^{1,a)}, Donald Hamelberg

^{1}and J. Andrew McCammon

^{1}

### Abstract

Accelerated dynamics is applied to entropy calculations on a set of toy and molecular systems and is found to enhance the rate of convergence.

Support has been provided by the NSF (MCB-0506593 and MCA-93S013 for J.A.M.), the NIH (GM31749 for J.A.M.), the Howard Hughes Medical Institute, the Center for Theoretical Biological Physics, the National Biomedical Computation Resource, San Diego Supercomputing Center, and Accelrys, Inc. D.D.L.M. was supported by the NIH Molecular Biophysics Training Grant at UCSD.

I. INTRODUCTION

II. METHODS

A. Accelerated dynamics

B. Entropy estimation

III. ONE-DIMENSIONAL DOUBLE WELLS

IV. TWO-DIMENSIONAL DOUBLE WELL

V. AN EIGHT-RESIDUE PEPTIDE

VI. CONCLUSIONS

### Key Topics

- Entropy
- 32.0
- Molecular dynamics
- 16.0
- Peptides
- 5.0
- Probability theory
- 5.0
- Brownian dynamics
- 4.0

## Figures

(Color) Quartic double-well potential with a barrier of (thick black line) and several biased potentials. The threshold boost energy is fixed at and is 5, 10, 15, 25, 50, or , in decreasing magnitude of boost.

(Color) Quartic double-well potential with a barrier of (thick black line) and several biased potentials. The threshold boost energy is fixed at and is 5, 10, 15, 25, 50, or , in decreasing magnitude of boost.

(Color) Quartic double-well potential with a barrier of (thick black line) and several biased potentials. The threshold boost energy is fixed at and is 5, 10, 15, 25, 50, or , in decreasing magnitude of boost.

(Color) Mean entropy as a function of elapsed time using normal and accelerated molecular dynamics on a quartic double-well potential with a barrier of . The numerical entropy of 4.05 is shown as a thick black line. Results from normal MD are shown by a dashed blue line and accelerated MD results are solid lines colored as in Fig. 1.

(Color) Mean entropy as a function of elapsed time using normal and accelerated molecular dynamics on a quartic double-well potential with a barrier of . The numerical entropy of 4.05 is shown as a thick black line. Results from normal MD are shown by a dashed blue line and accelerated MD results are solid lines colored as in Fig. 1.

(Color) Standard deviation of entropy estimates as a function of elapsed time using normal and accelerated molecular dynamics on a quartic double-well potential with a barrier of . Results from normal MD are shown by a dashed blue line and accelerated MD results are solid lines colored as in Fig. 1.

(Color) Standard deviation of entropy estimates as a function of elapsed time using normal and accelerated molecular dynamics on a quartic double-well potential with a barrier of . Results from normal MD are shown by a dashed blue line and accelerated MD results are solid lines colored as in Fig. 1.

(Color) Mean entropy as a function of elapsed time using normal and accelerated molecular dynamics on a quartic double-well potential with a barrier of . The numerical entropy of 3.26 is shown as a thick black line. Results from normal MD are shown by a dashed blue line and accelerated MD results are solid lines colored as in Fig. 2.

(Color) Mean entropy as a function of elapsed time using normal and accelerated molecular dynamics on a quartic double-well potential with a barrier of . The numerical entropy of 3.26 is shown as a thick black line. Results from normal MD are shown by a dashed blue line and accelerated MD results are solid lines colored as in Fig. 2.

(Color) Standard deviation of entropy estimates as a function of time using normal and accelerated molecular dynamics on a quartic double-well potential with a barrier of . Results from normal MD are shown by a dashed blue line and accelerated MD results are solid lines colored as in Fig. 2.

(Color) Standard deviation of entropy estimates as a function of time using normal and accelerated molecular dynamics on a quartic double-well potential with a barrier of . Results from normal MD are shown by a dashed blue line and accelerated MD results are solid lines colored as in Fig. 2.

(Color) Two-dimensional double-well potential with a barrier of . The principal components analysis eigenvectors from a normal molecular dynamics simulation are indicated by dashed white lines.

