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
|Control||22.535 ± 2.169||7.5 ± 0.5||2.35 ± 0.31||64.1|
|Hyper-osmotic||9.103 ± 0.455||4.5 ± 0.5||0.90 ± 1.09||69.1|
|High temperature||18.551 ± 0.943||9.0 ± 1.1||0.76 ± 0.67||22.4|