Genetics, Vol 121, 163-174, Copyright © 1989

INVESTIGATIONS

Evolutionarily Stable Mutation Rate in a Periodically Changing Environment

K. Ishii, H. Matsuda, Y. Iwasa and A. Sasaki
College of General Education, Nagoya University, Nagoya 464, Japan

Evolution of mutation rate controlled by a neutral modifier is studied for a locus with two alleles under temporally fluctuating selection pressure. A general formula is derived to calculate the evolutionarily stable mutation rate {mu}(ess) in an infinitely large haploid population, and following results are obtained. (I) For any fluctuation, periodic or random: (1) if the recombination rate r per generation between the modifier and the main locus is 0, {mu}(ess) is the same as the optimal mutation rate {mu}(op) which maximizes the long-term geometric average of population fitness; and (2) for any r, if the strength s of selection per generation is very large, {mu}(ess) is equal to the reciprocal of the average number T of generations (duration time) during which one allele is persistently favored than the other. (II) For a periodic fluctuation in the limit of small s and r, {mu}(ess)T is a function of sT and rT with properties: (1) for a given sT, {mu}(ess)T decreases with increasing rT; (2) for sT </= 1, {mu}(ess)T is almost independent of sT, and depends on rT as {mu}(ess)T {complex} 1.6 for rT << 1 and {mu}(ess)T {complex} 6/rT for rT >> 1; and (3) for sT >/= 1, and for a given rT, {mu}(ess)T decreases with increasing sT to a certain minimum less than 1, and then increases to 1 asymptotically in the limit of large sT. (III) For a fluctuation consisting of multiple Fourier components (i.e., sine wave components), the component with the longest period is the most effective in determining {mu}(ess) (low pass filter effect). (IV) When the cost c of preventing mutation is positive, the modifier is nonneutral, and {mu}(ess) becomes larger than in the case of neutral modifier under the same selection pressure acting at the main locus. The value of c which makes {mu}(ess) equal to {mu}(op) of the neutral modifier case is calculated. It is argued that this value gives a critical cost such that, so long as the actual cost exceeds this value, the evolution rate at the main locus must be smaller than its mutation rate {mu}(ess).

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