molecular dynamics simulations in statistical physics

Hiqmet Kamberaj

MOLECULAR DYNAMICS SIMULATIONS IN STATISTICAL PHYSICS Numerical Recipes for Computer Simulations September 1, 2016

Skopje

To the memory of my Mother and Father

Preface

Computer simulations are used very often to understand and solve practical problems in the area of statistical physics and biophysics. With good knowledge of Classical Mechanics, Thermodynamics and Statistical Physics, you will be able to understand and judge the content of this book. This book aims to be used as a numerical recipe for computer simulations with molecular dynamics techniques in statistical physics, where the main emphasizes are the macromolecular systems. Numerical methods are introduced in the form of computer algorithms which can then be implemented in computers using any desired computer programming language, such as, Fortran 90, C/C++, and others. Some of the numerical methods and their algorithms discussed in this book are implemented in existing computer programming software of macromolecular systems, such as CHARMM. In this book you will find out some advanced concepts of computer simulation techniques used in statistical physics and in particular understanding biological and physical systems. The molecular dynamics approach will be discussed in more details for understanding the use of this method in statistical physics problems. In the first part of the book I will introduce the principles of classical mechanics, thermodynamics and statistical physics, which are necessary concepts to know for understanding the real problems in different fields, such as physics, chemistry and biology, when we use the computer simulations for solving them. In Chapter 4, I introduce the use of statistical thermodynamics in understanding biological phenomena. In particular, in this chapter, I will describe the main theory used for understanding many useful techniques used in computer simulation, such as calculations by means of molecular dynamics simulations of the absolute free energy, solvation free energy, and binding

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free energy, which have a broad area of applications in physics, chemistry and biology. Then, in Chapter 5, I will introduce the molecular dynamics technique, such as, describing the equations of motion in different statistical ensembles of interest in order to mimic real experimental conditions. In the next chapter, I will introduce numerical techniques used to solve molecular dynamics equations of motion using Liouville formalism and Trotter factorization scheme. In additional, stability of numerical schemes will be discussed by applying to real physical systems for which the real solutions are known. In the following chapter, I will describe the use of molecular dynamics method in simulation of macromolecular systems. In particular, I will discuss some practical aspects of molecular dynamics simulations when used in large systems. The book is aimed to graduate students, researcher scientists working in the field of theoretical and computational biophysics, physics and chemistry. In additional, the book can be used from graduate students of other branches, such as, applied mathematics, computer sciences and bioinformatics.

Skopje, August 2016

Hiqmet Kamberaj

Contents

1

Principles of Classical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1 Mechanics of the System of Particles . . . . . . . . . . . . . . . . . . . . . .

1

1.2 Generalized Coordinates for Unconstrained Systems . . . . . . . .

8

1.3 Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5 Lagrange Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.5.1 Lagrangian for Unconstrained Systems . . . . . . . . . . . . . . 14 1.5.2 Lagrangian for Constrained Systems . . . . . . . . . . . . . . . . 17 1.6 Hamilton’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.6.1 Hamiltonian Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.6.2 Hamilton Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.7 Time Averages and Virial Theorem . . . . . . . . . . . . . . . . . . . . . . . 24 1.8 Canonical Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.8.1 The Symplectic Approach to Canonical Transformations 32 1.9 Poisson Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

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1.9.1 Equations of Motion in the Poisson Bracket Formulation 40 1.9.2 Conservation Theorems in the Poisson Bracket Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 1.9.3 Infinitesimal Canonical Transformations . . . . . . . . . . . . . 42 1.9.4 Liouville’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 1.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2

