From the Centre d'Etudes Métaboliques par Spectroscopie de
Résonance Magnétique (INSERM CRI 950102), Hôpital
Edouard Herriot, 69374 Lyon Cedex 03, France
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
-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.
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
-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 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
-ketoglutarate and therefore complete one tricarboxylic acid cycle
turn. In the presence of NH4+, 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
derived metabolite). The notations of the
proportions taking into account the recycling in the "glutamate
-ketoglutarate
glutamate" and "glutamate
glutamine
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.
[View Larger Version of this Image (23K GIF file)]
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
PEP1
Pyr1
OAA" cycle and (iii) the "OAA
PEP
Pyr
AcCoA1
Cit1
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
KG
Glu" cycle in which a
proportion (Glu
KG) of glutamate is recycled at each turn, (ii)
the "Glu
Gln
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
PEP
Pyr
OAA" cycle in which a
proportion (Pyr
OAA) of oxaloacetate is recycled at each turn, (v)
the "OAA
PEP
Pyr
AcCoA
Cit
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
KG
Glu" and
"Glu
Gln
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 +
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 +
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
KG
Glu" and "Glu
Gln
Glu" cycles and are noted {Glu
KG + Gln} and {
KG
Glu + Gln}, respectively (see Table I). The proportions of
glutamate, {vGlu
Glu}, and glutamine,
{vGlu
Gln}, accumulation and
-ketoglutarate
formation, {vGlu
KG}, from added glutamate are
equal to {Glu
KG + Gln} multiplied by (Glu
Glu), (Glu
Gln).(Gln
Gln) and (Glu
KG). The proportion of
oxaloacetate formed, {Cit
KG
OAA}, from
citrate-derived
-ketoglutarate is equal to {
KG
Glu+Gln} multiplied by (
KG
OAA).
[View Larger Version of this Image (22K GIF file)]
Glutamate is the metabolite common to the two other cycles that have
been introduced to improve the model; these are (iv) the "Glu
KG
Glu" cycle and (v) the "Glu
Gln
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.
[View Larger Version of this Image (15K GIF file)]
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
,
,
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 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
-ketoglutarate
glutamate" and "glutamate
glutamine
glutamate" cycles. Depending on which metabolite is recycled,
glutamate or
-ketoglutarate, the proportion of conversion is
multiplied either by {Glu
KG + Gln} or {
KG
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
-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 + 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.
[View Larger Version of this Image (27K GIF file)]
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
OAA
PEP
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.
[View Larger Version of this Image (17K GIF file)]
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.
[View Larger Version of this Image (23K GIF file)]
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 (
)
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 + 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
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
KG
Glu"
and "Glu
Gln
Glu" cycles is indicated by substituting the
parentheses () by {}.
Special symbols are used in the notations {vGlu
Met2} and {Cit
KG
OAA} to stress the fact that
only added glutamate and only citrate-derived
-ketoglutarate,
respectively, are concerned.
Let (AcCoA + OAA
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
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 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:
|
(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
KG) is the recycling factor of the "Glu
KG
Glu" cycle;
|
(Eq. 2)
|
where (Glu
KG) is the proportion of added glutamate
directly converted into
-ketoglutarate, and (
KG
Glu) is the
proportion of
-ketoglutarate directly converted into glutamate.
(ii) (Glu
Gln) is the recycling factor of the "Glu
Gln
Glu" cycle;
|
(Eq. 3)
|
where (Glu
Gln) is the proportion of glutamate directly
converted into glutamine and (Gln
Glu) and (Gln
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
-ketoglutarate or glutamine; therefore, the proportion of glutamate
formed which accumulated (Glu
Glu) is defined by
|
(Eq. 4)
|
In the following equations, the proportion of glutamate and
-ketoglutarate molecules that are metabolized and then resynthesized through the successive operation of "Glu
KG
Glu" and
"Glu
Gln
Glu" cycles over a theoretically infinite number
of turns are noted {Glu
KG + Gln} and {
KG
Glu + Gln}, respectively.
The resulting glutamate recycling through
-ketoglutarate and
glutamine, is
|
(Eq. 5)
|
(see also Fig. 2).
One can demonstrate that {
KG
Glu + Gln}, the resulting
-ketoglutarate recycling through glutamate and glutamine, is:
|
(Eq. 6)
|
(see also Fig. 2).
