previously (Leff & Harper, 1989) this provides an experimental test for the operation of conditions under which erroneous

Transcription

1 Br. J. Pharmaol. (199), 11, 55-6 If--" MamiUan Press Ltd, 199 Pharmaologial estimation of agonist affinity: detetion of errors that may be aused by the operation of reeptor isomerisation or ternary omplex mehanisms 'P. Leff, D. Harper, I.A. Dainty & I.G. Dougall Department of Pharmaology, Fisons pl, Bakewell Road, Loughborough, Leiestershire LEI 1 ORH 1 Reent theoretial studies have questioned the pharmaologial estimation of agonist affinity. They showed that when reeptor isomerisation or ternary omplex mehanisms operate, the reeptor inativation method an substantially overestimate affinity, whereas methods for partial agonist analysis are more aurate. We previously suggested that the operation of suh mehanisms and therefore the presene of errors ould be deteted by analysing the same partial agonist by the reeptor inativation and omparative methods. This paper desribes the pratial appliation of this test. 2 The ternary omplex mehanism was simulated for a partial agonist under various onditions relating reeptor (R) and transduer (T) onentrations, one of whih also orresponds to the reeptor isomerisation mehanism. The theoretial data so generated were then analysed by the inativation-and omparative methods to quantify the magnitude of error of affinity estimation that ould our. 3 This analysis showed that for a partial agonist with approximately 85% of the ativity of a full agonist, the inativation method ould produe an affinity (pka) estimate up to.7 log1o units higher than that produed by the omparative method. This differene would our when the total reeptor onentration ([RO]) is less than or equal to the total transduer onentration ([To]). It also showed that the overestimation of affinity by the inativation method was aompanied by drasti overestimation of Em, the maximal effet parameter. 4 The test was then exemplified using the musarini reeptor system in the guinea-pig isolated left atrial preparation, where there is evidene that a ternary omplex mehanism operates. The test agonist was piloarpine, whih produed on average 83% of the ativity of the full agonist, arbahol. Piloarpine was analysed in omparison with arbahol and by reeptor inativation in the same tissue resulting in small and statistially insignifiant differenes in Em (96.7% and 97.3% respetively) and pka (5.3 and 4.95 respetively). 5 In onlusion, in this experimental system, there was no evidene for the errors in agonist affinity estimation predited by theory. Although this onlusion only applies to this system and appliation of the test to others is neessary to establish the generality of the present results, further examination of the theoretial basis for the predited errors is required. Introdution In a number of reent artiles (Colquhoun, 1987; Makay, 1988; Kenakin, 1989; Leff & Harper, 1989) attention has been drawn to the theoretial unreliability of agonist affinity estimates obtained by pharmaologial methods due to the operation of reeptor isomerisation (del Castillo & Katz, 1957) and ternary omplex (De Lean et al., 1978) mehanisms. In the ase of the reeptor isomerisation mehanism it has been shown that it is theoretially impossible, by use of traditional pharmaologial methods (Stephenson, 1956; Furhgott, 1966; Barlow et al., 1967), to estimate agonist affinity independently of effiay, with the onsequene that affinity itself is overestimated (Colquhoun, 1987). The same preditions are made in the ase of the ternary omplex mehanism when the onentration of agonist-reeptor omplexes is similar to or less than the onentration of transduer units with whih they interat (Makay, 1988; Leff & Harper, 1989). However, it is predited for both mehanisms that the magnitude of overestimation of affinity will be larger in the ase of full agonist analysis by the reeptor inativation method of Furhgott (1966) than in the ase of partial agonist analysis by either the interation method (Stephenson, 1956) or the omparative method (Barlow et al., 1967). It follows, if these theoretial preditions are orret, that when the same agonist (whih would neessarily be a partial agonist) is analysed by the inativation method and either of the two latter methods, different estimates of affinity should be obtained. As suggested I Author for orrespondene. previously (Leff & Harper, 1989) this provides an experimental test for the operation of onditions under whih erroneous affinity estimates may be made. This test was originally proposed to detet the operation of the ternary omplex mehanism but it applies equally well to systems obeying the reeptor isomerisation mehanism. The present paper explains the pratialities of this test and illustrates its potential utility by its appliation to the analysis of a partial musarini agonist. Theory The estimation of agonist affinity by pharmaologial methods (Stephenson, 1956; Furhgott, 1966; Barlow et al., 1967) impliitly assumes the validity of traditional reeptor theory. With inreasing knowledge of reeptor mehanisms, it has beome neessary to question the utility of traditional reeptor theory as a basis for analysing agonist ation (Colquhoun, 1987). The entral issue is that the traditional theory assumes that affinity an be estimated independently of intrinsi effiay. Analysis of two apparently plausible mehanisms for the ativation of reeptors by agonists indiates that this assumption is not valid. Below are given the main preditions of these analyses. For full theoretial details the reader is referred to the original papers. Reeptor isomerisation mehanism This mehanism was originally proposed to explain how agonists exert their effets in ion-hannel-linked reeptor

2 56 P. LEFF et al. systems (del Castillo & Katz, 1957): KA E A + R e AR AR* R represents unoupied reeptors, AR represents oupied but unativated reeptors and AR* represents ativated reeptors. Thus, the generation of a pharmaologial effet by the agonist depends on its ability to bind to the reeptor and then to eliit its isomerisation into the ative state. In this sheme, agonist affinity is determined by the dissoiation onstant, KA, for the AR omplex, and intrinsi effiay is defined by the isomerisation onstant, E (see Colquhoun, 1987). Colquhoun (1987) gave the theoretial results of applying pharmaologial methods for the estimation of agonist affinity in suh systems. In the ase of the irreversible reeptor inativation method (Furhgott, 1966), it was predited that affinity would be overestimated, the estimated dissoiation onstant being: KAest = KA/(1 + E) (1) showing that the extent of overestimation would be proportional to the intrinsi effiay of the agonist. In ontrast, in the ase of the omparative (Barlow et al., 1967) or interation (Stephenson, 1956) methods for the analysis of partial agonists it was predited that the estimated affinity would be lose to the true value so long as the intrinsi effiay of the test partial agonist was muh less than that of the referene full agonist. Ternary omplex mehanism This mehanism was originally proposed to explain how agonists exert their effets in reeptor systems oupled to G- proteins (De Lean et al., 1978). In priniple, it may apply to any system in whih the reeptor interats with a transduer unit in order to initiate a response: KA LAR A + R - AR + T - ART As with the isomerisation mehanism, R represents unoupied reeptors, AR represents oupied but unativated reeptors, and ART represents reeptors in their ative state due to oupling with transduers. KA determines affinity in this mehanism as in the ase of the isomerisation mehanism, whereas effiay is now determined by the dissoiation onstant, KAR, for the ternary omplex, ART. In this ase, preditions relating to the appliation of pharmaologial methods for affinity estimation depend on the relative onentrations of R and T (Leff & Harper, 1989). Denoting the total onentrations of these as [RO] and [To] respetively, when [RO] > [TO], the reeptor inativation method as well as the omparative method provides a orret KA estimate. When [RO] << [TO], the omparative method provides a orret estimate but the reeptor inativation method overestimates affinity, the estimated dissoiation onstant being: KAMS, = KA/(1 + [TO]/KAR) (2) This result, whih was also given by Makay (1988), is analogous to the predition made for the isomerisation mehanism (equation 1), the extent of overestimation of affinity being dependent on intrinsi effiay, in this ase 1/KAR. When [RO] = [TO], both the inativation and omparative methods overestimate affinity but the size of the error is larger for the inativation method. A testfor errors in affinity estimation It follows, if these mehanisms operate, that the estimate of affinity for a partial agonist obtained using