## Abstract

Recently three equations for estimating gluconeogenesis in vivo have been proposed, two by J. A. Tayek and J. Katz [*Am. J. Physiol.* 270 (*Endocrinol. Metab.* 33): E709–E717, 1996, and *Am. J. Physiol.* 272 (*Endocrinol. Metab.* 35): E476–E484, 1997] and one by B. R. Landau, J. Wahren, K. Ekberg, S. F. Previs, D. Yang, and H. Brunengraber [*Am. J. Physiol.*274 (*Endocrinol. Metab.* 37): E954–E961, 1998]. Both groups estimate gluconeogenesis from cycling of [U-^{13}C]glucose to lactate and back to glucose, detected by mass spectrometry. Landau’s approach is based on analysis of labeled molecules, whereas Tayek and Katz’s is based on labeling of carbon atoms by use of the concept of “molar enrichment,” which weights each mass isotopomer by the number of labeled carbons. We derived an equation very similar to Landau’s using binomial probability. Our analysis demonstrates that the molecular-based approach is correct. Additionally, equations appropriate for ^{14}C studies are not appropriate for ^{13}C studies, because the method used to detect^{14}C, decay of atoms, differs from^{13}C mass isotopomers detected as labeled molecules. We conclude that the molar enrichment carbon-based approach is not useful in the derivation of equations for the polymerization of molecules detected by mass spectrometry of molecules, and we confirm the findings of Landau et al.

- glucose
- molecular condensation
- stable isotopes
- gas chromatography-mass spectrometry

readers of the *American Journal of Physiology: Endocrinology and Metabolism* may be aware of a recent series of papers proposing distinct equations for estimating gluconeogenesis in vivo after constant infusion of [U-^{13}C]glucose. The three key papers are by Tayek and Katz (4, 5) and Landau et al. (3). The issues raised in these studies are important, illustrating fundamental principles for the development of tracer methods. The papers cited have produced three different equations for fractional gluconeogenesis (Table 1). These equations are based on identical assumptions, yet they are algebraically different. This manuscript employs binomial probability to address this issue.

Deriving mathematical relationships that can be solved for useful information is fundamental to the study of metabolism. A derivation must be accurate; it should always produce a mathematically correct answer under the stated assumptions. Once a derivation has been presented, investigators may test it with experimental data. Often, as the history of gluconeogenesis estimates attests, application of the equation resulting from the derivation will fail to produce results consistent with experimental experience. This may occur because the assumptions underlying the derivation do not apply. In this case, a new derivation should be built on the previous one, differing only when the new set of assumptions dictates that a different relationship is required. Estimating gluconeogenesis by use of stable isotope tracers and mass or positional isotopomers is a relatively new field. The optimal method may not yet be in hand. Thus it is essential that correct forms of derivations are used in the earliest stages so that future studies may build on this framework.

In these studies, gas chromatography-mass spectrometry (GC-MS) is used to quantify the relative amount of each mass isotopomer of plasma glucose and lactate molecules. The terminology used here is consistent with the three published papers. The amount of each isotopomer is expressed as fractional abundance, that is, the amount of each isotopomer divided by the sum of all isotopomers. From our perspective, fractional abundances are required, because they are equivalent to probabilities. A different letter represents the isotopomers of each compound, and the subscript “*i*” indicates mass:*M _{i}
* for isotopomers of plasma lactate;

*M*for isotopomers of plasma glucose. We add “P

_{i}_{i}” for the isotopomers of phospho

*enol*pyruvate (PEP), “G

_{i}for gluconeogenic glucose, and “Q

_{i}” for lactate from glycolysis of plasma glucose. By this convention,

*m*

_{3}represents the fractional amount of plasma

^{13}C-

^{13}C-

^{13}C lactate. Tayek and Katz also use the term “molar enrichment,” which they define as the weighted sum of the isotopomer fractions of a molecule “

*x*” with

*n*carbons as Equation 1Accordingly, the molar enrichment of lactate =

*m*

_{1}+ 2 ×

*m*

_{2}+ 3 ×

*m*

_{3}. The protocol used in the studies is simple: a constant [U-

^{13}C]glucose infusion serves two purposes; it allows calculating the rate of appearance of plasma glucose (R

_{a}) as rate of infusion divided by

*M*

_{6}. It also provides a

^{13}C precursor for gluconeogenesis, labeled plasma lactate, that is easily sampled.

