CSB 2006 1

Efficient Computation of Minimum Recombination

With Genotypes (Not Haplotypes)

Yufeng Wu and Dan Gusfield

University of California, Davis

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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

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Haplotyping

0 1 1 1 0 0 1 1 0

1 1 0 1 0 0 1 0 0

2 1 2 1 0 0 1 2 0Genotype

Sites: 1 2 3 4 5 6 7 8 9

Haplotype

Haplotype Inference (HI) Problem: given a set of n genotypes, infer the real n haplotype pairs that form the given genotypes

2 1 2 1 0 0 1 2 0

0 1 1

1 1 0

0 1 0

1 1 1

Phasing the 2s

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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

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One-stage Approach

• What effect does the haplotyping inaccuracy has on downstream questions?

• Our work: directly use genotype data for downstream problems– Without fixing a choice for the HI solution– Minimum recombination problem

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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

110001111111001

000110000001111

Prefix

Suffix

11000 0000001111

breakpoint

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Kreitman’s Data (1983)00000000110000000011011101111000000000000000010000000000000001101110111100000000000000000000000000000000000000000000000001000010100000000000000001100000000000000000100110000001100010110011110000000000000000001000000001000000000000100000000000000101011100001000100000000000010000000000000111111010000001111100010111001000000000000011111101100000111110001011100100000000000001111110110000011111000101110010000000000000111111011000001111111110000101000010001000011111101000000

Question: what is the minimum number of recombinations needed to derive these sequences?Assume at most 1 mutation per site

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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

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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.

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0 0 0 1 01 0 0 1 00 0 1 0 01 0 1 0 00 1 1 0 00 1 1 0 10 0 1 0 1

1 2 3 4 5abcdefg

Incompatibility Graph (IG):A node each site, edgebetween incompatible pair

M

Lower Bound: Incompatibility

• 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

1 2 3 4 5

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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.

1 2 3 4 5

HK Lower Bound = 1

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IG for HI Solutions

01010101010020222200

0101001010101011010100000001010100010100

1 2 3 4 5

HK = 1

HI1

0101001010101011010100001001000000011100

1 2 3 4 5

HK = 3

HI2

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HK Bounds on Genotypes

• Known efficient algorithm for MinHK(G) (Wiuf, 2004).

• This paper: polynomial-time algorithm for MaxHK(G)

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Maximal Incompatibility Graph

• 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)

01010101010020222200

G

1 2 3 4 5

MIG(G)

E(G) = {12, 23, 35}

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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)|

• 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

Finding such an H

MIG(G)

Phasing G for Incompatibility

0101001010101011010100?0?00?0?0??001??00

0101001010101011010100?0?00?0?00?0011?00

010100101010101101010010?0000?0000011100

• No matter how a previous site p is phased, can always phase this site q to make p, q incompatible

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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

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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

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Result from haplotypes (Bafna and Bansal, 2005)

Result from original genotypes (this paper)

MS32 data (Jeffreys, et al. 2001)

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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

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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

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Thank You

• Software: available upon request