Phylogenetic networks: overview, subclasses and counting problems Philippe Gambette
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1 ANR-FWF-MOST meeting Wien Phylogenetic networks: overview, subclasses and counting problems Philippe Gambette
2 Outline An introduction to phylogenetic networks Classes of phylogenetic networks Counting problems on phylogenetic networks
3 Outline An introduction to phylogenetic networks Classes of phylogenetic networks Counting problems on phylogenetic networks
4 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
5 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) 5
6 Phylogenetic trees... and networks Phylogenetic tree of a set of species species tree S A B tokogeny of individuals C A B C
7 Genetic material transfers Transfers of genetic material between coexisting species: lateral gene transfer hybridization recombination A B C
8 Genetic material transfers Transfers of genetic material between coexisting species: lateral gene transfer hybridization recombination S A B C A B C
9 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
10 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
11 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
12 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.
13 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.
14 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.
15 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.
16 Abstract phylogenetic networks: split networks The boxes in split networks: incompatible splits open them as much as possible in SplitsTree!
17 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
18 Who is who in Phylogenetic Networks? Based on BibAdmin by Sergiu Chelcea + tag clouds, date histograms, journal lists, keyword definitions, co-author graphs (work of Tushar Agarwal)
19 Who is who in Phylogenetic Networks?
20 Who is who in Phylogenetic Networks?
21 Who is who in Phylogenetic Networks?
22 Keywords about phylogenetic networks
23 Keywords about phylogenetic networks
24 Who is who in Phylogenetic Networks? Software! from rooted trees Phylonet T-Rex from sequences Dendroscope from distances SplitsTree Based on BibAdmin by Sergiu Chelcea + tag clouds, date histograms, journal lists, keyword definitions, co-author graphs (work of Tushar Agarwal)
25 Who is who in Phylogenetic Networks? Keywords!
26 Who is who in Phylogenetic Networks? Keywords! input software
27 Who is who in Phylogenetic Networks? Keywords! input software problems
28 Who is who in Phylogenetic Networks? Keywords! input software problems algorithmic results
29 Explicit phylogenetic networks A gallery of explicit phylogenetic networks :
30 Phylogenetic network reconstruction espèce espèce espèce espèce espèce espèce espèce : : : : : : : 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
31 Phylogenetic network reconstruction Problem: methods are usually slow, especially with rapidly increasing sequence length. espèce espèce espèce espèce espèce espèce espèce : : : : : : : 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
32 Phylogenetic network reconstruction from trees espèce espèce espèce espèce espèce espèce espèce : : : : : : : 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
33 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
34 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!
35 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
36 Phylogenetic network reconstruction from trees espèce espèce espèce espèce espèce espèce espèce : : : : : : : 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
37 Phylogenetic network reconstruction from trees espèce espèce espèce espèce espèce espèce espèce : : : : : : : 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
38 Phylogenetic network reconstruction from trees espèce espèce espèce espèce espèce espèce espèce : : : : : : : 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
39 Outline An introduction to phylogenetic networks Classes of phylogenetic networks Counting problems on phylogenetic networks
40 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
41 Classes of phylogenetic networks: ISIPhyNC Information System on Inclusions of Phylogenetic Network Classes joint work with Maxime Morgado and Narges Tavassoli
42 Classes of phylogenetic networks: ISIPhyNC Information System on Inclusions of Phylogenetic Network Classes joint work with Maxime Morgado and Narges Tavassoli
43 Classes of phylogenetic networks: ISIPhyNC Information System on Inclusions of Phylogenetic Network Classes Inspired by ISGCI graphclasses.org joint work with Maxime Morgado and Narges Tavassoli
44 Classes of phylogenetic networks: inclusions joint work with Maxime Morgado and Narges Tavassoli
45 Classes of phylogenetic networks: inclusions joint work with Maxime Morgado and Narges Tavassoli
46 Classes of phylogenetic networks: inclusions joint work with Maxime Morgado and Narges Tavassoli
47 Classes of phylogenetic networks: inclusions level = maximum number of reticulation vertices among all bridgeless components in the network 2 2 joint work with Maxime Morgado and Narges Tavassoli
48 Classes of phylogenetic networks: inclusions cluster = set of leaves below a vertex distinct-cluster = all vertices (except reticulation vertices with their child) have distinct clusters joint work with Maxime Morgado and Narges Tavassoli
49 Classes of phylogenetic networks: inclusions cluster = set of leaves below a vertex distinct-cluster = all vertices (except reticulation vertices with their child) have distinct clusters not distinctcluster joint work with Maxime Morgado and Narges Tavassoli
50 Classes of phylogenetic networks: new inclusions joint work with Maxime Morgado and Narges Tavassoli
51 Who is who in Phylogenetic Networks? input software classes E problems algorithmic properties
52 Classes of phylogenetic networks: problems joint work with Maxime Morgado and Narges Tavassoli
53 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
54 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!
