Seeing the wood for the trees: Analysing multiple alternative phylogenies

Transcription

1 Seeing the wood for the trees: Analysing multiple alternative phylogenies Tom M. W. Nye, Newcastle University Isaac Newton Institute, 17 December 2007

2 Multiple alternative phylogenies Phylogenetic analysis often produces many possible trees Variability in data / uncertainty in inferred trees: - ML bootstrap trees - Bayesian posterior samples Different trees for different genes How can we summarize / represent this information? How can we compare different alternative tree topologies?

3 Multiple alternative phylogenies Phylogenetic analysis often produces many possible trees Variability in data / uncertainty in inferred trees: - ML bootstrap trees - Bayesian posterior samples Different trees for different genes How can we summarize / represent this information? How can we compare different alternative tree topologies?

4 Representing collections of trees Existing approaches include... Consensus trees Consensus networks But also... Multi-dimensional scaling (Hillis et al, Systematic Biology 2005) Clustering (Stockham et al, Bioinformatics 2002)

5 Representing collections of trees Existing approaches include... Consensus trees Consensus networks But also... Multi-dimensional scaling (Hillis et al, Systematic Biology 2005) Clustering (Stockham et al, Bioinformatics 2002)

6 Toy example How are these trees related? A B A B A C A C E C C E E B D E D D D B

7 Toy example Relate trees by a tree of trees : A B A B A C E D C E D C E D B A B A C A C C D E B D E D B E

8 Why use a tree of trees? Tree-space does not have a tree-like structure Advantages: Cluster similar trees together edges represent gain/loss of topological features Conflicting histories show up as separate clades on the meta-tree: - different modes in a distribution - outliers Convenient form of visualisation

9 Meta-trees Definition Given a fixed set of trees T 1, T 2,..., T n all having leaf-set L, a meta-tree ˆT is an unrooted tree with n leaves such that every vertex ˆv in ˆT has associated to it a species tree Tˆv with leaf set L, and the leaf vertices of ˆT are associated to the trees T 1,..., T n Aim to find meta-trees with minimum score Tree score = sum of edge scores score (ê) = d(tˆv1, Tˆv2 ) for an edge ê between vertices ˆv 1, ˆv 2 Different metrics d(, ) are available

10 Splits A split is a bi-partition of the leaves L induced by cutting a branch: B A C E D AB CDE Trees consist of sets of compatible splits e.g. AB CDE and AC BDE cannot both be in a tree The majority consensus of T 1,..., T n is the tree consisting of all splits in strictly greater than n/2 trees The Robinson-Foulds metric is defined by: d(t a, T b ) = (number of splits in T a \ T b ) + (number of splits in T b \ T a )

11 Analogy with parsimony for DNA trees Suppose we use the Robinson-Foulds metric Represent tree topologies by strings of 0 s and 1 s for presence / absence of splits Could we apply DNA parsimony algorithms to these strings to build an optimal meta-tree? No because strings must always represent trees General problem: replaced set of characters {A, C, G, T } with the set of trees with leaf-set L Meta-tree construction equivalent to Steiner tree problem

12 Analogy with parsimony for DNA trees Suppose we use the Robinson-Foulds metric Represent tree topologies by strings of 0 s and 1 s for presence / absence of splits Could we apply DNA parsimony algorithms to these strings to build an optimal meta-tree? No because strings must always represent trees General problem: replaced set of characters {A, C, G, T } with the set of trees with leaf-set L Meta-tree construction equivalent to Steiner tree problem

13 Analogy with parsimony for DNA trees Suppose we use the Robinson-Foulds metric Represent tree topologies by strings of 0 s and 1 s for presence / absence of splits Could we apply DNA parsimony algorithms to these strings to build an optimal meta-tree? No because strings must always represent trees General problem: replaced set of characters {A, C, G, T } with the set of trees with leaf-set L Meta-tree construction equivalent to Steiner tree problem

14 Analogy with parsimony for DNA trees Suppose we use the Robinson-Foulds metric Represent tree topologies by strings of 0 s and 1 s for presence / absence of splits Could we apply DNA parsimony algorithms to these strings to build an optimal meta-tree? No because strings must always represent trees General problem: replaced set of characters {A, C, G, T } with the set of trees with leaf-set L Meta-tree construction equivalent to Steiner tree problem

15 Analogy with parsimony for DNA trees Suppose we use the Robinson-Foulds metric Represent tree topologies by strings of 0 s and 1 s for presence / absence of splits Could we apply DNA parsimony algorithms to these strings to build an optimal meta-tree? No because strings must always represent trees General problem: replaced set of characters {A, C, G, T } with the set of trees with leaf-set L Meta-tree construction equivalent to Steiner tree problem

16 Majority consensus and optimality Consider a meta-tree with the star topology, central node T 0 : Meta-tree score = T 1 T n T 2 T n 1 splits p T 3 ( number of edges with p at one end but not at the other end ) Score minimised by majority consensus: majority consensus is a median tree NB: optimisation performed for each split independently

17 Majority consensus and optimality Consider a meta-tree with the star topology, central node T 0 : Meta-tree score = T 1 T n T 2 T n 1 splits p T 3 ( number of edges with p at one end but not at the other end ) Score minimised by majority consensus: majority consensus is a median tree NB: optimisation performed for each split independently

18 Majority consensus and optimality Consider a meta-tree with the star topology, central node T 0 : Meta-tree score = T 1 T n T 2 T n 1 splits p T 3 ( number of edges with p at one end but not at the other end ) Score minimised by majority consensus: majority consensus is a median tree NB: optimisation performed for each split independently

