the purpose of this discussion in Physiology Forum was to compare two sets of equations for the measurement of the rate of gluconeogenesis (or the fractional gluconeogenetic rate), which had previously been introduced by Drs. J. Katz and J. A. Tayek (1, 6) and Dr. B. R. Landau (3). To summarize, the two expressions for fractional gluconeogenesis are
Equation 1 and
Equation 2As indicated in Ref. 5, *Eq. 1
* is predicated on the stoichiometry illustrated by the fact that each glucose molecule of mass *M* + 3 is formed from one lactate molecule of mass*m* + 3 and another of mass*m*. From this it is concluded that*M*
_{3}/*m*
_{3}= 1. However, by definition,*M*
_{3} is the fraction of glucose molecules of mass*M* + 3 and*m*
_{3} is the fraction of lactate molecules of mass*m* + 3. To form*n* glucose molecules of mass*M* + 3, then,*n* lactate molecules of mass*m* + 3 and*n* of mass*m* are needed. If*x*% of the glucose molecules are*M* + 3, then*x*%/2 lactate molecules are*m* + 3, because for each molecule of mass *m* + 3 an additional molecule of mass *m* is needed, diluting the fractional enrichment,*m*
_{3}, of lactate relative to that (*M*
_{3}) of glucose. Therefore,*M*
_{3}/*m*
_{3}= 2. There is a consensus, established by arguments such as the above (3, 4), as well as in Refs. 3 and 4 by use of chemical kinetic (5) or combinatorial (2, 5) approaches, that the stoichiometry illustrated here in fact leads to this relationship. On this basis,*Eq. 1
* must be rewritten as
Equation 3The remaining discussion will therefore consider only*Eqs. 2
* and *
3
*. Interestingly, the approaches yield identical results, so long as either (*i*) there is no interaction of the gluconeogenetic pathway with the tricarboxylic acid (TCA) cycle or (*ii*) the lactate substrate is labeled only as*m*
_{3}. In the first case, the TCA cycle dilution factor (1, 5, 6) will be 1, and only the dilution factor by unlabeled carbon (6) is retained.*Equation 3
* then becomes
Equation 3` Because each type of labeled molecule (*m*
_{1},*m*
_{2},*m*
_{3}) will be diluted to exactly the same extent by unlabeled molecules, under these circumstances
and therefore
*Equation3′* will then reduce to *Eq.2
*. If *condition 2*holds, *m*
_{1} =*m*
_{2} = 0, and*Eq. 3
* immediately reduces to*Eq. 2
*. It should be noted that the two equations were derived by assuming that *condition 2* in fact prevails, or at least that*m*
_{3} >> (*m*
_{2}+*m*
_{1}).

In summary, therefore, *Eq. 2
* is derived by considering only labeled and unlabeled molecules.*Equations 3
* and *
3′*, on the other hand, consider the dilutions of labeled carbons. Both approaches yield identical estimates for fractional gluconeogenesis when the assumptions made in the calculation are identical.

As soon as lactate is labeled as {*m*
_{1},*m*
_{2},*m*
_{3}}, and TCA cycle dilution takes place, *Eqs.2
* and *
3
* yield divergent results. The reason for this divergence is that both equations make implicit assumptions about the behavior of the different mass isotopomers of lactate in their transition to glucose. As already discussed (5), both equations assume (if they are to be exact in the mathematical sense) that there is no loss of labeled molecules. Essentially both equations use*m*
_{3} as a reference point. *Equation 3
* assumes that all of the {*M*
_{1},*M*
_{2},*M*
_{3}} arose from *m*
_{3}lactate and corrects accordingly, thus overestimating fractional gluconeogenesis. *Equation 2
* assumes that all lactate molecules labeled as (*m*
_{3},*m*
_{2},*m*
_{1}) are cleared at the same rate, the rate at which*m*
_{3} lactate is removed. That this is not true can be seen by considering simply the equilibration with fumarate. Some of the*m*
_{1} molecules will become unlabeled, but all of the*m*
_{3} molecules will remain labeled (to some degree) and therefore will remain countable. All, however, are assumed to disappear at the rate of the most slowly cleared *m*
_{3}.*Equation 2
* will therefore underestimate fractional gluconeogenesis to a small extent. This divergence is predicted to be small and, indeed, is completely consistent with the difference in the rates of fractional gluconeogenesis seen (e.g., Table 1, Ref. 5). In addition,*Eq. 3
* yields a consistently higher estimate of fractional gluconeogenesis than *Eq.2
*. This is also in line with the predicted relative overestimation of fractional gluconeogenesis by *Eq.3
* and the relative underestimation by*Eq. 2
*.

This difference, albeit small, is an expression of the limitations of the strictly molecular approach to calculating the fractional rates of gluconeogenesis by use of the [U-^{13}C]glucose paradigm. Stated differently, all lactate molecules behave the same way in their conversion to glucose. The different mass isotopomers of lactate, however, do not. This is because each positional isotopomer (different combination of carbons labeled) demonstrates its own kinetics, as is completely characterized by the matrix of*Eq. 8* in Ref. 5. This equilibration of glucose and lactate labels in a recirculating system, such as the body, also leads to the generation of*m*
_{0}. This is assumed to be zero in the development of both of the above equations but clearly cannot be, as discussed by all previous papers on this subject (and the present set of commentaries). Again, this is a manifestation of the divergence of the behavior of the unlabeled and labeled molecules. It can, as discussed in Ref. 5, be examined by using the properties of individual isotopomers: read, carbons.

