A method for accounting for maintenance costs in flux balance analysis improves the prediction of plant cell metabolic phenotypes under stress conditions



Flux balance models of metabolism generally utilize synthesis of biomass as the main determinant of intracellular fluxes. However, the biomass constraint alone is not sufficient to predict realistic fluxes in central heterotrophic metabolism of plant cells because of the major demand on the energy budget due to transport costs and cell maintenance. This major limitation can be addressed by incorporating transport steps into the metabolic model and by implementing a procedure that uses Pareto optimality analysis to explore the trade-off between ATP and NADPH production for maintenance. This leads to a method for predicting cell maintenance costs on the basis of the measured flux ratio between the oxidative steps of the oxidative pentose phosphate pathway and glycolysis. We show that accounting for transport and maintenance costs substantially improves the accuracy of fluxes predicted from a flux balance model of heterotrophic Arabidopsis cells in culture, irrespective of the objective function used in the analysis. Moreover, when the new method was applied to cells under control, elevated temperature and hyper-osmotic conditions, only elevated temperature led to a substantial increase in cell maintenance costs. It is concluded that the hyper-osmotic conditions tested did not impose a metabolic stress, in as much as the metabolic network is not forced to devote more resources to cell maintenance.


Flux balance analysis (FBA) is used to explore flux distributions within genome-scale reconstructions of plant metabolic networks (Sweetlove and Ratcliffe, 2011; Collakova et al., 2012; de Oliveira Dal'Molin and Nielsen, 2013). Applications of this computational method include predicting metabolic phenotypes, guiding metabolic engineering strategies (Schwender and Hay, 2012) and addressing questions in evolution (Bekaert et al., 2011). However, the utility of FBA depends on its ability to predict realistic flux distributions. FBA is a constraint-based modelling approach, and one of the main constraints imposed on such models is the requirement to synthesize biomass of appropriate composition for the tissue under consideration. In addition to tracking the flows of carbon and other elements (nitrogen, phosphorus, oxygen and sulfur) through the metabolic network for the synthesis of biomass, FBA also balances the utilization of energy and reducing power. Herein lies a major problem: the synthesis of biomass only consumes a small proportion of the total energy budget (Masakapalli et al., 2010). Thus, fluxes through the energy-transforming pathways, i.e. glycolysis, the oxidative pentose phosphate pathway (OPPP) and the tricarboxylic acid (TCA) cycle, are greatly under-estimated by FBA when solely constrained by biomass synthesis (Poolman et al., 2009).

Accurate prediction of these fluxes requires several other energy costs to be taken into account. First, synthesis of biomass depends on both the uptake of substrates into the cell and the transport of metabolites between subcellular compartments. Many of these transport steps are energy-requiring, being driven by electrochemical gradients that are maintained by expenditure of ATP. Second, there is the cost of maintaining the instantaneous steady state, irrespective of whether the system is growing, in the face of processes such as turnover of polymers and the inherent leakiness of membranes. Transport costs are relatively easy to account for in FBA by adding appropriate transporters with defined ATP stoichiometry to the metabolic model, although this is complicated by the subcellular compartmentation of the metabolic network. Maintenance costs, by contrast, are notoriously difficult to quantify (Penning de Vries, 1975; Amthor, 2000; Piques et al., 2009), and accounting for maintenance has recently been identified as a major challenge for FBA of genome-scale models of plant metabolism (Sweetlove et al., 2013). For microbial models, maintenance constraints corresponding to growth-associated and non-growth-associated maintenance may be conveniently obtained from chemostat measurements (Thiele and Palsson, 2010). This approach is not generally applicable to plant cells, which are rarely grown in chemostats, let alone to plants themselves, and so an alternative strategy is to introduce a generic ATPase into the model to account for the cost of cell maintenance (Thiele and Palsson, 2010). This method for assessing maintenance ATP costs has been adopted in FBA investigations of plant metabolism (Poolman et al., 2009; Hay and Schwender, 2011), and is based on varying the flux of a generic ATPase in the model until the carbon consumed by the model matches that measured experimentally (Poolman et al., 2009). The importance of including maintenance costs is demonstrated by the large effect of altered ATP demand on the flux distribution in central metabolism (Poolman et al., 2009). However, this method assumes that maintenance costs can be described entirely in terms of ATP expenditure, even though maintenance processes such as antioxidant metabolism and the re-synthesis of lipids require substantial amounts of reductant.

To address this, we have developed a method to account for both the ATP and reductant costs of cell maintenance within the framework of an extended and updated version of our previous Arabidopsis genome-scale model (Poolman et al., 2009). This new model includes a detailed description of subcellular compartmentation and the associated transport costs. We show that attributing the cell maintenance costs to consumption of both ATP and reductant has a substantial effect on the predicted flux distributions and improves the match to experimentally determined fluxes. Moreover, a comparison of Arabidopsis cells grown under control, elevated temperature and hyper-osmotic conditions highlights substantial differences in the ATP and reductant costs of cell maintenance. This leads to the suggestion that stress may be considered as any condition that leads to an increase in cell maintenance costs.


