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Basics HMDP Inference Results HDPM Results CSci 8980: Advanced Topics in Graphical Models Analysis of Genetic Variation Instructor: Arindam Banerjee November 26, 2007 Basics HMDP Inference Results HDPM Results Genetic Polymorphism

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Basics HMDP Inference Results HDPM Results

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

Instructor: Arindam Banerjee November 26, 2007

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Genetic Polymorphism

Single nucleotide polymorphism (SNP)

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Genetic Polymorphism

Single nucleotide polymorphism (SNP)

Two possible kinds of nucleotides at a single locus

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Genetic Polymorphism

Single nucleotide polymorphism (SNP)

Two possible kinds of nucleotides at a single locus Nucleotide can be one of {A, C, T, G}

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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

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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

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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

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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

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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

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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

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Genetic Polymorphism (Contd.)

Linkage disequilibrium (LD)

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Genetic Polymorphism (Contd.)

Linkage disequilibrium (LD)

Non-random association of alleles at different loci

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Genetic Polymorphism (Contd.)

Linkage disequilibrium (LD)

Non-random association of alleles at different loci Recombination decouples alleles, increase randomness, decrease LD

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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

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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

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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

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Haplotype Recombination and Inheritance

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Hidden Markov Process

Generative model for choosing recombination sites

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Hidden Markov Process

Generative model for choosing recombination sites Hidden Markov process

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Hidden Markov Process

Generative model for choosing recombination sites Hidden Markov process

Hidden states correspond to index over chromosomes

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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

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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

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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)

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Dirichlet Process Mixtures

We know the basics of DPMs

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Dirichlet Process Mixtures

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

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Dirichlet Process Mixtures

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

A pool of ancestor haplotypes or founders

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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

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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

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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

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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

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Dirichlet Process Mixtures (Contd.)

Hi = [Hi,1, . . . , Hi,T] haplotype over T SNPs, chromosome i

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Dirichlet Process Mixtures (Contd.)

Hi = [Hi,1, . . . , Hi,T] haplotype over T SNPs, chromosome i Ak = [Ak,1, . . . , Ak,T] ancestral haplotype, mutation rate θk

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Dirichlet Process Mixtures (Contd.)

Hi = [Hi,1, . . . , Hi,T] haplotype over T SNPs, chromosome i Ak = [Ak,1, . . . , Ak,T] ancestral haplotype, mutation rate θk Ci, inheritance variable, latent ancestor of Hi

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Dirichlet Process Mixtures (Contd.)

Hi = [Hi,1, . . . , Hi,T] haplotype over T SNPs, chromosome i Ak = [Ak,1, . . . , Ak,T] ancestral haplotype, mutation rate θk Ci, inheritance variable, latent ancestor of Hi Generative Model:

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Dirichlet Process Mixtures (Contd.)

Hi = [Hi,1, . . . , Hi,T] haplotype over T SNPs, chromosome i Ak = [Ak,1, . . . , Ak,T] ancestral haplotype, mutation rate θk Ci, inheritance variable, latent ancestor of Hi Generative Model:

Draw a first haplotype a1|DP(τ, Q0) ∼ Q0 h1 ∼ Ph(·|a1, θ1)

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Dirichlet Process Mixtures (Contd.)

Hi = [Hi,1, . . . , Hi,T] haplotype over T SNPs, chromosome i Ak = [Ak,1, . . . , Ak,T] ancestral haplotype, mutation rate θk Ci, inheritance variable, latent ancestor of Hi Generative Model:

Draw a first haplotype a1|DP(τ, Q0) ∼ Q0 h1 ∼ Ph(·|a1, θ1) For subsequent haplotypes ci|DP(τ, Q0) ∼

  • p(ci = cjfor some j < i|c1, . . . , ci−1) =

ncj i−1+α0

p(ci = cjfor all j < i|c1, . . . , ci−1) =

α0 i−1+α0

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Dirichlet Process Mixtures (Contd.)

Generative Model (contd)

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Dirichlet Process Mixtures (Contd.)

Generative Model (contd)

Sample the founder of haplotype i φci|DP(τ, Q0)

  • = {acj, θcj}ifci = cjfor somej < i

∼ Q(a, θ)ifci = cjfor allj < i

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Dirichlet Process Mixtures (Contd.)

Generative Model (contd)

Sample the founder of haplotype i φci|DP(τ, Q0)

  • = {acj, θcj}ifci = cjfor somej < i

∼ Q(a, θ)ifci = cjfor allj < i Sample the haplotype according to its founder hi|ci ∼ P(·|aci, θci)

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Dirichlet Process Mixtures (Contd.)

Generative Model (contd)

Sample the founder of haplotype i φci|DP(τ, Q0)

  • = {acj, θcj}ifci = cjfor somej < i

∼ Q(a, θ)ifci = cjfor allj < i Sample the haplotype according to its founder hi|ci ∼ P(·|aci, θci)

Assumes each haplotype originates from one ancestor

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Dirichlet Process Mixtures (Contd.)