An overview of the flow of tracer glucose in this protocol is described in Fig. 1. To make the diagram easy to follow, higher values for [U-^{13}C]glucose infusion and for all labeled isotopomers are shown than were used in the studies discussed here. Because the actual enrichment of plasma lactate is not high in vivo, generation of glucose from two labeled lactate is ignored in Fig. 1. Under these conditions, part of the isotopomer profile of glucose can be used for determining the R_{a}(*M*
_{6}) and part (*M*
_{1} +*M*
_{2} +*M*
_{3}) for gluconeogenesis. The model allows several sites for dilution or exchange of isotope. Unlabeled carbon enters plasma glucose via glycogenolysis and unlabeled lactate from muscle mixes in the plasma with lactate derived from glucose. The tricarboxylic acid (TCA) cycle exchanges labeled carbon atoms for unlabeled ones. Both groups use the same assumptions, which we summarize. *1*) The amount of [U-^{13}C]glucose infused is small relative to other fluxes, so that correction of R_{a} for tracer infusion may be ignored. *2*) The fractional abundance of labeled lactate is low, and the probability of glucose formed from two labeled lactate moieties is negligible. *3*) Data are corrected appropriately for natural abundances of heavy isotopes, and all isotopomer data are expressed as fractional enrichments. *4*) Labeled carbon enters the TCA cycle only via pyruvate through pyruvate carboxylation. If ^{13}C enters the TCA cycle via^{13}C acetyl-CoA or^{13}CO_{2}fixation, gluconeogenesis will be overestimated. An exception occurs if the data are corrected for^{13}CO_{2}fixation, as in Landau et al. (3). *5*) Gluconeogenesis from glycerol is negligible. *6*) Isotopic enrichment of intrahepatic pyruvate is equal to that of plasma lactate. *7*) Except for pyruvate carboxylation, unlabeled carbon enters the TCA cycle only at acetyl-CoA. Thus exchange with TCA cycle intermediates, via amino acid transamination, for example, is negligible. (The impact of*assumptions 4, 5, 6,* and*7* is discussed below). *8*) The exchange of labeled molecules with the TCA cycle yields PEP molecules with fewer labeled carbons but does not yield significant unlabeled molecules. Neither group offers an equation to correct for the formation of unlabeled PEP molecules from labeled lactate molecules. Landau et al. (3) state this assumption explicitly. Tayek and Katz state that the correction factor used for dilution of carbon in the TCA cycle does not correct for the formation of unlabeled PEP (5). Both groups justify this assumption because the tracer entering the TCA cycle is predominately*m*
_{3}, and the rate of pyruvate carboxylation relative to the TCA cycle is substantial, resulting in significant*M*
_{3} and*M*
_{2} glucose.

The major differences in the approach of the two groups are that Tayek and Katz use carbon-based calculations, estimating “recycling of glucose *carbon*” and “dilution by unlabeled *carbon*.” In contrast, Landau uses a molecular approach, calculating “fraction of glucose*molecules* recycled.” These differences are most apparent when the interaction with the TCA cycle is considered. The TCA cycle effectively reduces the number of labeled carbon atoms but produces no change in the number of labeled molecules (*assumption 8*). The derivations presented by Tayek and Katz employ factors to correct for the loss of labeled carbon in the TCA cycle (4) or recycling of carbon atoms (5). To quantify this loss of carbon, they use the concept of molar enrichment defined above. In contrast, Landau et al., using a molecule-based approach, state that no correction is required for loss of labeled carbon in the TCA cycle so long as labeled molecules are conserved. Thus the central issue defining the differences between the two groups is the use of carbon-based calculations that involve corrections for lost carbon vs. molecule-based calculations that do not use this correction. Molar enrichment is used for carbon-based calculations.

