Introduction to Phylogenetic Networks

Transcription

1 Introduction to Phylogenetic Networks Daniel H. Huson GCB 2006, Tübingen,, September 19, Contents 1. Phylogenetic trees 2. Consensus networks and super networks 3. Hybridization and reticulate networks 4. Recombination networks 5. Other 2 1

2 Overview of Existing Concepts Phylogenetic networks Splits networks 1 Phylogenetic trees Reticulate networks Other types of phylogenetic networks Median networks from sequences Consensus (super) networks from trees Hybridization networks Special case: Galled trees Recombination networks 5 Augmented trees Split decomposition, Neighbor-net from distances Ancestor recombination graphs Any graph representing evolutionary data 3 Two Different Kinds of Networks Phylogenetic networks Splits networks 1 Phylogenetic trees Reticulate networks Other types of phylogenetic networks Median networks Consensus Hybridization Recombination Augmented (super) networks networks trees networks Split decomposition, Neighbor-net Ancestor recombination graphs Implicit networks Any graph representing evolutionary data 4 2

3 Two Different Kinds of Networks Phylogenetic networks Splits networks 1 Phylogenetic trees Reticulate networks Other types of phylogenetic networks Median networks Consensus Hybridization Recombination Augmented (super) networks networks trees networks Split decomposition, Neighbor-net Ancestor recombination graphs Explicit networks Any graph representing evolutionary data 5 Part I 1. Phylogenetic trees 2. Consensus networks and super networks 3. Hybridization and reticulate networks 4. Recombination networks 5. Other 6 3

4 Phylogenetic Trees The evolution of species is usually described by a phylogenetic tree Charles Darwin Ernst Haeckel,Tree of Life, Phylogenetic Trees Let X = {x 1,...,x n } denote a set of taxa. A phylogenetic tree T (or X-tree) ) is given by labeling the leaves of a tree by the set X: Cow Fin Whale Blue Whale Habor Seal Rat Mouse Chimp Human Gorilla Taxa + tree phylogenetic tree 8 4

5 Unrooted vs Rooted Trees Unrooted tree most popular methods produce unrooted trees Rooted tree, rooted using Chicken as outgroup biologically relevant, defines clades of related taxa 9 Branch Lengths Each branch e of a phylogenetic tree T may be scaled to represent r t, r the rate of evolution r time t along e: Blue Whale Mouse Fin Whale Rat Cow Chicken Seal Human Chimp Gorilla 0.01 root 10 5

6 A Simple Model of Evolution Sequences evolve along a pre-given tree T, called the evolutionary -, model - or the true tree Two types of events: mutations and speciation events 11 A Simple Model of Evolution TAA G C CG T ACT C CG A T AC G C C time AC C C A G C T Evolutionary tree AC G A C C C T Sequence of common ancestor Mutations along branches Speciation events at nodes 12 6

7 Tree Reconstruction Problem TAA G C CG T AC T C CG A T AC G C C Tree? Evolutionary tree 13 Tree of Life Based on 16S rrna (Doolittle, 2000) 14 7

