Are the States That Occlude Rubidium Obligatory Intermediates of the Na+/K+-ATPase Reaction?*

In the Albers-Post model, occlusion of K+ in the E 2 conformer of the enzyme (E) is an obligatory step of Na+/K+-ATPase reaction. If this were so the ratio (Na+/K+-ATPase activity)/(concentration of occluded species) should be equal to the rate constant for deocclusion. We tested this prediction in a partially purified Na+/K+-ATPase from pig kidney by means of rapid filtration to measure the occlusion using the K+ congener Rb+. Assuming that always two Rb+ are occluded per enzyme, the steady-state levels of occluded forms and the kinetics of deocclusion were adequately described by the Albers-Post model over a very wide range of [ATP] and [Rb+]. The same happened with the kinetics of ATP hydrolysis. However, the value of the parameters that gave best fit differed from those for occlusion in such a way that the ratio (Na+/K+-ATPase activity)/(concentration of occluded species) became much larger than the rate constant for deocclusion when [Rb+] <10 mm. This points to the presence of an extra ATP hydrolysis that is not Na+-ATPase activity and that does not involve occlusion. A possible way of explaining this is to posit that the binding of a single Rb+ increases ATP hydrolysis without occlusion.

species occluded through this process have been studied extensively (see reviews by Forbush (7), Glynn and Richards (10), and Glynn and Karlish (11)). The release of K ϩ or its congeners from their occluded states (deocclusion) is considerably accelerated by binding of ATP at a site with low affinity relative to the active site and where ATP does not seem to undergo hydrolysis (6,12). This process is the cause of most of the increase in Na ϩ /K ϩ -ATPase activity with increasing ATP concentration (13). Occlusion of Na ϩ has also been detected (5,14), and there is evidence that the occlusion site for Na ϩ is in the same domain but in a different conformation, as the K ϩ site (7,15). Na ϩ occlusion seems to occur at a stage of the ATP hydrolysis cycle that is different from that which occludes K ϩ , suggesting that occlusions of the two cations are mutually exclusive events.
In the present study we expand the information about the Na ϩ /K ϩ -ATPase reaction, using the congener 86 Rb ϩ as a probe for K ϩ and performing direct measurements of the steady-state concentration levels of the intermediates containing occluded Rb ϩ (Rb occ ). In these experiments Rb occ was formed through the so-called physiological route, i.e. during the turnover of the Na ϩ /K ϩ -ATPase. In this condition K ϩ enters the enzyme presumably from the extracellular side to activate the catalysis of the hydrolysis of ATP (16). Brief reports of measurements of this type of occlusion have been given by Glynn and Richards (4), Glynn et al. (5), and Forbush (6). Rossi and Nørby (9) have studied the pre-steady-state kinetics of Tl ϩ occlusion via the physiological route using a similar technique as that in the present paper.
The measurements we are reporting here were made possible by the development of a new technique (see "Experimental Procedures" and Ref. 17) for the isolation in the millisecond time scale and quantitative determination of the intermediates containing occluded cations. The method is used during both steady-state turnover and in transient state experiments with the Na ϩ /K ϩ -pump. With the same enzyme preparations and under the same experimental conditions, we have also measured the steady-state ATPase activity, v t , and the rate coefficient, k deocc , for deocclusion of Rb ϩ from Rb occ . These can be analyzed as part of the Albers-Post model; a simplified version is shown in Fig. 1.
The results presented here show that the kinetics of the species containing occluded Rb ϩ , i.e. E occ and E occ ATP, could be quantitatively described by the scheme in Fig. 1. We showed that the intermediates holding occluded Rb ϩ were only kinetically obligatory for the ATPase reaction at saturating concentrations of Rb ϩ . We found, however, that even at concentrations of Rb ϩ that were sufficiently high as to drive the reaction totally away from the pathway that generates Na ϩ -ATPase activity, a significant part of the hydrolysis bypassed the oc-cluded forms. We show that a possible reason for this is that the phosphoenzyme containing only one Rb ϩ (E 2 PRb) dephosphorylates, without leading to occlusion of Rb ϩ , with a rate coefficient whose value lies between those for dephosphorylation of E 2 P and E 2 PRb 2 .
At this point, it seems convenient to include the theoretical grounds that underlie the experiments in which we tested whether the enzyme species containing occluded Rb ϩ are obligatory intermediates in the reaction cycle.
To illustrate this, let us take the elementary reaction A 7 B in Scheme 1 as one of the steps of the enzyme reaction. The possibility that the step A 7 B can be bypassed with the rate v other during the overall reaction is important for the argument to follow, whereas the order of addition of substrate(s) and the release of product(s) are not.
If n pathways participate in the A 7 B transition, then in steady state the overall rate (v t ) of the reaction catalyzed by the enzyme can be written as Equation 1.   Equation 3 shows that the ratio between the steady-state overall velocity, v t , and the total concentration, [A], of the intermediates at this step will be a linear, weighted combination of the rate constants involved in their transformation, plus a term related to alternative reaction pathways, v other /[A]. Equations 1-3 provide a criterion to test if a family of intermediates (here, the A i s) are obligatory participants of the enzyme reaction. If this were the case, then v other would be 0 and the ratio v t /[A] would be equal to the rate coefficient for the reaction A 3 B.
