SPECIAL COMMUNICATION
Improved estimation of
anaplerosis in heart using 13C
NMR
David M.
Cohen and
Richard N.
Bergman
Metabolic Research Unit, Department of Physiology and
Biophysics, University of Southern California School of Medicine,
Los Angeles, California 90033
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ABSTRACT |
Anaplerotic
enzymes, such as pyruvate carboxylase or malic enzyme, catalyze
reactions that fill up the pools of the citric acid cycle (CAC),
thereby increasing the total mass of CAC intermediates. Relative
anaplerosis (y) denotes the ratio of
anaplerotic flux to the flux catalyzed by citrate synthase. We examine
conventional methods [C. R. Malloy, A. D. Sherry, and F. M. H. Jeffrey. J. Biol. Chem. 263:6964-6971, 1988; C. R. Malloy, A. D. Sherry, and F. M. H. Jeffrey. Am. J. Physiol. 259 (Heart Circ. Physiol. 28): H987-H995,
1990] of measurement of y using
13C-labeled precursors and
analysis of
[13C]glutamate
labeling by nuclear magnetic resonance (NMR) spectroscopy. Through
mathematical analysis and computer simulation, we show that isotopic
enrichment of the pool of pyruvate that is substrate for anaplerosis
will severely decrease the accuracy of estimates of
y made with conventional methods no
matter how small the mass of the pool of pyruvate. Suppose that the
recycling parameter R denotes the
fraction of molecules of pyruvate that contain carbons derived from
intermediates of the CAC. Each means of estimation of relative
anaplerosis in the peer-reviewed literature assumes that
R = 0, although this assumption has
not been confirmed by experiment. We show that conventional formulas,
using either fractional enrichments of carbons or isotopomer analysis,
actually estimate at most
y · (1
R) instead of
y during administration of
[2-13C]acetate and
unlabeled pyruvate. Using a new formula for estimation of
y, we recalculate values of
y from the literature and find them
~50% too low. We assume that all anaplerosis is via pyruvate and
that the difference in isotopic enrichment between cytosolic and
mitochondrial malate is negligible.
Krebs cycle; citric acid cycle; tricarboxylic acid cycle; nuclear
magnetic resonance spectroscopy; pyruvate; metabolism; isotopomer
analysis; malate-aspartate shuttle
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INTRODUCTION |
THE CITRIC ACID CYCLE (CAC) is the common pathway for
oxidative metabolism of carbohydrate, protein, and fat. Maintenance of
concentrations of intermediary substrates within the cycle is
accomplished by means of reactions that do not belong to the CAC, e.g.,
reactions catalyzed by pyruvate carboxylase and by malic enzyme.
Chemical reactions that add to (or remove from) the mass of pools of
metabolic intermediates of the CAC are termed anaplerotic (or
cataplerotic, respectively). The functional significance of changes in
anaplerosis and cataplerosis is unclear. Anaplerotic reactions may
contribute to the maintenance of contractile function of the heart
under normal conditions (1, 24) or during ischemia (10, 15, 33). Even
under conditions in which the rates of anaplerosis and cataplerosis are
nearly equal, resulting in little net flux, these reactions can still
contribute to cellular regulation. For example, Newsholme and Stanley
(19) presented evidence that so-called substrate cycles increase the
sensitivity of net pathway flux to small changes in concentrations of
metabolic substrates. In addition, it has been suggested that
transitory increases in anaplerosis may increase the production of
energy by the CAC by increasing the concentrations of substrates of
enzymes that are operating near their Michaelis-Menten kinetic values
(21).
In this report, we examine the assumptions of formulas for relative
anaplerosis and demonstrate that conventional formulas, using
isotopomer analysis (17) or fractional enrichments of carbons of
glutamate (16), may underestimate its
value.1 We show that ignoring
the isotopic enrichment of intracellular pyruvate during administration
of [3-13C]pyruvate or
[1-13C]glucose will
cause underestimation of anaplerosis. Even if exogenous pyruvate is not
isotopically labeled, it is possible that increases in the enrichment
of mitochondrial oxaloacetate and malate (e.g., during metabolism of
[2-13C]acetate) will
enhance the fractional enrichment of pyruvate via cataplerotic fluxes.
We propose a new formula for estimation of relative anaplerosis during
administration of isotopically labeled substrate and demonstrate that
it is more accurate than currently available formulations.
Glossary
Fractional enrichment
of C-i: number of molecules
isotopically labeled at C-i divided by
number of molecules in the entire metabolic pool
a1, a2:
fractional enrichment of C-1 and C-2 of the acetyl moiety of acetyl-CoA
(where carbon C-2 is the methyl carbon)
g1, g2, g3, g4,
g5: fractional enrichment of carbons C-1, C-2, C-3, C-4, and C-5 of
glutamate
o1, o2, o3, o4: fractional enrichment of
carbons C-1, C-2, C-3, and C-4 of oxaloacetate
p1, p2,
p3: fractional enrichment of carbons C-1, C-2, and C-3 of
pyruvate
p4: fractional enrichment of the pool of
carbon dioxide that condenses with pyruvate in the anaplerotic
reactions catalyzed by pyruvate carboxylase and by malic enzyme
r: "reverse" flux catalyzed by
fumarase
y: ratio of anaplerotic flux to flux
catalyzed by citrate synthase
vTCA:
rate of flux catalyzed by
-ketoglutarate dehydrogenase
complex
R: recycling
parameter
PDH: pyruvate dehydrogenase complex
OAA: oxaloacetate
FE: fractional enrichment
These symbols do not refer to symbols used by other
investigators in describing metabolic fluxes, fractional enrichments, or recycling of metabolic intermediates.
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METHODS |
First we describe the model of the myocardial CAC and anaplerosis from
pyruvate (Fig. 1). Next we describe in
general terms our technique for deriving algebraic formulas related to
flux rates at isotopic and metabolic steady state. Details of all
derivations are given in APPENDIX A.
Finally, we describe the methods by which we tested the accuracy of
proposed formulas for relative anaplerosis
y, the ratio of anaplerotic flux to
the flux catalyzed by citrate synthase.

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Fig. 1.
Schematic of citric acid cycle and associated reactions in myocytes,
showing transfer of carbons. Chemical fluxes are denoted by
letters in boxes: a, citrate synthase;
b, aconitase and isocitrate
dehydrogenase; c, -ketoglutarate
dehydrogenase complex and succinyl-CoA synthetase;
d, fumarase;
e, fumarase (reverse);
f, aspartate aminotransferase;
g, aspartate aminotransferase
(reverse); h, malic enzyme and
pyruvate carboxylase (cataplerosis);
i, malic enzyme and pyruvate
carboxylase (reverse, anaplerosis); and
j, pyruvate dehydrogenase complex.
Fumarate and succinate are combined into 1 pool, as are malate and OAA.
Reversible fluxes (catalyzed by fumarase, aspartate aminotransferase,
malic enzyme, and pyruvate carboxylase) have been separated into
"forward" and "reverse" fluxes. Rate of reverse flux of
fumarase (e) equals
r in Fig. 6 and in
Eqs. A7-A14. Chemical structures have
been drawn so that canonical numbering of carbons proceeds from
top to bottom (top carbon is C-1) in each
compound. To facilitate tracing the fate of individual carbons, carbons
of OAA are numbered 1-4, and carbons of the acetyl moiety of
acetyl-CoA are labeled a and
b. On compounds other than OAA and
acetyl-CoA, numerals and letters represent the locations to which the
corresponding carbons of OAA and acetyl-CoA, respectively, are
transferred by actions of enzymes of citric acid cycle.
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Biochemical Reactions and Flux Rates
The model of the CAC of myocytes used in our investigations (Fig. 1) is
a slight modification of the model used by Chance et al. (3) for a
Langendorff perfusion of the heart of fed rats. Malate and OAA are
combined into a single pool because of the extremely small content of
the latter compound (<0.7% of the malate pool) (3). We used flux
rates and pool sizes of CAC intermediates estimated by Chance et al.
