1. Estimating and Reconstructing Recombination in Populations: Problems in Population Genomics Dan Gusfield UC Davis Different parts of this work are joint with Satish Eddhu, Charles Langley, Dean Hickerson, Yun Song, Yufeng Wu, Z. Ding INCOB06, December 20, 2006, New Delhi, India

2. What is population genomics? The Human genome “sequence” is done. Now we want to sequence many individuals in a population to correlate similarities and differences in their sequences with genetic traits (e.g. disease or disease susceptibility). Presently, we can’t sequence large numbers of individuals, but we can sample the sequences at SNP sites.

3. SNP Data A SNP is a Single Nucleotide Polymorphism - a site in the genome where two different nucleotides appear with sufficient frequency in the population (say each with 5% frequency or more). Hence binary data. SNP maps are being compiled with a density of about 1 site per 1000. SNP data is what is mostly collected in populations - it is much cheaper to collect than full sequence data, and focuses on variation in the population, which is what is of interest.

4. Haplotype Map Project: HAPMAP NIH lead project ($100M) to find common SNP haplotypes (“SNP sequences”) in the Human population. Association mapping: HAPMAP used to try to associate genetic-influenced diseases with specific SNP haplotypes, to either find causal haplotypes, or to find the region near causal mutations. The key to the logic of Association mapping is historical recombination in populations. Nature has done the experiments, now we try to make sense of the results.

5. Our work: Reconstructing the Evolution of SNP Sequences I: Clean mathematical and algorithmic results: Galled-Trees, near-uniqueness, graph-theory lower bound, and the Decomposition theorem II: Practical computation of Lower and Upper bounds on the number of recombinations needed. Construction of (optimal) phylogenetic networks; uniform sampling; haplotyping with ARGs III: Extension to Gene Conversion IV: Applications

6. Perfect Phylogeny: Where it all starts

7. The Perfect Phylogeny Model for the History of SNP sequences 00000 1 2 4 3 5 10100 10000 01011 00010 01010 12345 sites Ancestral sequence Extant sequences at the leaves Site mutations on edges The tree derives the set M: 10100 10000 01011 01010 00010 Only one mutation per site allowed.

8. Classic NASC: Arrange the sequences in a matrix. Then (with no duplicate columns), the sequences can be generated on a unique perfect phylogeny if and only if no two columns (sites) contain all four pairs: 0,0 and 0,1 and 1,0 and 1,1 This is the 4-Gamete Test When can a set of sequences be derived on a perfect phylogeny?

9. A richer model 00000 1 2 4 3 5 10100 10000 01011 00010 01010 12345 10100 10000 01011 01010 00010 10101 added Pair 4, 5 fails the four gamete-test. The sites 4, 5 ``conflict”. Real sequence histories often involve recombination. M

10. 10100 01011 5 10101 The first 4 sites come from P (Prefix) and the sites from 5 onward come from S (Suffix). P S Sequence Recombination A recombination of P and S at recombination point 5. Single crossover recombination

11. Network with Recombination 00000 1 2 4 3 5 10100 10000 01011 00010 01010 12345 10100 10000 01011 01010 00010 10101 new 10101 The previous tree with one recombination event now derives all the sequences. 5 P S M

12. A Phylogenetic Network or ARG 00000 5 2 3 3 4S p P S 1 4 a:00010 b:10010 c:00100 10010 01100 d:10100 e:01100 00101 01101 f:01101 g:00101 00100 00010

13. Minimizing recombinations in Phylogenetic networks Problem: given a set of sequences M, find a phylogenetic network generating M, minimizing the number of recombinations used to generate M. The minimization objective is a rough, but useful, reflection of the true number of ``observable” recombinations that have occurred in the derivation of M.

14. Minimization is an NP-hard Problem There is no known efficient solution to this problem and there likely will never be one. What we can do: Solve special cases optimally with efficient algorithms (galled-trees); Solve small data-sets optimally with algorithms that are not provably efficient but work well in practice; Efficiently compute lower and upper bounds on the number of needed recombinations (HapBound, Shrub);

15. Galled-Trees: an efficient special case

16. Definition: Recombination Cycle In a Phylogenetic Network, with a recombination node x, if we trace two paths backwards from x, then the paths will eventually meet. The cycle specified by those two paths is called a ``recombination cycle”.