The experimental investigation of this criterion will be in many cases hampered by difficulties in the measurement of the concentration of the enzyme intermediates and in the direct determination of the rate constants of their reactions. In the case of the intermediates that hold occluded K ϩ or its congeners, these difficulties can be circumvented because newly available procedures (Refs. 9 and 17 and "Experimental Procedures" of this paper) make it possible to estimate with accuracy their steady-state levels and to follow the time courses of their release of occluded Rb ϩ .
The ratio between steady-state ATPase activity (v t ) and the total concentration of the conformer(s) of the enzyme holding the occluded Rb ϩ , [E occT ] ϭ [E occ ] ϩ [E occ ATP], would be given by (see Fig. 1

and Equation 3) Equation 4
: where v occ is the activity through the occlusion producing pathway, and under irreversibility conditions, it can be calculated as the sum of the rates of the deocclusion pathways as shown in Equation 5: A well known example of the eventual contribution to v t of the pathways that do not include E occT (v other in Equation 6) is the Na ϩ -ATPase activity of the Na ϩ /K ϩ -ATPase (see e.g. Ref. 13).
In the absence of Rb ϩ , hydrolysis of ATP proceeds with a rate that is about 5-10% of V max for Na ϩ /Rb ϩ -ATPase. Referring to Fig. 1 and the conditions of the present study, this rate will be v other ϭ v t ϭ k 40 ϫ [E 2 P] and almost all the enzyme will be in the form of E 2 P. At subsaturating concentrations of Rb ϩ the species E 2 PRb is also present, and v other must be expressed as Equation 7: FIG. 1. A kinetic scheme for ATP hydrolysis by the Na ؉ /K ؉ -ATPase based on the Albers-Post model. E 1 is the conformer of the enzyme that binds noncovalently ATP to the active site to form the enzyme-substrate complex, E 1 ATP. E 1 P and E 2 P are two conformational states of a phosphoenzyme formed by the covalent binding of the terminal phosphate of ATP to the enzyme. Occluded Rb ϩ is held in enzyme forms with (E occ ATP) and without (E occ ) noncovalently bound ATP. K ATP is the equilibrium constant for the dissociation of ATP from E occ ATP, and K Rb1 and K Rb2 are the equilibrium constants for the dissociation of Rb ϩ from E 2 PRb and E 2 PRb 2 , respectively. The lowercase ks are rate constants for the elementary steps indicated by the neighboring arrows. For the analysis of the behavior of the model, Na ϩ -ATPase will be considered as the activity that takes place through the dephosphorylation of E 2 P. As [Rb ϩ ] increases, Na ϩ -ATPase activity tends to 0, whereas the reaction is shifted toward pathways involving the dephosphorylation of E 2 PRb and E 2 PRb 2 (Na ϩ /Rb ϩ -ATPase activity). Only the dephosphorylation of E 2 PRb 2 leads to the formation of E occ . SCHEME I At saturating Rb ϩ concentrations, v other ϭ 0 because both [E 2 P] and [E 2 PRb] become 0, and all the reaction flux proceeds through the formation and breakdown of the occluded forms. Under this condition (see Equation 8), Thus, at saturating [Rb ϩ ], v t /[E occT ] as a function of [ATP] should be a rectangular hyperbola, increasing from k 0 to k ϱ as [ATP] goes from 0 to ϱ, respectively, with half-maximal effect for [ATP] ϭ K ATP . Note that K ATP will be larger than or equal to the equilibrium constant for the dissociation of ATP from E occ ATP, depending on whether rapid equilibrium holds or not for this reaction.
A preliminary report of some of the results given here has been published (18).
Reagents and Reaction Conditions-[␥-32 P]ATP was synthesized using the procedure of Glynn and Chappell (20), except that no unlabeled orthophosphate (P i ) was added. Carrier-free [ 32 P]P i was from the Comisión Nacional de Energía Atómica (Argentina).
[ 86 Rb]RbCl was from NEN Life Science Products. ATP, enzymes, and reagents for the synthesis of [␥-32 P]ATP were from Sigma. All other reagents were of analytical grade.
All incubations were performed at 25°C in media containing 150 mM NaCl, enough MgCl 2 to give a final concentration of free Mg 2ϩ of 0.5 mM, 25 mM imidazole-HCl, pH 7.4 (at 25°C), and the amounts of RbCl and ATP indicated for each experiment. In all cases before starting a reaction, the components to be mixed were diluted in these media.
Determination of ATPase Activity-ATPase activity was determined according to Schwarzbaum et al. (21), measuring the amount of [ 32 P]P i released from [␥-32 P]ATP. The enzyme concentration was never lower than 4 g/ml in order to approach that used in the determination of occluded rubidium ions. For this reason incubation times were short (between 30 s and 1 min) to avoid the hydrolysis of more than 10% of the ATP during the assay and to ensure initial rate conditions. Control experiments showed that activity was independent of the enzyme concentration from 4 to 200 g/ml and that it was unaffected by the passage of the enzyme through the rapid mixing apparatus described below. Na ϩ /Rb ϩ -and Na ϩ -ATPase activities were calculated as the difference between total activity and the activity either in the same media but with 1 mM ouabain or in media in which all the Na ϩ was replaced by K ϩ . The latter procedure was applied when the reaction times were less than 1 min because at high [Rb ϩ ], the onset of the inhibition by ouabain is not fast enough to ensure complete inhibition. The inorganic phosphate detected under these conditions was practically identical to that measured in the reagent blanks indicating that the concentration of ATPases other than the Na ϩ /K ϩ -ATPase was negligible.
