Graph Triangulations and the Compatibility of Unrooted Phylogenetic Trees

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

Improved Lower Bounds on the Compatibility of 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 $r \ge 2$, there exists an incompatible set $C$ of $\lfloor\frac{r}{2}\rfloor\cdot\lceil\frac{r}{2}\rceil + 1$ $r$-state characters such that every proper subset of $C$ is compatible. Thus, $f(r) \ge \lfloor\frac{r}{2}\rfloor\cdot\lceil\frac{r}{2}\rceil + 1$ for every $r \ge 2$. This improves the previous lower bound of $f(r) \ge r$ given by Meacham (1983), and generalizes the construction showing that $f(4) \ge 5$ given by Habib and To (2011). We prove our result via a result on quartet compatibility that may be of independent interest: For every integer $n \ge 4$, there exists an incompatible set $Q$ of $\lfloor\frac{n-2}{2}\rfloor\cdot\lceil\frac{n-2}{2}\rceil + 1$ quartets over $n$ labels such that every proper subset of $Q$ is compatible. We contrast this with a result on the compatibility of triplets: For every $n \ge 3$, 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 upper bound is tight by exhibiting, for every $n \ge 3$, a set of $n-1$ triplets over $n$ taxa such that $R$ is incompatible, but every proper subset of $R$ is compatible

Pd@MWCNTs/GCE based voltammetric sensor for butachlor herbicide detection in soil samples

Butachlor is a herbicide that belongs to the acetanilide family. It is widely used as a granule-based post-emergence herbicide on rice in India. As a result of the ongoing usage of these synthetic substances, soil fertility and soil organisms are declining. Differential pulse voltammetry was used to determine butachlor herbicide in soil samples with a modified glassy carbon electrode voltammetric sensor with palladium-supported multiwalled carbon nanotubes (Pd@MWCNTs). Scanning electron microscopy, energy dispersive x-ray spectroscopy, and X-ray diffraction spectroscopy were used to investigate the morphology of Pd@MWCNTs, while cyclic and differential pulse techniques were used to investigate the voltammetric properties. The butachlor herbicide under voltammetric investigation involves irreversible, two-electron reduction based on the protonation of the carbonyl group (>C=O). The voltammetric method was developed for the determination of butachlor in phosphate buffer solution at pH 6.0 as a supporting electrolyte. A good linear response to butachlor in the concentration ranging from 0.10 μg⸳mL−1 to 32.0 μg⸳mL−1 was observed, and a limit of detection of 0.044 μg⸳mL−1 was obtained with the calculation based on signal/noise=3. The suggested method was efficaciously applied for the detection of butachlor in soil samples

Fixed parameter algorithms for compatible and agreement supertree problems

Biologists represent evolutionary history of species through phylogenetic trees. Leaves of a phylogenetic tree represent the species and internal vertices represent the extinct ancestors. Given a collection of input phylogenetic trees, a common problem in computational biology is to build a supertree that captures the evolutionary history of all the species in the input trees, and is consistent with each of the input trees. In this document we study the tree compatibility and agreement supertree problems. Tree compatibility problem is NP-complete but has been shown to be fixed parameter tractable when parametrized by number of input trees. We characterize the compatible supertree problem in terms of triangulation of a structure called the display graph. We also give an alternative characterization in terms of cuts of the display graph. We show how these characterizations are related to characterization given in terms of triangulation of the edge label intersection graph. We then give a characterization of the agreement supertree problem. In real world data, consistent supertrees do not always exist. Inconsistencies can be dealt with by contraction of edges or removal of taxa. The agreement supertree edge contraction (AST-EC) problem asks if a collection of k rooted trees can be made to agree by contraction of at most p edges. Similarly, the agreement supertree taxon removal (AST-TR) problem asks if a collection of k rooted trees can be made to agree by removal of at most p taxa. We give fixed parameter algorithms for both cases when parametrized by k and p. We study the long standing conjecture on the perfect phylogeny problem; there exists a function f (r) such that a given collection C of r-state characters is compatible if and only if every f (r) subset of C is compatible. We will show that for r ≥ 2, f (r) ≥ lceil (r/2) rceil * lfloor(r/2)rfloor + 1.</p

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. Tree 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. Here, we show that the latter characterization yields a 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

Exploring biological interaction networks with tailored weighted quasi-bicliques.

BACKGROUND: Biological 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. RESULTS: We 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. CONCLUSIONS: 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 biological networks

Scheduling two agents with controllable processing times

We consider several two-agent scheduling problems with controllable job processing times, where agents A and B have to share either a single machine or two identical machines in parallel while processing their jobs. The processing times of the jobs of agent A are compressible at additional cost. The objective function for agent B is always the same, namely a regular function fmax. Several different objective functions are considered for agent A, including the total completion time plus compression cost, the maximum tardiness plus compression cost, the maximum lateness plus compression cost and the total compression cost subject to deadline constraints (the imprecise computation model). All problems are to minimize the objective function of agent A subject to a given upper bound on the objective function of agent B. These problems have various applications in computer systems as well as in operations management. We provide NP-hardness proofs for the more general problems and polynomial-time algorithms for several special cases of the problems.Agent scheduling Controllable processing times Availability constraints Imprecise computation Total completion time Maximum tardiness Maximum lateness