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Genetic stochasticity & environments

Gene ExpressionBy Razib KhanJanuary 22, 2007 9:34 PM


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So I near the end of my survey of chapter 5 of Evolutionary Genetics: Concepts & Case Studies.^1 Today, we address environmental variation, but I think sometimes the end is the beginning, so I quote:

Random environment models have many technical aspects...that make them difficult to analyze. As a result, they have ben largely ignored in population genetics. This is unfortunate as it is clear that environments do change and that adaptive evolution is driven by these changes.

The last sentence made me think, "No shit sherlock." This is a pretty deep indictment of population genetics, since for many environmental fluctuation and it impact on allele frequencies is the heart of evolution. I don't know much about ecological genetics myself, so the formalism was somewhat unfamiliar to me, but I will offer what seems to be the most perplexing equation derived from a single locus diallelic model assuming two selection coefficients (i.e., each allele is randomly affected by the environment): E{Δp} = σ^2epq(1/2 - p)

[update - this was a major transcription error, I think the confusion in the comments will be cleared up now]

This models the mean change in allele frequency for p, with σ^2 representing the expected variance of the change, and q naturally being simply 1 - p. I'll let the text express the peculiarity of the equation:

...when p 0 and when p > 1/2 E{Δ} < 0. This indicates that selection pushes p toward 1/2, on average...E{Δ} suggests that random changes in the fitnesses of a model that does not maintain polymorphism will turn it into a model of balancing selection that does maintain polymorphism

The issue is that selection coefficients associated with the alleles represented by p and q are random, as opposed to an overdominant scenario where the heterozygote, e.g., A1A2, is more fit than A1A1 & A2A2. In this case the maintenance of polymorphism fits our intuition insofar as one would expect that both alleles would persist to maintain an optimal frequency of the heterozygote. But the assumptions that this model started out with was not a case where the heterozygote exhibited an advantange, rather, it was one compatible with positive directional selection, which exhausts genetic variation over time. The author, John Gillespie, finds the results curious and perplexing. One could make several inferences. Perhaps the model that, with its one locus and two alleles, is so simple that its assumptions deviate too far from the reality which it is trying to capture. The mathematics need further exploration and this may simply be a "quirk" which will be resolved later. Another possibility is that the model is telling us something real about nature, that we are missing a great deal in the population genetic models which are predicated on "bean bag genetics," that nature's contingent complexity can not be so easily parsed into a few elegant parameters. Fundamentally, I think the "salvation" lay in the empirical world, particular in computational genomics, which can expand beyond the over simplifications of one locus diallelic analytic models. We may lose the ability to define the world by a single equation, but the reality is that the biological world is riddled with so many exceptions that we may have to settle for a finite but reasonable numbef of sui generis models. 1- Previous posts: III, III, IV & V.

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