Topological Modules Research Articles

φ-Multipliers on Banach Algebras and Topological Modules

We prove some results concerning Arens regularity and amenability of the Banach algebraMφAof allφ-multipliers on a given Banach algebraA. We also considerφ-multipliers in the general topological module setting and investigate some of their properties. We discuss theφ-strict andφ-uniform topologies onMφA. A characterization ofφ-multipliers onL1G-moduleLpG, whereGis a compact group, is given.

Linearly topologized modules over a discrete valuation ring

In this paper certain aspects of the theory of linearly topologized modules over an arbitrary discrete valuation ring are studied. Among the main topics discussed in this work we may mention: the class of linearly topologized spaces, linearly topologized topological modules of continuous linear mappings, equicontinuity of sets of linear mappings, dual systems of modules and orthogonality, weakly continuous linear mappings.

A network-based classification model for deriving novel drug-disease associations and assessing their molecular actions.

The growing number and variety of genetic network datasets increases the feasibility of understanding how drugs and diseases are associated at the molecular level. Properly selected features of the network representations of existing drug-disease associations can be used to infer novel indications of existing drugs. To find new drug-disease associations, we generated an integrative genetic network using combinations of interactions, including protein-protein interactions and gene regulatory network datasets. Within this network, network adjacencies of drug-drug and disease-disease were quantified using a scored path between target sets of them. Furthermore, the common topological module of drugs or diseases was extracted, and thereby the distance between topological drug-module and disease (or disease-module and drug) was quantified. These quantified scores were used as features for the prediction of novel drug-disease associations. Our classifiers using Random Forest, Multilayer Perceptron and C4.5 showed a high specificity and sensitivity (AUC score of 0.855, 0.828 and 0.797 respectively) in predicting novel drug indications, and displayed a better performance than other methods with limited drug and disease properties. Our predictions and current clinical trials overlap significantly across the different phases of drug development. We also identified and visualized the topological modules of predicted drug indications for certain types of cancers, and for Alzheimer’s disease. Within the network, those modules show potential pathways that illustrate the mechanisms of new drug indications, including propranolol as a potential anticancer agent and telmisartan as treatment for Alzheimer’s disease.

DiME: A Scalable Disease Module Identification Algorithm with Application to Glioma Progression

Disease module is a group of molecular components that interact intensively in the disease specific biological network. Since the connectivity and activity of disease modules may shed light on the molecular mechanisms of pathogenesis and disease progression, their identification becomes one of the most important challenges in network medicine, an emerging paradigm to study complex human disease. This paper proposes a novel algorithm, DiME (Disease Module Extraction), to identify putative disease modules from biological networks. We have developed novel heuristics to optimise Community Extraction, a module criterion originally proposed for social network analysis, to extract topological core modules from biological networks as putative disease modules. In addition, we have incorporated a statistical significance measure, B-score, to evaluate the quality of extracted modules. As an application to complex diseases, we have employed DiME to investigate the molecular mechanisms that underpin the progression of glioma, the most common type of brain tumour. We have built low (grade II) - and high (GBM) - grade glioma co-expression networks from three independent datasets and then applied DiME to extract potential disease modules from both networks for comparison. Examination of the interconnectivity of the identified modules have revealed changes in topology and module activity (expression) between low- and high- grade tumours, which are characteristic of the major shifts in the constitution and physiology of tumour cells during glioma progression. Our results suggest that transcription factors E2F4, AR and ETS1 are potential key regulators in tumour progression. Our DiME compiled software, R/C++ source code, sample data and a tutorial are available at http://www.cs.bham.ac.uk/~szh/DiME.

Predicting disease-related proteins based on clique backbone in protein-protein interaction network.

Network biology integrates different kinds of data, including physical or functional networks and disease gene sets, to interpret human disease. A clique (maximal complete subgraph) in a protein-protein interaction network is a topological module and possesses inherently biological significance. A disease-related clique possibly associates with complex diseases. Fully identifying disease components in a clique is conductive to uncovering disease mechanisms. This paper proposes an approach of predicting disease proteins based on cliques in a protein-protein interaction network. To tolerate false positive and negative interactions in protein networks, extending cliques and scoring predicted disease proteins with gene ontology terms are introduced to the clique-based method. Precisions of predicted disease proteins are verified by disease phenotypes and steadily keep to more than 95%. The predicted disease proteins associated with cliques can partly complement mapping between genotype and phenotype, and provide clues for understanding the pathogenesis of serious diseases.

