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1 Modified Mincut Supertrees Roderic Page University of Glasgow.

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1 1 Modified Mincut Supertrees Roderic Page University of Glasgow

2 2 Tree of Life About 1.7 million species described. What we have so far: TreeBASE database (15,000 taxa) Ribosomal Database Project (RDP II) (20,000 sequences) The Tree of Life Project (11,000 taxa)

3 3 Recent interest in the Tree of Life Assembling the Tree of Life: Science, Relevance, and Challenges AMNH, New York, May 2002 $US 10 million to construct a phylogeny for the 1.7 million described species of Life announced February 15 th 2002 NSF sponsored Tree of Life workshops ( ) European initiative (ATOL) under FP6

4 4 Problem: how to build the tree of life Solutions: Find one or more magic markers that will allow us to recover the whole tree in one go (problems: combinability and complexity) Assemble big tree from many smaller trees derived from many kinds of data (supertrees)

5 5 Tree terminology abc d {a,b} {a,b,c} {a,b,c,d} root leaf internal node cluster edge

6 6 Nestings and triplets abc d {a,b} < T {a,b,c,d} {b,c} < T {a,b,c,d} (bc)d bc|d Nestings Triplets

7 7 Supertree abcbcd abc d supertree T 1 T 2 + =

8 8 Some desirable properties of a supertree method (Steel et al., 2000) The supertree can be computed in polynomial time A grouping in one or more trees that is not contradicted by any other tree occurs in the supertree

9 9 Homo sapiens11 1 Pan paniscus Gorilla gorilla Pongo pygmaeus1 0 0 Hylobates MRP (Matrix Representation Parsimony) NP-hard Can generate many solutions

10 10 Aho et al.s algorithm (OneTree) Aho, A. V., Sagiv, Y., Syzmanski, T. G., and Ullman, J. D Inferring a tree from lowest common ancestors with an application to the optimization of relational expressions. SIAM J. Comput. 10: Input: set of rooted trees 1. If set is compatible (i.e., will agree on a tree), output that tree. 2. If set is not compatible, stop!

11 11 abcbcd T 1 T 2 a b c d a, bda, b, c, d a b c a, b, cabc Aho et al.s OneTree algorithm supertree

12 12 Mincut supertrees Semple, C., and Steel, M A supertree method for rooted trees. Discrete Appl. Math. 105: Modifies OneTree by cutting graph Requires rooted trees (no analogue of OneTree for unrooted trees) Recursive Polynomial time

13 13 abcdeabcd T 1 T 2 a b c de {T 1,T 2 } S Semple and Steel (2000)

14 14 a b c de a,b c de {T 1,T 2 } S max S/E {T 1,T 2 }{T 1,T 2 } Collapsing the graph (Semple and Steel mincut algorithm) This edge has maximum weight

15 15 Cut the graph to get supertree abcde supertree a,b c de max S/E {T 1,T 2 }{T 1,T 2 }

16 16 My mincut supertree implementation darwin.zoology.gla.ac.uk/~rpage/supertree Written in C++ Uses GTL (Graph Template Library) to handle graphs (formerly a free alternative to LEDA) Finds all mincuts of a graph faster than Semple and Steels algorithm

17 17 A counter example: two input trees... a b c x 1 x 2 x 3 c b a y 1 y 2 y 3 y 4

18 18 Mincut gives this (strange) result c x 1 x 2 x 3 b a y 1 y 2 y 3 y 4 Disputed relationships among a, b, and c are resolved x1, x2, and x3 collapsed into polytomy

19 19 Problem: Cuts depend on connectivity (in this example it is a function of tree size) a x1 x2 y1 y3 y4 x3 y2 c b {T 1,T 2 } S

20 20 So, mincut doesnt work But, Semple and Steel said it did My program seems to work Argh!!! What is happening….?

21 21 What mincut does… …and does not do Mincut supertree is guaranteed to include any nesting which occurs in all input trees Makes no claims about nestings which occur in only some of the trees Does exactly what it says on the tin

22 22 Modifying mincut supertree Can we incorporate more of the information in the input trees? Three categories of information Unanimous (all trees have that grouping) Contradicted (trees explicitly disagree) Uncontradicted (some trees have information that no other tree disagrees with)

23 23 Uncontradicted information assume we have k input trees ab a and b co-occur in a tree a and b nested in a tree ab c n c - n = 0 uncontradicted (if c = k then unanimous) c - n > 0 contradicted

24 24 Uncontradicted information assume we have k input trees ab a and b co-occur in a tree a and b nested in a tree ab c n c - n -f = 0 uncontradicted (if c = k then unanimous) c - n - f > 0 contradicted ab a and b in a fan f

25 25 a b c x 1 x x 3 y 1 y 2 y 3 y 4 2 a b c y 1 y 3 y 4 x 1 x 2 x 3 y 2 Uncontradicted Uncontradicted but adjacent to contradicted Contradicted Classifying edges {T 1,T 2 } S

26 26 Modified mincut Species a, b, and c form a polytomy x1, x2, and x3 resolved as per the input tree modifiedmincut a b c x 1 x 2 x 3 y 1 y 2 y 3 y 4

27 (12)5 (45)1 (23)5 (34)1 If no tree contradicts an item of information, is that information always in the supertree?

28 No! Steel, Dress, & Böcker 2000 The four trees display (12)5, (23)5, (34)1, and (45)1 No tree displays (IK)J or (JK)I for any (IJ)K above Triplets are uncontradicted, but cannot form a tree

29 29 Future directions Improve handling of uncontradicted information Add support for constraints Visualising very big trees Better integration into phylogeny databases (www.treebase.org) darwin.zoology.gla.ac.uk/~rpage/supertree


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