1st Edition

Entropy and Free Energy in Structural Biology From Thermodynamics to Statistical Mechanics to Computer Simulation

By Hagai Meirovitch Copyright 2020
    396 Pages
    by CRC Press

    396 Pages
    by CRC Press

    Computer simulation has become the main engine of development in statistical mechanics. In structural biology, computer simulation constitutes the main theoretical tool for structure determination of proteins and for calculation of the free energy of binding, which are important in drug design. Entropy and Free Energy in Structural Biology leads the reader to the simulation technology in a systematic way. The book, which is structured as a course, consists of four parts:

    Part I is a short course on probability theory emphasizing (1) the distinction between the notions of experimental probability, probability space, and the experimental probability on a computer, and (2) elaborating on the mathematical structure of product spaces. These concepts are essential for solving probability problems and devising simulation methods, in particular for calculating the entropy.

    Part II starts with a short review of classical thermodynamics from which a non-traditional derivation of statistical mechanics is devised. Theoretical aspects of statistical mechanics are reviewed extensively.

    Part III covers several topics in non-equilibrium thermodynamics and statistical mechanics close to equilibrium, such as Onsager relations, the two Fick's laws, and the Langevin and master equations. The Monte Carlo and molecular dynamics procedures are discussed as well.

    Part IV presents advanced simulation methods for polymers and protein systems, including techniques for conformational search and for calculating the potential of mean force and the chemical potential. Thermodynamic integration, methods for calculating the absolute entropy, and methodologies for calculating the absolute free energy of binding are evaluated.

    Enhanced by a number of solved problems and examples, this volume will be a valuable resource to advanced undergraduate and graduate students in chemistry, chemical engineering, biochemistry biophysics, pharmacology, and computational biology.

    Contents

    Preface ..................................................................................................................................................... xv

    Acknowledgments ...................................................................................................................................xix

    Author .....................................................................................................................................................xxi

    Section I Probability Theory

    1. Probability and Its Applications ..................................................................................................... 3

    1.1 Introduction ............................................................................................................................. 3

    1.2 Experimental Probability ........................................................................................................ 3

    1.3 The Sample Space Is Related to the Experiment .................................................................... 4

    1.4 Elementary Probability Space ................................................................................................ 5

    1.5 Basic Combinatorics ............................................................................................................... 6

    1.5.1 Permutations ............................................................................................................. 6

    1.5.2 Combinations ............................................................................................................ 7

    1.6 Product Probability Spaces ..................................................................................................... 9

    1.6.1 The Binomial Distribution .......................................................................................11

    1.6.2 Poisson Theorem ......................................................................................................11

    1.7 Dependent and Independent Events ...................................................................................... 12

    1.7.1 Bayes Formula......................................................................................................... 12

    1.8 Discrete Probability—Summary .......................................................................................... 13

    1.9 One-Dimensional Discrete Random Variables ..................................................................... 13

    1.9.1 The Cumulative Distribution Function ....................................................................14

    1.9.2 The Random Variable of the Poisson Distribution ..................................................14

    1.10 Continuous Random Variables ..............................................................................................14

    1.10.1 The Normal Random Variable ................................................................................ 15

    1.10.2 The Uniform Random Variable .............................................................................. 15

    1.11 The Expectation Value ...........................................................................................................16

    1.11.1 Examples ..................................................................................................................16

    1.12 The Variance ..........................................................................................................................17

    1.12.1 The Variance of the Poisson Distribution ................................................................18

    1.12.2 The Variance of the Normal Distribution ................................................................18

    1.13 Independent and Uncorrelated Random Variables ............................................................... 19

    1.13.1 Correlation .............................................................................................................. 19

    1.14 The Arithmetic Average ....................................................................................................... 20

    1.15 The Central Limit Theorem .................................................................................................. 21

    1.16 Sampling ............................................................................................................................... 23

    1.17 Stochastic Processes—Markov Chains ................................................................................ 23

    1.17.1 The Stationary Probabilities ................................................................................... 25

    1.18 The Ergodic Theorem ........................................................................................................... 26

    1.19 Autocorrelation Functions .................................................................................................... 27

