Model Applicable to NMR Studies for Calculating Flux Rates in Five Cycles Involved in Glutamate Metabolism*

(Received for publication, February 28, 1996, and in revised form, October 24, 1996)

Guy Martin Dagger , Marie-France Chauvin and Gabriel Baverel

From the Centre d'Etudes Métaboliques par Spectroscopie de Résonance Magnétique (INSERM CRI 950102), Hôpital Edouard Herriot, 69374 Lyon Cedex 03, France

ABSTRACT
INTRODUCTION
THEORY
DISCUSSION
FOOTNOTES
REFERENCES


ABSTRACT

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 right-arrow phosphoenolpyruvate right-arrow pyruvate right-arrow oxaloacetate" cycle and the "oxaloacetate right-arrow phosphoenolpyruvate right-arrow pyruvate right-arrow acetyl-CoA right-arrow citrate right-arrow oxaloacetate" cycle but also the "glutamate right-arrow alpha -ketoglutarate right-arrow glutamate" and the "glutamate right-arrow glutamine right-arrow 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.


INTRODUCTION

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 13C and 14C 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 13C NMR, gas chromatography-mass spectrometry, and 14C 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. 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 alpha -ketoglutarate either by glutamate dehydrogenase or alanine aminotransferase or aspartate aminotransferase or phosphoserine aminotransferase. The alpha -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 CO2. 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 alpha -ketoglutarate and therefore complete one tricarboxylic acid cycle turn. In the presence of NH4+, part of this alpha -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 right-arrow derived metabolite). The notations of the proportions taking into account the recycling in the "glutamate right-arrow alpha -ketoglutarate right-arrow glutamate" and "glutamate right-arrow glutamine right-arrow 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.
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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 "OAA1 right-arrow PEP1 right-arrow Pyr1 right-arrow OAA" cycle and (iii) the "OAA right-arrow PEP right-arrow Pyr right-arrow AcCoA1 right-arrow Cit1 right-arrow OAA" cycle.1


Fig. 2. Schematic representation of the metabolic cycles operating during glutamate metabolism. 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 right-arrow alpha KG right-arrow Glu" cycle in which a proportion (Glu up-down-arrow  alpha KG) of glutamate is recycled at each turn, (ii) the "Glu right-arrow Gln right-arrow Glu" cycle in which a proportion (Glu up-down-arrow  Gln) of glutamate is recycled at each turn, (iii) the tricarboxylic acid cycle in which a proportion (TCA up-down-arrow ) of oxaloacetate is recycled at each turn, (iv) the "OAA right-arrow PEP right-arrow Pyr right-arrow OAA" cycle in which a proportion (Pyr up-down-arrow  OAA) of oxaloacetate is recycled at each turn, (v) the "OAA right-arrow PEP right-arrow Pyr right-arrow AcCoA right-arrow Cit right-arrow OAA" cycle in which a proportion (AcCoA up-down-arrow  OAA) of oxaloacetate is recycled at each turn. The proportions (TCA up-down-arrow ) and (AcCoA up-down-arrow  OAA) take also into account the recycling of alpha -ketoglutarate through glutamate and glutamine, resulting from the operation of the "Glu right-arrow alpha KG right-arrow Glu" and "Glu right-arrow Gln right-arrow Glu" cycles. The effects of the operation of these two latter cycles on the accumulation of glutamate and glutamine and on the formation of alpha -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 + <UNL>OAA</UNL> right-arrow 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 + <UNL>AcCoA</UNL> right-arrow 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 alpha -ketoglutarate which take into account the total recycling through both "Glu right-arrow alpha KG right-arrow Glu" and "Glu right-arrow Gln right-arrow Glu" cycles and are noted {Glu up-down-arrow  alpha KG + Gln} and {alpha KG up-down-arrow  Glu + Gln}, respectively (see Table I). The proportions of glutamate, {vGlu right-arrow &cjs1670;Glu}, and glutamine, {vGlu right-arrow &cjs1670;Gln}, accumulation and alpha -ketoglutarate formation, {vGlu right-arrow alpha KG}, from added glutamate are equal to {Glu up-down-arrow  alpha KG + Gln} multiplied by (Glu right-arrow &cjs1670;Glu), (Glu right-arrow Gln).(Gln right-arrow &cjs1670;Gln) and (Glu right-arrow alpha KG). The proportion of oxaloacetate formed, {Citalpha KG right-arrow OAA}, from citrate-derived alpha -ketoglutarate is equal to {alpha KG up-down-arrow  Glu+Gln} multiplied by (alpha KG right-arrow OAA).
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Glutamate is the metabolite common to the two other cycles that have been introduced to improve the model; these are (iv) the "Glu right-arrow alpha KG right-arrow Glu" cycle and (v) the "Glu right-arrow Gln right-arrow 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.


Fig. 3. Schematic representations of oxaloacetate formation during the first and second multicycle turns. This figure contains another representation of Fig. 2 (left panel) which allows us, as shown for the first and second multicycle turns (right panel), to calculate the total amount of oxaloacetate formed during an infinite number of multicycle turns.
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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 CO2 which are represented in Figs. 4 and 5, respectively.


Fig. 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 alpha , beta , gamma  MET, where MET represents any glutamate-derived metabolite and alpha , beta , and gamma  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 µmol/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 (OAAi) or 1 - (OAAi), as indicated in the figure. For other metabolic conversions it is necessary to take into account the effect of the recycling through the "glutamate right-arrow alpha -ketoglutarate right-arrow glutamate" and "glutamate right-arrow glutamine right-arrow glutamate" cycles. Depending on which metabolite is recycled, glutamate or alpha -ketoglutarate, the proportion of conversion is multiplied either by {Glu up-down-arrow  alpha KG + Gln} or {alpha KG up-down-arrow  Glu + 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 alpha -ketoglutarate from citrate is shown in this figure. To take into account the recycling of alpha -ketoglutarate through glutamate and glutamine, it is necessary to multiply by {alpha KG up-down-arrow  Glu + 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 2·(OAAi) of the oxaloacetate formed by the pyruvate carboxylase reaction equilibrates with fumarate; half of this oxaloacetate, equal to (OAAi), gives rise to oxaloacetate molecules having an inverted labeling pattern.
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Fig. 5. Metabolic fate of the CO2 carbon fixed during glutamate metabolism in rabbit kidney tubules. Based on the same principle as that used in Fig. 4, this figure shows the fate of unlabeled oxaloacetate and of oxaloacetate labeled on its carbon 1 or 4 because the CO2 carbon which is fixed by pyruvate carboxylase at each turn of the "Pyr right-arrow OAA right-arrow PEP right-arrow Pyr" cycle yields oxaloacetate labeled either on its carbon 1 or 4. Oxaloacetate carbons directly labeled from the CO2 carbon are represented in brackets. The arrows have the same meaning as those used in Fig. 4.
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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.


Fig. 6. Schematic representations of the formation of the C-1, C-2, C-3, and C-4 of oxaloacetate from the C-3 of glutamate. This figure, derived from Fig. 4, allows the calculation of the amounts of the C-1, C-2, C-3, and C-4 of oxaloacetate formed during an infinite number of multicycle turns. The notations of different proportions are explained in the text and presented in Table I.
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Notations

Let us call [CyMET]CzGlu the amount of the metabolite (MET) formed labeled on its carbon y (where 1 <=  y <=  6) arising from glutamate labeled on its carbon z, where z is equal to 1, 3, or 5 because we used [1,2-13C]-, [1-14C]-, [3-13C]-, [5-13C]-, [1,5-14C]-, and [U-14C]glutamate as labeled substrates and also because it is assumed that the C-2 and C-3 of glutamate had the same metabolic fate as the C-5 and C-4, respectively.

And let [CyMET]Glu+*CO2 be the amount of the metabolite (MET) labeled on its carbon y (where 1 <=  y <=  6) arising from glutamate plus labeled CO2.

Similarly, let [(Cy+Cy')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 y'; let [(Cy,y')MET]CzGlu be the amount of the metabolite (MET) labeled simultaneously on its carbons y and y'.

Let [(Cy(MET1 + MET2)]CzGlu be the amount of the metabolite (MET1) labeled on its carbon y plus the amount of the metabolite (MET2) labeled on its carbon y.

Let [MET]Glu be the total amount of the metabolite (MET) formed as a result of glutamate metabolism.

When the amount of the metabolite accumulated is considered, a (&cjs1670;) sign is added on the left side inside the brackets [].

Let (Met1 up-down-arrow  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 (TCAup-down-arrow ) is used.

Let (Met1 up-down-arrow  Met2 + 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 right-arrow 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 right-arrow alpha KG right-arrow Glu" and "Glu right-arrow Gln right-arrow Glu" cycles is indicated by substituting the parentheses () by {}.

Special symbols are used in the notations {vGlu right-arrow Met2} and {Citalpha KG right-arrow OAA} to stress the fact that only added glutamate and only citrate-derived alpha -ketoglutarate, respectively, are concerned.

Let (AcCoA + OAA right-arrow 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 + AcCoA right-arrow 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 &cjs3675; 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.

Table I.

Definition of the parameters of the model


Proportion of Converted to Parameter notation

Proportions of the direct conversion
  OAA Citrate (OAA right-arrow Cit)
  OAA PEP (OAA right-arrow PEP)
  OAA Asp (OAA right-arrow Asp)
  PEP Pyr (PEP right-arrow Pyr)
  PEP 3P-glycerate (PEP right-arrow 3PG)
  Pyr OAA (Pyr right-arrow OAA)
  Pyr Ala (Pyr right-arrow Ala)
  Pyr AcCoA (Pyr right-arrow AcCoA)
  Pyr Lac (Pyr right-arrow Lac)
  3P-glycerate Glc (3PG right-arrow Glc)
  3P-glycerate Ser (3PG right-arrow Ser)
  alpha -KG OAA (alpha KG right-arrow OAA)
Proportions taking into account the recycling through "Glu right-arrow alpha KG right-arrow Glu" and "Glu right-arrow Gln right-arrow Glu" cycles
  Added Glu  alpha KG {&cjs1671;Glu right-arrow alpha KG}
  Added Glu Accumulated Glu {&cjs1671;Glu right-arrow &cjs1670;Glu}
  Added Glu Accumulated Gln {&cjs1671;Glu right-arrow &cjs1670;Gln}
  Citrate-derived alpha KG Glu {Citalpha KG right-arrow Glu}
  Citrate-derived alpha KG OAA {Citalpha KG right-arrow OAA}
Other parameters Parameter notation

OAA inversion (OAAi)
Recycling factor through
  ·Tricarboxylic acid cycle (TCA up-down-arrow ) = (OAA right-arrow Cit)·{Citalpha KG right-arrow OAA}
  ·"OAA right-arrow PEP right-arrow Pyr right-arrow OAA" cycle (Pyr up-down-arrow  OAA) = (OAA right-arrow PEP)·(PEP right-arrow Pyr)·(Pyr right-arrow OAA)
  ·"OAA right-arrow PEP right-arrow Pyr right-arrow AcCoA" (AcCoA up-down-arrow  OAA) = 
  right-arrow Cit right-arrow OAA" cycle (OAA right-arrow PEP)·(PEP right-arrow Pyr)·(Pyr right-arrow AcCoA)·{Citalpha KG right-arrow OAA}
Recycling ratio through "Glu right-arrow alpha KG right-arrow Glu" and "Glu right-arrow Gln right-arrow Glu" cycles of
  ·Glu {Glu up-down-arrow  alpha KG+Gln} = 1/[1-(Glu up-down-arrow  alpha KG) - (Glu up-down-arrow  Gln)]
  ·alpha KG {alpha KG up-down-arrow  Glu+Gln} = [1-(Glu up-down-arrow  Gln)]/[1-(Glu up-down-arrow  alpha KG)·(Glu up-down-arrow  Gln)]
Glu-derived AcCoA condensed with OAA not derived from Glu (AcCoA + <UNL>OAA</UNL> right-arrow Cit)
Glu-derived OAA condensed with AcCoA not derived from Glu (OAA + <UNL>AcCoA</UNL> right-arrow Cit)

The amount (in µmol/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:
(<UP>RR</UP>)=<LIM><OP>∑</OP><LL>n=0</LL><UL>∞</UL></LIM> (<UP>RF</UP>)<SUP>n</SUP>=1/[1−(<UP>RF</UP>)] (Eq. 1)
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 up-down-arrow  alpha KG) is the recycling factor of the "Glu right-arrow alpha KG right-arrow Glu" cycle;
(<UP>Glu</UP> ⇅ &agr;<UP>KG</UP>)=(<UP>Glu</UP> → &agr;<UP>KG</UP>) · (&agr;<UP>KG</UP> → <UP>Glu</UP>) (Eq. 2)
where (Glu right-arrow alpha KG) is the proportion of added glutamate directly converted into alpha -ketoglutarate, and (alpha KG right-arrow Glu) is the proportion of alpha -ketoglutarate directly converted into glutamate.