Generative Model (contd)

Sample the founder of haplotype i φci|DP(τ, Q0)

  • = {acj, θcj}ifci = cjfor somej < i

∼ Q(a, θ)ifci = cjfor allj < i Sample the haplotype according to its founder hi|ci ∼ P(·|aci, θci)

Assumes each haplotype originates from one ancestor

Valid only for short regions in chromosome

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Dirichlet Process Mixtures (Contd.)

Generative Model (contd)

Sample the founder of haplotype i φci|DP(τ, Q0)

  • = {acj, θcj}ifci = cjfor somej < i

∼ Q(a, θ)ifci = cjfor allj < i Sample the haplotype according to its founder hi|ci ∼ P(·|aci, θci)

Assumes each haplotype originates from one ancestor

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

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Hidden Markov Dirichlet Process

Nonparametric Bayesian HMM

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Hidden Markov Dirichlet Process

Nonparametric Bayesian HMM Sample a DP to form the support of the infinite state space

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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

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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

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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 Q0 with balls of K colors, nk of color k

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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 Q0 with balls of K colors, nk of color k HMM-urns Q1, . . . , QK for prior and transition probabilities

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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 Q0 with balls of K colors, nk of color k HMM-urns Q1, . . . , QK for prior and transition probabilities Let mj,k be the number of balls of color k in urn Qj

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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 Q0 with balls of K colors, nk of color k HMM-urns Q1, . . . , QK for prior and transition probabilities Let mj,k be the number of balls of color k in urn Qj HDPM can be simulated by sampling from the urn hierarchy

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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 Q0 with balls of K colors, nk of color k HMM-urns Q1, . . . , QK for prior and transition probabilities Let mj,k be the number of balls of color k in urn Qj HDPM can be simulated by sampling from the urn hierarchy

Hierarchical DPM Q0|α, F ∼ DP(α, F) Qj|τ, Q0 ∼ DP(τ, Q0)

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Hidden Markov Dirichlet Process (Contd.)

Each color corresponds to ancestor configuration φk = {ak, θk}

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Hidden Markov Dirichlet Process (Contd.)

Each color corresponds to ancestor configuration φk = {ak, θk} For n random draws from Q0 φn|φ−n ∼

K

  • k=1

nk n − 1 + αδφk(φn) + α n − 1 + αF(φn)

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Hidden Markov Dirichlet Process (Contd.)

Each color corresponds to ancestor configuration φk = {ak, θk} For n random draws from Q0 φn|φ−n ∼

K

  • k=1

nk n − 1 + αδφk(φn) + α n − 1 + αF(φn) Conditioned on Q0, the marginal configs from Qj φmj|φ−mj ∼

  • k

mj,k + τ

nk n−1+α

mj − 1 + tau + τ mj − 1 + τ α n − 1 + αF(φmj)

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Hidden Markov Dirichlet Process (Contd.)

Each color corresponds to ancestor configuration φk = {ak, θk} For n random draws from Q0 φn|φ−n ∼

K

  • k=1

nk n − 1 + αδφk(φn) + α n − 1 + αF(φn) Conditioned on Q0, the marginal configs from Qj φmj|φ−mj ∼

  • k

mj,k + τ

nk n−1+α

mj − 1 + tau + τ mj − 1 + τ α n − 1 + αF(φmj)

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HMDP for Recombination and Inheritance

Priors for the conditional model parameters F(A, θ) = p(A)p(θ)

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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

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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 Ci = [Ci,1, . . . , Ci,T] ancestral index for chromosome i

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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 Ci = [Ci,1, . . . , Ci,T] ancestral index for chromosome i With no recombination, Ci,t = k, ∀t for some k

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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 Ci = [Ci,1, . . . , Ci,T] ancestral index for chromosome i With no recombination, Ci,t = k, ∀t for some k Non-recombination is modeled by Poisson point process P(Ci,t+1 = Ci,t = k) = exp(−dr) + (1 − exp(−dr))πkk

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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 Ci = [Ci,1, . . . , Ci,T] ancestral index for chromosome i With no recombination, Ci,t = k, ∀t for some k Non-recombination is modeled by Poisson point process P(Ci,t+1 = Ci,t = k) = exp(−dr) + (1 − exp(−dr))πkk

d is the distance between the two loci

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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 Ci = [Ci,1, . . . , Ci,T] ancestral index for chromosome i With no recombination, Ci,t = k, ∀t for some k Non-recombination is modeled by Poisson point process P(Ci,t+1 = Ci,t = k) = exp(−dr) + (1 − exp(−dr))πkk

d is the distance between the two loci r is the rate of recombination per unit distance

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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 Ci = [Ci,1, . . . , Ci,T] ancestral index for chromosome i With no recombination, Ci,t = k, ∀t for some k Non-recombination is modeled by Poisson point process P(Ci,t+1 = Ci,t = k) = exp(−dr) + (1 − exp(−dr))πkk

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(Ci,t = k, Ci,t+1 = k′) = (1 − exp(dr))πkk′

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HMDP for Recombination and Inheritance (Contd.)

Hi is a mosaic of multiple ancestral chromosomes

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HMDP for Recombination and Inheritance (Contd.)

