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^{13}Clabeled precursors and analysis of [^{13}C]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 ofy 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 peerreviewed literature assumes thatR = 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 mosty ⋅ (1 −R) instead ofy during administration of [2^{13}C]acetate and unlabeled pyruvate. Using a new formula for estimation ofy, we recalculate values ofy 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
 malateaspartate shuttle
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 socalled 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 MichaelisMenten 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^{13}C]pyruvate or [1^{13}C]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^{13}C]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 Ci: number of molecules isotopically labeled at Ci divided by number of molecules in the entire metabolic pool
a1, a2: fractional enrichment of C1 and C2 of the acetyl moiety of acetylCoA (where carbon C2 is the methyl carbon)
g1, g2, g3, g4, g5: fractional enrichment of carbons C1, C2, C3, C4, and C5 of glutamate
o1, o2, o3, o4: fractional enrichment of carbons C1, C2, C3, and C4 of oxaloacetate
p1, p2, p3: fractional enrichment of carbons C1, C2, and C3 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
v_{TCA}: rate of flux catalyzed by αketoglutarate 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.
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 . Finally, we describe the methods by which we tested the accuracy of proposed formulas for relative anaplerosisy, the ratio of anaplerotic flux to the flux catalyzed by citrate synthase.
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 ). We simulated the effect of [2^{13}C]acetate or [3^{13}C]pyruvate on the labeling of metabolites of the CAC. In order to change as few parameters as possible during simulated perfusion with [3^{13}C]pyruvate, the pool sizes were maintained at the same levels as during simulated perfusion with [2^{13}C]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 and7), 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 (v _{TCA}) catalyzed by αketoglutarate dehydrogenase complex in the perfused rat heart was 4.13 (r/v _{TCA}= 4.13). Except where noted, simulations presented in this report were run with ther/v _{TCA}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 (socalled “instant randomization of carbons of OAA”) (35).^{3} In simulations with different values of relative anaplerosis (increasingy from 0.1 to 1 to 10), the value ofr remains the same to simulate activation of the anaplerotic enzymes rather than increased concentrations of substrates, which would affect the value ofr as well as the value ofy.
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 (v _{ANA} =v _{CATA}).
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 Table1), 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 steadystate fractional enrichment as pyruvate, e.g., alanine and lactate (see Fig. 2).^{4}Cataplerotic reactions add molecules to this pool of pyruvate (seeEqs. 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,v _{CATA}). We define the recycling parameter R as the fraction of substrate for anaplerosis that contains one or more carbons exported via cataplerosis. This definition ofR 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 isR =v _{CATA}/( pyrInflux + v _{CATA}).
Suppose pyruvate and [2^{13}C]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 (see Eq. EA28 ), anaplerotic flux can become considerably labeled because (for example) molecules of OAA doubly labeled at C2 and C3 are produced by the chemical reactions of the CAC. In the other extreme, if the influx into pyruvate is extremely large compared withv _{CATA}(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 parameterR do not depend on a particular model of intracellular pyruvate compartmentation but solely on the fractional enrichment of C2 and C3 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 . 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 ofv _{A} andv _{B} moles of C per second, respectively (Fig. 4). Letv _{C} 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 equationv _{A} +v _{B} =v _{C}.
Suppose that examination of the chemical reactions shows that carbon position C1 of A and carbon position C2 of B are transferred to carbon position C3 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., ^{13}C) 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 C1 of A, C2 of B, and C3 of C be denoted f_{A}, f_{B}, and f_{C}, respectively. Then the rate of conversion of labeled C1 of A to C3 of C equals f_{A} ⋅ v
_{A}, and the rate of conversion of labeled C2 of B to C3 of C equals f_{B} ⋅ v
_{B}. Equating the rate of production of labeled C3 of C with the rate of removal of label from C3 of compound C, we obtain the following equation
In the formulas presented below, it is assumed that each variable representing fractional enrichment of a compound refers to a single, wellmixed 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. Inappendix , we derive formulas to correct for a pool of glutamate that is metabolically inactive (we assume all acetylCoA 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) and2) by means of a computer simulation of a syntactic model (5). The dependent variables in the differential equations written for MLAB (see appendix ) 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^{13}C]glutamate and [1,3^{13}C]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 ofEqs. EA22, EA25, or EA26 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 timedependent changes in concentrations of positional isotopomers of metabolic intermediates (57). It is a stochastic simulation of chemical kinetics that relies on a rulebased 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 firstorder rate constant. Because of its stochastic nature, the pool sizes fluctuate somewhat around the steadystate 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.
