Limits to selection on standing variation in an asexual population

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 z₁,…,z_N, which are drawn from a distribution ψ; the fitness of individual i is proportional to [Formula: see text] . For illustration, we consider the Laplace and Gaussian distributions, which are parametrised only by the variance V₀, and show that for large N, there is a scaling limit which depends on a single parameter NV₀. When selection is weak relative to drift (NV₀≪1), the variance decreases exponentially at rate 1/N, and the expected ultimate gain in log fitness (scaled by V₀), is just NV₀, which is the same as Robertson's (1960) prediction for a sexual population. In contrast, when selection is strong relative to drift (NV₀≫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∼1/V₀ of an allele with value z is P(z), with mean P¯, the winning allele is the fittest of NP¯ survivors drawn from a distribution ψP/P¯. The expected ultimate change is ∼2log(1.15NV₀) for a Gaussian distribution, and ∼-12log0.36NV₀-log-log0.36NV₀ 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 ∼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.