths . ............... 321

12.10 Protein Folding: Dynamics and Structure Prediction . ...... 322 12.11 Sum mary ... .... ......... . .... .... . .... .... ... . .. . . . . . ... . ..... 323 Further Reading ........ ......... . .... . .. . .. .. ... ... .. .. .... ... . ..... 323

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