The effect of random selection coefficients on populations of finite size – some particular models

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Abstract
SUMMARY: The model of random selection coefficients is considered in the context of a finite population of diploids. The selection coefficients of the homozygotes are allowed to vary with equal variance while the fitness of the heterozygote is kept fixed. Steady-state solutions are found in the case of equal two-way mutation rates with particular reference to the expected heterozygosity. Increasing the variance of the selection coefficients of the homozygotes is found to uniformly increase the heterozygosity for all values of the average selection coefficients and its effect is largest when the selection coefficients of the homozygotes are fully correlated. The fate of mutant genes is also considered in the case of random selection coefficients by looking at the probability of ultimate fixation and the mean times to fixation and extinction. The errors in previous calculations (e.g. Kimura, 1954; Ohta, 1972) are pointed out. It is found that a small average heterozygote advantage together with a reasonable degree of variance in the coefficients can cause an unexpectedly large amount of heterozygosity to be maintained. It is also seen that probabilities of fixation and mean times to boundaries are usually increased by increasing the variance showing that it in fact helps to keep the population heterozygous for much longer than the non-random case. This is in contradiction to some conclusions of Karlin & Levikson (1974) because their haploid results are not easily extendable to the consideration of this sort of diploid model.