Abstract
We developed novel methods for analyzing the concentrationresponse curve of an agonist to estimate the product of observed affinity and intrinsic efficacy, expressed relative to that of a standard agonist. This parameter, termed intrinsic relative activity (RA_{i}), is most applicable for the analysis of responses at G proteincoupled receptors. RA_{i} is equivalent to the potency ratios that agonists would exhibit in a hypothetical, highly sensitive assay in which all agonists behave as full agonists, even those with little intrinsic efficacy. We investigated muscarinic responses at the M_{2} receptor, including stimulation of phosphoinositide hydrolysis through G_{α15} in HEK 293T cells, inhibition of cAMP accumulation through G_{i} in Chinese hamster ovary (CHO) cells, and stimulation of cAMP accumulation through G_{s} in CHO cells treated with pertussis toxin. The RA_{i} values of carbachol, oxotremorineM, and the enantiomers of aceclidine were approximately the same in the three assay systems. In contrast, the activity of 4[[N[3chlorophenyl]carbamoy]oxy2butynyl]trimethylammonium chloride (McNA343) was ∼10fold greater at M_{2} receptors coupled to G_{α15} in HEK 293T cells compared with M_{2} receptors coupled to G_{i} in the same cells or in CHO cells. Our results show that the RA_{i} estimate is a useful measure for quantifying agonist activity across different assay systems and for detecting agonist directed signaling.
Drug discovery often involves testing compounds in isolated tissues and high throughput assays to determine activity at target receptors. In the case of agonists, the measured parameters are usually EC_{50} and E_{max}, whereas the parameters of greater relevance to drug design are observed affinity and intrinsic efficacy.^{1} If the receptor is a ligandgated ion channel, then receptor activation can be measured directly as whole cell current, and the EC_{50} and E_{max} are reasonable estimates of observed affinity and intrinsic efficacy provided that desensitization is not excessive. It would seem that, in most instances, the observed affinity and intrinsic efficacy of an agonist for a ligandgated ion channel would be constant, regardless of the tissue or cell in which the receptor is expressed.
The situation is more complex for a G proteincoupled receptor (GPCR). First, GPCRs are inactive in isolation and must interact with a G protein to elicit a response. It is possible that the G protein with which the receptor interacts selects for a receptor conformation having a unique agonist profile and that the observed affinity and intrinsic efficacy of the agonistreceptor complex may be G proteinspecific or influenced by other proteins interacting with the receptor (Leff et al., 1997; Berg et al., 1998). Second, because it is difficult to measure receptor activation directly, most assays involve measuring a downstream response, and the corresponding EC_{50} and E_{max} values may vary, depending on the point in the signaling cascade at which the experimenter measures the response. Therefore, the magnitude of the response elicited by a GPCR is usually not proportional to receptor activation, and EC_{50} and E_{max} are not equivalent to observed affinity and intrinsic efficacy. Rather, EC_{50} and E_{max} depend on the receptor, proteins interacting with the receptor, and a long list of downstream elements that affect the sensitivity of the response. Because the number of nonreceptor entities that affect observed affinity and intrinsic efficacy is small compared with the long list of cellular components affecting EC_{50} and E_{max}, it would seem that observed affinity and intrinsic efficacy would be useful parameters for the characterization of the agonistGPCR complex across different assays. The estimation of these parameters, however, requires additional data not obtained in high throughput screens. Fortunately, it is possible to estimate the product of observed affinity and intrinsic efficacy of an agonist relative to that of a standard agonist just through analysis of the concentrationresponse curves (Ehlert et al., 1999; Ehlert and Griffin, 2001). This product is equivalent to the potency ratio that an agonist would exhibit relative to a standard agonist in a hypothetical, highly sensitive assay in which both agonists behave as full agonists, even if one or both have very little observed intrinsic efficacy. When viewed from this perspective, estimation of the product of observed affinity and efficacy provides a means of converting potential differences in both the EC_{50} and E_{max} values of agonists into a single, relative measurement, analogous to the potency ratio. Therefore, the product of observed affinity and intrinsic efficacy should be a useful, receptordependent parameter for measuring agonist activity at GPCRs.
Here we describe methods for analyzing agonist concentrationresponse curves to estimate the product of observed affinity and intrinsic efficacy expressed relative to that of a standard agonist. We refer to this product as intrinsic relative activity (RA_{i}). The calculations are the same as those used in the analysis of allosterism in which the control agonist concentrationresponse curve is compared with one whose properties of affinity and efficacy have been modified allosterically (Ehlert, 1988, 2005). This situation is analogous to comparing the concentrationresponse curves of two different agonists with differing affinities and efficacies. We have estimated the RA_{i} values of a group of agonists for eliciting responses through the human M_{2} receptor coupled to G_{i},G_{α15}, and G_{s} to determine whether the G protein with which the M_{2} receptor interacts influences the activity of the selected agonists. Although most of the agonists investigated exhibited similar RA_{i} values for triggering M_{2} muscarinic responses through the different G proteins, compound McNA343 exhibited ∼10fold greater activity when stimulating M_{2}G_{α15} responses compared with M_{2}G_{i} responses. Our results show that RA_{i} is a useful parameter for the estimation of agonist activity across different assays.
Materials and Methods
Cell Culture
Chinese hamster ovary (CHO) cells stably expressing the human M_{2} muscarinic receptor (CHO M_{2} cells) were obtained from Acadia Pharmaceuticals (San Diego, CA) and cultured in Dulbecco's modified Eagle's medium with high glucose plus lglutamine (DMEM) supplemented with 10% fetal calf serum, penicillinstreptomycin (100 units/ml) and G 418 (0.3 mg/ml) at 37°C in a humidified atmosphere with 5% CO_{2}. HEK 293T cells stably expressing G_{α15} (HEK G_{α15}) were kindly provided by Dr. Olivier Civelli (University of California, Irvine, Irvine, CA) and were grown in DMEM containing 10% fetal calf serum, penicillinstreptomycin (100 units/ml), G 418 (0.4 mg/ml), and puromycin (0.625 μg/ml). A plasmid containing the human M_{2} receptor (hM_{2} pCD clone) was kindly provided by Dr. Tom Bonner (National Institutes of Health, Bethesda, MD) and subcloned into a hygro()pcDNA3.1 vector as described previously (Griffin et al., 2003). HEK G_{α15} cells were transiently transfected with the human M_{2} receptor (HEK G_{α15} M_{2} cells) using Lipofectamine 2000 and 15 μg of the M_{2} vector. HEK 293T cells were stably transfected with the human M_{2} receptor in the same manner, and clones of stably transfected cells (HEK M_{2}) were isolated in DMEM containing 10% fetal calf serum, penicillinstreptomycin (100 units/ml), hygromycin (0.2 mg/ml), and G 418 (0.4 mg/ml).
