Abstract
We describe a modification of receptor theory that enables the estimation of relative affinity constants for the inactive state of a G proteincoupled receptor. Our approach includes the traditional parameters of observed affinity (K_{obs}) and efficacy (fraction of ligandreceptor complex in the active state, ε) and introduces the concept of the fraction of the ligandreceptor complex in the inactive state (intrinsic inactivity, ε_{i}). The relationship between receptor activation and the ligand concentration is known as the stimulus, and the operational model expresses the response as a logistic function of the stimulus. The latter function includes K_{obs} and the parameter τ, which is proportional to ε. We introduce the parameter τ_{i}, which is proportional to ε_{i}. We have previously shown that the product, K_{obs}τ, of one agonist, expressed relative to that of another (intrinsic relative activity, RA_{i}), is a relative measure of the affinity constant for the active state of the receptor. In this report, we show that the product, K_{obs}τ_{i}, of one agonist, expressed relative to that of another (intrinsic relative inactivity, RI_{i}), is a relative measure of the affinity constant for the inactive state of the receptor. We use computer simulation techniques to verify our analysis and apply our method to the analysis of published data on agonist activity at the M_{3} muscarinic receptor. Our method should have widespread application in the analysis of agonist bias in drug discovery programs and in the estimation of a more fundamental relative measure of efficacy (RA_{i}/RI_{i}).
Introduction
An abundance of evidence indicates that receptors evolved to act as molecular switches that undergo a conformational change into an active state when occupied by an endogenous agonist. Some of the strongest evidence for this view comes from the results of singlechannel recordings at ligandgated ion channels, which show an abrupt quantal increase in conductance upon binding of agonist with no evidence of gradual conductance changes (Colquhoun and Sakmann, 1985). At G proteincoupled receptors, the active state of the receptor presumably interacts with a conformation of the heterotrimeric G protein that has the guanine nucleotidebinding pocket on the ras domain of Gα opened up for rapid GDPGTP exchange (Oldham and Hamm, 2008).
The existence of quantal receptor states is not inconsistent with an agonist having a continuum of observed affinities for a G proteincoupled receptor, depending upon the concentration of G protein in the plasma membrane, the type of G protein with which the receptor interacts, and the concentration of guanine nucleotide. These factors can change the observed affinity of the agonistreceptor complex over a broad interval bounded on the high and low ends by the microscopic affinity constants of the agonist for the active and inactive states of the receptor, respectively (Ehlert, 2008). Thus, two states of the receptor can give rise to a continuum of observed affinities and efficacies. There may also be multiple active receptor states that exhibit differential selectivity for signaling proteins (i.e., G proteins or G proteincoupled receptor kinases), giving rise to biased agonism (Kenakin, 2011).
Although the estimation of observed affinity and relative efficacy provides an accurate estimate of how an agonist interacts with the receptor population, these macroscopic constants can vary for the same agonistreceptor complex as just described. However, the affinity constants of the agonist for the ground (K_{a}) and active (K_{b}) states of the receptor are invariant and represent a more fundamental estimate of the agonistreceptor interaction because a change in the equilibrium between these states gives rise to variation in observed affinity and efficacy and not a change in the affinity of the states themselves. We have previously described how to estimate the product of observed affinity and efficacy of an agonist through analysis of the concentrationresponse curve (Ehlert et al., 1999; Ehlert, 2008) and have recently shown that this estimate is proportional to the microscopic affinity constant of the active state of the receptor (Tran et al., 2009).
Given the inherent symmetry of ligandreceptor interactions, we reasoned that if the product of observed affinity (K_{obs}) and the fraction of the agonistreceptor complex in the active state (i.e., efficacy) is proportional to K_{b}, then it seems likely that the product of observed affinity and the fraction of the agonistreceptor complex in the inactive state (1 − efficacy) should be proportional to the microscopic affinity constant of the agonist for the inactive state (K_{a}). In the present report, we present analytical proof and computer simulation analysis showing that this postulate is correct and analyze published agonist concentrationresponse curves from the literature to illustrate our method for determining a relative estimate of the affinity constant of the agonist for the ground state of the receptor. We also show how to estimate the affinity constant of the agonist for the inactive state of the receptor in units of inverse molarity (M^{−1}) from functional data.
Materials and Methods
Simulation of Agonist ConcentrationResponse Curves.
We generated theoretical agonist concentrationresponse curves and analyzed them using the method describe under Results to determine whether it were possible to estimate the relative values of the agonist affinity constants for active and inactive states of the receptor that were used to generate the data in the first place.
