INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION NOTES APPENDIX REFERENCES

Competitive inhibitors with specificity for a target protein are useful probes of the kinetic mechanisms of enzyme-catalyzed reactions and membrane transport processes. The general approach to the study of such protein-inhibitor interactions involves the analysis of the decrease in reaction velocity that occurs in the presence of inhibitor as compared to inhibitor-free control conditions. Alternatively, a more direct approach relies on the interpretation of the kinetics of inhibitor binding, usually measured using radiolabeled inhibitors in the presence of some or all of the physiological effectors. In practice, the general approach has mostly been restricted to enzyme reactions for which linear rates of product formation can be measured on a time scale of seconds to minutes in the absence of inhibitor (Morrison and Walsh, 1988). By contrast, the direct approach has been widely applied to a number of transport systems on the rationale that the kinetic equations accounting for inhibitor binding are usually simpler and easier to test than those derived for substrate uptake because fewer carrier forms and translocation steps need to be considered (Turner and Silverman, 1980).

The current concepts describing the reversible interactions between a competitive inhibitor (I) and its target protein have been established within the context of steady-state kinetics of unireactant enzymes (E). These theoretical studies involved the analysis of an elementary reaction scheme in which the formation of the enzyme-substrate complex (ES) is rapid (rapid equilibrium assumption) whereas the overall rate of catalysis is limited by the breakdown of this complex to form product (Morrison and Walsh, 1988). Accordingly, the reaction rate is linear over the time period during which the initial rate assumptions are satisfied (see Note 1). Competitive inhibitors for S are usually referred to as classical or slow-binding inhibitors, defined where the rates of association with and dissociation from E are fast or slow, respectively, relative to the reaction velocity (Morrison and Walsh, 1988). In the presence of classical inhibitors, a steady-state concentration of EI is also rapidly established and the initial rate remains linear, hence their apparent fast-acting behavior. In contrast, the reaction rates in the presence of slow-binding inhibitors demonstrate an initial burst of high reaction rate followed by a slow decrease to a lower steady-state level. Two kinetic mechanisms have been proposed that may account for slow-binding inhibition (Morrison and Walsh, 1988). In mechanism A, the inhibitor binding step itself represents the overall rate-limiting step in the interaction because of barriers that the inhibitor encounters when binding at the active site of the enzyme. For mechanism B, it is assumed that inhibitor binding involves the rapid formation of an initial collision complex, but is followed by a slow isomerization reaction. The burst kinetics described above thus reflect the slow establishment of either the equilibrium between E, I, and EI (mechanism A), or between the two enzyme-inhibitor complexes (mechanism B).

It should be emphasized that the distinction between fast-acting and slow-binding inhibitors is in no way the result of these mechanistic considerations (see Note 2) but simply rests on the assumption that the two classes of inhibitors can be identified through the application of steady-state kinetic methods. Therefore, an obvious limitation to the steady-state formalism is that these methods may fail to detect the occurrence of slow-binding inhibition if the time required to achieve equilibrium binding is much larger than the time period over which the initial rate assumptions are satisfied (see Note 3). Such a limitation is particularly relevant to membrane transport processes for which it may prove difficult to record linear initial rates of transport using isolated membrane preparations or cells for time periods in excess of a few seconds (Berteloot and Semenza, 1990; Chenu and Berteloot, 1993; Kimmich, 1990). This problem is best illustrated by phlorizin, a reversible and highly selective competitive inhibitor of the sodium-coupled glucose transport (SGLT) systems of intestinal and renal tissues, that is not transported by the SGLT proteins (Diedrich, 1966; Aronson, 1978; Semenza et al., 1984; Kimmich, 1990; Wright, 1993). Using a fast-sampling rapid-filtration apparatus (Berteloot et al., 1991) and rabbit renal brush-border membrane vesicles, our recent studies demonstrated that the initial period during which glucose transport rates are constant occurs on a time scale of 0-9 s in the presence or absence of phlorizin (Oulianova and Berteloot, 1996). According to the steady-state formalism, these results should suffice to classify phlorizin as a fast-acting (classical) inhibitor, in which case, one would expect to observe constant binding of labeled inhibitor over a time period during which steady-state reaction rates are recorded (see Note 4). This prediction clearly conflicts with the slow rates of phlorizin binding usually reported using similar preparations, where up to 5 min incubation with radiolabeled phlorizin may be required to reach equilibrium (Glossmann and Neville, 1972; Chesney et al., 1974; Aronson, 1978; Turner and Silverman, 1981; Koepsell et al., 1990). This situation will be referred to as the fast-acting slow-binding paradigm in the present paper.

