University of Utah, Salt Lake City, UT, 84112, USA.
J Math Biol. 2020 Jun;80(7):2301-2325. doi: 10.1007/s00285-020-01496-9. Epub 2020 Apr 22.
Many events in biology are triggered when a diffusing searcher finds a target, which is called a first passage time (FPT). The overwhelming majority of FPT studies have analyzed the time it takes a single searcher to find a target. However, the more relevant timescale in many biological systems is the time it takes the fastest searcher(s) out of many searchers to find a target, which is called an extreme FPT. In this paper, we apply extreme value theory to find a tractable approximation for the full probability distribution of extreme FPTs of diffusion. This approximation can be easily applied in many diverse scenarios, as it depends on only a few properties of the short time behavior of the survival probability of a single FPT. We find this distribution by proving that a careful rescaling of extreme FPTs converges in distribution as the number of searchers grows. This limiting distribution is a type of Gumbel distribution and involves the LambertW function. This analysis yields new explicit formulas for approximations of statistics of extreme FPTs (mean, variance, moments, etc.) which are highly accurate and are accompanied by rigorous error estimates.
在生物学中，当扩散搜索者找到目标时，许多事件就会被触发，这被称为首次通过时间（FPT）。绝大多数 FPT 研究都分析了单个搜索者找到目标所需的时间。然而，在许多生物系统中，更相关的时间尺度是最快的搜索者（多个搜索者中的一个）找到目标所需的时间，这被称为极端 FPT。在本文中，我们应用极值理论来寻找扩散的极端 FPT 完整概率分布的可处理近似。该近似可以轻松应用于许多不同的场景，因为它仅取决于单个 FPT 生存概率的短时间行为的几个特性。我们通过证明当搜索者数量增加时，极端 FPT 的仔细缩放在分布上收敛来找到该分布。这种极限分布是一种 Gumbel 分布，涉及 LambertW 函数。该分析产生了极端 FPT 统计量（平均值、方差、矩等）的新的显式近似公式，这些公式具有高度的准确性，并附有严格的误差估计。
