Eran Halperin November 10, 2009 COMPUTATIONAL HUMAN GENETICS - SEARCHING FOR RELATIONS BETWEEN GENES, DISEASES, AND POPULATIONS Eran Halperin November 10, 2009

Genetic Factors Complex disease Environmental Factors Multiple genes may affect the disease. Therefore, the effect of every single gene may be negligible.

The Human Chromosomes April 05’

………ACCAGGACGA…… ………ACCAGGACGA…… Each chromosome ‘is’ a sequence over the alphabet {A,G,C,T} (base pairs) Copy from mother ………ACCAGGACGA…… ………ACCAGGACGA…… Copy from father

Facts about our genome 23 pairs of chromosomes. X and Y are the sex chromosomes (XX for women, XY for men). 3,300,000,000 base pairs in the human genome

The Human Genome Project “What we are announcing today is that we have reached a milestone…that is, covering the genome in…a working draft of the human sequence.” “But our work previously has shown… that having one genetic code is important, but it's not all that useful.” “I would be willing to make a predication that within 10 years, we will have the potential of offering any of you the opportunity to find out what particular genetic conditions you may be at increased risk for…” Washington, DC June, 26, 2000

The Vision of Personalized Medicine Genetic and epigenetic variants + measurable environmental/behavioral factors would be used for a personalized treatment and diagnosis

Paradigm shifts in medicine

Example: Warfarin An anticoagulant drug, useful in the prevention of thrombosis.

Example: Warfarin Warfarin was originally used as rat poison. Optimal dose varies across the population Genetic variants (VKORC1 and CYP2C9) affect the variation of the personalized optimal dose.

Association Studies Genetic variants such as Single Nucleotide Polymorphisms (SNPs) are tested for association with the trait.

Where should we look first? SNP = Single Nucleotide Polymorphism person 1: ….AAGCTAAATTTG…. person 2: ….AAGCTAAGTTTG…. person 3: ….AAGCTAAGTTTG…. person 4: ….AAGCTAAATTTG…. person 5: ….AAGCTAAGTTTG…. Each common SNP has only two possible letters (alleles).

Disease Association Studies SNP = Single Nucleotide Polymorphism Cases: Associated SNP (high Relative Risk) AGAGCAGTCGACAGGTATAGCCTACATGAGATCGACATGAGATCGGTAGAGCCGTGAGATCGACATGATAGCC AGAGCCGTCGACATGTATAGTCTACATGAGATCGACATGAGATCGGTAGAGCAGTGAGATCGACATGATAGTC AGAGCAGTCGACAGGTATAGTCTACATGAGATCGACATGAGATCGGTAGAGCCGTGAGATCGACATGATAGCC AGAGCAGTCGACAGGTATAGCCTACATGAGATCAACATGAGATCGGTAGAGCAGTGAGATCGACATGATAGCC AGAGCCGTCGACATGTATAGCCTACATGAGATCGACATGAGATCGGTAGAGCCGTGAGATCAACATGATAGCC AGAGCCGTCGACATGTATAGCCTACATGAGATCGACATGAGATCGGTAGAGCAGTGAGATCAACATGATAGCC AGAGCCGTCGACAGGTATAGCCTACATGAGATCGACATGAGATCGGTAGAGCAGTGAGATCAACATGATAGTC AGAGCAGTCGACAGGTATAGCCTACATGAGATCGACATGAGATCTGTAGAGCCGTGAGATCGACATGATAGCC Controls: Associated SNP (lower Relative Risk) AGAGCAGTCGACATGTATAGTCTACATGAGATCGACATGAGATCGGTAGAGCAGTGAGATCAACATGATAGCC AGAGCAGTCGACATGTATAGTCTACATGAGATCAACATGAGATCTGTAGAGCCGTGAGATCGACATGATAGCC AGAGCAGTCGACATGTATAGCCTACATGAGATCGACATGAGATCTGTAGAGCCGTGAGATCAACATGATAGCC AGAGCCGTCGACAGGTATAGCCTACATGAGATCGACATGAGATCTGTAGAGCCGTGAGATCGACATGATAGTC AGAGCCGTCGACAGGTATAGTCTACATGAGATCGACATGAGATCTGTAGAGCCGTGAGATCAACATGATAGCC AGAGCAGTCGACAGGTATAGTCTACATGAGATCGACATGAGATCTGTAGAGCAGTGAGATCGACATGATAGCC AGAGCCGTCGACAGGTATAGCCTACATGAGATCGACATGAGATCTGTAGAGCCGTGAGATCGACATGATAGCC AGAGCCGTCGACAGGTATAGTCTACATGAGATCAACATGAGATCTGTAGAGCAGTGAGATCGACATGATAGTC

