Positive feedback exists and drives the glucose-mediated repression in Escherichia coli

Abstract

The expression of the lac operon of E. coli is subject to positive feedback during growth in the presence of gratuitous inducers, but its existence in the presence of lactose remains controversial. The key question in this debate is: Do the lactose enzymes, Lac permease and β-galactosidase, promote accumulation of allolactose? If so, positive feedback exists since allolactose does stimulate synthesis of the lactose enzymes. Here, we addressed the above question by developing methods for determining the intracellular allolactose concentration as well as the kinetics of enzyme induction and dilution. We show that, during lac induction in the presence of lactose, the intracellular allolactose concentration increases with the lactose enzyme level, which implies that lactose enzymes promote allolactose accumulation, and positive feedback exists. We also show that, during lac repression in the presence of lactose + glucose, the intracellular allolactose concentration decreases with the lactose enzyme levels, which suggests that, under these conditions, the positive feedback loop turns in the reverse direction. The induction and dilution rates derived from the transient data show that the positive feedback loop is reversed due to a radical shift of the steady-state induction level. This is formally identical to the mechanism driving catabolite repression in the presence of TMG + glucose.

Significance

The lac operon of E. coli exhibits bistability in the presence of TMG but not lactose. To explain this observation, it has been hypothesized that positive feedback exists in the presence of TMG but not lactose, and this is because the lactose enzymes promote the accumulation of intracellular TMG but not allolactose, i.e., the intracellular TMG level increases with, but the intracellular allolactose level remains independent of, the lactose enzyme levels. In contrast, our data show that positive feedback exists even in the presence of lactose because the allolactose level and lac expression rate always increase with the lactose enzyme levels. Positive feedback also plays a crucial role in glucose-mediated repression because the repression disappears when positive feedback is abolished.

Introduction

The catabolite repression of the lac operon of E. coli in the presence of glucose is usually attributed to cAMP-mediated inhibition of lac transcription and EIIA^glc-mediated inhibition of lactose transport, referred to hereafter as cAMP regulation and inducer exclusion, respectively (1,2). Due to growing evidence that cAMP regulation, by itself, cannot account for the repression, it has been postulated that catabolite repression is entirely due to inducer exclusion (3,4). However, as shown in the prequel to this work (5), inducer exclusion represses lac expression by no more than 6-fold, and the 900-fold repression observed in the presence of glucose plus small concentrations of ¹⁴C-methyl-β-D-thiogalactoside (TMG) is primarily due to positive feedback which amplifies the small effects of cAMP-mediated regulation and inducer exclusion.

We have proposed that the same mechanism also drives glucose-mediated lac repression in the presence of lactose (6, 7, 8, 9). In this case, positive feedback occurs because lactose permease (LacY) and β-galactosidase (LacZ), referred to hereafter as lactose enzymes, promote the accumulation of allolactose, which in turn stimulates the synthesis of even more lactose enzymes. In the presence of pure lactose, this positive feedback loop results in the progressive accumulation of allolactose and the lactose enzymes. However, upon addition of glucose to a culture growing on lactose, the positive feedback loop turns in the reverse direction, thus leading to progressive depletion of allolactose and the lactose enzymes. More precisely, upon the addition of glucose, the lac expression rate decreases due to cAMP regulation and inducer exclusion. The resultant decrease of the lactose enzyme levels causes the decline of the allolactose level, which in turn leads to further reduction of the lactose enzyme levels. The repetition of this “reversed cycle” results in the progressive depletion of both lactose enzyme and allolactose levels, ultimately yielding almost complete repression. Thus, lac induction and repression result from the turning of the same positive feedback loop, but in opposite directions.

Our model is formally identical to the models proposed for lac induction (10, 11, 12) and glucose-mediated repression in the presence of the nonmetabolizable inducer TMG (5). However, theoretical studies have argued that the positive feedback loop postulated by our model exists only in the presence of nonmetabolizable inducers—it does not exist in the presence of the metabolizable substrate lactose (13, 14, 15). Specifically, these studies acknowledge that both TMG and allolactose stimulate synthesis of the lactose enzymes, but that these enzymes promote the accumulation of only TMG, and not allolactose. The mathematical basis of this claim is detailed in section S1, but it can be explained intuitively by considering the effect of increasing lactose enzyme levels on the intracellular TMG and allolactose concentrations, which can be assumed to be in quasi-steady state since they evolve much faster than the lactose enzyme levels. First, consider the case of cells exposed to TMG, which is accumulated in the cell by the permease, and depleted by passive diffusion (Fig. S1). If the permease level of such cells is increased, the accumulation rate of intracellular TMG is enhanced. Since intracellular TMG reaches quasi-steady state within minutes, this increase of the TMG accumulation rate must be rapidly followed by an equal increase of its depletion rate by passive diffusion. This can be achieved only if the intracellular TMG level increases by an amount proportional to the original increment of the permease level. Thus, the quasi-steady-state intracellular TMG level is proportional to the permease level—a precise expression of the informal claim that the lactose enzymes promote the accumulation of TMG. Next, consider the case of cells exposed to intracellular lactose, which is accumulated by the permease, and depleted by β-galactosidase rather than passive diffusion (Fig. S2). If the permease level of such cells is increased, the lactose uptake rate increases. However, since permease and β-galactosidase are coordinately synthesized, any increase of the permease level is accompanied by a proportional increase of the β-galactosidase level. Consequently, the increase of the lactose uptake rate is automatically balanced by an equal increase of the lactose removal rate, and quasi-steady state is achieved without any change in the intracellular lactose level. The quasi-steady-state intracellular lactose concentration is therefore independent of the lactose enzyme levels. Since allolactose is synthesized and depleted by the very same enzyme, β-galactosidase (Fig. S2), it follows a fortiori that the quasi-steady-state allolactose level is also independent of the lactose enzyme level. Thus, the lactose enzymes fail to promote any accumulation of allolactose, and there is no positive feedback at all.

The purported absence of positive feedback in the presence of lactose is based on two model assumptions that are inconsistent with the data and relaxing them restores the positive feedback. First, the models neglect the significant excretion of intracellular lactose and allolactose (16), which can restore the strong dependence of the quasi-steady-state allolactose concentrations on the prevailing lactose enzyme levels. To see this, observe that the drastically different behavior in the presence of TMG and lactose stems from the contrasting model assumptions regarding the fate of TMG and lactose: TMG is excreted but not metabolized, whereas lactose is metabolized but not excreted (17). Since lactose is not only metabolized but also excreted, the behavior in the presence of lactose can approximate that in the presence of TMG (18). This intuitive argument is corroborated by mathematical analysis (section S1), which shows furthermore that excretion of allolactose can also generate positive feedback. Second, the models assume that the LacY:LacZ ratio is always constant due to their coordinate synthesis, but this is true only in the absence of inducer exclusion. In the presence of glucose, some of the LacY, but not LacZ, molecules are inactivated by inducer exclusion, so that the LacY:LacZ ratio is not constant, but increases with the LacZ level (5). Under this condition, the intracellular lactose and allolactose levels increase with the LacZ level, and positive feedback can exist in the presence of lactose (section S1). As a side note, we may also observe that, since LacY, but not LacZ, is partially inactivated in the presence of glucose, the activity of LacZ, rather than LacY, provides a more accurate measure of lac expression. Henceforth, the terms induction level and induction rate will refer to the specific activity and synthesis rate of LacZ, respectively.

