An n Locus Multiallele Model for Gene Substitution

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We discuss the conceptual conflict between a slow series of gene substitutions as the mechanism of evolutionary change, and the apparent need for rapid and coordinated changes at many loci simultaneously in producing complex adaptations. To improve on the limitations of classical theory and accommodate the enormous amount of variability disclosed by electrophoretic studies, we develop a model that can deal with gene substitution at n loci, with numerous alleles at each locus. Fitness is treated somewhat differently from the usual way by allowing it to vary between zero and the number of offspring an individual of a particular species can produce. As maximum fitnesses, we chose five as typical of large mammals, 100 for insects like Drosophila, and 1000 for very prolific species. When our model is applied to the classical problem of determining the number of generations required to change the gene frequency from 0.0001 to 0.9999 (but for 100 loci rather than one), we find that it requires 22,899 generations when maximum fitness is five, 7,984 generations when maximum fitness is 100 and 5,333 generations when it is 1000. This is something of an improvement over the 300,000 generations calculated by Haldane (1957). By allowing the fitnesses in our model to be explicitly frequency dependent, these results are reduced considerably. In addition, allowing varying proportions of the population to inbreed reduces the number of generations required for the classical problem by as much as 50%. We also point out that, given the large amount of observed genetic variation, evolutionary change may not be so much a matter of classical gene substitution as it is of changing from one array of alleles to another. With our model, the array (0.5, 0.15, 0.2, 0.1, 0.05) can be changed to (0.03, 0.1, 0.2, 0.17, 0.5) at 1000 loci in 6,043, 2,108, or 1,408 generations, depending on whether the maximum fitness is five, 100, or 1000. Finally, we note that it is possible to substitute one array for another while continuously favoring heterozygotes.

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