Determination of Phosphorylated Intermediates-This was performed according to Schwarzbaum et al. (21).
Determination of Rubidium Occlusion-The technique was similar to that developed by Rossi and Nørby (9) except that 86 Rb ϩ was used instead of 204 Tl ϩ and by Rossi et al. (17). The arrangement that is described in detail in Ref. 17 is shown in (Fig. 2). It includes a rapid mixing apparatus (RMA) 1 connected to a quenching-and-washing chamber (QWC) through a suitable polyethylene tubing. Reactions took place at 25°C in the RMA and are quenched after the appropriate time by injecting the reaction mixture into the QWC at a flow rate of 1-5 ml s Ϫ1 . During the injection process, the fluid was mixed with an ice-cold washing solution flowing at a rate of 30 -40 ml/s and then filtered through a Millipore filter (AA, 0.8 m pore size) placed in the QWC in FIG. 2. A scheme of the arrangement used to measure occluded Rb ؉ . S1-S4, syringes 1-4; M1-M3, mixers 1-3; DL1-DL3, delay lines. DL3 is an external tube that has the additional function of transferring the reaction mixture from the RMA into the QWC. The rapid mixing apparatus was an SFM4 from Bio-Logic (France), with four syringes, each of its plungers being driven individually by a stepped motor. This allows us to perform precise and accurate dilution in the machine by changing the relative velocity of the plungers (mixing ratios). Additionally, the flow can be interrupted for several seconds (or minutes) to permit the evolution of the reactions in the delay lines. The RMA was thermostated at 25°C. The QWC is made of Plexiglas and consists of two pieces, the chamber piece and the funnel piece. The funnel piece holds a 55-mm diameter Millipore filter. A spherical joint pinch clamp keeps the chamber and funnel pieces tightened together, with the Millipore filter "sandwiched" in between, providing the sealing of the chamber. The pressure difference (about 745 mm Hg) driving the flow through the filter is created by a water-jet pump. A detailed description of the QWC and its properties has been published separately (17). Fig. 1 Equations for [E occT ] and v t were obtained for model in Fig. 1 assuming that (i) ATP and Na ϩ concentrations are high enough to saturate the formation E 1 ATP, E 1 P, and the occlusion of Na ϩ in E 1 P; (ii) phosphorylation and dephosphorylation are irreversible because of the absence of products; (iii) deocclusion is irreversible because all the experiments were performed in media with high concentrations of Na ϩ (see "Experimental Procedures"); (iv) the binding of Rb ϩ to E 2 P and to E 2 PRb and of ATP to E occ take place in rapid equilibrium.

TABLE I The equations for the steady-state levels of [E occT ] and v t according to the scheme in
order to retain the enzyme-containing membrane fragments. To ensure that the initial temperature in the QWC was 1-2°C and that the flow was constant, about 50 ml of washing solution was allowed to run through the filter prior to the injection of the reaction mixture, and 240 ml of washing solution was applied to the filter from that moment. The composition of the washing solution was 30 mM KCl, 20 mM imidazole-HCl, pH 7.4 at 0°C. The procedure is devised to slow down instantaneously the reactions by means of the sudden drop in both the temperature and the concentration of ligands, like ATP, that might accelerate deocclusion. The quenching time was 3-4 ms, even when the reactions were performed in media with 2.5 mM ATP (17). Furthermore, the washing effectively removes the unbound 86 Rb ϩ . After the washing solution was drained, the filter was removed, dried under an incandescent lamp, and counted for radioactivity in a ␤-scintillation counter. Blanks were estimated from the amount of radioactive Rb ϩ retained by the filters in experiments where occlusion was prevented by omitting ATP during the enzyme reactions. Otherwise the conditions were the same. As shown by Rossi et al. (17), the blanks were linearly related to the Rb ϩ concentration from 0 to 10 mM; they rarely exceeded 10% of the amount of occluded 86 Rb ϩ (and generally were much smaller) and were independent of the mass of enzyme employed.
Determination of Steady-state Levels and the Rate of Release of Occluded Rubidium Ions-The steady-state amount of occluded Rb ϩ was estimated by mixing equal volumes of a suspension of Na ϩ /K ϩ -ATPase in a buffer containing 150 mM NaCl, 0.5 mM free Mg 2ϩ , 25 mM imidazole-HCl, pH 7.4 (at 25°C), with the same buffer containing ATP and 86 Rb ϩ . Generally, the reaction mixture was incubated for 3 s before being injected into the QWC. Control experiments using aging times of 5-10 s showed no variation in the level of occluded rubidium with respect to the results obtained at 3 s.
To measure the rate of release of occluded Rb ϩ , Na ϩ /K ϩ -ATPase containing occluded rubidium ions was formed as described above. It was then mixed with enough of a similar solution, but lacking 86 Rb ϩ to cause a 20-fold isotopic dilution of 86 Rb ϩ . After the appropriate aging times, the resulting mixture was injected into the QWC, and the occluded 86 Rb ϩ remaining in the enzyme was measured as described above. To attain the isotopic dilution of 86 Rb ϩ , 1 volume of 86 Rb ϩcontaining enzyme was mixed either with 20 volumes of a solution with unlabeled Rb ϩ at the same concentration or with 1 volume of a solution with 20 times higher concentration of unlabeled Rb ϩ . Both procedures gave similar results for the rate of deocclusion, provided that the rest of the conditions remained unchanged. Note that 150 mM NaCl was present in all solutions.
