Fixation of beneficial mutations in the presence of epistatic interactions

We investigate the effect of deleterious mutations on the process of fixation of new advantageous mutants in an asexual population. In particular we wish to study the dependence of the process on the strength of the deleterious mutations. We suppose the existence of epistatic interaction between the genes. We study the model by means of branching process theory and also by numerical simulations. Our results show the occurrence of two distinct regimes of behavior for the probability of fixation of these variants. The occurrence of either regime depends on the ratio between the selective advantage of the beneficial mutation sb and on the selective parameter for deleterious mutations sd. In the former, which takes place for sb/sd< or =1, the probability of fixation increases with the epistasis parameter , whereas for sb/sd>>1 the probability of fixation is a complex function of and the mutation rate U. Surprisingly, we find that for the multiplicative landscape (alpha=1) the probability of fixation Pfix is given by Pfix= pib)e-U/sd where pi(sb) is the probability of fixation for the two-allele model in the absence of mutations as calculated by Haldane (1927, Proc. Camb. Phil. Soc., 26, 220-230) and Kimura (1962, Genetics, 47, 713-719).