The computer model was validated in vitro and used to ex amine the branch-point kinetics in detail and to obtain insights into the kinetic controls of methionine and threonine synthesis
Trang 1A kinetic model of the branch-point between the methionine
Gilles Curien, Ste´phane Ravanel and Renaud Dumas
Laboratoire de Physiologie Cellulaire Ve´ge´tale DRDC/CEA-Grenoble, France
This work proposes a model of the metabolic branch-point
between the methionine and threonine biosynthesis
path-ways in Arabidopsis thaliana which involves kinetic
compe-tition for phosphohomoserine between the allosteric enzyme
threonine synthase and the two-substrate enzyme
cysta-thionine c-synthase Threonine synthase is activated by
S-adenosylmethionine and inhibited by AMP
Cystathio-nine c-synthase condenses phosphohomoserine to cysteine
via a ping-pong mechanism Reactions are irreversible and
inhibited by inorganic phosphate The modelling procedure
included an examination of the kinetic links, the
determin-ation of the operating conditions in chloroplasts and the
establishment of a computer model using the enzyme rate
equations To test the model, the branch-point was
recon-stituted with purified enzymes The computer model showed
a partial agreement with the in vitro results The model was
subsequently improved and was then found consistent with
fluxpartition in vitro and in vivo Under near physiological
conditions, S-adenosylmethionine, but not AMP, modulates the partition of a steady-state fluxof phosphohomoserine The computer model indicates a high sensitivity of cysta-thionine fluxto enzyme and S-adenosylmecysta-thionine concen-trations Cystathionine fluxis sensitive to modulation of threonine fluxwhereas the reverse is not true The cysta-thionine c-synthase kinetic mechanism favours a low sensi-tivity of the fluxes to cysteine Though sensisensi-tivity to inorganic phosphate is low, its concentration conditions the dynamics of the system Threonine synthase and cystathio-nine c-synthase display similar kinetic efficiencies in the metabolic context considered and are first-order for the phosphohomoserine substrate Under these conditions out-flows are coordinated
Keywords: allosteric activation; branch-point; kinetic com-petition; ping-pong; sensitivity coefficient
Metabolic branch-points display a very large diversity in
terms of the number of the enzymes involved, the kinetic
mechanisms of the competing enzymes and the number as
well as the nature of the allosteric controls Whether such
diversity in the organization of the branch-points reflects
differences in the branch-point kinetics is not well known
Indeed, detailed models that take into account the
individ-ual enzyme kinetic properties in their metabolic context are
scarce Fluxpartition at the dividing point of several pathways has been studied both theoretically [1–3] and experimentally [2,4–7] Some studies used the framework of metabolic control analysis for this purpose [6,7] However, the allosteric controls of the branch-point enzymes are not taken into account in these experimental studies Also the occurrence of branch-point two-substrate enzymes and the consequence of their kinetic mechanisms for the partition of fluxin the systems studied previously have not been considered
The present paper proposes a computer model of the branch-point between the methionine and threonine biosynthesis pathways in Arabidopsis thaliana (Fig 1) The computer model was validated in vitro and used to ex amine the branch-point kinetics in detail and to obtain insights into the kinetic controls of methionine and threonine synthesis in plants
The branch-point between the methionine and threo-nine biosynthesis pathways (Fig 1) involves a two-substrate enzyme (cystathionine c-synthase, CGS) and an allosteric enzyme (threonine synthase, TS) These enzymes compete kinetically for their common substrate, phosphohomoserine (Phser), in chloroplasts [9–11] CGS catalyses the formation
of cystathionine, the precursor of methionine, by condensa-tion of Phser and cysteine The reaccondensa-tion follows a ping-pong mechanism [12] In the competing branch, TS catalyses the formation of threonine from Phser In plants, TS is sti-mulated in vitro by S-adenosylmethionine (AdoMet) in an allosteric manner [10,13–16] AdoMet is a direct derivative
Correspondence to G Curien, Laboratoire de Physiologie Cellulaire
Ve´ge´tale DRDC/CEA-Grenoble, 17 rue des Martyrs,
38054 Grenoble Cedex9, France.
Fax : + 33 4 38 78 50 91, Tel.: +33 4 38 78 23 64,
E-mail: gcurien@cea.fr
Abbreviations: AdoMet, S-adenosylmethionine; CGS, cystathionine
c-synthase; Phser, phosphohomoserine; TS, threonine synthase.
Enzymes: cystathionine c-synthase (EC 4.2.99.9; Swiss Prot entry
P55217); cystathionine b-lyase (EC 4.4.1.8; Swiss Prot entry P53780);
homoserine kinase (EC 2.7.1.39; Swiss Prot entry Q8L7R2); threonine
deaminase (EC 4.2.1.16; Swiss Prot entry Q9ZSS6); threonine synthase
(EC 4.2.99.2; Swiss Prot entry Q9S7B5); lactate dehydrogenase
(EC 1.1.1.27, Swiss Prot entry P13491).
Note: The mathematical model described here has been submitted to
the Online Cellular Systems Modelling Database and can be accessed
at http://jjj.biochem.sun.ac.za/database/curien/index.html free of
charge.
