Renewal theory for Brownian motion with stochastically gated targets

TL;DR

This paper develops a renewal theory framework for analyzing first passage time problems involving Brownian particles interacting with stochastically gated targets, extending previous models to include gating dynamics.

Contribution

It introduces a renewal equation approach for stochastic gating in FPT problems, unifying adsorption/desorption and gating mechanisms in a mathematical framework.

Findings

Renewal theory effectively models gated FPT problems.

The framework decomposes paths into diffusion and gating events.

Examples demonstrate the versatility of the approach.

Abstract

There are a wide range of first passage time (FPT) problems in the physical and life sciences that can be modelled in terms of a Brownian particle binding to a reactive surface (absorption). However, prior to absorption, the particle may undergo several rounds of surface attachment (adsorption), detachment (desorption) and diffusion. Alternatively, the surface may be stochastically gated so that absorption can only occur when the gate is open. In both cases one can view each return to the surface as a renewal event. In this paper we develop a renewal theory for stochastically gated FPT problems along analogous lines to previous work on adsorption/desorption processes. We proceed by constructing a renewal equation that relates the joint probability density for particle position and the state of a gate (or multiple gates) to the probability density and FPT density for a totally absorbing (non-gated) boundary. This essentially decomposes sample paths into an alternating sequence of bulk diffusion and instantaneous adsorption/desorption events, which is terminated when adsorption coincides with an open gate. Through a variety of examples, we show how renewal theory provides a general mathematical framework for incorporating stochastic gating into FPT problems.