Abstract
In prior work, we have shown that it is possible to estimate the product of observed affinity and intrinsic efficacy of an agonist expressed relative to that of a standard agonist simply through the analysis of their respective concentrationresponse curves. In this report, we show analytically and through mathematical modeling that this product, termed intrinsic relative activity (RA_{i}), is equivalent to the ratio of microscopic affinity constants of the agonists for the active state of the receptor. We also compared the RA_{i} estimates of selected muscarinic agonists with a relative estimate of the product of observed affinity and intrinsic efficacy determined independently through the method of partial receptor inactivation. There was good agreement between these two estimates when agonistmediated inhibition of forskolinstimulated cAMP accumulation was measured in Chinese hamster ovary cells stably expressing the human M_{2} muscarinic receptor. Likewise, there was good agreement between the two estimates when agonist activity was measured on the ileum from M_{2} muscarinic receptor knockout mice, a convenient assay for M_{3} receptor activity. The RA_{i} estimates of agonists in the mouse ileum were similar to those estimated at the human M_{3} receptor with the exception of 4(mchlorophenylcarbamoyloxy)2butynyltrimethylammonium (McNA343), which is known to be an M_{1} and M_{4}selective muscarinic agonist. Additional experiments showed that the response to McNA343 in the mouse ileum included a nonM_{3} muscarinic receptor component. Our results show that the RA_{i} estimate is a useful receptordependent measure of agonist activity and liganddirected signaling.
Drug discovery often involves testing compounds in highthroughput screens to determine their activity at specific receptors. The process not only identifies useful drugs but also helps to explain how variation in the structure of a compound alters its pharmacological activity. With regard to agonists at G proteincoupled receptors (GPCRs), the most common measurements of functional activity are the maximal response (E_{max}) and the concentration of agonist required for halfmaximal response (EC_{50}). These parameters can vary for the same agonist, however, depending on the coupling protein through which the receptor signals (e.g., G protein) and the nature of the response being measured. What is needed is a measure of agonist activity that is dependent solely on the agonistreceptor interaction and not on downstream elements in the signaling cascade.
To identify such a measure, it is useful to consider that the activity of an agonist can be analyzed at different, internally consistent, hierarchical levels as summarized in Fig. 1. Ultimately, agonist action depends on the microscopic affinity constants of the agonist for the ground and active states of the receptor (Colquhoun, 1998; Ehlert, 2000). These parameters have been estimated at ligandgated ion channels through the analysis of singlechannel activity (Colquhoun and Sakmann, 1985), but it is impossible to estimate all of these parameters through the analysis of a downstream response at GPCRs. If we take a less detailed view of agonist action and consider the activity of a population of receptors, it is possible to determine the relationship between the agonist concentration and the fraction of the receptor population in the active state. At a ligandgated ion channel, this measurement represents the ensemble average or wholecell current elicited by the agonist. At a GPCR, this activation function is known as the stimulus, and it can be estimated through the analysis of a downstream response using the method of partial receptor inactivation (Furchgott, 1966). The analysis yields estimates of the concentration of agonist required for halfmaximal receptor activation (observed dissociation constant) and the maximal level of receptor activation at 100% receptor occupancy (observed intrinsic efficacy). Affinity and efficacy are not fundamental constants unique to the specific agonistreceptor complex; rather, these parameters are complex functions of the microscopic affinity constants of the ground and active states of the receptor as well as other constants (Ehlert, 2000, 2008). This complexity is manifest, in part, by their dependence on the concentration of GTP and on other proteins that physically interact with the receptor (e.g., G proteins). Thus, although observed affinity and intrinsic efficacy are more invariant than the empirical parameters, EC_{50} and E_{max}, they are not solely dependent on the agonistreceptor complex. In addition, the requisite data for estimating observed affinity and intrinsic efficacy are rarely obtained in highthroughput screens.
In the present report, we show analytically and through mathematical modeling that the microscopic affinity constant of an agonist for the active state of the receptor is proportional to the product of its observed affinity and intrinsic efficacy, and that this relationship holds when there are different active states of the receptor signaling through different G proteins. We have shown previously that it is possible to estimate the product of observed affinity and intrinsic efficacy of an agonist expressed relative to that of a standard agonist simply through the analysis of their respective concentrationresponse curves (Griffin et al., 2007; Ehlert, 2008). This estimate is known as the intrinsic relative activity (RA_{i}) of the agonist. In this report, we also show that the RA_{i} values of agonists, estimated from their concentrationresponse curves, are equivalent to the product of observed affinity and intrinsic efficacy determined through the method of partial receptor inactivation. These assays were carried out on Chinese hamster ovary cells stably expressing the human M_{2} muscarinic receptor (CHO hM_{2} cells) and on the ileum from M_{2} muscarinic receptor knockout mice (M_{2} KO), which is a convenient assay system for M_{3} activity. Thus, although observed affinity and intrinsic efficacy are complex functions of microscopic constants, their product expressed relative to that of a standard agonist yields a single fundamental parameter: namely, the microscopic affinity constant of the active state of the receptor expressed relative to that of the standard agonist. This constant is solely dependent on agonistreceptor complex and is easily estimated from the agonist concentrationresponse curve using global nonlinear regression analysis.
Materials and Methods
Mice. The muscarinic M_{2} receptor knockout (M_{2} KO) and the M_{2}/M_{3} double receptor knockout (M_{2}/M_{3} KO) were generated previously by Matsui et al. (2000, 2002) in C57BL/6 mice. Only male knockout mice were used in our studies.