RESULTS
Accuracy of Eq. EA18
As predicted from theoretical analysis of its relative error with the assumption of instant equilibration of OAA and fumarate (seeappendix , Eq.EA32 ), Eq. EA18 consistently underestimates the true value of relative anaplerosisy (Tables2, 3, and4). During administration of [2^{13}C]acetate, the accuracy in Eq. EA18 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. EA20 ) indicate that the value obtained from Eq. EA18 equals (1 −R) ⋅ y, instead of y, during administration of [2^{13}C]acetate and unlabeled pyruvate.
If the exogenously administered substrate was [3^{13}C]pyruvate, estimates of relative anaplerosis by use of Eq.EA18 gave values indistinguishable from zero (slightly negative) for all values of y and all values of recycling parameter R (Table2). Simulations in which exogenous pyruvate was 50% unlabeled and 50% [3^{13}C]pyruvate gave exactly the same estimates for y by use of Eq. EA18 , but the fractional enrichments of each carbon of glutamate and pyruvate were reduced by 50% (data not shown). Increased enrichment of acetylCoA from the decreased rate of acetylCoA turnover improved the estimate ofy by use of Eq.EA18 ; the estimated value ofy 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. EA32 ). The value of y estimated byEq. EA18 will be negative whenever the fractional enrichment of C2 of glutamate exceeds the fractional enrichment of C4 of glutamate, or, equivalently, whenever the sum of fractional enrichments of C2 and C3 of pyruvate (i.e., p2 + p3) exceeds that of glutamate C2 plus C3 (i.e., g2 + g3). The accuracy of Eq. EA18 increases as (p2 + p3)/(g2 + g3) decreases (see Eq. EA32 ).
Accuracy of Eq. EA19
Equation EA19 , a formula for measuring relative anaplerosis from fractional enrichments of glutamate C1 and C3 (Table 1), underestimates the value ofy. As the dilution of the pyruvate pool by exogenous pyruvate increases (i.e., decreasingR), the ratio of fractional enrichments of C1 of pyruvate to C1 of glutamate decreases (Table 2), thereby decreasing the magnitude of the relative error iny incurred by use ofEq. EA19 (see Eq.EA33 ).
For administration of [2^{13}C]acetate, the accuracy of Eq. EA19 improves with decreasing recycling R (Tables 2 and3). During administration of exogenous [3^{13}C]pyruvate atR = 0.1, Eq.EA19 underestimated relative anaplerosis (y) by 68% wheny = 0.1, 40% wheny = 1, and by 70% wheny = 10 (Tables 2 and 3). At values of recycling parameter R of 0.5 or 0.9,Eq. EA19 underestimatedy by ∼65%. Computer simulations in which exogenous pyruvate was 50% unlabeled and 50% [3^{13}C]pyruvate gave exactly the same estimates of y by use of Eq. EA19 , but the fractional enrichment of each carbon of glutamate and pyruvate was reduced by 50% (data not shown). Increasing the enrichment of acetylCoA by decreasing its turnover rate decreased the accuracy of Eq.EA19 for y = 0.1 but had no effect for higher values of y(Table 4).
Accuracy of Eq. EA10
We propose a new formula (Eq. EA10 ) for estimation of relative anaplerosis. EquationEA10 is completely accurate when the CAC is simulated by solving a system of differential equations numerically during administration of either [2^{13}C]acetate or [3^{13}C]pyruvate (Table2). Computer simulations in which exogenous pyruvate was 50% unlabeled and 50% [3^{13}C]pyruvate gave exactly the same estimates of y by use of Eq. EA19 , but the fractional enrichment of each carbon of glutamate and pyruvate was reduced by 50% (data not shown). During administration of [3^{13}C]pyruvate, increasing the enrichment of acetylCoA by decreasing its turnover rate likewise resulted in zero error in estimation ofy by use of Eq.EA10 (Table 4). The relative error in the estimate ofy obtained from Eq.EA10 is <0.0005 under all conditions tested.