Steady-state Solutions for the Model in Fig. 1-These were obtained using the Solve routine of Mathematica ® , version 2.2.1 for Windows ® .
Data Analysis by Nonlinear Regression-Where indicated, results of steady-state ATPase activity and steady-state occluded rubidium were adjusted by nonlinear regression using commercial programs (Excel 5.0 for Windows and Sigma-Plot 2.0 for Windows). Equations were derived from the solutions for [E occT ] and v t in Table I (Equations 9 -11). These are shown in Equations 18 and 19, for ATPase activity. Notice that the mathematical form of these equations corresponds to that of Equations 9 and 10, respectively, after dividing both numerator and denominator by D 1,2 , and rearranging the numerators by lumping E T with rate and equilibrium constants into coefficients a i,j and b i . Additionally, in Equation 18 it is assumed that Rb occ is strictly proportional to [E occT ], i.e. that rubidium is occluded into the enzyme with a fixed stoichiometry (cf. Equation 9).
After inspection of duplicate errors in large samples of data, we used statistical weights 1/v t 2 for v t and 1 for Rb occ . When values of parameters were nonsignificantly different from 0 (i.e. whose mean Ϯ S.E. interval included 0), their corresponding terms were eliminated from the equation, giving rise to equations with less number of parameters. To evaluate the significance of eliminating parameters and choose among different equations, Akaike Information Criterion (AIC ϭ N ϫ ln(S) ϩ 2 ϫ P, with N ϭ number of data, P ϭ number of parameters, and S ϭ sum of weighted square errors of residuals) (22) was applied. The best fitting equation was considered the one that showed the lowest AIC value.

RESULTS
The present paper investigates the role of occluded Rb ϩ (acting as a congener of K ϩ ) in the reaction mechanism of Na ϩ /K ϩ-ATPase. Our fast quenching procedure with a time resolution of a few milliseconds has allowed us to determine quantitatively the concentration of occluded Rb ϩ (Rb occ ) formed via the physiological route, that is by the Rb ϩ -activated dephosphorylation of the phosphoenzyme E 2 P. We have used the procedure to measure the steady-state level of Rb occ and the rate of deocclusion of Rb ϩ at 25°C. We compared these measurements with estimations of steady-state Na ϩ /Rb ϩ -ATPase activity carried out in parallel experiments under exactly the same reaction conditions. To complete the information, we also measured the Na ϩ -ATPase activity in steady state. In all the experiments, [NaCl] was 150 mM, [free Mg 2ϩ ] was 0.5 mM, and the pH was kept at 7.4 with 20 mM imidazole HCl. The concentrations of Rb ϩ and ATP depended on each experiment and are given in the legend of each figure.
The results were analyzed using the scheme in Fig. 1, which shows in more detail that part of the Albers-Post model involved in the formation and breakdown of the intermediates of the Na ϩ /K ϩ -ATPase that are supposed to hold occluded Rb ϩ . As it is clear from the inspection of Fig. 1, the amount of enzyme containing occluded Rb ϩ (E occ and E occ ATP) is a function of the concentrations of Rb ϩ and ATP. The effect of ATP is 2-fold since the nucleotide not only provides the phosphate of the phosphoenzyme intermediates, but it also accelerates deocclusion by binding to E occ . Table I shows the equations generated by the analytical solution of the scheme in Fig. 1, for the particular conditions of our experiments, and the meaning of each of its coefficients in terms of rate and/or equilibrium constants. By using the equations for [E occT ] and v t in Table I, the overall ratio v t /[E occT ] will be as indicated in Equations 20 and 21:  Fig. 3, A and B. Second, the same measurements were focused in the low ATP concentration range (1-10 M, Fig. 4) to cover in more detail this region. These experiments were performed in media with 0.5-20 mM Rb ϩ and allowed us to obtain reliable, extrapolated values for Rb occ at [ATP] ϭ 0 (see below). Third, the high [ATP] concentration range was explored in media with up to 2.5 mM ATP and up to 10 mM Rb ϩ to procure dependable extrapolated values for Rb occ when [ATP] 3 ϱ. The results of these experiments are shown in Fig. 5.
As shown in Fig. 3A for concentrations of ATP above 10 M, Rb occ increased with [Rb ϩ ] tending to saturation along sigmoidal curves. The dotted and the continuous curves are solutions for the scheme in Fig. 1 with the parameters given in Table II, columns 3 and 4, respectively. It can be seen that the experimental data can be quantitatively accounted for by the scheme in Fig. 1 (see Table I, Equations 9 and 11). The difference between the two set of parameters lies mainly in the value of the rate constant k 41 , which is taken as equal to k 40 for drawing the dotted line and larger than k 40 for drawing the continuous lines (the reason for checking both alternatives will be explained below).
When plotted as a function of [ATP] (Fig. 3B), it becomes apparent that Rb occ decreases as [ATP] rises tending to constant positive values (see Fig. 5 for similar but more extended data with another enzyme preparation). The fact that Rb occ does not tend to 0 as [ATP] 3 ϱ indicates that the rate of deocclusion does not increase indefinitely with [ATP] but rather that it reaches a value comparable to that of at least one of the other elementary steps in the reaction sequence.
