Article

# Returnability in complex directed networks (digraphs)

Linear Algebra and its Applications (Impact Factor: 0.97). 04/2009; 430:1886-1896. DOI: 10.1016/j.laa.2008.09.033

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**ABSTRACT:**The study of quantitative structure-property relationships (QSPR) is important to study complex networks of chemical reactions in drug synthesis or metabolism or drug-target interactions networks. A difficult but also possible goal is the prediction of drug absorption, distribution, metabolism, and excretion (ADME) process with a single QSPR model. For this QSPR models need to use flexible structural parameters useful for the description of many different systems at different structural scales (multi-scale parameters). They also need to use also powerful analytical methods able to link in a single multi-scale hypothesis structural parameters of different target systems (multi-target modeling) with different experimental properties of these systems (multi-output models). In this sense, the QSPR study of complex bio-molecular systems may benefit substantially from the combined application of spectral moments of graph representations of complex systems with perturbation theory methods. On one hand, spectral moments are almost universal parameters that can be calculated to many different matrices used to represent the structure of the states of different systems. On the other hand, perturbation methods can be used to add "small" variation terms to parameters of a known state of a given system in order to approach to a solution for another state of the same or similar system with unknown properties. Here we present one state-of-art review about the different applications of spectral moments to describe complex bio-molecular systems. Next, we give some general ideas and formulate plausible linear models for a general-purpose perturbation theory of QSPR problems of complex systems. Last, we develop three new QSPR-Perturbation theory models based on spectral moments for three different problems with multiple in-out boundary conditions that are relevant to bio-molecular sciences. The three models developed correctly classify more than pairs 115,600; 48,000; 134,900 cases of the effects of in-out perturbations in intra-molecular carbolithiations, drug ADME process, or self-aggregation of micelle nanoparticles of drugs or surfactants. The Accuracy (Ac), Sensitivity (Sn), and Specificity (Sp) of these models were >90% in all cases. The first model predicts variations in the yield or enantiomeric excess due to structural variations or changes in the solvent, temperature, temperature of addition, or time of reaction. The second model predict changes in >18 parameters of biological effects for >3000 assays of ADME properties and/or interactions between 31,723 drugs and 100 targets (metabolizing enzymes, drug transporters, or organisms). The third model predicts perturbations due to changes in temperature, solvent, salt concentration, and/or structure of anions or cations in the self-aggregation of micelle nanoparticles of drugs and surfactants.Current Drug Metabolism 09/2014; 15(4):470-488. · 4.41 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The concept of network returnability is reformulated as an equilibrium constant for a reaction network. Using this concept we study the atmospheric reaction networks of Earth, Mars, Venus and Titan. We found that the reaction network in the Earth’s atmosphere has the largest disequilibrium, followed by that of Titan which is still far from the most returnable atmospheres of Mars and Venus. We find that the chemical species with null or very low returnability are those in the highest disequilibrium in their respective atmospheres mainly due to physical, biogenic and/or anthropogenic mechanisms.Journal of Mathematical Chemistry 01/2012; 50(6). · 1.23 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Many of the chemical reactions that take place within a living cell are irreversible. Due to evolutionary pressures, the number of allowable reactions within these systems are highly constrained and thus the resulting metabolic networks display considerable asymmetry. In this paper, we explore possible evolutionary factors pertaining to the reduced symmetry observed in these networks, and demonstrate the important role environmental variability plays in shaping their structural organization. Interpreting the returnability index as an equilibrium constant for a reaction network in equilibrium with a hypothetical reference system, enables us to quantify the extent to which a metabolic network is in disequilibrium. Further, by introducing a new directed centrality measure via an extension of the subgraph centrality metric to directed networks, we are able to characterise individual metabolites by their participation within metabolic pathways. To demonstrate these ideas, we study 116 metabolic networks of bacteria. In particular, we find that the equilibrium constant for the metabolic networks decreases significantly in-line with variability in bacterial habitats, supporting the view that environmental variability promotes disequilibrium within these biochemical reaction systems.Journal of Mathematical Chemistry 01/2014; 52(2):675-688. · 1.23 Impact Factor

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