Our approach for the simulations is based on two assumptions: 1) activation of a G proteincoupled receptor by an agonist is equivalent to the fraction of the agonistreceptor complex in the active state associated with the G proteinguanine nucleotide complex (DR_{s}*GX) (Ehlert and Rathbun, 1990; Ehlert, 2000) and 2) the operational model accurately describes the relationship between receptor activation and the measured response (Black and Leff, 1983). Thus, we used the following equation to generate agonist concentrationresponse curves: In this equation, m represents the transducer slope factor, M_{sys} represents the maximal response of the system, K_{E} represents the sensitivity constant of the transduction mechanism, and [DR_{s}*GX] represents the amount of the active state of the agonist receptor complex (DR_{s}*) in a quaternary complex with G protein (G) bound with guanine nucleotide (X). The model is essentially equivalent to that described by Black and Leff (1983), but with the concentration of agonistreceptor complex replaced with [DR_{s}*GX]. In addition, we have used the variable M_{sys} instead of E_{m}.
The model used to simulate the formation of the quaternary complex (DRGX) is shown in Fig. 1. The central square of equilibrium expressions in Fig. 1a represents the ternary complex model of De Lean et al. (1980), and the outer square incorporates a guanine nucleotidebinding step. Each receptor complex shown in Fig. 1a represents the summation of active and inactive states. For example, the DRGX complex is equivalent to the sum of the active (DR_{s}*GX) and inactive (DR_{s}GX) states. Thus, the complete model includes two layers of equilibrium expressions, each like that shown in Fig. 1a but undergoing an interconversion between active (R_{s}*) and inactive (R_{s}) states of the receptor complex as shown in Fig. 1b. A complete description of the constants in the model is given under Appendix as well as the equation used to generate the active state of the quaternary complex (eq. 30).
The model shown in Fig. 1b also enables the calculation of the amount of free active receptor in a complex with the G protein and guanine nucleotide (R_{s}*GX) (eq. 51). This species represents constitutive receptor activity. In all of our simulations, we used parameter estimates that yielded insignificant constitutive activity such that the response in the absence of agonist was less than 0.65% of the maximal response of the system.
Analysis of Theoretical and Experimental Data.
We used a modification of the operational model to analyze both theoretical and experimental agonist concentrationresponse curves. For estimation of the RA_{i} and K_{obs} values of agonist, we used global nonlinear regression analysis to fit eq. 2 to the concentrationresponse curve of the most efficacious agonist while simultaneously fitting eq. 3 to the concentrationresponse curves of the other agonists: In these equations, M_{sys} represents the maximal response of the system, m represents the transducer slope factor, D represents the concentration of agonist, and K_{obs} represents the affinity constant of the agonist. The symbol, ′, is used to denote the parameters of the standard agonist. R represents the product of K_{obs}′ and τ′, and τ is defined by eq. 20. RA_{i} is a relative estimate of the product of affinity and efficacy of the agonist and its definition is given by eq. 24. All of the agonist concentrationresponse curves were fitted simultaneously, sharing the estimates of M_{sys} and m among the curves and obtaining a unique estimate of R for the most efficacious agonist and unique estimates of K_{obs} and RA_{i} for the other agonists. The details of the fitting procedure have been described previously (Ehlert, 2008).
For the estimation of RI_{i} and K_{a} values, each agonist concentrationresponse curve was fitted to the following equation by nonlinear regression analysis: In this equation, ε represents efficacy, and K_{Eobs} is a measure of the sensitivity of the signaling pathway as described below in connection with eq. 20. Equation 4 was derived by taking the operational model (eq. 18), substituting in eq. 20 for τ, and expressing the parameters in logarithmic form. During regression analysis, the values of M_{sys}, K_{obs}, and m were constrained to the estimates obtained from the RA_{i} analysis. Repetitive regression analyses were done, each time constraining K_{Eobs} to a constant so that the upper limit of the domain of K_{Eobs} values that yielded a leastsquares fit was identified. The estimate of log ε was obtained for each agonist by regression analysis with K_{Eobs} constrained to the maximal value within the domain of K_{Eobs} values that yielded a leastsquares fit for the most efficacious agonist. More specific details of the analysis are described under Results. Having an estimate of ε enabled the estimation of RI_{i} and K_{a} using eqs. 8, 21, 23, and 26 as described under Results.
Results
Receptor Theory
A Relative Estimate of Agonist Affinity for the Inactive State of the Receptor.
Our thesis is that the product of K_{obs} and the fraction of the agonistreceptor complex in the active state is proportional to the microscopic affinity constant of the agonist for the inactive state of the receptor. If correct, this hypothesis should be easy to prove for a simple system, consisting of a receptor in equilibrium between active (R_{s}*) and inactive states (R_{s}). This simple model is shown in Scheme 1, in which the affinity constants of the agonist (D) for the active and inactive states are denoted by K_{b} and K_{a}, respectively, and K_{c} denotes the unimolecular constant describing the equilibrium between the receptor states. Our analysis applies to the condition in which there is little constitutive receptor activity (K_{c} ≪ 1).