The current hypothesis for the slow binding of phlorizin is the recruitment concept (referred to as mechanism C in this text), whereby the inhibitor is titrating a carrier conformation in which a substrate binding site becomes accessible to the inhibitor molecules (Aronson, 1978). More precisely, it is proposed that the sugar (phlorizin) binding site on the carrier has a predominant inward orientation, which is therefore shielded from access. A slow translocation (conformational change) to an outward-facing configuration would then explain the slow kinetics of phlorizin binding to this newly available site. To our knowledge, no theoretical justification has ever been provided to support the validity of the recruitment concept, which rests primarily on circumstantial pieces of evidence (Aronson, 1978; Toggenburger et al., 1978, 1982; Turner and Silverman, 1981; Restrepo and Kimmich, 1986). Moreover, the kinetics of phlorizin binding have never been considered with regard to the applicability of mechanisms A and B discussed above. In this respect, it should be noted that the early studies of Turner and Silverman (1980) were only concerned with the kinetics of inhibitor binding at equilibrium. Because all of mechanisms A-C above predict Scatchard-like kinetics relative to inhibitor concentrations under these conditions, this approach does not provide the information that allows us to assess which of these models might explain the fast-acting slow-binding paradigm.

The present studies are aimed at deriving kinetic equations that may best characterize inhibitor binding conforming to each of mechanisms A-C. The proposed theoretical approach takes on a quite general significance and involves models of arbitrary dimension on the rationale that realistic transport and enzyme mechanisms are usually more complex than the elementary reaction schemes used by Morrison and Walsh (1988) to describe the steady-state approach. Moreover, because the limitations associated with the steady-state formalism to assess inhibitor binding mechanisms apply to both transport processes (see Note 3) and enzyme reactions (Morrison and Walsh, 1988), the question of protein-inhibitor interactions is addressed with regard to the direct measurement of inhibitor binding to a protein. We therefore assume throughout this paper: 1) that an adequate assay has been found to measure, in a time-dependent way, the fraction of inhibitor bound to the relevant protein-inhibitor complexes, 2) that the progress of inhibitor binding can be satisfactorily described at the experimental level by a monoexponential function, and 3) that the binding data to be analyzed have been adequately corrected for nonspecific binding. Using this formalism, it is demonstrated that each of mechanisms A-C can be readily identified according to the position of the inhibitor binding step relative to the rate-limiting step in the inhibitor binding sequence. This key structural feature is the main determinant of unique kinetic properties that should allow anyone to determine unambiguously whether the inhibitor binding step either (A) represents, (B) precedes, or (C) follows the rate-limiting step. It is further shown that the relevance of mechanism C to slow-binding inhibition is conditioned by the turnover rate of the catalytic cycle under symmetrical conditions, and that mechanism B represents the only alternative to explain the fast-acting slow-binding paradigm of phlorizin.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION NOTES APPENDIX REFERENCES

Time scale separation hypothesis and basic model

To reduce the complexity of the mathematics involved in deriving the characteristic equations for models of arbitrary dimension, it is appropriate to introduce approximations based on the principle that some reaction steps are faster than others, so that the time course of inhibitor binding to be observed will be governed by the slowest steps (Wierzbicki et al., 1990). In the following, we limit our analyses to the case where the progress of inhibitor binding can be satisfactorily described at the experimental level by a monoexponential function. This simplifying assumption allows us to compare our results with those of the steady-state approach pioneered by Morrison and Walsh (1988).

