The cell's cytoplasm is characterized by a high concentration of macromolecules 14 resulting in a limited solvent capacity for the allocation of metabolic enzymes. More precisely, given that the enzyme molecules have a finite molar volume v_i only a finite number of them fit in a given cell volume V. Indeed, if n_i is the number of moles of the i^th enzyme, then

∑ i = 1 N v i n i ≤ V , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaWaaabmaeaacqWG2bGDdaWgaaWcbaGaemyAaKgabeaakiabd6gaUnaaBaaaleaacqWGPbqAaeqaaaqaaiabdMgaPjabg2da9iabigdaXaqaaiabd6eaobqdcqGHris5aOGaeyizImQaemOvayLaeiilaWcaaa@3C54@

where the inequality sign accounts for the volume of other cell components and the free volume necessary for cellular transport as well. Dividing by cell mass M we can reformulate this inequality in terms of the enzyme concentrations E_i = n_i/M (moles/unit mass), resulting in

∑ i = 1 N v i E i ≤ 1 C , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaWaaabmaeaacqWG2bGDdaWgaaWcbaGaemyAaKgabeaakiabdweafnaaBaaaleaacqWGPbqAaeqaaaqaaiabdMgaPjabg2da9iabigdaXaqaaiabd6eaobqdcqGHris5aOGaeyizImAcfa4aaSaaaeaacqaIXaqmaeaacqWGdbWqaaGccqGGSaalaaa@3D74@

where C = M/V is the cytoplasmic density. An enzyme concentration E_i results in a flux f_i = b_iE_i over reaction i, where the parameter b_i is determined by the reaction mechanism, kinetic parameters, and metabolite concentrations. Therefore, the enzyme concentration constraint (Eq. 2) is reflected in the metabolic flux constraint

∑ i = 1 N a i f i ≤ 1 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaWaaabmaeaacqWGHbqydaWgaaWcbaGaemyAaKgabeaakiabdAgaMnaaBaaaleaacqWGPbqAaeqaaaqaaiabdMgaPjabg2da9iabigdaXaqaaiabd6eaobqdcqGHris5aOGaeyizImQaeGymaeJaeiilaWcaaa@3BD5@

where

a i = C v i b i . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemyyae2aaSbaaSqaaiabdMgaPbqabaGccqGH9aqpjuaGdaWcaaqaaiabdoeadjabdAha2naaBaaabaGaemyAaKgabeaaaeaacqWGIbGydaWgaaqaaiabdMgaPbqabaaaaOGaeiOla4caaa@3858@

Since the coefficients a_i (units of inverse flux) quantifies the contribution to the overall crowding by reaction i we refer to them as the 'crowding coefficients'.

To understand the relevance of the constraint (Eq. 3) at physiological growth conditions we first estimate the crowding coefficients (Eq. 4) using data from experimental reports. The E. coli cytoplasmic density of macromolecules is C = 0.34 g/ml 15, while the molar volumes of proteins are proportional to their molar masses 16. The coefficient of proportionality represents the specific volume and it is about 0.73 ml/g. This empirical law allows us to compute the molar volumes of E. coli enzymes from their molar masses. As a first approximation we estimate b_i, the coefficient of proportionality between reaction rate and enzyme concentration, from the enzyme's turnover numbers. Data obtained from the BRENDA data base 17 for about hundred E. coli enzymes (Additional file 1) shows that the turnover numbers vary over five orders of magnitude (Fig. 1a), from 10^-2 to 10² 1/s. Using these parameter estimates we compute the crowding coefficients a_i for about a hundred E. coli enzymes (Fig. 1b), resulting in an average and standard deviation of 0.014 and 0.009 1/[mmol/g/h], respectively. Because of the large enzyme turnover variations the crowding coefficients are distributed over a wide range as well, from 10^-6 to 10⁰ 1/[mmol/g/h] (Fig. 1b).

Having estimated the crowding coefficients we next evaluate the relevance of the solvent capacity constraint (Eq. 3) at physiological growth conditions. To this end we utilize a FBA model of E. coli MG1655 metabolic network that takes into account this constraint referred to as 'flux balance analysis with molecular crowding' (FBAwMC) 12. Under conditions of aerobic growth in a glucose-limited medium, FBAwMC predicts a saturation of the glucose uptake rate and the growth rate (Fig. 2a,b) with increasing the glucose uptake capacity. The predicted maximum glucose uptake rate (~15 mmol/g/h) and maximum growth rate (~0.7 h^-1) are within the range of experimentally determined values 18, corroborating our previous report 12 that the solvent capacity constraint (Eq. 3) is relevant at physiological conditions.

