Targeted Mollified Impulse: A Multiscale Stochastic Integrator for Long Molecular Dynamics Simulations

Molecular dynamics (MD) is widely used in simulations of biomolecular systems such as DNA and proteins, systems which are multiscale in nature. However, current time stepping integrators are not able to address the time scale problems. Multiscale integrators, in which the presence of "fast" modes does not affect the time integration of "slow" modes, are pressingly needed in light of the fast growing biological data generated from the many genome sequencing projects. In this paper, we present a new multiple time stepping (MTS) multiscale integrator with stochasticity built in for constant temperature molecular dynamics simulations, called the targeted mollified impulse method (TM). TM combines the mollified impulse method, which is a stabler version of Verlet-I/r-RESPA ({\bf r}eversible {\bf RE}ference {\bf S}ystem {\bf P}ropagator {\bf A}lgorithm), and a self-consistent dissipative leapfrog integrator commonly used in dissipative particle dynamics. TM introduces the Langevin coupling in a targeted manner ...

[1]  G. Martyna,et al.  Electrostatic calculations and multiple time scales in molecular dynamics simulation of flexible molecular systems , 1998 .

[2]  J. Koelman,et al.  Dynamic simulations of hard-sphere suspensions under steady shear , 1993 .

[3]  J. M. Sanz-Serna,et al.  Numerical Hamiltonian Problems , 1994 .

[4]  I. Pagonabarraga,et al.  Dissipative particle dynamics for interacting systems , 2001, cond-mat/0105075.

[5]  M. Karplus,et al.  Proteins: A Theoretical Perspective of Dynamics, Structure, and Thermodynamics , 1988 .

[6]  B. Leimkuhler,et al.  A reversible averaging integrator for multiple time-scale dynamics , 2001 .

[7]  P. Kollman,et al.  Pathways to a protein folding intermediate observed in a 1-microsecond simulation in aqueous solution. , 1998, Science.

[8]  Klaus Schulten,et al.  Generalized Verlet Algorithm for Efficient Molecular Dynamics Simulations with Long-range Interactions , 1991 .

[9]  Jesús A. Izaguirre,et al.  An impulse integrator for Langevin dynamics , 2002 .

[10]  C. Sagui,et al.  Multigrid methods for classical molecular dynamics simulations of biomolecules , 2001 .

[11]  A. Ko,et al.  MDSimAid : AN AUTOMATIC RECOMMENDER FOR OPTIMIZATION OF FAST ELECTROSTATIC ALGORITHMS FOR MOLECULAR SIMULATIONS , 2002 .

[12]  Stephen Wolfram,et al.  A New Kind of Science , 2003, Artificial Life.

[13]  M. Levitt,et al.  Molecular dynamics of native protein. I. Computer simulation of trajectories. , 1983, Journal of molecular biology.

[14]  Angelo Vulpiani,et al.  On the effects of an uncertainty on the evolution law in dynamical systems , 1989 .

[15]  B. Berne,et al.  Novel methods of sampling phase space in the simulation of biological systems. , 1997, Current opinion in structural biology.

[16]  Mark E. Tuckerman,et al.  Reversible multiple time scale molecular dynamics , 1992 .

[17]  K. Schulten,et al.  Difficulties with multiple time stepping and fast multipole algorithm in molecular dynamics , 1997 .

[18]  Thierry Matthey,et al.  Overcoming Instabilities in Verlet-I/r-RESPA with the Mollified Impulse Method , 2002 .

[19]  M. A. López-Marcos,et al.  Explicit Symplectic Integrators Using Hessian-Vector Products , 1997, SIAM J. Sci. Comput..

[20]  J. Izaguirre.  Longer Time Steps for Molecular Dynamics , 1999 .

[21]  Jesús A. Izaguirre,et al.  Verlet-I/R-RESPA/Impulse is Limited by Nonlinear Instabilities , 2003, SIAM J. Sci. Comput..

[22]  T. Schlick,et al.  Overcoming stability limitations in biomolecular dynamics. I. Combining force splitting via extrapolation with Langevin dynamics in LN , 1998 .

[23]  H. G. Petersen,et al.  An algorithm for the simulation of condensed matter which grows as the 3/2 power of the number of particles , 1988 .

[24]  Tamar Schlick,et al.  A Family of Symplectic Integrators: Stability, Accuracy, and Molecular Dynamics Applications , 1997, SIAM J. Sci. Comput..

[25]  Jesús A. Izaguirre,et al.  Long time step molecular dynamics using targeted Langevin stabilization , 2003, SAC '03.

[26]  Robert D. Skeel,et al.  Symplectic Integration with Variable Stepsize , 1994 .

[27]  Qun Ma,et al.  Novel multiscale algorithms for molecular dynamics , 2003 .

[28]  T. Darden,et al.  Particle mesh Ewald: An N⋅log(N) method for Ewald sums in large systems , 1993 .

[29]  Laxmikant V. Kale,et al.  NAMD2: Greater Scalability for Parallel Molecular Dynamics , 1999 .