the reeptor inativation method will, in general, be higher than that obtained by the omparative method. Only in the ase of the ternary omplex mehanism, with [RO] > [TO], are the estimates predited to be the same, as would be assumed by traditional theory. This provides a test whih may help to detet the operation of mehanisms whih do not aord with traditional reeptor theory and in doing so assist in avoiding erroneous affinity estimation. Alternatively, the test may serve to indiate that the traditional theory holds and that the assoiated affinity estimates are valid. In the first part of this study, theoretial data are generated orresponding to the two mehanisms, and analysed in order to evaluate the magnitude of differene in affinity estimates that is likely to our in pratie. Having established the detetion limits of the test in this way, it is then applied to experimental data obtained in a musarini reeptor system. Methods Computer simulations Agonist-onentration effet, E/[A], urves were generated with the formulation of the ternary omplex model given by Leff & Harper (1989) and the reader is referred to that paper for full theoretial derivations; only brief details will be given here. Aording to theory, the equilibrium onentration of ative ART omplexes is given by a quadrati funtion of agonist onentration: [ART]2 + b[art] + =O (3) in whih b = -KAR(1 + KJ[A]) + [TO] + [R] = [Ro][TO] [ART] has values defined by the roots of this equation. The pharmaologially meaningful one is: [ART] = (-b-)/2 (4) E/[A] urves were simulated assuming that E equated with [ART], hoosing parameter values to represent the onditions [RO] >, = or 4 [To]. Under the ondition [RO] < [TO], the [ART]/[A] relation an be written learly (Leff & Harper, 1989): [ART] [R]=KA K + +([ROI+FT[A] (1 + TT)[A](5 with IT defined as [To]/KAR- In the ase of the isomerisation mehanism, the relation between [AR*] and [A] an be written (rearranging equation (3) in Colquhoun (1987)): [R]=KA + (1 + E)[A] (6) [AR*] K I[R+]E[A] whih is learly analogous to equation (5), the only differene being the mehanisti definition of effiay. This means that results obtained simulating equation (4) under the ondition [RO] < [TO] an be taken to represent either the ternary omplex or isomerisation mehanisms. All simulations were arried out on a Compaq 386/25 using the ommerially available spreadsheet pakage, Symphony, produed by Lotus. Guinea-pig isolated left atria (5) Male Dunkin-Hartley guinea-pigs (34 g) were killed by a blow to the bak of the head. The left atria were quikly removed and suspended in 2 ml organ baths ontaining Krebs solution of the following omposition (mm): NaCl , KC 4.69, MgSO4 1.18, KH2PO4 1.18, gluose 11.1, NaHCO and CaCl This was maintained at 37C and ontinually gassed with 5% CO2 in 2. The atria were

3 DETECTING ERRORS IN AGONIST AFFINITY ESTIMATION 57 attahed at the base to tissue-hook eletrodes by otton thread and onneted at the top to isometri fore transduers (Ormed Beam) with steel lips. The diastoli (resting) tension was set at an initial value of 1 gwt. The tissues were eletrially stimulated with pulses of 5 ms duration, at a frequeny of 2Hz and a voltage whih was 1.5 times that required for a threshold response (twith). Generated fore was reorded on a Devies M 19 or an Advane Bryans flat bed reorder. Experimental protools General All tissues were allowed a 6min period of equilibration before the addition of any drugs. Agonist onentration-effet, E/[A], urves were onstruted by umulative additions of either arbahol or piloarpine at.5 logl unit inrements. Up to 3 E/[A] urves were performed in eah tissue. Responses were reorded as perentage inhibitions of the stimulated twith. Carbahol effiay estimation by the reeptor inativation method In eah tissue a arbahol E/[A] urve was onstruted. After washing, tissues were inubated with the irreversible antagonist phenoxybenzamine (Pbz) (3 yim or 1,uM) or vehile (4% polyethylene glyol) for 3 min. Following removal of the irreversible antagonist by several hanges of the organ bath Krebs solution over a 4 min period, a seond arbahol E/[A] urve was performed. Piloarpine affinity estimation by the omparative and irreversible reeptor inativation methods In eah tissue a arbahol E/[A] urve was onstruted, followed, after washing, by a piloarpine E/[A] urve. Tissues were subsequently washed and then inubated with Pbz (.3 pm,.5 fm or 1 pm) or vehile