Article

# Returnability in complex directed networks (digraphs)

Linear Algebra and its Applications (Impact Factor: 0.97). 01/2009; 430:1886-1896. DOI: 10.1016/j.laa.2008.09.033

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**ABSTRACT:**Motivated by applications in quantum chemistry and solid state physics, we apply general results from approximation theory and matrix analysis to the study of the decay properties of spectral projectors associated with large and sparse Hermitian matrices. Our theory leads to a rigorous proof of the exponential off-diagonal decay ("nearsightedness") for the density matrix of gapped systems at zero electronic temperature in both orthogonal and non-orthogonal representations, thus providing a firm theoretical basis for the possibility of linear scaling methods in electronic structure calculations for non-metallic systems. We further discuss the case of density matrices for metallic systems at positive electronic temperature. A few other possible applications are also discussed.03/2012; - [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 03/2013; · 1.23 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**A number of key regulators of mouse embryonic stem (ES) cell identity, including the transcription factor Nanog, show strong expression fluctuations at the single-cell level. The molecular basis for these fluctuations is unknown. Here we used a genetic complementation strategy to investigate expression changes during transient periods of Nanog downregulation. Employing an integrated approach that includes high-throughput single-cell transcriptional profiling and mathematical modelling, we found that early molecular changes subsequent to Nanog loss are stochastic and reversible. However, analysis also revealed that Nanog loss severely compromises the self-sustaining feedback structure of the ES cell regulatory network. Consequently, these nascent changes soon become consolidated to committed fate decisions in the prolonged absence of Nanog. Consistent with this, we found that exogenous regulation of Nanog-dependent feedback control mechanisms produced a more homogeneous ES cell population. Taken together our results indicate that Nanog-dependent feedback loops have a role in controlling both ES cell fate decisions and population variability.Nature Cell Biology 10/2012; · 20.76 Impact Factor

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