Association mapping for mendelian, and complex disorders July 15Bafna, BfB

Association mapping for mendelian, and complex disorders

Apr 19, 2023 Bafna, BfB

UG Bioinformatics specialization at UCSD

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Abstraction of a causal mutation

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Looking for the mutation in populations

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A possible strategy is to collect cases (affected) and control individuals, and look for a mutation that consistently separates the two classes. Next, identify the gene.

Looking for the causal mutation in populations

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Case

Control

Problem 1: many unrelated common mutations, around one every 1000bp

Case

Control

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Looking for the causal mutation in populations

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Case

Control

Problem 2: We may not sample the causal mutation.

How to hunt for disease genes

• We are guided by two simple facts governing these mutations1. Nearby mutations are correlated2. Distal mutations are not

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Case

Control

This lecture

1. The bottom line: How do these facts help in finding disease genes?

2. The genetics: why should this happen?3. The computation4. Challenge of complex diseases.

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Case

Control

1. Nearby mutations are correlated2. Distal mutations are not

The basics of association mapping

• Sample a population of individuals at variant locations across the genome. Typically, these variants are single nucleotide polymorphisms (SNPs).

• Create a new bi-allelic variant corresponding to cases and controls, and test for correlations.

• By our assumptions, only the proximal variants will be correlated.

• Investigate genes near the correlated variants.

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Case

Control

00001

11

1

So, why should the proximal SNPs be correlated, and distal SNPs not?

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A bit of evolution

• Consider a fixed population (of chromosomes) evolving in time.

• Each individual arises from a unique, randomly chosen parent from the previous generation

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Time

Genealogy of a chromosomal population

Current (extant) population

Time

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Adding mutations

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Infinite sites assumption: A mutation occurs at most once at a site.

SNPs

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The collection of acquired mutations in the extant population describe the SNPs

Fixation and elimination

• Not all mutations survive.• Some mutations get fixed, and are no

longer polymorphic

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Removing extinct genealogies

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Removing fixed mutations

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The coalescent

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Disease mutation

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• We drop the ancestral chromosomes, and place the mutations on the internal branches.

Disease mutation

• A causal mutation creates a clade of affected descendants.

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Disease mutation

• Note that the tree (genealogy) is hidden. • However, the underlying tree topology

introduces a correlation between each pair of SNPs

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What have we learnt?

• The underlying genealogy creates a correlation between SNPs.

• By itself, this is not sufficient, because distal SNPs might also be correlated.

• Fortunately, for us the correlation between distal SNPs is quickly destroyed.

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Recombination

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Recombination

• In our idealized model, we assume that each individual chromosome chooses two parental chromosomes from the previous generation

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Multiple recombination change the local genealogy

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A bit of evolution

• Proximal SNPs are correlated, distal SNPs are not.• The correlation (Linkage disequilibirium) decays

rapidly after 20-50kb

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Basic statistics

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Testing for correlation

• In the absence of correlation

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Testing for correlation

• When correlated

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Assigning confidence

2 2

2 2

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X

X 4 0

0 4

X

X

Expected Observed

Assigning confidence

2 2

2 2

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X

X 4 0

0 4

X

X

Expected Observed

Assigning confidence

2.5 2.5

1.5 1.5

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X

3 2

1 2

XExpected Observed

X X

Statistical tests of association

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Tests for association: Pearson

• Case-control phenotype:– Build a 3X2 contingency table– Pearson test (2df)=

Cases Controls

mm

Mm

MM O1 O2

O3 O4

O6O5

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The χ2 test

Cases Controls

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Mm

MM O1

O5

O3 O4

O2

O6

• The statistic behaves like a χ2 distribution.

• A p-value can be computed directly

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Χ2 distribution properties

A related distribution is the F-distribution

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Likelihood ratio

• Another way to check the extremeness of the distribution is by computing a (log) likelihood ratio.

• We have two competing hypothesis. Let N be the total number of observations

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LLR

• An LLR value close to 0, implies that the null hypothesis is true. Asymptotically, the LLR statistic also follows the chi-square distribution.

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Exact test

• The chi-square test does not work so well when the numbers are small.

• How can we compute an exact probability of seeing a specific distribution of values in the cells?

• Remember: we know the marginals (# cases, # controls,

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Fischer exact test

Cases Controls

mm

Mm

MM a

e

c d

b

f

• Num: #ways of getting configuration (a,b,c,d,e,f)

• Den: #ways of ensuring that the row sums and column sums are fixed

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Fischer exact test

• Remember that the probability of seeing any specific values in the cells is going to be small.

• To get a p-value, we must sum over all similarly extreme values. How?

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Test for association: Fisher exact test

• Here P is the probability of seeing the exact count.• The actual significance is computed by summing over

all such tables that are at least this extreme.

Cases Controls

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Mm

MM a

e

c d

b

f

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Test for association: Fisher exact test

Cases Controls

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e

c d

b

f

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Continuous outcomes

• Instead of discrete (Case/control) data, we have real-valued phenotypes– Ex: Diastolic Blood Pressure

• In this case, how do we test for association

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Continuous outcome ANOVA

• Often, the phenotypes are not offered as case-controls but like a continuous variable– Ex: blood-pressure measurements

• Question: Are the mean values of the two groups significantly different?

MM mm

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Two-sided t-test

• For two categories, ANOVA is also known as the t-test

• Assume that the variables from the two sets are drawn from Normal distributions– Different means, equal variances

• Null hypothesis is that they are both from the same distribution

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t-test continued

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Two-sample t-test

• As the variance is not known, we use an estimate S, defined by

• The T-statistic is given by

• Significant deviations from 0 are used to reject the Null hypothesisApr 19, 2023 Bafna, BfB

Two-sample t-test (unequal variances)

• If the variances cannot be assumed to be equal, we use

• The T-statistic is given by

• Significant deviations from 0 are used to reject the Null hypothesisApr 19, 2023 Bafna, BfB

Confounding Association

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Confounding association

• Association tests can be confounded in many ways.

• We will explore a few of these, at a high level, and point to a few algorithmic problems.

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Confounding association with population substructure

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If the cases and controls are from different subpopulations, then sites with differing allele frequencies will confound association

The algorithmic problem

• Given a collection of individual genotypes, separate them into sub-populations.

• Idea: take markers that are very far apart so that no LD is possible.

• LD indicates structure.• Problem: Partition individuals into sub-populations so

that all correlation across pairs of distant markers is minimized. Penalty for increasing sub-populations?

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Confounding associations with genotypes

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A recombination event

Distinct haplotypes can create identical genotypes confounding association

Confounding association with interactions

• Individually, the markers do not correlate.• Together, they perfectly predict genes.• Find interacting partners that associate

with genes

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Confounding association with rare variants

• Not only can we have multiple interacting SNPs, each SNP individually occurs with very low frequency (< 1%).

• Can you detect associations with rare variants?

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Other problems

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• Can we reconstruct the phlogeny?• Useful for computing recombination

bounds.

Conclusion

• As individual genomes are sequenced, the association of variations with phenotypes presents many confounding challenges.

• Some of these challenges can be modeled as algorithmic problems.

• Population genetics should be part of a bioinformatics undergraduate curriculum.

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Thank you

• Homework (due Monday, March 15)– Describe an algorithm to detect associations

of interacting, rare-variants with a complex disease phenotype, in the presence of population substructure in the case-control population.

Apr 19, 2023 Bafna, BfB