Feedback regulation in the lactose operon: a mathematical modeling study and comparison with experimental data

Abstract

A mathematical model for the regulation of induction in the lac operon in Escherichia coli is presented. This model takes into account the dynamics of the permease facilitating the internalization of external lactose; internal lactose; beta-galactosidase, which is involved in the conversion of lactose to allolactose, glucose and galactose; the allolactose interactions with the lac repressor; and mRNA. The final model consists of five nonlinear differential delay equations with delays due to the transcription and translation process. We have paid particular attention to the estimation of the parameters in the model. We have tested our model against two sets of beta-galactosidase activity versus time data, as well as a set of data on beta-galactosidase activity during periodic phosphate feeding. In all three cases we find excellent agreement between the data and the model predictions. Analytical and numerical studies also indicate that for physiologically realistic values of the external lactose and the bacterial growth rate, a regime exists where there may be bistable steady-state behavior, and that this corresponds to a cusp bifurcation in the model dynamics.

FIGURE 1

Schematic representation of the lactose operon regulatory system.

FIGURE 2

The region in the space where a nonnegative steady state can exist as a function of external lactose levels L_e for the model when all parameters are held at the estimated values in Table 1 and when The shaded area shows the region where a steady state is not defined, whereas the solid line is the locus of values satisfying the steady state. The inset box shows that at large values of L_e, there is still a separation of the line for the steady state from the region where steady states are not defined. Notice that for these values of the parameters, there is a range of L_e values for which there are three coexisting steady-state values of allolactose A. The asterisk located at the right-most kink locates the minimal concentration of extracellular lactose required for induction, and our calculations indicate that it should be on the order of 62.0 μM.

FIGURE 3

β-galactosidase activity versus time when L_e = 8.0 × 10⁻² mM. The experimental data sets were taken from Knorre (1968) for E. coli ML30 (○) and from Pestka et al. (1984) for E. coli 294 (♦). The model simulation (solid line) was obtained using the parameters of Table 1 with a growth rate The selection of initial conditions is described in the text.

FIGURE 4

Oscillation in β-galactosidase activity in response to periodic phosphate feeding with period T = 100 min, which is the culture doubling time. The experimental data (^*) together with the model simulation (solid line) using the parameters of Table 1 and are presented. The experimental data are taken from Goodwin (1969). In the numerical simulation, periodic phosphate feeding was imitated by choosing a periodic function given by Eq. 7. The simulation was calculated by numerically solving the system of delay differential equations given by Eqs. 2–6. The initial conditions are the same as those values given in Table 2 for L_e = 8.0 × 10⁻² mM.

FIGURE 5

Semilog plot of β-galactosidase activity versus time (min) showing bifurcation in the numerical simulation with the parameters of Table 1 for five initial conditions and when L_e = 3.0 × 10⁻² mM, which is in the range of lactose concentration for the existence of three steady states. The selection of the five initial conditions is described in the text.

FIGURE 6

Semilog plot of β-galactosidase activity versus time showing effects of selection of the initial condition for t ∈[−τ, 0] in the numerical simulation with the parameters of Table 1 and various initial values of mRNA and allolactose oscillating around the unstable steady-state values corresponding to the middle branch of Fig. 2 when and L_e = 4.0 × 10⁻² mM (which is in the range of lactose concentration for coexistence of three steady states). The solid lines show the β-galactosidase activity when the initial allolactose functions are (n = 1, 2, …10), t ∈[−τ_M, 0], and the other variables are at the steady-state values on the middle branch. (Here is the unstable steady-state value of A on the middle branch). The dotted lines depict the temporal changes in β-galactosidase activity when (n = 1, 2, …10) for t ∈[−(τ_B + τ_P),0]. Again all the other variables are at the steady-state values when L_e = 4.0 × 10⁻² mM and is also the steady-state value of A on the middle branch. The steady-state values are , , , , and , when L_e = 4.0 × 10⁻² mM.