Effect of DNA looping on the induction kinetics of the lac operon.

The induction of the lac operon follows cooperative kinetics. The first mechanistic model of these kinetics is the de facto standard in the modeling literature [Yagil, G., Yagil, E., 1971. On the relation between effector concentration and the rate of induced enzyme synthesis. Biophys. J. 11, 11-17]. Yet, subsequent studies have shown that the model is based on incorrect assumptions. Specifically, the repressor is a tetramer with four (not two) inducer-binding sites, and the operon contains two auxiliary operators (in addition to the main operator). Furthermore, these structural features are crucial for the formation of DNA loops, the key determinants of lac repression and induction. Indeed, the repression is determined almost entirely (>95%) by the looped complexes [Oehler, S., Eismann, E.R., Krämer, H., Müller-Hill, B., 1990. The three operators of the lac operon cooperate in repression. EMBO J. 9(4), 973-979], and the pronounced cooperativity of the induction curve hinges upon the existence of the looped complexes [Oehler, S., Alberti, S., Müller-Hill, B., 2006. Induction of the lac promoter in the absence of DNA loops and the stoichiometry of induction. Nucleic Acids Res. 34(2), 606-612]. Here, we formulate a model of lac induction taking due account of the tetrameric structure of the repressor and the existence of looped complexes. We show that: (1) The kinetics are significantly more cooperative than those predicted by the Yagil and Yagil model. The cooperativity is higher because the formation of looped complexes is easily abolished by repressor-inducer binding. (2) The model provides good fits to the repression data for cells containing wild-type tetrameric or mutant dimeric repressor, as well as the induction curves for 6 different strains of Escherichia coli. It also implies that the ratios of certain looped and non-looped complexes are independent of inducer and repressor levels, a conclusion that can be rigorously tested by gel electrophoresis. (3) Repressor overexpression dramatically increases the cooperativity of the induction curve. This suggests that repressor overexpression can induce bistability in systems, such as growth of E. coli on lactose, that are otherwise monostable.