cAMP Accumulation
The effects of muscarinic agonists on forskolin or isoproterenolstimulated cAMP accumulation was measured in CHO M_{2} and HEK M_{2} cells using a modification of the [^{3}H]adenine prelabeling method of Schultz et al. (1972) and the chromatography procedure of Salomon et al. (1974). Confluent cell monolayers grown in T75 flasks were washed with DMEM media and then incubated in 9 ml of DMEM containing [^{3}H]adenine (60 μCi) and adenine (3 μM) for 1 h at 37°C in 5% CO_{2}. Cells were washed twice with DMEM and harvested using trypsin. The resulting cell suspensions were centrifuged for 10 min at 350g, suspended in KrebsRinger bicarbonate (KRB) 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) at pH 7.4 and centrifuged a second time. Ultimately, cells were suspended in KRB buffer (12 ml) containing isobutylmethylxanthine (0.5 mM) and incubated for an additional 10 min at 37°C before use in the cAMP assay. Muscarinic agonistmediated inhibition of cAMP accumulation was carried out in plastic tubes containing intact cells, forskolin (10 μM), isobutylmethylxanthine (0.5 mM), and various concentrations of a muscarinic agonist in a final volume of 0.35 ml of KRB buffer. The reaction was started by the addition of an aliquot (300 μl) of cells and was stopped by addition of an aliquot (200 μl) of icecold 30% (w/v) trichloroacetic acid. After at least 30 min on ice, the tubes were centrifuged for 10 min at 3000g, and an aliquot (0.5 ml) from each tube was applied to a Dowex 1.5ml column (AG50WX4, 200–400 mesh) and washed with two aliquots of water (1.25 ml each) to remove [^{3}H]ATP. The [^{3}H]cAMP was eluted onto a column of neutral alumina (0.6 g) with 4 ml of water and then eluted into scintillation vials with 4 ml of 0.1 M imidazole HCl (pH 7.5). The samples were counted by liquid scintillation spectroscopy. In experiments in which muscarinic agonistmediated stimulation in cAMP accumulation was measured, cells were first treated overnight (16–18 h) with pertussis toxin (0.1 μg/ml; final concentration) before labeling with [^{3}H]adenine.
Phosphoinositide Hydrolysis
Muscarinic agonistmediated stimulation of phosphoinositide hydrolysis was measured in suspensions of HEK G_{α15} and HEK G_{α15} M_{2} cells using a modification of the [^{3}H]inositol prelabeling method of Berridge et al. (1982) and the extraction method of Kendall and Hill (1990). Confluent cell monolayers grown in T75 flasks were washed with DMEM and then incubated in 9 ml of DMEM containing [^{3}H]inositol (45 μCi) for 16 to 18 h at 37°C in 5% CO_{2}. Cells were washed twice with DMEM and harvested using trypsin as described above. Ultimately, the cell pellet was suspended in KRB buffer (15 ml) containing LiCl (10 mM) and incubated at 37°C for 15 min. Muscarinic agonistmediated stimulation of [^{3}H]inositol phosphate accumulation was performed in plastic tubes containing intact cells, LiCl (10 mM), and various concentrations of muscarinic agonist in a final volume of 333 μl of KRB buffer. The reaction was started with the addition of an aliquot (300 μl) of cell suspension, and the mixture was gassed with O_{2}CO_{2}, capped with a rubber stopper, and incubated at 37°C for 15 min. The reaction was stopped with 5% perchloric acid (200 μl), and the tubes were placed on ice. [^{3}H]Inositol phosphates were isolated as described previously (Tran et al., 2006).
In some experiments, cells were treated with pertussis toxin (0.1 μg/ml final concentration) overnight during the incubation with [^{3}H]inositol. In experiments using 4DAMP mustard, AFDX 116 was added to cells after labeling with [^{3}H]inositol to give a final concentration of 4 μM. After a 10min incubation (37°C and 5% CO_{2}), a small volume of 4DAMP mustard was added (final concentration 40 nM), and the cells were allowed to incubate for 1 h at 37°C and 5% CO_{2}. Immediately before use, 0.4 mM 4DAMP mustard was cyclized to the aziridinium ion by a 30min incubation at 37°C in 10 mM NaKPO_{4}, pH 7.4 (Thomas and Ehlert, 1992). After 4DAMP mustard treatment, cell monolayers were washed 3 times with DMEM to remove AFDX 116 and unreacted 4DAMP mustard and its transformation products.
Preliminary Analysis of Agonist ConcentrationResponse Curves
The maximal response (E_{max}), concentration of agonist eliciting a halfmaximal response (EC_{50}), and the Hill slope (n) were estimated from the agonist concentrationresponse curve by nonlinear regression analysis using Prism (GraphPad Software, Inc., San Diego, CA). The data for agonistmediated inhibition of cAMP accumulation were fitted to the following equation: in which P denotes cAMP accumulation elicited by forskolin or isoproterenol in the absence of muscarinic agonist and E_{max} denotes the maximal percent inhibition of cAMP accumulation. The agonist concentrationresponse curve for stimulation of phosphoinositide hydrolysis and enhancement of forskolinstimulated cAMP accumulation in pertussis toxin treated cells was fitted to the following equation: In this equation, P denotes the basal value of phosphoinositide hydrolysis measured in the absence of agonist. The dissociation constant of an antagonist (K_{I}) was estimated on the basis of its ability to antagonize the response to an agonist competitively. The following equation was used to estimate K_{I}: In this equation, I denotes the concentration of the antagonist used in the assay, and CR denotes the EC_{50} value of the agonist measured in the presence of the antagonist divided by that measured in its absence.
