Sudheer Vakati

Exploring biological interaction networks with tailored weighted quasi-bicliques

BackgroundBiological networks provide fundamental insights into the functional characterization of genes and their products, the characterization of DNA-protein interactions, the identification of regulatory mechanisms, and other biological tasks. Due to the experimental and biological complexity, their computational exploitation faces many algorithmic challenges.ResultsWe introduce novel weighted quasi-biclique problems to identify functional modules in biological networks when represented by bipartite graphs. In difference to previous quasi-biclique problems, we include biological interaction levels by using edge-weighted quasi-bicliques. While we prove that our problems are NP-hard, we also describe IP formulations to compute exact solutions for moderately sized networks.ConclusionsWe verify the effectiveness of our IP solutions using both simulation and empirical data. The simulation shows high quasi-biclique recall rates, and the empirical data corroborate the abilities of our weighted quasi-bicliques in extracting features and recovering missing interactions from biological networks.

Graph triangulations and the compatibility of unrooted phylogenetic trees

Abstract We characterize the compatibility of a collection of unrooted phylogenetic trees as a question of determining whether a graph derived from these trees — the display graph — has a specific kind of triangulation, which we call legal. Our result is a counterpart to the well-known triangulation-based characterization of the compatibility of undirected multi-state characters.

Incompatible quartets, triplets, and characters

We study a long standing conjecture on the necessary and sufficient conditions for the compatibility of multi-state characters: There exists a function f(r) such that, for any set C of r-state characters, C is compatible if and only if every subset of f(r) characters of C is compatible. We show that for every r≥2, there exists an incompatible set C of Ω(r2)r-state characters such that every proper subset of C is compatible. This improves the previous lower bound of f(r)≥r given by Meacham (1983), and f(4)≥5 given by Habib and To (2011). For the case when r=3, Lam, Gusfield and Sridhar (2011) recently showed that f(3)=3. We give an independent proof of this result and completely characterize the sets of pairwise compatible 3-state characters by a single forbidden intersection pattern.Our lower bound on f(r) is proven via a result on quartet compatibility that may be of independent interest: For every n≥4, there exists an incompatible set Q of Ω(n2) quartets over n labels such that every proper subset of Q is compatible. We show that such a set of quartets can have size at most 3 when n=5, and at most O(n3) for arbitrary n. We contrast our results on quartets with the case of rooted triplets: For every n≥3, if R is an incompatible set of more than n−1 triplets over n labels, then some proper subset of R is incompatible. We show this bound is tight by exhibiting, for every n≥3, a set of n−1 triplets over n taxa such that R is incompatible, but every proper subset of R is compatible.

Fixed-Parameter Algorithms for Finding Agreement Supertrees

We study the agreement supertree approach for combining rooted phylogenetic trees when the input trees do not fully agree on the relative positions of the taxa. We consider two ways to deal with such conflict. The first is to contract a set of edges in the input trees so that the resulting trees have an agreement supertree. We show that this problem is NP-complete and give a fixed-parameter tractable (FPT) algorithm for the problem parameterized by the number of input trees and the number of edges contracted. The second approach is to remove a set of taxa from the input trees so that the resulting trees have an agreement supertree. Guillemot and Berry (2010) gave an FPT algorithm for this problem when the input trees are all binary. We give an FPT algorithm for the more general case where the input trees are allowed to have arbitrary degree.

Mining biological interaction networks using weighted quasi-bicliques

Biological network studies can provide fundamental insights into various biological tasks including the functional characterization of genes and their products, the characterization of DNA-protein interactions, and the identification of regulatory mechanisms. However, biological networks are confounded with unreliable interactions and are incomplete, and thus, their computational exploitation is fraught with algorithmic challenges. Here we introduce quasi-biclique problems to analyze biological networks when represented by bipartite graphs. In difference to previous quasi-biclique problems, we include biological interaction levels by using edge-weighted quasi-bicliques. While we prove that our problems are NP-hard, we also provide exact IP solutions that can compute moderately sized networks. We verify the effectiveness of our IP solutions using both simulation and empirical data. The simulation shows high quasi-biclique recall rates, and the empirical data corroborate the abilities of our weighted quasi-bicliques in extracting features and recovering missing interactions from the network.

