Numerical computation of effective thermal equilibrium in Stochastically Switching Langevin Systems

TL;DR

The quasipotential is derived that defines this effective equilibrium for a general overdamped Langevin system with a force switching according to a continuous-time Markov chain process for stability and transitions in switching systems.

Abstract

Stochastically switching force terms appear frequently in models of biological systems under the action of active agents such as proteins. The interaction of switching force and Brownian motion can create an "effective thermal equilibrium" even though the system does not obey a potential function. In order to extend the field of energy landscape analysis to understand stability and transitions in switching systems, we derive the quasipotential that defines this effective equilibrium for a general overdamped Langevin system with a force switching according to a continuous-time Markov chain process. Combined with the string method for computing most-probable transition paths, we apply our method to an idealized system and show the appearance of previously unreported numerical challenges. We present modifications to the algorithms to overcome these challenges, and show validity by demonstrating agreement between our computed quasipotential barrier and asymptotic Monte Carlo transition times in the system.

Figures (5)

• Figure 1: (Color Online) Comparison of Quasipotential and Escape Time Asymptotics for three different affinity functions. (A) Comparison of quasipotential along the string with the deterministic average, illustrating that the deterministic average would not agree with Monte Carlo statistics. (B) Average Monte Carlo escape times (points) showing linear behavior whose slope is predicted by the quasipotential string.
• Figure 2: (Color Online) (A) Trajectories taken by three beads over the first 20000 timesteps of a simulation. (B) Plot of pairwise distances between the three beads over time. Observe that at any time, one distance is small (between the two currently bound beads) and the other two distances are large, with which pair is bound rapidly switching. (C) Illustration of a transition between two bound states. The bound beads separate, arranging into a line. The line then morphs into a triangle. Finally, two beads again approach and enter a bound state.
• Figure 3: (Color Online) (A) Maximum change over all images from previous iteration, showing convergence only in the case of 10 images. (B) Quasipotential barrier height over iterations, showing that the $n=10$ converges to a barrier height of approximately $0.011$.
• Figure 4: (Color Online) Visualization of escape path computed using quasipotential string descent, connecting bound state (\ref{['fig:3bead_model']}C) to saddle leading to line (\ref{['fig:3bead_model']}D). Note that this transition happens entirely on the y-axis. (A) Trajectory shown in 2 dimensions. (B) Plot of just the y-coordinate varying along the string. Note that the behavior is not simply linear, especially visible in the case of bead 2 (green).
• Figure 5: (Color Online) (A) Quasipotential (solid) and deterministic energy (dotted) along transition path from \ref{['fig:string_main']}. Quasipotential barrier height is approximately $0.011$. (B) Comparison of asymptotic escape times computed via Monte Carlo simulation to slope taken from quasipotential barrier height.