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

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Table of Contents
Preface
Acknowledgments
Author
Section I: Probability Theory
	1: Probability and Its Applications
		1.1 Introduction
		1.2 Experimental Probability
		1.3 The Sample Space Is Related to the Experiment
		1.4 Elementary Probability Space
		1.5 Basic Combinatorics
			1.5.1 Permutations
			1.5.2 Combinations
		1.6 Product Probability Spaces
			1.6.1 The Binomial Distribution
			1.6.2 Poisson Theorem
		1.7 Dependent and Independent Events
			1.7.1 Bayes Formula
		1.8 Discrete Probability—Summary
		1.9 One-Dimensional Discrete Random Variables
			1.9.1 The Cumulative Distribution Function
			1.9.2 The Random Variable of the Poisson Distribution
		1.10 Continuous Random Variables
			1.10.1 The Normal Random Variable
			1.10.2 The Uniform Random Variable
		1.11 The Expectation Value
			1.11.1 Examples
		1.12 The Variance
			1.12.1 The Variance of the Poisson Distribution
			1.12.2 The Variance of the Normal Distribution
		1.13 Independent and Uncorrelated Random Variables
			1.13.1 Correlation
		1.14 The Arithmetic Average
		1.15 The Central Limit Theorem
		1.16 Sampling
		1.17 Stochastic Processes—Markov Chains
			1.17.1 The Stationary Probabilities
		1.18 The Ergodic Theorem
		1.19 Autocorrelation Functions
			1.19.1 Stationary Stochastic Processes
		Homework for Students
		A Comment about Notations
		References
Section II: Equilibrium Thermodynamics and Statistical Mechanics
	2: Classical Thermodynamics
		2.1 Introduction
		2.2 Macroscopic Mechanical Systems versus Thermodynamic Systems
		2.3 Equilibrium and Reversible Transformations
		2.4 Ideal Gas Mechanical Work and Reversibility
		2.5 The First Law of Thermodynamics
		2.6 Joule’s Experiment
		2.7 Entropy
		2.8 The Second Law of Thermodynamics
			2.8.1 Maximal Entropy in an Isolated System
			2.8.2 Spontaneous Expansion of an Ideal Gas and Probability
			2.8.3 Reversible and Irreversible Processes Including Work
		2.9 The Third Law of Thermodynamics
		2.10 Thermodynamic Potentials
			2.10.1 The Gibbs Relation
			2.10.2 The Entropy as the Main Potential
			2.10.3 The Enthalpy
			2.10.4 The Helmholtz Free Energy
			2.10.5 The Gibbs Free Energy
			2.10.6 The Free Energy, , H.(T,µ)
		2.11 Maximal Work in Isothermal and Isobaric Transformations
		2.12 Euler’s Theorem and Additional Relations for the Free Energies
			2.12.1 Gibbs-Duhem Equation
		2.13 Summary
		Homework for Students
		References
		Further Reading
	3: From Thermodynamics to Statistical Mechanics
		3.1 Phase Space as a Probability Space
		3.2 Derivation of the Boltzmann Probability
		3.3 Statistical Mechanics Averages
			3.3.1 The Average Energy
			3.3.2 The Average Entropy
			3.3.3 The Helmholtz Free Energy
		3.4 Various Approaches for Calculating Thermodynamic Parameters
			3.4.1 Thermodynamic Approach
			3.4.2 Probabilistic Approach
		3.5 The Helmholtz Free Energy of a Simple Fluid
		Reference
		Further Reading
	4: Ideal Gas and the Harmonic Oscillator
		4.1 From a Free Particle in a Box to an Ideal Gas
		4.2 Properties of an Ideal Gas by the Thermodynamic Approach
		4.3 The chemical potential of an Ideal Gas
		4.4 Treating an Ideal Gas by the Probability Approach
		4.5 The Macroscopic Harmonic Oscillator
		4.6 The Microscopic Oscillator
			4.6.1 Partition Function and Thermodynamic Properties
		4.7 The Quantum Mechanical Oscillator
		4.8 Entropy and Information in Statistical Mechanics
		4.9 The Configurational Partition Function
		Homework for Students
		References
		Further Reading
	5: Fluctuations and the Most Probable Energy
		5.1 The Variances of the Energy and the Free Energy