A common feature of the presentations of both groups is the use of glucose cycling. Both groups produce equations to represent the fraction of glucose carbon or molecules that recycles. These cycling equations are then used to derive fractional gluconeogenesis. To bring a fresh view to this discussion, we develop a derivation without involving cycling. Instead of working with the cycling of glucose, we simply take the fractional abundances of lactate as known. The resulting derivation illustrates principles for developing equations for the rate of production of molecules formed by condensation of two or more identical precursors. The derivation will be built as a two-step modeling process.

## FIRST MODEL: 2 LACTATE → GLUCOSE

We begin with the simpler model. Assume that lactate travels directly to glucose, bypassing oxaloacetate and the TCA cycle. TCA cycle has no effect on isotopomer labeling. Ignore the fact that the tracer enters as [U-^{13}C]glucose infusion, and consider the synthesis to begin with a population of isotopomers of lactate that are all either*m*
_{0} or*m*
_{3}. Note that*m*
_{0} and*m*
_{3} represent fractional enrichment values, so that*m*
_{0} +*m*
_{3} = 1. Thus*m*
_{0} and*m*
_{3} equal the probability that a randomly selected molecule from the gluconeogenic lactate pool is labeled, =*m*
_{3}, or unlabeled, = *m*
_{0}. In this simple model, gluconeogenesis is the polymerization of two lactate molecules. The probability distribution for the various possible labeled forms of glucose is derived from binomial probability by expanding the polynomial representing this dimer
Equation 2Accordingly, the probability of each isotopomer of gluconeogenic glucose (G_{i}) is
Equation 3
Equation 4
Equation 5These simple expressions provide the key to understanding the relationship between precursor and product molecules.

No comparable equation exists for carbon atoms.*Equation 4
* is most important, because G_{3} is produced solely by gluconeogenesis. The isotopomers of newly synthesized glucose enter plasma glucose by mixing with glucose derived from glycogenolysis and tracer infusion. When plasma glucose is considered, glycogenolysis will contribute to *M*
_{0}and tracer [U-^{13}C]glucose infusion will contribute to*M*
_{6}, leaving*M*
_{3} as the glucose isotopomer supplied only by gluconeogenesis. The fractional contribution of gluconeogenesis to the plasma glucose is
Equation 6This equation simply states that*M*
_{3} glucose will be directly proportional to the fraction of glucose derived from the condensation of an*m*
_{0} and*m*
_{3} lactate. An alternative statement of this relationship based on probability is
Equation 6a
*Equation 6a
* emphasizes the fact that expressing the amounts of isotopomers as fractional abundances is equivalent to probabilities. For this reason, fractional abundances are required for this type of analysis. Terminology based on mole or atom percent excess cannot be used. Before moving on, we emphasize that the binomial probability equation allowing this simple derivation for fractional gluconeogenesis is a property of molecules and not carbon atoms.

## SECOND MODEL: 2 LACTATE → TCA CYCLE → PEP → GLUCOSE

We now consider the interaction of the labeled lactate with the TCA cycle and introduce hepatic PEP as the immediate gluconeogenic precursor (Fig. 1). The net effect of the TCA cycle is that some labeled atoms are lost in the TCA cycle carbon exchange. Lactate, largely *m*
_{3}, enters the TCA cycle as oxaloacetate via intrahepatic pyruvate, and P_{3}, P_{2}, P_{1}, or even P_{0} PEP emerges. However, P_{0} PEP produced by this process is negligible (*assumption 8*). Tayek and Katz clearly and correctly state that their equation for TCA cycle carbon dilution does not correct for P_{0} (see p. E481 of Ref. 5). They also indicate that much of the carbon loss observed is due to conversion of *m*
_{3}lactate to *M*
_{2}glucose, a process that does not produce loss of labeled molecules (see p. E714 of Ref. 4). The example shown in Fig. 1 tests the effects of loss of labeled carbon but conservation of the fraction of labeled molecules between lactate and glucose. Thus the fraction of labeled lactate molecules equals that of PEP, (*m*
_{1} +*m*
_{2} +*m*
_{3}) or (9/90) = (P_{1} + P_{2} + P_{3}) or (3/30). To test the carbon vs. molecular approach, this example includes loss of labeled carbon via the TCA cycle. The molar enrichment of PEP (6/30) is less than that of plasma lactate (24/90 = 8/30).

