1. Vineet Bafna/Pavel Pevzner
CSE 291: Advanced Topics in Computational Biology L3: population sub-structure/ admixture mapping
Vineet Bafna/Pavel Pevzner

2. Ancestral Recombination Graph

3. Review: Coalescent theory applications
Coalescent simulations allow us to test various hypothesis. The coalescent/ARG is usually not inferred, unlike in phylogenies.

4. 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.

5. 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.

6. 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?

7. 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

8. 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

9. 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.

10. 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?

11. 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?

12. Population Sub-structure

13. 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)
0 .. 1
0 .. 0
1 .. 1
1 .. 0
Pop. A
Pop. B
p1=0.1
q1=0.9
P11=0.1
D=0.01
p1=0.9
q1=0.1

14. 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 .. 0
1 .. 1
Pop. A+B
p1=0.5
q1=0.5
P11=0.1
D= =0.15
1 .. 0
0 .. 0
1 .. 1

15. 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.

16. 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
0 .. 1
0 .. 0
1 .. 1
1 .. 0

17. Iterative algorithm for population sub-structure
Define
N = number of individuals (each has a single chromosome)
k = number of sub-populations.
Z  {1..k}N is a vector giving the sub-population.
Zi=k’ => 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

18. Example Ex: consider the following assignment P1,1,0 = 0.92
0 .. 1
0 .. 0
1 .. 1
1 .. 0
0 .. 0
1 .. 1

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

20. Algorithm:Structure 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?

21. 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 1 2
0 .. 1
0 .. 0
1 .. 1
1 .. 0
0 .. 0
1 .. 1

22. 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) 1 2
0 .. 1
0 .. 0
1 .. 1
1 .. 0
0 .. 0
1 .. 1

23. Sampling Z Pr[Z1 = 1] = Pr[”01” belongs to population 1]?
We know that each position should be in linkage equilibrium and independent.
Pr[”01” |Population 1] = p1,1,0 * p1,2,1 =(4/10)*(6/10)=(0.24)
Pr[”01” |Population 2] = p2,1,0 * p2,2,1 = (6/10)*(4/10)=0.24
Pr [Z1 = 1] = 0.24/( ) = 0.5
Assuming, HWE, and LE

24. Sampling Suppose, during the iteration, there is a bias.
Then, in the next step of sampling Z, we will do the right thing
Pr[“01”| pop. 1] = p1,1,0 * p1,2,1 = 0.7*0.7 = 0.49
Pr[“01”| pop. 2] = p2,1,0 * p2,2,1 =0.3*0.3 = 0.09
Pr[Z1 = 1] = 0.49/( ) = 0.85
Pr[Z6 = 1] = 0.49/( ) = 0.85
Eventually all “01” will become 1 population, and all “10” will become a second population 1 2
0 .. 1
0 .. 0
1 .. 1
1 .. 0
0 .. 0
1 .. 1

25. 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))

26. Estimating Z (admixture case)
Instead of estimating Pr(Z(i)=k|X,P,Q), (origin of individual i is k), we estimate Pr(Z(i,j,l)=k|X,P,Q)
i,1
i,2 j

27. Results on admixture prediction

28. 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.

29. 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
Africa
America

30. Population sub-structure:research 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

31. Admixture mapping