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Trapping diffusive particles at surfaces is a key step in many systems in chemical and biological physics. Trapping often occurs via reactive patches on the surface and/or the particle. The theory of boundary homogenization has been used in many prior works to estimate the effective trapping rate for such a system in the case that either (i) the surface is patchy and the particle is uniformly reactive or (ii) the particle is patchy and the surface is uniformly reactive. In this paper, we estimate the trapping rate for the case that the surface and the particle are both patchy. In particular, the particle diffuses translationally and rotationally and reacts with the surface when a patch on the particle contacts a patch on the surface. We first formulate a stochastic model and derive a five-dimensional partial differential equation describing the reaction time. We then use matched asymptotic analysis to derive the effective trapping rate, assuming that the patches are roughly evenly distributed and occupy a small fraction of the surface and the particle. This trapping rate involves the electrostatic capacitance of a four-dimensional duocylinder, which we compute using a kinetic Monte Carlo algorithm. We further use Brownian local time theory to derive a simple heuristic estimate of the trapping rate and show that it is remarkably close to the asymptotic estimate. Finally, we develop a kinetic Monte Carlo algorithm to simulate the full stochastic system and then use these simulations to confirm the accuracy of our trapping rate estimates and homogenization theory. 

Abstract Women are born with hundreds of thousands to over a million primordial ovarian follicles (PFs) in their ovarian reserve. However, only a few hundred PFs will ever ovulate and produce a mature egg. Why are hundreds of thousands of PFs endowed around the time of birth when far fewer follicles are required for ongoing ovarian endocrine function and only a few hundred will survive to ovulate? Recent experimental, bioinformatics, and mathematical analyses support the hypothesis that PF growth activation (PFGA) is inherently stochastic. In this paper, we propose that the oversupply of PFs at birth enables a simple stochastic PFGA mechanism to yield a steady supply of growing follicles that lasts for several decades. Assuming stochastic PFGA, we apply extreme value theory to histological PF count data to show that the supply of growing follicles is remarkably robust to a variety of perturbations and that the timing of ovarian function cessation (age of natural menopause) is surprisingly tightly controlled. Though stochasticity is often viewed as an obstacle in physiology and PF oversupply has been called “wasteful,” this analysis suggests that stochastic PFGA and PF oversupply function together to ensure robust and reliable female reproductive aging. 

Abstract A variety of systems in physics, chemistry, biology, and psychology are modeled in terms of diffusing ‘searchers’ looking for ‘targets’. Examples range from gene regulation, to cell sensing, to human decision-making. A commonly studied statistic in these models is the so-called hitting probability for each target, which is the probability that a given single searcher finds that particular target. However, the decisive event in many systems is not the arrival of a given single searcher to a target, but rather the arrival of the fastest searcher to a target out of many searchers. In this paper, we study the probability that the fastest diffusive searcher hits a given target in the many searcher limit, which we call the extreme hitting probability. We first prove an upper bound for the decay of the probability that the searcher finds a target other than the closest target. This upper bound applies in very general settings and depends only on the relative distances to the targets. Furthermore, we find the exact asymptotics of the extreme hitting probabilities in terms of the short-time distribution of when a single searcher hits a target. These results show that the fastest searcher always hits the closest target in the many searcher limit. While this fact is intuitive in light of recent results on the time it takes the fastest searcher to find a target, our results give rigorous, quantitative estimates for the extreme hitting probabilities. We illustrate our results in several examples and numerical solutions.