An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part I: Two-Dimensional Domains

Pillay, Samara; Ward, Michael J.; Peirce, Anthony; Kolokolnikov, Théodore

doi:10.1137/090752511

Cited by 172 publications

(230 citation statements)

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“…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). The basic idea is to construct the asymptotic solution for the MFPT in the limit ε → 0 using the method of matched asymptotic expansions.…”

Section: Narrow Escape Problemsmentioning

confidence: 99%

“…Now introduce stretched coordinates z = z/ε and s = (s − s j )/ε, and write the solution to the inner problem as τ (x) = w( z, s). Neglecting terms of O(ε), it can be shown that w satisfies the homogeneous equation (Pillay et al, 2010) …”

Section: Narrow Escape Problemsmentioning

confidence: 99%

“…Examples include an ion looking for an open ion channel within the cell membrane (Grigoriev et al, 2002), the transport of newly transcribed mRNA from the nucleus to the cytoplasm via nuclear pores (Gorski et al, 2006;Mistelli, 2008), the confinement of neurotransmitter receptors within a synapse of a neuron (Holcman and Schuss, 2004), and the confinement of calcium and other signaling molecules within sub cellular compartments such as dendritic spines (Biess et al, 2011). This has led to recent interest in using Green's function and asymptotic methods to solve the so-called narrow escape problem Grigoriev et al, 2002;Holcman and Schuss, 2004;Pillay et al, 2010;Schuss et al, 2007;Singer et al, 2006a,b). 3.…”

Section: Introductionmentioning

confidence: 99%

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Stochastic models of intracellular transport

2013

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The interior of a living cell is a crowded, heterogenuous, fluctuating environment. Hence, a major challenge in modeling intracellular transport is to analyze stochastic processes within complex environments. Broadly speaking, there are two basic mechanisms for intracellular transport: passive diffusion and motor-driven active transport. Diffusive transport can be formulated in terms of the motion of an over-damped Brownian particle. On the other hand, active transport requires chemical energy, usually in the form of ATP hydrolysis, and can be direction specific, allowing biomolecules to be transported long distances; this is particularly important in neurons due to their complex geometry. In this review we present a wide range of analytical methods and models of intracellular transport. In the case of diffusive transport, we consider narrow escape problems, diffusion to a small target, confined and single-file diffusion, homogenization theory, and fractional diffusion. In the case of active transport, we consider Brownian ratchets, random walk models, exclusion processes, random intermittent search processes, quasisteady-state reduction methods, and mean field approximations. Applications include receptor trafficking, axonal transport, membrane diffusion, nuclear transport, protein-DNA interactions, virus trafficking, and the self-organization of subcellular structures.

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Section: Narrow Escape Problemsmentioning

confidence: 99%

Section: Narrow Escape Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Stochastic models of intracellular transport

2013

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“…The other main ingredient has been the use of special functions and of transcendental constants. Some related works with similar problems that have also been analysed with asymptotic analysis techniques can be found in [12], [13], [16] and [17] and in the references therein.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Following [13], [17] and references therein, in the outer region far away from the hole, the solution should have an expansion of the form,…”

Section: Resultsmentioning

confidence: 99%

On an asymptotic formula for the maximum voltage drop in a on-chip power distribution network

Aguareles

Haro²,

Rius³

et al. 2011

Eur. J. Appl. Math

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Abstract. We present a new asymptotic formula for the maximum static voltage in a simplified model for on-chip power distribution networks of array bonded integrated circuits.In this model the voltage is the solution of a Poisson equation in an infinite planar domain whose boundary is an array of circular pads of radius ε, and we deal with the singular limit ε → 0 case. In comparison with approximations that appear in the electronic engineering literature, our formula is more complete since we have obtained terms up to order ε 15 . A procedure will be presented to compute all the successive terms, which can be interpreted as using multipole solutions of equations involving spatial derivatives of δ-functions. To deduce the formula we use the method of matched asymptotic expansions. Our results are completely analytical and we make an extensive use of special functions and of the Gauss constant G.

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