(Received 2 September 2003, accepted 23 September 2003)
Trang 2of methionine (Fig 1) and can be considered as the
end-product of the pathway AdoMet binding to TS increases the
enzyme’s catalytic constant and decreases the Michaelis–
Menten constant for the Phser substrate [15] CGS and TS
activities are inhibited by inorganic phosphate (Pi), a
by-product of the reaction [12,17] TS activity is inhibited by
AMP in vitro [16,17] and AMP competes with AdoMet for its
binding site on the enzyme [16]
Although the individual properties of CGS and TS are
known in detail and equation rates are available [12,15], the
equivalent data for when CGS and TS compete for their
common substrate in a metabolic context remain to be
determined For example, the effect on branch-point
partition of TS activity modifiers, AdoMet (allosteric
activation) and AMP (inhibition) and the concentration
ranges exhibiting this effect are unknown We also ignore
how cysteine, the second substrate for CGS, modulates
Phser distribution and to what extent changes in the
concentration of the inhibitor Pi alters the Phser flux
partition Due to the numerous interactions in the system, a
mathematical model of the branch-point could be
instru-mental in finding answers to these questions Such a model
could be built without any assumptions as detailed enzyme
rate equations and kinetic parameters are known
In this paper we first describe the procedure followed to build a mathematical model of the branch-point The model was then validated in vitro For this purpose, the branch-point was reconstituted with purified enzymes and partition
of a constant fluxof Phser was measured as a function of the concentration of AdoMet under conditions as close as possible to those thought to prevail in vivo in the chloroplast
of an illuminated leaf cell The model was subsequently improved and used to calculate the sensitivity of the fluxes
to the different input variables using the framework of metabolic control analysis The computer model was finally used to examine the consequences of TS allosteric activa-tion, Piinhibition and CGS ping-pong mechanism on the branch-point properties The analysis provides insights into the mechanisms of control of methionine and threonine syntheses in plants
The mathematical model described here has been submitted to the Online Cellular Systems Modelling Data-base and can be accessed at http://jjj.biochem.sun.ac.za/ database/curien/index.html free of charge
Materials and methods
Chemicals ATP, Hepes, homoserine, NADH, AdoMet, lactate dehy-drogenase (Rabbit Muscle type IV) were from Sigma Cysteine was from Fluka Phser was prepared according to [14] and AdoMet was purified as reported in [15]
Proteins ArabidopsisCGS, TS, cystathionine b-lyase and threonine deaminase were purified to homogeneity as described previously [12,15,18,19] Mature Arabidopsis homoserine kinase devoid of its transit peptide sequence was cloned, overexpressed in Escherichia coli and purified to homogen-eity for the present work (G Curien and R Dumas, unpublished results) Purified protein concentration was determined by absorbance measurements at 205 nm [20] Protein concentrations are expressed on a monomer basis Modelling procedure
Figure 1 maps all the kinetic links identified from previous studies carried out in vitro on the enzymes of the aspartate-derived amino acid pathway in plant This map indicates that (a) homoserine kinase, which provides Phser, catalyses
an irreversible reaction [21] and is not inhibited by its product Phser in planta [22]; (b) CGS and TS catalyse irreversible reactions [9,13]; (c) CGS activity depends on the concentration of Phser and cysteine and is not subject to allosteric control in the plant [12]; (d) TS activity is stimulated by AdoMet [10,13–15] and inhibited by AMP
in vitro[16,17]; (e) Piinhibits the activity of both CGS and
TS [12,17]; (f) the enzymatic products cystathionine and threonine do not inhibit the activities of CGS [9,12] and TS [13,16] and (g) finally and importantly, Phser is not an allosteric effector of upstream enzymatic activity Indeed, the concentration of Phser was shown to vary to a large extent (20-fold increase) in transgenic plants with reduced levels of CGS [23] Therefore, the concentration of Phser
Fig 1 Phser branch-point in the aspartate-derived amino acid
biosyn-thetic pathway in plants In plants and microorganisms, aspartate
serves as a precursor for the synthesis of lysine, methionine and
thre-onine Threonine is a precursor for isoleucine synthesis and methionine
is a direct precursor of S-adenosylmethionine (AdoMet) In plants, the
branching between the methionine and threonine biosynthesis
path-ways occurs at the level of phosphohomoserine (Phser) and involves
cystathionine c-synthase (CGS) and threonine synthase (TS) CGS is a
two-substrate enzyme that catalyses the condensation of Phser and
cysteine The production of the aspartate-derived amino acid in plants
is thought to be controlled by numerous allosteric controls identified
in vitro and represented in the figure as dotted lines The dashed square
indicates the limits of the Phser branch-point system analysed in the
present paper In microorganisms branching between the methionine
and threonine biosynthesis pathways occurs at the level of homoserine
and involves different enzymes and different allosteric patterns [8].
