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P-type ATPases, as major consumers of cellular ATP in eukaryotic cells, are characterized by the formation of a phosphorylated enzyme intermediate (E2P), a process that is allosterically coupled to translocation of cations against an electrochemical gradient. The catalytic cycle comprises binding of Mg-ATP at the nucleotide-binding domain, phosphorylation of the E1 state (E1), conformational transition to the E2P state, and dephosphorylation through the actuator domain and re-establishment of the E1 state. Recently, it has been suggested that, for several P-type ATPases, Mg-ATP binds to the phosphorylated enzyme, thereby accelerating the transition to the E1 state, before then becoming the enzyme's catalytic substrate. Here, we provide evidence supporting this viewpoint. We employed kinetic models based on steady-state kinetics in the presence and absence of the reversible inhibitor orthovanadate. Vanadate is generally considered to be a conformational probe that specifically binds to the E2 state, arresting the enzyme in a state analogous to the E2P state. Hydrolytic H+-ATPase activities were measured in inside-out plasma membrane vesicles isolated from roots and shoots of maize plants. For root enzymes, kinetic models of vanadate inhibition that allow simultaneous binding of Mg-ATP and vanadate to the same enzyme state were most plausible. For shoot enzymes, application of the competitive inhibitor Mg-free ATP attenuated vanadate inhibition, which is consistent with a model in which either Mg-free ATP or Mg-ATP is bound to the enzyme when vanadate binds. Therefore, data from roots and shoots indicate that binding of ATP species before transition to the E1 state plays an important role in the catalytic cycle of plant plasma membrane H+-ATPase.
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It was reported almost two decades ago that maize plasma membrane (PM) H+-ATPase (EC 184.108.40.206) exposes an additional ATP-binding site with high affinity in the presence of detergent . The authors speculated that this site could be the ATP-binding site in the E1 state that is typical of other P-type ATPases, such as Na+/K+-ATPase. They suggested that, under physiological conditions, this site may be inaccessible to the substrate. Since then, several reports have provided evidence for the existence of different ATP-binding sites in maize PM H+-ATPase, because the kinetic data suggested positive or negative cooperativity [2, 3]. In 1997, Roberts and Beaugé  presented four models of substrate binding to PM H+-ATPase of maize, one of which they called a ‘minimal’ model, which does not require two binding sites to explain cooperativity. In this model, ATP sequentially plays a regulatory role and a catalytic role. The model has recently been revived by a review on the mechanism of allosteric effects of ATP on the kinetics of P-type ATPases by Clarke . Clarke provided ample evidence for his statement that there would be no structural or functional need for two different ATP sites in P-type ATPases in order to translocate ions . He summarized several investigations indicating that the low-affinity ATP-binding site in the E2 state is transformed to a high-affinity binding site in the E1 state. Under physiological conditions Mg-ATP would enter the catalytic cycle in the E2 state. Substantial support for this notion comes from investigations of Na+/K+-ATPase. For this enzyme, it was clearly demonstrated that Mg-ATP at ≥ 100 μm binds to the phosphorylated enzyme intermediate (E2P) , subsequently serving as the substrate for the next catalytic cycle. The regulatory role of ATP does not require association with Mg2+. Binding of Mg-free ATP to the E2 state was proven to be an allosteric activator for the E2 to E1 transition of Na+/K+-ATPase  and for dephosphorylation of sarcoplasmic reticulum Ca2+-ATPase . For the latter enzyme, several crystal structures have been determined in which ATP or its analog adenosine 5′-(β,γ-methylene)triphosphate (AMPPCP) are bound to the E2 state in a cavity between the nucleotide domain, phosphorylation domain, and actuator domain [8-10]. One of these structures [Protein Data Bank (PDB) code 3FGO] shows bound AMPPCP in the transition state of enzyme dephosphorylation .
