Abstract
The theoretical basis for analyzing the effects of an allosteric modulator on the response to an agonist is described. The effects of an allosteric modulator on the concentrationresponse curve to an agonist can be attributed to changes in the observed dissociation constant and intrinsic efficacy of the agonistreceptor complex. Each of these two changes can be represented by a coefficient or factor. It is possible to estimate the ratio of the coefficient of change in agonist efficacy divided by that for the agonist dissociation constant. This ratio is designated as the relative activity (RA) of the agonist in the presence of the allosteric modulator. The RA value can be estimated for each concentration of allosteric modulator by nonlinear regression analysis, regardless of the shape of the concentrationresponse curve. Regression analysis of the RA values against the concentration of allosteric modulator yields estimates of the dissociation constant (K_{A}) of the allosteric modulator and the maximal RA value. If the concentrationresponse curve to the agonist obeys a logistic function and the allosteric modulator influences the maximal response, it is possible to distinguish between the maximal change in affinity from that of efficacy. If the agonist concentrationresponse curve obeys a logistic equation with a Hill slope of 1, the RA values can be estimated easily from the agonist EC_{50} and E_{max} values. This analysis illustrates the utility of the RA value in quantifying allosteric effects.
Drugs that act allosterically to modify receptor function have unique advantages over those that bind directly to the primary recognition site of a receptor. The latter agents produce a continuous effect, whereas allosteric drugs only tune up or tune down signaling when the endogenous neurotransmitter is released (Burgen, 1981; Ehlert, 1986; Christopoulos, 2002). Thus, allosteric drugs preserve the temporal pattern of signaling across the synapse or neuroeffector junction. In addition, with overdose, high concentrations of a directly acting agonist or antagonist can cause excessive receptor activation or inhibition, whereas the maximal effect of an allosteric drug has a ceiling depending upon the nature of its allosteric effect. Thus, protection against overdose can be achieved with allosteric drugs.
There are three pharmacodynamic properties that determine the effects of an allosteric drug on the action of an agonist or endogenous neurotransmitter. These properties are 1) the affinity (K_{D}) of the allosteric drug for its site on the receptor, 2) its modulatory effect on the affinity of the agonistreceptor complex (cooperativity), and 3) its modulatory effect on the intrinsic efficacy of the agonistreceptor complex. Several investigators have described methods for measuring the affinities and cooperative effects of allosteric drugs in radioligand binding assays (Stockton et al., 1983; Ehlert, 1988a; Lazareno and Birdsall, 1995). However, to date, there have been few attempts to discriminate between allosteric effects on agonist affinity and intrinsic efficacy in functional assays. Moreover, it is often assumed that the methods presently available for investigating allosterism in functional assays are valid only if the allosteric agent is without effect on the intrinsic efficacy of the agonistreceptor complex. However, a null method (Ehlert, 1988a) was described quite some time ago for estimating allosteric ligand affinities in functional assays under conditions where the allosteric agent modulates either the affinity or intrinsic efficacy of the agonist or both. The method was based on the assumption that the intrinsic efficacy of the agonistreceptorallosteric drug complex (XRA) could be different from that of the agonistreceptor complex (XR). By deriving the mathematical equation for the combined stimulus elicited by both receptor complexes (i.e., XR and XRA), it was possible rearrange the latter equation to derive a null equation for comparing equivalent tissue responses in the presence and absence of an allosteric drug (Ehlert, 1988a). In the previous report, simple methods for applying the null equation to evaluate allosteric changes in agonist affinity and efficacy were described for highly efficacious agonists exhibiting a large receptor reserve or for situations with little receptor reserve only involving a change in affinity. In principle, it should be possible to apply this mathematics to all conditions using nonlinear regression analysis.
In the present report, the theory for investigating allosteric interactions in functional assays is described. This theory is applied to predict the behavior of the concentrationresponse curve of an agonist in the presence of allosteric drugs that modify agonist affinity and efficacy. Three mathematical methods are described and applied to analyze the theoretical data. These are 1) a null method, which requires no assumption regarding the nature of the stimulusresponse relationship; 2) a method based on a logistic relationship between the stimulus and response (i.e., operational model); and 3) a special case of the former method, in which the Hill slope of the concentrationresponse curve equals 1. Finally, methods based on the operational model are used to analyze experimental data from the literature. The results show that it is possible to estimate the affinity of an allosteric drug in functional assays regardless of whether it influences the affinity and intrinsic efficacy of the agonist. Under all conditions, it is possible to measure the product of the allosteric changes in efficacy and affinity (i.e., reciprocal of the dissociation constant). If the allosteric agent influences the maximal response of the agonist, it is possible to distinguish the allosteric change in affinity from the change in efficacy.
