G. Curien, Laboratoire de Physiologie Cellulaire Végétale DRDC/CEA-Grenoble, 17 rue des Martyrs, 38054 Grenoble Cedex 9, France. Fax: + 33 4 38 78 50 91, Tel.: +33 4 38 78 23 64, E-mail: firstname.lastname@example.org
This work proposes a model of the metabolic branch-point between the methionine and threonine biosynthesis pathways in Arabidopsis thaliana which involves kinetic competition for phosphohomoserine between the allosteric enzyme threonine synthase and the two-substrate enzyme cystathionine γ-synthase. Threonine synthase is activated by S-adenosylmethionine and inhibited by AMP. Cystathionine γ-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 determination 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 reconstituted 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 flux partition in vitro and in vivo. Under near physiological conditions, S-adenosylmethionine, but not AMP, modulates the partition of a steady-state flux of phosphohomoserine. The computer model indicates a high sensitivity of cystathionine flux to enzyme and S-adenosylmethionine concentrations. Cystathionine flux is sensitive to modulation of threonine flux whereas the reverse is not true. The cystathionine γ-synthase kinetic mechanism favours a low sensitivity of the fluxes to cysteine. Though sensitivity to inorganic phosphate is low, its concentration conditions the dynamics of the system. Threonine synthase and cystathionine γ-synthase display similar kinetic efficiencies in the metabolic context considered and are first-order for the phosphohomoserine substrate. Under these conditions outflows are coordinated.
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 individual enzyme kinetic properties in their metabolic context are scarce. Flux partition 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 flux in 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 examine 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 threonine biosynthesis pathways (Fig. 1) involves a two-substrate enzyme (cystathionine γ-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 condensation of Phser and cysteine. The reaction follows a ping-pong mechanism . In the competing branch, TS catalyses the formation of threonine from Phser. In plants, TS is stimulated in vitro by S-adenosylmethionine (AdoMet) in an allosteric manner [10,13–16]. AdoMet is a direct derivative of 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 . 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 .
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 instrumental 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 flux of 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 activation, Pi inhibition 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.
ATP, Hepes, homoserine, NADH, AdoMet, lactate dehydrogenase (Rabbit Muscle type IV) were from Sigma. Cysteine was from Fluka. Phser was prepared according to  and AdoMet was purified as reported in .
Arabidopsis CGS, TS, cystathionine β-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 homogeneity for the present work (G. Curien and R. Dumas, unpublished results). Purified protein concentration was determined by absorbance measurements at 205 nm . Protein concentrations are expressed on a monomer basis.
Figure 1 maps all the kinetic links idengified 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  and is not inhibited by its product Phser in planta; (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 ; (d) TS activity is stimulated by AdoMet [10,13–15] and inhibited by AMP in vitro[16,17]; (e) Pi inhibits 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 . Therefore, the concentration of Phser depends 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 flux rates and the concentrations of AdoMet, cysteine, Pi and 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 µm for AdoMet (averaged from  and ), and about 15 µm for cysteine  can be calculated. The concentration of Pi in the spinach chloroplast stroma was shown to be about 10 mm. 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 , the soluble protein content in the chloroplast is about 400 mg·mL−1, the content of CGS monomer is thus approximately 36 µg·mL−1, that is 0.7 µm (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 µm in the chloroplast stroma (7 × 0.7 µm). The value of the flux of Phser in vivo is unknown for Arabidopsis and thus data from Lemna were used. In this plant, cystathionine and threonine flux rates 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 , Phser flux rate is about 8.9 nmol per frond per doubling time. With a doubling time of 41 h , a mean frond cellular volume of 0.509 µL  and assuming that Phser is restricted to the chloroplast (9.5% of cellular volume ), where it is produced and used, a value of 1 µm·s−1 can be calculated for the flux of Phser.
Modelling of the Phser branch-point at steady-state
The rate equations of CGS and TS published in  and  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).
Where, [CGS] is the CGS monomer concentration, is the apparent catalytic constant for CGS (Eqn 2) and is the apparent Michaelis–Menten constant for CGS with respect to Phser (Eqn 3).
Where, [Pi] is the concentration of Pi. Pi competitively inhibits Phser binding to CGS [12,17] and in 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].
Expressed in this form, apparent kinetic parameters and are defined as functions of [Phser] and [Pi] by Eqn (5) and Eqn (6), respectively:
TS catalytic rate depends hyperbolically on the concentration of Phser at any concentration of AdoMet  (Eqn 7).