Lessons from the modular organization of the transcriptional regulatory network of Bacillus subtilis

BackgroundThe regulation of gene expression at the transcriptional level is a fundamental process in prokaryotes. Among the different kind of mechanisms modulating gene transcription, the one based on DNA binding transcription factors, is the most extensively studied and the results, for a great number of model organisms, have been compiled making it possible the in silico construction of their corresponding transcriptional regulatory networks and the analysis of the biological relationships of the components of these intricate networks, that allows to elucidate the significant aspects of their organization and evolution.ResultsWe present a thorough review of each regulatory element that constitutes the transcriptional regulatory network of Bacillus subtilis. For facilitating the discussion, we organized the network in topological modules. Our study highlight the importance of σ factors, some of them acting as master regulators which characterize modules by inter- or intra-connecting them and play a key role in the cascades that define relevant cellular processes in this organism. We discussed that some particular functions were distributed in more than one module and that some modules contained more than one related function. We confirm that the presence of paralogous proteins confers advantages to B. subtilis to adapt and select strategies to successfully face the extreme and changing environmental conditions in which it lives.ConclusionsThe intricate organization is the product of a non-random network evolution that primarily follows a hierarchical organization based on the presence of transcription and σ factor, which is reflected in the connections that exist within and between modules.

On the Banach-Steinhaus theorem

A Banach-Steinhaus theorem for sets of continuous linear mappings on topological modules which are Baire spaces is proved, and some consequences of it are derived.

Design of High Current Two-Way DC/DC Converter in Lithium-Ion Battery Formation Equipment

As the critical process in battery production, formation is related to the quality of battery and directly affects production cost. A low voltage, high current DC/DC converter is designed for large capacity lithium battery formation equipment. Design of basic topological structure, feedback module and peripheral steering circuit are introduced. Besides, design of PWM driving circuit and parameters of high- frequency transformer are presented in detail. This two-way DC/DC converter has features of high stability, high reliability, small size, light weight and its efficiency of buck-boost can be up to 84% and 83% respectively.

Maximal independent, analytic sets in Abelian polish groups

Abstract The paper deals with the questions: (a)whether a topological module admits maximal linearly independent subsets that are analytic(b)whether an Abelian topological group admits maximal independent subsets that are analytic(c)whether a topological field extension admits transcendence bases that are analytic.

Research Resource: Interactome of Human Embryo Implantation: Identification of Gene Expression Pathways, Regulation, and Integrated Regulatory Networks

A prerequisite for successful embryo implantation is adequate preparation of receptive endometrium and the establishment and maintenance of a viable embryo. The success of implantation further relies upon a two-way dialogue between the embryo and uterus. However, molecular bases of these preimplantation and implantation processes in humans are not well known. We performed genome expression analyses of human embryos (n = 128) and human endometria (n = 8). We integrated these data with protein-protein interactions in order to identify molecular networks within the endometrium and the embryo, and potential embryo-endometrium interactions at the time of implantation. For that, we applied a novel network profiling algorithm HyperModules, which combines topological module identification and functional enrichment analysis. We found a major wave of transcriptional down-regulation in preimplantation embryos. In receptive-stage endometrium, several genes and signaling pathways were identified, including JAK-STAT signaling and inflammatory pathways. The main curated embryo-endometrium interaction network highlighted the importance of cell adhesion molecules in the implantation process. We also identified cytokine-cytokine receptor interactions involved in implantation, where osteopontin (SPP1), leukemia inhibitory factor (LIF) and leptin (LEP) pathways were intertwining. Further, we identified a number of novel players in human embryo-endometrium interactions, such as apolipoprotein D (APOD), endothelin 1 (END1), fibroblast growth factor 7 (FGF7), gastrin (GAST), kringle containing trnasmembrane protein 1 (KREMEN1), neuropilin 1 (NRP1), serpin peptidase inhibitor clade A member 3 (SERPINA3), versican (VCAN), and others. Our findings provide a fundamental resource for better understanding of the genetic network that leads to successful embryo implantation. We demonstrate the first systems biology approach into the complex molecular network of the implantation process in humans.