Estimation of RA_{i}
To reduce error in the estimation of RA_{i}, we always tested the standard agonist carbachol together with the test agonist in the same experiment. In most instances, all five agonists were tested simultaneously. Our methods for the estimation of RA_{i} represent a further refinement in our previously published methods (Ehlert et al., 1999; Ehlert and Griffin, 2001). We define the RA_{i} value of test agonist B relative to standard agonist A as the product of the observed intrinsic efficacy and affinity of agonist B divided by that for agonist A: In this equation, ϵ_{A} and ϵ_{B} denote the observed intrinsic efficacies, and K_{A} and K_{B} denote the observed dissociation constants of agonists A and B, respectively. In the first step of this analysis, the concentrationresponse curves of the test agonist B and the standard agonist A are analyzed by nonlinear regression analysis according to eqs. 1 or 2 to estimate the EC_{50}, E_{max}, and Hill slope values. If the E_{max} values of the two agonists are the same, then the RA_{i} value of agonist B expressed relative to A can be calculated as (Ehlert et al., 1999): In this equation, subscripts are used to denote the EC_{50} values of A and B. When the E_{max} value of the test agonist is different from that of the standard agonist, the analysis is more complicated, and three different methods have been developed to estimate RA_{i} in this situation, each with its own advantages and disadvantages. These are 1) a null method, which is independent of the relationship between occupancy and response, 2) a method based on the operational model, assuming a logistic relationship between occupancy and response, and 3) a special case of the operational model in which the Hill slope of the agonist concentrationresponse curve is equivalent to 1.
Null Method. The null method involves comparing equiactive concentrations of the test agonist B and the standard agonist A.In our description of the method, we designate the standard agonist as the agonist with the larger E_{max}. An advantage of this method is that it is suitable for the analysis of any shape of concentrationresponse curve as long as the curve is a continuous, increasing function of the agonist concentration. The method involve three steps: 1) estimation of pairs of equiactive concentrations of agonist A and B, 2) nonlinear regression analysis of these equiactive concentrations according to eq. 8, and 3) calculation of the RA_{i} value using eq. 9, which requires parameter estimates from step 2.
Pairs of equiactive agonist concentrations (A_{i} and B_{i}) are estimated as follows. The values of B_{i} are simply those of the concentrationresponse curve of test agonist B, with the subscript i denoting the different concentrations. Each corresponding A_{i} value is estimated from the concentrationresponse curve of agonist A by interpolating an A_{i} value that yields a response equivalent to that elicited by B_{i}. This interpolation is shown graphically in Fig. 3. Different methods of interpolation can be used, depending upon the shape of the concentrationresponse curve of A. In our experiments, the responses to all of the agonists were consistent with eqs. 1 and 2. Consequently, we rearranged these equations to solve for the agonist concentration (X) as a function of the response (R). Thus, eq. 1 was rearranged into the following form to interpolate A_{i} values from the concentrationresponse curve for inhibition of cAMP accumulation: In this equation, R_{iB} denotes the response elicited by the ith concentration of B (B_{i}), and EC_{50}_{A}, E_{max}_{A}, and n_{A} denote the bestfitting parameter estimates of the concentrationresponse curve of the standard agonist A. For agoniststimulated phosphoinositide hydrolysis in HEKG_{α15} cells and agonistmediated enhancement of forskolinstimulated cAMP accumulation in pertussis toxintreated CHO M_{2} cells, eq. 2 was rearranged to the following to enable an interpolation of A_{I} values: Thus, by substituting in the response values of the concentrationresponse curve to B into the corresponding eqs. 6 and 7, it was possible to estimate a set of A_{i} values equiactive to those of B_{i}. The resulting pairs of equiactive agonist concentrations were fitted to the following equation by nonlinear regression analysis: The derivation of this equation is given under Appendix, and it represents the log form of eq. 19. The parameters p and q are defined under Appendix (see eqs. 20 and 21) and are equivalent to the observed affinity and efficacy of B relative to A, respectively. As described under Results, there are an infinite number of parameter estimates that yield the leastsquares fit; however, the solution set is unique in that the ratio q/p is constant. Thus, it was possible to estimate q/p by fixing the parameter K_{A} to an arbitrarily high value and determining the values of q and p that minimized the residual sum of squares. Additional details of the regression analysis are given under Results, and a method for obtaining initial parameter estimates for the iterative procedure is given in the last section under Appendix. The RA_{i} value can be calculated as Substitution of eqs. 20 and 21 for p and q shows that eq. 9 reduces to eq. 4. It is possible to estimate the dissociation constant of the test agonist (K_{B})as in which K_{A} denotes the constant to which K_{A} was fixed during regression analysis.
Operational Model. When the agonist concentrationresponse curve is consistent with a logistic equation (eq. 1 or 2), it is possible to estimate the RA_{i} value of a partial agonist B relative to a more efficacious agonist A by first fitting the concentrationresponse curves of A and B to the operational model of Black and Leff (1983) by nonlinear regression analysis: in which R denotes the response to the agonist, X denotes the concentration of A or B, and τ is defined in eq. 25 under Appendix. For the experiments involving agonist stimulation of phosphoinositide hydrolysis and stimulation of cAMP accumulation in CHO M_{2} cells treated with pertussis toxin, the response denotes the measurement minus the basal value observed in the absence of agonist. For the experiments involving agonistmediated inhibition of forskolin or isoproterenolstimulated cAMP accumulation, the response represents the percentage inhibition of cAMP accumulation. Equation 11 is essentially equivalent to eq. 24 under Appendix. The subscript j is used to denote different agonists. Thus, K_{j} denotes the dissociation constants of A and B (K_{A} and K_{B}), whereas τ_{j} denotes the τ values of A and B (τ_{A} and τ_{B}), respectively. The concentrationresponse curves to the standard agonist A and the test agonist B are analyzed simultaneously by global nonlinear regression analysis, sharing the estimates of M_{sys} and m between the curves and allowing for individual estimation of τ_{j} and K_{j} for the standard agonist (τ_{A} and K_{A}) and the test agonist (τ_{B} and K_{B}). If the standard agonist is a full agonist with a receptor reserve, then an infinite number of parameter estimates will yield a leastsquares fit as described under Results. However, the solution set does yield the best estimates of m, M_{sys}, τ_{B}, K_{B}, and the ratio τ_{A}/K_{A}. Thus, it may be necessary to fix K_{A} to an arbitrarily high value and determine the values of the other parameters that yield a leastsquares fit as described under Results. Additional details of the regression analysis are given under Results, and a method for obtaining initial parameter estimates for the iterative procedure is given in the last section under Appendix. The RA_{i} value of B relative to A is calculated as By making the appropriate substitutions for τ (eq. 25 under Appendix), it can be shown that eq. 12 reduces to eq. 4.