We have dwelled at some length on the complementary nature of the molecular and carbon approaches of studying the fluxes of labeled molecules, because this aspect appears somewhat neglected by all of the other commentators. That careful interpretations are needed can be illustrated by considering some of the numerical examples that have been offered as demonstrations of the accuracy of particular formulas (albeit within a defined setting). Thus Fig. 1 of Ref. 4 shows an example of a system with no label exchange in the TCA cycle. When this example is used, identical answers (60.8%) are obtained using*Eqs. 2
* and *
3′*. Figure 2 of Ref. 4 and Fig. 1 of Ref. 2 illustrate the situation when a formalized exchange process is introduced into the model. In Fig. 2 of Ref. 4, one-half of*m*
_{3} is converted into *m*
_{1}, with the preexisting *m*
_{1}remaining intact; i.e., the label exchange takes place before the conversion of lactate to glucose. Application of *Eq.2
* indeed yields the predicted 100% fractional rate of gluconeogenesis, whereas that of *Eq. 3
*yields 154%. The caveat here is that this application of*Eq. 3
* is not appropriate, because it is derived on the basis of dilution of lactate label and the subsequent dilution of label in the oxaloacetate pool. Because the latter process does not occur, the TCA cycle-based dilution factor should not be included. When the relevant part of the formula (that is,*Eq. 3′*) is used, a 100.1% contribution of gluconeogenesis is again obtained. More relevant to the comparison, if the label exchange takes place during the conversion of lactate to glucose, we have (Ref. 4)*m*
_{1} = 0.052 and*m*
_{3} = 0.938. Under these circumstances, *Eqs. 2
* and *
3
* yield, respectively, 100 and 104%, illustrating the slight (and predicted) overestimation by*Eq. 3
*.

Furthermore, both Fig. 2 of Ref. 4 and Fig. 3 of Ref. 2 illustrate the source of the (small) differences in the two equations. If*m*
_{3} →*M*
_{1} (4) or →*M*
_{1}+*M*
_{2}(2), then, by exactly the same metabolic processes,*m*
_{2} will be converted to *M*
_{1}and *M*
_{0} and*M*
_{1} will be converted to *M*
_{0}. These conversions are not taken into account in these examples, because*m*
_{2} and*m*
_{1} are conserved, whereas *m*
_{3} is not. If, for example,*m*
_{1} → 0.33*m*
_{1}(phospho*enol*pyruvate) in Fig. 2 of Fig. 4, then *Eq. 2
* yields (approximately) 88% and *Eq. 3
*, 92%. This illustrates both the underestimation by both equations due to the disappearance of the labels in exchanges and the (relative) overestimation of *Eq. 3
*. This example is illustrative only; a quantitative estimation would have to involve the transition matrix of Ref. 5. Exactly the same considerations would hold for Fig. 1 of Ref. 2.

To avoid such interpretational problems, the following set of guidelines is offered for the development of numerical illustrations and also for the application of the equations in an appropriate context. • A steady state must be maintained. This must also be with respect to each isotopomer. For example, if*m*
_{1} and*m*
_{2} are present in the system, they must come either from metabolic processing of*m*
_{3} (i.e., the TCA cycle is present) or from exogenous administration. Net accumulation or depletion of any isotopomer should not occur. • Metabolic consistency must be maintained. For example, if *m*
_{3}→ *m*
_{2} or*m*
_{1}, then, in a parallel fashion, the same metabolic processes (e.g., TCA cycle interactions, fumarate equilibration) will convert*m*
_{2} →*m*
_{1} and*m*
_{1} →*m*
_{0}. The enrichment of one isotopomer should not be changed without changing the others in a consistent way. •The relevant formula must be applied. For example, if*m*
_{1} and*m*
_{2} are introduced only exogenously or lactate does not undergo exchange with the TCA cycle, do not apply a TCA cycle correction.

The application of these guidelines is demonstrated above. They furthermore illustrate the complexity of the system. In this commentary we have refrained from addressing the details of the other comments but have emphasized points that may perhaps be underplayed in the developments considered. In summary:*1*) when the stoichiometry of the lactate ↔ glucose interaction is taken into account, the formulas presented in Refs. 1 and 6 (as corrected above) and Refs. 3 and 4 are identical when applied in situations when exchange with the TCA cycle can be neglected or only*m*
_{3} is present.*2*) When the latter exchange comes into play or a real system with equilibration among the three isotopomers {*m*
_{1},*m*
_{2},*m*
_{3}} is dealt with, the two formulas diverge. Strictly molecular considerations can no longer explain the differences, because carbon (or positional isotopomer) exchanges must be considered.*3*) The same carbon-based label losses contribute to at least some of the underestimation of fractional gluconeogenesis by both these formulas, because*M*
_{0} is generated from {*m*
_{1},*m*
_{2},*m*
_{3}}.

- Copyright © 1999 the American Physiological Society