Following the binomial expansion approach, we derive a second relationship for fractional gluconeogenesis. Because intrahepatic PEP cannot be sampled in humans, (*m*
_{1} +*m*
_{2} +*m*
_{3}) is used in place of the equivalent (P_{1} + P_{2} + P_{3}). We lump together all labeled molecules such that the synthesis of a dimer, glucose, is described by the combination of labeled and unlabeled lactate molecules
Equation 7Again, newly synthesized glucose will have the following distribution
Equation 8
Equation 9
Equation 10G_{1}+ G_{2} + G_{3} will appear in glucose only as a result of gluconeogenesis, so that
or
Equation 11The substitution of lactate for the true gluconeogenic precursor, PEP, in *Eq. 11
* is valid provided (*m*
_{1} +*m*
_{2} +*m*
_{3}) equals (P_{1} + P_{2} + P_{3}). This requirement is related to *assumptions 4–7* above. If^{13}C enters the TCA cycle via^{13}C acetyl-CoA or^{13}CO_{2}fixation (*assumption 4*), gluconeogenesis will be overestimated. Overestimation results because (*m*
_{1}+*m*
_{2} +*m*
_{3})*m*
_{0} will be less than (P_{1}+ P_{2} + P_{3}) P_{0}, and the smaller lactate terms in the denominator of *Eq. 11
* lead to erroneously high fractional gluconeogenesis values. In contrast, if unlabeled carbon enters the gluconeogenic pathway beyond plasma lactate (*assumptions 5–7*), gluconeogenesis will be underestimated because (*m _{1}+ m_{2} + m_{3}) m_{0} will be greater than (P_{1}
*+ P

_{2}+ P

_{3}) P

_{0}.

## COMPARING PUBLISHED MODELS

The binomial expansion equation (*Eq.11
*) is compared with the published equations (Table1). *Equation 11
* very nearly equals Landau’s equation, differing only in the presence of the term*m*
_{0}, representing the fractional contribution of unlabeled lactate. This term is part of the binomial expansion. However, for these in vivo experiments where the [U-^{13}C]glucose infusion is a small fraction of R_{a},*m*
_{0} approaches 1. We include the *m*
_{0}term to be mathematically correct and to remind investigators that it might be significantly less than 1 in some situations. The effect of omitting *m*
_{0} from the equation is independent of the value for fractional gluconeogenesis and varies linearly with*m*
_{0} (Fig. 2). In Landau’s experiments*m*
_{0} was approximately equal to 0.97, signifying that Landau’s estimates of gluconeogenesis should be increased by the factor 1.03. In Table 1, the larger numerical difference between *Eq.11
* (binomial probability) and Landau’s result is due to the relatively low value of*m*
_{o} (0.9) in the example (Fig. 1). Both equations of Tayek and Katz differ from the derivation by binomial expansion and fail to produce the correct answer for the test case shown in Fig. 1. The*m*
_{o} term is also missing from the denominator of both of their equations.