Materials and Methods
Theory
Allosterism. An allosteric drug binds at a secondary allosteric site on the receptor, distinct from the site at which the agonist binds. The scheme showing the interaction of an allosteric drug (A) with an agonist (X) at a receptor (R) is as follows: in which K_{X} denotes the dissociation constant of the agonistreceptor complex, K_{A} denotes the dissociation constant of the allosteric drugreceptor complex, and αK_{A} and αK_{X} denote the corresponding dissociation constants when the receptor is occupied by both ligands. Receptor theory posits that the response to an agonist is a function (f) of the stimulus, with the stimulus being equivalent to the product of receptor occupancy and intrinsic efficacy (Stephenson, 1956; Furchgott, 1966). If it is assumed that the RA complex is inactive, then the stimulus (S) of an agonist in the presence of an allosteric drug is given by the following (Ehlert, 1988a): in which ϵ and ϵ′ denote the intrinsic efficacies of the XR and XRA complexes, respectively. As described previously (Ehlert, 1988a), making the appropriate substitutions yields the following: in which R_{T} denotes the total amount of receptors. This equation simplifies to the following: in which An equivalent form of eq. 3 has been described by Christopoulos (see eq. 1.21.8; Christopoulos, 2000). In addition, the activation state model of Hall (2000) for allosteric interactions is entirely consistent with eq. 3 under the condition where little receptor is active in the absence of agonist and the RA complex is inactive. Equation 3 shows that, in the absence of the allosteric modulator (i.e., A = 0 and therefore p = 1 and q = 1), the stimulus function resembles a onesite model with a maximum proportional to ϵR_{T} and the concentration of agonist eliciting a halfmaximal stimulus equivalent to K_{X}. The effect of the allosteric modulator is to cause a concentrationdependent modification in either the observed dissociation constant or the maximal stimulus or both by the coefficients p and q, respectively. If it were possible to measure the stimulus directly—such as in voltageclamp experiments on a ligandgated ion channel with a simple onesite occupancy curve— one could use regression analysis to fit eqs. 4 and 5 to the estimates of the relative change in potency and maximal current, respectively, to obtain estimates of K_{X}, α, and β. Similarly, in enzyme assays, where the activity of the enzyme is measured directly, one could use regression analysis to fit eqs. 4 and 5 to the estimates of the allosteric change in the maximum velocity (V_{max}) and Michaelis constant (K_{m}) to obtain the allosteric parameters K_{X}, α, and β. However, in most pharmacological assays of function, it is impossible to measure the stimulus directly. Rather, a consequent, downstream response is usually measured. In this situation, the response can be described by the following equation: in which f denotes the unknown stimulusresponse function. Below, three different approaches for estimating the influence of an allosteric modulator on the affinity and efficacy of the agonistreceptor complex are described. The first involves eliminating the stimulusresponse function from the analysis through the use of a null method. The second involves the use of the operational model to describe the stimulusresponse function, and the third involves a special case of the operational model in which the Hill slope of the concentrationresponse curve equals 1.
Null Method. One approach to estimate allosteric effects in functional assays is to compare equivalent tissue responses in the presence and absence of the allosteric drug (Ehlert 1988a) so that the unknown relationship between the stimulus and response is eliminated. Using this approach, the relationship describing equivalent tissue responses in the absence and presence of an allosteric drug is given by the following: in which X and X′ denote the equiactive concentrations of agonist in the absence and presence of the allosteric drug, respectively. This equation simplifies to the following: Equation 9 can also be rearranged in the form It is useful to define the term relative activity of the agonist in the presence of the allosteric modulator (RA), which represents the ratio of the allosteric change in intrinsic efficacy (q) divided by the allosteric change in the dissociation constant of the agonist (p): Making the appropriate substitutions for p (eq. 4) and q (eq. 5) yields the following: in which The method for using eqs. 9, 10, 11, and 12 for the analysis of allosteric effects in functional assays is described below under Results.
Operational Model. Although the null method just described has the advantage of being applicable for any type of stimulusresponse function (i.e., any shape of concentrationresponse curve), it is widely observed that agonists usually exhibit logistic concentrationresponse curves. Indeed, a curvefitting procedure based on the following logistic equation is the most common computational method that investigators use to estimate the maximal response (E_{max}) and concentration of agonist eliciting halfmaximal response (EC_{50}): in which n denotes the Hill slope. Several investigators have shown that if the input to the stimulusresponse function (f) is the stimulus (i.e., product of receptor occupancy and intrinsic efficacy) and the output obeys the logistic function just described, then the stimulusresponse function must be the following (Furchgott, 1966; Mackay, 1981; Kenakin and Beek, 1982; Black and Leff, 1983): in which M_{sys} denotes the maximum response of the system, K_{E} denotes the sensitivity of the stimulusresponse function, and m denotes the transducer slope factor. This exponent is related to, but not identical to, the Hill slope of the agonist concentrationresponse curve. Substitution of eq. 3 for S in eq. 15 above followed by simplification yields the following: in which It is useful to define the term relative activity (RA) of the agonist in the presence of the allosteric drug as follows: in which τ_{obs}′ and K_{obs}′ denote the τ_{obs} and K_{obs} values of the agonist in the absence of the allosteric drug. By making the appropriate substitutions for K_{obs} and τ_{obs} (eqs. 17 and 18, respectively), it can be shown that eq. 20 simplifies to eq. 12. In addition, it can be shown by substitution of eq. 4 for p in eq. 17 that the observed dissociation constant of the agonist (K_{obs}) in the presence of the allosteric drug divided by that measured in its absence (K_{obs}′) is equivalent to the following: In situations where the exponent m in eq. 16 equals 1, the Hill slope is also equal to 1 (n = 1), and eq. 16 reduces to the following: When the Hill slope is equal to 1, it is possible to estimate the RA of the agonist in the presence of the allosteric drug as follows: in which EC_{50}′ and E_{max}′ denote the EC_{50} and E_{max} values of the agonist in the absence of the allosteric drug, and EC_{50} and E_{max} denote those measured in the presence of the allosteric drug. By making the appropriate substitutions for the various EC_{50} and E_{max} values (eqs. 23 and 24, respectively), it can be shown that eq. 25, like eq. 20, also simplifies to eq. 12.
The use of eqs. 20, 21, and 25 in the analysis of allosteric effects in functional assays is described under Results.
Results
Simulation of Allosterism in Functional Assays
The influence of allosteric modulators on the agonist concentrationresponse curve has been simulated by Kenakin (1997) and Christopoulos (2000) for a variety of conditions. These simulations and the theory for allosterism (Ehlert, 1988a) show that an allosteric modulator causes a concentrationdependent change in the agonist concentrationresponse curve. The magnitude of this change approaches a limit with high concentrations of the modulator. Allosteric changes in affinity are manifest as a parallel shift in the agonist concentrationresponse curve along the abscissa, whereas a modulation in efficacy is usually manifest as a shift or both a shift and change in the maximal response, depending upon the receptor reserve. Below, four simulations are shown to provide examples for the analysis of allosteric interactions.
