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Proc Natl Acad Sci U S A. 2007 July 31; 104(31): 12663–12668. Published online 2007 July 24. doi: 10.1073/pnas.0609845104. | PMCID: PMC1937523 |
Copyright © 2007 by The National Academy of Sciences of the USA Statistics, Biophysics Intracellular crowding defines the mode and sequence of substrate uptake by Escherichia coli and constrains its metabolic activity Q. K. Beg,* A. Vazquez,†‡ J. Ernst,§ M. A. de Menezes,¶ Z. Bar-Joseph,§ A.-L. Barabási,‖ and Z. N. Oltvai*** *Department of Pathology, University of Pittsburgh, Pittsburgh, PA 15261; †The Simons Center for Systems Biology, Institute for Advanced Study, Princeton, NJ 08540; §Machine Learning Department, Carnegie Mellon University, Pittsburgh, PA 15217; ¶Instituto de Física, Universidade Federal Fluminense, 24210, Rio de Janeiro, Brazil; and ‖Department of Physics and Center for Complex Networks Research, University of Notre Dame, Notre Dame, IN 46556 Received November 7, 2006. The influence of the high intracellular concentration of macromolecules on cell physiology is increasingly appreciated, but its impact on system-level cellular functions remains poorly quantified. To assess its potential effect, here we develop a flux balance model of Escherichia coli cell metabolism that takes into account a systems-level constraint for the concentration of enzymes catalyzing the various metabolic reactions in the crowded cytoplasm. We demonstrate that the model's predictions for the relative maximum growth rate of wild-type and mutant E. coli cells in single substrate-limited media, and the sequence and mode of substrate uptake and utilization from a complex medium are in good agreement with subsequent experimental observations. These results suggest that molecular crowding represents a bound on the achievable functional states of a metabolic network, and they indicate that models incorporating this constraint can systematically identify alterations in cellular metabolism activated in response to environmental change. Keywords: flux balance analysis, metabolic networks, systems biology An important aim of systems biology is the identification of the organizing principles and fundamental constraints that characterize the function of molecular interaction networks and the limits of an organism's phenotypic diversity ( 1– 3). Flux balance-based modeling approaches, combining the constraints imposed by the metabolic network's structure with, e.g., mass- or energy-conservation principles ( 3– 6), are especially successful in providing experimentally testable predictions on an organism's metabolic flux state and growth rate. A relative shortcoming of these approaches, however, is that they do not take into account the physical and spatial constraints resulting from the cell's unique intracellular milieu ( 7– 9). For example, ≈20–30% of the Escherichia coli cytoplasm is occupied by macromolecules, many of them enzymes, whose cytoplasmic concentration cannot be further increased without drastically affecting protein folding, protein–protein association rates, biochemical reaction kinetics, and transport dynamics within a cell ( 9, 10). This suggests that constraint-based modeling approaches, such as flux balance analysis (FBA) ( 3, 11), could be improved if we take into account that the enzymes catalyzing each reaction compete for the available cytoplasmic space ( 12, 13), potentially limiting the attainable flux rates. Current flux balance-based modeling approaches also have limited ability to predict substrate uptake from the environment. Extensive experimental data indicates that when grown in complex medium bacterial cells use the available substrates either preferentially or simultaneously depending on the growth condition (see, e.g., refs. 14– 17). Efforts to model mixed-substrate growth have assumed specific kinetic expressions for substrate uptake and biomass growth rates, and their predictions are formulated in terms of known model parameters ( 15, 18). Similarly, FBA predictions are based on previous knowledge of the maximum uptake rates in the corresponding medium (the actual variables one aims to predict), and, in contrast to empirical evidence, FBA in itself predicts the simultaneous utilization of all carbon sources from a mixed-substrate growth medium. One way to overcome this deficiency is the superposition of regulatory mechanisms (in the form of mRNA expression signatures) on the FBA model, assessing which substrates are taken up and which are not ( 19). Yet regulatory mechanisms appear during the course of evolution because they result in a selective advantage for the cell. This selective advantage results from better use of the available resources within the metabolic constraints of the organism. Therefore, the metabolic constraint can be considered as the primary cause, whereas the