8 Aligned Sequences A set of taxa X = {x 1,...,x n } may be given as an alignment of molecular sequences, e.g.: Human Chimp Gorilla Harbor Seal Cow Fin Whale Blue Whale Rat Mouse fqtpmviilqaimgsatlamtliiftiiiiltvhdtnttvptmitpmllt fqtpmiiifqaimgsatlaltliiftiiviltvhdtntavpttitpmllt lqtpmviifqaimgsatlamtliiftvimiltvhetnttvptmiapmllt fqlpmviifqaiiggatlalafitftiiifltvhdtdtstlimilsmilt fqtpmviifqaiiggatlalalitftiiifmtvhdtdtstltmilsmflt lqtfmviifqaimgettlalafitftiaifltvhdtdtsmlltilsmllt lqtfmviifqaimgettlvlaiitftiaifltvhdtdtstlltilsmllt fqismiiifqaimggatlvlatitfiilvfltvhdtdtstfitiissmat fqismiiifqaimggatlvlatitfiilifltvhdtdtstfitiissmit Usually obtained from some gene or locus that all taxa have in common. 17 Tree Reconstruction Problem Given an alignment that evolved along some evolutionary tree T, can we reconstruct the tree? Human fqtpmviilqaimgsatlamtliift Chimp fqtpmiiifqaimgsatlaltliift Gorilla lqtpmviifqaimgsatlamtliift Seal fqlpmviifqaiiggatlalafitft Cow fqtpmviifqaiiggatlalalitft Fin Whale lqtfmviifqaimgettlalafitft Blue Whale lqtfmviifqaimgettlvlaiitft Rat fqismiiifqaimggatlvlatitfi Mouse fqismiiifqaimggatlvlatitfi Chicken pqismiaffqaimggatlfaatitfi? Blue Whale Fin Whale Seal Cow Chicken root Challenges: 1. determine the unrooted topology of T, 2. estimate the branch lengths of T, and 3. infer the position of the root in T. Mouse Rat Chimp Human Gorilla 18 8

9 Tree Reconstruction Methods Sequence-based methods search for a tree that optimally explains the given sequence data: Maximum Parsimony [10], Maximum Likelihood [11], and Bayesian Inference [26]. Distance-based methods infer a distance matrix and construct a tree from it: UPGMA [45], Neighbor-Joining [43] and its variants Bio-NJ [15] and Weighbor [4]. Tree-based methods infer a tree from a set of trees Consensus tree (e.g. strict, majority, loose) Super tree (if trees are defined on overlapping subsets of taxa) 19 Software Selection of programs that build phylogenetic trees: PAUP* [49], a program for performing phylogenetic analysis using parsimony, maximum likelihood and other methods, Phylip [12], a package for phylogenetic inference, MrBayes [25], a program for Bayesian inference of trees, Mesquite [38], a modular system for evolutionary analysis, PAL [9], an object-oriented oriented programming library for molecular evolution and phylogenetics, and SplitsTree4 [27,28], an integrated program for estimating phylogenetic trees and networks. 26 9

10 Part II 1. Phylogenetic trees 2. Consensus networks and super networks 3. Hybridization and reticulate networks 4. Recombination networks 5. Other 27 Overview Will include additional evolutionary events that are not considered in simple tree models. Fundamental observation: gene trees differ. How to represent conflicting signals using a consensus network or super network. Some other methods that use a network to represent conflicting signals

11 Additional Evolutionary Events Models as discussed above represent the evolution of a single gene. When studying more than one gene simultaneously, one must also consider that: individual genes may be born, duplicated or lost. Moreover, biological mechanisms such as recombination, hybridization, or horizontal gene transfer may be involved. But even when the data evolved on a tree, networks can help to understand problems due to sampling or model-specification error 29 Gene Trees Can Differ Consider a model in which the sequence of a gene evolves via mutations, but we also allow gene duplication and loss: x 1 x 2 x 3 A A B x x x A A B x 1 x 2 x 3 Gene duplication A B G Gene Tree Species Tree 30 11

12 Gene Trees vs Species Trees Differing gene trees give rise to mosaic sequences Gene A Gene B Gene C Gene D 31 The Consensus of Different Gene Trees For a given set of species, we can build evolutionary trees based on different genes How to form a consensus of the trees? Consensus trees Consensus networks Consensus super networks 32 12

13 The Splits of a Tree Every edge of a tree defines a split of the taxon set X: x 6 x 1 x 4 x 8 e x 5 x 2 x 7 x 3 x 1,x 3,x 4,x 6,x 7 vs x 2,x 5,x 8 33 The Split Encoding of a Tree Tree T: c d a b Split encoding Σ(T): e 5 trivial splits: 2 non-trivial splits: 34 13

