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Calculation of Absolute Metabolic Flux and the Elucidation of the Pathways of Glutamate Labeling in Perfused Rat Heart by 13C NMR Spectroscopy and Nonlinear Least Squares Analysis (∗)

  • John C. Chatham
    Correspondence
    To whom correspondence should be addressed: Dept. of Radiology, Division of NMR Research, 217 Traylor Bldg., The Johns Hopkins University School of Medicine, 720 Rutland Ave., Baltimore, MD 21205. Tel.: 410-955-7491; Fax: 410-955-7923
    Affiliations
    (1) Department of Radiology and Radiological Sciences, Division of NMR Research, The Johns Hopkins University School of Medicine, Baltimore, Maryland 21205
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  • John R. Forder
    Affiliations
    (1) Department of Radiology and Radiological Sciences, Division of NMR Research, The Johns Hopkins University School of Medicine, Baltimore, Maryland 21205
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  • Jerry D. Glickson
    Affiliations
    (1) Department of Radiology and Radiological Sciences, Division of NMR Research, The Johns Hopkins University School of Medicine, Baltimore, Maryland 21205
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  • Edwin M. Chance
    Affiliations
    (2) the Department of Biochemistry and Molecular Biology, University College London, Gower Street, London, WC1E 6BT, United Kingdom
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  • Author Footnotes
    ∗ This work was supported by Grants HL30579, CA51950, and CA51935 from the National Institutes of Health and grant-in-aids from the American Heart Association (to J. C. C. and J. R. F.) and a NATO collaborative grant (to E. M. C.). The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore by hereby marked “advertisement” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Open AccessPublished:April 07, 1995DOI:https://doi.org/10.1074/jbc.270.14.7999
      Absolute metabolic fluxes in isolated perfused hearts have been determined by a nonlinear least squares analysis of glutamate labeling kinetics from [1-13C]glucose, [4-13C]β-hydroxybutyrate, or [2-13C]acetate using 13C NMR spectroscopy. With glucose as substrate, the malate-aspartate shuttle flux was too slow to account for the reducing equivalents generated by glycolysis and to predict the observed oxygen consumption rate. For acetate and β-hydroxybutyrate, the malate-aspartate shuttle had to be reversed for the network to agree with the observed oxygen consumption and glutamate labeling. Thus, an additional redox shuttle was required to reoxidize the NADH produced by cytoplasmic malate dehydrogenase. Using this model there was good agreement between the experimentally determined oxygen consumption and glutamate labeling and the calculated values of these parameters from the model for all substrates. The contribution of exogenous substrate to the overall tricarboxylic acid (TCA) cycle flux, 89.6 ± 6.5% (mean ± S.D.) as measured in the tissue extracts compared well with 91.4 ± 4.2% calculated by the model. The ratio of TCA cycle flux to oxygen consumption for acetate, was 2.2 ± 0.1, indicating that NADH production is principally accounted for by TCA cycle flux. For glucose or β-hydroxybutyrate, this ratio was 2.9 ± 0.2, consistent with the existence of other NADH producing reactions (e.g. glycolysis, β-hydroxybutyrate oxidation).

      INTRODUCTION

      13C NMR spectroscopy has been used to study intermediary metabolism in a wide range of biological systems including isolated cells(
      • Dahlen J.V.
      • Porter J.W.
      ,
      • den Hollander J.A.
      • Behar K.L.
      • Shulman R.G.
      ,
      • Dickinson J.R.
      • Dawes I.W.
      • Boyd A.S.F.
      • Baxter R.L.
      ,
      • Lyon R.C.
      • Cohen J.S.
      • Faustino P.J.
      • Megnin F.
      • Myers C.E.
      ,
      • Jans A.W.H.
      • Leibfritz D.
      ,
      • Kuchel P.W.
      • Berthon H.A.
      • Bubb W.A.
      • Bulliman B.T.
      • Collins J.G.
      ), organs (both in vitro(
      • Bailey I.A.
      • Gadian D.G.
      • Matthews P.M.
      • Radda G.K.
      • Seeley P.J.
      ,
      • Malloy C.R.
      • Sherry A.D.
      • Jeffrey F.M.H.
      ,
      • Hoekenga D.E.
      • Brainard J.R.
      • Hutson J.Y.
      ,
      • Weiss R.G.
      • Chacko V.P.
      • Glickson J.D.
      • Gerstenblith G.
      ,
      • Lewandowski E.D.
      ,
      • Chatham J.C.
      • Forder J.R.
      ) and in vivo(
      • Reo N.V.
      • Siegfried B.A.
      • Ackerman J.J.H.
      ,
      • Rothman D.L.
      • Behar K.L.
      • Hetherington H.P.
      • Den Hollander J.A.
      • Bendall M.R.
      • Petroff O.A.C.
      • Shulman R.G.
      ,
      • Fitzpatrick S.M.
      • Hetherington H.P.
      • Behar K.L.
      • Shulman R.G.
      ,
      • Robitaille P.M.L.
      • Rath D.P.
      • Abduljalil A.M.
      • O'donnell J.M.
      • Jiang Z.
      • Zhang H.
      • Hamlin R.L.
      )), and most recently in humans(
      • Jue T.
      • Rothman D.L.
      • Shulman G.I.
      • Tavitian B.A.
      • DeFronzo R.A.
      • Shulman R.G.
      ,
      • Beckman N.
      • Turkalj I.
      • Seelig J.
      • Keller U.
      ). As 13C-enriched substrates are metabolized, the 13C label is transferred to various metabolic intermediates. The specific location of the label is dependent on the chemistry of the enzyme reactions, and the rate of incorporation is determined by the flux through particular metabolic pathways. If the concentration of the intermediates is high enough (approximately 10−3M), the time course of 13C label incorporation can be determined using 13C NMR spectroscopy. In many biological systems the intermediates in glycolysis and the TCA
      The abbreviations used are: TCA
      tricarboxylic acid
      MVO2
      oxygen consumption
      flux
      TCA cycle flux
      RFO2
      ratio of MVO2 to flux.
      cycle are below the limit of detection by NMR. However, glutamate is present in sufficient concentration for detection by NMR and is in exchange with the TCA cycle intermediate α-ketoglutarate, via two transaminase reactions. There have been many studies that have analyzed the steady state 13C labeling of glutamate to determine the relative contribution of various substrates to the overall TCA cycle flux(
      • Malloy C.R.
      • Sherry A.D.
      • Jeffrey F.M.H.
      ,
      • Lewandowski E.D.
      ,
      • Malloy C.R.
      • Sherry A.D.
      • Jeffrey F.M.H.
      ,
      • Malloy C.R.
      • Sherry A.D.
      • Jeffrey F.M.H.
      ,
      • Malloy C.R.
      • Thompson J.R.
      • Jeffrey F.M.H.
      • Sherry A.D.
      ). These approaches, however, are limited in that they do not provide estimates of absolute metabolic fluxes, which can be obtained by analysis of 13C labeling kinetics(
      • Chance E.M.
      • Seeholzer S.H.
      • Kobayashi K.
      • Williamson J.R.
      ).
      If glutamate is in rapid exchange with the TCA cycle, measurement of the time course of 13C label incorporation into glutamate by 13C NMR spectroscopy may provide a means for noninvasively determining the oxygen consumption of an organ in vivo. However, such an analysis is critically dependent on knowing the mechanism by which the glutamate pool is labeled, since only a small fraction of the total glutamate pool is in exchange with α-ketoglutarate. Consequently, there has been much interest in analyzing the kinetics of isotopic incorporation in order to estimate TCA cycle fluxes in vivo. Such calculations of absolute fluxes cannot be made from the measurement of steady state enrichment data and external fluxes alone, due to the complexity of the reaction networks involved.
      In this study, we have built upon the earlier work by Chance et al.(
      • Chance E.M.
      • Seeholzer S.H.
      • Kobayashi K.
      • Williamson J.R.
      ) who reported that modeling of 13C NMR kinetic data from rat hearts perfused with [2-13C]acetate and [3-13C]pyruvate lead to accurate estimates of the TCA cycle flux and the aspartate and alanine aminotransferase reaction rates. We have developed a mathematical description encompassing both glycolysis and the TCA cycle and analyzed the rate of labeling of glutamate in the isolated perfused heart with three different labeled substrates. Analysis of the kinetics of glutamate labeling in combination with measured values for pool size, fractional enrichments, and oxygen consumption have enabled us to show how small unlabeled influxes affect the end point enrichments and increase the accuracy of the calculated fluxes, yielding insight into the nature of the network of significant reactions that occur in the intact heart.

