^{1}and Peter G. Bolhuis

^{1}

### Abstract

The multiple state transition path sampling method allows sampling of rare transitions between many metastable states, but has the drawback that switching between qualitatively different pathways is difficult. Combination with replica exchange transition interface sampling can in principle alleviate this problem, but requires a large number of simultaneous replicas. Here we remove these drawbacks by introducing a single replica sampling algorithm that samples only one interface at a time, while efficiently walking through the entire path space using a Wang-Landau approach or, alternatively, a fixed bias. We illustrate the method on several model systems: a particle diffusing in a simple 2D potential, isomerization in a small Lennard Jones cluster, and isomerization of the alanine dipeptide in explicit water.

The authors thank David Swenson for a careful reading of the manuscript. This work is part of the research programme VICI 700.58.442, which is financed by the Netherlands Organization for Scientific Research (NWO).

I. INTRODUCTION

II. THEORETICAL BACKGROUND

A. Multiple state transition interface sampling

B. Single replica MSTIS

C. Wang-Landau biasing in single replica MSTIS

D. The Wang-Landau MSTIS algorithm

1. Initialization phase

2. Main loop

E. Fixed bias in single replica MSTIS

F. Adaptive sampling

III. RESULTS AND DISCUSSION

A. Langevin dynamics in a simple 2D potential

B. Adaptive sampling on the LJ7 cluster

C. Alanine dipeptide

IV. CONCLUSIONS

### Key Topics

- Electron densities of states
- 5.0
- Free energy
- 5.0
- Peptides
- 5.0
- Databases
- 4.0
- Isomerization
- 4.0

## Figures

Cartoon of the exchange of interfaces and state swaps for a 4-state system, where each state has two interfaces. The initial path in the left top corner crosses not only the current (black) interface, but also the outermost interface so an exchange is allowed. A shooting move creates a path toward another state, so that state swap is possible. The current interface is then the outermost interface of the new state. Another interface exchange and a subsequent shot creates the red path in the lower left corner cartoon.

Cartoon of the exchange of interfaces and state swaps for a 4-state system, where each state has two interfaces. The initial path in the left top corner crosses not only the current (black) interface, but also the outermost interface so an exchange is allowed. A shooting move creates a path toward another state, so that state swap is possible. The current interface is then the outermost interface of the new state. Another interface exchange and a subsequent shot creates the red path in the lower left corner cartoon.

Concept of the adaptive Wang-Landau MSTIS scheme. Sampling the outermost interface of the initial state (left) creates an activated path that finds another basin, which results in a very long trial trajectory (middle). This path is analyzed, and if the new state is stable it is treated as a new state with its own set of interfaces. This process is repeated, and new states are added to the database as needed (right).

Concept of the adaptive Wang-Landau MSTIS scheme. Sampling the outermost interface of the initial state (left) creates an activated path that finds another basin, which results in a very long trial trajectory (middle). This path is analyzed, and if the new state is stable it is treated as a new state with its own set of interfaces. This process is repeated, and new states are added to the database as needed (right).

The 2D potential with the 6 minima plotted in the x,y plane. Samples of the Wang-Landau MSTIS paths at β = 4 are drawn on top. The dynamically unbiased paths connect all states.

The 2D potential with the 6 minima plotted in the x,y plane. Samples of the Wang-Landau MSTIS paths at β = 4 are drawn on top. The dynamically unbiased paths connect all states.

Decorrelation in the Wang-Landau MSTIS sampling. Top: The state index as a function of simulation time measured in number of shooting moves. The dotted line shows the results when the DOP is initialized to zero and solid line shows the result for a converged DOP. Note that the decorrelation is almost equal. Bottom: The decorrelation between stable state minima. The main window shows the histogram of interval lengths between stable state visits (replica 0). The inset shows the histogram of interval lengths between stable state visits for the whole ensemble. Note that this interval is naturally much shorter than the one between replica 0, as the replica has to diffuse through the whole interface space.

Decorrelation in the Wang-Landau MSTIS sampling. Top: The state index as a function of simulation time measured in number of shooting moves. The dotted line shows the results when the DOP is initialized to zero and solid line shows the result for a converged DOP. Note that the decorrelation is almost equal. Bottom: The decorrelation between stable state minima. The main window shows the histogram of interval lengths between stable state visits (replica 0). The inset shows the histogram of interval lengths between stable state visits for the whole ensemble. Note that this interval is naturally much shorter than the one between replica 0, as the replica has to diffuse through the whole interface space.

Left: Convergence of the DOP in the course of the Wang-Landau MSTIS simulation. The replica index is defined as Im + i. Middle: Comparison of the logarithmic crossing probability ln P(λ) for the six states as a function of λ between the replica exchange MSTIS (dotted curves) and Wang-Landau MSTIS (solid curves) simulations. For each state the two approaches result in an identical crossing probability within the error bar (shown for 2 points). Right: The density of paths in the Wang-Landau algorithm converges to crossing probability.