Equation EA10 may be inaccurate when applied to the output of a simulation of a syntactic model (Table 3). If y equals ∼0.1 and [2^{13}C]acetate is administered, Eq. EA10 overestimates the true value by 17% for R = 0.5 and underestimates the true value by 3% forR = 0.1 (Table 3). Ify equals 0.1 and [3^{13}C]pyruvate is administered, then Eq. EA10 estimatesy 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^{13}C]acetate or [3^{13}C]pyruvate (Table3). If the relative anaplerosis is increased to ∼10.0, the magnitude of the relative error is <10% during administration of [2^{13}C]acetate but increases to 25% (40%) during administration of [3^{13}C]pyruvate withR equal to 52% (90%), respectively. Nevertheless, with a single exception (y = 0.1,R = 0.1, [3^{13}C]pyruvate),Eq. EA10 exhibits better accuracy thanEqs. EA19 or EA18 for all cases simulated with the syntactic model (Table 3).
The source of inaccuracy of Eq. EA10 is as follows. As the denominator in this formula fory 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 C2 (C3) of pyruvate and C3 (C2) of glutamate approaches zero, leading to a numerically unstable calculation (division by a very small number).
Accuracy of Eqs. EA22, EA25, andEA26
Malloy et al. (16, 17) derived formulas to estimate relative anaplerosis from the multiplets of the^{13}C NMR spectrum of glutamate. We selected one of their formulas (Eq.EA22 ) for the triplet resonance (C3T) of C3 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^{13}C]acetate or [3^{13}C]pyruvate,Eq. EA22 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. EA26 ) confirm that the value estimated by Eq. EA22 equals (1 −R) ⋅ yinstead of y.
Equation EA25 was consistently more accurate than Eq. EA22 yet showed considerable error when y = 0.1 andy = 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 byEq. EA26 is mathematically equivalent to Eq. EA25 yet estimatesy more accurately for all simulations (Table 3). To apply Eq. EA26 , 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 ofR could be made, thenEq. EA25 could be replaced by an equation (i.e., Eq. EA26 ) that is mathematically equivalent but more precise. On the other hand, computer simulations indicate that if relative anaplerosis (y) were to be estimated independently ofR, then Eq.EA26 would not be useful in estimatingR 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.EA26 could be used to estimateR, given the limitations of precision of physical measurements.
Effects of MalateAspartate 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 Ci in cytosolic malate (x3) to the fractional enrichment of Ci in mitochondrial malate (x4) under conditions of continuous administration of isotopically labeled acetate and pyruvate unlabeled at Ci(p_{0} = 0; see Fig.5):x3/x4 = [v _{2} ⋅ (v _{0}+v _{3})]/(v _{0} ⋅ v _{2}+v _{0} ⋅ v _{3}+v _{2} ⋅ v _{3}). If we allow administration of pyruvate isotopically labeled at Ci(p_{0} > 0 in Fig. 5) as well as arbitrarily labeled acetate, we can derive the following formula:x3 −x4 = (p_{0} −x1) ⋅ (v _{0} /v _{2}), where x1 is the fractional enrichment of Ci of cytosolic pyruvate (Fig. 5). From these formulas^{7} it is seen that, during administration of pyruvate unlabeled at Ci, 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 (v _{0}) or the rate of the flux catalyzed by cytosolic malic enzyme (v _{3}) is much slower than the rate of the malateaspartate shuttle (v _{2}) (see Fig.5): v _{0} <<v _{2} orv _{3} <<v _{2}. Furthermore, even if pyruvate that is administered is isotopically labeled at Ci, sufficiently large flux through the malateaspartate shuttle (v _{2}) compared with influx of pyruvate into the cytosolic pool (v _{0}) will ensure that x3 ≅x4 (i.e., cytosolic and mitochondrial malate have the same fractional enrichment).