In this article, we provide further evidence for Clarke's viewpoint that, at physiological substrate concentrations, it is the E2 state of P-type ATPases to which the Mg-ATP binds. Our investigation was performed with PM vesicles isolated from the roots of two maize genotypes and from the young shoot of one of these genotypes. The vesicle preparations represent a cocktail of different isoforms of the enzyme, which may differ in kinetic properties [11-13]. For example, the preparation from Pioneer root had a lower apparent Km than that from the shoot . The kinetic approach was based on measuring the steady-state kinetics of PM H+-ATPase in the presence and absence of two inhibitors: vanadate as a conformational probe for the E2 state [15-17], and Mg-free ATP as a competitive inhibitor of P-type ATPases [14, 18-20]. Cantley et al.  were the first to propose ‘that vanadate forms a trigonal bipyramidal structure analogous to the transition state for phosphate hydrolysis’ from the phosphorylated enzyme. Their conclusion was based on kinetic investigations showing a high-affinity binding site for vanadate in the E2 state. They presented a model in which vanadate traps the enzyme in a state analogous to the phosphorylated E2 state, with vanadate bound instead of the phosphoryl group. Later, electron microscopy maps of P-type ATPases treated with vanadate confirmed that the enzyme was in the conformation characteristic of the E2P state . Vanadate has been used as a conformational probe for the E2 state in several studies on the relevance of specific amino acid residues and of the K+ activity for the coupling efficiency of P-type ATPases [17, 22]. In the remarkable identification of the H+-binding residue of plant PM H+-ATPase AHA2, vanadate was employed to demonstrate that, after substitution of the residue, the catalytic cycle was blocked before the vanadate-sensitive step of enzyme dephosphorylation .
Following the rationale of the above-mentioned vanadate applications, it is possible to use vanadate for investigating the question of whether Mg-ATP binds to P-type ATPases in the E2 state: if Mg-ATP binds to the E1 state, vanadate inhibition will require preceding Mg-ATP binding. With a model based on Michaelian kinetics, vanadate inhibition would then be uncompetitive. Furthermore, because, with Mg-free ATP instead of Mg-ATP, the vanadate-sensitive E2P state will not be established, a complex of enzyme, Mg-free ATP and vanadate will not form. These are the two hypotheses tested in this investigation. For the investigated enzymes, Mg-free ATP has been confirmed as a competitive inhibitor in an earlier publication . As the possibility for Mg-ATP binding in the E1 state as well as in the E2 state is provided by the minimal model suggested by Roberts and Beaugé , their ‘two-role model’ serves as a versatile starting point for kinetic modeling in this article.
Results and Discussion
Kinetics in the absence of inhibitors
The hydrolytic activities of maize PM H+-ATPase from root and shoot were in accordance with simple models assuming Michaelian kinetics, as is evident from the good fit of the hyperbolic curves shown in Fig. 1A,B. Nevertheless, as several researchers have found that the kinetics of this enzyme may resemble cooperativity [1-4], our data were also analyzed with Eadie–Scatchard plots (Fig. 1C,D), which are ‘most sensititive to detect deviations from Michaelian behavior’ . In these plots, the enzymes from both root and shoot showed a slight deviation from the linear (Michaelian) relationship, particularly at the lowest tested ATP concentration of 45 μm, which may indicate more complex kinetics. The single-site model of Roberts and Beaugé  gives an example of a catalytic cycle in which Mg-ATP binds to the dephosphorylated E2 state of maize PM H+-ATPase (Fig. 2A). The model integrates two overlapping pathways of ATP hydrolysis: a slow-turnover path, in which the substrate binds with high affinity to the E1 state; and an accelerated path, in which the substrate binds with low affinity to the E2 state. For Na+/K+-ATPase, it has been established that regulatory binding of ATP to the E2 state in exchange for released ADP accelerates the E2–E1 conformational transition and drives substrate turnover at physiological ATP concentrations [5, 6]. In Fig. 2A, the transition from this two-role model to a Michaelian model is illustrated by the clamp box surrounded by the dashed line: The processes within the box comprise substrate binding, internal phosphorylation, and formation of the final product, which is the phosphate group released from the enzyme. Under steady-state conditions with negligible turnover in the slow pathway, the reactions in the box can be modeled as a single reaction, which then resembles Michaelian kinetics, but with substrate binding in the E2 state (pseudo-Michaelian kinetics). Figure 2D shows another single-site two-role model, which accounts for the case in which the substrate binds to a phosphorylated form of the enzyme. On the assumption that the rapid dephosphorylation of plant PM H+-ATPase  is an inherent property that is not related to regulatory ATP binding, the two-role model shown in Fig. 2A was selected for addressing the question of deviations from Michaelian kinetics. Reactions 2 and 3 of Fig. 2A are irreversible in the absence of ADP and phosphate , and were merged in a single step (Fig. 2C, reactions 1–5). The resulting condensed model was programmed with the kinetics software dynafit. In order to validate our dynafit script, we confirmed that, with reasonable kinetic constants, it can simulate Eadie–Scatchard plots resembling negative cooperative behavior (Fig. 3A, black line). When the slow pathway (Fig. 2C) is set to a very low-turnover rate constant, the programmed two-role model yields the Eadie–Scatchard plot of a Michaelis–Menten model (Fig. 3A, gray line). This illustrates that the observation of Michaelian kinetics does not necessarily imply substrate binding in the E1 state of PM H+-ATPase.
