Fig 1.
FBA uses genome-scale stoichiometry and measured extracellular fluxes to constrain fluxes, which are finally determined by assuming growth rate maximization. 13C MFA measures fluxes by using the highly informative 13C labeling experimental data along with central carbon stoichiometry and measured extracellular fluxes. 2S-13C MFA calculates fluxes by using the 13C labeling experimental data and the measured extracellular fluxes to constrain fluxes for a genome-scale model.
Fig 2.
Algorithm flow diagram for 2S-13C MFA showing a recursive procedure to achieve self-consistent results. The full model consists of a genome-scale model (iJR904 in this case) to which information on carbon transitions for the core sets of reactions is added (blue box on the left). The genome-scale model carries the measured extracellular fluxes information as upper and lower bounds (ubj and lbj). Carbon transitions (example line below the blue box) indicate the fate of each carbon atom in the reaction. The first step in the algorithm involves limiting the amount of flux that flows into the core set of metabolites and reactions, so as to enforce the two-scale approximation (i.e. that non-core contributions to labeling are negligible). The second step involves finding the set of fluxes that best fit the experimentally observed data, ignoring the non-core contributions. The final step tests that the error incurred by ignoring non-core reactions is negligible through External Labeling Variability Analysis (ELVA). If the ELVA does not indicate that the non-core contributions are negligible, the core set and the EMU model are expanded and the procedure repeated. When a self-consistent result is found, flux ranges compatible with the experimental data are obtained through 13C Flux Variability Analysis.
Fig 3.
A) 2S-13C MFA models microbial metabolism at two different scales of resolution, hence minimizing the computational effort to explain the experimental data. While stoichiometric balances are taken into account for the full genome-scale model (iJR904 in this case), metabolite labeling originating from the 13C feed in the labeling experiments is only tracked for the core set of reactions responsible for the main fraction of metabolite labeling (green box). The two-scale approximation assumes that non-core metabolites do not directly affect core metabolite labeling. The core set is expandable through the recursive procedure shown in Fig 2. B) Exemplary network of 20 reactions that illustrates the two-scale approximation and the approach. Measured data involves the MDV for metabolites A, C and E and extracellular fluxes for reactions producing metabolites T, U, Y and Z. The initial core set involves reactions and metabolites in the green box. The fit involves finding fluxes which best match the measured labeling and the values of the measured extracellular fluxes, where only the contribution of reactions inside the green box is taken into account to fit the labeling of metabolites A, C and E. However, the metabolite balance is global. In this way the fluxes are not overconstrained by e.g. NADPH balance: any excess NADPH can be balanced by the non-core fluxes that consume NADPH. C) Right lower panel illustrates External Labeling Variability Analysis (ELVA) for the exemplary network. ELVA gauges the effect of non-core reactions by considering only the core network and simulating the impact of non-core metabolite labeling through inflow metabolites (inflowD, inflowE, inflowF). The ELVA optimization problem (Eqs 9–15) finds the maximum impact that the unknown inflow metabolite labeling can have on the measured labeling pattern.
Fig 4.
External Labeling Variability Analysis (ELVA, see methods) shows how the impact of ignored reactions diminishes by expanding the core set of reactions. Each point corresponds to an m value of the Mass Distribution Vector (MDV) for each of the metabolites considered. The inset provides the same information for malate in a more intuitive form (red for experimental data, blue for computational fits), see S1–S3 Figs. Horizontal error bars indicate experimental CE-TOFMS error obtained from the instrument. Vertical error bars indicate computational errors obtained from the ELVA. These computational error bars indicate the maximum effect that non-core reactions (whose contribution to the carbon labeling is being ignored) could possibly have. The initial core set (left) shows a large computational error for malate (mal-L, green dots). By expanding the core set, the computational errors collapse to levels comparable with the experimental error as can be seen in the right panel. Hence, the method is self-consistent by ensuring that the final result meets the approximation used to calculate it.
Fig 5.
Relative effect of constraints.
Confidence intervals for pentose phosphate pathway (left panel) and glycolysis (right panel) fluxes calculated using FBA constrained by measured extracellular fluxes (through Flux Variability Analysis, FVA [64], in red), for FBA with constraints derived from the two-scale approximation through the “Limit flux to core” step in Fig 3 (FVA, in grey), and for 2S-13C MFA derived through Eqs 16–23 are shown (blue). Constraints induced by the two-scale approximation are not strong, hence justifying the use of this approximation. However, constraints induced by the 13C labeling data are dominant. A similar pattern can be observed in S13 Fig.
