Molecular Dynamics Simulations in Statistical Physics: Theory and Applications

Preface
Acknowledgments
Contents
About the Author
1 Principles of Classical Mechanics
	1.1 Mechanics of the System of Particles
	1.2 Generalized Coordinates for Unconstrained Systems
	1.3 Phase Space
	1.4 Dynamical Systems
	1.5 Lagrange Equations
		1.5.1 Lagrangian for Unconstrained Systems
		1.5.2 Lagrangian for Constrained Systems
	1.6 Hamilton's Principle
	1.7 Hamiltonian Method
		1.7.1 Hamilton Equations
	1.8 Time Averages and Virial Theorem
	1.9 Canonical Transformations
		1.9.1 The Symplectic Approach to CanonicalTransformations
	1.10 Poisson Brackets
		1.10.1 Equations of Motion in the Poisson BracketFormulation
		1.10.2 Conservation Laws in the Poisson BracketFormulation
		1.10.3 Infinitesimal Canonical Transformations
		1.10.4 Liouville's Theorem
	1.11 Invariant Measure
2 Principles of Classical Thermodynamics
	2.1 Introduction
	2.2 Microscopic and Macroscopic Views
	2.3 Some Definitions of Thermodynamics
	2.4 The First Law of Thermodynamics and Equilibrium
	2.5 The Second Law of Thermodynamics
	2.6 Thermal Equilibrium and Temperature
	2.7 Legendre Transformation
	2.8 Maxwell Relations
	2.9 Extensive Functions
	2.10 Intensive Functions
	2.11 Stability of Thermodynamic Systems
3 Principles of Statistical Mechanics
	3.1 Systems
	3.2 Ensembles
	3.3 Microcanonical Partition Function
	3.4 Canonical Partition Function
	3.5 Entropy, Free Energy and Internal Energy
	3.6 Thermodynamic Potentials
	3.7 Generalized Ensembles
	3.8 Isothermal-Isobaric Ensemble
	3.9 Grand Canonical Ensemble
	3.10 Grand Isothermal-Isobaric Ensemble
	3.11 Fluctuations
		3.11.1 Canonical Ensemble
		3.11.2 Generalized Ensemble
		3.11.3 Isothermal-Isobaric Ensemble
		3.11.4 Grand Canonical Ensemble
4 Thermodynamics of Biological Phenomena
	4.1 Introduction
	4.2 Stability of Macromolecular Conformations
	4.3 Gibbs Free Energy of the Transition
	4.4 The Binding Free Energy
	4.5 Theoretical Models
	4.6 Energy Function
5 Free Energy Calculation Methods Used in Computer Simulations
	5.1 The Free Energy Calculations
		5.1.1 Thermodynamic Perturbation Method
		5.1.2 Thermodynamic Integration Method
		5.1.3 The Slow Growth Method
	5.2 Free Energy Decomposition
	5.3 Implicit Models for Free Energy Calculations
		5.3.1 Empirical Solvation Models
			5.3.1.1 Implicit Nonpolar Solvation Free Energy
			5.3.1.2 Implicit Dielectric Surface Boundary Free Energy Term
			5.3.1.3 Implicit van der Waals Free Energy Term
			5.3.1.4 The Poisson-Boltzmann Model
			5.3.1.5 Generalized Born Model
6 Molecular Dynamics Methods in Simulations of Macromolecules
	6.1 Introduction
	6.2 Equations of Motion
		6.2.1 Microcanonical Ensemble
		6.2.2 Canonical Ensemble
			6.2.2.1 Isokinetic Dynamics Method
			6.2.2.2 Berendsen Thermostat
			6.2.2.3 Nosé-Hoover Thermostat
			6.2.2.4 Nosé-Hoover Chain of Thermostats
			6.2.2.5 Nosé-Poincaré Thermostats
			6.2.2.6 Nosé-Poincaré Chain of Thermostats
			6.2.2.7 Gaussian Thermostat
			6.2.2.8 Langevin Dynamics
			6.2.2.9 Remarks
		6.2.3 Isothermal-Isobaric Ensemble
			6.2.3.1 Nosé-Andersen Method
			6.2.3.2 Nosé-Andersen Chain of Thermostats Method
			6.2.3.3 Nosé-Poincaré-Andersen Method
			6.2.3.4 Nosé-Poincaré-Andersen Chain of Thermostats
		6.2.4 Grand Canonical Ensemble
			6.2.4.1 Nosé-Hoover Thermostats for Grand Canonical Ensemble
			6.2.4.2 Nosé-Hoover Chain of Thermostats for Grand Canonical Ensemble
		6.2.5 Grand Isothermal-Isobaric Ensemble