For Pioneer root, the measured activity data (Fig. 1A) were used to compare the fit of the two-role model to the fit of the Michaelis–Menten model. Indeed, at low ATP concentrations, the residuals became smaller, assuming two ATP roles (Fig. 3B, circles versus squares). For Pioneer shoot, fitting with the two-role model did not reduce the residuals. For Amadeo root, the lowest tested ATP concentration was 100 μm, which makes it impossible to detect high-affinity binding of ATP with slow turnover. Because, for Pioneer shoot and for Amadeo root, the datasets were not sufficient to ‘train’ the two-role model, subsequent analyses of these samples were performed with the Michaelian model. As explained in the previous paragraph, the rate constants for substrate binding calculated with the Michaelian model may, in fact, describe binding of the substrate to the E2 state. We will return to this issue of pseudo-Michaelian kinetics in the following sections.
Inhibition by vanadate
In order to explain the hydrolytic activities in the presence of vanadate, the basic inhibition models known from Michaelian kinetics were taken into consideration, as illustrated in Fig. 2B. For Pioneer root, the reactions of the two-role model were extended by combining them with reactions with vanadate before and after substrate binding in the E2 state, as shown in Fig. 2C. For the names of the inhibition types, quotation marks are used when they refer to the two-role model and not to the Michaelian model. As Km and the catalytic rate constants had already been determined from assays in the absence of vanadate, fitting was solely based on the dissociation rate constants of the enzyme–vanadate complexes. The two-role model with ‘linear mixed inhibition’ (all of the reactions shown in Fig. 2C) provided the best fit (dashed line in Fig. 4A) of all tested models, as is evident from the smallest sum of squared residuals in Table 1. Model discrimination analysis, which additionally takes into account how many kinetic constants are required by a model to describe the measured data (see 'Experimental procedures'), confirmed a favorable Delta of 0 (Table 1) for ‘linear mixed inhibition’. We conclude that this is the most plausible model for describing the kinetics of PM H+-ATPase from Pioneer root, with the limitation that it does not include reactions with Mg-free ATP. It should be noted that the Michaelian models with the different vanadate inhibition types (Fig. 2B) were included in the comparison of models for Pioneer root. The Delta of the Michaelian model with uncompetitive vanadate inhibition was similar to that of the best two-role model (Table 1). The fit of the Michaelian models is shown in Fig. 4B. For all samples, the model with the worst fit was the model in which Mg-ATP and vanadate compete for the same binding site.
Table 1. Plausibility of vanadate inhibition models for PM H+-ATPase from maize. V, vanadate; SSR, sum of squared residuals multiplied by 107. The number of fitting parameters was 1, except for linear mixed inhibition with three parameters. Interpretation of Delta : between 0 and 2, ‘substantial’ support for the model; between 4 and 7, ‘considerably less’ support; and > 10, ‘essentially no’ support. Fitted curves of the models are shown in the indicated Figs. Michaelian models highlighted with bold letters were extended for investigating interactions between vanadate and Mg-free ATP (Fig. 5D–E). For a further explanation of Delta, see 'Experimental procedures'
For PM H+-ATPase of Amadeo roots, the Michaelian models with uncompetitive, competitive and linear mixed inhibition yielded comparable, good fits at 10 μm vanadate (Fig. 4C). However, at 100 μm vanadate, the measured data deviated considerably from the uncompetitive inhibition model (Fig. 4D, solid black line), whereas a good fit was obtained with linear mixed inhibition. This inhibition type includes all reactions shown in Fig. 2B, in which E then must be understood to represent an enzyme state that can react with vanadate. Remembering the linearization of the Eadie–Scatchard plots of the two-role model with decreasing velocity in the slow pathway (Fig. 3A), it now becomes clear that the Michaelian model appears to be the approximation for a two role-model, in which the slow conversion from the E2 state without bound substrate to the E1 state and substrate binding in the E1 state (Fig. 2C, reaction 1) are negligible (with respect to catalytic turnover). Thus, linear mixed inhibition implies that the substrate binds to the E2 state both before and after vanadate. For Pioneer shoot, the kinetics were unambiguously consistent with uncompetitive inhibition at 10 μm vanadate (Fig. 4E) and at 100 μm vanadate (Fig. 4F).
