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Evaluation of a new tool for use in association mapping Structure Reinhard Simon, 2002/10/29

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Structure 2.0 http://pritch.bsd.uchicago.edu Pritchard JK, Stephens M, Donelly P (2000): Inference of population structure using multilocus genotype data. Genetics, 155: 945-959 Software

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Associations – the ideal CasesControls

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Test for association A diploid locus: Pearsons Chi-square test

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Example: Contingency table Frequency ofTotal # of alleles (diploid) AllelesAA* Casesqaqa 1-q a 2m a Controlsqoqo 1-q o 2m o totalnAnA n A* 2m

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Associations – the less ideal CasesControls

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Associations – simple admixture CasesControls

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Associations – admixture complications CasesControls

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Associations – admixture complications CasesControls High frequency of associated loci may indicate problems with underlying population structure (=stratification).

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Associations – accounted for CasesControls

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Questions Is there a stratification? If so: - how many subpopulations - which individual belongs to which subpopulation

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Test for stratification - principle Summarizing over all loci: Xi is Chi-square at i-th locus Null hypothesis: no differences between allele frequencies over all loci df equal to sum of df at individual locus Pritchard: 1999

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Test for stratification – ctd. Observations: strong positive selection requires increase of #loci subgroup specific markers decrease number of necessary loci Pritchard: 1999

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How to group individuals? Based on distance measures Based on models

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Pair wise distance measures Jaccard Nei & Li Sokal & Michener

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Model based Bayesian inference Bayesean statistics: Uncertainty is modeled using probabilities probability statements are made about model parameters Advantages: very general framework assumptions are made explicit and are quantified

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Bayesian inference – how? Bayesian inference centers on the posterior distribution p(theta|X), e.g. a genetic model of the distribution of allele frequencies However, analytic evaluation is seldom possible....

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Bayesian inference - methods Alternatives: Numerical evaluation approximation simulation, e.g. Markov Chain Monte Carlo Methods

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Simulation methods for Bayesian inference - general Generate random samples from a probability distribution (e.g. normal) Construct histogram If sample is large enough, this allows to calculate mean, variance,... MCMC allows to generate large samples from any probability distribution

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Markov Chain behaviour Reaches an equilibrium (basic MCMC theorem) and the present state depends only on the preceding: “The future depends on the past only through the present.”

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MCMC - strengths freedom in inference (e.g. simultaneous estimation, estimation of arbitrary functions of model parameters like ranks or threshold exceedence) Coherently integrates uncertainty Only available method for complex problems

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MCMC – contra computational intensive requires often specialized software

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Inferring population structure X = genotypes of sampled inviduals unknown: Z = population of origin P = allele frequencies in all populations Q = proportion of genome that originates from population k Pr(Z, P, Q|X) ~ Pr(Z) * Pr(P) * Pr(Q) * Pr(X|Z,P,Q) Solution: Using MCMC for Bayesian inference; simultaneous estimation of Q, Z and P.

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Basic MCMC algorithm – no admixture (Q) Initialize: Random values for Z (pop), e.g. from Pr(z) = 1/k Repeat for m=1,2,... 1. Sample P(m) from Pr(P|X, Z(m-1) (estimate allele frequencies) 2. Sample Z(m) from Pr(Z|X, P(m)) (estimate population of origin for each indiv.)

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Basic MCMC algorithm – with admixture (Q) Initialize: Random values for Z (pop), e.g. from Pr(z) = 1/k Repeat for m=1,2,... 1. Sample P(m), Q(m) from Pr(P, Q|X, Z(m-1) (estimate allele frequencies) 2. Sample Z(m) from Pr(Z|X, P(m), Q(m)) 3. Update alpha (admixture proportion)

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Program – parameters: MCMC

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Program – parameters: Q

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Program – parameters: P

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Program – parameters: Z, K

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Program – data types - marker: SNP, microsatellites AFLP, RFLP,... (biallelic) - ploidy: >1 -extra optional information for inclusion: - prior knowledge on groups (e.g. geographic location) - genetic map location of marker

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Program – data format

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Example – S.t. tuberosum vs andigena Other: 1 st 30 genotypes from tuberosum 2 nd 20 genotypes from andigena

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Example – S.t. tuberosum vs andigena PNA:

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Example – S.t. tuberosum vs andigena PNA: Estimation of k Simulation # k Pr(k)

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Example – S.t. tuberosum vs andigena PNA: assignment 1 = tbr; 2 = adg genotypes #31-#3: adg from India genotype #49: adg from Ecuador

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Example – S.t. tuberosum vs andigena Parameter change: allow admixture Ancestry Model Info Use Admixture Model * Infer Alpha * Initial Value of ALPHA (Dirichlet Parameter for Degree of Admixture): 1.0 * Use Same Alpha for all Populations * Use a Uniform Prior for Alpha ** Maximum Value for Alpha: 10.0 ** SD of Proposal for Updating Alpha: 0.025 Frequency Model Info Allele Frequencies are Independent among Pops * Infer LAMBDA ** Use a Uniform Lambda for All Population ** Initial Value of Lambda: 1.0

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Example – S.t. tuberosum vs andigena Parameter change: allow admixture

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Example – S.t. tuberosum vs andigena Parameter change: allow admixture

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Example – S.t. tuberosum vs andigena Parameter change: allow admixture

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Example – andigena

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Example – andigena: data

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Example – andigena K = 2

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Example – andigena K = 3

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Example – andigena K = 3

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Example – andigena: genetic distance K = 3

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Example – andigena: geographic distribution - 1 K = 3

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Example – andigena: geographic distribution - 2 K = 3

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Example – andigena: geographic distribution - 3 K = 3

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Example – I. batatas

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Example – I. batatas: settings

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Example – I. batatas: k K = 2

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Example – I. batatas: k = 2 1=PAN, 2=HON, 3=GTM, 4=NIC, 5=MEX, 6=COL, 7=VEN, 8=ECU, 9=PER

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Example – I. batatas: k = 3 1=PAN, 2=HON, 3=GTM, 4=NIC, 5=MEX, 6=COL, 7=VEN, 8=ECU, 9=PER

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Example – I. batatas: k = 4 1=PAN, 2=HON, 3=GTM, 4=NIC, 5=MEX, 6=COL, 7=VEN, 8=ECU, 9=PER

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Example – I. batatas: genetic distance

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Example – S. paucissectum

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Example – paucissectum: data

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Example – paucissectum: configuration

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Example – paucissectum: results: k =2

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Example – paucissectum: results: k =3

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Example – paucissectum: results: k =4

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Example – paucissectum: results: k =5

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Example – paucissectum: results: k =6

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Summary The software was tested with population data from diploid, tetraploid and hexaploid species with microsatellite and biallelic marker The algorithm seems stable and delivers sensible results under a variety of settings Great advantage: assigns each individual a probability of being a member of a certain subgroup

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Thanks for your attention!

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Lab 7. Estimating Population Structure

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