      EXPERIMENTAL PROCEDURES

      Heart Perfusion

      Randomly fed, male Sprague-Dawley rats, 300-350 g (Charles River, Laboratories, Wilmington MA) were heparinized (500 units/100 g, intraperitoneal) and anesthetized with sodium pentobarbital (10 mg/kg, intraperitoneal). Hearts were rapidly excised and the aorta cannulated for retrograde perfusion with a Krebs-Henseleit bicarbonate buffer at 37°C. The buffer contained (in mM): NaCl, 118; KCl, 6; Mg2SO4, 1.2; CaCl2,1.25; Na2HCO3, 25; and was equilibrated with 95% O2/5% CO2 to saturate the perfusate with oxygen and maintain a pH of 7.4. A balloon was inserted into the left ventricle through the mitral valve for continuous measurement of left ventricular pressure. Balloon volume was adjusted to obtain a left ventricular end-diastolic pressure of 5 mmHg. An additional pressure transducer was connected to the aortic perfusion line for continuous monitoring of coronary perfusion pressure, and flow was adjusted to maintain a constant perfusion pressure of 76-80 mmHg. Left ventricular pressures and coronary perfusion pressure were recorded on a Gould 3400 recorder (Gould, Inc., Billerica, MA).
      A total of six hearts were perfused under one of four substrate conditions: 1) glucose, 5 mM plus insulin (0.05U/ml), n = 2; 2) β-hydroxybutyrate, 11 mM plus glucose, 5 mM and insulin (0.05 unit/ml), n = 2; 3) acetate, 11 mM, n = 1 or 4) acetate, 11 mM plus glucose, 5 mM, and insulin (0.05 unit/ml), n = 1. After approximately 30-min equilibration with unlabeled substrates the perfusate was switched to that containing 13C-labeled substrates: 1) [1-13C]glucose, 5 mM; 2) [4-13C]β-hydroxybutyrate, 11 mM plus glucose, 5 mM; 3) [2-13C]acetate, 11 mM; or 4) [2-13C]acetate, 11 mM plus glucose, 5 mM. In all experiments the composition of the perfusate before and after switching to labeled substrates was identical except for the presence of the 13C label. In each group the 13C-labeled substrates were 99% enriched. In the groups with two substrates (i.e. β-hydroxybutyrate plus glucose and acetate plus glucose) the glucose was unlabeled. The 13C-labeled glucose and acetate were purchased from Cambridge Isotope Laboratories (Woburn, MA), and the [4-13C]β-hydroxybutyrate was purchased from Isotech Inc. (Miamisburg, OH).

      Measurement of Oxygen Consumption

      In a parallel set of experiments cardiac oxygen consumption was determined over a range of work loads (rate pressure product varied between 5,000 and 35,000 mmHg min−1). Effluent from the pulmonary artery was directed through a micro-flow-through electrode (model DO-166FT, Lazar Research Laboratories, Inc., Los Angeles, CA) for determination of oxygen content. Oxygen consumption was calculated from the difference between the oxygen content in the aortic cannula and the pulmonary artery, normalized to the coronary flow and wet weight of the heart. From these experiments a linear relationship between rate pressure product and oxygen consumption was obtained. This relationship was used to calculate the oxygen consumption of the hearts used in the NMR experiments. We found no differences in this relationship between the various substrates. The relationship between oxygen consumption and the rate pressure product determined in this study was very similar to that reported by Weiss et al.(
      • Weiss R.G.
      • Gloth S.T.
      • Kalil-Filho R.
      • Chacko V.P.
      • Stern M.D.
      • Gerstenblith G.
      ) in acetate-perfused hearts.

      NMR Spectroscopy

      The perfused hearts were placed in a remotely switched dual-tuned (13C and 31P) 20-mm commercial probe, and spectra were obtained with a Bruker AM 360-WB NMR spectrometer (8.5T, 89-mm diameter bore). Magnetic field homogeneity was optimized by observing the water signal using the 1H decoupling coil. 31P NMR spectra were recorded at the beginning and end of each experiment to assess the bioenergetic status of the hearts. 31P spectra were collected with bi-level 1H decoupling as described previously (
      • Chatham J.C.
      • Forder J.R.
      ,
      • Chatham J.C.
      • Cousins J.P.
      • Glickson J.D.
      ,
      • Chatham J.C.
      • Glickson J.D.
      ) using a 2-s relaxation delay and a 60° pulse in 5 min and stored in 1K data blocks. After four natural abundance 13C NMR spectra were recorded, hearts were perfused with buffer containing 13C-labeled substrates, and 13C spectra were acquired continuously for 40-55 min. 13C NMR spectra were collected in 5-min intervals using a 1-s delay and a 60° pulse and stored in 2 K data blocks.
      High resolution 13C NMR spectra of heart extracts were collected using a Bruker MSL-500 NMR spectrometer equipped with an 11.74T magnet (89-mm diameter bore) and a commercial 10-mm probe. Extract spectra were obtained under fully relaxed conditions (8-s total interpulse delay, 12 μs (60°) pulse) with a 25 KHz sweep width, and stored in 32K data blocks. Spectra were proton decoupled using composite pulse decoupling during data acquisition only (to avoid nuclear Overhauser enhancement).