regulatory processes represent the specific molecular mechanism developed to cope with this constraint. This fact opens the possibility for a FBA model that, after imposing the relevant constraints, predicts the selective advantage of implementing a regulatory mechanism. Here, we develop a modified FBA model that incorporates a solvent capacity constraint for the attainable enzyme concentrations within the crowded cytoplasm. Using this model, we predict the maximum growth rate of E. coli MG1655 wild-type and mutant strains on single carbon sources and for the dynamic patterns of substrate utilization from a mixed-substrate growth medium. We test the model predictions by using growth rate measurements and microarray and substrate concentration temporal profiles, and we obtain a good agreement between model predictions and experimental measurements. Taken together, these results suggest that macromolecular crowding indeed imposes a physiologically relevant constraint on bacterial metabolic activity and that incorporating this constraint allows for improved modeling of cell metabolism from system-level principles. FBA with Molecular Crowding. In the flux balance model of cellular metabolism a cell's metabolic network is mathematically represented by the stoichiometric matrix, Smi, providing the stoichiometric coefficient of metabolite m ( m = 1, …, M) in reaction i ( i = 1, …, N) ( 3, 20), where M and N are the number of metabolites and reactions, respectively. The cell is assumed to be in a steady state, where the concentration of each intracellular metabolite (other than the metabolites that constitute the biomass) remains constant in time. Thus, the stationary reaction rates (fluxes) consuming and producing a metabolite should balance,
where fi denotes the flux of reaction i. The study of the solution space defined by Eq. 1 together with maximum capacity constraints for the uptake rates of extracellular substrates constitutes the basis of FBA ( 3). We extend this framework to consider the physical and spatial constraints resulting from the very high intracellular concentration of macromolecules ( 7– 9). Given that the enzyme molecules have a finite molar volume vi, we can fit only a finite number of them in a given volume V. Indeed, if ni is the number of moles of the ith enzyme, then
Eq. 2 represents a constraint on the enzyme levels ni, potentially affecting their maximum attainable values and relative abundance. Dividing by cell mass M, we can reformulate this constraint in terms of the enzyme concentrations Ei = ni/ M (moles per unit mass), resulting in
where C = M/ V ≈ 0.34 g/ml is the E. coli cytoplasmatic density ( 21). Eq. 3 imposes a constraint on the maximum attainable enzyme concentrations and, therefore, we refer to it as the enzyme concentration constraint. This constraint is reflected in the metabolic fluxes as well. Indeed, an enzyme concentration Ei results in a flux fi = biEi over reaction i, where the parameter bi is determined by the reaction mechanism, kinetic parameters, and metabolite concentrations. Therefore, the enzyme concentration constraint ( Eq. 3) is reflected in the metabolic flux constraint
where ai = Cvi/ bi, affecting the maximum attainable fluxes and the flux distribution among different metabolic reactions. From here on, we refer to this mathematical framework as “flux balance analysis with molecular crowding” (FBAwMC). Furthermore, because the coefficient ai quantifies the contribution to the overall crowding by reaction i we refer to it as the “crowding coefficient of reaction i,” or simply “crowding coefficient.” Finally, we note that the enzyme concentration constraint is not the only additional constraint that could potentially restrict the metabolic capabilities of E. coli (for example, transporter capacities may be similarly limiting). Yet, our aim here is to test the predictive value of a model that assumes that the enzyme concentration constraint is indeed a main factor limiting the maximal metabolic capabilities of E. coli. FBAwMC Predicts the Relative Maximum Growth of E. coli Growing on Single Carbon Sources. To examine the validity of macromolecular crowding as a constraint on a cell's metabolic activity, and to test the predictive capability of the FBAwMC framework, we first examined the phenotypic consequences of extracellular substrate availability during growth in single carbon-limited medium with oxygen being in abundance, focusing on the maximum growth rate. The FBAwMC contains as a free parameter the average crowding coefficient  a , and the model predictions for the maximum growth rate are proportional to a. We first assumed that a is a constant independent of the substrates. In this case it is possible to make predictions for the maximum growth rate in different substrates in arbitrary units. To obtain the maximum growth rates in