Search processes with stochastic resetting and partially absorbing targets

TL;DR

This paper extends the theoretical analysis of search processes with stochastic resetting to include partially absorbing targets, revealing how absorption rate, resetting rate, and geometry influence mean first passage times in biological and physical systems.

Contribution

It introduces a generalized framework for partially absorbing targets in stochastic search models, connecting absorption rate with optimal resetting strategies and target geometry effects.

Findings

MFPT decreases monotonically with absorption rate κ

Optimal resetting rate r_opt exists and depends on κ and geometry

Results unify with previous models for fully absorbing targets as κ approaches infinity

Abstract

We extend the theoretical framework used to study search processes with stochastic resetting to the case of partially absorbing targets. Instead of an absorption event occurring when the search particle reaches the boundary of a target, the particle can diffuse freely in and out of the target region and is absorbed at a rate $\kappa$ when inside the target. In the context of cell biology, the target could represent a chemically reactive substrate within a cell or a region where a particle can be offloaded onto a nearby compartment. We apply this framework to a partially absorbing interval and to spherically symmetric targets in $\R^d$. In each case, we determine how the mean first passage time (MFPT) for absorption depends on $\kappa$, the resetting rate $r$, and the target geometry. For the given examples, we find that the MFPT is a monotonically decreasing function of $\kappa$, whereas it is a unimodal function of $r$ with a unique minimum at an optimal resetting rate $r_{\rm opt}$. The variation of $r_{\rm opt}$ with $\kappa$ depends on the spatial dimension $d$, decreasing in sensitivity as $d$ increases. For finite $\kappa$, $ r_{\rm opt}$ is a non-trivial function of the target size and distance between the target and the reset point. We also show how our results converge to those obtained previously for problems with totally absorbing targets and similar geometries when the absorption rate becomes infinite. Finally, we generalize the theory to take into account an extended chemical reaction scheme within a target.