To prove our hypothesis for the model in Scheme 1, we first derive expressions for observed affinity and the fraction of the agonistreceptor complex in the active state (observed efficacy, ε). It has been shown that the function describing the amount of agonistreceptor complex in the active state (DR_{s}*) is given by Tran et al. (2009): In this equation D denotes the concentration of agonist, R_{T} denotes the total concentration of receptors, ε denotes the efficacy of the agonist, and K_{obs} denotes the observed affinity constant of the agonist (reciprocal of the observed dissociation constant). This function is also known as the stimulus (S) as described by Furchgott (1966) and Stephenson (1956) in nearly equivalent forms. K_{obs} for the receptor is given by the following equation as described previously (Tran et al., 2009): Efficacy is equivalent to the fraction of the agonistreceptor complex in the active state. This can be determined at 100% receptor occupancy by taking the limit of the receptor activation function as the agonist concentrations approaches saturating levels as described previously (Tran et al., 2009): The fraction of the agonistreceptor complex in the inactive state (ε_{i}) is equivalent to We define ε_{i} as the intrinsic inactivity of the agonistreceptor complex. Substituting in eq. 7 for ε in eq. 8 followed by simplification yields The product of K_{obs} and ε_{i} of one agonist expressed relative to that of another standard agonist is equivalent to in which the parameters of the standard agonist are denoted with ′. The right side of this equation was derived by taking the expression on the left side and substituting in eqs. 6 and 9 for K_{obs} and ε_{i}, respectively. Further simplification yields This equation shows that if a receptor conforms to Scheme 1, then the product of observed affinity and intrinsic inactivity of one agonist expressed relative to that of another is equivalent to the corresponding ratio of microscopic affinity constants for the inactive state of the receptor. We define this term as the intrinsic relative inactivity of the agonist (RI_{i}). Figure 2 illustrates the relationship between RI_{i} and the product K_{obs}ε_{i} for two agonists. Also shown is the relationship between RA_{i} and the product of K_{obs}ε, which is described by eq. 24.
Estimation of the K_{a} Value in Units of Inverse Molarity.
If the receptor lacks appreciable constitutive activity, then it is usually possible to obtain an accurate estimate of the K_{a} value. In such instances, the K_{a} value is approximately equal to the product of K_{obs} and ε_{i} as shown by eqs. 12 to 15: The right side of this equation was derived by substituting in eqs. 6 and 9 for K_{obs} and ε_{i} on the left side, respectively. Rearrangement yields When there is little constitutive activity, K_{c} ≪ 1, and, consequently, eq. 13 can be approximated by This equation reduces to
Modification of the Operational Model for Estimation of RI_{i} and K_{a}.
Most assays for agonist activity at G proteincoupled receptors involve the measurement of a response downstream from receptor activation and not receptor activation itself. To estimate RI_{i} or K_{a}, therefore, it is necessary to incorporate the theory described above into an equation that expresses the response as a function of the parameters K_{obs} and ε. We use the reverse engineering approach described by Black and Leff (1983). These investigators showed that if the initial input to a receptor transduction mechanism is consistent with the receptor activation function (eq. 5) and the output (response, y) is a logistic function similar to then the equation that expresses the response (y) as a function of the stimulus (eq. 5) is
In eq. 16, n represents the slope factor of the concentrationresponse curve, E_{max} represents the maximal response, and EC_{50} represents the concentration of agonist eliciting a halfmaximal response. In eq. 17, S denotes the active agonistreceptor complex (DR*), M_{sys} denotes the maximal response of the system, K_{E} denotes the sensitivity constant of the system, and m denotes the transducerslope factor. Substituting in eq. 5 for S in eq. 17 yields the following equation after simplification: in which This equation for τ can be written in a simpler form by substituting in K_{Eobs} for the ratio K_{E}/R_{T}: We define τ_{i} as It follows that the product of K_{obs}τ_{i} of one agonist expressed relative to that of a standard agonist is equivalent to the corresponding ratio of products of observed affinity and intrinsic inactivity: Again, the ′ is used to designate the parameters of the standard agonist. From eq. 11, it follows that We have previously shown an analogous relationship between the product of observed affinity and efficacy and the ratio of microscopic affinity constants of the active state of the receptor (Tran et al., 2009): This ratio is known as the intrinsic relative activity of the agonist (RA_{i}).
When there is little constitutive activity, it is possible to estimate the K_{a} value of the agonist from the parameters of the operational model as shown by the next two equations: The right side of eq. 25 was generated from the left side by substituting in eq. 21 for τ_{i} and simplifying. When there is little constitutive activity, the right side of the equation can be replaced with K_{a} as shown by eq. 15:
Analysis of Simulated Agonist ConcentrationResponse Curves.