      Analysis of NMR Spectra

      13C NMR spectra of the perfused heart were processed on a SUN computer using a nonlinear least squares fitting routine in the time domain developed by Barker and Sibisi(
      • Barker P.B.
      • Sibisi S.
      ). Prior to analysis, natural abundance 13C spectra were subtracted from those obtained following perfusion with labeled substrate; thus, the resonance intensities obtained were those that increased as a result of metabolism of the labeled substrate.
      High resolution spectra of extracts were analyzed using standard integration routines. Spectra were zero-filled to 64 or 128K and filtered with a 2-Hz line broadening prior to Fourier transformation.

      Tissue Extracts

      At the end of the NMR experiments each heart was freeze-clamped between liquid nitrogen-cooled aluminum blocks. Each heart was pulverized into a fine powder under liquid nitrogen with a pestle and mortar. The powder was then homogenized with 3 volumes of 6% perchloric acid using a Tekmar Tissumizer (Tekmar Company, Cincinnati OH) for 5 min. The homogenate was centrifuged at 17,500 × g for 15 min at 4°C. The supernatant was removed from the protein pellet and neutralized with a mixture of KHCO3 (3 M) and KOH (1 M). An aliquot (1 ml) of the neutralized supernatant was frozen for subsequent assay of total glutamate based on the conversion of NADH to NAD+ via glutamate dehydrogenase(
      ). The remainder of the supernatant was mixed with chelating resin (Sigma) to remove paramagnetic metal ion contamination and centrifuged once more. The supernatant was freeze-dried and the lyophilized heart extract dissolved in D2O (99.9%). The pH was adjusted to 7.0 using 5 M HCl, the sample treated with chelating resin once more and filtered through a 0.2-μm filter, and high resolution NMR spectra of the extracts were recorded as described above.

      Formulation of Model

      The approach used was similar to that described in detail previously(
      • Chance E.M.
      • Seeholzer S.H.
      • Kobayashi K.
      • Williamson J.R.
      ). To the basic reactions of the TCA cycle, we have added malic enzyme, cytoplasmic transaminase, and malate-aspartate shuttle reactions. Since metabolite pool sizes remain constant over the time course of these experiments, the number of unknowns reduces to the number of endogenous and exogenous influxes and the flux through the transaminase reactions. The TCA cycle reaction networks used to analyze the 13C incorporation experiments are shown in Figure 1:, Figure 2:, Figure 3:. Oxygen consumption (MVO2) was calculated from the algebraic sum of the fluxes of oxidizing and reducing equivalents (i.e. the sum of the fluxes through the TCA cycle, malate-aspartate shuttle, glycerol phosphate shuttle, glycolysis, β-oxidation, and proteolysis) divided by two.
      Figure thumbnail gr1
      Figure 1:Reaction network used to analyze the kinetics of glutamate labeling for one of the experiments with [1-13C]glucose as substrate and the calculated steady state enzyme velocities (glucose Exp. 1 from ). The abbreviations of steady state fluxes and flux relationships are as follows: Fg = total influx from glycolysis; Fpcp = pyruvate influx; Fms = malate-aspartate shuttle flux; Fs = proteolysis influx; Fb = flux from endogenous substrates; MVO2 = oxygen consumption; RFO2 = MVO2/TCA cycle flux; +2e/−2e = two electron reduction/oxidation. The abbreviations of steady state enzyme velocities are defined in . Reactions with double-headed arrows indicate that the reaction is near equilibrium with a net flux in the direction of the double-headed arrows. Note that an additional redox shuttle was required to provide a hydrogen balance between the cytoplasm and the mitochondria; see “Results” and “Discussion” for more details.
      Figure thumbnail gr2
      Figure 2:Reaction network used to analyze the kinetics of glutamate labeling for one of the experiments with [4-13C]β-hydroxybutyrate and unlabeled glucose as substrates and the calculated steady state enzyme velocities (β-HB Exp. 1 from ). Fat = total β-hydroxybutyrate influx; see legend to for other definitions.
      Figure thumbnail gr3
      Figure 3:Reaction network used to analyze the kinetics of glutamate labeling with [2-13C]acetate only as substrate and the calculated steady state enzyme velocities (see ). Fat = total acetate influx; see legend to for other definitions.
      The model requires that intermediate pool sizes be defined in order for fluxes to be calculated. Although it was evident that the glutamate pool size would effect the flux calculations, it was not clear what effect differences in TCA cycle intermediate pool sizes would have on these calculations. An extensive sensitivity analysis was carried out to determine the effect of changes in these pools on the calculated fluxes. We found that if the intermediate pool sizes were small relative to glutamate, changes in pool size had no effect on the calculated fluxes. Since these intermediates were below the limits of detection in the 13C NMR spectra from both the intact heart and the tissue extracts, it was felt appropriate to assume that they were small relative to glutamate. Consequently the pool sizes for most of the TCA cycle intermediates were taken from the earlier work of Chance et al.(
      • Chance E.M.
      • Seeholzer S.H.
      • Kobayashi K.
      • Williamson J.R.
      ) and converted to micromoles/g of wet weight. For TCA cycle intermediates that were not measured in that paper, such as succinate, fumarate, and oxaloacetate, an arbitrary value of 0.1 μmol/g of wet weight was used.
      One of the other uncertainties in the model was the distribution of metabolites between the mitochondrial and cytosolic compartments. To determine whether this distribution was important in calculating the fluxes the ratio of these compartments was varied between 5 and 40%. This covers the range of values for the fraction of the total cell volume occupied by the mitochondria in skeletal and cardiac muscle (
      • Bers D.M.
      ). Analysis indicated that the relative sizes of these two compartments over this range did not have a significant effect on the calculated fluxes.
      The reactions used in construction of the model along with the number of differential equations necessary to describe each reaction are listed in Table 1. The total number of differential equations required to describe the network of reactions is 340. The reason for this large number of equations is that one equation is used to describe the reaction between two specific labeled intermediates. The number of possible labeled species for any intermediate is 2n, where n is the number of carbon atoms. For example, a total of 32 differential equations are required to describe the interconversion between α-ketoglutarate and glutamate via one aminotransferase reaction.