Allosteric Modulator Causing a Decrease in Both Agonist Affinity and Efficacy. The behavior of allosterism in functional assays can be appreciated by using eq. 16 to generate agonist concentrationresponse curves in the presence of an allosteric drug having differential effects on agonist affinity and intrinsic efficacy. Figure 1, a to c, shows the effect of an allosteric drug causing a maximal 90% reduction in intrinsic efficacy (β = 0.1) and observed affinity (10fold increase in K_{obs}; α = 10) for a response exhibiting a Hill slope (n) of 1 (m = n = 1). When there is little receptor reserve, increasing concentrations of the allosteric inhibitor cause a shift to the right and a decrease in the maximal response of the agonist concentrationresponse curve (Fig. 1a). However, with an intermediate receptor reserve (Fig. 1b) there is less of a decline in E_{max} and a greater shift to the right in the agonist concentrationresponse curve. With a large receptor reserve (Fig. 1c), there is practically no decrease in E_{max} and a maximal 100fold shift to the right in the agonist concentrationresponse curve. The EC_{50} and E_{max} values of the agonist in the presence of different concentrations of the allosteric inhibitor were estimated by nonlinear regression analysis of the data according to eq. 14, and these values are listed in Table 1 for the data shown in Fig. 1, a to c.
To investigate the influence of the Hill slope on the behavior of allosterism, the effects of an allosteric drug essentially identical to that described for Fig. 1, a to c, were simulated, but with the response of the agonist exhibiting a Hill slope (n) greater than 1. This change was accomplished by setting the transducer slope faction in the operational model (m) equal to 2. As shown in Fig. 2, a to c, behavior generally consistent with that observed in Fig. 1, a to c, was observed. The effects of the allosteric inhibitor on the EC_{50} and E_{max} values of the agonist concentrationresponse curves shown in Fig. 2 are listed in Table 2.
Allosteric Modulator Causing an Increase in Both Agonist Affinity and Efficacy. The effects of an allosteric modulator causing both an increase in affinity and intrinsic efficacy of the agonist are shown in Fig. 3, a to c. In these simulations the maximal increases in affinity and efficacy were both 10fold (i.e., α = 0.1 and β = 10), and the response exhibited a Hill slope (n)of1(m = n = 1). When there is little receptor reserve, increasing concentrations of the allosteric enhancer cause a shift to the left and an increase in the maximal response of the agonist concentrationresponse curve (Fig. 3a). However, with an intermediate receptor reserve (Fig. 3b) there is a very small increase in E_{max} and a greater increase in potency. With a large receptor reserve (Fig. 3c), there is essentially no change in E_{max} and a maximal 100fold shift to the left in the agonist concentrationresponse curve. The effects of the allosteric enhancers on the EC_{50} and E_{max} values for all of the curves show in Fig. 3, a to c, are listed in Table 3.
Allosteric Modulator Having Opposite Effects on Agonist Affinity and Efficacy. The effects of an allosteric modulator causing equal and opposite modulation of agonist affinity and intrinsic efficacy were simulated. Figure 4, a to c, shows the effects of an allosteric modulator causing a maximal 10fold increase in agonist affinity (α = 0.1) and a maximal 90% reduction in agonist efficacy (β = 0.1) with the response exhibiting a Hill slope (n) of 1 (m = n = 1). When the receptor reserve is low (Fig. 4a), the allosteric modulator causes a concentrationdependent decrease in both the E_{max} and EC_{50} values, so that the concentrationresponse curve shifts to the left with a lowered maximum. Similar effects are observed when the receptor reserve is intermediate. However, when the receptor reserve is great, the allosteric inhibitor has little effect on the concentrationresponse curve because the reduction in efficacy is offset by an equivalent increase in affinity. The effect of the allosteric modulator on the EC_{50} and E_{max} values of the data simulated in Fig. 4 is listed in Table 4.
Methods for the Analysis of Allosterism in Functional Assays
The methods for the analysis of allosterism presented here can be divided into two main categories, based on the application of the operational model and the null method described under Materials and Methods. These approaches are described below in sections 1 and 2, respectively, where they are applied in the analysis of the simulated data in Figs. 1, 2, 3, 4. In section 3, a set of experimental data from the literature is analyzed.
Section 1: Operational Model. The application of the operational model to the analysis of allosterism in functional assays involves three steps: 1) estimation of the RA values of the agonist in the presence of different concentrations of the allosteric modulator, 2) estimation of the K_{ob}_{s}/K_{obs}′ ratios in the presence of different concentrations of the allosteric modulator, and 3) nonlinear regression analysis of the foregoing RA values and K_{ob}_{s}/K_{obs}′ ratios according to eqs. 12 and 21, respectively. This analysis ultimately yields estimates of K_{A} and γ (eq. 12) as well as α (eq. 21), and hence β, since β = γ/α. In the case where the Hill slopes of the agonist concentrationresponse curves do not differ significantly from 1, it is possible to estimate the RA values from the E_{max} and EC_{50} values of the concentrationresponse curves. Thus, in this instance, the concentrationresponse curves can be analyzed independently, using a nonlinear regression program (Prism; GraphPad Software Inc., San Diego, CA) to estimate the EC_{50} and E_{max} values of each curve by fitting the standard equation for a sigmoidal doseresponse curve having a Hill slope of 1 (eq. 22) to the data. In situations where the Hill slopes differ from 1, the RA values are estimated from the τ_{obs} and K_{obs} values. These latter values can only be estimated by global nonlinear regression analysis according to eq. 16, which involves analyzing the all of the curves measured at different amounts of allosteric modulator simultaneously, sharing estimates of M_{sys} and m among the curves. Similarly, estimation of the cooperativity factor α always involves global nonlinear regression analysis of all of the concentration response curves, regardless of the estimate of the Hill slope.
