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Partial rectangular metric spaces and fixed point theorems.

ABSTRACT: The purpose of this paper is to introduce the concept of partial rectangular metric spaces as a generalization of rectangular metric and partial metric spaces. Some properties of partial rectangular metric spaces and some fixed point results for quasitype contraction in partial rectangular metric spaces are proved. Some examples are given to illustrate the observed results.

SUBMITTER: Shukla S

PROVIDER: S-EPMC3926289 | 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.

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Partial rectangular metric spaces and fixed point theorems.

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