    1.19.1 Stationary Stochastic Processes .............................................................................. 28

    Homework for Students .................................................................................................................... 28

    A Comment about Notations ............................................................................................................ 28

    References ........................................................................................................................................ 29

    Section II Equilibrium Thermodynamics and Statistical Mechanics

    2. Classical Thermodynamics ........................................................................................................... 33

    2.1 Introduction ........................................................................................................................... 33

    2.2 Macroscopic Mechanical Systems versus Thermodynamic Systems .................................. 33

    2.3 Equilibrium and Reversible Transformations ....................................................................... 34

    2.4 Ideal Gas Mechanical Work and Reversibility ..................................................................... 34

    2.5 The First Law of Thermodynamics ...................................................................................... 36

    2.6 Joule’s Experiment ................................................................................................................ 37

    2.7 Entropy .................................................................................................................................. 39

    2.8 The Second Law of Thermodynamics .................................................................................. 40

    2.8.1 Maximal Entropy in an Isolated System..................................................................41

    2.8.2 Spontaneous Expansion of an Ideal Gas and Probability ....................................... 42

    2.8.3 Reversible and Irreversible Processes Including Work ........................................... 42

    2.9 The Third Law of Thermodynamics .................................................................................... 43

    2.10 Thermodynamic Potentials ................................................................................................... 43

    2.10.1 The Gibbs Relation ................................................................................................. 43

    2.10.2 The Entropy as the Main Potential ......................................................................... 44

    2.10.3 The Enthalpy ........................................................................................................... 45

    2.10.4 The Helmholtz Free Energy .................................................................................... 45

    2.10.5 The Gibbs Free Energy ........................................................................................... 45

    2.10.6 The Free Energy, H(T,μ) ........................................................................................ 46

    2.11 Maximal Work in Isothermal and Isobaric Transformations ............................................... 47

    2.12 Euler’s Theorem and Additional Relations for the Free Energies ........................................ 48

    2.12.1 Gibbs-Duhem Equation .......................................................................................... 49

    2.13 Summary ............................................................................................................................... 49

    Homework for Students .................................................................................................................... 49

    References ........................................................................................................................................ 49

    Further Reading ................................................................................................................................ 49

    3. From Thermodynamics to Statistical Mechanics ........................................................................51

    3.1 Phase Space as a Probability Space .......................................................................................51

    3.2 Derivation of the Boltzmann Probability ............................................................................. 52

    3.3 Statistical Mechanics Averages ............................................................................................ 54

    3.3.1 The Average Energy ................................................................................................ 54

    3.3.2 The Average Entropy .............................................................................................. 54

    3.3.3 The Helmholtz Free Energy .................................................................................... 55

    3.4 Various Approaches for Calculating Thermodynamic Parameters ...................................... 55

    3.4.1 Thermodynamic Approach ..................................................................................... 55

    3.4.2 Probabilistic Approach ........................................................................................... 56

    3.5 The Helmholtz Free Energy of a Simple Fluid ..................................................................... 56

    Reference .......................................................................................................................................... 58

    Further Reading ................................................................................................................................ 58

    4. Ideal Gas and the Harmonic Oscillator ....................................................................................... 59

    4.1 From a Free Particle in a Box to an Ideal Gas ...................................................................... 59

    4.2 Properties of an Ideal Gas by the Thermodynamic Approach ............................................. 60

    4.3 The chemical potential of an Ideal Gas ................................................................................ 62

    4.4 Treating an Ideal Gas by the Probability Approach ............................................................. 63

    4.5 The Macroscopic Harmonic Oscillator ................................................................................ 64

    4.6 The Microscopic Oscillator .................................................................................................. 65

    4.6.1 Partition Function and Thermodynamic Properties ............................................... 66

    4.7 The Quantum Mechanical Oscillator ................................................................................... 68

    4.8 Entropy and Information in Statistical Mechanics ............................................................... 71

    4.9 The Configurational Partition Function ................................................................................ 71

    Homework for Students .................................................................................................................... 72

    References ........................................................................................................................................ 72

    Further Reading ................................................................................................................................ 72

    5. Fluctuations and the Most Probable Energy ............................................................................... 73

