An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part II: The Sphere

Abstract

Abstract. The mean first passage time (MFPT) is calculated for a Brownian particle in a spherical domain in R 3 that contains N small nonoverlapping absorbing windows, or traps, on its boundary. For the unit sphere, the method of matched asymptotic expansions is used to derive an explicit three-term asymptotic expansion for the MFPT for the case of N small locally circular absorbing windows. The third term in this expansion, not previously calculated, depends explicitly on the spatial configuration of the absorbing windows on the boundary of the sphere. The threeterm asymptotic expansion for the average MFPT is shown to be in very close agreement with full numerical results. The average MFPT is shown to be minimized for trap configurations that minimize a certain discrete variational problem. This variational problem is closely related to the well-known optimization problem of determining the minimum energy configuration for N repelling point charges on the unit sphere. Numerical results, based on global optimization methods, are given for both the optimum discrete energy and the arrangements of the centers {x 1 , . . . , x N } of N circular traps on the boundary of the sphere. A scaling law for the optimum discrete energy, valid for N 1, is also derived.Key words. narrow escape, mean first passage time, matched asymptotic expansions, surface Neumann Green's functions, discrete variational problem, logarithmic switchback terms AMS subject classifications. 35B25, 35C20, 35P15, 35J05, 35J08DOI. 10.1137/100782620 1. Introduction. The narrow escape problem concerns the motion of a Brownian particle confined in a bounded domain Ω ∈ R d (d = 2, 3) whose boundary ∂Ω = ∂Ω r ∪ ∂Ω a is almost entirely reflecting (∂Ω r ), except for small absorbing windows, or traps, labeled collectively by ∂Ω a , through which the particle can escape. Denoting the trajectory of the Brownian particle by X(t), the mean first passage time (MFPT) v(x) is defined as the expectation value of the time τ taken for the Brownian particle to become absorbed somewhere in ∂Ω a starting initially fromThe calculation of v(x) becomes a narrow escape problem in the limit when the measure of the absorbing set |∂Ω a | = O(ε d−1 ) is asymptotically small, where 0 < ε 1 measures the dimensionless radius of an absorbing window. Since the MFPT diverges as ε → 0, the calculation of the MFPT v(x) constitutes a singular perturbation problem.The narrow escape problem has many applications in biophysical modeling (see [2], [16], [19], [39] and the references therein). For the case of a two-dimensional domain, the narrow escape problem has been studied with a variety of analytical methods in [19], [42], [43], [20], and Part I of this paper [30]. In this paper, we use

Stochastic models of intracellular transport

“…Since the MFPT diverges as ε → 0, the calculation of τ (x) requires solving a singular perturbation problem. There have been a number of studies of the narrow escape problem using a combination of singular perturbation theory and Green's function methods for a variety of geometries in two and three dimensions Chevalier et al, 2011;Cheviakov et al, 2010;Holcman and Schuss, 2004;Pillay et al, 2010;Schuss et al, 2007;Singer et al, 2006a,b). Here we will review the particular approach of Ward and collaborators (Pillay et al, 2010;Ward, 2000).…”

“…That is, setting φ(x) = 1 − c(x) in steady-state, it follows that φ satisfies Laplace's equation with φ = 1 on the target boundary and φ = 0 at infinity, so that φ is equivalent to the electrostatic potential generated by a perfectly conducting object Ω held at unit potential. Moreover, the steady-state reaction rate k = 4πDQ where Q is the total charge on the surface of the conductor, which for a unit potential is equal to the capacitance, Q = C. Thus, determining the reaction rate for a general 3D target is equivalent to finding the capacitance of a perfect conductor with the same shape; see also (Cheviakov et al, 2010).…”

First Passage Statistics for the Capture of a Brownian Particle by a Structured Spherical Target with Multiple Surface Traps

“…. , N. The analysis to determine C 0 is related to that done for the narrow escape problem in [12] in that it relies on detailed knowledge of a certain surface Neumann Green's function together with the introduction of certain logarithmic switchback terms (cf. [30]) that arise from the subdominant logarithmic singularity of this Green's function.…”

“…The effectiveness of the diffusion mechanism in light of these factors can be understood by studying the first passage time statistics of Brownian walkers to small stationary targets. In many biological settings, the number of individual molecules is typically very large, and so the mean first passage time (MFPT) is an important quantity and the focus of many recent studies [47,24,9,43,1,32,38,37,46,15,10,12,16].…”

The Mean Escape Time for a Narrow Escape Problem with Multiple Switching Gates

“…There have been several significant works deriving the leading order and higher order terms of the asymptotic expansions of MET in two and three dimensions [3,5,8,9,10,12,14,15,16,17,18,19]. These work deal with the narrow escape problem when there are one or several gates and these gates are not fluctuating in time, in other words, the gates are open all the time.…”