Model applicable to NMR studies for calculating flux rates in five cycles involved in glutamate metabolism.

Based on the same principles as those utilized in a recent study for modeling glucose metabolism (Martin, G., Chauvin, M. F., Dugelay, S., and Baverel, G. (1994) J. Biol. Chem. 269, 26034-26039), a method is presented for determining metabolic fluxes involved in glutamate metabolism in mammalian cells. This model consists of five different cycles that operate simultaneously. It includes not only the tricarboxylic acid cycle, the “oxaloacetate → phosphoenolpyruvate → pyruvate → oxaloacetate” cycle and the “oxaloacetate → phosphoenolpyruvate → pyruvate → acetyl-CoA → citrate → oxaloacetate” cycle but also the “glutamate → α-ketoglutarate → glutamate” and the “glutamate → glutamine → glutamate” cycles. The fates of each carbon of glutamate, expressed as ratios of integrated transfer of this carbon to corresponding carbons in subsequent metabolites, are described by a set of equations. Since the data introduced in the model are micrograms of atom of traced carbon incorporated into each carbon of end products, the calculation strategy was determined on the basis of the most reliable parameters determined experimentally. This model, whose calculation routes offer a large degree of flexibility, is applicable to data obtained by 13C NMR spectroscopy, gas chromatography − mass spectrometry, or 14C counting in a great variety of mammalian cells.

In the accompanying paper (16), we have conducted a study on glutamate metabolism in isolated rabbit kidney tubules. For the interpretation of the data obtained, we have constructed a mathematical model that is based on the incorporation of 13 C and 14 C into various metabolites and allows the calculation of reaction rates of gluconeogenesis, tricarboxylic acid cycle, and the pathways of glutamate and glutamine synthesis and degradation occurring simultaneously in mammalian cells. This model, which is applicable to data obtained by 13 C NMR, gas chromatography-mass spectrometry, and 14 C counting, is described in the present paper. THEORY Schematic Representation of Glutamate Metabolism-A general representation of glutamate metabolism is given in Fig. 1. This figure shows the main pathways of glutamate metabolism, as well as the main products accumulated during glutamate metabolism. Fig. 2 shows five metabolic cycles that are functioning simultaneously during glutamate metabolism. Oxaloacetate is the only metabolite common to three of these cycles that were referred to as a multicycle in a previous study (1) and are (i) the tricarboxylic acid cycle, (ii) the "OAA 1 3 PEP 1 3 Pyr 1 3 OAA" cycle and (iii) the "OAA 3 PEP 3 Pyr 3 AcCoA 1 3 Cit 1 3 OAA" cycle. 1 Glutamate is the metabolite common to the two other cycles that have been introduced to improve the model; these are (iv) the "Glu 3 ␣KG 3 Glu" cycle and (v) the "Glu 3 Gln 3 Glu" cycle. Fig. 3, derived from Fig. 2, allows the calculation of the total amount of oxaloacetate formed from glutamate during 1 h of incubation and, subsequently, the calculation of the amount of the different intermediates and end products formed from glutamate during the same incubation time. From these data, fluxes can be calculated since a flux through a given enzyme is taken as the formation of one product of the reaction catalyzed by this enzyme during 1 h of incubation.
In our model, the calculation of the proportions of each metabolite converted into the next one(s) is based on the fates of individual carbons 3, 5, and 1 of the glutamate molecule together with the fate of the incorporated labeled CO 2 which are represented in Figs. 4 and 5, respectively. Fig. 6 shows the successive proportions allowing us to calculate the amount of labeled oxaloacetate formed from labeled glutamate. These proportions are related to the substrates and not to the products of the reactions.
And let [C y MET] Gluϩ*CO2 be the amount of the metabolite (MET) labeled on its carbon y (where 1 Յ y Յ 6) arising from glutamate plus labeled CO 2 .
Similarly, let [(C y ϩC yЈ )MET] CzGlu be the amount of the metabolite (MET) labeled on its carbon y plus the amount of the metabolite (MET) labeled on its carbon yЈ; let [(C y , yЈ )MET] CzGlu be the amount of the metabolite (MET) labeled simultaneously on its carbons y and yЈ.
Let When the amount of the metabolite accumulated is considered, a ( ) sign is added on the left side inside the brackets [].
