Beyond Galled Trees Decomposition and Computation of Galled Networks

Transcription

1 Beyond Galled Trees Decomposition and Computation of Galled Networks Daniel H. Huson & Tobias H.Kloepper RECOMB

2 Copyright (c) 2008 Daniel Huson. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license can be found at

3 Two Main Types of Phylogenetic Networks Implicit networks: visualization of incompatible signals Explicit networks: explicitly describe a evolutionary scenario involving reticulate events 2

4 Implicit Networks Visualization of incompatible signals Eg split network from binary characters: Haplotype data Split network Data: Cassens et al., 2003, Dusky dolphins 3

5 Explicit Networks Explicitly describe a evolutionary scenario involving reticulate events such as hybridization, HGT or recombination A B C D E F G H r 1 r 2 r 3 root 4

6 A Simple Model of Reticulate Evolution A B H C D P Hybridization HGT Recombination: Uneven Mixture mixture Order of matters of genomes reticulate events Q speciation events By C.R. Linder Ancestral genome By C.R. Linder mutations 5

7 Data For Reticulate Networks Hybridization or HGT: Different genes have different histories Input: Two or more gene trees T 1,T 2, Output: Network N that explains T 1,T 2, A B C D E A B C D E A B C D E N T 1 T 2 6

8 Data For Reticulate Networks Recombination: Recombination of closely related sequences Input: Alignment M of binary sequences Output: Network N that explains M Alignment M A: B: R: C: D: O: Additional annotation Network N 2 1, R: B: C: A: D: , O:

9 Combinatorial Approach Can be formulated in terms of splits: Every edge e of a tree T defines a split of the taxon set X: F A H e A,C,D,F,G vs B,E,H D E B G C Split encoding Σ(T) 8

10 Splits From Sequences Every non-constant binary character induces a split of the taxon set X: Alignment M A: B: C: D: E: F: ACDF vs BE Multiple columns may map to the same split Define Σ(M): set of all splits induced by M 9

11 Combinatorial Approach Hybridization or HGT: Input trees T 1,T 2, represented by splits :=Σ(T 1 ) Σ(T 2 ) (Information loss: which splits occur together in same input tree?) Recombination: Binary alignment M represented by splits :=Σ(M) (Information loss: order along sequence) 10

12 Reticulate Networks And Splits For a reticulate network N, how to define Σ(N)? Extract tree by deleting one reticulate edge for each reticulate node For each tree edge e: Obtain split from tree: A B C D E F G H r 1 r 2 r 3 e A,B,C,D,E,H vs F,G root Σ(N): set of all splits thus obtained 11

13 Parsimonious Reticulate Network Problem Input: Set of splits on a taxon set X. Output: A reticulate network N with: 1. Σ(N) 2. N contains a minimum number of reticulate nodes Such an N always exists (Baroni & Steel, 2005) To find one is NP-hard in general (Wang et al, 2001, Borderwich & Semple, 2006) Special case: N is a galled tree (Gusfield et al, ) 12

14 The Galled Tree Property Dan Gusfield et al ( ): If a solution exists that has the galled tree property, then it can be computed efficiently 13

15 The Galled Tree Property A reticulation is a gall, if it is cycle disjoint to all others P R Q A B C D E F Reticulation at P is a gall, at Q is a gall Addition of R destroys gall property for Q Gall property is fragile 14

16 The Loose Gall Property A reticulation is a loose gall, if it has a cycle whose backbone consists only of tree edges P R Q A B C D E F P, Q and R are loose galls Not fragile: Adding taxa doesn t destroy property 15

17 The Galled Network Property New definition: A reticulate network is a galled network, if all reticulations are loose galls. How to compute them? The Decomposition Theorem 16

18 Input: Computing A Galled Network Set of splits on X={A,B,,I} that comes from a network, either via trees or binary sequences, e.g.: G H I A B C D E F 17

19 Computing A Galled Network Assume we know G,H,I are reticulate taxa Where to attach G, H, I? H I G A B C D E F Induced splits Extended splits X-{G,H,I} X-{H,I} Orient edges to show where splits place G Attach G to ends of target path 18

20 Computing A Galled Network Assume we know G,H,I are reticulate taxa Where to attach G, H, I? G I H A B C D E F Induced splits Extended splits X-{G,H,I} X-{G,I} Orient edges to show where splits place H Attach H to ends of target path 19

21 Computing A Galled Network Assume we know G,H,I are reticulate taxa Where to attach G, H, I? G H I A B C D E F I Induced splits Extended splits X-{G,H,I} X-{G,H} Orient edges to show where splits place I Attach I to ends of target path 20

22 Computing A Galled Network Assume we know G,H,I are reticulate taxa Where to attach G, H, I? G H A B C D E F I If Σ(N), then return N 21

23 Algorithm Input: Set of splits on X, parameter k In increasing order of size k: Consider a set of taxa R X If X-R is compatible: Attempt to attach each r R to T( X-R ) If successful, construct network N If Σ(N), return N Return fail FPT, for fixed maximum size k of R 22

24 Decomposition Conjecture (Dan Gusfield) Input trees Split network Minimal reticulate network 23

25 Decomposition Conjecture Each incompatibility component can be considered independently: 1. component 2. component (Gusfield et al. 2005) (Huson et al. 2005) 24

26 The Decomposition Theorem Let of be a set of splits. If there exists a galled network N with Σ(N), then there exists a minimal network N min that has the decomposition property. To compute N min we can consider each component separately 25

27 Proof Easy, assuming non-degenerate: For every tree node v there exists a path of tree edges from v down to some leaf w. A B C D E F G H v non-degenerate root Degenerate node, no tree path to leaf 26

28 Consider any reticulation cycle A Proof a X Y R b B Any split S Σ(e) is incompatible with all S Σ(a) or S Σ(b): S contains either AXR BY or AX BYR S contains AR XYB S contains AXY BR e 27

29 Implementation Available in the latest version of SplitsTree4 Interactive program for phylogenetic analysis using trees and networks (Huson and Bryant, MBE, 2006) 28

30 Reticulate Network with 4 reticulations Data: Kumar et al, Restriction map of the rdna cistron, culicine mosquitos 29

31 Reticulate cladogram 30

32 Conclusion & Outlook Galled networks go beyond galled trees A user-friendly implementation is available in the latest version of SplitsTree4 Decomposition Conjecture unsolved in general All current methods based on combinatorics, thus are sensitive to false-positive splits More robust methods for computation of phylogenetic networks required 31

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