Estimation of K_{A} and γ when the Hill slope is equivalent to 1. Since the agonist concentrationresponse curves shown in Fig. 1, a to c, all exhibit Hill slopes of 1, accurate estimates of the RA of the agonist can be estimated from the E_{max} and EC_{50} values using eq. 25. These RA values are listed in Table 1 for the different concentrations of allosteric modulator. Figure 5a shows the log RA values plotted against the log of the concentration of allosteric modulator for the data from Fig. 1a. In the presence of maximally effective concentrations of the allosteric modulator, the plot approaches an asymptote corresponding to the maximal combined change in observed affinity (α) and intrinsic efficacy (β) of the agonist. This combined effect on affinity and efficacy is denoted as γ and is equivalent to β/α (eq. 13). The log form of eq. 12 was fitted to the data by nonlinear regression analysis, which yielded estimates of 10^{5} M and 0.01 for the K_{A} and γ, respectively. These estimates are essentially identical to those used in simulating the data in the first place, demonstrating that the nonlinear regression method provides accurate estimates of the allosteric parameters of the modulator. The same procedure was used to analyze the simulated data in Fig. 1, b and c, and identical results were obtained. This result was expected since the only difference among the plots in Fig. 1, a to c, is the receptor reserve (τ), and not the parameters of the allosteric modulator.
The functional data in Fig. 3 for the allosteric modulator causing an increase in agonist affinity and efficacy can be analyzed in a manner analogous to that described above for the allosteric inhibitor in Fig. 1 because both sets of data exhibit Hill slopes of 1. Figure 5c shows a plot of the RA values (Table 3) of the agonist plotted against the log of the allosteric modulator concentration for the data in Fig. 3, a to c. Regression analysis of the data according to eq. 12 yielded estimates of 10^{5} M and 100 for K_{A} and γ, respectively. These values are identical to those used in the simulation of the data.
The functional data in Fig. 4, a to c, can also be analyzed in a manner similar to that described for Fig. 1, a to c. Since the Hill slopes of the curves are equal to 1, the RA values can be calculated according to eq. 25, and these values are listed in Table 4. Figure 5e shows that the plot of log RA against log A yield a horizontal line at RA = 1. Thus, it is impossible to estimate K_{A} from this plot, since any value of K_{A} will yield a best fit of eq. 12 to the data as long as γ = 1. In this case where α/β = 1, the K_{A} value can be estimated through the analysis of the K_{obs} values as described in the next section.
Resolution of γ into its α and β components when the Hill slope is equivalent to 1. To discriminate between the modulatory effects of the allosteric agent on the observed dissociation constant and intrinsic efficacy of the agonist receptor complex, it is necessary to fit the operational model (eq. 16) to all of the agonist concentrationresponse curves with varying allosteric modulator simultaneously. With regard to the data in Fig. 1, regression analysis is done sharing the estimate of M_{sys} among all of the curves and estimating unique values of K_{obs} and τ_{obs} for each curve. Since all of the curves have Hill slopes of 1, the exponent m can be removed from eq. 16 or constrained at a constant value of 1. Global nonlinear regression analysis of the data in Fig. 1a yielded an estimate of 1.0 for M_{sys}, essentially the same as that used in the simulation of the data. The values of τ_{obs} and K_{obs} for the various concentrations of allosteric modulator are listed in Table 1. The RA values of the agonist can also be estimated from the τ_{obs} and K_{obs} values for each concentration of allosteric modulator using eq. 20, and these are also listed in Table 1. These RA values are essentially identical to those estimated from the E_{max} and EC_{50} values of the agonist (compare the last two columns in Table 1). Figure 5b shows a plot of the log ratio (K_{ob}_{s}/K_{obs}′) against the log concentration of the allosteric modulator. It can be seen that this ratio reaches a plateau corresponding to α at high concentrations of allosteric modulator. Nonlinear regression analysis was used to fit the log form of eq. 21 to the data, which yielded estimates of 10^{5} M and 10 for K_{A} and α, respectively. This estimate of K_{A} is identical to that estimated from the RA values shown in Fig. 5a and that used in the simulations, and the estimate of α is equivalent to that used in the simulations. Knowing γ and α, it is possible to estimate β by rearrangement of eq. 13 (γ = β/α), which yields an estimate of 0.1 for β. This value is identical to that used in the simulations.
A similar analysis was done to estimate the τ_{obs} and K_{obs} values for the concentrationresponse curves shown in Fig. 1b, and these parameters are also listed in Table 1. The corresponding RA values were estimated (Table 1) and found to be identical to those estimated from the curves in Fig. 1a. Consequently, plots identical to those shown in Fig. 5, a and b, were derived for the curves in Fig. 1b, and the same values for K_{A}, α, β, and γ were estimated.
Under normal circumstances, where actual experimental data are analyzed, it would be difficult to obtain unique values of τ_{obs} and K_{obs} for each curve shown in Fig. 1c because all of the E_{max} values of the curves are approximately the same. However, with the simulated data in Fig. 1c, it is possible to obtain these estimates because the data have no error, and there are actually small differences in the E_{max} values. Regardless, in the situation of little change in E_{max},it is always possible to obtain accurate estimates of the ratio of the two parameters (i.e., τ_{obs}/K_{obs}), even though estimation of the individual values may be impossible. The nonlinear regression method for obtaining this ratio is described below in connection with Fig. 6. As described in the next section, it is possible to estimate the RA values from these ratios, and these estimates are listed in Table 1. The estimates of τ_{obs} and K_{obs} for the data in Fig. 1c have been omitted from Table 1 because they would be difficult to estimate in experiments with data having moderate experimental error.
The approach described for the estimation of α from the data in Fig. 1 was applied to the data in Figs. 3 and 4, and the results are summarized in Tables 3 and 4. Figure 5d shows the ratio of K_{obs} values plotted against the log of the allosteric enhancer concentration for the data in Fig. 3, a and b. Regression analysis (eq. 21) yielded estimates of 10^{5} M and 0.1 for K_{A} and α, respectively. Knowing γ and α, it is possible to estimate a value of 10 for β using eq. 13. Figure 5f shows a plot of the log ratio (K_{obs}/K_{obs}′) against log A for the data in Fig. 4, a and b. Regression analysis according to eq. 21 yielded estimates of 10^{5} M and 0.1 for K_{A} and α, respectively. Knowing that the horizontal plot in Fig. 5e with RA = 1 implies that γ = 1, it is possible to estimate that β = 0.1 by rearrangement of eq. 13. It is impossible to estimate any allosteric parameters from the data in Fig. 4c because the allosteric modulator has little or no effect on the concentrationresponse curve. This situation occurs because the allosteric effects on affinity and intrinsic efficacy are equal and opposite. When there is a large receptor reserve, the 10fold reduction in intrinsic efficacy is manifest as a 10fold shift to the right in the concentrationresponse curve, whereas the 10fold increase in affinity causes the opposite effect, resulting in no net effect.