Inhibition by Mg-free ATP
Rate constants for the dissociation of Mg-free ATP from the enzyme–ATP complex had to be determined, as a prerequisite for simulating the interaction between Mg-free ATP and vanadate (see ‘Interaction of inhibitors’, below). The measured activity data of Pioneer root and shoot H+-ATPases in the presence of Mg-free ATP under the given experimental conditions have already been shown in Figs 1B and 2A of Hanstein et al. . Unlike in the previous investigation, in the present study calculation of kinetic constants for substrate binding and turnover was solely based on the part of the experiment in which Mg-free ATP did not occur (Fig. 1A,B). In a second step, a model based on competitive inhibition by Mg-free ATP was fitted to the activity data measured in the presence of Mg-free ATP, which then provided the rate constant for the dissociation of Mg-free ATP.
Interaction of inhibitors
We hypothesized that Mg-free ATP and vanadate are not simultaneously bound to the enzyme. This hypothesis was addressed with a set of three kinetic models for inhibitor interactions (Fig. 5D,E). The analysis was performed for those samples in which the most plausible vanadate inhibition type of the Michaelian models was uncompetitive, as indicated with bold letters in Table 1 (Pioneer root and shoot).
Uncompetitive vanadate inhibition (Fig. 2B) and substrate binding in the E1 state would be consistent with a model in which molecular interactions between the inhibitors are impossible, because both inhibitors bind at different stages. Mg-ATP and Mg-free ATP would bind to the E1 state. With Mg-ATP, the enzyme would hydrolyze ATP and proceed to the vanadate-sensitive E2P state. With Mg-free ATP, the vanadate-sensitive stage would not be reached. These requirements are fulfilled by the model in Fig. 5D. The activity curve simulated with this model for Pioneer root is indicated by the thick gray line in Fig. 5A, showing an acceptable fit to the measured data. Assuming that both inhibitors bind to the same enzyme state without influencing each other (compatibility model; Fig. 5E, model code pMuc0), the simulated activity was considerably lower than the measured activity (Fig. 5A, black solid line). The model of simultaneous binding was further elaborated by allowing the affinity for vanadate binding to decrease after binding of Mg-free ATP to the enzyme (Fig. 5E: allowing K5 to be different from K3). With this model (interference model; model code pMuc1), it is possible to obtain a transient increase in enzyme activity as soon as Mg-free ATP accumulates (Fig. 5A, dashed line). The vanadate dissociation constant increased by a factor of four when Mg-free ATP was bound to the enzyme instead of Mg-ATP (Table 2). It should be noted that subsequent Mg2+ binding to the complex of enzyme, vanadate and Mg-free ATP (Fig. 5E, K6) was essential to achieve the transient rise in activity at ATP concentrations around 6 mm. The interference model demonstrates that the hydrolytic activity is stimulated just when Mg-free ATP comes into play as an antagonist of vanadate, as long as it can attract some Mg2+ to the enzyme–ATP complex.
Table 2. Vanadate dissociation constants for the reaction with H+-ATPase from Pioneer root and shoot. The vanadate dissociation constant Kd1 was determined in the absence of Mg-free ATP from the data shown in Fig. 4, assuming Michaelian kinetics and uncompetitive vanadate inhibition. Kd2 was obtained from the activity data in the presence of vanadate (V) and Mg-free ATP (in Fig. 5 at total ATP concentrations > 3 mm) with the inhibitor interference model in Fig. 5E. This model assumes that: (a) substrate binding precedes vanadate binding (uncompetitive vanadate inhibition); (b) Mg-ATP and Mg-free ATP bind to the enzyme in the E2 state; and (c) binding of Mg-free ATP to the enzyme instead of Mg-ATP modifies the vanadate dissociation constant. Kd2 represents the vanadate dissociation constant for the complex of enzyme with Mg-free ATP, and Kd1 is the vanadate dissociation constant for the complex of enzyme with Mg-ATP. SD, standard deviation of the nonlinear regression analysis, which was performed with the average activities of three vesicle preparations
For Pioneer shoot, the interaction was studied at two vanadate concentrations. Figure 5B shows that, again, the interference model (Fig. 5E, pMuc1) was most appropriate for describing the experimental data at 10 μm vanadate. The dissociation constant for vanadate increased by a factor of eight when Mg-free ATP was bound to the enzyme instead of Mg-ATP (Table 2). At 100 μm vanadate, measured activities in the presence of Mg-free ATP (at concentrations of total ATP exceeding 5 mm) were definitely higher than expected from the models with binding at different stages (Fig. 5C, thick gray line) and with compatible binding (thin black line). Again, the best fit was obtained with the interference model, although this model could not explain the continuous increase in activity with increasing concentrations of Mg-free ATP.