    5.1 The Variances of the Energy and the Free Energy ............................................................... 73

    5.2 The Most Contributing Energy E* ....................................................................................... 74

    5.3 Solving Problems in Statistical Mechanics .......................................................................... 76

    5.3.1 The Thermodynamic Approach .............................................................................. 77

    5.3.2 The Probabilistic Approach .................................................................................... 78

    5.3.3 Calculating the Most Probable Energy Term .......................................................... 79

    5.3.4 The Change of Energy and Entropy with Temperature .......................................... 80

    References ........................................................................................................................................ 81

    6. Various Ensembles ......................................................................................................................... 83

    6.1 The Microcanonical (petit) Ensemble .................................................................................. 83

    6.2 The Canonical (NVT) Ensemble ........................................................................................... 84

    6.3 The Gibbs (NpT) Ensemble .................................................................................................. 85

    6.4 The Grand Canonical (μVT) Ensemble ................................................................................ 88

    6.5 Averages and Variances in Different Ensembles .................................................................. 90

    6.5.1 A Canonical Ensemble Solution (Maximal Term Method) .................................... 90

    6.5.2 A Grand-Canonical Ensemble Solution .................................................................. 91

    6.5.3 Fluctuations in Different Ensembles....................................................................... 91

    References ........................................................................................................................................ 92

    Further Reading ................................................................................................................................ 92

    7. Phase Transitions ........................................................................................................................... 93

    7.1 Finite Systems versus the Thermodynamic Limit ................................................................ 93

    7.2 First-Order Phase Transitions ............................................................................................... 94

    7.3 Second-Order Phase Transitions ........................................................................................... 95

    References ........................................................................................................................................ 98

    8. Ideal Polymer Chains ..................................................................................................................... 99

    8.1 Models of Macromolecules ................................................................................................... 99

    8.2 Statistical Mechanics of an Ideal Chain ............................................................................... 99

    8.2.1 Partition Function and Thermodynamic Averages ............................................... 100

    8.3 Entropic Forces in an One-Dimensional Ideal Chain..........................................................101

    8.4 The Radius of Gyration ...................................................................................................... 104

    8.5 The Critical Exponent ν ...................................................................................................... 105

    8.6 Distribution of the End-to-End Distance ............................................................................ 106

    8.6.1 Entropic Forces Derived from the Gaussian Distribution .................................... 107

    8.7 The Distribution of the End-to-End Distance Obtained from the Central Limit Theorem .... 108

    8.8 Ideal Chains and the Random Walk ................................................................................... 109

    8.9 Ideal Chain as a Model of Reality .......................................................................................110

    References .......................................................................................................................................110

    9. Chains with Excluded Volume .....................................................................................................111

    9.1 The Shape Exponent ν for Self-avoiding Walks ..................................................................111

    9.2 The Partition Function .........................................................................................................112

    9.3 Polymer Chain as a Critical System ....................................................................................113

    9.4 Distribution of the End-to-End Distance .............................................................................114

    9.5 The Effect of Solvent and Temperature on the Chain Size .................................................115

    9.5.1 θ Chains in d = 3 ...................................................................................................116

    9.5.2 θ Chains in d = 2 ...................................................................................................116

    9.5.3 The Crossover Behavior Around θ.........................................................................117

    9.5.4 The Blob Picture ....................................................................................................118

    9.6 Summary ..............................................................................................................................119

    References .......................................................................................................................................119

    Section III Topics in Non-Equilibrium Thermodynamics

    and Statistical Mechanics

    10. Basic Simulation Techniques: Metropolis Monte Carlo and Molecular Dynamics .............. 123

    10.1 Introduction ......................................................................................................................... 123

    10.2 Sampling the Energy and Entropy and New Notations ...................................................... 124

    10.3 More About Importance Sampling ..................................................................................... 125

    10.4 The Metropolis Monte Carlo Method ................................................................................. 126

    10.4.1 Symmetric and Asymmetric MC Procedures ....................................................... 127

    10.4.2 A Grand-Canonical MC Procedure ...................................................................... 128

    10.5 Efficiency of Metropolis MC .............................................................................................. 129

