 Research article
 Open Access
 Published:
Redox balance is key to explaining full vs. partial switching to lowyield metabolism
BMC Systems Biology volume 6, Article number: 22 (2012)
Abstract
Background
Lowyield metabolism is a puzzling phenomenon in many unicellular and multicellular organisms. In abundance of glucose, many cells use a highly wasteful fermentation pathway despite the availability of a highyield pathway, producing many ATP molecules per glucose, e.g., oxidative phosphorylation. Some of these organisms, including the lactic acid bacterium Lactococcus lactis, downregulate their highyield pathway in favor of the lowyield pathway. Other organisms, including Escherichia coli do not reduce the flux through the highyield pathway, employing the lowyield pathway in parallel with a fully active highyield pathway. For what reasons do some species use the highyield and lowyield pathways concurrently and what makes others downregulate the highyield pathway? A classic rationale for metabolic fermentation is overflow metabolism. Because the throughput of metabolic pathways is limited, influx of glucose exceeding the pathway's throughput capacity is thought to be redirected into an alternative, lowyield pathway. This overflow metabolism rationale suggests that cells would only use fermentation once the highyield pathway runs at maximum rate, but it cannot explain why cells would decrease the flux through the highyield pathway.
Results
Using flux balance analysis with molecular crowding (FBAwMC), a recent extension to flux balance analysis (FBA) that assumes that the total flux through the metabolic network is limited, we investigate the differences between Saccharomyces cerevisiae and L. lactis that downregulate the highyield pathway at increasing glucose concentrations, and E. coli, which keeps the highyield pathway functioning at maximal rate. FBAwMC correctly predicts the metabolic switching mode in these three organisms, suggesting that metabolic network architecture is responsible for differences in metabolic switching mode. Based on our analysis, we expect gradual, "overflowlike" switching behavior in organisms that have an additional energyyielding pathway that does not consume NADH (e.g., acetate production in E. coli). Flux decrease through the highyield pathway is expected in organisms in which the highyield and lowyield pathways compete for NADH. In support of this analysis, a simplified model of metabolic switching suggests that the extra energy generated during acetate production produces an additional optimal growth mode that smoothens the metabolic switch in E. coli.
Conclusions
Maintaining redox balance is key to explaining why some microbes decrease the flux through the highyield pathway, while other microbes use "overflowlike" lowyield metabolism.
Background
One of the key steps in energy metabolism is to transfer the energy carried by sugars, including glucose, to the biological "energy currency" adenosine triphosphate (ATP). The number of ATP molecules generated by metabolizing one molecule of glucose—the ATP yield—is one of the most basic measures of an organism's energy efficiency. One would perhaps expect that evolution has selected organisms for the ability to extract energy from their food at optimal efficiency by maximizing ATP yield. Yet surprisingly, many organisms switch between a highyield pathway, e.g., aerobic respiration that yields more than thirty moles of ATP per mole glucose, and a highly inefficient, lowyield fermentation pathway that yields only two or three moles ATP per mole of glucose. This effect is known as the Crabtreeeffect in the baker's yeast Saccharomyces cerevisiae. S. cerevisiae turns glucose into CO_{2} in aerobic, glucoselimited conditions. But in abundance of glucose, glucose is converted into ethanol [1], even if oxygen levels do not limit aerobic metabolism. Many bacteria also use a highyield metabolic pathway in glucoselimited conditions and a lowyield pathway in excess of glucose. Examples are Escherichia coli[2], Bacillus subtilis[3] and lactic acid bacteria, e.g., Lactobacillus plantarum and Lactococcus lactis[4, 5]. The effect is also found in multicellular eukaryotes, including human cancer cells, where it is called the Warburg effect [6]. Muscle cells switch to lowyield metabolism during heavy exercise [7], fermenting glucose into lactic acid. Why cells would produce less ATP per glucose molecule than they can is a longstanding question in biology [8–12].
