Olivier Gascuel Arbres formels et Arbre de la Vie Conférence ENS Cachan, septembre Arbres formels et Arbre de la Vie.

Transcription

1 Arbres formels et Arbre de la Vie Olivier Gascuel Centre National de la Recherche Scientifique LIRMM, Montpellier, France 10 permanent researchers 2 technical staff 3 postdocs, 10 PhDs 1

2 Phylogenetics Tree building algorithms and software (PhyML) Tree combinatorics (triplets, quartets, minimum evolution) Supertrees, gene trees/species trees Probabilistic modeling of substitutions Branch testing

3 Text algorithmics Alignment Searching for repeats, motifs, tags Comparative genomics New generation sequencing Machine learning and clustering Expression data analysis Hidden Markov Models Mining pathogens Plasmodium falciparum (malaria) HIV, Flu Biodiversity Arbres formels et Arbre de la Vie A bit of history and biology Definitions Numbers Topological distances Consensus Random models Algorithms to build trees 3

4 Charles Darwin Charles Darwin

5 Charles Darwin Charles Darwin

6 Arbre d Haeckel ~1875 Jakob Steiner

7 Watson et Crick 1962 Replication DNA duplicates Séquence Transcription RNA synthesis Translation Protein synthesis Protein folding Fonction 7

8 Replication DNA duplicates Mutations Transcription RNA synthesis Translation Protein synthesis Protein folding Sélection Theodosius Dobzhansky Nothing in Biology Makes Sense Except in the Light of Evolution 8

9 The Tree(s) of Life The Tree(s) of Life 9

10 HIV subtype A Eastern & Southern Europe The Tree(s) of Life 10

11 Arbres formels et Arbre de la Vie A bit of history and biology Definitions Numbers Topological distances Consensus Random models Algorithms to build trees A rooted phylogenetic tree Molecular clock assumption 11

12 A rooted phylogenetic tree Outgroup species OutG. An unrooted phylogenetic tree Inferred by most methods Topology Branch lengths Binary 12

13 An unrooted phylogenetic tree Inferred by some methods Topology Branch lengths Unresolved Graphs (vertices, leaves, labels, edges, weights) Adjacency table A B C Ngbr 1 A A B C C Ngbr 2 2 A 5 Ngbr 3 B C B A B C 13

14 Graphs (vertices, leaves, labels, edges, weights) A B C Ngbr1 A A B C C Lgth1 W1 W2 W3 W4 W5 W1 W3 W4 Ngbr2 2 A 5 Lgth2 W2 Wab W5 Ngbr3 B C B Lgth3 Computer representation Wab Wbc Wbc W1 A Wab B Wbc W4 W2 W3 C W5 Parentheses (expressions, newick format) + ((1 X 2) + ((3 X (4 5))) = -1 X X _

15 Parentheses (expressions and recursions) Expression Value ( LeftExp Operator RightExp ) compute (Exp) If Exp = Value, then return: Value Else Exp = (LeftExp Op RightExp) x = compute (LeftExp) y = compute (RightExp) Return: x Op y Parentheses (expressions and recursions) Value Expression Recursive procedure ( LeftExp Strongly Operator connected RightExp to ) trees Widely used E.g. with parsimony, ML compute (Exp) If Exp = Value, then return: Value Else Exp = (LeftExp Op RightExp) x = compute (LeftExp) y = compute (RightExp) Return: x Op y 15

16 Parentheses (expressions, newick format) (( )( ( ))) Parentheses (expressions, newick format) (( ) ( tax5)) ( ( ( Tax 5))) ( (( ) )). 16

17 Parentheses (expressions, newick format) ((:W1, :W2):Wab, :W3, ((:W4, :W5):Wbc); Output format W1 A Wab B Wbc W4 W2 W3 C W5 Bipartitions (binary characters, splits) Tree building and comparison Topology { {, } {,, } } {,, } {, } {} {,,, } {} L - {}, {} L - {} {} L - {}, {} L - {} 17

18 Bipartitions (binary characters, splits) A topology defines a bipartition set Given a bipartition set, it s easy to check that it defines a unique topology, using a local condition: A BandA' B' are tree compatible iff one of A A', A B', B A', B B' is empty A B A A topology is equal to its bipartition set Bipartitions (binary characters, splits) A phylogenetic tree defines a bipartition set Given a bipartition set, it s easy to check that it defines a unique topology, which may be unresolved { {, } {,, } {} L - {} } 18

19 Bipartitions (binary characters, splits) L = {eagle, duck, dog, mouse, kiwi) wings = {eagle, duck, kiwi} {mouse, dog} fly = {eagle, duck} {kiwi, mouse, dog} {eagle, duck} {mouse, dog} = eagle Maximizing character compatibility is hard! mouse duck kiwi dog Four Characters Suffice Katharina Huber, The Swedish University of Agricultural Sciences, and The Linnaeus Centre for Bioinformatics, Uppsala University, Sweden Vincent Moulton, The Linnaeus Centre for Bioinformatics, and Mike Steel, University of Canterbury, New Zealand 19

20 Phylogenetic trees Throughout this talk, we let X denote a finite set (of taxa). A tree T= (V,E ) together with a map f : X V is called a phylogenetic tree (on X) if f is a bijection of X onto the leaf set of T and all interior vertices of T have degree X = {1,2,,6} Partitions from characters We can associate a collection of partitions of X to any given collection of characters defined on X. 1 A T C G C T C 2 A T G C C G C 3 A G C T A G A 4 T C C A G T A