Characterizing compatibility and agreement of unrooted trees via cuts in graphs

BackgroundDeciding whether there is a single tree —a supertree— that summarizes the evolutionary information in a collection of unrooted trees is a fundamental problem in phylogenetics. We consider two versions of this question: agreement and compatibility. In the first, the supertree is required to reflect precisely the relationships among the species exhibited by the input trees. In the second, the supertree can be more refined than the input trees.Testing for compatibility is an NP-complete problem; however, the problem is solvable in polynomial time when the number of input trees is fixed. Testing for agreement is also NP-complete, but it is not known whether it is fixed-parameter tractable. Compatibility can be characterized in terms of the existence of a specific kind of triangulation in a structure known as the display graph. Alternatively, it can be characterized as a chordal graph sandwich problem in a structure known as the edge label intersection graph. No characterization of agreement was known.ResultsWe present a simple and natural characterization of compatibility in terms of minimal cuts in the display graph, which is closely related to compatibility of splits. We then derive a characterization for agreement.ConclusionsExplicit characterizations of tree compatibility and agreement are essential to finding practical algorithms for these problems. The simplicity of the characterizations presented here could help to achieve this goal.

Fixed-Parameter algorithms for finding agreement supertrees

We study the agreement supertree approach for combining rooted phylogenetic trees when the input trees do not fully agree on the relative positions of the taxa. Two approaches to dealing with such conflicting input trees are considered. The first is to contract a set of edges in the input trees so that the resulting trees have an agreement supertree. We show that this problem is NP-complete and give an FPT algorithm for the problem parameterized by the number of input trees and the number of edges contracted. The second approach is to remove a set of taxa from the input trees so that the resulting trees have an agreement supertree. An FPT algorithm for this problem when the input trees are all binary was given by Guillemot and Berry (2010). We give an FPT algorithm for the more general case when the input trees have arbitrary degree.

On compatibility and incompatibility of collections of unrooted phylogenetic trees

Abstract Let P = { T 1 , … , T k } be a collection of phylogenetic trees over various subsets of a set of species. For each i ∈ { 1 , … , k } , let L ( T i ) denote the set of species in tree T i . A supertree for P is a phylogenetic tree with species set ⋃ i = 1 k L ( T i ) . The tree compatibility problem asks whether there exists a supertree T for P such that, for each i ∈ { 1 , … , k } , T i can be obtained from T | L ( T i ) — the minimal subtree of T spanning L ( T i ) — by zero or more contractions of internal edges. If the answer is “yes”, then P is said to be compatible; otherwise, P is incompatible. We characterize compatibility via graph triangulations and tree decompositions. We then study how to make an incompatible collection of trees compatible through edge contraction and tree deletion. Finally, we introduce the notion of a phylogenetic minor to study under which conditions edge contraction, tree removal, and species removal/renaming operations preserve compatibility.

Characterizing Compatibility and Agreement of Unrooted Trees via Cuts in Graphs

Deciding whether there is a single tree —a supertree— that summarizes the evolutionary information in a collection of unrooted trees is a fundamental problem in phylogenetics. We consider two versions of this question: agreement and compatibility. In the first, the supertree is required to reflect precisely the relationships among the species exhibited by the input trees. In the second, the supertree can be more refined than the input trees.

Improved lower bounds on the compatibility of quartets, triplets, and multi-state characters

We study a long standing conjecture on the necessary and sufficient conditions for the compatibility of multi-state characters: There exists a function f(r) such that, for any set C of r-state characters, C is compatible if and only if every subset of f(r) characters of C is compatible. We show that for every r≥2, there exists an incompatible set C of