Time integrators for molecular dynamics

Entropy

Volumn 16, Issue 1, 2014, Pages 138-162

  Bou Rabee, Nawaf ^a  

^a RUTGERS UNIVERSITY   (United States)

Author keywords

Ergodicity; Explicit integrators; Metropolis algorithm; Weak accuracy

Indexed keywords

EID: 84893205677     PISSN: None     EISSN: 10994300     Source Type: Journal    
DOI: 10.3390/e16010138     Document Type: Article

References (116)

1. Allen, M.P.; Tildesley, D.J. Computer Simulation of Liquids; Clarendon Press: Oxford, UK, 1987.
2. Frenkel, D.; Smit, B. Understanding Molecular Simulation: From Algorithms to Applications, 2nd ed.; Academic Press: Waltham, MA, USA, 2002.
3. Rapaport, D.C. The Art of Molecular Dynamics Simulation; Cambridge University Press: Cambridge, UK, 2004.
4. Tuckerman, M. Statistical Mechanics and Molecular Simulations; Oxford University Press: Oxford, UK, 2008.
5. Schlick, T. Molecular Modeling and Simulation: An Interdisciplinary Guide; Springer: Berlin/Heidelberg, Germany, 2010; Volume 21.
6. Rahman, A. Correlations in the motion of atoms in liquid argon. Phys. Rev. 1964, 136, A405.
7. Verlet, L. Computer "experiments" on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules. Phys. Rev. 1967, 159, 98-103.
8. Alder, B.J.; Wainwright, T.E. Velocity autocorrelations for hard spheres. Phys. Rev. Lett. 1967, 18, 988-990.
9. Alder, B.J.; Gass, D.M.; Wainwright, T.E. Studies in molecular dynamics. VIII. The transport coefficients for a hard-sphere fluid. J. Chem. Phys. 1970, 53, 3813-3826.
10. Harp, G.D.; Berne, B.J. Time-correlation functions, memory functions, and molecular dynamics. Phys. Rev. A 1970, 2, 975-996.
11. Rahman, A.; Stillinger, F.H. Molecular dynamics study of liquid water. J. Chem. Phys. 1971, 55, 3336-3359.
12. Stillinger, F.H.; Rahman, A. Improved simulation of liquid water by molecular dynamics. J. Chem. Phys. 1974, 60, 1545-1557.
13. Stillinger, F.H. Water revisited. Science 1980, 209, 451-457.
14. McCammon, A.J.; Gelin, B.R.; Karplus, M. Dynamics of folded proteins. Nature 1977, 267, 585-590.
15. Van Gunsteren,W.F.; Berendsen, H.J.C. Algorithms for macromolecular dynamics and constraint dynamics. Mol. Phys. 1977 34, 1311-1327.
16. McCammon, J.A.; Karplus, M. Simulation of protein dynamics. Annu. Rev. Phys. Chem. 1980, 31, 29-45.
17. Van Gunsteren, W.F.; Karplus, M. Protein dynamics in solution and in a crystalline environment: A molecular dynamics study. Biochemistry 1982, 21, 2259-2274.
18. Karplus, M.; McCammon, A.J. Dynamics of proteins: Elements and function. Annu. Rev. Biochem. 1983, 52, 263-300.
19. Van Gunsteren, W.F.; Berendsen, H.J.C. Computer simulation of molecular dynamics: Methodology, applications, and perspectives in chemistry. Angew. Chem. Int. Ed. Engl. 1990, 29, 992-1023.
20. Karplus, M.; McCammon, A.J. Molecular dynamics simulations of biomolecules. Nat. Struct. Mol. Biol. 2002, 9, 646-652.
21. Case, D.A. Molecular dynamics and NMR spin relaxation in proteins. Acc. Chem. Res. 2002, 35, 325-331.
22. Adcock, A.S.; McCammon, A.J. Molecular dynamics: Survey of methods for simulating the activity of proteins. Chem. Rev. 2006, 106, 1589-1615.
23. Van Gunsteren, W.F.; Dolenc, J.; Mark, A.E. Molecular simulation as an aid to experimentalists. Curr. Opin. Struct. Biol. 2008, 18, 149-153.
24. Kapral, R.; Ciccotti, G. Molecular dynamics: An Account of its Evolution. In Theory and Applications of Computational Chemistry: The First Forty Years; Elsevier: New York, NY, USA, 2005; pp. 425-441.