Binomial expansion (*Eqs. 6
* and *
11
*) clearly includes a factor of 2 in the denominator. This factor was omitted by Tayek and Katz in 1996 (4). Landau was keenly aware of the need for the factor of 2 in the denominator of his equation. He explained the requirement for the factor of 2 “because one-half the triose units forming glucose molecules of masses*M*
_{1},*M*
_{2}, and*M*
_{3} are unlabeled and are not derived from [U-^{13}C]glucose.” In view of our reliance on binomial probability, we would rather assert that the factor of 2 is required because it is the coefficient in the binomial expansion. As such, it represents the two chances for making glucose one-half*m*
_{0} and one-half*m*
_{3},*m*
_{0} −*m*
_{3} and*m*
_{3} −*m*
_{0}. It should be noted that, in 1997, Tayek and Katz added a factor of 2 in the denominator (5). However, their 1997 equation was still not equivalent to that derived by binomial probability. We conclude that the equations of Tayek and Katz for estimating gluconeogenesis are not appropriate, because they fail to produce a result consistent with the binomial expansion.

It is interesting that the binomial expansion derivation is not step by step identical to Landau’s and yet produces an almost identical equation. This demonstrates that the glucose cycling properties of the model are not essential to the measurement of gluconeogenesis. Finding a different derivation, yielding essentially the same relationship, serves as a verification that the molecular approach of Landau is correct. For those who prefer examples to derivations, the values in Fig. 1 could be altered to test the validity of the molecular approach. Glycogenolysis, tracer infusion, unlabeled lactate flux, and TCA cycle activity could each be changed. As long as the system is in steady state, and the assumptions are not violated, *Eq.11
* will yield the correct result for gluconeogenesis. Alternatively, changing the values of the molar enrichment of lactate and PEP, retaining the number of labeled molecules, will have no effect on gluconeogenesis and will not affect the estimates when*Eq. 11
* or Landau’s equation is used, but it will change the values calculated with the equations of Tayek and Katz.

## THE CARBON VS. MOLECULAR APPROACH FOR CONDENSATION POLYMERIZATION EQUATIONS

Other than the factor of 2 discussed above, the remaining difference between Tayek and Katz (1996) and Landau et al. (1998) is the fact that Tayek and Katz employ a correction for the loss of some labeled carbon from labeled molecules, analogous to^{14}C studies. The difficulty with analyzing ^{13}C studies as analogous to ^{14}C studies stems directly from the carbon, rather than molecular approach and the use of molar enrichment to compensate for loss of carbon. In the 1996 paper, Tayek and Katz utilize molar enrichment to correct for TCA cycle dilution (4). In their 1997 paper, they state that the estimate of gluconeogenesis is not dependent on TCA cycle dilution (5). However, they continue to use molar enrichment to calculate “dilution by unlabeled carbon,” which again introduces a consideration of carbon rather than molecules.

The key to understanding the molecular approach is that no correction is required for the loss of some^{13}C atoms within a molecule if the number of ^{13}C-labeled molecules is conserved. The fractional abundances for molecules with 1–3^{13}C atoms are combined in the equations (see *Eqs.7-11
*). Thus it is of no consequence whether a labeled molecule has one, two, or three labeled carbons. Applying this concept to gluconeogenesis represents an important contribution of Landau and co-workers. Mass spectrometry of molecules detects each mass isotopomer with identical efficiency. It counts each molecule detected once regardless of the number of labeled carbon atoms. Put another way, the probability of detecting a^{13}C-labeled molecule by mass spectrometry is not decreased by a decrease in the number of labeled atoms/molecule. (We are not concerned here with signal-to-noise issues, which may decrease the precision of detecting specific mass isotopomers). Thus ^{13}C detected by mass spectrometry is different from^{14}C detected by liquid scintillation counting. Starting with [U-^{14}C]lactate, the probability of detecting a labeled glucose molecule is directly proportional to the number of labeled atoms that survive to reach the product. A correction factor analogous to that used by Tayek and Katz for dilution of labeled carbon is appropriate for^{14}C detected by liquid scintillation counting but not for^{13}C detected by mass spectrometry of molecules.

