Limits to selection on standing variation in an asexual population

Abstract

We consider how a population responds to directional selection on standing variation, with no new variation from recombination or mutation. Initially, there are N individuals with trait values z1, ..., zN ; the fitness of individual i is proportional to [Formula]. The initial values are drawn from a distribution{psi} with variance V0; we give examples of the Laplace and Gaussian distributions. When selection is weak relative to drift [Formula], variance decreases exponentially at rate 1/N ; since the increase in mean in any generation equals the variance, the expected net change is just NV0, which is the same as Robertsons (1960) prediction for a sexual population. In contrast, when selection is strong relative to drift [Formula], the net change 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 [Formula] of an allele with value z is P (z), with mean [Formula], the winning allele is the fittest of [Formula] survivors drawn from a distribution [Formula]. When N is large, there is a scaling limit which depends on a single parameter [Formula]; the expecte d ultimate change is [Formula] for a Gaussian distribution, and [Formula] for a Laplace distribution (where [W]is the product log function). This approach also reveals 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.