specific units we fit a to minimize the mean-square deviation between the predicted and measured growth rates, resulting in a = 0.0040 ± 0.0005 h·g/mmol, in which g is grams of dry weight. We have obtained an independent estimate of ai for ≈100 E. coli enzymes as well [ supporting information (SI) Datasets 1 and 2], resulting in values between 10 −6 and 10 −1 and most probable values between 10 −5 and 10 −2 (in units of h·g/mmol). The obtained a is, therefore, within the expected range. Using the reconstructed E. coli MG1655 metabolic network ( 22) ( SI Dataset 1), we first tested the maximal growth rate of E. coli MG1655 cells in various single carbon-limited media and compared the results with the theoretically predicted growth rates ( a). In most cases the line of perfect agreement falls within the standard deviation, implying an overall good agreement between the model predictions and the measured maximum growth rates. For glucose and glycerol, the line of perfect agreement is outside the standard deviation, indicating that our assumption of a substrate-independent a is not valid for these two substrates. E. coli is better adapted to growth on glucose, suggesting a smaller average crowding coefficient than in any of the other carbon sources. Indeed, the average crowding coefficient necessary to obtain a perfect agreement for glucose is smaller: a = 0.0031 ± 0.0001 h·g/mmol. In contrast, in some carbon-limited media E. coli reaches its predicted maximal growth rate only after a period of adaptive evolution ( 23, 24), suggesting a higher average crowding coefficient before metabolic adaptation. Indeed, the average crowding coefficient necessary to obtain a perfect agreement for glycerol is larger: a = 0.0053 ± 0.0001 h·g/mmol. The FBAwMC framework also allows us to predict the maximal growth rate of microbial strains with deleted metabolic enzymes, by simply removing the corresponding metabolic reaction from the FBAwMC model and recomputing the maximal growth rate. To test the power of this predictive capability we experimentally determined the maximal growth rate of several E. coli MG1655 single gene deletion mutants grown in glucose-limited medium. As shown in b, the agreement between predicted and measured maximal growth rates is remarkably good for various E. coli mutants, providing further evidence for the validity of our approach. It is worth noting that, as with FBA alone ( 25, 26), this analysis is not limited to single-enzyme mutants, but can be carried out for any combination of two or more enzyme deletions as well. Substrate Hierarchy Utilization by E. coli Cells Growing in Mixed Substrates. Extensive experimental data indicate that when grown in complex media bacterial cells use the available substrates either preferentially or simultaneously (see, e.g., refs. 14, 15, and 17), depending on the growth conditions. To further assess the role of an enzyme concentration limit on cellular metabolism we next examined the substrate utilization of E. coli cells in a mixed carbon-limited medium, and we compared the results to the FBAwMC E. coli model-predicted substrate uptake and utilization (). We grew E. coli MG1655 for 12 h in a batch culture containing an equal concentration (0.04% each) of five different carbon sources (galactose, glucose, maltose, glycerol, and lactate) ( a and SI Fig. 8). These substrates are taken up by E. coli through substrate-specific transport mechanisms and enter the central carbon metabolism through various substrate intermediates ( metabolic pathways). Note, that in single carbon-limited medium, maximum growth rates of E. coli in glucose (0.74 h −1) was higher; whereas the experimentally measured maximal growth rates in glycerol (0.41 h −1) and lactate (0.38 h −1) were lower than the model predictions ( a). In contrast, the maximal growth rates obtained on maltose (0.61 h −1) and galactose (0.51 h −1) were in good agreement with the FBAwMC-predicted values ( a). As typically seen in batch culture, initially E. coli cells showed minimal growth (lag phase) followed by rapid population expansion between 2 and 8 h (exponential growth phase) with no further growth afterward (stationary phase) (a). Parallel with this, the growth rate rapidly increased with the start of the logarithmic growth phase, reaching its maximum between 3 and 3.5 h. Thereafter the growth rate steadily declined, becoming negligible to zero after 8 h (a). Of the five supplied carbon sources, in the first 3.5 h of growth only glucose was used (phase 1); it was depleted from the medium within the first 4 h ( b). This “exclusive glucose use” phase coincided with the initial explosive growth and the maximal attained growth rate of the culture ( a). At 3.5–4 h E. coli cells started to use all four remaining carbon sources, albeit at different rates. Galactose, lactate, and maltose were preferentially