1. Yufeng Wu and Dan Gusfield University of California, Davis
Efficient Computation of Minimum Recombination With Genotypes (Not Haplotypes)
Yufeng Wu and Dan Gusfield
University of California, Davis
CSB 2006

2. Haplotypes/Genotypes
Diploid organisms have two copies of (not identical) chromosomes. A single copy is a haplotype, vector of 0,1. The mixed description is a genotype, vector of 0,1,2. At each site,
If both haplotypes are 0, genotype is 0
If both haplotypes are 1, genotype is 1
If one is 0 and the other is 1, genotype is 2
Key fact: easier to collect genotypes, but many downstream applications work better with haplotypes

3. Haplotyping
Sites:
Phasing the 2s
Haplotype
Genotype
Haplotype Inference (HI) Problem: given a set of n genotypes, infer the real n haplotype pairs that form the given genotypes

4. Two-stage Approach
Given a set of genotypes G, we are interested in downstream problems
Many HI solutions for G
Two stage: first infer the “correct” HI solution from the genotypes, then do the downstream analysis with the inferred haplotypes
Haplotype inference: extensively studied and believed to be accurate to certain extent

5. One-stage Approach
What effect does haplotyping inaccuracy have on downstream questions?
Our work: directly use genotype data for downstream problems
Without fixing a choice for the HI solution
Minimum recombination problem

6. Recombination: Single Crossover
Recombination is one of the principle genetic force shaping variation within species
Two equal length sequences generate a third equal length sequence
Prefix
11000
breakpoint
Suffix

7. Kreitman’s Data (1983)
Question: what is the minimum number of recombinations needed to derive these sequences?
Assume at most 1 mutation per site

8. Minimizing Recombination
Compute the minimum number of recombinations (Rmin) for deriving a set of haplotypes, assuming at most 1 mutation per site
NP-hard in general
Heuristics
Lower bounds on Rmin

9. Lower Bounds on Genotypes
For a particular recombination lower bound method L, what is the range of possible bounds for L over all possible HI solutions?
MinL(G): minimum L over all HI solutions for G.
MaxL(G): maximum L over all HI solutions for G.
This paper: HK bound, connected component bound and relaxed haplotype bound.
Polynomial-time algorithms for MaxHK, MinCC.
Heuristic method for relaxed haplotype bound.

10. Lower Bound: Incompatibility
Incompatibility Graph (IG):
A node each site, edge
between incompatible pair a b c d e f g M
Two sites (columns) p, q are incompatible if columns p,q contains all four ordered pairs (gametes): 00, 01, 10, 11
Sites p,q are incompatible  A recombination must occur between p,q

11. HK Bound (1985)
Arrange the nodes of the incompatibility graph on the line in order that the sites appear in the sequence.
HK bound = maximum number of non-overlapping edges in incompatibility graph (IG).
Easy to compute for haplotype data.
HK Lower Bound = 1

12. IG for HI Solutions HK = 1 HI1 1 2 3 4 5 01010 10101 00202 22200
00000
00101
01000
10100
HK = 1
HI1
01010
10101
00202
22200
01010
10101
00001
00100
00000
11100
HK = 3
HI2

13. HK Bounds on Genotypes
Known efficient algorithm for MinHK(G) (Wiuf, 2004).
This paper: polynomial-time algorithm for MaxHK(G)

14. Maximal Incompatibility Graph
MIG(G)
01010
10101
00202
22200
E(G) = {12, 23, 35}
An edge between sites p and q if there is a phasing of p, q so p and q are incompatible
Each pair of sites is considered independently
E(G): a maximum-sized set of non-overlapping edges in MIG(G)

15. MaxHK(G) Claim: MaxHK(G) = |E(G)| MaxHK(G)  |E(G)|
MIG(G): supergraph of IG(H) for any HI solution H
If we can find an HI solution H, whose every pair of sites in E(G) is incompatible, then HK(H)  |E(G)|
Together, MaxHK(G) = |E(G)|

16. Finding such an H MIG(G) Phase sites from left to right.
Each component in E(G) is a simple path
Each site only constrained by at most one site to the left

17. Phasing G for Incompatibility
01010
10101
00?0?
0??00
1??00
01010
10101
00?0?
00?00
11?00
01010
10101
0010?
0000?
00000
11100
No matter how a previous site p is phased, can always phase this site q to make p, q incompatible

18. Haplotyping With Minimum Number of Recombinations
Compute Rmin(G)
Haplotyping on a network with fewest recombinations
NP-hard
This paper: A branch and bound method computing exact Rmin(G) for data with small number of sites
APOE data: 47 non-trivial genotypes, 9 sites
Our method: 2 minutes, Rmin(G) = 5

19. Application: Recombination Hotspot
Recombination hotspot: regions where recombination rate is much higher than neighboring regions
Previous study (Bafna and Bansal, 2005): a recombination lower bound with inferred haplotypes were used to identify recombination hotspots
Our work: compute the exact Rmin(G) with genotypes for a sliding window of a small number of SNPs to detect recombination hotspots

20. MS32 data (Jeffreys, et al. 2001)
Result from haplotypes (Bafna and Bansal, 2005)
Result from original genotypes (this paper)

21. Other Applications Finding true Rmin from genotypes G
Two stage approach: run PHAS to get an HI solution H, and compute Rmin(H)
One stage approach: directly compute Rmin(G)
Accuracy of haplotype inference on a minimum network
Simulation results: comparable, slightly weaker and non-conclusive

22. Summary
Main goal of this paper: develop computational tools for the minimum recombination problem with genotypes
Polynomial-time algorithm for MaxHK and MinCC problems
Practical heuristics for other problems
Simulation results to several application questions are not conclusive
Our tools facilitate the study of these problems

23. Thank You
Software: available upon request