(3) during administration of 5.0 mM glucose and 5.0 mM acetate (see
Fig. 1 and APPENDIX A). We
simulated the effect of
[2-13C]acetate
or [3-13C]pyruvate on
the labeling of metabolites of the CAC. In order to change as few
parameters as possible during simulated perfusion with
[3-13C]pyruvate, the
pool sizes were maintained at the same levels as during simulated
perfusion with
[2-13C]acetate.2
We did not simulate alanine aminotransferase because alanine is
believed to be at equilibrium with pyruvate (12, 13, 22) and its
omission will not change the enrichment of pyruvate or other
metabolites at steady state. The rate of flux catalyzed by the pyruvate
dehydrogenase complex was set to approximately the value estimated (3)
during administration of glucose and pyruvate, and to 10% of that
value during acetate administration.
In the current model (unlike our earlier model described in Refs. 5 and
7), we include the reversibility of the chemical flux catalyzed by
fumarase (reaction e, Fig. 1). Indeed,
using radioisotopes, Nuutinen et al. (20) estimated that the ratio of
the flux (r) from malate to fumarate
to the flux
(vTCA)
catalyzed by
-ketoglutarate dehydrogenase complex in the perfused
rat heart was 4.13 (r/vTCA = 4.13). Except where noted, simulations presented in this report were
run with the
r/vTCA
value equal to 10, representing a compromise between the value measured
in perfused rat heart and the infinitely large value assumed in the
classic model (so-called "instant randomization of carbons of
OAA") (35).3 In simulations
with different values of relative anaplerosis (increasing
y from 0.1 to 1 to 10), the value of
r remains the same to simulate
activation of the anaplerotic enzymes rather than increased
concentrations of substrates, which would affect the value of
r as well as the value of
y.
We assume that anaplerotic fluxes are catalyzed predominantly by malic
enzyme and, to a lesser extent, by pyruvate carboxylase (28). In
perfused rat heart, with glucose as the sole energy source, anaplerosis
is almost entirely (>93%) via production of malate and/or
OAA, with significantly less (<8%) via metabolism of propionate
formed from oxidation of endogenous amino acids (21). We assume that
cataplerotic reactions use malate or OAA as substrate and are catalyzed
by the same enzymes as the anaplerotic reactions (Fig.
2). At metabolic steady state, the masses
of individual pools of intermediates of the CAC are not changing, and
therefore the rate of anaplerosis equals cataplerosis
(vANA = vCATA).

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Fig. 2.
Schematic of fluxes into and out of acetyl-CoA (AcCoA) and pool of
pyruvate that is substrate for anaplerotic reactions. Box labeled
"pyruvate" includes those pools of metabolites (e.g., alanine)
having the same fractional enrichment of corresponding carbons as
pyruvate at isotopic steady state. At metabolic steady state
anaplerosis equals cataplerosis
(vCATA = vANA). Not
shown: chemical fluxes between malate and fumarate, between OAA and
citrate, and between OAA and aspartate. Fluxes:
pyrInflux, influx of pyruvate from
precursor pool; pyrEfflux, loss of
pyruvate via other pathways; pDH, flux catalyzed by pyruvate
dehydrogenase complex;
vANA, anaplerotic
flux from pyruvate;
vCATA,
cataplerotic flux into pyruvate; acInflux, influx into acetyl-CoA
(e.g., from -oxidation of fatty acids); acEfflux, efflux from
acetyl-CoA (e.g., catalyzed by carnitine acetyltransferase). Recycling
parameter R is fraction of pool of
pyruvate that is substrate for anaplerosis containing carbons exported
from citric acid cycle (via cataplerosis). In this diagram, formula for
R is
R = vCATA/(vCATA + pyrInflux).
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Model of pyruvate.
To invoke a minimal number of hypotheses in our demonstration of the
effects of recycling of pyruvate on the accuracy of formulas for
estimation of relative anaplerosis, we have adopted a simple model of
pyruvate production and disposal in myocytes (Fig. 2). For the purpose
of evaluating alternative algebraic formulas (see Table
1), we hypothesize a single pool of
"pyruvate" that is substrate for anaplerosis and in which we
include those metabolic pools the carbons of which have the same
steady-state fractional enrichment as pyruvate, e.g., alanine and
lactate (see Fig. 2).4
Cataplerotic reactions add molecules to this pool of pyruvate (see
Eqs. A1-A15). We consider an influx
("pyrInflux") of pyruvate (e.g., from exogenous pyruvate or from
phosphoenolpyruvate via pyruvate
kinase) and an efflux ("pyrEfflux") out of this particular pool
of pyruvate. Effects observed in this simple model can be created in
more complicated models as well (e.g., Refs. 13 and 22).
It is convenient to have a parameter that indicates whether the
pyruvate pool is more or less composed of newly arrived molecules (from
pyrInflux in Fig. 2) or molecules that came from the CAC (via cataplerosis,
vCATA). We
define the recycling parameter R as
the fraction of substrate for anaplerosis that contains one or more
carbons exported via cataplerosis. This definition of R does not depend on a particular
model of pyruvate metabolism. For the purposes of simulating a specific
model of metabolism with a single anaplerotic pool and ignoring
mitochondrial/cytosolic compartmentation (for the present), the formula
for R at metabolic steady state is
R = vCATA/( pyrInflux + vCATA).
Suppose pyruvate and
[2-13C]acetate are
provided exogenously. In the extreme case of zero influx of pyruvate
from external sources (R = 1), at
isotopic steady state the enrichment of pyruvate would equal that of
OAA, and flux catalyzed by pyruvate dehydrogenase complex would equal
zero (see Fig. 2). As we show in APPENDIX A (see Eq. A28),
anaplerotic flux can become considerably labeled because (for
example) molecules of OAA doubly labeled at C-2 and C-3 are produced by
the chemical reactions of the CAC. In the other extreme, if the influx
into pyruvate is extremely large compared with
vCATA
(R
0), then the dilution of the
pyruvate pool by unlabeled molecules of pyruvate would render it
effectively unlabeled.
Our investigation of the effect of "recycling" of the anaplerotic
pool of pyruvate is necessarily speculative, because the disposition of
pyruvate into kinetically discernible metabolic pools is not well
understood.5 We illustrate the
effects (see RESULTS) that partial
sequestration of intracellular pyruvate and consequent recycling of
labeled carbons would have on the accuracy of classic formulas for
estimation of relative anaplerosis (y). It is
important to note that our formulas (see Eqs.
A1-A15) that do not involve the recycling parameter R do not depend on a particular model
of intracellular pyruvate compartmentation but solely on the fractional
enrichment of C-2 and C-3 of the pool of pyruvate that is substrate for
anaplerosis.
Derivation of Algebraic Formulas
We begin with the metabolic pathways shown in Figs. 1 and 2, from which
details of the transfer of carbons from reactants to products were
obtained (Fig. 3). To simplify the
presentation of mathematical material, we have gathered all the
derivations of formulas into APPENDIX
A. Next we illustrate our method for deriving formulas
relating rates of metabolic flux to fractional enrichments of carbons
of pools of metabolic intermediates by use of the principle of
conservation of mass. For example, assume that compounds A and B are
converted to compound C at the rate of
vA and
vB moles of C per
second, respectively (Fig. 4). Let vC equal the rate
of conversion of C to other compounds, which might even include A or B. If no compounds other than A or B are converted to C, then at metabolic
steady state we can equate the rate of production of C with the rate of
disposal of C by means of the equation
vA + vB = vC.

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Fig. 3.
Illustration of transfer of carbons within pools of citric acid cycle
and between those pools and aspartate, pyruvate, and glutamate. Carbon
positions C-i within molecules are
denoted by the following symbols:
AcCoA-i, acetyl-CoA;
-KGi, -ketoglutarate;
citi, citrate;
succi, succinate;
pyri, pyruvate;
aspi, aspartate;
glui, glutamate;
OAAi, OAA. As in Fig. 1, fumarate and
succinate are combined into a single pool, as are malate and OAA.
Carbon exchange catalyzed by aspartate aminotransferase between C-4
(C-5) of -ketoglutarate and C-4 (C-5) of glutamate is not shown.
Alanine aminotransferase does not alter fractional enrichments of
carbons of glutamate or -KG and is therefore not included.
vTCA, Rate of
flux through citric acid cycle;
vANA, rate of
anaplerotic flux;
vTA, rate of flux
catalyzed by aspartate aminotransferase.
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Fig. 4.