Hi is a mosaic of multiple ancestral chromosomes Model is a time-inhomogenous infinite HMM

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HMDP for Recombination and Inheritance (Contd.)

Hi is a mosaic of multiple ancestral chromosomes Model is a time-inhomogenous infinite HMM With r → ∞, we get stationary HMM

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HMDP for Recombination and Inheritance (Contd.)

Hi 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 p(ht|at, θ) = θI(ht=at) 1 − θ |B| − 1 I(ht=at)

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Haplotype Recombination and Inheritance

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HMDP for Recombination and Inheritance (Contd.)

Conditional probability of haplotype list h p(h|c, a) =

  • k
  • θk
  • i,t|ci,t=k

p(hi,t|ak,t, θk)Beta(θk|αh, βh)dθk =

  • k

Γ(αh + βh) Γ(αh)Γ(βh) Γ(αh + ℓk)Γ(βh + ℓ′

k)

Γ(αh + βh + ℓk + ℓ′

k)

  • 1

|B| − 1 ℓ′

k

where ℓk =

  • i,t

I(hi,t = ak,t)I(ci,t = k) ℓ′

k =

  • i,t

I(hi,t = ak,t)I(ci,t = k)

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Inference

Gibbs sampler proceeds in two steps

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Inference

Gibbs sampler proceeds in two steps

Sample inheritance {Ci,k} given h and a

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Inference

Gibbs sampler proceeds in two steps

Sample inheritance {Ci,k} given h and a Sample ancestors a = {a1, . . . , aK} given h, C

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Inference

Gibbs sampler proceeds in two steps

Sample inheritance {Ci,k} given h and a Sample ancestors a = {a1, . . . , aK} given h, C

Improve mixing for sampling inheritance

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Inference

Gibbs sampler proceeds in two steps

Sample inheritance {Ci,k} given h and a Sample ancestors a = {a1, . . . , aK} given h, C

Improve mixing for sampling inheritance

By Bayes rule p(ct+1 : t + δ|c−, h, a) ∝

t+δ

  • j=t

p(cj+1|cj, m, n)

t+δ

  • j=t+1

p(hj|acj,j, ℓcj)

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Inference

Gibbs sampler proceeds in two steps

Sample inheritance {Ci,k} given h and a Sample ancestors a = {a1, . . . , aK} given h, C

Improve mixing for sampling inheritance

By Bayes rule p(ct+1 : t + δ|c−, h, a) ∝

t+δ

  • j=t

p(cj+1|cj, m, n)

t+δ

  • j=t+1

p(hj|acj,j, ℓcj) Assume probability of having two recombinations is small p(ct+1 : t + δ|c−, h, a) ∝ p(ct′|ct′−1, m, n)p(ct+δ+1|ct+δ = ct′, m, n)

t

  • j=
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Inference (Contd.)

Assuming d, r to be small, λ = 1 − exp(−dr) ≈ dr p(ct′ = k|ct′−1 = k, m, n, r, d) =

  • λπk,k′ + (1 − λ)δ(k, k′)fork′ ∈ {1

λπk,K+1 fork′ = K + 1

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Inference (Contd.)

Assuming d, r to be small, λ = 1 − exp(−dr) ≈ dr p(ct′ = k|ct′−1 = k, m, n, r, d) =

  • λπk,k′ + (1 − λ)δ(k, k′)fork′ ∈ {1

λπk,K+1 fork′ = K + 1 Terms can be replaced in original equation to get sampler

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Inference (Contd.)

Assuming d, r to be small, λ = 1 − exp(−dr) ≈ dr p(ct′ = k|ct′−1 = k, m, n, r, d) =

  • λπk,k′ + (1 − λ)δ(k, k′)fork′ ∈ {1

λπk,K+1 fork′ = K + 1 Terms can be replaced in original equation to get sampler Posterior distribution for ancestors p(ak,t|c, h) ∝ Γ(αh + βh) Γ(αh)Γ(βh) Γ(αh + ℓk,t)Γ(βh + ℓ′

k,t)

Γ(αh + βh + ℓk,t + ℓ′

k,t)

  • 1

|B| − 1 ℓ′

k,t

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Single Population Data

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

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Two Population Data

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Hierarchical DPM for Haplotype Inference

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Hierarchical DPM for Haplotype Inference (Contd.)

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Experiments: Hapmap Data

SNP genotypes from four populations

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Experiments: Hapmap Data

SNP genotypes from four populations

CEPH, Utah residents with northern/weatern European ancestry, 60

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Experiments: Hapmap Data

SNP genotypes from four populations

CEPH, Utah residents with northern/weatern European ancestry, 60 YRI, Yoruba in Ibadan, Nigeria, 60

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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

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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

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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 (∼ 102 − 103) SNPs

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Short SNP Sequences

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Basics HMDP Inference Results HDPM Results

Long SNP Sequences

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SLIDE 91

Basics HMDP Inference Results HDPM Results

Mutation Rates and Diversity

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SLIDE 92

Basics HMDP Inference Results HDPM Results

Mutation Rates and Diversity (Contd.)