Isolated Ileum. Mice were euthanized with CO_{2}, and the ileum was dissected out and mounted in an organ bath containing KrebsRingerbicarbonate buffer (124 mM NaCl, 5 mM KCl, 1.3 mM MgCl_{2}, 26 mM NaHCO_{3}, 1.2 mM KH_{2}PO_{4}, 1.8 mM CaCl_{2}, and 10 mM glucose) gassed with O_{2}/CO_{2} (19:1). Contractions were measured and recorded as described previously (Matsui et al., 2003). For each tissue, the contractile responses were normalized relative to the contraction elicited by KCl (50 mM). Competitive muscarinic antagonists were allowed to incubate with the tissue 30 min before measuring contractile response to an agonist. When N(2chloroethyl)4piperidinyl diphenylacetate (4DAMP mustard) was used, it was first cyclized at 37°C for 30 min to allow the formation of the aziridinium ion as described previously (Thomas et al., 1992). Isolated ileum was incubated with 4DAMP mustard (10 nM) for 20 to 40 min depending on the agonist. We always estimated the dissociation constant of the standard agonist in the same experiment in which that of a test agonist was estimated. The entire process was repeated for each test agonist.
cAMP Accumulation. CHO cells stably transfected with the human M_{2} muscarinic receptor (CHO hM_{2}) were provided by Acadia Pharmaceuticals (San Diego, CA) and cultured in Dulbecco's modified Eagle's medium containing 10% fetal bovine serum, penicillinstreptomycin (100 U/ml), and G418 (0.3 mg/ml). The cells were grown in a humidified atmosphere at 37°C with 5% CO_{2}. We used the [^{3}H]adenine prelabeling method of Schultz et al. (1972) to measure cAMP accumulation in detached CHO cells, essentially as described previously (Griffin et al., 2007). The incubations with agonist were carried out at 37°C for 12 min in KrebsRinger bicarbonate buffer containing isobutyl methylxanthine gassed with O_{2}/CO_{2} (19:1). In due course, [^{3}H]cAMP was separated from [^{3}H]ATP using a method similar to that described by Salomon et al. (1974). Before use, 4DAMP mustard was cyclized as described above in experiments on the ileum. The dissociation constant of each agonist was estimated in separate experiments using the method of partial receptor inactivation with 4DAMP mustard. For each test agonist, a control concentrationresponse curve to the standard agonist carbachol was measured in the same experiment.
Estimation of Observed Affinity and Relative Efficacy. A modification of Furchgott's method of partial receptor inactivation (Furchgott, 1966) was used to estimate the dissociation constants of agonists. After partial receptor inactivation with 4DAMP mustard, agonist concentrations were interpolated on the control concentrationresponse curve (X_{i}) corresponding to the responses () of the concentrationresponse curve measured after partial receptor inactivation. For experiments on cAMP accumulation in CHO hM_{2} cells, the agonist concentrations were interpolated using the following equation: where T, E_{max},EC_{50}, and n represent the parameters of the control concentrationresponse curve. These are defined as the amount of cAMP accumulation stimulated by forskolin in the absence of agonist (T), the maximal percentage of inhibition of cAMP accumulation elicited by the agonist (E_{max}), the concentration of agonist causing halfmaximal inhibition of cAMP accumulation (EC_{50}), and the Hill slope (n). These parameters were estimated from the concentrationresponse curve by nonlinear regression analysis using the doseresponse function in GraphPad Prism 4.0 (GraphPad Software Inc., San Diego, CA). For experiments on the ileum from M_{2} KO mice, the following equation was used to interpolate agonist concentrations: in which E_{max} and EC_{50} denote the maximal response and concentration of agonist eliciting halfmaximal response for the control concentrationresponse curve, n denotes the Hill slope, and B denotes the resting tension measured in the absence of agonist. After determining pairs of equiactive agonist concentrations from the concentrationresponse curves under control (X_{i}) and 4DAMP mustard treated conditions (), the data were fitted to the following equation by nonlinear regression analysis (Ehlert, 1987): in which K_{obs} denotes the observed dissociation constant of the agonist and q denotes the fraction of residual, active receptors after inactivation with 4DAMP mustard. The relative efficacy values of agonists were estimated using the principles outlined by Furchgott and Bursztyn (1967). Knowing the dissociation constants of the agonists, it is possible to plot response against log receptor occupancy for each assay system (i.e., CHO M_{2} and M_{2} KO ileum). The responseoccupancy plots of the standard agonist and each test agonist for a given assay system were fitted simultaneously by global nonlinear regression analysis to the following logistic equation: in which O denotes receptor occupancy, m denotes the transducer slope factor, M_{sys} denotes the maximum response of the system, and τ is a parameter in the operational model (Black and Leff, 1983) related to intrinsic efficacy (ϵ), receptor density (R_{T}) and the sensitivity of the signaling pathway (K_{E}) (i.e., τ = ϵR_{T}/K_{E}). Regression analysis was done sharing the estimate of M_{sys} and m among the curves and obtaining unique estimates of τ for each agonist. The efficacy of the test agonist X (ϵ_{X}) expressed relative to that of a standard (ϵ_{Y}) is simply calculated as
Estimation of RA_{i}. Two methods were used to estimate RA_{i}:a null method, and a method based on the operational model. The former is independent of the relationship between occupancy and response, and the latter is based on a logistic relationship between the two. The theoretical basis for estimation of RA_{i} is given by Griffin et al. (2007), and stepbystep instructions for its estimation are given by Ehlert (2008). For the null method, equiactive agonist concentrations for the standard (Y) and test (X) agonists are determined using a procedure similar to that described above for the Furchgott analysis. The logarithms of the equiactive agonist concentrations were fitted to the following equation by nonlinear regression analysis: in which log(K_{Y}) denotes the log dissociation constant of the standard agonist, log(P) denotes the log ratio of the dissociation constant of the test agonist divided by that of the standard [log(K_{X}/K_{Y})] and log RA_{i} is defined as the log of the product of the observed affinity (1/K_{X}) and intrinsic efficacy (ϵ_{X}) of the test agonist divided by that of the standard agonist (product of 1/K_{Y} and ϵ_{Y}, respectively): As described previously (Griffin et al., 2007), if the standard agonist is a full agonist, then there are an infinite number of parameter estimates that give a leastsquares fit. This infinite solution set consists of a single estimate of log(RA_{i}) and an infinitely large, inversely correlated set of values for log(K_{Y}) and log(P). To obtain the leastsquares fit, log(K_{Y}) is set to an arbitrarily high constant (e.g., 1) during regression analysis. Regression analysis yields the best estimate of log(RA_{i}) and an estimate of log(P) that is perturbed from its true value depending on the arbitrary constant to which K_{Y} was set during regression analysis. Regardless, it is possible to obtain an accurate estimate of K_{X} by simply multiplying the constant to which K_{Y} was fixed during regression analysis by the estimate of P. Therefore, using logarithms, log(K_{X}) is calculated as In summary, if the standard agonist is a full agonist, regression analysis yields estimates of log(RA_{i}) and, ultimately, the log dissociation constant of the test agonist [log(K_{X})].
