#### Traditional vs. half-reaction model: three theoretical cases

To illustrate the difference in ^{13}C-labeling distribution when using either traditional reactions or half-reactions to model the nonoxidative branch of the PPP, three simplified metabolic networks were formulated (cases 1–3). Note that the three networks are oversimplified and are solely used to clarify the difference in ^{13}C-label distribution that can occur between the two different modeling approaches.

For case 1 consider the traditional model of Fig. 3, but now containing only the conventional nonoxidative PPP reactions (Fig. 3I^{A}). The reversibilities of the three bidirectional nonoxidative PPP reactions and the three bidireactional glycolytic reactions are set at zero, such that the PPP overall converts three p5p molecules (i.e. a pentose pool consisting of ribulose 5-phosphate, ribose 5-phosphate and xylulose 5-phosphate) into two f6p molecules and one g3p molecule. Consequently, only the forward reactions of the PPP (v_{8f}, v_{9f}, v_{14f}) and the glycolysis (v_{2f}, v_{3f}, v_{5f}) are active. Analogous to the traditional model, only the forward glycolytic reactions are included in the half-reaction model (n_{5f}, n_{3f} and n_{2f}). Using the relations in Appendix I, the active nonoxidative PPP reaction rates in the traditional model are converted to the corresponding rates in the half-reaction model, resulting in substantial throughput for half-reactions n_{8f}, n_{9b}, n_{10b}, n_{11b} and n_{12f}. Investigation of the acceptor and the donor of the C_{2} fragment in both models shows that the traditional model contains two C_{2} fragment pools created by reactions v_{8f} and v_{9f}, while the half-reaction model by definition contains one single C_{2} fragment pool that is solely formed by reaction n_{8f}(Fig. 4). However, both C_{2} fragment pools in the traditional model are formed by the cleavage of p5p and can thus be lumped into a single pool, resulting in identical C_{2} fragment pools for both modeling approaches. Examination of the origin of the C_{3} fragment pools shows that both models contain only one C_{3} fragment-producing reaction, both with s7p as the donor (v_{14f}, n_{12f}). So, in essence, both models described in this case contain one C_{2} and one C_{3} fragment pool. As a result, the redistribution of ^{13}C atoms in the PPP is identical for both models.

For case 2 consider the same traditional model as used in case 1, supplemented with the stoichiometric neutral exchange reaction for e4p and f6p (v_{12} in Fig. 3I^{B}). In the half-reaction model this means an increase in n_{9f} and n_{9b} (see Appendix I). As a result of this additional reaction, C_{2} fragments are now also produced from f6p, thus increasing the number of C_{2} fragment pools in the traditional model to three (Fig. 4). The absence of bidirectional reactions makes it impossible for the three C_{2} fragment pools, originating from either p5p or f6p, to efface their labeling differences. A different labeling of f6p (in comparison to p5p) therefore by necessity leads to two unique C_{2} fragment pools in the traditional model. The half-reaction model inherently contains one single C_{2} fragment pool that comprises all distinct C_{2} fragment pools of the traditional model, as shown in Fig. 4. From this single pool a C_{2} fragment is randomly retrieved and attached to any suitable acceptor. Consequently, the top two carbon atoms of s7p synthesized in the half-reaction model can originate from either f6p or p5p, while in the traditional model they can only originate from p5p. In a ^{13}C-labeling experiment with 100%^{13}C_{1} glucose this will result in the synthesis of unlabeled and ^{13}C_{1}-labeled s7p for the half-reaction model, in contrast to only unlabeled s7p for the traditional model.

For case 3 consider the same traditional and half-reaction model as used in case 2, but now with all bidirectional reactions set at maximum reversibility (99.9%). Owing to this reversibility assumption, the number of C_{2} fragment-producing reactions in the traditional model increases from two to four (v_{8f}, v_{8b}, v_{9f} and v_{9b}). However, the high reversibility of the bidirectional reactions also ensures that the label distributions of the C_{2} fragment pools (and also the C_{3} fragment pools) are fully exchanged, effacing the differences in labeling pattern amongst the separate pools. As a result, no difference in isotopomer distribution is observed between the two models under conditions of high reversibility.