For concentrations of ATP between 1 and 10 M and of Rb ϩ between 0.5 to 20 mM, Rb occ remained approximately constant at a value that is close to that of maximal occlusion (Fig. 4). This is expected from the properties of the model in Fig. 1 because, under these conditions (see Table II), the rate of occlusion is much higher than the rate of deocclusion. The agreement between the model and the experiments is shown by the good fit to the data points of the dotted and the continuous lines in Fig. 4, which were calculated using the constants in Table II only adjusting the value for Rb occ(max) .
In the media with the concentration of NaCl (150 mM) used in our experiments, no occlusion of Rb ϩ is detectable in the absence of ATP, so that no Rb occ is formed by the direct route described by Glynn Table  II, columns 3 (Table I) and the values for the rate constants given in Table II, columns 3 Fig. 1 using Equations 9 and 11 and the values for the rate constants given in Table II, column 3 or 4, respectively. All the equilibrium constants and the maximal amount of Rb occ , Rb occ(max) , were fitted by nonlinear regression. 4, and 5 were obtained by simulations of the reaction scheme in Fig. 1. It can be seen that the curves give an adequate description of the experimental results, indicating that models like those in Fig. 1 are able to predict the response of steady-state Rb occ to a wide range of ATP and Rb ϩ concentrations. The simulation in Fig. 3 was obtained assuming a fixed stoichiometry of occlusion. The numerical value of this stoichiometry was determined independently by measuring the ratio between the maximum amount of occluded rubidium (Rb occ(max) ) and the total amount of enzyme (E T ). Rb occ(max) was obtained by extrapolation to [ATP] ϭ 0 and [Rb ϩ ] 3 ϱ of the best fitting values obtained by nonlinear regression of the data in Fig. 3. An independent estimate of E T was obtained by measuring the maximum level of phosphorylation (EP T ) in the absence of Rb ϩ (see "Experimental Procedures"). In this situation, EP T Х E T because the rate constant for phosphorylation (k 2 in Fig. 1) is much larger than the rate constant for dephosphorylation (k 40 in Fig. 1). For the preparation used in Fig. 3, Rb occ(max) was 5.22-5.44 nmol mg Ϫ1 (Table II) and EP T was 2.7 nmol mg Ϫ1 , thus the ratio Rb occ(max) /E T was not significantly different from 2, indicating that two Rb ϩ are occluded in each enzyme molecule. This finding agrees with those of other authors (for references see 7, 11, and 23) as well as with previous results obtained by us in transient kinetic experiments using the K ϩ congener Tl ϩ (9). It also accounts for the sigmoidal shape of the Rb occ versus [Rb ϩ ] curves in Fig. 3A. For these reasons in the rest of this paper, [E occT ] was taken to be equal to Rb occ /2. E occ and E occ ATP as Obligatory Intermediates of the Na ϩ / K ϩ -ATPase Reaction-As described in detail in the Introduction (Equations 3, 6, and 8), it is possible to test if E occT is an obligatory intermediate in the Na ϩ /K ϩ -ATPase reaction by comparing the ratio v t /[E occT ] with the directly measured rate coefficient for deocclusion, k deocc , when all the parameters are measured under identical conditions. In the previous section we have described our data on Rb occ (and hence of E occT ). In this section we will present the results of experiments in which we measured v t and k deocc for enzyme preparations suspended in media containing from 1 to 2500 M ATP and from 0.25 to 20 mM Rb ϩ . We will describe first the results at "low" (Յ10 M) and then those at "high" (Ն100 M) ATP concentrations.
The results of measurements of steady-state Na ϩ /Rb ϩ -ATPase activity in media containing from 1 to 10 M [ATP], from 0.5 to 20 mM [Rb ϩ ], and 150 mM Na ϩ are given in Fig. 6. For this ATP concentration range the high affinity site for ATP will be saturated, and [ATP] will be much smaller than the apparent dissociation constant at the low affinity site (K ATP in Fig. 1 and Table II). Under these conditions it is easy to show that the plots of activity versus [ATP] will be linear functions of [ATP] at all the Rb ϩ concentrations tested (see also Equations 10 and 11 in Table I). Likewise, as the value of k 0 is smaller than the values of k 40 and k 41 , the model in Fig. 1 predicts that the ordinate intercepts corresponding to [ATP] ϭ 0 should decrease toward a lower value at sufficiently high [Rb ϩ ]. This is what we observe.
The data in Fig. 6A are plotted as a function of [Rb ϩ ] in Fig.  6B. It is apparent that for all ATP concentrations tested the activity decreased until it reached a constant value. Hence at this concentration range of ATP, Rb ϩ acted as a partial inhibitor of the ATPase. Inhibitory effects of the ATPase by K ϩ and its congeners at micromolar ATP have been found by many authors since they were discovered in 1969 by Neufeld and Levy (24). That this inhibition is partial has been shown previously by Rossi and Garrahan (25).
Direct measurements of the rate coefficient of deocclusion, k deocc , were performed from determinations of the amount of 86 Rb ϩ that remained occluded in the Na ϩ /K ϩ -ATPase during various lengths of time following a 20-fold isotopic dilution of the 86 Rb ϩ in the incubation media (see "Experimental Procedures"). Results of experiments of this kind are given in Fig. 7 as semilog plots of the fraction of 86 Rb ϩ remaining occluded versus incubation time in media with 0.5 M ATP and either 0.5 or 10 mM unlabeled Rb ϩ . It can be seen that Rb occ decayed along a single exponential function of time (Equation 24 in legend to Fig. 7) from which k deocc can be estimated. Fig. 7 also shows that the rate of deocclusion was not affected by the concentration of Rb ϩ in the deocclusion media.