Preliminary Definitions SNP – single nucleotide polymorphism. A genetic variant which may carry different ‘value’ for different individuals. Allele – the variant’s value: A,G,C, or T. Most SNPs are bi-allelic. There are only two observed alleles in the populations. Risk allele – the allele which is more common in cases than in controls (denoted R) Nonrisk allele – the allele which is more common in the controls (denoted N)

Relative Risk Risk=G Nonrisk=A Chances of developing type II diabetes: 30% Risk=G Chances of developing type II diabetes: 20% Nonrisk=A Relative Risk: Pr(D|R)/Pr(D|N) = 1.5

Other Structural Variants Inversion Copy number variant Deletion

Published Genome-Wide Associations through 6/2009, 439 published GWA at p < 5 x 10-8 NHGRI GWA Catalog www.genome.gov/GWAStudies

Public Genotype Data Growth HapMap Phase 2 5,000,000+ SNPs 600,000,000+ genotypes 2006 2001 Daly et al. Nature Genetics 103 SNPs 40,000 genotypes Gabriel et al. Science 3000 SNPs 400,000 genotypes 2002 TSC Data Nucleic Acids Research 35,000 SNPs 4,500,000 genotypes 2003 Perlegen Data Science 1,570,000 SNPs 100,000,000 genotypes 2004 NCBI dbSNP Genome Research 3,000,000 SNPs 286,000,000 genotypes 2005

Chance or Real Association? Cases: AGAGCAGTCGACAGGTATAGCCTACATGAGATCGACATGAGATCGGTAGAGCCGTGAGATCGACATGATAGCC AGAGCCGTCGACATGTATAGTCTACATGAGATCGACATGAGATCGGTAGAGCAGTGAGATCGACATGATAGTC AGAGCAGTCGACAGGTATAGTCTACATGAGATCGACATGAGATCGGTAGAGCCGTGAGATCGACATGATAGCC AGAGCAGTCGACAGGTATAGCCTACATGAGATCAACATGAGATCGGTAGAGCAGTGAGATCGACATGATAGCC AGAGCCGTCGACATGTATAGCCTACATGAGATCGACATGAGATCGGTAGAGCCGTGAGATCAACATGATAGCC AGAGCCGTCGACATGTATAGCCTACATGAGATCGACATGAGATCGGTAGAGCAGTGAGATCAACATGATAGCC AGAGCCGTCGACAGGTATAGCCTACATGAGATCGACATGAGATCGGTAGAGCAGTGAGATCAACATGATAGTC AGAGCAGTCGACAGGTATAGCCTACATGAGATCGACATGAGATCTGTAGAGCCGTGAGATCGACATGATAGCC Controls: Associated SNP (lower Relative Risk) AGAGCAGTCGACATGTATAGTCTACATGAGATCGACATGAGATCGGTAGAGCAGTGAGATCAACATGATAGCC AGAGCAGTCGACATGTATAGTCTACATGAGATCAACATGAGATCTGTAGAGCCGTGAGATCGACATGATAGCC AGAGCAGTCGACATGTATAGCCTACATGAGATCGACATGAGATCTGTAGAGCCGTGAGATCAACATGATAGCC AGAGCCGTCGACAGGTATAGCCTACATGAGATCGACATGAGATCTGTAGAGCCGTGAGATCGACATGATAGTC AGAGCCGTCGACAGGTATAGTCTACATGAGATCGACATGAGATCTGTAGAGCCGTGAGATCAACATGATAGCC AGAGCAGTCGACAGGTATAGTCTACATGAGATCGACATGAGATCTGTAGAGCAGTGAGATCGACATGATAGCC AGAGCCGTCGACAGGTATAGCCTACATGAGATCGACATGAGATCTGTAGAGCCGTGAGATCGACATGATAGCC AGAGCCGTCGACAGGTATAGTCTACATGAGATCAACATGAGATCTGTAGAGCAGTGAGATCGACATGATAGTC

How does it work? For every SNP we can construct a contingency table: Total Cases a b Controls c d