Extending and updating the Arabidopsis genome-scale metabolic model

The previous reconstruction of the Arabidopsis metabolic network (Poolman et al., 2009) was extended to include a description of the subcellular compartmentation of the network using known, experimentally validated locations for the enzymes of central metabolism. Transporters with specified ATP-use stoichiometry were added to the plasma membrane to account for the cost of nutrient uptake from the extracellular medium, and mitochondrial, peroxisomal and tonoplast membrane transporters were added to account for intracellular metabolite transport costs (Data S1). The model was also revised to account for new genes added to the Arabidopsis genome annotation. The updated model contains 2769 reactions, comprising 2039 in the cytosol, 292 in the plastid, 97 in the mitochondrion, 36 in the peroxisome and one in the vacuole, plus 192 intracellular transport reactions and 112 other reactions that include uptake of nutrients, biomass production and maintenance reactions. A total of 1860 reactions are associated with at least one gene product, and a total of 2857 genes are linked to reactions in the model. Model files are provided in SBML format in Data S2 and ScrumPy format in Data S3. The compartmentation of the model is more extensive than that of the AraGEM model (de Oliveira Dal'Molin et al., 2010) but less than that in a recently published Arabidopsis genome-scale model that also included compartmentation predictions of the reactions of secondary metabolism (Mintz-Oron et al., 2012).

Adding energy-linked transporters for nutrient uptake and for intracellular transport of metabolites between membrane-bound subcellular compartments facilitated a more comprehensive analysis of the energy costs of biomass synthesis in the model (Data S1). The model also accounted for transport of proteins into organelles by assuming that 50% of all proteins are located in organelles (Mintz-Oron et al., 2012) and that the energy cost for translocation is 1000 ATP molecules per protein translocated (Alder and Theg, 2003). Allowing for the cost of ion, metabolite and protein transport increased the ATP expenditure for the synthesis of biomass by 64% (from 61.8 mmol ATP day−1 L−1 to 101.2 mmol ATP day−1 L−1).

In the iterative process of model curation and analysis, metabolic fluxes expressed in units of mmol day−1 L−1 were simulated for an Arabidopsis heterotrophic cell culture using the same biomass production and nutrient uptake constraints as before (Williams et al., 2010) and an objective function of minimization of total flux, unless otherwise stated.

The source of NADPH for biosynthesis

Analysing the NADPH requirements for maintenance required some improvements to the way in which the model accounts for NADPH production. In the flux predictions from the previous version of the Arabidopsis model, the OPPP carried no flux and NADPH was mainly produced by NADP-glyceraldehyde dehydrogenase (GAPDH) (Poolman et al., 2009; Williams et al., 2010). Similarly, in a flux balance model of Brassica napus, a variety of plastidial dehydrogenases were used to generate NADPH in preference to using the OPPP (Hay and Schwender, 2011). Nevertheless, the OPPP is known to be an important source of NADPH in plant cells (Kruger and von Schaewen, 2003; Schwender et al., 2003). Moreover, it is unclear whether the dehydrogenases used as an alternative source of NADPH in these models function in the direction of NADPH production in vivo.

The NADP-GAPDH and NADP-malate dehydrogense (MDH) reactions generate NADPH by participating in transhydrogenase cycles that allow the transfer of reducing power from either plastidial or cytosolic NADH to plastidial NADP with no net changes in the NAD(H) and NADP(H) pool sizes (Figure 1 and Table S1). However, thermodynamic considerations suggest that such transhydrogenase cycles are unlikely to act as an alternative source of plastidial NADPH. In plant cells, the NAD(H) pool is more oxidized than the NADP(H) pool in both the light and the dark (Heineke et al., 1991; Igamberdiev and Gardeström, 2003), so it is thermodynamically implausible for cytosolic NADH to drive the provision of plastidic NADPH without an energy input. A proton-translocating transhydrogenase may achieve this (Pedersen et al., 2008), but although such a protein exists in most prokaryotes and in animal mitochondria, the Arabidopsis genome does not appear to contain the relevant gene (Arkblad et al., 2001). Thus, for subsequent modelling, the plastidial NADP-dependent dehydrogenases were set to operate in the direction of NADPH consumption. In this scenario, and in marked contrast to earlier FBA predictions for the heterotrophic Arabidopsis network (Poolman et al., 2009; Williams et al., 2010), the OPPP carries flux and is an important source of NADPH (Table S1).

Figure 1.

Trans-hydrogenase cycles involving plastidial NADP-GAPDH and NADP-MDH.

