5th course: phylogeny and chordal graphs - irifhabib/documents/phylogeny.pdf · 2011. 10. 11. ·...

Introduction An example of 4-State Full Characters Chordal Sandwich Graph

5th Course:Phylogeny and chordal graphs

Michel [email protected]

http://www.liafa.jussieu.fr/~habib

Chevaleret, october 2011

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Michel Habib [email protected] http://www.liafa.jussieu.fr/~habib 5th Course: Phylogeny and chordal graphs

http://www.liafa.jussieu.fr/~habibhttp://www.liafa.jussieu.fr/~habib


Schedule

1 Introduction

2 An example of 4-State Full Characters

3 Chordal Sandwich Graph

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Phylogenetics

Objective: reconstruct evolutionary history of species frombiological data of present-day species.Models: phylogenetic trees, networks (rooted or unrooted)

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An unrooted phylogenetic tree on the set of speciesL = {x , y , z , t, u, v}

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Perfect Phylogeny and chordal graphs

The data is an incidence matrix from a set of species S1, . . . ,Sn toa set of attributes F1, . . . ,Fm.Attributes can have different values (integers or colors).These species are modern species and we aim at reconstructing aplausible historical evolution tree T , finding ancestral species.In T vertices are species, and its leaves correspond to S1, . . . ,Snand every value of an attribute must define a connected subtree.

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We define a graph G whose vertices are the value of the attributes.Two values are joined by an edge if they are both present in somespecie.Then a specie corresponds to a clique in G .When G is chordal, then T is a maximal clique tree of G havingS1, . . . ,Sn as set of leaves.Such a situation is called a Perfect phylogeny.

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Many researches in this field to cope with uncertainity, bad data orpartial data.One model is using partial partitions of the species such as triplets,quadruplets . . . .

One major problem:

For which data there exists a unique phylogenetic tree ?

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An example

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Discussion

1 X ,Y the new maximal cliques of G + β2γ3 are plausibleancestral species.

2 G could have been also completed as a chordal graph, byadding the edge α1β1, but then this would imply that F1 isnot properly defined since α1, β1 are values of F1.

3 Therefore attributes must correspond to independent sets ofG (or colors).

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More precisely

Input:A species set LA character set C on L

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L A B C DA: a 3-state characterC : a 4-state full character.

Output:A phylogenetic tree

(ternary) on L on which allcharacters of C are convex

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More precisely


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A is convex on T

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More precisely


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B is convex on T

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More precisely


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C is convex on T

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More precisely


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D is convex on T

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More precisely


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⇒ A,B,C ,D are compatible

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More precisely


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A character not convex on T

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Previous work

Theorem (Fitch 75, Meacham 83)

For any r ≥ 2, there exists a set of r -state full characters in whichevery r − 1 characters are compatible but the whole set is notcompatible.

Theorem (Meacham 83, Lam and Gusfield 09)

For r = 2, 3, a set of r -state full characters is compatible iff every rcharacters are compatible.

Question: ∀r ≥ 2, a set of r -state full characters is compatible iffevery r characters are compatible? [Lam and Gusfield 09]

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Intersection partition graph

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L A B C D

C = {A,B,C ,D}A = {x , u}|{z , t}|{y} = A0|A1|A2B = {x , y}|{t, v}|{z} = B0|B1|B2C = {y , z}|{u, v}|{t}|{v} = C0|C1|C2|C3D = {x , u}|{y , z}|{t}|{v} = D0|D1|D2|D3

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of C

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Proper chordal completion

A graph is chordal iff every chordless cycle is a triangle, i.e. haslength 3.A chordal completion of a graph G = (V ,E ) is a chordal graphG ′ = (V ,E ′) such that E ⊆ E ′.A proper chordal completion of a vertex-coloured graph G is achordal completion of G without connecting any pair of vertices ofthe same colour.

Theorem (Meacham 83, Steel 92)

C is compatible ⇔ the intersectionpartition graph of C has a properchordal completion

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4-State Full Characters

A = {x , u}|{z , t}|{y}|{v} = A0|A1|A2|A3B = {x , y}|{t, v}|{z}|{u} = B0|B1|B2|B3C = {y , z}|{u, v}|{x}|{t} = C0|C1|C2|C3D = {x , u}|{y , z}|{t}|{v} = D0|D1|D2|D3E = {z , t}|{u, v}|{x}|{y} = E0|E1|E2|E3

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A,B,C ,D,E are notcompatible

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A,C ,D,E are compati-ble

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A,B,C ,D are compati-ble

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A,B,C ,E are compati-ble

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A,B,D,E are compati-ble

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B,C ,D,E are compati-ble

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To determine the compatibility of set of a 4-statecharacters, it is not sufficient to test thecompatibility of every 4 characters

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Notations

P = [V ,E ,F ] is a sandwichproblem of the graphG = (V ,E ), whereF ⊆ V × V \ E .

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GS = (V ,ES) is called aΠ-sandwich graph of P ifE ⊆ ES ⊆ V × V \ F and GSsatisfies property Π.

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Chordal Sandwich Graph

By taking F = {(u, v)|u, v have the same colour}, the problem ofchordal completion of vertex-coloured graph is transformed tochordal sandwich graph.

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Consequences

Perfect phylogeny is NP-complete [Bodlaender et al. 92, Steel92], since chordal sandwich problem is NP-hard.

Unique Perfect phylogeny is NP-hard [Habib, Stacho 11] alsounique minimal chordal sandwich problem is NP-hard.

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Consequences

Perfect phylogeny is NP-complete [Bodlaender et al. 92, Steel92], since chordal sandwich problem is NP-hard.

Unique Perfect phylogeny is NP-hard [Habib, Stacho 11] alsounique minimal chordal sandwich problem is NP-hard.

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IntroductionAn example of 4-State Full CharactersChordal Sandwich Graph

5th course: phylogeny and chordal graphs - irifhabib/documents/phylogeny.pdf · 2011. 10. 11. ·...

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