Two-timescale evolution on a singular landscape

TL;DR

This paper investigates the stochastic escape times from infinite potential peaks in a Wright-Fisher model, extending classical escape formulas to non-Gaussian distributions and providing a comprehensive landscape for evolutionary dynamics.

Contribution

It introduces an analytical approximation for escape times in singular landscapes, extending Kramers' formula to non-Gaussian contexts in evolutionary models.

Findings

Extended Kramers' escape formula to non-Gaussian distributions.

Numerical and analytical results for escape times in bi-stable cases.

Constructed a global landscape describing evolutionary dynamics.

Abstract

Under the effect of strong genetic drift, it is highly probable to observe gene fixation or gene loss in a population, shown by infinite peaks on a coherently constructed potential energy landscape. It is then important to ask what such singular peaks imply, with or without the effects of other biological factors. We studied the stochastic escape time from the infinite potential peaks in the Wright-Fisher model, where the typical two-scale diffusion dynamics was observed via computer simulations. We numerically found the average escape time for all the bi-stable cases and analytically approximated the results under weak mutations and selections by calculating the mean first passage time (MFPT) in singular potential peak. Our results showed that Kramers' classical escape formula can be extended to the models with non-Gaussian probability distributions, overcoming constraints in previous methods. The constructed landscape provides a global and coherent description for system's evolutionary dynamics, allowing new biological results to be generated.