14 Compatibility Two splits A 1 B 1 and A 2 B 2 of X are compatible, if {A 1 A 2,A 1 B 2,B 1 A 2,B 1 A 2 } Two compatible splits: A 1 B 1 x 4 A 2 B 2 x 2 x 3 x 7 x 8 x 1 x 5 x 6 x 9 X 35 Compatibility Two splits A 1 B 1 and A 2 B 2 of X are compatible, if {A 1 A 2,A 1 B 2,B 1 A 2,B 1 A 2 } Two incompatible splits: A 2 A 1 B 1 x 4 x 5 B 2 x 6 x 2 x 3 x 1 x 7 X 36 14

15 Compatibility Theorem A set of splits Σ can be represented by a tree T, if and only if all pairs of splits are compatible. 37 Representing Incompatible Trees Consider the following two trees T 1 and T 2, for which the splits are incompatible: e e e c c p q c p d q d d b b b a a a T 1 T + 2 SN(Σ) The splits network SN(Σ) ) represents the incompatible set of splits Σ:= :=Σ (T 1 ) Σ (T 2 ), using bands of parallel edges for incompatible splits

16 Consensus of Trees Given trees T 1,,T k Define Σ(p):={S Σ all : {i: S Σ(TS i )} >pk} Strict consensus: Σ strict = Σ * (1/1) Majority consensus: Σ maj =Σ(1/2) In general, Σ(1/(d+1) 1/(d+1)) ) defines a set of consensus splits for d d 0 39 Six gene trees: Consensus of Trees Σ(1/2): majority consensus: splits contained in more than 50% of trees Σ(1/6): splits contained in more than one tree Σ(0): splits contained in at least one tree 40 16

17 Consensus Networks A consensus network [22] is obtained by computing the consensus splits Σ(1/(d+1) 1/(d+1)) ) for some value d 0. The parameter d determines the maximum dimensionality of the corresponding network: for d=1 the network will be 1-dimensional, 1 a tree, for d=2 the network may contain parallelograms, and in general it may contain cubes of dimension d. 41 Consensus of Partial Gene Trees For a given set of species, we may build evolutionary trees based on many different genes But: : not every species has every gene, or some sequences may be unavailable How to deal with partial trees,, i.e. trees that do not mention all species? Answer: Compute a super-network 42 17

18 Example of A Super Network (Plants) Partial trees for five plant genes Super network 43 Z-Closure Method [29] Idea: Extend partial splits. Z-rule: A 1 A 2 A 1, A 1 A 2 B 1 B 2 B 1 B 2 B 2 Repeatedly apply to completion. A 2 Return all full splits. B 1 A 1 B

19 Example Five fungal trees from (Pryor 2000) and (Pryor 2003) Trees: ITS (two trees) SSU (two trees) Gpd (one tree) Numbers of taxa differ: partial trees 45 Individual Gene Trees ITS00 46 taxa 46 19

20 Individual Gene Trees ITS03 40 taxa 47 Individual Gene Trees SSU00 29 taxa 48 20

21 Individual Gene Trees SSU03 40 taxa 49 Individual Gene Trees Gpd03 40 taxa 50 21

22 Gene Trees as Super Network Z-closure: a fast super-network method 51 Gene Trees as Super Network ITS00+ ITS

23 Gene Trees as Super Network ITS03+ SSU00 53 Gene Trees as Super Network ITS00+ ITS00+ SSU

24 Gene Trees as Super Network ITS00+ ITS03+ SSU03+ Gpd03 55 Gene Trees as Super Network ITS00+ ITS03+ SSU00+ SSU03+ Gpd

25 Sequence & Distance-Based Splits Networks So, incompatible splits arise naturally in the context of multiple trees. There also exist a number of methods that generate incompatible splits directly from characters (a multiple sequence alignment), or a distance matrix. 58 Sequences to Split Network I. Cassens et al.,, The phylography of dusky dolphins: a critical If characters examination have only of 2 network states and methods not too and conflicting: rooting procedures, interpret ret columns as splits and Molecular draw full Ecology splits (2003) network 12: (median 1792 network) 59 25