The above theoretical arguments show that, even in the presence of lactose, positive feedback can exist and therefore drive glucose-mediated repression. However, since there is no empirical evidence for this, we set out to experimentally test the validity of our model hypotheses:

• 1.Positive feedback exists and drives glucose-mediated catabolite repression.
• 2.Positive feedback exists because allolactose stimulates synthesis of the lactose enzymes (19), which in turn promote accumulation of allolactose.

To this end, we developed methods for determining the intracellular allolactose concentration as well as the LacZ induction and dilution rates during the course of induction in the presence of lactose and repression in the presence of lactose + glucose. We verified the first hypothesis by showing that the induction rate increases with the induction level; moreover, the induction and dilution rates in the presence of glucose + lactose are formally similar to those observed in the presence of glucose + TMG (5) where the existence of positive feedback is widely accepted (20). To verify the second hypothesis, we measured the intracellular allolactose levels during induction in the presence of lactose, and repression in the presence of lactose + glucose. In both cases, the intracellular allolactose level increased with the induction level, which verifies our second hypothesis and contradicts the conclusion derived from mathematical models described above.

Materials and methods

Growth conditions

The wild-type strain E. coli K12 MG1655 was obtained from the Coli Genetic Stock Center at Yale University. Single colonies stored on LB Agar plates were inoculated in LB growth medium, grown for 6–8 h at 37°C, and then grown overnight at 37°C after a 1:1000 dilution to M9 medium (21) supplemented with the appropriate carbon source. In the induction experiment, cells were pregrown in maltose (5.6 mM), and transferred to prewarmed medium containing maltose (2.8 mM) + lactose (4 mM). We used this mixture instead of pure lactose to minimize the heterogeneity of the cell population that occurs during growth on pure lactose (22). In the repression experiment, cells were pregrown on maltose (2.8 mM) + lactose (4 mM), and transferred to prewarmed medium containing glucose (2.2 mM) + lactose (4 mM). The cell density (gdw/L) was followed by measuring the optical density at 600 nm (OD₆₀₀), which provided the cell density with relative error <2% via the expression gdw/L = 0.35 × OD₆₀₀. To ensure saturating substrate levels throughout the experiment, partial removal of cells was often necessary. To this end, a part of the culture was centrifuged and the supernatant was reintroduced into the culture medium.

Sample preparation

For each time point, three aliquots of 0.5–2 mL each were withdrawn from the experimental cultures. The first aliquot was used to measure the OD₆₀₀. The second aliquot, which was used to measure the β-galactosidase activity, was stored in an ice-water bath until the assay. The third aliquot, which was used to measure extracellular cAMP and allolactose concentrations, was rapidly filtered through a 0.22 μm nylon or polyethersulfone filter, and transferred to a boiling water bath to deactivate any residual enzymes. These filtered samples (also referred to as medium samples) were appropriately diluted in deionized water and stored at −20°C until cAMP and/or sugar analysis. For measuring the total allolactose concentrations (referred to as total samples in Fig. 1), a fourth aliquot was also drawn, which was rapidly diluted in boiling water to deactivate enzymes and release the intracellular allolactose into the medium.

Figure 1.

There is substantial efflux of allolactose and cAMP during growth of E. coli K12 MG1655 on lactose + maltose. (A) Evolution of extracellular cAMP (circles) and allolactose (squares) concentrations after inoculation of two shake flasks, each containing 4 mM lactose and 2.8 mM maltose, with uninduced cells grown on maltose (open symbols) and preinduced cells grown on lactose + maltose (solid symbols). The arrows show the times at which some of the cells were removed to decrease the cell density. (B) The difference between the mean allolactose concentrations in total and medium samples, based on three samples, is on the order of the measurement error. The allolactose concentrations in the total and medium samples were measured 6 min after inoculation in five independent experiments of the type described in (A). The inset shows that the intracellular concentrations calculated by the difference method suffer from unacceptably high error (the error bars represent the standard error stemming from three or more measuments). In this calculation, the cellular water volume was assumed to be 2.7 mL gdw⁻¹ (23).

Measurement of β-galactosidase activity

The specific enzyme activity of β-galactosidase was assayed using Miller's method (21), which yielded error of <3% in our hands.

Measurement of sugar concentration

To quantify sugar concentrations, we used high-performance anion-exchange chromatography with pulsed amperometric detection in a Dionex ICS3000 system (Dionex, Sunnyvale, CA) (20). Sugars were separated at 30°C on a CarboPac PA1 column (4 × 250 mm) connected to a CarboPac PA1 guard column (Dionex) using eluent containing 10 mM NaOH and 2 mM barium acetate (Eluent A) to precipitate the bicarbonate (24). Appropriately diluted samples were freshly thawed and sparged with N₂ for >45 s. Manual injections (100 μL loop) were used to ensure reproducibility and minimize sample usage. Peaks obtained were integrated using Chromeleon 6.8 software. The relative error was invariably <3%.

Measurement of cAMP concentration

The Dionex ICS3000 system was also used for measuring the extracellular cAMP concentrations. The chromatographic separation and quantification method were adapted from Bhattacharya et al. (25). High-performance anion-exchange chromatography with variable wavelength detector was used. Separation of cAMP was carried out at 30°C on an IonPac AS11-HC column (4 × 250 mm) connected to an IonPac AS11 guard column (Dionex) using gradient elution with a flow rate of 1.0 mL/min. Combinations of 100 mM sodium hydroxide (Eluent B) and deionized water (Eluent C) were used to create the gradients. Initially, 7% B and 93% C were mixed (7 mM NaOH) and maintained for 10 min. The gradient of B was increased linearly in 2 min—50% B and 50% C (50 mM NaOH) —and maintained for 6 min. This was followed by restoring the eluent concentration to 7 mM in 2 min. The column was restabilized with initial elution conditions for 5–8 min before the next injection. Samples were freshly thawed and sparged with N₂ for >45 s.

Generation of induction curve for IPTG

LacY⁻ mutants were isolated using UV mutagenesis and tested for the absence of LacY activity by measuring the accumulation of TMG (21). When exposed to ¹⁴C-TMG, these mutants developed equal concentrations on either side of the cellular membrane (5). To generate the induction curve for IPTG, the LacY⁻ cells were grown for at least 10 generations on a medium containing maltose (11 mM) or glycerol (43 mM), and various levels of IPTG, after which the specific β-galactosidase activity was measured using Miller's method.