(ii) (Glu up-down-arrow  Gln) is the recycling factor of the "Glu right-arrow Gln right-arrow Glu" cycle;
(<UP>Glu</UP> ⇅ <UP>Gln</UP>)=(<UP>Glu</UP> → <UP>Gln</UP>) · (<UP>Gln</UP> → <UP>Glu</UP>)=(<UP>Glu</UP> → <UP>Gln</UP>) · [1(<UP>Gln</UP> → &cjs1670;<UP>Gln</UP>)] (Eq. 3)
where (Glu right-arrow Gln) is the proportion of glutamate directly converted into glutamine and (Gln right-arrow Glu) and (Gln right-arrow &cjs1670;Gln) are the proportions of glutamine converted into glutamate and glutamine accumulated, respectively.

Fig. 1 shows that glutamate is either accumulated or con- verted into alpha -ketoglutarate or glutamine; therefore, the proportion of glutamate formed which accumulated (Glu right-arrow &cjs1670;Glu) is defined by
(<UP>Glu</UP> → &cjs1670;<UP>Glu</UP>)=1−(<UP>Glu</UP> → &agr;<UP>KG</UP>)−(<UP>Glu</UP> → <UP>Gln</UP>) (Eq. 4)
In the following equations, the proportion of glutamate and alpha -ketoglutarate molecules that are metabolized and then resynthesized through the successive operation of "Glu right-arrow alpha KG right-arrow Glu" and "Glu right-arrow Gln right-arrow Glu" cycles over a theoretically infinite number of turns are noted {Glu up-down-arrow  alpha KG + Gln} and {alpha KG up-down-arrow  Glu + Gln}, respectively.

The resulting glutamate recycling through alpha -ketoglutarate and glutamine, is
{<UP>Glu</UP> ⇅ &agr;<UP>KG</UP>+<UP>Gln</UP>}=<LIM><OP>∑</OP><LL>n=0</LL><UL>∞</UL></LIM>[(<UP>Glu</UP> ⇅ &agr;<UP>KG</UP>)+(<UP>Glu</UP> ⇅ <UP>Gln</UP>)]<SUP>n</SUP>=1/[1 (Eq. 5)
−(<UP>Glu</UP> ⇅ &agr;<UP>KG</UP>)−(<UP>Glu</UP> ⇅ <UP>Gln</UP>)]
(see also Fig. 2).

One can demonstrate that {alpha KG up-down-arrow  Glu + Gln}, the resulting alpha -ketoglutarate recycling through glutamate and glutamine, is:
{&agr;<UP>KG</UP> ⇅ <UP>Glu</UP>+<UP>Gln</UP>}=[1−(<UP>Glu</UP> ⇅ <UP>Gln</UP>)]/[1−(<UP>Glu</UP> ⇅ &agr;<UP>KG</UP>)−(<UP>Glu</UP> ⇅ <UP>Gln</UP>)] (Eq. 6)
(see also Fig. 2).

Therefore, from the latter equation and Equation 3, it follows that
{&agr;<UP>KG</UP> ⇅ <UP>Glu</UP>+<UP>Gln</UP>}=[1−(<UP>Glu</UP> ⇅ <UP>Gln</UP>)] · {<UP>Glu</UP> ⇅ &agr;<UP>KG</UP>+<UP>Gln</UP>} (Eq. 7)
={<UP>Glu</UP> ⇅ &agr;<UP>KG</UP>
+<UP>Gln</UP>}−{&cjs1671;<UP>Glu</UP> → <UP>Gln</UP>}+{&cjs1671;<UP>Glu</UP> → &cjs1670;<UP>Gln</UP>}
(see also Fig. 2)

where {vGlu right-arrow Gln} and {vGlu right-arrow &cjs1670;Gln} represent the proportion of added glutamate not recycled or recycled only in the "Glu right-arrow alpha KG right-arrow Glu" and "Glu right-arrow Gln right-arrow Glu" cycles which was converted in glutamine formed and glutamine accumulated, respectively.

The parameters {vGlu right-arrow Gln} and {vGlu right-arrow &cjs1670;Gln}, as defined above (see "Notations"), are equal to {Glu up-down-arrow  alpha KG + Gln} multiplied by (Glu right-arrow Gln) and (Glu right-arrow &cjs1670;Gln), respectively.

(iii) (Pyr up-down-arrow  OAA) is the recycling factor of the "OAA right-arrow PEP right-arrow Pyr right-arrow OAA" cycle, i.e. the proportion of oxaloacetate resynthesized at each turn of this cycle. (Pyr up-down-arrow  OAA) = (OAA right-arrow PEP)·(PEP right-arrow Pyr)·(Pyr right-arrow OAA) (see Fig. 2). Where (OAA right-arrow PEP), (PEP right-arrow Pyr), and (Pyr right-arrow OAA) are the proportions of oxaloacetate, phosphoenolpyruvate, and pyruvate converted into phosphoenolpyruvate, pyruvate, and oxaloacetate, respectively (see Fig. 2).

(iv) (TCA up-down-arrow ) 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 alpha -ketoglutarate, noted (Cit right-arrow alpha KG) is considered to be 1. 
(<UP>TCA</UP>⇅)=(<UP>OAA</UP> → <UP>Cit</UP>) · {<SUB><UP>Cit</UP></SUB>&agr;<UP>KG</UP> → <UP>OAA</UP>} (Eq. 8)
where (OAA right-arrow Cit) and {Citalpha KG right-arrow OAA} are the proportions of oxaloacetate converted into citrate and citrate-derived alpha -ketoglutarate converted into oxaloacetate, respectively (see Fig. 2).

(v) (AcCoA up-down-arrow  OAA) is the recycling factor of the "OAA right-arrow PEP right-arrow Pyr right-arrow AcCoA right-arrow Cit right-arrow OAA" cycle, i.e. the proportion of oxaloacetate resynthesized at each turn of this cycle (see Fig. 2). The proportion of acetyl-CoA yielding citrate, noted (AcCoA right-arrow Cit) is equal to 1. 
(<UP>AcCoA</UP> ⇅ <UP>OAA</UP>)=(<UP>OAA</UP> → <UP>PEP</UP>) (Eq. 9)
 · (<UP>PEP</UP> → <UP>Pyr</UP>) · (<UP>Pyr</UP> → <UP>AcCoA</UP>) · {<SUB><UP>Cit</UP></SUB>&agr;<UP>KG</UP> → <UP>OAA</UP>}
In the latter 2 equations (TCA up-down-arrow ) and (AcCoA up-down-arrow  OAA) take into account the recycling of alpha -ketoglutarate through glutamate and glutamine since {Citalpha KG right-arrow OAA} = (alpha KG right-arrow OAA)·{alpha KG up-down-arrow  Glu + Gln} (see Fig. 2).

The amount of glutamine formed directly from the substrate glutamate is given by X·{vGlu right-arrow Gln}, where X is the amount of glutamate utilized and {vGlu right-arrow Gln} is the proportion of the substrate glutamate converted into glutamine. As already mentioned, this proportion takes into account glutamate recycling through alpha -ketoglutarate and glutamine.

The corresponding amount of glutamine accumulated is given by X·{vGlu right-arrow &cjs1670;Gln} = X·(Glu right-arrow &cjs1670;Gln)·{Glu up-down-arrow  alpha KG + Gln} where (Glu right-arrow &cjs1670;Gln) = (Glu right-arrow Gln)·(Gln right-arrow &cjs1670;Gln) and (Gln right-arrow &cjs1670;Gln) is the proportion of glutamine formed which accumulated (see also Equation 3).

It can be seen in Fig. 4 that X·{vGlu right-arrow &cjs1670;Gln} is given by C5Gln accumulated from C5Glu: X·{vGlu right-arrow &cjs1670;Gln} = [&cjs1670;C5Gln]C5Glu·X·{vGlu right-arrow &cjs1670;Gln} is also equal to [&cjs1670;C3Gln]C3Glu - [&cjs1670;C2Gln]C3Glu (see Fig. 4).

Figs. 2 and 4 allow us to calculate the amount X·{vGlu right-arrow alpha KG}·(alpha KG right-arrow OAA) of glutamate-derived oxaloacetate: X·{vGlu right-arrow alpha KG}·(alpha KG right-arrow OAA) = [*CO2]C1Glu where {vGlu right-arrow alpha KG} is equal to (Glu right-arrow alpha KG)·{Glu up-down-arrow  alpha KG + Gln}.

[C2Glu]C3Glu/[C2Gln]C3Glu and [C1Glu]C5Glu/[C1Gln]C5Glu is equal to the ratio {vGlu right-arrow &cjs1670;Glu}/{vGlu right-arrow &cjs1670;Gln} which, after simplification by the factor {Glu up-down-arrow  alpha KG + Gln}, is also equal to (Glu right-arrow &cjs1670;Glu)/(Glu right-arrow &cjs1670;Gln) (see above, Figs. 2 and 4 and Table I).

From X·{vGlu right-arrow &cjs1670;Gln} and {vGlu right-arrow &cjs1670;Glu}/{vGlu right-arrow &cjs1670;Gln}, one can calculate X·{vGlu right-arrow &cjs1670;Glu}.

Fig. 2 shows that the glutamate utilized (X) is either accumulated as glutamate, X·{vGlu right-arrow &cjs1670;Glu}, or glutamine, X·{vGlu right-arrow &cjs1670;Gln}, or converted into oxaloacetate, X·{vGlu right-arrow alpha KG}·(alpha KG right-arrow OAA), indicating that X = X·{vGlu right-arrow &cjs1670;Glu} + X·{vGlu right-arrow &cjs1670;Gln} + X·{vGlu right-arrow alpha KG}·(alpha KG right-arrow OAA).

