### Abstract

Numerical resonance artifacts have become recognized recently as a limiting factor to increasing the timestep in multiple-timestep (MTS) biomolecular dynamics simulations. At certain timesteps correlated to internal motions (e.g., 5 fs, around half the period of the fastest bond stretch,T_{min}), visible inaccuracies or instabilities can occur. Impulse-MTS schemes are vulnerable to these resonance errors since large energy pulses are introduced to the governing dynamics equations when the slow forces are evaluated. We recently showed that such resonance artifacts can be masked significantly by applying extrapolative splitting to stochastic dynamics. Theoretical and numerical analyses of force-splitting integrators based on the Verlet discretization are reported here for linear models to explain these observations and to suggest how to construct effective integrators for biomolecular dynamics that balance stability with accuracy. Analyses for Newtonian dynamics demonstrate the severe resonance patterns of the Impulse splitting, with this severity worsening with the outer timestep, Δt; Constant Extrapolation is generally unstable, but the disturbances do not grow with Δt. Thus, the stochastic extrapolative combination can counteract generic instabilities and largely alleviate resonances with a sufficiently strong Langevin heat-bath coupling (γ), estimates for which are derived here based on the fastest and slowest motion periods. These resonance results generally hold for nonlinear test systems: a water tetramer and solvated protein. Proposed related approaches such as Extrapolation/Correction and Midpoint Extrapolation work better than Constant Extrapolation only for timesteps less thanT_{min}/2. An effective extrapolative stochastic approach for biomolecules that balances long-timestep stability with good accuracy for the fast subsystem is then applied to a biomolecule using a three-class partitioning: the medium forces are treated byMidpoint Extrapolationvia position Verlet, and the slow forces are incorporated byConstant Extrapolation. The resulting algorithm (LN) performs well on a solvated protein system in terms of thermodynamic properties and yields an order of magnitude speedup with respect to single-timestep Langevin trajectories. Computed spectral density functions also show how the Newtonian modes can be approximated by using a small γ in the range of 5-20 ps^{-1}.

Original language | English (US) |
---|---|

Pages (from-to) | 74-113 |

Number of pages | 40 |

Journal | Journal of Computational Physics |

Volume | 151 |

Issue number | 1 |

DOIs | |

State | Published - May 1 1999 |

### Fingerprint

### Keywords

- Force splitting
- Molecular dynamics
- Multiple timesteps
- Numerical resonance
- Symplecticness

### ASJC Scopus subject areas

- Computer Science Applications
- Physics and Astronomy(all)

### Cite this

**Masking Resonance Artifacts in Force-Splitting Methods for Biomolecular Simulations by Extrapolative Langevin Dynamics.** / Sandu, Adrian; Schlick, Tamar.

Research output: Contribution to journal › Article

*Journal of Computational Physics*, vol. 151, no. 1, pp. 74-113. https://doi.org/10.1006/jcph.1999.6202

}

TY - JOUR

T1 - Masking Resonance Artifacts in Force-Splitting Methods for Biomolecular Simulations by Extrapolative Langevin Dynamics

AU - Sandu, Adrian

AU - Schlick, Tamar

PY - 1999/5/1

Y1 - 1999/5/1

N2 - Numerical resonance artifacts have become recognized recently as a limiting factor to increasing the timestep in multiple-timestep (MTS) biomolecular dynamics simulations. At certain timesteps correlated to internal motions (e.g., 5 fs, around half the period of the fastest bond stretch,Tmin), visible inaccuracies or instabilities can occur. Impulse-MTS schemes are vulnerable to these resonance errors since large energy pulses are introduced to the governing dynamics equations when the slow forces are evaluated. We recently showed that such resonance artifacts can be masked significantly by applying extrapolative splitting to stochastic dynamics. Theoretical and numerical analyses of force-splitting integrators based on the Verlet discretization are reported here for linear models to explain these observations and to suggest how to construct effective integrators for biomolecular dynamics that balance stability with accuracy. Analyses for Newtonian dynamics demonstrate the severe resonance patterns of the Impulse splitting, with this severity worsening with the outer timestep, Δt; Constant Extrapolation is generally unstable, but the disturbances do not grow with Δt. Thus, the stochastic extrapolative combination can counteract generic instabilities and largely alleviate resonances with a sufficiently strong Langevin heat-bath coupling (γ), estimates for which are derived here based on the fastest and slowest motion periods. These resonance results generally hold for nonlinear test systems: a water tetramer and solvated protein. Proposed related approaches such as Extrapolation/Correction and Midpoint Extrapolation work better than Constant Extrapolation only for timesteps less thanTmin/2. An effective extrapolative stochastic approach for biomolecules that balances long-timestep stability with good accuracy for the fast subsystem is then applied to a biomolecule using a three-class partitioning: the medium forces are treated byMidpoint Extrapolationvia position Verlet, and the slow forces are incorporated byConstant Extrapolation. The resulting algorithm (LN) performs well on a solvated protein system in terms of thermodynamic properties and yields an order of magnitude speedup with respect to single-timestep Langevin trajectories. Computed spectral density functions also show how the Newtonian modes can be approximated by using a small γ in the range of 5-20 ps-1.

AB - Numerical resonance artifacts have become recognized recently as a limiting factor to increasing the timestep in multiple-timestep (MTS) biomolecular dynamics simulations. At certain timesteps correlated to internal motions (e.g., 5 fs, around half the period of the fastest bond stretch,Tmin), visible inaccuracies or instabilities can occur. Impulse-MTS schemes are vulnerable to these resonance errors since large energy pulses are introduced to the governing dynamics equations when the slow forces are evaluated. We recently showed that such resonance artifacts can be masked significantly by applying extrapolative splitting to stochastic dynamics. Theoretical and numerical analyses of force-splitting integrators based on the Verlet discretization are reported here for linear models to explain these observations and to suggest how to construct effective integrators for biomolecular dynamics that balance stability with accuracy. Analyses for Newtonian dynamics demonstrate the severe resonance patterns of the Impulse splitting, with this severity worsening with the outer timestep, Δt; Constant Extrapolation is generally unstable, but the disturbances do not grow with Δt. Thus, the stochastic extrapolative combination can counteract generic instabilities and largely alleviate resonances with a sufficiently strong Langevin heat-bath coupling (γ), estimates for which are derived here based on the fastest and slowest motion periods. These resonance results generally hold for nonlinear test systems: a water tetramer and solvated protein. Proposed related approaches such as Extrapolation/Correction and Midpoint Extrapolation work better than Constant Extrapolation only for timesteps less thanTmin/2. An effective extrapolative stochastic approach for biomolecules that balances long-timestep stability with good accuracy for the fast subsystem is then applied to a biomolecule using a three-class partitioning: the medium forces are treated byMidpoint Extrapolationvia position Verlet, and the slow forces are incorporated byConstant Extrapolation. The resulting algorithm (LN) performs well on a solvated protein system in terms of thermodynamic properties and yields an order of magnitude speedup with respect to single-timestep Langevin trajectories. Computed spectral density functions also show how the Newtonian modes can be approximated by using a small γ in the range of 5-20 ps-1.

KW - Force splitting

KW - Molecular dynamics

KW - Multiple timesteps

KW - Numerical resonance

KW - Symplecticness

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U2 - 10.1006/jcph.1999.6202

DO - 10.1006/jcph.1999.6202

M3 - Article

VL - 151

SP - 74

EP - 113

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 1

ER -