Left: Convergence of the DOP in the course of the Wang-Landau MSTIS simulation. The replica index is defined as Im + i. Middle: Comparison of the logarithmic crossing probability ln P(λ) for the six states as a function of λ between the replica exchange MSTIS (dotted curves) and Wang-Landau MSTIS (solid curves) simulations. For each state the two approaches result in an identical crossing probability within the error bar (shown for 2 points). Right: The density of paths in the Wang-Landau algorithm converges to crossing probability.

The RPE contains both information about the free energy and the kinetics. Left: FE projections of the RPE in the x, y plane. Note that this projection is naturally equal to the potential V(x, y). Right: Projection of the committor p 3 in the x,y plane. Purple denotes p 3 = 0 and white denotes p 3 = 1.

The RPE contains both information about the free energy and the kinetics. Left: FE projections of the RPE in the x, y plane. Note that this projection is naturally equal to the potential V(x, y). Right: Projection of the committor p 3 in the x,y plane. Purple denotes p 3 = 0 and white denotes p 3 = 1.

Convergence of the equilibrium populations as a function of simulation time. Left: Fixed bias. Right: Wang-Landau MSTIS scheme.

Convergence of the equilibrium populations as a function of simulation time. Left: Fixed bias. Right: Wang-Landau MSTIS scheme.

The configurations corresponding to the stable states in the LJ7 cluster. From left to right: the pentagonal bipiramid, the capped octahedron, bicapped trigonal bipyramid, and tricapped tetrahedron. 38,39

The configurations corresponding to the stable states in the LJ7 cluster. From left to right: the pentagonal bipiramid, the capped octahedron, bicapped trigonal bipyramid, and tricapped tetrahedron. 38,39

Buildup of the density of paths for the LJ7 cluster using the adaptive Wang-Landau MSTIS scheme. The index is defined as Im + i.

Buildup of the density of paths for the LJ7 cluster using the adaptive Wang-Landau MSTIS scheme. The index is defined as Im + i.

Converged crossing probability (curves) and density of paths (symbols) for the LJ7 cluster.

Converged crossing probability (curves) and density of paths (symbols) for the LJ7 cluster.

Tetrahedral projection of the trajectories consisting of the 4 square distance sum order parameters. The projection is such that the corners represent exact correspondence to the target structure, whereas the center is infinitely away from all structures.

Tetrahedral projection of the trajectories consisting of the 4 square distance sum order parameters. The projection is such that the corners represent exact correspondence to the target structure, whereas the center is infinitely away from all structures.

The molecular structure of alanine dipeptide rendered in licorice. Carbons in cyan, oxygen in red, nitrogen in blue, and hydrogen in white. The two order parameters describing the metastable states are the dihedral angles ϕ and ψ.

The molecular structure of alanine dipeptide rendered in licorice. Carbons in cyan, oxygen in red, nitrogen in blue, and hydrogen in white. The two order parameters describing the metastable states are the dihedral angles ϕ and ψ.

The free energy surface from a replica exchange MD simulation obtained by projecting the logarithm of the probability to find a (ϕ, ψ) pair. 43

The free energy surface from a replica exchange MD simulation obtained by projecting the logarithm of the probability to find a (ϕ, ψ) pair. 43

Logarithm of crossing probability against interface for states β, α, E, and F. The red, green, and blue lines are the last three DOP biasing functions, while the black line with circles is the final logarithm of the crossing probability.

Logarithm of crossing probability against interface for states β, α, E, and F. The red, green, and blue lines are the last three DOP biasing functions, while the black line with circles is the final logarithm of the crossing probability.

## Tables

Sorted squared distance sum for the four minima.

Sorted squared distance sum for the four minima.

(a) Outermost interface transition matrix for the four states. Subscripts denote the error in the last digits. Columns are leaving states and rows are arriving states. (b) Flux through the first interface for the four states.

(a) Outermost interface transition matrix for the four states. Subscripts denote the error in the last digits. Columns are leaving states and rows are arriving states. (b) Flux through the first interface for the four states.

Rate matrix for the four states. Subscripts denote the error in the last digits. Columns are leaving states and rows are arriving states.

Rate matrix for the four states. Subscripts denote the error in the last digits. Columns are leaving states and rows are arriving states.

Definition of interfaces for states β, α, E, and F.

Definition of interfaces for states β, α, E, and F.

Rate constant data. (a) Flux through first interface, crossing probability from the first to the last interface, and flux at the outermost interface for each state. (b) Transition probability for states β, α, E, and F at their respective outer interfaces. The rows denote the leaving state and the columns denote the arriving state. (c) Rate constant matrix from single replica MSTIS for states β, α, E, and F.

Rate constant data. (a) Flux through first interface, crossing probability from the first to the last interface, and flux at the outermost interface for each state. (b) Transition probability for states β, α, E, and F at their respective outer interfaces. The rows denote the leaving state and the columns denote the arriving state. (c) Rate constant matrix from single replica MSTIS for states β, α, E, and F.

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