The correlation between architecture and mRNA abundance in the genetic regulatory network of Escherichia coli.

BACKGROUND: Two aspects of genetic regulatory networks are the static architecture that describes the overall connectivity between the genes and the dynamics that describes the sequence of genes active at any one time as deduced from mRNA abundances. The nature of the relationship between these two aspects of these networks is a fundamental question. To address it, we have used the static architecture of the connectivity of the regulatory proteins of Escherichia coli to analyse their relationship to the abundance of the mRNAs encoding these proteins. In this we build on previous work which uses Boolean network models, but impose biological constraints that cannot be deduced from the mRNA abundances alone. RESULTS: For a cell population of E. coli, we find that there is a strong and statistically significant linear dependence between the abundance of mRNA encoding a regulatory protein and the number of genes regulated by this protein. We use this result, together with the ratio of regulatory repressors to promoters, to simulate numerically a genetic regulatory network of a single cell. The resulting model exhibits similar correlations to that of E. coli. CONCLUSION: This analysis clarifies the relationship between the static architecture of a regulatory network and the consequences for the dynamics of its pattern of mRNA abundances. It also provides the constraints on the architecture required to construct a model network to simulate mRNA production.

A thermodynamic view of networks.

Networks can be described by the frequency distribution of the number of links associated with each node (the degree of the node). Of particular interest are the power law distributions, which give rise to the so-called scale-free networks, and the distributions of the form of the simplified canonical law (SCL) introduced by Mandelbrot, which give what we shall call the Mandelbrot networks. Many dynamical methods have been obtained for the construction of scale-free networks, but no dynamical construction of Mandelbrot networks has been demonstrated. Here we develop a systematic technique to obtain networks with any given distribution of the degrees of the nodes. This is done using a thermodynamic approach in which we maximise the entropy associated with degree distribution of the nodes of the network subject to certain constraints. These constraints can be chosen systematically to produce the desired network architecture. For large networks we therefore replace a dynamical approach to the stationary state by a thermodynamical viewpoint. We use the method to generate scale-free and Mandelbrot networks with arbitrarily chosen parameters. We emphasise that this approach opens the possibility of insights into a thermodynamics of networks by suggesting thermodynamic relations between macroscopic variables for networks.

Networks as constrained thermodynamic systems.

We show how a network of interconnections between nodes can be constructed to have a specified distribution of nodal degrees. This is achieved by treating the network as a thermodynamic system subject to constraints and then rewiring the system to maintain the constraints while increasing the entropy. The general construction is given and illustrated by the simple example of an exponential network. By considering the constraints as a cost function analogous to an internal energy, we obtain a characterisation of the correspondence between the intensive and extensive variables of the network. Applied to networks in living organisms, this approach may lead to macroscopic variables useful in characterising living systems.