Hypothesis testing Null hypothesis: Pr(R|case) = Pr(R|control) Alternative hypothesis: Pr(R|case) ≠ Pr(R|control) The model assumes that all individuals are independent (unrelated), and therefore our sample is a random sample from a Binomial distribution Cases sampled from distribution X~B(n,Pr(R|cases)) Controls sampled from distribution Y~B(n,Pr(R|controls))

Hypothesis testing, cont. When n is large, B(n,p) ~ N(np, np(1-p)). Under the null hypothesis:

P-value Z is called a test-statistic (z-score in this case). We can calculate Z* for our data, and then calculate (using the normal approximation): p-value = Pr(|Z| > |Z*|) Often we take , which is

Results: Manhattan Plots

The curse of dimensionality – corrections of multiple testing In a typical Genome-Wide Association Study (GWAS), we test millions of SNPs. If we set the p-value threshold for each test to be 0.05, by chance we will “find” about 5% of the SNPs to be associated with the disease. This needs to be corrected.

Bonferroni Correction If the number of tests is n, we set the threshold to be 0.05/n. A very conservative test. If the tests are independent then it is reasonable to use it. If the tests are correlated this could be bad: Example: If all SNPs are identical, then we lose a lot of power; the false positive rate reduces, but so does the power.

Challenge 1 Population Substructure

Population Substructure Imagine that all the cases are collected from Africa, and all the controls are from Europe. Many association signals are going to be found The vast majority of them are false; Why ??? Different evolutionary forces: drift, selection, mutation, migration, population bottleneck.

Evolution Theory Mutations add to genetic variation Natural Selection controls the frequency of certain traits and alleles Genetic drift

Mutations AGAGCAGTCGACAGGTATAGCCTACATGAGATCGACATGAGA AGAGCAGTCCACAGGTATAGCCTACATGAGATCGACATGAGA Estimated probability of a mutation in a single generation is 10^-8

Other ‘mutations’ - recombination Copy 1 Copy 2 Probability ri (~10^-8) for recombination in position i. child chromosome

Natural Selection Example: being lactose telorant is advantageous in northern Europe, hence there is positive selection in the LCT gene different allele frequencies in LCT

Genetic Drift Even without selection, the allele frequencies in the population are not fixed across time. Consider the case where we assume Hardy- Weinberg Equilibrium (HWE), that is, individuals are mating randomly in the population. If at the first generation the allele frequencies are p0 (of a) and q0=1-p0 (of A). Under HWE, E[pk+1]=pk, but V[pk+1] > 0, so the next generation will have pk+1≠p0.

The rate of the drift N – effective population size (if all individuals are entirely unrelated than N is the total population size). Under an assumption of constant population size, if Xk counts the number of occurrences of a at generation k, then Xk+1 ~ B(N,pk). E[pk+1] = E[Xk+1]/N = pk. Var[pk+1] = pk(1-pk)/N. The effect of genetic drift depends on the time and the effective populations size. Small population increases the effect.

Bottleneck effect Effective population size Time Genetic drift’s rate is higher.

The Wright-Fisher Model Generation 1 Allele frequency 1/9

The Wright-Fisher Model Generation 2 Allele frequency 1/9

The Wright-Fisher Model Generation 3 Allele frequency 1/9

The Wright-Fisher Model Generation 4 Allele frequency 1/3

The Wright-Fisher Model

The Wright-Fisher Model

Ancestral population

Ancestral population migration

different allele frequencies Ancestral population Genetic drift

Population Substructure Imagine that all the cases are collected from Africa, and all the controls are from Europe. Many association signals are going to be found The vast majority of them are false; What can we do about it?

Jakobsson et al, Nature 421: 998-103

Principal Component Analysis Dimensionality reduction Based on linear algebra (Singular Value Decomposition) Intuition: find the ‘most important’ features of the data – project the data on the axis with the largest variance.

Principal Component Analysis Plotting the data on a one dimensional line for which the spread is maximized.

Principal Component Analysis In our case, we want to look at two dimensions at a time. The original data has many dimensions – each SNP corresponds to one dimension.

Ancestry Inference To what extent can population structure be detected from SNP data? What can we learn from these inferences? Can we build the tree of life? How do we analyze complex populations (mixed)? Novembre et al., Nature, 2008

Challenge 2 Modeling Correlation

A typical associated region

Linkage Disequilibrium

Haplotype Data in a Block (Daly et al., 2001) Block 6 from Chromosome 5q31

Phasing - haplotype inference Haplotypes Genotype þ ý ü î í ì A C CG G T ATCCGA AGACGC mother chromosome father chromosome Cost effective genotyping technology gives genotypes and not haplotypes. ATACGA AGCCGC Possible phases: AGACGA ATCCGC ….