Trang 3depends exclusively on the flux of Phser and on CGS and TS
activity As a consequence it is possible to model the
branch-point kinetics if one knows the CGS and TS rate equations,
Phser fluxrates and the concentrations of AdoMet, cysteine,
Piand the two enzymes in a metabolic context
To determine the values of the input variables, we
considered the metabolic state of an illuminated plant leaf
cell chloroplast Some data were already available from
previous studies and these were completed with data from
the present work Assuming a homogeneous distribution in
Arabidopsis leaf cells, concentrations of about 20 lM for
AdoMet (averaged from [24] and [25]), and about 15 lMfor
cysteine [26] can be calculated The concentration of Piin
the spinach chloroplast stroma was shown to be about
10 mM [27] We assumed a similar concentration for
Arabidopsis The concentration of CGS in the chloroplast
can be estimated as follows: CGS represents 1/11000 of the
soluble proteins in the spinach chloroplast [28], the soluble
protein content in the chloroplast is about 400 mgÆmL)1
[29], the content of CGS monomer is thus approximately
36 lgÆmL)1, that is 0.7 lM (on a 52-kDa monomer mass
basis) Such data are lacking for TS, however, the ratio
[CGS]/[TS] can be calculated as follows: ELISA assays were
carried out using rabbit antibodies raised against the
recombinant proteins [12,14] and purified proteins as
standards We measured that an extract of soluble proteins
from Arabidopsis contains 1500 ng TS and 210 ng CGS per
mg protein (data not shown), corresponding to a [CGS]/
[TS] ratio of about 1/7 Thus, [TS] is approximately 5 lMin
the chloroplast stroma (7· 0.7 lM) The value of the fluxof
Phser in vivo is unknown for Arabidopsis and thus data from
Lemna [30] were used In this plant, cystathionine and
threonine fluxrates are about 1 and 7.9 nmol per frond per
doubling time, respectively As Phser has no other fate in
plant than the synthesis of cystathionine and threonine [31],
Phser fluxrate is about 8.9 nmol per frond per doubling
time With a doubling time of 41 h [30], a mean frond
cellular volume of 0.509 lL [32] and assuming that Phser is
restricted to the chloroplast (9.5% of cellular volume [33]),
where it is produced and used, a value of 1 lMÆs)1can be
calculated for the fluxof Phser
Modelling of the Phser branch-point at steady-state
The rate equations of CGS and TS published in [12] and [15]
required to model the branch-point kinetics are expressed
here as hyperbolic functions of Phser concentration These
forms are equivalent to those previously published but they
suit our modelling purpose better (see later)
The CGS rate equation is defined by Eqn (1):
mcystathionine ¼k
app catCGS½CGS½Phser
KappmCGSþ ½Phser ð1Þ Where, [CGS] is the CGS monomer concentration, kcatCGSapp
is the apparent catalytic constant for CGS (Eqn 2) and
KmCGSapp is the apparent Michaelis–Menten constant for CGS
with respect to Phser (Eqn 3)
kappcatCGS¼ kcatCGS
1þ K
Cys mCGS
Cys
KappmCGS¼ K
Phser mCGS
1þ K
Cys mCGS
Cys
KPi
iCGS
! ð3Þ
Where, [Pi] is the concentration of Pi Pi competitively inhibits Phser binding to CGS [12,17] and KPi
iCGSin Eqn (3)
is the CGS inhibition constant for Pi
An equivalent mathematical form of the CGS rate equation can also be derived (Eqn 4) and will be used in the Discussion In this equation, the enzyme velocity is expressed as a function of [Cys] instead of [Phser]
mcystathionine¼k
appCys catCGS½CGS½Cys
KappCysmCGS þ ½Cys ð4Þ Expressed in this form, apparent kinetic parameters kappCyscatCGS and KmCGSappCys are defined as functions of [Phser] and [Pi] by Eqn (5) and Eqn (6), respectively:
kappCyscatCGS¼ kcatCGS
1þ KPhsermCGS
Phser
1 þ ½Pi
KPiiCGS
KappCysmCGS ¼ k
Cys mCGS
1þ KPhsermCGS
Phser
1 þ
Pi
KiCGS
TS catalytic rate depends hyperbolically on the concen-tration of Phser at any concenconcen-tration of AdoMet [15] (Eqn 7)
mThr¼½TS k
app catTS ½Phser
Where, [TS] is TS monomer concentration, kcatTSapp is the TS apparent catalytic constant and KmTSapp is the apparent Michaelis–Menten constant for TS with respect to Phser
kappcatTS and KmTSapp are complexfunctions depending on the concentration of AdoMet [15] as defined by Eqn (8) and Eqn (9), respectively
kappcatTS¼ k
noAdoMet catTS þ kAdoMet
catTS ½AdoMetK 2
1 K2
1þ½AdoMetK 2
1 K2
0
@
1
KappmTS¼
2501þ
½AdoMet
0:5
1þ ½AdoMet
1:1
1þ½AdoMet140 2
0 B
1 C A 1 þK½PPii
iTS
! ð9Þ
Where, knoAdoMet
catTS and kAdoMet
catTS are the TS catalytic constant
in the absence and presence of a saturating concentration of AdoMet, respectively K1K2 is the product of the binding constants for the association of the first and the second molecule of AdoMet with the TS dimer
Picompetitively inhibits Phser binding to TS [17] KiTSPi is the TS inhibition constant for Pi KPi
iTSis independent of the concentration of AdoMet (G Curien and R Dumas, unpublished results) Numerical values in the expression of
KmTSapp (expressed in lM) correspond to groups of kinetic constants explaining the effect of AdoMet when present at
Trang 4low concentrations (< 2 lM [15]) Values of the kinetic
parameters for CGS and TS are summarized in Table 1
The mechanism of inhibition of TS by AMP is unclear,
and some kinetic parameters are lacking However, as will
be shown below (Results), the AMP effect on partition is
negligible under physiological conditions and for this reason
the inhibition was not taken into account in the present
model
A simple mathematical procedure was developed to
simulate the steady-state of a two-enzyme branch-point [2]
Three conditions allowed us to use this procedure for the
simulation of the Phser branch-point kinetics First, the
enzymes homoserine kinase, CGS and TS catalyse
irrevers-ible reactions Second, Phser fluxis an external variable
(Phser concentration does not determine Phser flux, see
above) and third, Phser substrate saturation curves for CGS
and TS are hyperbolic (Eqns 1 and 7) The mathematical
treatment of LaPorte et al [2] is reproduced here for the
Phser branch-point
When the branch-point is in steady-state, the fluxof Phser
(JPhser) is equal to the sum of the fluxof cystathionine
(Jcystathionine) and the fluxof threonine (JThr) (Eqn 10)
JPhser¼ Jcystathionineþ JThr ð10Þ
Jcystathionineand JThr in Eqn (10) can be replaced by CGS
and TS Michaelis–Menten equations (Eqns 1 and 7)
yielding the following quadratic equation (Eqn 11)
½Phser2ðJPhser kappcatCGS kappcatTSÞ
þ ðKappmCGSðJPhser kappcatTSÞ
þ KappmTSðJPhser kappcatCGSÞÞ½Phser
þ ðJPhserKappmCGSKappmTSÞ ¼ 0 ð11Þ
Solving Eqn (11) yields an expression for [Phser]steady-state
that can be introduced back into Eqns (1 and 7) yielding
expressions for the output fluxes at steady-state Such
calculations, based on the integration of independent kinetic
data, are authorized because the initial velocity