The values for k deocc , measured by means of experiments like those in Fig. 7  ] Ͻ 10 mM is strong evidence for the existence of routes for ATP hydrolysis that do not pass through E occT . As it will be shown in the following section, this feature is also apparent at high ATP concentrations.
We will turn now to the experiments performed at higher ATP concentrations. In these, the ATPase activity, v t , and the  (Fig. 1), and maximal amounts of occluded rubidium used in Table I (Fig. 4)  The data for the rate of hydrolysis presented in Fig. 9 show the well known fact that Rb ϩ is a potent activator of the ATPase when [ATP] Ն 100 M. Activation is predicted by the model in Fig. 1 since, in the presence of enough ATP, acceleration by Rb ϩ of dephosphorylation is not limited by deocclusion, the next step in the cycle. The data were analyzed by nonlinear regression using Equation 19 under "Experimental Procedures," which predicts that the curves are hyperbolas with a positive intercept at the y axis. As shown in Fig. 9, there was a satisfactory correspondence between the curves and the data. The values of the coefficients corresponding to the denominator of Equation 19 are listed in Table III together with those obtained by a similar, independent analysis of the data for Rb occ in Fig. 5 using Equation 18. The comparison between the values of the two sets of coefficients allows us to test if the experimental results satisfy the theoretical prediction that both v t and [E occT ] share the same denominator (see Equations 9 and 10 in Table I). The fact that both sets of values (and thus the denominators) are (approximately) the same satisfies the requirement for the two steady-state quantities as expressions of the functioning of the Na ϩ /K ϩ -ATPase and justifies us to

86
Rb ϩ remaining occluded after a 20-fold dilution of the label is plotted as a function of time after dilution. Since dilution leaves 5% of the initial specific activity of the media, 86 Rb occ,ϱ ϭ 86 Rb occ,0 /20 was subtracted from each measurement. The concentration of ATP and Rb ϩ in the media in which occlusion took place was 10 M and 0.5 mM, respectively. 86 Rb ϩ release was followed in media with 0.5 M ATP, 0.5 mM Mg 2ϩ , 150 mM Na ϩ , and either 0.5 (E) or 10 mM (q) Rb ϩ . Equation 24 ͑ 86 Rb occ,t Ϫ 86 Rb occ,ϱ ͒/͑ 86 Rb occ,0 Ϫ 86 Rb occ,ϱ ͒ ϭ e Ϫkdeocct t (Eq. 24) was adjusted to the data to obtain the best fitting values of k deocc which were 0.201 Ϯ 0.007 s Ϫ1 (E) and 0.185 Ϯ 0.009 s Ϫ1 (q).  Table II, columns 3  calculate the ratio between v t and [E occT ], where the denominator cancels out (see Table I, Equations 9 -11, and below). Fig. 10 shows the results of experiments in which the rate coefficient for deocclusion, k deocc , was measured in media with the same ATP concentrations as in the experiments in Figs. 5 and 9 using the same procedure as that employed in the experiments with low ATP concentrations (Fig. 7). In additional experiments using 0.5 mM ATP, both during the occlusion and the deocclusion phases (not shown), we observed that k deocc did not change if Rb ϩ concentration was varied from 0.5 to 25 mM in the media used for occlusion or for deocclusion or in both. This extends to high [ATP] the observations at low [ATP] about the absence of effect of Rb ϩ on the rate of deocclusion. In the experiment in Fig. 10, like that in Fig. 7, the loss of 86 Rb ϩ followed single exponential functions of time, with rate coefficients that increased with the concentration of the nucleotide as expected from the model in Fig. 1.
In Fig. 11A, the ratio v t /[E occT ] calculated from the data in Figs. 9 and 5 and the directly measured value of k deocc obtained from Fig. 10 are plotted together against the concentration of ATP. Two features of the plots seem to be worth mentioning. First, the v t /[E occT ] versus [ATP] curves at low [Rb ϩ ] are positioned above those at higher [Rb ϩ ], and as [Rb ϩ ] increases, the curves become superimposable. This is comparable to what happens at micromolar [ATP] (Fig. 7) except that v t /[E occT ] in Fig. 11A seems to be higher than k deocc even at saturating [Rb ϩ ]. Second, v t /[E occT ] for any given [Rb ϩ ] can be fitted by a rectangular hyperbola that starts from a small positive value at [ATP] ϭ 0, which corresponds to that obtainable from Fig. 8A. This is in accordance with the principles outlined in Equation  6, see Introduction. The maximal value of the hyperbola ((v t / [E occT ]) [ATP]3ϱ ), decreases with increasing [Rb ϩ ] as expected from Equations 20 and 21. However, a feature of the hyperbolas that fit the data of v t /[E occT ] versus [ATP] that is not predicted by the model in Fig. 1 is that their values of K 0.5 decrease with increasing [Rb ϩ ] (inset in Fig. 11A). Fig. 11A also shows that the deocclusion rate coefficient, k deocc , also follows a hyperbola (see legend to   (18) and (19) after independent, non-linear regression analysis of the data in Figs. 5 and 9, respectively We followed the procedure detailed under "Experimental Procedures." Both for Rb occ and v t the values of d 0,0 and d 0,1 were nonsignificantly different from zero (NS).

