A simple framework to describe the regulation of gene expression in prokaryotes

Based on the bimolecular mass action law and the derived mass conservation laws, we propose a mathematical framework in order to describe the regulation of gene expression in prokaryotes. It is shown that the derived models have all the qualitative properties of the activation and inhibition regulatory mechanisms observed in experiments. The basic construction considers genes as templates for protein production, where regulation processes result from activators or repressors connecting to DNA binding sites. All the parameters in the models have a straightforward biological meaning. After describing the general properties of the basic mechanisms of positive and negative gene regulation, we apply this framework to the self-regulation of the trp operon and to the genetic switch involved in the regulation of the lac operon. One of the consequences of this approach is the existence of conserved quantities depending on the initial conditions that tune bifurcations of fixed points. This leads naturally to a simple explanation of threshold effects as observed in some experiments.