Where, [TS] is TS monomer concentration, is the TS apparent catalytic constant and is the apparent Michaelis–Menten constant for TS with respect to Phser. and are complex functions depending on the concentration of AdoMet  as defined by Eqn (8) and Eqn (9), respectively.
Where, and 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.
Pi competitively inhibits Phser binding to TS . is the TS inhibition constant for Pi. is independent of the concentration of AdoMet (G. Curien and R. Dumas, unpublished results). Numerical values in the expression of (expressed in µm) correspond to groups of kinetic constants explaining the effect of AdoMet when present at low concentrations (< 2 µm). Values of the kinetic parameters for CGS and TS are summarized in Table 1.
Table 1. CGS and TS kinetic parameters.
CGS kinetic parameters
TS kinetic parameters
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 . 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 irreversible reactions. Second, Phser flux is 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.  is reproduced here for the Phser branch-point.
When the branch-point is in steady-state, the flux of Phser (JPhser) is equal to the sum of the flux of cystathionine (Jcystathionine) and the flux of threonine (JThr) (Eqn 10).
Jcystathionine and 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).
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 measurements of purified CGS and TS were carried out under similar physicochemical conditions (30 °C, 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 flux of Phser was obtained with purified homoserine kinase in the presence of saturating concentrations 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 dehydrogenase. Threonine deaminase transforms threonine into oxobutyrate that is further reduced by lactate dehydrogenase in the presence of NADH. Cystathionine flux was measured with cystathionine β-lyase and lactate dehydrogenase. Cystathionine β-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 spectrophotometer (decrease in absorbance at 340 nm). Steady-state fluxes can be determined in the two branches in independent reactions containing either threonine deaminase or cystathionine β-lyase mixed with homoserine kinase, CGS, TS and lactate dehydrogenase.
Experiments were carried out in a thermoregulated quartz cuvette (30 °C) and in a total volume of 150 µL. Twenty microlitres of protein mix (0.15 µm homoserine kinase, 0.7 µm CG, 5 µm TS, 2 µm lactate dehydrogenase, and 2 µm threonine deaminase or 0.7 µm cystathionine β-lyase) were added to a 120-µL solution containing: 50 mm Hepes KOH (pH 8.0), 10 mm KPi (pH 8.0), 2 mm l-homoserine, 200 µm NADH, 250 µm l-cysteine and 0–100 µm AdoMet (final concentrations). The reaction was started by addition of ATP-Mg (10 µL, final concentration 2 mm ATP, 10 mm Mg-Acetate). In the absence of threonine deaminase or cystathionine β-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 β-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 µm), this reaction contributed significantly to the production of pyruvate under our conditions, where the concentrations of cysteine and cystathionine β-lyase are high. Thus, a correction had to be made to obtain the actual flux of cystathionine. The side reaction of cystathionine β-lyase exhibited first-order kinetic behaviour with respect to cysteine concentration under our conditions (not shown). The rate was calculated with the following relation, v = k.[Cystathionine β-lyase][Cys] with k = 2.2 10−4 µm−1·s−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 β-lyase side reaction from the total rate of NADH oxidation yielded the actual rate of cystathionine production.
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 idengified 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).
Secondly, 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 calculations can be found in Methods. Results are shown in Table 2.
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 photosynthetic leaf tissue.
Concentration (mm) [Pi]
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 previously for the study of the isocitrate branch-point in E. coli 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 flux by the ‘upstream’ enzyme (homoserine kinase). The fluxes of cystathionine and threonine (Jcystathionine and JThr) were measured with the enzymes that occur downstream of CGS and TS, namely cystathionine β-lyase and threonine 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 flux had to be set at one third of its estimated value in the chloroplast of an illuminated leaf cell to minimize substrate consumption. In addition, the concentration of cysteine was set at 250 µm rather than 15 µm (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 µm 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 steady-state fluxes could not have been obtained. The experiment was carried out for different AdoMet concentrations 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 µm, 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 µm. 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 µm, 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 Jcystathionine and JThr as a function of the concentration 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 concentration 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 complex dependence 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 (nH = 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 . However, the model indicates that when TS operates at the branch-point, Phser concentration is low ([Phser] << ). Moreover, the presence of Pi prevents 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  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 complex concentration 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 µm (no AdoMet) to 5 µm (100 µm 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 concentration at any AdoMet concentration. So the complicated mathematical expression of TS velocity (Eqns 7–9) could be simplified to a linear equation for Phser concentration (Eqn 12).
where, kTS is TS apparent specificity constant for Phser (). 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 µm). The experimental results (not shown) were fitted to a Hill equation thus giving the following empirical equation for kTS (Eqn 13).
