Renewal theory for Brownian motion with stochastically gated targets

Abstract

There are a wide range of first passage time (FTP) 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 FTP 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 FTP problems.

Figures (8)

• Figure 1: a) Single-particle diffusion in a bounded domain $\Omega$ containing a target ${\mathcal{U}}\subset \Omega$. The exterior boundary $\partial \Omega$ is totally reflecting whereas the target boundary $\partial {\mathcal{U}}$ is absorbing. (b) Stochastically gated adsorption. The particle randomly switches between a reactive and non-reactive state, and can only be absorbed at $\partial {\mathcal{U}}$ when in the reactive state otherwise it is reflected. Equivalently, the surface $\partial {\mathcal{U}}$ switches between an open (absorbing) state and a closed (reflecting) state. (c) When a particle reaches the surface it may temporarily bind to $\partial {\mathcal{U}}$ (adsorb) and then either unbind and restart bulk diffusion (desorb) or be permanently removed (absorb).
• Figure 2: Brownian particle in the half-line $[0,\infty)$ with a stochastically gated boundary at $x=0$. The gate is open if $\sigma(t)=1$ and closed if $\sigma(t)=-1$. Transitions $\sigma(t)\rightarrow -\sigma(t)$ are given by a two-state Markov chain with transition rates $\alpha,\beta$. The gate could represent a conformational state of the particle (shown) or the boundary. (We consider a 2D domain for illustrative purposes.)
• Figure 3: Example sample paths arising in the first three terms of the iterative expansion of the renewal equation (\ref{['itrenewal']}). (The time axis in successive diagrams is rescaled for illustrative purposes.) Each black (red) circle represents hitting the boundary when the gate is closed (open). If the gate is closed then the reflected particle is infinitesimally displaced by an amount $\epsilon$ (see insert).
• Figure 4: Schematic illustration comparing (a) partial absorption through an open gate and (b) partial adsorption prior to encountering the gate. Reduction of a thick arrow indicates partial reflection.
• Figure 5: Plots of the MFPTs ${\mathcal{T}}_{\pm 1}(x_0)$ as a function of the resetting rate $r$ for various values of $\epsilon$: (a) $\alpha =1$ and (b) $\alpha =0.5$. Other parameter values are $D=1$ and $x_0=1$.