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1 Daniel H. Huson Stockholm, May 28, 2005

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 1. Phylogenetic trees 2. Consensus networks and super networks 3. Hybridization and reticulate networks 4. Recombination networks

4 Splits networks Phylogenetic trees Reticulate networks Other types of phylogenetic networks Median networks Consensus (super) networks Hybridization networks Special case: Galled trees Recombination networks Augmented trees Split decomposition, Neighbor-net Ancestor recombination graphs Any graph representing evolutionary data

5 Splits networks 1 Phylogenetic trees Reticulate networks Other types of phylogenetic networks Median networks Consensus (super) networks Hybridization networks Special case: Galled trees Recombination networks Augmented trees Split decomposition, Neighbor-net Ancestor recombination graphs Any graph representing evolutionary data

6 1. Phylogenetic trees 2. Consensus networks and super networks 3. Hybridization and reticulate networks 4. Recombination networks

7 Ernst Haeckel, Tree of Life 1866

8 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 F i n Wha l e B l ue Wha l e Habo r Se a l Ra t Mous e Ch i mp Human Go r i l l a Taxa + tree phylogenetic tree

9 Unrooted tree mathematically and algorithmically easier to deal with Rooted tree, rooted using Chicken as outgroup biologically relevant, defines clades of related taxa

10 Each branch e of a phylogenetic tree T may be scaled to represent r t, the rate of evolution r time t along e: 0.01 Chicken Seal Blue Whale Mouse Fin Whale Seal Cow Rat Chimp Human Gorilla root

11 Sequences evolve along a pre-given tree T, called the evolutionary -, model - or the true tree Two types of events: mutations and speciation events

12 time Evolutionary tree Sequence of common ancestor Mutations along branches Speciation events at nodes

13 Tree? Evolutionary tree

14 (Doolittle, 2000)

15 1. Phylogenetic trees 2. Consensus networks and super networks 3. Hybridization and reticulate networks 4. Recombination networks

16 Also allow gene duplication and loss: x 1 x 2 x 3 x x x A A B x 1 x 2 x 3 Gene duplication Gene Tree Species Tree

17 Differing gene trees give rise to mosaic sequences Gene A Gene B Gene C Gene D

18 For a given set of species, different genes lead to different trees How to form a consensus of the trees? Consensus trees Consensus networks Consensus super networks

19 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

20 Tree T: Split encoding (T): 5 trivial splits: 2 non-trivial splits:

21 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

22 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 splits: A 1 B 1 x 4 x 5 A 2 B 2 x 6 x 2 x 1 x 7 x 3 X

23 Consider the following two trees T 1 and T 2, for which the splits are incompatible: T 1 + T 2 SN( ) The splits network SN( ) represents the incompatible set of splits := (T 1 ) (T 2 ), using bands of parallel edges for incompatible splits.

24 Given trees T 1,,T k Define (p):={s all : {i: S (T i )} >pk} Strict consensus: strict = * (1/1) Majority consensus: maj = (1/2) In general, ( 1/(d+1)) ) defines a set of consensus splits for d 0

25 Six gene trees: 1/2): majority consensus: splits contained in more than 50% of trees 1/6): splits contained in more than one tree 0): splits contained in at least one tree

26 Partial trees for five plant genes Joint work with Kim McBreen and Pete Lockhart Super network

27 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 B 1 Return all full splits. A 1 [Huson, Dezulian, Kloepper and Steel, 2004] B 2

28 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

30 ITS00 46 taxa

31 ITS03 40 taxa

32 SSU00 29 taxa

33 SSU03 40 taxa

34 Gpd03 40 taxa

35 Z-closure: a fast super-network method

36 ITS00+ ITS03

37 ITS03+ SSU00

38 ITS00+ ITS00+ SSU03

39 ITS00+ ITS03+ SSU03+ Gpd03

40 ITS00+ ITS03+ SSU00+ SSU03+ Gpd03

41 1. Phylogenetic trees 2. Consensus networks and super networks 3. Hybridization and reticulate networks 4. Recombination networks

42 Occurs when two organisms from different species interbreed and combine their genomes Copyright 2003 University of Illinois Copyright 2003 University of Illinois Copyright 2003 University of Illinois Water hemp Hybrid Pigs weed

43 There are a number of known mechanisms by which bacteria can exchange genes Transformation Conjugation transduction

44 b 1 a h c b 3 P Q Tree for gene g 1 g 1 Ancestral genome

45 b 1 a c b 3 P Q g 1 -tree is P -variant g 1

46 b 1 a c b 3 g 1 -tree is P -variant

47 b 1 a h c b 3 P Q Tree for gene g 2 g 2

48 b 1 a h c b 3 P Q g2-tree is Q -variant g 2

49 b 1 a h c b 3 g2-tree is Q -variant

50 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 P-tree N Q-tree

51 Note, however that the two choices P i and Q i can lead to the same tree topology: Here, both induced trees are of the form: ((a,h),(b,c)).

