Renewal theory for Brownian motion with stochastically gated targets

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 target surface and initiating a downstream event (absorption). However, prior to absorption, the particle may undergo several rounds of surface attachment (adsorption), detachment (desorption) and diffusion. That is, the surface is effectively ‘sticky’. 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 attachment to the surface as a renewal event. In this paper we develop a renewal theory for stochastically gated target problems along analogous lines to previous work on sticky targets. We proceed by constructing a first renewal equation that relates the joint probability density for particle position and the state of a gate 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. In order to ensure that diffusion restarts in a state that avoids immediate re-adsorption, we assume that whenever the particle reaches a closed boundary it is instantan eously shifted a distance ϵ from the boundary (desorption-induced resetting). We explicitly solve the renewal equation in the one-dimensional case and show how the solution to the original gated FPT problem is recovered in the limit ϵ → 0. We then calculate the MFPT for absorption (assuming it exists) and determine its dependence on ϵ and the switching rate of the gate. We also show how spectral methods can be used to solve the renewal equation in higher spatial dimensions. We thus establish renewal theory as a general mathematical framework for modelling both sticky and stochastically gated targets