Therefore the following parameters can be calculated:
{&cjs1671;<UP>Glu</UP> → &agr;<UP>KG</UP>} · (&agr;<UP>KG</UP> → <UP>OAA</UP>)=X · {&cjs1671;<UP>Glu</UP> → &agr;<UP>KG</UP>} · (&agr;<UP>KG</UP> → <UP>OAA</UP>)/X (Eq. 10)
{&cjs1671;<UP>Glu</UP> → &cjs1670;<UP>Glu</UP>}=X · {&cjs1671;<UP>Glu</UP> → &cjs1670;<UP>Glu</UP>}/X
{&cjs1671;<UP>Glu</UP> → &cjs1670;<UP>Gln</UP>}=X · {&cjs1671;<UP>Glu</UP> → &cjs1670;<UP>Gln</UP>}/X
Fig. 4 allows us to calculate that
[<UP>C</UP><SUB>1</SUB><UP>OAA</UP>]<SUP><UP>C</UP><SUB>5</SUB><UP>Glu</UP></SUP>=X{&cjs1671;<UP>Glu</UP> → &agr;<UP>KG</UP>} · (&agr;<UP>KG</UP> → <UP>OAA</UP>) · (1/2) ·  (Eq. 11)
<LIM><OP>∑</OP><LL>n=0</LL><UL>∞</UL></LIM>([1−(<UP>OAA</UP><SUB><UP>i</UP></SUB>)] · (<UP>Pyr</UP> ⇅ <UP>OAA</UP>))<SUP>n</SUP>
=X · {&cjs1671;<UP>Glu</UP> → &agr;<UP>KG</UP>} · (&agr;<UP>KG</UP> → <UP>OAA</UP>) · (1/2)
/(1−[1−(<UP>OAA</UP><SUB><UP>i</UP></SUB>)] · (<UP>Pyr</UP> ⇅ <UP>OAA</UP>))
where (OAAi) is the proportion of oxaloacetate inverted as a result of its equilibration with fumarate, a symmetrical molecule (see also Table I), and where, as mentioned above (Pyr up-down-arrow  OAA) = (OAA right-arrow PEP)·(PEP right-arrow Pyr)·(Pyr right-arrow OAA) (see also Fig. 2 and Table I).
[<UP>C</UP><SUB>4</SUB><UP>OAA</UP>]<SUP><UP>C</UP><SUB>5</SUB><UP>Glu</UP></SUP>=X · {&cjs1671;<UP>Glu</UP> → &agr;<UP>KG</UP>} · (&agr;<UP>KG</UP> → <UP>OAA</UP>)/2 (Eq. 12)
+(<UP>OAA</UP><SUB><UP>i</UP></SUB>) · (<UP>Pyr</UP> ⇅ <UP>OAA</UP>) · [<UP>C</UP><SUB>1</SUB><UP>OAA</UP>]<SUP><UP>C</UP><SUB>5</SUB><UP>Glu</UP></SUP>=(1−[1
−2 · (<UP>OAA</UP><SUB><UP>i</UP></SUB>)] · (<UP>Pyr</UP> ⇅ <UP>OAA</UP>)) · [<UP>C</UP><SUB>1</SUB><UP>OAA</UP>]<SUP><UP>C</UP><SUB>5</SUB><UP>Glu</UP></SUP>
Then, [C4OAA]C5Glu = X·{vGlu right-arrow alpha KG}·(alpha KG right-arrow OAA)·(1/2)·(1 - [1 - 2·(OAAi)]·(Pyr up-down-arrow  OAA))/(1 - [1 - (OAAi)]·(Pyr up-down-arrow  OAA)). And, [(C1 + C4)OAA]C5Glu = X·{vGlu right-arrow alpha KG}·(alpha KG right-arrow OAA)·[1 + [(OAAi)/2]/(1 - [1 - (OAAi)]·(Pyr up-down-arrow  OAA))].

From Fig. 4 one can also deduce that
[<UP>C</UP><SUB>1</SUB><UP>OAA</UP>]<SUP><UP>C</UP><SUB>3</SUB><UP>Glu</UP></SUP>=([<UP>C</UP><SUB>2</SUB><UP>OAA</UP>]<SUP><UP>C</UP><SUB>3</SUB><UP>Glu</UP></SUP> · [(<UP>AcCoA</UP> ⇅ <UP>OAA</UP>)/2]+[<UP>C</UP><SUB>3</SUB><UP>OAA</UP>]<SUP><UP>C</UP><SUB>3</SUB><UP>Glu</UP></SUP> ·  (Eq. 13)
[(<UP>TCA</UP> ⇅)/2]) · <LIM><OP>∑</OP><LL>n=0</LL><UL>∞</UL></LIM> ([1−(<UP>OAA</UP><SUB><UP>i</UP></SUB>)] · (<UP>Pyr</UP> ⇅ <UP>OAA</UP>))<SUP>n</SUP>
where, as mentioned in Equation 9 (AcCoA up-down-arrow  OAA) = (OAA right-arrow PEP).(PEP right-arrow Pyr)·(Pyr right-arrow AcCoA)·{Citalpha KG right-arrow OAA} (see also Fig. 2 and Table I) and ([C2OAA]C3Glu·[(AcCoA up-down-arrow  OAA)/2] + [C3OAA]C3Glu·[(TCAup-down-arrow )/2]) represents the C1OAA formed directly from the C-2 and C-3 of oxaloacetate.

Fig. 6 shows that
[(<UP>C</UP><SUB>2</SUB>+<UP>C</UP><SUB>3</SUB>)<UP>OAA</UP>]<SUP><UP>C</UP><SUB>3</SUB><UP>Glu</UP></SUP> (Eq. 14)
=X · {&cjs1671;<UP>Glu</UP> → &agr;<UP>KG</UP>} · (&agr;<UP>KG</UP> → <UP>OAA</UP>) · <LIM><OP>∑</OP><LL>n=0</LL><UL>∞</UL></LIM>((<UP>Pyr</UP> ⇅ <UP>OAA</UP>)+[(<UP>TCA</UP>⇅)
+(<UP>AcCoA</UP> ⇅ <UP>OAA</UP>)]/2)<SUP>n</SUP>=X · {&cjs1671;<UP>Glu</UP> → <UP>&agr;KG</UP>} · (<UP>&agr;KG</UP> → <UP>OAA</UP>)/(1
−(<UP>Pyr</UP> ⇅ <UP>OAA</UP>)−[(<UP>TCA</UP>⇅)+(<UP>AcCoA</UP> ⇅ <UP>OAA</UP>)]/2)
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
[<UP>C</UP><SUB>2</SUB><UP>OAA</UP>]<SUP><UP>C</UP><SUB>3</SUB><UP>Glu</UP></SUP>=[<UP>C</UP><SUB>3</SUB><UP>OAA</UP>]<SUP><UP>C</UP><SUB>3</SUB><UP>Glu</UP></SUP>= (Eq. 15)
X · {&cjs1671;<UP>Glu</UP> → &agr;<UP>KG</UP>} · (&agr;<UP>KG</UP> → <UP>OAA</UP>) · (1/2)/(1−(<UP>Pyr</UP> ⇅ <UP>OAA</UP>)
−[(<UP>TCA</UP>⇅)+(<UP>AcCoA</UP> ⇅ <UP>OAA</UP>)]/2)
Using Equation 14, Equation 13 can be rewritten as
[<UP>C</UP><SUB>1</SUB><UP>OAA</UP>]<SUP><UP>C</UP><SUB>3</SUB><UP>Glu</UP></SUP>=([(<UP>C</UP><SUB>2</SUB>+<UP>C</UP><SUB>3</SUB>)<UP>OAA</UP>]<SUP><UP>C</UP><SUB>3</SUB><UP>Glu</UP></SUP> · [(<UP>TCA</UP>⇅) (Eq. 16)
+(<UP>AcCoA</UP> ⇅ <UP>OAA</UP>)]/4)/(1−[1−(<UP>OAA</UP><SUB><UP>i</UP></SUB>)] · (<UP>Pyr</UP> ⇅ <UP>OAA</UP>))
Then,
[<UP>C</UP><SUB>1</SUB><UP>OAA</UP>]<SUP><UP>C</UP><SUB>3</SUB><UP>Glu</UP></SUP> (Eq. 17)
=X · {&cjs1671;<UP>Glu</UP> → &agr;<UP>KG</UP>} · (&agr;<UP>KG</UP> → <UP>OAA</UP>) · (1/2) · ([(<UP>TCA</UP>⇅)
+(<UP>AcCoA</UP> ⇅ <UP>OAA</UP>)]/2)/[(1−(<UP>Pyr</UP> ⇅ <UP>OAA</UP>)−[(<UP>TCA</UP>⇅)
+(<UP>AcCoA</UP> ⇅ <UP>OAA</UP>)]/2) · (1−[1−(<UP>OAA</UP><SUB><UP>i</UP></SUB>)] · (<UP>Pyr</UP> ⇅ <UP>OAA</UP>))]
And,
[<UP>C</UP><SUB>4</SUB><UP>OAA</UP>]<SUP><UP>C</UP><SUB>3</SUB><UP>Glu</UP></SUP>=([(<UP>C</UP><SUB>2</SUB>+<UP>C</UP><SUB>3</SUB>)<UP>OAA</UP>]<SUP><UP>C</UP><SUB>3</SUB><UP>Glu</UP></SUP> · [(<UP>TCA</UP>⇅)+(<UP>AcCoA</UP> ⇅ <UP>OAA</UP>)]/4) (Eq. 18)
+(<UP>OAA</UP><SUB><UP>i</UP></SUB>) · (<UP>Pyr</UP> ⇅ <UP>OAA</UP>) · [<UP>C</UP><SUB>1</SUB><UP>OAA</UP>]<SUP><UP>C</UP><SUB>3</SUB><UP>Glu</UP></SUP>=(1−[1
−2 · (<UP>OAA</UP><SUB><UP>i</UP></SUB>)] · (<UP>Pyr</UP> ⇅ <UP>OAA</UP>)) · [<UP>C</UP><SUB>1</SUB><UP>OAA</UP>]<SUP><UP>C</UP><SUB>3</SUB><UP>Glu</UP></SUP>
=X · {&cjs1671;<UP>Glu</UP> → &agr;<UP>KG</UP>} · (&agr;<UP>KG</UP> → <UP>OAA</UP>) · (1/2) · [([(<UP>TCA</UP>⇅)
+(<UP>AcCoA</UP> ⇅ <UP>OAA</UP>)]/2)/(1−(<UP>Pyr</UP> ⇅ <UP>OAA</UP>)−[(<UP>TCA</UP>⇅)
+(<UP>AcCoA</UP> ⇅ <UP>OAA</UP>)]1/2)] · [(1−[1−2 · (<UP>OAA</UP><SUB><UP>i</UP></SUB>)] · (<UP>Pyr</UP> ⇅ <UP>OAA</UP>))/(1
−[1−(<UP>OAA</UP><SUB><UP>i</UP></SUB>)] · (<UP>Pyr</UP> ⇅ <UP>OAA</UP>))]
And,
[(<UP>C</UP><SUB>1</SUB>+<UP>C</UP><SUB>4</SUB>)<UP>OAA</UP>]<SUP><UP>C</UP><SUB>3</SUB><UP>Glu</UP></SUP> (Eq. 19)
=X · {&cjs1671;<UP>Glu</UP> → &agr;<UP>KG</UP>} · (&agr;<UP>KG</UP> → <UP>OAA</UP>) · [([(<UP>TCA</UP>⇅)
+(<UP>AcCoA</UP> ⇅ <UP>OAA</UP>)]/2)/(1−(<UP>Pyr</UP> ⇅ <UP>OAA</UP>)−[(<UP>TCA</UP>⇅)
+(<UP>AcCoA</UP> ⇅ <UP>OAA</UP>)]/2)] · [1+[(<UP>Pyr</UP> ⇅ <UP>OAA</UP>)/2]/(1
−[1−(<UP>OAA</UP><SUB><UP>i</UP></SUB>)] · (<UP>Pyr</UP> ⇅ <UP>OAA</UP>))]
Therefore,
[<UP>C</UP><SUB>4</SUB><UP>OAA</UP>]<SUP><UP>C</UP><SUB>3</SUB><UP>Glu</UP></SUP>/[<UP>C</UP><SUB>4</SUB><UP>OAA</UP>]<SUP><UP>C</UP><SUB>5</SUB>Glu</SUP>=([(<UP>TCA⇅</UP>)<UP>+</UP>(<UP>AcCoA⇅OAA</UP>)]<UP>/2</UP>)<UP>/</UP>(<UP>1−</UP> (Eq. 20)
(Pyr⇅OAA)−[(TCA⇅)+(AcCoA⇅OAA)]/2)
Let us call A the Latter Ratio, then one can deduce from Fig. 4 that
  [<UP>C</UP><SUB>4</SUB><UP>OAA</UP>]<SUP><UP>C</UP><SUB>3</SUB><UP>Glu</UP></SUP>/[<UP>C</UP><SUB>4</SUB><UP>OAA</UP>]<SUP><UP>C</UP><SUB>5</SUB><UP>Glu</UP></SUP>=[<UP>C</UP><SUB>1</SUB><UP>Glx</UP><SUP>1</SUP>]<SUP><UP>C</UP><SUB>3</SUB><UP>Glu</UP></SUP>/[<UP>C</UP><SUB>1</SUB><UP>Glx</UP>]<SUP><UP>C</UP><SUB>5</SUB><UP>Glu</UP></SUP>=A (Eq. 21)
and
<UP>  </UP>[<UP>*CO</UP><SUB><UP>2</UP></SUB>]<SUP><UP>C</UP><SUB><UP>3</UP></SUB></SUP><SUP><UP>Glu</UP></SUP><UP>/</UP>[<UP>*CO</UP><SUB><UP>2</UP></SUB>]<SUP><UP>C</UP><SUB><UP>5</UP></SUB></SUP><SUP><UP>Glu</UP></SUP><UP>=</UP>[<UP>C<SUB>1</SUB>Ala</UP><SUP><UP>1</UP></SUP>]<SUP><UP>C</UP><SUB><UP>3</UP></SUB></SUP><SUP><UP>Glu</UP></SUP><UP>/</UP>[<UP>C<SUB>1</SUB>Ala</UP>]<SUP><UP>C</UP><SUB><UP>5</UP></SUB></SUP><SUP><UP>Glu</UP></SUP><UP>=A</UP> (Eq. 22)
From Fig. 4, one can deduce that
  [<UP>C</UP><SUB>3</SUB><UP>OAA</UP>]<SUP><UP>C</UP><SUB>3</SUB><UP>Glu</UP></SUP>/[<UP>C</UP><SUB>1</SUB><UP>OAA</UP>]<SUP><UP>C</UP><SUB>5</SUB><UP>Glu</UP></SUP>=[<UP>C</UP><SUB>3</SUB><UP>Ala</UP>]<SUP><UP>C</UP><SUB>3</SUB><UP>Glu</UP></SUP>/[<UP>C</UP><SUB>1</SUB><UP>Ala</UP>]<SUP><UP>C</UP><SUB>5</SUB><UP>Glu</UP></SUP>=B (Eq. 23)
where the value B of the latter 2 ratios can be calculated from Equations 11 and 15: B = [1 - (Pyr up-down-arrow  OAA) + (OAAi)·(Pyr up-down-arrow  OAA)]/(1 - (Pyr up-down-arrow  OAA) - [(TCA up-down-arrow ) + (AcCoA up-down-arrow  OAA)]/2).

