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# The stochasticity of genetic drift

Gene ExpressionNovember 15, 2006 10:00 PM

I have spoken of the probability of extinction and the rate of substitution once past extinction, but now to something more prosaic, genetic drift. My post is based on John Gillespie's treatment in Evolutionary Genetics: Concepts & Case Studies. Like R.A. Fisher he does not think much of this process in evolutionary dynamics, and deemphasizes its salience. The easiest way to think of drift is simply as sample variance over generations, the expected deviation from the mean as one moves through time. In a diallelic model the deviation between generation n and generation n + 1 is: σ = √(pq)/(2N), where p = allele 1 q = allele 2 (and is 1 - p) N = population size From this, you can see that as N → ∞, σ → 0. The larger the population size the lower the proportionate sample variance across generations. Intuitively this should make sense, you know that the more data points you have from a population the more likely you are to approximate its true character. The more flips of a coin you make the closer your will approach to a 50/50 heads vs. tails rate over all of your tosses. The formalism above implies that the power of drift is inversely proportional to the size of the population. Sewall Wright showed that the rate of the loss of genetic variation is proportional to 1/(2N), following the equation above. As the population increases the rate of the loss of genetic variation decreases as the gene pool is less significantly buffeted by the stochastic influence of random genetic drift. Gillespie makes much of the fact that the decrease in heterozygosity of a population between generations is: ΔH = (σ^2/N)H Where σ^2 is variance (the square of the deviation above), and N is the population size and H heterozygosity. What is heterozygosity? Simply stated is the fraction of individuals who are heterozygous at a locus. In the model above, that means an individuals who has allele 1 and allele 2, the 2pq in the Hardy-Weinberg Equilibrium. In a two allele system eterozygosity is maximized when p = q, and since it is a diallelic model it follows that p and q are both 0.5.^1 Any deviation from this equilibrium reduces heterozygosity, and as we know that genetic drift shifts the frequencies away from their initial state, either toward fixation (100%) or extinction, H is always hurtling downward sans other evolutionary dynamics such as selection (e.g., heterosis) or mutation to maintain or replenish variation. Compared to the earlier processes Gillespie, like Fisher, maintains that genetic drift driven by population size is a relatively weak long term influence on evolutionary dynamics. The probability of extinction in one generation is 1/3, but the time for a neutral mutation to move to fixation is 4N generations! A lot happens between t = 0 and t = 4N.... Addendum: Please play with this random genetic drift simulator if you haven't before. It'll give you a general "feel" for how powerful (or not) this force is depending upon the parameters you pop into it. Update: In response to RPM's comment, yes, I'm assuming HWE, random mating and all that. In response to Larry's comment, I do not deny the relevance of the Neutral Theory (and model) on the molecular scale, I think Gillespie's point would be that "boundary process" (i.e., when a mutation is at very low frequency in generation t < 10) is far more important than simply random genetic drift over the long term in terms of stochasticity. Note: Adapted from chapter 5 of Evolutionary Genetics: Concepts & Case Studies. 1 - Confirm it numerically using a Hardy-Weinberg model, or, just think of it as calclus where f(x) = 2x(1-x), where x = p, and set f(x)^' = 2 - 4x, and when the rate of change is 0 you get x = 0.5.

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