From binary to multivalued to continuous models: the lac operon as a case study

Using the lac operon as a paradigmatic example for a gene regulatory system in prokaryotes, we demonstrate how qualitative knowledge can be initially captured using simple discrete (Boolean) models and then stepwise refined to multivalued logical models and finally to continuous (ODE) models. At all stages, signal transduction and transcriptional regulation is integrated in the model description. We first show the potential benefit of a discrete binary approach and discuss then problems and limitations due to indeterminacy arising in cyclic networks. These limitations can be partially circumvented by using multilevel logic as generalization of the Boolean framework enabling one to formulate a more realistic model of the lac operon. Ultimately a dynamic description is needed to fully appreciate the potential dynamic behavior that can be induced by regulatory feedback loops. As a very promising method we show how the use of multivariate polynomial interpolation allows transformation of the logical network into a system of ordinary differential equations (ODEs), which then enables the analysis of key features of the dynamic behavior.