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

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

3. The Human Chromosomes
April 05’

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

5. 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

6. 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

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

8. Paradigm shifts in medicine

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

10. 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.

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

12. 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).

13. 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

14. 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)

15. 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

16. Other Structural Variants
Inversion
Copy number variant
Deletion

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

19. 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

20. 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

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

22. 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))

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

24. 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

25. Results: Manhattan Plots

26. 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.

27. 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.

28. Challenge 1
Population Substructure

29. 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.

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

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

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

33. 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

34. 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.

35. 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.

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

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

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

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

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

41. The Wright-Fisher Model

42. The Wright-Fisher Model

43. Ancestral population

44. Ancestral population
migration

45. different allele frequencies
Ancestral population
Genetic drift

46. 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?

47. Jakobsson et al, Nature 421: 998-103

48. 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.

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

50. 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.

51. 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

52. Challenge 2
Modeling Correlation

53. A typical associated region

54. Linkage Disequilibrium

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

56. 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 ….

57. 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

58. 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.

59. 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

60. Phasing via Maximum Likelihood
Soft decision: Hard decision:

61. An iterative algorithm
Data: 1 0 h h 1
h h
1 h h 1 1
0.4
0.6
0.75
0.25
/12
/12
/12
/12
/12
/12
/12
/12
/12

62. An iterative algorithm
Data: 1 0 h h 1
h h
1 h h 1 1
/12
/12
/12
/12
/12
/12
/12
/12
/12
0.4
0.6
0.75
0.25
0.6
0.4

63. An iterative algorithm
Data: 1 0 h h 1
h h
1 h h 1 1 /6 /2 /6 /6 1 1 1

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

65. EM rationale
Lemma:. Proof: First, note that

67. QED

68. 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”) 

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

70. EM for phasing

71. This is maximized for:

72. 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.

73. 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.

74. 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