Fixation Probability in a Two-Locus Model by the Ancestral Recombination–Selection Graph

Abstract

We use the ancestral influence graph (AIG) for a two-locus, two-allele selection model in the limit of a large population size to obtain an analytic approximation for the probability of ultimate fixation of a single mutant allele A. We assume that this new mutant is introduced at a given locus into a finite population in which a previous mutant allele B is already segregating with a wild type at another linked locus. We deduce that the fixation probability increases as the recombination rate increases if allele A is either in positive epistatic interaction with B and allele B is beneficial or in no epistatic interaction with B and then allele A itself is beneficial. This holds at least as long as the recombination fraction and the selection intensity are small enough and the population size is large enough. In particular this confirms the Hill-Robertson effect, which predicts that recombination renders more likely the ultimate fixation of beneficial mutants at different loci in a population in the presence of random genetic drift even in the absence of epistasis. More importantly, we show that this is true from weak negative epistasis to positive epistasis, at least under weak selection. In the case of deleterious mutants, the fixation probability decreases as the recombination rate increases. This supports Muller's ratchet mechanism to explain the accumulation of deleterious mutants in a population lacking recombination.T HE Hill-Robertson (HR) effect (Hill and Robertson 1966) is often mentioned as one of the main arguments in favor of the evolution of recombination. In short, it predicts that beneficial mutant alleles arising at different loci in a finite population are more likely to fix in the population as the recombination rate increases even when selection acts independently upon the loci.Since the early works of Fisher (1930) and Muller (1932), it is generally believed that an evolutionary advantage of recombination is to bring together beneficial mutant alleles arising at different loci. Accordingly the effect of recombination should be to increase the rate of evolution of the population (Crow and Kimura 1965). However, it has been shown that recombination has no effect on this rate in an infinite population if there is initial linkage equilibrium and absence of epistasis so that linkage equilibrium is maintained thereafter in the population (Felsenstein 1965;Maynard Smith 1968).If recombination can have an effect on the rate of evolution only by breaking down linkage disequilibrium in absolute value, then the effect should be to increase this rate only when linkage disequilibrium in the population is negative (NLD). In the case of a two-locus model, this happens when the frequency of the double mutant is strictly smaller than the product of the frequencies of the mutant alleles. This situation is arguably likely to happen in the view that beneficial mutations are very rare (Crow and Kimura 1969).On the other hand, NLD could be produced by negative epistasis (NE), with the double mutant being l...

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