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 $ε$, where $ερ$ 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 $ρ\rightarrow ρ-\widetildeΨ(1/ρ)$, where $\widetildeΨ$ is the Laplace transform of the stopping local time distribution.
Several classic problems for particles diffusing outside an arbitrary configuration of non-overlapping partially reactive spherical traps in three dimensions are revisited. For this purpose, we describe the generalized method of separation of variables for solving boundary value problems of the associated modified Helmholtz equation. In particular, we derive a semi-analytical solution for the Green function that is the key ingredient to determine various diffusion-reaction characteristics such as the survival probability, the first-passage time distribution, and the reaction rate. We also pres…
We investigate restricted diffusion in a bounded domain towards a small partially reactive target in three- and higher-dimensional spaces. We propose a simple explicit approximation for the principal eigenvalue of the Laplace operator with mixed Robin-Neumann boundary conditions. This approximation involves the harmonic capacity and the surface area of the target, the volume of the confining domain, the diffusion coefficient and the reactivity. The accuracy of the approximation is checked by using a finite-elements method. The proposed approximation determines also the mean first-reaction time…
A probabilistic framework for studying single-particle diffusion in partially absorbing media has recently been developed in terms of an encounter-based approach. The latter computes the joint probability density (generalized propagator) for particle position $\X_t$ and a Brownian functional ${\mathcal U}_t$ that specifies the amount of time the particle is in contact with a reactive component $\calM$. Absorption occurs as soon as $\calU_t$ crosses a randomly distributed threshold (stopping time). Laplace transforming the propagator with respect to $\calU_t$ leads to a classical boundary value…
We develop an encounter-based approach for describing restricted diffusion with a gradient drift towards a partially reactive boundary. For this purpose, we introduce an extension of the Dirichlet-to-Neumann operator and use its eigenbasis to derive a spectral decomposition for the full propagator, i.e., the joint probability density function for the particle position and its boundary local time. This is the central quantity that determines various characteristics of diffusion-influenced reactions such as conventional propagators, survival probability, first-passage time distribution, boundary…
In this paper we develop a hybrid version of the encounter-based approach to diffusion-mediated absorption at a reactive surface, which takes into account stochastic switching of a diffusing particle's conformational state. For simplicity, we consider a two-state model in which the probability of surface absorption depends on the current particle state and the amount of time the particle has spent in a neighborhood of the surface in each state. The latter is determined by a pair of local times $\ell_{n,t}$, $n=0,1$, which are Brownian functionals that keep track of particle-surface encounters …
The path decomposition expansion represents the propagator of the irreversible reaction as a convolution of the first-passage, last-passage and rebinding time probability densities. Using path integral technique, we give an elementary, yet rigorous, proof of the path decomposition expansion of the Green's functions describing the non-reactive case and the irreversible reaction of an isolated pair of molecules. To this end, we exploit the connection between boundary value problems and interaction potential problems with $δ$- and $δ'$-function perturbation. In particular, we employ a known exact…
In this paper we develop an encounter-based model of reaction-subdiffusion in a domain $Ω$ with a partially absorbing interior trap $\calU\subset Ω$. We assume that the particle can freely enter and exit $\calU$, but is only absorbed within $\calU$. We take the probability of absorption to depend on the amount of time a particle spends within the trap, which is specified by a Brownian functional known as the occupation time $A(t)$. The first passage time (FPT) for absorption is identified with the point at which the occupation time crosses a random threshold $\widehat{A}$ with probability dens…
The effects of Poissonian resetting at a constant rate $r$ on the reaction time between a Brownian particle and a stochastically gated target are studied. The target switches between a reactive state and a non-reactive one. We calculate the mean time at which the particle subject to resetting hits the target for the first time, while the latter is in the reactive state. The search time is minimum at a resetting rate that depends on the target transition rates. When the target relaxation rate is much larger than both the resetting rate and the inverse diffusion time, the system becomes equivale…
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 $κ$ 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 sp…
We study diffusion of particles on the surface of a sphere toward a partially reactive circular target with partly reversible binding kinetics. We solve the coupled diffusion-reaction equations and obtain the exact expressions for the time-dependent concentration of particles and the total diffusive flux. Explicit asymptotic formulas are derived in the small target limit. This study reveals the strong effects of reversible binding kinetics onto diffusion-mediated reactions that may be relevant for many biochemical reactions on cell membranes.
Optimization of multisite reactions in complex compartmentalized media
Laboratoire Ondes et Matière d’Aquitaine, CNRS/University of Bordeaux（阿基坦波与物质实验室，法国国家科学研究中心/波尔多大学）；Centre de physique théorique, CNRS / Aix-Marseille university（理论物理中心，法国国家科学研究中心/艾克斯-马赛大学）
Diffusion-mediated surface phenomena are crucial for human life and industry, with examples ranging from oxygen capture by lung alveolar surface to heterogeneous catalysis, gene regulation, membrane permeation and filtration processes. Their current description via diffusion equations with mixed boundary conditions is limited to simple surface reactions with infinite or constant reactivity. In this letter, we propose a probabilistic approach based on the concept of boundary local time to investigate the intricate dynamics of diffusing particles near a reactive surface. Reformulating surface-pa…