Illustration of method for deriving equations for flux rates (see
text). Metabolic intermediates A and B are independently converted to
compound C at rates
vA and
vB (mol/min),
respectively. The sum of rates of removal of compound C is
vC (mol/min),
including chemical reactions and transport of molecules out of this
metabolic pool of C. It is assumed that no other chemical intermediates
(besides A and B) are converted directly to C.
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Suppose that examination of the chemical reactions shows that carbon
position C-1 of A and carbon position C-2 of B are transferred to
carbon position C-3 of compound C. (Note that A and B compete for the
same carbon position in compound C.) Let a source of labeled carbon
(i.e., 13C) be introduced,
resulting in the isotopic labeling of compounds A, B, and C. At
metabolic and isotopic steady state, compounds A, B, and C will
maintain constant concentrations and constant fractional enrichments of
individual carbons. Let the fractional enrichments of C-1 of A, C-2 of
B, and C-3 of C be denoted fA, fB, and
fC, respectively. Then the rate of
conversion of labeled C-1 of A to C-3 of C equals
fA · vA,
and the rate of conversion of labeled C-2 of B to C-3 of C equals
fB · vB.
Equating the rate of production of labeled C-3 of C with the rate of
removal of label from C-3 of compound C, we obtain the following
equation
All
of the formulas in APPENDIX A are
derived with this principle.
In the formulas presented below, it is assumed that each variable
representing fractional enrichment of a compound refers to a single,
well-mixed pool and excludes any molecules that are metabolically
inactive (i.e., that do not exchange carbons with constituent metabolic
pools of the CAC). In practice, this assumption may not hold. In
APPENDIX B, we derive formulas to
correct for a pool of glutamate that is metabolically inactive (we
assume all acetyl-CoA is metabolically active).
Testing Algebraic Formulas
We tested the accuracy of formulas for relative anaplerosis (Table 1)
by simulating the CAC in two ways:
1) as a system of ordinary
differential equations to be solved by use of MLAB (2) and
2) by means of a computer simulation
of a syntactic model (5). The dependent variables in the differential
equations written for MLAB (see APPENDIX
A) were the fractional enrichments of individual
carbons of the metabolic intermediates (Fig. 3). Because several of the
formulas to be tested rely on knowledge of positional isotopomers, we
also employed a computer simulation of a syntactic model of these
reactions. (A positional isotopomer of a compound is an isomer that is
determined by the positions of isotopes within the compound, e.g.,
[1,2-13C]glutamate and
[1,3-13C]glutamate are
positional isotopomers of glutamate.) Use of MLAB to obtain the
abundance of positional isotopomers of glutamate would have required
solving more than 180 ordinary differential equations simultaneously.
We therefore did not use MLAB to test the accuracy of
Eqs. A22, A25, or A26 but used a computer simulation of
a syntactic model instead. From these simulations, we were able to
independently confirm the results obtained with differential equations.
The syntactic approach provides a convenient means of predicting the
time-dependent changes in concentrations of positional isotopomers of
metabolic intermediates (5-7). It is a stochastic simulation of
chemical kinetics that relies on a rule-based description of the
transfer of atoms among chemical intermediates of metabolic pathways
(5, 7). Instead of describing the changes over time with differential
equations, we specify the transfer of carbons from reactants to
products by use of syntactic rules. The time course of changes in the
simulation is determined by a Monte Carlo simulation algorithm (6).
Flux rates are determined at any time by the product of concentrations
of substrates and a first-order rate constant. Because of its
stochastic nature, the pool sizes fluctuate somewhat around the
steady-state levels, just as one would expect in the cell. Because the
molecules in each metabolic pool are simulated as if a miniature
replica of the CAC existed inside the computer, one can examine the
concentration of positional isotopomers of any pool at any time. This
allows for the testing of mathematical relationships among flux rates,
pool sizes, isotopomer abundances, and fractional enrichments.
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RESULTS |
Accuracy of Eq. A18
As predicted from theoretical analysis of its relative error with the
assumption of instant equilibration of OAA and fumarate (see
APPENDIX A, Eq. A32), Eq. A18
consistently underestimates the true value of relative anaplerosis
y (Tables
2, 3, and
4). During administration of
[2-13C]acetate, the
accuracy in Eq. A18 increases with
increasing dilution of pyruvate by molecules of exogenous
pyruvate (i.e., decreasing R). In
fact, computer simulation (Tables 2 and 3) and mathematical analysis
(Eq. A20) indicate that the value
obtained from Eq. A18 equals (1
R) · y,
instead of y, during administration of
[2-13C]acetate and
unlabeled pyruvate.
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Table 2.
Estimates of relative anaplerosis (y) and values of fractional
enrichments of carbons C-i of glutamate and pyruvate for different
values of y and of recycling parameter R
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Table 3.
Estimation of y by use of syntactic simulation (5) to obtain
steady-state fractional enrichments of carbons and of positional
isotopomers
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Table 4.
Estimates of y and values of fractional enrichments of carbons C-i of
glutamate and pyruvate for different values of y and R
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If the exogenously administered substrate was
[3-13C]pyruvate,
estimates of relative anaplerosis by use of Eq. A18 gave values indistinguishable from zero (slightly
negative) for all values of y and all
values of recycling parameter R (Table
2). Simulations in which exogenous pyruvate was 50% unlabeled and 50%
[3-13C]pyruvate gave
exactly the same estimates for y by
use of Eq. A18, but the fractional
enrichments of each carbon of glutamate and pyruvate were reduced by
50% (data not shown). Increased enrichment of acetyl-CoA from the
decreased rate of acetyl-CoA turnover improved the estimate of
y by use of Eq. A18; the estimated value of
y increased to the range of positive
values (Table 4). The increased accuracy (underestimation by 61, 81, and 98% for y = 0.1, 1, and 10, respectively, and R = 0.1) can be
explained by the increase in enrichment of carbons of glutamate
relative to pyruvate (see Eq. A32).
The value of y estimated by
Eq. A18 will be negative whenever the
fractional enrichment of C-2 of glutamate exceeds the fractional
enrichment of C-4 of glutamate, or, equivalently, whenever the sum of
fractional enrichments of C-2 and C-3 of pyruvate (i.e., p2 + p3)
exceeds that of glutamate C-2 plus C-3 (i.e., g2 + g3). The accuracy
of Eq. A18 increases as (p2 + p3)/(g2 + g3) decreases (see Eq. A32).
Accuracy of Eq. A19
Equation A19, a formula for measuring
relative anaplerosis from fractional enrichments of glutamate C-1 and
C-3 (Table 1), underestimates the value of
y. As the dilution of the pyruvate pool by exogenous pyruvate increases (i.e., decreasing
R), the ratio of fractional
enrichments of C-1 of pyruvate to C-1 of glutamate decreases (Table 2),
thereby decreasing the magnitude of the relative error in
y incurred by use of
Eq. A19 (see Eq. A33).
For administration of
[2-13C]acetate, the
accuracy of Eq. A19 improves with
decreasing recycling R (Tables 2 and
3). During administration of exogenous
[3-13C]pyruvate at
R = 0.1, Eq. A19 underestimated relative anaplerosis (y) by 68% when
y = 0.1, 40% when
y = 1, and by 70% when
y = 10 (Tables 2 and 3). At values of
recycling parameter R of 0.5 or 0.9, Eq. A19 underestimated
y by ~65%. Computer simulations in which exogenous pyruvate was 50% unlabeled and 50%
[3-13C]pyruvate gave
exactly the same estimates of y by use
of Eq. A19, but the fractional
enrichment of each carbon of glutamate and pyruvate was reduced by 50%
(data not shown). Increasing the enrichment of acetyl-CoA by decreasing
its turnover rate decreased the accuracy of Eq. A19 for y = 0.1 but
had no effect for higher values of y (Table 4).
Accuracy of Eq. A10
We propose a new formula (Eq. A10)
for estimation of relative anaplerosis. Equation A10 is completely accurate when the CAC is simulated by
solving a system of differential equations numerically during
administration of either
[2-13C]acetate or
[3-13C]pyruvate (Table
2). Computer simulations in which exogenous pyruvate was 50% unlabeled
and 50%
[3-13C]pyruvate gave
exactly the same estimates of y by use
of Eq. A19, but the fractional
enrichment of each carbon of glutamate and pyruvate was reduced by 50%
(data not shown). During administration of
[3-13C]pyruvate,
increasing the enrichment of acetyl-CoA by decreasing its turnover rate
likewise resulted in zero error in estimation of
y by use of Eq. A10 (Table 4). The relative error in the estimate of
y obtained from Eq. A10 is <0.0005 under all conditions tested.