for 3 min. Pbz was then removed by several washes over a 4 min period and a further piloarpine E/[A] urve onstruted. Drugs The following drugs were used: arbahol hloride (Sigma Chemial Company), piloarpine hydrohloride (Evans Medial Ltd), phenoxybenzamine hydrohloride (Smith, Kline and Frenh Laboratories), and atropine sulphate (Sigma Chemial Company). Carbahol, piloarpine and atropine were dissolved in distilled water. Phenoxybenzamine was dissolved in 4% polyethylene glyol. Data analysis Operational model fitting Computer-generated and experimental data were fitted using the operational model of agonism (Blak & Leff, 1983; Blak et al., 1985): E = Emtn[A]7 (KA + [A])n + TI[A]n in whih E and [A] are the pharmaologial effet and the onentration of the agonist, respetively; E. is the maximum possible effet; KA is the agonist dissoiation onstant (this was estimated as the negative logarithm, that is, pka); r is the effiay of the agonist (estimated as a logarithm); n determines the steepness of the oupany-effet relation. The fitting proedures employed are desribed in detail elsewhere (Leff et al., 1989); only brief details will be given here. As a multiple urve design was used, eah tissue provided an estimate of KA by eah of the omparative and inativation methods. In the ase of the omparative method the partial agonist urve data were fitted to the operational model (equation 7) simultaneously to fitting the full agonist data to a logisti funtion of the form: Em[A]n (8) [A5]n + [A]n where Em and n are as defined above and [A5] is the loation parameter for the full agonist urve. (Loation parameters were estimated as negative logarithms (p[a5o])). This allows KA as well as T for the partial agonist to be estimated as well as Em and n from eah pair of urves. In the ase of the inativation method, the E/[A] urve data for the agonist (full or partial) before and after reeptor inativation were fitted to equation (7) providing a ommon estimate of Em, n and KA, and a value of T for eah urve in the pair. Average values of the parameter estimates are quoted with standard errors orresponding to between tissue variation. Comparisons of parameter estimates obtained by the omparative and inativation methods were made by paired t test. Standard errors derived from the fitting proedure orresponding to internal fitting error (a measure of goodness-offit) are also quoted. Logisti urve fitting In ontrol experiments designed to determine any effet of drug vehile on the response to arbahol and piloarpine, individual E/[A] urves were fitted to a logisti funtion of the form: EA= E= a[a]' [A5]m + [A]tm (9) where a, [A5] and m are the asymptote, loation and slope parameters respetively (any vehile effet would be shown by signifiant hanges in one or more of these parameters). All data fitting proedures were arried out with the statistial pakage, BMDP, and a Vax 11/78 mainframe omputer. Results Analysis of theoretial datafor a partial agonist Figure 1 shows theoretial E/[A] urves generated using equation (4). Parameter values were hosen to represent the onditions [RO] > [TO] (panel (a)), [RO] = [TO] (panel (b)) and [RO] << [TO] (panel ()) in the ternary omplex mehanism, the last ondition also representing the isomerisation mehanism. Under eah ondition, an intrinsi effiay (l/kar) value was hosen to simulate a partial agonist whih an generate approximately 85% of the maximum effet. The same agonist was simulated using a lower value of [RO] representing the effets of reeptor inativation, and a full agonist urve was also generated in order to analyse the partial agonist by the omparative method. 'Data' points overing a realisti experimental range were sampled from the urves at.5 loglo unit intervals. These data are shown as symbols in Figure 1. These 'data' points refer to onentrations of [ART]. For urve fitting, [ART] was assumed to be diretly proportional to effet, and the onentrations of [ART] were saled up to over a range of to 1. Em for the system is equivalent to the maximum theoretial onentration of [ART] that an be ahieved. For eah ondition, the partial agonist urve was analysed by the omparative method and by the inativation method. The resulting estimates of affinity and of the other operational parameters are ollated in Table 1. This analysis showed that, under the ondition [RO] > [TO] aurate estimates of pka were obtained by both methods. Also, it is to be noted that Em was aurately estimated by both methods and the standard errors assoiated with the estimated model parameters were small and similar. However, under the onditions [RO] = [TO] and method overestimated pka by [RO] < [TO], the inativation some.7 log1o units, whilst the omparative method was