Microbial species show remarkable differences in their metabolic switching strategies. At low glucose concentrations and low growth rates, E. coli uses highyield metabolism, aerobically converting glucose into CO_{2} and water. At higher glucose concentrations and fast growth rates, it redirects part of the glucose influx into a lowyield fermentation pathway, keeping oxidative phosphorylation fully active [2]. S. cerevisiae uses highyield, aerobic respiration at slow growth rates; at fast growth rates it ferments most glucose into ethanol, and downregulates aerobic respiration, keeping aerobic respiration active at a much lower rate. Although L. lactis does not have an aerobic respiration pathway, it still performs a metabolic switch. At fast growth rates it makes a full switch to lactic acid fermentation [4], which yields about 50% less ATP than the higheryield mixed acid fermentation pathway, that produces formate, acetate and ethanol.
A plausible explanation for metabolic switching is "overflow metabolism". It assumes that organisms only switch to lowyield metabolism if the highyield pathway is operating at maximum rate and cannot process any more molecules [13, 14]. The remainder would then spill into the lowyield pathway. This explanation requires the lowyield pathway to operate at a faster rate than the highyield pathway, which is likely the case [8, 15, 16]. Thus overflow metabolism plausibly explains concurrent use of highyield and lowyield pathways, as in E. coli. However, a problem with overflow metabolism is that it does not explain why organisms like S. cerevisiae or L. lactis would partly switch off their highyield pathways at high growth rates.
Recent studies have suggested that the limited amount of metabolic enzymes fitting inside the cell may be key to lowyield metabolism [12, 17, 18]. Simply because cells can host only a finite number of metabolic enzymes, they may need to trade off investment into the bulky enzymatic machinery required for lowthroughput, highyield metabolism, or alternatively to invest into many more "lean" glycolytic enzymes producing a highthroughput, low yield metabolism. Thus, according to this view, high glucose uptake rate should correlate with low yield metabolism, and vice versa. Indeed, this is observed in comparative studies of metabolism in yeast species of the Saccharomyces clade [19] and in comparative studies of glucose metabolism of various bacterial species [20].
If cells need to trade off fast metabolism and highyield metabolism, then why do we still observe overflow metabolism, as in E. coli? We address this question by comparing the optimal metabolic switching strategies of L. lactis, S. cerevisiae, and E. coli as predicted by a genomescale computational model. These three organisms use different pathways to metabolize glucose. In Figure 1 a simplified reaction scheme of the most important glucose degrading pathways in these three organisms is presented. E. coli can use oxidative phosphorylation, lactate fermentation, ethanol fermentation and acetate fermentation. L. lactis can use mixedacid fermentation, producing formate, acetate and ethanol, or lactate fermentation. S. cerevisiae can use oxidative phosphorylation, ethanol fermentation and acetate fermentation.
To predict the metabolic switches these three organisms can perform, we make use of a variant of Flux Balance Analysis (FBA), a method that calculates fluxes through metabolic networks given constraints on the network and given an objective function to maximize. By maximizing growth rate, FBA often correctly predicts cellular metabolism, including uptake, excretion and growth rates of cells [2, 21]. However, because the glucose uptake rate is fixed in these simulations, growth yield (defined as the growth rate divided by the glucose uptake rate) is effectively maximized [16]. Therefore, FBA cannot satisfactorily predict lowyield metabolism.
For this reason, we use an extension of FBA, Flux Balance Analysis with Molecular Crowding (FBAwMC) [17, 22]. In contrast to FBA, FBAwMC calculates the optimal flux distribution through a metabolic network under the physiologicallyplausible constraint that only a finite number of metabolic enzymes fit into a cell. Because each of the enzymes has a maximum turnover number (k_{ cat } ), molecular crowding naturally results in a constraint on the total metabolic flux through the network:
with f_{ i } being the flux through reaction i, V_{ prot } the volume fraction of macromolecules devoted to metabolic enzymes and c_{ i } the "crowding coefficient" of reaction i. The crowding coefficient of reaction i is defined as the volume that needs to be occupied with enzymes to reach unit flux through reaction i and is given by ${c}_{i}\equiv \frac{M{v}_{i}}{V{b}_{i}},$where M is the cell mass, V the cell volume, v_{ i } the molar volume of the enzyme catalyzing reaction i, and b_{ i } a variable describing the proportionality between enzyme concentration and flux through reaction i[22]. Intuitively, the crowding coefficient can be seen as the protein cost of a reaction: enzymes with low crowding coefficients have small molecular volume or catalyse fast reactions. FBAwMC correctly predicts lowyield metabolism: e.g., growth curves of E. coli[17, 22], and the Warburg effect in cancer cells [18]. Therefore, FBAwMC is well suited for our aim: to unravel the metabolic differences between microbes that decrease the flux through the highyield pathway at high growth rates and those that keep the highyield pathway always fully active.
