On the computation of zone and double zone diagrams

Classical objects in computational geometry are defined by explicit relations. Several years ago the pioneering works of T. Asano, J. Matousek and T. Tokuyama introduced "implicit computational geometry", in which the geometric objects are defined by implicit relations involving sets. An important member in this family is called "a zone diagram". The implicit nature of zone diagrams implies, as already observed in the original works, that their computation is a challenging task. In a continuous setting this task has been addressed (briefly) only by these authors in the Euclidean plane with point sites. We discuss the possibility to compute zone diagrams in a wide class of spaces and also shed new light on their computation in the original setting. The class of spaces, which is introduced here, includes, in particular, Euclidean spheres and finite dimensional strictly convex normed spaces. Sites of a general form are allowed and it is shown that a generalization of the iterative method suggested by Asano, Matousek and Tokuyama converges to a double zone diagram, another implicit geometric object whose existence is known in general. Occasionally a zone diagram can be obtained from this procedure. The actual (approximate) computation of the iterations is based on a simple algorithm which enables the approximate computation of Voronoi diagrams in a general setting. Our analysis also yields a few byproducts of independent interest, such as certain topological properties of Voronoi cells (e.g., that in the considered setting their boundaries cannot be "fat").Comment: Very slight improvements (mainly correction of a few typos); add DOI; Ref [51] points to a freely available computer application which implements the algorithms; to appear in Discrete & Computational Geometry (available online

Preferential regulation of miRNA targets by environmental chemicals in the human genome

<p>Abstract</p> <p>Background</p> <p>microRNAs (miRNAs) represent a class of small (typically 22 nucleotides in length) non-coding RNAs that can degrade their target mRNAs or block their translation. Recent disease research showed the exposure to some environmental chemicals (ECs) can regulate the expression patterns of miRNAs, which raises the intriguing question of how miRNAs and their targets cope with the exposure to ECs throughout the genome.</p> <p>Results</p> <p>In this study, we comprehensively analyzed the properties of genes regulated by ECs (EC-genes) and found miRNA targets were significantly enriched among the EC-genes. Compared with the non-miRNA-targets, miRNA targets were roughly twice as likely to be EC-genes. By investigating the collection methods and other properties of the EC-genes, we demonstrated that the enrichment of miRNA targets was not attributed to either the potential collection bias of EC-genes, the presence of paralogs, longer 3'UTRs or more conserved 3'UTRs. Finally, we identified 1,842 significant concurrent interactions between 407 miRNAs and 497 ECs. This association network of miRNAs-ECs was highly modular and could be separated into 14 interconnected modules. In each module, miRNAs and ECs were closely connected, providing a good method to design accurate miRNA markers for ECs in toxicology research.</p> <p>Conclusions</p> <p>Our analyses indicated that miRNAs and their targets played important roles in cellular responses to ECs. Association analyses of miRNAs and ECs will help to broaden the understanding of the pathogenesis of such chemical components.</p