Heterozygosity is variance. How Drift Affects Heterozygosity. Decay of heterozygosity in Buri s two experiments

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eterozygosity is variance ow Drift Affects eterozygosity Alan R Rogers September 17, 2014 Assumptions Random mating Allele A has frequency p N diploid individuals Let X 0,1, or 2) be the number of

X is binomial with N = 2 and p equal to the frequency of A within the population Variance of a binomial is Np1 p) Variance of X is 2p1 p) Same as heterozygosity We begin with data from an experiment,

1 eterozygosity is variance ow Drift Affects eterozygosity Alan R Rogers September 17, 2014 Assumptions Random mating Allele A has frequency p N diploid individuals Let X 0,1, or 2) be the number of copies of A within a random individual What is the probability distribution of X? What is the variance of X? X is binomial with N = 2 and p equal to the frequency of A within the population Variance of a binomial is Np1 p) Variance of X is 2p1 p) Same as heterozygosity We begin with data from an experiment, described by Peter Buri in 1956 ene frequency in small populations of mutant Drosophila, Evolution, 10: ) Buri s drift experiment I Each generation: 107 bottles, each w/ 8 male & 8 female fruit flies 0: all flies heterozygous Rows show distribution of allele frequency in 19 successive generations Peter Buri, 1956 Decay of heterozygosity in Buri s two experiments 050 Buri s drift experiment II Each generation: 105 somewhat larger bottles Otherwise like experiment I Peter Buri, Experiment I Experiment II eterozygosity ) starts at 05 Declines to about 02 Why?

2 As heterozygosity declined w/i bottles, the variance among them increased As heterozygosity declined w/i bottles, the variance among them increased Computer Simulations of enetic Drift The Urn Metaphor 10 Imagine two urns: metaphors for a population in two successive generations Urn 1 has 50 balls, some red, some white, representing parental gene copies Urn 2 is empty until urn 1 has reproduced as follows: p 05 = 100 = 20 1 Examine a random ball from urn 1 2 Put a ball of the same color into urn 2 3 Replace the ball from urn 1 4 Repeat until there are 50 balls in urn s The number of red balls in urn 2 is likely to differ from that in urn 1, because of random sampling This metaphor is used as a model of genetic drift Decay of eterozygosity: Notation The urn model behaves a lot like genetic drift in real populations: 1 variation between populations increases 2 variation within populations decreases Yet real organisms don t reproduce as our urns do The best urn model is unlikely to be one in which the number of balls matches the number of gene copies N = # of diploid individuals in population = # of gene copies in population = Probability that two random gene copies, drawn with replacement from generation t, are copies of the same allele = same thing in the generation t + 1

3 Decay of eterozygosity: Logic Two gene copies either are or are not copies of the same parental gene copy Two gene copies may be identical in state either because 1 they are copies of the same parental gene copy, or 2 they are copies of distinct parental gene copies, which happen to be identical in state Two genes are copies of the same parental gene with probability 1/, and of distinct parental genes with probability / Event Prob Individual carries 2 copies of same parental gene 1/ Event Individual carries copies of 2 distinct parental genes, which are themselves identical Prob /) Explanation: 1 First draw a random gamete from among those produced by the parental generation This gamete is equally likely to have been produced by any of the parental genes 2 Next draw another gamete It is equally likely to have been produced by any of the parental genes There is 1 chance in that the second gene is a copy of the same parental gene as the first Explanation: 1 The two random gene are copies of distinct parental genes with probability / 2 These distinct parental genes are copies of the same allele with probability that is the definition of 3 Both things are true with probability: ) In short, the two genes are identical if they are copies either of 1 the same parental gene probability 1/), or of 2 distinct but identical genes probability /)) Altogether, = 1 + ) To see where this goes, it is easier to work with the probability that the two genes are copies of different alleles, ie with the heterozygosity, = 1 = ) after some algebra) Can you supply the algebra?

4 The Time-path of eterozygosity Example 1 = 2 = = t = ) 0 ) 1 ) 2 0 ) t 0 In Peter Buri s experiment, 1 = 1/2 because half the population were heterozygotes after the first generation of random mating 18 generations later: But what is? 19 = 1 2 ) 18 where 0 is the original heterozygosity and t is the heterozygosity in generation t eterozygosity: Buri s experiment I vs urn model eterozygosity: Buri s experiment I vs urn model = 18 = 32 = = 32 There were 32 gene copies in each bottle Yet = 32 provides a poor fit to data Better fit with = is the effective population size F ST measures variation among populations Model fits after setting N = N e Data from Buri 1956) Data from Buri 1956) Series I Series II theory data Series I e = Series II e =

5 Effects of Drift and Mutation on ene Diversity and ene Identity) ow Mutation Affects the Decay of eterozygosity Alan R Rogers Adding mutation to the equation describing decay of gene diversity Example: Prehistoric elk from the Emeryville Shellmound September 17, 2014 Two models of mutation ow drift affects gene identity The mutation rate is u per gamete per generation Infinite alleles K alleles Each mutant is an allele never seen before When allele i mutates, the mutant is equally likely to be any allele other than i There are K 1 possibilities, each with probability 1/K 1) We ll focus on the model of infinite alleles Without mutation With mutation = 1 + ) = 1 u) 2 [ 1 + ) ] Approximations Numerical example 1 u) 2 approximation 1 u) 2 1 2u 1 2u 1 u 1 u) 2 1 2u

Numerical example 1 u)/) approximation u 1 2u)/) 1/) 00001 10 00999800 010000 00001 100 00099980 001000 00001 1000 00009998 000100 00001 10000 00001000 000010 00001 100000 00000100 000001 Before

6 Numerical example 1 u)/) approximation u 1 2u)/) 1/) Before approximations After = 1 u) 2 [ 1 + At equilibrium =, so ) ] = 1 + ) 2u 1 Ĝ = 4Nu + 1 Ĥ = 1 Ĝ = 4Nu 4Nu + 1 for model of infinite alleles The hats indicate that these are equilibrium values If 4Nu is large, Ĥ 1 Model of 2 alleles Emeryville archaeological site If 4Nu is large, Ĥ 1/2 Ĥ = 4Nu 8Nu + 1 on SF Bay, near Berkeley deposit spans 1000 years dramatic decline in numbers of elk Decline in Elk at Emeryville Broughton 1999) Artiodactyl/Carnivore Artiodactyl/Sea otter Stratum Stratum Left panel: Frequency of artiodactyls relative to small carnivores Right panel: Frequency of artiodactyls relative to sea otters ow would this decline in elk have affected gene diversity at a mitochondrial locus with u = 1/1000?

7 A guess The initial population was large, say 10,000 alf of these would be females, so the number of genes would be = 5000 With biallelic model of mutation 4Nu = /1000 = 10 0 = 4Nu 8Nu + 1 = = 0476 The later population would be tiny, say 200, so = 100 and 4Nu = /1000 = 02 1 = = 0143 An enormous decline in heterozygosity Implication Populations of different size should differ enormously in heterozygosity Is this really true? eterozygosity at enzyme polymorphisms vs log 10 grams of body wgt Wooten & Smith 1985) Range: 3 g to 4000 kg Puzzles Why is there so little heterozygosity? Small animals have large populations and should have high heterozygosity Why don t they? Much of the rest of this course is about these questions

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