Neutral mutations in ideal populations. The null model in population genetics The Fisher-Wright population model

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1 eutral mutations in ideal populations The null model in population genetics The Fisher-Wright population model ardy-weinberg equilibrium Stochastic changes in allele frequency The decay of heterozygosity over time Identity by descent Predicting levels of variation Infinite allele model Infinite sites model Identity in state Equilibrium heterozygosity Effective population size Variation in reproductive success Change in population size over time Differences between the sexes Continuous models Diffusion theory Wright s distribution of allele frequencies Ewens sampling theory Copyright: ilean McVean,

2 The null model in population genetics Let us suppose othing interesting ever happens in biology Alleles in populations behave like beans in a beanbag t t+ What would patterns of genetic variability look like? Do natural populations actually look like this? Copyright: ilean McVean,

3 Sample variance and evolutionary stochasticity Parameters Population Sample e u e u ASK iven samples, can we detect differences in the underlying parameters Copyright: ilean McVean, 3

4 The Fisher-Wright population model 3 t Assumptions ermaphroditism with possibility of selfing Random mating with respect to space and genotype on-overlapping generations Constant diploid population of size ( alleles) o migration to or from other populations o selection Binomial sampling Pr{ j i}! j!( j)! i j i j Copyright: ilean McVean, 4

5 Alleles and genotypes: ardy-weinberg equilibrium t ( ) x x -x x -x x -x enotype + Probability x x(-x) (-x) Punnett square of expected genotype frequencies A a A AA Aa a Aa aa Copyright: ilean McVean, 5

6 Applications of W theory Testing for equilibrium χ Population African Americans ( Obs Exp) Exp MM M χ.3 European Americans ative Americans 3 7. Estimating allele frequencies in recessive disease e.g. cystic fibrosis in 5 autosomal recessive enotype Phenotype Frequency AA ormal (-x) Aa ormal x(-x) aa Disease x x /5 x. Carriers.4 Copyright: ilean McVean, 6

7 Changes in allele frequency due to random sampling time First complete solution by Kimura (955) Copyright: ilean McVean, 7

8 The decay of heterozygosity over time t xt ( xt ) Probability that two alleles picked at random are different from each other.6 t E[ t+ ] t ( ) time E[ t ] ( ) t Copyright: ilean McVean, 8

9 Copyright: ilean McVean, 9 Identity by descent Same parent Pr / Different parents Pr - / ) ( ) ( by descent} Pr{identical t + + ) ( ) ( ) )( ( Pr{different by descent} t t ) ( t t Probability different by descent at time t

10 Adding mutations The infinite allele model for allozyme mutation mutation All mutations create alleles not previously present in the population The infinite sites model for DA mutation All mutations occur at sites at which mutations are not currently segregating in the population Every new mutation creates a new haplotype Copyright: ilean McVean,

11 ow much variation? Identity in state * t+ Same parent Different parents Pr ) ( u Pr ( * ) ) t ( u At equilibrium If u << and ~ * + 4u >> * * t+ t ~ * 4u + 4u Probability two alleles picked at random are different in state Copyright: ilean McVean,

12 Variation in the infinite alleles model E + u u Variation in the infinite sites model E π u Copyright: ilean McVean,

13 Estimating from polymorphism eterozygosity.4.3. Drosophila umans u enetic divergence Time since split uman allozyme Mutation rate u.5-6 per gene per generation enerations per year Proportion electrophoretic changes umans 6, Drosophila, BUT census >> estimated for both Copyright: ilean McVean, 3

14 Effective population size Many biologically realistic complications of the Fisher-Wright model result in populations that behave in the same way, but with a smaller EFFECTIVE population size e t+ t ( e ) Variation in breeding success Pr{ alleles from same parent} ki ( ki ) i σk If population size constant e σ k Copyright: ilean McVean, 4

15 More on effective population size Fluctuations in population size 4 8 Arithmetic mean 6 4 armonic mean e t i Different numbers of reproductive males and females Males Females f m e 4 m 3.5 m f + f Copyright: ilean McVean, 5

16 Variation in the infinite alleles model E + e u e u Variation in the infinite sites model E π e u Copyright: ilean McVean, 6

17 Phylogenetic constancy of e Estimated e Census Testing the effective population size model Frequency ASK A A A3 LA-A eterozygosity.86 A4 Aw9 A A6 A8 Aw3 Aw3 A3 A5 Aw3 Blank Aw33 Aw43 Allele Are the number of alleles and their frequency distribution compatible with the standard neutral model? Copyright: ilean McVean, 7

18 Diffusion theory Frequency among populations Assume xδx x x+δx Allele frequency A) Changes in allele frequency due to drift, mutation, migration or selection are small M δx - ux B) The only source of variance in the change in allele frequency over generations is random sampling x( x) Vδ x C) The mean and variance of allele frequency change are sufficient to describe the whole population Copyright: ilean McVean, 8 e

19 The equilibrium allele frequency distribution φ( x) θx ( x) θ u θ 4 e log[φ(x)] θ. Most alleles at low frequency Kimura and Crow (964) Allele frequency Ewens sampling formula Expected number of alleles in a sample of size n n + Ewens (97) Copyright: ilean McVean, 9

20 Testing the neutral hypothesis Frequency A A A3 LA-A A4 Aw9 A A6 A8 Aw3 Aw3 A3 A5 Aw3 Blank Aw33 Aw43 Sample size n 8 o. alleles k 4 Pr{n, n. n k k, n} k! l n! n n... Allele k n k Ewens (97) Test statistic B x i ln( x i ) i Pr{B -.9} Copyright: ilean McVean,