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} = σ2epq(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 σ2 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} 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.
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The easy part: you should read Roff's book called Life History Evolution. I just started it, but it is a back-burner book for now. You've read Evol Quant Gen, and I assume you've seen trade-offs before, so you can skip the first 3 chapters. The first meaty chapter is about evolution in constant environments, so you could skip that too. Chapter 5 is about evolution & maintenance of genetic & phenotypic variation in "stochastic environments," although I don't know if that's what Gillespie means by random.
The annoying part: as the equation is written, I don't see how E{D} can be negative -- it is the product of three non-negative terms. Back-of-the-envelope work shows that the slope of the right-hand side is 0 when p = 1/2 (and is a max), positive when p is less than 1/2, and negative when p is greater than 1/2. Is there supposed to be a "first derivative of" on the right-hand side?
I find the conclussion that termporal variation in selection coefficients can result in a stable equilibrium unsurprising. But, yeah, where does the negative E{D} come from?
And does he address spacially varying selection coefficients?
hey guys, busy right now but i will closely at gillespie's logic here tonight, situation permitting. i omitted some stuff where he recommended the reader plug & chug.
oh, re: spatial, no, but i think the same formalism would do the trick, space & time are the same since they are one independent variable on two dependent variables (the random selection coefficients which affect the alternative alleles).
Razib's equation is talking about the expected change in "p". Thus it is written as E(delta-P).
Gillespie is trying to say that when "p" is small, there is an increase (i.e., positive) in the frequency of "p" and when "p" is large, there is a decrease (i.e., negative) in the frequency of "p". Since allele frequency will always be between 0 and 1, he's trying to model the change in this frequency when genotypes take on selection coefficients that are random variables.
The selection coefficients are treated as random variables and though he doesn't mention it in this paper, they are always greater than -1 (negative one).
Given varying selection coefficients, the expected change in an allele frequency "p" is E(delta-p) = pq * (var in selection coefficients) * (0.5 - p). This is the equation that I see in Gillespie's chapter. The variance is given as Var(delta-p) = 0.5 * (variance in selection coefficients) * p^2 * q^2.
His basic idea is to show that if you look only at the expected change, you might think that balancing selection will occur, but in fact this is really a model of shifting directional selection (he assigned fitness values to genotypes such that the heterozygote was intermediate between the two homozygotes, but the homozygotes fitnesses--being random--will tend to be unequal, resulting in a standard ordering of genotypes: aa > aA > AA, or directional selection). Thus, fluctuating environments with shifting directional selection will maintain a polymorphism. The expected change in p moves "p" toward 0.5 and the expected variance tends to push "p" to either 0 or 1, thereby maintaining the "polymorphism."
(I think that's what he means!?)
btw, for the record, i wasn't too surprised about polymorphism maintenance either...variation in env. is one of the ways i was taught that balancing selection occurred....
...in the hurry to get the HTML entity for delta to work i transcribed incorrectly. it shoud work out now. sorry about that :-)