Therefore, from the latter equation and Equation 3, it follows
that
|
(Eq. 7)
|
(see also Fig. 2)
where {vGlu
Gln} and {vGlu
Gln} represent the proportion of added glutamate not recycled or
recycled only in the "Glu
KG
Glu" and "Glu
Gln
Glu" cycles which was converted in glutamine formed and glutamine
accumulated, respectively.
The parameters {vGlu
Gln} and {vGlu
Gln}, as defined above (see "Notations"), are equal to
{Glu
KG + Gln} multiplied by (Glu
Gln) and (Glu
Gln), respectively.
(iii) (Pyr
OAA) is the recycling factor of the "OAA
PEP
Pyr
OAA" cycle, i.e. the proportion of oxaloacetate
resynthesized at each turn of this cycle. (Pyr
OAA) = (OAA
PEP)·(PEP
Pyr)·(Pyr
OAA) (see Fig. 2). Where (OAA
PEP),
(PEP
Pyr), and (Pyr
OAA) are the proportions of oxaloacetate,
phosphoenolpyruvate, and pyruvate converted into phosphoenolpyruvate,
pyruvate, and oxaloacetate, respectively (see Fig. 2).
(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
KG) is considered to be 1.
|
(Eq. 8)
|
where (OAA
Cit) and {Cit
KG
OAA} are the
proportions of oxaloacetate converted into citrate and citrate-derived
-ketoglutarate converted into oxaloacetate, respectively (see Fig.
2).
(v) (AcCoA
OAA) is the recycling factor of the "OAA
PEP
Pyr
AcCoA
Cit
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
Cit) is
equal to 1.
|
(Eq. 9)
|
In the latter 2 equations (TCA
) and (AcCoA
OAA) take into
account the recycling of
-ketoglutarate through glutamate and
glutamine since {Cit
KG
OAA} = (
KG
OAA)·{
KG
Glu + Gln} (see Fig. 2).
The amount of glutamine formed directly from the substrate glutamate is
given by X·{vGlu
Gln}, where
X is the amount of glutamate utilized and
{vGlu
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.
The corresponding amount of glutamine accumulated is given by
X·{vGlu
Gln} = X·(Glu
Gln)·{Glu
KG + Gln} where (Glu
Gln) = (Glu
Gln)·(Gln
Gln) and (Gln
Gln) is the proportion of
glutamine formed which accumulated (see also Equation 3).
It can be seen in Fig. 4 that X·{vGlu
Gln} is given by C5Gln accumulated from
C5Glu: X·{vGlu
Gln} = [
C5Gln]C5Glu·X·{vGlu
Gln} is also equal to
[
C3Gln]C3Glu
[
C2Gln]C3Glu
(see Fig. 4).
Figs. 2 and 4 allow us to calculate the amount
X·{vGlu
KG}·(
KG
OAA) of
glutamate-derived oxaloacetate: X·{vGlu
KG}·(
KG
OAA) = [*CO2]C1Glu
where {vGlu
KG} is equal to (Glu
KG)·{Glu
KG + Gln}.
[C2Glu]C3Glu/[C2Gln]C3Glu
and
[C1Glu]C5Glu/[C1Gln]C5Glu
is equal to the ratio {vGlu
Glu}/{vGlu
Gln} which, after
simplification by the factor {Glu
KG + Gln}, is also
equal to (Glu
Glu)/(Glu
Gln) (see above, Figs. 2 and 4
and Table I).
From X·{vGlu
Gln} and
{vGlu
Glu}/{vGlu
Gln},
one can calculate X·{vGlu
Glu}.
Fig. 2 shows that the glutamate utilized (X) is either
accumulated as glutamate, X·{vGlu
Glu}, or glutamine, X·{vGlu
Gln}, or converted into oxaloacetate,
X·{vGlu
KG}·(
KG
OAA),
indicating that X = X·{vGlu
Glu} + X·{vGlu
Gln} + X·{vGlu
KG}·(
KG
OAA).
Therefore the following parameters can be calculated:
|
(Eq. 10)
|
Fig. 4 allows us to calculate that
|
(Eq. 11)
|
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
OAA) = (OAA
PEP)·(PEP
Pyr)·(Pyr
OAA) (see also Fig. 2
and Table I).
|
(Eq. 12)
|
Then,
[C4OAA]C5Glu = X·{vGlu
KG}·(
KG
OAA)·(1/2)·(1
[1
2·(OAAi)]·(Pyr
OAA))/(1
[1
(OAAi)]·(Pyr
OAA)).