		5.2 The Most Contributing Energy E*
		5.3 Solving Problems in Statistical Mechanics
			5.3.1 The Thermodynamic Approach
			5.3.2 The Probabilistic Approach
			5.3.3 Calculating the Most Probable Energy Term
			5.3.4 The Change of Energy and Entropy with Temperature
		References
	6: Various Ensembles
		6.1 The Microcanonical (petit) Ensemble
		6.2 The Canonical (NVT) Ensemble
		6.3 The Gibbs (NpT) Ensemble
		6.4 The Grand Canonical (µVT) Ensemble
		6.5 Averages and Variances in Different Ensembles
			6.5.1 A Canonical Ensemble Solution (Maximal Term Method)
			6.5.2 A Grand-Canonical Ensemble Solution
			6.5.3 Fluctuations in Different Ensembles
		References
		Further Reading
	7: Phase Transitions
		7.1 Finite Systems versus the Thermodynamic Limit
		7.2 First-Order Phase Transitions
		7.3 Second-Order Phase Transitions
		References
	8: Ideal Polymer Chains
		8.1 Models of Macromolecules
		8.2 Statistical Mechanics of an Ideal Chain
			8.2.1 Partition Function and Thermodynamic Averages
		8.3 Entropic Forces in an One-Dimensional Ideal Chain
		8.4 The Radius of Gyration
		8.5 The Critical Exponent ν
		8.6 Distribution of the End-to-End Distance
			8.6.1 Entropic Forces Derived from the Gaussian Distribution
		8.7 The Distribution of the End-to-End Distance Obtained from the Central Limit Theorem
		8.8 Ideal Chains and the Random Walk
		8.9 Ideal Chain as a Model of Reality
		References
	9: Chains with Excluded Volume
		9.1 The Shape Exponent ν for Self-avoiding Walks
		9.2 The Partition Function
		9.3 Polymer Chain as a Critical System
		9.4 Distribution of the End-to-End Distance
		9.5 The Effect of Solvent and Temperature on the Chain Size
			9.5.1 θ Chains in d = 3
			9.5.2 θ Chains in d = 2
			9.5.3 The Crossover Behavior Around
			9.5.4 The Blob Picture
		9.6 Summary
		References
Section III: Topics in Non-Equilibrium Thermodynamics and Statistical Mechanics
	10: Basic Simulation Techniques: Metropolis Monte Carlo and Molecular Dynamics
		10.1 Introduction
		10.2 Sampling the Energy and Entropy and New Notations
		10.3 More About Importance Sampling
		10.4 The Metropolis Monte Carlo Method
			10.4.1 Symmetric and Asymmetric MC Procedures
			10.4.2 A Grand-Canonical MC Procedure
		10.5 Efficiency of Metropolis MC
		10.6 Molecular Dynamics in the Microcanonical Ensemble
		10.7 MD Simulations in the Canonical Ensemble
		10.8 Dynamic MD Calculations
		10.9 Efficiency of MD
			10.9.1 Periodic Boundary Conditions and Ewald Sums
			10.9.2 A Comment About MD Simulations and Entropy
		References
	11: Non-Equilibrium Thermodynamics—Onsager Theory
		11.1 Introduction
		11.2 The Local-Equilibrium Hypothesis
		11.3 Entropy Production Due to Heat Flow in a Closed System
		11.4 Entropy Production in an Isolated System
		11.5 Extra Hypothesis: A Linear Relation Between Rates and Affinities
			11.5.1 Entropy of an Ideal Linear Chain Close to Equilibrium
		11.6 Fourier’s Law—A Continuum Example of Linearity
		11.7 Statistical Mechanics Picture of Irreversibility
		11.8 Time Reversal, Microscopic Reversibility, and the Principle of Detailed Balance
		11.9 Onsager’s Reciprocal Relations
		11.10 Applications
		11.11 Steady States and the Principle of Minimum Entropy Production
		11.12 Summary
		References
	12: Non-equilibrium Statistical Mechanics
		12.1 Fick’s Laws for Diffusion
			12.1.1 First Fick’s Law
			12.1.2 Calculation of the Flux from Thermodynamic Considerations
			12.1.3 The Continuity Equation
			12.1.4 Second Fick’s Law—The Diffusion Equation
			12.1.5 Diffusion of Particles Through a Membrane
			12.1.6 Self-Diffusion
		12.2 Brownian Motion: Einstein’s Derivation of the Diffusion Equation
		12.3 Langevin Equation
