^{1,a)}, Ernesto E. Borrero

^{1,a)}and Fernando A. Escobedo

^{1,b)}

### Abstract

Forward flux sampling (FFS) simulations were used to study the kinetics of alanine dipeptide both in vacuum and in explicit solvent. The recently proposed FFS least-squares estimation approach and an algorithm that optimizes the position of the interfaces were implemented to determine a reaction coordinate that adequately describes the transition dynamics. A new method is also introduced to try to ensure that the ensemble of “starting points” (for the trial trajectories) is properly sampled. The rate constant estimates for the transition of alanine dipeptide in vacuum were used to demonstrate the consistency between Monte Carlo and molecular dynamics (MD) simulations. FFS-MD simulations were then performed for the study of the transition in explicit solvent. The kinetic results for both systems in vacuum and explicit solvent are in general agreement with previous experimental and computational studies for this peptide. In vacuum, an additional dihedral angle besides the one typically used as order parameter is identified as a significant variable in the reaction coordinate model. In solution, several dihedral angles and variables that describe the solvent action on the molecule’s dynamics are found to play a significant role in the description of the system’s dynamics.

The authors are grateful for support from the National Science Foundation, Award No. 0553719, and from an ACS-PRF grant.

I. INTRODUCTION

II. METHODS

A. Forward flux sampling

B. FFS-LSE algorithm

C. Adaptive staging optimization algorithm

1. Interfaces 1 to

2. Interface 0

D. Stochastic thermostat for the FFS-MD simulations

III. SIMULATION DETAILS

A. Alanine dipeptide in vacuum

1. MC scheme

2. MD scheme

B. Alanine dipeptide in explicit solvent

IV. RESULTS

A. Alanine dipeptide in vacuum

B. Alanine dipeptide in water

V. CONCLUSIONS

### Key Topics

- Solvents
- 35.0
- Monte Carlo methods
- 32.0
- Peptides
- 29.0
- Reaction rate constants
- 19.0
- Free energy
- 9.0

## Figures

A model for alanine dipeptide. Also shown are the main dihedral angles: , , , and . Carbon, oxygen, nitrogen, and hydrogen atoms are depicted in light green, red, blue, and gray, respectively.

A model for alanine dipeptide. Also shown are the main dihedral angles: , , , and . Carbon, oxygen, nitrogen, and hydrogen atoms are depicted in light green, red, blue, and gray, respectively.

A schematic view of the generation of branched paths (thick lines) using the BG sampling method. The first stage involves the simulation run in the basin shown by a dotted line. Starting points for the subsequent generation of branched paths are marked with a black circle at . The second stage corresponds to the trial runs fired from ; those that reached the next interface are shown by a thick line and those which failed to reach are shown by a dotted line. For this example, having , , and , the value for point 1 at is then obtained recursively from Eq. (3): .

A schematic view of the generation of branched paths (thick lines) using the BG sampling method. The first stage involves the simulation run in the basin shown by a dotted line. Starting points for the subsequent generation of branched paths are marked with a black circle at . The second stage corresponds to the trial runs fired from ; those that reached the next interface are shown by a thick line and those which failed to reach are shown by a dotted line. For this example, having , , and , the value for point 1 at is then obtained recursively from Eq. (3): .

(a) Distribution for the center of mass velocity of water molecules for MD simulations using (−) thermostat A and (●) Nosé–Hoover thermostat. (b) Time progression of the VACFs for thermostat A (−) and (●) Nosé–Hoover thermostat.

(a) Distribution for the center of mass velocity of water molecules for MD simulations using (−) thermostat A and (●) Nosé–Hoover thermostat. (b) Time progression of the VACFs for thermostat A (−) and (●) Nosé–Hoover thermostat.

Free energy landscape for blocked alanine dipeptide in vacuum at 300 K. The color scheme for the visited states changes from highest (green) to lowest (gray/blue) elevations.

Free energy landscape for blocked alanine dipeptide in vacuum at 300 K. The color scheme for the visited states changes from highest (green) to lowest (gray/blue) elevations.

Free energy landscape for alanine dipeptide in explicit solvent at 300 K. The color scheme for the visited states changes from highest (gray/light blue) to lowest (black/red) elevations.

Free energy landscape for alanine dipeptide in explicit solvent at 300 K. The color scheme for the visited states changes from highest (gray/light blue) to lowest (black/red) elevations.

Results for the FFS-MC simulations in vacuum at 300 K. (a) Free energy profile along the dihedral angle as order parameter. The dotted line corresponds to the value of at the TS. (b) Free energy landscape ( plane) where the color scheme for the visited states changes from highest (gray/light blue) to lowest (black/dark blue) elevations. The solid (black) lines correspond to the initial order parameter (state upper limit), 125 (TS), and 150 (state lower limit). The dotted (red) lines correspond to the isocommittor surface.