Another approach for proving that the product K_{obs}ε_{i} is proportional to the microscopic affinity constant of the agonist for the inactive state of the receptor is to generate theoretical agonist concentrationresponse curves using the operational model and to determine whether it is possible to obtain accurate estimates for the RI_{i} and K_{a} values from the theoretical data. A robust way to simulate the data is to generate the receptoractivation function using the ternary complex model with guanine nucleotide (quaternary complex model) defined at the level of receptor states. We have previously explained that the fraction of the agonistreceptor complex in the active state associated with the heterotrimeric G protein bound with guanine nucleotide (DR_{s}*GX, active quaternary complex) is proportional to receptor activation (Ehlert and Rathbun, 1990; Ehlert, 2000). Thus, we have used this model to generate the theoretical stimulus for the operational model as described under Appendix. The resulting output (agonist concentrationresponse curve) was submitted to nonlinear regression analysis to obtain estimates of K_{obs}ε_{i} for each agonist. A random error with a range of ±5% was added to the theoretical data to ensure that our estimation procedure was feasible. The remainder of this section describes how to estimate RI_{i} (relative estimate of K_{obs}ε_{i}) and K_{a} from agonist concentrationresponse curves.
Analysis of Agonist ConcentrationResponse Curves When the Intracellular Concentration of GTP Is High and the Transducer Slope Factor Equals One (m = 1).
We simulated curves for agonist receptor activation (DR_{s}*GX) under conditions in which the concentration of guanine nucleotide is high (X = 10^{−3} M), because GTP is often present in cells at concentrations that saturate G proteins. For these simulations, the affinity constant of each agonist for the inactive state (K_{a}) was set to a constant value (K_{a} = 10^{5} M^{−1}) and that for the active state (K_{b}) was varied to yield K_{b}/K_{a} ratios of 100,000, 10,000, 1000, 100, and 10 for agonists A to E, respectively. The amount of agonist bound in the form of quaternary complex was estimated using eq. 30 as described under Appendix. The resulting receptor activation curves (stimulus function) are shown in Fig. 3, a and b. The complete set of parameters for the simulations is listed in the legend to Fig. 3. With these parameters, the amount of spontaneous receptor activation was very low in the absence of agonist (R_{s}*GX = 0.013%), which yielded a basal response of only 0.65%. The maximal fractional amount of the DR_{s}*GX complex is equivalent to ε, and the reciprocal of the concentration of agonist causing a halfmaximal formation of DR_{s}*GX is equivalent to K_{obs}. The values of K_{a,} K_{b}, K_{b}/K_{b}′, K_{obs}, and a relative estimate of ε are listed for each agonist in Table 1.
The simulated stimulus curves were used as input to the operational model, and theoretical agonist concentrationresponse curves were generated using eq. 1, with M_{sys} = 100%, K_{Eobs} = 0.02, and m = 1. A random error (±5%) was added to the simulated data, and the resulting curves are illustrated in Fig. 3c. The EC_{50} and E_{max} values of the agonists were estimated by nonlinear regression analysis of the data using eq. 16, and these estimates are listed in Table 2.
The first step in the analysis of the simulated data involves estimation of the agonist RA_{i} values by global nonlinear regression analysis using eqs. 2 and 3 as described previously (Ehlert, 2008). This analysis also yields estimates of the K_{obs} values of the partial agonists. The log RA_{i} values were first estimated relative to the most efficacious agonist (0.00, −1.05, −1.99, −3.01, and −3.99 for agonists A to E, respectively). These values are listed in Table 1, normalized to the least efficacious agonist. For each agonist, there is reasonable agreement between log RA_{i} and the log ratio of its K_{b} value expressed relative to that of the least efficacious agonist (K_{b}′). This regression analysis also yielded estimates of M_{sys} (98.6%) and m (1.00).
The estimation of RI_{i} and K_{a} requires estimates of the K_{obs} values of the full agonists. We used the theoretical values for the full agonists A and B, but with real experimental data, it would be necessary to estimate K_{obs} using the method of partial receptor inactivation.
The first step in the estimation of RI_{i} and K_{a} involves determination of the maximal value of K_{Eobs} that yields a leastsquares fit of eq. 4 to the concentrationresponse curve of the most efficacious agonist (agonist A). Equation 4 is essentially the operational model with τ expressed as the ratio ε/K_{Eobs}. Regression analysis is done with the log K_{obs} value of agonist A constrained to its estimated value (theoretical value of 5.0 in this case). The parameters M_{sys} and m are constrained to the values estimated in the RA_{i} analysis described above. The parameter log K_{Eobs} is constrained to an arbitrarily low value (e.g., −4). The parameter ε is constrained to the range 0 < ε < 1.0 (i.e., log ε < 0), because for a real receptor, ε can only assume values between 0 and 1. Regression analysis is initiated, and the estimate of log ε that yields a leastsquares fit is determined. This process is repeated iteratively, constraining log K_{Eobs} to higher values for each regression until the maximal value of log K_{Eobs} that yields a leastsquares fit is determined.
A summary of this iterative procedure is shown in Fig. 3d for agonist A. The plot shows that the residual sum of squares (RSS) for the regression is at a minimum whenever log K_{Eobs} is constrained to a value less than −1.55.
This process was also done for agonists B to D, and the results are shown in Fig. 4. The vertical dashed lines in Fig. 4, a to c, correspond to the maximal value of log K_{Eobs} that yields a leastsquares fit for agonist A (log K_{Eobs} = −1.55). The estimates of log ε when log K_{Eobs} is constrained to its upper limit (−1.55) were −0.02, −0.23, −0.84, −1.77, and −2.78 for agonists A to E, respectively.
