Dealing with Metabolic Complexity
Chapter 2 Chapter 2 Analysis of Metabolic Reaction Rates
2.3 Example
2.3.3 Reducing Equivalents Mass Balance
The approach studied next was the one that considers the mass balance of the reducing equivalents from N A D H and N A D P H (equation (2.48)). Two different studies were again performed: one without and one with the constraint of the C 0 2 mass balance.
The results are presented in Figure 5. The oxygen uptake rate used for the mass balance equation (2.48) as a first estimate was the same as the experimental one.
This first solution suggested that the value of the flux v l j was approximately 15%
of twice the value of the oxygen uptake rate. The problem was solved again using a value for the oxygen uptake rate of 85% of the experimentally determined one. This second solution gave again a value for the v15 that was approximately 15% of twice the value of the experimentally determined oxygen uptake rate over the whole range of the dilution rates. Therefore, the iteration stopped and the "corrected" value for the oxygen uptake was used in both cases (without and with the constraint for the C 0 2 mass balance).
Even in the absence of the constraint of the C 0 2 mass balance, the estimated RQ is very close to the experimental value (Figure 5.A). This is in contrast to the previous approaches with which, even when the constraint for the C 0 2 mass balance included, the estimated RQ was significantly different from the experimental value (Figure 3.A). In general, both cases were found t o be in excellent agreement with each other (Figures 5.C-5.F). However, the flux vl was estimated to be zero over the whole range of the dilution rates when the constraint for the C02 mass balance was not included, but it had a positive value when the C02 mass balance was considered.
This difference strongly suggests that C 0 2 mass balance should be included in any flux analysis. The value for vl in this second case was found to be approximately equal t o 0.1 over the whole range of dilution rates (Figure 5.B), a value that is very close
t o the biosynthetic requirements for R 5 P and E 4 P . Therefore, it appears that the the R 5 P and E 4 P required for biosynthesis are coming from GGP via reaction step 1. However, this solution is qualitatively very different from the one found with the approach using the NADPH mass balance. There, for low dilution rates vl was zero below a certain value for the dilution rate, and above that it increased monotonically up t o 0.25. The fact that no assumption concerning the value of
4
in the sum (2.66) was made in this last approach suggests that this last result is more reliable.In order to evaluate this approach with respect to the previous one, the ratio:
was calculated based on the estimated fluxes. A value for this ratio had been assumed and used as the sum (2.66) constraint in the previous approach. The calculated ratio is presented in Figure 6. For low dilution rates the ratio is higher than one suggesting that an excess of NADPH is produced. This is probably recycled back to NADH. However, for dilution rates higher than 0.15h-' , this ratio is lower than unity suggesting that the NADPH produced does not fulfill the biosynthetic requirements and that the additional amount needed is provided from the excess NADH produced.
Interestingly enough, when this ratio is equal to or higher than 0.8, the value for the flux vl is positive, while the previous approach suggested that, as this ratio decreases, flux vl becomes zero.
The Euclidean norm of the estimated fluxes was also calculated for every case in the two approaches (Figures 2.D, 3.D, 4.D, and 5.D). I t appears that the last ap- proach has the lowest norm, especially when the constraint of the COz mass balance is included, forcing the fluxes considered t o produce the minimum amount of en- tropy. If our thermodynamic suggestion is right, then the last approach also satisfies also the thermodynamic optimality criteria. However, the main uncertainty of this thermodynamic consideration still holds; i.e., the question of which fluxes should be considered in calculating the Euclidean norm. Another interesting difference between the Euclidean norms of the fluxes estimated from the two approaches is their trend
with respect t o dilution rates. In the first approach the norm decreases with increas- ing dilution rate (Figures 2.D and 3.D), whereas, in the last approach, the norm increases with increasing dilution rates. In general, the latter is more reasonable, since for increasing dilution rates the specific growth rate, the specific glucose and oxygen uptake rates all increase, indicating that the biocatalytic machinery of the cell should operate at higher rates.
Similar results were also obtained when the stoichiometric coefficient p for reaction 15 was considered (see equation (2.48)). The oxygen uptake rate used in the mass balance was "corrected" following the same procedure as in the last approach. For a value of p equal to 0.5 the oxygen uptake rate was reduced by 7%. That corresponds closely to the 16% correction for the last approach when p was equal to zero.
In conclusion, this last approach is more attractive since it requires the fewest assumptions. No assumption has been made with respect t o energetics of the cell.
The correction of the oxygen uptake rate used in the mass balance equation (2.48) was not based on a parameter-fitting approach but simply on the consistency of the value of one flux, vl5, with respect to the corrected value of the oxygen uptake rate.
Metabolic flux analysis has enjoyed a lot of attention over the last five years. Various metabolic systems have been analyzed, and useful insights resulted from those analy- ses. However, because most metabolic systems are underdetermined (more metabolic reactions than metabolic species), various assumptions have been used to make the systems determined, and little attention has been paid on the effects of these assump- tions to the final conclusions.
In this chapter flux analyses for the bacterium B. subtilis were performed. A systematic algorithmic procedure was proposed that can take metabolic constraints into account. Some of the commonly used assumptions were considered and the effects of these assumptions on the resulting estimated fluxes have been studied. Two main conclusions were drawn for aerobically grown bacterial systems:
1. Any flux analysis that employs assumptions about cell energetics should be considered as a qualitative description of the trends of the fluxes under different growing conditions. A sensitivity analysis of the results with respect to the assumptions should always be performed.
2. The mass balance of COz should be included in the analysis even though it is not linearly independent from the rest of the mass balances. I t appears t o improve the estimation of the fluxes by making the system less sensitive t o assumptions and by integrating additional experimental information with the analysis.
A novel procedure for flux analysis has been suggested and applied to the exarnple metabolic system. This procedure does not employ any assumptions regarding the energetics of the cell and therefore does not bias the results. Application of the pro- cedure to the example system and comparison of the results with those of approaches that use assumptions regarding cell energetics have shown that this approach is more advantageous.
It has been suggested that the common approach of minimizing the Euclidean norm of the fluxes as an additional criterion for choosing an estimate for the metabolic fluxes from an infinite number of possible solutions when the metabolic system is un- derdetermined is related to thermodynamic optimality criteria based on the evolution theory. However, further investigation is required before a definite connection between these criteria and flux analysis is asserted.
Finally, it should be stressed that flux analysis is a mathematical modeling method that integrates the available biochemical knowledge in order t o provide further insight on the behavior of biological systems. I t provides an estimate for the values of the metabolic fluxes and can be used t o compare relative changes in the fluxes under dif- ferent conditions that do not significantly influence the stoichiometry of the metabolic networks, such as different growth rates or changes in enzyme amounts arising from mutation or genetic engineering. As with every mathematical method used in biotech- nology, it should be used in an iterative way: the initial information it provides based on preliminary experimental data will suggest the next experimental approach. The
results of this new experiment will be used for a second mathematical analysis or even for possible reformulations of the stoichiometric model, that will again suggest the next experimental approach.