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A general topic of current interest is the analysis of diffusion problems insingularly perturbed domains with small interior targets or traps (the narrowcapture problem). One major application is to intracellular diffusion, wherethe targets typically represent some form of reactive biochemical substrate.Most studies of the narrow capture problem treat the target boundaries astotally absorbing. In this paper, we analyze the three-dimensional narrowcapture problem in the more realistic case of partially reactive targetboundaries. 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 eachreactive surface in powers of $\epsilon$, where $\epsilon \rho$ is a giventarget size. In turn, the fluxes determine the splitting probabilities fortarget absorption. We then extend our analysis to more general types ofreactive targets by combining matched asymptotic analysis with anencounter-based formulation of diffusion-mediated surface reactions. That is,we derive an asymptotic expansion of the joint probability density for particleposition and the boundary local time. The effects of surface reactions are thenincorporated via an appropriate stopping condition for the boundary local time.Finally, we illustrate the theory by exploring how the leading-ordercontributions to the splitting probabilities depend on the choice of surfacereactions. In particular, we show that there is an effective renormalization ofthe target radius of the form $\rho\rightarrow \rho-\widetilde{\Psi}(1/\rho)$,where $\widetilde{\Psi}$ is the Laplace transform of the stopping local timedistribution.

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