For estimating RA_{i} using the operational model, the concentrationresponse curves of the standard and test agonists were analyzed simultaneously by global nonlinear regression analysis using the following two equations:
In these equations, M denotes the maximum response of the system, N denotes the transducer slope factor, and R denotes the ratio τ_{Y}/K_{Y}. Global nonlinear regression analysis is done fitting eq. 9 to the concentrationresponse curve of the standard agonist and eq. 10 to those of the test agonists. As described previously (Griffin et al., 2007), if the standard agonist is a full agonist, then there are an infinite number of parameter estimates that give a leastsquares fit. This infinite solution set, however, consists of a single estimate of log(RA_{i}) and an infinite set of log(K_{Y}) values bounded by the range, log(K_{Y}) is greater than or equal to its actual value. Therefore, it is possible to obtain a leastsquares fit by setting log(K_{Y}) to an arbitrarily high constant (e.g., 1) for the global nonlinear regression analysis. During regression analysis, the estimates of M and N are shared among the curves, and unique estimates of log(R), log(K_{X}), and log(RA_{i}) are obtained.
Materials. The muscarinic agonists including carbachol, oxotremorineM, oxotremorine, arecoline, pilocarpine, bethanechol, and McNA343, as well as isobutylmethylxanthine, tetrodotoxin, atropine, G 418, adenine and neutral alumina, were obtained from SigmaAldrich (St. Louis, MO). Other reagents were obtained from the following sources: pirenzepine (Research Biochemicals International, Natick, MA); Dulbecco's modified Eagle's medium and penicillinstreptomycin (Invitrogen, Carlsbad, CA); [^{3}H]adenine (PerkinElmer Life and Analytical Sciences, Waltham, MA); forskolin (Calbiochem, San Diego, CA); and Dowex AG 50WX4 (BioRad Laboratories, Hercules, CA). (S)Aceclidine was synthesized as described previously by Ringdahl et al. (1979).
Results
Mathematical Modeling
Relationship between RA_{i} and the Microscopic Affinity Constants of Agonists. As described previously, the RA_{i} value is equivalent to the product of observed affinity and intrinsic efficacy of an agonist expressed relative to that of a standard agonist. All that is required for estimation of RA_{i} are the concentrationresponse curves of the agonists. First, we show analytically that the product of observed affinity and intrinsic efficacy of an agonist, expressed relative to that of another agonist, is equivalent to the corresponding ratio of microscopic affinity constants of the agonists for the active state of the receptor. In our analysis, we assume that the receptor is in equilibrium between ground (R) and active states (R^{*} and R^{**}) as shown in Fig. 2. Two active states of the receptor were considered so that it would be possible to address the question of liganddirected signaling, which involves the preferential coupling of different active states to different coupling proteins (e.g., G proteins). The details of our solution are given under the Appendix, and a schematic summary of our results is shown in Fig. 3. The figure shows the active state of the agonistreceptor complex plotted against the concentration of agonist. Curves for two agonists, A and B, are shown. The maximum of their receptor activation functions is equivalent to observed intrinsic efficacy (ϵ), and the concentration of agonist required for halfmaximal formation of the active receptor complex is equivalent to the observed dissociation constant (K_{obs}). The mathematics described under Appendix show that the product of observed affinity (1/K_{obsB}) and intrinsic efficacy (ϵ_{B}) of agonist B divided by the corresponding product for A [(1/K_{obsA})ϵ_{A}] is equivalent to the microscopic affinity constant of agonist B for the active state (K_{b}) divided by that for agonist A ().
Next, we simulated agonist concentrationresponse curves and estimated the RA_{i} value of an agonist relative to a standard agonist. From this analysis, it is possible to determine the dependence of the RA_{i} value on the microscopic affinity constants of the agonists. Our model is based on the assumption that the stimulus (i.e., product of receptor occupancy and observed intrinsic efficacy) (Furchgott, 1966) is proportional to the amount of active, agonistreceptor complex in the form of a quaternary complex consisting of agonist, receptor, G protein, and guanine nucleotide (AR^{*}GX) (Ehlert and Rathbun, 1990; Ehlert, 2000). We used methods described previously to simulate the amount of agonist complex in the AR^{*}GX complex based on theoretical values for the microscopic affinity constants of the agonist for different states of the receptor (Ehlert, 2008). To broaden the relevance of the model, we considered a receptor with two different active states, each interacting with a different G protein. This condition accounts for the phenomenon of liganddirected signaling (Leff et al., 1997). A pictorial representation of the model is shown in Fig. 4, and the details of the calculations and definitions of the parameters are given in Ehlert (2008). Additional details of the model for a single active state are described in Ehlert (2000), and a description of the equation used to do the modeling is listed in the Appendix.