The kinetic mechanism shown in Fig. 1 A represents the simplest scheme that satisfies the above requirements, and the question of its relevance to more realistic kinetic mechanisms will be addressed in the Discussion. This model assumes that the binding of an arbitrary number of A_i effector molecules to a protein N can be described by a linear array of elementary reactions among which a slow isomerization (conformational change) with rate constants k_on and k_off represents the only rate-limiting step within the reaction sequence. This hypothesis is equivalent to stating that all the rate constants governing the association and dissociation of the A_i effectors with the protein are fast compared to k_on and k_off. Therefore, it is possible to define two blocks of elementary reactions called X and Y, in which all the chemical species can be considered to be in equilibrium with each other (rapid equilibrium assumption) both before and during the time-dependent slow interconversion between blocks X and Y (Cha, 1968; Wierzbicki et al., 1990). This, in turn, allows us to introduce the dissociation constants K_i^X and K_i^Y to describe each of the A_i binding steps within blocks X and Y, respectively. Note that the inhibitor binding step has not been included at this stage of the analysis to initially establish the most general solution describing the time-dependent interconversion (relaxation) between blocks X and Y, and to avoid redundancy in the derivation of more specific solutions applying to the different mechanisms of inhibitor binding, which will be considered later.

View larger version (9K): [in this window] [in a new window]   FIGURE 1   Basic model used for detailed kinetic analyses. (A) The structure of the basic model is composed of a linear array of elementary reactions, only one of which may be considered to be rate-limiting for the relaxation process with association and dissociation rate constants kon and koff, respectively. All the other reactions thus satisfy the rapid equilibrium assumption, which allows us to define the two blocks of elementary reactions X and Y. The Ni species may represent any protein under its free form (N1) or following complexation with Ai effectors, each complex formation being described by a dissociation constant Ki. (B) Reduced basic model according to Cha's formalism (Cha, 1968) in which the apparent rate constants k*on and k*off defined in Eqs. 1 and 2 in the text now replace the true rate constants kon and koff.

Time-dependent solution of the basic model

In the following, it is postulated that the concentrations of all of the A_i effectors present in the incubation media are constant over time and represent their total concentrations. For simplicity at this point, one may assume that none of the A_i effectors in the X and Y blocks is a substrate involved in a catalytic process, a hypothesis that will be relaxed in the Discussion.

According to Cha's rules (Cha, 1968), the general scheme depicted in Fig. 1 A can be reduced to its equivalent form shown in Fig. 1 B where the apparent rate constants k^*_on and k^*_off are defined as

(1)

(2)

and now replace the true rate constants k_on and k_off. The latter are weighted by the factors f_p^X and f₁^Y, whose mathematical expressions,

(3)

(4)

clearly indicate that they represent the fractional concentrations of the chemical species N_p^X and N₁^Y within blocks X and Y, respectively. The denominators of Eqs. 3 and 4 can be expressed relative to the N_p^X and N₁^Y species as

(5)

(6)

where algebraic expressions of the quantities L^X and L^Y

(7)

(8)

are conditioned by the rapid equilibrium assumption. Note that the two blocks, X and Y, are also linked through the conservation equation,

(9)

The time-dependent interconversion between blocks X and Y can be described by the differential equation

(10)

in which the right-hand-side expression results from the consideration of Eqs. 3-6 and can be transformed as

(11)

by incorporating Eq. 9. Equation 11 can be further rearranged as

(12)

with the definitions for k_obs (apparent first-order rate of the relaxation process)

(13)

and Y_e (equilibrium concentration of Y at the end of the relaxation process),

(14)

The integration of Eq. 12 over time is straightforward and leads to equivalent monoexponential functions