Ideals in the Ring of Colombeau Generalized Numbers

The structure of the ideals in the ring of Colombeau generalized numbers is investigated. Connections with the theories of exchange rings, Gelfand rings and lattice-ordered rings are given. Characterizations for prime, projective, pure, and topologically closed ideals are given, answering in particular the questions about prime ideals in [1]. Also z-ideals [23] are characterized. It is shown that the quotient rings modulo maximal ideals are canonically isomorphic with nonstandard fields of asymptotic numbers and that the Hahn–Banach extension property does not hold for a large class of topological modules over the ring of Colombeau generalized numbers.

Detecting communities of triangles in complex networks using spectral optimization

The study of the sub-structure of complex networks is of major importance to relate topology and functionality. Many efforts have been devoted to the analysis of the modular structure of networks using the quality function known as modularity. However, generally speaking, the relation between topological modules and functional groups is still unknown, and depends on the semantic of the links. Sometimes, we know in advance that many connections are transitive, and as a consequence, triangles have a specific meaning. Here we propose the study of the modular structure of networks considering triangles as the building blocks of modules. The method generalizes the standard modularity and uses spectral optimization to find its maximum. We compare the partitions obtained with those resulting from the optimization of the standard modularity in several real networks. The results show that the information reported by the analysis of modules of triangles complements the information of the classical modularity analysis.

Modular and Hierarchically Modular Organization of Brain Networks

Brain networks are increasingly understood as one of a large class of information processing systems that share important organizational principles in common, including the property of a modular community structure. A module is topologically defined as a subset of highly inter-connected nodes which are relatively sparsely connected to nodes in other modules. In brain networks, topological modules are often made up of anatomically neighboring and/or functionally related cortical regions, and inter-modular connections tend to be relatively long distance. Moreover, brain networks and many other complex systems demonstrate the property of hierarchical modularity, or modularity on several topological scales: within each module there will be a set of sub-modules, and within each sub-module a set of sub-sub-modules, etc. There are several general advantages to modular and hierarchically modular network organization, including greater robustness, adaptivity, and evolvability of network function. In this context, we review some of the mathematical concepts available for quantitative analysis of (hierarchical) modularity in brain networks and we summarize some of the recent work investigating modularity of structural and functional brain networks derived from analysis of human neuroimaging data.

Centralized Modularity of N-Linked Glycosylation Pathways in Mammalian Cells

Glycosylation is a highly complex process to produce a diverse repertoire of cellular glycans that are attached to proteins and lipids. Glycans are involved in fundamental biological processes, including protein folding and clearance, cell proliferation and apoptosis, development, immune responses, and pathogenesis. One of the major types of glycans, N-linked glycans, is formed by sequential attachments of monosaccharides to proteins by a limited number of enzymes. Many of these enzymes can accept multiple N-linked glycans as substrates, thereby generating a large number of glycan intermediates and their intermingled pathways. Motivated by the quantitative methods developed in complex network research, we investigated the large-scale organization of such N-linked glycosylation pathways in mammalian cells. The N-linked glycosylation pathways are extremely modular, and are composed of cohesive topological modules that directly branch from a common upstream pathway of glycan synthesis. This unique structural property allows the glycan production between modules to be controlled by the upstream region. Although the enzymes act on multiple glycan substrates, indicating cross-talk between modules, the impact of the cross-talk on the module-specific enhancement of glycan synthesis may be confined within a moderate range by transcription-level control. The findings of the present study provide experimentally-testable predictions for glycosylation processes, and may be applicable to therapeutic glycoprotein engineering.

Asymptotic extension of topological modules and algebras

Given a topological R–module or algebra E and an asymptotic scale M⊂R Λ, we define a natural M–extended topology on the sequence space E Λ, and the M–extension of E as the Hausdorff space associated with the subspace of sequences for which multiplication is continuous with respect to this topology. Colombeau's generalized functions and similar constructions are obtained as special cases, but this new approach also allows the iteration of the construction, which was not possible with previously existing theories. We use only the topology, i.e. neigbourhoods of zero, but not its explicit definition in terms of seminorms, inductive or projective limits etc., which is particularly useful in non-metrizable spaces. Many ideas commonly used in generalized function spaces (functoriality, association, sheaf structure, algebraic analysis, …) can be applied to a large extent; but reasoning on a category-theoretic level allows the establishment of several results so far only known for particular cases, to the whole class of such spaces.