Inferring Haplotypes From Trios 1??11? 1100?? 0100?? 1?0??? 10?11? 11?11? 1100?? 0100?? 100??? 110??? 10011? 11111? 11000? 01001? 1??11? ?100?? 1?0??? Parent 1 122112 Parent 2 210022 120222 Child Assumption: No recombination

Maximum Likelihood Until now we discussed the case of two hypotheses (null, and alternative). In some cases we are interested in many hypotheses and we search for the best one. Normally a hypothesis will be defined by a set of parameters θ. The likelihood of θ is .We are interested in the hypothesis that maximizes the likelihood.

Soft assignment Compute probabilities P={ph} for all possible haplotypes. For each genotype g, we do not assign one pair of haplotypes, but a distribution of possible pairs. The set of pairs of haplotypes compatible with g is denoted as C(g). In soft assignment, a pair is explaining g with probability

Phasing via Maximum Likelihood Soft decision: Hard decision:

An iterative algorithm Data: 1 0 h h 1 h 0 0 1 h 1 h h 1 1 0.4 0.6 0.75 0.25 0 0 0 1 0 1/12 0 0 0 1 1 1/12 1 0 0 0 1 1/12 1 0 0 1 0 1/12 1 0 0 1 1 3/12 1 0 1 0 1 1/12 1 0 1 1 1 2/12 1 1 0 1 1 1/12 1 1 1 1 1 1/12 1 0 0 0 1 1 0 1 1 1 1 0 0 1 1 1 0 1 0 1 ¼ 0 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 0 ¼ 1 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 ¼

An iterative algorithm Data: 1 0 h h 1 h 0 0 1 h 1 h h 1 1 0 0 0 1 0 .125 0 0 0 1 1 .042 1 0 0 0 1 .067 1 0 0 1 0 .042 1 0 0 1 1 .325 1 0 1 0 1 .1 1 0 1 1 1 .067 1 1 0 1 1 .067 1 1 1 1 1 .1 0 0 0 1 0 1/12 0 0 0 1 1 1/12 1 0 0 0 1 1/12 1 0 0 1 0 1/12 1 0 0 1 1 3/12 1 0 1 0 1 1/12 1 0 1 1 1 2/12 1 1 0 1 1 1/12 1 1 1 1 1 1/12 1 0 0 0 1 1 0 1 1 1 1 0 0 1 1 1 0 1 0 1 ¼ 0.4 0.6 0 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 0 ¼ 0.75 0.25 1 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 ¼ 0.6 0.4

An iterative algorithm Data: 1 0 h h 1 h 0 0 1 h 1 h h 1 1 0 0 0 1 0 1/6 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1 1 1/2 1 0 1 0 1 1/6 1 0 1 1 1 0 1 1 0 1 1 0 1 1 1 1 1 1/6 1 0 0 0 1 1 0 1 1 1 1 0 0 1 1 1 0 1 0 1 ¼ 1 0 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 0 ¼ 1 1 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 ¼ 1

Expectation Maximization (EM) D – given data Θ– parameters that need to be estimated Z – Latent missing variables

EM rationale Lemma:. Proof: First, note that

QED

MLE from Incomplete Data Finding MLE parameters: nonlinear optimization problem log P(x| ) E ’[log P(x,y| )] Expectation Maximization (EM): Use “current point” to construct alternative function (which is “nice”) 

MLE from Incomplete Data log P(x| ) E ’[log P(x,y| )] 

EM for phasing

This is maximized for:

Phasing summary Expectation maximization is easy to implement, works reasonably well in practice. We can use other models (tree models) to improve the accuracy of the phasing prediction.

Human Genetics – where to? We can typically explain 5%-15% of the heritability of common diseases. Where is the missing heritability? Rare variants Gene-gene interactions Gene-environment interactions Creative computational methods are key to the discovery of the missing heritability.

Course: Computational Human Genetics Semester bet More background in human genetics, statistics, and machine learning. Studying genetics of human disease Privacy and forensics Analysis of new technologies (sequencing) Population genetics – detecting selection, mutation rate, recombination rates, etc. Reconstructing human history