In the FBA model, NADP-GAPDH and NADP-MDH were set to be irreversible in the direction of NADPH consumption as shown in the diagram. This forces the trans-hydrogenase cycles (represented by curved rectangles) to proceed in the direction of NADPH consumption, generating NADH in the plastid or the cytosol at the expense of plastidic NADPH. Setting either or both NADP-GAPDH and NADP-MDH to be reversible allowed the trans-hydrogenase cycles to proceed in the direction of NADPH production, which led to altered flux predictions. Abbreviations: GAP, glyceraldehyde 3–phosphate; 1,3–DPGA, glycerate 1,3–bisphosphate; 3–PGA, glycerate 3–phosphate; OAA, oxaloacetate. Enzymes (circled numbers): (1) cytosolic NAD-dependent GAPDH, (2) cytosolic phosphoglycerate kinase, (3) plastidial NADP-GAPDH, (4) plastidial NAD-dependent GAPDH, (5) plastidial phosphoglycerate kinase, (6) plastidial NAD-dependent MDH, (7) plastidial NADP-MDH, and (8) cytosolic NAD-dependent MDH.

Estimating maintenance costs

When transport costs were included, FBA predicted that synthesis of biomass by the Arabidopsis cell culture under control conditions required a glucose consumption rate of 84.5% of the experimentally determined value. It can be assumed that the remaining glucose was consumed to meet maintenance energy demands through provision of ATP and NADPH. Our approach to account for these maintenance costs was to add generic ATPase and NADPH oxidase steps to the model. These steps were iteratively assigned different flux values, generating a set of model solutions. The glucose consumption rate in the model increased with increasing flux through the generic steps, and, for a subset of ATP and NADPH maintenance values, the total glucose consumption in the model matched the measured value. However, because ATP and NADPH are generated by different metabolic processes that compete for the same hexose phosphate precursor, a maintenance cost trade-off operates, i.e. it is only possible to produce more ATP at the expense of NADPH production and vice versa.

This trade-off was explored by calculating the Pareto front (Vo et al., 2004; Bordbar et al., 2011) of ATP and NADPH maintenance costs that result in the experimentally measured glucose consumption constraint being satisfied (Figure 2). The Pareto optima that defined the Pareto front corresponded to different flux distributions in the central metabolic network, and it was observed that the flux ratio between the oxidative steps of the OPPP (generating NADPH) and glycolysis (leading to ATP generation) decreased from 1.37 to 0.38 as the maintenance demand switched from NADPH to ATP (Figure 2). This suggests that this flux ratio could be a good indicator of the actual ratio of NADPH to ATP maintenance. The measured value for this flux ratio for the Arabidopsis cell culture under control conditions (Data S4) was used to determine the point on the Pareto front that corresponds to the in vivo balance of ATP and NADPH maintenance costs (Figure 2). From this, it was possible to deduce that maintenance in the Arabidopsis cell culture under control conditions required 54.5 mmol ATP day−1 L−1 culture, corresponding to 33% of the total ATP produced, and 19.2 mmol NADPH day−1 L−1 culture, corresponding to 50% of the total NADPH produced.

Figure 2.

Pareto front for the trade-off between ATP and NADPH maintenance in a heterotrophic Arabidopsis cell culture under control conditions.

Dotted lines represent estimates of the in vivo ATP and NADPH maintenance costs (mmol day−1 L−1 culture) that lead to the flux ratio between the oxidative steps of the OPPP and glycolysis determined by 13C–MFA (point 2).

The effect of transport and maintenance costs on flux predictions in central metabolism

The effect of including transport and maintenance costs in the FBA of the Arabidopsis cell culture grown under control conditions was investigated by comparing the predicted flux solutions with the fluxes in central metabolism obtained by 13C metabolic flux analysis (MFA) (Figure 3). Using just biomass as a constraint, i.e. not considering transport and maintenance costs, produced a relatively poor prediction of the fluxes through central metabolism, with the glycolytic, OPPP and TCA cycle fluxes being substantially under-estimated. Including transport costs improved the predicted fluxes through glycolysis and the TCA cycle, and adding NADPH and ATP maintenance costs also altered the predicted flux distribution. At the highest NADPH maintenance demand (Figure 2, point 1), there was a large flux through the OPPP and very little TCA cycle flux (Figure 3), whereas at the highest ATP maintenance demand (Figure 2, point 3), the fluxes through glycolysis and the TCA cycle were higher compared to the other two scenarios, and there was very little OPPP flux (Figure 3). However, when the balance between the NADPH and ATP maintenance costs was set by constraining the flux ratio between the oxidative steps of the OPPP and glycolysis to the measured value (Figure 2, point 2), the flux distribution was qualitatively similar to that obtained by 13C-MFA (Figure 3).

Figure 3.

Comparison of MFA and FBA fluxes for Arabidopsis thaliana cells under control conditions.

Fluxes of central carbon metabolism from 13C–MFA (top left); flux prediction with biomass constraint but no transport and maintenance cost constraints (top middle); flux prediction with biomass and transport cost constraints but no maintenance cost constraint (top right); flux predictions with biomass, transport and maintenance cost constraints (bottom row). In each case, maintenance costs in terms of ATP and/or NADPH are indicated by the numbers in black circles, which correspond to the numbers on the Pareto front indicated in Figure 2. The arrow thickness is proportional to flux through the reaction. The mean flux through the TCA cycle is shown on each flux map. For all FBA, minimization of fluxes was used as the objective function.

