From graph classes to phylogenetic networks Philippe Gambette
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1 40 années d'algorithmique de graphes 40 Years of Graphs and Algorithms 11/10/ Paris From graph classes to phylogenetic networks Philippe Gambette
2 Outline Discovering graph classes with Michel An introduction to phylogenetic networks Classes of phylogenetic networks The Tree Containment Problem Counting phylogenetic networks
3 Outline Discovering graph classes with Michel An introduction to phylogenetic networks Classes of phylogenetic networks The Tree Containment Problem Counting phylogenetic networks
4 Discovering graph classes with Michel Master thesis supervised by Michel Habib about 2-interval graphs Reading about graph classes, learning about graph decompositions
5 Discovering graph classes with Michel Master thesis supervised by Michel Habib about 2-interval graphs Reading about graph classes, learning about graph decompositions Obtaining results on balanced 2-interval graphs and number of arc-annotated sequences A balanced 2-interval graph G 5 1 G A balanced 2-interval realization of G a vertex balanced 2-intervals = union of 2 intervals of same length an edge non empty intersection of the two 2-intervals
6 Discovering graph classes with Michel Master thesis supervised by Michel Habib about 2-interval graphs Reading about graph classes, learning about graph decompositions Obtaining results on balanced 2-interval graphs and number of arc-annotated sequences Understanding what is important about the obtained results
7 Outline Discovering graph classes with Michel An introduction to phylogenetic networks Classes of phylogenetic networks The Tree Containment Problem Counting phylogenetic networks
8 Phylogenetic trees Phylogenetic tree of a set of species Classify them depending on common characters classification Describe their evolution Lamarck Histoire naturelle des animaux sans vertèbres (1815) 4
9 Phylogenetic trees Phylogenetic tree of a set of species Classify them depending on common characters Describe their evolution modelization animals green plants fungi Woese, Kandler, Wheelis Towards a natural system of organisms proposal for the domains Archaea, Bacteria, and Eucarya, Proceedings of the National Academy of Sciences, 87(12), (1990) 9
10 Phylogenetic trees... and networks Phylogenetic tree of a set of species species tree S A B tokogeny of individuals C A B C
11 Genetic material transfers Transfers of genetic material between coexisting species lateral gene transfer hybridization recombination A B C
12 Genetic material transfers Transfers of genetic material between coexisting species lateral gene transfer hybridization recombination S A B C A B C
13 Genetic material transfers Transfers of genetic material between coexisting species lateral gene transfer hybridization recombination species tree S A B C gene G1 A A B network N C A B C B C
14 Genetic material transfers Transfers of genetic material between coexisting species lateral gene transfer hybridization recombination species tree S gene G1 A A B C B C incompatible gene trees gene G2 A B network N C A B C A B C
15 Phylogenetic networks Phylogenetic network network representing evolution data abstract / implicit / data-display phylogenetic networks to classify, visualize data minimum spanning network split network median network SplitsTree Network explicit phylogenetic networks to model evolution TCS synthesis diagram galled network Simplistic level-2 network Dendroscope HorizStory
16 Abstract phylogenetic networks split networks Willems, M., Lord, E., Laforest, L., Labelle, G., Lapointe, F.-J., Di Sciullo, A. M. & Makarenkov, V. (2016). Using hybridization networks to retrace the evolution of indo-european languages. BMC Evolutionary Biology, 16(1), 180.
17 Abstract phylogenetic networks split networks a split Willems, M., Lord, E., Laforest, L., Labelle, G., Lapointe, F.-J., Di Sciullo, A. M. & Makarenkov, V. (2016). Using hybridization networks to retrace the evolution of indo-european languages. BMC Evolutionary Biology, 16(1), 180.
18 Abstract phylogenetic networks split networks another (compatible) split a split Willems, M., Lord, E., Laforest, L., Labelle, G., Lapointe, F.-J., Di Sciullo, A. M. & Makarenkov, V. (2016). Using hybridization networks to retrace the evolution of indo-european languages. BMC Evolutionary Biology, 16(1), 180.
19 Abstract phylogenetic networks split networks another (incompatible) split a split Willems, M., Lord, E., Laforest, L., Labelle, G., Lapointe, F.-J., Di Sciullo, A. M. & Makarenkov, V. (2016). Using hybridization networks to retrace the evolution of indo-european languages. BMC Evolutionary Biology, 16(1), 180.
20 Abstract phylogenetic networks split networks The boxes in split networks incompatible splits open them as much as possible in SplitsTree!
