Quasi Linkage Equilibrium under Weak Two-Locus Viability Selection: II. Loci with Multiple Alleles

A model of weak viability selection at two multiple-allele loci in a haploid population is considered. It is assumed that population is randomly mating, the generations do not overlap, and the genetic structure dynamics is modeled by ordinary differential equations. Weak selection is considered as a perturbation of the model without selection. The quasi-equilibrium estimates for the linkage nonequilibrium coefficient D obtained earlier at diallelic loci are generalized to the case of multiple alleles. Moreover, the interpretation of estimates in terms of average effects in quantitative genetics turned out to be independent of the number of alleles. The first approximation for the quasi-equilibrium of \({{D}_{{{{i}_{1}}{{i}_{2}}}}}\) is \(D_{{{{i}_{{\text{1}}}}{{i}_{{\text{2}}}}}}^{ * }\) = \({{\varepsilon }_{{{{i}_{{\text{1}}}}{{i}_{{\text{2}}}}}}}(\mathbf{p})\frac{\mu }{r}{{x}_{{{{i}_{1}}}}}{{y}_{{{{i}_{2}}}}},\) where μ is the intensity of selection; r is the coefficient of recombination; \({{\varepsilon }_{{{{i}_{{\text{1}}}}{{i}_{{\text{2}}}}}}}(\mathbf{p})\) is the index of epistasis nonadditivity for the viabilities \({{{v}}_{{{{i}_{{\text{1}}}}{{i}_{{\text{2}}}}}}}\) of haploid genotypes i₁i₂; and \({{x}_{{{{i}_{{\text{1}}}}}}},\)\({{y}_{{{{i}_{{\text{2}}}}}}}\) are the frequencies of alleles constituting the genotype with the numbers i₁, i₂ for the first and second locus, respectively. The specificity of the perturbed model is that it distinguishes fast (the coefficient of disequilibrium \({{D}_{{{{i}_{1}}{{i}_{{\text{2}}}}}}}\)) and slow (allele frequencies) variables, which characterizes the model as singularly perturbed. This makes it possible, without solving the equations, to distinguish two dynamic stages. At the first stage, fast variables converge to quasi-equilibrium values, and at the second stage, the evolution of slow variables proceeds under the condition of quasi-equilibrium of the fast ones. The equations obtained by the delta method with respect to the quasi-equilibrium values \({{D}_{{{{i}_{k}}{{i}_{m}}}}}\) give the correct result from the point of view of the expansion in powers of a small parameter of the singularly perturbed equation solutions. The problems of measuring epistasis and decomposition of the coefficient of epistasis into elementary components are discussed, and the two-locus coefficients of epistasis are analyzed as characteristics of a multilocus model. The generalization of the results obtained for the haploid population to the diploid case is briefly indicated.