Kinetic coefficient
Units Rb occ (ϮS.E.) (Fig. 5,  Equation 18) v t (ϮS.E.) (Fig. 9 Equation 22 gives the sum of the Na ϩ -ATPase activity and the activity when only one Rb ϩ is bound to E 2 P. From the measured activity of the Na ϩ -ATPase activity, we have estimated k 40 Х 2.5 s Ϫ1 . In our initial simulations on the Rb occ data (dotted curves in Figs. [3][4][5], we took arbitrarily k 41 ϭ k 40 (column 3, ) and E T to be adjusted by the regression algorithm resulted in a new set of parameters that is listed in the 4th column of Table II. These were used to draw the full curves in Figs. 3-5. Hence as it will be analyzed in more detail under "Discussion," a possible explanation for the data in Figs. 8B and 11B is that E 2 PRb dephosphorylates with a rate constant Х50 s Ϫ1 that is between that for E 2 P (Х2.5 s Ϫ1 ) and E 2 PRb 2 (Х250 s Ϫ1 ) without leading to the formation or Rb occ . DISCUSSION We have studied, during steady-state turnover of the enzyme, the dependence on [ATP] and on [Rb ϩ ] of the amount of Rb ϩ occluded in the Na ϩ /K ϩ -ATPase as well as the rate coefficient for release of Rb ϩ from this state. In all cases Rb occ was formed via the physiological route. We were thus able to get a previously unavailable, comprehensive set of measurements of Rb occ and k deocc over a very wide range of ATP and Rb ϩ concentrations.
We also performed parallel measurements of ATPase activity as a function of both [Rb ϩ ] and [ATP] using the same enzyme preparation and in media with the same composition and temperature as those used to measure Rb occ .
Direct measurements of steady-state Rb occ formed by the physiological route have been obtained by other workers who used either the passage of the incubation media through a cation exchange resin (4) or rapid filtration procedures (6). In neither case, however, were these studies performed in the range of concentrations of Rb ϩ and ATP or with the accuracy required for a quantitative treatment as that presented in this paper. Moreover, neither of these techniques are able to isolate Rb occ in the millisecond time scale. In a few cases, Forbush (6) employed the filtration technique to measure occluded Rb ϩ in enzyme retained by filters, but in most of his experiments filtration was used to follow the time course of release of 86 Rb into the filtrate. By using media in which both [ATP] and [Rb ϩ ] were saturating, Forbush (6) concluded that E occT is an obligatory intermediate of the overall ATPase reaction. Glynn and Richards (4) reported a sigmoidal response of Rb occ to [Rb ϩ ] in media with less than 10 M [ATP] and showed that, as [ATP] increased, Rb occ first increased and then fell. Although these results agree with those shown in this paper, a quantitative comparison is not possible because the method employed by Glynn and Richards (4) does not allow to accurately control the actual concentration of the nucleotide in the vicinity of the enzyme.
The Kinetics of Rb ϩ Occlusion-Comparison between simulations and the experimental data indicated that the response of the steady-state level of Rb occ to [ATP] and [Rb ϩ ] agreed with the predictions of reaction schemes like that shown in Fig. 1 in the following: (i) the stoichiometry of occlusion is constant; (ii) Rb ϩ binds to a conformer of the enzyme that is different from that which binds ATP; (iii) the release of occluded Rb ϩ is strongly accelerated by ATP binding at the conformer that occludes the cation; and (iv) the numerical values of the rate and equilibrium constants required to fit the model to the experimental data are close to those usually considered to govern the elementary steps of this model (for references, see Ref. 21).
The model also predicts adequately most of the properties of   Fig. 1.
The independence of k deocc with the concentration of Rb ϩ is consistent with results obtained by Glynn and Richards (4) and Forbush (6) who investigated the effects of the addition of various cations in the absence of ATP. Forbush (6) showed that in media with 4 mM ATP (but no Mg 2ϩ ), there is a saturable increase of k deocc with the concentration of either K ϩ or Na ϩ . Since the maximum value for k deocc for both cations is the same and is reached at about 150 mM Na ϩ or K ϩ , the phenomenon would remain unobservable under our experimental conditions. E occT as an Obligatory Intermediate for the Hydrolysis of ATP by Na ϩ /K ϩ -ATPase-Since the properties of the scheme in Fig.  1 that are satisfied by the experimental results on Rb occ ϭ ƒ([ATP], [Rb ϩ ]) are those of the Albers-Post model of the Na ϩ / K ϩ -ATPase (here Na ϩ /Rb ϩ -ATPase), our results suggest that this model provides a satisfactory description of the steadystate kinetics of Rb ϩ occlusion. This, however, is not sufficient to prove that under all conditions E occT is an obligatory intermediate of the Na ϩ /Rb ϩ -ATPase reaction. To analyze this, we compared the response to ATP and Rb ϩ of the ratio v t /[E occT ] with that of k deocc . At this point, it seems important to notice that the term E T k 2 k 3 of the numerator and the denominators of the steady-state equations for v t and [E occT ] are canceled out when v t is divided into [E occT ] (cf. Equations 9 and 10 in Table  I). The cancellation of coefficients determines that v t /[E occT ] will only depend on the rate and equilibrium constants that participate directly in the turnover of the occluded species, being totally unaffected by changes in any of the other elementary steps that participate in the reaction of ATP hydrolysis. The theoretical prediction of the cancellation of the denominator was actually fulfilled by our experiments, since the values of the denominators of the functions that best fitted v t and [E occT ] were not significantly different (Table III), thus validating the use the v t to [E occT ] ratio in our results.