21 Abstract phylogenetic networks split networks The boxes in split networks incompatible splits open them as much as possible in SplitsTree! Philippe Gambette & Daniel H. Huson, Improved Layout of Phylogenetic Networks, IEEE/ACM TCBB 5(3), p
22 Explicit phylogenetic networks A gallery of explicit phylogenetic networks http//phylnet.univ-mlv.fr/recophync/networkdraw.php
23 Phylogenetic network reconstruction AATTGCAG ACCTGCAG GCTTGCCG ATTTGCAG TAGCCCAAAAT TAGACCAAT TAGACAAGAAT AAGACCAAAT TAGACAAGAAT ACTTGCAG TAGCACAAAAT ACCTGGTG TAAAAT G1 G2 {gene sequences} distance methods Bandelt & Dress Legendre & Makarenkov 2000 Bryant & Moulton Chan, Jansson, Lam & Yiu Willems, Tahiri & Makarenkov 2014 parsimony methods Hein Kececioglu & Gusfield Jin, Nakhleh, Snir, Tuller Park, Jin & Nakhleh Kannan & Wheeler, Kelk, Pardio, Scornavacca & van Iersel, 2017 likelihood methods network N Snir & Tuller Jin, Nakhleh, Snir, Tuller 2009 Velasco & Sober Meng & Kubatko 2009
24 Phylogenetic network reconstruction Problem methods are usually slow, especially with rapidly increasing sequence length AATTGCAG ACCTGCAG GCTTGCCG ATTTGCAG TAGCCCAAAAT TAGACCAAT TAGACAAGAAT AAGACCAAAT TAGACAAGAAT ACTTGCAG TAGCACAAAAT ACCTGGTG TAAAAT G1 G2 {gene sequences} distance methods Bandelt & Dress Legendre & Makarenkov 2000 Bryant & Moulton Chan, Jansson, Lam & Yiu Willems, Tahiri & Makarenkov 2014 parsimony methods Hein Kececioglu & Gusfield Jin, Nakhleh, Snir, Tuller Park, Jin & Nakhleh Kannan & Wheeler, Kelk, Pardio, Scornavacca & van Iersel, 2017 likelihood methods network N Snir & Tuller Jin, Nakhleh, Snir, Tuller 2009 Velasco & Sober Meng & Kubatko 2009
25 Phylogenetic network reconstruction from trees AATTGCAG ACCTGCAG GCTTGCCG ATTTGCAG TAGCCCAAAAT TAGACCAAT TAGACAAGAAT AAGACCAAAT TAGACAAGAAT ACTTGCAG TAGCACAAAAT ACCTGGTG TAAAAT G1 G2 {gene sequences} Reconstruction of a tree for each gene present in several species Guindon & Gascuel, SB, 2003 T1 {trees} T2 HOGENOM Database Dufayard, Duret, Penel, Gouy, Rechenmann & Perrière, BioInf, 2005 Tree reconciliation or consensus explicit network optimal super-network N - contains the input trees - has the smallest number of reticulations http//doua.prabi.fr/databases/hogenom/home.php?contents=hogenom4
26 Phylogenetic network reconstruction from trees The hybridization network problem given 2 trees, find the smallest network containing both of them with the minimum number of hybrid vertices Easy to find a network containing the two trees! T1 T2 a b c d a b c d
27 Phylogenetic network reconstruction from trees The hybridization network problem given 2 trees, find the smallest network containing both of them with the minimum number of hybrid vertices Easy to find a network containing the two trees! T1 N T2 a b c d a b c d add a root above the two trees, glue the leaves together a b c d But n hybrid vertices for trees with n leaves not optimal!