Principles of Classical Thermodynamics . . . . . . . . . . . . . . . . . . 51 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.2 Microscopic and Macroscopic Views . . . . . . . . . . . . . . . . . . . . . . . 52 2.3 Some Definitions of Thermodynamics . . . . . . . . . . . . . . . . . . . . . 52 2.4 The first law of thermodynamics and equilibration . . . . . . . . . . 53 2.5 The second law of thermodynamics . . . . . . . . . . . . . . . . . . . . . . . 55 2.6 Thermal equilibrium and temperature . . . . . . . . . . . . . . . . . . . . . 60 2.7 The equation of state of an ideal gas . . . . . . . . . . . . . . . . . . . . . . 63 2.8 Heat capacity or specific heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.9 Calculation of specific heats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.10 Isothermal and adiabatic expansion . . . . . . . . . . . . . . . . . . . . . . . 70 2.11 Legendre transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.12 Maxwell Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 2.13 Extensive Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 2.14 Intensive Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 2.15 Stability of thermodynamics systems . . . . . . . . . . . . . . . . . . . . . . 85 2.15.1 Multiple phase Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 85

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2.16 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3

Principles of Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 99 3.1 Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.2 Canonical partition function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.3 Entropy, free energy and internal energy . . . . . . . . . . . . . . . . . . . 101 3.4 Thermodynamic potentials and corresponding ensembles . . . . 103 3.5 Generalized ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.5.1 Isothermal-isobaric ensemble . . . . . . . . . . . . . . . . . . . . . . . 105 3.5.2 Grand canonical ensemble . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.5.3 Grand isothermal-isobaric ensemble . . . . . . . . . . . . . . . . . 107 3.6 Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4

Thermodynamics of Biological Phenomena . . . . . . . . . . . . . . . 115 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.2 Stability of Macromolecular Conformations . . . . . . . . . . . . . . . . 117 4.3 Gibbs Free Energy of the Transition . . . . . . . . . . . . . . . . . . . . . . 124 4.4 The Binding Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.5 Theoretical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.5.1 Energy Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.5.2 Solvation Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.6 Empirical Solvation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.6.1 Implicit Nonpolar Solvation Free Energy . . . . . . . . . . . . 134 4.6.2 The Poisson-Boltzmann Model . . . . . . . . . . . . . . . . . . . . . 144

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4.6.3 Generalized Born Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 4.7 The Free Energy Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 4.7.1 Thermodynamic Perturbation Method . . . . . . . . . . . . . . 170 4.7.2 Thermodynamic Integration Method . . . . . . . . . . . . . . . . 179 4.7.3 The Slow Growth Method . . . . . . . . . . . . . . . . . . . . . . . . . 182 5

Molecular Dynamics Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 5.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 5.2.1 Microcanonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . 185 5.2.2 Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 5.2.3 Isothermal-Isobaric Ensemble . . . . . . . . . . . . . . . . . . . . . . 201 5.2.4 Grand Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . 217 5.2.5 Generalized Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

6

Molecular Dynamics of Rigid Bodies . . . . . . . . . . . . . . . . . . . . . . 229 6.1 Equations of motion in relevant ensembles . . . . . . . . . . . . . . . . . 229 6.1.1 Microcanonical ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . 229 6.1.2 Canonical ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 6.1.3 Isothermal-isobaric ensemble . . . . . . . . . . . . . . . . . . . . . . . 239

7

Symplectic and time reversible integrators . . . . . . . . . . . . . . . . 243 7.1 Symplectic and Hamiltonian splitting methods . . . . . . . . . . . . . 243 7.1.1 Symplectic maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

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7.1.2 Symplecticness of Hamiltonian flow-maps . . . . . . . . . . . . 244 7.1.3 Phase-space area preservation for d = 1 . . . . . . . . . . . . . 245 7.1.4 Time-reversal symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 7.1.5 Hamiltonian splitting methods . . . . . . . . . . . . . . . . . . . . . 247 7.2 The Liouville formalism and Trotter formula . . . . . . . . . . . . . . . 248 7.3 Backward error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 8

Symplectic and time reversible integrators . . . . . . . . . . . . . . . . 253 8.1 Symplectic and Hamiltonian splitting methods . . . . . . . . . . . . . 253 8.1.1 Symplectic maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 8.1.2 Symplecticness of Hamiltonian flow-maps . . . . . . . . . . . . 254 8.1.3 Phase-space area preservation for d = 1 . . . . . . . . . . . . . 255 8.1.4 Time-reversal symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 8.1.5 Hamiltonian splitting methods . . . . . . . . . . . . . . . . . . . . . 257 8.2 The Liouville formalism and Trotter formula . . . . . . . . . . . . . . . 258 8.3 Microcanonical ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 8.4 Canonical ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 8.5 Isothermal-isobaric ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

