Slowest first passage times, redundancy, and menopause timing

TL;DR

This paper develops mathematical tools to analyze the distribution of the slowest search times in biological systems, with applications to ovarian aging and menopause timing, revealing the importance of extreme search times in biological reliability.

Contribution

The paper introduces rigorous approximations for the distribution and moments of slowest first passage times using extreme value theory, applicable to various stochastic search models.

Findings

Approximations are accurate for any number of searchers in typical scenarios.

Slowest FPTs influence biological processes like menopause timing and cell signaling.

Mathematical framework applies to diffusive, subdiffusive, and mortal search models.

Abstract

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