Stochastic Narrow Escape in Molecular and Cellular Biology
01 Jan 2015-
About: The article was published on 2015-01-01. It has received 85 citations till now.
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TL;DR: Recent developments in the non-standard asymptotics of the narrow escape problem are reviewed, which are based on several ingredients: a better resolution of the singularity of Neumann's function,resolution of the boundary layer near the small target by conformal mappings of domains with bottlenecks, and the breakup of composite domains into simpler components.
Abstract: The narrow escape problem in diffusion theory is to calculate the mean first passage time of a diffusion process to a small target on the reflecting boundary of a bounded domain. The problem is equivalent to solving the mixed Dirichlet--Neumann boundary value problem for the Poisson equation with small Dirichlet and large Neumann parts. The mixed boundary value problem, which goes back to Lord Rayleigh, originates in the theory of sound and is closely connected to the eigenvalue problem for the mixed problem and for the Neumann problem in domains with bottlenecks. We review here recent developments in the non-standard asymptotics of the problem, which are based on several ingredients: a better resolution of the singularity of Neumann's function, resolution of the boundary layer near the small target by conformal mappings of domains with bottlenecks, and the breakup of composite domains into simpler components. The new methodology applies to two- and higher-dimensional problems. Selected applications are r...
175 citations
TL;DR: A self-consistent approximation is developed to derive for Tε a general expression, akin to the celebrated Collins-Kimball relation in chemical kinetics, being minimal for the ones having an intermediate extent, neither too concentrated on the boundary nor penetrating too deeply into the bulk.
Abstract: We study the mean first exit time (Te) of a particle diffusing in a circular or a spherical micro-domain with an impenetrable confining boundary containing a small escape window (EW) of an angular size e. Focusing on the effects of an energy/entropy barrier at the EW, and of the long-range interactions (LRIs) with the boundary on the diffusive search for the EW, we develop a self-consistent approximation to derive for Te a general expression, akin to the celebrated Collins–Kimball relation in chemical kinetics and accounting for both rate-controlling factors in an explicit way. Our analysis reveals that the barrier-induced contribution to Te is the dominant one in the limit e → 0, implying that the narrow escape problem is not “diffusion-limited” but rather “barrier-limited”. We present the small-e expansion for Te, in which the coefficients in front of the leading terms are expressed via some integrals and derivatives of the LRI potential. Considering a triangular-well potential as an example, we show that Te is non-monotonic with respect to the extent of the attractive LRI, being minimal for the ones having an intermediate extent, neither too concentrated on the boundary nor penetrating too deeply into the bulk. Our analytical predictions are in good agreement with the numerical simulations.
86 citations
TL;DR: The physical models, the mathematical analysis and the new paradigm of setting the scale to be the shortest time for activation that clarifies the role of population redundancy in selecting and accelerating transient cellular search processes are reviewed.
70 citations
TL;DR: In this article, the mean first passage time and first passage times for two-channel Markov additive diffusion in a 3-dimensional spherical domain were investigated and it was shown that the first passage statistics at long times do not display Poisson-like behavior if none of the phases has a vanishing diffusion coefficient.
Abstract: We present rigorous results for the mean first passage time and first passage time statistics for two-channel Markov additive diffusion in a 3-dimensional spherical domain. Inspired by biophysical examples we assume that the particle can only recognise the target in one of the modes, which is shown to effect a non-trivial first passage behaviour. We also address the scenario of intermittent immobilisation. In both cases we prove that despite the perfectly non-recurrent motion of two-channel Markov additive diffusion in 3 dimensions the first passage statistics at long times do not display Poisson-like behaviour if none of the phases has a vanishing diffusion coefficient. This stands in stark contrast to the standard (one-channel) Markov diffusion counterpart. We also discuss the relevance of our results in the context of cellular signalling.
54 citations
TL;DR: Asymptotic laws for the probability density function of the first and second arrival times of a large number N of i.i.d. Brownian trajectories to a small target in 1, 2 and 3-dimensions are derived and applied to activation of biochemical pathways in cellular biology.
Abstract: The first of N identical independently distributed (i.i.d.) Brownian trajectories that arrives to a small target sets the time scale of activation, which in general is much faster than the arrival to the target of a single trajectory only. Analytical asymptotic expressions for the minimal time is notoriously difficult to compute in general geometries. We derive here asymptotic laws for the probability density function of the first and second arrival times of a large number N of i.i.d. Brownian trajectories to a small target in 1, 2 and 3-dimensions and study their range of validity by stochastic simulations. The results are applied to activation of biochemical pathways in cellular biology.
44 citations