In summary, for Pioneer shoot the activities measured in the presence of Mg-free ATP (at total ATP concentrations > 5 mm) are higher than simulated with the Michaelian model. The vanadate dissociation constant used for the simulations was determined from the activities at total ATP concentrations < 3 mm in the same experiment. This means that the concentrations of vanadate or possibly oligo-anions of vanadate were identical over the complete concentration range, ruling out the possibility that an increase in oligo-vanadate can explain the underestimation of enzyme activity in the presence of Mg-free ATP. The inhibitor interaction model (pMuc1) shows that a two-role model is more flexible and can be adjusted to follow the higher activities in the presence of Mg-free ATP. These results have two implications, as follows. (a) They support the view that the Michaelian model is not satisfactory to explain the mechanism of vanadate inhibition for maize PM H+-ATPase. Vanadate and Mg-free ATP appear to be simultaneously bound to the same enzyme state, either in close proximity (see below) or with allosteric interaction. As Mg-free ATP acts as a competitive inhibitor, as demonstrated under the same experimental conditions , these data indicate that Mg-ATP is bound to the E2 state as well. It cannot be ruled out that Mg-ATP binds even before enzyme dephosphorylation, which would require models of the kind shown in Fig. 2D. The structural requirement for a substrate-binding site in a phosphorylated P-type ATPase was demonstrated by Olesen et al. . The authors identified a nucleotide-binding pocket with an arginine in the E2P state of sarcoplasmic reticulum Ca2+-ATPase. This residue is not conserved among P-type ATPases. (b) At first glance, interference of Mg-free ATP with vanadate binding, and Mg-ATP and vanadate not competing for the same binding site, appear to be contradictory. However, Mg2+ could make the difference: for the E2 state of sarcoplasmic reticulum Ca2+-ATPase, a crystal structure has been described with bound ATP in which Mg2+ pulls the γ-phosphate away from the residue that is phosphorylated during the catalytic cycle . Figure 6 shows another crystal structure at a slightly earlier stage (during dephosphorylation), and illustrates how the γ-phosphate of Mg-free ATP could approach the phosphorylated site and subsequently affect vanadate binding by electrostatic repulsion.
Mechanistic relevance of ATP binding to the E2 state
For sarcoplasmic reticulum Ca2+-ATPase, it was suggested that allosteric binding of Mg-ATP to the E2 state could keep the enzyme in a compact closed conformation throughout the enzymatic cycle, thereby avoiding great structural rearrangements [5, 7]. Another advantage of allosteric binding of Mg-ATP to the E2 state could be related to the sensitivity of the E1P state to ADP, which causes regeneration of ATP at high Ca2+ concentrations in the sarcoplasmic reticulum. Olesen et al.  suggested that ATP binding to the E2P state prevents a smooth reversal to the ADP-sensitive E1P state. The mechanistic relevance of substrate binding in the E2 state may also be related to the particular way in which proton backflow is prevented in plant PM H+-ATPases. Probably, this is accomplished by a built-in counterion (arginine) that blocks the aspartate from which the proton was released . The built-in counterion is thought to stimulate dephosphorylation. The intrinsic capability for fast dephosphorylation has been suggested as an explanation for the observation that PM H+-ATPases, unlike other P-type ATPases, cannot be phosphorylated by inorganic phosphate (Pi) . On the other hand, the electrostatic attraction between the aspartate and the counterion ‘counteracts’ the conformational transition from the E2 state to the E1 state. ATP bound to the dephosphorylated E2 state is likely to provide energy for this transition through the increase in ATP affinity from the E2 to the E1 state .
