Descendent Subtrees Comparison of Phylogenetic Trees with Applications to Co-evolutionary Classifications in

Bacterial Genome

Yaw-Ling Lin 1 Tsan-Sheng Hsu2

1 Dept Computer Sci. & Info. Management,Providence University, Taichung, Taiwan.

2 Institute of Information ScienceAcademia Sinica, Taipei, Taiwan

Yaw-Ling Lin, Providence, Taiwan 2

Motivation – Where the problems

come from?

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Two-Component System

• Two-component systems (2CS):– Sensor histidine kinase– response regulator

• The major controlling machinery in order for bacteria to encounter a diverse and often hostile environment

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2CS in Pseudomonas aeruginosa PAO1

http://www.pseudomonas.com/

“Complete genome sequence of Pseudomonas aeruginosa PAO1, an opportunistic pathogen.” Nature. 2000 Aug 31;406(6799):947-8. by Stover CK, Pham XQ, Erwin AL, et al.

• Genome: 6.3M bp• predicted genes: 5570• 123 genes were classif

ied as 2CSs.

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2CS in PAO1

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2CS in PAO1

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2CS in PAO1

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2CS in PAO1

• There are 123 annotated 2CS genes in PAO1.• Use systemic analysis of the evolutionary relations

hips between the sensor kinase and response regulator of a 2CS.

• Construct phylogenic trees using Clustal-W for 54 sensor kinases and 59 response regulators.

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2CS in PAO1 -- Sensor Tree

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2CS: Regulator Tree

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Subtrees Analysis of 2CS

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Co-evolution subtree Analysis

Sensor Tree Regulator Tree

versus

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Different Trees• Different phylogenetic trees inference methods:

- Maximum parsimony

- Maximum likelihood

- Distance matrix fitting

- Quartet based methods

• Comparing the same set of species w.r.t. different biological sequences or different genes, hence obtaining various trees.

• How to find the largest set of items on which the trees agree ?

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Previous Results• Measuring the similarity / difference between trees:

- Symmetric difference [Robinson 1979]

- Robinson and Foulds (RF) metric [Robinson 1981]

- Nearest-neighbor interchange [Waterman 1978]

- Subtree transfer distance [Allen 2001]

- Quartet metric [Estabrook 1985]

• Inferring the consensus tree: maximum agreement subtree problem (MAST) ; a.k.a the maximum homeomorphic agreement subtree

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MAST: Maximum Agreement Subtree

• Problem: given a set of rooted trees whose leaves are drawn from the same set of items of size n, find the largest subset of these items so that the portions of the trees restricted to the subset are isomorphic.

• [Amir and Keselman 1997]: NP-hard even for 3 unbounded degree trees.

• [Hein 1995]: the MAST for 3 trees with unbounded degree is hard to be approximated.

• [Amir et al 1997] Polynomial time algorithms for three or more bounded degree trees, but the time complexity is exponential in the bound for the degree.

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MAST: Maximum Agreement Subtree

• [Farach and Thorup 1997]: O(n1. 5 log n) time algorithm for two arbitrary degree trees.

• [Cole et al 2002]: MAST of two binary trees can be found in O(n log n) time; MAST of two degree d trees can be found in time.2(min{ log , log log })O n d n nd n d

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Problem Definition• A phylogenetic tree with n leaves is a (rooted) t

ree such that all the leaf nodes are uniquely labelled from 1 to n.

• The descendent subtree of a phylogenetic tree T is the subtree composed by all edges and nodes of T descending from a vertex.

• Given a set of n-leaf phylogenetic trees, we wish to explore the descendent subtrees relationships within these trees.

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Normalized cluster distance between two sets

• Symmetric set difference:

• Normalized cluster distance:

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All Pairs Subtrees Comparison – A naïve O(n3) algorithm

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All Pairs Subtrees Comparison – Property

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All Pairs Subtrees Comparison – an O(n2) algorithm

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Lowest Common Ancestor

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Confluent subtree

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Confluent subtree – Illustration

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Consructing confluent subtree

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Nearest subtree

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Nearest subtree: reasoning

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Nearest subtree: Algorithm

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Leaf-agree / Isomorphic Subtrees

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leaf-agreement – Two Trees

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All-agreement: Illustration

XY

z

x yy’=Lca(Y)

T1

X

z’=Lca(x’, y’)

Y

x’=Lca(X)

T2

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All-agreement Method

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leaf-agreement – k Trees

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Isomorphic Descendent Subtrees

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Isomorphic Descendent Subtrees (2)

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Conclusion• Computing all pairs normalized cluster distances between a

ll paired subtrees of two trees can be computationally optimally done in O(n2) time

• Finding nearest subtrees for a collection of pairwise disjointed subsets of leaves can be done in O(n) time.

• Finding all descendent subtrees consisting of the same set of leaves in a set of (unbounded-degree) trees is solvable in time linear to the size of the input trees.

• Finding all isomorhpic descendent subtrees in a set of (unbounded-degree) trees is solvable in time linear to the size of the input trees.

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Future Research

• Clustering analysis of 2CS for functional prediction of uncharacterized genes

• Co-evolutionary analysis of 2CS

• (Rooted / unrooted) phylogenetic trees comparison: when edges are labeled with (likelihood, log-odds) distances.

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The End

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What Date is Today?

• Magic Number:– 4/4, 6/6, 8/8, 10/10, 12/12– 7/11, 9/5 [also 11/7, 5/9]– 3/0? [implying 2/28, 2/0 = 1/31]

• Extension:– 365 = 52 * 7 + 1– Leap Year?

• 2003: 5 ; 2004: 7 ; 2005: 1 ; 2005:2