BIOMOLECULAR DYNAMICS AT LONG TIMESTEPS:
Bridging the Timescale Gap Between Simulation and Experimentation

Tamar Schlick, Eric Barth & Margaret Mandziuk

Abstract

An exciting range of numerical methods has been offered during the last decade for solving on modern computers the classical Newtonian equations that govern the motion of macromolecules. Molecular motions hold the key to many biological processes and hence are of great interest as well as practical importance. The many innovative techniques are attempting to reduce the severity of the timestep problem, namely the requirement that the integration timestep be sufficiently small to resolve the fastest components of the motion (bond vibrations) and thus guarantee numerical stability. This problem has turned to be much more challenging than originally imagined if one strictly seeks faster methods with the same all-atom resolution at small timesteps. Mathematical techniques that have worked well in other multiple-timescale contexts --- where the fast motions are rapidly decaying or largely decoupled from the others --- have not been as successful for biomolecules, where vibrational coupling is strong. Indeed, progress in increasing the timespan covered by all-atom biomolecular simulations has been slow relative to improvements in computer power during the last decade.

In this article, general issues that limit the timestep are discussed, and specialized techniques for biomolecules described and assessed with respect to physical reliability and computational demands. These methods include constrained dynamics, reduced-variable formulations, implicit schemes, symplectic schemes, multiple-timestep methods, and normal-mode-based schemes. We also describe our dual timestep ($\Delta \tau, \Delta t$) method termed ``LN'' (for its origin in a Langevin/Normal Modes algorithm) which provides speedup for biomolecules (e.g., factor of 4). LN relies on an approximate linearization of the equations of motion every $\Delta t$ interval (5 fs or less); this system is explicitly integrated using an inner timestep $\Delta \tau$ (such as 0.5 fs). Since this subintegration process does not require new force evaluations, as in every step of standard molecular dynamics integration, LN can be computationally competitive. Furthermore, since the harmonic approximation is quite good over the short interval $\Delta t$, results are in good agreement with small-timestep simulations. We also include a section comparing results of the different integration methods discussed in this article on a model dipeptide and assess physical and numerical performance of each one. Results indicate the strengths of each method and suggest corresponding asymptotic and expected speedup.

These collective algorithmic efforts are certainly helping fill the gap between the time range that can be simulated on modern computers and the times of major biological interest (milliseconds and longer). Still, it is likely that only a hierarchy of models and methods for dynamics and conformational sampling, as well as improvements in experimentational resolution, will ultimately give theoretical modeling the status of partner with experiment.