With these estimates of ε, it is possible to estimate ε_{i} using eq. 8. Then, τ_{i} is estimated from ε_{i} using eq. 21. Finally, RI_{i} and K_{a} are estimated from τ_{i} and K_{obs} using eqs. 23 and 26, respectively. The relationship between log RI_{i} and log K_{Eobs} is also shown in Fig. 3d (agonist A) and Fig. 4 (agonists B to D). For agonists B to D, the estimate of log RI_{i} is approximately equal to the true value (log RI_{i} = 0) when log K_{Eobs} = −1.55. For agonists C and D, the estimate of log RI_{i} is constant over the range log K_{Eobs} ≤ −1.55. For all the agonists, the estimate of log RI_{i} is equivalent to the corresponding ratio of K_{obs} values when log K_{Eobs} is very low.
The estimates of log RI_{i} and log K_{a} are listed in Table 1. There is general agreement between these estimates and the theoretical values used to generate the data (i.e., log RI_{i} = 0 and log K_{a} = 5). Although the estimates of log RI_{i} and log K_{a} are accurate, the values of log K_{Eobs}, ε, and ε_{i} are unreliable.
The lengthy iteration procedure summarized in Figs. 3d and 4 was described to verify and explain the relationship between log K_{Eobs} and the estimates of log RI_{i} and log K_{a}. However, the maximal estimate of log K_{Eobs} for the most efficacious agonist can be determined more quickly by regression analysis with eq. 4 with the parameters constrained as described above except that ε is constrained to 1.0 (log ε = 0) and log K_{Eobs} is unconstrained. The resulting estimate of log K_{Eobs} is a little larger than the maximal value that yields a leastsquares fit (−1.55). The latter can be determined by constraining log K_{Eobs} to nearby smaller values as described above in connection with Fig. 3d. It is unnecessary to do the regression with the less efficacious agonists.
When the transducer slope factor in the operational model (m) is equivalent to one, the slope factor of the agonist concentrationresponse curve (n) is also equivalent to one, and there are simple relationships among the EC_{50} and E_{max} values of agonists and the parameters RA_{i}, K_{obs}, and relative efficacy (eqs. 27–39). Under this condition (i.e., m = 1), RA_{i} is described by the following equation (Ehlert et al., 1999): The ′ is used to designate the parameters of the standard agonist. It can also be shown that the K_{obs} value of an agonist can be estimated by (see Appendix, eq. 56): Finally, the relative efficacy of agonists can be estimated by (see Appendix, eqs. 57–59): Practically speaking, eqs. 28 and 29 can only be applied to the analysis of partial agonists, because there is little difference between E_{max} and M_{sys} for full agonists. These equations yielded reasonably accurate estimates of the corresponding parameters for the partial agonists as illustrated in Table 2.
Analysis of Agonist ConcentrationResponse Curves When the Intracellular Concentration of GTP is Low and the Transducer Slope Factor Equals One (m = 1).
We also considered the condition in which the concentration of guanine nucleotide is low. These simulations were performed as described above for the data in Fig. 3 except that the concentration of guanine nucleotide (X) was set to a lower value of 1.8 μM. Figure 5 shows the results of these simulations. The reduction in X caused an increase in the value of K_{obs} and a decrease in the maximal amount of DR_{s}*GX for each agonist (Fig. 5a; Table 3). The reduction in efficacy was offset by the increase in affinity such that there was little change in the agonist concentrationresponse curves except for an overall decrease in E_{max} (Fig. 5b). The EC_{50} and E_{max} values of the curves are listed in Table 4. The RA_{i} values of the agonists were estimated by global nonlinear regression analysis using eqs. 2 and 3, and these estimates are listed in Table 3.
The agonistconcentration response curves were also analyzed by regression analysis using eq. 4 to estimate the RI_{i} and K_{a} values of the agonists using the procedure described above. A leastsquares fit was obtained for the most efficacious agonist when the estimate of K_{Eobs} was constrained over the domain K_{Eobs} ≤ −0.50. The concentrationresponse curves of the less efficacious agonists were analyzed by eq. 4 with K_{Eobs} constrained to −0.50. The resulting estimates of ε for each agonist were used to estimate RI_{i} and K_{a} as described above. These values are listed in Table 3. There is general agreement between the RA_{i}, RI_{i}, and K_{a} estimates and the corresponding theoretical values (K_{b}/K_{b}′, K_{a}/K_{a}′, and K_{a}) used to generate the data. For the partial agonists, there was also good agreement between the theoretical values of RA_{i}, K_{obs}, and relative efficacy and the corresponding estimates determined from the EC_{50} and E_{max} values using eqs. 27 to 29 (Table 4).
Analysis of Agonist ConcentrationResponse Curves When the Intracellular Concentration of GTP Is High and the Transducer Slope Factor Greater Than One.