And, [(C1 + C4)OAA]C5Glu = X·{vGlu
KG}·(
KG
OAA)·[1 + [(OAAi)/2]/(1
[1
(OAAi)]·(Pyr
OAA))].
From Fig. 4 one can also deduce that
|
(Eq. 13)
|
where, as mentioned in Equation 9 (AcCoA
OAA) = (OAA
PEP).(PEP
Pyr)·(Pyr
AcCoA)·{Cit
KG
OAA} (see also Fig. 2 and Table I) and
([C2OAA]C3Glu·[(AcCoA
OAA)/2] + [C3OAA]C3Glu·[(TCA
)/2])
represents the C1OAA formed directly from the C-2 and C-3
of oxaloacetate.
Fig. 6 shows that
|
(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
|
(Eq. 15)
|
Using Equation 14, Equation 13 can be rewritten as
|
(Eq. 16)
|
Then,
|
(Eq. 17)
|
And,
|
(Eq. 18)
|
And,
|
(Eq. 19)
|
Therefore,
|
(Eq. 20)
|
Let us call A the Latter Ratio, then one can deduce from Fig. 4
that
|
(Eq. 21)
|
and
|
(Eq. 22)
|
From Fig. 4, one can deduce that
|
(Eq. 23)
|
where the value B of the latter 2 ratios can be calculated from
Equations 11 and 15: B = [1
(Pyr
OAA) + (OAAi)·(Pyr
OAA)]/(1
(Pyr
OAA)
[(TCA
) + (AcCoA
OAA)]/2).
Combining Equations 21 and 23, yields (A + 1)/(B
A
1) = [1
(Pyr
OAA)]/[(OAAi)·(Pyr
OAA)].
Let C = [1
(Pyr
OAA)]/[(OAAi)·(Pyr
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
OAA))·[(OAA
Cit)
(TCA
)]/[(OAA
PEP)·(PEP
Pyr)·(Pyr
Ala)]. Where (Pyr
Ala) is the proportion of
pyruvate directly converted to alanine.
From Fig. 5 it is also possible to calculate that
|
(Eq. 24)
|
Combining the latter equation and Equation 24, we obtain
|
(Eq. 25)
|
Let us call E = [C1Glx]Glu + *CO2/[C1Ala]Glu + * CO2
where
|
(Eq. 26)
|
(see Fig. 5) and
|
(Eq. 27)
|
(see Fig. 5). Combining Equations 24 and 26, we obtain
|
(Eq. 28)
|
Let us call F = D/E
|
(Eq. 29)
|
Therefore, i = F·(1 + C)/[2 + C·(1 + F)]. Since C was
defined as [1
(Pyr
OAA)]/[(OAAi)·(Pyr
OAA)] (see above), one can
calculate that (Pyr
OAA) = 1/[1 + (OAAi) ·C].
Let us call G = 1/A; then,
|
(Eq. 30)
|
and
|
(Eq. 31)
|
From Fig. 6, it can be deduced that
|
(Eq. 32)
|
where
|
(Eq. 33)
|
And Fig. 4 shows that
|
(Eq. 34)
|
where (Pyr
Lac) is the proportion of pyruvate directly
converted to lactate.
Combining Equations 32 and 34, it follows that
(OAA
PEP) · (PEP
Pyr) · (Pyr
Lac) = [(C2 + C3)Lac]C3Glu · H/[X · {vGlu
KG} · (
KG
OAA)].
Similarly,
|
(Eq. 35)
|
Then,
[C2OAA]C3Glu and
[C3OAA]C3Glu
can be calculated:
|
(Eq. 36)
|
and similarly,
|
(Eq. 37)
|
Furthermore,
[C2OAA]C3Glu and
[C3OAA]C3Glu
are theoretically identical (see Figs. 4 and 6).
The proportion (OAA
Asp1) of oxaloacetate directly
converted into aspartate is given by (OAA
Asp) = [C3Asp]C3Glu/[C3OAA]C3Glu
(see Fig. 4).
