CSE 291: Advanced Topics in Computational Biology L3: population substructure admixture mapping

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CSE 291 Advanced Topics in Computational
BiologyL3 population sub-structure/admixture
mapping

• Vineet Bafna/Pavel Pevzner

www.cse.ucsd.edu/classes/sp05/cse291-a
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Ancestral Recombination Graph
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Review Coalescent theory applications

• Coalescent simulations allow us to test various
  hypothesis. The coalescent/ARG is usually not
  inferred, unlike in phylogenies.

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Coalescent theory example

• Ex 1400bp at Sod locus in Dros.
• 10 taxa
• 5 were identical. The other 5 had 55 mutations.
• Q Is this a chance event, or is there selection
  for this haplotype.

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Coalescent application

• 10000 coalescent simulations were performed on 10
  taxa.
• 55 mutations on the coalescent branches
• Count the number of times 5 lineages are
  identical
• The event happened in 1.1 of the cases.
• Conclusion selection, or some other mechanism
  explains this data.

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Coalescent example Out of Africa hypothesis

• Looking at lineage specific mutations might help
  discard the candelabra model. How?
• How do we decide between the multi-regional and
  Out-of-Africa model? How do we decide if the
  ancestor was African?

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Coalescent simulation

• Example 2
• Sample from a region. What is the recombination
  rate in this region?
• Gabriel et al. Science 2002.
• 3 populations were sampled at multiple regions
  spanning the genome
• 54 regions (Average size 250Kb)
• SNP density 1 over 2Kb
• 90 Individuals from Nigeria (Yoruban)
• 93 Europeans
• 42 Asian
• 50 African American

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Population specific recombination

• D was used as the measure between SNP pairs.
• SNP pairs were in one of the following
• Strong LD
• Strong evidence for recombination
• Others (13 of cases)
• Can this be used for Out-Of-Africa hypothesis?

Gabriel et al., Science 2002
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Haplotype Blocks

• A haplotype block is a region of low
  recombination.
• Define a region as a block if less than 5 of the
  pairs show strong recombination
• Much of the genome is in blocks.
• Distribution of block sizes vary across
  populations.

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Testing Out-of-Africa

• Generate simulations with and without migration.
• Check size of haplotype blocks.
• Does it vary when migrations are allowed?
• When the new population has a bottleneck?
• If there were a bottleneck that created European
  and Asian populations, can we say anything about
  frequency of alleles that are African specific?
• Should they be high frequency, or low frequency
  in African populations?

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Haplotype Block implications

• The genome is mostly partitioned into haplotype
  blocks.
• Within a block, there is extensive LD.
• Is this good, or bad, for association mapping?

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Population Sub-structure
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Population sub-structure can increase LD

• Consider two populations that were isolated and
  evolving independently.
• They might have different allele frequencies in
  some regions.
• Pick two regions that are far apart (LD is very
  low, close to 0)

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Recent ad-mixing of population

• If the populations came together recently (Ex
  African and European population), artificial LD
  might be created.
• D 0.15 (instead of 0.01), increases 10-fold
• This spurious LD might lead to false associations
• Other genetic events can cause LD to arise, and
  one needs to be careful

0 .. 1 0 .. 1 0 .. 0 1 .. 1 0 .. 1 0 .. 1 0 ..
1 0 .. 1 0 .. 1
Pop. AB
p10.5 q10.5 P110.1 D0.1-0.250.15
1 .. 0 1 .. 0 0 .. 0 1 .. 1 1 .. 0 1 .. 0 1 ..
0 1 .. 0 1 .. 0
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Determining population sub-structure

• Given a mix of people, can you sub-divide them
  into ethnic populations.
• Turn the problem of spurious LD into a clue.
• Find markers that are too far apart to show LD
• If they do show LD (correlation), that shows the
  existence of multiple populations.
• Sub-divide them into populations so that LD
  disappears.

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Determining Population sub-structure

• Same example as before
• The two markers are too similar to show any LD,
  yet they do show LD.
• However, if you split them so that all 0..1 are
  in one population and all 1..0 are in another, LD
  disappears

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Iterative algorithm for population sub-structure

• Define
• N number of individuals (each has a single
  chromosome)
• k number of sub-populations.
• Z ? 1..kN is a vector giving the
  sub-population.
• Zik gt individual i is assigned to population
  k
• Xi,j allelic value for individual i in position
  j
• Pk,j,l frequency of allele l at position j in
  population k

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Example

• Ex consider the following assignment
• P1,1,0 0.9
• P2,1,0 0.1

0 .. 1 0 .. 1 0 .. 0 1 .. 1 0 .. 1 0 .. 1 0 ..
1 0 .. 1 0 .. 1 0 .. 1
1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2
1 .. 0 1 .. 0 0 .. 0 1 .. 1 1 .. 0 1 .. 0 1 ..
0 1 .. 0 1 .. 0 1 .. 0
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Goal

• X is known.
• P, Z are unknown.
• The goal is to estimate Pr(P,ZX)
• Various learning techniques can be employed.
• maxP,Z Pr(XP,Z) (Max likelihood estimate)
• maxP,Z Pr(XP,Z) Pr(P,Z) (MAP)
• Sample P,Z from Pr(P,ZX)
• Here a Bayesian (MCMC) scheme is employed to
  sample from Pr(P,ZX). We will only consider a
  simplified version

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AlgorithmStructure

• Iteratively estimate
• (Z(0),P(0)), (Z(1),P(1)),.., (Z(m),P(m))
• After convergence, Z(m) is the answer.
• Iteration
• Guess Z(0)
• For m 1,2,..
• Sample P(m) from Pr(P X, Z(m-1))
• Sample Z(m) from Pr(Z X, P(m))
• How is this sampling done?

