Mathematical Model of the lac Operon: Inducer Exclusion, Catabolite Repression, and Diauxic Growth on Glucose and Lactose Patrick Wong, Stephanie Gladney, and J. D. Keasling* Department of Chemical Engineering, University of California, Berkeley, California 94720-1462 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 -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 -galac- tosidase-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. Introduction The lactose (lac) operon encodes the genes in the pathway for the import of lactose into the cell and its transformation to glucose and galactose. Due to the large amount of information that has been acquired, this operon has served as a model for genetic control. It has also served as a workhorse for expression of heterologous genes. The use of the lac repressor and promoter for heterologous gene expression has motivated the formula- tion of mathematical models to describe induction of the lac operon by the inducer IPTG (Lee and Bailey, 1984a,b; Laffend and Shuler, 1994a). However, relatively few models have been developed to describe induction of the lac operon by lactose (Van Dedem and Moo-Young, 1975) or growth on mixed substrates (Straight and Ramkrish- na, 1991; Ramakrishna et al., 1996), and no models have considered the effects of catabolite repression and inducer exclusion on lac operon induction by lactose. Mathemati- cal models of lac operon function during growth on glucose and lactose could give insight into the natural function of the various control mechanisms involved in lac operon regulation. The lac operon is induced when lactose is transported into the cell, and a fraction of the lactose is transformed by -galactosidase into the inducer allolactose. Allolac- tose binds to the lac repressor thereby derepressing expression of the genes for lac permease, transacetylase, and -galactosidase. The remainder of the lactose and the allolactose are hydrolyzed to galactose and glucose, which then enter glycolysis and the Krebs cycle. The lac operon is controlled by glucose at two levels: inducer exclusion and catabolite repression. Transport of lactose by the lac permease is inhibited by glucose, a phenomenon known as inducer exclusion. The second mechanism by which glucose affects the lactose operon is through catabolite repression. In the absence of glucose, cyclic AMP (cAMP) accumulates inside the cell where it binds to the cAMP receptor protein (CRP). The cAMP-CRP complex binds to the CRP binding region near the lac promoter and enhances transcription. In the absence of binding of the cAMP-CRP complex to the DNA, transcription of genes under control of catabolite repression is significantly reduced (Saier et al., 1996). We present a model to describe regulation of the lac operon in Escherichia coli during growth on lactose and glucose. The kinetic information centers on the control of the lac operon, the transport of lactose into the cell by lac permease, the formation of allolactose, the consump- tion of lactose and allolactose by -galactosidase, and cell growth on an intermediate of glucose and lactose catabo- lism, glucose 6-phosphate (Glu6P). The model incorpo- rates catabolite repression and inducer exclusion. Where possible, literature estimates for model parameters were used. * Corresponding author: FAX, (510) 642-4778; phone, (510) 642- 4862; e-mail, keasling@socrates.berkeley.edu. 132 Biotechnol. Prog. 1997, 13, 132-143 S8756-7938(97)00003-9 CCC: $14.00 © 1997 American Chemical Society and American Institute of Chemical Engineers