Fixed-density boundary conditions in overdamped Langevin simulations of diffusion in channels

TL;DR

This paper introduces a new algorithm for simulating overdamped Langevin particles in channels with fixed concentration boundary conditions, avoiding previous complexities and accurately matching theoretical results.

Contribution

The authors develop a boundary condition implementation method for overdamped Langevin simulations that does not require reservoirs, velocity variables, or adjustable parameters.

Findings

Algorithm accurately reproduces theoretical concentration profiles

Simulations match theoretical flux in nonequilibrium conditions

Applicable to complex ionic channels and similar models

Abstract

We consider the numerical integration of Langevin equations for particles in a channel, in the presence of boundary conditions fixing the concentration values at the ends. This kind of boundary condition appears for instance when considering the diffusion of ions in molecular channels, between the different concentrations at both sides of the cellular membrane. For this application the overdamped limit of Brownian motion (leading to a first order Langevin equation) is most convenient, but in previous works some difficulties associated with this limit were found for the implementation of the boundary conditions. We derive here an algorithm that, unlike previous attempts, does not require the simulation of particle reservoirs or the consideration of velocity variables or adjustable parameters. Simulations of Brownian particles in simple cases show that results agree perfectly with theory, both for the local concentration values and for the resulting particle flux in nonequilibrium situations. The algorithm is appropriate for the modeling of more complex ionic channels and, in general, for the treatment of analogous boundary conditions in other physical models using first order Langevin equations. ***This version corrects misprints in two equations of the published paper***