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 flux of Phser under physiological conditions. The partition was measured in the conditions of Fig. 2B in the presence of 20 µm AdoMet and a physiological concentration of AMP (100 µm). 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 flux of 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.
Consistency with data collected in planta
Measurements in planta indicated that Jcystathionine and JThr represent 11% and 89% of the flux of 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 µm AdoMet. Considering that flux partition 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 concentration of about 80 µm in 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 µm. The Phser content in A. thaliana leaves is about 6.6 nmol·g−1 fresh weight . The concentration was calculated assuming that Phser is restricted to the chloroplast (60 µL·mg−1 chlorophyll  and 1.3 mg chlorophyll per gram fresh weight ). 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 description 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.
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 input 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 vivo for 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 flux response coefficients as defined in the framework of metabolic control analysis [35–40]. The results are displayed in Table 3. The changes in flux and their sensitivities are explained by variations in the concentration of Phser. For this reason, the concentration of Phser calculated for each of the situations analysed are indicated in Fig. 4.
Table 3. Flux response coefficients. The values of the flux response coefficients ( = (ΔJ/J)/(ΔI/I)) where J stands for flux and 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. = α means that a 1% change in I around a given value promotes an α percent change in flux J. A negative value means that input variable and flux vary in opposite directions.
Input variable (I)
Input variable physiological value (illuminated leaf cell)
From the results in Table 3 one can verify that the summation relationship  between control coefficients is satisfied in the three enzyme system, thus, showing an internal consistency of the model. Indeed
( is the homoserine kinase control coefficient over cystathionine flux). The same relation is obtained for JThr.
Next, we analysed the sensitivity of the flux of cystathionine and threonine to Pi, cysteine, AdoMet, CGS and TS concentrations as well as to Phser input flux in the three enzyme system.
Sensitivity to Pi. The sensitivity of the system to Pi was considered because the concentration of Pi in the chloroplast is high and variable (from 5 to 30 mm depending on the physiological state of the cell ). The calculations indicate that the flux response coefficients for Pi are very low (Table 3). Figure 3A also shows that Jcystathionine and JThr are virtually unmodified despite important changes in the concentration of Pi. Indeed, KiPi values 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 µm). This was not possible in the experiments used for Fig. 2B (see above). Table 3 indicates that the flux response coefficients for the cystathionine and threonine fluxes at 15 µm cysteine are low (0.18 and −0.03, respectively). Also, Fig. 3B shows that when the concentration of cysteine is increased above 15 µm, the fluxes are modified only slightly. This result indicates that the partition experimentally determined in Fig. 2B at 20 µm AdoMet would not have been different if cysteine concentration had been set at 15 µm instead of 250 µm. 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 concentration of AdoMet determines Phser flux partition in a much more sensitive manner than do cysteine and Pi. At 20 µm AdoMet, JThr is larger than Jcystathionine in accordance with the in vivo situation (see above). Therefore, although AdoMet-mediated changes in JThr promote quantitatively equivalent opposite changes in Jcystathionine, relative changes (flux response coefficient), are larger for Jcystathionine than for JThr (Table 3). In the model, Jcystathionine is about six times more sensitive to AdoMet than JThr for AdoMet at 20 µm. These calculations highlight an asymmetry in the branch-point. JThr and Jcystathionine are 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 flux in the corresponding branch and a quantitatively equivalent but opposite change in the flux in the other branch (Fig. 3D,E). However, as observed for AdoMet, and as a consequence of the flux imbalance, 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 ≈ 5 µm and CGS ≈ 0.7 µm). By contrast, Jcystathionine is about six times more sensitive in the same conditions.
