Genetic associations under mixed mating systems: the Bennett-Binet effect

Using elementary algebraic geometry and computational commutative algebra, supported by the program Macaulay2, we studied and developed operators that define the zygotic and gametic evolution under a mixed-mating system with parameters s selfing rate, r recombination rate, and g relative fitness of inbreeders, for any possible combination of initial zygotic or gametic frequencies with two alleles at each of two loci. We found that (i) the allelic frequencies are preserved in every generation; (ii) the gametic frequencies converge to values that depend exclusively on the allelic frequencies; (iii) every zygotic population converges to a population in equilibrium with double heterozygotes equally frequent; (iv) the rate of convergence decreases to arbitrary small values with sufficiently small values of r or with sufficiently large values of s and (v) as g decreases, the maximal 'association between the two loci' occurs with higher values of selfing. We also found generalizations for the case of several alleles at each locus.