We also considered theoretical data derived from the operational model with a transducer slope factor greater than one (i.e., m = 1.8). The theoretical microscopic constants for these simulations are listed in Table 5, and the corresponding receptor activation functions are shown in Fig. 6, a and b. The theoretical concentrationresponse curves were generated using the operational model with a K_{Eobs} value of 0.02, and the resulting curves are shown in Fig. 6c. The EC_{50}, E_{max}, and slope factors (n) of the agonists were estimated from the theoretical concentrationresponse curves, and these values are listed in Table 6. The K_{obs}, RA_{i}, RI_{i}, and K_{a} values were estimated by nonlinear regression analysis using the approach described above for the analysis of the data in Fig. 3, and the resulting parameter estimates are listed in Table 5. There was general agreement between these estimates and the corresponding theoretical values that were used to generate the theoretical data. The largest error (approximately 3fold) was in the estimate of K_{a} and RI_{i} for the most efficacious agonist.
Other Simulations.
We also examined a variety of other conditions including a change in the ratio of G protein to receptor and a variation in the sensitivity constant of the transducer function (K_{E}). In each case, we obtained reliable estimates of RA_{i}, RI_{i}, and K_{a} of all but the most efficacious agonist in a series.
Analysis of Experimental Data
We analyzed some of our prior data on muscarinic agonist stimulation of phosphoinositide hydrolysis in Chinese hamster ovary cells transfected with the human M_{3} muscarinic receptor (Ehlert et al., 1999) (Fig. 7a). The EC_{50} and E_{max} values of the agonists were estimated by nonlinear regression analysis using eq. 16 with the slope factor constrained to a value of one (Table 8). There was no significant reduction in residual error when the slope factors of the agonists were allowed to differ from one during regression analysis (F_{9, 155} = 0.35; P = 0.96).
We used global nonlinear regression analysis to estimate the K_{obs} and RA_{i} values of the agonists using eqs. 2 and 3 with oxotremorineM used as the standard agonist. These values were subsequently normalized relative to the least efficacious agonist as listed in Table 7. Next, we fitted eq. 4 to each agonist concentrationresponse curve with the values of M_{sys} and m constrained to the values obtained from the RA_{i} analysis (i.e., 59.5fold increase in [^{3}H]inositol phosphates and 1.08, respectively) and the value of K_{Eobs} constrained to the maximal value that yielded a leastsquares fit for the most efficacious agonist, oxotremorineM (log K_{Eobs} = −1.0). From this regression, the values of RI_{i} and K_{a} were estimated for each agonist as described above, and the estimates are listed in Table 7. The estimate of the K_{a} of oxotremorineM was highly dependent on the value of K_{Eobs}. When log K_{Eobs} was reduced from the maximal value that yielded a leastsquares fit (−1.0) to −1.3, the estimate of the log K_{a} of oxotremorineM increased from 4.09 to 5.18. At lower values of K_{Eobs}, the estimate of log K_{a} changed very little and approached the limiting value of K_{obs} (5.46). Thus, it was impossible to estimate the K_{a} of oxotremorineM accurately.
We also used eqs. 27 to 29 to estimate the RA_{i}, K_{obs}, and relative efficacy values of the agonists because these equations are applicable when the slope factors of the concentrationresponse curves are equivalent to one. These estimates are listed in Table 8 for all of the agonists. There was little difference between the parameter values when estimated using the latter method or by regression analysis using eq. 4 (Table 7).
A plot of the log RA_{i}, RI_{i}, and RA_{i}/RI_{i} ratios of each agonist expressed relative to that of the least efficacious agonist [4(mchlorophenylcarbamoyloxy)2butynyltrimethylammonium (McNA343)] is shown in Fig. 7c. For oxotremorineM, the log K_{a} and RI_{i} are not given, because of the error in the estimation of K_{a}. There was much greater variation in the RA_{i} compared with the RI_{i}. The ratio RA_{i}/RI_{i} is a relative measure of the selectivity of the agonist for the active state relative to the inactive state. This estimate for the most efficacious agonist oxotremorineM was at least 200fold greater than that for McNA343. Whereas bethanechol has approximately 10fold lower affinity for the active and inactive states of the M_{3} receptor compared with carbachol, it maintains a comparable degree of selectivity for the active state.
Discussion
In this report, we introduce the concept of the fraction of the agonistreceptor complex in the inactive state (intrinsic inactivity, ε_{i}) and show that the product of ε_{i} and K_{obs} is proportional to the microscopic affinity constant of an orthosteric ligand for the inactive state of the receptor (K_{a}) when there is little constitutive activity. We also show that, among a group of agonists, it is possible to estimate the K_{a} value of any agonist having an efficacy less than onethird that of the most efficacious agonist.