(15)

in which Y₀ represents the zero-time concentration of Y at the start of the relaxation. Note that the evaluation of Y₀ shall depend on the specific kinetic mechanisms to be considered later. A last quantity that may be measured experimentally is the initial rate of interconversion (Y_i), the mathematical expression of which can be readily obtained from the first derivative of Eq. 15 at t = 0, as

(16)

Note that Eq. 10 could have been made homogeneous relative to X to describe the same phenomenon with the differential equation

(17)

as a result, the integration of which leads to

(18)

where the quantity X₀ (zero-time concentration of X) should reflect the boundary conditions applying to each specific mechanism while X_e (equilibrium concentration of X) represents the solution of Eq. 17 when dX/dt = 0,

(19)

Inhibitor-binding mechanisms and further assumptions

If inhibitor binding is now considered in the kinetic schemes shown in Fig. 1, it is readily apparent that only three possibilities may be viewed whereby the inhibitor binding step either represents (mechanism A), precedes (mechanism B) or follows (mechanism C) the rate-limiting step. As depicted in Fig. 2, these three mechanisms are associated with the assumptions 1) there is only one inhibitor binding site, so that inhibitor binding at equilibrium should conform to Scatchard kinetics relative to the free concentration of inhibitor [I], 2) the inhibitor binding site is accessible through only one of the N_i species, and 3) the total concentration of inhibitor (I_T) far exceeds the total concentration of binding sites (N_T), so that [I]  I_T and [I] can be considered constant during the time interval over which the binding assay is performed. Note that the relaxation of assumptions 1 and 2 will be considered in the Discussion whereas assumption 3 excludes from consideration the so-called tight-binding inhibitors for which there is not a single rate equation for the time course, but rather a pair of parametric equations that describe the progress curves at different inhibitor concentrations (Sculley et al., 1996). Also, with the current hypothesis that none of the A_i effectors is a substrate involved in a catalytic process, mechanisms A-C in Fig. 2 are quite general and may describe the kinetics of inhibitor (or any other molecule) binding to any protein independent of the time scale of the observation, which is conditioned by the nature of the binding assay and limited by the absolute values of k^*_on and k^*_off. The question of the applicability of mechanisms A-C to fast-acting and slow-binding inhibitions is addressed in the Discussion.

View larger version (18K): [in this window] [in a new window]   FIGURE 2   The three kinetic mechanisms that may account for slow inhibitor binding. (A) Extension to a dimensionless model of previous mechanism A (Morrison and Walsh, 1988), in which inhibitor binding may itself represent the rate-limiting step. (B) Extension to a dimensionless model of previous mechanism B (Morrison and Walsh, 1988), in which fast inhibitor binding with dissociation constant Kd is followed by a slow isomerization step. Before inhibitor addition, all the Ni species are constrained within subblock X1, so that X00 = X1 = NT. Once added, the inhibitor promotes fast redistribution of the NiX1 species between subblocks X1 and X2, forming the new X block, so that X0 = NT. The relaxation process between blocks X and Y follows thereafter. (C) Application to a dimensionless model of the recruitment hypothesis (Aronson, 1978; Turner and Silverman, 1980), in which a slow conformational change controls the rate of inhibitor binding with dissociation constant Kd. Before inhibitor binding, there is a true equilibrium between block X and subblock Y1. Once added, the inhibitor promotes fast redistribution of the NiY1 species between subblocks Y1 and Y2, forming the new Y block, and the relaxation process between blocks X and Y follows thereafter. When not given in details in mechanisms A-C, the structure of the elementary reactions occurring in blocks X and Y is identical to that shown in Fig. 1 A. The apparent rate constants k*on and k*off are defined by Eqs. 1 and 2 in the text except for mechanism A, in which Eq. 22 should be substituted for Eq. 1.