17. Galled-Trees A phylogenetic network where no recombination cycles share an edge is called a galled tree. A cycle in a galled-tree is called a gall. Problem: If Haplotype matrix M cannot be generated on a true tree, can it be generated on a galled-tree?

18. 4 1 3 25 a: 00010 b: 10010 d: 10100 c: 00100 e: 01100 f: 01101 g: 00101 2 4 p s p s 13 4 25 Incompatibility Graph

19. Results about galled-trees Theorem: Efficient (provably polynomial-time) algorithm to determine whether or not any haplotype set H can be derived on a galled-tree. Theorem: A galled-tree (if one exists) produced by the algorithm minimizes the number of recombinations used over all possible phylogenetic-networks. Theorem: If M can be derived on a galled tree, then the Galled-Tree is ``nearly unique”. This is important for biological conclusions derived from the galled-tree. Gusfield et al. papers from 2003-2005.

20. Elaboration on Near Uniqueness Theorem: The number of arrangements (permutations) of the sites on any gall is at most three, and this happens only if the gall has two sites. If the gall has more than two sites, then the number of arrangements is at most two. If the gall has four or more sites, with at least two sites on each side of the recombination point (not the side of the gall) then the arrangement is forced and unique. Theorem: All other features of the galled-trees for M are invariant.

21. Efficient Bounding Algorithms We cannot efficiently compute the exact minimum number of needed recombinations, in general, but we can efficiently compute close lower and upper bounds on the minimum number. The bounds and the computations to obtain them have many practical applications.

22. The general composite lower bound method (S. Myers 2002) Given a set of intervals on the line, and for each interval I, a number N(I), which is a (local) lower bound on the number of recombinations needed in interval I, define Vmin as the minimum number vertical lines needed so that every interval I intersects at least N(I) of the vertical lines. Vmin is a valid lower bound on the total number of recombinations needed in the whole data. Vmin is a called a composite bound. Vmin is easy to compute by a left-to-right myopic algorithm.

23. The Composite Method (Myers & Griffiths 2003) M 1. Given a set of intervals, and Composite Problem: Find the minimum number of vertical lines so that every I intersects at least N(I) vertical lines. 2 1 2 2 2 3 1 2. for each interval I, a number N(I) 8

24. Haplotype (local) Lower Bound (S. Myers) Rh = Number of distinct sequences (rows) - Number of distinct sites (columns) -1 <= minimum number of recombinations needed (folklore) Generally Rh is really bad bound, often negative, when used on large intervals, but Very Good when used as local bounds on small intervals with the Composite Method, and other methods.

25. Composite Subset Method (Myers) Let S be a subset of sites, and Rh(S) be the haplotype bound computed on the input sequences restricted to the sites in S. If the leftmost site in S is L and the rightmost site in S is R, then use Rh(S) as a local bound N(I) for interval I = [S,L]. Compute Rh(S) on many subsets, and then solve the composite problem to find a valid composite bound.

26. RecMin (S. Myers) Computes local bounds using subsets of sites, but limits the size and the span of the subsets. Default parameters are s = 6, w = 15 (s = size, w = span). Generally, impractical to set s and w large, so generally one doesn’t know if increasing the parameters would increase the composite bound. Still, RecMin often gives a bound more than three times the HK bound. Example LPL data: HK gives 22, RecMin gives 75.

27. Optimal RecMin Bound (ORB) The Optimal RecMin Bound is the lower bound that RecMin would produce if both parameters were set to their maximum possible values. In general, RecMin cannot compute the ORB in practical time. We have developed a practical program, HAPBOUND, based on integer linear programming that guarantees to compute the ORB, and have incorporated ideas that lead to even higher lower bounds than the ORB.

28. HapBound: The general approach For an interval of sites I, let H(I) be the largest haplotype lower bound obtained from any subset of sites in I. We have shown that we can efficiently compute H(I) by using integer linear programming. We set N(I) = H(I) in the composite method, and the resulting composite bound is the ORB.

