The stochastic edge in adaptive evolution

TL;DR

This paper extends the stochastic edge model of adaptive evolution to high speeds, overcoming previous limitations by employing new extrapolation methods, and confirms the results through analytical comparisons and numerical simulations.

Contribution

It introduces a generalized method for analyzing the stochastic edge at high adaptation speeds, improving upon Desai and Fisher's original approach.

Findings

The extended model applies to rapid adaptation scenarios.

Results align with alternative analytical approaches.

Numerical simulations validate the extended model.

Abstract

In a recent article, Desai and Fisher (2007) proposed that the speed of adaptation in an asexual population is determined by the dynamics of the stochastic edge of the population, that is, by the emergence and subsequent establishment of rare mutants that exceed the fitness of all sequences currently present in the population. Desai and Fisher perform an elaborate stochastic calculation of the mean time $\tau$ until a new class of mutants has been established, and interpret $1/\tau$ as the speed of adaptation. As they note, however, their calculations are valid only for moderate speeds. This limitation arises from their method to determine $\tau$: Desai and Fisher back-extrapolate the value of $\tau$ from the best-fit class' exponential growth at infinite time. This approach is not valid when the population adapts rapidly, because in this case the best-fit class grows non-exponentially during the relevant time interval. Here, we substantially extend Desai and Fisher's analysis of the stochastic edge. We show that we can apply Desai and Fisher's method to high speeds by either exponentially back-extrapolating from finite time or using a non-exponential back-extrapolation. Our results are compatible with predictions made using a different analytical approach (Rouzine et al. 2003, 2007), and agree well with numerical simulations.