### abstract ###
Gene regulatory networks consist of direct interactions but also include indirect interactions mediated by metabolites and signaling molecules.
We describe how these indirect interactions can be derived from a model of the underlying biochemical reaction network, using weak time-scale assumptions in combination with sensitivity criteria from metabolic control analysis.
We apply this approach to a model of the carbon assimilation network in Escherichia coli.
Our results show that the derived gene regulatory network is densely connected, contrary to what is usually assumed.
Moreover, the network is largely sign-determined, meaning that the signs of the indirect interactions are fixed by the flux directions of biochemical reactions, independently of specific parameter values and rate laws.
An inversion of the fluxes following a change in growth conditions may affect the signs of the indirect interactions though.
This leads to a feedback structure that is at the same time robust to changes in the kinetic properties of enzymes and that has the flexibility to accommodate radical changes in the environment.
### introduction ###
The adaptation of bacteria to changes in their environment involves adjustments in the expression of genes coding for enzymes, regulators, membrane transporters, etc. CITATION CITATION.
These adjustments are controlled by gene regulatory networks ensuring the coordinated expression of clusters of functionally related genes.
The interactions in the network may be direct, as in the case of a gene coding for a transcription factor regulating the expression of another gene.
Most of the time, however, regulatory interactions are indirect, e.g. when a gene encodes an enzyme producing a transcriptional effector CITATION .
A gene regulatory network can thus not be reduced to its transcriptional regulatory interactions: by ignoring indirect interactions mediated by metabolic and signaling pathways we may miss crucial feedback loops in the system.
The network controlling carbon uptake in the bacterium Escherichia coli is a good example because it integrates metabolism, signal transduction, and gene expression.
At the level of gene expression, the network includes intricate feedback loops that arise from indirect interactions between the subsystems.
Global regulators like Crp control expression of enzymes in carbon metabolism CITATION CITATION, while intermediates of the latter pathways control the expression of global regulators.
For instance, the phosphorylation of EIIA activates adenylate cyclase to produce cAMP which is required for the activation of Crp CITATION, CITATION .
The aim of this paper is to develop a method for the systematic derivation of direct and indirect interactions in a gene regulatory network from the underlying biochemical reaction network.
Due to the complexity of the intermediate metabolic and signaling networks, determining indirect interactions is difficult in general.
We show that model reduction based on quasi-steady-state approximations expressing weak assumptions on time-scale hierarchies in the system CITATION CITATION, together with sensitivity criteria from metabolic control analysis CITATION, CITATION, are able to uncover such interactions.
Indeed the MCA formalism uniquely allows to relate systemic sensitivities with the sensitivities of individual reactions to reactants and effectors CITATION, CITATION.
It therefore provides a proper framework for investigating metabolic effects in gene regulation.
We apply our approach to a model of the upper part of the carbon assimilation network in E. coli, consisting of the glycolysis and gluconeogenesis pathways and their genetic and metabolic regulation.
The analysis of the derived gene regulatory network leads to three new insights.
First, contrary to what is often assumed, the network is densely connected due to numerous feedback loops resulting from indirect interactions.
This additional complexity is an important issue for the correct interpretation of data from genome-wide transcriptome studies.
Second, the derived gene regulatory network for carbon assimilation in E. coli is sign-determined, in the sense that the signs of interactions are essentially fixed by weak information on flux directions of biochemical reactions, without explicit specification of kinetic rate laws or parameter values.
Therefore the feedback structure is robust to changes in kinetic properties of enzymes and other biochemical reactions species.
Third, a change in environmental conditions may invert fluxes, and thus the signs of indirect interactions, resulting in a dynamic rewiring of the regulatory network.
