COMPUTATIONAL HUMAN GENETICS SEARCHING FOR RELATIONS BETWEEN GENES, DISEASES, AND POPULATIONS

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COMPUTATIONAL HUMAN GENETICS - SEARCHING FOR
RELATIONS BETWEEN GENES, DISEASES, AND POPULATIONS

• Eran Halperin
• November 10, 2009

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Environmental Factors
Genetic Factors
Complexdisease
Multiple genes may affect the disease.
Therefore, the effect of every single gene may
be negligible.
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The Human Chromosomes
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Each chromosome is a sequence over the
alphabet A,G,C,T (base pairs)
Copy from mother
ACCAGGACGA
ACCAGGACGA
Copy from father
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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

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The Human Genome Project
What we are announcing today is that we have
reached a milestonethat is, covering the genome
ina 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
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The Vision of Personalized Medicine
Genetic and epigenetic variants measurable
environmental/behavioral factors would be used
for a personalized treatment and diagnosis
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Paradigm shifts in medicine
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Example Warfarin
An anticoagulant drug, useful in the prevention
of thrombosis.
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Example Warfarin
Warfarin was originallyused as rat poison.
Optimal dose variesacross the
population Genetic variants (VKORC1 and CYP2C9)
affect the variation of the personalized optimal
dose.
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Association Studies

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

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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).
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Disease Association Studies
SNP Single Nucleotide Polymorphism
Cases
AGAGCAGTCGACAGGTATAGCCTACATGAGATCGACATGAGATCGGTAGA
GCCGTGAGATCGACATGATAGCC AGAGCCGTCGACATGTATAGTCTACA
TGAGATCGACATGAGATCGGTAGAGCAGTGAGATCGACATGATAGTC AG
AGCAGTCGACAGGTATAGTCTACATGAGATCGACATGAGATCGGTAGAGC
CGTGAGATCGACATGATAGCC AGAGCAGTCGACAGGTATAGCCTACATG
AGATCAACATGAGATCGGTAGAGCAGTGAGATCGACATGATAGCC AGAG
CCGTCGACATGTATAGCCTACATGAGATCGACATGAGATCGGTAGAGCCG
TGAGATCAACATGATAGCC AGAGCCGTCGACATGTATAGCCTACATGAG
ATCGACATGAGATCGGTAGAGCAGTGAGATCAACATGATAGCC AGAGCC
GTCGACAGGTATAGCCTACATGAGATCGACATGAGATCGGTAGAGCAGTG
AGATCAACATGATAGTC AGAGCAGTCGACAGGTATAGCCTACATGAGAT
CGACATGAGATCTGTAGAGCCGTGAGATCGACATGATAGCC
Associated SNP (lower Relative Risk)
Controls
AGAGCAGTCGACATGTATAGTCTACATGAGATCGACATGAGATCGGTAGA
GCAGTGAGATCAACATGATAGCC AGAGCAGTCGACATGTATAGTCTACA
TGAGATCAACATGAGATCTGTAGAGCCGTGAGATCGACATGATAGCC AG
AGCAGTCGACATGTATAGCCTACATGAGATCGACATGAGATCTGTAGAGC
CGTGAGATCAACATGATAGCC AGAGCCGTCGACAGGTATAGCCTACATG
AGATCGACATGAGATCTGTAGAGCCGTGAGATCGACATGATAGTC AGAG
CCGTCGACAGGTATAGTCTACATGAGATCGACATGAGATCTGTAGAGCCG
TGAGATCAACATGATAGCC AGAGCAGTCGACAGGTATAGTCTACATGAG
ATCGACATGAGATCTGTAGAGCAGTGAGATCGACATGATAGCC AGAGCC
GTCGACAGGTATAGCCTACATGAGATCGACATGAGATCTGTAGAGCCGTG
AGATCGACATGATAGCC AGAGCCGTCGACAGGTATAGTCTACATGAGAT
CAACATGAGATCTGTAGAGCAGTGAGATCGACATGATAGTC
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Preliminary Definitions

• SNP single nucleotide polymorphism. A genetic
  variant which may carry different value for
  different individuals.
• Allele the variants 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)

