The narrow escape problem revisited

The time needed for a particle to exit a confining domain through a small window, called the narrow escape time (NET), is a limiting factor of various processes, such as some biochemical reactions in cells. Obtaining an estimate of the mean NET for a given geometric environment is therefore a requisite step to quantify the reaction rate constant of such processes, which has raised a growing interest in the last few years. In this Letter, we determine explicitly the scaling dependence of the mean NET on both the volume of the confining domain and the starting point to aperture distance. We show that this analytical approach is applicable to a very wide range of stochastic processes, including anomalous diffusion or diffusion in the presence of an external force field, which cover situations of biological relevance. The first-passage time (FPT), namely the time it takes a random walker to reach a given target site is known to be a key quantity to quantify the dynamics of various processes of practical interest [1, 2, 3]. Indeed, biochemical reactions [4, 5, 6, 7], foraging strategies of animals [8, 9, 10], the spread of sexually transmitted diseases in a human social network or of viruses through the world wide web [11] are often controlled by first encounter events. Among first-passage processes, the case where the target is a small window on the boundary of a confining domain , defined as the narrow escape problem, has proved very recently to be of particular importance [12]. The narrow escape time (NET) gives the time needed for a random walker trapped in a confining domain with a single narrow opening to exit the domain for the first time (see fig.1). The relevance of the NET is striking in cellular biology, since it gives for instance the time needed for a reactive particle released from a specific organelle to activate a given protein on the cell membrane [13]. Further examples are given by biochemical reactions in cellular microdomains, like dentritic spines, synapses or microvesicles to name a few [12, 13]. These submicrom-eter domains often contain a small amount of particles which must first exit the domain in order to fulfill their biological function. In these examples, the NET is therefore a limiting quantity whose quantization is a first step in the modeling of the process. An important theoretical advance has been made recently by different groups [12, 14, …