Sensitivity to JPhser. Individual output fluxes are expected to present a different sensitivity on JPhser depending on the absolute and relative degree of saturation of CGS and TS by the common substrate Phser. Figure 3F indicates that the flux of threonine depends in a quasi-linear manner on JPhser whereas the flux of cystathionine displays a slight downward curvature for the same range of JPhser values. When a larger range for JPhser is considered (not shown) the curve for threonine flux displays an upward curvature. Accordingly, the sensitivity of JThr is slightly higher than unity (1.03, Table 3), and the sensitivity of Jcystathionine is lower ( = 0.8) for the physiological state considered. Figure 4F indicates that the Phser steady-state concentration 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-state increases hyperbolically and reaches infinity as JPhser gets closer to the sum of CGS and TS maximal catalytic rates. In the next part this response of the system to JPhser will be related to the enzyme individual properties, but the important point here is the following: Fig. 3F indicates that, as JPhser is 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 flux in the range 0–2 µm·s−1 taking 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 flux of Phser, small errors in the estimation of its in vivo value would not change the conclusions. Also partition measured in Fig. 2 with Phser flux set at 0.3 µm·s−1 would not be different at 1 µm·s−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 mechanistic 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 µm (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 flux of 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 for 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 Km for 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 µm Phser the apparent Km for cysteine is 2.5 µm. Thus, at 15 µm cysteine (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 Km for cysteine is lower than for Phser (M and = 2500 µm, Table 1); Secondly, this difference is amplified in the presence of Pi which increases the apparent Km for Phser and decreases the apparent Km for 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  of the branch-point system (τ) is about 20 times higher in the presence of 10 mm Pi (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 (τ) is defined by the following equation:
where, kCGS and kTS are CGS and TS specificity constants. Considering that following a perturbation the steady-state is reached after approximately 5× τ, methionine and threonine metabolisms are rather slow, with the kinetic controls potentially operating in a time scale of at least 10 min].
Prior to the present study, the only model available for the branch-point between the methionine and threonine biosynthesis 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 . As TS activity is inhibited by AMP in vitro some authors denied a physiological importance for the allosteric activation of TS by AdoMet . 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 flux of Phser in the branch-point when present at a physiological concentration. This result thus strongly suggests that TS allosteric activation by AdoMet is physiologically significant. Our results validate the qualitative model in Fig. 1 and strongly suggest that there is indeed a single allosteric control at the Phser branch-point in plants. The concentration of AdoMet determines the partition of flux between the cystathionine and threonine synthesis pathways. However, the model shows that, as a consequence of an imbalance in the partition of Phser flux (threonine flux is much more important than cystathionine flux), the cystathionine flux (but not threonine flux) is highly sensitive to changes in AdoMet concentration. The interaction of AdoMet with TS is therefore consistent with AdoMet being part of a negative feedback mechanism for methionine synthesis, as an increase in AdoMet concentration decreases cystathionine flux in a highly sensitive manner. Nevertheless, a definitive answer concerning the function of the allosteric activation of TS can still not be given. Indeed, in the three-enzyme system of the present study (in vitro and in the computer model) homoserine kinase is not inhibited by its product  and therefore necessarily controls the overall flux (Jcystathionine + JThr). This is however, not true in the complete system of the aspartate pathway where AdoMet and threonine potentially control the flux of Phser (Fig. 1). These molecules act, respectively, on lysine/AdoMet-sensitive aspartate kinase  and on threonine-sensitive aspartate kinase-homoserine dehydrogenase . Sensitivities of the fluxes of cystathionine and threonine to AdoMet may thus be modified when the branch-point is embedded in the aspartate system. Two scenarios were previously proposed : in the first, the activating interaction of AdoMet with TS may attenuate the changes in the flux of threonine due to a modification of the level of AdoMet. Indeed, upon an increase in the level of AdoMet, TS is activated but Phser flux may simultaneously decrease via the inhibition of AdoMet/lysine-sensitive aspartate kinase by AdoMet. In the second more frequently proposed scenario, AdoMet mediates an indirect negative feedback for the synthesis of methionine, via the activation of TS by AdoMet, followed by the inhibition of bifunctional aspartate kinase-homoserine dehydrogenase by threonine. In this case, it is implicitly assumed that the interaction of AdoMet with AdoMet/lysine-sensitive aspartate kinase does not control Phser flux. If one assumes that regulatory mechanisms need to be sensitive to be efficient, the low sensitivity of the threonine flux to AdoMet indicated by our model together with the existence of a large pool of threonine in plant cell (about 1 mm) may argue against the indirect negative feedback mechanism scenario. As in vivo experiments  failed to distinguish between the two scenarios, a computer model taking into account the properties of the enzymes upstream the Phser branch-point is required to solve this question definitively.