Figure 5 illustrates the results of our simulations, which were done with the concentration of GTP set at a nearly saturating value (1 mM). The parameters of the model where chosen so that agonist A stimulates signaling through the two G proteins, G_{1} and G_{2}, to the same extent, whereas agonist B exhibits a preference for signaling through G_{1}. Figure 5, a to d, show theoretical predictions for the two agonists (A and B) acting on a receptor in a dynamic equilibrium with G_{1} and G_{2} (dynamic equilibrium case). In this example, the microscopic affinity constants of agonist A and B for the ground state of the receptor ( and K_{a}, respectively) were set to the same value (i.e., 10^{5}). Likewise, the microscopic affinity constants of A and B for the active state (R^{*}) that preferentially interacts with G_{1} are also set to the same value ( = K_{b} = 10^{9}). In contrast, the microscopic affinity constant of agonist A for the active state (R^{**}) that preferentially interacts with G_{2} () was set to 10^{9}, whereas the corresponding constant for agonist B was assigned a lower value (K_{c} = 10^{8}). Using these microscopic constants and others described under Appendix (independent variables), it is possible to simulate the amount of agonistreceptorG protein complex in the active state bound with guanine nucleotide, as well as the downstream concentrationresponse curve (dependent variables). This output was generated using equations described under Appendix. This output also yields the dependent parameters, observed affinity and intrinsic efficacy, as well as the EC_{50} and E_{max} values of the downstream, concentrationresponse curve. Figure 5, a and b, shows the output from the system through the G_{1} pathway. The amount of the active state (R^{*}) of the receptor in the form of quaternary complex is shown in Fig. 5a for agonists A and B (AR^{*}G_{1}X and BR^{*}G_{1}X, respectively). The maximal amount of active quaternary complex formed by agonist B (56%) is greater than that of A (37%) even though the selectivity of agonists A and B for the R^{*} state is the same ( = K_{b}/K_{a} = 10^{4}). The maximum is proportional to observed intrinsic efficacy of the agonistreceptor complex for signaling through G_{1} (ϵ_{1}). The different ϵ_{1} values of the agonists is caused by competition of the two G proteins with the two different active states of the receptor (R^{*} and R^{**}). With regard to agonist A, this competition is equal because = 10^{4}. In contrast, with agonist B, the competition is shifted in favor of G_{1} because > . The EC_{50} values of the agonists for halfmaximal formation of the quaternary complex with G_{1} are equivalent to the observed dissociation constant (K_{obs}). When expressed as negative logarithms (pK_{obs}) the values for agonist A and B are 5.65 and 5.47, respectively. K_{obs} is a function of the microscopic constants (K_{a}, K_{b}, and K_{c}) and other parameters. Because the K_{c} value of agonist B is 10fold lower than that of A (), then the K_{obs} value of agonist B exhibits lower potency than that of A. The plot of quaternary complex as a function of the agonist concentration represents the stimulus, and this function was substituted into the operational model (Black and Leff, 1983) to generate a theoretical concentrationresponse curve for each agonist (Fig. 5b). These were generated with an operational model having a moderately sensitive signaling cascade (K_{E} = 0.03; Fig. 5), resulting in a receptor reserve. Even though the stimuli generated by the agonists differ, the resulting concentrationresponse curves for signaling through G_{1} are identical, with EC_{50} values of 0.2 μM and E_{max} values of 100%. The indistinguishable curves yield an RA_{i} value of agonist B relative to A of 1.0. This value is equivalent to the ratio of the microscopic affinity constant of agonist B for the active state of the receptor (K_{b}) divided by the corresponding constant () for A (i.e., K_{b}/ = 10^{9}/10^{9} = 1.0). It can also be shown that the RA_{i} is equivalent to the product of affinity (1/K_{obsB}) and intrinsic efficacy (ϵ_{1B}) of agonist B divided by the corresponding product for agonist A. Relative to agonist A, the higher intrinsic efficacy of agonist B is offset by a lower observed affinity, resulting in an RA_{i} value of 1.0. Table 1 summarizes the dependent parameters of the simulated data in Fig. 5, a and b.
Figure 5, c and d, summarizes the theoretical curves for responses mediated through the R^{**} active conformation of the receptor, which preferentially signals through a different G protein (G_{2}). Figure 5c shows the theoretical curves for the active quaternary complex of each agonist (AR^{**}G_{2}X and BR^{**}G_{2}X) plotted against the agonist concentration. Because the receptor is in a dynamic equilibrium with two G proteins, the K_{obs} values of the agonists are the same as those shown in Fig. 5a for signaling through G_{1}. In contrast the maximal amount of quaternary complex formed by agonist B (BR^{**}G_{2}X_{max}) is much less than that shown in Fig. 5a for the corresponding G_{1} complex (BR^{*}G_{1}X_{max}), which correlates with the lower K_{c}/K_{a} ratio (10^{3}) compared with K_{b}/K_{a} (10^{4}). Figure 5d shows the theoretical concentrationresponse curves of the two agonists for eliciting a response downstream from G_{2}. The lower activity of agonist B is readily apparent from the figure, and its RA_{i} value relative to agonist A was estimated to be 0.1. This RA_{i} value accurately predicts the ratio of the microscopic affinity constant of agonist B for the active state (K_{c}) relative to that of agonist A () (i.e., K_{c}/ = 10^{8}/10^{9} = 0.1). The theoretical curves shown in Fig. 5, a to d, show that agonist B directs signaling through G_{1} relative to G_{2} and that this selectivity is accurately reflected in its higher RA_{i} value for the G_{1} response relative to that of G_{2}.
These simulations were repeated with the same parameters, but with the equilibrium between the receptor and G proteins segregated into two distinct equilibriums: one for R and G_{1} (Fig. 5, e and f) and another for R and G_{2} (Fig. 5, g and h) (segregated equilibrium case). The results were qualitatively similar to those shown for the dynamic equilibrium case. One difference is that the K_{obs} value of agonist B for eliciting responses through G_{1} is different from that for eliciting responses through G_{2}. Nonetheless, the RA_{i} values of agonist B for eliciting responses through G_{1} are the same in the both dynamic equilibrium and segregation cases (Fig. 5, b and f), and the same is true for the RA_{i} values for G_{2} responses (Fig. 5, d and h).
Table 1 summarizes the results of the simulations shown in Fig. 5. In each case, it can be shown that the product of observed affinity and intrinsic efficacy of agonist B expressed relative to A is equivalent to RA_{i}. In addition, the latter estimate is equivalent to the ratio of the microscopic affinity constant of agonist B for the active state expressed relative to that of A, and that the RA_{i} value is unaffected by segregation of the G proteins into two separate pools.
Summary of Simulations Using a Diverse Range of Parameter Values. We investigated a wide range of parameter values (microscopic constants) for the model shown in Fig. 5 to ensure that our conclusions were not dependent on the particular parameters used in Fig. 5. These additional simulations showed the same result: namely, that the RA_{i} value is equivalent to the ratio of microscopic affinity constants of the agonists for the active state of the receptor. In these simulations, we kept the level of constitutive activity to a minimum and the affinity of guanine nucleotide for the G protein much lower when the activated receptor is associated with it compared with the inactive receptor. This condition results in a high degree of negative cooperativity between the binding of guanine nucleotide (GDP or GTP) and a highly efficacious agonist with the receptorguanine nucleotide complex, which is a basic requirement for agonistinduced guanine nucleotide exchange. With these two constraints, we found that RA_{i} was equivalent to the ratio of microscopic affinity constants of the two agonists for the active state of the receptor regardless of the parameter values, including a variation in the concentration of GTP.
