Stochastic processes with resetting

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• Paul Bressloff (Imperial College London)
• Friday 24 November 2023, 13:30-14:30
• Seminar Room 2, Newton Institute.

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MMV - Mathematics of movement: an interdisciplinary approach to mutual challenges in animal ecology and cell biology

A topic of considerable interest within nonequilibrium statistical physics is the study of stochastic search processes with resetting, whereby the current position of the particle or searcher is reset to a fixed location over a random sequence of times. In many cases, the mean first passage time (MFPT) for finding a target has a unique minimum as a function of the resetting rate. In this talk we review the basic theory of random search processes with resetting, and then consider two applications.  (A) Cytoneme-mediated morphogenesis. Cytonemes are long, thin actin-rich cell protrusions varying in length from 10-200 mm. They allow for the active transport of morphogens or their cognate receptors by establishing direct physical contacts between cells. Cytonemes are also thought to play a role in cell-to-cell viral spread. We formulate the rapid growth and shrinkage of cytonemes prior to finding a target cell as a FPT problem with resetting and determine the MFPT . We also indicate how queuing theory can be used to determine the accumulation of resources across an array of target cells.  (B) Transition path theory (TPT). Many chemical reactions can be formulated in terms of particle diffusion in a complex energy landscape. TPT is a theoretical framework for describing the direct (reaction) pathways from reactant to product states within this energy landscape, and calculating the effective reaction rate. It is now the standard method for analyzing rare events between long lived states. We consider a completely different application of TPT , namely, a dual-aspect diffusive search process in which a particle alternates between collecting cargo from a source domain A and then delivering it to a target domain B. The rate of resource accumulation at the target, kAB, is determined by the statistics of direct paths from A to B. Stochastic resetting introduces two non-trivial problems in the application of TPT . First, the process is not time-reversal invariant. Second, calculating kAB involves determining the total probability flux of direct transport paths across a dividing surface S between A and B, which includes discontinuous jumps due to resetting.

This talk is part of the Isaac Newton Institute Seminar Series series.

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