Hill Slope Equals 1. When the Hill slope of the agonist concentrationresponse curve is equal to 1, the RA_{i} value of agonist B relative to standard agonist A is calculated as If the definition of τ (eq. 25) is substituted into eqs. 27 and 28, and these latter equations are then substituted for the E_{max} and EC_{50} values into eq. 13, then it can be shown that eq. 13 reduces to eq. 4.
Drug and Chemicals
Drugs and chemicals were obtained from the following sources: [^{3}H]adenine and [^{3}H]inositol, PerkinElmer Life and Analytical Sciences (Boston, MA); DMEM and trypsinEDTA (Invitrogen, Carlsbad, CA); carbachol, McNA343, and oxotremorineM (SigmaAldrich, St. Louis, MO); and AFDX 116 (Boehringer Ingelheim Pharmaceutical, Ridgefield, CT). 4DAMP mustard was synthesized as described previously (Thomas et al., 1992). The enantiomers of aceclidine were synthesized and resolved as described by Ringdahl et al. (1979).
Results
Phosphoinositide Hydrolysis. We investigated the ability of muscarinic agonists to elicit phosphoinositide hydrolysis through the M_{2} muscarinic receptor in HEK G_{α15} cells. The highly efficacious muscarinic agonist, oxotremorineM, elicited a small phosphoinositide response in HEK G_{α15} cells, presumably through activation of an endogenous muscarinic receptor. When the latter cells were transiently transfected with the human M_{2} muscarinic receptor, the resulting cells (HEK G_{α15} M_{2}) exhibited a robust phosphoinositide response to oxotremorineM (Fig. 1a). The response was resistant to treatment with pertussis toxin (0.1 μg/ml, overnight incubation; data not shown). To identify the endogenous receptor mediating phosphoinositide hydrolysis in HEK G_{α15} cells, we characterized the antagonism of the response by subtypeselective muscarinic antagonists. When measured in the presence of either AFDX 116 (10 μM) or pirenzepine (1.58 μM), the concentrationresponse curve to oxotremorineM shifted to the right 17 and 14fold, respectively. These antagonistinduced shifts correspond to calculated pK_{I} values of 6.20 ± 0.095 and 6.89 ± 0.19 for AFDX 116 and pirenzepine, respectively (see eq. 3). Collectively, these pK_{I} values agree with the corresponding binding affinities (pK_{D} values) of AFDX 116 and pirenzepine at the human M_{3} muscarinic receptor (6.10 ± 0.06 and 6.59 ± 0.03) but not those of the M_{1} (6.24 ± 0.03 and 7.77 ± 0.03), M_{2} (7.27 ± 0.05 and 5.96 ± 0.05), M_{4} (6.96 ± 0.12 and 7.23 ± 0.02), or M_{5} muscarinic receptor (5.29 ± 0.11 and 6.55 ± 0.06). Consequently, we conclude that the M_{3} subtype mediates phosphoinositide hydrolysis in HEK G_{α15} cells.
To eliminate this M_{3} response, we treated cells with the irreversible muscarinic antagonist 4DAMP mustard in combination with the competitive M_{2} selective antagonist AFDX 116 for 60 min followed by washing. This treatment has been shown to cause 93% inactivation of M_{3} receptors while only inactivating M_{2} by 22% (Griffin et al., 2003). As shown in Fig. 1b, 4DAMP mustard treatment completely eliminated the response to the endogenous M_{3} receptor in HEK G_{α15} cells while having little inhibitory effect on that measured in HEK G_{α15} M_{2} cells.
Having developed a simple method for investigating the signaling of the M_{2} receptor through G_{α15}, we measured the ability of a group of agonists with varying potencies and intrinsic efficacies to stimulate phosphoinositide hydrolysis in HEK G_{α15} M_{2} cells, which had been previously treated with 4DAMP mustard. We also treated these cells with pertussis toxin (0.1 μg/ml) overnight to prevent M_{2} receptor signaling through G_{i/o}. Figure 2a shows the results of these experiments. OxotremorineM, carbachol, and Saceclidine all elicited potent responses with similar E_{max} values, whereas Raceclidine and McNA343 behaved as partial agonists. Table 1 lists the EC_{50}, E_{max}, and Hill slope values for these experiments. We also investigated the ability of the same group of agonists to stimulate phosphoinositide hydrolysis in HEK G_{α15} cells that had not been transfected with the M_{2} receptor nor treated with 4DAMP mustard (Fig. 2b). Under these conditions, the agonists were less potent and Raceclidine and McNA343 produced barely detectable responses. Table 2 summarizes the results of these experiments.