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Relative Risk
Chances of developing type II diabetes 30
RiskG
Chances of developing type II diabetes 20
NonriskA
Relative Risk Pr(DR)/Pr(DN) 1.5
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Other Structural Variants
Inversion
Deletion
Copy number variant
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Published Genome-Wide Associations through
6/2009, 439 published GWA at p lt 5 x 10-8
NHGRI GWA Catalog www.genome.gov/GWAStudies
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Public Genotype Data Growth
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Chance or Real Association?
Cases
AGAGCAGTCGACAGGTATAGCCTACATGAGATCGACATGAGATCGGTAGA
GCCGTGAGATCGACATGATAGCC AGAGCCGTCGACATGTATAGTCTACA
TGAGATCGACATGAGATCGGTAGAGCAGTGAGATCGACATGATAGTC AG
AGCAGTCGACAGGTATAGTCTACATGAGATCGACATGAGATCGGTAGAGC
CGTGAGATCGACATGATAGCC AGAGCAGTCGACAGGTATAGCCTACATG
AGATCAACATGAGATCGGTAGAGCAGTGAGATCGACATGATAGCC AGAG
CCGTCGACATGTATAGCCTACATGAGATCGACATGAGATCGGTAGAGCCG
TGAGATCAACATGATAGCC AGAGCCGTCGACATGTATAGCCTACATGAG
ATCGACATGAGATCGGTAGAGCAGTGAGATCAACATGATAGCC AGAGCC
GTCGACAGGTATAGCCTACATGAGATCGACATGAGATCGGTAGAGCAGTG
AGATCAACATGATAGTC AGAGCAGTCGACAGGTATAGCCTACATGAGAT
CGACATGAGATCTGTAGAGCCGTGAGATCGACATGATAGCC
Associated SNP (lower Relative Risk)
Controls
AGAGCAGTCGACATGTATAGTCTACATGAGATCGACATGAGATCGGTAGA
GCAGTGAGATCAACATGATAGCC AGAGCAGTCGACATGTATAGTCTACA
TGAGATCAACATGAGATCTGTAGAGCCGTGAGATCGACATGATAGCC AG
AGCAGTCGACATGTATAGCCTACATGAGATCGACATGAGATCTGTAGAGC
CGTGAGATCAACATGATAGCC AGAGCCGTCGACAGGTATAGCCTACATG
AGATCGACATGAGATCTGTAGAGCCGTGAGATCGACATGATAGTC AGAG
CCGTCGACAGGTATAGTCTACATGAGATCGACATGAGATCTGTAGAGCCG
TGAGATCAACATGATAGCC AGAGCAGTCGACAGGTATAGTCTACATGAG
ATCGACATGAGATCTGTAGAGCAGTGAGATCGACATGATAGCC AGAGCC
GTCGACAGGTATAGCCTACATGAGATCGACATGAGATCTGTAGAGCCGTG
AGATCGACATGATAGCC AGAGCCGTCGACAGGTATAGTCTACATGAGAT
CAACATGAGATCTGTAGAGCAGTGAGATCGACATGATAGTC
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How does it work?

• For every SNP we can construct a contingency
  table

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Hypothesis testing

• Null hypothesis Pr(Rcase) Pr(Rcontrol)
• Alternative hypothesis Pr(Rcase) ?
  Pr(Rcontrol)
• 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
  XB(n,Pr(Rcases))
• Controls sampled from distribution
  YB(n,Pr(Rcontrols))

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Hypothesis testing, cont.

• When n is large, B(n,p) N(np, np(1-p)).
• Under the null hypothesis

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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-valu
  e Pr(Z gt Z)
• Often we take , which is

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Results Manhattan Plots
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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.

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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.

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Challenge 1

• Population Substructure

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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.
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Evolution Theory

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

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Mutations
AGAGCAGTCGACAGGTATAGCCTACATGAGATCGACATGAGA
AGAGCAGTCCACAGGTATAGCCTACATGAGATCGACATGAGA
Estimated probability of a mutation in a single
generation is 10-8
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Other mutations - recombination
Copy 1
Copy 2
Probability ri (10-8) for recombination in
position i.
child chromosome
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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

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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 q01-p0 (of A).
• Under HWE, Epk1pk, but Vpk1 gt 0, so the
  next generation will have pk1?p0.

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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 Xk1 B(N,pk).
• Epk1 EXk1/N pk.
• Varpk1 pk(1-pk)/N.
• The effect of genetic drift depends on the time
  and the effective populations size. Small
  population increases the effect.

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Bottleneck effect
Effective population size
Time
Genetic drifts rate is higher.
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The Wright-Fisher Model
Generation 1 Allele frequency 1/9
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The Wright-Fisher Model
Generation 2 Allele frequency 1/9
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The Wright-Fisher Model
Generation 3 Allele frequency 1/9
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The Wright-Fisher Model
Generation 4 Allele frequency 1/3
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The Wright-Fisher Model
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The Wright-Fisher Model
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Ancestral population
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Ancestral population
migration
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• different allele frequencies

Ancestral population
Genetic drift
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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?
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Jakobsson et al, Nature 421 998-103
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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.

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

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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 complexpopulations (mixed)?

Novembre et al., Nature, 2008
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Challenge 2

• Modeling Correlation

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A typical associated region
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Linkage Disequilibrium
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Haplotype Data in a Block
(Daly et al., 2001) Block 6 from Chromosome 5q31
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Phasing - haplotype inference
Haplotypes
ATCCGA AGACGC

• Cost effective genotyping technology gives
  genotypes and not haplotypes.

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Inferring Haplotypes From Trios
Parent 1
122112
Parent 2
210022
120222
Child
Assumption No recombination
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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 ?.
• The likelihood of ? is
  .We are interested in the hypothesis that
  maximizes the likelihood.

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Soft assignment

• Compute probabilities Pph 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

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Phasing via Maximum Likelihood

• Soft decision
• Hard decision

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An iterative algorithm
Data1 0 h h 1 h 0 0 1 h 1 h h 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 ¼ ¼ ¼
0 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 0
¼ ¼ ¼ ¼
1 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1
¼ ¼ ¼ ¼
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An iterative algorithm
Data1 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 ¼ ¼ 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 ¼ ¼ 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 ¼ ¼ ¼
0.6 0.4
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An iterative algorithm
Data1 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 ¼ ¼ ¼
0 1
0 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 0
¼ ¼ ¼ ¼
1 0
1 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1
¼ ¼ ¼ ¼
1 0
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Expectation Maximization (EM)

• D given data
• T parameters that need to be estimated
• Z Latent missing variables

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EM rationale

• Lemma.
• Proof First, note that

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QED
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MLE from Incomplete Data

• Finding MLE parameters nonlinear optimization
  problem

log P(x ?)
E ?log P(x,y ?)
?
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MLE from Incomplete Data
log P(x ?)
E ?log P(x,y ?)
?
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EM for phasing
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• This is maximized for

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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.

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Human Genetics where to?

• We can typically explain 5-15of the
  heritability of commondiseases.
• 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.

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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