Estimation of allosteric parameters when the Hill slope differs from 1 and there is a modulation of E_{max}. To analyze the data shown in Fig. 2 according to the operational model, it is inaccurate to estimate agonist RA from their EC_{50} and E_{max} values according to eq. 25 because the Hill slopes of the concentrationresponse curve differ from 1. Consequently, eq. 16 was fitted to all of the concentration responsecurves simultaneously, sharing the estimate of M_{sys} and the exponent m among the curves and estimating unique values of τ_{obs} and K_{obs} for each curve. The global estimates of M_{sys} and m from regression analysis were 1.0 and 2.0 for M_{sys} and m, respectively. These values are identical to those used in the simulations. The estimates of τ_{obs} and K_{obs} were used to calculate the agonist RA values (eq. 20). These RA estimates are listed in Table 2 and are essentially the same as those listed in Table 1 for the curves shown in Fig. 1, a and b. This result was expected because the only difference between the parameters for the data in Figs. 1, a to c, and 2, a to c, is the value of m, but not any of the other parameters, including the RA values. Consequently, analysis of the RA values for Fig. 2, a and b, yields plots essentially identical to those shown in Fig. 5, a and b, and the K_{A}, α, and β estimates for the allosteric drug were 10^{5} M, 10, and 0.1, respectively.
Estimation of allosteric parameters when the Hill slope differs from 1 and there is no modulation of E_{max}. Since the receptor reserve is great for Fig. 2c and all of the agonist concentrationresponse curves exhibit nearly the same E_{max}, numerous combinations of τ_{obs} and K_{obs} all yield the same leastsquares fit. Nevertheless, it is possible to estimate the ratio of τ_{obs}/K_{obs} for each curve shown in Fig. 2c. A leastsquares fit can be obtained when the value of K_{obs} is equal to or greater than the larger of the two constants, K_{X} or αK_{X}. Thus, the best fit can be obtained by constraining K_{obs} as a constant at any arbitrary value equal to or greater than the true value of αK_{X} (10^{4} M) and estimating unique values of τ_{obs} for each of the curves. A summary of this analysis is illustrated in Fig. 6, which shows the best fitting parameter estimates for various values of K_{obs} between 10^{8} and 10^{1} M as well as the residual sum of squares (RSS). The ordinate scale on the left corresponds to the various parameter estimates, whereas that on the right corresponds to RSS. The data in Fig. 6 are from the concentrationresponse curve simulated in the presence of 10^{4} M A; analogous plots can be made for the data simulated at the other concentrations of A. The estimate of RSS in the figure is based on all of the curves, not just the one simulated at A = 10^{4} M. As the constrained value of K_{obs} increases to 10^{4}, RSS approaches a minimum plateau and remains at this best fitting level with a further increase in K_{obs} no matter how large. Thus, it can be seen that a leastsquares fit is obtained once the value of K_{obs} is equal to or greater than αK_{X} (10^{4}). As the value of K_{obs} increases, the estimate of τ_{obs} increases in a proportional manner so that the ratio of the two estimates (τ_{obs}/K_{obs}) is constant (Fig. 6). Thus, the best fitting estimate of the ratio τ_{obs}/K_{obs} can be obtained by constraining K_{obs} to an arbitrarily high constant value and estimating the other parameters that minimize RSS. Using this approach, the ratio of τ_{obs}/K_{obs} was estimated for each of the curves shown in Fig. 2c. Alternatively, it is possible to set τ_{obs} as a constant and estimate the value of K_{obs} that gives the leastsquares fit (analysis not shown).
Knowing the ratio of τ_{obs}/K_{obs} for each of the curves shown in Fig. 2c, it is possible to estimate their respective RA values by dividing the ratio estimated in the presence of a given concentration of A by the corresponding ratio obtained for the control curve. In other words, if the τ_{obs}/K_{obs} ratio of one curve is divided by the corresponding ratio for the control curve (τ_{obs}′/K_{obs}′), then the following ratio (τ_{obs}′K_{obs}′/τ_{obs}′K_{obs}) is obtained, which is equivalent to the RA value (eq. 20). The RA values for the data in Fig. 2c were estimated in this manner and are listed in Table 2. These values are identical to those estimated for the data in Figs. 1, a to c, and 2, a and b; consequently, regression analysis of these data according to eq. 20 yields a plot identical to that shown in Fig. 5a with K_{A} and γ estimates of 10^{5} M and 0.01, respectively. Since it was impossible to estimate unique K_{obs} values for each of the curves shown in Fig. 2c, it was impossible to separate γ into its α and β components. As described above, it is impossible to make this distinction when the allosteric inhibitor has no influence on the E_{max} of the concentrationresponse curve.
It is informative to estimate the RA values of the data shown in Fig. 2 from the E_{max} and EC_{50} values according to eq. 25. As described above, this equation is only appropriate for conditions when the Hill slopes of the agonist concentrationresponse curve are equal to 1. When these estimate were made, and the data analyzed according to the plots shown in Fig. 5, a and b, estimates of 5.5 × 10^{6} M and 0.00073 (Fig. 2a) and 9.21 × 10^{6} M and 0.0037 (Fig. 2b) were made for K_{A} and γ, respectively. Thus, there is substantial error in these parameters estimates, which should equal 10^{5} M and 0.01. However, when this method was used to analyze the data in Fig. 2c, values of 10^{5} M and 0.0090 were estimated for K_{A} and γ, respectively. Note that the latter value of K_{A} is correct, whereas the estimate of γ is only off by 10%. Thus, when the allosteric inhibitor has no influence on E_{max} it is possible to estimate the RA values according to eq. 25, even if the Hill slopes differ from 1. This conclusion seems reasonable because the two different methods for calculating the RA value (eqs. 20 and 25) yield similar estimates for the data in Fig. 2c but not for those of Fig. 2, a and b (compare the last two columns of Table 2). Note that when the allosteric inhibitor has no effect on the E_{max}, the estimation of RA from eq. 25 can be simplified to the following: Figure 7 shows the ratio of EC_{50} values of the agonist in the presence of the allosteric drug divided by that measured in its absence plotted against the log of the allosteric drug concentration for the data from Fig. 2c. Nonlinear regression analysis of the data according to the log form of eq. 12 yielded estimates of 10^{5} M and 0.009 for K_{A} and γ, respectively, as described above.