Our analysis is also appropriate for a receptor system exhibiting substantial constitutive receptor activity, because the basis of our approach rests on the agonistinduced response above basal activity. Using our method, however, it would not be possible to compare the activity of an agonist with that of an inverse agonist. Nonetheless, it would be possible to use an analogous approach to compare the activity of a series of inverse agonists with a standard inverse agonist in a system with constitutive receptor activity. In this instance, if the response were defined as the inhibition of basal activity, then the corresponding measure of RA_{i} would be equivalent to a relative estimate of the microscopic affinity constant of the ground state of the receptor.
A potential criticism of our modeling is the use of a simple equilibrium constant to describe the interaction between receptor and G protein within the membrane. This type of constant is usually used to define the relationship between the concentrations of bound and free ligand and receptor in solution. We do not envision this constant in the same light, but rather as a simple constant describing a reversible interaction between two proteins in the membrane. The twodimensional constraints of the membrane and the involvement of potential scaffolding proteins in the interaction raises the issue of a possible limiting supply of G protein in the local environment. We explored this issue by taking into account depletion in the concentration of G protein as described previously (Ehlert 2000) and explored a range of ratios of G protein to receptor, including very low ratios (0.1). Under this condition, we also varied the other parameters described above, and in each case, found that the RA_{i} estimate was essentially equivalent to the ratio of microscopic affinity constants of the two agonists for the active state of the receptor.
Biological Data
In this section, we show that the product of observed affinity and intrinsic efficacy of an agonist, estimated by Furchgott's method of partial receptor inactivation, is equivalent to the RA_{i} value estimated from the agonist concentrationresponse curve. These studies investigated the human M_{2} receptor expressed in CHO cells and the mouse M_{3} receptor in the ileum from M_{2} muscarinic receptor KO mice.
CHOM_{2} Cells. CHO hM_{2} cells were used as a model system for studying the activity of muscarinic agonists at the human M_{2} muscarinic receptor. All of the agonists tested elicited a concentrationdependent inhibition of forskolinstimulated cAMP accumulation (Fig. 6a). The average maximal inhibition ± S.E.M. caused by carbachol was 64.6 ± 1.2% of the stimulation elicited by forskolin (10 μM). Most of the agonists tested behaved as full agonists and elicited maximal responses ranging from 95.1 to 107.9% that of carbachol. OxotremorineM and oxotremorine were the most potent followed by carbachol, (S)aceclidine, arecoline, pilocarpine, bethanechol, and McNA343. Pilocarpine and McNA343 behaved as partial agonists (E_{max} values, 70.6 ± 2.5 and 36.7 ± 2.7% of that of carbachol, respectively). The EC_{50} and E_{max} values of the agonists are listed in Table 2. The Hill slope of pilocarpine seemed unusually steep. We have no explanation for this behavior and assume that it is due to experimental error.
To estimate the observed dissociation constants of the agonists, we used a variant of Furchgott analysis (Ehlert, 1987) to examine the relationship between equivalent tissue response before and after partial receptor inactivation with the irreversible muscarinic antagonist 4DAMP mustard. Figure 6b shows an example of the effect of 4DAMP mustard treatment on responses elicited by carbachol and oxotremorine. Incubation of CHO hM_{2} cells with 4DAMP mustard (40 nM) for 20 min followed by washing caused an increase in the EC_{50} and a decrease in the E_{max} values of all of the agonists except pilocarpine and McNA343 (Table 2). The responses to the latter agonists were completely inhibited by 4DAMP mustard treatment. To determine the affinities of the full agonists, we interpolated agonist concentrations on the control concentrationresponse curve corresponding to equivalent responses on the curve measured after 4DAMP mustard treatment. The average equiactive agonist concentrations are plotted in Fig. 6c for all of the full agonists. Regression analysis was used to fit eq. 3 to the corresponding data from each experiments to yield an estimate of the dissociation constant of the agonist and that of the residual proportion of receptors not inactivated by 4DAMP mustard (q). The average values of these estimates for each agonist are listed in Table 2. The effectiveness of 4DAMP mustard in inactivating the response varied in different experiments with the different agonists as manifest as variation in the q values. We assume that this variation is due to experimental error and variation in the concentration of the aziridinium ion of 4DAMP mustard in each experiment on a different agonist. The dissociation constants of the partial agonists pilocarpine and McNA343 were estimated through simultaneous analysis of their data with those of carbachol using RA_{i} analysis described under Materials and Methods. Knowing the affinities of the muscarinic agonists, it is possible to estimate receptor occupancy and, hence, to establish the relationship between occupancy and response as shown in Fig. 6d for all of the agonists. The efficacy of each agonist expressed relative to that of carbachol was estimated from this type of plot using nonlinear regression analysis (eq. 4) followed by substitution of the corresponding τ values into eq. 5 (Table 2). Regression analysis was done for each agonist using its own control occupancyresponse relationship for carbachol. The estimates of m and M_{sys}, expressed as the percentage of inhibition of forskolinstimulated cAMP accumulation, did not differ significantly among the agonists, and the average estimates ± S.E.M. were M_{sys}, 69.6 ± 4.1%, and m, 0.98 ± 0.06.
Isolated Ileum. The isolated ileum from M_{2} KO mice was used as an assay system for M_{3} muscarinic receptor activity. It is well known that the M_{3} receptor elicits a direct contractile response in the ileum from rodents (Eglen, 1997; Ehlert et al., 1997a). This tissue also contains an abundance of M_{2} receptors, which mediate contractile responses contingent upon activation of other receptors, including the M_{3} (Ehlert, 2003). To avoid a possible contribution of the M_{2} receptor, we measured contractile activity in ileum from M_{2} KO mice (Fig. 7a). The data from each experiment were first normalized relative to the contractile response elicited by 50 mM KCl. The average ± S.E.M. for the E_{max} of carbachol was 205.6 ± 7.1% relative to the KCl response. The data were normalized further by expressing the contractile response relative to the E_{max} of carbachol. A summary of the data can be found in Table 3, which lists EC_{50}, E_{max}, and Hill slopes. Carbachol, oxotremorineM, oxotremorine, and (S)aceclidine behaved as full agonists (E_{max} values, 98.4123% that of carbachol) whereas McNA343 behaved as a partial agonist (E_{max} value, 17.9 ± 1.4% that of carbachol).