4 58 P. LEFF et al. a 1-8- x N b 1 - (D x F 4- r- x 2 - Affinity pka pka full partial (a) (b) () [Ro] >> [To] Table 1 Analysis of theoretial data for a partial agonist by omparative and inativation methods l log Agonist (M) [Rol = [To] I I Ị I. ' log Agonist (M) ] [Ro] << [To] [Ro] > [To] Em n log T pka [Ro] = [To] Em n log T pka Comparative Method Standard Estimate error Inativation Method Standard Estimate error [RO] 4 [To] (also represents the isomerisation mehanism) Em n log T pka In the last fit, standard errors ould not be omputed due to degeneraies enountered during the fitting proedure. The 'orret' estimates of pka and Em were 5. and 1. respetively. example under the ondition [RO] = [To] the standard error on Em was about 5%, whereas under the ondition [RO] > [To] it was less than 1%. This is an indiation of a poor quality of fit, whih is antiipated sine the theoretial data should not aord with the operational model under the onditions [RO] < [TO]. Indeed, when [RO] << [To] standard errors ould not be estimated due to poorness of fit. Intrinsi effiay pkar pkar full partial r 4- r-lar-- Analysis of experimental datafor thefull agonist, arbahol r In order to apply the omparative method to the analysis of.. 6m the partial agonist, piloarpine (see below), it was neessary I 6 4 log Ago first to ensure that the referene agonist, arbahol, ats as a nist (M full agonist in the guinea-pig left atrial preparation. E/[A] Figure 1 Theoretial E/[A] urves underrvarious onditions relating urves for arbahol were obtained before and after Pbz- Panels (a), (b) treatment (3pM or 1pM, 3min). Control experiments reeptor ([RO]) and transduer ([To]) oonentrations. and () represent the onditions [RO] > [To], [RO] = [To] and showed that onseutive arbahol E/[A] urves obtained in [RO] << [To] respetively. The latter oi isomerisation mehanism. Under eah odition is equivalent to the the absene of Pbz treatment were effetively superimposed, ondition, theoretial urves onfirming the validity of the paired urve analysis. The mean were generated orresponding to a full a] gonist (). a partial agonist (-) with approximately 85% of the ativi logisti parameters (± s.e.; n = 5) of these urves were as ty of the full agonist and the follows: 1st urve: a = 86.9 (± 1.7); m = 1.35 (±.7); same partial agonist after reeptor inative tion (A). The simulation parameters used were a'sfollows: p[a5] 6.68 (±.4). 2nd = urve: a 85.8 (± 1.5); = m 1.4 = r4l"- (+.6); p[a5] = 6.62 (±.4). Figure 2 illustrates typial results obtained in a single Pbztreated tissue. The lines drawn through the data are the results [Ro] [Ro] of operational model-fitting. The substantial rightward shift [To] initial inativated and depression of the arbahol E/[A] urve produed by Pbz treatment indiated that the agonist has a high effiay in this tissue. Analysis of 7 experiments gave an average estimate of effiay (X) of 72.4 (log X = 1.86 (±.11)) and an estimate of affinity (pka) of 4.73 (±.1). The high effiay onfirmed The symbols represent the 'data' points t] hat were sampled from om- that arbahol was indeed a full agonist in this system. This is puter generated urves and analysed by perational model-fitting (see in agreement with the findings of other workers (Furhgott, Table 1). 1966). The reliability of this analysis and that with piloarpine requires that Pbz ats purely to derease [RO] in this system. This was onfirmed by experiments in whih oinubation aurate to within.1 log1o units. Also, Em was overestimated with the reversible ompetitive antagonist atropine was shown by up to 5% by the inativation mlethod but aurately esti- to protet the tissues from the effets of Pbz (data not shown, mated to within 5% by the omparative method under both n = 3). Thus, E/[A] urves for arbahol obtained following onditions. Furthermore, the standa*rd errors assoiated with inubation of tissues with atropine (O.1 Mm) and Pbz (1pM) model fitting were onsiderably la] rger for the inativation then washout were indistinguishable from urves obtained fol- (see Table 1). For lowing atropine treatment method than for the omparative rxmethod alone.