Equation A10 may be inaccurate when
applied to the output of a simulation of a syntactic model (Table 3).
If y equals ~0.1 and
[2-13C]acetate is
administered, Eq. A10 overestimates
the true value by 17% for R = 0.5 and
underestimates the true value by 3% for R = 0.1 (Table 3). If
y equals 0.1 and
[3-13C]pyruvate is
administered, then Eq. A10 estimates
y as 0.21 (overestimation by 114%)
when R = 0.1. When the value of
relative anaplerosis is ~1.0, the estimate is within 10% of the
correct value for all values of R and
with application of either
[2-13C]acetate or
[3-13C]pyruvate (Table
3). If the relative anaplerosis is increased to ~10.0, the magnitude
of the relative error is <10% during administration of
[2-13C]acetate but
increases to 25% (40%) during administration of [3-13C]pyruvate with
R equal to 52% (90%), respectively.
Nevertheless, with a single exception (y = 0.1, R = 0.1, [3-13C]pyruvate),
Eq. A10 exhibits better accuracy than
Eqs. A19 or A18 for all cases simulated with the
syntactic model (Table 3).
The source of inaccuracy of Eq. A10 is
as follows. As the denominator in this formula for
y approaches zero (see fractional enrichments in Table 2), experimental error and noise decrease the
precision of the estimate. The simulation of the syntactic model adds
noise to the system both by reason of the stochasticity of the
algorithm and because of the finite size of the simulated metabolic
pools (5, 6). For larger R (less
dilution by exogenous pyruvate), the difference in enrichment of C-2
(C-3) of pyruvate and C-3 (C-2) of glutamate approaches zero, leading
to a numerically unstable calculation (division by a very small
number).
Accuracy of Eqs. A22, A25, and
A26
Malloy et al. (16, 17) derived formulas to estimate relative
anaplerosis from the multiplets of the
13C NMR spectrum of glutamate. We
selected one of their formulas (Eq. A22) for the triplet resonance (C3T) of C-3 of
glutamate and compared its accuracy with the classical formulas and
with our formulas for anaplerosis. To evaluate the accuracy of formulas that rely on C3T, we simulated each of the conditions in Table 2, using
a syntactic model (5, 6) of the CAC and anaplerosis (see Table 3).
During administration of either
[2-13C]acetate or
[3-13C]pyruvate,
Eq. A22 underestimated the value of
relative anaplerosis, with the magnitude of the error increasing with
increasing recycling (R) of
pyruvate. In fact, computer simulations as well as mathematical analysis (Table 3, Eq. A26) confirm
that the value estimated by Eq. A22
equals (1
R) · y
instead of y.
Equation A25 was consistently more
accurate than Eq. A22 yet showed
considerable error when y = 0.1 and
y = 10 (Table 3). We traced the source
of the error to the numerical instability of the formula, in which the
denominator became almost zero. This can be seen by observing that the
formula for y given by
Eq. A26 is mathematically equivalent
to Eq. A25 yet estimates
y more accurately for all simulations
(Table 3). To apply Eq. A26, the
recycling parameter R is needed; we
obtained R from the simulation by
examining the rate of the influx into pyruvate from exogenous sources
and the rate of anaplerosis. We infer that if a precise measurement of
R could be made, then
Eq. A25 could be replaced by an
equation (i.e., Eq. A26) that is
mathematically equivalent but more precise. On the other hand, computer
simulations indicate that if relative anaplerosis
(y) were to be estimated independently of
R, then Eq. A26 would not be useful in estimating
R because of the extreme sensitivity
of the estimate to small errors in the measurement of g4 and/or
C3T (data not shown). Thus it is unlikely that Eq. A26 could be used to estimate
R, given the limitations of precision of physical measurements.
Effects of Malate-Aspartate Shuttle on Estimation of Relative
Anaplerosis
In a model of the metabolic fluxes across the mitochondrial membrane
(at steady state),6 it is possible to
derive the following equation for the ratio of the fractional enrichment of carbon
C-i in cytosolic malate (x3) to the fractional enrichment of
C-i in mitochondrial malate (x4) under conditions of continuous
administration of isotopically labeled acetate and pyruvate unlabeled
at C-i
(p0 = 0; see Fig. 5):
x3/x4 = [v2 · (v0+v3)]/(v0 · v2 + v0 · v3 + v2 · v3).
If we allow administration of pyruvate isotopically labeled at
C-i
(p0 > 0 in Fig. 5) as well as
arbitrarily labeled acetate, we can derive the following formula:
x3
x4 = (p0
x1) · (v0 /v2), where x1 is the fractional enrichment
of C-i of cytosolic pyruvate (Fig. 5).
From these formulas7 it is seen
that, during administration of pyruvate unlabeled at
C-i, cytosolic and mitochondrial
malate will have approximately equal fractional enrichments at steady
state (i.e., x3
x4) if and only if either pyruvate
influx into the cytosolic pool
(v0) or the
rate of the flux catalyzed by cytosolic malic enzyme
(v3) is much
slower than the rate of the malate-aspartate shuttle
(v2) (see Fig.
5): v0 <<
v2 or
v3 <<
v2.
Furthermore, even if pyruvate that is administered is isotopically
labeled at C-i, sufficiently large
flux through the malate-aspartate shuttle
(v2) compared with influx of pyruvate into the cytosolic pool
(v0) will
ensure that x3
x4 (i.e., cytosolic and mitochondrial
malate have the same fractional enrichment).

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|
Fig. 5.
Diagram of cytosolic vs. mitochondrial compartmentation of metabolic
pools germane to our study. Metabolic fluxes shown are influx into
cytosolic pyruvate pool that is substrate for anaplerosis
(v0); transport
of pyruvate across the mitochondrial membranes
(v1,
vIR);
malate-aspartate shuttle (malate- -ketoglutarate exchange,
aspartate-glutamate electrogenic exchange,
v2); cytosolic
malic enzyme (v3,
v4);
mitochondrial malic enzyme
(v5);
mitochondrial pyruvate carboxylase
(v6); efflux of
pyruvate from the cytosolic pool that is substrate for anaplerosis
(v7); cytosolic
aspartate aminotransferase
(v8);
mitochondrial aspartate aminotransferase
(v9);
propionyl-CoA carboxylase, methylmalonyl-CoA racemase,
methylmalonyl-CoA isomerase
(v10); pyruvate
dehydrogenase complex
(v11); and the
citric acid cycle
(vTCA). Flux
v10 represents
metabolism of propionate (from fatty acids) and of methionine,
isoleucine, and valine (18). For a given carbon position, fractional
enrichment of molecules produced by flux
v0 equals
p0. In deriving equations for the
fractional enrichment of carbons of chemical intermediates at metabolic
and isotopic steady state, it was assumed that
vIR 0, v3 v4,
v5 = v6 + v10, and
v8 v9. Not shown is
one-half of the malate-aspartate shuttle, flux of -ketoglutarate
from mitochondrion into cytosol (coupled to malate transport), and flux
of glutamate from cytosol into mitochondrion (coupled to aspartate
transport). The cytosolic form of
NADP+-dependent malic enzyme in
rat or rabbit heart catalyzes both anaplerotic and cataplerotic
reactions, whereas the mitochondrial form of enzyme catalyzes
cataplerosis almost exclusively (27). In addition, there is substantial
activity (in vitro) of pyruvate carboxylase in mitochondria isolated
from rat heart (28), which may be important under specific conditions,
such as increases in concentration of acetyl-CoA. Parametrization of
metabolic diagram represents a simple case in which stoichiometry of
exchange of -ketoglutarate and malate matches exchange of glutamate
and aspartate [as reported in perfused rat heart during perfusion
with glucose and insulin at metabolic steady state (25)].
|
|
 |
DISCUSSION |
Our theoretical investigations show that current formulas using ratios
of isotopic enrichments of carbons of glutamate or of isotopomers of
glutamate (16, 17) will generally underestimate relative anaplerosis
( y) in
heart.8 These formulas derive
from the pioneeringwork done by Weinman et al. (30) on the
metabolism of
[14C]acetate in
hepatocytes. They assumed that anaplerotic flux was completely
unlabeled and that differences in enrichment of mitochondrial and
cytosolic metabolic pools were inconsequential. We reasoned that if the
metabolic source of anaplerosis were isotopically labeled, these
formulas would no longer give the correct value of relative anaplerosis
(in heart). Indeed, our investigations support this conjecture. We
further observed that it might be possible for the chemical reactants
of anaplerosis (pyruvate in our examples) to become isotopically
labeled by the actions of the CAC. If this were true, then an
endogenous source of labeling of anaplerosis would amplify the problems
caused by exogenous sources (e.g.,
[3-13C]pyruvate or
[1-13C]glucose) and
might even reduce the accuracy of formulas for anaplerosis during
administration of compounds that do not directly label the source of
anaplerosis (e.g.,
[2-13C]acetate).