21 Convexity A partition P is called convex on a phylogenetic tree T if for distinct parts A, B of P the subtrees T A and T B (i.e. the minimal subtrees of T containing A, B, respectively) have no vertex in common. Consider the partition T T {35} Defining trees A collection of partitions P defines a phylogenetic tree T if P is convex on T, and T is the only phylogenetic tree with this property T 3 { , , } defines T 21

22 but if P = { , , } then P is convex on both of the following trees T 2 5 T' Question How many partitions suffice to define any given phylogenetic tree? 22

23 2 suffices for caterpillars Three partitions do not suffice

24 ..but at most five do! Theorem (Semple and Steel, 2002) Every phylogenetic tree can be defined by at most five partitions. Four does suffice! Theorem (Huber, Moulton, Steel, 2003) Any phylogenetic tree can be defined by at most four partitions. 24

25 Quartets (4-trees) Topology Quartets (4-trees) Topology { 12 34, 12 35, 12 45, 13 45, } Tree building and comparison 25

26 Quartets (4-trees) A complete quartet set: for every quadruple {i, j, k, l} we have one resolved 4-tree, eg ij kl A binary topology defines a complete quartet set It easy to check that a complete quartet set is tree compatible, and then defines a unique tree. A tree is equal to its quartet set. Quartets (4-trees) It s easy to infer 4-trees for all quadruples (eg ML) But: 4-trees are not reliable It is computationally hard to check that an incomplete quartet set is tree compatible It is computationally hard to select the maximum number of compatible 4-trees Heuristics needed! 26

27 Additive distances Tree with branch lengths Tree building (and comparison) Additive distances 17 i = 1, j = 2, k = 3, l = 4 11 A tree with lengths defines an additive distance. A distance is additive iff it satisfies the local 4-point condition: For every quadruple i, j, k, l, the two largests of ij kl ik jl il jk 2,, are equal

28 Additive distances A tree with lengths defines an additive distance. A distance is additive iff it satisfies the local 4-point condition, which is easily checked. An additive distance defines a unique tree, which is easily built. A tree is equal to its path length distance Additive distances Estimating evolutionary distances between all taxon pairs is easy (ML) But these distances are never 100% additive This induces hard optimization problems Numerous approaches and heuristics 28

29 Summary of Definitions Numerous definitions of trees (graphs, parentheses, bipartitions, characters, quartets, distances) Used to represent, compare and infer trees These definitions involve easy (polynomial) algorithms to recognize trees and change of representation But hard problems to infer trees from data Arbres formels et Arbre de la Vie A bit of history and biology Definitions Numbers Topological distances Consensus Random models Algorithms to build trees 29

30 Numbers Number of edges in a binary tree with n taxa: 2 tax: 1, 3 tax: 3, 4 tax: 5, 5 tax : 7 n tax: e(n) = e(n-1) + 2 = 2n -3 n Numbers Number of unrooted binary trees with n taxa: : 2 tax: 1, 3 tax: 1, 4 tax: 3, 5 tax : 15, 5 tax : tax: atoms in the universe n tax: t(n) = t(n - 1) x e(n -1) = (2n 5)(2n 7) hard optimization problems! 30

31 Numbers Number of rooted binary trees with n taxa: 2 tax: 1, 3 tax: 3, 4 tax: 15, 5 tax: 105 n tax: r(n) = t(n) x e(n) = t(n+1) = (2n 3) (2n 5) root root Arbres formels et Arbre de la Vie A bit of history and biology Definitions Numbers Topological distances Consensus Random models Algorithms to build trees 31

32 Topogical distance Measure the distance between two topologies with the same taxon set To analyze alternative trees (e.g. with parsimony) To compare reconstruction methods with simulated data To infer horizontal gene transfers Robinson & Foulds topogical distance Number of moves to transform one tree into the other Moves = edge contraction, unresolved node expansion Tree1 32

33 Robinson & Foulds topogical distance Number of moves to transform one tree into the other Moves = edge contraction, unresolved node expansion Tree2 Robinson & Foulds topogical distance Number of moves to transform one tree into the other Moves = edge contraction, unresolved node expansion contraction 33

34 Robinson & Foulds topogical distance Number of moves to transform one tree into the other Moves = edge contraction, unresolved node expansion expansion Tree1 R&F(Tree1, Tree2) = 2 Bipartition distance Number of bipartitions in one tree but not the other Bipartition and R&F distances are equal (easy calculation) Tree1: {12 345, } Tree2: {12 345, } 34

35 Quartet distance Number of 4-trees in one tree but not the other Easy to compute, more refined than R&F distance Tree1: {12 34, 12 35, 12 45, 13 45, 23 45} Tree2: {12 34, 12 35, 12 45, 14 35, 24 35} QD = 4 Horizontal gene transfers and SPR distance Species tree 35

36 Horizontal gene transfers and SPR distance Gene transfer Horizontal gene transfers and SPR distance Gene tree Subtree (, ) is Pruned and Regraft 1 HGT 1 SPR 36

37 Horizontal gene transfers and SPR distance SPR distance: minimum number of SPR moves required to tansform one tree into the other. Biologically relevant: number of HGTs Very hard to compute! Exercice Compute the RF and SPR distance between: 37

38 Arbres formels et Arbre de la Vie A bit of history and biology Definitions Numbers Topological distances Consensus Random models Algorithms to build trees Consensus We aim at estimating the consensus of a family of trees with the same taxon set. Most consensus problems are hard (think to elections ) But it s easy to define and compute the majority rule consensus tree 38

39 Majority rule consensus tree n trees with the same taxon set every tree t defines a bipartition set Bt = {b} collect B = { b seen in > n/2 sets Bt } any pair b, b is seen in at least one common set Bt therefore, b and b are tree compatible and B defines a unique tree! Exercice: Compute the majority consensus tree between 39