measure-ments of purified CGS and TS were carried out under
similar physicochemical conditions (30C, pH 7.5–8)
The simulations were carried out with KALEIDAGRAPH
(Abelbeck Software, Reading, PA, USA) A series of
constant or changing values were generated for the different
input variables and the calculations were done using the
appropriate equations
Reconstitution of the branch-point
A constant fluxof Phser was obtained with purified
homoserine kinase in the presence of saturating
concentra-tions of ATP and homoserine Two different coupling
systems were used in order to measure threonine and
cystathionine flux Threonine flux was measured using purified threonine deaminase and commercial lactate dehy-drogenase Threonine deaminase transforms threonine into oxobutyrate that is further reduced by lactate dehydroge-nase in the presence of NADH Cystathionine fluxwas measured with cystathionine b-lyase and lactate dehydro-genase Cystathionine b-lyase transforms cystathionine into homocysteine and pyruvate Pyruvate is reduced by lactate dehydrogenase in the presence of NADH The achievement
of the steady-states can be followed with a spectrophoto-meter (decrease in absorbance at 340 nm) Steady-state fluxes can be determined in the two branches in independent reactions containing either threonine deaminase or cysta-thionine b-lyase mixed with homoserine kinase, CGS, TS and lactate dehydrogenase
Experiments were carried out in a thermoregulated quartz cuvette (30C) and in a total volume of 150 lL Twenty microlitres of protein mix(0.15 lM homoserine kinase, 0.7 lMCG, 5 lMTS, 2 lMlactate dehydrogenase, and 2 lM threonine deaminase or 0.7 lMcystathionine b-lyase) were added to a 120-lL solution containing: 50 mMHepes KOH (pH 8.0), 10 mMKPi(pH 8.0), 2 mM L-homoserine, 200 lM NADH, 250 lM L-cysteine and 0–100 lM AdoMet (final concentrations) The reaction was started by addition of ATP-Mg (10 lL, final concentration 2 mM ATP, 10 mM Mg-Acetate) In the absence of threonine deaminase or cystathionine b-lyase, the rate of NADH oxidation was undetectable Background NADH oxidation was negligible
in the presence of threonine deaminase when homoserine or ATP were omitted However, cystathionine b-lyase was shown to catalyse the degradation of cysteine into pyruvate Though certainly a minor quantitative contribution in vivo where the concentration of cysteine is low (15 lM), this reaction contributed significantly to the production of pyruvate under our conditions, where the concentrations of cysteine and cystathionine b-lyase are high Thus, a correc-tion had to be made to obtain the actual fluxof cystathionine The side reaction of cystathionine b-lyase exhibited first-order kinetic behaviour with respect to cysteine concentra-tion under our condiconcentra-tions (not shown) The rate was calculated with the following relation, v¼ k.[Cystathionine b-lyase] [Cys] with k¼ 2.2 10)4lM )1Æs)1 The concentration
of cysteine at each time point was estimated to be equal to the initial concentration of cysteine minus the concentration of NAD+at time, t A small error is made in this calculation as
a consequence of the time delay in the enzymatic chain Subtraction of the rate of the cystathionine b-lyase side reaction from the total rate of NADH oxidation yielded the actual rate of cystathionine production
Results
Modelling procedure
In order to model the branch-point between the methionine and threonine biosynthesis pathways in Arabidopsis the following procedure was used
Firstly, the kinetic links inside the branch-point and between the branch-point system and the rest of the pathway were identified explicitly We used some of our previous results concerning CGS and TS enzymes as well as other works for this purpose (Fig 1 and Materials and methods)
Table 1 CGS and TS kinetic parameters.
k catCGS 30 s)1 k catTSnoAdoMet 0.42 s)1
K mCGSPhser 2500 l M K 1 K 2 73 l M2
Trang 5Secondly, as the model aimed to describe a physiological
situation, we characterized the in vivo operating conditions
of the system in terms of input flux, enzyme concentrations
and external metabolite concentrations (AdoMet, cysteine,
Pi, AMP) We chose to consider the metabolic state of an
illuminated chloroplast leaf cell as many data were available
for this state in Arabidopsis or other plants that can be
considered equivalent We also determined the in vivo
concentration of CGS and TS in A thaliana Details
concerning the sources of the information and the
calcula-tions can be found in Methods Results are shown in
Table 2
Thirdly, the rate equations of CGS and TS [12,15] were
used to create a computer model of the steady-state in the
branch-point The mathematical procedure published
pre-viously for the study of the isocitrate branch-point in E coli
[2] was adapted to model the Phser branch-point kinetics
(see Materials and methods) Finally, prior to its use for the
examination of branch-point kinetics, the model was
validated in vitro
Validation of the computer model
The model was derived from initial velocity measurements
carried out with low enzyme concentrations and high
substrate concentrations, that is, under conditions exactly
opposite to those found in the physiological situation In
order to estimate the validity of the computer model, the
branch-point was reconstituted with purified enzymes and
allowed to reach a steady-state, under conditions as close
as possible to those thought to occur in vivo Phser was
delivered in fluxby the upstream enzyme
(homo-serine kinase) The fluxes of cystathionine and threonine
(Jcystathionine and JThr) were measured with the enzymes
that occur downstream of CGS and TS, namely
cystathi-onine b-lyase and threcystathi-onine deaminase, respectively,
coupled to lactate dehydrogenase Under these conditions,
CGS and TS were operating in vitro at physiological
concentration, with Phser concentration set by the system
and in the presence of the reaction products, neighboring
enzymes and salts (K+and Mg2+) Phser fluxhad to be
set at one third of its estimated value in the chloroplast of
an illuminated leaf cell to minimize substrate
consump-tion In addition, the concentration of cysteine was set at
250 lM rather than 15 lM (physiological concentration)
Indeed, it was difficult to achieve a constant concentration
of cysteine However, as will be detailed later, CGS
velocity was saturated by cysteine in these conditions and
Jcystathionine was not affected by the consumption of
cysteine The time courses of the reactions in the presence
of 20 l AdoMet are displayed in Fig 2A, showing that
the fluxes reached a steady-state in about 600 s Results in Fig 2A confirmed that CGS was saturated by cysteine throughout the time course of the reactions, otherwise
Table 2 Estimated values of the input variables in a leaf cell chloroplast.