Because crowding coefficients for most metabolic enzymes are unknown, previous studies proposed a range of strategies to estimate them. Beg et al.[22] fitted an average crowding coefficient 〈c〉 in order to obtain a good match between predicted and measured growth rates. Shlomi et al.[23] obtained 15% of crowding coefficients from experimental data and assigned the median of the known crowding coefficient values to the remaining unknown crowding coefficients. Vazquez et al.[17] sampled crowding coefficients randomly from a range of physiologicallyplausible values obtained from online, biochemical databases, and presented averages and variations of the metabolic fluxes predicted for a large random sample of crowding coefficients.
Although the study of an estimated, specific set of crowding coefficients or an average can provide some insight, in reality metabolic networks may operate under an entirely different set of crowding coefficients. Therefore, in the absence of accurate, experimental estimates of crowding coefficients, FBAwMC cannot decide on one real situation. Studying growth yield predictions for large samples of biochemicallyplausible sets of crowding coefficients can give more robust insights into the metabolic network than studies with single crowding coefficient estimates, because it reveals what growth yields are most plausible and what are the alternative behaviors of the network.
Our analysis suggests that mechanisms to maintain NAD^{+}/NADH ratio are key to the metabolic differences between the two types of metabolic switches. Organisms in which both the highyield and lowyield pathways reduce NADH may downregulate highyield metabolism at high growth rates. If organisms have an additional energyyielding pathway that does not consume NADH (e.g., acetate production in E. coli), it is optimal to keep both the lowyield and highyield pathways active at high growth rates.
Results
Predicted yield distributions reflect metabolic switching strategy
Using genomescale stoichiometric networks of L. lactis[24], E. coli[25], and S. cerevisiae[26] we first confirmed that our implementation of the FBAwMC method reproduces the correct growth curves (Figure 2). Indeed, FBAwMC qualitatively reproduces both the metabolic switch of S. cerevisiae and L. lactis and the overflow metabolism of E. coli. Following Vazquez and coworkers [17], crowding coefficients were chosen at random from a distribution of crowding coefficients based on published molar volumes (Metacyc [27]) and turnover numbers (Brenda [28]). Thus the growth curves that FBAwMC predicts are the average behavior for 1000 randomly sampled sets of crowding coefficients. Figure 2 reports the average and standard deviations of waste product formation or oxygen consumption for the set of simulations resulting in lowyield metabolism (growth yield < 0.3 gr dry weight/gr glucose). This represents 74% of all simulations for E. coli, 44% for L. lactis and 32% for S. cerevisiae. As reflected by the growth curves' large variation, the optimal growth curve that FBAwMC predicts depends strongly on the particular sample of crowding coefficients used. Because a crowding coefficient c_{ i } reflects the enzymatic cost of producing a metabolic flux f_{ i } , we asked if the relation between crowding coefficient selections and the predicted optimal fluxes could yield new biological insight.
Figure 3 shows the predicted, optimal growth yields at unconstrained glucose influx for a sample of 1000 sets of randomly selected crowding coefficients. Interestingly, computationally obtained growth yields in L. lactis and S. cerevisiae distribute bimodally, with only few simulations predicting intermediate growth yields. By contrast, the predicted growth yields of E. coli are distributed more uniformly. Strikingly, the predicted growth yield distributions for a random sampling of crowding coefficients correlate well with the organisms' switching strategies: L. lactis and S. cerevisiae, which downregulate their highyield metabolism, display a bimodal distribution of predicted growth yields, whereas E. coli, which has overflowlike metabolism, displays a uniform distribution of predicted growth yields.