The quantitative match between the fluxes predicted with the various transport and maintenance estimates and the experimentally determined fluxes for the Arabidopsis cells under control conditions was assessed by calculating the weighted Euclidean distance between them. Details of the mapping between the MFA and flux balance models (which are not identical) are given in Data S5. Because flux solutions in FBA are also influenced by the objective function, flux solutions were generated using five objective functions: minimization of overall flux, maximization of biomass, minimization of glucose consumption, maximization of ATP production and maximization of NADPH production. In total, 25 combinations of objective function and maintenance energy/other constraints were tested. Flux solutions for each of these combinations (Data S6) were compared to the fluxes measured by 13C–MFA, and the Euclidean distances are shown in Figure 4.

Figure 4.

Weighted Euclidean distances between flux predictions from 25 objective function/constraint combinations and the 13C–MFA flux map for the heterotrophic Arabidopsis cell culture under control conditions.

Each column corresponds to a combination of objective function and one or more constraints, where the constraints applied are indicated by shading. The Euclidean distance is a measure of the closeness of fit between the FBA flux prediction and the 13C–MFA flux map, with a small value being a close fit. Objective functions: minimization of total flux (Min Flux), minimization of glucose consumption (Min Glucose), maximization of biomass production including transport costs (Max Biomass), maximization of ATP yield (Max ATP), and maximization of NADPH yield (Max NADPH). Constraints: biomass production excluding transport costs (Biomass), transport costs (Transport), glucose consumption rate (Glucose), estimated ATP maintenance cost from the Pareto optimization (MATP), and estimated NADPH maintenance cost from the Pareto optimization (MNADPH).

When transport and maintenance costs (both ATP and NADPH) were included as constraints, the FBA gave flux predictions that were very close to the 13C–MFA flux map, irrespective of the objective function. Generally, ATP maintenance had a quantitatively larger effect than NADPH maintenance, although including both reduced the Euclidean distance further and thus improved the flux prediction. Inclusion of transport costs improved the flux predictions, except when maximization of ATP was the objective function. In this case, the predicted flux distribution with the biomass and glucose consumption constraints was unaffected by the extra ATP expenditure on transport because the transport constraint had no bearing on the ATP-generating capacity of the network.

Although the use of different objective functions had little effect on prediction of the 15 compared fluxes, there may be advantages to choosing one objective function over the others. For example, minimization of flux eliminates unconstrained cycles that may otherwise carry infinite fluxes in solutions obtained using the other objective functions. This was confirmed by a flux variability analysis: no reactions with an infinite flux range (unconstrained reactions) were found using minimization of flux as an objective function, whereas 36 unconstrained metabolic reactions were found for the other four objective functions. Unconstrained reactions are often involved in parallel metabolic pathways in different compartments, e.g. glycolysis in the cytosol and the plastid. Reactions involved in parallel pathways were resolved into finite flux ranges by minimization of flux objective function, but not by any of the other objective functions investigated. For example, it was possible to predict, using minimization of flux, the extent to which glycolysis occurred in both the cytosol and the plastid, and that most of the flux through the TCA cycle was mitochondrial and not cytosolic (Figure 5).

Figure 5.

Flux ranges for glycolysis, aconitase and fumarase predicted using the objective of minimization of total flux for the heterotrophic Arabidopsis cell culture under control conditions.

The thickness of the arrow represents the mean flux value for each reaction. Flux ranges are shown beside the arrows, with the top and bottom values being the minimum and maximum flux values, respectively. Glycolysis was predicted to operate in both the cytosol and the plastid, whereas fluxes through aconitase and fumarase primarily occur in the mitochondria. Abbreviations: DHAP, dihydroxyacetone phosphate; GAP, glyceraldehyde 3–phosphate; 3–PGA, glycerate 3–phosphate; PEP, phosphoenolpyruvate.

Comparison of MFA and FBA under stress conditions

Figure 6 shows that the proposed method for determining the maintenance costs and constraining the FBA predictions is effective under both control and stress conditions. Comparison of the 13C–MFA flux maps and the FBA predictions shows that using the ATP and NAPDH maintenance costs based on the experimentally determined flux ratio between the oxidative steps of the OPPP and glycolysis (Table 1) to constrain the FBA resulted in excellent agreement between the predicted and experimental flux maps under both elevated temperature (29°C) and hyper-osmotic (171 mm mannitol) conditions (Figure 6).

Table 1. OPPP to glycolysis flux ratio, ATP and NADPH maintenance costs, and ATP and NADPH requirements for biosynthesis in Arabidopsis cell suspension cultures under control and stress conditions
  1. Values predicted from FBA and deduced from MFA flux maps are indicated by [FBA] and [MFA], respectively. Maintenance costs computed from FBA expressed as a percentage of total ATP or NADPH produced are shown in parentheses. Maintenance costs were estimated from MFA flux maps using the procedure described by Masakapalli et al. (2010), and details of the calculations are shown in Data S9.