From the latter equation and Equation 37, it follows that (OAA
Asp)
is equal to
[C3Asp]C3Glu·[(OAA
PEP)·(PEP
Pyr)·(Pyr
Ala)]/[C3Ala]C3Glu.
Oxaloacetate is either converted into citrate or phosphoenolpyruvate or
aspartate (see Fig. 4) with the proportion (OAA
Cit), (OAA
PEP)
or (OAA
Asp), respectively.
Therefore, it follows that (OAA
Cit) + (OAA
PEP) + (OAA
Asp) = 1 which yields
|
(Eq. 38)
|
From Figs. 2 and 4, it can be deduced that
|
(Eq. 39)
|
Thus, (TCA
) and (AcCoA
OAA) can be calculated from
Equations 31 and 39:
|
(Eq. 40)
|
Let us call J = [(C6 + C1 + C5 + C2)Glc1]C3Glu and
L = [(C3 + C2)Ser1]C3Glu.
Then J + L = (OAA
PEP)·(PEP
3PG)·[(C2 + C3)OAA]C3Glu (see
Fig. 4). Where (PEP
3PG) is the proportion of
phosphoenolpyruvate directly converted to 3-phosphoglycerate.
From Equations 14 and 33, it follows that
|
(Eq. 41)
|
From Eqs. 38 and 39, and knowing (AcCoA
OAA)/(TCA
) and
[(OAA
Cit)+(OAA
PEP)],we obtain:
|
(Eq. 42)
|
Phosphoenolpyruvate is either converted into pyruvate or
3-phosphoglycerate to yield glucose and serine (see Figs. 1 and 4) with
the proportion (PEP
Pyr) or (PEP
3PG), respectively. Therefore,
(PEP
Pyr) + (PEP
3PG) = 1 which yields:
|
(Eq. 43)
|
Pyruvate is either converted into oxaloacetate, lactate, alanine,
or acetyl-CoA (see Figs. 1 and 4) with the proportion (Pyr
OAA),
(Pyr
Lac), (Pyr
Ala), or (Pyr
AcCoA), respectively. Therefore,
|
(Eq. 44)
|
Combining the latter equation and Equations 42 and 43, one can
deduce that
|
(Eq. 45)
|
Where, as indicated above, (Pyr
OAA) = (OAA
PEP).(PEP
Pyr)·(Pyr
OAA).
Then (PEP
3PG) = (OAA
PEP)·(PEP
3PG)/(OAA
PEP,
and, since (PEP
Pyr) + (PEP
3PG) = 1, (PEP
Pyr) = 1
(PEP
3PG).
The parameter (OAA
Cit) can be calculated from Equation 38:
|
(Eq. 46)
|
Since the recycling factor in the tricarboxylic acid cycle
(TCA
), which accounts also for
-ketoglutarate recycling through glutamate and glutamine, is equal to (OAA
Cit)·{Cit
KG
OAA} (see Equation 8), it can be
calculated that
|
(Eq. 47)
|
From Equation 7 and since as shown in Fig. 1 (Glu
KG) + (Glu
Glu) + (Glu
Gln) = 1 (see Equation 4), it follows
that
|
(Eq. 48)
|
Thus,
|
(Eq. 49)
|
Then, multiplying the 2 members of the latter equation by (
KG
OAA) and rearranging, we obtain
|
(Eq. 50)
|
Therefore, the
-ketoglutarate recycling through glutamate and
glutamine {
KG
Glu + Gln} is equal to {
KG
Glu + Gln} = {Cit
KG
OAA}/(
KG
OAA).
Then, {
Glu
KG} can be calculated from Equations 10 and 50.
Equation 39 yields (OAA
PEP)·(PEP
Pyr)·(Pyr
AcCoA) = [(AcCoA
OAA)/(TCA
)]·(OAA
Cit).
The proportion (Pyr
AcCoA) of pyruvate converted into acetyl-CoA is
given by (Pyr
AcCoA) = (OAA
PEP)·(PEP
Pyr)·(Pyr
AcCoA)/[(OAA
PEP)·(PEP
Pyr)].