      The 13C-labeled substrates enter the network as influxes and are assumed to be the principle sources of energy and of 13C label in the system. Due to the cleavage of the [1-13C]glucose and [4-13C]β-hydroxybutyrate molecules into 3- and 2-carbon fragments, respectively, the maximum possible fractional enrichment at acetyl-CoA C-2 is 50% (in contrast to that of [2-13C]acetate, which is 100%). In order to account for our observation that end point enrichments of glutamate were significantly less than the maximum possible enrichment, additional unlabeled endogenous influxes had to be included in the network. Isotopic dilution at glutamate C-4 originates from influxes into the acetyl-CoA pool; this is a common end point of many different pathways. Two principal pathways that could be responsible for isotopic dilution of acetyl-CoA are glycolysis (Fgl) and β-oxidation of endogenous fatty acids (Fb). Depending on the perfusion conditions, Fgl could arise from unlabeled glucose or from glycogen breakdown; it is not possible to distinguish between these pathways in our network.
      Entry of unlabeled substrates into the TCA cycle other than via acetyl-CoA, such as metabolism of amino acids, was represented by a single unlabeled influx (Fs) at succinyl-CoA. It should be noted that since succinyl thiokinase is freely reversible, both succinate and succinyl-CoA were treated as a single pool in our model. Although Fs has been restricted to influx at succinyl-CoA, the network cannot discriminate between unlabeled substrate influx here or via other anaplerotic pathways; the net result is the same i.e. dilution of glutamate C-3 enrichment relative to C-4
      In order to maintain constant pool size in the TCA cycle intermediates and to allow the possibility of achieving a steady state, a branch point in the network at the malate pool via malic enzyme had to be included. Malic enzyme has been shown to be active in rat heart mitochondria by a number of workers(
      • Newsholme E.A.
      • Williams T.
      ,
      • Brdiczka D.
      • Pette D.
      ,
      • Nolte J.
      • Brdiczka D.
      • Pette D.
      ,
      • Nagel W.O.
      • Dauchy R.T.
      • Sauer L.A.
      ). In terms of the reaction network, the branching opens an alternative pathway from malate to citrate with different fates for the individual carbon atoms. It would be impossible to maintain constant pool sizes if the pathway from mitochondrial malate to mitochondrial pyruvate were omitted from the network.
      Examination of the networks in Figure 1:, Figure 2:, Figure 3: indicates that the absence of malic enzyme in combination with proteolysis would lead to the flux into acetyl-CoA being less than cycle flux and the flux into oxaloacetate being greater than TCA cycle flux. The result of this would be a decrease in acetyl-CoA and malate pools and an increase in oxaloacetate and citrate. A depletion of acetyl-CoA would ultimately lead to a cessation of the TCA cycle due to the lack of entry of 2-carbon fragments and thus a decline in function and ultimately an imbalance in the demand and supply of ATP. However, cardiac function and high energy phosphates are stable in our experiments. It is possible that the additional acetyl-CoA moieties could be supplied by β-oxidation; if this were the case then the fractional enrichment of glutamate C-4 would be significantly less than we measured. Furthermore, if there was an accumulation of citrate this would be evident in the NMR spectra as a resonance at approximately 45 ppm, which was not observed. It therefore seems justified to include the malic enzyme in the network.
      All the reactions in the network are considered to be irreversible with the exception of the transaminases and fumarase. The reverse flux of the cytosolic transaminase was varied as an unknown parameter. The best agreement between the experimental and calculated results was obtained when the reverse reaction was slow compared with the forward reaction, indicating that the cytosolic transaminase is out of equilibrium. High forward and backward rates for fumarase relative to the TCA cycle were required to maintain a positive velocity and to achieve and maintain the fumarate and malate pool sizes at their equilibrium values.
      Although there is evidence of pyruvate carboxylase activity in the heart(
      • Malloy C.R.
      • Sherry A.D.
      • Jeffrey F.M.H.
      ,
      • Davis E.J.
      • Spydevolt Ø.
      • Bremer J.
      ,
      • Peuhkurinen K.J.
      • Hassinen I.E.
      ,
      • Peuhkurinen K.J.
      • Nuutinen E.M.
      • Pietiläinen E.P.
      • Hiltunen J.K.
      • Hassinen I.E.
      ), this pathway has not been included in the network. Flux through this enzyme is usually activated in response to an increase in TCA cycle intermediate pool sizes as occurs following potassium arrest or alterations in available substrates(
      • Peuhkurinen K.J.
      • Hassinen I.E.
      ,
      • Peuhkurinen K.J.
      • Nuutinen E.M.
      • Pietiläinen E.P.
      • Hiltunen J.K.
      • Hassinen I.E.
      ,
      • Nuutinen E.M.
      • Peuhkurinen K.J.
      • Pietiläinen E.P.
      • Hiltunen J.K.
      • Hassinen I.E.
      ). In our experiments metabolite pool sizes are constant; thus, it is unlikely that there is significant flux through pyruvate carboxylase. Furthermore, even if there was a small flux through pyruvate carboxylase, in the experiments with unlabeled glucose plus 13C-labeled acetate or β-hydroxybutyrate, this flux would be accounted for by Fs, as the net result would be a dilution of glutamate C-2 and C-3 enrichment. Although metabolism of [1-13C]glucose via pyruvate carboxylase would initially lead to an inequality in the enrichments of glutamate C-2 and C-3, the high forward and backward rates of fumarase will result in complete randomization in the labeling of glutamate C-2 and C-3. Therefore, a small contribution from pyruvate carboxylase to the overall TCA cycle flux cannot be ruled out. However, even if there is a small flux through pyruvate carboxylase, this would not affect the calculated MVO2, the exchange of reducing equivalents between the mitochondria and the cytosol or the labeling of the glutamate pool via the malate-aspartate shuttle.