26 Split Decomposition The Split Decomposition method [2] computes a set of weighted X-splits X Σ decomp such that the sum of weights of all splits that separate two taxa x,y X approximates the distance D(x,y). It produces a tree,, whenever the distance matrix fits a tree, and else produces a network that displays different and incompatible signals. 60 Distances to Split Network Split Decomposition or Neighbor-Net Net produces network from distances 61 26

27 Neighbor-Net Net Split Decomposition is useful for visualizing conflicting signals in a data set. However, it is sensitive to noise and only has good resolution for small or clean data sets. The Neighbor-Net Net [5] method is a hybrid of Neighbor-Joining and Split Decomposition. It is applicable to data sets containing hundreds of taxa. However, it tends to produce spider-webs. 64 Neighbor-Net Net Splits network computed via Neighbor-net from distances between human mtdna sequences

28 Software SplitsTree4 [24] provides implementations of all methods described in this chapter, including a number of different algorithms for constructing networks from splits. SpectroNet [23] provides an algorithm for constructing a splits network (a special case, namely the median network) and some related methods 66 Implicit vs Explicit Networks Two fundamentally different types of phylogenetic networks: Implicit networks aim at displaying incompatible signals Example: split networks Explicit networks aim at providing an explicit model of reticulate evolution Example: hybridization and recombination networks 67 28

29 Part III 1. Phylogenetic trees 2. Consensus networks and super networks 3. Hybridization and reticulate networks 4. Recombination networks 5. Other 68 Overview Hybrid speciation. A simple model of evolution that incorporates gene trees and reticulation events. Reticulate networks and some approaches for inferring them from gene trees. Software

30 Hybridization Occurs when two organisms from different species interbreed and combine their chromosomes Copyright 2003 University of Illinois Copyright 2003 University of Illinois Copyright 2003 University of Illinois Water hemp Hybrid Pigs weed 70 Speciation by Hybridization 1 In allopolyploidization,, two different lineages produce a new species that has the complete nuclear genomes of both parental species [41]: 71 30

31 Speciation by Hybridization 1 Two parents X and Y each pass on their whole diploid genomes, with 2n 1 and 2n 2 chromosomes, respectively, to produce a polyploid offspring Z with (2n 1 +2n 2 ) chromosomes. Subsequently, it can happen that the genome reduces to half its size and is then a mosaic of genes from both ancestors. 72 Speciation by Hybridization 2 In diploid (or homoploid) hybrid speciation, each of the parents produces normal gametes (haploid) to produce a normal diploid hybrid [41]: 73 31

32 Speciation by Hybridization 2 Although diploid hybridization is more common, the ability of the hybrid to backcross with the parent species usually prevents that a new species will arise. Although less common, allopolyploidization is believed to produce more new species. Hybridization is usually restricted to plants, frogs and fish. 74 Horizontal Gene Transfer There are a number of known mechanisms by which bacteria can exchange genes Transformation Conjugation transduction

33 A Simple Model of Reticulate Evolution b 1 a h c b 3 P Q Tree for gene g 1 g 1 Ancestral genome 76 A Simple Model of Reticulate Evolution b 1 a h c b 3 P Q g 1 -tree is P -variant g

34 A Simple Model of Reticulate Evolution b 1 a h c b 3 g 1 -tree is P -variant 78 A Simple Model of Reticulate Evolution b 1 a h c b 3 P Q Tree for gene g 2 g

35 A Simple Model of Reticulate Evolution b 1 a h c b 3 P Q g2-tree is Q -variant g 2 80 A Simple Model of Reticulate Evolution b 1 a h c b 3 g2-tree is Q -variant 81 35

36 Reticulate Networks and Trees The evolutionary history associated with any given gene is a tree A network N with k reticulations gives rise to 2 k different gene trees b 1 a h c b 3 b 1 a h c b 3 P Q b 1 a h c b 3 N P-tree Q-tree 82 Reticulate Networks and Trees Note, however that the two choices P i and Q i can lead to the same tree topology: a h b c P i r i Q i Here, both induced trees are of the form: ((a,h),(b,c))