Determination of growth, enzyme synthesis, and metabolite efflux kinetics

We are interested in the variation of the specific rates of growth $\mu$, β-galactosidase synthesis $r_{E_g}$, cAMP efflux $r_{C,d}^{-}$, and allolactose efflux $r_{A,d}^{-}$ during the course of lac induction and repression (excretion of allolactose and cAMP assumed to be via diffusion, see below). These rates were obtained by measuring the evolution of the concentrations of biomass $X\left(t\right)$, β-galactosidase $E_g\left(t\right)$, extracellular cAMP $C_e\left(t\right)$, and extracellular allolactose $A_e\left(t\right)$, and then appealing to the following mass balances describing their evolution:

$$\frac{dX}{dt}=\mu X\Rightarrow \mu =\frac{1}{X}\frac{dX}{dt},$$ (1)

$$\frac{d{E}_g}{dt}=r_{E_g}-\mu E_g\Rightarrow r_{E_g}=\frac{d{E}_g}{dt}+\mu E_g,$$ (2)

$$\frac{d{C}_e}{dt}=(r_{C,d}^{-}-r_C^{+})X\Rightarrow r_{C,d}^{-}=\frac{1}{X}\frac{d{C}_e}{dt}+r_C^{+},$$ (3)

$$\frac{d{A}_e}{dt}=(r_{A,d}^{-}-r_A^{+})X\Rightarrow r_{A,d}^{-}=\frac{1}{X}\frac{d{A}_e}{dt}+r_A^{+},$$ (4)

where $\mu E_g$ denotes the rate of β-galactosidase dilution (by growth), and $r_C^{+}$, $r_A^{+}$ denote the specific rates of influx of intracellular cAMP and allolactose, respectively. Since we measured $X\left(t\right)$, $E_g\left(t\right)$, $C_e\left(t\right)$, and $A_e\left(t\right)$, we could calculate $\mu$, $d{E}_g/dt$, and the net specific efflux rates of cAMP $\left(1/X\right)\left(d{C}_e/dt\right)$ and allolactose $\left(1/X\right)\left(d{A}_e/dt\right)$. Now, given $\mu$, $E_g$, and $d{E}_g/dt$, we calculated the dilution rate $\mu E_g$, and hence, the induction rate $r_{E_g}$ by appealing to Eq. 2. Also, given $\left(1/X\right)\left(d{C}_e/dt\right)$ and $\left(1/X\right)/\left(d{A}_e/dt\right)$, we calculated the efflux rates $r_{C,d}^{-}$ and $r_{A,d}^{-}$ from Eqs. 3 and 4 provided the uptake rates $r_C^{+}$ and $r_A^{+}$ were known. We show below that there is no cAMP uptake ($r_C^{+}=0$), and that the allolactose uptake rate $r_A^{+}$ can be determined from suitable experiments.

We assume that cAMP and allolactose are excreted from the cell by diffusion, and that their extracellular concentrations $C_e$, $A_e$ are negligible compared with the corresponding intracellular concentrations denoted $C$, $A$, i.e.,

$$r_{C,d}^{-}=k_{C,d}^{-}\left(C-C_e\right)\approx k_{C,d}^{-}C,r_{A,d}^{-}=k_{A,d}^{-}\left(A-A_e\right)\approx k_{A,d}^{-}A,$$ (5)

where $k_{C,d}^{-}$, $k_{A,d}^{-}$ denote the diffusivities of cAMP and allolactose across the cell membrane. Epstein and co-workers showed that the specific efflux rate of cAMP was proportional to its intracellular concentration (26). We show below that the same is also true for allolactose.

Data analysis

Data were processed using Microsoft Office Excel 2007 and MATLAB 2016b (The MathWorks, Natick, MA). In particular, the transient data were fitted by the least-square cubic B-spline method (27) using splinetool in MATLAB, and the derivatives of the fitted splines were obtained using the fnder command within the splinetool toolbox.

Results

Measurement of intracellular allolactose by direct methods is prone to error

Two methods are generally used to quantify the intracellular concentration of a metabolite, namely the cell-separation method in which the concentration is measured after separating the cells from the medium, and the difference method in which the concentration is determined (without separating the cells from the medium) by subtracting the amount of metabolite in the medium from the total amount in the culture sample containing both cells and medium (28, 29, 30, 31, 32). However, both methods are prone to large errors when significant quantities of the metabolite accumulate in the medium (33,34). We show below that, since there is significant accumulation of allolactose in the medium, the above methods lead to unacceptably large errors in the measurement of intracellular allolactose concentrations.

Since the existing cell-separation methods involve the use of chemical quenching agents that invariably cause metabolite leakage (30), we devised a thermal quenching method by isolating a mutant that expressed temperature-sensitive LacZ. When this mutant was diluted in preheated medium at 60°C, LacZ was completely inactivated within 2 s (Fig. S3 A). However, the allolactose concentrations, measured after filtering the quenched cells and extracting their intracellular allolactose in ∼20 s, were extremely variable. This was probably due to rapid efflux of intracellular allolactose because when wild-type cells preloaded with [¹⁴C]TMG were diluted into prewarmed medium at 60°C, 50% of the [¹⁴C]TMG was lost within 20 s (Fig. S3 B).

The efflux of intracellular allolactose also precluded the difference method. Indeed, when induced or noninduced cells of E. coli were grown in the presence of maltose + lactose, allolactose accumulated in the medium (Fig. 1 A). Although the extracellular allolactose concentrations were not especially large (≤20 μM), the intracellular concentrations could not be determined precisely by the difference method. Fig. 1 B shows that, even when the mean concentration of extracellular allolactose is only ∼1 μM, the difference between the mean concentrations of total allolactose (in medium + cells) and extracellular allolactose (in medium) is $\lesssim 0.1$ μM, which is comparable with the error in the measurement of the mean concentrations. The difference method is therefore susceptible to unacceptably large errors because it involves calculation of the difference between two measured, and hence error-prone, quantities of almost equal magnitudes, namely the total and extracellular allolactose concentrations.

There is significant consumption of extracellular allolactose, but not cAMP

The inherent errors of the methods described above led us to consider an alternative method for measuring the intracellular allolactose concentration. It turns out that the intracellular concentrations of several excreted compounds, such as cAMP and synthetic galactosides, are proportional to their specific efflux rates (26,35, 36, 37). This led us to devise a new method involving measurement of the specific allolactose efflux rate $r_{A,d}^{-}$ as a surrogate for the intracellular allolactose concentration.