The distribution of growth yields only gives information of the metabolic behavior at maximal growth rates. Next we tested if individual simulations show overflowlike metabolism or not. Figure 4 shows a frequency diagram of the reduction of the flux through the highyield pathway at maximal growth rate, as a percentage of the maximal flux through the highyield pathway. For E. coli we observe that most sets of crowding coefficients lead to little flux decrease through the highyield pathway (Figure 4C). However, for L. lactis, most sets of crowding coefficients produce a complete halt of the highyield pathway (Figure 4A). For S. cerevisiae, for 40% of crowding coefficients the highyield pathway is repressed more than twofold at high growth rates, whereas for 42% the highyield pathway is reduced at most by 10% (Figure 4B). Thus, at high growth rates FBAwMC predicts overflowlike metabolism in E. coli, whereas it predicts that the flux through the highyield pathway is likely to be downregulated for L. lactis and S. cerevisiae.
Acetate excretion makes E. coli use overflowlike switching
What could explain that S. cerevisiae and L. lactis downregulate highyield metabolism at high growth rates, whereas E. coli uses overflowlike metabolism? Because we sampled from sets of prokaryotic crowding coefficients for E. coli and L. lactis, and from a eukaryotic dataset for S. cerevisiae, we first checked if the speciesspecific sets of crowding coefficients were responsible for our observations. We performed simulations with the E. coli network using the eukaryotic set of crowding coefficients and vice versa, and found that this did not affect our results. Therefore we conclude that the key difference between the three models is in the speciesspecific topology of the metabolic networks and their behavior in the presence of crowding, not in the specific values of crowding coefficients.
A key difference that sets E. coli apart from L. lactis and S. cerevisiae is shown in Figure 1. After converting glucose to pyruvate, L. lactis and S. cerevisiae either convert it into a waste product (ethanol) or further metabolize pyruvate to yield extra ATP. L. lactis converts two acetylcoA into acetate and ethanol in parallel to retain redox balance, gaining (at most) one additional ATP per mole glucose. S. cerevisiae feeds pyruvate into the citric acid cycle and oxydative phosphorylation, gaining (at most) twentyeight additional ATP and two GTP per mole glucose. Interestingly, E. coli has three choices: it can convert pyruvate into the waste product ethanol, it can metabolize pyruvate in the citric acid cycle, or it can gain one extra ATP in the conversion of acetylcoA into acetate.
Although acetate production is a "cheap" way—in terms of the number of enzymes required—to produce additional ATP from pyruvate, it poses an additional challenge to E. coli. During formation of waste products (i.e., lactate or ethanol) the NADH produced in glycolysis or in the conversion from pyruvate to acetylcoA is reduced back to NAD^{+}. Thus such waste product formation is a "fast" way to restore a sufficiently high NAD^{+}/NADHratio. Acetate production does not restore the NAD^{+}/NADHratio, so acetate production might deplete the available NAD^{+} in the cell. So, E. coli might keep oxidative phosphorylation running at fast growth rates (and consume oxygen) in order to profit from the extra ATP yield in acetate formation and restore the NAD^{+}/NADH ratio.
This analysis suggests that, if E. coli ferments glucose into lactate or ethanol, its oxygen consumption will be reduced. We therefore studied the excretion patterns belonging to different sets of crowding coefficients more carefully. We found that, if a set of crowding coefficients results in acetate fermentation, without any ethanol or lactate fermentation, the cells continue to consume oxygen consumption during the switch. If a set of crowding coefficients results in lactate or ethanol fermentation, the consumption of oxygen is often reduced (Figure 5). Of the fraction of simulations of E. coli in which oxygen consumption is reduced during the metabolic switch (see Figure 4C) most produce ethanol and lactate, not acetate (data not shown). Thus, together these simulations are in agreement with the fact that E. coli needs to consume oxygen as an external electron acceptor during acetate fermentation in order to maintain a sufficiently high NAD^{+}/NADH ratio [30]. If E. coli is genetically engineered to eliminate the TCA cycle and all NADHreducing fermentation pathways (i.e., the strain cannot produce lactate or ethanol), it needs dissolved oxygen to grow [30].