A mechanistic relationship between low-affinity substrate binding (to the E2 state) and the ability to pump protons despite a steep electrochemical H+ gradient is indicated by physiological experiments. For example, the enzyme dephosphorylation step is coupled to the closing of the proton exit pathway to the apoplast , which is particularly important when the apoplastic proton activity is high. For Pioneer root, cultivation in nutrient solution with pH 3.5 increases the apparent Km of the isoform cocktail . As already noted, the isoforms from Pioneer shoot showed a higher Km than those from Pioneer root after cultivation in standard nutrient solution with neutral to slightly alkaline pH. In leaf tissue, the apoplastic pH was overestimated for a long time, until ion-sensitive microelectrodes revealed a low pH, between 4 and 5 [25, 26]. There may be a mechanistic link between the ability to release protons to an acidic environment, modulatory (regulatory) subtrate binding to the phosphorylated enzyme, and a higher Km. Seen in these perspectives, the various possibilities for ATP binding before return to the E1 state are mechanistically highly interesting, as they may support high rates of proton pumping in plant and fungal P-type ATPases at high membrane potentials and/or steep pH gradients .
A catalytic cycle in which either Mg-ATP or Mg-free ATP binds before the enzyme is in the E1 state has important implications for structural research on inhibition by Mg-free ATP: The search for structural determinants, in particular isoforms that may counteract binding of Mg-free ATP without compromising binding of Mg-ATP, should not focus only on the E1 state, but should also include the investigation of amino acids with modulatory nucleotide contacts in the E2 state.
Plant material and assay conditions
The investigation was performed with root and shoot material from Zea mays L. cv. Pioneer 3906. Root investigations were extended by including a second cultivar, Z. mays L. cv. Amadeo (KWS, Kleinwanzleben, Germany). Cultivation of the maize plants in nutrient solution in the climate chamber has been described previously . PM vesicles were isolated according to Yan et al.  for roots and according to Zörb et al.  for shoots. All assays were performed as described by Hanstein et al. , at pH 6.5 with 150 mm K+ and 0.02% (w/v) Brij-58. ATP was regenerated from ADP . Unspecific ATP hydrolysis by trace amounts of other enzymes was suppressed with 100 mm nitrate, 1 mm azide, and 1 mm molybdate.
Nonlinear regression analysis
General fitting procedures and model discrimination analysis
The measured activity data were subjected to nonlinear regression analysis as described by Hanstein et al. . dynafit was used to fit different mechanistic models to the measured enzyme activities . All calculations were based on average activities of three biological replicates (vesicles from replicate plant containers). The dynamic fitting method  was used with a mixing delay time of 20 s. In all fitting procedures, the rate constants for association were used as constants and based on literature values, and the rate constants for dissociation were used for fitting. The fitting procedure requires that initial values are specified that were also derived from literature values as far as possible. In general, parameters were sequentially fitted with the scripts provided as Supporting information in the following manner: (a) on the basis of assays in the absence of inhibitors, rate constants for substrate dissociation and catalytic rate constants were determined (Table S1); (b) with these constants, rate constants for inhibitor dissociation from the enzyme–inhibitor complex were derived from activities in the presence of the respective inhibitor, i.e. for vanadate dissociation from assays with only vanadate (Table S2) and for Mg-free ATP from assays with only Mg-free ATP (Table S3); and (c) assays with both inhibitors were used for fitting the rate constant for vanadate dissociation from the complex of enzyme with both inhibitors in the interference model (Fig. 5E; see below). In this last step, all other rate constants were set to the values determined in steps (a) and (b) (Table S4). For comparison of the probabilities of different models, Akaike's information criterion (AIC) was computed with dynafit. The difference between the AIC of the model with the best fit and the AIC of the model with the worse fit (termed Delta by dynafit) was used to estimate the plausibility of the latter model . A Delta value between 0 and 2 lends ‘substantial’ support to a model, a value between 4 and 7 lends ‘considerably less’ support, and a value > 10 lends ‘essentially no’ support . Model discrimination analysis provides a method that also rates the complexity of a model. For example, when different models achieve the same fit to the experimental data (sum of squared residuals), model discrimination analysis calculates the smallest Delta for the simplest model, i.e. the model with the smallest number of parameters (reactions).
Rate constants and models for substrate binding
The reaction Mg2+ + Mg-free ATP ↔ Mg-ATP was included in all models (not shown in Figs 2 and 5: Mg + A ↔ S). The rate constant for Mg-ATP association of 10 μm−1·s−1 was according to Phillips , and the rate constant for dissociation was calculated by using a Kd of 28 μm . The same values were used for the reaction between free Mg2+ and the enzyme–ATP complex, as Pilotelle-Bunner et al. (2009) have found, for the E1 state of Na+/K+-ATPase, that Mg2+ is primarily bound by the bound ATP . The rate constant for vanadate–enzyme association was set to 10 μm−1·s−1. The rate constants for enzyme–substrate association were 50 μm−1·s−1 with respect to the high-affinity binding site as reported by Clarke for Na+/K+-ATPase , and 20 μm−1·s−1 with respect to the low-affinity binding site as used by Roberts and Beaugé for maize PM H+-ATPase . The latter value was employed in all models that do not differentiate between a catalytic role and a regulatory role of the substrate (Michaelian models).
