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Diffusive Search for Diffusing Targets with Fluctuating Diffusivity and Gating

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Abstract

The time that it takes a diffusing particle to find a small target has emerged as a critical quantity in many systems in molecular and cellular biology. In this paper, we extend the theory for calculating this time to account for several ubiquitous biological features which have largely been ignored in the mathematics and physics literature on this problem. In particular, we allow (i) targets to diffuse on the two-dimensional boundary of the three-dimensional domain, (ii) targets to diffuse in the interior of the domain, (iii) the diffusivities of the searcher particle and the targets to stochastically fluctuate, (iv) targets to be stochastically gated, and (v) the transition times between fluctuations in diffusivity and gating to have effectively any probability distribution. In this general framework, we analytically calculate the leading order behavior of the mean first passage time and splitting probability for the searcher to reach a target as the target size decays, which is the so-called narrow escape limit. To make these extensions, we use a generalized Itô’s formula to derive a system of coupled partial differential equations which are satisfied by statistics of the process, where the size of the system and its spatial dimension can be arbitrarily large. We apply matched asymptotic analysis to this system and verify our analytical results by numerical simulation. Our results reveal several new features and generic principles of diffusive search for small targets.

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Notes

  1. Itô’s formula is the stochastic version of the chain rule (Oksendal 2003). The generalized Itô’s formula applies to SDEs with random switching. For more information, see Lemma 3 on page 104 of [59] or Lemma 1.9 on page 49 of Mao and Yuan (2006).

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Acknowledgements

SDL was supported by the National Science Foundation (DMS-1814832 and DMS-1148230). The support and resources from the Center for High Performance Computing at the University of Utah are gratefully acknowledged.

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  1. Department of Mathematics, University of Utah, Salt Lake City, UT, 84112, USA

    Sean D. Lawley

  2. Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012, USA

    Christopher E. Miles

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  1. Sean D. Lawley
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Correspondence to Sean D. Lawley.

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Communicated by Dr. Paul Newton.

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Lawley, S.D., Miles, C.E. Diffusive Search for Diffusing Targets with Fluctuating Diffusivity and Gating. J Nonlinear Sci 29, 2955–2985 (2019). https://doi.org/10.1007/s00332-019-09564-1

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