Combining Equations 21 and 23, yields (A + 1)/(B - A - 1) = [1 - (Pyr up-down-arrow  OAA)]/[(OAAi)·(Pyr up-down-arrow  OAA)].

Let C = [1 - (Pyr up-down-arrow  OAA)]/[(OAAi)·(Pyr up-down-arrow  OAA)].

Let D = [C1Glx]C5Glu/[C1Ala]C5Glu

From Fig. 4 and Equations 11 and 12, one can deduce that D = (1 - [1 - 2·(OAAi)]·(Pyr up-down-arrow  OAA))·[(OAA right-arrow Cit) - (TCA up-down-arrow )]/[(OAA right-arrow PEP)·(PEP right-arrow Pyr)·(Pyr right-arrow Ala)]. Where (Pyr right-arrow Ala) is the proportion of pyruvate directly converted to alanine.

From Fig. 5 it is also possible to calculate that
[<UP>C</UP><SUB>4</SUB><UP>OAA</UP>]<SUP><UP>Glu+*CO</UP><SUB>2</SUB></SUP>=[<UP>OAA</UP>]<SUP><UP>Glu</UP></SUP> · [1−[<UP>OAA</UP><SUB><UP>i</UP></SUB>)] · (<UP>Pyr</UP> ⇅ <UP>OAA</UP>)
+(<UP>OAA</UP><SUB><UP>i</UP></SUB>) · (<UP>Pyr</UP> ⇅ <UP>OAA</UP>) · [<UP>C</UP><SUB>1</SUB><UP>OAA</UP>]<SUP><UP>Glu+*CO</UP><SUB>2</SUB></SUP>
[<UP>C</UP><SUB>1</SUB>OAA]<SUP><UP>Glu+*CO</UP><SUB>2</SUB></SUP>=[<UP>OAA</UP>]<SUP><UP>Glu</UP></SUP> · (<UP>OAA</UP><SUB><UP>i</UP></SUB>) · (<UP>Pyr</UP> ⇅ <UP>OAA</UP>) · <LIM><OP>∑</OP><LL>n=0</LL><UL>∞</UL></LIM>([1 (Eq. 24)
−(<UP>OAA</UP><SUB><UP>i</UP></SUB>)] · (<UP>Pyr</UP> ⇅ <UP>OAA</UP>))<SUP>n</SUP>=[<UP>OAA</UP>]<SUP><UP>Glu</UP></SUP> · (<UP>OAA</UP><SUB><UP>i</UP></SUB>) · (<UP>Pyr</UP> ⇅ <UP>OAA</UP>)/(1−[1−(<UP>OAA</UP><SUB><UP>i</UP></SUB>)] · (<UP>Pyr</UP> ⇅ <UP>OAA</UP>))
Combining the latter equation and Equation 24, we obtain
[<UP>C<SUB>4</SUB>OAA</UP>]<SUP><UP>Glu+*CO</UP><SUB><UP>2</UP></SUB></SUP><UP> = </UP>[<UP>OAA</UP>]<SUP><UP>Glu</UP></SUP><UP> · </UP>[[<UP>1 − </UP>(<UP>OAA</UP><SUB><UP>i</UP></SUB>)]<UP> · </UP>(<UP>Pyr ⇅ OAA</UP>)<UP> +</UP> (Eq. 25)
((OAA<SUB>i</SUB>) · (Pyr ⇅ OAA))<SUP>2</SUP>/(1 − [1 − (OAA<SUB>i</SUB>)] · (Pyr ⇅ OAA))]
Let us call E = [C1Glx]Glu + *CO2/[C1Ala]Glu + * CO2

where
[<UP>C</UP><SUB>1</SUB><UP>Glx</UP>]<SUP><UP>Glu+*CO</UP><SUB>2</SUB></SUP>=[<UP>C</UP><SUB>4</SUB><UP>OAA</UP>]<SUP><UP>Glu + * CO</UP><SUB>2</SUB></SUP> · [(<UP>OAA</UP> → <UP>Cit</UP>)−(<UP>TCA</UP>⇅)] (Eq. 26)
(see Fig. 5) and
[<UP>C</UP><SUB>1</SUB><UP>Ala</UP>]<SUP><UP>Glu+*CO</UP><SUB>2</SUB></SUP>=[<UP>C</UP><SUB>1</SUB><UP>OAA</UP>]<SUP><UP>Glu+*CO</UP><SUB>2</SUB></SUP> · [(<UP>OAA</UP> →  (Eq. 27)
<UP>PEP</UP>) · (<UP>PEP</UP> → <UP>Pyr</UP>) · (<UP>Pyr</UP> → <UP>Ala</UP>)]
(see Fig. 5). Combining Equations 24 and 26, we obtain
E=([1−(<UP>Pyr</UP>⇅<UP>OAA</UP>)]/(<UP>OAA</UP><SUB>i</SUB>)+2·(<UP>Pyr</UP>⇅<UP>OAA</UP>)−1)·[(<UP>OAA</UP>→<UP>Cit</UP>)− (Eq. 28)
(<UP>TCA</UP>⇅)]/[(<UP>OAA</UP>→<UP>PEP</UP>)·(<UP>PEP</UP>→<UP>Pyr</UP>)·(<UP>Pyr</UP>→<UP>Ala</UP>)]
Let us call F = D/E
<UP>F</UP>=(1−[1−2 · (<UP>OAA</UP><SUB><UP>i</UP></SUB>)] · (<UP>Pyr</UP> ⇅ <UP>OAA</UP>))/([1−(<UP>Pyr</UP> → <UP>OAA</UP>)]/(<UP>OAA</UP><SUB><UP>i</UP></SUB>) (Eq. 29)
+2 · (<UP>Pyr</UP> ⇅ <UP>OAA</UP>)−1)=(2+<UP>C</UP>) · (<UP>OAA</UP><SUB><UP>i</UP></SUB>)/(<UP>C</UP> · [1−(<UP>OAA</UP><SUB><UP>i</UP></SUB>)]+1)
Therefore, i = F·(1 + C)/[2 + C·(1 + F)]. Since C was defined as [1 - (Pyr up-down-arrow  OAA)]/[(OAAi)·(Pyr up-down-arrow  OAA)] (see above), one can calculate that (Pyr up-down-arrow  OAA) = 1/[1 + (OAAi) ·C].

Let us call G = 1/A; then,
<UP>  G</UP>=[1−(<UP>Pyr</UP>⇅<UP>OAA</UP>)]<UP>/</UP>([(<UP>TCA</UP>)+(<UP>AcCoA</UP>⇅<UP>OAA</UP>)]<UP>/2</UP>)−1 (Eq. 30)
and
[(<UP>TCA</UP> ⇅) + (<UP>AcCoA</UP> ⇅ <UP>OAA</UP>)]=2 · [1−(<UP>Pyr</UP> ⇅ <UP>OAA</UP>)]/(1+<UP>G</UP>) (Eq. 31)
From Fig. 6, it can be deduced that
[(<UP>C</UP><SUB>2</SUB>+<UP>C</UP><SUB>3</SUB>)<UP>OAA</UP>]<SUP><UP>C</UP><SUB>3</SUB><UP>Glu</UP></SUP>=X · {&cjs1671;<UP>Glu</UP> → &agr;<UP>KG</UP>} · (&agr;<UP>KG</UP> → <UP>OAA</UP>)<UP>/H</UP> (Eq. 32)
where
   <UP>H</UP>=1−(<UP>Pyr</UP> ⇅ <UP>OAA</UP>)−[(<UP>TCA</UP>⇅)+(<UP>AcCoA</UP> ⇅ <UP>OAA</UP>)]/2 (Eq. 33)
And Fig. 4 shows that
[(<UP>C<SUB>2</SUB>+C</UP><SUB><UP>3</UP></SUB>)<UP>Lac</UP><SUP><UP>1</UP></SUP>]<SUP><UP>C<SUB>3</SUB>Glu</UP></SUP><UP> = </UP>[(<UP>C</UP><SUB>2</SUB>+<UP>C</UP><SUB>3</SUB>)<UP>OAA</UP>]<SUP><UP>C<SUB>3</SUB>Glu</UP></SUP>·(<UP>OAA</UP>→<UP>PEP</UP>)·(<UP>PEP</UP>→<UP>Pyr</UP>)·(<UP>Pyr</UP>→<UP>Lac</UP>) (Eq. 34)
where (Pyr right-arrow Lac) is the proportion of pyruvate directly converted to lactate.