[30]  Robert D. Skeel,et al.  Multiple grid methods for classical molecular dynamics , 2002, J. Comput. Chem..

[31]  W. L. Jorgensen,et al.  Comparison of simple potential functions for simulating liquid water , 1983 .

[32]  J. Valverde.  Molecular Modelling: Principles and Applications , 2001 .

[33]  P. B. Warren,et al.  DISSIPATIVE PARTICLE DYNAMICS : BRIDGING THE GAP BETWEEN ATOMISTIC AND MESOSCOPIC SIMULATION , 1997 .

[34]  P. P. Ewald.  Die Berechnung optischer und elektrostatischer Gitterpotentiale , 1921 .

[35]  C. Brooks.  Computer simulation of liquids , 1989 .

[36]  P. Español,et al.  Statistical Mechanics of Dissipative Particle Dynamics. , 1995 .

[37]  A. Grzybowski,et al.  Ewald summation of electrostatic interactions in molecular dynamics of a three-dimensional system with periodicity in two directions , 2000 .

[38]  Ernst Hairer,et al.  Long-Time Energy Conservation of Numerical Methods for Oscillatory Differential Equations , 2000, SIAM J. Numer. Anal..

[39]  Robert D. Skeel,et al.  Long-Time-Step Methods for Oscillatory Differential Equations , 1998, SIAM J. Sci. Comput..

[40]  B. Berne,et al.  A Multiple-Time-Step Molecular Dynamics Algorithm for Macromolecules , 1994 .

[41]  Ignacio Pagonabarraga,et al.  Self-consistent dissipative particle dynamics algorithm , 1998 .

[42]  B. Berne,et al.  A new molecular dynamics method combining the reference system propagator algorithm with a fast multipole method for simulating proteins and other complex systems , 1995 .

[43]  T. Schlick,et al.  Extrapolation versus impulse in multiple-timestepping schemes. II. Linear analysis and applications to Newtonian and Langevin dynamics , 1998 .

[44]  Robert D. Skeel,et al.  Integration Schemes for Molecular Dynamics and Related Applications , 1999 .

[45]  D. Case,et al.  Optimized particle-mesh Ewald/multiple-time step integration for molecular dynamics simulations , 2001 .

[46]  J. Marsden,et al.  Asynchronous Variational Integrators , 2003 .

[47]  David Fincham,et al.  Optimisation of the Ewald Sum for Large Systems , 1994 .

[48]  Jesús A. Izaguirre,et al.  The Five Femtosecond Time Step Barrier , 1999, Computational Molecular Dynamics.

[49]  Michael R. Shirts,et al.  Mathematical analysis of coupled parallel simulations. , 2001, Physical review letters.

[50]  Timothy R. Forester,et al.  On Multiple Time-step Algorithms and the Ewald Sum , 1994 .

[51]  Thierry Matthey,et al.  ProtoMol, an object-oriented framework for prototyping novel algorithms for molecular dynamics , 2004, TOMS.

[52]  P. Procacci,et al.  Taming the Ewald sum in molecular dynamics simulations of solvated proteins via a multiple time step algorithm , 1996 .

[53]  M. Levitt.  Protein folding by restrained energy minimization and molecular dynamics. , 1983, Journal of molecular biology.

[54]  Klaus Schulten,et al.  Elastic Rod Model of a DNA Loop in the Lac Operon , 1999 .

[55]  J. Marsden,et al.  Symplectic-energy-momentum preserving variational integrators , 1999 .

[56]  Edward D Harder,et al.  Efficient multiple time step method for use with Ewald and particle mesh Ewald for large biomolecular systems , 2001 .

[57]  T. Schlick,et al.  New splitting formulations for lattice summations , 2001 .

[58]  T. Schlick,et al.  Masking Resonance Artifacts in Force-Splitting Methods for Biomolecular Simulations by Extrapolative Langevin Dynamics , 1999 .

[59]  J. Mccammon,et al.  Molecular Dynamics Simulations of a Polyalanine Octapeptide under Ewald Boundary Conditions: Influence of Artificial Periodicity on Peptide Conformation , 2000 .

[60]  Karttunen,et al.  Towards better integrators for dissipative particle dynamics simulations , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[61]  M. Levitt,et al.  Computer simulation of protein folding , 1975, Nature.

[62]  Robert D. Skeel,et al.  Practical Construction of Modified Hamiltonians , 2001, SIAM J. Sci. Comput..

[63]  Todd R. Littell,et al.  Error Analysis of Symplectic Multiple Time Stepping , 1997 .

[64]  R. Skeel,et al.  Langevin stabilization of molecular dynamics , 2001 .

[65]  Thierry Matthey,et al.  Framework Design, Parallelization and Force Computation in Molecular Dynamics , 2002 .

[66]  M. Karplus,et al.  Stochastic boundary conditions for molecular dynamics simulations of ST2 water , 1984 .

[67]  H. Dufner,et al.  Ewald summation versus direct summation of shifted‐force potentials for the calculation of electrostatic interactions in solids: A quantitative study , 1997 .