To further confirm the hypothesis that at high growth rates overflow metabolism is optimal in E. coli due to acetate excretion, we blocked acetate excretion in the FBAwMC model of E. coli (by setting the maximum efflux to zero) such that acetate production stalled, and recalculated the distribution of optimal growth yields. As Figure 6 demonstrates, in this simulation experiment the growth yields become bimodally distributed over the crowding coefficient samples, suggesting that E. coli can switch bimodally. Also, there are more sets of crowding coefficients that result in a decrease in flux through the highyield pathway (Additional file 1: Figure S1). Apparently, after blocking the route for producing one extra ATP, it again becomes optimal to restore the NAD^{+}/NADH ratio by producing the alternative waste products lactate or ethanol. This model observation agrees with experiments by De Mey et al.[31] who report increased lactate and ethanol excretion after reducing the carbon flow to acetate.
To confirm that an additional ATPproducing pathway can indeed lead to an additional optimal growth mode, we developed a simplified metabolic network model [16], illustrated in Figure 7A. The simplified model has five reactions that represent glycolysis, lactate/ethanol excretion, acetate excretion, the TCAcycle, and oxidative phosphorylation. Using FBAwMC we predicted the optimal yields for a sample of crowding coefficients. In this model, we found four metabolic modes (Figure 7B). After knocking out the acetate pathway, we found only two metabolic modes, a highyield and a lowyield pathway (Figure 7C). Thus, also in this simplified model, acetate production introduces intermediateyield metabolic modes.
Discussion
We have computationally compared metabolic switching at high growth rates in E. coli with L. lactis and S. cerevisiae. E. coli shows overflow metabolism, meaning that at high growth rates it increases its metabolic rate by activating lowyield metabolic pathways in addition to the highyield oxidative phosphorylation pathway. Instead, L. lactis and S. cerevisiae show metabolic switching: they suppress the flux through their highyield pathways at high growth rates, relying mostly on lowyield metabolism. Our analysis suggests that a key difference between the two groups is the number of metabolic pathways yielding ATP, and the effect of these pathways on the NAD^{+}/NADHratio. L. lactis and S. cerevisiae have two alternative pathways, an efficient, highyield pathway—glycolysis followed by mixed acid fermentation or oxidative phosphorylation—and an inefficient, lowyield pathway—glycolysis followed by lactic acid or ethanol production. In both the lowyield and highyield pathways, the NADH resulting from glycolysis is oxidized back to NAD^{+}. In addition to lactate and ethanol fermentation and oxidative phosphorylation, E. coli has a second lowyield pathway: the conversion of pyruvate to acetate. This pathway yields one extra ATP over lactate fermentation, but it does not oxidize NADH, so the NAD^{+}/NADHratio must be restored elsewhere. Our model suggests that in this case it is optimal to keep oxidative phosphorylation running, instead of calling in lowyield pathways to reduce NADH, e.g., lactate production.
To test the idea that acetate production is the cause of overflow metabolism in E. coli, we blocked acetate fermentation in the FBAwMC model. The distribution of predicted growth yields became more bimodal, and the proportion of cells that downregulated their highyield pathway increased. Should we hence expect E. coli to downregulate its highyield pathway at high growth rates, if its acetate production pathway were blocked experimentally? Note that FBAwMC predicts optimal growth rates. Thus it predicts the growth rates for organisms that have already evolved towards optimality. Our results would therefore suggest that mutated E. coli strain with blocked acetate fermentation would evolve downregulation of its highyield pathway after selection for growth rate in cell culture experiments.
Our model results suggest that restoring the redox balance is key in metabolic switching, agrees with experimental observation. Vemuri et al.[32] overexpressed both NADH oxidase (NOX) and alternative oxidase (AOX) in S. cerevisiae and found that glycerol (for NOX) or ethanol formation (for AOX) were diminished. In another study, Vemuri et al.[33] increased oxidation of NADH by overexpressing NOX in E. coli and studied the effect on overflow metabolism. They found that overexpression of NOX strongly diminished acetate fermentation.