Combining Equations 32 and 34, it follows that (OAA right-arrow PEP) · (PEP right-arrow Pyr) · (Pyr right-arrow Lac) = [(C2 + C3)Lac]C3Glu · H/[X · {vGlu right-arrow alpha KG} · (alpha KG right-arrow OAA)].

Similarly,
(<UP>OAA → PEP</UP>)·(<UP>PEP</UP>→<UP>Pyr</UP>)·(<UP>Pyr</UP>→<UP>Ala</UP>)
=[(<UP>C<SUB>2</SUB>+C</UP><SUB><UP>3</UP></SUB>)<UP>Ala</UP>]<SUP><UP>C</UP><SUB><UP>3</UP></SUB></SUP><SUP><UP>Glu</UP></SUP><UP>·H/</UP>[<IT>X</IT><UP>·</UP>{<UP>&cjs1671;Glu→&agr;KG</UP>}<UP>·</UP>(<UP>&agr;KG→OAA</UP>)] (Eq. 35)
Then, [C2OAA]C3Glu and [C3OAA]C3Glu can be calculated:
[<UP>C</UP><SUB>2</SUB><UP>OAA</UP>]<SUP><UP>C</UP><SUB>3</SUB><UP>Glu</UP></SUP>=[<UP>C</UP><SUB>2</SUB><UP>Ala</UP>]<SUP><UP>C</UP><SUB>3</SUB><UP>Glu</UP></SUP>/[(<UP>OAA</UP> → <UP>PEP</UP>) · (<UP>PEP</UP> → <UP>Pyr</UP>) · (<UP>Pyr</UP> → <UP>Ala</UP>)] (Eq. 36)
and similarly,
[<UP>C</UP><SUB>3</SUB><UP>OAA</UP>]<SUP><UP>C</UP><SUB>3</SUB><UP>Glu</UP></SUP>=[<UP>C</UP><SUB>3</SUB><UP>Ala</UP>]<SUP><UP>C</UP><SUB>3</SUB><UP>Glu</UP></SUP>/[(<UP>OAA</UP> → <UP>PEP</UP>) · (<UP>PEP</UP> → <UP>Pyr</UP>) · (<UP>Pyr</UP> → <UP>Ala</UP>)] (Eq. 37)
Furthermore, [C2OAA]C3Glu and [C3OAA]C3Glu are theoretically identical (see Figs. 4 and 6).

The proportion (OAA right-arrow Asp1) of oxaloacetate directly converted into aspartate is given by (OAA right-arrow Asp) = [C3Asp]C3Glu/[C3OAA]C3Glu (see Fig. 4).

From the latter equation and Equation 37, it follows that (OAA right-arrow Asp) is equal to [C3Asp]C3Glu·[(OAA right-arrow PEP)·(PEP right-arrow Pyr)·(Pyr right-arrow Ala)]/[C3Ala]C3Glu.

Oxaloacetate is either converted into citrate or phosphoenolpyruvate or aspartate (see Fig. 4) with the proportion (OAA right-arrow Cit), (OAA right-arrow PEP) or (OAA right-arrow Asp), respectively.

Therefore, it follows that (OAA right-arrow Cit) + (OAA right-arrow PEP) + (OAA right-arrow Asp) = 1 which yields
  (<UP>OAA</UP> → <UP>Cit</UP>)+(<UP>OAA</UP> → <UP>PEP</UP>)=[1−(<UP>OAA</UP> → <UP>Asp</UP>)] (Eq. 38)
From Figs. 2 and 4, it can be deduced that
[<UP>C</UP><SUB>4</SUB><UP>Glx</UP>]<SUP><UP>C</UP><SUB>3</SUB><UP>Glu</UP></SUP>/[<UP>C</UP><SUB>2</SUB><UP>Glx</UP>]<SUP><UP>C</UP><SUB>3</SUB><UP>Glu</UP></SUP>=(<UP>AcCoA</UP> ⇅ <UP>OAA</UP>)/(<UP>TCA</UP>⇅) (Eq. 39)
=(<UP>OAA</UP> → <UP>PEP</UP>) · (<UP>PEP</UP> → <UP>Pyr</UP>) · (<UP>Pyr</UP> → <UP>AcCoA</UP>)/(<UP>OAA</UP> → <UP>Cit</UP>)
Thus, (TCA up-down-arrow ) and (AcCoA up-down-arrow  OAA) can be calculated from Equations 31 and 39:
(<UP>TCA⇅</UP>)<UP>=</UP>[(<UP>TCA⇅</UP>)<UP>+</UP>(<UP>AcCoA⇅OAA</UP>)]<UP>/</UP>[<UP>1+</UP>(<UP>AcCoA⇅OAA</UP>)<UP>/</UP>(<UP>TCA⇅</UP>)]<UP> and</UP> (Eq. 40)
(AcCoA⇅OAA)=[(TCA⇅)+(AcCoA⇅OAA)]−(TCA⇅)
Let us call J = [(C6 + C1 + C5 + C2)Glc1]C3Glu and L = [(C3 + C2)Ser1]C3Glu. Then J + L = (OAA right-arrow PEP)·(PEP right-arrow 3PG)·[(C2 + C3)OAA]C3Glu (see Fig. 4). Where (PEP right-arrow 3PG) is the proportion of phosphoenolpyruvate directly converted to 3-phosphoglycerate.

From Equations 14 and 33, it follows that
(<UP>OAA→PEP</UP>)<UP>·</UP>(<UP>PEP→3PG</UP>)<UP>=</UP>(<UP>J+L</UP>)<UP>·H/</UP>[<IT>X</IT><UP>·</UP>{<SUP><UP>v</UP></SUP><UP>Glu→&agr;KG</UP>}<UP>·</UP>(<UP>&agr;KG→OAA</UP>)] (Eq. 41)
From Eqs. 38 and 39, and knowing (AcCoAup-down-arrow OAA)/(TCAup-down-arrow ) and [(OAAright-arrowCit)+(OAAright-arrowPEP)],we obtain:
[(<UP>AcCoA</UP> ⇅ <UP>OAA</UP>)/(<UP>TCA</UP>⇅)] · [1−(<UP>OAA</UP> → <UP>Asp</UP>)] (Eq. 42)
=(<UP>OAA</UP> → <UP>PEP</UP>) · ((<UP>PEP</UP> → <UP>Pyr</UP>) · (<UP>Pyr</UP> → <UP>AcCoA</UP>)
+(<UP>AcCoA</UP> ⇅ <UP>OAA</UP>)/(<UP>TCA</UP>⇅))
Phosphoenolpyruvate is either converted into pyruvate or 3-phosphoglycerate to yield glucose and serine (see Figs. 1 and 4) with the proportion (PEP right-arrow Pyr) or (PEP right-arrow 3PG), respectively. Therefore, (PEP right-arrow Pyr) + (PEP right-arrow 3PG) = 1 which yields:
(<UP>OAA</UP> → <UP>PEP</UP>)=(<UP>OAA</UP> → <UP>PEP</UP>) · (<UP>PEP</UP> → <UP>Pyr</UP>) (Eq. 43)
+(<UP>OAA</UP> → <UP>PEP</UP>) · (<UP>PEP</UP> → 3<UP>PG</UP>)
Pyruvate is either converted into oxaloacetate, lactate, alanine, or acetyl-CoA (see Figs. 1 and 4) with the proportion (Pyr right-arrow OAA), (Pyr right-arrow Lac), (Pyr right-arrow Ala), or (Pyr right-arrow AcCoA), respectively. Therefore,
(<UP>Pyr → OAA</UP>)<UP> + </UP>(<UP>Pyr → Lac</UP>)<UP> + </UP>(<UP>Pyr → Ala</UP>)<UP> + </UP>(<UP>Pyr → AcCoA</UP>)<UP> = 1</UP> (Eq. 44)
Combining the latter equation and Equations 42 and 43, one can deduce that
(<UP>OAA → PEP</UP>)<UP> = </UP>((<UP>Pyr ⇅ OAA</UP>)<UP> +</UP> (Eq. 45)
 (OAA → PEP) · (PEP → Pyr) · (Pyr → Lac) + 
(OAA → PEP) · (PEP → Pyr) · (Pyr → Ala) +
[(AcCoA ⇅ OAA)/(TCA ⇅ )] · [1 − (OAA → Asp)] +
(OAA → PEP) · (PEP → 3PG))/(1 + 
[(AcCoA ⇅ OAA)/(TCA ⇅ )])
Where, as indicated above, (Pyrup-down-arrow OAA) = (OAA right-arrow PEP).(PEP right-arrow Pyr)·(Pyr right-arrow OAA).

Then (PEP right-arrow 3PG) = (OAA right-arrow PEP)·(PEP right-arrow 3PG)/(OAA right-arrow PEP, and, since (PEP right-arrow Pyr) + (PEP right-arrow 3PG) = 1, (PEP right-arrow Pyr) = 1 - (PEP right-arrow 3PG).

The parameter (OAA right-arrow Cit) can be calculated from Equation 38:
(<UP>OAA → Cit</UP>)<UP> = </UP>[<UP>1 − </UP>(<UP>OAA → Asp</UP>)<UP> − </UP>(<UP>OAA → PEP</UP>)] (Eq. 46)
Since the recycling factor in the tricarboxylic acid cycle (TCAup-down-arrow ), which accounts also for alpha -ketoglutarate recycling through glutamate and glutamine, is equal to (OAA right-arrow Cit)·{Citalpha KG right-arrow OAA} (see Equation 8), it can be calculated that
{<SUB><UP>Cit</UP></SUB><UP>&agr;KG → OAA</UP>}<UP> = </UP>(<UP>TCA ⇅ </UP>)<UP>/</UP>(<UP>OAA → Cit</UP>)<UP>.</UP> (Eq. 47)
From Equation 7 and since as shown in Fig. 1 (Glu right-arrow alpha KG) + (Glu right-arrow &cjs1670;Glu) + (Glu right-arrow Gln) = 1 (see Equation 4), it follows that
<UP> </UP>{<SUP><UP>v</UP></SUP><UP>Glu → &agr;KG</UP>}<UP> + </UP>{<SUP><UP>v</UP></SUP><UP>Glu → &cjs1670;Glu</UP>}<UP> +</UP> (Eq. 48)
{<SUP>v</SUP>Glu → &cjs1670;Gln} = {&agr;KG ⇅ Glu + Gln}
Thus,
[{<SUP><UP>v</UP></SUP><UP>Glu → &cjs1670;Glu</UP>}<UP> + </UP>{<SUP><UP>v</UP></SUP><UP>Glu → &cjs1670;Gln</UP>}]<UP> = </UP>{<UP>&agr;KG ⇅ Glu +</UP> (Eq. 49)
Gln} - {<SUP>v</SUP>Glu → &agr;KG}
Then, multiplying the 2 members of the latter equation by (alpha KG right-arrow OAA) and rearranging, we obtain
(&agr;<UP>KG</UP> → <UNL><UP>OAA</UP></UNL>)=[{<SUB><UP>Cit</UP></SUB>&agr;<UP>KG</UP> → <UP>OAA</UP>} (Eq. 50)
−{&cjs1671;<UP>Glu</UP> → &agr;<UP>KG</UP>} · (&agr;<UP>KG</UP> → <UP>OAA</UP>)]/[{&cjs1671;<UP>Glu</UP> → &cjs1670;<UP>Glu</UP>}
+{&cjs1671;<UP>Glu</UP> → &cjs1670;<UP>Gln</UP>}]
Therefore, the alpha -ketoglutarate recycling through glutamate and glutamine {alpha KG up-down-arrow  Glu + Gln} is equal to {alpha KG up-down-arrow  Glu + Gln} = {Citalpha KG right-arrow OAA}/(alpha KG right-arrow OAA).