Let (Met1 ) Met2) be the proportion of any metabolite resynthesized at each turn of the metabolic cycle that is identified by the two characteristic metabolites Met1 and Met2. For the tricarboxylic acid cycle, the simple notation (TCA)) is used.
Let (Met1 ) Met2 ϩ Met3) be the proportion of any metabolite resynthesized at each turn of the metabolic cycle which is identified by the two characteristic metabolites Met1 and Met2 while a second metabolic cycle identified by Met2 and Met3 is performing an infinite number of turns.
Let (Met1 3 Met2) be the proportion of Met1 converted into Met2 by either one or a succession of enzymatic reactions but without any recycling. The corresponding proportion taking into account the recycling over an infinite number of turns of exclusively the "Glu 3 ␣KG 3 Glu" and "Glu 3 Gln 3 Glu" cycles is indicated by substituting the parentheses () by {}.
Special symbols are used in the notations { v Glu 3 Met2} and { Cit ␣KG 3 OAA} to stress the fact that only added glutamate and only citrate-derived ␣-ketoglutarate, respectively, are concerned.
Let (AcCoA ϩ OAA 3 Cit) be the proportion of acetyl-CoA related to glutamate metabolism that is condensed to oxaloacetate not related to glutamate metabolism to yield direct synthesis of citrate at each turn of the TCA cycle.
Similarly, let (OAA ϩ AcCoA 3 Cit) the proportion of oxaloacetate related to glutamate metabolism which is condensed to acetyl-CoA not related to glutamate metabolism to yield direct synthesis of citrate at each turn of the TCA cycle.
Let [Met1 ® Met2] the flux of conversion of Met1 to Met2. The flux through an enzyme E can also be indicated by [E].
Calculations of the Parameters of the Model-The different notations employed in the figures and in the text to characterize the parameters of our model are also defined in Table I.
The amount (in mol/g dry wt/h) of any given intermediate or end product formed from the substrate (glutamate) can be calculated by multiplying the amount of the substrate removed by g dry wt (X) by the successive proportions of intermediates passing through the different pathways leading to the intermediate or end product of interest. Fig. 1 shows the metabolic pathways involved in glutamate metabolism which include five main metabolic cycles as presented in Fig. 2.
Let's consider individually each metabolic cycle; the recycling factor (RF) is taken as the proportion of any metabolite resynthesized after each complete turn of the cycle of interest.
The corresponding recycling ratio (RR) is taken as the sum of the successive proportions of any metabolite resynthesized during an infinite number of cycle turns plus 1, the proportion of this metabolite present at the beginning of the first cycle turn: A complete turn of the metabolic cycle is considered to have occurred as soon as the metabolite of interest of the cycle is resynthesized once. Fig. 2 shows the recycling factor of each metabolic cycle.
(i) (Glu ) ␣KG) is the recycling factor of the "Glu 3 ␣KG 3 Glu" cycle; where (Glu 3 ␣KG) is the proportion of added glutamate directly converted into ␣-ketoglutarate, and (␣KG 3 Glu) is the proportion of ␣-ketoglutarate directly converted into glutamate.
