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

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

2 Genetic Polymorphism Single nucleotide polymorphism (SNP)

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

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

5 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

6 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

7 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

8 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

9 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 Inherited as a unit, if there is no recombination

10 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 Inherited as a unit, if there is no recombination Repeated recombinations between ancestral haplotypes

11 Genetic Polymorphism (Contd.) Linkage disequilibrium (LD)

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

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

14 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

15 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

16 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 Analyze genetic variation to reconstruct evolutionary history

17 Haplotype Recombination and Inheritance

18 Hidden Markov Process Generative model for choosing recombination sites

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

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

21 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

22 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

23 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 Implemented using a Hidden Markov Dirichlet Process (HMDP)

24 Dirichlet Process Mixtures We know the basics of DPMs

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

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

27 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

28 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

29 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 Hidden variables is prohibitively large

30 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 Hidden variables is prohibitively large Hard to perform inference of ancestral features

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

32 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

33 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

34 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:

35 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 )

36 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

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

38 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

39 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 )

40 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

41 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 Valid only for short regions in chromosome

42 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 Valid only for short regions in chromosome Long regions will have recombination

43 Hidden Markov Dirichlet Process Nonparametric Bayesian HMM

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

45 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

46 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

47 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

48 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

49 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

50 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 HDPM can be simulated by sampling from the urn hierarchy

51 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 HDPM can be simulated by sampling from the urn hierarchy Hierarchical DPM Q 0 α, F DP(α, F ) Q j τ, Q 0 DP(τ, Q 0 )

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

53 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)

54 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 )

55 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 )

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

57 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

58 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

59 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

60 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

61 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

62 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 r is the rate of recombination per unit distance

63 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 r is the rate of recombination per unit distance The transition probability to state k is P(C i,t = k, C i,t+1 = k ) = (1 exp(dr))π kk

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

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

66 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

67 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

68 Haplotype Recombination and Inheritance

69 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)

70 Inference Gibbs sampler proceeds in two steps

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

72 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

73 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

74 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 )

75 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=

76 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

77 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

78 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

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

80 Two Population Data

81 Hierarchical DPM for Haplotype Inference

82 Hierarchical DPM for Haplotype Inference (Contd.)

83 Experiments: Hapmap Data SNP genotypes from four populations

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

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

86 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

87 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

88 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 Experiments on short ( 10) and long ( ) SNPs

89 Short SNP Sequences

90 Long SNP Sequences

91 Mutation Rates and Diversity

92 Mutation Rates and Diversity (Contd.)