55 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?
56 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
57 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
58 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)
59 Classes of phylogenetic networks and the T.C.P. State of the art in 2011: available proofs
60 Classes of phylogenetic networks and the T.C.P. State of the art in 2011: available proofs + results deduced from inclusions?
61 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
62 Strategy to get a quadratic time algorithm for T.C.P. Given N, a phylogenetic network with n leaves and the input tree T of the T.C.P. Theorem 1: If N is reticulation-visible then: #{reticulation vertices of N} 4(n-1) #{vertices of N} 9n Theorem 2: If N is nearly-stable then #{reticulation vertices of N} 12(n-1) Theorem 3: Considering a longest path in N (nearly stable), it is possible, in constant time: either to realize that T is not contained in N or to build a network N with less arcs than N such that T contained in N if and only if T contained in N Gambette, Gunawan, Labarre, Vialette & Zhang, RECOMB 2015
63 Number of reticulations of a reticulation-visible network Decompose N into 2n-2 paths: remove one reticulation arc per reticulation, ensuring we get no «dummy leaf», to get a tree T with n leaves summarize T into a rooted binary tree T with n leaves... and 2n-2 arcs We can prove (technical) that: each path contains at most 2 reticulation vertices N Gambette, Gunawan, Labarre, Vialette & Zhang, RECOMB 2015
64 Number of reticulations of a reticulation-visible network Decompose N into 2n-2 paths: remove one reticulation arc per reticulation, ensuring we get no «dummy leaf», to get a tree T with n leaves summarize T into a rooted binary tree T with n leaves... and 2n-2 arcs We can prove (technical) that: each path contains at most 2 reticulation vertices T Gambette, Gunawan, Labarre, Vialette & Zhang, RECOMB 2015
65 Number of reticulations of a reticulation-visible network Decompose N into 2n-2 paths: remove one reticulation arc per reticulation, ensuring we get no «dummy leaf», to get a tree T with n leaves summarize T into a rooted binary tree T with n leaves... and 2n-2 arcs We can prove (technical) that: each path contains at most 2 reticulation vertices T T Gambette, Gunawan, Labarre, Vialette & Zhang, RECOMB 2015
66 Number of reticulations of a reticulation-visible network Decompose N into 2n-2 paths: remove one reticulation arc per reticulation, ensuring we get no «dummy leaf», to get a tree T with n leaves summarize T into a rooted binary tree T with n leaves... and 2n-2 arcs We can prove (technical) that: each path contains at most 2 reticulation vertices N contains at most 4(n-1) reticulation vertices Gambette, Gunawan, Labarre, Vialette & Zhang, RECOMB 2015
67 Classes of phylogenetic networks and the T.C.P. Recent results (since 2015): Bordewich & Semple, Advances in Applied Mathematics, 2016 Gunawan, DasGupta & Zhang, RECOMB 2017 Gunawan, AlCoB 2018 Weller, RECOMB-CG 2018
68 Fête de la Science at UPEM (science festival) Photos : Campus numérique de l'upem
69 Outline An introduction to phylogenetic networks Classes of phylogenetic networks Counting problems on phylogenetic networks
70 Outline An introduction to phylogenetic networks Classes of phylogenetic networks Counting problems on phylogenetic networks
71 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
72 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
73 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
74 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
75 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
76 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
77 Counting labeled unrooted level-1 networks Multivariate generating function: Using Theorem 2.23 of : Joint work with Mathilde Bouvel and Marefatollah Mansouri
78 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
79 Counting labeled unrooted level-2 networks Joint work with Mathilde Bouvel and Marefatollah Mansouri
80 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
81 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
82 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
83 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
84 Counting labeled level-k networks Open problem on level-1 networks = galled trees: Counting galled tree shapes (= unlabeled networks) Upper bound Upper levels... Chang, Hon & Thankachan, Data Compression Conference 2018
85 Other phylogenetic network counting results Tree-child, normal networks: McDiarmid, Semple & Welsh 2015 Fuchs, Gittenberger & Mansouri 2018 Galled networks: Louxin Zhang 2018
86 Thank you for your attention! Reticulogram of the 100 most frequent words in the titles of articles published in the last 10 editions of ANALCO (red: most recent) Built with TreeCloud + Trex (treecloud.org, trex.uqam.ca)
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