      Numerical Methods

      The nonlinear least squares analysis of the kinetics of glutamate labeling was carried out using FACSIMILE (ARC Scientific, Oxford, United Kingdom), a computer program specially designed for the solution of large networks of simultaneous, ordinary differential equations(
      • Chance E.M.
      • Curtis A.R.
      • Jones I.P.
      • Kirby C.R.
      ). The use of FACSIMILE overcomes many of the problems inherent in the solution of large networks of differential equations. It uses the laws of mass action to derive the differential equations based on the network provided by the user. Furthermore since the program uses vectors of variables, operations of large arrays can be executed as a single instruction. This approach also enables changes in the network to be carried out with ease. Previously, the solution of such networks of differential equations required a main frame, large scale, sequential processor(
      • Chance E.M.
      • Seeholzer S.H.
      • Kobayashi K.
      • Williamson J.R.
      ); however, as a result of advances in both hardware and software, the analyses presented here were carried out on a 486-DX2, 50 MHz laptop in less than 3 min.

      RESULTS

      A typical series of 13C NMR spectra recorded from a heart perfused with [4-13C]β-hydroxybutyrate is shown in Fig. 4. The label first appeared in the C-4 position of glutamate and subsequently in the C-2 and C-3 positions; no other resonances were observed. With [1-13C]glucose as substrate the only differences were the additional incorporation of the 13C label into lactate, alanine, and glycogen. The time course of labeling of glutamate C-4 and C-3 when perfused with [1-13C]glucose, [4-13C]-β-hydroxybutyrate plus unlabeled glucose, or [2-13C]acetate as substrates is shown in Fig. 5. Since the enrichment time curves for glutamate C-2 and C-3 were indistinguishable, only the C-3 data were used in the model. The optimized fit of the mathematical model to the experimental data is plotted along with the experimentally determined values.
      Figure thumbnail gr4
      Figure 4:Series of 13C NMR spectra from isolated perfused rat heart with [4-13C]β-hydroxybutyrate as substrate for 50 min. The spectra have been expanded to show the glutamate region only.
      Figure thumbnail gr5
      Figure 5:Time course of labeling of glutamate C-4 and C-3 with [1-13C]glucose as substrate from the same experiment as the data in (a); with [4-13C]β-hydroxybutyrate plus unlabeled glucose as substrate (data from ) (b); and with [2-13C]acetate as substrate (data from ) (c). The data points are the experimentally determined values and the lines have been calculated by the reaction network.
      The unknown fluxes and rates of oxygen uptake (which to some extent depend on these fluxes) calculated by nonlinear least squares analysis for all six experiments are given in Table 2. The topology of fluxes, which includes those fluxes that depend on the net influxes, is presented schematically in Figure 1:, Figure 2:, Figure 3: for each perfusion condition.
      Although proteolysis was not well determined by the network, in five of the six experiments the flux was significantly greater than zero (Table 2). That is, although the confidence limits may be relatively large, omission of Fs from the network resulted in a poorer fit between the calculated glutamate labeling kinetics and the experimental data. The calculated rate of proteolysis was between 0.02-0.12 μmol/min/g of wet weight, which is in good agreement with other studies of proteolysis in the isolated perfused rat heart(
      • Takala T.
      • Hiltunen J.K.
      • Hassinen I.E.
      ,
      • Peuhkurinen K.J.
      • Takala T.E.S.
      • Nuutinen E.M.
      • Hassinen I.E.
      ,
      • Pisarenko O.I.
      • Solomatina E.S.
      • Studneva I.M.
      ), which report rates of 0.01-0.05 μmol/min/g of wet weight. In one experiment the influx from proteolysis was not significantly different from zero. In other words, it was not possible to determine differences in end point enrichments between glutamate C-4 and C-3.
      This could be due to greater noise in the data in this experiment, or it could be because there was no proteolysis in this experiment. The results from the model were also consistent with the contribution of anaplerosis to the overall TCA cycle determined from the ratio of glutamate C-3/C-4 intensities in the tissue extracts(
      • Malloy C.R.
      • Sherry A.D.
      • Jeffrey F.M.H.
      ). The calculated values for proteolysis ranged from 0 to 5.4% of the total TCA cycle flux compared with 1.9-8.3% calculated from the high resolution 13C NMR spectra for all six experiments. The mean of experimentally measured values of proteolysis for the glucose (2.9%) and acetate (2.4%) experiments compared well with those calculated by the model (2.7 and 2.1%, respectively), despite the fact that these values were not statistically well determined. In the ketone body experiments the model underestimated proteolysis at a mean of 2.3% compared with 7.7% calculated from the high resolution 13C NMR spectra of the extracts.
      Previous 13C NMR experiments with the rat heart showed that alanine is labeled only on pyruvate or glucose perfusion and not on acetate perfusion(
      • Chance E.M.
      • Seeholzer S.H.
      • Kobayashi K.
      • Williamson J.R.
      ). This was confirmed in the current study, as we did not observe any alanine or lactate labeling with mitochondrial substrates. This observation suggests that the only extramitochondrial pathway is the labeling of cytoplasmic glutamate by the malate-aspartate shuttle. This agrees with the previous results from 14C-tracer experiments (
      • Mowbray J.
      • Ottaway J.H.
      ,
      • Peuhkurinen K.J.
      • Hiltunen J.K.
      • Hassinen I.E.
      ) and suggests that in these experiments the pathway from propionyl-CoA to pyruvate via malic enzyme is located entirely in the mitochondria. Although extramitochondrial activity has been observed in rat heart(
      • Brdiczka D.
      • Pette D.
      ), the observations mentioned above suggest that extramitochondrial malic enzyme is not active in these perfusions and therefore was not included in the analysis. In contrast to the earlier study (
      • Chance E.M.
      • Seeholzer S.H.
      • Kobayashi K.
      • Williamson J.R.