We have assumed that the fraction of the agonistreceptor complex in the active state coupled with the guanine nucleotidebound form of the G protein (DR_{s}*GX) is an accurate measure of receptor activation for responses mediated through G proteins. The maximal amount of this quaternary complex can be substantially less than the amount of receptor, depending on the concentration of GTP, the relative amount of G protein to receptor, and the nature of their interaction. For the example shown in Fig. 3, the theoretical maximal amount of DR_{s}*GX that could be formed by an agonist having a very large K_{b}/K_{a} ratio is only 71% of the total amount of receptor. For the example in Fig. 5, in which the concentration of GTP is only 1.8 μM, the maximum possible amount of DR_{s}*GX is only approximately 6% of the amount of receptor. Thus, at G proteincoupled receptors, the maximal value that ε can attain is probably substantially less than one.
When agonist receptor activation approaches the maximal limit of the system, then our method for estimating the parameter K_{Eobs} in eq. 4 provides an accurate means of estimating the K_{a} value of any agonist. For example, the maximal amount of receptor activation caused by agonist A (67%) in the simulation shown in Fig. 3 is almost equivalent to the maximal limit of this theoretical simulation (71%). Using the approach described in the text, it was possible to obtain a reasonable estimate of K_{a} through regression analysis of the concentrationresponse curve using eq. 4 with the value of log K_{Eobs} constrained to the maximal value that yields a leastsquares fit (i.e., log K_{Eobs} = −1.55). Accurate estimates of the K_{a} values of the less efficacious agonists were also obtained when log K_{Eobs} was constrained to −1.55.
However, with real data, it could be possible that the most efficacious agonist in a series might behave like agonist C, for example, which only causes 11% receptor activation. When its concentrationresponse curve was analyzed, the maximal value of log K_{Eobs} that yielded a leastsquares fit was much larger (−0.8). With log K_{Eobs} constrained to −0.8, the corresponding estimate of log K_{a} (4.32) is substantially different from the true value (5.00), although the estimates of the log K_{a} values of the less efficacious agonists (D and E) are accurate. Thus, without knowledge of the level of receptor activation caused by an agonist, we would be uncertain whether our estimate of the K_{a} of the most efficacious agonist is accurate.
A possible solution is to examine the activity of agonists in a broken cell preparation with low concentrations of GTP and GDP (e.g., agonistinduced [^{35}S]GTPγS binding). Under this condition, highly efficacious agonists might cause full receptor activation as shown for the theoretical example in Fig. 5 even though the same agonists exhibit substantially different relative efficacies when the concentration of GTP is high (e.g., compare the relative efficacies of agonists A–C for the concentrationresponse curves in Figs. 3c and 5b). Thus, for a given receptorG protein pair, if efficacious agonists exhibit different relative efficacies in an intact cell assay, but the same efficacy in a [^{35}S]GTPγS binding assay with a low concentration of guanine nucleotides (GDP + GTP), then this result suggests that the estimate of the K_{a} of the most efficacious agonist in the [^{35}S]GTPγS assay would be accurate. This information could then be used to assign an appropriate K_{a} value for the most efficacious agonist in the intact cell assay.
Our inability to estimate the K_{a} value of oxotremorineM in the phosphoinositide assay on CHO cells expressing the M_{3} muscarinic receptor illustrates the problem of estimating K_{a} for the most efficacious agonist in a series. Given the colossal ratio of K_{b}/K_{a} that is required for agonist A to elicit a near maximal formation of DR_{s}*GX in the simulation shown in Fig. 3, it seems likely that highly efficacious agonists with more reasonable (i.e., smaller) K_{b}/K_{a} ratios would not cause substantial receptor activation in the presence of the normally high concentration of GTP in the cytosol. Under this condition, the K_{a} would be approximately equal to K_{obs}. Strange (2008) has also suggested that the K_{a} value of an agonist at a G proteincoupled receptor is probably approximately equivalent to K_{obs} because of the high concentration of cytosolic GTP. Nonetheless, our demonstration that it is only possible to estimate a range of K_{a} values for the most efficacious agonist in a series indicates the uncertainty in assuming that K_{a} is equivalent to K_{obs}. We have noted that the functional estimates of the K_{obs} values of efficacious muscarinic agonists for M_{2} and M_{3} receptors expressed in CHO cells and the mouse ileum, respectively, are unusually high, suggesting a substantial deviation of K_{obs} from K_{a} (Tran et al., 2009).
The observed affinity and efficacy of an agonistreceptor complex can change, depending on the concentration of GTP and how the G protein interacts with the receptor (Ehlert and Rathbun, 1990; Ehlert, 2000). When a given agonistreceptor complex is compared across systems, a reduction in the concentration of GTP has the effect of reducing agonist efficacy, because the maximal amount of DR_{s}*GX formed is less. When an agonist is compared with other agonists in a series, however, reducing the concentration of GTP has the effect of increasing the efficacy of agonists relative to the most efficacious agonist. In addition, a decrease in GTP causes the efficacy of the most efficacious agonist to achieve, or to come closer to achieving, the maximal level of receptor activation for the system. A plot of relative efficacy against the ratio of K_{b}/K_{a} illustrates this relationship for the theoretical examples shown in Figs. 3 and 5 (Fig. 8a). Even though agonists A to E have the same microscopic affinity constants for the active (K_{b}) and inactive (K_{a}) states of the receptors in the two simulations, more of the agonists exhibit maximal relative efficacy for the system when the concentration of GTP is low. Figure 8b shows the corresponding plot for the experimental data shown in Fig. 7, a and b.
