Characterizing Block Graphs in Terms of their Vertex-Induced Partitions

Given a finite connected simple graph $G=(V,E)$ with vertex set $V$ and edge set $E\subseteq \binom{V}{2}$, we will show that $1.$ the (necessarily unique) smallest block graph with vertex set $V$ whose edge set contains $E$ is uniquely determined by the $V$-indexed family ${\bf P}_G:=\big(\pi_0(G^{(v)})\big)_{v \in V}$ of the various partitions $\pi_0(G^{(v)})$ of the set $V$ into the set of connected components of the graph $G^{(v)}:=(V,\{e\in E: v\notin e\})$, $2.$ the edge set of this block graph coincides with set of all $2$-subsets $\{u,v\}$ of $V$ for which $u$ and $v$ are, for all $w\in V-\{u,v\}$, contained in the same connected component of $G^{(w)}$, $3.$ and an arbitrary $V$-indexed family ${\bf P}p=({\bf p}_v)_{v \in V}$ of partitions $\pi_v$ of the set $V$ is of the form ${\bf P}p={\bf P}p_G$ for some connected simple graph $G=(V,E)$ with vertex set $V$ as above if and only if, for any two distinct elements $u,v\in V$, the union of the set in ${\bf p}_v$ that contains $u$ and the set in ${\bf p}_u$ that contains $v$ coincides with the set $V$, and $\{v\}\in {\bf p}_v$ holds for all $v \in V$. As well as being of inherent interest to the theory of block graphs, these facts are also useful in the analysis of compatible decompositions and block realizations of finite metric spaces

Modeling the effect of copper availability on bacterial denitrification

When denitrifying bacteria such as Paracoccus denitrificans respire anaerobically they convert nitrate to dinitrogen gas via a pathway which includes the potent greenhouse gas, nitrous oxide (NO). The copper-dependent enzyme Nitrous Oxide reductase (Nos) catalyzes the reduction of NO to dinitrogen. In low-copper conditions, recent experiments in chemostats have demonstrated that Nos efficiency decreases resulting in significant NO emissions. For the first time, a chemostat-based mathematical model is developed that describes the anaerobic denitrification pathway based on Michaelis-Menten kinetics and published kinetic parameters. The model predicts steady-state enzyme levels from experimental data. For low copper concentrations, the predicted Nos level is significantly reduced, whereas the levels for the non copper-dependent reductases in the pathway remain relatively unaffected. The model provides time courses for the pathway metabolites that accurately reflect previously published experimental data. In the absence of experimental data purely predictive analyses can also be readily performed by calculating the relative Nos level directly from the copper concentration. Here, the model quantitatively estimates the increasing level of emitted NO as the copper level decreases. We have developed a mathematical model for the denitrification pathway based on existing experimental results, Michaelis-Menten kinetics and experimentally obtained kinetic constants. This is the first such model to incorporate the copper concentration in order to predict emissions of the potent greenhouse gas, nitrous oxide (NO), as well as the other nitrogenous compounds in the pathway. The model predicts increasing NO emissions as the copper level is lowered, in agreement with experimental observations in chemostats. © 2013 The Authors. MicrobiologyOpen published by John Wiley & Sons Ltd.

Representing Partitions on Trees

In evolutionary biology, biologists often face the problem of constructing a phylogenetic tree on a set X of species from a multiset Π of partitions corresponding to various attributes of these species. One approach that is used to solve this problem is to try instead to associate a tree (or even a network) to the multiset ΣΠ consisting of all those bipartitions {A,X − A} with A a part of some partition in Π. The rational behind this approach is that a phylogenetic tree with leaf set X can be uniquely represented by the set of bipartitions of X induced by its edges. Motivated by these considerations, given a multiset Σ of bipartitions corresponding to a phylogenetic tree on X, in this paper we introduce and study the set P(Σ) consisting of those multisets of partitions Π of X with ΣΠ = Σ. More specifically, we characterize when P(Σ) is non-empty, and also identify some partitions in P(Σ) that are of maximum and minimum size. We also show that it is NP-complete to decide when P(Σ) is non-empty in case Σ is an arbitrary multiset of bipartitions of X. Ultimately, we hope that by gaining a better understanding of the mapping that takes an arbitrary partition system Π to the multiset ΣΠ, we will obtain new insights into the use of median networks and, more generally, split-networks to visualize sets of partitions

Finding sRNA generative locales from high-throughput sequencing data with NiBLS.

Journal ArticleCopyright © 2010 MacLean et al; licensee BioMed Central Ltd.BACKGROUND: Next-generation sequencing technologies allow researchers to obtain millions of sequence reads in a single experiment. One important use of the technology is the sequencing of small non-coding regulatory RNAs and the identification of the genomic locales from which they originate. Currently, there is a paucity of methods for finding small RNA generative locales. RESULTS: We describe and implement an algorithm that can determine small RNA generative locales from high-throughput sequencing data. The algorithm creates a network, or graph, of the small RNAs by creating links between them depending on their proximity on the target genome. For each of the sub-networks in the resulting graph the clustering coefficient, a measure of the interconnectedness of the subnetwork, is used to identify the generative locales. We test the algorithm over a wide range of parameters using RFAM sequences as positive controls and demonstrate that the algorithm has good sensitivity and specificity in a range of Arabidopsis and mouse small RNA sequence sets and that the locales it generates are robust to differences in the choice of parameters. CONCLUSIONS: NiBLS is a fast, reliable and sensitive method for determining small RNA locales in high-throughput sequence data that is generally applicable to all classes of small RNA.Gatsby Charitable Foundatio