used during the next 2 h (phase 2), all three of them being depleted from the growth medium by the sixth hour ( b). During this “mixed carbon utilization” phase lactate was used up at the fastest rate, followed by maltose and galactose. A small amount of glycerol was also taken up during this time interval, but its predominant utilization occurred only after 6 h, and it was completely depleted from the medium by 7.5 h ( b). The concentration of acetate, a well known byproduct of rapid E. coli aerobic growth ( 27, 28), increased steadily, reaching its peak concentration in the growth medium at 6 h of growth. Thereafter, the process was reversed, and acetate, along with glycerol, was rapidly consumed and was depleted from the medium by 8 h ( b), denoting a “late carbon utilization” phase in the culture (phase 3). Of note, as single carbon source, acetate provides a lower maximal growth rate (0.24 h −1) than any of the five supplied carbon sources ( a). Taken together, the sequential order of carbon substrates' uptake in the batch culture experiment only partially correlates with the maximal growth rate they individually provide: it appears earlier for lactate and later for maltose and glycerol. Subsequently, we tested FBAwMC E. coli model on the mixed-substrate conditions. In contrast with FBA ( 3, 11), which predicts the simultaneous utilization of all carbon sources, we find a remarkably good correlation between the mode and sequence of FBAwMC-predicted and measured substrate uptake and consumption ( b and c). There are, however, two notable differences. First the FBAwMC predicts a lesser excretion of acetate. In turn the substrates are consumed faster in vivo ( b), because a larger fraction of the carbon source is diverted toward the excretion of acetate. As a consequence the different phases of substrate consumption are shifted to the right (longer times) for the model predictions. The second major discrepancy is the delayed consumption of galactose in the model predictions ( c). Yet, overall FBAwMC correctly predicts the existence of the three experimentally observed phases of substrate consumption: initial consumption of glucose, intermediate mixed-substrate consumption, and late consumption of glycerol and acetate. As surrogate markers of cellular metabolism, during the batch culture experiments we also traced the changes in pH and oxygen concentrations in the growth medium. We observed a steady decline in pH during the first 6 h, followed by a slight increase then decrease between 6 and 7 h, and finally an increase between 7 and 8 h ( a and SI Fig. 8). There was also an accelerating decline in the dissolved oxygen concentration (pO 2) in the medium during the first 7 h, followed by a rapid stepwise increase during the next 30 min. However, the decline phase (indicating aerobic respiration in an increasingly acidic environment because of acetate excretion) was consistently interrupted by rapid upswings in pO 2 concentration ( a and SI Fig. 8). These spikes indicate brief pauses in aerobic metabolism likely due to switches in predominant substrate use. Indeed, the first of these spikes, at ≈4 h, correlates with the depletion of glucose and initiation of mixed-substrate utilization; the second, at ≈5 h, with the depletion of lactate and increased utilization of maltose; and the third, at ≈6 h, with the start of joint glycerol and acetate utilization. Similarly, the first rapid increase in oxygen concentration at ≈7 h correlates with the near-depletion of glycerol, followed by a final increase after 30 min corresponding with the depletion of acetate from the medium ( b). The Mode and Sequence of Substrate Utilization Correlate with the Expression of Genes Participating in the Uptake Modules. We also prepared mRNA from samples obtained at 30-min intervals between 2 and 8 h and processed them for microarray analysis presented as SI Dataset 3. At the level of substrate uptake pathways () it is evident that the expression of ptsG, the gene encoding the glucose transporter PtsG/Crr, was at a high level from the first time point up until the depletion of glucose from the growth medium, and ptsG expression was rapidly turned off afterward. Similarly, the expression of the gene ( lldP) encoding the lactate transporter, LldP, was high during the first 4.5 h of growth and was turned off rapidly thereafter, in agreement with the earlier than predicted utilization of lactate. In contrast, the expression of gene products responsible for maltose and galactose uptake and utilization were turned on much later and peaked at 4.5–5.5 h, corresponding with the period of their maximal uptake. The expression of gene products responsible for glycerol uptake and utilization peaked in two waves, the smaller one between 4.5 and 5 h and the larger one at 6.5–7 h, the latter corresponding to maximal glycerol consumption