If the cells do not consume the extracellular allolactose, Eq. 4 implies that $r_{A,d}^{-}$ equals $\left(1/X\right)/\left(d{A}_e/dt\right)$, which can be easily determined by measuring the concentrations of biomass and extracellular allolactose (see supplement for the complete list of notations). However, Fig. 1 A suggests that there is significant consumption of extracellular allolactose because its concentration increases rather slowly compared with the exponentially rising extracellular cAMP concentration. This becomes more transparent if we plot the instantaneous concentrations of extracellular cAMP and allolactose of preinduced cells (Fig. 1 A) against the corresponding biomass concentrations. To see this, observe that the slope of these $C_e\left(t\right)$ vs. $X\left(t\right)$ and $A_e\left(t\right)$ vs. $X\left(t\right)$ phase plots, commonly referred to as differential plots, are given by the relations

$$\frac{d{C}_e}{dX}=\frac{r_{C,d}^{-}-r_C^{+}}{\mu },\frac{d{A}_e}{dX}=\frac{r_{A,d}^{-}-r_A^{+}}{\mu },$$ (6)

obtained by dividing Eqs. 3 and 4 by Eq. 1. Since preinduced cells have been growing exponentially for several generations, they undergo balanced growth, so that $r_{C,d}^{-}$, $r_{A,d}^{-}$, and $\mu$ are constant throughout the experiment (38). Now, at $t=0$, $r_C^{+}$ and $r_A^{+}$ are zero because there is no cAMP and allolactose in the medium. Hence, Eq. 6 implies that the slopes of the differential plots are $d{C}_e/dX=r_{C,d}^{-}/\mu$ and $d{A}_e/dX=r_{A,d}^{-}/\mu$ at $t=0$, and they remain constant at these initial values only if uptake remains negligible compared with efflux. Fig. 2 shows that in preinduced cells, the slope of the differential plot for cAMP remains constant (solid circles). Hence, cAMP uptake is negligible ($r_C^{+}\ll r_{C,d}^{-}$) throughout the experiment, and it follows from Eq. 3 that $r_{C,d}^{-}\approx \left(1/X\right)\left(d{C}_e/dt\right)$ can be easily calculated from the biomass and extracellular cAMP concentrations. In contrast, the slope of the differential plot for allolactose decreases markedly and approaches zero at the end of the experiment (solid squares). Hence, there is significant allolactose uptake during the experiment, and Eq. 4 implies that the specific allolactose efflux rate $r_{A,d}^{-}$ cannot be calculated from the concentrations of biomass and extracellular allolactose unless the specific allolactose uptake rate $r_A^{+}$ is determined.

Figure 2.

There is substantial uptake of allolactose, but not cAMP, during growth of E. coli K12 MG1655 on maltose + lactose. Differential plots of the data in Fig. 1A showing the variation of the extracellular cAMP (circles) and allolactose (squares) concentrations with the cell density in cultures inoculated with preinduced cells. The differential plots for cAMP are linear throughout the experiment; however, the slopes of the differential plots for allolactose declines substantially toward the end.

Development of a method for quantifying the specific allolactose uptake rate

The determination of $r_A^{+}$ is facilitated by the observation that allolactose uptake is mediated primarily by Lac permease because $r_A^{+}$ decreases 30-fold in lacY^– mutants (section S2). Hence, assuming allolactose uptake follows Michaelis-Menten kinetics, $r_A^{+}$ can be represented by the expression

$$r_A^{+}=k_A^{+}{E}_p\frac{A_e}{K_A^{+}+A_e},$$ (7)

where $E_p$ denotes the instantaneous specific LacY activity. In particular, the uptake rate of preinduced cells is

$$r_A^{+}=V_A^{+}\frac{A_e}{K_A^{+}+A_e},V_A^{+}=k_A^{+}{E}_{p,m},$$ (8)

where $E_{p,m}$ denotes the specific LacY activity of preinduced cells. We show below that $V_A^{+}$ and $K_A^{+}$ can be estimated from the data for preinduced cells in Fig. 1 A, and this allows us to estimate $r_A^{+}$ in cells induced to any level.

Since $r_{A,d}^{-}$ is constant in preinduced cells, $V_A^{+}$ and $K_A^{+}$ can be estimated by fitting the data for preinduced cells in Fig. 1 A to the mass balance equation

$$\frac{1}{X}\frac{d{A}_e}{dt}=r_{A,d}^{-}-V_A^{+}\frac{A_e}{K_A^{+}+A_e},$$ (9)

obtained by substituting Eq. 8 in Eq. 4. More precisely, we fitted the biomass and extracellular allolactose concentration data using cubic B-splines (data analysis), and calculated the instantaneous values of $A_e$ and $\left(1/X\right)\left(d{A}_e/dt\right)$ shown in Fig. 3 A. Upon plotting these values against each other (Fig. 3 B) and fitting this graph to Eq. 9, we obtained a good fit with the parameter values $r_{A,d}^{-}=88$ μmol gdw⁻¹ h⁻¹, $V_A^{+}=$ 117 μmol gdw⁻¹ h⁻¹, and $K_A^{+}=10.6$ μM.

Figure 3.

The specific allolactose uptake rate $r_{A,d}^{-}$ of preinduced cells of E. coli K12 MG1655 increases hyperbolically with the extracellular allolactose concentration. (A) The net specific allolactose uptake rate $\left(1/X\right)/\left(d{A}_e/dt\right)=r_{A,d}^{-}-r_A^{+}$ (□) was determined from the measured concentrations of biomass $X$ (not shown) and extracellular allolactose $A_e$ (▪). Since $r_{A,d}^{-}$ is constant in preinduced cells, the specific allolactose uptake rate $r_A^{+}\left(t\right)$ equals the decline of $\left(1/X\right)/\left(d{A}_e/dt\right)$ from its initial value $r_{A,d}^{-}$, as indicated by the double-headed arrow. (B) Determination of the kinetic parameters for allolactose uptake and efflux by preinduced cells. The open squares are obtained by plotting the instantaneous values of $\left(1/X\right)/\left(d{A}_e/dt\right)$ in (A) against the corresponding values of $A_e\left(t\right)$. The solid curve shows the graph obtained when these data are fitted to Eq. 9. The dashed lines show the best-fit parameter values of $r_{A,d}^{-}=88$μmol gdw⁻¹ h⁻¹, $V_A^{+}=$ 117 μmol gdw⁻¹ h⁻¹, and $K_A^{+}$ = 10.6 μM.

Given $V_A^{+}$ and $K_A^{+}$, we can estimate $r_A^{+}$ in cells induced to any induction level. To see this, observe that Eq. 7 can be written as

$$r_A^{+}=V_A^{+}\frac{E_p}{E_{p,m}}\frac{A_e}{K_A^{+}+A_e}.$$ (10)

Since we measured the specific activity of LacZ, rather than LacY, it is convenient to rewrite this equation as

$$r_A^{+}=V_A^{+}\frac{E_g}{E_{g,m}}\left(\frac{E_p/E_{p,m}}{E_g/E_{g,m}}\right)\frac{A_e}{K_A^{+}+A_e},$$ (11)

where $E_{g,m}$ denotes the specific LacZ activity of fully induced cells, and $\left(E_p/E_{p,m}\right)/\left(E_g/E_{g,m}\right)$ is a measure of the LacY:LacZ ratio. This ratio is a function of the relative induction level $E_g/E_{g,m}$ which can be determined from the inducer exclusion data in Fig. 2 of (5)—it is equal to 1 in the absence of glucose, and $<1$ in the presence of glucose. Consequently, Eq. 11 can be used to calculate that the specific allolactose excretion rate $r_A^{+}$ at any given relative induction level $E_g/E_{g,m}$ and extracellular allolactose concentration $A_e$.