The concept of gluconeogenesis as the condensation of two precursor molecules is fundamental for understanding how gluconeogenesis is estimated from tracers, both ^{14}C and ^{13}C. The binomial probability equation (*Eq. 11
*) underlies^{14}C calculations as well. Working with ^{14}C is disadvantageous because it deals with atoms; liquid scintillation counting measures atoms by emitted radiation, yielding disintegrations per minute (dpm). The dpm detected may be used to convert atoms to molecules by use of the specific activity of the traced compound, dpm/(mole of the compound). However, in doing this, one must carefully account for each atom of ^{14}C lost in the path from precursor to glucose and multiply by a dilution factor that effectively increases the observed dpm to account for the missing^{14}C. This is the basis of the elaborate ^{14}C equations developed by Katz (1) and Kelleher (2) in the 1980s to calculate dilution factors. The calculations for ^{13}C mass isotopomers are less complicated than for^{14}C dpm data, because the labeling information is readily obtained in the form in which it is used in the equations, as the fraction of labeled molecules.

The differences in the ^{14}C and^{13}C calculations described above are not differences inherent in the type of tracer. Rather, they are consequences of the detection method used. To illustrate this point, compare two variations of the protocol described in Fig. 1. First, consider a hypothetical experiment replacing the [U-^{13}C]glucose with [U-^{14}C]glucose in the example shown as Fig. 1. GC-MS could be used to detect^{14}C-labeled glucose and lactate molecules, just as with the ^{13}C experiments. The isotopomer would occur as*M*
_{6} and*M*
_{12}, but the data could be analyzed as shown here with binomial probability. Because GC-MS detects labeling of molecules, no correction would be required for loss of ^{14}C-labeled carbon atoms within labeled molecules. Alternatively, consider a^{13}C study isolating glucose and lactate and combusting the molecules to CO_{2}. Isotope ratio MS (IRMS) of CO_{2} could measure the labeling of carbon atoms. This method, like radioactivity, measures carbon atoms and requires a correction to relate the number of labeled carbons to the number of labeled molecules of glucose or lactate. These examples illustrate the importance of using corrections appropriately to reflect both the type of data collected and the mathematics of the underlying derivation. Biosynthesis of polymers by condensation of precursors is a molecular process. To estimate its rate, equations must deal with the labeling of molecules. Mass spectrometry of molecules presents us with the data in the correct form. The carbon-based molar enrichment approach converts molecular isotopomer data to carbon data, leading to errors. For this reason, the equations of Tayek and Katz are not useful building blocks for the future of this field.

One may ask whether molar enrichment may be used to correct for the loss of labeled molecules in the TCA cycle. Neither group directly accounts for this possibility. It is possible to derive equations to correct for loss of labeled PEP molecules. These equations are functions of the rate of pyruvate carboxylation relative to TCA cycle flux (“*y*”) and of the fumarase equilibrium. The resulting equations can be expressed in terms of*m*
_{1},*m*
_{2} and*m*
_{3.} However, these equations do not contain terms that multiply the amount of each mass isotopomer by the number of labeled carbons, as dictated by molar enrichment. Molar enrichment is a concept that has only one obvious use in our experience. It may be used to predict the dpm to be found if a^{13}C study is repeated with^{14}C as tracer or as a combustion IRMS study. Other uses for molar enrichment should be carefully justified.

Finally, our analysis agrees with that of Landau et al. (3), supporting their conclusion that gluconeogenesis is underestimated by the [U-^{13}C]glucose technique. Investigations using the carbon-based molar enrichment approach failed to detect this underestimation. As Landau and co-workers pointed out, correct equations can yield physiologically implausible answers (3). The reason must be that some of the assumptions are not valid or that we have overlooked some issue entirely. It is now the task of interested researchers to build on derivations consistent with binomial probability to learn why the application of correct equations and seemingly reasonable assumptions do not yield expected values.

## Acknowledgments

This article was solicited by the Journal to resolve the differences in the formulas developed in Refs. 3, 4, and 5. It was supported by National Institute of Diabetes and Digestive and Kidney Diseases Grant DK-45160.

- Copyright © 1999 the American Physiological Society