^{1}, Daan Frenkel

^{1}and Pieter Rein ten Wolde

^{1,a)}

### Abstract

We present three algorithms for calculating rate constants and sampling transition paths for rare events in simulations with stochastic dynamics. The methods do not require *a priori* knowledge of the phase-space density and are suitable for equilibrium or nonequilibrium systems in stationary state. All the methods use a series of interfaces in phase space, between the initial and final states, to generate transition paths as chains of connected partial paths, in a ratchetlike manner. No assumptions are made about the distribution of paths at the interfaces. The three methods differ in the way that the transition path ensemble is generated. We apply the algorithms to kinetic Monte Carlo simulations of a genetic switch and to Langevin dynamics simulations of intermittently driven polymer translocation through a pore. We find that the three methods are all of comparable efficiency, and that all the methods are much more efficient than brute-force simulation.

The authors thank Peter Bolhuis, Eduardo Sanz, Chantal Valeriani, and Patrick B. Warren for many useful discussions. We also thank Chantal Valeriani for her very helpful reading of the manuscript and Ovidiu Radulescu for pointing out to us Ref. 15. This work is part of the research program of the “Stichting voor Fundamenteel Onderzoek der Materie (FOM),” which is financially supported by the “Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO).” One of the authors (R.J.A.) was funded by the European Union Marie Curie program.

I. INTRODUCTION

II. THEORETICAL BACKGROUND

III. ALGORITHMS

A. The forward flux sampling method

B. The branched growth method

C. The Rosenbluth method

D. Pruning

IV. APPLICATIONS

A. Genetic switch

B. Driven polymer translocation through a pore

V. DISCUSSION

### Key Topics

- Polymers
- 48.0
- Proteins
- 12.0
- Genetic switches
- 11.0
- Monte Carlo methods
- 5.0
- Reaction rate constants
- 5.0

## Figures

Schematic illustration of the definition of regions and and the interfaces (here, ). Three transition paths are shown.

Schematic illustration of the definition of regions and and the interfaces (here, ). Three transition paths are shown.

The first (a) and second (b) stages of the FFS method. The distribution of points at the interfaces depends on the history of the paths, as illustrated by the dashed lines in (b). The circles are members of the collection of points at the interfaces .

The first (a) and second (b) stages of the FFS method. The distribution of points at the interfaces depends on the history of the paths, as illustrated by the dashed lines in (b). The circles are members of the collection of points at the interfaces .

Schematic illustration of the extraction of the transition path ensemble from the FFS procedure. All partial paths that reach the subsequent interface are shown. Partial paths that do not contribute to the TPE are shown by dotted lines. The solid lines correspond to the TPE; the width of the line indicates the weight of the contribution of a particular partial path to the TPE.

Schematic illustration of the extraction of the transition path ensemble from the FFS procedure. All partial paths that reach the subsequent interface are shown. Partial paths that do not contribute to the TPE are shown by dotted lines. The solid lines correspond to the TPE; the width of the line indicates the weight of the contribution of a particular partial path to the TPE.

A schematic view of the generation of a branched path (thick lines) using the branched growth sampling method. The simulation run in the basin is shown by a dotted line. Trial runs which fail to reach are shown by thin lines. The generation of the initial point for the next path is also shown.

A schematic view of the generation of a branched path (thick lines) using the branched growth sampling method. The simulation run in the basin is shown by a dotted line. Trial runs which fail to reach are shown by thin lines. The generation of the initial point for the next path is also shown.

A schematic view of the generation of a transition path using the Rosenbluth sampling method. The simulation run in the basin is shown by a dotted line. The transition path is shown by bold lines. Trial runs which do not form part of the transition path are shown by thin lines. The generation of the next starting point at is also illustrated.

A schematic view of the generation of a transition path using the Rosenbluth sampling method. The simulation run in the basin is shown by a dotted line. The transition path is shown by bold lines. Trial runs which do not form part of the transition path are shown by thin lines. The generation of the next starting point at is also illustrated.

Two transition paths generated by the Rosenbluth method. The bottom path must be reweighted by a factor of 9 relative to the top path.

Two transition paths generated by the Rosenbluth method. The bottom path must be reweighted by a factor of 9 relative to the top path.

Reaction scheme for the genetic switch. Proteins A and B can dimerize and bind to the DNA at the operator site, . When is bound to , is not produced, and when is bound to , is not produced. Both proteins are degraded in the monomer form. Forward and backward rate constants and are identical for A and B.

Reaction scheme for the genetic switch. Proteins A and B can dimerize and bind to the DNA at the operator site, . When is bound to , is not produced, and when is bound to , is not produced. Both proteins are degraded in the monomer form. Forward and backward rate constants and are identical for A and B.

as a function of time (in units of ) for a typical simulation run.

as a function of time (in units of ) for a typical simulation run.

(a) An illustration of the polymer simulation. (b) A “zoomed in” view, showing the three regions used to define , in Eq. (13).

(a) An illustration of the polymer simulation. (b) A “zoomed in” view, showing the three regions used to define , in Eq. (13).

(a) as a function of time (in units of ) for a typical brute-force simulation run.

(a) as a function of time (in units of ) for a typical brute-force simulation run.

Illustration of the division of a path into partial paths. (a) A path which begins in and reaches . Points 2–6 belong to the partial path , points 7–17 to , and points 18–42 to . (b) A path which begins and ends in . Partial paths are coded as follows: open circles, squares, and triangles.

Illustration of the division of a path into partial paths. (a) A path which begins in and reaches . Points 2–6 belong to the partial path , points 7–17 to , and points 18–42 to . (b) A path which begins and ends in . Partial paths are coded as follows: open circles, squares, and triangles.

## Tables

Path sampling and brute-force (BF) results for , , and . The brute-force result is obtained by fitting as described in the text. is the approximate number of simulation steps performed in obtaining the result in the table.

Path sampling and brute-force (BF) results for , , and . The brute-force result is obtained by fitting as described in the text. is the approximate number of simulation steps performed in obtaining the result in the table.

Path sampling and brute-force results for , , and . Units of and are . The brute-force result is obtained by fitting as described in the text. The errors represent the standard error in the mean of a series of independent estimates. is the approximate number of simulation steps performed in arriving at the result given in the table.

Path sampling and brute-force results for , , and . Units of and are . The brute-force result is obtained by fitting as described in the text. The errors represent the standard error in the mean of a series of independent estimates. is the approximate number of simulation steps performed in arriving at the result given in the table.

Path sampling and brute-force results for , and , for the polymer translocation problem with the altered parameter set. Units of and are . The errors represent the standard error in the mean of a series of independent estimates. is the approximate number of simulation steps performed in arriving at the result given in the table.

Path sampling and brute-force results for , and , for the polymer translocation problem with the altered parameter set. Units of and are . The errors represent the standard error in the mean of a series of independent estimates. is the approximate number of simulation steps performed in arriving at the result given in the table.

FFS and brute-force results for , , and , for the polymer translocation problem of Sec. IV B, with pruning probability at all interfaces. Units of and are . The errors represent the standard error in the mean of a series of independent estimates. is the approximate number of simulation steps performed in arriving at the result given in the table

FFS and brute-force results for , , and , for the polymer translocation problem of Sec. IV B, with pruning probability at all interfaces. Units of and are . The errors represent the standard error in the mean of a series of independent estimates. is the approximate number of simulation steps performed in arriving at the result given in the table

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