Renewal theory for Brownian motion across a stochastically gated interface

Abstract Stochastically gated interfaces play an important role in a variety of cellular transport processes, including diffusion through membrane ion channels and intercellular gap junctions. Most studies of stochastically-gated interfaces are based on macroscopic models that track the particle concentration averaged with respect to different realisations of the gate dynamics. In this paper we develop a novel probabilistic model of single-particle Brownian motion (BM) through a stochastically gated interface. We proceed by constructing a renewal equation for one-dimensional BM with an interface at the origin, which effectively sews together a sequence of BMs on the half-line with a totally absorbing boundary at x = 0. Each time the particle is absorbed, the stochastic process is immediately restarted according to the following rule: if the gate is closed then BM restarts on the same side of the interface, whereas if the gate is open then BM restarts on either side of the interface with equal probability. In order to ensure that diffusion restarts in a state that avoids immediate re-absorption. we assume that whenever the particle reaches the interface it is instantaneously shifted a distance ε from the origin. We explicitly solve the renewal equation for ε > 0 and show how the solution of a corresponding forward Kolmogorov equation is recovered in the limit ϵ → 0 . However, the renewal equation provides a more general mathematical framework for modelling a stochastically gated interface by explicitly separating the first passage time problem of detecting the gated interface (absorption) and the subsequent rule for restarting BM. We illustrate this by calculating the non-equilibrium stationary state across an interface in the presence of stochastic resetting. We conclude by discussing some of the mathematical challenges in extending the theory to higher-dimensional interfaces.