The Ottawa Hospital and, University of Ottawa, Ottawa, Ontario, Canada K1Y 4E9, Wai-Nang Paul Lee, Department of Pediatrics, Harbor-University of California, Los Angeles Medical Center, Torrance, California 90502
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
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REFERENCES
and
(1)
As 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
M3/m3 = 1. However, by definition,
M3 is the
fraction of glucose molecules of mass M + 3 and
m3 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,
m3, of lactate
relative to that
(M3) of
glucose. Therefore,
M3/m3 = 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
(2)
The 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
m3. 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
(3)
Because
each type of labeled molecule
(m1,m2,m3)
will be diluted to exactly the same extent by unlabeled molecules,
under these circumstances
(3`)
and
therefore
Equation 3' will then reduce to Eq. 2. If condition 2 holds, m1 = m2 = 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
m3 >>
(m2+m1).
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 {m1,m2,m3}, 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 m3 as a reference point. Equation 3 assumes that all of the {M1,M2,M3} arose from m3 lactate and corrects accordingly, thus overestimating fractional gluconeogenesis. Equation 2 assumes that all lactate molecules labeled as (m3,m2,m1) are cleared at the same rate, the rate at which m3 lactate is removed. That this is not true can be seen by considering simply the equilibration with fumarate. Some of the m1 molecules will become unlabeled, but all of the m3 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 m3. 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-13C]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 m0. 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 m3 is converted into m1, with the preexisting m1 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) m1 = 0.052 and m3 = 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
m3 M1 (4) or
M1+M2
(2), then, by exactly the same metabolic processes,
m2 will be
converted to M1
and M0 and
M1 will be
converted to M0.
These conversions are not taken into account in these examples, because
m2 and
m1 are conserved, whereas m3 is
not. If, for example,
m1
0.33 m1
(phosphoenolpyruvate) 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
m1 and
m2 are present in
the system, they must come either from metabolic processing of
m3 (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 m3
m2 or
m1, then, in a
parallel fashion, the same metabolic processes (e.g., TCA cycle
interactions, fumarate equilibration) will convert m2
m1 and
m1
m0. 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
m1 and
m2 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
m3 is present.
2) When the latter exchange comes
into play or a real system with equilibration among the three
isotopomers {m1,m2,m3}
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
M0 is generated
from
{m1,m2,m3}.
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FOOTNOTES |
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Address for correspondence and reprint requests: J. Radziuk, Ottawa Hospital (Civic Site), 1053 Carling Ave, Ottawa, ON, Canada K1Y 4E9 (E-mail: jradziuk{at}ottawahospital.on.ca).
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REFERENCES |
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1.
Katz, J.,
and
J. A. Tayek.
Recycling of glucose and the determination of the Cori cycle and gluconeogenesis.
Am. J. Physiol.
277 (Endocrinol. Metab. 40):
E000-E000,
1999.
2.
Kelleher, J. K.
Estimating gluconeogenesis with [U-13C]glucose: molecular condensation requires a molecular approach.
Am. J. Physiol.
277 (Endocrinol. Metab. 40):
E000-E000,
1999.
3.
Landau, B. R.
Limitations in the use of [U-13C6]glucose to estimate gluconeogenesis: a commentary.
Am. J. Physiol.
277 (Endocrinol. Metab. 40):
E000-E000,
1999.
4.
Landau, B. R.,
J. Wahren,
K. Ekberg,
S. F. Previs,
D. Yang,
and
H. Brunengraber.
Limitations in estimating gluconeogenesis and Cori cycling from mass isotopomer distributions using [U-13C6]glucose.
Am. J. Physiol.
274 (Endocrinol. Metab. 37):
E954-E961,
1998
5.
Radziuk, J.,
and
W.-N. P. Lee.
Measurement of gluconeogenesis and mass isotopomer analysis based on [U-13C]glucose.
Am. J. Physiol.
277 (Endocrinol. Metab. 40):
E199-E207,
1999
6.
Tayek, J. A.,
and
J. Katz.
Glucose production, recycling, and gluconeogenesis in normals and diabetics: a mass isotopomer [U-13C]glucose study.
Am. J. Physiol.
270 (Endocrinol. Metab. 33):
E709-E717,
1996