25. Levitt, M. Molecular dynamics of native protein: I. Computer simulation of trajectories. J. Mol. Biol. 1983, 168, 595-617.
26. Levitt, M.; Sharon, R. Accurate simulation of protein dynamics in solution. Proc. Natl. Acad. Sci. USA 1988, 85, 7557-7561.
27. Daggett, V.; Levitt, M. A model of the molten globule state from molecular dynamics simulations. Proc. Natl. Acad. Sci. USA 1992, 89, 5142-5146.
28. Li, A.; Daggett, V. Investigation of the solution structure of chymotrypsin inhibitor 2 using molecular dynamics: Comparison to X-Ray crystallographic and NMR data. Protein Eng. 1995, 8, 1117-1128.
29. Duan, Y.; Kollman, P.A. Pathways to a protein folding intermediate observed in a 1-microsecond simulation in aqueous solution. Science 1998, 282, 740-744.
30. Freddolino, P.L.; Liu, F.; Gruebele, M.; Schulten, K. Ten-microsecond molecular dynamics simulation of a fast-folding WW domain. Biophys. J. 2008, 94, 75-77.
31. Shaw, D.E.; Maragakis, P.; Lindorff-Larsen, K.; Piana, S.; Dror, R.O.; Eastwood, M.P.; Bank, J.A.; Jumper, J.M.; Salmon, J.K.; Shan, Y. Atomic-level characterization of the structural dynamics of proteins. Science 2010, 330, 341-346.
32. Quentrec, B.; Brot, C. New method for searching for neighbors in molecular dynamics computations. J. Comput. Phys. 1973, 13, 430-432.
33. Ryckaert, J.P.; Ciccotti, G.; Berendsen, H.J.C. Numerical integration of the Cartesian equations of motion of a system with constraints: Molecular dynamics of n-Alkanes. J. Comput. Phys. 1977, 23, 327-341.
34. Schneider, T.; Stoll, E. Molecular dynamics study of a three-dimensional one-component model for distortive phase transitions. Phys. Rev. B 1978, 17, 1302-1322.
35. Brünger, A.; Brooks, C.L.; Karplus, M. Stochastic boundary conditions for molecular dynamics simulations of ST2 water. Chem. Phys. Lett. 1984, 105, 495-500.
36. Brooks, B.R.; Bruccoleri, R.E.; Olafson, B.D.; States, D.J.; Swaminathan, S.; Karplus, M. CHARMM: A program for macromolecular energy, minimization, and dynamics calculations. J. Comput. Chem. 1983, 4, 187-217.
37. Nosé, S. A unified formulation for constant temperature molecular dynamics methods. J. Chem. Phys. 1984, 81, 511-519.
38. Hoover, W.G. Canonical dynamics: Equilibrium phase-space distributions. Phys. Rev. A 1985, 31, 1695-1697.
39. Greengard, L.; Rokhlin, V. A fast algorithm for particle simulations. J. Comput. Phys. 1987, 73, 325-348.
40. Tuckerman, M.E.; Berne, B.J.; Martyna, G. Reversible multiple time scale molecular dynamics. J. Chem. Phys. 1992, 97, 1990-2001.
41. Skeel, R.D.; Izaguirre, J. An impulse integrator for Langevin dynamics. Mol. Phys. 2002, 100, 3885-3891.
42. Ricci, A.; Ciccotti, G. Algorithms for Brownian dynamics. Mol. Phys. 2003, 101, 1927-1931.
43. Bussi, G.; Parrinello, M. Accurate sampling using Langevin dynamics. Phys. Rev. E 2007, 75, 056707.
44. Milstein, G.N.; Tretyakov, M.V. Quasi-symplectic methods for Langevin-type equations. IMA J. Num. Anal. 2003, 23, 593-626.
45. Milstein, G.N.; Tretyakov, M.V. Computing ergodic limits for Langevin equations. Physica D 2007, 229, 81-95.
46. Yao, Z.; Wang, J.S.; Liu, G.R.; Cheng, M. Improved neighbor list algorithm in molecular simulations using cell decomposition and data sorting method. Comput. Phys. Commun. 2004, 161, 27-35.
