Evolution in the weak-mutation limit: Stasis periods punctuated by fast transitions between saddle points on the fitness landscape

Yuri Bakhtin,Yuri I. Wolf,Mikhail I. Katsnelson,Eugene V. Koonin

doi:10.1073/pnas.2015665118

Abstract

A mathematical analysis of the evolution of a large population under the weak-mutation limit shows that such a population would spend most of the time in stasis in the vicinity of saddle points on the fitness landscape. The periods of stasis are punctuated by fast transitions, in lnNe/s time (Ne , effective population size; s, selection coefficient of a mutation), when a new beneficial mutation is fixed in the evolving population, which accordingly moves to a different saddle, or on much rarer occasions from a saddle to a local peak. Phenomenologically, this mode of evolution of a large population resembles punctuated equilibrium (PE) whereby phenotypic changes occur in rapid bursts that are separated by much longer intervals of stasis during which mutations accumulate but the phenotype does not change substantially. Theoretically, PE has been linked to self-organized criticality (SOC), a model in which the size of "avalanches" in an evolving system is power-law-distributed, resulting in increasing rarity of major events. Here we show, however, that a PE-like evolutionary regime is the default for a very simple model of an evolving population that does not rely on SOC or any other special conditions.

Highlights

Phyletic gradualism, that is, evolution occurring via a succession of mutations with infinitesimally small fitness effects, is a central tenet of Darwin’s theory [1]
We analyze a simple mathematical model of population evolution on fitness landscapes and show that, for a large population in the weak-mutation limit, the process of adaptive evolution consists of extended periods of stasis, which the population spends around saddle points on the landscape, interrupted by rapid transitions to new saddle points when a beneficial mutation is fixed
It has to be stressed that this model is entirely within the classical framework of population genetics which includes estimates of mutation fixation times and the waiting times between fixation events [42, 43]

Summary

Results

We derive the PE-like evolutionary regime from several reasonable assumptions on the geometry of the graph, the fitness function, population size, mutation rates, and the initial state. We denote I = I(x(0)) for brevity and note that, given the absence of mutations, our stochastic model and ODE [1] are defined on the simplex ΔI = {x ∈ RI+ : ∑xi = 1}. This simplex is i∈I the convex hull of its vertices e(i), i ∈ I, corresponding to pure states where only one type is present: e(ki).

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Discussion