Intracellular allolactose levels and induction/dilution rates during lac induction

We are now ready to test the existence of the hypothesized allolactose-mediated positive feedback loop by determining if the intracellular allolactose concentration and induction rate increase with the induction level during the course of lac induction. To this end, we exposed noninduced cells to a mixture of lactose (4 mM) and maltose (2.8 mM), and measured the subsequent evolution of $X\left(t\right)$, $A_e\left(t\right)$, and $E_g\left(t\right)$. As shown below, these measurements allowed us to calculate not only the specific allolactose efflux rate $r_{A,d}^{-}\left(t\right)$, but also the induction and dilution rates, $r_{E_g}\left(t\right)$, $\mu \left(t\right)E_g\left(t\right)$.

The instantaneous specific allolactose efflux rates $r_{A,d}^{-}\left(t\right)$ were estimated by two methods.

• 1.Semiempirical method: the net specific allolactose efflux rate $\left(1/X\right)\left(d{A}_e/dt\right)$ was first calculated from the $X$ vs. $t$ and $A_e$ vs. $t$ data (open triangles and open squares, respectively, in Fig. 4 A). Next, the specific allolactose uptake rate $r_A^{+}\left(t\right)$ was estimated from the $A_e$ vs. $t$ and $E_g$ vs. $t$ data (open squares and open circles, respectively, in Fig. 4 A) by using Eq. 11 with $\left(E_p/E_{p,m}\right)/\left(E_g/E_{g,m}\right)=1$. Finally, the values of $\left(1/X\right)\left(d{A}_e/dt\right)$ and $r_A^{+}\left(t\right)$ thus obtained, which are represented in Fig. 4 B by $\times$ and $+$, respectively, were added, in accordance with Eq. 4, to determine the specific allolactose efflux rate $r_{A,d}^{-}$ (open squares in Fig. 4 B). We refer to this as the semiempirical method for determining $r_{A,d}^{-}$ since Eq. 11 appeals to the assumption $r_A^{+}\propto E_p$, which is plausible but not supported by direct evidence.
• 2.Empirical method: to validate the above semiempirical method, we also determined $r_{A,d}^{-}\left(t\right)$ at various instants, denoted S1, S2, and S3 in Fig. 4 A, by the initial rate method. At these instants, some of the cells in the culture were transferred to a shake flask containing prewarmed fresh medium and lactose (4 mM) + maltose (2.8 mM), but no allolactose. Thereafter, $X$ and $A_e$ were measured for 10–20 min (solid triangles and solid squares, respectively, in Fig. 4 A). Under this condition, there is almost no allolactose uptake ($r_A^{+}\approx 0$), and the specific allolactose efflux rate is given by the expression $r_{A,d}^{-}\approx \left(1/X\right)\left(d{A}_e/dt\right)$. The specific allolactose efflux rates thus determined, which are represented in Fig. 4 B by solid squares, were almost identical to those obtained by the semiempirical method (open squares in Fig. 4 B).

Figure 4.

Determination of the instantaneous specific allolactose efflux rate $r_{A,d}^{-}\left(t\right)$, induction rate $r_{E_g}\left(t\right)$, and dilution rate $\mu \left(t\right)E_g\left(t\right)$ during growth of an initially uninduced culture of E. coli K12 MG1655 on lactose + maltose. (A) Evolution of the cell density $X$ (▵), extracellular allolactose concentration $A_e$ (□), and specific β-galactosidase activity $E_g$ (○). To keep the cell density sufficiently low, the culture was centrifuged at the points labeled R1 and R2, and a fraction of the cells was reintroduced into the same medium. To determine the initial allolactose uptake rates, the cells were transferred to fresh medium at the time points labeled S1, S2, and S3, after which $X$ (▲), $A_e$ (▪), and $E_g$ (●)were measured for 10–20 min. (B) Evolution of the semiempirical (□) and empirical (▪) $r_{A,d}^{-}\left(t\right)$ derived from the data in (A). (C) Evolution of $r_{E_g}\left(t\right)$ and $\mu \left(t\right)E_g\left(t\right)$ derived from the data in (A).

Since both methods yielded similar values of $r_{A,d}^{-}$, this quantity was estimated by the more convenient semiempirical method in all subsequent experiments.

The instantaneous induction and dilution rates of β-galactosidase, $r_{E_g}\left(t\right)$ and $\mu \left(t\right)E_g\left(t\right)$, were calculated as follows from the $X$ vs. $t$ and $E_g$ vs. $t$ data in Fig. 4 A. We first determined the instantaneous specific growth rate $\mu \left(t\right)=\left(1/X\right)/\left(dX/dt\right)$, β-galactosidase dilution rate $\mu \left(t\right)E_g\left(t\right)$, and rate of change of the induction level $d{E}_g/dt$. We then calculated $r_{E_g}\left(t\right)$ from the relation

$$r_{E_g}\left(t\right)=\frac{d{E}_g}{dt}+\mu E_g\left(t\right)$$ (12)

obtained from Eq. 2. The induction and dilution rates thus obtained are shown in Fig. 4 C.

We gain insight into the variation of the intracellular allolactose level with the induction level by plotting the specific efflux rate $r_{A,d}^{-}\left(t\right)$ in Fig. 4 B against the corresponding induction level $E_g\left(t\right)$ in Fig. 4 A (Fig. 5 A). It can be seen that $r_{A,d}^{-}$, which is shown later to be proportional to $A$, increases in a biphasic manner. Initially, it increases rapidly while $E_g$ is essentially constant, but thereafter $r_{A,d}^{-}$ and $E_g$ increase simultaneously. The rapid initial increase is presumably due to the burst of allolactose produced immediately after lactose enters the uninduced cells. After this initial transient relaxes, the intracellular allolactose level, which is now in quasi-steady state, increases with the induction level. These data contradict the claim that the quasi-steady-state intracellular allolactose concentration is independent of the induction level (13, 14, 15). Instead, they support the hypothesis that lactose enzymes promote the accumulation of intracellular allolactose. Since intracellular allolactose is known to stimulate synthesis of the lactose enzymes (19), the potential for positive feedback clearly exists.

Figure 5.

