Selection at two sex-linked loci

Recurrence equations are derived for selection at two sex-linked or haplodiploid loci, each with two alleles. Equilibrium equations containing only the female gametic frequencies are obtained by using a transformation that absorbs the male gametic frequencies. It is shown that gametic equilibria occur at a stationary point of the geometric mean viability if, and only if, ½ r f [ x 1 x 4 ( v 11 + v 44 ) − x 2 x 3 ( v 22 + v 33 )] w 22 = 0, where r f is the recombination fraction in females, w 22 is the viability of the double heterozygote, x 1 , x 2 , x 3 , x 4 are the frequencies of the female gametes AB, Ab, aB, ab respectively and v 11 , v 22 , v 33 , v 44 are the corresponding male viabilities. Therefore, in general, linkage equilibrium is neither sufficient nor necessary for the mean viability to be at a stationary point and there will always be some linkage disequilibrium in the system, either in females or males or both. Symmetrical selection in females is analysed in some detail and it is shown that unless there is also a particular type of symmetrical selection in males then only asymmetrical equilibria can occur. The hitchhiking effect at sex-linked and autosomal loci is compared, and it is found that with no recombination in males at autosomal loci the effect can be stronger than at sex-linked loci.