OPPP:glycolysis flux ratio [MFA]0.5910.6370.563
ATP maintenance cost [FBA] (% of total ATP produced)54.532 (33.3)7.046 (13.0)252.608 (79.0)
ATP maintenance cost [MFA]47.0524.271262.741
NADPH maintenance cost [FBA] (% of total NADPH produced)19.167 (49.6)5.881 (41.8)48.786 (89.7)
NADPH maintenance cost [MFA]18.5305.48248.915
NADPH maintenance cost/ATP maintenance cost [FBA]0.3510.8350.193
NADPH maintenance cost/ATP maintenance cost [MFA]0.3941.2840.186
Figure 6.

Comparison of measured and predicted fluxes for Arabidopsis thaliana cells under control conditions, hyper-osmotic stress and elevated temperature (29°C).

Fluxes were either determined experimentally by 13C–MFA or predicted by FBA using the same biomass constraints, a detailed analysis of transport costs, the measured value of the flux ratio between the oxidative steps of the OPPP and glycolysis, and minimization of fluxes as the objective function.

Analysis of the predicted maintenance costs (Table 1) shows that the proportion of the available ATP used for maintenance was low under hyper-osmotic stress relative to the control, and high at elevated temperature, and that elevated temperature also increased the proportion of the available NADPH that was used for maintenance. These predictions, which are in good agreement with the MFA data, show that the relative contribution of NADPH and ATP to the maintenance cost is strongly dependent on the conditions imposed on the cell culture, with predicted values for the NADPH:ATP maintenance ratio of 0.35 under control conditions, 0.84 under hyper-osmotic conditions, and 0.19 at elevated temperature. In contrast, the relative contribution of NADPH and ATP to the cost of biosynthesis was very similar under the three conditions, reflecting the similarity in the biomass composition of the cells.


Accounting for energy expenditure in a flux balance model

The utility of flux balance models depends on the extent to which realistic flux distributions can be predicted from them. While the biomass constraint has proved effective in predicting fluxes in microbes (Feist and Palsson, 2010), it is less clear whether this constraint provides sufficient bounds for flux prediction in more complex organisms such as plants. Even though it has been previously shown that there is significant agreement in the flux predictions for heterotrophic plant metabolism and those estimated by 13C–MFA (Williams et al., 2010; Hay and Schwender, 2011), certain fluxes, such as those in the central pathways of glycolysis, the TCA cycle, and particularly the OPPP, were not well matched.

The main issue is that these central pathways, in addition to providing carbon skeletons for the synthesis of biomass components, are also the main routes for energy transformation. Although synthesis of biomass consumes energy, there are other substantial energy drains in the cell, including the cost of transporting ions, metabolites and macromolecules, and the cost of cell maintenance. Both of these costs are potentially higher in plants than in microbial systems, and could therefore have a substantial effect on predicted flux distributions in central metabolism. In fact, under control conditions, the analysis presented here shows that synthesis of biomass (including metabolite transport) accounts for only 67% of the total ATP budget in a heterotrophic Arabidopsis cell culture under control conditions, with the remainder being used for maintenance (Table 1).

Transport costs can be included in FBA by specifying co-transport of a defined number of protons per transported molecule in most cases, and the ATP requirement is then defined by the proton:ATP stoichiometry of the ATPases that generate transmembrane proton gradients. Some transport costs have been included in previous flux balance models of plant metabolism (e.g. Hay and Schwender, 2011), but these have only considered transport across the plasma membrane. As shown here, adding costs associated with intracellular transport and protein translocation improved the prediction of the glycolytic and TCA cycle fluxes significantly, but had no effect on the OPPP flux prediction (Figure 3). Thus adding transport costs to the usual biomass constraint is not sufficient to optimize the predictive accuracy of FBA in a plant metabolic network.

Maintenance costs are more difficult to account for in FBA because maintenance covers many ATP- and NADPH-dependent processes that are difficult to quantify. While some aspects of maintenance, such as macromolecule turnover, may be measurable experimentally, the techniques rely on numerous assumptions and there is a substantial range of estimates for such costs (Scheurwater et al., 2000; Piques et al., 2009). Others, such as membrane leak rate, recycling of unwanted metabolites (Seaver et al., 2012) and antioxidant NADPH expenditure have yet to be experimentally quantified and are not rigorously accounted for, even in microbial FBA studies. This major limitation in FBA may be solved by the method introduced in this paper, which leads to a substantial improvement in the accuracy of the flux predictions for the pathways of central metabolism in heterotrophic Arabidopsis cells (Figure 6).