(ii) (Glu ) Gln) is the recycling factor of the "Glu 3 Gln 3 Glu" cycle; where (Glu 3 Gln) is the proportion of glutamate directly converted into glutamine and (Gln 3 Glu) and (Gln 3 Gln) are the proportions of glutamine converted into glutamate and glutamine accumulated, respectively. Fig. 1 shows that glutamate is either accumulated or con-FIG. 1. Pathways of glutamate metabolism in rabbit kidney tubules. Glutamate which enters the cell can be accumulated or converted by glutamine synthetase into glutamine which can accumulate or be reconverted into glutamate by glutaminase. Glutamate can also be converted into ␣-ketoglutarate either by glutamate dehydrogenase or alanine aminotransferase or aspartate aminotransferase or phosphoserine aminotransferase. The ␣-ketoglutarate formed is either reconverted into glutamate mainly by glutamate dehydrogenase or enters the tricarboxylic acid cycle to give oxaloacetate after having lost one carbon as CO 2 . The oxaloacetate formed after transamination with glutamate by aspartate aminotransferase yields aspartate. Oxaloacetate can also be converted into phosphoenolpyruvate, thanks to the phosphoenolpyruvate carboxykinase reaction, or condense with acetyl-CoA to give citrate and, after decarboxylation, regenerate ␣-ketoglutarate and therefore complete one tricarboxylic acid cycle turn. In the presence of NH 4 ϩ , part of this ␣-ketoglutarate may be reconverted into glutamate resulting, as already mentioned, in accumulation of glutamate or glutamine. The phosphoenolpyruvate formed may be converted into pyruvate by pyruvate kinase, or into glucose by the gluconeogenic pathway, or into serine. Pyruvate, after transamination with glutamate by alanine aminotransferase, yields alanine. Pyruvate can also be accumulated as lactate by lactate dehydrogenase or, after decarboxylation by pyruvate decarboxylase, converted into acetyl-CoA. This acetyl-CoA together with acetyl-CoA originating from endogenous sources and from exogenous acetate, when added to the incubation medium, is condensed to oxaloacetate to give citrate. The amount per g dry wt of added glutamate and added acetate utilized during 1 h of incubation are designed by X and Y, respectively. The notations of the proportion of a metabolite directly converted into the subsequent one(s) is simply given by the figure and can be represented by (precursor metabolite 3 derived metabolite). The notations of the proportions taking into account the recycling in the "glutamate 3 ␣-ketoglutarate 3 glutamate" and "glutamate 3 glutamine 3 glutamate" cycles are presented under "Notation" and in Table I. The non-volatile end products of glutamate metabolism have been underlined and the flux calculation method is indicated in the text. verted into ␣-ketoglutarate or glutamine; therefore, the proportion of glutamate formed which accumulated (Glu 3 Glu) is defined by In the following equations, the proportion of glutamate and ␣-ketoglutarate molecules that are metabolized and then resynthesized through the successive operation of "Glu 3 ␣KG 3 Glu" and "Glu 3 Gln 3 Glu" cycles over a theoretically infinite number of turns are noted {Glu ) ␣KG ϩ Gln} and {␣KG ) Glu ϩ Gln}, respectively.
The resulting glutamate recycling through ␣-ketoglutarate and glutamine, is Ϫ ͑Glu ) ␣KG͒ Ϫ ͑Glu ) Gln͔͒ (Eq. 5) (see also Fig. 2). One can demonstrate that {␣KG ) Glu ϩ Gln}, the resulting ␣-ketoglutarate recycling through glutamate and glutamine, is: Therefore, from the latter equation and Equation 3, it follows that  This figure shows five cycles that are functioning simultaneously; oxaloacetate is the only metabolite common to three of these cycles. The five cycles are as follows: (i) the "Glu 3 ␣KG 3 Glu" cycle in which a proportion (Glu ) ␣KG) of glutamate is recycled at each turn, (ii) the "Glu 3 Gln 3 Glu" cycle in which a proportion (Glu ) Gln) of glutamate is recycled at each turn, (iii) the tricarboxylic acid cycle in which a proportion (TCA )) of oxaloacetate is recycled at each turn, (iv) the "OAA 3 PEP 3 Pyr 3 OAA" cycle in which a proportion (Pyr ) OAA) of oxaloacetate is recycled at each turn, (v) the "OAA 3 PEP 3 Pyr 3 AcCoA 3 Cit 3 OAA" cycle in which a proportion (AcCoA ) OAA) of oxaloacetate is recycled at each turn. The proportions (TCA )) and (AcCoA ) OAA) take also into account the recycling of ␣-ketoglutarate through glutamate and glutamine, resulting from the operation of the "Glu 3 ␣KG 3 Glu" and "Glu 3 Gln 3 Glu" cycles. The effects of the operation of these two latter cycles on the accumulation of glutamate and glutamine and on the formation of ␣-ketoglutarate and oxaloacetate are shown on the right part of the figure. A citrate molecule is obtained from the condensation of one oxaloacetate and one acetyl-CoA molecule. The main proportion of acetyl-CoA molecules derived from glutamate metabolism is condensed with glutamate-derived oxaloacetate whereas the remaining proportion, (AcCoA ϩ OAA 3 Cit), is condensed with oxaloacetate not derived from glutamate, i.e. derived from endogenous sources. The main proportion of oxaloacetate molecules derived from glutamate metabolism, (OAA ϩ AcCoA 3 Cit), is condensed with acetyl-CoA not derived from glutamate, i.e. derived from endogenous sources and acetate (when added as substrate). The remaining proportion is condensed with glutamate-derived acetyl-CoA. The right part of this figure presents the resulting proportions of added glutamate and citrate-derived ␣-ketoglutarate which take into account the total recycling through both "Glu 3 ␣KG 3 Glu" and "Glu 3 Gln 3 Glu" cycles and are noted {Glu ) ␣KG ϩ Gln} and {␣KG ) Glu ϩ Gln}, respectively (see Table I (iv) (TCA )) is the recycling factor of the tricarboxylic acid cycle, i.e. the proportion of oxaloacetate resynthesized at each turn of this cycle. The proportion of citrate converted into ␣-ketoglutarate, noted (Cit 3 ␣KG) is considered to be 1.