37 Rooted Reticulate Network Definition Let X be a set of taxa. A rooted reticulate network N on X is a connected, directed acyclic graph with: precisely one node of indegree 0, the root, all other nodes are tree nodes of indegree 1, or reticulation nodes of indegree 2, every edge is a tree edge joining two tree nodes, or a reticulation edge from a tree node to a reticulation node, and the set of leaves consists of tree nodes and is labeled by X. 84 Rooted Reticulate Network a b c d e f g h r 1 r 3 r 2 root 85 37

38 Reconstruction of Reticulate Networks Given a set of trees T ={T 1,...,T m }, want to determine the reticulate network N from which the trees were sampled with T = T(N). This form of the problem is not always solvable, e.g. if some of the 2 k possible trees are missing. Thus we consider the following: 86 Reconstruction of Reticulate Networks Most Parsimonious Network Problem: Given a set of trees T,, determine a reticulate network N such that T T(N) and N contains a minimum number of reticulation nodes. In fully generality, this is known to be a computationally hard problem [50]. We now discuss a special case that can be solved efficiently

39 Independent Reticulations Two reticulation nodes r i, r j in N are independent of each other, if they are not contained in any common simple cycle. r 1 r 2 r 3 Here, r 1 is independent of r 2 and r 3, whereas r 2 and r 3 are not independent of each other, as the highlighted cycle shows. 88 Galled Trees A reticulation that is independent of all others is also called a gall [18]. A network N in which all reticulations are galls is also called a galled tree [18] or gt-network [41]

40 SPR's and Independent Reticulations Observation [39]: If N contains only a single reticulation r, then it corresponds to a sub-tree prune and regraft operation: Reticulate network N: r SPR 90 SPR-Based Algorithm [39] Given two bifurcating trees, compute their SPR distance If the distance is 0, return a tree If the distance is 1, return a network Else, return fail This approach has been generalized to networks with multiple independent reticulations [41] 91 40

41 Challenge Unfortunately, on real data, such algorithms will often return fail". Please note: : All current approaches aim at solving a combinatorial puzzle: does there exist a network that induces the given set of trees? (below: does there exist a network that induces the given alignment of binary sequences?) One challenge is to produce useful output in the case of imperfect data. 92 Splits-Based Approach A new splits-based approach [29]: gene tree1 gene tree2 splits network of all splits reticulate network 93 41

42 Multiple Independent Reticulations Two reticulations four different gene trees all splits Reticulate network that induces all input trees 94 Overlapping Reticulations Current splits-based methods can resolve components in which the reticulation cycles overlap along a common path [31]: 95 42

43 Multiple and Overlapping Reticulations Input trees all splits Reticulate network that induces all input trees 96 Decomposition Theorem There exists a one-to to-one one correspondence between [17,29]: the connected components of the incompatibility graph, the netted regions" of the splits network and the tangles" of dependent reticulations of the reticulate network 98 43

44 Splits-Based Algorithm This leads to the following approach: Determine the set of all input splits Determine the connected components of the incompatibility graph or splits network Analyze each component C separately: If C can be explained by a reticulate network N(C), then locally replace C by N(C) 99 Application to Real Data New Zealand Ranunculus (buttercup) species JSA region in chloroplast ITS region in nuclear genome

45 Application to Real Data JSA ITS Split network representing both trees simultaneously Not explicit 101 Split network for ITS & JSA trees Filter splits [53] Hybridization network Two cases of hybridization Application to Real Data explicit

46 Details of Splits-Based Approach A reticulation corresponds to a subtree that attaches at two places: B 1 B 2 B 3 B 4 A X X C 103 Details of Splits-Based Approach A reticulation corresponds to a subtree that attaches at two places: B 1 B 2 B 3 B 4 A X X C

