CSci 8980: Advanced Topics in Graphical Models Analysis of Genetic Variation

CSci 8980: Advanced Topics in Graphical Models Analysis of Genetic Variation Instructor: Arindam Banerjee November 26, 2007

Genetic Polymorphism Single nucleotide polymorphism (SNP)

Genetic Polymorphism Single nucleotide polymorphism (SNP) Two possible kinds of nucleotides at a single locus

Genetic Polymorphism Single nucleotide polymorphism (SNP) Two possible kinds of nucleotides at a single locus Nucleotide can be one of {A, C, T, G}

Genetic Polymorphism Single nucleotide polymorphism (SNP) Two possible kinds of nucleotides at a single locus Nucleotide can be one of {A, C, T, G} Most genetic human variation are related to SNPs

Genetic Polymorphism Single nucleotide polymorphism (SNP) Two possible kinds of nucleotides at a single locus Nucleotide can be one of {A, C, T, G} Most genetic human variation are related to SNPs Each variant is called an allele Haplotype List of alleles in a local region of a chromosome

Genetic Polymorphism (Contd.) Linkage disequilibrium (LD)

Genetic Polymorphism (Contd.) Linkage disequilibrium (LD) Non-random association of alleles at different loci

Genetic Polymorphism (Contd.) Linkage disequilibrium (LD) Non-random association of alleles at different loci Recombination decouples alleles, increase randomness, decrease LD

Genetic Polymorphism (Contd.) Linkage disequilibrium (LD) Non-random association of alleles at different loci Recombination decouples alleles, increase randomness, decrease LD Infer chromosomal recombination hotspots Help understand origin and characteristics of genetic variation

Haplotype Recombination and Inheritance

Hidden Markov Process Generative model for choosing recombination sites

Hidden Markov Process Generative model for choosing recombination sites Hidden Markov process

Hidden Markov Process Generative model for choosing recombination sites Hidden Markov process Hidden states correspond to index over chromosomes

Hidden Markov Process Generative model for choosing recombination sites Hidden Markov process Hidden states correspond to index over chromosomes Transition probabilities correspond to recombination rates Emission model corresponds to mutation process that give descendants

Dirichlet Process Mixtures We know the basics of DPMs

Dirichlet Process Mixtures We know the basics of DPMs Haplotype modeling using an infinite mixture model

Dirichlet Process Mixtures We know the basics of DPMs Haplotype modeling using an infinite mixture model A pool of ancestor haplotypes or founders

Dirichlet Process Mixtures We know the basics of DPMs Haplotype modeling using an infinite mixture model A pool of ancestor haplotypes or founders The size of the pool is unknown

Dirichlet Process Mixtures We know the basics of DPMs Haplotype modeling using an infinite mixture model A pool of ancestor haplotypes or founders The size of the pool is unknown Standard coalescence based models

Dirichlet Process Mixtures (Contd.) H i = [H i,1,..., H i,t ] haplotype over T SNPs, chromosome i

Dirichlet Process Mixtures (Contd.) H i = [H i,1,..., H i,t ] haplotype over T SNPs, chromosome i A k = [A k,1,..., A k,t ] ancestral haplotype, mutation rate θ k

Dirichlet Process Mixtures (Contd.) H i = [H i,1,..., H i,t ] haplotype over T SNPs, chromosome i A k = [A k,1,..., A k,t ] ancestral haplotype, mutation rate θ k C i, inheritance variable, latent ancestor of H i Generative Model: Draw a first haplotype a 1 DP(τ, Q 0 ) Q 0 h 1 P h ( a 1, θ 1 ) For subsequent haplotypes { p(c i = c j for some j < i c 1,..., c i 1 ) = c i DP(τ, Q 0 ) p(c i c j for all j < i c 1,..., c i 1 ) = nc j i 1+α 0 α0 i 1+α 0

Dirichlet Process Mixtures (Contd.) Generative Model (contd)

Dirichlet Process Mixtures (Contd.) Generative Model (contd) Sample the founder of haplotype i { = {a cj, θ cj }ifc i = c j for somej < i φ ci DP(τ, Q 0 ) Q(a, θ)ifc i c j for allj < i

Dirichlet Process Mixtures (Contd.) Generative Model (contd) Sample the founder of haplotype i { = {a cj, θ cj }ifc i = c j for somej < i φ ci DP(τ, Q 0 ) Q(a, θ)ifc i c j for allj < i Sample the haplotype according to its founder h i c i P( a ci, θ ci ) Assumes each haplotype originates from one ancestor

Hidden Markov Dirichlet Process Nonparametric Bayesian HMM

Hidden Markov Dirichlet Process Nonparametric Bayesian HMM Sample a DP to form the support of the infinite state space

Hidden Markov Dirichlet Process Nonparametric Bayesian HMM Sample a DP to form the support of the infinite state space Conditioned on each state, sample a DP with the same support

Hidden Markov Dirichlet Process Nonparametric Bayesian HMM Sample a DP to form the support of the infinite state space Conditioned on each state, sample a DP with the same support Hierarchical Urns

Hidden Markov Dirichlet Process Nonparametric Bayesian HMM Sample a DP to form the support of the infinite state space Conditioned on each state, sample a DP with the same support Hierarchical Urns Stock urn Q 0 with balls of K colors, n k of color k HMM-urns Q 1,..., Q K for prior and transition probabilities Let m j,k be the number of balls of color k in urn Q j

Hidden Markov Dirichlet Process (Contd.) Each color corresponds to ancestor configuration φ k = {a k, θ k }

Hidden Markov Dirichlet Process (Contd.) Each color corresponds to ancestor configuration φ k = {a k, θ k } For n random draws from Q 0 φ n φ n K k=1 n k n 1 + α δ α φ k (φ n ) + n 1 + α F (φ n)