The values of the input variables were derived as indicated in Materials
and methods from measurements carried out on illuminated
photo-synthetic leaf tissue.
Input
variable
J Phser
(l M Æs)1)
Concentration (l M )
Concentration (m M ) [P i ] [CGS] [TS] [AdoMet] [Cys]
Fig 2 Phser branch-point kinetic behaviour in vitro (A) establishment
of the steady-state The fluxof cystathionine (lower curve) was measured with cystathionine b-lyase and lactate dehydrogenase and threonine flux(upper curve) was measured with threonine deaminase and lactate dehydrogenase The fluxof Phser was generated with homoserine kinase in conditions where substrates were saturating Phser flux, 0.3 l M Æs)1; AdoMet, 20 l M ; cysteine, initial concentration,
250 l M ; P i , 10 m M ; CGS, 0.7 l M ; TS, 5 l M The rate of NADH oxi-dation at each time point was calculated from the absorbance time curves (A 340 ) with a Dt of 20 s (B) Steady-state fluxof cystathionine (m) and threonine (d) in the reconstituted branch-point as a function
of the concentration of AdoMet Experimental conditions were as in (A) The total flux(h) is the sum of the fluxes of cystathionine and threonine at steady-state The experimental points were fitted to Hill equations The thick curves are fluxvalues calculated with the com-puter model using CGS and TS mechanistic equations Input variables were set at the value they have in the experiment (C) The experimental results in (B) were compared with the predictions using the improved version of the numerical model (bold curves; details in the text).
Trang 6steady-state fluxes could not have been obtained The
experiment was carried out for different AdoMet
concen-trations and outflow values measured at steady-state were
plotted as a function of AdoMet concentration (Fig 2B)
Jcystathionine and JThr summed to a constant value, thus
confirming that steady-state had been reached (For
[AdoMet] < 5 lM, the time constant of the system was
high and steady-state may not be entirely reached.)
Figure 2B shows that Jcystathionine and JThr are strongly
dependent on the AdoMet concentration, in the range
0–100 lM The fluxes showed a sigmoidal dependence on
the concentration of AdoMet with Jcystathionine decreasing
and JThr increasing as the concentration of AdoMet was
increased Half changes in Jcystathionine and JThr are
obtained for a concentration in AdoMet of about
15 lM, i.e for a value close to the estimated cellular
concentration
In order to determine whether the properties of isolated
CGS and TS, as defined by their mechanistic equations
(Eqns 1–9, Materials and methods), could explain the
observed behaviour in Fig 2B, the computer model
described in the Materials and methods was used to
calculate Jcystathionineand JThras a function of the
concen-tration of AdoMet with the remaining input variables set at
the experimental values used to obtain Fig 2B As shown in
Fig 2B, the experimental fluxes depend on the
concentra-tion of AdoMet in a manner similar to that predicted by the
computer model [The small bumps in the theoretical curves
barely discernable at low AdoMet concentration originate
from the complexdependence of TS Km for Phser on
AdoMet at low concentration (Eqn 9) This effect is either
too subtle to be detected in the present experiments or
irrelevant to the present experimental conditions.] However,
despite good agreement, the model was not entirely
satisfying Indeed, when experimental and predicted curves
are fitted with Hill equations, the Hill number thus obtained
is much higher in the first case (nH¼ 2.7) than in the second
(1.8)
Improvement of the computer model
We anticipated that the discrepancy between the computer
model and the experimental data originated from an
inadequacy of the TS mechanistic equation This equation
correctly describes the interaction between TS and AdoMet
in the presence of high concentrations of Phser [15]
However, the model indicates that when TS operates at
the branch-point, Phser concentration is low ([Phser] <<
KmTSapp) Moreover, the presence of Piprevents the binding of
Phser on the enzyme and contributes to a decrease in the
concentration of the enzyme-substrate complex Under
these conditions, AdoMet binds on the enzyme which is
virtually free of substrate We showed previously [15] that a
synergy exists between Phser and AdoMet for their binding
to TS The Hill number calculated for the free enzyme/
AdoMet binding curve was about three and only about two
for the enzyme–substrate/AdoMet binding curve As a
consequence, a new equation had to be derived for AdoMet
binding to TS under the present conditions where the
enzyme-substrate complexconcentration was low For this
purpose, it was first observed that, when TS operates at the
branch-point, the calculated concentration of Phser ranged
from 1000 lM (no AdoMet) to 5 lM (100 lM AdoMet) (Fig 4C) Under these conditions we observed graphically (not shown) that TS catalytic rate at the branch-point is approximately first-order with respect to Phser concentra-tion at any AdoMet concentraconcentra-tion So the complicated mathematical expression of TS velocity (Eqns 7–9) could be simplified to a linear equation for Phser concentration (Eqn 12)
mThr¼ ½TS kTS
1þ½Pi
KPiiTS
where, kTS is TS apparent specificity constant for Phser (kcatTSapp =KmTSapp) kTS is a function of the concentration of AdoMet that can be determined experimentally In order
to obtain this function, TS velocity (TS alone) was measured as a function of the concentration of AdoMet
in the physicochemical environment of the experiments of Fig 2 Threonine deaminase and lactate dehydrogenase were used as the coupling system and TS activity was measured in the presence of a low concentration of Phser (500 lM) The experimental results (not shown) were fitted
to a Hill equation thus giving the following empirical equation for kTS(Eqn 13)
