T HE P ROBLEM OF R ECONSTRUCTING K - ARTICULATED P HYLOGENETIC N ETWORK Supervisor : Dr. Yiu Siu Ming Second Examiner : Professor Francis Y.L. Chin Student : Vu Thi Quynh Hoa

C ONTENTS 1. Introduction  Motivation  Related Work  Project Plan 2. Problem Definitions 3. Algorithms  1-articulated Network Algorithm  2-articulated Network Algorithm

I NTRODUCTION – M OTIVATION To model the evolutionary history of species, phylogenetic network is a powerful approach to represent the articulation events Level-x network : the time complexity of all existing algorithms increases exponentially when x gets higher k-articulated network is a more naturally biological model which can capture complex scenarios of articulation events with a smaller value of k E.g. level-4 network vs. 2-articulated network

R ELATED W ORK The problem of constructing phylogenetic networks has been worked under many approaches using different input types Nakhleh et al. proposed an algorithm constructing a level-1 network from two trees in polynomial time Huynh et al. with a polynomial -running-time algorithm building a galled network from a set of trees Bryant and Moulton developed NeighborNet method to construct a network from a distance matrix Jansson, Nguyen and Sung with O ( n 3 ) running time to construct a galled network given a set of triplets Extending to level-2 network, Van Iersel et al. provided an O ( n 8 ) algorithm

S CHEDULES – P ROJECT P LAN ObjectivesTime Round 1 1.Reconstructing restricted 1-articulated network from a set of binary phylogenetic trees 30 Sep Reconstructing restricted 2-articulated network from a set of binary phylogenetic trees 15 Nov 2011 Round 2 3.Reconstructing 1-articulated network from a Distance Matrix 15 Feb Implementation one of the three problems 31 Mar 2012

D EFINITIONS Phylogenetic Tree A rooted, unordered tree with distinctly labeled leaves representing each strain of the species Phylogenetic Network A rooted, directed acyclic graph in which: One node has indegree 0 (the root ), and all other nodes have indegree 1 or 2 All nodes with indegree 2 must have outdegree 1 ( hybrid nodes ) All other nodes with indegree 1 have outdegree 0 or 2 Nodes with outdegree 0 are leaves which are distinctly labeled Node s is called a split node of a hybrid node h if s can be reached using two disjoint paths from the children of s

P HYLOGENETIC N ETWORK

D EFINITIONS k-articulated network a phylogenetic network in which every split node corresponds to at most k hybrid nodes A level- k network is a k -articulated network A k- articulated network can model a level- x network (x > k) Level-2 network 1-articulated network

D EFINITIONS A network is non-skew if all paths from any split node to its hybrid node have a length ≥ 2 A network is safe if the siblings of all hybrid nodes are not hybrid nodes A network is restricted if it is non-skew and safe

D EFINITIONS Given a hybrid node h and its parents p and q, a cut on edge ( p, h ) means removing the edge ( p, h ) from the network, and then for every node with indegree 1 and outdegree less than 2, contracting its outgoing edge A network N is compatible with phylogenetic tree T if N can be converted to T by performing a series of cuts one by one. h p q h p q

P ROBLEM D EFINITION Reconstructing a restricted k-articulated network (where k = 1, 2 ) from a set of binary trees Given a set of phylogenetic binary trees T i, i = 1, 2, …, k, with the same leaf label set, construct a restricted k-articulated network N (where k = 1, 2) with minimum number of hybrid nodes compatible with each tree T i

A LGORITHM Divide and Conquer Technique Dividing BipartitionTripartitionQuadripartition Conquering ?

1- ARTICULATED N ETWORK A LGORITHM Case 1: Each input tree is a single node – Base case Case 2: Input tree set admits a leaf set bipartition Case 3: Input tree set admits a leaf set tripartition

1- ARTICULATED N ETWORK A LGORITHM Case 1: Each input tree is a single node – Base case – O (1) Return a network which is a single node with the same label

1- ARTICULATED N ETWORK A LGORITHM Case 2: Input tree set admits a leaf set bipartition – O ( kn ) T1T1 T2T2 TkTk N1N1 N2N2 r N Combination

r 1- ARTICULATED N ETWORK A LGORITHM Case 3: Input tree set admits a leaf set tripartition – O ( kn ) T1T1 T2T2 TkTk N1N1 N2N2 NhNh x y It takes O ( kn ) to find nodes x in N 1 and y in N 2

2- ARTICULATED N ETWORK A LGORITHM Case 1: Each input tree is a single node – Base case Case 2: Input tree set admit a leaf set bipartition Case 3: Input tree set admit a leaf set tripartition Case 4: Input tree set admit a leaf set quadripartition

r 4- ARTICULATED N ETWORK A LGORITHM Case 4: Input tree set admits a leaf set quadripartition – O ( kn ) T1T1 T2T2 TkTk N h1 x1x1 y1y1 It takes O ( kn ) to find nodes x 1 & x 2 in N 1 and y 1 & y 2 in N 2 N h1 x2x2 y2y2 N2N2 N1N1

r 4- ARTICULATED N ETWORK A LGORITHM Case 4: Input tree set admits a leaf set quadripartition – O ( kn ) T1T1 T2T2 TkTk N h1 x1x1 y1y1 It takes O ( kn ) to find nodes x 1 & x 2 in N 1 and y 1 & y 2 in N 2 N h1 x2x2 y2y2 N2N2 N1N1

T IME C OMPLEXITY Time complexity of the Algorithms in reconstructing a restricted k -articulated network, in both cases when k = 1, 2: Each recursive step takes O ( kn ) running time to check whether the input tree set admit a leaf set bipartition or tripartition, and then combine the subnetworks returned The number of nodes in the restricted 1-articulated network is O ( n ) Therefore, the total time complexity is O ( kn 2 )

T HANK Y OU !