INTRODUCTION

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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
v_TCA: rate of flux catalyzed by -ketoglutarate dehydrogenase complex
R: recycling parameter
PDH: pyruvate dehydrogenase complex
OAA: oxaloacetate
FE: fractional enrichment

	METHODS

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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.

View larger version (28K): [in this window] [in a new window]   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.

Biochemical Reactions and Flux Rates

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 (v_TCA) catalyzed by -ketoglutarate 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 the r/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 (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 (v_ANA = v_CATA).

View larger version (18K): [in this window] [in a new window]   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).

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).

Derivation of Algebraic Formulas

View larger version (28K): [in this window] [in a new window]   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.

View larger version (7K): [in this window] [in a new window]   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.

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 f_A, f_B, and f_C, respectively. Then the rate of conversion of labeled C-1 of A to C-3 of C equals f_A · v_A, and the rate of conversion of labeled C-2 of B to C-3 of C equals f_B · v_B. 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

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.

	RESULTS

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Accuracy of Eq. A18

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

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

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

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

View larger version (22K): [in this window] [in a new window]   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

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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.

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