We demonstrated that the isotopic enrichment of pyruvate must be
included in a general formula that accurately estimates relative anaplerosis from pyruvate, even with the assumption that
cytosolic/mitochondrial compartmentation does not affect the estimate.
By mathematical analysis and computer simulation, we showed that if
[3-13C]pyruvate (or
equivalently,
[1-13C]glucose) or
[2-13C]acetate is
administered, then conventional formulas (16, 17) using the C1/C3,
C2/C4 or isotopomer ratios of glutamate (Eqs. A18,
A19, and A22)
underestimate relative anaplerosis (Table 1). We propose a formula
(Eq. A10) that is accurate during
administration of
[2-13C]acetate,
[3-13C]pyruvate,
and/or
[1-13C]glucose,
provided it is possible to measure the fractional enrichment of carbons
C-2 and C-3 of pyruvate that is substrate for anaplerotic reactions. We
recognize the difficulty in making this measurement, but the formula is
extremely important in defining the relevant variables that must be
considered in estimating the relative rate of anaplerosis. Our study
demonstrates that enrichment of the pool of pyruvate that is substrate
for anaplerosis will severely decrease the accuracy in estimates of
relative anaplerosis by use of conventional methods (16, 17), no matter
how small the mass of the pool of pyruvate.
Isotopomer distributions are more difficult to analyze than ratios of
fractional enrichment of individual carbons. The assumptions underlying
contemporary isotopomer analysis in heart (16) are 1) instant equilibration of OAA with
fumarate and 2) no reincorporation of
[13C]CO2
by anaplerotic reactions. An additional assumption is not inherent in
the equations (16) but is a simplification adopted in all analyses (16,
17), namely, 3) if molecules of
exogenous pyruvate are not doubly labeled at C-2 and C-3, then
anaplerosis from pyruvate does not contain molecules that are doubly
labeled at both carbons C-2 and C-3. However, our simulations (see
Table 3) and mathematical analysis (see
Eq. A28) demonstrate that molecules of OAA doubly labeled at C-2 and C-3 are produced by the actions of the
CAC during administration of
[2-13C]acetate or
[3-13C]pyruvate. From
published steady-state data (16) and our Eq. A29, we calculate that 65.5% (46.3%) of molecules of
OAA in the perfused rat heart are doubly labeled at C-2 and C-3 during
administration of
[2-13C]acetate (or
[3-13C]pyruvate,
respectively). Therefore, one cannot ignore anaplerosis of doubly
labeled molecules possessing 13C
at both positions C-2 and C-3, unless one assumes that recycling of
carbons of pyruvate does not occur (i.e., one assumes that R = 0). In fact, we derive a formula
(Eq. A31) for the fraction of
molecules of pyruvate doubly labeled at C-2 and C-3 at steady state
(specifically referring to the pool of pyruvate that is substrate for
anaplerosis). We show that reincorporation of labeled carbon fragments
into the CAC (cataplerosis followed by anaplerosis) can have a
significant impact on the accuracy of the formulas used to estimate
relative anaplerosis.
The accuracy of Eq. A10 (Table 1) for
estimating relative anaplerosis is not affected by recycling of carbons
between pyruvate and pools of the CAC. The assumptions underlying
Eq. A10 follow. Reactions of the CAC
and the transfer of carbons among metabolic pools are given in Figs.
1-3. Mitochondrial/cytosolic compartmentation of metabolic pools
may be ignored, i.e., corresponding pools in the mitochondria and in
the cytosol are assumed to have the same fractional enrichments. We
restrict our consideration to anaplerotic reactions that solely use
pyruvate as substrate. This excludes oxidation of fatty acids having an
odd number of carbons, because the terminal 3-carbon fragment of
-oxidation is metabolized to succinyl-CoA (18). However, metabolism
of fatty acids such as palmitate that contain an even number of carbons
is allowed. In the current analysis, we assume that there is no influx
into pools of the CAC other than that shown in Fig. 1. For example, we
do not consider the exchange of molecules of the pool of glutamate with
plasma glutamate. Further work will extend our analysis to the more
general case.
Two factors complicate measurement of the rate of anaplerosis in heart.
First, it has been observed (13) that there may exist a pool of
glutamate in heart that is metabolically inactive (i.e., whose
molecules do not exchange carbons with
-ketoglutarate that is a
constituent of the CAC). We suggest two methods for estimating the true
fractional enrichment of carbons of glutamate (i.e., excluding the
metabolically inactive pool), either use of isotopomer analysis of
glutamate or use of the fractional enrichment of C-2 of mitochondrial
acetyl-CoA and C-4 of glutamate (see APPENDIX B). This is admittedly a difficult problem for which
we do not have a complete solution at the present time. Nonetheless, it
is still meaningful to discuss theoretical results in terms of the
fractional enrichments of the carbons of glutamate or of the positional
isotopomers of glutamate. Such discussion can easily be translated into
the case in which the true enrichment of glutamate can be estimated
only imprecisely. The second difficulty in modeling the CAC is the
presence of multiple pools of pyruvate in the cytosol (13, 22).
However, there is experimental evidence that cytosolic alanine has the
same fractional enrichment as the cytosolic pool of pyruvate that feeds
the mitochondrial pyruvate (12, 13, 22, 26), which may be useful in
formulas for y that require knowledge
of the fractional enrichment of this pool of cytosolic pyruvate.
It is of interest to apply Eq. A10 to
published data on the isolated mammalian heart perfused with
[3-13C]pyruvate and to
recalculate the value of y. In the
isolated rabbit heart perfused with 2.5 mM
[3-13C]pyruvate and no
glucose (13), our formula yields 18% as the estimate of
y (after correction for a pool of
metabolically inactive glutamate) compared with the value of 10%
obtained with Eq. A18. In the isolated
rat heart perfused with 1 mM
[3-13C]pyruvate and 5 mM glucose (3), the value of y
obtained with our formula is 14% compared with the value obtained with
the classical formula (Eq. A18) of 4% (correction for
metabolically inactive glutamate pool could not be made). As predicted
from simulations (Table 2), the classical formula (Eq. A18) that uses the ratio of enrichments of C-2 to C-4
of glutamate underestimates the true value of relative anaplerosis by
~50%. In our calculations using Eq. A10, we assumed that alanine and pyruvate that is
substrate for anaplerosis are in equilibrium and that C-2 of these
compounds is unenriched (12, 13, 22). Therefore, the values obtained
with Eq. A10 in these examples are
lower bounds, which would be increased if the fractional enrichment of
C-2 of pyruvate were greater than zero. As we have mentioned, we do not
make any assumptions concerning the fractional enrichment of carbons of
lactate.
The malate-aspartate shuttle may affect estimates of relative
anaplerosis, even at isotopic steady state. Several investigators have
provided evidence that, in the perfused rat heart, the rate of the
malate-aspartate shuttle is fast enough that analysis of the dynamics
of enrichment of glutamate could be used to estimate the absolute rate
of the CAC as well as the rate of anaplerosis, without regard for
compartmentation (3, 4, 32). Other investigators have shown that
the malate-aspartate shuttle is rate limiting for transport of
-ketoglutarate and glutamate between the mitochondrion and cytosol,
thereby affecting the dynamics of labeling of glutamate (4, 34).