47. Bussi, G.; Donadio, D.; Parrinello, M. Canonical sampling through velocity rescaling. J. Chem. Phys. 2007, 126, 014101.
48. Samoletov, A.A.; Chaplain, M.A.; Dettmann, C.P. Thermostats for "Slow" configurational modes. J. Stat. Phys. 2008, 128, 1321-1336.
49. Leimkuhler, B.; Noorizadeh, E.; Theil, F. A gentle stochastic thermostat for molecular dynamics. J. Stat. Phys. 2009, 135, 261-277.
50. Leimkuhler, B.; Reich, S. A Metropolis adjusted Nosé-Hoover thermostat. Math. Model. Num. Anal. 2009, 43, 743-755.
51. Scheraga, H.A.; Khalili, M.; Liwo, A. Protein-folding dynamics: Overview of molecular simulation techniques. Annu. Rev. Phys. Chem. 2007, 58, 57-83.
52. Dror, R.O.; Dirks, R.M.; Grossman, J.P.; Xu, H.; Shaw, D.E. Biomolecular simulation: A computational microscope for molecular biology. Annu. Rev. Biophys. 2012, 41, 429-452.
53. Lane, T.J.; Shukla, D.; Beauchamp, K.A.; Pande, V.S. To milliseconds and beyond: Challenges in the simulation of protein folding. Curr. Opin. Struct. Biol. 2013, 23, 58-65.
54. Nielsen, S.O.; Lopez, C.F.; Srinivas, G.; Klein, M.L. Coarse grain models and the computer simulation of soft materials. J. Phys. Condens. Matter 2004, 16, R481.
55. Tozzini, V. Coarse-grained models for proteins. Curr. Opin. Struct. Biol. 2005, 15, 144-150.
56. Clementi, C. Coarse-grained models of protein folding: Toy models or predictive tools? Curr. Opin. Struct. Biol. 2008, 18, 10-15.
57. Sherwood, P.; Brooks, B.R.; Sansom, M.S.P. Multiscale methods for macromolecular simulations. Curr. Opin. Struct. Biol. 2008, 18, 630-640.
58. Weinan, E. Principles of Multiscale Modeling; Cambridge University Press: Cambridge, UK, 2011.
59. Weinan, E.; Vanden-Eijnden, E. Metastability, Conformation Dynamics, and Transition Pathways in Complex Systems. In Multiscale Modelling and Simulation; Attinger, S., Koumoutsakos, P., Eds.; Lecture Notes in Computational Science and Engineering; Springer: Berlin/Heidelberg, Germany, 2004; pp. 35-68.
60. Vanden-Eijnden, E.; Venturoli, M. Markovian milestoning with Voronoi tessellations. J. Chem. Phys. 2009, 130, 194101.
61. Vanden-Eijnden, E.; Venturoli, M. Exact rate calculations by trajectory parallelization and twisting. J. Chem. Phys. 2009, 131, 044120.
62. Weinan, E.; Vanden-Eijnden, E. Transition-path theory and path-finding algorithms for the study of rare events. Annu. Rev. Phys. Chem. 2010, 61, 391-420.
63. Nelson, M.T.; Humphrey, W.; Gursoy, A.; Dalke, A.; Kalé, L.V.; Skeel, R.D.; Schulten, K. NAMD: A parallel, object-oriented molecular dynamics program. Int. J. High Perform. Comput. Appl. 1996, 10, 251-268.
64. Scott, W.R.P.; Hünenberger, P.H.; Tironi, I.G.; Mark, A.E.; Billeter, S.R.; Fennen, J.; Torda, A.E.; Huber, T.; Krüger, P.; van Gunsteren, W.F. The GROMOS biomolecular simulation program package. J. Phys. Chem. A 1999, 103, 3596-3607.
65. Talay, D. Stochastic Hamiltonian systems: Exponential convergence to the invariant measure, and discretization by the implicit Euler scheme. Markov Process. Relat. Fields 2002, 8, 1-36.
66. Higham, D.J.; Mao, X.; Stuart, A.M. Strong convergence of Euler-type methods for nonlinear stochastic differential equations. IMA J. Num. Anal. 2002, 40, 1041-1063.
67. Milstein, G.N.; Tretyakov, M.V. Numerical integration of stochastic differential equations with nonglobally Lipschitz coefficients. IMA J. Num. Anal. 2005, 43, 1139-1154.