Similarly, the proportions (Pyr
Lac), (Pyr
Ala), and (Pyr
OAA) corresponding to the proportions of pyruvate transformed into
lactate, alanine, and oxaloacetate, respectively, are obtained as
follows. (Pyr
Lac) = (OAA
PEP)·(PEP
Pyr)·(Pyr
Lac)/(OAA
PEP)·(PEP
Pyr); (Pyr
Ala) = (OAA
PEP)·(PEP
Pyr)·(Pyr
Ala)/(OAA
PEP)·(PEP
Pyr);
(Pyr
OAA) = (Pyr
OAA)/(OAA
PEP)·(PEP
Pyr); where (Pyr
OAA) = (OAA
PEP)·(PEP
Pyr)·(Pyr
OAA).
The proportion (3PG
Glc) of 3-phosphoglycerate which yields glucose
is given by (3PG
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
OAA), (TCA
) and (AcCoA
OAA) represent the oxaloacetate recycled in the "OAA
PEP
Pyr
OAA," the tricarboxylic acid and the "OAA
PEP
Pyr
AcCoA
Cit
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
KG
Glu" and "Glu
Gln
Glu" cycles.
Let us call (AcCoA +
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 +
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 +
Cit)·{Cit
KG
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
PEP
Pyr
AcCoA
Cit
OAA" cycle is (AcCoA +
Cit)·(AcCoA
OAA), while the proportion of
oxaloacetate formed by the "OAA
PEP
Pyr
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) + (TCA
) + (AcCoA +
Cit)·(AcCoA
OAA), which represents the proportion of oxaloacetate recycled at each turn of the
multicycle presented in Fig. 2.
|
(Eq. 51)
|
Then
|
(Eq. 52)
|
In the latter equation, 1/[1
(Pyr
OAA)
(TCA)
(AcCoA +
Cit)·(AcCoA
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 +
Cit) can be obtained by
calculating first [1
(AcCoA +
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,
|
(Eq. 53)
|
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:
|
(Eq. 54)
|
Similarly, one can demonstrate that
|
(Eq. 55)
|
where (OAA +
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
|
(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.
|
(Eq. 57)
|
and
|
(Eq. 58)
|
In view of the fact that [OAA]Glu and (AcCoA +
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
-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
[Glu]Glu is
derived directly from the glutamate utilized which undergoes a
recycling through
-ketoglutarate and glutamine:
Glu
[Glu]Glu = (glutamate
utilized)·(recycling ratio of glutamate through
-ketoglutarate and
glutamine). From Equation 5, it follows that
|
(Eq. 59)
|
(ii) The glutamate noted
Cit
[Glu]Glu is formed from molecules of
-ketoglutarate or glutamine which are derived from citrate
molecules:
|
(Eq. 60)
|
where Cit
[Gln]Glu = Cit
[Glu]Glu·(Glu
Gln) (see Figs. 1
and 2).
Combining the latter two equations, we obtain
|
(Eq. 61)
|
Replacing (Glu
Gln)·(Gln
Glu) by (Glu
Gln), as
indicated in Equation 3, it follows that:
|
(Eq. 62)
|
where Cit
[
KG]Glu = (Cit
formed)·(recycling ratio of
-ketoglutarate through glutamate and
glutamine).
From Equation 6 one can calculate that
|
(Eq. 63)
|
Since {Glu
KG + Gln} = 1/[1
(Glu
KG)
(Glu
Gln)] (see Equation 5), one can deduce from
Equations 62 and 63 that
|
(Eq. 64)
|
The total amount of glutamate involved in the metabolism is
obtained from Equations 59 and 64
|
(Eq. 65)
|
but it cannot be calculated because X·{Glu
KG + Gln} and (
KG
Glu)·{Glu
KG + 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
[
KG]Glu (see Equation 59), and the
KG derived from citrate,
Cit
[
KG]Glu (see Equation 63) (see
also Figs. 2 and 4).
|
(Eq. 66)
|
where Glu
[
KG]Glu = Glu
[Glu]Glu·(Glu
KG) and
{vGlu
KG} = (Glu
KG)·{Glu
KG+Gln}.
From Equation 65 and Fig. 2, unidirectional flux of glutamate to
-ketoglutarate, noted [Glu
KG], can be calculated as
follows:
|
(Eq. 67)
|
From Equation 66 and Fig. 2, unidirectional flux of
-ketoglutarate to glutamate, noted [
KG
Glu], can also be
expressed as:
|
(Eq. 68)
|
where {Cit
KG
Glu} is the proportion of
citrate-derived
-ketoglutarate converted to glutamate which takes
into account the total recycling through the "Glu
KG
Glu" and "Glu
Gln
Glu" cycles (see under
"Notation").