However, for the purposes of this paper, only the steady-state behavior
is relevant; we did not consider the dynamics of isotopic labeling.
With the assumption that the source of cytosolic glutamate is either
export from mitochondria or transamination with cytosolic
-ketoglutarate, the fractional enrichment of glutamate would be
identical in mitochondria and cytosol at steady state, and equal to the
fractional enrichment of (mitochondrial)
-ketoglutarate. Indeed,
Lewandowski et al. (14), using NMR spectroscopy, found that in extracts
from perfused rabbit hearts the fractional enrichment of C-4 of
-ketoglutarate and that of glutamate were equal.
We investigated the effect of the rate of the malate-aspartate shuttle
on formulas for estimation of relative anaplerosis which assume that
differences in fractional enrichment between cytoplasmic and
mitochondrial metabolic pools of the CAC are negligible. Analysis of a
model of the metabolic fluxes shows that if the mass flux of the
malate-aspartate shuttle is much greater than the rate of influx of
cytosolic pyruvate
(v2 >>
v0 in Fig. 5), then the difference in enrichment of cytosolic and mitochondrial pools
of malate is negligible and the formulas may be applied. Alternatively,
if the influx of pyruvate is unlabeled (and the influx of acetate is
isotopically labeled), then it suffices that the mass flux of the
malate-aspartate shuttle greatly exceed the mass flux catalyzed by
cytosolic malic enzyme
(v2 >>
v3 in Fig. 5).
Thus we have derived sufficient conditions for the application of
models of the CAC that are used to derive algebraic formulas for
estimation of relative anaplerosis.
Recommendations
In acid extracts obtained from glucose-perfused or acetate-perfused
isolated mammalian hearts, the lack of NMR resonances for pyruvate can
be explained by the low concentration of pyruvate under these
conditions (23). We suggest that the enrichment of pyruvate is
important, although perhaps difficult to measure. Instead, either
conditions that enlarge the pool of pyruvate ought to be selected (so
that an assessment of the fractional enrichment of pyruvate can be
made),9 or an indirect
estimation of the fractional enrichments of carbons of pyruvate ought
to be made, i.e., fractional enrichment of carbons of alanine (12, 13,
22, 26). Use of acetate should be discouraged under conditions that
cause the concentrations of pyruvate and alanine to become too small to
be observed with NMR spectroscopy. Administration of
[3-13C]lactate,
[3-13C]pyruvate, or
[1-13C]glucose may
allow for estimation of the fractional enrichment of carbons of
pyruvate from the NMR spectrum of alanine (13, 31). Together with the
fractional enrichment of carbons of glutamate, this may enable relative
anaplerosis (y) to be estimated with greater
accuracy (Eq. A10).
 |
APPENDIX A. MATHEMATICAL EQUATIONS AND DERIVATIONS |
Formulas for Estimating Rates of Flux at Metabolic and Isotopic
Steady State
Our mathematical analysis presumes the metabolic pathways shown in
Figs. 1 and 2 and the details concerning transfer of carbons within the
relevant pathways shown in Fig. 3. One can show (see Fig. 3) that at
isotopic steady state the following equations hold
|
(A1-A6)
|
where oi is the the
fractional enrichment of C-i of OAA;
gi is the fractional enrichment of
C-i of glutamate;
ai is the fractional enrichment of C-i
of acetyl-CoA (the carbonyl carbon is C-1); and c6 is the fractional
enrichment of the carboxyl carbon of citrate that is covalently bonded
to the carbon to which is attached to the -OH group (cit6 in Fig. 3;
same as the carbon position of citrate that receives C-1 of OAA in Fig.
1). All fractional enrichments pertain to pools of metabolites that are
metabolically active (see APPENDIX
B).
The equations describing the fractional enrichments of carbons of the
pools of the CAC are derived from examination of Fig. 6
|
(A7)
|
|
(A8)
|
|
(A9)
|
where s2 equals the fractional enrichment of carbons C-2 and
C-3 of succinate, and pi equals the
fractional enrichment of C-i of the
pool of pyruvate that is substrate for anaplerotic enzymes. Letting
y equal the ratio of
vANA to
vTCA in Fig. 2, one can derive the following from Eqs.
A1-A9
|
(A10)
|
provided
g2 + g3
p2
p3
0, and
|
(A11)
|
where
r is the rate of the reverse flux
catalyzed by fumarase (from malate to fumarate).

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Fig. 6.
Model of atomic (carbon) flow between selected pools of citric acid
cycle that is useful for illustrating derivation of
Eqs. A7-A9. As in Fig. 1, succinate
and fumarate are combined into a single compartment, as are malate and
OAA. Each box represents the set of carbons at a given position (e.g.,
C-4 glutamate) or positions (e.g., C-2 and C-3, succinate and
fumarate). vTCA,
Rate of flux catalyzed by -ketoglutarate dehydrogenase complex;
r, rate of flux from OAA to fumarate,
catalyzed by malate dehydrogenase and by fumarase;
vANA, rate of
anaplerosis (equal to rate of cataplerosis).
|
|
Examination of carbons C-1 and C-4 of succinate and OAA in Fig. 3 and
of the description of flux rates in Fig. 6 allows for the derivation of
the following equations
|
(A12)
|
|
(A13)
|
|
(A14)
|
where s1 equals the fractional enrichment of the carbons C-1
and C-4 of succinate, and p4 equals the fractional enrichment of
CO2 that is substrate for
anaplerotic reactions. From Eqs. A12-A14 and A1-A6, one
can derive the following formula involving p4 (provided that
y
0)
|
(A15)
|
Formula for Recycling Parameter R
We assume metabolic steady state, so that pool sizes are unchanging and
hence anaplerosis and cataplerosis are equal
(vTCA = vCATA). If
Qi equals the fractional enrichment of
carbon C-i of the pyruvate influx
(Fig. 2), then at isotopic steady state, for
i = 1, 2, and 3,
|
(A16)
|
where
oi and
pi are the fractional enrichments of
carbon C-i of OAA and pyruvate,
respectively. From Eq. A16 and the
definition of R (see Fig. 2), one can
derive the following formula for R (provided that oi
Qi)
|
(A17)
|
Equation A17 also holds if oi,
pi, and
Qi denote fractional enrichments of
certain positional isotopomers of OAA, intracellular pyruvate, and
exogenous pyruvate, e.g.,
[2,3-13C]oxaloacetate,
[2,3-13C]pyruvate, and
exogenous
[2,3-13C]pyruvate,
respectively (see Eq. A30 below).
Classical Formulas That Assume Instant Randomization of OAA
Suppose that molecules of OAA are instantly equilibrated with molecules
of fumarate. Then, at isotopic steady state, regardless of the isotopic
labeling of exogenous substrate, the following equalities hold: o1 = o4
and o2 = o3; and hence, by Eqs. A1-A4, g1 = c6 and g2 = g3. If one assumes that anaplerosis is unlabeled at
C-2 and C-3, so that p2 = 0 and p3 = 0, one can derive the following formula by substituting into Eqs.
A10 and A15 and
rearranging
|
(A18)
|
If
we assume that anaplerosis is unlabeled at C-1 (p1 = 0), that
reincorporation of
[13C]CO2
does not occur (p4 = 0), and that acetyl-CoA is not isotopically labeled at C-1 (g5 = 0, see Eq. A5),
then we can derive the following formula
|
(A19)
|
Equations
A18 and A19 were
originally derived for hepatic metabolism (30).
In our model of the CAC (which does not include instant equilibration),
we may derive a formula similar to Eq. A18. If we assume that molecules of exogenous pyruvate
are not labeled at either C-2 or C-3, then by examination of Fig. 3 we
may infer that g2 = g3, and from Eqs.
A10 and A17 we may
derive the following equation
|
(A20)
|
Formula for y With Positional
Isotopomer Analysis
The formula for the C-3 triplet is the following (17)
|
(A21)
|
and
therefore
|
(A22)
|
where
C3T is the fraction of molecules of glutamate that are (triply) labeled
at C-2, C-3, and C-4 divided by the fractional enrichment of glutamate
C-3
|
(A23)
|
This
fraction can be obtained by dividing the area of the triplet at C-3 of
glutamate by the area of the entire C-3 resonance of glutamate in the
13C NMR spectrum.
