Three parallel algorithms for classical molecular dynamics are presented. The first assigns each processor a fixed subset of atoms; the second assigns each a fixed subset of inter-atomic forces to compute; the third assigns each a fixed spatial region. The algorithms are suitable for molecular dynamics models which can be difficult to parallelize efficiently—those with short-range forces where the neighbors of each atom change rapidly. They can be implemented on any distributed-memory parallel machine which allows for message-passing of data between independently executing processors. The algorithms are tested on a standard Lennard-Jones benchmark problem for system sizes ranging from 500 to 100,000,000 atoms on several parallel supercomputers--the nCUBE 2, Intel iPSC/860 and Paragon, and Cray T3D. Comparing the results to the fastest reported vectorized Cray Y-MP and C90 algorithm shows that the current generation of parallel machines is competitive with conventional vector supercomputers even for small problems. For large problems, the spatial algorithm achieves parallel efficiencies of 90% and a 1840-node Intel Paragon performs up to 165 faster than a single Cray C9O processor. Trade-offs between the three algorithms and guidelines for adapting them to more complex molecular dynamics simulations are also discussed.

Fast parallel algorithms for short-range molecular dynamics

NAMD is a parallel molecular dynamics code designed for high-performance simulation of large biomolecular systems. NAMD scales to hundreds of processors on high-end parallel platforms, as well as tens of processors on low-cost commodity clusters, and also runs on individual desktop and laptop computers. NAMD works with AMBER and CHARMM potential functions, parameters, and file formats. This article, directed to novices as well as experts, first introduces concepts and methods used in the NAMD program, describing the classical molecular dynamics force field, equations of motion, and integration methods along with the efficient electrostatics evaluation algorithms employed and temperature and pressure controls used. Features for steering the simulation across barriers and for calculating both alchemical and conformational free energy differences are presented. The motivations for and a roadmap to the internal design of NAMD, implemented in C++ and based on Charm++ parallel objects, are outlined. The factors affecting the serial and parallel performance of a simulation are discussed. Finally, typical NAMD use is illustrated with representative applications to a small, a medium, and a large biomolecular system, highlighting particular features of NAMD, for example, the Tcl scripting language. The article also provides a list of the key features of NAMD and discusses the benefits of combining NAMD with the molecular graphics/sequence analysis software VMD and the grid computing/collaboratory software BioCoRE. NAMD is distributed free of charge with source code at www.ks.uiuc.edu.

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Scalable molecular dynamics with NAMD

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Scalable Molecular Dynamics with NAMD

Hamiltonian dynamics can be used to produce distant proposals for the Metropolis algorithm, thereby avoiding the slow exploration of the state space that results from the diffusive behaviour of simple random-walk proposals. Though originating in physics, Hamiltonian dynamics can be applied to most problems with continuous state spaces by simply introducing fictitious "momentum" variables. A key to its usefulness is that Hamiltonian dynamics preserves volume, and its trajectories can thus be used to define complex mappings without the need to account for a hard-to-compute Jacobian factor - a property that can be exactly maintained even when the dynamics is approximated by discretizing time. In this review, I discuss theoretical and practical aspects of Hamiltonian Monte Carlo, and present some of its variations, including using windows of states for deciding on acceptance or rejection, computing trajectories using fast approximations, tempering during the course of a trajectory to handle isolated modes, and short-cut methods that prevent useless trajectories from taking much computation time.

MCMC using Hamiltonian dynamics

Hamiltonian dynamics can be used to produce distant proposals for the Metropolis algorithm, thereby avoiding the slow exploration of the state space that results from the diffusive behaviour of simple random-walk proposals. Though originating in physics, Hamiltonian dynamics can be applied to most problems with continuous state spaces by simply introducing fictitious “momentum” variables. A key to its usefulness is that Hamiltonian dynamics preserves volume, and its trajectories can thus be used to define complex mappings without the need to account for a hard-to-compute Jacobian factor — a property that can be exactly maintained even when the dynamics is approximated by discretizing time. In this review, I discuss theoretical and practical aspects of Hamiltonian Monte Carlo, and present some of its variations, including using windows of states for deciding on acceptance or rejection, computing trajectories using fast approximations, tempering during the course of a trajectory to handle isolated modes, and short-cut methods that prevent useless trajectories from taking much computation time.

