Protection of Recessive and Dominant Traits in a Subdivided Population with General Migration Structure

The problem of migration-selection interaction in a subdivided population affecting a recessive allele is studied. Explicit conditions for protection of a recessive allele are determined, accessible for any migration structure. More specifically, consider allele a recessive to allele A distributed over N demes endowed with fitness parameters w_i, 1, 1 for genotypes aa, Aa, and AA operating in deme i, respectively. For an arbitrary migration structure between demes summarized by the backward migration matrix, M = |m_ij|^N_i,j=1 (m_ij = the proportion per generation of the population in deme i whose parents came from deme j), let $$\underline{\xi}$$ = (ξ₁,..., ξ_N) be the unique left eigenvector of M corresponding to eigenvalue 1, whose sum is normalized to unity. The exact necessary and sufficient condition for protection of the recessive allele, a, is that the "weighted average" of the aa-genotype fitness values, namely ∑^N_i=1 ξ_iwi, exceed unity. The result applies whether soft or hard selection is assumed. It is further established that, with a dominant or recessive trait, fixation of alleles a and A cannot coexist as stable alternatives. Interpretations and further applications of the conditions are given for some special migration patterns, including the stepping-stone migration mode and the generalized Island model gene flow.