Estimation of Agonist RA_{i} Values for Phosphoinositide Hydrolysis in HEK Cells. We estimated the RA_{i} values of agonists from the data shown in Fig. 2a using the procedures described under Materials and Methods. Figure 3 illustrates our use of the null method to estimate the RA_{i} value of McNA343 relative to carbachol. First, predicted concentrations of carbachol that elicit responses equivalent to those of the concentrationresponse curve to McNA343 were interpolated as shown in Fig. 3a using eq. 7. These interpolated concentrations are plotted against the corresponding equiactive concentrations of McNA343 in Fig. 3b. Equation 8 was fitted to the data by nonlinear regression analysis to estimate p, q, and K_{A}; however, an infinite number of parameter values yield the same leastsquares fit. Consequently, it is impossible to find a solution set consisting of single values of these parameters. When the value of K_{A} was fixed as a constant; however, it was possible to obtain a leastsquares fit corresponding to unique values of p and q and the constant K_{A}. As the value of log K_{A} was set at various values between 7.5 and an arbitrarily high value of 1, the bestfitting values of p and q decreased proportionately so that their ratio remained constant as well as the residual sum of squares (RSS). Thus, it was possible to estimate the ratio p/q even though it was impossible to estimate unique values of K_{A}, p, and q. The RA_{i} of McNA343, expressed relative to carbachol is simply calculated as the ratio q/p (see eq. 9). The RA_{i} values of S and Raceclidine were estimated using a similar strategy. Because the E_{max} value of carbachol was less than those of oxotremorineM and Saceclidine, we first calculated the RA_{i} values of carbachol and Saceclidine relative to oxotremorineM and then expressed the resulting RA_{i} values relative to carbachol. A summary of these estimates is given in Table 1. We estimated the dissociation constants of the partial agonists Raceclidine (pK = 3.81 ± 0.37) and McNA343 (pK = 4.63 ± 0.15) using eq. 10. We also used the operational model to estimate the RA_{i} values of the agonists. First, we fitted the concentrationresponse curves shown in Fig. 2a to eq. 11 using global nonlinear regression analysis, sharing the estimates of M_{sys} and m among the curves and estimating unique values of τ and K for each curve. It was possible to obtain unique estimates of all of the parameters using this method; however, the estimates of τ and K for the full agonists were poorly defined and exhibited huge standard errors as well as the estimate of M_{sys}. This limitation arises with full agonists or, in other words, whenever the E_{max} value is approximately equal to the maximum response of the system (M_{sys}). Nevertheless, the ratio of τ/K for each agonist was well defined. Thus, it was possible to estimate the RA_{i} value of each agonist by taking the ratio of the τ/K of each agonist relative to that of carbachol as shown by eq. 12. The RA_{i} estimate of each agonist is listed in Table 1 together with the values estimated by the null method. It can be seen that there is close agreement between the two methods. It was also possible to estimate the dissociation constants of the partial agonists, Raceclidine (pK = 3.64 ± 0.46) and McNA343 (pK = 4.93 ± 0.12).
To illustrate the unique relationship among parameter estimates in the operational model, we fixed the log K_{A} value of the standard agonist carbachol (K_{carb}) at various values between 6 and 1 and used nonlinear regression analysis to estimate the bestfitting values of the other parameters. Figure 4b shows the results of this regression analysis for the estimation of the RA_{i} value of McNA343 relative to carbachol. It can be seen that once the log K_{carb} value is ∼≥–5, the RSS drops to a minimum and remains at this low value no matter how large the increase in K_{carb}. Over this range of K_{carb} values, the estimates of m, M_{sys}, τ_{McN}, and K_{McN} are constant. As log K_{carb} increases from 5 to 1, there is a proportional increase in τ_{carb} such that the ratio of τ_{carb}/K_{carb} is constant (Fig. 4c). Figure 4c illustrates that the ratio of τ_{McN}/K_{McN} is also constant over the domain log K_{carb} >5 as well as the estimate of RA_{i}. In summary, the results in Fig. 4 illustrate that it is possible to estimate the RA_{i} value of a partial agonist relative to a full agonist by setting the K_{A} value of the full agonist to an arbitrarily high value and using nonlinear regression analysis to estimate the other parameters in the operational model (eq. 11). The RA_{i} value can then be calculated from these parameters using eq. 12. This analysis for the agonists in Fig. 2 also yielded estimates of m (0.721 ± 0.044), M_{sys} (104 ± 4.7), and the dissociation constants of the partial agonists as listed in the previous paragraph. Figure 4a shows the best fit of the operational model to the concentrationresponse curves of carbachol and McNA343 using this method.
We also estimated the RA_{i} values of the agonists for stimulation of phosphoinositide hydrolysis in HEK G_{α15} M_{2} cells using the simple calculation given in eq. 13, which is valid provided that the Hill slopes are similar to 1. The calculation is also reasonably accurate if the Hill slopes differ from 1, and the E_{max} values of the agonists are not less than 50% that of a full agonist. Table 1 lists these estimates. It can be seen that there is close agreement between the estimates using eq. 13 and those estimated with the null method and operational model for all of the agonists exhibiting similar E_{max} values (i.e., oxotremorineM, carbachol, Saceclidine, and Raceclidine). However, the simple estimate of the RA_{i} for McNA343 is ∼3fold greater than those estimated using the other two methods, which illustrates the flaw in using the simple method when the Hill slopes differ from 1 and there is a large difference in E_{max} between the standard and test agonist.
We also estimated the RA_{i} values of carbachol, oxotremorineM, Saceclidine, Raceclidine, and McNA343 for their effects on phosphoinositide hydrolysis in HEK G_{α15} cells that had not been transfected with the M_{2} receptor, and these results are summarized in Table 2.
AgonistMediated Inhibition of cAMP Accumulation in CHO M_{2} Cells. To determine whether the RA_{i} values of agonist varied depending upon the G protein with which the M_{2} receptor interacts, we measured agonistmediated inhibition of forskolinstimulated cAMP accumulation in CHO cells stably transfected with the human M_{2} muscarinic receptor (Fig. 5a). Carbachol, oxotremorine–M, and the enantiomers of aceclidine behaved as full agonists and caused a maximal inhibition of cAMP accumulation of ∼73%, whereas that of McNA343 was only 32%. There was a tendency for the maximal effect of Raceclidine to exceed that of the other full agonists. As described below, Raceclidine exhibited a nonspecific inhibition of cAMP accumulation at high concentrations (i.e., 1.0 mM). Consequently, we shared the estimate of the E_{max} among the full agonists to prevent an overestimation in the E_{max} of Raceclidine. The E_{max},EC_{50}, and Hill slope values of these data are summarized in Table 3. The RA_{i} values of the full agonists were calculated according to eq. 13 and are also listed in Table 3. Note that when the standard agonist and the test agonist exhibit the same E_{max}, eqs. 5 and 13 yield the same result. The RA_{i} value of McNA343 was estimated using eq. 13 as well as the null method and operational model, and these estimates are also shown in Table 3. In this instance, eq. 13 gave results similar to those for the other two methods, probably because the Hill slopes of carbachol and McNA343 were close to 1. The estimates of the negative logarithm of the dissociation constant of McNA343 using the null method (pK = 4.16 ± 0.24) and the operational model (pK = 4.07 ± 0.16) were approximately the same.