47 Detecting a Reticulation u 1 u 2 u 3 u 4 A X B 1 B 2 B 3 B 4 C d 1 d 2 d 3 d 4 A B 1 B 2 B 3 B 4 X C 105 Detecting a Reticulation The associated splits network B 1 B 2 B 3 B 4 A u 1 d 1 u 2 d 2 u 3 d 3 u 4 d 4 C d 1 u 4 d 2 u 3 d 3 u 2 d 4 u 1 X

48 Splits Network to Reticulate Network The associated splits network B 1 B 2 B 3 B 4 A C Delete all internal edges X 107 Splits Network to Reticulate Network The associated splits network& & the reticulate network B 1 B 2 B 3 B 4 A C Delete all internal edges X Note: Algorithm operates directly on the set of splits, not on the splits network

49 Details of Splits-Based Approach Multiple reticulations can overlap along a path: B 1 B 2 B 3 B 4 A X Y C 109 Details of Splits-Based Approach Multiple reticulations can overlap along a path: B 1 B 2 B 3 B 4 A X Y Y X C

50 Software SplitsTree4 [28] contains an implementation of the splits-based algorithm [31] that can handle overlapping reticulations. In [41] a program SPNet is described for galled trees, but it is not available for download. 111 Part IV 1. Phylogenetic trees 2. Consensus networks and super networks 3. Hybridization and reticulate networks 4. Recombination networks 5. Other

51 Overview Consider an alignment of binary sequences that have evolved under a model of mutation-,, speciation- and recombination events We will look at the problem of reconstructing the underlying reticulate network Software 113 Recombination (Sexual) recombination is studied in population genetics [24, 20,16, 46, 47, 48] and there ancestor recombination graphs (ARGs)) are used for statistical purposes. [41]

52 Chromosomal Recombination We will study the combinatorial aspects of chromosomal (meiotic) recombination and thus consider recombination networks rather than ARGs. Simplifying assumptions: all sequences have a common ancestor, and any position can mutate at most once. 115 Example of a Recombination Network r: b: a: c: d: ,11 Alignment A: a: b: r: c: d: o: , outgroup root

53 Recombination Network For an alignment A of binary sequences of length n, a recombination network R is a reticulate network N, together with [7]: a labeling of all nodes by binary sequences of length n, such that the leaves of R are labeled by A, a labeling of each tree edge e by the positions that mutate along e, and a labeling of each reticulation node r determining the recombination at r. 117 Non-Uniqueness of Mutations The placement of mutations on edges is not uniquely defined. Here, the mutation at position 5 can happen along two different edges: a: r: b: a: r: b: (a) 3, , (b) Current algorithms [18, 30] place such ambiguous mutations outside of the reticulation cycle, as in (a)

54 Recombination Network Tree-based approach [18, 17] for computing galled trees: Determine components of incompatibility graph For each component: Determine restricted dataset Determine whether removing one taxon produces a perfect phylogeny If so, arrange taxa in gall Return description of network 119 Recombination Network Splits-based approach [30] for computing overlapping networks: Determine a reticulate network as described earlier. Compute the labeling of nodes and edges

55 Computing a Labelling o: , ,11 a: b: r: c: Labelling of splits network is easy to compute d: o: ,5 a: Copy labelling to recombination network , b: c: ,11 r: d: Example 1, Data Fungus Fusarium, 37 strains reported in [52] Locus TRI101 known to undergone intragenic recombination

