The narrow capture problem: an encounter-based approach to partially reactive targets

TL;DR

This paper develops an asymptotic analysis for the narrow capture problem with partially reactive targets, extending classical models to include realistic surface reactions and deriving formulas for splitting probabilities based on boundary local time.

Contribution

It introduces a new encounter-based framework for analyzing diffusion to partially reactive targets, generalizing previous models that assumed totally absorbing boundaries.

Findings

Derived asymptotic expansion for flux into reactive surfaces.

Extended analysis to general reactive targets using boundary local time.

Showed effective renormalization of target radius due to surface reactions.

Abstract

A general topic of current interest is the analysis of diffusion problems in singularly perturbed domains with small interior targets or traps (the narrow capture problem). One major application is to intracellular diffusion, where the targets typically represent some form of reactive biochemical substrate. Most studies of the narrow capture problem treat the target boundaries as totally absorbing. In this paper, we analyze the three-dimensional narrow capture problem in the more realistic case of partially reactive target boundaries. We begin by considering classical Robin boundary conditions. Matching inner and outer solutions of the single-particle probability density, we derive an asymptotic expansion of the Laplace transformed flux into each reactive surface in powers of $\epsilon$, where $\epsilon \rho$ is a given target size. In turn, the fluxes determine the splitting probabilities for target absorption. We then extend our analysis to more general types of reactive targets by combining matched asymptotic analysis with an encounter-based formulation of diffusion-mediated surface reactions. That is, we derive an asymptotic expansion of the joint probability density for particle position and the boundary local time. The effects of surface reactions are then incorporated via an appropriate stopping condition for the boundary local time. Finally, we illustrate the theory by exploring how the leading-order contributions to the splitting probabilities depend on the choice of surface reactions. In particular, we show that there is an effective renormalization of the target radius of the form $\rho\rightarrow \rho-\widetilde{\Psi}(1/\rho)$, where $\widetilde{\Psi}$ is the Laplace transform of the stopping local time distribution.