To check whether our model is consistent with these experiments, we mimicked them in the FBAwMC models for E. coli and S. cerevisiae. We introduced the reactions that NOX and AOX catalyze to the metabolic model and enforced a lower bound on their fluxes to mimic the effect of overexpression. We performed simulations of S. cerevisiae with ten thousand sets of crowding coefficients, of which only 110 crowding coefficient selections resulted in excretion of both ethanol and glycerol. We went on with these sets of crowding coefficients, because they best mimicked the wildtype phenotype that Vemuri et al.[33] used. The simulated overexpression of NOX or AOX reduced excretion of ethanol and glycerol in practically all of these 110 simulations (Table 1). Thus, in agreement with experiments, the FBAwMC model of S. cerevisiae suggests that excessive NADH breakdown reduces ethanol and glycerol fermentation. In the FBAwMC model of E. coli we found that increased oxidation of NADH could either result in a decreased or in an increased production of acetate, depending on the selection of crowding coefficients. Thus, the experiments of Vemuri et al.[33] do not corroborate nor falsify our model. The reason is that FBAwMC identifies optimal fluxes. In E. coli, during acetate fermentation some NADH is formed in the conversion of pyruvate to acetylcoA. During NOX overexpression, the cells must boost NADH production to maintain optimal growth; this can be done either using the TCAcycle or using acetate fermentation. Depending on the crowding coefficients, either way is optimal.
The computational results presented in this paper are contingent on two underlying, biological assumptions of FBAwMC that may limit the applicability of our approach to strains growing in wellmixed, nutrientrich lab conditions: a) evolution optimizes cells' growth rates instead of yields, and b) a solvent constraint (i.e., the number of enzymes "fitting" inside the cell) puts selective pressure on cells to evolve mechanisms to rapidly produce or remove enzymes for alternative metabolic pathways [22]. Thus FBAwMC implicitly assumes that evolution has shaped cells to make the optimal choice between alternative metabolic pathways.
The optimality assumption is not necessarily correct in all environments. Apart from the fact that evolution does not always lead to optimality [34], game theory suggests that spatial or seasonal environments favor maximization of growth yield [35, 36]. Optimization of growth rate, as implicitly assumed in FBAwMC, is more likely applicable to homogeneous, nonseasonal environments, i.e., a chemostat [35, 37]. Thus our simulations apply primarily to laboratory strains, which are adapted to wellmixed, nutrientrich laboratory conditions. For natural strains, the maximization of growth yield that standard FBA assumes might be better applicable [20].
The second key assumption of FBAwMC, namely that cells have evolved regulation mechanisms to activate production of enzymatic machinery for the pathway giving optimal growth rate [22], relates closely to the explanation proposed by Molenaar et al.[12]. Using a minimal model of a selfreplicator, they showed how a tradeoff between the metabolic efficiency of a pathway, and the cost associated with producing the enzymes for that payway, can lead to a switch in metabolic strategy. Very recently, Zhuang et al.[38] proposed instead that competition for membrane space between glucose transporters and respiratory chain enzymes could be responsible for the metabolic switch between respirative and respirofermentative metabolism in E. coli. They introduced an alternative extension of FBA to accommodate for this effect. We anticipate that our results would hold if we used Zhuang et al.'s modification of FBA instead of FBAwMC. A requirement for our results is that the sum of a set of key metabolic fluxes is constrained. In FBAwMC this constraint is proposed to be due to the limited enzyme solvent capacity in the cytosol [12, 17]. Mathematically, Zhuang et al.[38] propose a very similar constraint, but argue it is due to competition for membrane space between glucose enzymes and respiratory chain enzymes. In fact, the explanation proposed by Vazquez et al.[17] may be more generally applicable, because Zhuang et al.'s rationale would not hold if glucose transporters and respiratory chain enzymes do not share the same membranes as, e.g., in eukaryotes.
Conclusions
Why, at high rates, do some microbes use lowyield metabolism in addition to the highyield pathway—overflow metabolism—whereas other microbes downregulate their highyield pathways? Here we show that maintaining redox balance is key to understanding overflow metabolism in E. coli. Microbes that use lowyield pathways converting NADH back to NAD, including L. lactis and S. cerevisiae, are expected to downregulate their highyield pathways at high growth rates. E. coli can get one extra ATP using acetate secretion; doing so it must keep the oxidative phosphorylation pathway running to restore redox balance, giving rise to "overflowlike" metabolism.
Methods
Flux balance analysis with molecular crowding
We have used FBAwMC [17, 22] to predict growth, uptake and excretion rates in S. cerevisiae, E. coli and L. lactis, using the genomescale metabolic models published in [26, 25] and [24], respectively. We downloaded the E. coli and S. cerevisiae model from the BiGG database [39]http://bigg.ucsd.edu/. The L. lactis model was downloaded from the Supplementary Materials in Oliveira et al.[24].