To estimate the observed affinity constants of the agonists, we used the method of partial receptor inactivation as described above. Figure 7b shows an example of the effect of 4DAMP mustard treatment (10 nM) on responses elicited by carbachol and (S)aceclidine after incubation with the mustard for 40 and 20 min, respectively. Treatment with 4DAMP mustard caused an increase in the EC_{50} and a decrease in the E_{max} values of all agonists except for McNA343 (Table 3), whose responses were completely eliminated by 4DAMP mustard. Equiactive agonist concentrations before and after 4DAMP mustard treatment were estimated for each agonist as described above, and the average values are plotted in Fig. 7c. Equation 3 was fitted to the corresponding data from each experiment using nonlinear regression analysis to yield estimates of the affinity constant of the agonist and the residual fraction of receptors (q). The average values for each agonist are listed in Table 3. The affinity of McNA343 was estimated through simultaneous analysis of its data together with those of carbachol using the RA_{i} analysis as described above. Knowing the affinities of the muscarinic agonists, it is possible to plot the response against receptor occupancy (Fig. 7d). The efficacies of all the agonists relative to that of carbachol were estimated from this type of plot using eqs. 4 and 5 as described above (Table 3).
In most instances, the EC_{50} values of the agonists after 4DAMP mustard treatment were larger than the corresponding observed dissociation constants. This behavior is consistent with the existence of a threshold for contraction in the ileum (Furchgott, 1966). Knowing the relative efficacy of McNA343 (0.089) and that 4DAMP mustard treatment (10 nM, 20 min; q = 0.46) eliminated the response to McNA343, we estimate the minimum value of this threshold to be approximately 4% of the maximal stimulus elicited to carbachol.
Comparison of RA_{i} with the Product of Affinity and Efficacy. As explained previously, it is possible to estimate the product of the affinity and efficacy of an agonist expressed relative to that of a standard agonist, simply through the analysis of their respective concentrationresponse curves (Ehlert et al., 1999; Griffin et al., 2007). This estimate is known as intrinsic relative activity (RA_{i}). We estimated the RA_{i} values of agonists from the control concentrationresponse curves measured in CHO hM_{2} cells and in the ileum from M_{2} KO mice. We used two different methods for estimation of RA_{i}. The first is a null method, which lacks any assumption about the relationship between the stimulus and response, and the second is based on a logistic relationship between stimulus and response (operational model; Black and Leff, 1983). Because all of the concentrationresponse curves resembled symmetrical logistic functions, the condition for the use of the operational model seems to have been met, and we would expect little difference in the estimates of RA_{i} using both methods. The RA_{i} values of all of the agonists were estimated using the two methods, and these are listed in Tables 4 and 5 for M_{2} and M_{3} assays, respectively.
A relative estimate of the product of affinity and efficacy was calculated for each agonist. This was done by multiplying the affinity constant of each agonist by its relative efficacy and dividing this product by the corresponding product for carbachol. These estimates are listed in Tables 4 (M_{2} assay, i.e., CHO hM_{2}) and 5 (M_{3} assay, i.e., ileum). Figure 8a shows a histogram of data from the CHO hM_{2} assay, comparing the RA_{i} estimate of each agonist with its relative estimate of the product of affinity and efficacy. Two estimates of RA_{i} are shown for each agonist, corresponding to the two methods of estimation. There is general agreement between the two estimates of RA_{i} for each agonist, both of which are approximately equal to the estimate of the product of affinity and efficacy. We did not include the product of affinity and efficacy for pilocarpine and McNA343 in Fig. 8 because these estimates were made through analysis of the same data used to estimate RA_{i}. It can be shown that the regression equations used to estimate the affinity of partial agonists are degenerate forms of those used to estimate the RA_{i} of partial agonists. The agreement between RA_{i} and the product of affinity and efficacy for pilocarpine and McNA343 shown in Tables 4 and 5, therefore, is trivial. Rather, Fig. 8 illustrates that when the RA_{i} is calculated from a concentrationresponse curve, the estimate is similar to the product of affinity and efficacy calculated from a different set of data consisting of responses measured before and after partial receptor inactivation.
We also compared the RA_{i} estimates of the muscarinic agonists in the mouse M_{2} KO ileum (M_{3} assay) with those estimated previously in studies on the phosphoinositide response in CHO cells transfected with the hM_{3} receptor (Fig. 8b). There is general agreement among all of the estimates of RA_{i} for each agonist with the exception of McNA343, whose RA_{i} is substantially greater in the mouse ileum compared with that of the CHO hM_{3} cell (Ehlert et al., 1999). These data suggest that at least part of the response to McNA343 in the mouse ileum is mediated through a muscarinic receptor distinct from the M_{3}. Evidence presented in the Supplemental Data support this hypothesis.
Discussion
Our overall hypothesis is that it is possible to calculate a relative estimate of the product of observed affinity and intrinsic efficacy of an agonist simply through the analysis of its concentrationresponse curve and that this estimate is a relative measure of the microscopic affinity constant of the agonist for the active state of the receptor. When calculated in CHO hM_{2} cells by the method of partial receptor inactivation, our estimates of the observed affinity constants of the agonists oxotremorine and oxotremorineM were moderately higher (pK_{obs} values: 6.54 and 6.15, respectively) than those estimated previously on homogenates of the rabbit myocardium (5.66 and 5.12) in the presence of GTP (0.1 mM) (Ehlert, 1987). The mammalian myocardium is known to express an abundance of M_{2} muscarinic receptors (Waelbroeck et al., 1986). Increasing the concentration of GTP reduces the observed affinity of agonists at the M_{2} receptor and increases the maximal amount of GDPGTP exchange at G_{i}, which should increase agonist efficacy (Ehlert and Rathbun, 1990). Perhaps the higher affinity observed here may indicate that the concentration of GTP in the cytosol of CHO cells is lower than 0.1 mM. When the concentration of GTP is lower, it is easier for the agonist to induce the active conformation of the receptor, and under such conditions, the most efficacious agonists achieve maximal receptor activation. This condition may have been met in the present study because the relative efficacy values of arecoline, carbachol, oxotremorineM, and oxotremorine seem to vary randomly around the value of the highly efficacious standard, carbachol (relative efficacy = 1; see Table 2). In the rabbit myocardium, however, the agonists exhibited the following rank order for relative efficacy: oxotremorineM (3.6) > carbachol (2.3) > oxotremorine (1.2) (Ehlert, 1985). Because GTP has opposite effects on observed affinity and intrinsic efficacy (Ehlert and Rathbun, 1990), a variation in the concentration of GTP should have little effect on the product of these two parameters and, hence, on the estimate of RA_{i}.
