Another Proof of the Total Positivity of the Discrete Spline Collocation Matrix

AbstractWe provide a different proof for Morken's result on necessary and sufficient conditions for a minor of the discrete B-spline collocation matrix to be positive and supply intuition for those conditions

Seeing the Forest for the Trees: Using the Gene Ontology to Restructure Hierarchical Clustering

Motivation: There is a growing interest in improving the cluster analysis of expression data by incorporating into it prior knowledge, such as the Gene Ontology (GO) annotations of genes, in order to improve the biological relevance of the clusters that are subjected to subsequent scrutiny. The structure of the GO is another source of background knowledge that can be exploited through the use of semantic similarity. Results: We propose here a novel algorithm that integrates semantic similarities (derived from the ontology structure) into the procedure of deriving clusters from the dendrogram constructed during expression-based hierarchical clustering. Our approach can handle the multiple annotations, from different levels of the GO hierarchy, which most genes have. Moreover, it treats annotated and unannotated genes in a uniform manner. Consequently, the clusters obtained by our algorithm are characterized by significantly enriched annotations. In both cross-validation tests and when using an external index such as protein–protein interactions, our algorithm performs better than previous approaches. When applied to human cancer expression data, our algorithm identifies, among others, clusters of genes related to immune response and glucose metabolism. These clusters are also supported by protein–protein interaction data. Contact: [email protected] Supplementary information: Supplementary data are available at Bioinformatics online.Lynne and William Frankel Center for Computer Science; Paul Ivanier center for robotics research and production; National Institutes of Health (R01 HG003367-01A1

Seeing the forest for the trees: using the Gene Ontology to restructure hierarchical clustering

Motivation: There is a growing interest in improving the cluster analysis of expression data by incorporating into it prior knowledge, such as the Gene Ontology (GO) annotations of genes, in order to improve the biological relevance of the clusters that are subjected to subsequent scrutiny. The structure of the GO is another source of background knowledge that can be exploited through the use of semantic similarity

Biological Process Linkage Networks

BACKGROUND. The traditional approach to studying complex biological networks is based on the identification of interactions between internal components of signaling or metabolic pathways. By comparison, little is known about interactions between higher order biological systems, such as biological pathways and processes. We propose a methodology for gleaning patterns of interactions between biological processes by analyzing protein-protein interactions, transcriptional co-expression and genetic interactions. At the heart of the methodology are the concept of Linked Processes and the resultant network of biological processes, the Process Linkage Network (PLN). RESULTS. We construct, catalogue, and analyze different types of PLNs derived from different data sources and different species. When applied to the Gene Ontology, many of the resulting links connect processes that are distant from each other in the hierarchy, even though the connection makes eminent sense biologically. Some others, however, carry an element of surprise and may reflect mechanisms that are unique to the organism under investigation. In this aspect our method complements the link structure between processes inherent in the Gene Ontology, which by its very nature is species-independent. As a practical application of the linkage of processes we demonstrate that it can be effectively used in protein function prediction, having the power to increase both the coverage and the accuracy of predictions, when carefully integrated into prediction methods. CONCLUSIONS. Our approach constitutes a promising new direction towards understanding the higher levels of organization of the cell as a system which should help current efforts to re-engineer ontologies and improve our ability to predict which proteins are involved in specific biological processes.Lynn and William Frankel Center for Computer Science; the Paul Ivanier center for robotics research and production; National Science Foundation (ITR-048715); National Human Genome Research Institute (1R33HG002850-01A1, R01 HG003367-01A1); National Institute of Health (U54 LM008748

Another Proof of the Total Positivity of the Discrete Spline Collocation Matrix

We provide a different proof for Morken's result on necessary and sufficient conditions for a minor of the discrete B-spline collocation matrix to be positive and supply intuition for those conditions

Sleeved CoClustering

A coCluster of a m × n matrix X is a submatrix determined by a subset of the rows and a subset of the columns. The problem of finding coClusters with specific properties is of interest, in particular, in the analysis of microarray experiments. In that case the entries of the matrix X are the expression levels of m genes in each of n tissue samples. One goal of the analysis is to extract a subset of the samples and a subset of the genes, such that the expression levels of the chosen genes behave similarly across the subset of the samples, presumably reflecting an underlying regulatory mechanism governing the expression level of the genes. We propose to base the similarity of the genes in a co-Cluster on a simple biological model, in which the strength of the regulatory mechanism in sample j is Hj, and the response strength of gene i to the regulatory mechanism is Gi. In other words, every two genes participating in a good coCluster should have expression values in each of the participating samples, whose ratio is a constant depending only on the two genes. Noise in the expression levels of genes is taken into account by allowing a deviation from the model, measured by a relative error criterion. The sleeve-width of the coCluster reflects the extent to which entry i, j in the coCluster is allowed to deviate, relatively, from being expressed as the product GiHj. We present a polynomial-time Monte-Carlo algorithm which outputs a list of coClusters whose sleeve-widths do not exceed a prespecified value. Moreover, we prove that the list includes, with fixed probability, a coCluster which is nearoptimal in its dimensions. Extensive experimentation with synthetic data shows that the algorithm performs well

Determining a singleton attractor of a boolean network with nested canalyzing functions.

In this article, we study the problem of finding a singleton attractor for several biologically important subclasses of Boolean networks (BNs). The problem of finding a singleton attractor in a BN is known to be NP-hard in general. For BNs consisting of n nested canalyzing functions, we present an O(1.799(n)) time algorithm. The core part of this development is an O(min(2(k/2) · 2(m/2), 2(k)) · poly(k, m)) time algorithm for the satisfiability problem for m nested canalyzing functions over k variables. For BNs consisting of chain functions, a subclass of nested canalyzing functions, we present an O(1.619(n)) time algorithm and show that the problem remains NP-hard, even though the satisfiability problem for m chain functions over k variables is solvable in polynomial time. Finally, we present an o(2(n)) time algorithm for bounded degree BNs consisting of canalyzing functions