Comparison of the data in Tables 1 and 3 shows that whereas the RA_{i} values of oxotremorineM, carbachol, Saceclidine, and Raceclidine are similar, the value for McNA343 is ∼10fold greater in HEK G_{α15} M_{2} cells relative to that measured in CHO M_{2} cells. This increase in the RA_{i} value of McNA343 was associated with an average 4.5fold increase in observed affinity (decrease in dissociation constant), indicating a small increase in observed intrinsic efficacy (∼2fold) as well. To verify that this difference can be attributed to different G proteins and not to the cells in which the M_{2} receptor was expressed, we investigated M_{2} receptormediated inhibition of isoproterenolstimulated cAMP accumulation in HEK T cells stably expressing the human M_{2} muscarinic receptor (HEK T M_{2}). Figure 5b shows the results of these experiments using carbachol and McNA343 as agonists. The results of these experiments are summarized in Table 4 where it can be seen that the RA_{i} value of McNA343 is similar to that measured in CHO M_{2} cells but only ∼1/10 that measured in HEK G_{α15} M_{2} cells. These results suggest that McNA343 preferentially directs signaling of the M_{2} receptor through G_{α15} relative to G_{i}.
AgonistMediated Enhancement of ForskolinStimulated cAMP Accumulation. Michal et al. (2001) have shown that the M_{2} receptor has biphasic effects on cAMP accumulation in CHO cells when forskolin is present. At low concentrations, muscarinic agonists cause a potent inhibition of cAMP accumulation, whereas at high concentrations, an enhancement of cAMP accumulation occurs. The inhibitory effect is mediated via G_{i} and is prevented by pertussis toxin treatment, whereas the stimulatory effect is mediated by G_{s} (Michal et al., 2001, 2007). After pertussis toxin treatment, only the less potent stimulatory phase is observed. Consequently we measured agonist concentrationresponse curves for stimulating cAMP accumulation in the presence of forskolin (10 μM) in CHO M_{2} cells that had been previously treated with pertussis toxin to prevent G_{i}mediated inhibition of adenylyl cyclase (Fig. 6a). Carbachol, oxotremorineM, and Saceclidine all exhibited a similar shaped sigmoid concentrationresponse curves with similar E_{max} values, whereas Raceclidine exhibited a bellshaped curve with a lower maximum. McNA343 actually caused a concentrationdependent inhibition of cAMP accumulation. Because both McNA343 and Raceclidine inhibited cAMP accumulation at high concentrations, we measured their effects on forskolinstimulated cAMP accumulation in CHO cells that had not been transfected with the M_{2} muscarinic receptor (Fig. 6b). Both agonists caused a concentrationdependent inhibition of forskolinstimulated cAMP accumulation in untransfected CHO cells, suggesting that the compounds interfere with adenylyl cyclase through a nonmuscarinic receptor mechanism at high concentrations. In contrast, carbachol had no such effect. Thus, in analyzing the concentrationresponse curve to Raceclidine, we only used the data over the log molar concentration range of 5 to 3.5 Raceclidine. The estimates of the RA_{i} values of carbachol, oxotremorineM, Saceclidine, and Raceclidine for stimulation of cAMP accumulation were calculated using the three different methods, and these estimates are listed in Table 4. There is agreement among the three different estimates for a given agonist. Also the RA_{i} values estimated for stimulation of cAMP accumulation agree generally with those estimated in the other assays.
Analysis of the Loss of Activity of McNA343 in Stimulating cAMP Accumulation. The lack of effect of McNA343 in enhancing forskolinstimulated cAMP accumulation in CHO M_{2} cells treated with pertussis toxin may indicate that the RA_{i} value of McNA343 in this assay is less than that observed in the two other assays. Alternatively, it may be that this response is so insensitive that the relatively weak activity of McNA343 at the M_{2} receptor simply does not register in this assay even though its RA_{i} value may be similar to those observed in the other assays. To discriminate between these two possibilities we determined the loss in sensitivity to carbachol that occurs when stimulation of cAMP accumulation (CHO M_{2} G_{s} assay) is measured instead of inhibition of cAMP accumulation (CHO M_{2} G_{i} assay), and determined whether an equivalent loss in sensitivity could eliminate the response to McNA343 in the CHO M_{2} G_{s} assay, assuming no change in its RA_{i} value. Figure 7a shows that the concentrationresponse curve of carbachol in the CHO M_{2} G_{s} assay is shifted to the right 26fold compared with that for the CHO M_{2} G_{i} assay. From the perspective of the operational model, this loss in sensitivity can be attributed to a reduction in τ to a value only 3.9% of that measured in the CHO M_{2} G_{i} assay. If we apply the same loss in sensitivity to the concentrationresponse curve of McNA343 in the CHO M_{2} G_{i} assay, the resulting curve only reaches an E_{max} of 1.7% as shown in Fig. 8a by the theoretical curve for McNA343 in the CHO M_{2} G_{s} assay. This level of stimulation would have been impossible to detect, given the nonspecific inhibitory effect of McNA343 on cAMP accumulation. The apparent reduction in the τ value of carbachol between the two assays can be estimated in the same manner that one estimates RA_{i}, except that in this instance, we are estimating the activity of carbachol in the CHO M_{2} G_{s} assay relative to its activity in the CHO M_{2} G_{i} assay. Because carbachol behaves as a full agonist in both assays, its τ value in the CHO M_{2} G_{s} assay relative to that of the CHO M_{2} G_{i} assay is simply equivalent to the ratio of its EC_{50} value in the G_{i} assay expressed relative to that for the G_{s} assay. We also carried out a similar type of analysis for the loss in sensitivity to carbachol in the CHO M_{2} G_{s} assay compared with the HEK G_{α15} M_{2} assay (Fig. 8b). In the latter case, the loss in sensitivity to carbachol corresponded to a reduction in τ of 77%, and this change nearly, but not completely, eliminated the theoretical response to McNA343 in the CHO M_{2} G_{s} assay, assuming no change in RA_{i} between the two assays. The E_{max} of the theoretical response was 7%, and this level of stimulation would have been difficult to detect given the small nonspecific inhibitory effects of McNA343 over the same concentration range. Thus, we conclude that the decreased sensitivity of the CHO M_{2} G_{s} assay can account for the loss in the response to McNA343, and it is impossible to tell from the data whether the actual RA_{i} value of McNA343 in the CHO M_{2} G_{s} assay is any less than those observed in the HEK G_{α15} M_{2} and CHO M_{2} G_{i} assays.