Using a Pareto optimization, it was found that the balance between ATP and NADPH maintenance had a substantial effect on the distribution of fluxes in central metabolism (Figure 3), and this highlighted the importance of setting the correct division of maintenance energy consumption. It was therefore necessary to introduce a constraint on the relative activities of the generic ATPase and NADPH oxidase that were used to account for maintenance costs. The branch point where hexose phosphates enter either glycolysis or the OPPP is likely to be a key determinant of the commitment to ATP and NADPH production. Using the value for the ratio of fluxes at this branch point taken from 13C–MFA measurements as a constraint substantially improved the flux prediction and produced the closest overall match to the 13C–MFA flux distribution for central metabolism irrespective of the objective function used to generate the flux predictions.

Thus, addition of a single extra constraint, the flux ratio of glycolysis to OPPP, allows ATP and NADPH maintenance costs to be accounted for and produces a more realistic prediction of the flux distribution through the central metabolic network. However, we note that, in systems such as developing oilseeds, where the OPPP is a less significant source of NADPH (Schwender et al., 2003), additional flux ratios may be required as constraints. In this study, the required flux ratio was taken from 13C–MFA measurements on the same cell suspension culture. In practice, 13C–MFA flux maps may not always be available, and the requirement to generate one would negate the benefit of flux prediction by FBA. However, there are a number of simpler labelling experiments that may give access to this flux ratio. For example, the labelled CO2 generated from positionally labelled 14C–glucose gives an estimate of the OPPP flux relative to other pathways of carbohydrate oxidation (Malone et al., 2006). A more direct measure of the OPPP flux can be obtained by feeding [1–14C]gluconate, which enters the OPPP as 6–phosphogluconate and is released as 14CO2 (Garlick et al., 2002; Zhao et al., 2008). Alternatively, mass isotopomers of downstream metabolic products of [1,2–13Cglucose] allow estimation of the OPPP flux (Dusick et al., 2007). While these methods may require some optimization, in principle these relatively straightforward labelling experiments allow the crucial flux ratio to be estimated.

The effect of different constraints and objective functions

The effect of different constraints and objective functions on the accuracy of the flux distribution predicted by FBA was investigated. The most important conclusion is that comprehensively accounting for energy costs in the system is more important in terms of predicting a realistic flux distribution than the choice of objective function.

A similar systematic evaluation of objective functions and constraints in Escherichia coli (Schuetz et al., 2007) gave a rather different conclusion. In E. coli, inclusion of ATP and NADPH maintenance costs actually reduced the predictive fidelity of the FBA, producing a poorer match to MFA-determined fluxes. This curious result may have been caused either by handling the NADPH and ATP costs separately, or by the relatively low maintenance costs in E. coli. In an E. coli flux balance model, the maintenance ATP demand was 7.6 mmol ATP g DW−1 h−1, which is only 14% of the 55.2 mmol ATP g DW−1 h−1 generated in the system (Varma and Palsson, 1994). This contrasts with the 33% of total ATP produced in heterotrophic Arabidopsis cells that was used for maintenance under control conditions (Table 1). Nevertheless, it is difficult to see why an FBA solution that does not include maintenance costs would generate a closer match to the MFA-determined fluxes than one that includes maintenance, unless the included maintenance values were substantial over-estimates. This is possible, as estimates of E. coli maintenance costs (in terms of ATP) vary from 1.9 mmol ATP g DW−1 h−1 under glucose limitation to 16.8 and 30.8 mmol ATP g DW−1 h−1 under nitrogen and sulfur limitation, respectively (Farmer and Jones, 1976). This highlights the problem of applying experimentally determined maintenance values obtained under one set of conditions to FBA models of metabolism under different conditions.

Although the choice of objective function had little effect on the accuracy with which the 15 compared fluxes were predicted by FBA, the choice of objective function is nevertheless still an important consideration. For example, minimization of flux eliminates futile cycles from the predicted flux map, and this is not necessarily in agreement with experimental measurements. Some 13C–MFA measurements have suggested that sucrose cycling consumes a surprisingly large proportion of the available ATP (Rontein et al., 2002; Alonso et al., 2007), although theoretical analysis has identified an assumption in these steady-state analyses that has yet to be verified and is quite likely to be unjustifiable (Kruger et al., 2007). In any case, biological systems have evolved to operate over a range of conditions that may require competing objectives, and this competition has been explored in E. coli metabolism using the Pareto optimality concept (Schuetz et al., 2012). It was found that the Pareto surface formed from three objective functions (minimization of total flux, maximization of ATP yield and maximization of biomass yield) best described the fluxes measured from 13C–MFA. Thus, Pareto optimality is a versatile tool for FBA, allowing exploration of competing objective functions (Schuetz et al., 2012) or, as shown here, competing constraints.