    10.6 Molecular Dynamics in the Microcanonical Ensemble ......................................................131

    10.7 MD Simulations in the Canonical Ensemble ...................................................................... 134

    10.8 Dynamic MD Calculations ..................................................................................................135

    10.9 Efficiency of MD .................................................................................................................135

    10.9.1 Periodic Boundary Conditions and Ewald Sums .................................................. 136

    10.9.2 A Comment About MD Simulations and Entropy................................................ 136

    References ...................................................................................................................................... 137

    11. Non-Equilibrium Thermodynamics—Onsager Theory .......................................................... 139

    11.1 Introduction ......................................................................................................................... 139

    11.2 The Local-Equilibrium Hypothesis .................................................................................... 139

    11.3 Entropy Production Due to Heat Flow in a Closed System ................................................ 140

    11.4 Entropy Production in an Isolated System...........................................................................141

    11.5 Extra Hypothesis: A Linear Relation Between Rates and Affinities ..................................142

    11.5.1 Entropy of an Ideal Linear Chain Close to Equilibrium .......................................143

    11.6 Fourier’s Law—A Continuum Example of Linearity ......................................................... 144

    11.7 Statistical Mechanics Picture of Irreversibility ...................................................................145

    11.8 Time Reversal, Microscopic Reversibility, and the Principle of Detailed Balance ............147

    11.9 Onsager’s Reciprocal Relations ...........................................................................................149

    11.10 Applications ........................................................................................................................ 150

    11.11 Steady States and the Principle of Minimum Entropy Production .....................................151

    11.12 Summary ..............................................................................................................................152

    References .......................................................................................................................................152

    12. Non-equilibrium Statistical Mechanics ......................................................................................153

    12.1 Fick’s Laws for Diffusion ....................................................................................................153

    12.1.1 First Fick’s Law ......................................................................................................153

    12.1.2 Calculation of the Flux from Thermodynamic Considerations ............................ 154

    12.1.3 The Continuity Equation ........................................................................................155

    12.1.4 Second Fick’s Law—The Diffusion Equation ...................................................... 156

    12.1.5 Diffusion of Particles Through a Membrane ........................................................ 156

    12.1.6 Self-Diffusion ........................................................................................................ 156

    12.2 Brownian Motion: Einstein’s Derivation of the Diffusion Equation .................................. 158

    12.3 Langevin Equation .............................................................................................................. 160

    12.3.1 The Average Velocity and the Fluctuation-Dissipation Theorem .........................162

    12.3.2 Correlation Functions.............................................................................................163

    12.3.3 The Displacement of a Langevin Particle ............................................................. 164

    12.3.4 The Probability Distributions of the Velocity and the Displacement ................... 166

    12.3.5 Langevin Equation with a Charge in an Electric Field ..........................................168

    12.3.6 Langevin Equation with an External Force—The Strong Damping Velocity .......168

    12.4 Stochastic Dynamics Simulations .......................................................................................169

    12.4.1 Generating Numbers from a Gaussian Distribution by CLT .................................170

    12.4.2 Stochastic Dynamics versus Molecular Dynamics................................................171

    12.5 The Fokker-Planck Equation ...............................................................................................171

    12.6 Smoluchowski Equation.......................................................................................................174

    12.7 The Fokker-Planck Equation for a Full Langevin Equation with a Force...........................175

    12.8 Summary of Pairs of Equations ...........................................................................................175

    References .......................................................................................................................................176

    13. The Master Equation ....................................................................................................................177

    13.1 Master Equation in a Microcanonical System .....................................................................177

    13.2 Master Equation in the Canonical Ensemble.......................................................................178

    13.3 An Example from Magnetic Resonance ............................................................................. 180

    13.3.1 Relaxation Processes Under Various Conditions ...................................................181

    13.3.2 Steady State and the Rate of Entropy Production ................................................. 184

    13.4 The Principle of Minimum Entropy Production—Statistical Mechanics Example............185

    References .......................................................................................................................................186

    Section IV Advanced Simulation Methods: Polymers

    and Biological Macromolecules

    14. Growth Simulation Methods for Polymers .................................................................................189