Discussion
We have described a method for estimating the product of observed affinity and intrinsic efficacy of an agonist for a receptor expressed relative to that of another using only a single concentrationresponse curve for each agonist. This measure (RA_{i}) provides a means of converting EC_{50} and E_{max} values into a single estimate, enabling one to compare the activity of an agonist across different assays, regardless of whether the agonist behaves as a full or partial agonist. Our results show that carbachol, oxotremorineM, and the enantiomers of aceclidine have similar activity across the different M_{2} assays, whereas McNA343 exhibits ∼10fold greater activity at the M_{2} receptor signaling through G_{α15} compared with G_{i} (Fig. 8a). This ability of McNA343 to select for M_{2}G_{α15} responses is important in drug screening because G_{α15} is often used to transduce signals through GPCRs.
If an agonist induces or selects a receptor conformation that preferentially interacts with G_{15}, then the agonist should exhibit a larger RA_{i} value for eliciting G_{15} responses relative to a standard agonist that lacks this selectivity. Presumably, this phenomenon explains the selectivity of McNA343 for G_{15} responses relative to G_{i} responses. This selectivity for G_{15} responses should be manifest when the agonist is assayed in different cells, each expressing a single G protein (e.g., G_{15} or G_{i}). If both G proteins are expressed within the same cell and the receptor is in a dynamic equilibrium with both, then the RA_{i} value of an agonist for responses mediated through either G protein should be the same and equal to a weighted average of those observed in different cells (Leff et al., 1997). Bearing this hypothesis in mind, we treated HEK G_{α15} M_{2} cells with pertussis toxin to prevent M_{2} receptor signaling through G_{i} when G_{α15} responses in HEK G_{α15} M_{2} cells were measured so that our RA_{i} estimate would reflect only the M_{2}G_{α15} interaction. Nevertheless we also performed the same experiments without pertussis toxin treatment and observed essentially the same results for McNA343 and the other agonists (data not shown). Perhaps the selectivity of the McNA343M_{2} receptor complex for G_{15} is so great that G_{i} does not compete effectively and causes little perturbation of the M_{2} receptorG_{15} interaction. Alternatively, these findings may suggest that different pools of M_{2} receptors are dedicated to either G_{15} or G_{i} and that the M_{2} receptor is not in equilibrium with both G proteins at the same time in HEK G_{α15} M_{2} cells. We speculate that this compartmentalization might be maintained by scaffolding or accessory proteins as established for other systems (Ostrom et al., 2000).
It is possible that the M_{2} receptormediated enhancement in forskolinstimulated cAMP accumulation has little physiological relevance. We found that muscarinic agonists cause little or no increase in cAMP accumulation in pertussis toxintreated CHO cells unless forskolin is present. Presumably, forskolin enhances the weak activation of G_{s} by muscarinic agonists in CHO cells so that a substantial activation of adenylyl cyclase can be measured in cells treated with pertussis toxin. It has been shown that type VI adenylyl cyclase is abundantly expressed in CHO cells (Varga et al., 1998) and that this isozyme exhibits synergistic interactions between forskolin and G_{s} (Sutkowski et al., 1994). Thus, although the M_{2} receptorG_{s} interaction may be too weak to subserve a physiological role, it nonetheless provides an interesting opportunity to test whether the nature of the G protein with which the receptor interacts influences agonist activity.
The RA_{i} values that we reported for muscarinic agonists in the CHO M_{2} G_{i} assay are consistent with the activity of these agonists reported in the electrically driven guinea pig left atrium (Fig. 8b). This preparation is a standard assay for the M_{2} muscarinic receptor signaling through a G_{i} mechanism. We estimated the RA_{i} values of the agonists in this preparation from published reports of the EC_{50} and E_{max} values (Eltze et al., 1993; Barocelli et al., 2000) using eq. 13 and also using regression analysis (eq. 11) of the concentrationresponse curves for carbachol and McNA343 published by Christopolous and Mitchelson, 1997). The good agreement between the two sets of data illustrate the usefulness of the RA_{i} estimate in predicting agonist activity at the same receptorG protein signaling mechanisms across different assays.
One way to appreciate the significance of RA_{i} is to consider a series of agonists in a hypothetical, highly sensitive assay in which the agonists would behave as full agonists, even those with little intrinsic efficacy. In this situation, the RA_{i} values would be equivalent to the ratio of EC_{50} values of the agonists expressed relative to a standard agonist. To illustrate this point, we show a comparison of the RA_{i} values of muscarinic agonists for stimulating contractions in the guinea pig ileum with those for stimulating phosphoinositide hydrolysis in CHO cells transfected with the human M_{3} receptor (Fig. 8c). In the former assay, most of the agonists behave as full agonists, whereas in the latter, most behaved as partial agonists (Ehlert et al., 1999). There is exceptional agreement in the RA_{i} values of agonists across these two M_{3} receptorG_{q} responses.
An interesting property of RA_{i} is related to the initial slope of the agonist concentrationresponse curve. By taking the first derivative (dy/dX) of the operational model (eq. 11) with respect to the agonist concentration (X) and evaluating its limit as the agonist concentration approaches 0, it is possible to estimate the initial slope of an agonist concentrationresponse curve. If the initial slope for agonist B is divided by that of a standard agonist A, the result is RA ^{m}_{i}, where which m denotes the transducer slope factor in the operational model: in which In these equations, X denotes the concentration of agonist with subscripts for test agonist (B) and standard agonist (A). Thus, when considered from this perspective, the RA ^{m}_{i} is a relative measure of the sensitivity of receptorG protein complexes for different agonists. The initial slope of the plot of output against signal intensity is a common means of expressing the sensitivity of a transducer in electronics and engineering.
GPCRs are unique in that observed affinity is influenced by the concentration of guanine nucleotide in the cytosol, particularly for G_{i} and G_{s} linked receptors. As the concentration of GTP and GDP increases, observed affinity decreases. This change in affinity occurs with a corresponding opposite change in intrinsic efficacy such that the product of observed affinity and intrinsic efficacy remains constant (Ehlert and Rathbun, 1990; Ehlert, 2000). Consequently the RA_{i} estimate is unperturbed by changes in the guanine nucleotide concentration, making it an ideal measurement of activity in cells with different concentrations of GTP or in broken cell assays in which the guanine nucleotide concentration is manipulated [e.g., agonist stimulated guanosine 5′O(3[^{35}S]thio)triphosphate binding].