5 DETECTING ERRORS IN AGONIST AFFINITY ESTIMATION :LI 6-._-.Q 4-._ g 2 a _l 6 T 4 C I ~~~~~~~~~~ I* I. * I. I log Carbahol (M) Figure 2 Effet of phenoxybenzamine (Pbz) treatment on arbahol E/[A] urves. Carbahol E/[A] urves were obtained before () and following 3min exposure to 1pUM () Pbz. The symbols represent the data from a single tissue. The lines drawn through the data are the results of operational model fitting using the reeptor inativation method. For this partiular tissue the estimated model parameters were as follows (standard errors in parentheses): E. = 92.6 (± 1.3); n = 1.43 (±.8); rt (ontrol) = 58.9 (log T, = 1.77 (±.7)); T2 (1pUM Pbz) =.8 (log T2 = -.8 (±.4)); pka = 4.84 (±.7). Analysis ofexperimental datafor the partial agonist, piloarpine Piloarpine E/[A] urves were obtained before and after Pbz treatment, following onstrution of an E/[A] urve for arbahol. As above, ontrol experiments showed no differenes between onseutive piloarpine E/[A] urves obtained in the absene of Pbz treatment. The mean logisti parameters (± s.e.; n = 6) were as follows: 1st urve: ax = 67.9 (± 4.8); m = 1.21 (±.1); p[a5] = 5.47 (±.7). 2nd urve: ax = 68.6 (+ 2.9); m = 1.4 (±.13); p[a5] = 5.42 (±.5). Figure 3 illustrates a typial set of data obtained in a single Pbz-treated left atrium. Eight suh experiments were onduted. In eah instane, piloarpine was analysed in omparison with arbahol and by the inativation method. The lines drawn through the data are the results of operational model-fitting. Table 2 shows the resulting parameter estimates and statistial analysis of the pka and Em values obtained. Evidently, for eah of these parameters the differene between the estimates provided by the two methods was very small and learly not signifiant. Also, fitting errors assoiated with the pka and Em estimates were small, on average 3% and 8% for the inativation method and 2% and 2% for the omparative method (individual errors not shown). Disussion The possibility that errors may our in the estimation of agonist affinity by pharmaologial methods makes it important that tests are available for their detetion. This paper illustrates one suh test. Theoretial studies (Colquhoun, 1987; Makay, 1988; Leff & Harper, 1989) of two plausible reeptor models, the isomerisation (del Castillo & Katz, 1957) and the ternary omplex (de Lean et al., 1978) mehanisms, had already shown that the reeptor inativation method (Furhgott, 1966) is likely to provide higher estimates of agonist affinity than the inter- b 1 A, 8 C.) '.- 6._ -C log Agonist (M) log Piloarpine (M) Figure 3 Analysis of piloarpine E/[A] urve data by the omparative and inativation methods. The data shown are from a single tissue for arbahol (), piloarpine () and piloarpine after 3min exposure to 1 jm phenoxybenzamine (Pbz, A). The lines drawn through the data are the results of operational model-fitting. Panel (a) illustrates the omparative analysis of piloarpine using arbahol as the referene full agonist. The model parameter estimates were as follows (standard errors in parentheses): Em = 11. (± 2.1); n = 1.8 (±.5); X = 5.1 (log X =.71 (±.5)); pka = 5.6 (±.6). Panel (b) illustrates the analysis of piloarpine by reeptor inativation. The model parameter estimates were as follows (standard errors in parentheses): E. = 15.6 (± 7.); n = 1.6 (±.8);,r (ontrol) = 4.3 (log t, =.63 (±.14));?2 (1puM Pbz) =.4 (log T2 = -.4 (±.7)); pka = 5.12 (±.13). For omplete results see Table 2. ation (Stephenson, 1956) or omparative (Barlow et al., 1967) methods. It had also been suggested (Leff & Harper, 1989) that this differene ould serve as the basis for a test to detet the operation of suh reeptor mehanisms and, therefore, to assess the validity of agonist affinity estimates