Department of Physiology, The George Washington University School of
Medicine and Health Sciences, Washington, District of Columbia
20037
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-13C]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 14C studies are
not appropriate for 13C studies,
because the method used to detect
14C, decay of atoms, differs from
13C 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
 |
INTRODUCTION |
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-13C]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:
Mi for
isotopomers of plasma lactate;
Mi for
isotopomers of plasma glucose. We add
"Pi" for the isotopomers of phosphoenolpyruvate (PEP),
"Gi for gluconeogenic glucose, and "Qi" for lactate from
glycolysis of plasma glucose. By this convention,
m3 represents the
fractional amount of plasma 13C-13C-13C
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
|
(1)
|
Accordingly,
the molar enrichment of lactate = m1 + 2 × m2 + 3 × m3. The
protocol used in the studies is simple: a constant [U-13C]glucose
infusion serves two purposes; it allows calculating the rate of
appearance of plasma glucose
(Ra) as rate of infusion divided
by M6. It also
provides a 13C 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-13C]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
Ra
(M6) and part
(M1 + M2 + M3) 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-13C]glucose infused
is small relative to other fluxes, so that correction of
Ra 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 13C enters the TCA cycle via
13C acetyl-CoA or
13CO2
fixation, gluconeogenesis will be overestimated. An exception occurs if
the data are corrected for
13CO2
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
m3, and the rate
of pyruvate carboxylation relative to the TCA cycle is substantial,
resulting in significant M3 and
M2 glucose.

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Fig. 1.
Gluconeogenesis from
[U-13C]glucose
infusion (     ) and cycling at steady state. Glucose
enters the plasma compartment from three sources:
[U-13C]glucose
infusion, glycogenolysis, and gluconeogenesis. Fractional abundances of
each isotopomer are indicated to right
of symbol. Isotopomers of lactate derived from plasma glucose
(compartment Qi) are as
predicted from plasma glucose.
Mi, isotopomers
of plasma glucose. Pi,
isotopomers of phosphoenolpyruvate
(PEP). As lactate travels back to glucose, the tricarboxylic acid (TCA)
cycle reduces the number of labeled carbons per molecule but retains
the number of labeled molecules. Gluconeogenic glucose
(Gi) isotopomers are formed, as
indicated in text, by use of binomial probability. For example,
(G1 + G2 + G3) = 2 × P0 × (P1 + P2 + P3).
Other refers to formation of glucose
from 2 labeled PEP precursors (see Eq. 10). When values here are used, fractional
gluconeogenesis = 15/30 = 0.5.
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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-13C]glucose
infusion, and consider the synthesis to begin with a population of
isotopomers of lactate that are all either
m0 or m3. Note that
m0 and
m3 represent
fractional enrichment values, so that
m0 + m3 = 1. Thus
m0 and
m3 equal the
probability that a randomly selected molecule from the gluconeogenic
lactate pool is labeled, = m3, or unlabeled, = m0. 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
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(2)
|
Accordingly,
the probability of each isotopomer of gluconeogenic glucose
(Gi) is
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(3)
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(4)
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(5)
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These
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
G3 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 M0
and tracer
[U-13C]glucose
infusion will contribute to
M6, leaving
M3 as the glucose isotopomer supplied only by gluconeogenesis. The fractional
contribution of gluconeogenesis to the plasma glucose is
|
(6)
|
This equation simply states that
M3 glucose will
be directly proportional to the fraction of glucose derived from the
condensation of an
m0 and
m3 lactate. An
alternative statement of this relationship based on probability
is
|
(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 m3,
enters the TCA cycle as oxaloacetate via intrahepatic pyruvate, and P3,
P2,
P1, or even
P0 PEP emerges. However,
P0 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
P0 (see p. E481 of Ref. 5). They
also indicate that much of the carbon loss observed is due to
conversion of m3
lactate to M2
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,
(m1 + m2 + m3) or (9/90) = (P1 + P2 + P3) 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,
(m1 + m2 + m3) is used in
place of the equivalent (P1 + P2 + P3). 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
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(7)
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Again,
newly synthesized glucose will have the following distribution
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(8)
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(9)
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(10)
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G1 + G2 + G3 will appear in glucose only
as a result of gluconeogenesis, so that
or
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(11)
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The substitution of lactate for the true gluconeogenic
precursor, PEP, in Eq. 11 is valid
provided (m1 + m2 + m3) equals
(P1 + P2 + P3). This requirement is related
to assumptions 4-7 above. If
13C enters the TCA cycle via
13C acetyl-CoA or
13CO2
fixation (assumption 4),
gluconeogenesis will be overestimated. Overestimation results because
(m1+
m2 + m3)
m0 will be less
than (P1+
P2 + P3)
P0, 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
(m1+ m2 + m3)
m0 will be greater than (P1+ P2 + P3)
P0.

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Fig. 2.
Effect of omitting
m0 from equation
for fractional gluconeogenesis is an underestimate. The fraction of the
correct value resulting from ignoring
m0 term in
denominator is plotted.
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COMPARING PUBLISHED MODELS |
The binomial expansion equation (Eq. 11) is compared with the published equations (Table
1). Equation 11 very nearly equals Landau's equation, differing only in the presence of the term m0, representing
the fractional contribution of unlabeled lactate. This term is part of
the binomial expansion. However, for these in vivo experiments where
the [U-13C]glucose
infusion is a small fraction of
Ra,
m0 approaches 1. We include the m0
term to be mathematically correct and to remind investigators that it
might be significantly less than 1 in some situations. The effect of
omitting m0 from
the equation is independent of the value for fractional gluconeogenesis
and varies linearly with
m0 (Fig. 2). In
Landau's experiments
m0 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
mo (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
mo 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
M1,
M2, and
M3 are unlabeled and are not derived from
[U-13C]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
m0 and one-half
m3, m0
m3 and
m3
m0. 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
14C studies. The difficulty with
analyzing 13C studies as analogous
to 14C 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
13C atoms within a molecule if the
number of 13C-labeled molecules is
conserved. The fractional abundances for molecules with 1-3
13C 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
13C-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 13C detected by
mass spectrometry is different from
14C detected by liquid
scintillation counting. Starting with
[U-14C]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
14C detected by liquid
scintillation counting but not for
13C 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 14C
and 13C. The binomial probability
equation (Eq. 11) underlies
14C calculations as well. Working
with 14C 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 14C 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
14C. This is the basis of the
elaborate 14C equations developed
by Katz (1) and Kelleher (2) in the 1980s to calculate dilution
factors. The calculations for 13C
mass isotopomers are less complicated than for
14C 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 14C and
13C 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-13C]glucose with
[U-14C]glucose in the
example shown as Fig. 1. GC-MS could be used to detect
14C-labeled glucose and lactate
molecules, just as with the 13C
experiments. The isotopomer would occur as
M6 and
M12, 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 14C-labeled carbon
atoms within labeled molecules. Alternatively, consider a
13C study isolating glucose and
lactate and combusting the molecules to
CO2. Isotope ratio MS (IRMS) of
CO2 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
m1,
m2 and m3. 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
13C study is repeated with
14C 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-13C]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.
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.
Address for correspondence and reprint requests: J. K. Kelleher, Dept.
of Physiology, George Washington Univ. Medical Center, 2300 Eye St. NW,
Washington, DC 20037 (E-mail: phyjkk{at}gwumc.edu).