The diffusive influx and carrier efflux have a strong effect on the   bistability of the lac operon in Escherichia coli
  In the presence of gratuitous inducers, the lac operon of Escherichia coli exhibits bistability. Most models in the literature assume that the inducer enters the cell via the carrier (permease), and exits by a diffusion-like process. The diffusive influx and carrier efflux are neglected. However, analysis of the data shows that in non-induced cells, the diffusive influx is comparable to the carrier influx, and in induced cells, the carrier efflux is 7 times the diffusive efflux. Since bistability entails the coexistence of steady states corresponding to both non-induced and induced cells, neither one of these fluxes can be ignored. Here, we formulate a model accounting for both fluxes. We show that: (a) The thresholds of bistability are profoundly affected by both fluxes. The diffusive influx reduces the on threshold by enhancing inducer accumulation in non-induced cells. The carrier efflux increases the off threshold by decreasing inducer accumulation in induced cells. (b) Simulations of the model with experimentally measured parameter values are in good agreement with the data for IPTG. However, there are discrepancies with respect to the data for TMG. They are most likely due to two features missing from the model, namely, the variation of the inducer exclusion effect and the specific growth rate with the lactose enzyme levels. (c) The steady states and thresholds obtained in the presence of both fluxes are well approximated by simple analytical expressions, which may be useful for the preliminary design of the lac genetic switch in synthetic biology. 
J. T. Noel, S. S. Pilyugin, Atul Narang