			12.3.1 The Average Velocity and the Fluctuation-Dissipation Theorem
			12.3.2 Correlation Functions
			12.3.3 The Displacement of a Langevin Particle
			12.3.4 The Probability Distributions of the Velocity and the Displacement
			12.3.5 Langevin Equation with a Charge in an Electric Field
			12.3.6 Langevin Equation with an External Force—The Strong Damping Velocity
		12.4 Stochastic Dynamics Simulations
			12.4.1 Generating Numbers from a Gaussian Distribution by CLT
			12.4.2 Stochastic Dynamics versus Molecular Dynamics
		12.5 The Fokker-Planck Equation
		12.6 Smoluchowski Equation
		12.7 The Fokker-Planck Equation for a Full Langevin Equation with a Force
		12.8 Summary of Pairs of Equations
		References
	13: The Master Equation
		13.1 Master Equation in a Microcanonical System
		13.2 Master Equation in the Canonical Ensemble
		13.3 An Example from Magnetic Resonance
			13.3.1 Relaxation Processes Under Various Conditions
			13.3.2 Steady State and the Rate of Entropy Production
		13.4 The Principle of Minimum Entropy Production—Statistical Mechanics Example
		References
Section IV: Advanced Simulation Methods: Polymers and Biological Macromolecules
	14: Growth Simulation Methods for Polymers
		14.1 Simple Sampling of Ideal Chains
		14.2 Simple Sampling of SAWs
		14.3 The Enrichment Method
		14.4 The Rosenbluth and Rosenbluth Method
		14.5 The Scanning Method
			14.5.1 The Complete Scanning Method
			14.5.2 The Partial Scanning Method
			14.5.3 Treating SAWs with Finite Interactions
			14.5.4 A Lower Bound for the Entropy
			14.5.5 A Mean-Field Parameter
			14.5.6 Eliminating the Bias by Schmidt’s Procedure
			14.5.7 Correlations in the Accepted Sample
			14.5.8 Criteria for Efficiency
			14.5.9 Locating Transition Temperatures
			14.5.10 The Scanning Method versus Other Techniques
			14.5.11 The Stochastic Double Scanning Method
			14.5.12 Future Scanning by Monte Carlo
			14.5.13 The Scanning Method for the Ising Model and Bulk Systems
		14.6 The Dimerization Method
		References
	15: The Pivot Algorithm and Hybrid Techniques
		15.1 The Pivot Algorithm—Historical Notes
		15.2 Ergodicity and Efficiency
		15.3 Applicability
		15.4 Hybrid and Grand-Canonical Simulation Methods
		15.5 Concluding Remarks
		References
	16: Models of Proteins
		16.1 Biological Macromolecules versus Polymers
		16.2 Definition of a Protein Chain
		16.3 The Force Field of a Protein
		16.4 Implicit Solvation Models
		16.5 A Protein in an Explicit Solvent
		16.6 Potential Energy Surface of a Protein
		16.7 The Problem of Protein Folding
		16.8 Methods for a Conformational Search
			16.8.1 Local Minimization—The Steepest Descents Method
			16.8.2 Monte Carlo Minimization
			16.8.3 Simulated Annealing
		16.9 Monte Carlo and Molecular Dynamics Applied to Proteins
		16.10 Microstates and Intermediate Flexibility
			16.10.1 On the Practical Definition of a Microstate
		References
	17: Calculation of the Entropy and the Free Energy by Thermodynamic Integration
		17.1 “Calorimetric” Thermodynamic Integration
		17.2 The Free Energy Perturbation Formula
		17.3 The Thermodynamic Integration Formula of Kirkwood
		17.4 Applications
			17.4.1 Absolute Entropy of a SAW Integrated from an Ideal Chain Reference State
			17.4.2 Harmonic Reference State of a Peptide
		17.5 Thermodynamic Cycles
			17.5.1 Other Cycles
			17.5.2 Problems of TI and FEP Applied to Proteins
		References
	18: Direct Calculation of the Absolute Entropy and Free Energy
		18.1 Absolute Free Energy from 
		18.2 The Harmonic Approximation
		18.3 The M2 Method
		18.4 The Quasi-Harmonic Approximation
		18.5 The Mutual Information Expansion
		18.6 The Nearest Neighbor Technique