Then, {&cjs1671;Glu right-arrow alpha KG} can be calculated from Equations 10 and 50.

Equation 39 yields (OAA right-arrow PEP)·(PEP right-arrow Pyr)·(Pyr right-arrow AcCoA) = [(AcCoA up-down-arrow  OAA)/(TCA up-down-arrow )]·(OAA right-arrow Cit).

The proportion (Pyr right-arrow AcCoA) of pyruvate converted into acetyl-CoA is given by (Pyr right-arrow AcCoA) = (OAA right-arrow PEP)·(PEP right-arrow Pyr)·(Pyr right-arrow AcCoA)/[(OAA right-arrow PEP)·(PEP right-arrow Pyr)].

Similarly, the proportions (Pyr right-arrow Lac), (Pyr right-arrow Ala), and (Pyr right-arrow OAA) corresponding to the proportions of pyruvate transformed into lactate, alanine, and oxaloacetate, respectively, are obtained as follows. (Pyr right-arrow Lac) = (OAA right-arrow PEP)·(PEP right-arrow Pyr)·(Pyr right-arrow Lac)/(OAA right-arrow PEP)·(PEP right-arrow Pyr); (Pyr right-arrow Ala) = (OAA right-arrow PEP)·(PEP right-arrow Pyr)·(Pyr right-arrow Ala)/(OAA right-arrow PEP)·(PEP right-arrow Pyr); (Pyr right-arrow OAA) = (Pyr up-down-arrow  OAA)/(OAA right-arrow PEP)·(PEP right-arrow Pyr); where (Pyr up-down-arrow  OAA) = (OAA right-arrow PEP)·(PEP right-arrow Pyr)·(Pyr right-arrow OAA).

The proportion (3PG right-arrow Glc) of 3-phosphoglycerate which yields glucose is given by (3PG right-arrow Glc) = J/(J + 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 up-down-arrow  OAA), (TCA up-down-arrow ) and (AcCoA up-down-arrow  OAA) represent the oxaloacetate recycled in the "OAA right-arrow PEP right-arrow Pyr right-arrow OAA," the tricarboxylic acid and the "OAA right-arrow PEP right-arrow Pyr right-arrow AcCoA right-arrow Cit right-arrow OAA" cycles, respectively (see Fig. 2 and Table I).

It should be stressed that the proportions (TCA up-down-arrow ) and (AcCoA up-down-arrow  OAA) take also into account the recycling in the "Glu right-arrow alpha KG right-arrow Glu" and "Glu right-arrow Gln right-arrow Glu" cycles.

Let us call (AcCoA + <UNL>OAA</UNL> right-arrow 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 + <UNL>OAA</UNL> right-arrow 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 + <UNL>OAA</UNL> right-arrow Cit)·{Citalpha KG right-arrow 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 right-arrow PEP right-arrow Pyr right-arrow AcCoA right-arrow Cit right-arrow OAA" cycle is (AcCoA + <UNL>OAA</UNL> right-arrow Cit)·(AcCoA up-down-arrow  OAA), while the proportion of oxaloacetate formed by the "OAA right-arrow PEP right-arrow Pyr right-arrow OAA" cycle and by the tricarboxylic acid cycle are (Pyr up-down-arrow  OAA) and (TCAup-down-arrow ), 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 up-down-arrow  OAA) + (TCAup-down-arrow ) + (AcCoA + <UNL>OAA</UNL> right-arrow Cit)·(AcCoA up-down-arrow  OAA), which represents the proportion of oxaloacetate recycled at each turn of the multicycle presented in Fig. 2.
[<UP>OAA</UP>]<SUP><UP>Glu</UP></SUP>=X · {&cjs1671;<UP>Glu</UP> → &agr;<UP>KG</UP>} · (&agr;<UP>KG</UP> → <UP>OAA</UP>) · <LIM><OP>∑</OP><LL>n=0</LL><UL>∞</UL></LIM>[(<UP>Pyr</UP> ⇅ <UP>OAA</UP>) (Eq. 51)
+(<UP>TCA</UP>⇅)+(<UP>AcCoA + <UNL>OAA</UNL> → Cit</UP>) · (<UP>AcCoA</UP> ⇅ <UP>OAA</UP>)]<SUP>n</SUP>
Then
[<UP>OAA</UP>]<SUP><UP>Glu</UP></SUP>−X · {&cjs1671;<UP>Glu</UP> → &agr;<UP>KG</UP>} · (&agr;<UP>KG</UP> → <UP>OAA</UP>)/[1−(<UP>Pyr</UP> ⇅ <UP>OAA</UP>) (Eq. 52)
−(<UP>TCA</UP>⇅)−(<UP>AcCoA + <UNL>OAA</UNL> → Cit</UP>) · (<UP>AcCoA</UP> ⇅ <UP>OAA</UP>)]
In the latter equation, 1/[1 - (Pyr up-down-arrow  OAA) - (TCA) - (AcCoA + <UNL>OAA</UNL> right-arrow Cit)·(AcCoA up-down-arrow  OAA)] represents the proportion of the oxaloacetate formation over an infinite number of multicycle turns, i.e. the oxaloacetate turnover. Using the NMR data obtained with [3-13C]glutamate as substrate, the value of (AcCoA + <UNL>OAA</UNL> right-arrow Cit) can be obtained by calculating first [1 - (AcCoA + <UNL>OAA</UNL> right-arrow Cit)], which represents the proportion of acetyl-CoA molecules derived from added glutamate (noted [AcCoA]Glu) that has been condensed with oxaloacetate molecules also derived from added glutamate (noted [OAA]Glu).

This proportion can be assessed by the ratio of [2-13C]Acetyl-CoA, which condenses with [2-13C]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,
[1−(<UP>AcCoA + <UNL>OAA</UNL> → Cit</UP>)]=[<UP>OAA</UP>]<SUP><UP>Glu</UP></SUP> · (<UP>OAA</UP> → <UP>Cit</UP>) · [1 (Eq. 53)
−(&agr;<UP>KG</UP> → <UP>OAA</UP>)] · [{&cjs1671;<UP>Glu</UP> → &cjs1670;<UP>Glu</UP>}
+{&cjs1671;<UP>Glu</UP> → &cjs1670;<UP>Gln</UP>}] · [&cjs1670;<UP>C</UP><SUB>3</SUB><UP>C</UP><SUB>4</SUB><UP>Glx</UP>]<SUP><UP>C</UP><SUB>3</SUB><UP>Glu</UP></SUP>/([&cjs1670;<UP>C</UP><SUB>4</SUB><UP>Glx</UP>]<SUP><UP>C</UP><SUB>3</SUB><UP>Glu</UP></SUP> · [&cjs1670;<UP>C</UP><SUB>3</SUB><UP>Glx</UP>]<SUP><UP>C</UP><SUB>3</SUB><UP>Glu</UP></SUP>)
Since, as mentioned above, the C-3 and C-2 of Glx are formed in equal amounts from [3-13C]Glu (see Fig. 4), it follows that:
[<UP>1−</UP>(<UP>AcCoA + <UNL>OAA</UNL> → Cit</UP>)]<UP>=</UP>[<UP>OAA</UP>]<SUP><UP>Glu</UP></SUP><UP>·</UP>(<UP>OAA→Cit</UP>)<UP>·</UP>[<UP>1−</UP> (Eq. 54)
(&agr;KG→OAA)]·[{&cjs1671;Glu→&cjs1670;Glu}+{<SUP>v</SUP>Glu→
&cjs1670;Gln}]·[&cjs1670;C<SUB>3</SUB>C<SUB>4</SUB>Glx]<SUP>C<SUB>3</SUB>Glu</SUP>/([&cjs1670;C<SUB>4</SUB>Glx]<SUP>C<SUB>3</SUB>Glu</SUP>·[&cjs1670;C<SUB>2</SUB>Glx]<SUP>C<SUB>3</SUB>Glu</SUP>)
Similarly, one can demonstrate that
[1−(<UP>OAA + <UNL>AcCoA</UNL> → Cit</UP>)]=[<UP>OAA</UP>]<SUP><UP>Glu</UP></SUP> · [<UP>OAA</UP> → <UP>Cit</UP>) · [1 (Eq. 55)
−(&agr;<UP>KG</UP> → <UP>OAA</UP>)] · [·{&cjs1671;<UP>Glu</UP> → &cjs1670;<UP>Glu</UP>}
+{&cjs1671;<UP>Glu</UP> → &cjs1670;<UP>Gln</UP>}] · [&cjs1670;<UP>C</UP><SUB>3</SUB><UP>C</UP><SUB>4</SUB><UP>Glx</UP>]<SUP><UP>C<SUB>3</SUB>Glu</UP></SUP><UP>/</UP>([<UP>&cjs1670;C</UP><SUB>2</SUB><UP>Glx</UP>]<SUP><UP>C<SUB>3</SUB>Glu</UP></SUP>)<SUP><UP>2</UP></SUP>
where (OAA + <UNL>AcCoA</UNL> right-arrow 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
(<UP>OAA + <UNL>AcCoA</UNL> → Cit</UP>)−1−
[1−(<UP>AcCoA + <UNL>OAA</UNL> → Cit</UP>)] · (<UP>AcCoA</UP> ⇅ <UP>OAA</UP>)/(<UP>TCA</UP>⇅) (Eq. 56)
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.
[<UP>OAA</UP>]<SUP><UP>nonGlu</UP></SUP>=[<UP>AcCoA</UP>]<SUP><UP>Glu</UP></SUP> · (<UP>AcCoA + <UNL>OAA</UNL></UP> → <UP>Cit</UP>)=[<UP>OAA</UP>]<SUP><UP>Glu</UP></SUP> (Eq. 57)
and
  [<UP>AcCoA</UP>]<SUP><UP>nonGlu</UP></SUP><UP>=</UP>[<UP>OAA</UP>]<SUP><UP>Glu</UP></SUP><UP>·</UP>(<UP>OAA→Cit</UP>)<UP>·</UP>(<UP>OAA+<UNL>AcCoA</UNL>→Cit</UP>) (Eq. 58)
In view of the fact that [OAA]Glu and (AcCoA + <UNL>OAA</UNL> right-arrow Cit) are not independent parameters, they should be calculated by iterations.