    14.1 Simple Sampling of Ideal Chains ........................................................................................189

    14.2 Simple Sampling of SAWs .................................................................................................. 190

    14.3 The Enrichment Method ..................................................................................................... 192

    14.4 The Rosenbluth and Rosenbluth Method ............................................................................ 193

    14.5 The Scanning Method ......................................................................................................... 195

    14.5.1 The Complete Scanning Method .......................................................................... 195

    14.5.2 The Partial Scanning Method ............................................................................... 196

    14.5.3 Treating SAWs with Finite Interactions ................................................................ 197

    14.5.4 A Lower Bound for the Entropy ........................................................................... 197

    14.5.5 A Mean-Field Parameter ....................................................................................... 198

    14.5.6 Eliminating the Bias by Schmidt’s Procedure ...................................................... 199

    14.5.7 Correlations in the Accepted Sample ................................................................... 200

    14.5.8 Criteria for Efficiency ........................................................................................... 201

    14.5.9 Locating Transition Temperatures ........................................................................ 202

    14.5.10 The Scanning Method versus Other Techniques .................................................. 203

    14.5.11 The Stochastic Double Scanning Method ............................................................ 204

    14.5.12 Future Scanning by Monte Carlo .......................................................................... 204

    14.5.13 The Scanning Method for the Ising Model and Bulk Systems ............................. 205

    14.6 The Dimerization Method .................................................................................................. 206

    References ...................................................................................................................................... 208

    15. The Pivot Algorithm and Hybrid Techniques ............................................................................211

    15.1 The Pivot Algorithm—Historical Notes ..............................................................................211

    15.2 Ergodicity and Efficiency ....................................................................................................211

    15.3 Applicability ........................................................................................................................212

    15.4 Hybrid and Grand-Canonical Simulation Methods .............................................................213

    15.5 Concluding Remarks ............................................................................................................214

    References .......................................................................................................................................214

    16. Models of Proteins .........................................................................................................................217

    16.1 Biological Macromolecules versus Polymers ......................................................................217

    16.2 Definition of a Protein Chain ...............................................................................................217

    16.3 The Force Field of a Protein ................................................................................................218

    16.4 Implicit Solvation Models ....................................................................................................219

    16.5 A Protein in an Explicit Solvent ......................................................................................... 220

    16.6 Potential Energy Surface of a Protein ................................................................................ 221

    16.7 The Problem of Protein Folding ......................................................................................... 222

    16.8 Methods for a Conformational Search ................................................................................ 222

    16.8.1 Local Minimization—The Steepest Descents Method ........................................ 223

    16.8.2 Monte Carlo Minimization ................................................................................... 224

    16.8.3 Simulated Annealing ............................................................................................ 225

    16.9 Monte Carlo and Molecular Dynamics Applied to Proteins .............................................. 225

    16.10 Microstates and Intermediate Flexibility ........................................................................... 226

    16.10.1 On the Practical Definition of a Microstate .......................................................... 227

    References ...................................................................................................................................... 227

    17. Calculation of the Entropy and the Free Energy by Thermodynamic Integration ................231

    17.1 “Calorimetric” Thermodynamic Integration ...................................................................... 232

    17.2 The Free Energy Perturbation Formula .............................................................................. 232

    17.3 The Thermodynamic Integration Formula of Kirkwood ................................................... 234

    17.4 Applications ........................................................................................................................ 235

    17.4.1 Absolute Entropy of a SAW Integrated from an Ideal Chain Reference State ..... 235

    17.4.2 Harmonic Reference State of a Peptide ................................................................ 237

    17.5 Thermodynamic Cycles ...................................................................................................... 237

    17.5.1 Other Cycles .......................................................................................................... 240

    17.5.2 Problems of TI and FEP Applied to Proteins ....................................................... 240

    References ...................................................................................................................................... 241

    18. Direct Calculation of the Absolute Entropy and Free Energy ................................................ 243

    18.1 Absolute Free Energy from <exp[+E/kBT]> ...................................................................... 243

    18.2 The Harmonic Approximation ........................................................................................... 244

    18.3 The M2 Method .................................................................................................................. 245