from the medium ( b). Finally, the expression of acs, whose gene product catalyzes acetate uptake toward the citric acid cycle, peaked between 6 and 8 h of growth, corresponding to the maximal uptake of previously secreted acetate from the growth medium. We note that all these changes are in good agreement with the FBAwMC model-predicted uptake of the corresponding substrates (, black tracings). To assess the quality of the microarray profiles and to identify genes with expression patterns that are similar to those of genes encoding enzymes of the uptake pathways we used TimeSearcher ( 29). We find that most genes displaying expression patterns similar to those of the query genes are colocalized with them in the same operon ( SI Figs. 10–15). For example, for the maltose uptake module genes ( malEFGK, malQ, and glk), TimeSearcher identified several other genes ( lamB, malM, malP, malS, and malZ) with similar expression profiles. These genes are part of various operons within the maltose regulon ( 30), although not all of them directly participate in maltose uptake. Similarly, for glycerol metabolism several related glycerol utilization genes ( glpA, glpB, glpC, glpD, glpQ, and glpT) displayed expression patterns that were similar to those of the three genes responsible for glycerol uptake ( glpF, glpK, and gpsA). The products of these genes are part of the pathway for glycerol catabolism after its uptake. Activation of Stress Programs upon Switching Metabolic Phases. To assess the global state of E. coli transcriptome during the various metabolic phases of the time course experiment, we used three different data analysis methods to analyze the full microarray data. These methods included hierarchical clustering with optimal leaf ordering ( 31, 32) (), principal component analysis (PCA) ( 33) ( SI Fig. 9), and a probabilistic clustering method based on hidden Markov models (HMMs) ( 34) ( SI Figs. 19–26). It is evident that during the exclusive glucose utilization phase there are similar expression profiles in all samples collected between 2 and 3.5 h, followed by transition in the transcriptome state at the beginning of mixed utilization phase after 4 h (). Within the latter phase, the up-regulation of, e.g., the genes of the maltose regulon is clearly evident (). Samples obtained during the mixed-substrate utilization phase (5 and 5.5 h) and the late carbon utilization phase (6.5 h) display similar global expression profiles (), interrupted by a significant alteration in the expression profile at 6 h that denotes the switch from acetate secretion to acetate utilization (). Interestingly, the transcriptome at 6 h displays substantial similarity to that characterizing E. coli cells at the (near) exhaustion of all substrates from the medium (7–8 h), a phase that is characterized by generic stress response ( and SI Fig. 27). Similarly, many of the genes up-regulated at the end of the last phase are also up-regulated to a lesser extent at 3.5 h, the stage of switching from exclusive glucose utilization to a mixed-substrate utilization phase. To further characterize the time-point-specific expression profiles, we also prepared mRNA samples from individual mid-logarithmic batch culture E. coli cells (OD 600 ≈ 0.2) grown in glucose-, maltose-, glycerol-, acetate-, lactate, or galactose-limited medium, processed them for microarray analysis (presented as SI Datasets 4 and 5), and compared the obtained transcriptome profiles with those of the individual time points ( Lower) from mixed-substrate experiment. It is evident that the transcriptome profiles during the glucose-only and mixed-substrate utilization phases display the highest correlation to that of glucose- and maltose-limited cultures, especially at the initial time points, whereas the late carbon utilization profiles are most similar to that of glycerol- and, especially, acetate-limited cultures. Highly notable is the transcriptome profile of galactose-limited cultures, which shows some similarity to that of cells at the stage of switching from exclusive glucose utilization to a mixed-substrate-utilization phase (3.5 h), and an even higher similarity to the transcriptome profiles of cells when all carbon sources are depleted (8 h). Thus, E. coli cells display a partial adaptation/stress response at each major metabolic transition, followed by a generic stress response ( SI Fig. 27) and implementation of a foraging program ( 35) at complete exhaustion of all extracellular substrates that seems to be most primed for acetate and galactose catabolism. A key aim of systems biology is the identification of the organizing principles and fundamental constraints that characterize the function of molecular interaction networks, including those that define cellular metabolism. In the present work we have focused on the identification of principles