Using the same model of the CAC but including the effects of labeled
anaplerosis from pyruvate, we have derived the following formula (see
Eqs. A28 and A29)
|
(A24)
|
where
p23 is the mole fraction of the pool of pyruvate that is doubly labeled
at C-2 and C-3 and is substrate for anaplerotic reactions. Solving for
y, one obtains
|
(A25)
|
provided
that (g3 · C3T
g4 · p23)
0. As we show (see below) for the 1-pool model of pyruvate
metabolism (Fig. 2), in the special case that exogenously administered
pyruvate is not doubly labeled at both C-2 and C-3 (although
endogenously generated [2,3-13C]pyruvate and
[1,2,3-13C]pyruvate
are allowed), our formula simplifies to the following
|
(A26)
|
where
R is the recycling parameter for
pyruvate described above and (0
R < 1). Comparing Eq. A24 with
Eq. A21 and Eq. A26 with Eq. A22, we
conclude that, in the formulas derived from isotopomer analysis, there
are implicit assumptions that p23 = 0 and that R = 0.
Generation of Doubly Labeled OAA
We next derive a formula for o23, the fraction of molecules of OAA that
are labeled at both C-2 and C-3 at isotopic steady state. These
molecules derive from molecules of
-ketoglutarate that are
isotopically labeled at C-3 and C-4 or from molecules of pyruvate that
are doubly labeled at C-2 and C-3 (see Fig. 3). Assume instant
equilibration of OAA with malate, fumarate, and succinate. From the
balance of inflow and outflow of molecules of OAA (see Fig. 2), we
derive the following equation
|
(A27)
|
Substituting
for a2 and o2 (by Eqs. A2 and A6), the following formula is
obtained
|
(A28)
|
where
y = vANA/vTCA.
Equation A28 demonstrates that o23
g3 · g4/(1+y),
establishing a lower bound for o23.
Derivation of Eq. A24
By definition of C3T, g3 · C3T equals the fraction of
molecules of glutamate that are isotopically labeled at C-2, C-3, and C-4. The atoms in positions C-2, C-3, and C-4 of a molecule of glutamate derive from carbons C-3 and C-2 of OAA and from C-2 of
acetyl-CoA, respectively (see Fig. 3). Therefore, recalling that a2 = g4 (see Eq. A6), one may write the
following equation to describe isotopic steady state
|
(A29)
|
Solving
for y in Eqs.
A28 and A29, one
obtains Eq. A24.
Derivation of Eq. A26
If exogenously labeled pyruvate is not doubly labeled at C-2 and C-3,
then by Eq. A17
|
(A30)
|
Eliminating
o23 from Eqs. A29 and A30, and solving for p23, one obtains
|
(A31)
|
With
use of Eq. A31 to substitute for p23
in Eq. A25, one derives
Eq. A26.
Analysis of Errors in Formulas for y
We define the relative error of an estimate to be the difference
between the estimate and the actual value divided by the actual value,
i.e., relative error = (estimated value
actual value)/actual
value. The advantage of using this measure of error is that it is
independent of the actual value of the estimate, thereby allowing
comparison of estimates of y for
widely differing values of y. We
obtain the following formulas for the relative error (Rel Err) in
Eqs. A18 and A19
|
(A32)
|
|
(A33)
|
Equation A33 assumes that OAA is instantly equilibrated with
fumarate, whereas Eq. A32 compares the
accuracy of Eq. A18 with a formula
that applies to the general case of incomplete equilibration, i.e., g2
(p2) may be unequal to g3 (p3). If exogenously administered pyruvate is
not labeled at either C-2 or C-3, then pyruvate derives its label
solely from cataplerosis, and hence p2 = p3 (at isotopic steady state)
and g2 = g3, by Eq. A11. Therefore,
for this special case, and with the assumption of the 1-pool model of
pyruvate metabolism (Fig. 2), the relative error in
Eq. A18 equals
p2/g3, which
equals
R (by Eq. A17).
The formula for the relative error in Eq. A22, obtained by comparison of that equation with
Eq. A24 (which we take to be the
"correct" formula, assuming instant equilibration of OAA and
succinate) is Rel Err (Eq. A22) =
p23/o23. If exogenously administered molecules of pyruvate are
not doubly labeled at C-2 and C-3 (see discussion of
Eq. A17), then the relative error in
Eq. A22 equals
R.
Differential Equations for Fractional Enrichments of Carbons
We next present the parameters of the ordinary differential equations
that were solved using MLAB (2) for Tables 2 and 4 (see Figs.
1-3). The equations were run under conditions of metabolic steady
state until isotopic steady state was reached. The rates of influx
(efflux) of acetyl-CoA and pyruvate from their respective pools are
acInf and pyrInf (acEff, pyrEff), respectively. The rates of CAC,
anaplerosis (equal to cataplerosis), and flux catalyzed by aspartate
aminotransferase and by the pyruvate dehydrogenase (PDH) complex are
denoted by vTCA,
vANA,
vTA, and PDH,
respectively. The rate of flux from malate to fumarate catalyzed by
fumarase is denoted by r. Values for
the flux rates (in µmol · g dry
wt
1 · min
1)
are vTCA (8.28),
vTA (22.9),
vANA
(y · vTCA,
where y is the desired relative
anaplerosis), PDH (0.8 or 8.0 for administration of
[2-13C]acetate or
[3-13C]pyruvate,
respectively), and acInf (8.28 or 0.28 for Tables 2 and 4,
respectively). The remaining parameters are computed by the following
equations (see Fig. 2)
|
(A34)
|
|
(A35)
|
|
(A36)
|
|
(A37)
|
where
vANA = vCATA (with
assumption of metabolic steady state), and
R > 0 (with assumption that
vCATA > 0). If
vCATA = 0, then
the formula for pyrEff (Eq. A34)
would not be required (and in this case pyrEff is a free variable). To
ensure a feasible simulation, we must have a nonnegative value of the
rate (pyrEff) of efflux from the pyruvate pool, implying that the
following condition must hold: (1
R)/R
PDH/vANA,
which is equivalent to R
vANA/(PDH + vANA). This
upper bound for the recycling parameter R causes some values of
R to determine simulations that are
not feasible for given rates of anaplerosis and of flux catalyzed by
PDH (see Tables 2-4).
The (constant) parameters representing the concentrations of metabolic
pools are Mcit (citrate),
Makg (
-ketoglutarate),
Mfum (fumarate and succinate, combined
pools), Mmal (malate and OAA, combined
pools), Mpyr (pyruvate),
Maccoa (acetyl-CoA),
Masp (aspartate), and
Mglu (glutamate). The parameter
representing the enrichment of carbon
C-i in the influx into the pyruvate
pool is pyrCiIN, e.g., pyrC1IN, and
into the pool of acetyl-CoA is acCiIN
(e.g., acC1IN). The parameter representing the fractional enrichment of
CO2 that is substrate for
anaplerotic reactions is p4 (We let p4 = 0 in our simulations). The
values for the pool sizes (in µmol/g dry weight) are
Mcit (2.03),
Makg (0.16),
Mfum (0.32), Mmal (0.3223),
Mpyr (0.2),
Maccoa (0.02),
Masp (2.02), and
Mglu (20.87).
