Hitchhiking as Indirect Selection: I. Diallelic Loci

The indirect pressure of one locus selection on another, neutral locus, is considered. Models of haploid and diploid populations with non-overlapping generations are analyzed. The model of a haploid population is formulated as a system of difference equations linking the values of the concentration of the favorable allele, the concentration of the allele of the neutral locus, and the coefficient of linkage disequilibrium of the considered diallelic loci in adjacent generations. Equilibria are considered, and qualitative properties of the dynamics of the genetic state of the population are studied. Under directional selection, the favorable allele monotonically displaces the alternative one. The concentration of the neutral allele can either increase or decrease monotonically. Its ultimate cumulative change is practically insignificant, and the convergence to the limit is much faster than for the favorable allele. These results are illustrated by graphs. Then the model under consideration is approximated by a system of ordinary differential equations, and its analysis confirms the conclusions of the discrete model, both qualitatively and quantitatively. Linkage disequilibrium can first increase and then decrease to zero if the selection coefficient is greater than the recombination one. Otherwise, the convergence to zero is monotonic, and in any case, the sign of the disequilibrium does not change. A generalization of the haploid population model is given, allowing for the influence of pairwise random interactions of haplotypes on viability. It leads to equations corresponding to viability selection in a diploid population, where the appearance of a polymorphic stable equilibrium partitions the interval of admissible states (0, 1) into two positively invariant parts, on each of which the dynamics of the allele concentration under selection pressure is monotonic. This allows us to reduce the corresponding problems of the analysis of a diploid population to those already studied in the haploid case.