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Biological events are often initiated when a random "searcher" finds a"target," which is called a first passage time (FPT). In some biologicalsystems involving multiple searchers, an important timescale is the time ittakes the slowest searcher(s) to find a target. For example, of the hundreds ofthousands of primordial follicles in a woman's ovarian reserve, it is theslowest to leave that trigger the onset of menopause. Such slowest FPTs mayalso contribute to the reliability of cell signaling pathways and influence theability of a cell to locate an external stimulus. In this paper, we use extremevalue theory and asymptotic analysis to obtain rigorous approximations to thefull probability distribution and moments of slowest FPTs. Though the resultsare proven in the limit of many searchers, numerical simulations reveal thatthe approximations are accurate for any number of searchers in typicalscenarios of interest. We apply these general mathematical results to models ofovarian aging and menopause timing, which reveals the role of slowest FPTs forunderstanding redundancy in biological systems. We also apply the theory toseveral popular models of stochastic search, including search by diffusive,subdiffusive, and mortal searchers.

Slowest first passage times, redundancy, and menopause timing