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Example

• Choose Z at random, so each individual is
  assigned to be in one of 2 populations. See
  example.
• Now, we need to sample P(1) from Pr(P X, Z(0))
• Simply count
• Nk,j,l number of people in pouplation k which
  have allele l in position j
• pk,j,l Nk,j,l / N

0 .. 1 0 .. 1 0 .. 0 1 .. 1 0 .. 1 0 .. 1 0 ..
1 0 .. 1 0 .. 1 0 .. 1
1 2 2 1 1 2 1 2 1 2 1 2 2 1 1 2 1 2 2 1
1 .. 0 1 .. 0 0 .. 0 1 .. 1 1 .. 0 1 .. 0 1 ..
0 1 .. 0 1 .. 0 1 .. 0
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Example

• Nk,j,l number of people in population k which
  have allele l in position j
• pk,j,l Nk,j,l / Nk,j,
• N1,1,0 4
• N1,1,1 6
• p1,1,0 4/10
• p1,2,0 4/10
• Thus, we can sample P(m)

0 .. 1 0 .. 1 0 .. 0 1 .. 1 0 .. 1 0 .. 1 0 ..
1 0 .. 1 0 .. 1 0 .. 1
1 2 2 1 1 2 1 2 1 2 1 2 2 1 1 2 1 2 2 1
1 .. 0 1 .. 0 0 .. 0 1 .. 1 1 .. 0 1 .. 0 1 ..
0 1 .. 0 1 .. 0 1 .. 0
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Sampling Z

• PrZ1 1 Pr01 belongs to population 1?
• We know that each position should be in linkage
  equilibrium and independent.
• Pr01 Population 1 p1,1,0 p1,2,1
  (4/10)(6/10)(0.24)
• Pr01 Population 2 p2,1,0 p2,2,1
  (6/10)(4/10)0.24
• Pr Z1 1 0.24/(0.240.24) 0.5

Assuming, HWE, and LE
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Sampling

• Suppose, during the iteration, there is a bias.
• Then, in the next step of sampling Z, we will do
  the right thing
• Pr01 pop. 1 p1,1,0 p1,2,1 0.70.7
  0.49
• Pr01 pop. 2 p2,1,0 p2,2,1 0.30.3
  0.09
• PrZ1 1 0.49/(0.490.09) 0.85
• PrZ6 1 0.49/(0.490.09) 0.85
• Eventually all 01 will become 1 population, and
  all 10 will become a second population

0 .. 1 0 .. 1 0 .. 0 1 .. 1 0 .. 1 0 .. 1 0 ..
1 0 .. 1 0 .. 1 0 .. 1
1 1 1 2 1 2 1 2 1 1 2 2 2 1 2 2 1 2 2 1
1 .. 0 1 .. 0 0 .. 0 1 .. 1 1 .. 0 1 .. 0 1 ..
0 1 .. 0 1 .. 0 1 .. 0
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Allowing for admixture

• Define qi,k as the fraction of individual i that
  originated from population k.
• Iteration
• Guess Z(0)
• For m 1,2,..
• Sample P(m),Q(m) from Pr(P,Q X, Z(m-1))
• Sample Z(m) from Pr(Z X, P(m),Q(m))

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Estimating Z (admixture case)

• Instead of estimating Pr(Z(i)kX,P,Q), (origin
  of individual i is k), we estimate
  Pr(Z(i,j,l)kX,P,Q)

i,1
i,2
j
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Results on admixture prediction
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Results Thrush data

• For each individual, q(i) is plotted as the
  distance to the opposite side of the triangle.
• The assignment is reliable, and there is evidence
  of admixture.

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Population Structure

• 377 locations (loci) were sampled in 1000 people
  from 52 populations.
• 6 genetic clusters were obtained, which
  corresponded to 5 geographic regions (Rosenberg
  et al. Science 2003)

Oceania
Eurasia
East Asia
America
Africa
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Population sub-structureresearch problem

• Systematically explore the effect of admixture.
  Can admixture be predicted for a locus, or for an
  individual
• The sampling approach may or may not be
  appropriate. Formulate as an optimization/learning
  problem
• (w/out admixture). Assign individuals to
  sub-populations so as to maximize linkage
  equilibrium, and hardy weinberg equilibrium in
  each of the sub-populations
• (w/ admixture) Assign (individuals, loci) to
  sub-populations

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Admixture mapping
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