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 [^{14}C]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^{13}C]pyruvate or [1^{13}C]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^{13}C]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^{13}C]pyruvate (or equivalently, [1^{13}C]glucose) or [2^{13}C]acetate is administered, then conventional formulas (16, 17) using the C1/C3, C2/C4 or isotopomer ratios of glutamate (Eqs. EA18,EA19, and EA22 ) underestimate relative anaplerosis (Table 1). We propose a formula (Eq. EA10 ) that is accurate during administration of [2^{13}C]acetate, [3^{13}C]pyruvate, and/or [1^{13}C]glucose, provided it is possible to measure the fractional enrichment of carbons C2 and C3 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) are1) instant equilibration of OAA with fumarate and 2) no reincorporation of [^{13}C]CO_{2}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 C2 and C3, then anaplerosis from pyruvate does not contain molecules that are doubly labeled at both carbons C2 and C3. However, our simulations (see Table 3) and mathematical analysis (seeEq. EA28 ) demonstrate that molecules of OAA doubly labeled at C2 and C3 are produced by the actions of the CAC during administration of [2^{13}C]acetate or [3^{13}C]pyruvate. From published steadystate data (16) and our Eq.EA29 , we calculate that 65.5% (46.3%) of molecules of OAA in the perfused rat heart are doubly labeled at C2 and C3 during administration of [2^{13}C]acetate (or [3^{13}C]pyruvate, respectively). Therefore, one cannot ignore anaplerosis of doubly labeled molecules possessing ^{13}C at both positions C2 and C3, unless one assumes that recycling of carbons of pyruvate does not occur (i.e., one assumes thatR = 0). In fact, we derive a formula (Eq. EA31 ) for the fraction of molecules of pyruvate doubly labeled at C2 and C3 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. EA10 (Table 1) for estimating relative anaplerosis is not affected by recycling of carbons between pyruvate and pools of the CAC. The assumptions underlyingEq. EA10 follow. Reactions of the CAC and the transfer of carbons among metabolic pools are given in Figs.13. 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 3carbon fragment of βoxidation is metabolized to succinylCoA (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 αketoglutarate 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 C2 of mitochondrial acetylCoA and C4 of glutamate (see appendix ). 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. EA10 to published data on the isolated mammalian heart perfused with [3^{13}C]pyruvate and to recalculate the value of y. In the isolated rabbit heart perfused with 2.5 mM [3^{13}C]pyruvate and no glucose (13), our formula yields 18% as the estimate ofy (after correction for a pool of metabolically inactive glutamate) compared with the value of 10% obtained with Eq. EA18 . In the isolated rat heart perfused with 1 mM [3^{13}C]pyruvate and 5 mM glucose (3), the value of yobtained with our formula is 14% compared with the value obtained with the classical formula (Eq. EA18 ) of 4% (correction for metabolically inactive glutamate pool could not be made). As predicted from simulations (Table 2), the classical formula (Eq.EA18 ) that uses the ratio of enrichments of C2 to C4 of glutamate underestimates the true value of relative anaplerosis by ∼50%. In our calculations using Eq.EA10 , we assumed that alanine and pyruvate that is substrate for anaplerosis are in equilibrium and that C2 of these compounds is unenriched (12, 13, 22). Therefore, the values obtained with Eq. EA10 in these examples are lower bounds, which would be increased if the fractional enrichment of C2 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 malateaspartate 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 malateaspartate 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 malateaspartate shuttle is rate limiting for transport of αketoglutarate 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 steadystate 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 αketoglutarate, the fractional enrichment of glutamate would be identical in mitochondria and cytosol at steady state, and equal to the fractional enrichment of (mitochondrial) αketoglutarate. Indeed, Lewandowski et al. (14), using NMR spectroscopy, found that in extracts from perfused rabbit hearts the fractional enrichment of C4 of αketoglutarate and that of glutamate were equal.
We investigated the effect of the rate of the malateaspartate 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 malateaspartate shuttle is much greater than the rate of influx of cytosolic pyruvate (v _{2} >>v _{0} 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 malateaspartate shuttle greatly exceed the mass flux catalyzed by cytosolic malic enzyme (v _{2} >>v _{3} 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 glucoseperfused or acetateperfused 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^{13}C]lactate, [3^{13}C]pyruvate, or [1^{13}C]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. EA10 ).