Hidden Markov Dirichlet Process (Contd.) Each color corresponds to ancestor configuration φ k = {a k, θ k } For n random draws from Q 0 φ n φ n K k=1 n k n 1 + α δ α φ k (φ n ) + n 1 + α F (φ n) Conditioned on Q 0, the marginal configs from Q j φ mj φ mj k n k m j,k + τ n 1+α m j 1 + tau + τ α m j 1 + τ n 1 + α F (φ m j )

HMDP for Recombination and Inheritance Priors for the conditional model parameters F (A, θ) = p(a)p(θ)

HMDP for Recombination and Inheritance Priors for the conditional model parameters F (A, θ) = p(a)p(θ) p(a) is assumed uniform, p(θ) is assumed beta

HMDP for Recombination and Inheritance Priors for the conditional model parameters F (A, θ) = p(a)p(θ) p(a) is assumed uniform, p(θ) is assumed beta C i = [C i,1,..., C i,t ] ancestral index for chromosome i With no recombination, C i,t = k, t for some k Non-recombination is modeled by Poisson point process P(C i,t+1 = C i,t = k) = exp( dr) + (1 exp( dr))π kk d is the distance between the two loci

HMDP for Recombination and Inheritance (Contd.) H i is a mosaic of multiple ancestral chromosomes

HMDP for Recombination and Inheritance (Contd.) H i is a mosaic of multiple ancestral chromosomes Model is a time-inhomogenous infinite HMM

HMDP for Recombination and Inheritance (Contd.) H i is a mosaic of multiple ancestral chromosomes Model is a time-inhomogenous infinite HMM With r, we get stationary HMM

HMDP for Recombination and Inheritance (Contd.) H i is a mosaic of multiple ancestral chromosomes Model is a time-inhomogenous infinite HMM With r, we get stationary HMM Single locus mutation model for emission ( ) 1 θ I(ht at) p(h t a t, θ) = θ I(ht=at) B 1

Haplotype Recombination and Inheritance

HMDP for Recombination and Inheritance (Contd.) Conditional probability of haplotype list h p(h c, a) = p(h i,t a k,t, θ k )Beta(θ k α h, β h )dθ k k θ k i,t c i,t =k = Γ(α h + β h ) Γ(α h + l k )Γ(β h + l k ) ( ) 1 l k Γ(α h )Γ(β h ) Γ(α h + β h + l k + l k k ) B 1 where l k = i,t I(h i,t = a k,t )I(c i,t = k) l k = i,t I(h i,t a k,t )I(c i,t = k)

Inference Gibbs sampler proceeds in two steps

Inference Gibbs sampler proceeds in two steps Sample inheritance {C i,k } given h and a

Inference Gibbs sampler proceeds in two steps Sample inheritance {C i,k } given h and a Sample ancestors a = {a 1,..., a K } given h, C

Inference Gibbs sampler proceeds in two steps Sample inheritance {C i,k } given h and a Sample ancestors a = {a 1,..., a K } given h, C Improve mixing for sampling inheritance

Inference Gibbs sampler proceeds in two steps Sample inheritance {C i,k } given h and a Sample ancestors a = {a 1,..., a K } given h, C Improve mixing for sampling inheritance By Bayes rule t+δ p(c t+1 : t + δ c, h, a) p(c j+1 c j, m, n) j=t t+δ j=t+1 p(h j a cj,j, l cj ) Assume probability of having two recombinations is small p(c t+1 : t + δ c, h, a) p(c t c t 1, m, n)p(c t+δ+1 c t+δ = c t, m, n) t j=

Inference (Contd.) Assuming d, r to be small, λ = 1 exp( dr) dr { λπ k,k + (1 λ)δ(k, k )fork {1 p(c t = k c t 1 = k, m, n, r, d) = λπ k,k+1 fork = K + 1

Inference (Contd.) Assuming d, r to be small, λ = 1 exp( dr) dr { λπ k,k + (1 λ)δ(k, k )fork {1 p(c t = k c t 1 = k, m, n, r, d) = λπ k,k+1 fork = K + 1 Terms can be replaced in original equation to get sampler Posterior distribution for ancestors p(a k,t c, h) Γ(α h + β h ) Γ(α h + l k,t )Γ(β h + l k,t ) ( ) 1 l k,t Γ(α h )Γ(β h ) Γ(α h + β h + l k,t + l k,t ) B 1

Single Population Data Haplotype block boundaries HMDP (black solid), HMM (red dotted), MDL (blue dashed)

Two Population Data

Hierarchical DPM for Haplotype Inference

Hierarchical DPM for Haplotype Inference (Contd.)

Experiments: Hapmap Data SNP genotypes from four populations

Experiments: Hapmap Data SNP genotypes from four populations CEPH, Utah residents with northern/weatern European ancestry, 60

Experiments: Hapmap Data SNP genotypes from four populations CEPH, Utah residents with northern/weatern European ancestry, 60 YRI, Yoruba in Ibadan, Nigeria, 60

Experiments: Hapmap Data SNP genotypes from four populations CEPH, Utah residents with northern/weatern European ancestry, 60 YRI, Yoruba in Ibadan, Nigeria, 60 CHB, Han Chinese in Beijing, 45

Experiments: Hapmap Data SNP genotypes from four populations CEPH, Utah residents with northern/weatern European ancestry, 60 YRI, Yoruba in Ibadan, Nigeria, 60 CHB, Han Chinese in Beijing, 45 JPT, Japanese in Tokyo, 44

Short SNP Sequences

Long SNP Sequences

Mutation Rates and Diversity

Mutation Rates and Diversity (Contd.)