Thus, net flux of glutamate to
-ketoglutarate, noted
net[Glu
KG], can be obtained from the two latter
equations:
|
(Eq. 69)
|
where, as mentioned above, {vGlu
KG} = (Glu
KG)·{Glu
KG + Gln}
Since {Glu
KG + Gln} and {
KG
Glu + Gln} are equal
to 1/[1
(Glu
KG)
(Glu
Gln)] and [1
(Glu
Gln)]/[1
(Glu
KG)
(Glu
Gln)],
respectively (see Equations 5 and 6), and since, from Equations 2 and 3
(Glu
KG) = (Glu
KG)·(
KG
Glu) and (Glu
Gln) = (Glu
Gln)·(Gln
Glu), it follows that
|
(Eq. 70)
|
It can be verified by using the latter equation that, as expected
from Fig. 2, the net flux of glutamate to
-ketoglutarate is equal to
the flux of
-ketoglutarate dehydrogenase, noted [
KGdH], minus
the flux of citrate synthase, noted [CS]:
|
(Eq. 71)
|
with
|
(Eq. 72)
|
(see Figs. 2 and 4)
Fig. 2 and Equation 65 allow us to calculate the accumulation of
glutamate
|
(Eq. 73)
|
with
|
(Eq. 74)
|
Fig. 2 shows that citrate originates from oxaloacetate through two
possible pathways, namely the tricarboxylic acid cycle and the "OAA
PEP
Pyr
AcCoA
Cit
OAA" cycle. Therefore, the
latter equation can be rewritten as
|
(Eq. 75)
|
(see also the definition of (AcCoA +
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:
|
(Eq. 76)
|
Thus, the accumulation of glutamate plus glutamine is given by
|
(Eq. 77)
|
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 + glutamate synthesized, see Equation 76):
|
(Eq. 78)
|
The net utilization of glutamate can be obtained from Equations
70, 73, and 78 and is equal to net[Glu
KG] + [
Gln]Glu
[
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:
|
(Eq. 79)
|
and
|
(Eq. 80)
|
but, these fluxes cannot be calculated since labeled carbon data
don't allow us to obtain the value of {vGlu
Gln}.
Flux through pyruvate dehydrogenase, which is equal to
[AcCoA]Glu, can be derived from Fig. 2:
[AcCoA]Glu = [OAA]Glu·(OAA
PEP)·(PEP
Pyr)·(Pyr
AcCoA).
Flux through phosphoenolpyruvate carboxykinase is given by
[OAA]Glu·(OAA
PEP) (see Fig. 2).
Flux through pyruvate kinase is equal to [OAA]Glu·(OAA
PEP)·(PEP
Pyr) (see Fig. 2).
Flux through lactate dehydrogenase is obtained by multiplying flux
through pyruvate kinase by (Pyr
Lac).
Flux through phosphoglyceromutase is given by [Glc]Glu + [Ser]Glu and corresponds to the flux through
phosphoenolpyruvate carboxykinase multiplied by (PEP
3PG).
Flux through glucose-6-phosphatase is equal to
[OAA]Glu·(OAA
PEP)·(PEP
3PG)·(3PG
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
-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
PEP)·(PEP
Pyr)·(Pyr
Ala).
Net flux through aspartate aminotransferase is equal to aspartate
accumulation: [Asp]Glu = [OAA]Glu·(OAA
Asp).
Net flux through 3-phosphoglycerate dehydrogenase and phosphoserine
aminotransferase are equal to [OAA]Glu·(OAA
PEP)·(PEP
3PG)·(3PG
Ser).
Net flux through glutamate dehydrogenase is equal to the net flux of
glutamate conversion into
-ketoglutarate minus the net flux through
transaminases.
Flux through
-ketoglutarate dehydrogenase, [
KGdH], is given by
the amount of
-ketoglutarate converted into succinyl-CoA and
subsequently into oxaloacetate (see Equations 8, 9, 72, and 75).
|
(Eq. 81)
|
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·
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
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
-ketoglutarate. Note here
that Shulman and co-workers (11, 12, 15) were also able to calculate
the
-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.