Mutation+Selection=EvolutionEarly Criticism of DarwinMendelian InheritanceSimple case: no selectionThe Hardy-Weinberg LawThe Hardy-Weinberg LawThe Hardy-Weinberg LawHW application: Sickle cell anemiaApproach: Detect selection through comparison to neutral expectationNeutral Theory HistoryGenetic DriftNeutral allele diffusionEwens sampling formula (1972)Infinite alleles modelInfinite alleles modelExpected Site FrequenciesEwens Sampling formulaThe CoalescentThe CoalescentCoalescent inferenceTurning neutral models into tests of neutralityTurning neutral models into tests of neutralityTurning neutral models into tests of neutralityFrequency-based neutrality testsFrequency-based neutrality testsNeutral Expectation (no selection, no structure, constant population size)Positive Selection (Sweep)Balancing SelectionPopulation Structure/SubdivisionPopulation ExpansionPolymorphism vs. DivergenceHKA TestPolymorphism/divergence with a twist: site classesMK TestRate-based selection metric: dN/dSRate-based selection metric: dN/dSRate-based selection metric: dN/dSRate-based selection metric: dN/dSCorrelates with dN/dS (or just dN)Neutrality Tests SummaryThe future:MIT OpenCourseWare http://ocw.mit.edu 6.047 / 6.878 Computational Biology: Genomes, Networks, EvolutionFall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Computational Biology 6.04710/09/08 Guest Lecture:Molecular evolution: traditional tests of neutralityDr. Daniel NeafseyResearch Scientist, Broad InstituteMutation+Selection=EvolutionRelative importance of each for maintaining variation in population?Early Criticism of DarwinBlending inheritance, ‘gemmules’Fleeming Jenkin (1867):Var[X(t+1)] = ½ Var[X(t)]X=Mendelian Inheritancepublished 1865-66, rediscovered 1900Law of Segregation:• allelic variation• offspring receive 1 allele from each parent• dominance/recessivity• parental alleles ‘segregate’ to form gametesLaw of Independent AssortmentSimple case: no selectionThe Hardy-Weinberg Law (1908)Requires:• infinite population size• random mating• non-overlapping generations• no selection, mutation, or migrationThe Hardy-Weinberg LawGenotype: AA Aa aaFrequency at time 0: u0v0w0u0+ v0+ w0= 1frequency of A (p0) = u0+ v0/2frequency of a (q0) = w0+ v0/2p0+ q0= 1The Hardy-Weinberg LawGenotype: AA Aa aaFrequency at time 0: u0v0w0Mating Pair Frequency OffspringAA Aa aaAA x AA u02100AA x Aa u0v0½½0Aa x AA u0v0½½0Aa x Aa v02¼½¼Frequency of AA in next generation: u1 = u02 + u0v0 + 1/4 v02= (u0+ v0/2)2= p02The Hardy-Weinberg LawIf assumptions met:•allele frequencies don’t change•after a single generation of random mating, genotype frequencies are:u = p2v = 2pq w = q2•entire system characterized by one parameter (p)Deviation from expectations indicates failure of 1 or more assumptions—selection?HW application: Sickle cell anemia=SS=ssObserved ExpectedCounts CountsSS 834Ss 161ss 52pq *1000= 129p = √0.834 = 0.91q= √0.005 = 0.071Approach: Detect selection through comparison to neutral expectationKimura: neutral theoryEwens: sampling formulaCoalescenceNeutral Theory History• Motoo Kimura (1924-1994)• 1968: a large proportion of genetic change is not driven by selection• Adapted diffusion approximations to genetics• Dealt with finite popsGenetic Driftno driftinfinite popdriftfinite popNeutral allele diffusiont = 100t = 40t = 500t = 28t = 100.01.02.00.0 0.5 1.0A graph illustrating the process of the change in the distribution of genefrequencies with random fluctuation in the selection intensities.Gene frequencyRelative probabilityFigure by MIT OpenCourseWare, based on:Kimura, Motoo. "Process Leading to Quasi-Fixation of Genes in Natural Populations due to Random Fluctuation of Selection Intensities." Genetics 39, no. 3 (1954): 280-295.Ewens sampling formula (1972)• built on foundations of diffusion theory• extended idea of ‘identity by descent’ (ibd)• sample-based• shifted focus to inferential methods• introduced ‘infinite alleles’ modelInfinite alleles model• infinite number of states into which an allele can mutate, therefore each mutation assumed unique (protein-centric)•2Nμnew alleles introduced each generation, derived from existing alleles• initial allele frequency = 1/(2N)• every allele eventually lostInfinite alleles modelUnder diffusion, probability of an allele whose frequency is between x and x+δxis:11() (1 )fxx x x x−Θ−∂=Θ − ∂where4NuΘ=N = population sizeμ= mutation rateExpected Site Frequencies12345iiE(n)Ewens Sampling formulaProbability that a sample of n gene copies contains k alleles and that there are a1, a2, …, analleles represented 1,2, …,n times in the sample:121()!1(, ,..., )!jknnajnjnPa a aja=Θ=ΠΘ()(1)...( 1)nnΘ=ΘΘ+ Θ+−whereand ajis the number of alleles found in j copiesThe CoalescentAlternate, ‘backwards’ approach to generating expected allele frequency distributionsi = 1i = 2i = 3 or 1t4t3t2infer tree structure (genealogy), because tree structure dictates pattern of polymorphism in dataThe CoalescentHow far back in time did a sample share a common ancestor?time presentTpop≈4N generationsTsampCoalescent inference(pattern) (appropriate mutations | ) ( )GPP GPG=∑summary statistics obviate need to actually sum over all genealogiesSample of size 2:P(coal) = 1/2N2221()2tNfteN−=t21()11kPkΘ⎛⎞⎛⎞=⎜⎟⎜⎟Θ+ Θ+⎝⎠⎝⎠Probability of k mutation events before two sequences coalesceP(mutation|event) P(coalescence|event)Turning neutral models into tests of neutralityThree polymorphism summary statistics:S no. of segregating sites in sampleπavg. no. of pairwise differencesηino. of sites that divide the sample into iand n-i sequencesTurning neutral models into tests of neutralityS no. of segregating sites in sampleπavg. no. of pairwise differencesnino. of sites that divide the sample into i and n-i sequences/21niiSη==∑/211()2niiin inη=Π= −⎛⎞⎜⎟⎝⎠∑Turning neutral models into tests of neutrality111()niESi−==Θ∑()Eπ=Θ1()1nEnη=Θ−Θ Estimator111niSi−=∑π11nnη−Θ = 4ΝμFrequency-based neutrality testsTajima (1989) proposed:11111/1 where (/)niSaDaiVar S aππ−=−==−∑Fu and Li (1993) proposed:11111/*1/nSanDnSanηη−−=−−111*1nnFnnπηπη−−=−−Frequency-based neutrality tests111/1*/1*DSanDSannFnπηπη∝−−∝−−∝−S = η1η1maximizedπ minimizedS = η[n/2]η1minimizedπ