Although the regression methods that we used for estimating RA_{i} values using the null method (eq. 8) and the operational model (eq. 11) may seem complicated, a computer algorithm can be written to make these calculations automatic, like the estimation of EC_{50} and E_{max}. We think that the RA_{i} estimate has widespread application in comparing agonist activity across different assay systems and in detecting agonistdirected signaling. Analogous types of calculations can be used to analyze allosterism (Ehlert, 2005), sitedirected mutagenesis (Ehlert, 2000) and the loss of function in knockout mice when more than one receptor contributes to the function (Tran et al., 2006).
APPENDIX
Theoretical Basis for the Estimation of RA_{i}. This section describes the theoretical basis for the equations used to estimate the RA_{i} values of agonists. Our method for the estimation of RA_{i} is based on a further development of our previous methods (Ehlert et al., 1999; Ehlert and Griffin, 2001). In our analysis we assume that the amount of receptor activation to agonist X is proportional to the stimulus as defined by Furchgott (1966): in which ϵ_{X} denotes the intrinsic efficacy of X, R_{T} denotes the concentration of functional receptors, and K_{X} denotes the equilibrium constant. The response to agonist X can be represented as a function (f) of the stimulus: Below, three different approaches for estimating the RA_{i} value of an agonist relative to a standard agonist are described. The first involves eliminating the stimulusresponse function from the analysis through the use of a null method. The second involves the use of the operational model to describe the stimulusresponse function, and the third involves a special case of the operational model in which the Hill slope of the concentrationresponse curve equals 1.
Null Method. One approach to estimation of RA_{i} is to compare equivalent tissue responses to the two agonists so that the unknown relationship between the stimulus and response (f) is eliminated. By using this approach, the relationship describing equivalent tissue responses to agonist B and standard agonist A is given by in which A_{i} and B_{i} denote the equiactive concentrations of agonists A and B, respectively. This equation simplifies to in which As described under Materials and Methods (eq. 13), the RA_{i} value can be estimated from the ratio q/p.
Operational Model. Although the null method just described has the advantage of being applicable for any type of stimulusresponse function (i.e., any shape of concentrationresponse curve), it is widely observed that agonists usually exhibit logistic concentrationresponse curves. Indeed, a curvefitting procedure based on the following logistic equation is the most common computational method that investigators use to estimate the E_{max} and EC_{50}: in which n denotes the Hill slope. Several investigators have shown that if the input to the stimulusresponse function (f) is the stimulus (i.e., product of receptor occupancy and intrinsic efficacy) and the output obeys the logistic function just described, then the stimulusresponse function must be the following (Furchgott, 1966; Mackay, 1981; Kenakin and Beek, 1982; Black and Leff, 1983): in which M_{sys} denotes the maximum response of the system, K_{E} denotes the sensitivity of the stimulusresponse function, and m denotes the transducer slope factor. This exponent is related to, but not identical to, the Hill slope of the agonist concentrationresponse curve. Substitution of eq. 16 for S in eq. 23 above followed by simplification yields in which As described under Materials and Methods, the RA_{i} value of an agonist can be estimated from the K_{X} and τ_{X} values of the agonist (see eq. 12).
In situations in which the transducer slope factor m in eq. 24 equals 1, the Hill slope of eq. 22 is also equal to 1 (n = 1), and eq. 24 reduces to in which If eq. 25 is substituted into eqs. eq. 27 and eq. 28 for τ_{X} and then the resulting equations are substituted in for the E_{max} and EC_{50} values in eq. 13 under Materials and Methods, it can be shown that eq. 13 equals eq. 4. Thus, if the Hill slopes are equal to 1, it is possible to estimate the RA_{i} value of agonists from the E_{max} and EC_{50} values.
Initial Parameter Estimates for Nonlinear Regression Analysis. We used the Solver function in Excel (Microsoft Office, 2004) to obtain the leastsquares fit to eqs. 8 and 11 when RA_{i} was estimated by the null method and operational model, respectively. The approach is analogous to that described by Christopoulos and Mitchelson (1998), but the regression equations are changed appropriately for our analysis. This regression analysis requires initial parameter estimates, and below we describe a means to calculate these when analyzing a partial agonist (B) with a standard full agonist (A). For the null method, the value of K_{A} is fixed to an arbitrarily high value (e.g., log molar K_{A} = 1) and the initial estimates of p and q (p′ and q′, respectively) are estimated as follows: In eqs. 29 and 30, K_{A} denotes the arbitrarily high value to which K_{A} is set as a constant for the regression analysis. For the operational model, the following initial parameter estimates for τ_{A}, τ_{B}, K_{B}, m, and M_{sys} (τ_{A}′, τ_{B}′, K_{B}′, m′ and M_{sys}′, respectively) are used in situations in which the standard agonist was a full agonist: In eq. 31, K_{A} denotes the arbitrarily high value to which K_{A} is set as a constant for the regression analysis.
Acknowledgments
We thank Drs. Olivier Civelli and HansPeter Nothacker, Department of Pharmacology, University of California, Irvine, for providing us with HEK 293T cells stably transfected with G_{α15}.
Footnotes

↵1 Here and throughout the article, we use the term “observed intrinsic efficacy” to refer to Furchgott's definition of “intrinsic efficacy” (Furchgott, 1966), which denotes the amount of activated receptors. As described previously (Ehlert, 2000), it is useful to discriminate between the latter definition of observed intrinsic efficacy and the ratio of agonist affinity constants for ground and active conformations of the receptor (intrinsic efficacy).

This work was supported by National Institutes of Health Grant GM 69829 (to F.J.E.).

Article, publication date, and citation information can be found at http://jpet.aspetjournals.org.

doi:10.1124/jpet.107.120857.

ABBREVIATIONS: GPCR, G proteincoupled receptor; McNA343, 4[[N[3chlorophenyl]carbamoy]oxy2butynyl]trimethylammonium chloride; CHO, Chinese hamster ovary; DMEM, Dulbecco's modified Eagle's medium with high glucose and lglutamine; KRB, KrebsRinger bicarbonate; 4DAMP mustard, N2chloroethyl4piperidinyldiphenylacetate; AFDX 116, 11[[2[(diethylamino)methyl]1piperidinyl]acetyl]5,11dihydro6Hpyrido[2,3b][1,4]benzodiazepine6one; RSS, residual sum of squares.
 Received January 31, 2007.
 Accepted March 26, 2007.
 The American Society for Pharmacology and Experimental Therapeutics