The amount of glutamine formed directly from the substrate glutamate is given by X⅐{ v Glu 3 Gln}, where X is the amount of glutamate utilized and { v Glu 3 Gln} is the proportion of the substrate glutamate converted into glutamine. As already mentioned, this proportion takes into account glutamate recycling through ␣-ketoglutarate and glutamine.
It can be seen in Fig. 4 that X⅐{ v Glu 3 Gln} is given by 4. Metabolic fate of the C-5, C-3, and C-1 of glutamate in rabbit kidney tubules. This figure shows the metabolic fate of glutamate labeled either on its carbon 5, 3, or 1 which, for sake of simplicity, is represented as 5,3,1 GLU. Glutamate metabolites are represented as ␣, ␤, ␥ MET, where MET represents any glutamate-derived metabolite and ␣, ␤, and ␥ the labeled carbon of these metabolites when the labeled carbon of the glutamate added as substrate was 5, 3, or 1, respectively. Unlabeled carbons of glutamate metabolites are represented by a minus sign. The amount (in mol/g dry wt/h) of labeled glutamate utilized is represented by X. The proportion of the direct conversion of a metabolite to the next one is indicated by a simple arrow with no special mention. To take into account the fact that some reactions yield a metabolite labeled at two different positions, it is necessary to multiply the proportion of conversion by the proper factor 1/2 or (OAA i ) or 1 Ϫ (OAA i ), as indicated in the figure. For other metabolic conversions it is necessary to take into account the effect of the recycling through the "glutamate 3 ␣-ketoglutarate 3 glutamate" and "glutamate 3 glutamine 3 glutamate" cycles. Depending on which metabolite is recycled, glutamate or ␣-ketoglutarate, the proportion of conversion is multiplied either by {Glu ) ␣KG ϩ Gln} or {␣KG ) Glu ϩ Gln} (see Table I) which are represented by specific arrows consisting of a double line and a dash-stacked line, respectively. For sake of clarity, only direct formation of ␣-ketoglutarate from citrate is shown in this figure. To take into account the recycling of ␣-ketoglutarate through glutamate and glutamine, it is necessary to multiply by {␣KG ) Glu ϩ Gln}. The oxaloacetate recycled after one complete multicycle turn remains labeled only when the substrate glutamate is labeled on its carbon 3. Therefore, the fate of the C-3 of glutamate requires more than one multicycle turn to be defined. The synthesis of oxaloacetate resulting from the first multicycle turn is considered to represent the beginning of the second turn. The relative amount of substrate (labeled glutamate) transformed into any labeled intermediate or end product is obtained by multiplying the successive proportions found in the pathway from the substrate to the intermediate or end product of interest. The amount (named flux), expressed in C3 units of intermediate formed or end product accumulated during the incubation period (1 h), is obtained by multiplying the corresponding relative amount by the amount (X) of labeled glutamate utilized. It is assumed that the proportion 2⅐(OAA i ) of the oxaloacetate formed by the pyruvate carboxylase reaction equilibrates with fumarate; half of this oxaloacetate, equal to (OAA i ), gives rise to oxaloacetate molecules having an inverted labeling pattern.
