The dynamics of interlocus associations in the three locus hitchhiking model. 1. The three-way linkage disequilibrium function.

The dynamical behavior of the three-way linkage disequilibrium function is investigated within deterministic, three locus hitchhiking systems consisting of two neutral loci and one selected locus. The analysis is based on the case in which at least one of the neutral loci is initially in linkage equilibrium with the selected locus. It also applies approximately, whenever the product Dn1SDn2S, is negligible relative to the three-way association, where DniS denotes the pairwise linkage disequilibrium between the selected locus and neutral locus i. A complete qualitative description of the resulting behavior is given for several selection models as a function of the gene frequency at the selected locus, the strength of selection, the degree of recombination among the loci, and the relative position of the selected locus. A major finding is that, although not functionally equivalent, the dynamics are qualitatively like those found previously for the pairwise association between a neutral locus and a linked, selected locus (Asmussen and Clegg 1981). Ultimately, the three-way linkage disequilibrium will decay at a rate faster than expected for three neutral loci. The range of gene frequencies associated with the accelerated decay steadily expands as selection intensifies, and either grows or shrinks with increasing recombination among the loci, depending upon the mode of selection operating. In the early stages of a trajectory, the triple interaction may increase in magnitude only if the selection intensity exceeds the frequency with which a detectable recombination occurs between the middle and end loci, and a favored allele (at the selected locus) is rare. When it exists, the region of increase first enlarges, then shrinks as selection intensifies, and contracts with increasing recombination among the loci. An interesting position effect is that systems in which the selected locus is to one side of the two neutral loci may have a greater three-way interaction, and hence a greater hitchhiking effect, than equivalent systems in which the selected locus is between the neutral loci.