19 A local optimality condition Consider internal vertex ˆv on an optimal meta-tree: when does Tˆv contain a split p? p ˆv p p p ˆv p p (a) Score=1 if p Tˆv Score=2 if p / Tˆv (b) Score=2 if p Tˆv Score=1 if p / Tˆv

20 The Meta-NJ algorithm B Z 1 A Z 2 Z k Z 3 A X B Z 1 Z 2 Z Z k X Z 1 Z 2 Z Z k ˆT r Start with star phylogeny ˆT AB r At r-th step pick two nodes A, B to agglomerate ˆT r+1 Form new nodes X and Z that are the majority consensus of their neighbours: X = maj{a, B, Z} and Z = maj{x, Z 1,..., Z k } Calculate score for the resulting configuration ˆT AB r Try every pair A, B and pick the pair with min score

21 The Meta-NJ algorithm B Z 1 A Z 2 Z k Z 3 A X B Z 1 Z 2 Z Z k X Z 1 Z 2 Z Z k ˆT r Start with star phylogeny ˆT AB r At r-th step pick two nodes A, B to agglomerate ˆT r+1 Form new nodes X and Z that are the majority consensus of their neighbours: X = maj{a, B, Z} and Z = maj{x, Z 1,..., Z k } Calculate score for the resulting configuration ˆT AB r Try every pair A, B and pick the pair with min score

22 The Meta-NJ algorithm B Z 1 A Z 2 Z k Z 3 A X B Z 1 Z 2 Z Z k X Z 1 Z 2 Z Z k ˆT r Start with star phylogeny ˆT AB r At r-th step pick two nodes A, B to agglomerate ˆT r+1 Form new nodes X and Z that are the majority consensus of their neighbours: X = maj{a, B, Z} and Z = maj{x, Z 1,..., Z k } Calculate score for the resulting configuration ˆT AB r Try every pair A, B and pick the pair with min score

23 Features of the algorithm Simultaneous equations for X and Z can be solved (almost) uniquely: splits are considered independently Each vertex on resulting meta-tree is majority consensus of its neighbours Algorithm greedily constructs meta-trees with the local optimality condition Zero length branches are sometimes produced leads to multifurcations Ties in score: pick one agglomeration at random

24 Yeast data set Rokas et al, Genome-scale approaches to resolving incongruence, Nature 2003: Genomes from 8 species of yeast ML trees constructed for 106 orthologs 23 different topologies obtained Consensus network (Holland et al, Mol. Biol. and Evolution 2004):

25 Yeast data set results Link to web Topology 11, YDR484W Topology 15, repeated 5 Topology 1, repeated 41 Topology 6, repeated 2 Topology 10, repeated 4 Topology 7, repeated 9 Topology 8, repeated 8 Topology 18, YPL210C Topology 2, YDR531W Topology 20, YKL120W Topology 9, YGL225W Topology 5, repeated 2 Topology 21, repeated 4 Topology 3, repeated 6 Topology 17, repeated 4 Topology 12, repeated 2 Topology 13, repeated 3 Topology 14, YGL192W Topology 22, YJR068W Topology 16, repeated 4 Topology 23, repeated 2 Topology 19, YKL034W Topology 4, repeated 2

26 Yeast data set results

27 Yeast data set results

28 Yeast data set results split absent split present

29 Yeast data set results split present split absent

30 Fish data set 10 orthologous genes in 14 species of ray-finned fish Li et al, BMC Evolutionary Biology, 2007

31 Fish bootstrap analysis To what extent is incongruence caused by: (a) lack of phylogenetic signal in each gene sequence, or (b) genuine evolutionary differences? Generate 10 boostrap replicates for each gene Replicates for each gene form clusters distinct evolutionary histories Replicates scattered lack of signal in each gene

32 Fish bootstrap results

33 Fish bootstrap results sreb2 ENC1 plag12 Glyt

34 Fish bootstrap results Clade absent Gain / loss of clade (Danio, Ictalur, Ochor, Semiotil) Clade present

35 Summary Attempt to represent a collection of trees by a tree-of-trees or meta-tree Finding an optimal meta-tree is computationally hard Meta-NJ algorithm: heuristic approach that builds meta-trees by maintaining a local optimality condition: each vertex is the majority consensus of its neighbours Examples show typical insights meta-trees can provide

36 Summary Attempt to represent a collection of trees by a tree-of-trees or meta-tree Finding an optimal meta-tree is computationally hard Meta-NJ algorithm: heuristic approach that builds meta-trees by maintaining a local optimality condition: each vertex is the majority consensus of its neighbours Examples show typical insights meta-trees can provide

37 Summary Attempt to represent a collection of trees by a tree-of-trees or meta-tree Finding an optimal meta-tree is computationally hard Meta-NJ algorithm: heuristic approach that builds meta-trees by maintaining a local optimality condition: each vertex is the majority consensus of its neighbours Examples show typical insights meta-trees can provide

38 Summary Attempt to represent a collection of trees by a tree-of-trees or meta-tree Finding an optimal meta-tree is computationally hard Meta-NJ algorithm: heuristic approach that builds meta-trees by maintaining a local optimality condition: each vertex is the majority consensus of its neighbours Examples show typical insights meta-trees can provide

39 Acknowledgements Thanks to Wally Gilks Antonis Rokas (yeast data set) Chenhong Li (fish data set) Web site Software is available on line at: ntmwn/phylo comparison/multiple.html