Since the activity of malic enzyme is considered to be negligible in rabbit kidney tubules (2), one can write that [OAA]Glu = flux through pyruvate carboxylase + flux through alpha -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 right-arrow [Glu]Glu is derived directly from the glutamate utilized which undergoes a recycling through alpha -ketoglutarate and glutamine: Glu right-arrow [Glu]Glu = (glutamate utilized)·(recycling ratio of glutamate through alpha -ketoglutarate and glutamine). From Equation 5, it follows that
<SUB><UP>Glu</UP>→</SUB>[<UP>Glu</UP>]<SUP><UP>Glu</UP></SUP>=X · {<UP>Glu</UP> ⇅ &agr;<UP>KG</UP>+<UP>Gln</UP>} (Eq. 59)
(ii) The glutamate noted Cit right-arrow [Glu]Glu is formed from molecules of alpha -ketoglutarate or glutamine which are derived from citrate molecules:
<SUB><UP>Cit</UP>→</SUB>[<UP>Glu</UP>]<SUP><UP>Glu</UP></SUP>=<SUB><UP>Cit</UP>→</SUB>[&agr;<UP>KG</UP>]<SUP><UP>Glu</UP></SUP> · (&agr;<UP>KG</UP> → <UP>Glu</UP>)+<SUB><UP>Cit</UP>→</SUB>[<UP>Gln</UP>]<SUP><UP>Glu</UP></SUP> · (<UP>Gln</UP> → <UP>Glu</UP>) (Eq. 60)
where Cit right-arrow [Gln]Glu = Cit right-arrow [Glu]Glu·(Glu right-arrow Gln) (see Figs. 1 and 2).

Combining the latter two equations, we obtain
<SUB><UP>Cit→</UP></SUB>[<UP>Glu</UP>]<SUP><UP>Glu</UP></SUP><UP>·</UP>[<UP>1−</UP>(<UP>Glu→Gln</UP>)<UP>·</UP>(<UP>Gln→Glu</UP>)]<UP>=</UP><SUB><UP>Cit→</UP></SUB>[<UP>&agr;KG</UP>]<SUP><UP>Glu</UP></SUP><UP>·</UP>(<UP>&agr;KG → Glu</UP>) (Eq. 61)
Replacing (Glu right-arrow Gln)·(Gln right-arrow Glu) by (Glu up-down-arrow  Gln), as indicated in Equation 3, it follows that:
<SUB><UP>Cit</UP>→</SUB>[<UP>Glu</UP>]<SUP><UP>Glu</UP></SUP>=<SUB><UP>Cit</UP>→</SUB>[&agr;<UP>KG</UP>]<SUP><UP>Glu</UP></SUP> · (&agr;<UP>KG</UP> → <UP>Glu</UP>)/[1−(<UP>Glu</UP> ⇅ <UP>Gln</UP>)] (Eq. 62)
where Cit right-arrow [alpha KG]Glu = (Cit formed)·(recycling ratio of alpha -ketoglutarate through glutamate and glutamine).

From Equation 6 one can calculate that
 <SUB><UP>Cit</UP>→</SUB>[&agr;<UP>KG</UP>]<SUP><UP>Glu</UP></SUP>=[<UP>Cit</UP>]<SUP><UP>Glu</UP></SUP> · [1−(<UP>Glu</UP> ⇅ <UP>Gln</UP>)]/[1−(<UP>Glu</UP> ⇅ &agr;<UP>KG</UP>)
−(<UP>Glu</UP> ⇅ <UP>Gln</UP>)]=[<UP>Cit</UP>]<SUP><UP>Glu</UP></SUP> · {&agr;<UP>KG</UP> ⇅ <UP>Glu</UP>+<UP>Gln</UP>} (Eq. 63)
Since {Glu up-down-arrow  alpha KG + Gln} = 1/[1 - (Glu up-down-arrow  alpha KG) - (Glu up-down-arrow  Gln)] (see Equation 5), one can deduce from Equations 62 and 63 that
<SUB><UP>Cit</UP>→</SUB>[<UP>Glu</UP>]<SUP><UP>Glu</UP></SUP>=[<UP>Cit</UP>]<SUP><UP>Glu</UP></SUP> · (&agr;<UP>KG</UP> → <UP>Glu</UP>)/[1−(<UP>Glu</UP> ⇅ &agr;<UP>KG</UP>) (Eq. 64)
−(<UP>Glu</UP> ⇅ <UP>Gln</UP>)]=[<UP>Cit</UP>]<SUP><UP>Glu</UP></SUP> · (&agr;<UP>KG</UP> → <UP>Glu</UP>) · {<UP>Glu</UP> ⇅ &agr;<UP>KG</UP>+<UP>Gln</UP>}
The total amount of glutamate involved in the metabolism is obtained from Equations 59 and 64
[<UP>Glu</UP>]<SUP><UP>Glu</UP></SUP>=<SUB><UP>Glu</UP>→</SUB>[<UP>Glu</UP>]<SUP><UP>Glu</UP></SUP>+<SUB><UP>Cit</UP>→</SUB>[<UP>Glu</UP>]<SUP><UP>Glu</UP></SUP>=[X (Eq. 65)
+[<UP>Cit</UP>]<SUP><UP>Glu</UP></SUP> · (&agr;<UP>KG</UP> → <UP>Glu</UP>)] · {<UP>Glu</UP> ⇅ &agr;<UP>KG</UP>+<UP>Gln</UP>}
but it cannot be calculated because X·{Glu up-down-arrow  alpha KG + Gln} and (alpha KG right-arrow Glu)·{Glu up-down-arrow  alpha KG + Gln} cannot be derived from the labeled carbon data.

The total amount of alpha -ketoglutarate formed during glutamate metabolism is obtained from the sum of the alpha -ketoglutarate formed directly from the glutamate utilized, Glu right-arrow [alpha KG]Glu (see Equation 59), and the alpha KG derived from citrate, Cit right-arrow [alpha KG]Glu (see Equation 63) (see also Figs. 2 and 4).
[&agr;<UP>KG</UP>]<SUP><UP>Glu</UP></SUP>=<SUB><UP>Glu</UP>→</SUB>[&agr;<UP>KG</UP>]<SUP><UP>Glu</UP></SUP>+<SUB><UP>Cit</UP>→</SUB>[&agr;<UP>KG</UP>]<SUP><UP>Glu</UP></SUP>=X · {&cjs1671;<UP>Glu</UP> → &agr;<UP>KG</UP>} (Eq. 66)
+[<UP>Cit</UP>]<SUP><UP>Glu</UP></SUP> · {&agr;<UP>KG</UP> ⇅ <UP>Glu</UP>+<UP>Gln</UP>}
where Glu right-arrow [alpha KG]Glu = Glu right-arrow [Glu]Glu·(Glu right-arrow alpha KG) and {vGlu right-arrow alpha KG} = (Glu right-arrow alpha KG)·{Glu up-down-arrow  alpha KG+Gln}.

From Equation 65 and Fig. 2, unidirectional flux of glutamate to alpha -ketoglutarate, noted [Glu &cjs3675; alpha KG], can be calculated as follows:
[<UP>Glu</UP> &cjs3675; &agr;<UP>KG</UP>]=[<UP>Glu</UP>]<SUP><UP>Glu</UP></SUP> · (<UP>Glu</UP> → &agr;<UP>KG</UP>)=[X (Eq. 67)
+[<UP>Cit</UP>]<SUP><UP>Glu</UP></SUP> · (&agr;<UP>KG</UP> → <UP>Glu</UP>)]·{&cjs1671;<UP>Glu</UP> → &agr;<UP>KG</UP>}
From Equation 66 and Fig. 2, unidirectional flux of alpha -ketoglutarate to glutamate, noted [alpha KG &cjs3675; Glu], can also be expressed as:
[&agr;<UP>KG</UP> &cjs3675; <UP>Glu</UP>]=[&agr;<UP>KG</UP>]<SUP><UP>Glu</UP></SUP> · (&agr;<UP>KG</UP> → <UP>Glu</UP>)=[X · {&cjs1671;<UP>Glu</UP> → &agr;<UP>KG</UP>} (Eq. 68)
+[<UP>Cit</UP>]<SUP><UP>Glu</UP></SUP> · {&agr;<UP>KG</UP> ⇅ <UP>Glu</UP>+<UP>Gln</UP>}] · (&agr;<UP>KG</UP> → <UP>Glu</UP>)
=X · {&cjs1671;<UP>Glu</UP> → &agr;<UP>KG</UP>} · (&agr;<UP>KG</UP> → <UP>Glu</UP>)+[<UP>Cit</UP>]<SUP><UP>Glu</UP></SUP> · {<SUB><UP>Cit</UP></SUB>&agr;<UP>KG</UP> → <UP>Glu</UP>}
where {Citalpha KG right-arrow Glu} is the proportion of citrate-derived alpha -ketoglutarate converted to glutamate which takes into account the total recycling through the "Glu right-arrow alpha KG right-arrow Glu" and "Glu right-arrow Gln right-arrow Glu" cycles (see under "Notation").

Thus, net flux of glutamate to alpha -ketoglutarate, noted net[Glu &cjs3675; alpha KG], can be obtained from the two latter equations:
<SUB><UP>net</UP></SUB>[<UP>Glu</UP> &cjs3675; &agr;<UP>KG</UP>]=[<UP>Glu</UP> &cjs3675; &agr;<UP>KG</UP>]−[&agr;<UP>KG</UP> &cjs3675; <UP>Glu</UP>]=[X (Eq. 69)
+[<UP>Cit</UP>]<SUP><UP>Glu</UP></SUP> · (&agr;<UP>KG</UP> → <UP>Glu</UP>)] · {&cjs1671;<UP>Glu</UP> → &agr;<UP>KG</UP>}−[X · {&cjs1671;<UP>Glu</UP> → &agr;<UP>KG</UP>}
+[<UP>Cit</UP>]<SUP><UP>Glu</UP></SUP> · {&agr;<UP>KG</UP> ⇅ <UP>Glu</UP>+<UP>Gln</UP>}] · (&agr;<UP>KG</UP> → <UP>Glu</UP>)
where, as mentioned above, {vGlu right-arrow alpha KG} = (Glu right-arrow alpha KG)·{Glu up-down-arrow  alpha KG + Gln}

Since {Glu up-down-arrow  alpha KG + Gln} and {alpha KG up-down-arrow  Glu + Gln} are equal to 1/[1 - (Glu up-down-arrow  alpha KG) - (Glu up-down-arrow  Gln)] and [1 - (Glu up-down-arrow  Gln)]/[1 - (Glu up-down-arrow  alpha KG) - (Glu up-down-arrow  Gln)], respectively (see Equations 5 and 6), and since, from Equations 2 and 3 (Glu up-down-arrow  alpha KG) = (Glu right-arrow alpha KG)·(alpha KG right-arrow Glu) and (Glu up-down-arrow  Gln) = (Glu right-arrow Gln)·(Gln right-arrow Glu), it follows that
<SUB><UP>net</UP></SUB>[<UP>Glu&cjs3675;&agr;KG</UP>] (Eq. 70)
=X{<SUP>v</SUP>Glu→
&agr;KG}(&agr;KG→OAA)−[Cit]<SUP>Glu</SUP>·[1−{<SUB>Cit</SUB>&agr;KG→OAA}]
It can be verified by using the latter equation that, as expected from Fig. 2, the net flux of glutamate to alpha -ketoglutarate is equal to the flux of alpha -ketoglutarate dehydrogenase, noted [alpha KGdH], minus the flux of citrate synthase, noted [CS]:
<SUB><UP>net</UP></SUB>[<UP>Glu&cjs3675;&agr;KG</UP>]<UP>=</UP>[<UP>&agr;KGdH</UP>]<UP>−</UP>[<UP>CS</UP>] (Eq. 71)
with
[&agr;<UP>KGdH</UP>]=X · {&cjs1671;<UP>Glu</UP> → &agr;<UP>KG</UP>} · (&agr;<UP>KG</UP> → <UP>OAA</UP>)+[<UP>Cit</UP>]<SUP><UP>Glu</UP></SUP> · {<SUB><UP>Cit</UP></SUB>&agr;<UP>KG</UP> → <UP>OAA</UP>} (Eq. 72)
(see Figs. 2 and 4)