			6.2.5.1 Nosé-Andersen Thermostats for Grand Isothermal-Isobaric Ensemble
			6.2.5.2 Nosé-Andersen Chain of Thermostats for Grand Isothermal-Isobaric Ensemble
		6.2.6 Generalized Ensemble
			6.2.6.1 Separable Distribution Functions
7 Molecular Mechanics
	7.1 Simple Molecular Mechanics Force Field
	7.2 Features of Molecular Mechanics Force Fields
	7.3 Molecular Mechanics Force Field Parameters Calibration
		7.3.1 Parameterization by Trial and Error Assessment
		7.3.2 Least-Square Fitting
		7.3.3 Machine-Like Learning Approach
			7.3.3.1 Atomic Feature Descriptors
			7.3.3.2 Environment Descriptors
			7.3.3.3 Machine Learning Approaches
		7.3.4 Perspectives of Automated Force FieldParameterization
			7.3.4.1 Topological Data Analysis
			7.3.4.2 Hardware Versus Software Machine Learning Approach
	7.4 Classical Force Fields Interaction Types
		7.4.1 Bond Stretching
		7.4.2 Angle Bending
		7.4.3 Torsional Angle
		7.4.4 The van der Waals Potential
		7.4.5 Electrostatic Potential
8 Slow Collective Variables of Macromolecular Systems
	8.1 Normal Modes
		8.1.1 General Theory of Normal Modes
		8.1.2 Dynamical Behavior of System
		8.1.3 Time Averaged Properties
		8.1.4 Thermal Amplitudes
	8.2 Principal Components Analysis
		8.2.1 Diffusive Motion in a Protein
		8.2.2 Stability of PCA
	8.3 Equations of Motion of Collective Coordinates
		8.3.1 Bath of Harmonic Oscillators
		8.3.2 Dynamic Friction Coefficient
	8.4 Analyzing Slow Collective Variables Using Machine Learning Approach
		8.4.1 Bootstrapping Swarm Artificial Neural Network
		8.4.2 Related Work
		8.4.3 Time-Lagged Auto-encoder Approach
9 Information Theory and Statistical Mechanics
	9.1 Random Walks in Macromolecular Systems
	9.2 Optimization of Embedding Parameters
	9.3 Shannon and Relative Entropy
	9.4 Relationship with the Second Law of Thermodynamics
	9.5 Transfer Entropy
	9.6 Relationship Between Transfer Entropy and Thermodynamics
	9.7 A Statistical Mechanics Point of View of Transfer Entropy
	9.8 Mutual Information
	9.9 Symbolic Analysis
10 Practical Aspects of Molecular Dynamics Simulations
	10.1 Designing Constraints for Molecular Dynamics Simulations
	10.2 Initial Configuration
	10.3 Periodic Boundary Conditions
	10.4 Potential Cutoffs and the Minimum Image Convention
	10.5 Neighbor Lists
	10.6 Cell Lists
	10.7 Long-Range Forces
		10.7.1 Ewald Summation Method
		10.7.2 A Physical Perspective of Ewald Method
		10.7.3 Choice of Ewald Summation Parameters
		10.7.4 Improved Ewald Summation
		10.7.5 Particle-Particle Particle-Mesh Ewald
		10.7.6 Particle-Mesh Ewald
		10.7.7 Multipole Ewald Summation Methods
		10.7.8 Reaction Field Method
	10.8 Equilibration
11 Symplectic and Time Reversible Integrator
	11.1 Flow-Maps
		11.1.1 Symplectic Maps
		11.1.2 Symplecticness of Hamiltonian Flow-Maps
		11.1.3 Phase-Space Area Preservation for d = 1
		11.1.4 Time-Reversal Symmetry
	11.2 Symplectic and Hamiltonian Splitting Methods
		11.2.1 Hamiltonian Splitting Methods
	11.3 Liouville Formalism and Trotter Formula
	11.4 Microcanonical Ensemble
	11.5 Canonical Ensemble
	11.6 Isothermal-Isobaric Ensemble
	11.7 Multiple Time Step Integrator
12 Generalized Ensemble Molecular Dynamics Methods
	12.1 Multicanonical Sampling Method
	12.2 Tsallis Statistics Molecular Dynamics Method
	12.3 Swarm Particle-Like Molecular Dynamics Method
	12.4 Replica Exchange Method
	12.5 Swarm Particle-Like Replica Exchange Method
	12.6 Weighted Histogram Analysis Method
References
Index