Lecture 2: Population Structure Advanced Topics in Computa8onal Genomics

Transcription

1 Lecture 2: Population Structure Advanced Topics in Computa8onal Genomics 1

2 What is population structure? Popula8on Structure A set of individuals characterized by some measure of gene8c dis8nc8on A popula8on is usually characterized by a dis8nct distribu8on over genotypes Example Genotypes aa aa AA Popula8on 1 Popula8on 2 2

3 1000 Genome Projects 3

4 Motivation Reconstruc*ng individual ancestry: The Genographic Project hjps://genographic.na8onalgeographic.com/genographic/index.html Studying human migra*on Out of Africa Mul*- regional hypothesis Study of various traits Lactose intolerance Origins in Europe? Infer from Migra8on studies Muta8on studies in popula8ons 4

5 200,000 years ago 50,000 years ago 30,000 years ago 10,000 years ago hjps://genographic.na8onalgeographic.com/ genographic/index.html 5

6 Overview Background Hardy- Weinberg Equilibrium Gene8c driz Wright s F ST Inferring popula8on structure from genotype data Structure (Falush et al., 2003) Matrix factoriza8on/dimensionality reduc8on methods (Engelhardt & Stephens, 2010) 6

7 Hardy-Weinberg Equilibrium Hardy- Weinberg Equilibruim Under random ma8ng, both allele and genotype frequencies in a popula8on remain constant over genera8ons. Assump8ons of the standard random ma8ng Diploid organism Sexual reproduc8on Nonoverlapping genera8ons Random ma8ng Large popula8on size Equal allele frequencies in the sexes No migra8on/muta8on/selec8on Chi- square test for Hardy- Weinberg equilibrium 7

8 Hardy-Weinberg Equilibrium D, H, R: genotype frequencies for AA, Aa, aa, respec8vely. p q: allele frequencies of A and a 8

9 Hardy-Weinberg Equilibrium The genotype and allele frequencies of the offspring 9

10 Genetic Drift The change in allele frequencies in a popula8on due to random sampling Neutral process unlike natural selec8on But gene8c driz can eliminate an allele from the given popula8on. The effect of gene8c driz is larger in a small popula8on 10

11 Population Divergence Wright s F ST Sta8s8cs used to quan8fy the extent of divergence among mul8ple popula8ons rela8ve to the overall gene8c diversity Summarizes the average devia8on of a collec8on of popula8ons a way from the mean F ST = Var(p k )/p (1-p ) p : the overall frequency of an allele across all subpopulations p k :the allele frequency within population k 11

12 Scenarios of How Populations Evolve 12

13 Methods for Learning Population Structure from Genetic Markers Low- dimensional projec8on PCA- based methods (PaJerson et al., PLoS Gene8cs 2006) Clustering Distance- based (Bowcock et al., Nature 1994) Model- based STRUCTURE (Pritchard et al., Gene8cs 2000) mstruct (Shringarpure & Xing, Gene8cs 2008) 13

14 Probabilistic Models for Population Structure Mixture model Cluster individuals into K popula8ons Admixture model The genotypes of each individual are an admixture of mul8ple ancestor popula8ons Assumes alleles are in linkage equilibrium Linkage model Model recombina8on, correla8on in alleles across chromosome F model Model correla8on in alleles in ancestry 14

15 Mixture Model K popula8ons z (i) : popula8on of origin of individual i For each of the K popula8ons p klj : the frequency of allele j at locus l in popula8on k 15

16 Admixture Model Relax the assump8on of one ancestor per individual in mixture model Individuals can have ancestors in mul8ple different popula8ons q k (i) : propor8on of individual i s genome derived from popula8on k Alleles at different lock can come from different popula8ons 16

17 Structure Model Hypothesis: Modern popula8ons are created by an intermixing of ancestral popula8ons. An individual s genome contains contribu8ons from one or more ancestral popula8ons. The contribu8ons of popula8ons can be different for different individuals. Other assump8ons Hardy- weinberg equilbrium No linkage disequilbrium Markers are i.i.d (independent and iden8cally distributed) 17

18 Linkage Model From admixture model, replace the assump8on that the ancestry labels z il for individual i, locus l are independent with the assump8on that adjacent z il are correlated. Use Poisson process to model the correla8on between neighboring alleles d l : distance between locus l and locus l+1 r: recombina8on rate 18

19 Linkage Model As recombina8on rate r goes to infinity, all loci become independent and linkage model becomes admixture model. Recombina8on rate r can be viewed as being related to the number of genera8ons since admixture occurred. Use MCMC algorithm to fit the unkown parameters. 19

20 F Model Introduce correla8ons in allele frequencies among ancestral popula8ons p Al : allele frequencies in ancestral popula8ons modeled as symmetric Dirichlet distribu8on Subpopula8ons of the ancestral popula8on go through gene8c driz at different rate F k Individuals are admixture of those K popula8ons who went through gene8c driz from the common ancestral popula8on 20

