1. Recombination and Pedigrees Genealogies and Recombination: The ARG Recombination Parsimony The ARG and Data Pedigrees: Models and Data Pedigrees & ARGs Challenges Empirical Investigations: Open Questions

2. Recombination Histories & Global Pedigrees Acknowledgements Yun Song - Rune Lyngsø - Mike Steel Finding Minimal Recombination Histories 1 23 4 1 23 4 1 2 3 4 Global Pedigrees Finding Common Ancestors NOW

3. Hudson & Kaplan’s R M If you equate R M with expected number of recombinations, this could be used as an estimator. Unfortunately, R M is a gross underestimate of the real number of recombinations. 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1

4. Local Inference of Recombinations 00001110000111 00011010001101 00 10 01 11 Four combinations Incompatibility: Myers-Griffiths (2002): Number of Recombinations in a sample, N R, number of types, N T, number of mutations, N M obeys: 00110011 01010101 T... G T... C A... G A... C Recoding At most 1 mutation per column 0 ancestral state, 1 derived state

5. Finding Minimal Recombination Histories 1.J.J.Hein: Reconstructing the history of sequences subject to Gene Conversion and Recombination. Mathematical Biosciences. (1990) 98.185-200. 2.J.J.Hein: A Heuristic Method to Reconstruct the History of Sequences Subject to Recombination. J.Mol.Evol. 20.402-411. 1993 3.Hein,J.J., T.Jiang, L.Wang & K.Zhang (1996): "On the complexity of comparing evolutionary trees" Discrete Applied Mathematics 71.153-169 4.Song, Y.S. (2003) “On the combinatorics of rooted binary phylogenetic trees”. Annals of Combinatorics, 7:365–379 5.Song, Y.S. & Hein, J. (2005) Constructing Minimal Ancestral Recombination Graphs. J. Comp. Biol., 12:147–169Constructing Minimal Ancestral Recombination Graphs 6.Song, Y.S. & Hein, J. (2004) On the minimum number of recombination events in the evolutionary history of DNA sequences. J. Math. Biol., 48:160–186.On the minimum number of recombination events in the evolutionary history of DNA sequences. 7.Song, Y.S. & Hein, J. (2003) Parsimonious reconstruction of sequence evolution and haplotype blocks: finding the minimum number of recombination events, Lecture Notes in Bioinformatics, Proceedings of WABI'03, 2812:287–302. 8. Lyngsø, Song and Hein (2005) “Minimal Recombination Histories by Branch and Bound” WABI 64 Bodmer & Edwards: Parsimony defined as reconstruction principle 85 Hudson Kaplan uses minimal recombination histories as observed recombinations Attempts to find minimal histories of sequences Definition of recombination as Subtree Prune Regraft operations

6. Minimal Number of Recombinations Last Local Tree Algorithm: L 21 Data 2 n i-1i 1 Trees The Kreitman data (1983): 11 sequences, 3200bp, 43(28) recoded, 9 different How many neighbors? Bi-partitions How many local trees? Unrooted Coalescent

7. Two Adjacent Columns 2. Infinite Site Assumption: Local Trees must contain Local Bipartition 1. RecDist[T 1,T 2 ] is hard for large leaf number, but can be automatically calculated by adding Diam Rec trivial columns and only considering 1 recombination neighbors. 110011110110011110 000011111000011111 1 2 Diam Rec RecDist

8. Metrics on Trees based on subtree transfers. Pretending the easy problem (unrooted) is the real problem (age ordered), causes violation of the triangle inequality: Tree topologies with age ordered internal nodes Rooted tree topologies Unrooted tree topologies Trees including branch lengths

9. Observe that the size of the unit-neighbourhood of a tree does not grow nearly as fast as the number of trees Song (2003+) Due to Yun Song Tree Combinatorics and Neighborhoods Allen & Steel (2001)

10. 1 2 3 4 5 6 7

11. Methods # of rec events obtained Hudson & Kaplan (1985)5 Myers & Griffiths (2003)6 Song & Hein (2004). Set theory based approach.7 Song & Hein (2003). Tree scanning using DP Lyngsø, Song & Hein (2006). Massive Acceleration using Branch and Bound Algorithm. Lyngsø, Song & Hein (2006). Minimal number of Gene Conversions (in prep.) 7 5-2/6-1 The Minimal Recombination History for the Kreitman Data

12. - recombination 27 ACs 0 1 2 3 4 5 6 7 8 1 1 4 2 5 3 1 5 5 The Griffiths-Ethier-Tavare Recursions No recombination: Infinite Site Assumption Ancestral State Known History Graph: Recursions Exists No cycles Possible Histories without Recombination for simple data example + recombination 3*10 8 ACs

13. 1st 2nd Ancestral configurations to 2 sequences with 2 segregating sites mid-point heuristic

14. Counting Recursion + k k 1 (k 2 +1)+1 padded with “-” 1k+1 Summary statistic lumping configurations k1k1 k2k2 (k 2 +1)*k 1 +1 possible ancestral columns.

15. Counting + Branch and Bound Algorithm ? Exact length Lower bound Upper Bound 0 3 1 91 2 1314 3 8618 4 30436 5 62794 6 78970 7 63049 8 32451 9 10467 10 1727 289920 k-recombinatination neighborhood k

16. Time versus Spatial: Coalescent-Recombination (Griffiths, 1981; Hudson, 1983 - Wiuf & Hein, 1999) Temporal ProcessSpatial Process ii. The trees cannot be reduced to Topologies i. The process is non-Markovian * * =

17. Elston-Stewart (1971) -Temporal Peeling Algorithm: Lander-Green (1987) - Genotype Scanning Algorithm: Mother Father Condition on parental states Recombination and mutation are Markovian Mother Father Condition on paternal/maternal inheritance Recombination and mutation are Markovian Time versus Spatial 2: Pedigrees

18. Time versus Spatial 3: Phylogenetic Alignment Spatial: Temporal: Optimisation Algorithms indels of length 1 (David Sankoff, 1973) Spatial indels of length k (Bjarne Knudsen, 2003) Temporal Statistical Alignment

19. minARGs: Recombination Events & Local Trees True ARG Reconstructed ARG 1 23 45 123 45 ((1,2),(1,2,3)) ((1,3),(1,2,3)) n=7,  =10,  =75 Minimal ARG True ARG 0 4 Mb Hudson- Kaplan Myers- Griiths Song-Hein n=8,  =40 n=8,  =15 Mutation information on only one side Mutation information on both sides

20. Likelihood Calculations on the  -ARG 010 101 110 Example:

22. Reconstructing global pedigrees: Superpedigrees Steel and Hein, 2005 The gender-labeled pedigrees for all pairs, defines global pedigree k Gender-unlabeled pedigrees doesn’t!!

23. Reconstructing global pedigrees: Links and lassos Steel and Hein, 2005 Gender-labeled links and lassos determine the global pedigree.

24. All embedded phylogenies are observable Do they determine the pedigree? Genomes with  and  --> infinity  recombination rate,  mutation rate Benevolent Mutation and Recombination Process Counter example: Embedded phylogenies:

25. Summary and Future Presentation can be found at: http://mathgen.stats.ox.ac.uk/bioinformatics/ Minimal Recombination Histories Likelihood Calculations Global Pedigrees & Inferring Pedigrees from Genomes Recombination: Remove infinite site assumption Investigate MCMC algorithms Pedigrees: Data Analysis Algorithms

26. To Do Hudson Slide Neighbor trees Literature and History