52 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.

53 a b c d e f g h r 1 r 3 r 2 root

54 Given a set of trees ={T 1,...,T m }, want to determine the reticulate network N from which the trees were sampled with = = 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:

55 Given a set of trees, determine a reticulate network N such that T(N) and N contains a minimum number of reticulation nodes. In fully generality, this is known to be a computationally hard problem [Wang et al 2001].

56 Reticulation nodes r i, r j N are independent, if they are not contained in a common cycle: r 1 r 2 r 3 Independent reticulations also called galls and a network only containing galls is also called a galled tree [Gusfield et al. 2003]

57 Observation [Maddison 1997]: : If N contains only one reticulation r, then it corresponds to a sub-tree prune and regraft operation: Reticulate network N: r SPR

58 Given two bifurcating trees, compute their SPR distance: If = 0, return the tree If = 1, return the reticulate network Else, return fail Generalized to networks with multiple independent reticulations [Nakhleh et al 2004]

59 A new splits-based approach [Huson, Kloepper, Lockhart and Steel 2005]: gene tree1 gene tree2 splits network of all splits reticulate network

60 Two reticulations four different gene trees all splits Reticulate network that induces all input trees

61 Input trees all splits Reticulate network that induces all input trees

62 Each incompatibility component can be considered independently: 1. component 2. component [Gusfield & Bansal, 2005] [Huson, Kloepper, Lockhart & Steel, 2005]

63 Consider a component:

64 Find decomposition R B as a set of reticulate taxa and backbone taxa

65 Necessary condition: splits restricted to B must correspond to a tree

66 Consider all possible choices for R of size [Gusfield et al., 2003, 2004]: R={t 7 } not a tree, R not good

67 Consider all possible choices for R of size [Gusfield et al., 2003, 2004]: R={c} not a tree, R not good

68 Consider all possible choices for R of size [Gusfield et al., 2003, 2004]: R={t 5 } not a tree, R not good

69 Consider all possible choices for R of size : R={t 6,t 7 } not a tree, R not good

70 Consider all possible choices for R of size : R={b,c} is a,, R is a candidate

71 For R={b,c}, check that the reticulation cycles overlap correctly along a path:

72 Modify splits network to represent reticulations:

73 Input: Set of trees, not necessarily bifurcating, can be partial trees Parameter k Output: All reticulate networks N for which every incompatibility component can be explained by at most k overlapping reticulations Complexity: polynomial for fixed k

74 New Zealand Ranunculus (buttercup) species Nuclear ITS region Chloroplast J SA region

nivicola removal of five confusing may branches be a hybrid in the input splits and one taxon leads of trees the and

75 New Zealand Ranunculus (buttercup) species four splits here This Current splits algorithms network are However, interactive suggests sensitive that to false R.nivicola removal of five confusing may branches be a hybrid in the input splits and one taxon leads of trees the and evolutionary here initially to the detection of an lineages no reticulation the is left- and appropriate reticulation. right-hand detected. hand sides. Splits network for both genes Reticulate network

76 1. Phylogenetic trees 2. Consensus networks and super networks 3. Hybridization and reticulate networks 4. Recombination networks

77 Recombination is studied in population genetics [24, 20,16, 46, 47, 48] and there ancestor recombination graphs (ARGs) are used for statistical purposes.

78 We will study the combinatorial aspects of chromosomal (meiotic) recombination and will consider recombination networks rather than ARGs. Simplifying assumptions: all sequences have a common ancestor, and any position can mutate at most once.

79 : : : : : : b : r : c : d : o : outgroup root

80 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)

81 For an alignment A of binary sequences of length n, a recombination network R is a reticulation 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.