    18.4 The Quasi-Harmonic Approximation ................................................................................. 246

    18.5 The Mutual Information Expansion ................................................................................... 247

    18.6 The Nearest Neighbor Technique ....................................................................................... 248

    18.7 The MIE-NN Method ......................................................................................................... 249

    18.8 Hybrid Approaches ............................................................................................................. 249

    References ...................................................................................................................................... 249

    19. Calculation of the Absolute Entropy from a Single Monte Carlo Sample...............................251

    19.1 The Hypothetical Scanning (HS) Method for SAWs ...........................................................251

    19.1.1 An Exact HS Method .............................................................................................251

    19.1.2 Approximate HS Method ...................................................................................... 252

    19.2 The HS Monte Carlo (HSMC) Method .............................................................................. 253

    19.3 Upper Bounds and Exact Functionals for the Free Energy ................................................ 255

    19.3.1 The Upper Bound FB ............................................................................................ 255

    19.3.2 FB Calculated by the Reversed Schmidt Procedure ............................................. 256

    19.3.3 A Gaussian Estimation of FB ................................................................................ 257

    19.3.4 Exact Expression for the Free Energy .................................................................. 258

    19.3.5 The Correlation Between σA and FA ..................................................................... 258

    19.3.6 Entropy Results for SAWs on a Square Lattice .................................................... 259

    19.4 HS and HSMC Applied to the Ising Model ........................................................................ 260

    19.5 The HS and HSMC Methods for a Continuum Fluid ..........................................................261

    19.5.1 The HS Method ......................................................................................................261

    19.5.2 The HSMC Method ............................................................................................... 262

    19.5.3 Results for Argon and Water ................................................................................. 264

    19.5.3.1 Results for Argon .................................................................................. 264

    19.5.3.2 Results for Water .................................................................................. 266

    19.6 HSMD Applied to a Peptide ............................................................................................... 266

    19.6.1 Applications .......................................................................................................... 269

    19.7 The HSMD-TI Method ....................................................................................................... 269

    19.8 The LS Method ................................................................................................................... 270

    19.8.1 The LS Method Applied to the Ising Model ......................................................... 270

    19.8.2 The LS Method Applied to a Peptide ................................................................... 272

    References .......................................................................................................................................274

    20. The Potential of Mean Force, Umbrella Sampling, and Related Techniques ........................ 277

    20.1 Umbrella Sampling ............................................................................................................. 277

    20.2 Bennett’s Acceptance Ratio ................................................................................................ 278

    20.3 The Potential of Mean Force .............................................................................................. 281

    20.3.1 Applications .......................................................................................................... 284

    20.4 The Self-Consistent Histogram Method ............................................................................. 285

    20.4.1 Free Energy from a Single Simulation.................................................................. 286

    20.4.2 Multiple Simulations and The Self-Consistent Procedure.................................... 286

    20.5 The Weighted Histogram Analysis Method ....................................................................... 289

    20.5.1 The Single Histogram Equations .......................................................................... 290

    20.5.2 The WHAM Equations ..........................................................................................291

    20.5.3 Enhancements of WHAM .................................................................................... 293

    20.5.4 The Basic MBAR Equation .................................................................................. 295

    20.5.5 ST-WHAM and UIM ............................................................................................ 296

    20.5.6 Summary ............................................................................................................... 296

    References ...................................................................................................................................... 297

    21. Advanced Simulation Methods and Free Energy Techniques ................................................. 301

    21.1 Replica-Exchange ............................................................................................................... 301

    21.1.1 Temperature-Based REM ..................................................................................... 301

    21.1.2 Hamiltonian-Dependent Replica Exchange .......................................................... 305

    21.2 The Multicanonical Method ............................................................................................... 308

    21.2.1 Applications ...........................................................................................................311

    21.2.2 MUCA-Summary ..................................................................................................312

    21.3 The Method of Wang and Landau .......................................................................................312

    21.3.1 The Wang and Landau Method-Applications ........................................................314

    21.4 The Method of Expanded Ensembles ..................................................................................315

    21.4.1 The Method of Expanded Ensembles-Applications ..............................................317

    21.5 The Adaptive Integration Method .......................................................................................317