An issue of interest with respect to flux partition at a branch-point concerns the relative degree of dependence of the diverging pathways. The model shows that cystathionine flux is sensitive to threonine flux but that the reverse is not true. Therefore, the model suggests that threonine flux is relatively independent of what happens on the cystathionine side. The Phser branch-point combines the divergence of two fluxes (fluxes of cystathionine and threonine) with the convergence of two fluxes (fluxes of Phser and cysteine). Interestingly, the properties of CGS are such that the flux of cystathionine and, as a consequence the flux of threonine, present a low sensitivity to the CGS second substrate cysteine. According to the model, a large increase in the concentration of cysteine, to sustain a larger demand for glutathione for example, may occur without major effects on the fluxes of cystathionine and threonine. CGS properties thus confer independence between the cysteine and the cystathionine/threonine fluxes. The nature of the kinetic mechanism of CGS (a ping-pong mechanism) is particularly favourable to such an effect. For a sequential mechanism (ternary complex mechanism) the apparent Km for one substrate does not stringently depend on the concentration of the second substrate. The same performance with a two-substrate enzyme following a sequential mechanism would therefore require either a very high concentration of cysteine or a very low Km for cysteine. The constraints imposed by two-substrate enzyme kinetic mechanisms may thus be important to consider when one plans to modify or create a branch-point in a living organism for industrial purposes.
In addition to the control of partition due to kinetic interactions, the model shows that partition is determined by the relative abundance of CGS and TS enzymes. At 20 µm AdoMet, the imbalance of the fluxes results only from the difference in protein concentrations, as CGS and TS catalytic efficiencies are similar (Fig. 5). Partition is thus determined in these conditions by the regulatory processes which control the enzymes' abundance. Interestingly, whereas no mechanisms that would change the concentration of TS could be idengified in planta, AdoMet was shown to control CGS mRNA abundance in plants [31,44–47]. This mechanism involves the N-terminal part of CGS. The time constant of this control is unknown. If this time constant is much larger than the time constant of the kinetic controls (about 100 s) then the separation of the kinetic and genetic controls (an implicit assumption in our model) would be jusgified.
The characteristics of the Phser branch-point described in Fig. 5 clearly distinguishes this branch-point from the isocitrate branch-point in E. coli[2,4,5]. In the latter, isocitrate dehydrogenase is saturated by the common substrate isocitrate whereas the competing enzyme (isocitrate lyase) exhibits first-order kinetics for this substrate concentration. This organization allows the isocitrate branch-point to operate as a switch. Upon growth on acetate, the flux of isocitrate increases and isocitrate dehydrogenase is inhibited by phosphorylation. The flux through isocitrate lyase thus increases 300-fold, switching on the glyoxylate shunt. The Phser branch-point with its two enzymes operating in the first-order range with respect to the common substrate concentration cannot display such a switch property, but instead allow flux coordination. Outflows may increase to a similar extent as Phser flux increases. Such a change in the input flux with the other variables left unchanged may correspond to an increase in carbon supply in vivo (upon increase in light intensity for example).
The extent to which the Phser branch-point can serve as a model for the other two-partner branch-points is hard to establish. It would be necessary to determine the physiological operating conditions and obtain kinetic data of physiological significance before a valid comparison is possible. However, one can hypothesize that the enzyme kinetic properties in the other two-partner branch-points of the aspartate-derived amino-acids pathway and aromatic amino-acids pathway in plant and in microorganisms are such that flux coordination could also be obtained. The distribution of the carbon skeleton toward the various end-products would not be affected when supply increases or decreases in these conditions. If this is true then, as shown here for TS, limited in vitro kinetic characterization of the allosteric enzymes involved at these branch-point would be required to obtain equations and parameters to model the behaviour of the branch-points. This possibility may be of special interest to simplify the characterization of branch-points where the enzyme activities are controlled by numerous allosteric interactions.
We wish to thank Marie-Christine Butikofer and Valérie Verne for the ELISA assays. We thank Pr. Roland Douce and Dr Michel Matringe and Mickaëla Hoffman for critical reading of the manuscript. Special thanks to Maighread Gallagher for the correction of the English. This work was supported by BayerCropScience 14–20 Rue Pierre Baizet 69263 Lyon cedex 09 (France).