We let the following variables denote the fractional enrichment of
carbon C-i of citrate,
-ketoglutarate, fumarate, malate, pyruvate, acetyl-CoA, aspartate,
and glutamate, respectively: citCi,
akgCi,
fumCi,
malCi,
pyrCi,
accoaCi,
aspCi, and
gluCi. In the following equations, the
independent variable (time t) is
shown for the derivatives [e.g.,
citC1'(t)] but not for
the dependent variables (e.g., citC1). The initial value of each of these variables is zero in our simulations. These ordinary differential equations were solved using MLAB (2)
 |
APPENDIX B. CORRECTION FOR A POOL OF GLUTAMATE THAT DOES NOT
CONTRIBUTE TO MITOCHONDRIAL METABOLISM |
In the isolated, perfused rat heart metabolizing pyruvate, lower
fractional enrichment (FE) of C-4 of glutamate compared with the methyl
carbon of acetyl-CoA has been observed, indicating the existence of a
pool of glutamate that does not mix with the mitochondrial glutamate
produced from
-ketoglutarate by aminotransferase reactions
(13). For purposes of exposition, these glutamate molecules shall be
called "metabolically inactive" (with respect to exchange of
carbons with
-ketoglutarate, which is a constituent pool of the
CAC). From the mechanism of transfer of carbons in the CAC (5), it can
be shown that the FE of C-2 of acetyl-CoA (the methyl carbon, denoted
a2 in APPENDIX A) equals the FE of
C-4 of glutamate (denoted g4 in APPENDIX
A), if all acetyl-CoA molecules are available for the
reaction catalyzed by citrate synthase and all glutamate molecules are
in exchange with mitochondrial
-ketoglutarate. In other words, the
ratio factive of the mass of
metabolically active glutamate to the mass of total glutamate
(metabolically active as well as inactive glutamate) equals
g4obs/a2, where a2 equals the true
FE of C-2 of acetyl-CoA and g4obs
equals the observed FE of C-4 of glutamate (ratio of moles of glutamate
possessing 13C at carbon C-4
divided by moles of total glutamate, including metabolically inactive
molecules). It is assumed the FE of carbons of acetyl-CoA can be
correctly determined. To obtain the actual fractional enrichment of
carbon positions or of positional isotopomers of glutamate, multiply
the observed FE by 1/factive.
Alternatively, provided there is sufficient enrichment of acetyl-CoA,
conventional isotopomer analysis (16) can be used to obtain
1/factive, the correction factor
for converting "observed" FEs of glutamate carbons or of
glutamate isotopomers to "true" FEs (whose values are not
affected by the amount of metabolically inactive glutamate). This
application of isotopomer analysis to the problem of correcting for
metabolically inactive glutamate is new and
original.10 In
APPENDIX A, we derive the following
equation for the C-3 triplet of glutamate (see Eq. A29), using the definitions of isotopomer fractions
as presented in Ref. 16
where o23 is
the fraction of molecules of OAA that possess
13C at both carbon positions C-2
and C-3. By similar analysis, one can derive the following equations
for the C-4 quartet (C4Q), a C-4 doublet (C4D34), the C-2 quartet
(C2Q), and a C-2 doublet of glutamate (C2D23)
where
Fc2 equals the fraction of molecules of acetyl-CoA labeled at C-2 but
not at C-3, and Fc3 equals the fraction of molecules of acetyl-CoA
labeled at both C-2 and C-3. In the last four equations, the values of
g2, g3, and g4 represent true values that would be obtained if all
glutamate molecules were metabolically active. However, we observe that
the ratio of two true FEs (e.g., g3/g4) equals the ratio of the
corresponding observed FEs (i.e.,
g3obs/g4obs), regardless of the amount of metabolically inactive glutamate. By
solving these four equations simultaneously, we obtain two expressions
for 1/factive
where
g2obs,
g3obs, and
g4obs are observed FEs obtained by
dividing the moles of glutamate molecules having
13C at position C-2 (or C-3, C-4,
respectively) by the moles of total glutamate (including glutamate
whose carbons do not exchange with
-ketoglutarate that is a
constituent pool of the CAC).
 |
ACKNOWLEDGEMENTS |
We thank Dr. Jang Youn, Dr. Ursula Sonnewald, and several
anonymous referees for helpful suggestions and comments on this manuscript.
 |
FOOTNOTES |
This study was supported by National Institutes of Health Grant
AM-27619 and the University of Southern California (USC) Faculty Research and Innovation Fund. D. M. Cohen was supported by National Institute on Aging Training Grant AG-00093. R. N. Bergman is supported by the USC Salerni Collegium.
1
We recognize that any model is founded on a set
of assumptions and is only as accurate as the assumptions are valid.
Our analysis points out the consequences of the isotopic labeling of
the pyruvate that is substrate for anaplerosis and suggests that it be
included in any formula for relative anaplerosis. Future experiments
may demonstrate the insufficiency of our assumptions and require yet additional assumptions to be incorporated in a model that is used to
estimate relative anaplerosis.
2
Note that efflux of AcCoA during simulated
administration of
[3-13C]pyruvate is 10 times greater than that during simulated administration of
[2-13C]acetate (Table
2). The former simulations were redone at zero efflux of AcCoA (Table
4).
3
Increasing
r/vTCA
in the model increases (decreases) the accuracy of Eq. A19 during administration of
[3-13C]pyruvate
([2-13C]acetate,
respectively) and has no effect on the accuracy of Eq. A18 during administration of either
[3-13C]pyruvate or
[2-13C]acetate (data
not shown).
4
Our assumption of equal FEs of carbons of
pyruvate, lactate, and alanine has been found to be valid in heart
extracts for experiments in vivo in the dog heart during infusion of
[3-13C]lactate (12).
However, experiments in the in vivo dog heart (12), as well as in the
isolated perfused rabbit heart (13), found that administration of
[3-13C]pyruvate caused
the fractional enrichment of C-3 of alanine to equal the fractional
enrichment of C-3 of intracellular pyruvate, but to exceed the
fractional enrichment of C-3 of intracellular lactate. Therefore, our
hypothetical 1-pool model of pyruvate metabolism may be realistic
during administration of
[3-13C]lactate but not
during administration of
[3-13C]pyruvate.
5
Several studies have found evidence that the
pool of pyruvate that is used as substrate for anaplerosis is distinct
from the pool of pyruvate that communicates with tissue lactate or with extracellular pyruvate (13, 22, 35).
6
Equations describing isotopic steady state
follow.
x1 · (v0 + v4) = x3 · v4 + p0 · v0;
x3 · (v2 + v4 + v8) = x5 · (v2 + v8) + x1 · v3;
x5 · (v2 + v8) = x3 · v8 + x6 · v2;
x6 · (v2 + v9) = x4 · (v2 + v9). From the
last three equations, the following is derived:
x3 · (v2 + v4) = x1 · v4 + x4 · v2.
From the last equation and the first equation, the formula for
x3/x4
is derived (assuming p0 = 0), and
(also) the formula for x3
x4 (not assuming p0 = 0).
7
Note that we assume equal exchanges of
glutamate/aspartate compared with
-ketoglutarate/malate in heart
(see Fig. 5), which has not been demonstrated in vitro or in vivo (11).
The
-ketoglutarate/malate exchange is rapidly reversible, but the
glutamate/aspartate exchange is not. Hence it is possible (in
principle) that, at metabolic steady state, the stoichiometries of the
two exchangers will be the same, but one must await experimental data.
The equation relating the fractional enrichment of malate in the
cytosol (x3) with the enrichment in the mitochondrion
(x4) applies to our simplified model (Fig. 5) and may need
to be modified as data are acquired.
8
For brevity, we have restricted our
investigations to closed form (algebraic) formulas for relative
anaplerosis from pyruvate in heart. Each "conventional" formula
in Table 1, i.e., Eqs. A18, A19, and
A22, is used in the interpretation of
13C NMR spectroscopic data in the
peer-reviewed literature. We exclude the excellent work done on the CAC
in liver, including analysis of labeling patterns of metabolic
intermediates and of 13C mass
isotopomers. Isotopomer analysis has recently been applied to oxidation
(via PDH complex) of pyruvate that is exported from the CAC when
anaplerosis occurs solely at succinyl-CoA (during metabolism of
propionate) (9). This approach, in which a numerical solution for
y is obtained (without an explicit
algebraic formula), may prove most practical when applied to the
problems described in this article. The existence of multiple pools of
pyruvate remains a formidable problem that will need to be addressed by
this advance in isotopomer analysis.
9
In rat hearts perfused with glucose and
insulin, the addition of acetate caused the ratio of moles of alanine
to moles of glutamate plus glutamine to be 3.5%, whereas the addition
of pyruvate caused the ratio to be 34% (8).
10
Our application of isotopomer analysis relaxes
several of the assumptions of conventional analysis (16). We allow for
recycling of pyruvate (R > 0),
finite reverse flux catalyzed by fumarase (finite
r/vTCA),
and reincorporation of
[13C]CO2
by anaplerotic reaction (p4 > 0, see APPENDIX
A).
Address for reprint requests: D. M. Cohen, Laboratory of Cerebral
Metabolism, Bldg. 36, Rm. 1A-07, National Institutes of Health,
Bethesda, MD 20892.
Received 2 April 1997; accepted in final form 16 September 1997.
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