28 Phylogenetic network reconstruction from trees The hybridization network problem given 2 trees, find the smallest network containing both of them with the minimum number of hybrid vertices NP-hard to minimize the number of hybrid vertices Bordewich & Semple (2007) Discrete Appl Math
29 Phylogenetic network reconstruction from trees AATTGCAG ACCTGCAG GCTTGCCG ATTTGCAG TAGCCCAAAAT TAGACCAAT TAGACAAGAAT AAGACCAAAT TAGACAAGAAT ACTTGCAG TAGCACAAAAT ACCTGGTG TAAAAT G1 {gene sequences} G2 T1 {trees} T2 triplet a ce {triplets} explicit network a b c d e optimal super-network N - contains the input triplets - has the smallest number of reticulations
30 Phylogenetic network reconstruction from trees AATTGCAG ACCTGCAG GCTTGCCG ATTTGCAG TAGCCCAAAAT TAGACCAAT TAGACAAGAAT AAGACCAAAT TAGACAAGAAT ACTTGCAG TAGCACAAAAT ACCTGGTG TAAAAT G1 {gene sequences} G2 T1 {trees} T2 {triplets} explicit network optimal super-network N - contains the input triplets - has the smallest number of reticulations still NP-hard
31 Phylogenetic network reconstruction from trees AATTGCAG ACCTGCAG GCTTGCCG ATTTGCAG TAGCCCAAAAT TAGACCAAT TAGACAAGAAT AAGACCAAAT TAGACAAGAAT ACTTGCAG TAGCACAAAAT ACCTGGTG TAAAAT G1 {gene sequences} G2 T1 {trees} T2 {triplets} ( dense set) explicit network super-network N - contains the input triplets - has level at most k (i.e. close to a tree) polynomial time algorithm for fixed k Habib & To, CPM 2009, JBCB 2012
32 Outline Discovering graph classes with Michel An introduction to phylogenetic networks Classes of phylogenetic networks The Tree Containment Problem Counting phylogenetic networks
33 Classes of phylogenetic networks level-k networks level = maximum number of reticulations per blob (=bridgeless component of the underlying undirected graph). a b c d e f g h i j k level-2 network level-1 network ( galled tree ) a b c d e f g h i j k
34 Classes of phylogenetic networks ISIPhyNC Information System on Inclusions of Phylogenetic Network Classes joint work with Maxime Morgado and Narges Tavassoli http//phylnet.univ-mlv.fr/isiphync/
35 Classes of phylogenetic networks inclusions joint work with Maxime Morgado and Narges Tavassoli
36 Classes of phylogenetic networks inclusions joint work with Maxime Morgado and Narges Tavassoli
37 Classes of phylogenetic networks inclusions joint work with Maxime Morgado and Narges Tavassoli http//phylnet.univ-mlv.fr/isiphync/network.php?id=4
38 Classes of phylogenetic networks inclusions level = maximum number of reticulation vertices among all bridgeless components in the network cluster = set of leaves below a vertex joint work with Maxime Morgado and Narges Tavassoli http//phylnet.univ-mlv.fr/isiphync/network.php?id=4
39 Classes of phylogenetic networks inclusions level = maximum number of reticulation vertices among all bridgeless components in the network cluster = set of leaves below a vertex joint work with Maxime Morgado and Narges Tavassoli http//phylnet.univ-mlv.fr/isiphync/network.php?id=4
40 Classes of phylogenetic networks new inclusions joint work with Maxime Morgado and Narges Tavassoli
41 Classes of phylogenetic networks problems joint work with Maxime Morgado and Narges Tavassoli
42 Classes of phylogenetic networks problems Problem easy to solve on class A easy to solve on subclass B Hard to solve on class B hard to solve on superclass A (similar to ISGCI) joint work with Maxime Morgado and Narges Tavassoli
43 Outline Discovering graph classes with Michel Classes of phylogenetic networks An introduction to phylogenetic networks The Tree Containment Problem Counting phylogenetic networks
44 Phylogenetic network reconstruction from trees Recall the hybridization network problem given 2 trees, find the smallest network containing both of them with the minimum number of hybrid vertices Even checking a solution is hard!
45 The Tree Containment Problem (T.C.P.) Input A binary phylogenetic network N and a tree T over the same set of taxa. Question Does N display T?
46 The Tree Containment Problem (T.C.P.) Input A binary phylogenetic network N and a tree T over the same set of taxa. Question Does N display T? Can we remove one incoming arc, for each vertex with >1 parent in N, such that the obtained tree is equivalent to T? T N a b c d a b c d
47 The Tree Containment Problem (T.C.P.) Input A binary phylogenetic network N and a tree T over the same set of taxa. Question Does N display T? Can we remove one incoming arc, for each vertex with >1 parent in N, such that the obtained tree is equivalent to T? T N a b c d a b c d
48 The Tree Containment Problem (T.C.P.) Input A binary phylogenetic network N and a tree T over the same set of taxa. Question Does N display T? NP-complete in general (Kanj, Nakhleh, Than & Xia, 2008) NP-complete for tree-sibling, time-consistent, regular networks (Iersel, Semple & Steel, 2010) Polynomial-time solvable for normal networks, for binary tree-child networks, and for level-k networks (Iersel, Semple & Steel, 2010)
49 Classes of phylogenetic networks and the T.C.P. State of the art in 2011
50 Classes of phylogenetic networks and the T.C.P. State of the art in 2011?
51 Reticulation-visible and nearly-stable networks A vertex u is stable if there exists a leaf l such that all paths from the root to l go through u. A phylogenetic network is reticulation-visible if every reticulation vertex is stable. A phylogenetic network is nearly-stable if for each vertex, either it is stable or its parents are. Gambette, Gunawan, Labarre, Vialette & Zhang, RECOMB 2015
52 Classes of phylogenetic networks and the T.C.P. Recent results (since 2015)
53 Classes of phylogenetic networks and the T.C.P. Recent results (since 2015) Spread-k networks leaves can be ordered such that the set of leaves below each vertex is a union of at most k intervals.