The formalism previously discussed and justified by Wierzbicki et al. (1990) for presteady-state kinetics can now be used to identify time-dependent solutions of inhibitor binding provided that the following assumptions, with regard to the conditions of the binding assays, are introduced. We first assume that the relevant preparations to be tested for inhibitor binding have been sufficiently resuspended in the uptake media to ensure that true equilibrium conditions have been reached with each of the A_i effector molecules at the time of the assay, so that it is possible to calculate the boundary conditions (X₀₀ and Y₀₀) prevailing before the start of the binding assay. We next assume that the binding assay is started by mixing the above preparations in identical media containing the inhibitor to be tested, so that there will be fast redistribution only of those N_i^X or N_i^Y species involved in rapid equilibrium reactions with the inhibitor. Accordingly, it is possible to calculate the boundary conditions (X₀ and Y₀) prevailing at the very start of the relaxation process when t = 0. Finally, because, in mechanisms B and C, the rates of association and dissociation of the inhibitor are fast compared to k^*_on and k^*_off, one may also need to consider that some binding assays, including the rapid filtration technique, may fail to detect most (if not all) of the inhibitor molecules bound to the N_i^X or N_i^Y species in each of these models, respectively.

To avoid redundancy in the writing of both similar equations and the definitions of similar parameters, reference is made throughout the text to the Scatchard or Michaelis-Menten equation

(20)

in which B represents the amount of inhibitor molecules bound to its specific site on the protein, (I) stands for the free concentration of inhibitor (but I  I_T, see above)), and B_max (apparent maximum number of binding sites) and K_d (apparent dissociation constant for inhibitor binding) are the usual kinetic parameters of interest to be determined. The indices 0, i, and e associated with these parameters consistently represent the kinetic situation that prevails, respectively, at the very start of the binding assay when t = 0, over the time period during which it can be assumed that true initial rates of binding can be measured, and at equilibrium when a steady-state plateau value has been reached. When necessary, the kinetic parameter

(21)

is introduced, which represents the dissociation constant characterizing the rate limiting step. Note that K_XY = K_d (intrinsic dissociation constant for inhibitor binding) in mechanism A only.

All calculations involving complex algebraic expressions have been performed using Mathematica software for Windows. For the more specific cases discussed in the Appendix, the initial velocity equations and the relevant distribution equations are derived using computer calculations described elsewhere (Falk et al., 1998).

	RESULTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION NOTES APPENDIX REFERENCES

Mechanism A: Inhibitor binding is the rate-limiting step

To solve mechanism A, the basic model depicted in Fig. 1 A only needs to be modified to include inhibitor binding to the N_p^X species as shown, under reduced form, in Fig. 2 A. Accordingly, Eqs. 2-9 still apply to this model while a correct expression for k^*_on should now read as follows.

(22)

Moreover, because all the N_i species in block Y contribute to the binding equation, the time course of inhibitor binding will be described by Eq. 15 in which B is substituted for Y, and k_obs is given by Eq. 13. The boundary conditions

(23)

are easily established from our assumptions regarding the binding assay conditions (see the section entitled "Inhibitor-binding mechanisms and further assumptions" under Materials and Methods). The quantities B_e = Y_e and B_i = Y_i can also be found from proper substitution into Eq. 14 and 16, respectively. A few arithmetic manipulations using the relevant equations above are necessary to derive the algebraic expressions of the different kinetic parameters shown in Tables 1 and 2.

Mechanism B: Binding of the inhibitor precedes the rate-limiting step

General considerations

(24)

(25)

(26)

(27)

The binding equation involves the N_i^Y species only: The occlusion case

(28)

The binding equation involves both the N_i^X and N_i^Y species: The general case

(29)

(30)

(31)

(32)

Mechanism C: Binding of the inhibitor follows the rate-limiting step

To solve this kinetic mechanism, the basic scheme depicted in Fig. 1 A also needs to be expanded to include an inhibitor-binding step within the Y block as shown in Fig. 2 C. Note that the Y₂ block, which represents the fraction of inhibitor molecules bound to the N_i^Y species, is not protected from free exchange with the external milieu in this particular case. Therefore, inhibitor binding may not be measurable using a radiotracer technique in combination with a rapid filtration assay. Subject to this experimental limitation, the following theoretical considerations are nevertheless right.

