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The Langevin/implicit-Euler/normal-mode scheme for molecular dynamics at
large time steps

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As molecular dynamics simulations continue to provide important insights
into biomolecular structure and function, a growing demand for increasing
the time span of the simulations is emerging. Our focus here is developing
a new algorithm, LIN (Langevin/implict-Euler/normal mode), that combines
normal-mode and implicit integration techniques, for large time step biomolecular
applications. In the normal-mode phase of LIN, we solve an approximate
linearized Langevin formulation to resolve the rapidly varying components
of the motion. In the implicit phase, we resolve the remaining components
of the motion by numerical integration with the implicit-Euler scheme.
Developments of the normal-mode phase of LIN are discussed in this paper.
Specifically, we solve two crucial issues of the method. The first involves
how to choose and how often to update the Hessian approximation for
the linearized Langevin equation. This approximation must be computationally
feasible and physically reasonable to capture the motion in the higher
end of the vibrational spectrum. Three such general Hessian approximations
are discussed. The related issue-the frequency of the Hessian update-is
analyzed by projecting the motion onto the different vibrational modes.
This analysis demonstrates that a one-picosecond interval is reasonable
for updating the Hessian in the model system examined here. In this connection,
we illustrate that the high-frequency motions are highly localized while
the low frequency motions are delocalized. We also show rigorously
that the mode amplitudes are inversely proportional to the
frequency (consistent with the equipartition theorem), with 90% of the
displacement fluctuations coming from a very small group of low frequency
modes. Anharmonic effects essentially influence the low frequency modes.
The second issue involves how to solve the linearized Langevin equation
at large timesteps correctly, where the usual discretized formation
of the random force is invalid. This is accomplished by using analytic
expressions for the distributions associated with positions and velocities
of the individual oscillators as a function of frequency, obtained as the
solution of the corresponding Fokker-Planck equation. We apply LIN with
these developments to the nucleic acid component deoxycytidine with timesteps
ranging from 100 to 1000 fs. We demonstrate that LIN is stable in these
simulations, with energies fluctuating about the same values-and possessing
overall similar dynamical features-in comparison to 1 fs explicit simulations,
though the fluctuations are significantly larger ar larger timesteps. Moreover
continuous dynamics is maintained, and pathway information can be
obtained. Computational performance is competitive only at very large time
steps: a gain factor of 3-4 is obtained for runs with 1000 fs time steps.
Larger gains may be achieved for biomolecules, where sparsity and parallelization
can be exploited significantly.

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