Acknowledgments
We thank Dr. Jang Youn, Dr. Ursula Sonnewald, and several anonymous referees for helpful suggestions and comments on this manuscript.
Footnotes

Address for reprint requests: D. M. Cohen, Laboratory of Cerebral Metabolism, Bldg. 36, Rm. 1A07, National Institutes of Health, Bethesda, MD 20892.

This study was supported by National Institutes of Health Grant AM27619 and the University of Southern California (USC) Faculty Research and Innovation Fund. D. M. Cohen was supported by National Institute on Aging Training Grant AG00093. 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^{13}C]pyruvate is 10 times greater than that during simulated administration of [2^{13}C]acetate (Table2). The former simulations were redone at zero efflux of AcCoA (Table4).

↵3 Increasingr/v _{TCA}in the model increases (decreases) the accuracy of Eq.EA19 during administration of [3^{13}C]pyruvate ([2^{13}C]acetate, respectively) and has no effect on the accuracy of Eq.EA18 during administration of either [3^{13}C]pyruvate or [2^{13}C]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^{13}C]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^{13}C]pyruvate caused the fractional enrichment of C3 of alanine to equal the fractional enrichment of C3 of intracellular pyruvate, but to exceed the fractional enrichment of C3 of intracellular lactate. Therefore, our hypothetical 1pool model of pyruvate metabolism may be realistic during administration of [3^{13}C]lactate but not during administration of [3^{13}C]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 ⋅ (v _{0}+ v _{4}) =x3 ⋅ v _{4}+ p_{0} ⋅ v _{0};x3 ⋅ (v _{2}+ v _{4} +v _{8}) =x5 ⋅ (v _{2}+ v _{8}) +x1 ⋅ v _{3};x5 ⋅ (v _{2}+ v _{8}) =x3 ⋅ v _{8}+x6 ⋅ v _{2};x6 ⋅ (v _{2}+ v _{9}) =x4 ⋅ (v _{2}+ v _{9}). From the last three equations, the following is derived:x3 ⋅ (v _{2}+ v _{4}) =x1 ⋅ v _{4}+x _{4} ⋅ v _{2}. From the last equation and the first equation, the formula forx3/x4 is derived (assuming p_{0} = 0), and (also) the formula for x3 −x4 (not assuming p_{0} = 0).

↵7 Note that we assume equal exchanges of glutamate/aspartate compared with αketoglutarate/malate in heart (see Fig. 5), which has not been demonstrated in vitro or in vivo (11). The αketoglutarate/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. EA18, EA19 , and EA22 , is used in the interpretation of^{13}C NMR spectroscopic data in the peerreviewed literature. We exclude the excellent work done on the CAC in liver, including analysis of labeling patterns of metabolic intermediates and of ^{13}C 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 succinylCoA (during metabolism of propionate) (9). This approach, in which a numerical solution fory 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 (finiter/v _{TCA}), and reincorporation of [^{13}C]CO_{2}by anaplerotic reaction (p4 > 0, see appendix ).