The three cases discussed above show that the difference in ^{13}C-label distribution amongst the two modeling approaches becomes more pronounced as the number of C_{2} and C_{3} fragment-producing reactions increases, while high reaction reversibilities diminish this difference. In reality the nonoxidative branch of the PPP contains multiple C_{2} and C_{3} fragment-producing reactions, thereby in essence creating different ^{13}C-label distributions. As shown in case 3, these differences can be alleviated by high reversibilities for the nonoxidative PPP reactions. Even though the reversibility of these reactions was argued by Follstad & Stephanopoulos[29], it remains questionable whether these reversibilities are high enough to efface the difference in ^{13}C-label distribution created by the multiple C_{2} and C_{3} fragment-producing reactions.

#### Application of the half-reaction model: flux patterns in *S. cerevisiae*

To investigate the actual difference in estimated flux patterns when applying either the traditional model or the half-reaction model shown in Fig. 3, measured mass isotopomers of ^{13}C-labeled primary metabolites [21] were used to refit the fluxes in the glycolysis and the PPP of *S. cerevisiae* CEN.PK113-7D. Similarly to the previously published fit, only measured mass isotopomer fractions larger than 0.03 were included. Figures 5I,II and Table 1 show the previously estimated flux patterns for the traditional model, as well as the newly estimated flux patterns using the half-reaction model. In order to facilitate the comparison of the two flux sets in Table 1, the flux estimates for the traditional model have been converted into their corresponding half-reaction rates using the equations given in Appendix I. The difference in flux pattern is evident, although, in general, not very large. As expected, the largest differences are found for the PPP split-ratio and the fluxes of the nonoxidative branch of the PPP.

Table 1. Comparison of the flux estimates for the traditional and half-reaction models presented in Fig. 5I,II. The pentose phosphate pathway fluxes in the traditional model have been converted to their corresponding fluxes in the half-reaction model using the equations in Appendix I. Reaction no. | Fluxes in the half-reaction model | Converted fluxes in the traditional model | Relative change (%) |
---|

n_{1} | 100 | 100 | 0 |

n_{2 net} | 26 | 26 | 0 |

n_{2 exchange} | 134 | 105 | 21 |

n_{3 net} | 56 | 50 | 11 |

n_{3 exchange} | > 1000 | > 1000 | – |

n_{4} | 65 | 63 | 3 |

n_{5 net} | 65 | 63 | 3 |

n_{5 exchange} | 221 | 194 | 13 |

n_{6} | 121 | 119 | 2 |

n_{7} | 18 | 24 | 36 |

n_{8 net} | 9 | 13 | 48 |

n_{8 exchange} | 4 | 10 | > 100 |

n_{9 net} | −3 | −5 | 67 |

n_{9 exchange} | 10 | 155 | > 100 |

n_{10 net} | −6 | −8 | 38 |

n_{10 exchange} | 124 | 4898 | > 100 |

n_{11 net} | −6 | −8 | 38 |

n_{11 exchange} | 25 | 24 | 4 |

n_{12 net} | 6 | 8 | 38 |

n_{12 exchange} | 0 | 0 | 0 |

The minimized covariance-weighted sum of squared residuals (SS_{res}) in these fits was calculated to be 20.9 and 6.5 for the half-reaction and traditional model, respectively. The SS_{res} is distributed according to a χ^{2}(n-p) distribution, with n-p being the degrees of freedom equal to the number of independent data points (*n* = 26) minus the number of free parameters (*P =* 14 and 9 for the traditional and the half-reaction model, respectively). Given the probabilities *P*[χ^{2}(12) > 6.5] = 0.89 and P[χ^{2}(17) > 20.9] = 0.23, it follows that within the 95% confidence interval both models give statistically acceptable flux estimates. Even though both models are statistically acceptable, it must be noted that the discrepancy between the measured and the fitted mass isotopomers (SS_{res}) is higher for the half-reaction model. One possible explanation for the higher SS_{res} in the half-reaction model is an overparameterization of the traditional model. In an overparameterized model, some parameters are actually used to fit measurement errors, thereby underestimating the true SS_{res}[50]. To determine the extent of this overparameterization, the estimated error variance () criterion can be used:

