Dynamics of the Linkage Disequilibrium Function under Models of Gene-Frequency Hitchhiking.

The dynamic behavior of the linkage disequilibrium (D) between a neutral and a selected locus is analyzed for a variety of deterministic selection models. The time-dependent behavior of D is governed by the gene frequency at the selected locus (p) and by the selection (s) and recombination (r) parameters. Thomson (1977) showed numerically that D may increase under certain initial conditions. We give exact conditions for D to increase in time, which require that the selection intensity exceed the recombination fraction (s > r) and that p be near zero or one. We conclude from this result that gene frequency hitchhiking is most likely to be important when a new favorable mutant enters a population. We also show that, for what can be a wide range of gene frequencies, D will decay at a faster rate than the neutral rate. Consequently, the hitchhiking effect may quickly diminish as the selected gene becomes more common.-The method of analysis allows a complete qualitative description of the dynamics of D as a function of s and r. Two major findings concern the range of gene frequencies at the selected locus for which D either increases over time or decays at a faster rate than under neutrality. For all models considered, the region where D increases (i) first enlarges then shrinks as selection intensifies, and (ii) steadily shrinks as r increases. In contrast, the region of accelerated decay constantly enlarges as the selection intensity increases. This region will either shrink or enlarge as r increases, depending upon the form of selection in force.