Computer simulation has become the main engine of development in statistical mechanics. In structural biology, computer simulation constitutes the main theoretical tool for structure determination of proteins and for calculation of the free energy of binding, which are important in drug design. Entropy and Free Energy in Structural Biology leads the reader to the simulation technology in a systematic way. The book, which is structured as a course, consists of four parts: Part I is a short course on probability theory emphasizing (1) the distinction between the notions of experimental probability, probability space, and the experimental probability on a computer, and (2) elaborating on the mathematical structure of product spaces. These concepts are essential for solving probability problems and devising simulation methods, in particular for calculating the entropy. Part II starts with a short review of classical thermodynamics from which a non-traditional derivation of statistical mechanics is devised. Theoretical aspects of statistical mechanics are reviewed extensively. Part III covers several topics in non-equilibrium thermodynamics and statistical mechanics close to equilibrium, such as Onsager relations, the two Fick's laws, and the Langevin and master equations. The Monte Carlo and molecular dynamics procedures are discussed as well. Part IV presents advanced simulation methods for polymers and protein systems, including techniques for conformational search and for calculating the potential of mean force and the chemical potential. Thermodynamic integration, methods for calculating the absolute entropy, and methodologies for calculating the absolute free energy of binding are evaluated. Enhanced by a number of solved problems and examples, this volume will be a valuable resource to advanced undergraduate and graduate students in chemistry, chemical engineering, biochemistry biophysics, pharmacology, and computational biology.

Table of Contents:
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