From Glu 3 Glu}. Fig. 2 shows that the glutamate utilized (X) is either accu- Therefore the following parameters can be calculated:

Fig. 4 allows us to calculate that
where (OAA i ) is the proportion of oxaloacetate inverted as a result of its equilibration with fumarate, a symmetrical molecule (see also Table I Table I). From Fig. 4 one can also deduce that where, as mentioned in Equation 9 (AcCoA ) OAA) ϭ (OAA 3 PEP).(PEP 3 Pyr)⅐(Pyr 3 AcCoA)⅐{ Cit ␣KG 3 OAA} (see also Fig. 2 and Table I) and 3 OAA] C3Glu ⅐[(TCA))/2]) represents the C 1 OAA formed directly from the C-2 and C-3 of oxaloacetate. Fig. 6 shows that Ϫ ͑Pyr ) OAA͒ Ϫ ͓͑TCA)͒ ϩ ͑AcCoA ) OAA͔͒/2͒ (Eq. 14) From Fig. 4, one can deduce that the C-3 of glutamate yields equal amounts of C-2 and C-3 of oxaloacetate, so that Using Equation 14, Equation 13 can be rewritten as Then, And, And,   where the value B of the latter 2 ratios can be calculated from From Fig. 5 it is also possible to calculate that Let Since C was defined as From Fig. 6, it can be deduced that And Fig. 4 shows that where (Pyr 3 Lac) is the proportion of pyruvate directly converted to lactate. Combining Equations 32 and 34, it follows that Similarly,  Fig. 4).
From the latter equation and Equation 37, it follows that Oxaloacetate is either converted into citrate or phosphoenolpyruvate or aspartate (see Fig. 4) with the proportion (OAA Since the recycling factor in the tricarboxylic acid cycle (TCA)), which accounts also for ␣-ketoglutarate recycling through glutamate and glutamine, is equal to (OAA 3 Cit)⅐{ Cit ␣KG 3 OAA} (see Equation 8), it can be calculated that (Eq. 47) From Equation 7 and since as shown in Fig. 1 Glu 3 ␣KG} (Eq. 49) Then, multiplying the 2 members of the latter equation by (␣KG 3 OAA) and rearranging, we obtain The proportion (3PG 3 Glc) of 3-phosphoglycerate which yields glucose is given by (3PG 3 Glc) ϭ J/(J ϩ L) (see also Fig. 4).
Calculations of the Enzymatic Fluxes-It should be stressed that, in this study, we did not calculate enzyme activities. Our model allowed us to calculate only mean fluxes in relation to glutamate metabolism. In this model, as already mentioned, a flux through a given enzyme is taken as the formation of one product per g dry wt and per unit of time (1 h in this study) of the reaction catalyzed by this enzyme. It should be pointed out that oxaloacetate is the only metabolite common to three of the metabolic cycles involved in glutamate metabolism. Therefore, a key step in the calculations of enzymatic fluxes is the determination of the amount of the oxaloacetate molecules that have been formed in relation to glutamate metabolism (noted [OAA] Glu ), these oxaloacetate molecules containing 1, 2, 3, 4, or 0 carbon atoms derived from glutamate. Fig. 3 gives a schematic representation providing the basic elements needed for such a determination. In the left panel of Fig. 3, which is derived from Fig. 2 (Pyr ) OAA), (TCA )) and (AcCoA ) OAA) represent the oxaloacetate recycled in the "OAA 3 PEP 3 Pyr 3 OAA," the tricarboxylic acid and the "OAA 3 PEP 3 Pyr 3 AcCoA 3 Cit 3 OAA" cycles, respectively (see Fig. 2 and Table I).
It should be stressed that the proportions (TCA )) and (AcCoA ) OAA) take also into account the recycling in the "Glu 3 ␣KG 3 Glu" and "Glu 3 Gln 3 Glu" cycles.
Let us call (AcCoA ϩ OAA 3 Cit) the proportion of the acetyl-CoA molecules derived from glutamate that have been condensed with oxaloacetate molecules of endogenous origin to give citrate. It is necessary to introduce this proportion (see Fig. 2) to calculate correctly the oxaloacetate formation from glutamate by avoiding to take into account twice the citrate molecules synthesized from an oxaloacetate and an acetyl-CoA molecules originating both from glutamate. The proportion (AcCoA ϩ OAA 3 Cit) also allows one to take into account the citrate molecules formed from an acetyl-CoA molecule derived from glutamate and an oxaloacetate molecule arising from endogenous substrates. Thus, the proportion of acetyl-CoA derived from glutamate and condensed with endogenous oxaloacetate to give oxaloacetate via the tricarboxylic acid cycle is equal to (AcCoA ϩ OAA 3 Cit)⅐{ Cit ␣KG 3 OAA} (Figs. 2 and 3). Then, at each turn of the multicycle, the additional proportion of oxaloacetate formed as a result of the operation of the "OAA 3 PEP 3 Pyr 3 AcCoA 3 Cit 3 OAA" cycle is (AcCoA ϩ OAA 3 Cit)⅐(AcCoA ) OAA), while the proportion of oxaloacetate formed by the "OAA 3 PEP 3 Pyr 3 OAA" cycle and by the tricarboxylic acid cycle are (Pyr ) OAA) and (TCA)), respectively.
