Lecture 8 - Interaction of selection and drift.doc; intro to evol

Lecture 8 - Interaction of selection and drift.doc; intro to evol

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1 Lecture 8.1 - The Interaction between Selection and Drift We have previously considered natural selection in an infinite population, and genetic drift at a neutral locus. Obviously, in real life the two forces occur together. A) Probability of loss of a new NEUTRAL mutation in the first generation? Pr(allele w/ frequency p leaves no copies in the next generation) = binomial probability Pr p 0 (1-p) 2N-0 0 2 N freq(new mutation) = 1/2N, so Pr (1/2N) 0 (1-1/2N) 2N = (1)(1)(1-1/2N) 2N 0 2 N As it so happens, (1 + x) e x when x is small (you can prove this by Taylor series expansion), so (1 - 1/2N) 2N (e -1/2N ) 2N = e -1 0.37 So, a new mutation has a > 1/3 chance of being lost in the first generation. Note that this result is independent of population size. Suppose that a new beneficial mutation confers a two-fold advantage relative to the wild-type allele (s = 0.5), and N e = 1000. So, the 1999 wild-type alleles leave one descendent on average, but there is a >1/3 chance any one of them leaves no descendents. The mutant leaves two offspring on average. How much less likely do you think it is that the new mutant is lost than a wild-type allele? B) The probability of fixation of a beneficial allele given its initial frequency p Δ Δ + Δ = p p p p p ) ( ) Pr( ) ( π In English, the probability of fixation of an allele with initial frequency p is the probability of a given change in p (i.e., Δ p ) multiplied by the probability of fixation of an allele with frequency p + Δ p , summed over all possible Δ p . The fixation probability is a function of the selection coefficient s , the coefficient of dominance, h , and (in principle) the effective population size, N e . For example, Prob( Δ p>0) is greater for beneficial alleles than for deleterious alleles and Prob( Δ p<0) is greater for deleterious alleles than for beneficial alleles.
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2 We will use another common approximation to derive the fixation probability, called the TAMH function: T hen A M iracle H appens (i.e., the Kolmogorov Backward Equation). The derivation of the fixation probability involves a branch of applied probability theory called "diffusion theory". It's the same set of equations physicists use to model heat diffusion. The math is somewhat sophisticated so we won't go through the derivation. Most important special case, for a new beneficial mutation, p = 1/2N π (1/2N) 2 hs when s is small (~ 0.2 or less) NOTICE THAT THE FIXATION PROBABILITY FOR A NEW BENEFICIAL MUTATION: 1) Is (approximately) independent of the population size 2) Is a function of the selective advantage of the heterozygote (specifically, is about twice the selective advantage of the het) 3) Is SMALL (e.g., for s = 0.2, h = 0.5, π = 0.2; for hs = 2%, π = 4%) DRIFT PREDOMINATES WHEN RARE, EVEN FOR BENEFICIAL ALLELES! "Think of all those great alleles out there that never made it out of the quagmire of rarity"
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This note was uploaded on 03/01/2010 for the course PCB 4683 taught by Professor Williams,j during the Spring '08 term at University of Central Florida.

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Lecture 8 - Interaction of selection and drift.doc; intro to evol

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