      ) where the pathway of glutamate labeling in the cytosol was by direct exchange, we have included metabolite transport across the mitochondrial membrane by the malate-aspartate shuttle.
      The magnitude of the specific fluxes in Table 2 is dependent on the work carried out by the heart for each experiment. For the six experiments in this study, the work load, defined as the rate pressure product (i.e. heart rate × developed pressure) ranged from 18,800 to 39,200 mmHg/min. In order for the network to adequately describe the experimental results, both the calculated kinetics of glutamate labeling and the calculated oxygen consumption had to agree with the experimentally measured values. Good agreement between the calculated and experimental MVO2 required inclusion of a redox shuttle in addition to the malate-aspartate shuttle to export reducing equivalents from the cytosol to the mitochondria (Fig. 6). In the absence of any evidence to the contrary, we have assumed that this additional redox shuttle is the glycerol phosphate shuttle; however, the precise nature of the shuttle is unimportant as long as the net effect is to transfer reducing equivalents from the cytosol to the mitochondria.
      Figure thumbnail gr6
      Figure 6:Comparison between the experimentally measured and the calculated oxygen consumption (MVO2) with and without the glycerol phosphate shuttle. Error bars represent 5 and 95% confidence limits of calculated values.
      In the case of the glucose experiment, the rate of the malate-aspartate shuttle as determined by the kinetics of glutamate labeling was too slow to account for the total reducing equivalents generated by the rate of glycolysis and too slow to produce the observed value of MVO2. For the acetate-perfused and the β-hydroxybutyrate plus glucose-perfused hearts, the malate-aspartate shuttle had to be reversed in order to produce a network which would: 1) maintain a constant metabolite pool size, 2) account for the observed labeling kinetics of glutamate, and 3) predict the observed MVO2. As a consequence, an additional hydrogen shuttle was required to reoxidize the NADH produced by cytoplasmic malate dehydrogenase. A comparison between the measured and calculated MVO2 in the presence and absence of the additional redox shuttle is shown in Fig. 6. It is clear from these data that inclusion of the glycerol phosphate shuttle significantly improves the agreement between the calculated and measured MVO2 for all the experiments.
      The fractional enrichment at glutamate C-4 that results from the metabolism of labeled substrates can be determined from the high resolution 13C NMR spectra of tissue extracts(
      • Malloy C.R.
      • Sherry A.D.
      • Jeffrey F.M.H.
      ). For glucose and β-hydroxybutyrate, the maximal fractional enrichment is 50% and for acetate it is 100%; enrichment less than these maximal values indicates entry of unlabeled substrate at acetyl-CoA. We compared the calculated fractional enrichment at α-ketoglutarate C-4 with the measured enrichment at glutamate C-4; at steady state these should be equal. The mean of the experimentally measured values of glutamate enrichment for the glucose (41%), ketone body (47.2%), and acetate (92.4%) experiments compared well with those calculated by the model (46.1, 47.2, and 86.7% respectively). The contribution of exogenous substrate to the overall TCA cycle flux for all six experiments was 89.6 ± 6.5% (mean ± S.D.) as measured in the tissue extracts which was in good agreement with the value 91.4 ± 4.2% calculated by the model.
      It is apparent from these data that there was significant dilution at glutamate C-4. Isotopic dilution at glutamate C-4 is a result of an influx of endogenous substrates into the acetyl-CoA pool. Two principal pathways that could be responsible for isotopic dilution of acetyl-CoA are glycolysis (Fgl) and β-oxidation of endogenous fatty acids (Fb). The best fit with the experimental data was obtained with the majority of this unlabeled influx entering via glycolysis rather than β-oxidation (i.e. Fgl > Fb).
      Inspection of the observed mathematical form of label incorporation into glutamate C-3 shows a hyperbolic form for the glucose experiments and a marked delay in β-hydroxybutyrate and acetate experiments. As a result, a sigmoidal curve for the labeling kinetics of glutamate C-3 provides a more accurate representation for hearts perfused with β-hydroxybutyrate plus glucose or acetate as sole substrate(s). In all cases labeling of glutamate C-4 showed a direct, hyperbolic response. On closer examination of the results, this was also observed in the earlier work by Chance et al.(
      • Chance E.M.
      • Seeholzer S.H.
      • Kobayashi K.
      • Williamson J.R.
      ). The sigmoidal response in the calculated curves was obtained when the cytoplasmic transaminase reaction was out of equilibrium. This was achieved by allowing the reverse flux to vary as an unknown parameter.
      Although the experimental data indicate that a similar situation may exist in the glucose perfused hearts, this observation is critically dependent on the early time points, which are poorly defined in this experiment. Thus, it is not possible to distinguish mathematically between a rectangular hyperbole or a sigmoid curve for the glutamate C-3 enrichment kinetics in the glucose experiments. In the studies using extracts(
      • Chance E.M.
      • Seeholzer S.H.
      • Kobayashi K.
      • Williamson J.R.
      ), earlier time points are better defined and clearly demonstrate the sigmoidal nature of the glutamate C-2 labeling with acetate. Fitting these data with the new model using both a fast reverse reaction and a slow reverse reaction is shown in Fig. 7. It is clear from these data that the fast reverse reaction overestimates all the observed values for glutamate C-3 (Fig. 7a). In contrast, with a slow back reaction, statistical analysis indicates that there is a random distribution of residuals and a lower sum of squares of residuals (Fig. 7b).
      Figure thumbnail gr7
      Figure 7:Time course of labeling of glutamate C-4 and C-3 with [2-13C]acetate as substrate from the original work by Chance et al.(
      • Chance E.M.
      • Seeholzer S.H.
      • Kobayashi K.
      • Williamson J.R.
      ) comparing the effect of rapid (a) and slow (b) exchange of the cytosolic transaminase reactions.