kTS¼ 5:4 105þ6:2 10
3½AdoMet2:9
322:9þ ½AdoMet2:9 ð13Þ When the branch-point behaviour was simulated with Eqns (12 and 13) instead of the TS mechanistic equations (Eqns 7–9) the computer model was in much better agreement with the experimental results (Fig 2C) These results confirm that TS velocity is first-order with respect to Phser concentration Moreover the agreement indicates that the branch-point behaviour is fully explained by the individual enzyme’s kinetic properties More complex phenomena such as protein–protein interactions, need not
be invoked to explain the behaviour of the system in response to changes in AdoMet concentration
AMP inhibition does not affect partition
As a kinetic mechanism for the inhibition of TS activity by AMP is unclear, and kinetic parameters are lacking, it was
of special interest to use the in vitro system to test the effect
of AMP on the partition of the fluxof Phser under physiological conditions The partition was measured in the conditions of Fig 2B in the presence of 20 lMAdoMet and a physiological concentration of AMP (100 lM [34]) Under these conditions, we observed that the partition was the same whether AMP was present or not (result not shown) indicating that AMP was efficiently displaced in these conditions [Measurements of TS initial catalytic rate showed that the binding of AMP to TS is efficiently displaced by AdoMet and Pi(G Curien and R Dumas, unpublished observations)] Our results suggest that the presence of AMP in vivo does not have any quantitative consequence on the partition of the fluxof Phser, at least under the physiological operating conditions defined in Table 2 As a consequence, the inhibitory effect is not taken into account in the model
Trang 7Consistency with data collected in planta
Measurements in planta [32] indicated that Jcystathionineand
JThr represent 11% and 89% of the fluxof Phser,
respectively The numerical model using the simplified TS
equation (Fig 2C) or the in vitro model give a value of 20–
30% for Jcystathionine (and 70–80% for JThr) at 20 lM
AdoMet Considering that fluxpartition is highly sensitive
to the concentrations of AdoMet and of the competing
enzyme concentrations (see later) and thus to small errors in
the estimation of their physiological values, the consistency
is satisfying The in vitro and numerical models are
consistent with JThr being larger than Jcystathionine in the
metabolic condition of a leaf cell Also, a Phser
concentra-tion of about 80 lMin A thaliana leaf chloroplast can be
derived from the measurements in planta, in good agreement
with the model which predicts a value of about 128 lM The
Phser content in A thaliana leaves is about 6.6 nmolÆg)1
fresh weight [23] The concentration was calculated
assu-ming that Phser is restricted to the chloroplast (60 lLÆmg)1 chlorophyll [33] and 1.3 mg chlorophyll per gram fresh weight [34]) Together, these data indicate that the model of the Phser branch-point is relevant to at least one metabolic situation and therefore provides a realistic, detailed descrip-tion of the branch-point between the methionine and threonine biosynthesis pathways In the following the model
is used to investigate the sensitivity of the two-enzyme system to the different input variables and to explain the behaviour of the branch-point in terms of the kinetic properties of CGS and TS
Sensitivity analysis
In a first analysis, fluxes of cystathionine and threonine (Fig 3) were calculated as a function of each input variable The fixed input variables were set at their physiological values (Table 2) Although the curves in Fig 3 are displayed for a large range of the changing
Fig 3 Calculated fluxes in the vicinity of the physiological operating point The steady-state fluxes were calculated with the improved version of the computer model in which the TS equation was the simplified empirical equation (Eqn 12) All input variables but one (indicated beneath the graphs abscissa) were set at their values in an illuminated leaf cell chloroplast (Table 2) The dotted lines in the graphs indicate the value of the changing input variable in the physiological context considered The flux response coefficients were calculated from these curves and are indicated in Table 3.
Trang 8input variables, the analysis has to be limited to the
vicinity of the physiological operating point, especially
when a high sensitivity to the changing variable is
predicted Indeed, the values of the input variables
in vivofor a metabolic context that is very different from
the one indicated in Table 2 are unknown In order to
describe the sensitivity of the system at the physiological
operating point in quantitative terms, the results in Fig 3
were used to calculate the fluxresponse coefficients as
defined in the framework of metabolic control analysis
[35–40] The results are displayed in Table 3 The changes
in fluxand their sensitivities are explained by variations in
the concentration of Phser For this reason, the
concen-tration of Phser calculated for each of the situations
analysed are indicated in Fig 4
From the results in Table 3 one can verify that the
summation relationship [35] between control coefficients is
satisfied in the three enzyme system, thus, showing an
internal consistency of the model Indeed
RJcystathionineþ RJcystathionineþ RJcystathionine¼ 1
(RJcystathionineJPhser is the homoserine kinase control coefficient over cystathionine flux) The same relation is obtained for JThr
Fig 4 Calculated Phser concentration for changing input variables Phser concentrations corresponding to the steady-state conditions calculated in Fig 3 are plotted as a function of the changing input variable with the other input variables set at their physiological values The dotted lines in the graphs indicate the value of the changing input variable in the physiological context considered.
