1. L4: Counting Recombination events

2. 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?

3. 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))

4. 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

5. Results on admixture prediction: simulated data

6. 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.

7. 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

8. NJ versus Structure:thrush data
Objective function is different in standard clustering algorithms!

9. 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

10. Admixture mapping

11. Estimating Recombination Rates

12. Recombination in human chromosome 22 (Mb scale)
Dawson et al.
Nature 2002
Q: Can we give a direct count of the number of the recombination
events?

13. Recombination hot-spots (fine scale)

14. Recombination rates (chimp/human)
Fine scale recombination rates differ between chimp and human
The six hot-spots seen in human are not seen in chimp

15. Combinatorial Bounds for estimating recombination rate
Recall that expected #recombinations =  log n
Procedure
Generate N random ARGs that results in the given sample
Compute mean of the number of recombinations
Alternatively, generate a summary statistic s from the population.
For each , generate many populations, and compute the mean and variance of s (This only needs to be done once).
Use this to select the most likely 
What is the correct summary statistic?
Today, we talk about the min. number of recombination events as a possible summary statistic. It is not the most natural, but it is the most interesting computationally.

16. The Infinite Sites Assumption & the 4 gamete condition 3 8 5
Consider a history without recombination. No pair of sites shows all four gametes 00,01,10,11.
A pair of sites with all 4 gametes implies a recombination event

17. Hudson & Kaplan
Any pair of sites (i,j) containing 4 gametes must admit a recombination event.
Disjoint (non-overlapping) sites must contain distinct recombination events, which can be summed! This gives a lower bound on the number of recombination events.
Based on simulations, this bound is not tight.

18. Myers and Griffiths’03: Idea 1
Let B(i,j) be a lower bound on the number of recombinations between sites i and j.
1=i1 i2 i3 i4 i5 i ik=n
Can we compute maxP R(P) efficiently?

19. The Rm bound

20. Improved lower bounds
The Rm bound also gives a general technique for combining local lower bounds into an overall lower bound.
In the example, Rm=2, but we cannot give any ARG with 2 recombination events.
Can we improve upon Hudson and Kaplan to get better local lower bounds?
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1

21. Hudson and Kaplan: Idea 2
Consider the history of individuals. Let Ht denote the number of distinct halotypes at time t
One of three things might happen at time t:
Mutation: Ht increase by at most 1
Recombination: Ht increase by at most 1
Coalescence: Ht does not increase

22. The RH bound Ex: R>= 8-3-1=4 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1

23. RH bound In general, RH can be quite weak:
consider the case when S>H
However, it can be improved
Partitioning idea: sum RH over disjoint intervals
Apply to any subset of columns. Ex: Apply RH to the yellow columns
(BB’05)

24. The Rs bound
Compute the minimum number of recombination events R in any ARG. Note that, we do not explicitly construct the ARG.
Consider a matrix with M with H rows and S columns.
The rows correspond to haplotypes.
Columns correspond to sites.

25. Rs bound: Observation I
Non-informative column: If a site contains at most one 1, or one 0, then in any history, it can be obtained by adding a mutation to a branch.
EX: if a is the haplotype containing a 1, It can simply be added to the branch without increasing number of recombination events
R(M) = R(M-{s}) : 1 a a b c

26. Redundant rows: If two rows h1 and h2 are identical, then
Rs bound: Observation 2
Redundant rows: If two rows h1 and h2 are identical, then
R(M) = R(M-{h1}) r1 r2 c

27. Rs bound: Observation 3
Suppose M has no non-informative columns, or redundant rows.
Then, at least one of the haplotypes is a recombinant.
There exists h s.t.
R(M) = R(M-{h})+1
Which h should you choose?

28. Rs bound (Procedural) Procedure Compute_Rs(M)
If  non-informative column s
return (Compute_Rs(M-{s}))
Else if  redundant row h
return (Compute_Rs(M-{h}))
Else
return (1 + minh(Compute_Rs(M-{h}))

29. Results

30. Additional results/problems
Using dynamic programming, Rs can be computed in 2^n poly(mn) time.
Also, Rs can be augmented to handle intermediates.
Are there poly. time lower bounds?
The number of connected components in the conflict graph is a lower bound (BB’04).
Fast algorithms for computing ARGs with minimum recombination.
Poly. Time to get ARG with 0 recombination
Poly. Time to get ARGs that are galled trees (Gusfield’03)

31. Underperforming lower bounds
Sometimes, Rs can be quite weak
An RI lower bound that uses intermediates can help (BB’05)

32. LPL data set
71 individuals, 9.7Kbp genomic sequence
Rm=22, Rh=70