Quasi Linkage Equilibrium under Weak Two-Locus Viability Selection: I. Haploid Population with Diallelic Loci

Abstract

A model of weak viability selection at two diallelic loci with standardization of approaches through the use of perturbation theory is examined. The estimate of the quasi-equilibrium value for the linkage disequilibrium coefficient D is analyzed, and results in terms of average effects in quantitative genetics and in terms of the theory of singular perturbations in mathematics are obtained. The approximation of a discrete-time model of a random mating population with non-overlapping generations under weak selection by ordinary differential equations is considered. Weak selection is considered as a perturbation of the model without selection. The resulting model is singularly perturbed; that is, fast (D) and slow (allele frequencies) variables can be distinguished. The first approximation equation for quasi-equilibrium of D is obtained using the first terms of the Taylor series expansion of the model functions. It coincides with the corresponding part of the system of the first approximation of the asymptotic series for solving singularly perturbed equations. The first approximation for quasi-equilibrium of D is D* = e(p) $$\frac{\mu }{r}$$ x(1 – x) y(1 – y), e(p) ≡ v11(p) – v12(p) – v21(p) + v22(p), where μ is the intensity of selection, r is the recombination coefficient, e(p) is the index of epistasis nonadditivity for the viabilities $${{{v}}_{{{{i}_{{\text{1}}}}{{i}_{{\text{2}}}}}}}$$ , x and y are the allele frequencies of the first and second locus, respectively. The evolution of a nonequilibrium state of the population actually takes place at quasi-equilibrium (as the result of the stage of fast variable D dynamics) and current values of slow x, y.