Eqs. 1, 3, 5, and 7, which all apply to the X block, are still valid as shown in this case, hence the algebraic expression of k^*_on appearing in Table 1. Similarly, Eqs. 2 and 4, referring to the Y block, still hold; however, Eq. 6 needs to be rewritten as

(33)

to account for the inhibitor binding step. The algebraic expression of k^*_off appearing in Table 1 follows from these considerations, whereas those of the new parameters appearing in Eq. 33 are given as

(34)

(35)

(36)

In this model, the binding function,

(37)

and the fractional concentration of the chemical species to which the inhibitor is bound in the Y block,

(38)

can be readily written from a visual inspection of Fig. 2 C. Because the latter relationships hold true at both t = 0 (when B₀ = f_b^YY₀) and equilibrium (when B_e = f_b^YY_e), the time course of inhibitor binding should thus be described by Eq. 15, in which B is substituted for Y, k_obs is given by Eq. 13, and the quantity B_e found from proper substitution into Eq. 14. A few arithmetic manipulations are necessary to get the algebraic expressions of the kinetic parameters shown in Table 2. A correct expression of B_i can be derived from Eq. 16 as

(39)

One thus needs to define the boundary conditions to determine the algebraic expression of the initial rate of binding. This is done as follows. Because there is a true equilibrium between the block X (= X₀₀) and the subblock Y₁ (= Y₀₀) before addition of the inhibitor (see the section "Inhibitor-binding mechanisms and further assumptions" under Materials and Methods), it is possible to write down the set of equations,

(40)

(41)

When solved in conjunction with Eqs. 1-5 and 33 (in which [I] = 0 before the start of the experiment), the algebraic expression,

(42)

can be obtained following a few arithmetic manipulations. Because Y₀ = Y₀₀ in this particular case (see the section "Inhibitor-binding mechanisms and further assumptions" under Materials and Methods), then B_o = f_b^YY₀₀ with f_b^Y and Y₀₀ as given by Eqs. 38 and 42, respectively. B₀ can be cast under the Scatchard form given in Table 1 with algebraic expressions of the kinetic parameters as reported in Table 2. The initial rate of binding can now be calculated from Eq. 39 and proper substitutions therein. B_i can be cast under the generic form shown in Table 1 with algebraic expressions of B_maxi and K_di as given in Table 2. The B_i versus [I] plot should clearly deviate from simple Scatchard kinetics, and the apparent Hill number value (n_H) that one may expect from a Hill plot analysis of the initial rate data can be estimated from the relationship

(43)

which was previously derived from a similar equation (Falk et al., 1998). The (I)_0.5 expression,

(44)

can be found by solving the B_i equation in Table 1 for (I) after setting B_i = B_maxi/2, whereas the formal development of the terms in brackets at the right-hand-side of Eq. 43 leads to

(45)

and then to n_H = 1.17 when Eqs. 44 and 45 are combined. Accordingly, the sigmoidicity predicted in the B_i versus [I] plot may not be easily detected at the experimental level, particularly because the inflexion point occurs in the very early part of the curve when (I) = K_do/2.

	DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION NOTES APPENDIX REFERENCES

Kinetic criteria aimed at model discrimination

The results presented in these studies clearly demonstrate that the three basic mechanisms of inhibitor binding depicted in Fig. 2 are associated with a set of kinetic features which could be easily investigated at the experimental level by the analysis of a family of binding time courses generated at different concentrations of the inhibitor as follows