Effect of hyper-osmotic stress and elevated temperature on maintenance costs

As well as greatly improving the predictive accuracy of FBA, constraining the analysis using the measured value of the flux ratio between the oxidative steps of the OPPP and glycolysis led to estimates of the ATP and NADPH costs of maintenance under control, elevated temperature and hyper-osmotic conditions (Table 1). This information may be interpreted in terms of the growth characteristics of the cell line under the three conditions (Table 2). Increasing the temperature increased the proportion of both ATP and NADPH used for maintenance by the cell culture, while at the same time causing a marked reduction in the carbon conversion efficiency. It is likely that the increased membrane leakage (Ruter, 1993) and oxidative stress (Schwarzländer et al., 2012) that occur at elevated temperature make major contributions to the increased ATP and NADPH maintenance costs, respectively. This conclusion is consistent with observations based on respiratory rates, suggesting that maintenance respiration rather than growth respiration increase with increased temperature (Amthor, 2000). In contrast, hyper-osmotic conditions caused a small decrease in the proportion of NADPH used for maintenance, and a substantial reduction in the proportion of ATP used for the same purpose, while at the same time reducing the growth rate and increasing the carbon conversion efficiency. The lower proportion of ATP used for maintenance may reflect the slower turnover of cellular components in the slower growing cells, but, more generally, the contrast with the markedly increased maintenance costs in the stressed cells at elevated temperature suggests that that the cells were not stressed under these hyper-osmotic conditions. This conclusion may be compared with a recent analysis in which several environmental stresses, including elevated temperature, were observed to increase the rate of mitochondrial membrane potential pulsing, in parallel with a marked change in mitochondrial redox status in the epidermal cells of Arabidopsis roots (Schwarzländer et al., 2012). Strikingly, hyper-osmotic conditions did not alter these parameters, again suggesting that these conditions do not disrupt normal cellular activity. Thus it appears that an increase in cell maintenance costs may be a metabolic signature of stress conditions, and that the slow growth of the Arabidopsis cells under hyper-osmotic conditions should be interpreted as the result of restricted glucose uptake rather than the result of perturbed metabolic processes.

Table 2. Effect of stress treatments on cell culture growth parameters
ConditionsGlucose consumption (mmol day−1 L−1)Biomass at day 5 (g DW L−1)Growth (g DW L−1 day−1)CCE (%)
  1. Glucose consumption rate, final biomass (after 5 days of growth), growth rate (biomass accumulation) and carbon conversion efficiency (CCE) of Arabidopsis cell suspension cultures grown under control conditions (21°C), under hyper-osmotic conditions (171 mm mannitol in culture medium) and under high temperature (29°C). Values are means ± SD of four replicates.

Control22.535 ± 2.1697.5 ± 0.52.35 ± 0.3164.1
Hyper-osmotic9.103 ± 0.4554.5 ± 0.50.90 ± 1.0969.1
High temperature18.551 ± 0.9439.0 ± 1.10.76 ± 0.6722.4

Experimental procedures

Model construction

A genome-scale metabolic model of Arabidopsis was constructed from AraCyc 9.0 (http://www.arabidopsis.org/biocyc/) using the ScrumPy metabolic modelling package (Poolman, 2006; Poolman et al., 2009). Reversibility and reaction direction were taken from the AraCyc database wherever possible. The model was manually curated for reaction stoichiometry, reversibility and direction, the presence or absence of reactions in Arabidopsis, and coenzyme specificity. Other than metabolites that consist solely of hydrogen and/or oxygen, no other unconserved metabolites were found, which means that the model is stoichiometrically consistent with respect to the other elements (Gevorgyan et al., 2008). As before (Poolman et al., 2009), two types of protons were defined in the model, energetic protons and stoichiometric protons, represented as ‘Pumped-PROTON’ and ‘PROTON’ respectively. Energetic protons are pumped across membranes to drive metabolite transport and/or ATP synthesis, and these protons were mass-balanced independently across the plasma membrane, the mitochondrial membrane and the tonoplast. Stoichiometric protons were set as external, and represent protons involved in the chemical changes in metabolic reactions. Based on information in the literature and the SUBA database (Heazlewood et al., 2007), reactions were manually assigned to five subcellular compartments, cytosol (‘_c’), plastid (‘_p’), mitochondrion (‘_m’), peroxisome (‘_x’) and vacuole (‘_v’), with the cytosol being the ‘default’ compartment (Data S2 and S3). Details for known intracellular transporters were obtained from Linka and Weber (2010) and elsewhere, and further transporters were added to enable biomass production from the given inputs (Data S2 and S3). Transport steps in the model were defined using the subscripts ‘_pc’, ‘_mc’, ‘_xc’, ‘_vc’, ‘_biomass’ and ‘_tx’ for plastidial, mitochondrial, peroxisomal and tonoplast transporters, biomass drains and extracellular transport, respectively.

