Saturday, February 04, 2006
Likely you all know that heterozygote advantange, where the fitness of the heterozygote (Aa) is greater than that of either homozygotes (aa, AA), is often given as a reason why genetic diversity (polymorphism) persists. This was one of the many ways that the Balance School in classical genetics, derived from Sewall Wright, argued that genetic variation was maintained.1 In the 1960s the emergence of dramatic polymoprhism revealed by molecular techniques did not vindicate the Balance School because the extent of variation was far greater than anticipated or explainable via balancing selective forces.
But back to stage one, the equilibrium frequency for a frequency of allele A, where genotype Aa is overdominant (its mean fitness is greater than the homozygotes) is:
Frequency A = (fitness[Aa] - fitness[aa])/(2*fitness[Aa] - fitness[AA] - fitness[aa])note - 2
This is the difference between the heterozygote fitness and that of the other allele, divided by the difference between twice the heterozygote fitness after subtraction of the sum of homozygote fitnesses.
Frequency a = 1 - A
A situation where the mean fitness of the homozygotes is the same but the heterozygote is higher will naturally result in frequencies of both alleles persisting at ~0.5 (the frequency of the heterozygote genotype is maximized via Hardy-Weinberg Equilibrium, p2 + 2pq + q2). The logic seems transparent and intuitive. But what about multiple alleles at a locus?
In the 1970s Richard Lewontin and others investigated these scenarios and they found that their initial models showed that increasing the number of alleles can lead to counterintuitive results. If you have a scenario where you have n alleles, and all heterozygotes are more fit than all homozygotes,3 this is neither necessary nor sufficient for persistence of genetic variation (polymorphism) over time. Remember, the mean fitness of a heterozygote is greater than that of any of the homozygotes. When fitness is assigned randomly to allelic combinations the researchers found that the parameter space allowing a stable polymorphism decreased as a function of the number of alleles. For example 0.1% randomly chosen viabilities allowed stable polymorphism at a gene with seven alleles.
The point here is that I am skeptical that this fits in with our intuition of what should occur as you increase the number of alleles and retain heterozygote advantage. This, along with the difficulty of detecting overdominance in nature, is why W.D. Hamilton appealed to long term frequency dependent selection for his "Red Queen" hypothesis (recall that immunity related loci, such as MHC, are highly polymorphic). But science doesn't stop until results are definitive. Here is a 1992 paper which tweaks the parameters to show how multiple alleles can persist.
In A Reason for Everything Marek Kohn recounts how Amotz Zahavi, proponent of the handicap principle, had difficulty getting along with the late J.M. Smith, because their modes of thinking were very different. Zahavi saw no reason to cede his intuitions in the face of Smith's analytic formalist methodologies ("doing the sums"). Verbal arguments and intuition are essential preconditions of science, nevertheless, as we explore the full sample space of reality, our intuitions tend to lose their way and we must put faith in the guard-rails of deduction and theory.
1 - The Balance School is to be contrasted by the Classical School which took after R.A. Fisher. While Fisher believed that genetic variation was only an epiphenomenon that derived from the transitions toward fixation of newly beneficial alleles, Wright contended that various balancing selective forces, heterozygote advantange, environmental fluctuation, gene-gene interactions (epistasis), resulted in a rugged adaptive landscape which allowed multiple stable fitness peaks.
2 - Adapted from Principles of Population Genetics.
3 - fitnessii is less than fitnessij, fitnessjj is less than fitnessij.