5.5 How an advantageous mutation spreads througha populationAlmost every probability textbook treats the basic setup of genetics – an indi-vidual has two genes at a site, inheriting one from each parent, this one chosenat random from the parent’s two. It is noteworthy that this is one of very fewinteresting real-world phenomena that (as far as scientists understand) really is“physically random” like coin tossing. Introductory textbooks typically treatthe dominant/recessive setting, for instance in simplified models of eye color.There theory gives predictions for genetic type of child in term of genetic type ofparents; and (via Bayes rule and empirical population frequencies) predictionsfor child’s eye color in terms of parents’ eye color.Instead of that familiar topic, I will describe toy models in population ge-netics. Let me approach the topic obliquely, via a little history. The fact thatempires have risen and fallen throughout human history is hardly controver-sial. People sometimes propose general explanations for how this happens –divine favor, racial superiority, class struggle, technological superiority, societalethics, ecological collapse – but none is not widely accepted, and indeed aregenerally taken to reflect prejudices of the era when they were formulated. Bya rough analogy, in the middle of the nineteenth century, once dinosaur andother fossils were being discovered, the proposition that life on Earth has beenin existence for a very long time, that earlier species had become extinct andthat other species had originated – this proposed fact wasn’t particularly con-troversial. But the point of Darwin’s idea of “evolution by natural selection”was that there is one explanation of this process – natural selection. Darwinand his nineteenth century followers did not have our current notion of geneticsand did not seek a mathematical formulation of their theory. And indeed theywere aware that there was a difficulty with the whole idea, if approached froma certain common sense view of heredity (“paint mixing”, below). Let me firstdescribe the difficulty, and then show how it is resolved in the correct theory ofgenetics.5.5.1 If heredity were like paint mixingObservation of animal breeding might suggest offspring are a mixture of par-ents, like a mixture of blue and yellow paint makes green paint. Of course thiscouldn’t be the whole story, or every individual in a population would be identi-cal by heredity, but (unaware of genetics) we might imagine heredity working as“mixture of parents, plus individual randomness”. And indeed this kind of “ad-ditive” model does correctly predict the behavior of some real-world quantitativecharacteristics, for instance height in humans ([15] sec. 25.3). However, let usconsider a model for how natural selection might work on a novel hereditabletrait, if heredity were like paint mixing. We’ll give a model that ignores ran-domness (both in number of offspring and assumed “individual randomness”),but incorporating randomness doesn’t change the conclusions.71A paint mixing model. One individual (in generation 0, say) has a newcharacteristic giving selective advantage α , meaning that the mean number ofoffspring reaching maturity is 2(1 + α) instead of 2. Each offspring (generation1) has only half of the characteristic (this is the “like paint mixing” assumption),so has selective advantage α/2, so each generation 1 offspring has mean number2(1 +12α) offspring in generation 2, and these generation 2 individuals has aquarter of the characteristic. So the “penetration” (sum over individuals of theirproportion of the characteristic) of the characteristic in successive generationsisgeneration 0 1 2mean number individuals 1 2(1 + α) 4(1 + α)(1 +12α)proportion of characteristic 11214penetration 1 1 + α (1 + α)(1 +12α)As time passes the mean penetration increases, not indefinitely but only to afinite limitβ(α)=∞!i=0(1 + 2−iα)which for small α is approximately 1 + 2α. This value doesn’t depend on thepopulation size (N , say). So the key conclusion is that the effect of a sin-gle appearance of a new characteristic would be, after many generations, thateach individual in the population gets a proportion around (1 + 2α)/N of thecharacteristic.This conclusion is bad news for a theory of natural selection, b ecause itimplies that to become “fixed” in a population, a new characteristic would haveto reappear many times – order N times – even when it provides a selectiveadvantage.5.5.2 The genetic modelHow does genetics really work? What I describe here is less realistic than theusual toy model (the Wright-Fisher model – see section 5.7.1) but leads to thesame formula. We consider genes rather than individuals, so there are 2N genesin each generation. On average, a gene has 1 copy in the next generation, withsome s.d. (= σ., say). For a new allele (the alleles are the possible forms of agiven gene) which confers a small selective advantage, we suppose the averagenumber of copies becomes µ = 1+α. This can only be true while the penetration(proportion of all genes which are that allele) is small, which is an unrealisticaspect of the model.So suppose a mutation creates a new allele with small selective advantage α.Then the number of copies behaves (while penetration is small, at least) as ajust supercritical Galton-Watson process described in section 5.6. In particular,either the new allele disappears from the p opulation quite quickly (extinction,in the Galton-Watson terminology) or the penetration grows and eventuallybecomes fixed in the p opulation – every gene is this allele. So the formula (5.3)72for survival probability of just supercritical Galton-Watson processes carriesover to the present settingFor a single mutation giving a gene with small selective advantage α, thechance that the gene becomes fixed in the population is about2ασ2.This conclusion is much better news for a theory of natural selection, becausenow the population size do esn’t matter. If the chance above were 1/10, say, thenan advantageous mutation needs to reappear only 10 or 20 times to be likely tobecome “fixed” in the population, regardless of how large the population sizeN is.The whole process of an allele becoming fixed in this way is called a selectivesweep. Once a sweep is under way, the number of copies grows at rate α pergeneration, and soduration of a selective sweep