Fig. 2 and Equation 65 allow us to calculate the accumulation of glutamate
[&cjs1670;<UP>Glu</UP>]<SUP><UP>Glu</UP></SUP>=[<UP>Glu</UP>]<SUP><UP>Glu</UP></SUP> · (<UP>Glu</UP> → &cjs1670;<UP>Glu</UP>)=(X+[<UP>Cit</UP>]<SUP><UP>Glu</UP></SUP> · (&agr;<UP>KG</UP> → <UP>Glu</UP>)) · {&cjs1671;<UP>Glu</UP> → &cjs1670;<UP>Glu</UP>} (Eq. 73)
with
<UP> </UP>[<UP>Cit</UP>]<SUP><UP>Glu</UP></SUP>=[<UP>OAA</UP>]<SUP><UP>Glu</UP></SUP> · (<UP>OAA</UP> → <UP>Cit</UP>)+[<UP>AcCoA</UP>]<SUP><UP>Glu</UP></SUP> · (<UP>AcCoA</UP>
+ <UNL><UP>OAA</UP></UNL><UP> → Cit</UP>) (Eq. 74)
Fig. 2 shows that citrate originates from oxaloacetate through two possible pathways, namely the tricarboxylic acid cycle and the "OAA right-arrow PEP right-arrow Pyr right-arrow AcCoA right-arrow Cit right-arrow OAA" cycle. Therefore, the latter equation can be rewritten as
[<UP>Cit</UP>]<SUP><UP>Glu</UP></SUP>=[<UP>OAA</UP>]<SUP><UP>Glu</UP></SUP> · [(<UP>OAA</UP> → <UP>Cit</UP>)+(<UP>OAA</UP> →<UP>PEP</UP>) · (<UP>PEP</UP> → <UP>Pyr</UP>)
 · (<UP>Pyr</UP> → <UP>AcCoA</UP>) · (<UP>AcCoA</UP> + <UNL><UP>OAA</UP></UNL><UP> → Cit</UP>)] (Eq. 75)
(see also the definition of (AcCoA + <UNL>OAA</UNL> right-arrow Cit) in Table I).

Note that Equation 75 takes into account the condensation of endogenous oxaloacetate molecules with labeled glutamate-derived acetyl-CoA molecules.

Fig. 2 and Equation 65 also allow to calculate the accumulation of glutamine:
[&cjs1670;<UP>Gln</UP>]<SUP><UP>Glu</UP></SUP>=[<UP>Glu</UP>]<SUP><UP>Glu</UP></SUP> · {&cjs1671;<UP>Glu</UP> → &cjs1670;<UP>Gln</UP>}=(X+[<UP>Cit</UP>]<SUP><UP>Glu</UP></SUP> · (&agr;<UP>KG</UP> → <UP>Glu</UP>)) · {&cjs1670;<UP>Glu</UP> → &cjs1670;<UP>Gln</UP>} (Eq. 76)
Thus, the accumulation of glutamate plus glutamine is given by
[&cjs1670;<UP>Glx</UP>]<SUP><UP>Glu</UP></SUP>=[&cjs1670;<UP>Glu</UP>]<SUP><UP>Glu</UP></SUP>+[&cjs1670;<UP>Gln</UP>]<SUP><UP>Glu</UP></SUP>=(X (Eq. 77)
+[<UP>Cit</UP>]<SUP><UP>Glu</UP></SUP> 
· (&agr;<UP>KG</UP> → <UP>Glu</UP>)) · ({&cjs1671;<UP>Glu</UP> → &cjs1670;<UP>Glu</UP>}+{&cjs1671;<UP>Glu</UP> → &cjs1670;<UP>Gln</UP>})
The net flux of glutamine accumulation, [Glu &cjs3675; &cjs1670;Gln], is given by the amount of glutamine accumulated from the glutamate available for the metabolism (added glutamate utilized + glutamate synthesized, see Equation 76):
[<UP>Glu</UP> &cjs3675; &cjs1670;<UP>Gln</UP>]=[&cjs1670;<UP>Gln</UP>]<SUP><UP>Glu</UP></SUP>=[<UP>Glu</UP>]<SUP><UP>Glu</UP></SUP> · (<UP>Glu</UP> → &cjs1670;<UP>Gln</UP>)=(X+[<UP>Cit</UP>]<SUP><UP>Glu</UP></SUP> (Eq. 78)
· (&agr;<UP>KG</UP> → <UP>Glu</UP>)) · {&cjs1671;<UP>Glu</UP> → &cjs1670;<UP>Gln</UP>}
The net utilization of glutamate can be obtained from Equations 70, 73, and 78 and is equal to net[Glu &cjs3675; alpha KG] + [&cjs1670;Gln]Glu - [&cjs1670;Glu]Glu; the value obtained can be compared with the glutamate utilization measured enzymatically.

The equations for unidirectional fluxes of glutamine synthetase, [GS], and glutaminase, [Glnase], obtained from Equation 65 and Fig. 1, are:
  [<UP>GS</UP>]=(X+[<UP>Cit</UP>]<SUP><UP>Glu</UP></SUP> · (&agr;<UP>KG</UP> → <UP>Glu</UP>)) · {&cjs1671;<UP>Glu</UP> → <UP>Gln</UP>} (Eq. 79)
and
[<UP>Glnase</UP>]= (Eq. 80)
(X+[<UP>Cit</UP>]<SUP><UP>Glu</UP></SUP>·(<UP>&agr;KG → Glu</UP>))·{<SUP><UP>v</UP></SUP><UP>Glu → Gln</UP>}·(<UP>Gln → Glu</UP>)
but, these fluxes cannot be calculated since labeled carbon data don't allow us to obtain the value of {vGlu right-arrow Gln}.

Flux through pyruvate dehydrogenase, which is equal to [AcCoA]Glu, can be derived from Fig. 2: [AcCoA]Glu = [OAA]Glu·(OAA right-arrow PEP)·(PEP right-arrow Pyr)·(Pyr right-arrow AcCoA).

Flux through phosphoenolpyruvate carboxykinase is given by [OAA]Glu·(OAA right-arrow PEP) (see Fig. 2).

Flux through pyruvate kinase is equal to [OAA]Glu·(OAA right-arrow PEP)·(PEP right-arrow Pyr) (see Fig. 2).

Flux through lactate dehydrogenase is obtained by multiplying flux through pyruvate kinase by (Pyr right-arrow Lac).

Flux through phosphoglyceromutase is given by [Glc]Glu + [Ser]Glu and corresponds to the flux through phosphoenolpyruvate carboxykinase multiplied by (PEP right-arrow 3PG).

Flux through glucose-6-phosphatase is equal to [OAA]Glu·(OAA right-arrow PEP)·(PEP right-arrow 3PG)·(3PG right-arrow Glc) and corresponds to 2·[Glc]Glu since fluxes are expressed in C3 units.

Flux through citrate synthase, [CS], is given by the amount of citrate formed as shown in Fig. 2 and Equation 75.

Net flux through transaminases resulting in a net conversion of glutamate into alpha -ketoglutarate is given by the sum of the net transaminase fluxes involved in alanine, aspartate, and serine formation.

Net flux through alanine aminotransferase is equal to alanine accumulation: [Ala]Glu = [OAA]Glu·(OAA right-arrow PEP)·(PEP right-arrow Pyr)·(Pyr right-arrow Ala).

Net flux through aspartate aminotransferase is equal to aspartate accumulation: [Asp]Glu = [OAA]Glu·(OAA right-arrow Asp).

Net flux through 3-phosphoglycerate dehydrogenase and phosphoserine aminotransferase are equal to [OAA]Glu·(OAA right-arrow PEP)·(PEP right-arrow 3PG)·(3PG right-arrow Ser).

Net flux through glutamate dehydrogenase is equal to the net flux of glutamate conversion into alpha -ketoglutarate minus the net flux through transaminases.

Flux through alpha -ketoglutarate dehydrogenase, [alpha KGdH], is given by the amount of alpha -ketoglutarate converted into succinyl-CoA and subsequently into oxaloacetate (see Equations 8, 9, 72, and 75).
[&agr;<UP>KGdH</UP>]=X · {&cjs1671;<UP>Glu</UP> → &agr;<UP>KG</UP>} · (&agr;<UP>KG</UP> → <UP>OAA</UP>)+
[<UP>OAA</UP>]<SUP><UP>Glu</UP></SUP> · [(<UP>OAA</UP> → <UP>Cit</UP>)+
(<UP>OAA</UP> →<UP>PEP</UP>) · (<UP>PEP</UP> → <UP>Pyr</UP>) · (<UP>Pyr</UP> →
<UP>AcCoA</UP>) · (<UP>AcCoA</UP> + <UNL><UP>OAA</UP></UNL><UP> → Cit</UP>)] ·{<SUB><UP>Cit</UP></SUB>&agr;<UP>KG</UP>
→ <UP>OAA</UP>} = X · {&cjs1671;<UP>Glu</UP> →
 &agr;<UP>KG</UP>} · (&agr;<UP>KG</UP> → <UP>OAA</UP>)+
[<UP>OAA</UP>]<SUP><UP>Glu</UP></SUP> · [(<UP>TCA</UP>⇅)+
(<UP>AcCoA</UP> + <UNL><UP>OAA</UP></UNL><UP> → Cit</UP>) · (<UP>AcCoA</UP> ⇅ <UP>OAA</UP>)] (Eq. 81)
Since oxaloacetate is produced either by pyruvate carboxylase or alpha -ketoglutarate dehydrogenase, flux through pyruvate carboxylase is equal to [OAA]Glu minus flux through alpha -ketoglutarate dehydrogenase (see Fig. 2).

As shown in a previous study (1), flux of oxaloacetate equilibration with fumarate is equal to [OAA]Glu· 2·(OAAi)/[1 - 2·(OAAi)].


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 (3-14) 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 13C 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 right-arrow 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 CO2 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 CO2 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 alpha -ketoglutarate. Note here that Shulman and co-workers (11, 12, 15) were also able to calculate the alpha -ketoglutarate - 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 metabolism and brings new insights into the complexity of such a metabolism in mammalian cells.


FOOTNOTES

*   The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Dagger    To whom correspondence should be addressed: Centre d'Etudes Métaboliques par Spectroscopie de Résonance Magnétique, Pavillon P, Hôpital Edouard Herriot, place d'Arsonval, 69374 Lyon Cedex 03, France. Tel.: (33) 04-78-77-86-65; Fax: (33) 04-78-77-87-39.
1    The abbreviations used are: OAA, oxaloacetate; Ac, acetate; AcCoA, acetyl-coenzyme A; Asp, aspartate; Cit, citrate; CS, citrate synthase; Glc, glucose; Gln, glutamine; Glnase, glutaminase; Glu, glutamate; Glx, glutamate + glutamine; GS, glutamine synthetase; alpha KG, alpha -ketoglutarate; alpha KGdH, alpha -ketoglutarate dehydrogenase; Lac, lactate; 3PG, 3-phosphoglycerate; PEP, phosphoenolpyruvate; Pyr, pyruvate; TCA, tricarboxylic acid, MET, metabolite; Ser, serine.

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