Paradoxical fixation of deleterious alleles in two-locus systems with epistasis

The dynamics of four deterministic models of interaction between two Mendelian loci are explored numerically. At one locus there are two hypostatic alleles: h, the wild type and H, a deleterious mutant that in either heterozygous or homozygous state (depending on the specifics of the model) produces an abnormal phenotype. At another arbitrarily linked epistatic locus there are two alleles: e, the wild type with no effect on the expression of the H locus, and E, an epistatic mutant that in heterozygous or homozygous state (again, depending on the specifics of the model) blocks the expression of H. The parameters are the initial gamete frequencies, the recombination fraction, the genotypic viabilities, and the forward and back mutation rates at each locus. The He haplotype is eventually eliminated (unless back mutation occurs) from the population. If mutation is ignored, the evolutionary outcome is determined by the initial gamete frequencies and is either (1) an edge equilibrium comprising one pair of haplotypes only (he and He, or he and hE, or hE and HE, or He and HE, or, if there is no recombination, he and HE or hE and He); or (2) a corner equilibrium consisting of a single gametic type. Given that the forward mutation rates at both loci are greater than the back mutation rates, then the outcome is always the corner equilibrium in which HE is universal (apart from transient perturbations by mutation). In the process of fixation, the phenotypic impact of the deleterious allele H becomes neutralized by the epistatic allele E. The rate at which the initially harmful gene replaces the wild type gene depends on the recombination fraction, the genotypic viabilities, and the mutation rates.