To summarize, we have developed a method of analysis to estimate the K_{a} value of any agonist, full or partial, having an efficacy less than onethird that of the most efficacious agonist in a series. The analysis depends on having accurate estimates of the K_{obs} values of full agonists, which can be determined using the method of partial receptor inactivation. The analysis provides accurate estimates of K_{a} even if K_{obs} > K_{a}. This condition can occur with a low concentration of GTP in assays on cellular homogenate, and, perhaps, in some intact cell preparations and tissues. Finally, we also show how to estimate the K_{obs} and relative efficacies of partial agonists from their EC_{50} and E_{max} values when the Hill slopes of their concentrationresponse curves are equal to one.
Authorship Contributions
Participated in research design: Ehlert, Griffin, and Suga.
Conducted experiments: Ehlert, Griffin, and Suga.
Performed data analysis: Ehlert, Griffin, and Suga.
Wrote or contributed to the writing of the manuscript: Ehlert.
Appendix
Simulation of the DR_{s}*GX Complex.
We used the model described in Fig. 1 to simulate the amount of agonist (D) bound in the form of an active quaternary complex (DR_{s}*GX) consisting of the active state of the receptor (R_{s}*), the G protein (G), and guanine nucleotide (X). The equation describing the amount of DR_{s}*GX as a function of the agonist concentration is as follows: in which R_{T} denotes the total amount of receptors. The affinity constants with lettered subscripts refer to the properties of the receptor states, and these are defined below by eqs. 44 to 50. The derivation of eq. 30 has been described previously (Ehlert and Rathbun, 1990; Ehlert, 2000). K_{DRGX} is given by In these equations, G_{T} and G denote the total and free concentrations of G protein. The cooperativity constants and microscopic constants for the receptor complexes are defined in Fig. 1a. As described previously (Ehlert, 2000, 2008), these are defined at the level of receptor states by the following equations: The fundamental affinity constants for the receptor states are
Simulation of the R_{s}*GX Complex.
The model in Fig. 1a was also used to simulate constitutive receptor activity, that is, the amount of the active state of the unoccupied receptor in a complex with G protein and guanine nucleotide (R_{s}*GX). The equation describing the amount of this complex as a function of the agonist concentration is as follows:
In this equation, G is defined as described above.
Estimation of K_{obs} and Relative Efficacy When the Slope Factor of the Agonist ConcentrationResponse Curve Is Equal to One (n = 1).
When the agonist concentrationresponse curve has a slope factor of 1, then the transducer slope factor in the operational model is also equal to one (m = 1), and there are simple relationships between observed affinity (K_{obs}) and relative efficacy and the EC_{50} and E_{max} values of agonists. To derive these relationships, we begin by expressing E_{max} and EC_{50} in terms of the parameters of the operational model. E_{max} can be derived by solving the limit of the operational model (eq. 18) as D approaches infinity when m = 1: Solving this limit yields
The EC_{50} value can be solved from the following relationship:
The left side of the equation represents the E_{max} multiplied by 0.5, with the E_{max} value defined by eq. 53. The right side of the equation represents the operational model with the agonist concentration replaced with by EC_{50}. This equation communicates the idea that the response is halfmaximal when the drug concentration equals the EC_{50}. Simplification yields
Having expressions for E_{max} and EC_{50} enables the proof of the equations for K_{obs} and relative efficacy (eqs. 28 and 29, respectively) described under Results. K_{obs} can be estimated from the expression The left side of this equation was derived from the right side by substituting in eqs. 53 and 55 for E_{max} and EC_{50}, respectively. Simplification yields eq. 28 under Results.
The efficacy of one agonist (ε), expressed relative to that of another (ε′), can be estimated from the equation The left side of this equation was derived from the right side by substituting in eq. 53 for E_{max}. The parameters of the agonist to which the efficacy of the other agonist is normalized are indicated with ′. Simplification yields Substituting in eq. 19 for τ yields Simplification yields eq. 29 under Results.
Footnotes
This work was supported by the National Institutes of Health National Institute of General Medical Sciences [Grant GM69829].
Article, publication date, and citation information can be found at http://jpet.aspetjournals.org.
doi:10.1124/jpet.111.179291.

ABBREVIATIONS:
 McNA343
 4(mchlorophenylcarbamoyloxy)2butynyltrimethylammonium
 RSS
 residual sum of squares
 GTPγS
 guanosine 5′O(3thio)triphosphate
 CHO
 Chinese hamster ovary.
 Received January 11, 2011.
 Accepted May 13, 2011.
 Copyright © 2011 by The American Society for Pharmacology and Experimental Therapeutics