    21.6 Methods Based on Jarzynski’s Identity ...............................................................................319

    21.6.1 Jarzynski’s Identity versus Other Methods for Calculating ΔF ........................... 323

    21.7 Summary ............................................................................................................................. 324

    References ...................................................................................................................................... 324

    22. Simulation of the Chemical Potential ..........................................................................................331

    22.1 The Widom Insertion Method .............................................................................................331

    22.2 The Deletion Procedure .......................................................................................................332

    22.3 Personage’s Method for Treating Deletion ......................................................................... 334

    22.4 Introduction of a Hard Sphere ............................................................................................ 336

    22.5 The Ideal Gas Gauge Method ............................................................................................. 337

    22.6 Calculation of the Chemical Potential of a Polymer by the Scanning Method .................. 338

    22.7 The Incremental Chemical Potential Method for Polymers ............................................... 340

    22.8 Calculation of μ by Thermodynamic Integration ................................................................341

    References .......................................................................................................................................341

    23. The Absolute Free Energy of Binding ........................................................................................ 343

    23.1 The Law of Mass Action ..................................................................................................... 343

    23.2 Chemical Potential, Fugacity, and Activity of an Ideal Gas............................................... 344

    23.2.1 Thermodynamics .................................................................................................. 344

    23.2.2 Canonical Ensemble.............................................................................................. 344

    23.2.3 NpT Ensemble ....................................................................................................... 345

    23.3 Chemical Potential in Ideal Solutions: Raoult’s and Henry’s Laws ................................... 345

    23.3.1 Raoult’s Law ......................................................................................................... 346

    23.3.2 Henry’s Law .......................................................................................................... 346

    23.4 Chemical Potential in Non-ideal Solutions ......................................................................... 346

    23.4.1 Solvent ................................................................................................................... 346

    23.4.2 Solute ..................................................................................................................... 347

    23.5 Thermodynamic Treatment of Chemical Equilibrium ....................................................... 347

    23.6 Chemical Equilibrium in Ideal Gas Mixtures: Statistical Mechanics ................................ 348

    23.7 Pressure-Dependent Equilibrium Constant of Ideal Gas Mixtures .................................... 349

    23.8 Protein-Ligand Binding ...................................................................................................... 350

    23.8.1 Standard Methods for Calculating ΔA0 .................................................................352

    23.8.2 Calculating ΔA0 by HSMD-TI .............................................................................. 354

    23.8.3 HSMD-TI Applied to the FKBP12-FK506 Complex: Equilibration ................... 356

    23.8.4 The Internal and External Entropies..................................................................... 357

    23.8.5 TI Results for FKBP12-FK506 ............................................................................. 360

    23.8.6 ΔA0 Results for FKBP12-FK506 .......................................................................... 360

    23.9 Summary ............................................................................................................................. 362

    References ...................................................................................................................................... 362

    Appendix ............................................................................................................................................... 367

    Index ...................................................................................................................................................... 369

    Biography

    Hagai Meirovitch is professor Emeritus in the Department of Computational and Systems Biology at the University of Pittsburgh School of Medicine. He earned an MSc degree in nuclear physics from the Hebrew University, a PhD degree in chemical physics from the Weizmann Institute, and conducted postdoctoral training in the laboratory of Professor Harold A. Scheraga at Cornell University. His research focused on developing computer simulation methodologies within the scope of statistical mechanics, as highlighted below. He devised novel methods for extracting the absolute entropy from Monte Carlo samples and techniques for generating polymer chains, which were used to study phase transitions in polymers, magnetic, and lattice gas systems. These methods, together with conformational search techniques for proteins, led to a free energy-based approach for treating molecular flexibility. This approach was used to analyze NMR relaxation data from cyclic peptides and to study structural preferences of surface loops in bound and free enzymes. He developed a new methodology for calculating the free energy of ligand/protein binding, which unlike standard techniques, provides the decrease in the ligand’s entropy upon binding. Dr Meirovitch conducted part of the research depicted above, and other studies, at the Supercomputer Computations Research Institute of the Florida State University, Tallahassee.