1.	As shown in Table 1, all curves should either go through the origin [mechanisms A and B (occluded)] or intercept the y axis at discrete B0 values [mechanisms B (general) and C]. In the latter situation, a B0 versus (I) plot should demonstrate Scatchard kinetics from which either the total [mechanisms B (general)] or apparent (mechanism C) number of binding sites, and the initial apparent dissociation constant for inhibitor binding (Kd0) can be determined (Table 2).
2.	The initial rate of binding (Bi in Table 1) should saturate at increasing inhibitor concentrations in mechanisms B (occluded) and C. For the latter, the binding data deviates from simple Scatchard kinetics (note that it might prove difficult to detect the sigmoidicity of the Bi versus [I] plot at the experimental level, see Eqs. 43-45), and the Kdi value estimated from its kinetic analysis should be identical to the Kd0 value. In contrast, Bi should increase linearly with [I] in mechanism A, and so, the initial rate data might be mistakenly thought to represent nonspecific binding. In mechanism B (general) too, the Bi versus [I] plot deviates from simple Scatchard kinetics and the data curve first increases to a maximum value and then decreases toward 0.
3.	When the inhibitor binding step either precedes (mechanism B) or itself represents (mechanism A) the rate-limiting step, the k*on value only, and not the k*off, is affected by [I], whereas the reverse situation holds true in mechanism C, where the inhibitor binds downstream of the rate-limiting step (Table 1). Consequently, according to Eq. 13, kobs should linearly increase and hyperbolically decrease in mechanisms A and C, respectively. In contrast, mechanism B predicts a Scatchard-like dependence of the kobs versus [I] plot with an intercept value on the y axis representing the apparent first-order rate for dissociation of the inhibitor. Note that the algebraic expression of k*on in Table 1 can be further rearranged as (46)

where (47)

to demonstrate that the half-saturation of kobs is achieved when [I] = Kdi (occluded case) = Kd0 (general case), thus providing an internal test of mechanism B. A similar rearrangement of the algebraic expression of k*off for mechanism C in Table 1 could be performed to show that the value of this parameter is reduced by half at [I] = Kd0.

4.

As expected from the assumption that there is only one inhibitor binding site, the Be versus [I] plot should saturate for all mechanisms, and a Scatchard analysis of the equilibrium data should allow one to determine the apparent dissociation constant of the inhibitor and the total number of binding sites except for mechanism B (occluded) in which Bmaxe < NT (Table 2). In the latter case, NT can only be calculated, and this is easily done using Eq. 47. A comparison of the algebraic expressions of Kde and of either Kdi or Kd0, shown in Table 2 for mechanism B, allows us to establish, (48)

that the apparent affinity for inhibitor binding estimated at equilibrium should always be higher than that observed during the initial phase of binding. In contrast, the relationship (49)

is always predicted in the case of mechanism C. It can be concluded that the value of the Kde/Kdi ratio is determined by kinetic properties that appear intrinsic to mechanisms B and C, so that this parameter takes on particular relevance for model discrimination.

Among these kinetic features, the dependence on inhibitor concentration of the apparent first-order rate of binding k_obs appears to be the most reliable indicator for diagnostic purposes. Its analysis should thus allow one to establish unambiguously whether the inhibitor binding step itself or a step that either precedes or follows inhibitor binding represents the overall rate-limiting step in a binding process. However, this conclusion raises the question of the predictive value of the basic schemes shown in Figs. 1 and 2 when applied to more complex models and to transport mechanisms in particular.

Validity of the predictions of the basic scheme with regard to more complex models and transport mechanisms

It could be argued that the simplistic nature of the basic models depicted in Figs. 1 and 2 may restrict the validity of our studies to just a few realistic kinetic mechanisms. This argument is refuted below, where it clearly appears that our results are, in fact, conditioned by the structure of the reduced kinetic scheme shown in Fig. 1 B.