When the kinetics indicated two substrate-binding sites with different affinities, the fit of a model with two substrate roles was compared with the fit of the Michaelian model. The original two-role model of Roberts and Beaugé  (model A in Fig. 2) was simplified by combining reaction 2 with reaction 3 (enzyme dephosphorylation), as shown in Fig. 2C. This reaction (dephosphorylation) is considered to occur rapidly in plant PM H+-ATPase . A similar set of reactions was used by Clarke and Kane  in their bicyclic model of Na+/K+-ATPase, but they assigned the Pi release to reactions 3 and 5. With their model, it is assumed that the substrate is already bound to the phosphorylated enzyme. For the dataset shown in Fig. 1A, the two-role model and the model with Pi release according to Clarke and Kane  were compared by using the same initial values for the fitting parameters. The fitting results were almost identical with both models. The fitting parameters and their initial values for the simplified two-role model according to Roberts and Beaugé  in Fig. 2C are as follows (see also Table S1). (a) The initial rate constant for substrate dissociation from the low-affinity site was 4000 s−1, as derived from the apparent Km for maize root PM H+-ATPase in the order of 200 μm  and a rate constant for enzyme–substrate association of 20 μm−1·s−1 (Fig. 2C, reaction 1). (b) Assuming continuous removal of ADP by the ATP-regenerating system and rapid dephosphorylation, enzyme phosphorylation, conformational transition and dephosphorylation were considered to be unidirectional , and were treated as a single step (reaction 2) with an initial rate constant of 200 s−1 [4, 5, 23, 35]. (c) The initial rate constant of the conformational transition from the E2 state to the E1 state was 1 s−1  (reaction 3). (d) The initial rate constant of substrate dissociation from the high-affinity site of 100 s−1 was derived from the Kd of 2 μm for the maize high-affinity site  and a rate constant for enzyme–substrate association of 50 μm−1·s−1 (reaction 4). (e) The initial rate constant of conformational transition from E2.S to E1.S was 100 s−1, following observations for Na+/K+-ATPase with rate constants between 65 s−1 and 90 s−1  (reaction 5). This transition was treated as unidirectional, which represents a simplification of the model of Roberts and Beaugé . P-type ATPases of subfamily III (to which PM H+-ATPases belong) differ from other P-type ATPases in that they cannot be phosphorylated by Pi . This is generally considered to reflect the fact that, without phosphorylation, no transition from the E1 to the E2 state occurs .
Two-role model and vanadate inhibition
For Pioneer root, in addition to the Michaelian model, a model with two ATP roles according to Roberts and Beaugé  was included in model comparison, and a combination with four types of reversible vanadate inhibition was tested, as shown in Fig. 2C. It is generally assumed that vanadate (as a transition state analog) binds during the dephosphorylation of the E2P state [15, 21], which keeps the enzyme in a state analogous to the E2P state, but with bound vanadate instead of the phosphoryl group. This implies that it acts as an inhibitor, although dephosphorylation occurs, because it inhibits progression to the next catalytic cycle. This widely assumed mode of vanadate inhibition provides the blueprint for the mode of substrate action when it binds to the E2 state: It stimulates phosphate release (in the next catalytic cycle), because it accelerates the E2 to E1 transition. The models in Fig. 2C represent very simple models in which vanadate and substrate can bind to the E2 state sequentially or simultaneously, including the possibility that substrate binding may precede vanadate binding. However, they do not reflect the fact that substrate binding may already take place at the phosphorylated enzyme , which is a reasonable view, because ADP is thought to dissociate during the E1P to E2P transition , leaving the space for Mg-ATP binding. Considerably more complex models would be required to represent substrate binding to the phosphorylated enzyme.