56 Example 1, Split Network Implicit network 123 Example 1, Recombination Network Explicit network

57 Example 2, Data Input: Restriction maps of the rdna cistron (length 10kb) of twelve species of mosquitoes using eight 6bp recognition restriction enzymes [35]: Aedes albopictus Aedes aegypti Aedes seatoi Aedes avopictus Aedes alcasidi Aedes katherinensis Aedes polynesiensis Aedes triseriatus Aedes atropalpus Aedes epactius Haemagogus equinus Armigeres subalbatus Culex pipiens Tripteroides bambusa Sabethes cyaneus Anopheles albimanus Example 2, Split Network This data set was analyzed using different tree- reconstruction methods with inconclusive results [35]. The associated splits network (or median network [3] in this context), with edges labeled by the corresponding mutations: Anopheles_albimanus root 10 Aedes_katherinensis Aedes_seatoi Aedes_alcasidi Aedes_flavopictus Aedes_albopictus 25 Aedes_polynesiensis 3,5,9,14-15,21, Tripteroides_bambusa ,23,26 Aedes_aegypti Sabethes_cyaneus Culex_pipiens Haemagogus_equinus Aedes_epactius Aedes_atropalpus Armigeres_subalbatus Aedes_triseriatus

58 Example 2, Subset Recombination scenarios based on the complete data set look unconvincing. However, removal of two taxa Aedes triseriatus and Armigeres subalbatus gives rise to a simpler splits network: Anopheles albimanus root Sabethes cyaneus 3,5,9,14-15,21,24 Haemagogus equinus Aedes epactius Aedes atropalpus Aedes aegypti 7 17,23,26 Aedes polynesiensis Culex pipiens Tripteroides bambusa Aedes katherinensis Aedes seatoi Aedes alcasidi Aedes albopictus Aedes flavopictus 127 Example 2, Recombination Network A possible recombination scenario is given by: Anopheles_albimanus root Sabethes_cyaneus Haemagogus_equinus Aedes_epactius 3,5,9,14-15,21, ,25 Aedes_atropalpus Aedes_aegypti 7 17,23,26 Culex_pipiens Tripteroides_bambusa Aedes_polynesiensis Aedes_katherinensis Aedes_seatoi Aedes_alcasidi Aedes_albopictus Aedes_flavopictus Here, Haemagogus equinus appears to arise by a single- crossover recombination, and a second such recombination leads to A.albopictus and A.avopictus

59 Recombination Network New branch-and and-bound approach [Lyngso,, Song and Hein, WABI 2005] Input: data and limit number of recombinations Branch: Starting from original data, consider all possible steps backward in time Bound: If recombinations used plus a lower bound on recombinations still needed exceeds the prescribed limit, do not pursue current configuration 129 Example 3 a b c d e f g h i haplotyped sites of the alcohol dehydrogenase locus from 11 chromosomes of D.melanogaster [Kreitman 1985] Recombination network with 7 events found using the branch-and and-bound method

60 Software Software for computing a recombination network from binary sequences: Software implementing the approach of Dan Gusfield et al. [18, 17] for constructing galled trees is available from: wwwcsif.cs.ucdavis.edu/~gusfield. SplitsTree4 [28] contains a method RecombinationNetwork for constructing galled trees and more general recombination networks [31, 30]. Beagle [Lunyso[ Lunyso,, Song and Hein, 2005] uses branch-and and-bound to compute network Part V 1. Phylogenetic trees 2. Consensus networks and super networks 3. Hybridization and reticulate networks 4. Recombination networks 5. Other

61 Augmented Trees: Reticulograms A reticulogram is a tree with additional short-cut edges. It is obtained from a distance matrix by first building a tree and then repeatedly adding new edges so as to optimize the least square fit of the graph distances to the matrix. Implemented in the program T-Rex [Makarenkov and Lengdre 2000, 2004] 133 Augmented Trees: Reticulograms Data: DNA sequences of 677 for honey bees. A.mellifer A.cerana A.dorsata A.andrenof A.koschev A.florea Reticulogram produced using T-RexT Bootstrap network [28] displays competing signals

62 Augmenting Species Trees by Gene Trees The goal here is to map a set of gene trees on to a given species tree, thus postulating a set of horizontal gene transfer events Implemented in the program lattrans [Hallet and Lagergren 2001] [Addario-Berry et al. 2003] 135 Augmenting Species Trees by Gene Trees A horizontal gene transfer scenario for the rbcl gene presented in [Hallet[ and Lagergren,, 2001]

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