Formal Representation of GRNs

G Vector of all transcriptional units (TUs) involved in the GRN (in terms, for example, of gene dosage per TU). 

R Vector of messenger RNA (mRNA), transfer RNA (tRNA), ribosomal RNA (rRNA), and other processed RNAs. 

An example of a GRN is given next to illustrate the formal representation just described. The example also demonstrates the art of modeling and reduction of the network into minimal mathematical models. 

The lac operon in the bacterium Escherichia coli is a well-studied GRN. This prokaryotic gene network has been the subject of numerous reviews it is discussed here primarily to illustrate the various aspects of GRN modeling, starting with the information on genome organization (operon structure) to knowledge on protein-DNA interactions, protein-protein interactions, and the influence of metabolites. 

New mathematical models and reviews on the lac operon have appeared recently. Yildirim and Mackey used delay differential equations to account for the transcriptional and translational steps that are missing in their model. An earlier detailed kinetic model was proposed and analyzed by Wong, Gladney and Keasling. Recently, Vilar, Guet, and Leibler used a four-variable model that captures many of the essential dynamics of the lac operon. Note that the Vilar-Guet-Leibler model is essentially a three-variable model. The bistability exhibited by the model was the explanation for the on-off behavior of the lac operon. 

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