(Color) Two-dimensional double-well potential with a barrier of . The principal components analysis eigenvectors from a normal molecular dynamics simulation are indicated by dashed white lines.

(Color) Mean entropy as a function of elapsed time using normal and accelerated molecular dynamics on a two-dimensional double-well potential with a barrier of . The entropy is calculated using a two-dimensional probability density estimate from a histogram. The numerical entropy of 5.59 is shown as a thick black line. Results from normal MD are shown by a dashed blue line and accelerated MD results are solid lines colored as in Fig. 2.

(Color) Mean entropy as a function of elapsed time using normal and accelerated molecular dynamics on a two-dimensional double-well potential with a barrier of . The entropy is calculated using a two-dimensional probability density estimate from a histogram. The numerical entropy of 5.59 is shown as a thick black line. Results from normal MD are shown by a dashed blue line and accelerated MD results are solid lines colored as in Fig. 2.

(Color) Standard deviation of entropy estimates as a function of time using normal and accelerated molecular dynamics on a two-dimensional double-well potential with a barrier of . The entropy is calculated using a two-dimensional probability density estimate from a histogram. Results from normal MD are shown by a dashed blue line and accelerated MD results are solid lines colored as in Fig. 2.

(Color) Standard deviation of entropy estimates as a function of time using normal and accelerated molecular dynamics on a two-dimensional double-well potential with a barrier of . The entropy is calculated using a two-dimensional probability density estimate from a histogram. Results from normal MD are shown by a dashed blue line and accelerated MD results are solid lines colored as in Fig. 2.

(Color) Mean entropy as a function of elapsed time using normal and accelerated molecular dynamics on a two-dimensional double-well potential with a barrier of . The entropy is calculated using two one-dimensional probability density estimates from histograms of projections into principal components analysis space. The numerical entropy of 5.59 is shown as a thick black line. Results from normal MD are shown by a dashed blue line and accelerated MD results are solid lines colored as in Fig. 2.

(Color) Mean entropy as a function of elapsed time using normal and accelerated molecular dynamics on a two-dimensional double-well potential with a barrier of . The entropy is calculated using two one-dimensional probability density estimates from histograms of projections into principal components analysis space. The numerical entropy of 5.59 is shown as a thick black line. Results from normal MD are shown by a dashed blue line and accelerated MD results are solid lines colored as in Fig. 2.

(Color) Standard deviation of entropy estimates as a function of time using normal and accelerated molecular dynamics on a two-dimensional double-well potential with a barrier of . The entropy is calculated using two one-dimensional probability density estimates from histograms of projections into principal components analysis space. Results from normal MD are shown by a dashed blue line and accelerated MD results are solid lines colored as in Fig. 2.

(Color) Standard deviation of entropy estimates as a function of time using normal and accelerated molecular dynamics on a two-dimensional double-well potential with a barrier of . The entropy is calculated using two one-dimensional probability density estimates from histograms of projections into principal components analysis space. Results from normal MD are shown by a dashed blue line and accelerated MD results are solid lines colored as in Fig. 2.

(Color online) Entropy estimates using normal and accelerated molecular dynamics on an eight-residue peptide. The entropy is calculated using one-dimensional probability density estimates from histograms of projections into principal components analysis space. From bottom to top, the entropy estimates are for normal MD, AMD applied to the peptide torsion angles, AMD applied to both the torsional angles and the total potential, and AMD for a larger boost on both.

(Color online) Entropy estimates using normal and accelerated molecular dynamics on an eight-residue peptide. The entropy is calculated using one-dimensional probability density estimates from histograms of projections into principal components analysis space. From bottom to top, the entropy estimates are for normal MD, AMD applied to the peptide torsion angles, AMD applied to both the torsional angles and the total potential, and AMD for a larger boost on both.

(Color online) Entropy estimates using normal and accelerated molecular dynamics on an eight-residue peptide. The entropy is calculated using quasiharmonic analysis. The key is as in Fig. 12.

(Color online) Entropy estimates using normal and accelerated molecular dynamics on an eight-residue peptide. The entropy is calculated using quasiharmonic analysis. The key is as in Fig. 12.

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