Positive feedback exists and drives induction by forward turning of the positive feedback loop. The phase plots are derived from the instantaneous $E_g\left(t\right)$ and $r_{A,d}^{-}\left(t\right)$, $r_{E_g}\left(t\right)$, and $\mu \left(t\right)E_g\left(t\right)$ obtained during growth of uninduced cells of E. coli K12 MG1655 on maltose (2.8 mM) + lactose (4.0 mM) (Fig. 4). The arrows indicate the direction of evolution in time. (A) After an initial burst, the specific allolactose efflux rate $r_{A,d}^{-}$ and the corresponding $A$ (estimated from Fig. 7B) increase with $E_g$, which implies that positive feedback can exist. (B) After an initial burst, the induction rate $r_{E_g}$ increases with $E_g$, which shows that positive feedback does exist. The steady-state induction level ${\tilde{E}}_g^{\text{mal}}$ (●) lies at the intersection of $r_{E_g}$ and $\mu E_g$. (C) The net induction rate $d{E}_g/dt=r_{E_g}-\mu E_g$ is positive for all $0\le E_g<{\tilde{E}}_g^{\text{mal}}$.

We confirmed that positive feedback does exist and leads to fully induced cells by plotting the induction and dilution rates, $r_{E_g}\left(t\right)$ and $\mu \left(t\right)E_g\left(t\right)$, in Fig. 4 C against the corresponding induction level $E_g\left(t\right)$ in Fig. 4 A (Fig. 5 B). The plot shows that $r_{E_g}$ mirrors the biphasic profile of $r_{A,d}^{-}$. Initially, $r_{E_g}$ increases rapidly while $E_g$ is essentially constant, which reflects the effect of the initial burst of allolactose synthesis. This is followed by a subsequent slow phase during which $r_{E_g}$, which has now relaxed to a quasi-steady state, increases with $E_g$, and therefore provides direct evidence of positive feedback—the higher the induction level, the faster the quasi-steady-state induction rate. Fig. 5 B also reveals how positive feedback drives the cells toward the fully induced steady state. Indeed, the steady-state induction level, denoted ${\tilde{E}}_g^{\text{mal}}$, lies at the intersection of the $r_{E_g}$ and $\mu E_g$ curves. Now, for all $0<E_g<{\tilde{E}}_g$, $r_{E_g}$ lies above $\mu E_g$, so that $d{E}_g/dt=r_{E_g}-\mu E_g>0$ (Fig. 5 C). Hence, the induction level of initially uninduced cells ($E_g\left(0\right)\approx 0$) increases, slowly but relentlessly, until it approaches the steady-state level ${\tilde{E}}_g^{\text{mal}}$. During this slow increase of the induction level, the quasi-steady-state intracellular allolactose level increases as described by Fig. 5 A. Thus, the quasi-steady concentrations both $E_g$ and $A$ increase simultaneously predicted by our model.

Intracellular allolactose levels and induction/dilution rates during lac repression

We have also hypothesized that glucose-mediated catabolite repression occurs primarily because the positive feedback loop, which turns forward during induction in the presence of lactose, reverses direction upon the addition of glucose.

To test this hypothesis, we allowed uninduced cells to grow on maltose (2.8 mM) + lactose (4 mM) until the induction level reached steady state, at which point these fully induced cells were transferred to a medium containing glucose (2.2 mM) + lactose (4 mM) ($t=0$ in Fig. S5). We determined the instantaneous concentrations $X\left(t\right)$, $A_e\left(t\right)$, and $E_g\left(t\right)$ during the experiment and calculated the corresponding instantaneous rates $r_{A,d}^{-}\left(t\right)$, $r_{E_g}\left(t\right)$, $\mu \left(t\right)E_g\left(t\right)$, and $d{E}_g/dt$ by the methods described above. Upon plotting these instantaneous rates against the corresponding values of $E_g\left(t\right)$, we obtained the graphs shown in Fig. 6 where the dashed and solid curves show the respective fits to the data obtained during induction of the uninduced cells in the presence of maltose + lactose, and subsequent repression of the fully induced cells upon their transfer to glucose + lactose.

Figure 6.

For a Figure360 author presentation of this figure, see https://doi.org/10.1016/j.bpj.2022.01.017.

Positive feedback drives repression by reversed turning of the positive feedback loop. The dashed (--) and solid (—) phase plots are derived from the instantaneous $E_g\left(t\right)$ and $r_{A,d}^{-}\left(t\right)$, $r_{E_g}\left(t\right)$, and $\mu \left(t\right)E_g\left(t\right)$ (Fig. S5) obtained during induction of uninduced cells of E. coli K12 MG1655 on maltose (2.8 mM) + lactose (4 mM), and repression of the induced cells thus obtained upon transfer to glucose (2.2 mM) + lactose (4 mM), respectively. The arrows indicate the direction of temporal evolution. (A) Upon transfer to glucose + lactose, the specific allolactose efflux rate $r_{A,d}^{-}$ (○) and the corresponding $A$ (estimated from Fig. 7B) decrease abruptly at constant $E_g$, which is followed by a gradual decline with $E_g$ (□). (B) Upon transfer to glucose + lactose, the dilution rate $\mu E_g$ (□) remains unchanged, and the induction rate $r_{E_g}$ shifts downward by ∼6-fold (○→□), but the steady-state induction level ${\tilde{E}}_g^{\text{glc}}$ (●→▪) is reduced by 1000-fold. (C) Upon transfer to glucose + lactose, $d{E}_g/dt=r_{E_g}-\mu E_g$ is negative for all ${\tilde{E}}_g^{\text{glc}}<E_g\le {\tilde{E}}_g^{\text{mal}}$ (□).

If $r_{A,d}^{-}$ is a surrogate for $A$, then it is clear from the solid curve of Fig. 6 A that immediately after transfer of the fully induced culture to lactose + glucose, $A$ decreases threefold from its steady-state value during growth on lactose + maltose, whereas $E_g$ remains nearly constant. This rapid decline of $A$ at constant $E_g$ is reminiscent of inducer exclusion: upon addition of glucose, permease, but not β-galactosidase, is partially inactivated, and $A$ responds by instantly decreasing to a lower quasi-steady-state level. After this rapid transient, the induction level $E_g$ decreases slowly, and so does the quasi-steady-state intracellular allolactose level $A$ as it continuously adjusts to the slowly decline of the induction level $E_g$.

It remains to explain the foregoing slow decline of the induction level in the presence of glucose + lactose, but this requires a closer look at the dynamics of the induction level which are governed by the LacZ induction and dilution rates (solid curves in Fig. 6 B). To this end, observe that, after the fully induced cells are transferred to a medium containing glucose, the dilution rate remains essentially unchanged (compare dashed lines and full lines in Fig. 6 B), but the induction rate shifts downward due to inducer exclusion and cAMP regulation (compare the dashed and full curves in Fig. 6 B). Although the induction rate declines by only 6-fold, the induction level ${\tilde{E}}_g^{\text{glc}}$ at the new steady state, which occurs at the intersection of the solid curves representing the induction and dilution rates (▪), is 1000-fold lower than the induction level ${\tilde{E}}_g^{\text{mal}}\approx 1100$ MU at the steady state before the addition of glucose (●). Moreover, for all ${\tilde{E}}_g^{\text{glc}}<E_g\le {\tilde{E}}_g^{\text{mal}}$, the dilution rate exceeds the induction rate, so that $d{E}_g/dt=r_{E_g}-\mu E_g<0$ (solid curve of Fig. 6 C). Hence, $E_g$ decreases slowly from its initial value ${\tilde{E}}_g^{\text{mal}}$ to the drastically lower new steady-state value ${\tilde{E}}_g^{\text{glc}}$.