      DISCUSSION

      The network described here results in excellent agreement between the measured and calculated metabolic parameters over a range of work loads and under different substrate conditions. The principal assumptions that we have made are: 1) that the reactions described by the network are consistent with the known biochemistry of the system and 2) that the glutamate pool is evenly distributed between the mitochondrial and cytosolic compartments. The relative size of these compartments had no impact on the calculated fluxes; varying the mitochondrial compartment between 5 and 40% of the total cell volume did not significantly affect the results. A sensitivity analysis on the effects of changes in intermediate pool sizes indicated that the only pool sizes that significantly affected the calculations were those large enough to be determined by NMR spectroscopy. Therefore the only metabolite pool size that was of significance in our experiments was the glutamate pool, which was determined by enzymatic analysis for each experiment.
      The main limitation of these results is the quality of the 13C kinetic data from the intact heart. The agreement between the experimental and the calculated labeling kinetics is not as good in the hearts perfused with labeled glucose as for those hearts perfused with either labeled acetate or β-hydroxybutyrate. This probably results from a 50% dilution of labeled acetyl-CoA units originating from the [1-13C]glucose combined with a relatively small glutamate pool. In the [4-13C]β-hydroxybutyrate experiments there is a similar dilution at acetyl-CoA (50%), but the glutamate concentration is increased 2-fold compared with hearts perfused with glucose as the sole substrate; thus, there is better agreement between the calculated and experimental data. The best agreement between the calculated and experimental data is observed with [2-13C]acetate as substrate, since 100% of the acetyl-CoA units originating from [2-13C]acetate are labeled.
      One of major differences between the three substrate conditions was the ratio of TCA cycle flux to MVO2 (i.e. RFO2). In acetate-perfused hearts RFO2 was approximately 2, which agrees well with earlier studies(
      • Weiss R.G.
      • Gloth S.T.
      • Kalil-Filho R.
      • Chacko V.P.
      • Stern M.D.
      • Gerstenblith G.
      ), indicating that the NADH produced is entirely accounted for by TCA cycle flux. However, in both glucose and β-hydroxybutyrate perfused hearts the RFO2 is significantly greater than that seen with acetate (Table 2), which is consistent with the existence of other NADH producing reactions such as glycolysis and oxidation of β-hydroxybutyrate.
      Based on current knowledge the only mechanism for labeling the cytoplasmic glutamate pool from a very small mitochondrial α-ketoglutarate pool is via the malate-aspartate shuttle. In all cases the calculated flux of this shuttle, determined by the kinetics of glutamate labeling, is too slow to account for the measured values of MVO2. With glucose as substrate the expected pathway from cytoplasmic NADH to mitochondrial NADH would be the malate-aspartate shuttle; however, the rate of the malate-aspartate shuttle is of the order of only 20% of the total flux of reducing power generated by glycolysis. In other words, the rate of cytoplasmic glutamate labeling was too slow to account for the rate of NADH production in the cytoplasm (i.e. Fgl > Fms). For the mitochondrial substrates, the rate of endogenous glycolysis, calculated from the dilution of label at glutamate C-4, was too small to account for the rate of glutamate labeling (i.e. Fgl < Fms). Consequently, as the labeling of cytoplasmic glutamate requires oxidation/reduction reactions, it was necessary to add an additional redox shuttle and to reverse the malate shuttle (compared with the direction when glucose was sole substrate).
      The precise nature of this additional redox shuttle is unimportant as long as the net effect is to transfer reducing equivalents from the cytosol to the mitochondria; however, a promising candidate for this reaction is the glycerol phosphate shuttle. Isaac et al.(
      • Isaacs G.H.
      • Sacktor B.
      • Murphy T.A.
      ) provided evidence that the glycerol phosphate shuttle was operational in cardiac tissue; however, Safer et al.(
      • Safer B.
      • Smith C.M.
      • Williamson J.R.
      ) suggested that hydrogen flux through the shuttle was limited under normal conditions by low concentrations of glycerol 3-phosphate. It is well known, however, that low concentrations of metabolites can be assayed by cycling reactions, and thus low concentrations of glycerol 3-phosphate need not preclude shuttle activity. Furthermore, it should be noted that the incorporation of 13C label from [1-13C]glucose into glycerol 3-phosphate has been observed under similar conditions(
      • Chacko V.P.
      • Weiss R.G.
      ), indicating that at least the cytoplasmic portion of this shuttle is operative.
      From our analysis we have shown that relatively subtle changes in the network can have a significant impact on the agreement between the experimental and calculated results. From the experimental data for acetate and β-hydroxybutyrate, the labeling kinetics of glutamate C-3 shows a lag compared with the labeling in the C-4 position. With the cytosolic transaminase at near equilibrium (like fumarase) the mathematical form of the curve does not adequately describe the experimental data; it systematically overestimates the experimental findings (Fig. 7a). Some lag is the result of the necessity of more than one complete turn of the TCA cycle in order for label to be incorporated into glutamate C-3. This is insufficient, however, to account for the experimental results. When the reverse flux of transaminase was varied as an unknown parameter, the best fit to the experimental data was obtained when this flux was exceedingly small (Fig. 7b). Although there also appears to be a lag time in the C-3 labeling kinetics in the glucose experiments, there are insufficient early time points to determine effectively the mathematical form of the glutamate C-3 curve.
      In order to account for the observed isotopic dilution at the glutamate C-4, influxes from unlabeled substrates via glycolysis (Fgl) or β-oxidation (Fb) were included in the network. In all the experiments where there was significant dilution of glutamate C-4, Fgl was significantly greater than Fb. For the mitochondrial substrates this flux from glycolysis could be attributed to the metabolism of exogenous unlabeled glucose or with acetate as the sole substrate resulting from glycogenolysis. In the glucose experiments glycogen synthesis was clearly evident (data not shown); thus, it is unlikely that the glutamate C-4 dilution is a result of glycogen breakdown. An alternative explanation is that the isotopic dilution is a result of metabolism via the pentose phosphate pathway. When 6-phosphogluconate is metabolized to 5-ribulose phosphate, the C-1 carbon of glucose is lost as CO2; thus, if the C-1 position is labeled, there will be a loss of label which will appear as an influx of unlabeled glucose into acetyl-CoA. The dilution observed in these glucose experiments (i.e. 6-7%) is in agreement with a recent study by Chatham et al.(
      • Chatham J.C.
      • Forder J.R.
      • Glickson J.D.
      ) which indicated a 5-6% dilution at alanine and acetyl-CoA in hearts perfused with [1-13C]glucose. Therefore it is possible that in the glucose experiments Fgl is a measure of pentose phosphate activity.
      The approach presented here is only one of many possible computational methods for modeling cardiac metabolism. For example, Cohen and Bergman (
      • Cohen D.M.
      • Bergman R.N.
      ,
      • Cohen D.M.
      • Bergman R.N.
      ) recently presented a syntactical approach which avoided the use of differential equations. The problems inherent in the solution of large networks of differential equations as described by Cohen and Bergman(
      • Cohen D.M.
      • Bergman R.N.
      ,
      • Cohen D.M.
      • Bergman R.N.
      ), were minimized in our study by using FACSIMILE, a program designed specifically for the solution of complex reaction networks(
      • Nuutinen E.M.
      • Peuhkurinen K.J.
      • Pietiläinen E.P.
      • Hiltunen J.K.
      • Hassinen I.E.
      ). This enabled us to construct a network with few assumptions that included both cytosolic and mitochondrial compartments as well as the malate-aspartate shuttle. Therefore, we have been able to construct a more complete mathematical description of cardiac metabolism than previously published. Furthermore, although the syntactical approach (
      • Cohen D.M.
      • Bergman R.N.
      ,
      • Cohen D.M.
      • Bergman R.N.
      ) may be computationally simpler, they required 1.5-14.5 h to execute a single simulation on a SPARCstation IPX. In contrast, using our more complete model a single simulation is obtained in a few seconds on a 486-DX2, 50 MHz laptop. In order to optimize the fit to the experimental data approximately 100 simulations are required and thus a complete solution was obtained within only a few minutes.
      In order to completely describe all the reactions included in the network a total of 340 differential equations are required. The complete set of differential equations enables the position of every carbon atom in the network to be accounted for at any one time. Clearly for any specific 13C-labeled substrate only a subset of equations will be relevant since not all carbon atoms in all the intermediates will become enriched. Thus, it would be possible to simplify the network for a specific set of conditions; however, extensive coding changes would be required. Consequently the time needed to simplify the network would be excessive, given that the full network can used to fit the experimental data in a matter of a few minutes on a personal computer. Furthermore, if such simplification was made, a new model would have to be constructed for each set of experimental of conditions. In contrast using the full network it is straight forward to use different substrates or combinations of labeled substrates.
      Using the mathematical system described above, we have obtained excellent agreement between the calculated and experimentally determined oxygen consumption rates for hearts perfused with glucose, with glucose plus β-hydroxybutyrate, and with acetate. The calculated time courses of enrichment of glutamate C-4 and C-3 also agree well with the experimental data. It should be emphasized that the network of labeling equations used to calculated these fluxes is highly constrained. The measured oxygen consumption rates, the kinetics of labeling of the different glutamate isotopomers as well as their different steady state enrichments all need to be satisfied by the model. Furthermore, the network must be consistent with the known biochemistry of the system and must maintain a steady state reflected by constant metabolite pool sizes. As a result of the steady state constraint, the internal fluxes (i.e. enzyme activities) depend on the external influxes, and the only unknowns are the exogenous and endogenous influxes and the reverse flux of the transaminases. Thus, despite the complexity of the network, the number of unknown parameters (i.e. total substrate influx; rate of proteolysis (Fs); magnitude of the flux of the malate-aspartate shuttle and reverse flux for the cytoplasmic transaminase) is relatively small. The elucidation of the mechanism of glutamate labeling offers the prospect of a noninvasive method for determining the rate of oxygen uptake in tissue, in general, and cardiac muscle in particular. This work provides a framework with which to examine the complex relationship between oxygen consumption, work, and substrate utilization.

