Abstract

The first arrivals among N Brownian particles is ubiquitous in the life sciences, as it often triggers cellular processes from the molecular level. We study here the case where stochastic particles, which represent proteins or molecules can switch between two states inside the non-negative real line. The switching process is modeled as a two-state Markov chain and particles can only escape in state 1. We estimate the fastest arrival time by solving asymptotically the Fokker–Planck equations for three different initial distributions: Dirac-delta, uniformly distributed and long-tail decay. The derived formulas reveal that the fastest particle avoids switching when the switching rates are much smaller than the diffusion time scale, but switches twice when the diffusion in state 2 is much faster than in state 1. The present results are compared to stochastic simulations revealing the range of validity of the derived formulas.

Graphic abstract

中文翻译：

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N个切换随机粒子中最快的到达时间

摘要

N中的第一个到达者布朗粒子在生命科学中无处不在，因为它经常从分子水平触发细胞过程。我们在这里研究代表蛋白质或分子的随机粒子可以在非负实线内的两种状态之间切换的情况。切换过程被建模为两态马尔可夫链，粒子只能在状态 1 中逃逸。我们通过渐近求解三种不同初始分布的 Fokker-Planck 方程来估计最快到达时间：Dirac-delta、均匀分布和长-尾巴腐烂。推导出的公式表明，当切换速率远小于扩散时间尺度时，最快的粒子会避免切换，但当状态 2 的扩散比状态 1 快得多时切换两次。

图形摘要