		18.7 The MIE-NN Method
		18.8 Hybrid Approaches
		References
	19: Calculation of the Absolute Entropy from a Single Monte Carlo Sample
		19.1 The Hypothetical Scanning (HS) Method for SAWs
			19.1.1 An Exact HS Method
			19.1.2 Approximate HS Method
		19.2 The HS Monte Carlo (HSMC) Method
		19.3 Upper Bounds and Exact Functionals for the Free Energy
			19.3.1 The Upper Bound FB
			19.3.2 FB Calculated by the Reversed Schmidt Procedure
			19.3.3 A Gaussian Estimation of FB
			19.3.4 Exact Expression for the Free Energy
			19.3.5 The Correlation Between sA and FA
			19.3.6 Entropy Results for SAWs on a Square Lattice
		19.4 HS and HSMC Applied to the Ising Model
		19.5 The HS and HSMC Methods for a Continuum Fluid
			19.5.1 The HS Method
			19.5.2 The HSMC Method
			19.5.3 Results for Argon and Water
				19.5.3.1 Results for Argon
				19.5.3.2 Results for Water
		19.6 HSMD Applied to a Peptide
			19.6.1 Applications
		19.7 The HSMD-TI Method
		19.8 The LS Method
			19.8.1 The LS Method Applied to the Ising Model
			19.8.2 The LS Method Applied to a Peptide
		References
	20: The Potential of Mean Force, Umbrella Sampling, and Related Techniques
		20.1 Umbrella Sampling
		20.2 Bennett’s Acceptance Ratio
		20.3 The Potential of Mean Force
			20.3.1 Applications
		20.4 The Self-Consistent Histogram Method
			20.4.1 Free Energy from a Single Simulation
			20.4.2 Multiple Simulations and The Self-Consistent Procedure
		20.5 The Weighted Histogram Analysis Method
			20.5.1 The Single Histogram Equations
			20.5.2 The WHAM Equations
			20.5.3 Enhancements of WHAM
			20.5.4 The Basic MBAR Equation
			20.5.5 ST-WHAM and UIM
			20.5.6 Summary
		References
	21: Advanced Simulation Methods and Free Energy Techniques
		21.1 Replica-Exchange
			21.1.1 Temperature-Based REM
			21.1.2 Hamiltonian-Dependent Replica Exchange
		21.2 The Multicanonical Method
			21.2.1 Applications
			21.2.2 MUCA-Summary
		21.3 The Method of Wang and Landau
			21.3.1 The Wang and Landau Method-Applications
		21.4 The Method of Expanded Ensembles
			21.4.1 The Method of Expanded Ensembles-Applications
		21.5 The Adaptive Integration Method
		21.6 Methods Based on Jarzynski’s Identity
			21.6.1 Jarzynski’s Identity versus Other Methods for Calculating ΔF
		21.7 Summary
		References
	22: Simulation of the Chemical Potential
		22.1 The Widom Insertion Method
		22.2 The Deletion Procedure
		22.3 Personage’s Method for Treating Deletion
		22.4 Introduction of a Hard Sphere
		22.5 The Ideal Gas Gauge Method
		22.6 Calculation of the Chemical Potential of a Polymer by the Scanning Method
		22.7 The Incremental Chemical Potential Method for Polymers
		22.8 Calculation of µ by Thermodynamic Integration
		References
	23: The Absolute Free Energy of Binding
		23.1 The Law of Mass Action
		23.2 Chemical Potential, Fugacity, and Activity of an Ideal Gas
			23.2.1 Thermodynamics
			23.2.2 Canonical Ensemble
			23.2.3 NpT Ensemble
		23.3 Chemical Potential in Ideal Solutions: Raoult’s and Henry’s Laws
			23.3.1 Raoult’s Law
			23.3.2 Henry’s Law
		23.4 Chemical Potential in Non-ideal Solutions
			23.4.1 Solvent
			23.4.2 Solute
		23.5 Thermodynamic Treatment of Chemical Equilibrium
		23.6 Chemical Equilibrium in Ideal Gas Mixtures: Statistical Mechanics
		23.7 Pressure-Dependent Equilibrium Constant of Ideal Gas Mixtures
		23.8 Protein-Ligand Binding
			23.8.1 Standard Methods for Calculating .A0
			23.8.2 Calculating .A0 by HSMD-TI
			23.8.3 HSMD-TI Applied to the FKBP12-FK506 Complex: Equilibration
			23.8.4 The Internal and External Entropies
			23.8.5 TI Results for FKBP12-FK506
			23.8.6 .A0 Results for FKBP12-FK506
		23.9 Summary
		References
Appendix
Index