FBAwMC assumes that the metabolic network is in steady state
where $\stackrel{\u20d7}{x}$ is a vector of all metabolites, $\stackrel{\u20d7}{f}$is a vector describing the metabolic flux through each reaction in the network, and S the stoichiometric matrix. S is defined as follows: if reaction i produces n metabolites of type j, then S_{ ij } ≡ n ; S_{ ij } ≡  n for consumption of metabolite j; otherwise S_{ ij } ≡ 0. FBAwMC attempts to find a solution F of Eq. 2 that maximizes an objective function, given a set of constraints. In this study, we always optimize for growth rate. We also incorporate constraints on the individual fluxes:
where f_{ lb, n } is the minimal flux and f_{ ub, n } the maximal flux through reaction n. Furthermore, a constraint on the total flux through the network is added to account for the limited amount of enzymes in any given cell, given by
Here ${c}_{n}\equiv \frac{M{v}_{n}}{V{b}_{n}}$is the "crowding coefficient", M the cell mass, V the cell volume, v_{ n } the molar volume of the enzyme catalysing reaction n and b_{ n } a parameter describing the proportionality between enzyme concentration and flux. For a derivation of Eq. 3 see Beg et al.[22]. V_{ prot } is a constant (0 ≤ V_{ prot } ≤ 1) representing the volume fraction of macromolecules devoted to metabolic enzymes. We fit V_{ prot } to the experimentally observed growth rate and glucose uptake rate. Additional file 2: Table S1 lists the results of this fitting procedure. Interestingly, V_{ prot } values are very similar across different organisms, ranging from 0.150.2. Note that Vazquez et al.[17] assumed that V_{ prot } = 1, which we believe is unrealistic, because not only metabolic enzymes fill the cell's cytoplasm. Linear programming efficiently solves this problem, but the solution is not necessarily unique.
Crowding coefficients
To obtain the crowding coefficients c_{ i } we adopted the approach of Vazquez et al.[17]. The molar volume v_{ i } can be estimated from the molar masses of the enzymes using a specific protein volume of 0.73 ml/g. b_{ i } depends on the concentration of metabolites and on the turnover numbers of the enzymes (for example in a MichaelisMenten way, where $b={V}_{max}\frac{S}{S+{K}_{M}},$here, we would estimate b = V_{ max } ). Following Vazquez et al.[17], we constructed a distribution of crowding coefficients from turnover numbers and enzyme masses. We obtained turnover numbers from enzyme database Brenda [28], enzyme masses from MetaCyc [27]. The distribution of crowding coefficients we then obtained using the relationship ${c}_{i}\equiv \frac{M{v}_{i}}{V{b}_{i}}.$The turnover numbers and enzyme masses used are given in Additional file 3: Figure S2.
As there is insufficient data for L. lactis we used E. coli crowding coefficients for this organism as well. The turnover numbers, both for E. coli and S. cerevisiae varied over orders of magnitudes. Importantly, a few enzymesubstrate combinations had extremely low turnover numbers that effectively stopped the reactions. Because these turnover numbers typically occurred for nonmetabolic reactions (e.g., DNA repair) or for nontypical substrates of metabolic enzymes, we only used turnover numbers of metabolic enzymes and for each enzyme we only kept the highest available turnover number and left out enzymes with turnover number smaller than 0.01/s. We used wildtype turnover numbers if reported. The resulting distribution of crowding coefficients for E. coli was similar to the distribution found by Vazquez et al.[17] (see Additional file 4: Figure S3). For S. cerevisiae we found a similar distribution as for E. coli (Additional file 5: Figure S4).
In silico growth experiments
We initiated each simulation with randomly select crowding coefficients from the obtained distributions. We assigned a crowding coefficient of 0 to nonenzymatic reactions. The COBRA Toolbox [40] was used to perform FBAwMC in Matlab, with the GNU Linear Programming Kit as linear programming solver http://www.gnu.org/software/glpk.