      REFERENCES

        • Dahlen J.V.
        • Porter J.W.
        Arch. Biochem. Biophys. 1968; 127: 207-223
        • den Hollander J.A.
        • Behar K.L.
        • Shulman R.G.
        Proc. Natl. Acad. Sci. U. S. A. 1981; 76: 2693-2697
        • Dickinson J.R.
        • Dawes I.W.
        • Boyd A.S.F.
        • Baxter R.L.
        Proc. Natl. Acad. Sci. U. S. A. 1983; 80: 5847-5851
        • Lyon R.C.
        • Cohen J.S.
        • Faustino P.J.
        • Megnin F.
        • Myers C.E.
        Cancer Res. 1988; 48: 870-877
        • Jans A.W.H.
        • Leibfritz D.
        NMR Biomed. 1989; 1: 171-176
        • Kuchel P.W.
        • Berthon H.A.
        • Bubb W.A.
        • Bulliman B.T.
        • Collins J.G.
        Biomed. Biochim. Acta. 1990; 49: 757-770
        • Bailey I.A.
        • Gadian D.G.
        • Matthews P.M.
        • Radda G.K.
        • Seeley P.J.
        FEBS Lett. 1981; 123: 315-318
        • Malloy C.R.
        • Sherry A.D.
        • Jeffrey F.M.H.
        FEBS Lett. 1987; 212: 58-62
        • Hoekenga D.E.
        • Brainard J.R.
        • Hutson J.Y.
        Circ. Res. 1988; 62: 1065-1074
        • Weiss R.G.
        • Chacko V.P.
        • Glickson J.D.
        • Gerstenblith G.
        Proc. Natl. Acad. Sci. U. S. A. 1989; 86: 6426-6430
        • Lewandowski E.D.
        Biochemistry. 1992; 31: 8916-8923
        • Chatham J.C.
        • Forder J.R.
        J. Mol. Cell. Cardiol. 1993; 25: 1203-1213
        • Reo N.V.
        • Siegfried B.A.
        • Ackerman J.J.H.
        J. Biol. Chem. 1984; 259: 13664-13667
        • Rothman D.L.
        • Behar K.L.
        • Hetherington H.P.
        • Den Hollander J.A.
        • Bendall M.R.
        • Petroff O.A.C.
        • Shulman R.G.
        Proc. Natl. Acad. Sci. U. S. A. 1985; 82: 1633-1637
        • Fitzpatrick S.M.
        • Hetherington H.P.
        • Behar K.L.
        • Shulman R.G.
        J. Cereb. Blood Flow Metab. 1990; 10: 170-179
        • Robitaille P.M.L.
        • Rath D.P.
        • Abduljalil A.M.
        • O'donnell J.M.
        • Jiang Z.
        • Zhang H.
        • Hamlin R.L.
        J. Biol. Chem. 1993; 268: 26296-26301
        • Jue T.
        • Rothman D.L.
        • Shulman G.I.
        • Tavitian B.A.
        • DeFronzo R.A.
        • Shulman R.G.
        Proc. Natl. Acad. Sci. U. S. A. 1989; 86: 4489-4491
        • Beckman N.
        • Turkalj I.
        • Seelig J.
        • Keller U.
        Biochemistry. 1991; 30: 6362-6366
        • Malloy C.R.
        • Sherry A.D.
        • Jeffrey F.M.H.
        J. Biol. Chem. 1988; 263: 6964-6971
        • Malloy C.R.
        • Sherry A.D.
        • Jeffrey F.M.H.
        Am. J. Physiol. 1990; 259: H987-H995
        • Malloy C.R.
        • Thompson J.R.
        • Jeffrey F.M.H.
        • Sherry A.D.
        Biochemistry. 1990; 29: 6756-6761
        • Chance E.M.
        • Seeholzer S.H.
        • Kobayashi K.
        • Williamson J.R.
        J. Biol. Chem. 1983; 258: 13785-13794
        • Weiss R.G.
        • Gloth S.T.
        • Kalil-Filho R.
        • Chacko V.P.
        • Stern M.D.
        • Gerstenblith G.
        Circ. Res. 1992; 70: 392-408
        • Chatham J.C.
        • Cousins J.P.
        • Glickson J.D.
        J. Mol. Cell. Cardiol. 1990; 22: 1187-1197
        • Chatham J.C.
        • Glickson J.D.
        Biochim. Biophys. Acta. 1992; 1138: 1-5
        • Barker P.B.
        • Sibisi S.
        Nonlinear Least Squares Analysis of in Vivo P NMR Data, Ninth Annual Scientific Meeting, August 18-24, 1990, New York. Society of Magnetic Resonance in Medicine, New York1990
      1. Bergmeyer H.U. Bergmeyer J. Grassl M. Methods of Enzymatic Analysis. VCH Verlagsgesellschaft, Weinheim, Germany1985
        • Bers D.M.
        Excitation-Contraction Coupling and Cardiac Contractile Force. Kluwer Academic Publishers, Norwell, MA1991
        • Newsholme E.A.
        • Williams T.
        Biochem. J. 1978; 176: 623-626
        • Brdiczka D.
        • Pette D.
        Eur. J. Biochem. 1971; 19: 546-551
        • Nolte J.
        • Brdiczka D.
        • Pette D.
        Biochim. Biophys. Acta. 1972; 284: 497-507
        • Nagel W.O.
        • Dauchy R.T.
        • Sauer L.A.
        J. Biol. Chem. 1980; 255: 3849-3854
        • Davis E.J.
        • Spydevolt Ø.
        • Bremer J.
        Eur. J. Biochem. 1980; 110: 255-262
        • Peuhkurinen K.J.
        • Hassinen I.E.
        Biochem. J. 1982; 202: 67-76
        • Peuhkurinen K.J.
        • Nuutinen E.M.
        • Pietiläinen E.P.
        • Hiltunen J.K.
        • Hassinen I.E.
        Biochem. J. 1982; 208: 577-581
        • Nuutinen E.M.
        • Peuhkurinen K.J.
        • Pietiläinen E.P.
        • Hiltunen J.K.
        • Hassinen I.E.
        Biochem. J. 1981; 194: 867-875
        • Chance E.M.
        • Curtis A.R.
        • Jones I.P.
        • Kirby C.R.
        FACSIMILE: A Computer Program for Flow and Chemistry Simulation and General Initial Value Problems. H. M. Stationary Office, London1977
        • Takala T.
        • Hiltunen J.K.
        • Hassinen I.E.
        Biochem. J. 1980; 192: 285-295
        • Peuhkurinen K.J.
        • Takala T.E.S.
        • Nuutinen E.M.
        • Hassinen I.E.
        Am. J. Physiol. 1983; 244: H281-H288
        • Pisarenko O.I.
        • Solomatina E.S.
        • Studneva I.M.
        Biochim. Biophys. Acta. 1986; 885: 154-161
        • Mowbray J.
        • Ottaway J.H.
        Eur. J. Biochem. 1973; 36: 369-379
        • Peuhkurinen K.J.
        • Hiltunen J.K.
        • Hassinen I.E.
        Biochem. J. 1983; 210: 193-198
        • Isaacs G.H.
        • Sacktor B.
        • Murphy T.A.
        Biochim. Biophys. Acta. 1969; 177: 196-203
        • Safer B.
        • Smith C.M.
        • Williamson J.R.
        J. Mol. Cell. Cardiol. 1971; 2: 111-124
        • Chacko V.P.
        • Weiss R.G.
        Am. J. Physiol. 1993; 264: C755-C760
        • Chatham J.C.
        • Forder J.R.
        • Glickson J.D.
        J. Mol. Cell. Cardiol. 1994; 26: CLXXVI
        • Cohen D.M.
        • Bergman R.N.
        Comput. Biomed. Res. 1994; 27: 130-147
        • Cohen D.M.
        • Bergman R.N.
        Am. J. Physiol. 1994; 266: E341-E350