54 Fête de la Science at UPEM (science festival) Photos Campus numérique de l'upem
55 Outline Discovering graph classes with Michel Classes of phylogenetic networks An introduction to phylogenetic networks The Tree Containment Problem Counting phylogenetic networks
56 Counting labeled unrooted level-1 networks Unrooted level-1 networks explicit formula for n leaves, c cycles, m edges involved in the cycles. Semple & Steel, TCBB, 2006
57 Counting labeled unrooted level-1 networks Unrooted level-1 networks explicit formula for n leaves, c cycles, m edges involved in the cycles. Pointing + bijection Bijection between labeled unrooted level-1 networks with n+1 leaves and labeled pointed level-1 networks with n leaves. Semple & Steel, TCBB, 2006
58 Counting labeled unrooted level-1 networks Unrooted level-1 networks explicit formula for n leaves, c cycles, m edges involved in the cycles. Pointing + bijection Bijection between labeled unrooted level-1 networks with n+1 leaves and labeled pointed level-1 networks with n leaves. Recursive decomposition of pointed level-1 networks with n leaves or or Exponential generating function G = z + 1 G² + 1 G² 2 2 (1-G) Semple & Steel, TCBB, 2006
59 Counting labeled unrooted level-1 networks Unrooted level-1 networks explicit formula for n leaves, c cycles, m edges involved in the cycles. Pointing + bijection Bijection between labeled unrooted level-1 networks with n+1 leaves and labeled pointed level-1 networks with n leaves. Recursive decomposition of pointed level-1 networks with n leaves or or Exponential generating function G = z + 1 G² + 1 G² 2 2 (1-G) Seq 2, any direction Semple & Steel, TCBB, 2006
60 Counting labeled unrooted level-1 networks Exponential generating function G = z + 1 G² + 1 G² 2 2 (1-G) Using the Singular Inversion Theorem (Theorem VI.6 of ) gn (1.8904)n nn-1 Joint work with Mathilde Bouvel and Marefatollah Mansouri
61 Counting labeled unrooted level-1 networks Exponential generating function G = z + 1 G² + 1 G² 2 2 (1-G) Using the Singular Inversion Theorem (Theorem VI.6 of ) gn (1.8904)n nn-1 Proof We write G = z φ(g), with φ(z) = Then gn n! φ(τ) 2φ''(τ) ρ n π n3 1 1 ½ z (1+1/(1-z)), with ρ = τ / φ(τ) and τ is the solution of φ(z)-zφ'(z)=0 Joint work with Mathilde Bouvel and Marefatollah Mansouri
62 Counting labeled unrooted level-2 networks Recursive decomposition of pointed level-2 networks with n leaves z 1 U² 2 1 U² 2 1-U Seq 1, any direction Seq 2, any direction or... Seq 1 Seq 2 simple edge sym edge symmetry horizontal symmetry with sym new orientation for lower edges
63 Counting labeled unrooted level-2 networks Joint work with Mathilde Bouvel and Marefatollah Mansouri
64 Counting labeled unrooted level-2 networks Recursive decomposition of pointed level-2 networks with n leaves Rewrite Joint work with Mathilde Bouvel and Marefatollah Mansouri
65 Counting labeled unrooted level-2 networks Recursive decomposition of pointed level-2 networks with n leaves Rewrite Lagrange inversion Taylor expansions of φn(z) number of leaves unrooted level Joint work with Mathilde Bouvel and Marefatollah Mansouri
66 Counting labeled unrooted level-2 networks Recursive decomposition of pointed level-2 networks with n leaves Rewrite Lagrange inversion Taylor expansions + Newton formula Joint work with Mathilde Bouvel and Marefatollah Mansouri
67 Counting labeled level-k networks Unrooted level-1 networks explicit formula for n leaves, c cycles, m edges involved in the cycles Semple & Steel, TCBB, asymptotic evaluation for n leaves (1.890)n nn-1 Rooted level-1 networks Explicit formula for n leaves, c cycles, m edges across cycles + asymptotic evaluation for n leaves (2.943)n nn-1 Unrooted level-2 networks Explicit formula for n leaves (n-1)! Σ 0 s q p k i n-1 j=n-1-i-k-p-q-s 0 i 0 number of leaves unrooted level-1 rooted level-1 unrooted level n+i-1 4i+j-1 i k p q -3 s 9 i -23 k p -10 q i j k p q s (-1) Joint work with Mathilde Bouvel and Marefatollah Mansouri
68 A message from Mathilde...
69 The academic phylogenetic network of Michel & friends A level-2 network... so far! Add yourself to https//tinyurl.com/michelacademicgenealogy
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