Genetic Drift - Kent Holsinger

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The effective size of a population is the size of an ideal population that has thesame properties with respect to genetic drift as our actual population does.What does that phrase “same properties with respect to genetic drift” mean? Well there aretwo ways it can be defined. 17Variance effective sizeYou may remember 18 that the variance in allele frequency in an ideal population isV ar(p t+1 ) = p t(1 − p t )2NSo one way we can make our actual population equivalent to an ideal population to make theirallele frequency variances the same. We do this by calculating the variance in allele frequencyfor our actual population, figuring out what size of ideal population would produce the samevariance, and pretending that our actual population is the same as an ideal population ofthe same size. To put that into an equation, 19 let ̂V ar(p) be the variance we calculate forour actual population. ThenN (v)e =p(1 − p)2 ̂V ar(p)is the variance effective population size, i.e., the size of an ideal population that has the sameproperties with respect to allele frequency variance as our actual population.Inbreeding effective sizeYou may also remember that we can think of genetic drift as analogous to inbreeding. Theprobability of identity by descent within populations changes in a predictable way in relationto population size, namelyf t+1 = 1 (2N + 1 − 1 )f t .2NSo another way we can make our actual population equivalent to an ideal population is tomake them equivalent with respect to how f changes from generation to generation. We dothis by calculating how the inbreeding coefficient changes from one generation to the next inour actual population, figuring out what size an ideal population would have to be to show17 There are actually more than two ways, but we’re only going to talk about two.18 You probably won’t, so I’ll remind you19 As if that will make it any clearer. Does anyone actually read these footnotes?.8
the same change between generations, and pretending that our actual population is the samesize at the ideal one. So suppose ˆf t and ˆf t+1 are the actual inbreeding coefficients we’d havein our population at generation t and t + 1, respectively. Thenˆf t+1 ==ˆf t+1 − ˆf t =12N e(f)( 1(1 − 1 )2N e(f))+2N e(f)( ) 12N (f)e(1 − ˆf t ) + ˆf t(1 − ˆf t )ˆf tN (f)e =1 − ˆf t2( ˆf t+1 − ˆf t ).In many applications it’s convenient to assume that ˆf t = 0. In that case the calculation getsa lot simpler:N e (f) = 12 ˆf.t+1We also don’t lose anything by doing so, because N e(f) depends only on how much f changesfrom one generation to the next, not on its actual magnitude.Comments on effective population sizesThose are nice tricks, but there are some limitations. The biggest is that N e(v) ≠ N e(f) if thepopulation size is changing from one generation to the next. 20 So you have to decide whichof these two measures is more appropriate for the question you’re studying.• N e(f) is naturally related to the number of individuals in the parental populations. Ittells you something about how the probability of identity by descent within a singlepopulation will change over time.• N e(v) is naturally related to the number of individuals in the offspring generation. Ittells you something about how much allele frequencies in isolated populations willdiverge from one another.20 It’s even worse than that. When the population size is changing, it’s not clear that any of the availableadjustments to produce an effective population size are entirely satisfactory. Well, that’s not entirely trueeither. Fu et al. [2] show that there is a reasonable definition in one simple case when the population sizevaries, and it happens to correspond to the solution presented below.9
Page 1 and 2: Genetic DriftIntroductionSo far in
Page 3 and 4: Now if we stare at those a little w
Page 5 and 6: • The population has no memory. 8
Page 7: 3. If the process is allowed to con
Page 11 and 12: With those facts in hand, we’re r
Page 13 and 14: Consider, for example, a population

Genetic Drift - Kent Holsinger

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