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Generalized contractive mappings and weakly α-admissible pairs in G-metric spaces.

ABSTRACT: The aim of this paper is to present some coincidence and common fixed point results for generalized (ψ, φ)-contractive mappings using partially weakly G-α-admissibility in the setup of G-metric space. As an application of our results, periodic points of weakly contractive mappings are obtained. We also derive certain new coincidence point and common fixed point theorems in partially ordered G-metric spaces. Moreover, some examples are provided here to illustrate the usability of the obtained results.

SUBMITTER: Hussain N

PROVIDER: S-EPMC4150471 | biostudies-other | 2014

REPOSITORIES: biostudies-other

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Project description:BackgroundIn the post-genomic era, a central and overarching question in the analysis of protein-protein interaction networks continues to be whether biological characteristics and functions of proteins such as lethality, physiological malfunctions and malignancy are intimately linked to the topological role proteins play in the network as a mathematical structure. One of the key features that have implicitly been presumed is the existence of hubs, highly connected proteins considered to play a crucial role in biological networks. We explore the structure of protein interaction networks of a number of organisms as metric spaces and show that hubs are non randomly positioned and, from a distance point of view, centrally located.ResultsBy analysing how the human functional protein interaction network, the human signalling network, Saccharomyces cerevisiae, Arabidopsis thaliana and Escherichia coli protein-protein interaction networks from various databases are distributed as metric spaces, we found that proteins interact radially through a central node, high degree proteins coagulate in the centre of the network, and those far away from the centre have low degree. We further found that the distribution of proteins from the centre is in some hierarchy of importance and has biological significance.ConclusionsWe conclude that structurally, protein interaction networks are mathematical entities that share properties between organisms but not necessarily with other networks that follow power-law. We therefore conclude that (i) if there are hubs defined by degree, they are not distributed randomly; (ii) zones closest to the centre of the network are enriched for critically important proteins and are also functionally very specialised for specific 'house keeping' functions; (iii) proteins closest to the network centre are functionally less dispensable and may present good targets for therapy development; and (iv) network biology requires its own network theory modelled on actual biological evidence and that simply adopting theories from the social sciences may be misleading.

Project description:BACKGROUND: We have recently shown by formally modelling human protein interaction networks (PINs) as metric spaces and classified proteins into zones based on their distance from the topological centre that hub proteins are primarily centrally located. We also showed that zones closest to the network centre are enriched for critically important proteins and are also functionally very specialised for specific 'house keeping' functions. We proposed that proteins closest to the network centre may present good therapeutic targets. Here, we present multiple pieces of novel functional evidence that provides strong support for this hypothesis. RESULTS: We found that the human PINs has a highly connected signalling core, with the majority of proteins involved in signalling located in the two zones closest to the topological centre. The majority of essential, disease related, tumour suppressor, oncogenic and approved drug target proteins were found to be centrally located. Similarly, the majority of proteins consistently expressed in 13 types of cancer are also predominantly located in zones closest to the centre. Proteins from zones 1 and 2 were also found to comprise the majority of proteins in key KEGG pathways such as MAPK-signalling, the cell cycle, apoptosis and also pathways in cancer, with very similar patterns seen in pathways that lead to cancers such as melanoma and glioma, and non-neoplastic diseases such as measles, inflammatory bowel disease and Alzheimer's disease. CONCLUSIONS: Based on the diversity of evidence uncovered, we propose that when considered holistically, proteins located centrally in the human PINs that also have similar functions to existing drug targets are good candidate targets for novel therapeutics. Similarly, since disease pathways are dominated by centrally located proteins, candidates shortlisted in genome scale disease studies can be further prioritized and contextualised based on whether they occupy central positions in the human PINs.

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Generalized contractive mappings and weakly α-admissible pairs in G-metric spaces.

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