Reduction of dynamical biochemical reaction networks in computational biology

Biochemical networks are used in computational biology, to model the static and dynamical details of systems involved in cell signaling, metabolism, and regulation of gene expression. Parametric and structural uncertainty, as well as combinatorial explosion are strong obstacles against analyzing the dynamics of large models of this type. Multi-scaleness is another property of these networks, that can be used to get past some of these obstacles. Networks with many well separated time scales, can be reduced to simpler networks, in a way that depends only on the orders of magnitude and not on the exact values of the kinetic parameters. The main idea used for such robust simplifications of networks is the concept of dominance among model elements, allowing hierarchical organization of these elements according to their effects on the network dynamics. This concept finds a natural formulation in tropical geometry. We revisit, in the light of these new ideas, the main approaches to model reduction of reaction networks, such as quasi-steady state and quasi-equilibrium approximations, and provide practical recipes for model reduction of linear and nonlinear networks. We also discuss the application of model reduction to backward pruning machine learning techniques

Combinatorial Gradient Fields for 2D Images with Empirically Convergent Separatrices

This paper proposes an efficient probabilistic method that computes combinatorial gradient fields for two dimensional image data. In contrast to existing algorithms, this approach yields a geometric Morse-Smale complex that converges almost surely to its continuous counterpart when the image resolution is increased. This approach is motivated using basic ideas from probability theory and builds upon an algorithm from discrete Morse theory with a strong mathematical foundation. While a formal proof is only hinted at, we do provide a thorough numerical evaluation of our method and compare it to established algorithms.Comment: 17 pages, 7 figure

Genomic regulatory architecture of human embryo retroviral LTR elements affecting evolution, development, and pathophysiology of Modern Humans

Two distinct families of pan-primate endogenous retroviruses, namely HERVL and HERVH, infected primates germline, colonized host genomes, and evolved into the global retroviral genomic regulatory dominion (GRD) operating during human embryogenesis (HE). HE retroviral GRD constitutes 8839 highly conserved fixed LTR elements linked to 5444 down-stream target genes forged by evolution into a functionally-consonant constellation of 26 genome-wide multimodular genomic regulatory networks (GRNs), each of which is defined by significant enrichment of numerous single gene ontology (GO)-specific traits. Locations of GRNs appear scattered across chromosomes to occupy from 5.5%-15.09% of human genome. Each GRN harbors from 529-1486 retroviral LTRs derived from LTR7, MLT2A1, and MLT2A2 sequences that are quantitatively balanced according to their genome-wide abundance. GRNs integrate activities from 199-805 down-stream target genes, including transcription factors, chromatin-state remodelers, signal-sensing and signal-transduction mediators, enzymatic and receptor binding effectors, intracellular complexes and extracellular matrix elements, and cell-cell adhesion molecules. GRNs compositions consist of several hundred to thousands smaller GO enrichment-defined genomic regulatory modules (GRMs) combining from a dozen to hundreds LTRs and down-stream target genes, which appear to operate on individuals life-span timescale along specific phenotypic avenues to exert profound effects on patterns of transcription, protein-protein interactions, developmental phenotypes, physiological traits, and pathological conditions of Modern Humans. Overall, this study identifies 69,573 statistically significant retroviral LTR-linked GRMs (Binominal FDR q-value threshold of 0.001), including 27,601 GRMs validated by the single GO-specific directed acyclic graph (DAG) analyses across six GO annotations.Comment: 66 pages, 16 figure

Homological tree-based strategies for image analysis

Homological characteristics of digital objects can be obtained in a straightforward manner computing an algebraic map φ over a finite cell complex K (with coefficients in the finite field F2={0,1}) which represents the digital object [9]. Computable homological information includes the Euler characteristic, homology generators and representative cycles, higher (co)homology operations, etc. This algebraic map φ is described in combinatorial terms using a mixed three-level forest. Different strategies changing only two parameters of this algorithm for computing φ are presented. Each one of those strategies gives rise to different maps, although all of them provides the same homological information for K. For example, tree-based structures useful in image analysis like topological skeletons and pyramids can be obtained as subgraphs of this forest

On the dimension of spline spaces on planar T-meshes

We analyze the space of bivariate functions that are piecewise polynomial of bi-degree \textless{}= (m, m') and of smoothness r along the interior edges of a planar T-mesh. We give new combinatorial lower and upper bounds for the dimension of this space by exploiting homological techniques. We relate this dimension to the weight of the maximal interior segments of the T-mesh, defined for an ordering of these maximal interior segments. We show that the lower and upper bounds coincide, for high enough degrees or for hierarchical T-meshes which are enough regular. We give a rule of subdivision to construct hierarchical T-meshes for which these lower and upper bounds coincide. Finally, we illustrate these results by analyzing spline spaces of small degrees and smoothness