Deterministic behavior of HLA haplotypes under symmetric overdominant selection

Abstract

he HLA haplotypes are modelled by considering two loci with n_A and n_B alleles. Two symmetric overdominant selection models are studied: (1) the fitness of an individual is positively correlated with the number of antigens in that person and (2) the fitness of an individual is higher when he/she is in heterozygous condition with respect to the haplotype combination. For the first model, both multiplicative and additive fitnesses are considered. Population dynamics of the haplotypes are studied by considering four groups of haplotypes, i.e., A_iB_j, A_iB_j (l≠j), AkB_j (k≠I) and AkBl (k≠i, l≠j) with respective frequencies X₁₁, X₁₀, X₀₁, and X00 among adults. It is shown that the equilibrium points for the three models are the same, with X₁₁= 1/n_An_B, X₁₀=(n_B-1) X₁₁, X₀₁=(n_A-1) X₁₁, X00=(n_A-1) (n_B-1)X₁₁, and D= X₁₁X00-X₁₀X₀₁=0. The Hardy-Weinberg equilibrium point is globally stable except for the multiplicative fitness in the first model where the stability holds only when the recombination fraction is larger than or equal to the product of segregation loads at the two loci. The asymptotic rate of approach to the equilibrium is also obtained. That rate for the second model is identical to that for a single locus-multiple allele model with symmetric overdominant selection.