68. Higham, D.J. Stochastic ordinary differential equations in applied and computational mathematics. IMA J. Appl. Math. 2011, 76, 449-474.
69. Hutzenthaler, M.; Jentzen, A.; Kloeden, P.E. Strong convergence of an explicit numerical method for SDEs with non-globally Lipschitz continuous coefficients. Ann. Appl. Probab. 2012, 22, 1611-1641.
70. Scemama, A.; Lelievre, T.; Stoltz, G.; Cancés, E.; Caffarel, M. An efficient sampling algorithm for variational Monte Carlo. J. Chem. Phys. 2006, 125, 114105.
71. Akhmatskaya, E.; Bou-Rabee, N.; Reich, S. A comparison of generalized hybrid Monte Carlo methods with and without momentum flip. J. Comput. Phys. 2009, 228, 2256-2265.
72. Bou-Rabee, N.; Vanden-Eijnden, E. Pathwise accuracy and ergodicity of Metropolized integrators for SDEs. Commun. Pure Appl. Math. 2010, 63, 655-696.
73. Bou-Rabee, N.; Vanden-Eijnden, E. A patch that imparts unconditional stability to explicit integrators for Langevin-like equations. J. Comput. Phys. 2012, 231, 2565-2580.
74. Metropolis, N.; Rosenbluth, A.W.; Rosenbluth, M.N.; Teller, A.H.; Teller, E. Equations of state calculations by fast computing machines. J. Chem. Phys. 1953, 21, 1087-1092.
75. Hastings, W.K. Monte-Carlo methods using Markov chains and their applications. Biometrika 1970, 57, 97-109.
76. Rossky, P.J.; Doll, J.D.; Friedman, H.L. Brownian dynamics as smart Monte Carlo simulation. J. Chem. Phys. 1978, 69, 4628.
77. Duane, S.; Kennedy, A.D.; Pendleton, B.J.; Roweth, D. Hybrid Monte-Carlo. Phys. Lett. B 1987, 195, 216-222.
78. Horowitz, A.M. A generalized guided Monte-Carlo algorithm. Phys. Lett. B 1991, 268, 247-252.
79. Kennedy, A.D.; Pendleton, B. Cost of the generalized hybrid Monte Carlo algorithm for free field theory. Nucl. Phys. B 2001, 607, 456-510.
80. Liu, J.S. Monte Carlo Strategies in Scientific Computing, 2nd ed.; Springer: Berlin/Heidelberg, Germany, 2008.
81. Akhmatskaya, E.; Reich, S. GSHMC: An efficient method for molecular simulation. J. Comput. Phys. 2008, 227, 4937-4954.
82. Lelievre, T.; Rousset, M.; Stoltz, G. Free Energy Computations: A Mathematical Perspective, 1st ed.; Imperial College Press: London, UK, 2010.
83. MATLAB, Version 8.0.0 (R2012b); The MathWorks Inc.: Natick, MA, USA, 2012.
84. Introducing MEX-Files. Available online: http://www.mathworks.com/help/matlab/ matlab external/introducing-mex-files.html (accessed on 19 September 2013).
85. Kloeden, P.E.; Platen, E. Numerical Solution of Stochastic Differential Equations; Springer: Berlin, Germany, 1992.
86. Hutzenthaler, M.; Jentzen, A.; Kloeden, P.E. Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients. Proc. R. Soc. A: Math. Phys. Eng. Sci. 2011, 467, 1563-1576.
87. Vanden-Eijnden, E.; Ciccotti, G. Second-order integrators for Langevin equations with holonomic constraints. Chem. Phys. Lett. 2006, 429, 310-316.
88. Leimkuhler, B.; Matthews, C. Robust and efficient configurational molecular sampling via Langevin dynamics. J. Chem. Phys. 2013, 138, 174102.
89. Evans, L. An Introduction to Stochastic Differential Equations; American Mathematical Society: Providence, RI, USA, 2013.
90. Bou-Rabee, N.; Owhadi, H. Long-run accuracy of variational integrators in the stochastic context. SIAM J. Numer. Anal. 2010 48, 278-297.
91. Leimkuhler, B.; Matthews, C.; Stoltz, G. The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics. 2013, arXiv:1308.5814.
