The Problem of Reconstructing k-articulated Phylogenetic Network

1. The Problem of Reconstructing k-articulated Phylogenetic Network Supervisor: Dr. YiuSiu Ming Second Examiner: Professor Francis Y.L. Chin Student: Vu ThiQuynhHoa

2. Contents • Introduction • Motivation • Related Work • Project Plan • Problem Definitions • Algorithms • 1-articulated Network Algorithm • 2-articulated Network Algorithm

3. Introduction – Motivation • 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

4. Related Work 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(n3) running time to construct a galled network given a set of triplets • Extending to level-2 network, Van Iersel et al. provided an O(n8) algorithm

5. Schedules – Project Plan

6. Definitions • 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

7. Phylogenetic Network

8. Definitions • 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-articulatednetwork • A k-articulatednetwork can model a level-x network (x > k) Level-2 network 1-articulated network

9. Definitions • A network is non-skewif 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

10. Definitions • 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. q q p p h h

11. Problem Definition Reconstructing a restricted k-articulated network (where k = 1, 2) from a set of binary trees Given a set of phylogenetic binary trees Ti , 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 Ti

12. Algorithm

13. 1-articulated Network Algorithm 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

14. 1-articulated Network Algorithm 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

15. 1-articulated Network Algorithm Case 2: Input tree set admits a leaf set bipartition – O(kn) T1 T2 Tk r N N1 N2 Combination

16. 1-articulated Network Algorithm • Case 3: Input tree set admits a leaf set tripartition – O(kn) T1 T2 Tk r It takes O(kn) to find nodes x in N1 and y in N2 N1 N2 x y Nh

17. 2-articulated Network Algorithm 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

18. 4-articulated Network Algorithm • Case 4: Input tree set admits a leaf set quadripartition – O(kn) T1 T2 Tk r It takes O(kn) to find nodes x1 & x2 in N1and y1 & y2in N2 N2 N1 x1 y1 y2 x2 Nh1 Nh1

19. 4-articulated Network Algorithm • Case 4: Input tree set admits a leaf set quadripartition – O(kn) T1 T2 Tk r It takes O(kn) to find nodes x1 & x2 in N1and y1 & y2in N2 N2 N1 x1 y1 y2 x2 Nh1 Nh1

20. Time Complexity 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(kn2)

21. Thank You!