In summary, it can be concluded that whenever an allosteric modulator is without effect on E_{max}, only K_{A} and γ can be estimated. Under these conditions, one can estimate the RA values from eq. 25 or 26 even if the Hill slope differs from 1. Equation 12 can be fitted to the data by nonlinear regression analysis to obtain estimates of K_{A} and γ. It is unnecessary to perform the complicated regression analysis summarized in Fig. 6 to estimate the RA values in this situation. Nevertheless, this analysis has been presented here to illustrate the relationship among the various parameter estimates in this situation.
Section 2: Null Method. An advantage of the null method for the analysis of allosteric interactions is that it can be applied to agonist concentrationresponse curves that deviate from logistic behavior. Nevertheless, the method will be used here to analyze the logistic curves shown in Figs. 1 and 3. The null method involves three steps: 1) estimation of equiactive concentrations of agonist in the absence (X) and presence (X′) of the various concentrations of the allosteric modulator; 2) global nonlinear regression analysis of the log equiactive agonist concentrations to obtain estimates of the ratio q/p (RA values; see eq. 11) of the agonist in the presence of different concentrations of the allosteric modulator; and 3) nonlinear regression analysis of the foregoing RA values according to eq. 12. This analysis yields estimates of K_{A} and γ.
Allosteric modulator causes a reduction in E_{max}. Figure 8 shows the application of this method for the data in Fig. 1a. Since the allosteric modulator causes a reduction in the E_{max} of the agonist, the agonist concentrations (X) are interpolated from the control concentrationresponse curve that yield responses equivalent to those elicited by the agonist concentrations (X′) used in the presence of the allosteric modulator (see Fig. 8a for the data simulated at A = 10^{3} M). These pairs of equiactive agonist concentrations are then plotted on a log scale as shown in Fig. 8b. Each curve in Fig. 8b corresponds to a different allosteric modulator concentration. These curves were analyzed simultaneously by global nonlinear regression analysis of the data according to the log form of eq. 9, with the estimate of K_{X} shared among all the curves and unique values of p and q estimated for each curve. However, it is possible to estimate numerous combinations of K_{X}, p, and q that yield the same leastsquares fit for each curve. Using an approach analogous to that described above in connection with Fig. 6, it is possible to estimate the ratio q/p (RA; see eq. 11), by constraining K_{X} to a constant and determining the values of the other parameters that minimize RSS. Figure 8c shows the results of regression analysis for the data simulated at A = 10^{3} M. Over the range of K_{X} values shown (10^{9}–10^{1}), RSS was insignificant and depended only on the criteria used for convergence of the nonlinear regression algorithm and not on K_{X}. Analogous behavior was observed for the data simulated at the other concentrations of A. Using this approach, various values of RA can be estimated at various the concentrations of the allosteric modulator (A). The resulting log RA values are plotted against log A and analyzed by nonlinear regression analysis according to eq. 12. The results yielded a plot identical to that shown in Fig. 5a, and values of 10^{5} M and 0.01 were estimated for K_{A} and γ, respectively. These values are identical to those used in the simulation of the data in Fig. 1a. The data in Fig. 1b were analyzed in a similar manner and the same estimates of γ and K_{A} were obtained (i.e., 0.01 and 10^{5} M, respectively).
It is also possible to use the same strategy for the simulated data in Fig. 1c; however, since the data exhibit little or no change in E_{max} with allosteric modulation, it is necessary to constrain both K_{X} and p as constants and analyze all of the data simultaneously by global nonlinear regression analysis (eq. 9) to obtain q values for each of the curves. By dividing the estimates of q value by the constant value of p, it is possible to obtain the RA estimates for each curve, and hence the values of γ and K_{A} by regression analysis according to eq. 12. When this was done, a plot identical to that shown in Fig. 5a was obtained as well as the same estimates of γ and K_{A} (0.01 and 10^{5} M, respectively). Regardless, this complicated analysis is probably unnecessary because the RA values can be estimated accurately from EC_{50} values according to eq. 26 when there is little change in E_{max} with the allosteric modulator. Presumably, this situation would apply to nonlogistic agonist concentrationresponse curves.
Allosteric modulator causes an increase in E_{max}. In situations where the allosteric modulator causes an increase in the E_{max} of the agonist, it is possible to analyze the data according to null eq. 10. However, in this instance, where the concentrationresponse curve in the presence of the allosteric modulator has a greater E_{max} than that measured in its absence, the X′ values are interpolated from the curve measured in the presence of the allosteric inhibitor that yield responses equivalent to those generated by the concentrations (X) of the control concentrationresponse curve. Using this approach and an overall strategy analogous to that just described in the preceding paragraph, it is possible to estimate the γ and K_{A} values for the allosteric enhancer shown in Fig. 3, a to c. However, an important difference is the range of values over which the parameter K_{X} can be constrained as a constant for global nonlinear regression analysis. This range was found to be K_{X} ≤ 10^{5} for the data in Fig. 3. When this analysis was done, plots identical to that shown in Fig. 5c were obtained, and values of 100 and 10^{5} M were estimated for γ and K_{A}, respectively, which are equivalent to those used in simulating the data.