Associated with the predicted saturation of E. coli metabolic rates, FBAwMC predicts a metabolic switch characterized by a change in the effective criteria of metabolic efficiency. At low growth rates the ratio between the biomass production rate and the glucose uptake rate is at a maximum but decreases with increasing the growth rate. In contrast, the ratio between the biomass production rate and the average reaction rate increases with increasing the growth rate, reaching a maximum at high growth rates. In agreement with our expectations, at low growth rates nutrients are scarce and the best strategy for a cell is to maximize the biomass production rate per unit of limiting nutrient (in this case, glucose) uptake rate. In contrast, at high growth rates the nutrients are abundant, the predicted metabolic rate is limited by the solvent capacity constraint (Eq. 3) and, therefore, the maximum growth rate is achieved by maximizing the biomass production rate per average reaction rate (Fig. 2c). The predicted change in metabolic efficiency objective is accompanied by a redistribution of the metabolic fluxes, including those of exchange fluxes. Indeed, a characteristic example is the predicted excretion of acetate at high growth rates (Fig. 2d) that is well supported by experimental observations 91920.

FBAwMC is also able to predict internal metabolic fluxes as a function of the growth rate. A subset of the FBAwMC-derived flux predictions in the central carbon metabolism are shown in Figure 3. In most cases the FBAwMC predicted fluxes are within the range of experimentally determined values. This is a striking result given that this implementation of FBAwMC does not contain any free paremeters. The only model parameters are the crowding coefficients, which were determined above using independent experimental results. We should also note that the observed wide variability around the average behavior (Fig. 3, experimental error bars) is not a shortcoming of our modeling framework but is due to our current inability to obtain a direct estimate for all crowding coefficients (due to incomplete information available on various kinetic parameters). Thus, further testing of our predictions will be necessary upon availability of better estimates for the crowding coefficients.

Limiting our analysis to the expected behavior, we observe a slope change for several fluxes when reaching the highest growth rates. The reactions of the glycolytic pathway, the flux towards the pentose-phosphate pathway via the reaction catalyzed by the gene product of zwf, and the acetate pathway switch at high growth rates to a faster flux increase with increasing the growth rate (Fig. 3). The experimental values corroborate this qualitative behavior, but the change is bigger for the ptsG-catalyzed reaction and even qualitatively different for the pykA-catalyzed reaction, both being part of the glycolytic pathway. A second noticeable effect is the predicted saturation of the TCA cycle flux at high growth rates. The experimentally measured values of the TCA cycle flux exhibit, however, a stronger effect characterized by a decreasing tendency at high growth rates (Fig. 3). Taken together these results indicate that while for most reactions the FBAwMC predictions are within the range of experimental measurements, a method for a more accurate estimate of the crowding coefficients on a network scale will be required to provide more precise predictions.

To examine if the changes in growth conditions and the corresponding adjustments in cellular metabolism can be traced by distinct molecular signatures we next measured the in vitro activity of eighteen selected enzymes (Additional file 2) that catalyze reactions in the central carbon metabolism of E. coli MG1655, and correlate their changes with those observed for the measured flux rates (Fig. 4). For several enzymes there is a good correlation between the measured enzyme- and flux activities (Pearson Correlation Coefficient, PCC, close to- or larger than 0.8). For example, with an increasing growth rate the enzyme activity of the ptsG and pfkA gene products follow the same increasing tendency as the fluxes of the corresponding metabolic reactions (PCC = 0.79 and 0.85, respectively). (The glycolytic flux is known to be controlled by the activity of these two enzymes while other reactions adjust their fluxes through changes in metabolite concentrations 21). In contrast, we found no significant correlation between the measured fluxes and enzyme activities of the TCA reactions (PCC = 0.64, 0.35, -0.03 and -0.28 for enzymes associated with gltA, sucA, fumA and mdh, respectively), implying that the TCA flux is controlled by the activity of enzymes catalyzing reactions outside this pathway. A possible candidate to exert this action is the acetate pathway. Indeed, an increase of the flux on the acetate pathway towards the production of acetate can balance both the increase in the flux originating from the glycolytic pathway through aceE and a decrease in the flux from Acetyl-CoA to the TCA cycle. This hypothesis is supported by the increase in the enzyme activity of phosphotransacetylase (pta) when the growth rate increases beyond 0.4 h^-1 (PCC = 0.98), which is exactly the growth rate threshold where the switch is taking place.

In parallel with the enzyme activity measurements we also prepared mRNA from samples obtained at all five dilution rates and processed them for microarray analysis. The full microarray data set is presented in the Additional file 3 and its detailed analysis is presented in Additional file 4 (SI text 5–8). In contrast to the observed overall correlation between measured fluxes and in vitro enzyme activities we do not observe a significant correlation between the measured metabolic fluxes and the relative changes in mRNA levels of enzyme-encoding genes (SI Fig. S5 in Additional file 4), implying that the switch and corresponding enzymatic functions are not predominantly controlled at the transcriptional level. Correspondingly, no significant correlation between the in vitro enzyme activities and the relative changes in mRNA levels of enzyme-encoding genes can be seen (SI Fig. S5 in Additional file 4). Taken together these results indicate that the metabolic switch is predominantly controlled by an increase in the enzyme activities of the end products of ptsG and pfkA controlling the glycolysis flux, and pta controlling the acetate pathway flux, respectively.