21 F Model Rela8onship between F k and F ST Designed to between closely related popula8ons with similar allele frequencies 21

22 Scenarios of How Populations Evolve 22

23 Unknown Parameters To Be Estimated q i : the admixture propor8ons of individual i p k : allele frequencies of popula8on k z i : popula8on label for each locus of individual i r : recombina8on rate F k : es8mate of popula8on divergence from the ancestral popula8on 23

24 Population Structure from Ancestry Proportion of Each Individual How to display popula8on structure? Ancestral proportion Africa Europe Mid- East Cent./S. Asia East Asia Oceania Genetic structure of Human Populations (Rosenberg et al., #( 2002 Science 24

25 Population of Origin Assignments of a Single Individual True origin Es8mated Origin (Phased data) Es8mated Origin (Unphased data) 25

26 Admixture vs Divergence 26

27 Posterior Distribution of Recombination Rate Using the original dataset AZer permu8ng the genotype loci 27

28 Distinguishing Between Two Closely Related Populations 28

29 Three Sources of Linkage Disequilibrium Mixture LD Due to varia8on in ancestry across individuals that induce correla8on among markers at different loci Modeled by admixture model Admixture LD Due to unbroken chunks of DNA derived from an ancestor popula8on. Modeled by linkage model Background LD Due to LD within popula8ons Decays at smaller scale 29

30 Low-dimensional Projections Gene8c data is very large Number of markers may range from a few hundreds to hundreds of thousands Thus each individual is described by a high- dimensional vector of marker configura8ons A low- dimensional projec8on allows easy visualiza8on Technique used Factor analysis Many sta8s8cal methods exist ICA, PCA, NMF etc. Principal Components Analysis (next slide) Allows projec8on of individuals into a low dimensional space Usually projected to 2 dimensions to allow visualiza8on 30

31 Principal Component Analysis Most common form of factor analysis The new variables/dimensions... Are linear combina8ons of the original ones Are uncorrelated with one another Orthogonal in original dimension space Capture as much of the original variance in the data as possible Are called Principal Components Demo at hjp:// 31

32 What are the new axes? Original Variable B PC 2 PC 1 Original Variable A Orthogonal direc8ons of greatest variance in data Projec8ons along PC1 discriminate the data most along any one axis 32

33 Principal Components First principal component is the direc8on of greatest variability (covariance) in the data Second is the next orthogonal (uncorrelated) direc8on of greatest variability So first remove all the variability along the first component, and then find the next direc8on of greatest variability And so on 33

34 Dimensionality Reduction Can ignore the components of lesser significance. You do lose some informa8on, but if the eigenvalues are small, you don t lose much n dimensions in original data calculate n eigenvectors and eigenvalues choose only the first p eigenvectors, based on their eigenvalues final data set has only p dimensions 34

35 PCA Analysis (Cavalli-sforza,1978) Plot of geographical distribu8on of 3 PCs (Intensity propor8onal to value of each component) First blue Second - green Third - red 35

36 Matrix Factorization and Population Structure Matrix factoriza8on for learning popula8on structure Genotype Data (NxP matrix) N: number of samples P: number of genotypes = Individuals ancestry propor8ons (NxK matrix) K: number of subpopula8ons x Subpopula8on Allele Frequencies (KxP matrix) 36

37 Unifying Framework of Matrix Factorization Admixture Based on probability models: rows of Λ and columns of F should sum to 1. Works well if the individuals are admixtures of discretely separated popula8ons PCA Based on eigen decomposi8on: columns of Λ are orthogonal, rows of F are orthnormal. Works well for the case of isola8on- by- distance (con8nuous varia8on of popula8ons among individuals) Sparse factor model Sparsity via automa8c relevance determina8on prior 37

38 Discrete/Admixed Populations Loading 1 Loading 2 Loading 3 SFA PCA Admixture 38

39 Isolation-by-Distance Models 39

40 SFA Clustered Populations in 1d Habitat Assume two popula8ons Assume five popula8ons Admixture Assume two popula8ons Assume five popula8ons PCA 40

41 Analysis of European Genotype Data PCA SFAm Admixture 41

42 Comparison of Different Methods Advantages PCA Sta8s8cal tests for significance of results (PaJerson et al. 2006) Easy visualiza8on Model- based Clustering Genera8ve process that explicitly models admixture Clustering is probabilis8c: it is possible to assign confidence level of clusters Disadvantages No intui8on about underlying processes Computa8onally more demanding Based on assump8ons of evolu8onary models: Structure: No models of muta8on, recombina8on Muta8on added in mstruct Recombina8on added in extension by Falush et al. 42