Limits to selection on standing variation in an asexual population

Abstract

We consider how a population of $N$ haploid individuals responds to directional selection on standing variation, with no new variation from recombination or mutation. Individuals have trait values $z_1,\dots ,z_N$, which are drawn from a distribution $\psi$; the fitness of individual $i$ is proportional to $e^{z_i}$. For illustration, we consider the Laplace and Gaussian distributions, which are parametrised only by the variance $V_0$, and show that for large $N$, there is a scaling limit which depends on a single parameter $N\sqrt{V_0}$. When selection is weak relative to drift ($N\sqrt{V_0}\ll 1$), the variance decreases exponentially at rate $1/N$, and the expected ultimate gain in log fitness (scaled by $\sqrt{V_0}$), is just $N\sqrt{V_0}$, which is the same as Robertson’s (1960) prediction for a sexual population. In contrast, when selection is strong relative to drift ($N\sqrt{V_0}\gg 1$), the ultimate gain can be found by approximating the establishment of alleles by a branching process in which each allele competes independently with the population mean and the fittest allele to establish is certain to fix. Then, if the probability of survival to time $t\sim 1/\sqrt{V_0}$ of an allele with value $z$ is $P\left(z\right)$, with mean $\overline{P}$, the winning allele is the fittest of $N\overline{P}$ survivors drawn from a distribution $\psi P/\overline{P}$. The expected ultimate change is $\sim \sqrt{2log(1.15N\sqrt{V_0})}$ for a Gaussian distribution, and $\sim -\frac{1}{\sqrt{2}}\left(log\left(\frac{0.36}{N\sqrt{V_0}}\right)-log\left(-log\left(\frac{0.36}{N\sqrt{V_0}}\right)\right)\right)$ for a Laplace distribution. This approach also predicts the variability of the process, and its dynamics; we show that in the strong selection regime, the expected genetic variance decreases as $\sim t^{-3}$ at large times. We discuss how these results may be related to selection on standing variation that is spread along a linear chromosome.