29. HapBound vs. RecMin on LPL from Clark et al. ProgramLower BoundTime RecMin (default)593s RecMin –s 25 –w 25757944s RecMin –s 48 –w 48No result5 days HapBound ORB7531s HapBound -S781643s 2 Ghz PC

30. Example where RecMin has difficulity in Finding the ORB on a 25 by 376 Data Matrix ProgramBoundTime RecMin default361s RecMin –s 30 –w 30423m 25s RecMin –s 35 –w 354324m 2s RecMin –s 40 –w 40432h 9m 4s RecMin –s 45 –w 454310h 20m 59s HapBound442m 59s HapBound -S4839m 30s

31. Constructing Optimal Phylogenetic Networks Optimal = minimum number of recombinations. Called Min ARG.

32. Kreitman’s 1983 ADH Data 11 sequences, 43 segregating sites Both HapBound and SHRUB took only a fraction of a second to analyze this data. Both produced 7 for the number of detected recombination events Therefore, independently of all other methods, our lower and upper bound methods together imply that 7 is the minimum number of recombination events.

33. A Min ARG for Kreitman’s data ARG created by SHRUB

34. The Human LPL Data (Nickerson et al. 1998) Our new lower and upper bounds Optimal RecMin Bounds (We ignored insertion/deletion, unphased sites, and sites with missing data.) (88 Sequences, 88 sites)

35. Application: Association Mapping Given case-control data M, uniformly sample the minimum ARGs (in practice for small windows of fixed number of SNPs) Build the ``marginal” tree for each interval between adjacent recombination points in the ARG Look for non-random clustering of cases in the tree; accumulate statistics over the trees to find the best mutation explaining the partition into cases and controls.

36. Input Data 00101 10001 10011 11111 10000 00110 Seqs 0-2: cases Seqs 3-5: controls sample One Min ARG for the data

37. Input Data 00101 10001 10011 11111 10000 00110 Seqs 0-2: cases Seqs 3-5: controls Tree The marginal tree for the interval past both breakpoints Cases

39. Haplotyping (Phasing) genotypic data using a Min ARG

40. Genotypes and Haplotypes Each individual has two “copies” of each chromosome. At each site, each chromosome has one of two alleles (states) denoted by 0 and 1 (motivated by SNPs) 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 0 Two haplotypes per individual Genotype for the individual Merge the haplotypes

41. Haplotyping Problem Biological Problem: For disease association studies, haplotype data is more valuable than genotype data, but haplotype data is hard to collect. Genotype data is easy to collect. Haplotyping (Phasing) Problem: Given a set of n genotypes, determine the original set of n haplotype pairs that generated the n genotypes. This is hopeless without a genetic model for the evolution of haplotype sequences.

42. Haplotyping by Minimizing Recombinations We want to haplotype genotypic data by finding those pairs of haplotypes (that explain the genotypes) and minimize the number of recombinations needed to derive the haplotypes. Minimizing recombination encodes the biology.

43. We have a branch and bound algorithm that finds the haplotypes minimizing the number of recombinations, building a Min ARG for deduced haplotypes. It works for genotype data with a small number of sites, but a larger number of genotypes.

44. Application: Detecting Recombination Hotspots with Genotype Data Bafna and Bansel (2005) uses recombination lower bounds to detect recombination hotspots with haplotype data. We apply our program on the genotype data –Compute the minimum number of recombinations for all small windows with fixed number of SNPs –Plot a graph showing the minimum level of recombinations normalized by physical distance –Initial results shows this approach can give good estimates of the locations of the recombination hotspots

45. Recombination Hotspots on Jeffreys, et al (2001) Data Result from Bafna and Bansel (2005), haplotype data Our result on genotype data

46. #Seq#Sites%missingAccuracy 20505 %96 % 205010 %95 % 205030 %93 % 32 5 %97 % 32 10 %96 % 32 30 %94 % 50205 %97 % 502010 %96 % 502030 %94 % #Seq#Sites%missingAccuracy 201005 %95 % 2010010 %95 % 2010030 %93 % 45 5 %98 % 45 10 %97 % 45 30 %96 % 100205 %97 % 1002010 %96 % 1002030 %95 % Datasets with about 1,000 entriesDataets with about 10,000 entries Application: Missing Data Imputation by Constructing near-optimal ARGs For  = 5.

47. Haplotyping genotype data via a minimum ARG Compare to program PHASE, in order to try to understand why Phase is so accurate. Experience shows PHASE may give solutions whose recombination is close to the minimum –Example: In all solutions of PHASE for three sets of case/control data from Steven Orzack, recombinatons are minimized. –Simulation results: PHASE’s solution minimizes recombination in 57 of 100 data (20 rows and 5 sites).

48. Papers and Software on wwwcsif.cs.ucdavis.edu/~gusfield I would like to thank the organizers of Incob 2006 for inviting me, and thank you for your attention.