9

Symplectic and time reversible integrators for rigid bodies 277 9.0.1 Hamiltonian splitting methods . . . . . . . . . . . . . . . . . . . . . 277 9.1 The Liouville formalism and Trotter formula . . . . . . . . . . . . . . . 278 9.2 Microcanonical ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

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9.3 Canonical ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 9.4 Isothermal-isobaric ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 10 Practical aspects of Molecular Dynamics Simulations . . . . . 303 10.1 Designing Constraints for Molecular Dynamics Simulations . . 303 10.2 Initial configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 10.3 Periodic boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 10.4 Truncating the potential and the minimum image convention 307 10.5 Neighbour lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 10.6 Long-range forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 10.6.1 The Ewald summation method . . . . . . . . . . . . . . . . . . . . . 310 10.6.2 The reaction field method . . . . . . . . . . . . . . . . . . . . . . . . . 312 10.7 Equilibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 11 Advanced Molecular Dynamics Methods . . . . . . . . . . . . . . . . . . 317 11.1 Current methods for enhancing conformational sampling . . . . 317 11.1.1 Multiple time step integrators . . . . . . . . . . . . . . . . . . . . . . 318 11.1.2 Generalized ensemble methods . . . . . . . . . . . . . . . . . . . . . 322 11.1.3 Metadynamics method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 11.1.4 Umbrella sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 11.1.5 Transition path sampling . . . . . . . . . . . . . . . . . . . . . . . . . . 334 11.1.6 Accelerated molecular dynamics . . . . . . . . . . . . . . . . . . . . 335 11.1.7 Conformational flooding . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 11.2 Discussion and prospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340

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12 Molecular Dynamics Method in Simulations of Macromolecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 12.1 Molecular Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 12.1.1 Simple Molecular Mechanics Force Field . . . . . . . . . . . . . 344 12.1.2 Features of Molecular Mechanics Force Fields . . . . . . . . 345 12.1.3 Molecular Mechanics Force Field Parameters Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 12.1.4 Bond Stretching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 12.1.5 Angle bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 12.1.6 Torsional angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 12.1.7 The van der Waals potential . . . . . . . . . . . . . . . . . . . . . . . 350 12.1.8 Modelling the van der Waals interactions . . . . . . . . . . . . 352 12.2 Normal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 12.2.1 General theory of normal modes . . . . . . . . . . . . . . . . . . . . 354 12.2.2 Dynamical behaviour of system . . . . . . . . . . . . . . . . . . . . 358 12.2.3 Time averaged properties . . . . . . . . . . . . . . . . . . . . . . . . . . 360 12.2.4 Thermal amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 12.2.5 Normal modes of proteins . . . . . . . . . . . . . . . . . . . . . . . . . 361 13 Applications of Molecular Dynamics in Classical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 13.1 Principal components analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 13.2 Diffusive motion in proteins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 13.3 Stability of the PCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 13.4 Root mean square deviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

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14 Coarse-grained models for biomolecular systems . . . . . . . . . . 377 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 14.2 Coarse-grained models of proteins . . . . . . . . . . . . . . . . . . . . . . . . 379 14.3 Coarse-grained models of nucleic acids . . . . . . . . . . . . . . . . . . . . 380 14.3.1 Worm-like chain model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 14.3.2 Elasticity theory framework . . . . . . . . . . . . . . . . . . . . . . . . 385 14.3.3 Three-bead DNA model . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 14.3.4 Martini coarse-grained model of DNA . . . . . . . . . . . . . . . 388 14.4 Coarse-graining strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 14.5 The many-body potential of mean force . . . . . . . . . . . . . . . . . . . 392 14.5.1 Native structure-based models . . . . . . . . . . . . . . . . . . . . . 393 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398