Results for the FFS-MC simulations in vacuum at 300 K. (a) Free energy profile along the dihedral angle as order parameter. The dotted line corresponds to the value of at the TS. (b) Free energy landscape ( plane) where the color scheme for the visited states changes from highest (gray/light blue) to lowest (black/dark blue) elevations. The solid (black) lines correspond to the initial order parameter (state upper limit), 125 (TS), and 150 (state lower limit). The dotted (red) lines correspond to the isocommittor surface.

Results for the FFS-MD simulations in vacuum at 300 K. (a) Free energy profile along the dihedral angle as order parameter. (b) Free energy landscape over the plane. Color and line schemes are the same as those indicated in the caption of Fig. 6.

Results for the FFS-MD simulations in vacuum at 300 K. (a) Free energy profile along the dihedral angle as order parameter. (b) Free energy landscape over the plane. Color and line schemes are the same as those indicated in the caption of Fig. 6.

Density map obtained from the TPE for several FFS-MD runs for the reaction at 300 K in explicit solvent. The color scheme for the visited states changes from most (red) to least (light blue) visited region. The solid (black) lines correspond to and . Three representative trajectories are also shown.

Density map obtained from the TPE for several FFS-MD runs for the reaction at 300 K in explicit solvent. The color scheme for the visited states changes from most (red) to least (light blue) visited region. The solid (black) lines correspond to and . Three representative trajectories are also shown.

Results for the optimization process of the positioning in the FFS-MD simulation for the reaction at 300 K in explicit solvent: (a) ACFs for the angle for states collected at , 125, 120, 115, and 110 and (b) (◆) , (●) (picoseconds), and (▲) (picoseconds) curves as a function of the location of .

Results for the optimization process of the positioning in the FFS-MD simulation for the reaction at 300 K in explicit solvent: (a) ACFs for the angle for states collected at , 125, 120, 115, and 110 and (b) (◆) , (●) (picoseconds), and (▲) (picoseconds) curves as a function of the location of .

Isocommittor surfaces obtained during the FFS-MD simulations for the reaction at 300 K in explicit solvent. The solid (black) lines correspond to and . The color scheme changes from highest (light blue) to lowest (red) elevations. The isocommittor surfaces (see Table VI) are shown for fixed values of and (solid red lines) and for fixed values of and (dotted black lines). For the data considered, is the average value observed, and are the [lower, upper] limits of the range of values observed.

Isocommittor surfaces obtained during the FFS-MD simulations for the reaction at 300 K in explicit solvent. The solid (black) lines correspond to and . The color scheme changes from highest (light blue) to lowest (red) elevations. The isocommittor surfaces (see Table VI) are shown for fixed values of and (solid red lines) and for fixed values of and (dotted black lines). For the data considered, is the average value observed, and are the [lower, upper] limits of the range of values observed.

## Tables

Optimized sets for vacuum and explicit solvent FFS-MC and FFS-MD simulations.

Optimized sets for vacuum and explicit solvent FFS-MC and FFS-MD simulations.

Optimized move set for MC simulation in vacuum (Ref. 25). Parameters for the automatic optimization of move sizes (ARM and DOMC) are also given.

Optimized move set for MC simulation in vacuum (Ref. 25). Parameters for the automatic optimization of move sizes (ARM and DOMC) are also given.

LSE parameters and ANOVA for the reaction coordinate model of the FFS-MC simulation in vacuum. The and angles are given in radians.

LSE parameters and ANOVA for the reaction coordinate model of the FFS-MC simulation in vacuum. The and angles are given in radians.

LSE parameters and ANOVA for the reaction coordinate model of the FFS-MD simulation in vacuum. The and angles are given in radians.

LSE parameters and ANOVA for the reaction coordinate model of the FFS-MD simulation in vacuum. The and angles are given in radians.

LSE parameters and ANOVA for the reaction coordinate model of the slower reaction of a FFS-MD simulation in explicit solvent. The and angles are given in radians. and are given in Å and kcal/mol, respectively.

LSE parameters and ANOVA for the reaction coordinate model of the slower reaction of a FFS-MD simulation in explicit solvent. The and angles are given in radians. and are given in Å and kcal/mol, respectively.

LSE parameters and ANOVA for the reaction coordinate model of the faster reaction from a FFS-MD simulation in explicit solvent. The and angles are given in radians. and are given in Å and kcal/mol, respectively.

LSE parameters and ANOVA for the reaction coordinate model of the faster reaction from a FFS-MD simulation in explicit solvent. The and angles are given in radians. and are given in Å and kcal/mol, respectively.

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