The names of the vanadate inhibition models were derived from the simple inhibition models based on Michaelian kinetics . They are given in quotation marks, as here they refer to kinetics that resemble Michaelian kinetics only in the steady state and with a very slow reaction 3 (Fig. 2C) (see ‘Kinetics in the absence of inbitors’, above). In the ‘uncompetitive’ model, vanadate binding was restricted to E2.S, whereas in the ‘noncompetitive’ model and in the ‘linear mixed’ model, vanadate binding to the E2 state was also permitted. The ‘noncompetitive’ model was obtained when the rate constants for reactions 6 and 8 were equal and when the rate constants for reactions 4 and 7 were equal. In the ‘linear mixed’ model, the rate constants for these reactions were allowed to be different. The ‘competitive’ inhibition model was based on competition with substrate binding in the E2 state. The number of fitting parameters was 1 for all models except for the ‘linear mixed’ model, in which reactions 6–8 were used for fitting.
Implications of Mg2+ activity in experiments with Mg-free ATP
Inhibitor interaction studies were performed in assays in which vanadate and Mg-free ATP were simultaneously present. The presence of Mg-free ATP implies that the enzyme assays were conducted at low Mg2+ activities. For example, if we consider a simple system with only Mg2+ at a total concentration of 5 mm and ATP, the Mg2+ activity will be 28 μm at 10 mm total ATP, and 14 μm at 15 mm total ATP. These values are obtained with a Kd of Mg-ATP of 28 μm . It has long been known that binding of Mg2+ and binding of vanadate are synergistic . For the E1 conformation of Na+/K+-ATPase, it has been demonstrated that the enzyme's Mg2+ affinity is determined by the Mg2+ affinity of ATP. If this also applied to the E2P state, then vanadate binding could be reduced, owing to a lack of Mg2+ at the enzyme. This would mimic an antagonistic action of both inhibitors, and rule out any conclusions about binding of both inhibitors to the same enzyme state. However, as Stokes and Green  recalled, a dramatic increase in Mg2+ affinity occurs during enzyme phosphorylation of sarcoplasmic reticulum Ca2+-ATPase: the Mg2+ release rate decreased after enzyme phosphorylation to < 1%, from 80 s−1 to 0.5 s−1. High-affinity binding of Mg2+ to the phosphoryl aspartate of the E2P state is thought to protect the anhydride bond from nucleophilic attack by water molecules . Assuming that the association rate is not changed, and that in the E1 state the dissociation constant is ~ 28 μm (see above), the dissociation constant in the E2P state will be < 1 μm. Therefore, it is likely that, at Mg2+ activities down to 10 μm, Mg-free ATP can be applied to an enzyme that still has the Mg2+ bound to the phosphoryl aspartate in the E2P state.
Models for studying inhibitor interactions
In order to investigate whether binding of Mg-free ATP interferes with vanadate binding, three interaction models were designed, as illustrated in Fig. 5D–E. In all three models, substrate binding precedes vanadate binding. The model ‘binding to different states’ reflects substrate binding to the E1 state, and subsequent transition to an intermediate enzyme state that can bind vanadate (E2P). Without vanadate binding, Pi is released from the intermediate state, and the enzyme returns to the E1 state. The model excludes simultaneous binding of both inhibitors, as is evident from the lack of vanadate binding to the enzyme with bound Mg-free ATP (denoted E.A). The other models were derived from the two-role model. The upper model of Fig. 5E shows the two-role model with substrate binding in the E1 state and in the E2 state. Substrate binding in the E2 state is an essential requirement for interactions between vanadate and ATP species to occur. In the model with compatible inhibitor binding, vanadate binding is not influenced by the presence of Mg-free ATP at the enzyme, which is expressed by a single vanadate dissociation constant (K3 = K5). The scheme includes competition between the substrate and Mg-free ATP in the E2 state. In the E1 state, this competition is omitted for simplicity. The transition from the E2 to the E1 state is considered to be very slow as compared with the transition from E2.S to E1.S. The slow reaction is only required to reach steady state. Under steady-state conditions, it plays almost no role in the velocity of phosphate release. Our experimental data are based on an incubation time of 30 min. Assuming that these data reflect the steady state, the kinetics were modeled with the simple set of reactions in the lower model of Fig. 5E. The model of inhibitor binding with interference (Fig. 5E: model code pMuc1) allows simultaneous binding and modification of the vanadate dissociation constant when Mg-free ATP is bound to the enzyme instead of Mg-ATP (fitting of the equilibrium constant K5).
This work was supported by the Natural Science Foundation of China (31071845) and by a China Scholarship Council (2007832114). Introduction to nonlinear regression analysis with dynafit by P. Friedhoff, Justus Liebig University, is gratefully acknowledged. We appreciate his helpful critical comments, and the advice to present the various reaction schemes in Figs 2 and 5 and to show the fitted curves of all models.