The E. coli intracellular density is C = 0.34 g/ml 15. The specific volume was estimated for several proteins using the molar volumes and masses reported in Ref. 16, resulting in average of 0.73 ml/g and standard deviation of 0.02 ml/g. The enzymes' turnover rates were obtained from the BRENDA database 17 for 102 E. coli enzymes.

The Flux Balance analysis with Molecular Crowding 12 is implemented by solving the following optimization problem: maximize the biomass production rate subject to the constraints: balance in the production and consumption of each metabolite (flux balance), the maximum capacity constraint for the carbon source uptake rate and the solvent capacity constraint (Eq. 3). After expressing the reaction's stoichiometric coefficients in units of mol/dry biomass, the maximum growth rate corresponds to the biomass production rate, where biomass production is an auxiliary reaction containing as substrates the cellular components in their relative concentrations and as product the cell's biomass. The crowding coefficients were modeled as noise, assigning them randomly selected values from a list of estimated values for about hundred E. coli enzymes (Additional file 1). The predictions for all fluxes are provided in Additional file 5.

The E. coli K12 strain MG1655 (F^- λ^- ilvG rfb50 rph1) was used throughout the work. In order to obtain biomass samples for flux measurements, 20-ml of the overnight grown culture (~8–10 h) of wild-type cells in LB-medium was inoculated in 980-ml M9 minimal medium (Sigma) containing 2 g/L glucose, where 90% was natural glucose and the remaining 10% was labeled glucose [1,2-¹³C₂]-glucose (with >99% purity and 99% isotope enrichment for each position, [Cambridge Isotope Laboratories, Andover, MA]). Cells were grown in a continuous growth mode at 5 different dilution rates (0.1, 0.25, 0.4, 0.55, and 0.72 L h^-1) in a Labfors bioreactor (Infors, Switzerland). The growth of the bacterial culture was regularly monitored at A_600nm to document steady state, as described in Additional file 4 (SI text 3 and SI Fig. S7)

For determining metabolic enzyme activities from three separate experiments, the cell pellets (collected at all five dilution rates) were first resuspended and washed in 100 mM Tris-HCl (pH 7.0) sonication buffer containing 20 mM KCl, 5 mM MnSO₄, 2 mM DTT and 0.1 mM EDTA, and then disrupted in a sonicator by 3 sonication cycles of 30 sec each. The cell debris was removed by centrifugation and the resulting cell extract (supernatant) was immediately used for enzyme assays or stored at -20°C. All subsequent steps were carried out on ice. Supernatants of the samples were used for quantitative assaying of endogenous enzyme activities by continuous spectrophotometric rate determination at 30°C in a thermostatically controlled spectrophotometer (Cary 500) with 1-cm light path, as described in the Additional file 4 (SI text 3). Supernatants of the samples were also used for determining total protein concentrations using standard Bradford's assay (BioRad).

For flux analysis, biomass (from ~100 ml culture) and supernatant samples were collected at all five dilution rates. These samples were immediately flash frozen in liquid nitrogen and stored at -80°C until further analysis. Flux rates were determined using a tracer-substrate based GC-MS and NMR metabolome mapping platform. The analyses included determining positional ¹³C tracer enrichment in multiple intermediary metabolites of glycolysis, glycogen synthesis, tricarboxylic acid cycle and their intracellular products from [1,2-¹³C₂]-D-glucose, as described in detail in Additional file 4 (SI text 4). The retention times and mass-to-charge (m/z) ion clusters of selected ions of bacterial and culture media metabolites were determined using mass isotopomer analysis (MIDA) 2728, and expressed as net fluxes by subtracting reverse fluxes from forward tracer incorporation patterns via reversible metabolic steps 29. Results were expressed as mmol/hr/g dry biomass glucose. Each experiment was carried out using triplicate cell cultures for each condition within each experiment, and the experiments were repeated once. Mass spectroscopic analyses were carried out by three independent automatic injections of 1 μl samples by the automatic sampler and accepted only if the standard sample deviation was less than 1% of the normalized peak intensity. Statistical analysis was performed using the Student's t-test for unpaired samples. Two-tailed significance at the 99% confidence interval (μ +/- 2.58σ), p < 0.01 indicated significant differences in glucose-derived fluxes. For some reversible reactions, we measured both forward and reverse fluxes and calculated net fluxes towards product synthesis.

At all five dilution rates, 10–20 ml of the cell culture was collected, mixed with 10% (v/v) of ice cold stop-solution (5% water-saturated phenol in absolute ethanol), and cell pellets were obtained by centrifugation at 4,500 × g for 5 min at 4°C, followed by flash-freezing of pellets with liquid nitrogen. Cell pellets were stored at -80°C until further use. RNA was isolated from the frozen cell pellets using Masterpure RNA isolation kit (Epicentre Biotechnologies, Madison, WI) and RNA samples were processed for transcriptome analysis using E. coli Affymetrix microarray chips, as described previously 12. Dchip method was used to analyze all microarray data as described in Additional file 4 (SI text 5–8).