Using the null approach, it is also possible to estimate the γ and K_{A} values of the other allosteric modulators simulated above, with the exception of the data in Fig. 4. In this case, the plot of RA against log A yields a horizontal line at RA = 1 as shown in Fig. 5e. As described above, it is impossible to estimate K_{A} from this figure, only γ. In summary, if the allosteric modulator causes a decrease in E_{max}, the approach describe in connection with Fig. 8 is used, whereas if an increase in E_{max} is observed, the approach described in the preceding paragraph is used.
Section 3: Experimental Data. The methods based on the operational model were applied to experimental data to determine the impact of experimental error on the estimates of allosteric parameters. Figure 9a shows the results of a prior study in which the influence of gallamine on muscarinic receptor mediated inhibition of adenylyl cyclase activity was measured in homogenates of the rat myocardium (Ehlert, 1988b). This preparation is known to contain a relatively homogeneous population of M_{2} muscarinic receptors (Hammer et al., 1986). Gallamine is a neuromuscular blocking agent long known to cause sinus tachycardia (Walton, 1950) through a mechanism involving an inhibition of postjunctional muscarinic receptors (Riker and Wescoe, 1951). The mechanism was later shown to be allosteric in functional studies on isolated beating hearts (Clark and Mitchelson, 1976) and in muscarinic receptor binding assays (Stockton et al., 1983).
As shown in Fig. 9a, gallamine caused a concentrationdependent shift to the right in the agonist concentrationresponse curve with little change in E_{max}. Regression analysis of each curve according to logistic eq. 14 yielded the estimates of EC_{50}, E_{max}, and n (Hill slope), and these values are listed in Table 5. The results show that the agonist concentrationresponse curves exhibit low Hill slopes (approximately 0.7–1.0), and that there is little change in E_{max}. In this situation, it is only possible to estimate K_{A} and γ; it is impossible to resolve γ into its α and β components from these data alone (see analysis of the data in Figs. 1c and 3c). Accordingly, the RA value of the agonist can be estimated easily by calculating the ratio of EC_{50} values measured in the presence of gallamine divided by that measured in its absence as described above in eq. 26. Nevertheless, the RA values were estimated using the operational model so that the influence of experimental error on these estimates might be appreciated. Normally, this method would only be required in situations where the allosteric modulator influences the E_{max}.
The data were analyzed by global nonlinear regression analysis according to eq. 16 using the procedure described above in connection with Fig. 6 for the estimation of RA values. Regression analysis was done sharing the estimates of M_{sys} and m among the curves and constraining K_{obs} as a constant so that the ratio τ_{obs}/K_{obs} could be estimated. A summary of the analysis of the data at 10^{4} M gallamine is shown in Fig. 9b where the logarithm of the parameter estimates are plotted against K_{obs}, which was constrained to various fixed values during regression analysis. A leastsquares fit was obtained over the range K_{obs} > 10^{3} as shown by the minimum value for log RSS. Over this range the estimates of M_{sys}, m, and the ratio τ_{obs}/K_{obs} were constant. This process was repeated for each curve in Fig. 9a, and a leastsquares fit was always obtained over the domain K_{obs} > 10^{3}. The estimates ± S.E. of M_{sys} and m were 55 ± 1.4% inhibition and 0.68 ± 0.037, respectively. The RA values were calculated from the τ_{obs}/K_{obs} ratios estimated with K_{obs} constrained to an arbitrarily high value (i.e., 10^{2}), and these estimates are also listed in Table 5 and plotted in Fig. 9b. Regression analysis of the data according to eq. 12 (Fig. 9c) yielded estimates of K_{A} and γ of 0.62 μM and 0.019, respectively (mean ± S.E.M.; pK_{A} = 6.21 ± 0.15; log γ = 1.72 ± 0.16).
For heuristic purposes, the RA values were also estimated using eq. 25. Generally, the use of eq. 25 is restricted to data with Hill slopes of 1, but as described above, it is applicable in situations where there is little change in E_{max}. These estimates were made for each experiment and the mean values of four experiments ± S.E.M. are listed in Table 5. The corresponding log RA values are also plotted against the log of the gallamine concentration in Fig. 9c. Regression analysis of the data according to eq. 12 yielded estimates of K_{A} and γ of 0.56 μM and 0.025, respectively (mean ± S.E.M.; pK_{A} = 6.25 ± 0.21; log γ = 1.60 ± 0.21).
Discussion
The study of allosterism in functional assays provides potential information about the allosteric modulation of the intrinsic efficacy of the agonistreceptor complex. This information is important because the therapeutic effects of allosteric modulators depend on their modulation of both the affinity and intrinsic efficacy of the endogenous ligand. The theoretical basis for the present analysis was described in a previous report (Ehlert, 1988a), where it was shown that simple graphical techniques can be used to estimate the affinity of the allosteric modulator and its combined effects on the affinity and intrinsic efficacy (γ) of the agonistreceptor complex when the receptor reserve is great. Given the widespread availability of software for nonlinear regression analysis, the time seemed appropriate to reinvestigate this method and to apply it to all conditions. To this end, the concept of relative activity of the agonist (RA) is developed here and shown to be a robust measure of allosteric effects.
The present report shows that it is always possible to estimate the influence of the allosteric modulator on the RA value of the agonist. Regression analysis of the RA values according to the logarithmic form of eq. 12: provides estimates of the dissociation constant (K_{A}) of the allosteric modulator as well as the product (γ) of its modulatory effects on the intrinsic efficacy (β) and affinity (1/α) of the agonistreceptor complex. In situations where the Hill slopes of the agonist concentrationresponse curves are equal to 1, the RA values can be estimated easily from the EC_{50} and E_{max} values (eq. 25). If the allosteric modulator has no influence on the E_{max} of the agonist, it is impossible to determine the individual components of γ (α and β) unless an independent estimate of α is made, perhaps through ligand binding analysis (Lazareno and Birdsall, 1995). If the allosteric modulator does influence the E_{max} of the agonist, then it is possible to estimate the individual components α and β as described under Results. The method involves global nonlinear regression analysis with the operational model (eq. 16) to obtain individual estimates of τ_{obs} and K_{obs}.