In the presence of small concentrations of TMG, positive feedback exists (11,12) and drives glucose-mediated catabolite repression due to the reversal of the positive feedback loop (5). Now, comparison of Fig. 6, A–C with Fig. 5, A–C of (5) shows that the kinetics observed upon addition of glucose to a culture growing on lactose are formally similar to those observed upon addition of glucose to a culture growing on glycerol and 100 μM TMG. Therefore, it is reasonable to conclude that the same mechanism, namely reversal of the positive feedback loop, drives glucose-mediated repression in the presence of both TMG and lactose.

In principle, the sixfold decline of the induction rate immediately after the addition of glucose (Fig. 6 B) could be due to both inducer exclusion and cAMP-mediated regulation, but the data show that inducer exclusion plays no role initially because at this time, $r_{E_g}$ is saturated with respect to $A$. To see this, observe that, even after the initial threefold decline of $A$ (Fig. 6 A), $r_{E_g}$ remains constant while $E_g$ decreases from 1100 to 500 MU (Fig. 6 B) and, hence, $A$ decreases from 80 to 30 μM (Fig. 6 A). It follows that the initial sixfold decline of $r_{E_g}$ must be due to cAMP-mediated regulation, which is consistent with the cAMP-mediated severe transient repression that is known to strongly inhibit lac expression for ∼1 doubling time after glucose addition, as opposed to the cAMP-mediated weak permanent repression that persists thereafter (1). However, later, when $E_g$ has fallen below 500 MU and $r_{E_g}$ is unsaturated with respect to $A$, inducer exclusion can also play a role in the decline of the induction rate. We also observed similar dynamics during glucose-mediated repression in the presence of small TMG concentrations (5).

The allolactose efflux rate is proportional to the intracellular allolactose level

Thus far, we have assumed that $r_{A,d}^{-}$ is proportional to $A$. Here, we show that this is indeed the case, and determine the corresponding proportionality constant.

To this end, we first determined the induction rate $r_{E_g}$ at various $r_{A,d}^{-}$ by calculating, as described above, the evolution of $r_{E_g}\left(t\right)$ and $r_{A,d}^{-}\left(t\right)$ during the growth of uninduced cells on lactose (4 mM) + maltose (2.8 mM) (crosses and open squares in Fig. S6). We then normalized $r_{E_g}\left(t\right)$ by its value in fully induced cells because the theory suggests (section S3), and the data discussed below confirm, that this normalization eliminates the effect of cAMP on the induction rate. Upon plotting this normalized induction rate ${\rho }_{E_g}\left(t\right)$ against $r_{A,d}^{-}\left(t\right)$, we obtained the ${\rho }_{E_g}$ vs. $r_{A,d}^{-}$ plot (open squares in Fig. 7 A), which will be referred to as the normalized induction curve with respect to $r_{A,d}^{-}$. Since ${\rho }_{E_g}$ is independent of the cAMP level, we may assume that it is completely determined by the intracellular allolactose concentration $A$. As we show next, this property enables the estimation of $A$ corresponding to any given $r_{A,d}^{-}$.

Figure 7.

The intracellular concentration of allolactose $A$ is proportional to its specific efflux rate $r_{A,d}^{-}$. (A) The dashed curve and open squares (□) show the variation of the normalized induction rate ${\rho }_{E_g}$ with $r_{A,d}^{-}$, which is derived from the data obtained during growth of uninduced cells on maltose + lactose (Fig. S6). The solid curve shows the variation of ${\rho }_{E_g}$ with the extracellular IPTG concentration [IPTG], which is derived from the steady-state induction rates obtained in lacY^– cells exposed to various concentrations of extracellular IPTG in the presence of glycerol (○) and maltose ($\times$). Given any $r_{A,d}^{-}$, the dashed arrows indicate the algorithm for determining the corresponding value of [IPTG], and hence $A$. (B) The extracellular IPTG or intracellular allolactose concentrations corresponding to various values of $r_{A,d}^{-}$. The dotted line (···) shows that the estimated intracellular allolactose concentration $A$ is proportional to $r_{A,d}^{-}$.

To estimate $A$ corresponding to any given $r_{A,d}^{-}$, one must ideally determine the normalized induction curve with respect to allolactose (${\rho }_{E_g}$ vs. $A$), which can be obtained by measuring and normalizing the induction rates obtained in lacY^– lacZ^– cells exposed to a noninducing carbon source, such as glycerol or maltose, and various concentrations of extracellular allolactose. Given the ${\rho }_{E_g}$ vs. $r_{A,d}^{-}$ and ${\rho }_{E_g}$ vs. $A$ curves, one can calculate the value of $A$ corresponding to any given $r_{A,d}^{-}$ as follows. First, find the value of ${\rho }_{E_g}$ corresponding to the given $r_{A,d}^{-}$ by using the ${\rho }_{E_g}$ vs. $r_{A,d}^{-}$ curve, and then find the value of $A$ that yields this value of ${\rho }_{E_g}$ by using the ${\rho }_{E_g}$ vs. $A$ curve. In other words, the value of $A$ corresponding to any given $r_{A,d}^{-}$ is that value of $A$ which provides the same normalized induction rate ${\rho }_{E_g}$ as that obtained from the given value of $r_{A,d}^{-}$.

We could not determine the ${\rho }_{E_g}$ vs. $A$ curve because allolactose is not commercially available, but this curve is essentially identical to the normalized induction curve with respect to IPTG (${\rho }_{E_g}$ vs. [IPTG]) obtained by measuring and normalizing the induction rates in lacY^– cells exposed to a noninducing carbon source and various IPTG concentrations. Indeed, the theory suggests that the ${\rho }_{E_g}$ vs. $A$ and ${\rho }_{E_g}$ vs. [IPTG] curves coincide if the dissociation constants for binding of Lac repressor to allolactose and IPTG are the same (section S3), but this is known to be the case from measurements of the dissociation constants (19). Hence, in the foregoing method for determining the value of $A$ corresponding to $r_{A,d}^{-}$, we can just as well use the ${\rho }_{E_g}$ vs. [IPTG], instead of the ${\rho }_{E_g}$ vs. $A$ curve.