The estimates of the dissociation constants and relative efficacies of carbachol, oxotremorine, and oxotremorineM in the M_{2} KO mouse are similar to those estimated in the isolated guinea pig ileum (Ringdahl and Jenden, 1983; Ringdahl, 1984, 1985). In addition, the RA_{i} values of muscarinic agonists are consistent with those measured in the CHO hM_{3} cell, except for McNA343. Our results with pirenzepine and tetrodotoxin indicate that the response to McNA343 includes activation of another muscarinic receptor subtype.
In our studies on CHO M_{2} cells and the mouse ileum, we estimated the RA_{i} values from the control concentrationresponse curve and from the individual parameters of observed affinity and relative efficacy using Furchgott analysis of the control data and that obtained after partial receptor inactivation. We found that the product of observed affinity and intrinsic efficacy was similar to the RA_{i} estimate as predicted by theory. From the perspective of validating the RA_{i} estimate, this approach may seem like a tautology because the model has only two degrees of freedom. If RA_{i} (i.e., the product) and the observed affinity constant (i.e., a factor) are calculated first, and the relative efficacy is estimated using the control concentrationresponse curve together with receptor occupancy based on observed affinity, then there is a natural tendency for the estimate of relative efficacy to equal the RA_{i} value divided by observed affinity, and hence, for the product of observed affinity and efficacy to equal the estimate of RA_{i}. The basis for this tautology, however, rests on the assumption that the theory on which the RA_{i} is based is valid in the first place. Our other reasons for estimating the observed affinities and intrinsic efficacies were to determine the individual components of the RA_{i} estimate for the specific agonists tested and the extent to which a practical application of the two methods yielded similar results. We found reasonable agreement between the two approaches.
In comparing the contractile activity of muscarinic agonists in the ileum from the mouse and guinea pig, it is important to note that the guinea pig ileum is much more sensitive. For example, the potency of oxotremorine in the guinea pig ileum (Ringdahl, 1985; pEC_{50}, 7.87) is approximately 30fold greater than that measured in the mouse M_{2} KO ileum (pEC_{50}, 6.41; see Table 4). This large difference cannot be attributed to the lack of the M_{2} receptor in the M_{2} KO mouse because there is little difference in the activity of carbachol in wildtype and M_{2} KO mouse ileum (Matsui et al., 2002). In contrast, the potencies and relative E_{max} values of McNA343 in the mouse M_{2} KO and guinea pig ilea are approximately the same (compare this study with Ehlert et al., 1999). Thus, the M_{3} contractile response of the mouse ileum is very insensitive, which explains perhaps why an M_{1} contractile mechanism for McNA343 is unmasked in this tissue (see Supplementary Data). The nature of the putative M_{1} response and its interaction with the M_{3} response is unclear, and it is impossible to estimate the relative contribution of a putative M_{1} component to the contractile response accurately from our antagonism studies with pirenzepine. It has been shown previously that the competitive inhibition of a response mediated by more than one receptor is complex, and the extent of the antagonism depends on the nature of the interaction between the two receptors (Ehlert, 2003). Nevertheless, our analysis indicates that pirenzepine causes a greater antagonism of the response to McNA343 relative to that of carbachol in the M_{2} KO mouse ileum (Supplementary Data). In addition, it seems that part of the response to McNA343 is neurogenic, as indicated by the small inhibitory effect of tetrodotoxin (Supplementary Data). It would seem, therefore, that our RA_{i} estimate for McNA343 is not representative of a pure M_{3} response but rather of a mixed response.
We also show that the estimate of RA_{i} is equivalent to the microscopic affinity constant of an agonist for the active state of the receptor expressed relative to that of the standard agonist (Table 5). For a G proteincoupled receptor, the active state exhibits selectivity for G proteins or other coupling proteins (e.g., G proteincoupled receptor kinase), and in some instances, it seems that agonists may select unique conformations that recruit different G proteins, resulting in the phenomenon of liganddirected signaling (Urban et al., 2007). Our mathematical modeling shows that the RA_{i} estimate accurately reflects the microscopic affinity constant of the agonist for the active state of the receptor under these conditions. Leff et al. (1997) originally proposed a model for liganddirected signaling, based on two different active conformations of the receptor that interact with two different G proteins. If an agonist exhibits a preference for one active state, it will tend to direct signaling through the corresponding G protein. Leff et al. (1997) showed that the stimulus function for a given pathway differs depending on whether the receptor is in a dynamic equilibrium with both G proteins at the same time or whether the two different receptorG protein signaling pathways are segregated in different cells. In the former case, an agonist that preferentially directs signaling through one pathway would exhibit the same observed affinity for the two pathways but a difference in intrinsic efficacy, whereas in the latter case, the same agonist would exhibit differences in both observed affinity and intrinsic efficacy. We show that the estimate of RA_{i} is unaffected by segregation or dynamic equilibrium, and in both cases, it accurately reflects a relative estimate of the microscopic affinity constant of the corresponding active state.
The phenomenon of liganddirected signaling has led some to conclude that the transduction pathway can determine the activity of the agonist (Urban et al., 2007). Of course, the nature of the stimulus and concentrationresponse curve elicited by a specific agonistreceptor complex can change under conditions of liganddirected signaling. Likewise, the RA_{i} estimate for an agonist that directs signaling can change at the same receptor depending on the G protein that mediates the response. Under this condition, however, it is important to note that the RA_{i} estimate accurately reflects the microscopic affinity constant of the agonist for the active receptor conformation eliciting the response, and hence, it is entirely receptordependent. Thus, rather than modifying signaling, G proteins simply provide a window for estimating the affinity constants of agonists for different effectorselective, active conformations. When viewed from this perspective, the phenomenon of liganddirected signaling is determined by the agonistreceptor complex yet is manifest through different coupling proteins.