This criterion minimizes the variance of the sum of squared residuals by dividing the SS_{res} of a model by its degrees of freedom. As the traditional model contains more parameters than the half-reaction model, this will result in a smaller denominator for , thus compensating for any possible overparameterization. Nevertheless, the traditional model gives an of 0.54 compared to 1.23 for the half-reaction model, implying that the traditional model performs better from a statistical point of view.

A second explanation for the higher SS_{res} found for the half-reaction model might be the presence of isoenzymes for TK. As stated above, the genome of *S. cerevisiae* contains two genes encoding a TK, which adds a second C_{2} fragment pool to the metabolic network model. To test whether the introduction of an isoenzyme for TK in the metabolic network model results in a better fit, the half-reaction model in Fig. 3 was expanded with a second set of TK half-reactions (r.10–12) and subsequently used to fit the measured mass isotopomer fractions of *S. cerevisiae*. Figure 5III shows the estimated reaction rates for the so-called ‘double TK’ half-reaction model. The SS_{res} for this model was 6.5, meaning that this model also adequately fitted the measured mass isotopomer fractions {*P*[χ^{2}(12) > 6.5] = 0.89}. Interestingly, exactly the same values for the minimized SS_{res} and the number of free parameters (14) were found for both the ‘double TK’ half-reaction and the traditional model, making it impossible to distinguish the two models using the criterion. Table 2 shows that the flux estimates for both models were also very similar. The resemblance between the two models can be understood when one realizes that both models, unlike the half-reaction model, have the ability to create separate C_{2} fragment pools. Considering the reported finding that *tkl1* encodes the majority of the TK activity in *S. cerevisiae* cells grown in synthetic mineral medium on glucose, it was not anticipated that the addition of a TK isoenzyme to the metabolic network model would result in an increased goodness-of-fit. It must be noted that the prevalence of the *tkl1*-encoded TK was measured under excess glucose conditions, while the ^{13}C-labeling experiment was performed in a chemostat under glucose-limiting conditions.

Table 2. Comparison of the flux estimates for the traditional and the ‘double transketolase’ (‘double TK’) half-reaction model presented in Fig. 5II,III. The two separate fluxes for the transketolase-catalyzed half-reactions in the ‘double TK’ half-reaction model have been summed to allow for comparison with the converted fluxes of the traditional model shown in Table 1. Reaction no. | Converted fluxes in the ‘double TK’ half-reaction model | Converted fluxes in the traditional model | Relative change (%) |
---|

n_{1} | 100 | 100 | 0 |

n_{2 net} | 26 | 26 | 0 |

n_{2 exchange} | 103 | 105 | 2 |

n_{3 net} | 50 | 50 | 1 |

n_{3 exchange} | > 1000 | > 1000 | – |

n_{4} | 63 | 63 | 0 |

n_{5 net} | 63 | 63 | 0 |

n_{5 exchange} | 199 | 194 | 3 |

n_{6} | 118 | 119 | 0 |

n_{7} | 25 | 24 | 2 |

n_{8 net} | 13 | 13 | 3 |

n_{8 exchange} | 10 | 10 | 0 |

n_{9 net} | −5 | −5 | 4 |

n_{9 exchange} | 100 | 155 | 55 |

n_{10 net} | − 8 | −8 | 3 |

n_{10 exchange} | 11 | 4898 | > 100 |

n_{11 net} | −8 | −8 | 3 |

n_{11 exchange} | 24 | 24 | 2 |

n_{12 net} | 8 | 8 | 3 |

n_{12 exchange} | 0 | 0 | 0 |