For heuristic purposes, this analysis has emphasized the utility of the RA estimate in understanding allosteric interactions. Consequently, the log RA value has been treated as the dependent variable in nonlinear regression analysis. However, this strategy might not be the best statistical approach when the RA value is estimated from the EC_{50} and E_{max} values in situations where the Hill slope is equivalent to 1. For example, if there is error in the estimates of the E_{max} and EC_{50} values of the control concentrationresponse curve, then this error will introduce a systematic error in the RA estimates for all the concentrationresponse curves measured in the presence of the allosteric modulator. A better strategy might be to calculate log ratio of the E_{max}/EC_{50} values measured in the absence and presence of the various concentrations of the allosteric modulator and use the following equation to analyze the data by nonlinear regression analysis: Regression analysis should yield estimate of γ, K_{A}, and P, the log ratio of E_{max}′/EC_{50}′ for the control agonist concentrationresponse curve. The latter equation can be derived from eqs. 12 and 25, and it only applies to situations where the Hill slope of the concentrations response curves are equivalent to 1. In addition, when the Hill slope is equivalent to 1 and the allosteric modulator has no influence on E_{max}, the abovementioned equation reduces to that described previously (Ehlert, 1988b): Regression analysis would yield estimates of γ, K_{A}, and the log control EC_{50} value. A similar statistical argument might seem appropriate when the RA values are estimated from K_{obs} and τ_{obs} (i.e., error in the estimates of K_{X} and τ will introduce a systematic error in the estimate of RA); however, this concern is unnecessary because the values of K_{X} and τ (K_{obs}′ and τ_{obs}′) are estimated by global nonlinear regression analysis (eq. 10) of all of the agonist concentrationresponse curves simultaneously. Thus, all of the data are considered in the estimation of K_{X} and τ and not just the control data. It should be apparent from the foregoing that it is possible to estimate the dissociation constant of the agonist (K_{X}) through the analysis of an allosteric modulator that influences the intrinsic efficacy of the agonist to such an extent that a change in E_{max} is observed.
The present report also describes a null method, whereby the values of K_{A} and γ were determined by nonlinear regression analysis of pairs of equiactive agonist concentrations in the absence and presence of various concentrations of the allosteric modulator. This type of approach should be applicable in situations where the response of the agonist deviates from logistic behavior.
The estimates of the K_{A} (0.56–0.62 μM) and γ (0.019–0.025) values for gallamine antagonism of the action of oxotremorineM at M_{2} muscarinic receptors made in this report by regression analysis of the log RA values are in good agreement with those (0.52 μM and 0.020) made in a previous analysis of the same data using regression analysis of the log EC_{50} values according to eq. 29 above (Ehlert, 1988b). This agreement shows the feasibility of estimating RA values from experimental data. The estimate of the K_{A} of gallamine reported here is also in good agreement with that (0.93 μM) estimated by antagonism of the negative inotropic effect of acetylcholine in the isolated, electrically driven, guinea pig atrium (Christopoulos, 2000) and by competitive inhibition of the binding of [^{3}H]Nmethylscopolamine to myocardial muscarinic receptors (Stockton et al., 1983; Ehlert, 1988b) (0.77 and 1.1 μM, respectively).
In a previous report on the antagonism of the action of the highly efficacious agonist oxotremorineM and the partial agonist BM5 at cardiac M_{2} muscarinic receptors by gallamine, the estimates of γ were calculated to be 0.020 and 0.011, respectively (Ehlert, 1988b). In both cases, gallamine had little or no effect on the E_{max} values of the agonists. Since BM5 behaved as a partial agonist, any gallamineinduced change in the intrinsic efficacy of BM5 should have been manifest as an alteration in E_{max}. The lack of this change indicates that β = 1 and α = 0.011, approximately, for the allosteric interaction between gallamine and BM5 (i.e., γ = α/β = 0.011). Additional proof for this hypothesis could be obtained if a similar estimate of the α value for the interaction between gallamine and BM5 was obtained in ligand binding studies using the technique described by Lazareno and Birdsall (1995).
The presumed lack of effect of gallamine on the intrinsic efficacy of BM5 has important implications with regard to the action of gallamine and the number of ground and active states of the M_{2} muscarinic receptor. In a simple twostate model, where there is only one active and one inactive state of the receptor, there should be a correlation between allosteric effects and the intrinsic efficacy of the primary ligand (Ehlert, 2000). However, gallamine has been shown to inhibit the binding of both agonists and antagonists to M_{2} muscarinic receptors. Moreover, if gallamine reduces the affinity of BM5 without influencing its intrinsic efficacy, this result implies the existence of another pair of ground and active states of the M_{2} receptor that are selected by gallamine and whose affinities for BM5 are both lower by the same amount so that the selectivity of BM5 for the two states is the same, and hence its efficacy unaltered (i.e., the ratio of microscopic affinity constants of BM5 for the ground and active states are the same). Alternatively, a more complicated hypothesis involving a group of active and inactive states might explain the data.
Part of the analysis described herein is analogous to a method previously described for analyzing a series of concentrationresponse curves to different agonists with varying affinity and intrinsic efficacy. In that previous analysis, the concept of intrinsic relative activity was developed (Ehlert et al., 1999; Ehlert and Griffin, 2001). This term denotes the product of the affinity and intrinsic activity of the agonist expressed relative to that of a standard agonist. This estimate is analogous to the RA value described here, which is a measure of the product of affinity and intrinsic efficacy of the agonist in the presence of the allosteric modulator relative to that measured in its absence. Accurate measurements of the components of allosterism are important and likely to be more significant as the number of allosteric drugs used in therapeutics and research increases.
Footnotes

This work was supported by National Institutes of Health Grant 69829. Article, publication date, and citation information can be found at http://jpet.aspetjournals.org.

doi:10.1124/jpet.105.090886.

ABBREVIATIONS: RA, relative activity; RSS, residual sum of squares; BM5, N(4pyrrolidino2butynyl)Nmethylacetamide.
 Received June 10, 2005.
 Accepted July 20, 2005.
 The American Society for Pharmacology and Experimental Therapeutics