We have shown that for E occT to be an obligatory intermediate of the Na ϩ /Rb ϩ -ATPase reaction, the ratio v t /[E occT ] must be equal to k deocc . In the particular case of our enzyme, in which the breakdown of E occT takes place through two different pathways controlled by the binding of ATP to a single class of sites in E occ , two additional predictions can be made as follows: (i) at any constant [Rb ϩ ], v t /[E occT ] will increase with [ATP] along rectangular hyperbolas whose K 0.5 values are independent of [Rb ϩ ]; and (ii) the values of v t /[E occT ] at 0 and at infinity [ATP] will be decreasing functions of [Rb ϩ ] that will tend to k 0 and to k ϱ , respectively, as the increase in [Rb ϩ ] shifts ATP hydrolysis away from Na ϩ -ATPase toward Na ϩ /Rb ϩ -ATPase activity (see Fig. 1).
Our results showed that, in media in which [Rb ϩ ] was sufficiently high as to make negligible the contribution of the Na ϩ -ATPase to the overall activity, v t /[E occT ] was much larger than k deocc and only approached or reached k deocc at Rb ϩ concentrations above 10 mM. At 1-10 M ATP, there was a complete convergence between v t /[E occT ] and k deocc . At nonlimiting [ATP], the convergence annulled about 90% of the difference between both parameters so that, albeit much reduced, v t / [E occT ] remained larger than k deocc at nonlimiting [Rb ϩ ]. It seems premature to consider this residual difference as the sign of a genuine phenomenon, since small systematic errors biasing the estimation of v t , [E occT ], and/or k deocc cannot yet be discarded. We have explored two possible sources of errors: one is the underestimation of [E occT ] (which would artificially increase v t /[E occT ]) and the other is an underestimation of k deocc . Deviations in any of these quantities should increase with the concentration of ATP. Control experiments (17) demonstrated that the loss of Rb ϩ from the occluded state using our filtration technique will give at most a 10% loss at the highest (2500 M) [ATP] tested. It is more difficult to conceive why the measurements of k deocc at high [ATP] should lead to underestimation of the real value, at least on the basis of a loss of occluded rubidium during the washing of the enzyme on the filter.
As already mentioned, we ruled out the possibility of a residual Na ϩ -ATPase activity as the only cause for the discrepancies between v t /[E occT ] and k deocc . This was based on the following quantitative evidence: (i) in the experiments performed at 1-10 M ATP, E occT had reached saturation with respect to [Rb ϩ ] at all [ATP] and [Rb ϩ ] tested, so that Na ϩ -ATPase activity should be practically zero; (ii) in the experiments performed at 100 -2500 M [ATP], the discrepancy between v t /[E occT ] and k deocc persisted after the maximal value of Na ϩ -ATPase activity was subtracted from each of the values of v t before calculating v t /[E occT ].
An additional, more qualitative, discrepancy between our experiments and the prediction of the model in Fig. 1 is that, although our results are in agreement with this model in the sense that, for all [Rb ϩ ], v t /[E occT ] ϭ ƒ([ATP]) were rectangular hyperbolas, the experimentally obtained values for K 0.5 decreased with [Rb ϩ ] instead of remaining constant, as should happen if no steps other than those shown in Fig. 1 governed the effect of ATP on k deocc . As before, we cannot yet discard that small deviations in the measurement of [E occT ] (now dependent on both [ATP] and [Rb ϩ ]) could explain this unexpected effect of Rb ϩ .
Before analyzing if the model in Fig. 1 has to be modified or replaced to account for discrepancies between theory and experiment, it seems reasonable to see if these are caused by the restrictive assumptions of fixed stoichiometry of occlusion and irreversibility of deocclusion which we used to obtain numerical simulations of our model. Let us, for the sake of the argument, disregard the abundant experimental evidence for fixed stoichiometry of occlusion and assume that the Na ϩ /Rb ϩ -ATPase activity also occurs through enzymes that occlude only one Rb ϩ . In this case E occT would vary between Rb occ and Rb occ /2, and if the species occluding one Rb ϩ had the same turnover than those occluding two Rb ϩ , v t /[E occT ] would be at most twice as large as k deocc at sufficiently low [Rb ϩ ]. Since this is too small to account for the effects we observed in our experiments, the variable stoichiometry hypothesis would require the additional postulate that enzymes occluding one Rb ϩ have a higher turnover than those occluding two Rb ϩ . An effect of this kind cannot be postulated, since it would have been detected as a decrease in the values of k deocc as high [Rb ϩ ] drove the putative state holding one occluded Rb ϩ to the state holding two occluded Rb ϩ .
An additional assumption we used to solve the model in Fig.  1 is that the reactions of deocclusion (E occ 3 E 1 and E occ ATP 3 E 1 ATP) are irreversible. If this were not the case, an increase in [Rb ϩ ] would decrease the net rate of hydrolysis (v t ) as the inverse reaction sets in, with the consequent decrease in v t / [E occT ]. In the context of the model in Fig. 1, this kind of effect on v t would always yield values of v t /[E occT ] smaller than those of k deocc . This is so because if deocclusion of Rb ϩ were reversible its net rate, and hence the overall Na ϩ /K ϩ -ATPase activity,