Mathematical Model of the <i>lac</i> Operon: Inducer Exclusion, Catabolite Repression, and Diauxic Growth on Glucose and Lactose

Abstract A mathematical model of the lactose ( lac ) operon was developed to study diauxic growth on glucose and lactose. The model includes catabolite repression, inducer exclusion, lactose hydrolysis to glucose and galactose, and synthesis and degradation of allolactose. Two models for catabolite repression were tested: (i) cyclic AMP (cAMP) synthesis inversely correlated with the external glucose concentration and (ii) synthesis inversely correlated with the glucose transport rate. No significant differences in the two models were observed. In addition to synthesis, degradation and secretion of cAMP were also included in the model. Two models for the phosphorylation of the glucose produced from lactose hydrolysis were also tested: (i) phosphorylation by intracellular hexokinase and (ii) secretion of glucose and subsequent phosphorylation upon transport back into the cell. The latter model resulted in weak catabolite repression when the glucose produced from lactose was transported out of the cell, whereas the former model showed no catabolite repression during growth on lactose. Parameter sensitivity analysis indicates the importance of key parameters to lac operon expression and cell growth: the lactose and allolactose transformation rates by β‐galactosidase and the glucose concentrations that affect catabolite repression and inducer exclusion. Large values of the allolactose hydrolysis rate resulted in low concentrations of allolactose, low‐level expression of the lac operon, and slow growth due to limited import and metabolism of lactose; small values resulted in a high concentration of allolactose, high‐level expression of the lac operon, and slow growth due to a limiting concentration of glucose 6‐phosphate formed from allolactose. Changes in the rates of all β‐galactosidase‐catalyzed reactions showed similar behavior, but had more drastic effects on the growth rate. Changes in the glucose concentration that inhibited lactose transport could extend or contract the diauxic growth period during growth in the presence of glucose and lactose. Moreover, changes in the glucose concentration that affected catabolite repression affected the cAMP levels and lac operon expression, but had a lesser effect on the growth rate.