The in silico growth media included the vitamins, nucleotides and minerals required for optimal growth. For the constraints on the reactions used in the simulations, we refer to Additional file 6: Table S2. Because L. lactis cannot synthesize many amino acids we must supply them in the in silico growth medium. In order to ensure that cells are not limited by amino acid uptake, we constrained the maximal amino acid uptake rates to the biomass content of that amino acid multiplied by twice the (experimentally observed) maximal growth rate. In this way, the maximal amino acid uptake rate suffices for twice the experimentally observed growth rate.
Matlab code to reproduce the simulations are included in Additional file 7. The COBRA Toolbox [40] and a linear programming solver are required.
Abbreviations
 AOX:

Alternative oxidase
 ATP:

Adenosine triphosphate
 BiGG:

Biochemical Genetic and Genomic knowledgebase of large scale metabolic reconstructions
 CO_{2}:

Carbon dioxide
 COBRA:

COnstraints Based Reconstruction and Analysis
 FBA:

Fluxbalance analysis
 FBAwMC:

Fluxbalance analysis with molecular crowding
 GTP:

Guanosine5'triphosphate
 NAD^{+}:

Nicotinamide Adenine Dinucleotide
 NADH:

Reduced form of NAD^{+}
 NOX:

NADH oxidase.
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Acknowledgements
We thank three anonymous referees for their constructive comments that have helped us to improve the manuscript. This work was cofinanced by the Netherlands Consortium for Systems Biology (NCSB), which is part of the Netherlands Genomics Initiative/Netherlands Organisation for Scientific Research.
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Authors' contributions
MvH designed the model and performed the simulations. RM and MvH conceived of the study and drafted the manuscript. All authors read and approved the final manuscript.
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Additional file 2:Table S1. Table describing the summary of fitting V_{ prot } to experimental growth rate and glucose uptake rate. For every organism, we varied V_{ prot } (volume fraction of macromolecules devoted to metabolic enzymes) between 0 and 1 and performed, for each value of V_{ prot } , 1000 simulations with random sets of crowding coefficients. For the simulations described in this paper, we used the value of V_{ prot } that minimized ((μ_{ max, fit }  μ_{ max, obs } )/μ_{ max, obs } )^{2}+((Gup_{ max, fit }  Gup_{ max, obs } )/Gup_{ max, obs } )^{2}. Here, μ_{ max, fit } , μ_{ max, obs } are the fitted and observed maximal growth rate and Gup_{ max, fit } , Gup_{ max, obs } are the fitter and observed maximal glucose uptake rate. In this table, P_{ ineff } indicates the fraction of the 1000 simulations that exhibits lowyield metabolism, which was defined as having a growth yield < 0.3 gr/gr glucose. Experimental data is from Hoek et al.[29]; Thomas et al.[4]; Varma and Palsson [2]. (PDF 28 KB)
Additional file 4:Figure S3. Histograms of turnover numbers (1/s) (A) and crowding coefficients (gram DW hr/mmol) (B) of E. coli. A. All turnover numbers of E. coli in BRENDA (Chang et al.[28]); B. Crowding coefficients resulting from all turnover numbers of E. coli in BRENDA (Chang et al.[28]); C. Turnover numbers of E. coli used for the simulations; D. Crowding coefficients of E. coli used in the simulations; E. Turnover numbers as used in Vazquez et al.[17]; F. Crowding coefficients as used in Vazquez et al.[17]. (PDF 13 KB)
Additional file 5:Figure S4. Histograms of turnover numbers (1/s) (A,C) and crowding coefficients (gram DW hr/mmol) (B,D) of S. cerevisiae. A. All turnover numbers of S. cerevisiae in BRENDA (Chang et al.[28]); B. Crowding coefficients resulting from all turnover numbers of S. cerevisiae in BRENDA (Chang et al.[28]); C. Turnover numbers of S. cerevisiae used for the simulations; D. Crowding coefficients of S. cerevisiae used in the simulations. (PDF 9 KB)
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van Hoek, M.J., Merks, R.M. Redox balance is key to explaining full vs. partial switching to lowyield metabolism. BMC Syst Biol 6, 22 (2012). https://doi.org/10.1186/17520509622
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Keywords
 Metabolic switching
 Genomescale metabolic model
 Flux Balance Analysis with Molecular Crowding
 Overflow metabolism
 Redox balance
 Escherichia coli
 Lactococcus lactis
 Saccharomyces cerevisiae