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MCMC Using Hamiltonian Dynamics

In this paper we show the possibility of using very mild stochastic damping to stabilize long time step integrators for Newtonian molecular dynamics. More specifically, stable and accurate integrations are obtained for damping coefficients that are only a few percent of the natural decay rate of processes of interest, such as the velocity autocorrelation function. Two new multiple time stepping integrators, Langevin Molly (LM) and Brunger–Brooks–Karplus–Molly (BBK–M), are introduced in this paper. Both use the mollified impulse method for the Newtonian term. LM uses a discretization of the Langevin equation that is exact for the constant force, and BBK–M uses the popular Brunger–Brooks–Karplus integrator (BBK). These integrators, along with an extrapolative method called LN, are evaluated across a wide range of damping coefficient values. When large damping coefficients are used, as one would for the implicit modeling of solvent molecules, the method LN is superior, with LM closely following. However, with mild damping of 0.2 ps−1, LM produces the best results, allowing long time steps of 14 fs in simulations containing explicitly modeled flexible water. With BBK–M and the same damping coefficient, time steps of 12 fs are possible for the same system. Similar results are obtained for a solvated protein–DNA simulation of estrogen receptor ER with estrogen response element ERE. A parallel version of BBK–M runs nearly three times faster than the Verlet-I/r-RESPA (reversible reference system propagator algorithm) when using the largest stable time step on each one, and it also parallelizes well. The computation of diffusion coefficients for flexible water and ER/ERE shows that when mild damping of up to 0.2 ps−1 is used the dynamics are not significantly distorted.

Langevin stabilization of molecular dynamics

Motivation: CompuCell is a multi-model software framework for simulation of the development of multicellular organisms known as morphogenesis. It models the interaction of the gene regulatory network with generic cellular mechanisms, such as cell adhesion, division, haptotaxis and chemotaxis. A combination of a state automaton with stochastic local rules and a set of differential equations, including subcellular ordinary differential equations and extracellular reaction--diffusion partial differential equations, model gene regulation. This automaton in turn controls the differentiation of the cells, and cell--cell and cell--extracellular matrix interactions that give rise to cell rearrangements and pattern formation, e.g. mesenchymal condensation. The cellular Potts model, a stochastic model that accurately reproduces cell movement and rearrangement, models cell dynamics. All these models couple in a controllable way, resulting in a powerful and flexible computational environment for morphogenesis, which allows for simultaneous incorporation of growth and spatial patterning.

Results: We use CompuCell to simulate the formation of the skeletal architecture in the avian limb bud.

Availability: Binaries and source code for Microsoft Windows, Linux and Solaris are available for download from http://sourceforge.net/projects/compucell/

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CompuCell, a multi-model framework for simulation of morphogenesis

In molecular dynamics one solves Newton''s equations of motion. This is a challenging problem because the number of particles is large (thousands of atoms) but more importantly because stability severely limits the length of time steps relative to the total length needed for simulations---time steps are in the order of femtoseconds ($10^{-15}$ seconds) whereas simulations of a few microseconds ($10^{-6}$ seconds) up to one second are most desired. This situation also holds for multiple time stepping (MTS) integrators which evaluate different parts of the force at different frequencies. This work is an attempt to construct MTS integrators which are stable for long time steps. This has been done by applying a seminal idea that appeared in ~\cite{GaSS98}: modify the potential by defining it at time-averaged positions $\mathcal{A}(x)$, where the time averaging $\mathcal{A}(x)$ takes into account the high frequency vibrational motion. The resulting family of numerical integrators is called the mollified impulse method or MOLLY. Several averagings suggested, but not tested, in~\cite{GaSS98} are evaluated in this dissertation, along with more successful new ones. Finally, an efficient implementation of these averagings is devised. One of them is included in a production molecular dynamics code (NAMD 2). It is shown that the limitations in current MOLLY methods stem from a deficient representation of the eigenspace of the fast part of the forces. An alternative, less rigorous, approach based on ``Langevin stabilization'''' is explored. A mild damping and random noise is added to the system to remove instabilities without significantly affecting its dynamics.

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Longer Time Steps for Molecular Dynamics

The best simple method for Newtonian molecular dynamics is indisputably the leapfrog Stormer-Verlet method. The appropriate generalization to simple Langevin dynamics is unclear. An analysis is presented comparing an ‘impulse method’ (kick; fluctuate; kick), the 1982 method of van Gunsteren and Berendsen, and the Brunger-Brooks-Karplus (BBK) method. It is shown how the impulse method and the van Gunsteren-Berendsen methods can be implemented as efficiently as the BBK method. Other considerations suggest that the impulse method is the best basic method for simple Langevin dynamics, with the van Gunsteren-Berendsen method a close contender.

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An impulse integrator for Langevin dynamics

In this paper we present the foundation of a unified, object-oriented, three-dimensional biomodelling environment, which allows us to integrate multiple submodels at scales from subcellular to those of tissues and organs. Our current implementation combines a modified discrete model from statistical mechanics, the Cellular Potts Model, with a continuum reaction–diffusion model and a state automaton with well-defined conditions for cell differentiation transitions to model genetic regulation. This environment allows us to rapidly and compactly create computational models of a class of complex-developmental phenomena. To illustrate model development, we simulate a simplified version of the formation of the skeletal pattern in a growing embryonic vertebrate limb.

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Jesús A. Izaguirre

Papers

Langevin stabilization of molecular dynamics

CompuCell, a multi-model framework for simulation of morphogenesis

Longer Time Steps for Molecular Dynamics

An impulse integrator for Langevin dynamics

On multiscale approaches to three-dimensional modelling of morphogenesis