that define the growth and substrate utilization mode of bacterial cells in complex environments. Our experimental results indicate the occurrence of three major metabolic phases during the growth of E. coli on one type of mixed-substrate medium. Glucose, which by itself provides the highest growth rate, is preferentially used by E. coli, followed by simultaneous utilization of maltose, l-lactate, and galactose. Glycerol and (secreted) acetate are used at a third and final stage of growth. In addition, global mRNA expression data indicate that the organism-level integration of cellular functions in part involves the appearance of partial stress response by E. coli at the boundaries of major metabolic phases, and, as previously shown ( 35), the activation of a foraging program upon exhaustion of substrates from the growth medium (). The simulation results show that the FBAwMC model introduced here successfully captures all main features of the examined metabolic activities. First, there is a significant correlation between in vivo relative maximal growth rates of E. coli in different carbon-limited media and the in silico predictions of the FBAwMC (). Second, the FBAwMC model predicts remarkably well the existence of three metabolic phases and hierarchical mode (i.e., single- or mixed-substrate utilization) of substrate utilization in mixed-substrate growth medium (–). In essence, our modeling approach indicates that when E. coli cells grow in conditions of substrate abundance their growth rate is determined by the solvent capacity of the cytoplasm; vice versa, the solvent capacity should be saturated at the maximal growth rate. Therefore, when growing in a mixture of abundant carbon sources E. coli cells should preferentially consume the carbon source resulting in the highest growth rate. At solvent capacity saturation, the synthesis of metabolic enzymes for the utilization of a second, less efficient, carbon source can take place only at the expenses of degrading metabolic enzymes involved in the consumption of the more efficient carbon source. However, this would result in a growth rate reduction and, therefore, cells preferentially using the more efficient carbon source would outgrow those that allow the simultaneous utilization of other carbon sources. We observe, however, two discrepancies of the FBAwMC model predictions: ( i) a higher than predicted amount of secreted acetate in the growth medium, and ( ii) a somewhat earlier uptake and consumption of various substrates from the medium compared with that predicted by the model. The first discrepancy is likely rooted on the contribution of other cell components apart from metabolic enzymes. With increasing growth rate higher concentrations of ribosomal proteins, mRNA, and DNA are required in addition to metabolic enzymes ( 36). This observation indicates that the FBAwMC model may underestimate the impact of macromolecular crowding and the resulting excretion of acetate. The second discrepancy is quite likely a consequence of the first one, as acetate secretion is generally correlated with an increased carbon source uptake rate ( 27). Taken together, our results show that in silico models incorporating flux balance and other physicochemical constraints can capture increasingly well the metabolic activity of bacterial cells, and that the maximum enzyme concentration is a key constraint shaping the hierarchy of substrate utilization in mixed-substrate growth conditions. Yet, while the metabolic capabilities of a cell are limited by such constraints, in reality any change in metabolic activity is controlled by regulatory mechanisms evolved in the context of these constraints. Therefore, constrained optimization approaches are also expected to help us better understand and/or uncover regulatory mechanisms acting in E. coli and other organisms. Mathematical Framework. The FBAwMC modeling framework has been established, as described in Results and as detailed in SI Text, S1 and S2. Growth Experiments, Carbon Substrate, and Microarray Analyses. The E. coli K12 strain MG1655 (F − λ − ilvG rfb50 rph1) was used throughout the work. Isogenic E. coli mutants ( pgk, atpC, gpmA, nuoA, gdhA, and pfkA) were obtained from F. Blattner (University of Wisconsin, Madison) ( 37). The experimental details of the growth rate measurements, substrate concentration assays and microarray analyses are detailed in SI Text, S3–S12. We thank N. Gerry and M. Lenburg (Boston University, Boston, MA) for their help with microarray experiments, and an anonymous reviewer for comments on the manuscript. Research at the University of Notre Dame and at the University of Pittsburgh was supported by the National Institutes of Health Grant U01 AI070499. | | FBA | flux balance analysis | | | FBAwMC | FBA with molecular crowding. |
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