The pattern of genetic hitchhiking under recurrent mutation

Genetic hitchhiking describes evolution at a neutral locus that is linked to a selected locus. If a beneficial allele rises to fixation at the selected locus, a characteristic polymorphism pattern (so-called selective sweep) emerges at the neutral locus. The classical model assumes that fixation of the beneficial allele occurs from a single copy of this allele that arises by mutation. However, recent theory (Pennings and Hermisson, 2006a; Pennings and Hermisson, 2006b) has shown that recurrent beneficial mutation at biologically realistic rates can lead to markedly different polymorphism patterns, so called soft selective sweeps. We extend an approach that has recently been developed for the classical hitchhiking model (Schweinsbergand Durrett, 2005; Etheridge, Pfaffelhuber, Wakolbinger, 2006) to study the recurrent mutation scenario. We show that the genealogy at the neutral locus can be approximated (to leading orders in the selection strength) by a marked Yule process with immigration. Using this formalism, we derive an improved analytical approximation for the expected heterozygosity at the neutral locus at the time of fixation of the beneficial allele.