Entropy and Free Energy in Structural Biology
portes grátis
Entropy and Free Energy in Structural Biology
From Thermodynamics to Statistical Mechanics to Computer Simulation
Meirovitch, Hagai
Taylor & Francis Ltd
04/2022
374
Mole
Inglês
9780367427450
15 a 20 dias
730
Descrição não disponível.
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,?) ........................................................................................ 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 (?VT) 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 ? ...................................................................................................... 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 ? 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 ? Chains in d = 3 ...................................................................................................116
9.5.2 ? Chains in d = 2 ...................................................................................................116
9.5.3 The Crossover Behavior Around ?.........................................................................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 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 ?A 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 ?F ........................... 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 ? 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 ?A0 .................................................................352
23.8.2 Calculating ?A0 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 ?A0 Results for FKBP12-FK506 .......................................................................... 360
23.9 Summary ............................................................................................................................. 362
References ...................................................................................................................................... 362
Appendix ............................................................................................................................................... 367
Index ...................................................................................................................................................... 369
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,?) ........................................................................................ 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 (?VT) 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 ? ...................................................................................................... 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 ? 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 ? Chains in d = 3 ...................................................................................................116
9.5.2 ? Chains in d = 2 ...................................................................................................116
9.5.3 The Crossover Behavior Around ?.........................................................................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 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 ?A 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 ?F ........................... 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 ? 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 ?A0 .................................................................352
23.8.2 Calculating ?A0 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 ?A0 Results for FKBP12-FK506 .......................................................................... 360
23.9 Summary ............................................................................................................................. 362
References ...................................................................................................................................... 362
Appendix ............................................................................................................................................... 367
Index ...................................................................................................................................................... 369
Este título pertence ao(s) assunto(s) indicados(s). Para ver outros títulos clique no assunto desejado.
Structural Biology;Partition Function;MC;Ising Model;Free Energy;FEP;NVT Ensemble;Ti;Helmholtz Free Energy;Absolute Entropy;Free Energy Difference;Ideal Chain;Local Equilibrium Hypothesis;Canonical Ensemble;Absolute Free Energy;Ideal Gas;Boltzmann Probability;Heat Bath;Detailed Balance Condition;Grand Canonical Ensemble;MD Simulation;Entropy Production;Intermediate Flexibility;Entropy Functional;Minimum Entropy Production
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,?) ........................................................................................ 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 (?VT) 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 ? ...................................................................................................... 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 ? 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 ? Chains in d = 3 ...................................................................................................116
9.5.2 ? Chains in d = 2 ...................................................................................................116
9.5.3 The Crossover Behavior Around ?.........................................................................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 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 ?A 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 ?F ........................... 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 ? 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 ?A0 .................................................................352
23.8.2 Calculating ?A0 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 ?A0 Results for FKBP12-FK506 .......................................................................... 360
23.9 Summary ............................................................................................................................. 362
References ...................................................................................................................................... 362
Appendix ............................................................................................................................................... 367
Index ...................................................................................................................................................... 369
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,?) ........................................................................................ 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 (?VT) 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 ? ...................................................................................................... 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 ? 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 ? Chains in d = 3 ...................................................................................................116
9.5.2 ? Chains in d = 2 ...................................................................................................116
9.5.3 The Crossover Behavior Around ?.........................................................................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 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 ?A 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 ?F ........................... 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 ? 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 ?A0 .................................................................352
23.8.2 Calculating ?A0 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 ?A0 Results for FKBP12-FK506 .......................................................................... 360
23.9 Summary ............................................................................................................................. 362
References ...................................................................................................................................... 362
Appendix ............................................................................................................................................... 367
Index ...................................................................................................................................................... 369
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Structural Biology;Partition Function;MC;Ising Model;Free Energy;FEP;NVT Ensemble;Ti;Helmholtz Free Energy;Absolute Entropy;Free Energy Difference;Ideal Chain;Local Equilibrium Hypothesis;Canonical Ensemble;Absolute Free Energy;Ideal Gas;Boltzmann Probability;Heat Bath;Detailed Balance Condition;Grand Canonical Ensemble;MD Simulation;Entropy Production;Intermediate Flexibility;Entropy Functional;Minimum Entropy Production
