The dynamics of interlocus associations in the three-locus hitchhiking model. 2. The pairwise linkage disequilibrium between two neutral loci

The hitchhiking effects of a selected locus upon the dynamics of the pairwise association, Dnn, between two neutral loci is examined analytically for the special case where at least one of the neutral loci is in linkage equilibrium with the selected locus. The results apply approximately whenever the product of the pairwise associations between the selected locus and each neutral locus is negligible with respect to the three-way linkage disequilibrium. It is shown that precisely four broad classes of trajectories are possible, whether the selected locus is between (nsn) or to one side (snn) of the neutral loci, and whatever the mode of selection operating. Dnn may: decay rapidly to zero, at a rate faster in each generation than that expected for two isolated neutral loci; monotonically decay to zero at a rate which is slower in every generation than under the usual neutral regime; increase initially and/or in intermediate periods before eventually slowly decaying to zero; or exhibit type 1 behavior in the first segment of the trajectory and either type 2 or 3 behavior in the subsequent generations, with the transition marked by a change in sign. The nature of a given trajectory is largely determined by the direction of gene frequency change at the selected locus, and the initial signs of both Dnn and three-way linkage disequilibrium. The single most important consequence of these results is that there is no simple relation between the amount of pairwise association between two neutral markers and the recombination fraction between them. Several factors influencing the magnitude of the hitchhiking effect are also examined. It is shown that, all else being equal, the greater the three-way linkage disequilibrium, the greater the departure of Dnn from the expected neutral dynamic. Increased recombination among the loci reduces the hitchhiking effect on Dnn. The dependence of the behavior upon the exact position of the selected locus is also determined both within and between nsn and snn chromosomal systems. An interesting discovery is that given equivalent nsn and snn systems, with each having the same recombination between their two neutral loci, Dnn will deviate more from the standard neutral dynamic in the snn system if its selected locus is sufficiently tightly linked to the neutral loci.