Equations similar in form to Eqs. 5 and 6 can be derived for any kinetic mechanism to which the rapid equilibrium assumption applies. This assertion follows from Cha's rule, stating that, when there is more than one pathway through which N_i may be converted to N_j in a rapid equilibrium segment, any one and only one of these pathways may be used for the evaluation of N_j relative to N_i (Cha, 1968). Accordingly, either one or both of the X and Y blocks shown in Fig. 1 B could include any number of cycles, random sequences of effector addition, and/or branched pathways. In addition, the rate-limiting step linking these two blocks could involve any of the N_i^X and/or N_i^Y species. In such cases, the algebraic expressions of L^X and L^Y might be more complex than those shown in Eqs. 7 and 8 and include both K_j/A_j and A_j/K_j terms for those effector molecules which, respectively, dissociate from or associate with the N_i^X and/or N_i^Y species linking the two blocks. Accordingly, it can be predicted that low upstream or high downstream effector concentrations relative to location of N_i^X would both act to decrease the apparent value of k_obs (increase the time constant of the relaxation process). Such a situation has been described for the slow binding of ³H-ouabain to Na⁺,K⁺-ATPase, which was found to be accelerated by Na⁺ and retarded by K⁺, thus suggesting that Na⁺ and K⁺ modulate glycoside interaction through an induction (Na⁺) or repression (K⁺) of the macromolecular conformation appropriate for glycoside binding (Schwartz et al., 1974). In this particular case, the inhibitor binding step itself represents the rate-limiting step of the ouabain binding process, a conclusion that was reached on the ground that k_obs increases linearly with glycoside concentrations, in agreement with the results of our studies using the basic model shown in Fig. 2 A.

More complex situations could arise if rate-limiting steps exist within cycles as may occur in transport mechanisms (see examples given in the Appendix) and other kinetic mechanisms showing random sequences of effector and/or inhibitor addition. In such cases, the predictive value of the basic models depicted in Fig. 2 will not be affected provided that:

1.

the rate-limiting steps isolate two clearly identifiable blocks. This condition is essential for applying the rule of additivity of parallel pathways proposed by Volkenstein and Goldstein (1966), which would allow one to reduce the kinetic mechanism to a scheme similar in form to that shown in Fig. 1 B. Because this rule is not restrictive as to the number of rate-limiting steps connecting the two blocks, the expressions, (50) (51)

	in which n represents the number of rate-limiting steps with rate constants (kon)i and (koff)i, only need to be substituted for Eqs. 1 and 2, respectively. Note that all the fiX or fiY fractions characterizing the Ni species involved in the relaxation process have identical denominators (Cha, 1968), so that their summation does not introduce [I]2 terms.
2.	all rate-limiting steps involve inhibitor binding, in which case the kinetic mechanism can be reduced to a scheme similar in form to that shown for mechanism A in Fig. 2, where the k*on expression, (52)

only needs to be substituted for Eq. 22. Alternatively, the inhibitor molecule may bind either upstream or downstream of the rate-limiting steps, in which case the corresponding kinetic mechanisms can be reduced to schemes similar in form to those shown for mechanisms B and C in Fig. 2, respectively. Clearly, then, all of these situations would preserve the predictive value of kobs for discrimination between mechanisms A and C (Table 1).

Note that the failure to satisfy condition 1 should result in binding time courses showing more than one relaxation constant; however, the B_e versus [I] plot should still conform to Scatchard kinetics if only one inhibitor binding site is involved. Also, the failure to satisfy condition 2 should lead to a situation where the k_obs versus [I] plot is more complex than predicted in spite of both monoexponential binding kinetics and of B_e versus [I] plots conforming to Scatchard kinetics.

Finally, Cha's rule and the additivity principle above may be combined (Cha, 1968) to reduce almost any kinetic mechanism to the basic structure shown in Fig. 1 B provided that our hypotheses (see Materials and Methods) and the restrictions discussed above are respected. Thus, equations similar in form to Eqs. 24 and 33 may be derived for kinetic mechanisms in which the inhibitor binds to more than one of the N_i species involved in blocks X or Y. Because our analysis was restricted to the case of one inhibitor binding site only, this would preclude [I]² terms in the above equations under rapid equilibrium conditions. However, should there be more than one binding site in block X or Y, the predictive value of k_obs would be preserved: this parameter should still decrease or increase with the concentration of inhibitor in models B or C, respectively. Such mechanisms may then be recognized from the non-Scatchard dependence of