The right panel of Fig. 3 summarizes the oxaloacetate formation from glutamate shown in more detail in the left panel of the same figure. It allows us to calculate the total amount of oxaloacetate derived from glutamate ([OAA] Glu ) using the following equations derived from Fig. 3, in which the repetitiveness of the formation of oxaloacetate by the operation of the multicycle is taken into account thanks to the parameter (Pyr ) OAA) ϩ (TCA)) ϩ (AcCoA ϩ OAA 3 Cit)⅐(AcCoA ) OAA), which represents the proportion of oxaloacetate recycled at each turn of the multicycle presented in Fig. 2.
This proportion can be assessed by the ratio of [2-13 C]Acetyl-CoA, which condenses with [2-13 C]oxaloacetate. This ratio, reflected by the proportion of the C-4 and C-3 of glutamate plus glutamine (noted Glx) found to be coupled on the NMR spectra, was corrected to take into account the total oxaloacetate formation from glutamate. Thus, Since, as mentioned above, the C-3 and C-2 of Glx are formed in equal amounts from [3-13 C]Glu (see Fig. 4), it follows that: Similarly, one can demonstrate that 2 (Eq. 55) where (OAA ϩ AcCoA 3 Cit) is the proportion of oxaloacetate molecules derived from added glutamate that have been condensed with acetyl-CoA molecules not derived from added glutamate.
Finally, the latter equation together with Equations 39 and 54 yield It is possible to determine the amount of oxaloacetate, [OAA] non-Glu , and acetyl-CoA, [AcCoA] non-Glu , not derived from added glutamate that condense with glutamate-derived acetyl-CoA and glutamate-derived oxaloacetate, respectively.
Since the activity of malic enzyme is considered to be negligible in rabbit kidney tubules (2), one can write that [OAA] Glu ϭ flux through pyruvate carboxylase ϩ flux through ␣-ketoglutarate dehydrogenase.
To determine correctly enzymatic fluxes during glutamate metabolism, one should know the total amount of glutamate involved in this process, noted [Glu] Glu , which, as indicated in Figs. 1 and 2, has two possible origins.
(i) The glutamate noted Glu 3 [Glu] Glu is derived directly from the glutamate utilized which undergoes a recycling through ␣-ketoglutarate and glutamine: Glu Equation 5), one can deduce from Equations 62 and 63 that Cit3 ͓Glu͔ Glu ϭ ͓Cit͔ Glu The total amount of glutamate involved in the metabolism is obtained from Equations 59 and 64 but it cannot be calculated because X⅐{Glu ) ␣KG ϩ Gln} and (␣KG 3 Glu)⅐{Glu ) ␣KG ϩ Gln} cannot be derived from the labeled carbon data.
The total amount of ␣-ketoglutarate formed during glutamate metabolism is obtained from the sum of the ␣-ketoglutarate formed directly from the glutamate utilized, Glu 3 [␣KG] Glu where Glu 3 From Equation 65 and Fig. 2 where { Cit ␣KG 3 Glu} is the proportion of citrate-derived ␣-ketoglutarate converted to glutamate which takes into account the total recycling through the "Glu 3 ␣KG 3 Glu" and "Glu 3 Gln 3 Glu" cycles (see under "Notation").
Thus, net flux of glutamate to ␣-ketoglutarate, noted net [Glu ® ␣KG], can be obtained from the two latter equations:   Fig. 2 shows that citrate originates from oxaloacetate through two possible pathways, namely the tricarboxylic acid cycle and the "OAA 3 PEP 3 Pyr 3 AcCoA 3 Cit 3 OAA" cycle. Therefore, the latter equation can be rewritten as (Eq. 75) (see also the definition of (AcCoA ϩ OAA 3 Cit) in Table I).