 Copyright © 1997 the American Physiological Society
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
The equations describing the fractional enrichments of carbons of the pools of the CAC are derived from examination of Fig.6
Examination of carbons C1 and C4 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
Formula for Recycling Parameter R
We assume metabolic steady state, so that pool sizes are unchanging and hence anaplerosis and cataplerosis are equal (v
_{TCA} =v
_{CATA}). If Qi equals the fractional enrichment of carbon Ci of the pyruvate influx (Fig. 2), then at isotopic steady state, fori = 1, 2, and 3,
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 C2 and C3, so that p2 = 0 and p3 = 0, one can derive the following formula by substituting into Eqs.EA10
and
EA15
and rearranging
In our model of the CAC (which does not include instant equilibration), we may derive a formula similar to Eq.EA18
. If we assume that molecules of exogenous pyruvate are not labeled at either C2 or C3, then by examination of Fig. 3 we may infer that g2 = g3, and from Eqs.EA10
and
EA17
we may derive the following equation
Formula for y With Positional Isotopomer Analysis
The formula for the C3 triplet is the following (17)
Using the same model of the CAC but including the effects of labeled anaplerosis from pyruvate, we have derived the following formula (seeEqs. EA28
and
EA29
)
Generation of Doubly Labeled OAA
We next derive a formula for o23, the fraction of molecules of OAA that are labeled at both C2 and C3 at isotopic steady state. These molecules derive from molecules of αketoglutarate that are isotopically labeled at C3 and C4 or from molecules of pyruvate that are doubly labeled at C2 and C3 (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
Derivation of Eq. EA24
By definition of C3T, g3 ⋅ C3T equals the fraction of molecules of glutamate that are isotopically labeled at C2, C3, and C4. The atoms in positions C2, C3, and C4 of a molecule of glutamate derive from carbons C3 and C2 of OAA and from C2 of acetylCoA, respectively (see Fig. 3). Therefore, recalling that a2 = g4 (see Eq. A6), one may write the following equation to describe isotopic steady state
Derivation of Eq. EA26
If exogenously labeled pyruvate is not doubly labeled at C2 and C3, then by Eq. EA17
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) inEqs. EA18
and
EA19
The formula for the relative error in Eq.EA22 , obtained by comparison of that equation withEq. EA24 (which we take to be the “correct” formula, assuming instant equilibration of OAA and succinate) is Rel Err (Eq. EA22 ) = −p23/o23. If exogenously administered molecules of pyruvate are not doubly labeled at C2 and C3 (see discussion ofEq. EA17 ), then the relative error inEq. EA22 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.13). The equations were run under conditions of metabolic steady state until isotopic steady state was reached. The rates of influx (efflux) of acetylCoA 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 v
_{TCA},v
_{ANA},v
_{TA}, 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 v
_{TCA} (8.28),v
_{TA} (22.9),v
_{ANA}(y ⋅ v
_{TCA}, where y is the desired relative anaplerosis), PDH (0.8 or 8.0 for administration of [2^{13}C]acetate or [3^{13}C]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)
The (constant) parameters representing the concentrations of metabolic pools are Mcit (citrate),Makg (αketoglutarate),Mfum (fumarate and succinate, combined pools), Mmal (malate and OAA, combined pools), Mpyr (pyruvate),Maccoa (acetylCoA),Masp (aspartate), andMglu (glutamate). The parameter representing the enrichment of carbon Ci in the influx into the pyruvate pool is pyrCiIN, e.g., pyrC1IN, and into the pool of acetylCoA is acCiIN (e.g., acC1IN). The parameter representing the fractional enrichment of CO_{2} 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) areMcit (2.03),Makg (0.16),Mfum (0.32),Mmal (0.3223),Mpyr (0.2),Maccoa (0.02),Masp (2.02), andMglu (20.87).
We let the following variables denote the fractional enrichment of carbon Ci of citrate, αketoglutarate, fumarate, malate, pyruvate, acetylCoA, 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)
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 C4 of glutamate compared with the methyl carbon of acetylCoA has been observed, indicating the existence of a pool of glutamate that does not mix with the mitochondrial glutamate produced from αketoglutarate by aminotransferase reactions (13). For purposes of exposition, these glutamate molecules shall be called “metabolically inactive” (with respect to exchange of carbons with αketoglutarate, 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 C2 of acetylCoA (the methyl carbon, denoted a2 in appendix ) equals the FE of C4 of glutamate (denoted g4 in appendix ), if all acetylCoA molecules are available for the reaction catalyzed by citrate synthase and all glutamate molecules are in exchange with mitochondrial αketoglutarate. In other words, the ratio f_{active} of the mass of metabolically active glutamate to the mass of total glutamate (metabolically active as well as inactive glutamate) equals g4_{obs}/a2, where a2 equals the true FE of C2 of acetylCoA and g4_{obs}equals the observed FE of C4 of glutamate (ratio of moles of glutamate possessing ^{13}C at carbon C4 divided by moles of total glutamate, including metabolically inactive molecules). It is assumed the FE of carbons of acetylCoA 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/f_{active}.
Alternatively, provided there is sufficient enrichment of acetylCoA, conventional isotopomer analysis (16) can be used to obtain 1/f_{active}, 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} Inappendix
, we derive the following equation for the C3 triplet of glutamate (see Eq.EA29
), using the definitions of isotopomer fractions as presented in Ref. 16