Entropy and Free Energy in Structural Biology

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

Contents

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

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

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

Section I Probability Theory

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Section II Equilibrium Thermodynamics and Statistical Mechanics

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Section III Topics in Non-Equilibrium Thermodynamics

and Statistical Mechanics

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Section IV Advanced Simulation Methods: Polymers

and Biological Macromolecules

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

18.1 Absolute Free Energy from E/kBT]> ...................................................................... 243

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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