Model analysis

Flux balance analysis was performed using the Gnu linear programming kit (http://www.gnu.org/software/glpk) and the ScrumPy metabolic modelling package (Poolman et al., 2009). The flux balance problem was formulated as a linear programming problem:

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Subject to:

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where v is the flux vector, Z is the objective function, c is the transpose of a vector of objective coefficients, S is the stoichiometry matrix, LB is the vector of flux lower bounds and UB is the vector of flux upper bounds. Five objective functions were used for the analysis: minimization of overall flux, maximization of biomass, minimization of glucose consumption, maximization of ATP production, and maximization of NADPH production (Data S7). Glucose consumption and biomass production constraints were identical to those used for 13C–MFA of a heterotrophic Arabidopsis cell culture under control, elevated temperature and hyper-osmotic conditions (Williams et al., 2010). The biomass and other constraints are listed in Data S8. Nitrogen input was set to be 50% from nitrate and 50% from ammonium (Masakapalli et al., 2013).

Flux variability analysis was implemented using the method described by Mahadevan and Schilling (2003). In brief, the primary objective was optimized, and then the optimal value was used as an additional constraint. For each step, the reaction flux was maximized and minimized to obtain a flux range that was compatible with the optimal value for the primary objective. This range is represented as (vmin, vmax), where vmin and vmax are the minimum and maximum feasible flux values, respectively. For comparison between MFA fluxes and the genome-scale model prediction, reactions in the genome-scale model were projected onto the MFA fluxes. Each mapped MFA reaction flux corresponds to a grouped reaction that consists of one or more reaction(s) in the genome-scale metabolic model (Data S5). The flux range of a grouped reaction was calculated by minimizing and maximizing the weighted sum of the reactions as described by Hay and Schwender (2011).

Pareto optimality analysis was performed by linear programming, optimizing for a weighted sum of all objectives of interest (Vo et al., 2004; Bordbar et al., 2011). The Pareto front, which is the set of Pareto optimal solutions, was identified by constraining the glucose consumption rate and the biomass production rate to experimental values and maximizing the fluxes through the generic ATPase and NADPH oxidase reactions. ATP maintenance was assumed to occur entirely in the cytosol, involving mainly energization of the plasma membrane and tonoplast, and re-synthesis of protein and DNA/RNA following turnover. A substantial proportion of the NADPH cost for maintenance is likely to be accounted for by the antioxidant ascorbate–glutathione cycle, which is present in the cytosol, mitochondria and plastid (Chew et al., 2003). As there is little information on the relative fluxes in this cycle in the three compartments, an NADPH oxidase reaction was defined for each compartment with the constraint that the three reactions carried the same flux (Data S8). This is equivalent to assuming that the NADPH maintenance cost is equally distributed between the three compartments. The Pareto optimal front was sampled by running the optimization 10 000 times, each with a different set of randomized weighting factors. Adjacent sampled points were connected to form the Pareto front.

To assess the match between the FBA flux solution and the 13C–MFA flux map, reactions in the FBA model were mapped to reactions in the 13C–MFA model. In 13C–MFA, multiple enzyme-catalysed reaction steps between branch points are reduced to a single step, so multiple reactions in the genome-scale FBA model must be mapped onto a single reaction in the 13C–MFA model as described in Data S5. For each mapped reaction corresponding to the 13C–MFA model, the flux range from the genome-scale metabolic model was calculated by maximizing and minimizing the genome-scale metabolic model reactions that are mapped to the 13C–MFA model reaction, with the objective function being the stoichiometry of the mapping described in Data S5. The mean values of the 15 fluxes provide feasible flux solutions for all 25 combinations of objective function and constraints, and these values were used for comparison with the 13C–MFA flux map. The weighted Euclidean distance between the two solutions was calculated as:

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where the subscripts MFA and FBA denote flux values obtained from 13C–MFA and FBA respectively, math formula is the mean flux value obtained using flux variability analysis, σMFA is the MFA experimental standard error and nmap (15 in this study) is the number of reactions mapped between MFA and FBA models.

Metabolic flux analysis

Metabolic fluxes in heterotrophic Arabidopsis cell cultures grown under control, elevated temperature (29°C) and hyper-osmotic conditions (171 mm mannitol) were estimated using 13C–FLUX version 20050329 (Wiechert et al., 2001). Stable isotope incorporation data and measurements of glucose consumption and biomass production were identical to those described previously (Williams et al., 2010), but with minor alterations with regard to the calculation of fluxes to biosynthetic precursors. The same metabolic model (control treatment, Williams et al., 2010) was used for all three conditions. The fitting process was initiated 300 times, low-residuum feasible solutions from multiple fits obtained by Monte Carlo variation of the labelling data were averaged, and this averaged flux solution was used as the starting point for a final round of fitting. Confidence intervals for the final flux values were determined using the EstimateStat component of 13C–FLUX.


C.Y.M.C. was supported by a studentship from the EPSRC University of Oxford Systems Biology Doctoral Training Centre, funded by the Clarendon Fund, and a Keble College Sloane–Robinson award. T.C.R.W. was supported by a Petrobras (Petroleo Brasileiro S.A.) post-doctoral grant. We thank W. Wiechert (Forschungszentrum Jülich GmbH, Germany) for permission to use 13C–FLUX, and P. Spelluci (Fachbereich Mathematik, Technische Universität Darmstadt, Germany) for developing the Donlp2 algorithm.