Fig 6.
Cofactor balances show how NADPH and NADH production and consumption change after pgi is knocked out. Arrows pointing inwards on the left indicate fluxes that produce the indicated metabolite and fluxes pointing outwards on the right indicate fluxes that consume it (in units of mMol/gdw/hr). Reaction names are per iJR904 model. Upper panels show NADPH balances for wild type (left) and pgi KO (right) at 5 and 21hr (equivalent growth points due to a lower growth rate in the pgi KO). Lower panels show NADH balances for wild type (left) and pgi KO (right) at the same time points. Note that, unlike FBA, 2S-13C MFA can provide confidence intervals bounded by the data from 13C labeling experiments. These are shown below the reaction name. For some cases (e.g. GND) the experimental data can very effectively constrain the flux value, even if the reaction is not in the core set over which labeling is being tracked (e.g. C181SN). For some others (e.g. THD2), the data can only constrain the flux value in a very limited fashion. Knocking out pgi radically changes NADPH and NADH supply and consumption patterns.
Fig 7.
Robustness with respect to measurement error in labeling profile.
30 different new labeling data sets were generated by randomly choosing new labeling values within the experimental error (see equation 13 in S2 Text). Fluxes were calculated through 2S-13C MFA for these new data sets and the standard deviation is shown for the PPP. Hence, the method is robust with respect to experiment accuracy in 13C labeling.
Fig 8.
Robustness with respect to genome reconstruction errors.
A reconstruction error was simulated by changing the NADPH dependence to NADH dependence for the glucose-6-phosphate dehydrogenase (with a large flux value of 2.9 mMol/gdw/hr in the initial 2S-13C MFA calculation). We calculated fluxes again through 2S-13C MFA and FBA (constrained by extracellular flux measurements), and the new fluxes are plotted for the TCA cycle. As can be observed, the change is much larger for FBA than 2S-13C MFA, showing that it is less robust to reconstruction errors (note that squares and circles are almost on top of each other for the 2S-13C MFA case). The transparency in the original flux profile for 2S-13C MFA indicates the confidence intervals. For the 2S-13C MFA case, the NADPH production shift is compensated entirely by the THD2 transhydrogenase. Since the flux value for SUCD1i is negative, the absolute value has been plotted. Similar figures for the glutamate dehydrogenase (GLUDy) and isocitrate dehydrogenase (ICDHyr) reactions are available as S16 Fig.
Fig 9.
Flux comparison with COBRA predictions for pyk KO at 5 hours.
The comparison of flux predictions through COBRA methods with fluxes measured through 2S-13C MFA shows how and why predictive methods fail. Left panel: Predicted fluxes for reactions in the pentose phosphate pathway (PPP) through FBA using maximum growth, FBA using maximum ATP production and 13C MOMA are compared with measurements through 2S-13C MFA (shading indicates confidence intervals). Fluxes predicted through maximum growth FBA offer a good qualitative description of fluxes but are quantitatively erroneous. 13C MOMA is a variation of MOMA that leverages 2S-13C MFA flux profiles and predicts fluxes more accurately. Center and right panels: Flux maps for two special cases (FBA and 2S-13C MFA). Maximum growth FBA overestimates flux into the PPP. Flux values are depicted in red: the upper value is the flux for the best fit and the lower values are the range of values compatible with data from 13C labeling experiments.
Fig 10.
Metabolite Labeling prediction.
The combination of 13C labeling data and genome-scale models with COBRA prediction methods produces predictions that cannot be obtained through either 13C MFA or FBA. The 13C MOMA predictions of fluxes shown in Fig 9 and S19 Fig are accurate enough that they can be used to predict the metabolite labeling (MDV, see Fig 4) for the pyk strain at 5 hours without using any data from the experiment on the pyk strain. A genome-scale model is needed to use 13C MOMA, and the accuracy provided by the 13C data is necessary to produce an accurate initial flux profile for 13C MOMA (see S2 Text). OF denotes the objective function: the average deviation of predicted labeling from the experimental value, measured in units of the experimental error. The prediction though standard MOMA, based on a FBA flux profile, is much less accurate (OF = 11.9) than the one obtained through 13C MOMA (OF = 2.2).