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Ovarian aging in women can be described as highly unpredictable within individuals but predictable across large populations. We showed previously that modeling an individual woman’s ovarian reserve of primordial follicles using mathematical random walks replicates the natural pattern of growing follicles exiting the reserve. Compiling many simulations yields the observed population distribution of the age at natural menopause (ANM). Here, we have probed how stochastic control of primordial follicle loss might relate to the distribution of the preceding menopausal transition (MT), when women begin to experience menstrual cycle irregularity. We show that identical random walk model conditions produce both the reported MT distribution and the ANM distribution when thresholds are set for growing follicle availability. The MT and ANM are shown to correspond to gaps when primordial follicles fail to grow for 7 and 12 days, respectively. Modeling growing follicle supply is shown to precisely recapitulate epidemiological data and provides quantitative criteria for the MT and ANM in humans. 

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.