92. Bou-Rabee, N.; Donev, A.; Vanden-Eijnden, E. Metropolized integration schemes for self-adjoint diffusions. 2013, arXiv:1309.5037.
93. Milstein, G.N.; Tretyakov, M.V. Stochastic Numerics for Mathematical Physics; Springer: Berlin, Germany, 2004.
94. Marsden, J.E; Ratiu, T.S. Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems; Springer: Berlin/Heidelberg, Germany, 1999.
95. Van Gunsteren, W.F.; Mark, A.E. Validation of molecular dynamics simulation. J. Chem. Phys. 1998, 108, 6109-6116.
96. Leimkuhler, B.; Reich, S. Simulating Hamiltonian Dynamics; Cambridge Monographs on Applied and Computational Mathematics; Cambridge University Press: Cambridge, UK, 2004.
97. Hairer, E.; Lubich, C.; Wanner, G. Geometric Numerical Integration; Springer: Berlin/Heidelberg, Germany, 2010.
98. Ikeda, N.; Watanabe, S. Stochastic Differential Equations and Diffusion Processes; North-Holland: Amsterdam, The Netherlands, 1989.
99. Klebaner, F.C. Introduction to Stochastic Calculus with Applications; Imperial College Press: London, UK, 2005.
100. Mengersen, K.L.; Tweedie, R.L. Rates of convergence of the Hastings and Metropolis algorithms. Ann. Stat. 1996, 24, 101-121.
101. Prato, G.D.; Zabczyk, J. Ergodicity for Infinite Dimensional Systems; Cambridge University Press: Cambridge, UK, 1996.
102. Haussman, U.G.; Pardoux, E. Time reversal for diffusions. Ann. Probab. 1986, 14, 1188-1205.
103. Kent, J. Time-reversible diffusions. Adv. Appl. Prob. 1978, 10, 819-835.
104. Ezra, G.S. Reversible measure-preserving integrators for non-Hamiltonian systems. J. Chem. Phys. 2006, 125, 034104.
105. Shardlow, T. Splitting for dissipative particle dynamics. SIAM J. Sci. Comput. 2003, 24, 1267-1282.
106. Serrano, M.; de Fabritiis, G.; Espanol, P.; Coveney, P.V. A stochastic Trotter integration scheme for dissipative particle dynamics. Math. Comput. Simulat. 2006, 72, 190-194.
107. Pavliotis, G.A.; Stuart, A.M.; Zygalakis, K.C. Calculating effective diffusivities in the limit of vanishing molecular diffusion. J. Comput. Phys. 2008, 228, 1030-1055.
108. Van Gunsteren, W.F.; Berendsen, H.J.C. Algorithms for Brownian dynamics. Mol. Phys. 1982, 45, 637-647.
109. Nummelin, E. General Irreducible Markov Chains and Non-Negative Operators; Cambridge University Press: New York, NY, USA, 1984.
110. Tierney, L. Markov chains for exploring posterior distributions. Ann. Stat. 1994, 22, 1701-1728.
111. Gelman, A.; Gilks, W.R.; Roberts, G.O. Weak convergence and optimal scaling of random walk Metropolis algorithms. Ann. Appl. Probab. 1997, 7, 110-120.
112. Roberts, G.O.; Rosenthal, J.S. Optimal scaling of discrete approximations to Langevin diffusions. J. R. Stat. Soc. Ser. B 1998, 60, 255-268.
113. Beskos, A.; Roberts, G.O.; Stuart, A.M. Optimal scalings for local Metropolis-Hastings chains on non-product targets in high dimensions. Ann. Appl. Probab. 2009, 19, 863-898.
114. Beskos, A.; Pillai, N.S.; Roberts, G.O.; Sanz-Serna, J.M.; Stuart, A.M. Optimal tuning of hybrid Monte-Carlo algorithm. 2010, arXiv:1001.4460.
115. Mattingly, J.C.; Pillai, N.S.; Stuart, A.M. Diffusion limits of the random walk Metropolis algorithm in high dimensions. Ann. Appl. Probab. 2012, 22, 881-930.
116. Kang, F.; Dao-Liu, W. Dynamical Systems and Geometric Construction of Algorithms. In Computational Mathematics in China; Contemporary Mathmatics, Volume 163; Shi, Z.-C., Yang, C.C., Eds.; American Mathmatical Society: New York, NY, USA,1994; pp. 1-32.