82 Note: 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 : , , Current algorithms [18, 30] place such ambiguous mutations outside of the reticulation cycle, as in (a)

83 Tree-based approach [Gusfield for computing galled trees: For each component: [Gusfield et al. 2003] If so, arrange taxa in gall Return description of network

84 Splits-based approach for computing overlapping networks: approach [Huson & Kloepper 2005] Determine a reticulate network as described above. Compute the labeling of nodes and edges.

85 o: , ,11 a: b: c: r: Labelling of splits network is easy to compute d: o: ,5 a: Copy labelling to recombination network , ,11 b: c: r: d:

86 Input: Presence (0) or absence (1) of a given restriction site in a 3.2kb region of variable chloroplast DNA in Pistacia [Parfitt & Badenes 1997] [Parfitt & Badenes 1997]: P. l en t i s cus P. we i nmann nn i f o l i a P. ch i nens i s P. i n t ege r r i ma P. t e r eb i n t hus P. a t l an t i c a P. me x i c ana a P. t e x ana P. kh i n j uk P. v e r a Sch i nus mo l l e

87 Load this data in to SplitsTree4 and select to obtain:

88 Combinatorically, this can be explained using only one single-crossover recombination: However: recombination of chloroplast is unlikely

89 Input: Restriction maps of the rdna cistron (length 10kb) of twelve species of mosquitoes using eight 6bp recognition restriction enzymes [Kumar et al,, 1998]: Aede s a l bop i c t us Aede s a egyp t i Aede s s e a t o i Aede s a vop i c t us Aede s a l c a s i d i Aede s k a t he r i nens i s Aede s po l yne s i ens i s Aede s t r i s e r i a t us Aede s a t r opa l pus Aede s epa c t i us Ha emagogus equ i nus A r m i ge r e s suba l ba t us Cu l e x p i p i ens T r i p t e r o i de s bambus a Sabe t he s c y aneus Anophe l e s a l b i manus s

90 This data set was analyzed using different tree- reconstruction methods with inconclusive results. The associated splits network (or median network [Bandelt in this context), with edges labeled by the corresponding mutations: Anopheles_albimanus [Bandelt et al,, 1995] 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 13 Culex_pipiens 19 Haemagogus_equinus Aedes_triseriatus Aedes_epactius Aedes_atropalpus Armigeres_subalbatus

91 Recombination scenarios based on the complete data set look unconvincing. However, trial-and-error removal of two taxa Aedes triseriatus and Armigeres subalbatus gives rise to a simpler splits network: Anopheles albimanus root 3,5,9,14-15,21,24 11 Sabethes cyaneus Aedes katherinensis Aedes seatoi Aedes alcasidi 13 Haemagogus equinus Aedes polynesiensis Aedes aegypti Aedes albopictus Aedes flavopictus Aedes epactius Aedes atropalpus 17,23,26 Culex pipiens Tripteroides bambusa

92 A possible recombination scenario is given by: Anopheles_albimanus root 3,5,9,14-15,21,24 Sabethes_cyaneus ,25 2 Haemagogus_equinus 7 19 Aedes_aegypti 17,23,26 Aedes_epactius Aedes_atropalpus 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.

93 19 restriction endonucleases were used to analyze patterns of cleavage site variation in the mtdna of Zonotrichia. 7 taxa, 122 characters [Zink et al,, 1991] ' Zono t r i c h i a_ qu e r u l a ' ' Z. _ a t r i c a p i l l a ' ' Z. _ l e u c oph r y s ' ' Z. _ a l b i c o l l i s ' ' Z. _c apens i s - - Bo l i v i a ' ' Z. _c apens i s - - Co s t a_r i c a ' ' J. _hy ema l i s ' Recombination of mtdna???

94 The unrooted splits network for a dataset of restriction sites in the mtdna of Zonotrichia:

95 The rooted splits network for the same data set, but suppressing all splits that are only supported by one site in the data:

96 Possible recombination scenario involving two non-independent reticulations:

97 Incompatible signals in gene trees can be usefully displayed using splits networks A reticulate network may be extracted by combinatorial analysis of individual components Implementations of many tree and network methods are available in

Splits and Phylogenetic Networks. Daniel H. Huson

Splits and Phylogenetic Networks Daniel H. Huson aris, June 21, 2005 1 2 Contents 1. Phylogenetic trees 2. Splits networks 3. Consensus networks 4. Hybridization and reticulate networks 5. Recombination