Fig. 7A shows the ${\rho }_{E_g}$ vs. [IPTG] data obtained with two different noninducing carbon sources, namely, maltose ($\times$) and glycerol (○). At any given concentration of IPTG, the induction rates obtained with maltose and glycerol are quite different; in particular, under fully induced conditions, the induction rates were 768 and 1308 MU h⁻¹, respectively, which presumably reflect the different intracellular cAMP levels during growth on maltose and glycerol (26). However, the corresponding normalized induction rates in Fig. 7 A are essentially the same, thus confirming that normalization eliminates the effect of cAMP.

The dashed lines of Fig. 7 A describe the process by which we obtain the value of $A$ corresponding to one particular value of $r_{A,d}^{-}$. Application of this process to several values of $r_{A,d}^{-}$ yields multiple pairs of the form $\left(r_{A,d}^{-},A\right)$, which were plotted to obtain the graph shown in Fig. 7 B. It is clear that $r_{A,d}^{-}$ is essentially proportional to $A$, which implies that the specific allolactose efflux rate is a legitimate surrogate for the intracellular allolactose concentration.

Discussion

Comparison with the data in the literature

We are aware of only one study reporting intracellular allolactose concentrations. Huber et al. used the difference method to measure the intracellular allolactose concentrations in lac-constitutive cells exposed to 60 mM lactose (16). They reported intracellular allolactose concentrations of ∼300 mM, which are much higher than our estimate of 200 μM in fully induced cells. This is probably due to the errors inherent in the difference method—Fig. 1 of their paper shows that the intracellular allolactose concentrations were frequently negative since the amount in the medium exceeded the total amount, which mirrors our data (Fig. 1 B). In another study, Huber et al. also reported the efflux rates of glucose, galactose, and allolactose in lac-constitutive cells exposed to 1 mM lactose (39). They observed that almost all the lactose taken by the cells was rapidly excreted as glucose, galactose, and allolactose. We observed the same phenomenon with induced cells for the first 15 min (Table 1), but after this initial period, there was no net efflux of glucose and galactose (Fig. S7). We also find that the specific allolactose efflux rates in our induced cells are 15-fold less than those observed by Huber et al. This may be due to the 15-fold smaller lactose concentrations (4 mM) used in our experiments.

Table 1.

Comparison of initial Rates of sugar Production obtained in this Study with the Results of Huber et al. (1980)

Compound	Initial Rate of Production in C-mmol h−1 gdw−1 (Calculated from (39))	Initial Rate of Production in C-mmol h−1 gdw−1 (This work)
Lactose	−70.68	−31.32
Galactose	+25.98	+13.68
Glucose	+26.28	+14.22
Allolactose	+21.36	+1.32
Total products	+73.62	+29.22

The negative rate of lactose production indicates consumption of lactose.

Evolutionary implication of the high affinity of Lac permease for allolactose

It is remarkable that permease-mediated allolactose uptake occurs even in the presence of 4 mM lactose. Indeed, we found that the apparent saturation constant for permease-mediated allolactose uptake, observed in the presence of 4 mM lactose, was 10 μM (Fig. 3 B), which is an order of magnitude smaller than the intrinsic saturation constant of 270 μM for permease-mediated lactose uptake (23,36). It seems likely that the intrinsic saturation constant for permease-mediated allolactose uptake is even lower. If we assume simple competitive kinetics with lactose as the competitor, we find that the intrinsic saturation constant for allolactose uptake is 1 μM, which is 250-fold smaller than the intrinsic saturation constant for lactose. This remarkably high affinity of Lac permease for allolactose suggests that the lac operon did not evolve for consumption of lactose. Egel made this claim earlier, but it was based on the observation that allolactose, rather than lactose, was the true inducer (40). Our data show that the permease has a much higher affinity for allolactose than lactose, which provides further evidence that the lac operon may have evolved to promote consumption of allolactose or a structurally similar compound.

Explaining the data adduced in support of the inducer exclusion hypothesis

In the prequel to this work (5), it was observed that several interventions that target inducer exclusion also abolish catabolite repression. However, we have also argued that positive feedback is the dominant mechanism for catabolite repression since inducer exclusion accounts only for the initial 3-fold decline of the allolactose level (as indicated by $r_{A,d}^{-}$), and the subsequent >30-fold decline is driven by the reversal of the positive feedback loop (Fig. 6, A and B). Given this fact, it is reasonable to ask: If positive feedback, rather than inducer exclusion, is the dominant mechanism of catabolite repression, why do interventions that target inducer exclusion succeed in abolishing catabolite repression?

The answer to the above question might be that the interventions that targeted inducer exclusion abolished not only inducer exclusion, but also positive feedback. To see this, recall that positive feedback can occur only if the inducer stimulates lactose enzyme synthesis and the lactose enzymes promote inducer accumulation, i.e., the induction rate has the functional form $r_{E_g}\left(I\left(E_g\right)\right)$, wherein $r_{E_g}$ increases with the inducer concentration $I$, which in turn increases with $E_g$. It follows that there is no positive feedback if

• 1.$r_{E_g}$ is independent of $I$, i.e., lac expression is independent of inducer levels as is the case in lacI^– cells.
• 2.$I$ is independent of $E_g$, i.e., inducer accumulation is independent of lactose enzyme levels. This is the case when surplus IPTG is present in the medium since, under these conditions, the main inducer is not allolactose, but IPTG, which accumulates inside cells primarily by the LacY-independent mechanism of passive diffusion (41,42).

In both cases, $r_{E_g}$ is independent of $E_g$, i.e., lac expression is constitutive. Thus, the foregoing interventions targeting inducer exclusion also abolish positive feedback, which might explain their efficacy in abolishing catabolite repression.

Conclusions

In this work, we addressed a controversial debate on the mechanism underlying the strong >500-fold lac repression observed when glucose is added to a culture growing in the presence of lactose. Specifically, we have proposed that the allolactose-mediated positive feedback loop amplifies the small effects of inducer exclusion and cAMP-mediated transcriptional repression, but mathematical models show that the proposed positive feedback loop does not exist since the allolactose concentration is independent of the induction level.

To address these questions, we have shown that:

• 1.The proposed positive feedback loop can exist in theory because the foregoing mathematical models are based on two assumptions—no lactose/allolactose excretion and constant LacY:LacZ ratio—that are inconsistent with the data, and relaxing these assumptions restores positive feedback.
• 2.The intracellular allolactose levels increase with the LacZ level, which contradicts the predictions of prevailing models, and implies that positive feedback can exist.
• 3.The LacZ induction rate always increases with the LacZ level, which implies that positive feedback does exist in the presence of lactose. Moreover, positive feedback drives massive repression by reducing the steady-state induction level by 1000-fold, thus forcing the positive feedback loop to turn in the reverse direction.

Therefore, it not just the molecular mechanisms, but also systemic dynamical properties of the regulatory network, such as positive feedback, that drive glucose-mediated lac repression.

Author contributions

R.K.A. and A.N. designed the study and experiments. R.K.A. obtained the experimental results. R.K.A. and A.N. analyzed the data. R.K.A. and A.N. wrote the manuscript.

Editor: Stanislav Shvartsman.