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

Abstract

Abstract. We study the first passage time problem for a diffusing molecule in an enclosed region to hit a small spherical target whose surface contains many small absorbing traps. This study is motivated by two examples of cellular transport. The first is the intracellular process through which proteins transit from the cytosol to the interior of the nucleus through nuclear pore complexes that are distributed on the nuclear surface. The second is the problem of chemoreception, in which cells sense their surroundings through diffusive contact with receptors distributed on the cell exterior. Using a matched asymptotic analysis in terms of small absorbing pore radius, we derive and numerically verify a high order expansion for the capacitance of the structured target which incorporates surface effects and gives explicit information on interpore interaction through a Coulomb-type discrete energy with additional logarithmic dependencies. In the large N dilute surface trap fraction limit, a single homogenized Robin boundary condition ∂nv + κv = 0 is derived in which κ depends on the total absorbing fraction, the characteristic pore scale, and parameters relating to interpore interactions.

Distribution of extreme first passage times of diffusion

“…which indeed violates (29). Therefore, if the limiting distribution is nondegenerate, it must be Gumbel.…”

“…where T 1,N := T N . While the distribution and statistics of a single FPT, τ 1 , are well understood in a variety of scenarios [26,27,28,29,30], studying the so-called extreme FPTs, T k,N , is notoriously difficult, both analytically and numerically [31,32,12,11,33,34]. The essential difficulty is that extreme FPTs depend on very rare events.…”

Universal formula for extreme first passage statistics of diffusion

“…Diffusion with space-dependent diffusivity and drift. For the diffusion process in(13), the thin trajectories are 15 typical paths which get pushed to the left by the drift. The thick blue curve shows that the fastest searcher is unaffected by the drift and moves almost deterministically to the target through regions of fast diffusion (grey regions) while avoiding regions of slow diffusion (white regions).…”

Reversible reactions controlled by surface diffusion on a sphere

“…The Legendre function P ν (x) of the first kind is the solution of the Legendre equation (21), which is regular at x = 1 and normalized such that P ν (1) = 1. For an integer ν, P ν (x) is the Legendre polynomial of degree ν.…”

“…Berg and Purcell estimated the reaction * Electronic address: denis.grebenkov@polytechnique.edu rate controlled by bulk diffusion toward multiple small patches evenly distributed over the surface of a sphere [14]. This seminal paper and the associated homogenization concept were further extended by many authors [15][16][17][18][19][20][21][22][23]. Shoup et al devised an efficient approximation to compute the reaction rate from the bulk to an active circular region on the surface of a sphere, even in the presence of rotational motion [24].…”