Note that Equation 75 takes into account the condensation of endogenous oxaloacetate molecules with labeled glutamatederived acetyl-CoA molecules. Fig. 2 and Equation 65 also allow to calculate the accumulation of glutamine: Thus, the accumulation of glutamate plus glutamine is given by The net flux of glutamine accumulation, [Glu ® Gln], is given by the amount of glutamine accumulated from the glutamate available for the metabolism (added glutamate utilized ϩ glutamate synthesized, see Equation 76): The net utilization of glutamate can be obtained from Equations 70, 73, and 78 and is equal to net [ and Since oxaloacetate is produced either by pyruvate carboxylase or ␣-ketoglutarate dehydrogenase, flux through pyruvate carboxylase is equal to [OAA] Glu minus flux through ␣-ketoglutarate dehydrogenase (see Fig. 2).
As shown in a previous study (1), flux of oxaloacetate equilibration with fumarate is equal to [OAA] Glu ⅐ 2⅐(OAA i )/[1 Ϫ 2⅐(OAA i )]. DISCUSSION Our model, which can be used at any time point, is based on proportions of metabolite conversion. It allows us to ignore the status of the system irrespective of whether or not it is in steady state since the resynthesis of the substrate carbons on which most of the calculations are based is small. Other models (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) are based on kinetic reaction rates but were applied under steady state conditions.
With our model, we calculated mean fluxes related to glutamate metabolism over 1 h of incubation; for this we divided the amount per g dry wt of the metabolite of interest (that was formed during the incubation) by the incubation time (1 h in this study). Similarly, it should also be underlined that our parameter values were not necessarily constant with time but were also mean values. For example, it is clear that, at early times of incubation, the 13 C atoms entering the glutamine pool were significantly diluted by the glutamine already present in the tubules at zero time. This resulted in a low proportion of glutamine converted into glutamate (Gln 3 Glu). However, since the glutamine present at zero time was only a small fraction (less than 10%) of the total glutamine found after 60 min of incubation, we may conclude that the impact of what happened during early times was limited when compared with what happened over a 60-min incubation period.
Most of the proportions and equations we used to calculate enzymatic fluxes were derived from the fate of the C-3 of glutamate, which provides more information about all the turns of the tricarboxylic acid cycle and the other cycles than that of the C-5 and the C-1 of glutamate; indeed the latter carbons are released as CO 2 and recovered in the non-volatile products of glutamate metabolism that accumulate only before the end of the first turn of the tricarboxylic acid cycle. In the present study, the data obtained with unlabeled glutamate plus labeled CO 2 as substrate were used to calculate the equilibration of oxaloacetate with fumarate.
It should be emphasized that many proportions could also have been calculated by using different sets of data, yielding similar results. This illustrates the flexibility of our mathematical model which can also be applied not only to glutamate metabolism in tissues other than the kidney but also to data obtained with substrates other than glutamate and under many physiopathological conditions. This model, which includes the simultaneous operation of five interdependent metabolic cycles, represents a significant progress when compared with our previous model of glucose metabolism which involved only three metabolic cycles (1). Indeed, in the present model, the glutamate resynthesized and further metabolized is taken into account. Moreover, this new model allows the calculation of the simultaneous synthesis and degradation of glutamate and ␣-ketoglutarate. Note here that Shulman and co-workers (11,12,15) were also able to calculate the ␣-ketoglutarate Ϫ glutamate exchange in rat and human brain in vivo. In addition, our model of glutamate metabolism allows us to calculate the simultaneous synthesis and degradation of glutamine that result from opposing unidirectional fluxes through glutamine synthetase and glutaminase. Such a more complex description of glutamate metabolism than previously described was made possible by the careful analysis of the labeling pattern and the amount of label recovered in glutamate and glutamine that accumulated after having passed through the tricarboxylic acid cycle.
It should be pointed out that in studies performed in vitro like the present one, it is possible to obtain detailed NMR data, which in turn calls for a highly detailed analysis in order to obtain as much information as possible. Studies performed entirely in vivo, in contrast, avoid physiological uncertainties associated with differences of metabolism in vivo and in vitro but yield much less detailed information due to reduced spectral resolution and limited averaging time.
Finally, depending on experimental data available, our model permits us to calculate either net or unidirectional enzymatic fluxes through the cycles involved in glutamate me-tabolism and brings new insights into the complexity of such a metabolism in mammalian cells.