Fitness Model for Gene Frequency in Populations

Imagine a single gene with two alleles, A and a, where the frequency of the A allele in the gene pool of an entire population is . Assuming that this is a large and isolated population, that there is random mating, and that there are fixed fitness coefficients for each of the three possible genotypes AA, Aa, and aa through the generations, one can predict the frequency of the A allele in the population after generations.
This Demonstration allows you to adjust the values of the fitness coefficients (drag their sliders to the right to improve the fitness of a genotype), and to adjust the initial frequency of the A allele in the gene pool (vertical slider). You may also control how many generations are viewed.



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You will find that in some cases the A allele will truly become dominant and the other allele will go extinct (e.g., see Snapshot 1). Even with the same fitness coefficients, if the initial allele frequency of A is sufficiently small, it will go extinct (Snapshot 2). And in the case of heterozygous advantage (when the Aa fitness coefficient is 100% and the others are less), a stable limiting value for the frequency of A will manifest itself (Snapshot 3). This limiting value is indicated by a green line. The most famous example of heterozygous advantage is that of sickle cell anemia in certain tropical populations.
In some other cases a red horizontal line appears. This is a repelling (rather than attracting) stable value for the allele frequencies. That is, if is less than this value, , while if is greater than this value, . Whether red or green, the limiting value is given by , where , , and are the fitness coefficients for aa, Aa, and AA, respectively.
The mathematics behind this Demonstration is quite simple and is widely documented. See, for example, chapter 6.4 of E. S. Allman and J. A. Rhodes, Mathematical Models in Biology: An Introduction, New York: Cambridge University Press, 2004.
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