The Evolution of Multilocus Systems under Weak Selection

The evolution of multilocus systems under weak selection is investigated. Generations are discrete and nonoverlapping; the monoecious population mates at random. The number of multiallelic loci, the linkage map, dominance, and epistasis are arbitrary. The genotypic fitnesses may depend on the gametic frequencies and time. The results hold for s << c(min), where s and c(min) denote the selection intensity and the smallest two-locus recombination frequency, respectively. After an evolutionarily short time of t(1) ~ (ln s)/ln(1 - c(min)) generations, all the multilocus linkage disequilibria are of the order of s [i.e., O(s) as s -> 0], and then the population evolves approximately as if it were in linkage equilibrium, the error in the gametic frequencies being O(s). Suppose the explicit time dependence (if any) of the genotypic fitnesses is O(s(2)). Then after a time t(2) ~ 2t(1), the linkage disequilibria are nearly constant, their rate of change being O(s(2)). Furthermore, with an error of O(s(2)), each linkage disequilibrium is proportional to the corresponding epistatic deviation for the interaction of additive effects on fitness. If the genotypic fitnesses change no faster than at the rate O(s(3)), then the single-generation change in the mean fitness is ΔW = W(-1)V(g) + O(s(3)), where V(g) designates the genic (or additive genetic) variance in fitness. The mean of a character with genotypic values whose single-generation change does not exceed O(s(2)) evolves at the rate ΔZ = W(-1)C(g) + O(s(2)), where C(g) represents the genic covariance of the character and fitness (i.e., the covariance of the average effect on the character and the average excess for fitness of every allele that affects the character). Thus, after a short time t(2), the absolute error in the fundamental and secondary theorems of natural selection is small, though the relative error may be large.