Revealing Modular Architecture of Human Brain Structural Networks by Using Cortical Thickness from MRI

Modularity, presumably shaped by evolutionary constraints, underlies the functionality of most complex networks ranged from social to biological networks. However, it remains largely unknown in human cortical networks. In a previous study, we demonstrated a network of correlations of cortical thickness among specific cortical areas and speculated that these correlations reflected an underlying structural connectivity among those brain regions. Here, we further investigated the intrinsic modular architecture of the human brain network derived from cortical thickness measurement. Modules were defined as groups of cortical regions that are connected morphologically to achieve the maximum network modularity. We show that the human cortical network is organized into 6 topological modules that closely overlap known functional domains such as auditory/language, strategic/executive, sensorimotor, visual, and mnemonic processing. The identified structure-based modular architecture may provide new insights into the functionality of cortical regions and connections between structural brain modules. This study provides the first report of modular architecture of the structural network in the human brain using cortical thickness measurements.

Topological modules of continuous homomorphisms

In this paper, we consider the notion of module homomorphisms in the general topological module setting and establish their linearity and continuity under some suitable conditions. We also introduce the strict and uniform topologies on the modules of continuous linear homomorphisms and study their various properties.

Modular co-evolution of metabolic networks.

BackgroundThe architecture of biological networks has been reported to exhibit high level of modularity, and to some extent, topological modules of networks overlap with known functional modules. However, how the modular topology of the molecular network affects the evolution of its member proteins remains unclear.ResultsIn this work, the functional and evolutionary modularity of Homo sapiens (H. sapiens) metabolic network were investigated from a topological point of view. Network decomposition shows that the metabolic network is organized in a highly modular core-periphery way, in which the core modules are tightly linked together and perform basic metabolism functions, whereas the periphery modules only interact with few modules and accomplish relatively independent and specialized functions. Moreover, over half of the modules exhibit co-evolutionary feature and belong to specific evolutionary ages. Peripheral modules tend to evolve more cohesively and faster than core modules do.ConclusionThe correlation between functional, evolutionary and topological modularity suggests that the evolutionary history and functional requirements of metabolic systems have been imprinted in the architecture of metabolic networks. Such systems level analysis could demonstrate how the evolution of genes may be placed in a genome-scale network context, giving a novel perspective on molecular evolution.

Yeast Protein Interactome topology provides framework for coordinated-functionality

The architecture of the network of protein–protein physical interactions in Saccharomyces cerevisiae is exposed through the combination of two complementary theoretical network measures, betweenness centrality and ‘Q-modularity’. The yeast interactome is characterized by well-defined topological modules connected via a small number of inter-module protein interactions. Should such topological inter-module connections turn out to constitute a form of functional coordination between the modules, we speculate that this coordination is occurring typically in a pairwise fashion, rather than by way of high-degree hub proteins responsible for coordinating multiple modules. The unique non-hub-centric hierarchical organization of the interactome is not reproduced by gene duplication-and-divergence stochastic growth models that disregard global selective pressures.

Structure of modules over the stereotype algebra ℒ(X) of operators

It is well known that every module M over the algebra ℒ(X) of operators on a finite-dimensional space X can be represented as the tensor product of X by some vector space E, M ≅ = E ⊗ X. We generalize this assertion to the case of topological modules by proving that if X is a stereotype space with the stereotype approximation property, then for each stereotype module M over the stereotype algebra ℒ (X) of operators on X there exists a unique (up to isomorphism) stereotype space E such that M lies between two natural stereotype tensor products of E by X, $$E \circledast X \subseteq M \subseteq E \odot X.$$ . As a corollary, we show that if X is a nuclear Frechet space with a basis, then each Frechet module M over the stereotype operator algebra ℒ(X) can be uniquely represented as the projective tensor product of X by some Frechet space E, \(M = E \widehat \otimes X\).