Table 3 Flux response coefficients The values of the fluxresponse coefficients (R Ji
I ¼ (DJ/J)/(DI/I)) where J stands for fluxand I for input variable) were calculated using the curves in Fig 3 for the estimated physiological environment of the Phser branch-point in Arabidopsis leaf chloroplast R J
I ¼ a means that a 1% change in I around a given value promotes an a percent change in flux J A negative value means that input variable and fluxvary in opposite directions.
Input variable (I)
Input variable physiological value (illuminated leaf cell) R I
Jcystathionine
R JThr I
J Phser 1 l M Æs)1 0.81 1.03
Trang 9Next, we analysed the sensitivity of the flux of
cystathi-onine and threcystathi-onine to Pi, cysteine, AdoMet, CGS and TS
concentrations as well as to Phser input fluxin the three
enzyme system
Sensitivity to Pi The sensitivity of the system to Piwas
considered because the concentration of Piin the chloroplast
is high and variable (from 5 to 30 mM depending on the
physiological state of the cell [27]) The calculations indicate
that the fluxresponse coefficients for Pi are very low
(Table 3) Figure 3A also shows that Jcystathionineand JThr
are virtually unmodified despite important changes in the
concentration of Pi Indeed, KiPivalues for CGS and TS are
similar and lower than the physiological concentration of Pi
Note that the linear dependence of Phser concentration on
the concentration of Pi(Fig 4A) is due to the competitive
nature of the inhibition
Sensitivity to cysteine An advantage of the computer
model is the possibility to vary the concentration of cysteine
around the estimated physiological concentration (15 lM)
This was not possible in the experiments used for Fig 2B
(see above) Table 3 indicates that the fluxresponse
coefficients for the cystathionine and threonine fluxes at
15 lMcysteine are low (0.18 and)0.03, respectively) Also,
Fig 3B shows that when the concentration of cysteine is
increased above 15 lM, the fluxes are modified only slightly
This result indicates that the partition experimentally
determined in Fig 2B at 20 lMAdoMet would not have
been different if cysteine concentration had been set at
15 lM instead of 250 lM Figure 3B also explains why
cysteine consumption left Jcystathionine unaffected in the
experiments described in Fig 2B This response of the
system to cysteine will be related to the kinetic mechanism of
CGS later
Sensitivity to AdoMet Figure 3C indicates that the
con-centration of AdoMet determines Phser fluxpartition in a
much more sensitive manner than do cysteine and Pi At
20 lMAdoMet, JThris larger than Jcystathioninein accordance
with the in vivo situation (see above) Therefore, although
AdoMet-mediated changes in JThr promote quantitatively
equivalent opposite changes in Jcystathionine, relative changes
(fluxresponse coefficient), are larger for Jcystathioninethan for
JThr(Table 3) In the model, Jcystathionineis about sixtimes
more sensitive to AdoMet than JThrfor AdoMet at 20 lM
These calculations highlight an asymmetry in the
branch-point JThrand Jcystathionineare not equivalent with respect to
changes in the concentration of AdoMet
Sensitivity to the concentration of CGS and TS In the
model, an increase or a decrease in the concentration of one
of the branch-point enzymes promotes an increase or a
decrease in the fluxin the corresponding branch and a
quantitatively equivalent but opposite change in the fluxin
the other branch (Fig 3D,E) However, as observed for
AdoMet, and as a consequence of the fluximbalance, an
asymmetry in the response is observed As indicated in
Table 3, JThr presents a low sensitivity to changes in the
concentration of the enzymes (for TS M and
CGS M) By contrast, Jcystathionineis about sixtimes
more sensitive in the same conditions
Sensitivity to JPhser Individual output fluxes are expected
to present a different sensitivity on JPhserdepending on the absolute and relative degree of saturation of CGS and TS by the common substrate Phser Figure 3F indicates that the fluxof threonine depends in a quasi-linear manner on JPhser whereas the fluxof cystathionine displays a slight downward curvature for the same range of JPhservalues When a larger range for JPhser is considered (not shown) the curve for threonine fluxdisplays an upward curvature Accordingly, the sensitivity of JThr is slightly higher than unity (1.03, Table 3), and the sensitivity of Jcystathionine is lower (RJJcystathioninePhser ¼ 0.8) for the physiological state considered Figure 4F indicates that the Phser steady-state concentra-tion depends in a quasi-linear manner on JPhser Using a larger scale for the abscissa (not shown) would reveal an upward curvature Indeed, [Phser]steady-stateincreases hyper-bolically and reaches infinity as JPhsergets closer to the sum
of CGS and TS maximal catalytic rates In the next part this response of the system to JPhserwill be related to the enzyme individual properties, but the important point here is the following: Fig 3F indicates that, as JPhseris increased and the concentration of Phser increases (Fig 4F), the outflows are modified in the same sense and to a similar extent The model thus predicts that changes in Phser fluxin the range 0–2 lMÆs)1taking place with no changes in the other input variables, would not modify partition In other words, changes of the output fluxes are coordinated in these conditions Note that as the simulations indicate that partition is not a sensitive function of the fluxof Phser, small errors in the estimation of its in vivo value would not change the conclusions Also partition measured in Fig 2 with Phser fluxset at 0.3 lMÆs)1would not be different at
1 lMÆs)1
Comparison of CGS and TS kinetic efficiencies under physiological operating conditions