Appendix
RA_{i} and the Microscopic Affinity Constant of the Active State of the Receptor. The first part of this Appendix describes the derivation of RA_{i} in terms of the microscopic affinity constant of the agonist for the active state of the receptor. In this analysis, we consider the case of liganddirected signaling, where there are two distinct active receptor states (R^{*} and R^{**}) that trigger responses through different G proteins. The model is shown schematically in Fig. 2. The parameter K_{a} denotes the microscopic affinity constant of agonist A for the ground state of the receptor (R), and K_{b} and K_{c} denote the microscopic affinity constants of the two active states. These affinity constants are defined in inverse molarity units (e.g., K_{a} = [AR]/[A][R]). K_{q} and K_{r} define the equilibrium between the free forms of the receptor (K_{q} = [R^{*}]/[R] and K_{r} = [R^{**}]/[R]).
We begin by deriving an equation expressing the fraction of occupied receptor in the active state (R^{*}) as a function of the agonist concentration. This function is equivalent to the stimulus as defined by Furchgott (1966). Its maximum is equivalent to intrinsic efficacy (ϵ), and the concentration of agonist eliciting halfmaximal formation of the active state is equivalent to the observed dissociation constant (K_{obs}). Then we solve the function for ϵ and K_{obs}, and substitute these functions into eq. 7, which defines RA_{i} in terms of observed affinity and intrinsic efficacy. We repeat this process for the other active state (R^{**}), as well as for the simple situation when there is only one active state.
The fractional amount of agonistreceptor complex in the active state R^{*} is defined as in which R_{T} denotes the total receptor population. Using the definitions of the microscopic affinity constants, it is possible to replace each agonistreceptor complex on the right side of the equation with an expression in terms of A, R, and microscopic affinity constants. For example AR^{*} = AK_{b}K_{q}R. Making these substitutions yields Simplifying yields
This equation can be arranged in the following form, which is equivalent to Furchgott's definition of the stimulus (Furchgott, 1966):
The variables stimulus_{1}, ϵ_{1}, and K_{obs1} denote the stimulus, observed intrinsic efficacy, and observed dissociation constant of the agonist for triggering a response through R^{*}, respectively.
Substituting in eqs. 16 and 15 for the observed affinity (K_{obs1}) and intrinsic efficacy (ϵ_{1}) of the test agonist and standard agonist into eq. 7 under Materials and Methods yields an equation expressing RA_{i} in terms of the microscopic affinity constants of the various receptor states:
In this equation, the microscopic constants of the test agonist are denoted in the normal manner (K_{a}, K_{b}, and K_{c}), whereas those of the standard agonist are denoted with an apostrophe (, , and ). This equation simplifies to This equation shows that RA_{i} value of an agonist for eliciting a response through the R^{*} state of the receptor (RA_{i1}) is simply equivalent to the ratio of the microscopic affinity constant of the test agonist for the active state of the receptor (R^{*}) divided by that of the standard agonist.
Using an analogous strategy for the R^{**} state, it can be shown that the fractional amount of agonist bound in the AR^{**} is given by
in which
In eqs. 20 to 22, stimulus_{2}, ϵ_{2}, and K_{obs2} denote the stimulus, observed intrinsic efficacy, and observed dissociation constant of the agonist for triggering a response through R^{**}, respectively. The foregoing equations for the observed affinity (K_{obs2}) and intrinsic efficacy (ϵ_{2}) are substituted into eq. 7 under Materials and Methods to yield an equation expressing RA_{i} in terms of the microscopic affinity constants of the various receptor states:
in which K_{a}, K_{b}, and K_{c} denote the microscopic constant of the test agonist, and , , and denote those of the standard agonist. This equation reduces to This relationship between RA_{i} and the microscopic affinity constants of the agonist for the active state of the receptor can also be shown to apply in the simple case where there is only one active conformation of the receptor (R^{*}). In this situation, the amount of agonistreceptor complex in the active state is given by This equation can be rearranged into the following form to define the stimulus in which Substituting these equations for ϵ and K_{obs} in to eq. 7 yields an equation for the RA_{i} value in terms of the microscopic constants of the test agonist (K_{a}, K_{b}) and standard agonist (, ):
In summary, eqs. 18, 24, and 30 demonstrate that the RA_{i} estimate of an agonist is equivalent to its microscopic affinity constant for the active state of the receptor divided by that of the standard agonist.
Simulation of LigandDirected Signaling. Here we list the equations used to simulate the stimulus functions illustrated in Fig. 5 (i.e., eqs. 31 and 32). The model is shown schematically in Fig. 4, which represents a receptor in equilibrium with two G proteins in the presence of GTP. The derivation of these mathematics has been described previously (Ehlert, 2008), and the relevant equations for generating the plots of the quaternary complex against the agonist concentration are listed below for convenience. In these equations, K_{a}, K_{b}, K_{c}, K_{q}, and K_{r} are defined as described above. The equations describing the amount agonist (A) bound in the form of the two quaternary complexes (AR^{*}G_{1}X and AR^{**}G_{2}X), consisting of agonist, the active state of the receptor (R^{*} and R^{**}), G protein (G_{1} and G_{2}), and guanine nucleotide (X) are
and R_{T} denotes the total amount of receptor and G_{1T} and G_{2T} denote the total amount of G_{1} and G_{2} in the membrane. The cooperativity constants and microscopic constants for the different receptor complexes are defined as
The microscopic constants describing the equilibrium between the various states of the receptor, the two G proteins (G_{1} and G_{2}) and guanine nucleotide (X) are
Footnotes

This work was supported by the National Institutes of Health [Grant GM 69829].

ABBREVIATIONS: GPCR, G proteincoupled receptor; RA_{i}, intrinsic relative activity; 4DAMP mustard, N(2chloroethyl)4piperidinyldiphenyl acetate; CHO, Chinese hamster ovary; McNA343, 4(mchlorophenylcarbamoyloxy)2butynyltrimethylammonium; KO, knockout.

↵ The online version of this article (available at http://molpharm.aspetjournals.org) contains supplemental material.
 Received August 11, 2008.
 Accepted November 6, 2008.
 The American Society for Pharmacology and Experimental Therapeutics