# Accelerating Kinetics with Time-Reversal Path Sampling

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Results

## 3. Numerical Results

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematics on accelerating kinetics with time reversibility. (

**a**) The construction of the path ensemble. A very long trajectory (blue line) is cut into some short paths (segments) using two cutting planes A and B located at the basin minima of the free-energy profile (brown line). The cutting points are marked with red crossings. Filled green circles represent the conformations sampled on the trajectory with a fixed time step ($dt$) and violet arrows indicate their velocities. (

**b**) A collection of paths crossing the transition state (TS). Starting from an initial conformation (bigger circles with red edge) (obeying equilibrium distribution) within the TS range $\left[T{S}_{-},T{S}_{+}\right]$ (indicated by dashed–dotted lines), forward and backward simulations with opposite initial velocities are conducted until they reach any cutting planes (A or B), and, with the time-reversal symmetry, they can be assembled to give a path passing the TS range.

**Figure 2.**The free-energy profiles of the protein CI2 as a function of the number of native contacts ($Q$) at different reduced temperatures (from top to bottom): $T=0.9,0.855,0.8\epsilon /{k}_{\mathrm{B}}$ (where $\epsilon $ is the native contact energy strength). Two choices of transition-state (TS) regions at around $Q$= 50 and $Q=80$ are indicated by thin black lines.

**Figure 3.**Accelerating folding/unfolding kinetics of CI2 with the tRPS method. Folding (right branches) and unfolding (left branches) rates were measured in a unit of ${\left(\Delta t\right)}^{-1}$, where $\Delta t$ is the MD time step. The temperature $T$ was measured in a unit of $\epsilon /{k}_{\mathrm{B}}$. The folding/unfolding rates of direct simulations were plotted in blue squares, while those obtained using tRPS (Equation (7)) were shown in red circles (with ${Q}_{\mathrm{TS}}\in \left[80,81\right]$) and violet triangles (with ${Q}_{\mathrm{TS}}\in \left[50,51\right]$). The sampling rates of transition paths were plotted in scattered green diamonds to demonstrate the acceleration effect. Each datapoint of direct simulations was averaged from about 400 folding/unfolding runs, while those of tRPS were each averaged from about 4000 paths.

**Figure 4.**Properties of transition paths. (

**a**) The average duration of transition paths (${t}_{\mathrm{TPath}}$) as a function of $1/T$, which obeys an exponential law (solid line). (

**b**–

**d**) The distributions of ${t}_{\mathrm{TPath}}$ (

**b**), the duration of a path spent within the TS region (${t}_{\mathrm{TS}}$) (

**c**) and the number of times a path crosses the TS region (

**d**) at the midpoint temperature $T=0.855$. Solid lines are a quadratic fit in (

**b**) and linear fit in (

**c**,

**d**).

**Figure 5.**Distribution of minimal/maximal $Q$ for A–A and B–B paths at $T=0.855$. ${Q}_{\mathrm{A}}=20$ and ${Q}_{\mathrm{B}}=110$ were used in cutting paths. Filled symbols represent datapoints from direct simulations, while open symbols for those from tRPS with ${Q}_{\mathrm{TS}}\in \left[50,51\right]$ (down triangles) and ${Q}_{\mathrm{TS}}\in \left[80,81\right]$ (up triangles).

**Figure 6.**Equilibrium thermodynamics and kinetics of protein folding/unfolding for acylphosphatase. (

**a**) The free-energy profiles at different reduced temperatures (from top to bottom): $T=0.95,0.925,0.913,0.9,0.875\epsilon /{k}_{\mathrm{B}}$. (

**b**) Accelerating kinetics with tRPS. The temperature $T$ was measured in a unit of $\epsilon /{k}_{\mathrm{B}}$. The folding/unfolding rates of direct simulations were plotted in filled squares/circles, while those obtained using tRPS were shown in open squares/circles (with ${Q}_{\mathrm{TS}}\in \left[100,102\right]$). The sampling rates of transition paths were plotted in scattered diamonds to demonstrate the acceleration effect.

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**MDPI and ACS Style**

Liu, Z.
Accelerating Kinetics with Time-Reversal Path Sampling. *Molecules* **2023**, *28*, 8147.
https://doi.org/10.3390/molecules28248147

**AMA Style**

Liu Z.
Accelerating Kinetics with Time-Reversal Path Sampling. *Molecules*. 2023; 28(24):8147.
https://doi.org/10.3390/molecules28248147

**Chicago/Turabian Style**

Liu, Zhirong.
2023. "Accelerating Kinetics with Time-Reversal Path Sampling" *Molecules* 28, no. 24: 8147.
https://doi.org/10.3390/molecules28248147