In order to detail the characteristics of the branch-point in terms of the individual enzyme properties, the kinetic efficiencies of CGS and TS (v/[E]) were calculated for the physiological context considered (Table 2) Results in Fig 5 show that, under these conditions, using either the mech-anistic or the simplified rate equations for TS (details in Fig legends), the saturation curves of CGS and TS by Phser are very similar in the concentration range investigated The concentration of Phser in the stroma is about 80 lM(see above) Under these conditions, the model suggests that CGS and TS have similar kinetic efficiencies in the in vivo context Moreover, both enzymes (and not only TS as indicated previously) operate in the first-order range for Phser concentrations under physiological conditions These two features explain the response of the system to the modifications of the fluxof Phser as indicated in Fig 3F
Consequences of CGS ping-pong kinetic mechanism
on the branch-point kinetic properties
As CGS follows a ping-pong mechanism, its specificity constant for Phser, in marked difference with a sequential mechanism, does not depend on the second substrate (cysteine) concentration (Eqns 2 and 3) Therefore, as the concentration of cysteine is increased, CGS velocity curve
Trang 10for low concentrations of Phser is not modified and thus
remains similar to the TS velocity curve as indicated in
Fig 5
Another property of the ping-pong mechanism is the
hyperbolic dependence of the apparent Kmfor one substrate
on the concentration of the other substrate (Eqns 3 and 6)
This explains why the flux of cystathionine is saturated for
low concentrations of cysteine (Fig 3B) Indeed, as the
concentration of Phser is low in the physiological conditions
considered, the Km for cysteine is low For example, at
80 lMPhser the apparent Kmfor cysteine is 2.5 lM Thus, at
15 lMcysteine (6· Km), CGS velocity is virtually maximal
(Fig 6) Though a similar relation exists for the apparent
Km for Phser and cysteine concentration (Eqn 3), the
situation is not symmetrical from a quantitative point of
view for two reasons: firstly, the maximal Kmfor cysteine
is lower than for Phser (KmCGSCys ¼ 460 lM and
KmCGSPhser ¼ 2500 lM, Table 1); Secondly, this difference is amplified in the presence of Piwhich increases the apparent
Kmfor Phser and decreases the apparent Kmfor cysteine (Eqns 3 and 6) Therefore, in the physiological context considered, CGS operates in the first-order range with respect to Phser (Fig 5), but is virtually saturated by cysteine in the same range of concentration (Fig 6) Time-constant of the branch-point system Physiological changes in the concentration of Pi do not modify the partition (Fig 3A) However, the presence of Pi considerably affects the dynamics of the system Indeed, in the presence of 10 mM Pi, the model indicates that the catalytic rates of CGS and TS are divided by a factor of 6 and 11, respectively, compared to a situation without Pi One can therefore calculate that the time constant [41] of the branch-point system (s) is about 20 times higher in the presence of 10 mMPi(102 s) than in its absence (4.8 s) [In the physiological operating condition considered, CGS and
TS are first-order with respect to their common substrate (Fig 5) Thus, the time constant of the branch-point (s) is defined by the following equation:
kCGS ½CGS þ kTS ½TS
where, kCGS and kTS are CGS and TS specificity constants Considering that following a perturbation the steady-state is reached after approximately 5· s [41], methionine and threonine metabolisms are rather slow, with the kinetic controls potentially operating in a time scale of at least 10 min]
Discussion
Prior to the present study, the only model available for the branch-point between the methionine and threonine bio-synthesis pathways in the plant was the qualitative model shown in Fig 1 The allosteric interaction of TS with AdoMet was observed in vitro with the enzyme isolated from the other enzymes of the system [10,13–16], suggesting that the allosteric interaction had a function in the control of Phser partition in vivo However, no experimental results, whether in vivo or in complete systems in vitro, supported this assumption [31] As TS activity is inhibited by AMP
in vitrosome authors denied a physiological importance for the allosteric activation of TS by AdoMet [16] In addition
to this controversy, the quantitative influences of the inhibitor phosphate and cysteine (CGS second substrate)
on the branch-point kinetics have never been considered
In order to solve these questions we established a computer model of the branch-point and validated it
in vitro A satisfying but imperfect agreement of the predictions with the experimental results lead us to improve the model with a simplification of the TS mechanistic equation The improved version of the numerical model was
in a very good agreement with the in vitro results and consistent with threonine and cystathionine syntheses
in vivo Our results show that although AMP is an inhibitor
of TS in vitro [16,17], this general metabolite has no effect on the partition of the fluxof Phser in the branch-point when present at a physiological concentration This result thus
Fig 6 CGS velocities calculated as a function of cysteine concentration.
P i concentration is 10 m M and Phser concentration is as indicated The
dotted vertical line indicates the physiological operating condition (leaf
chloroplast).
Fig 5 Comparison of the kinetic efficiencies of CGS and TS CGS and
TS velocities as a function of Phser concentration v/[CGS] (thin line)
was calculated using Eqns (1–3) v/[TS] was calculated using either the
mechanistic equation (Eqns 7–9), thick line, or TS empirical simplified
equation (Eqn 12), thick dotted line For the calculations
[cys-teine] ¼ 15 l M , [AdoMet] ¼ 20 l M and [P i ] ¼ 10 m M Under these
conditions, KmCGSapp ¼ 474 l M , kappcatCGS¼ 0.95 s)1, KmTSapp ¼ 1526 l M
and kappcatTS¼ 3.02 s)1 The dotted vertical line indicates the
physiolo-gical operating condition (leaf chloroplast).