obtained by pharmaologial methods. In pratie, suh a test an only be applied to partial agonists sine only these an be analysed by both the inativation and omparative methods. The first objetive of the present study was to establish the magnitude of the differene in affinity estimates that the two tests would be likely to provide. This was done using theoretial data for a partial agonist operating by the ternary omplex and isomerisation mehanisms and produing approximately 85% of the maximum possible effet. This effet level was hosen to be similar to that produed by the experimental example, piloarpine. This Table 2 Mean parameter estimates and statistial analysis of Em values and piloarpine pka values obtained from experimental data Mean (n = 8) 96.7 s.e. 3.3 paired t test Comparative method Em pka Inativation methods Em PKA Mean differene Em pka P > P >.5

6 6 P. LEFF et al. theoretial analysis showed that, other than under the ondition [RO] > [To] in the ternary omplex model, the inativation method was likely to produe an affinity estimate some five fold or.7 log1o units higher than the true value, whereas the omparative method was aurate to within.1 log1o units under all onditions. In addition, the overestimation of affinity by the inativation method was aompanied by drasti overestimation of Em. Therefore, in the ontext of the two mehanisms studied, it an be onluded that the operation of onditions unfavourable to affinity estimation an be deteted by differenes both in the values of KA and Em obtained by the two pharmaologial methods. In model-fitting terms, the overestimation of KA and Em are onneted. The result of overestimating affinity is that the loation of the ontrol urve, obtained in the absene of reeptor inativation, is nearer to the estimated KA than to the true KA. Therefore, the partial agonist is treated by the fitting proess as if it has lower effiay than it atually does. In other words, the maximum effet of the partial agonist is treated as if it is lower in relation to the maximum possible effet, Em, than it really is. Therefore, onversely, Em is overestimated. This also aounts for the rather large standard error assoiated with Em under the onditions [RO] = [To] and the inability to fit under the ondition [RO] 4 [To]. The lower the maximum effet of the partial agonist, the more the fitting proess will have diffiulty in projeting Em from the urve depression aused by reeptor inativation. The seond objetive of this study was to exemplify the test in a pratial example. The musarini reeptor system in the guinea-pig left atrium was hosen sine there is evidene that it is a G-protein-linked system (Eglen et al., 1987) and, therefore, ould be antiipated to operate by the ternary omplex mehanism. In this system, as demonstrated here, arbahol is a full agonist (see Figure 2 and analysis), permitting its use as the referene agonist in the omparative method. Piloarpine demonstrates partial agonism in this tissue, ahieving approximately 85% of the maximum effet of arbahol. Analysis of piloarpine by reeptor inativation and by omparison with Referenes BARLOW, R.B., SCOTT, N.C. & STEPHENSON, R.P. (1967). The affinity and effiay of onium salts on frog retus abdominis. Br. J. Pharmaol., 31, BLACK, J.W. & LEFF, P. (1983). Operational models of pharmaologial agonism. Pro. R. So. B, 22, BLACK, J.W., LEFF, P., SHANKLEY, N.P. & WOOD, J. (1985). An operational model of agonism: the effet of E/[A] urve shape on agonist dissoiation onstant estimation. Br. J. Pharmaol., 84, COLQUHOUN, D. (1987). Affinity, effiay, and reeptor lassifiation: Is the lassial theory still useful? In Perspetives on Reeptor Classifiation. ed. Blak, J.W., Jenkinson, D.H. & Gerskowith, V.P., pp New York: Alan R. Liss. DEL CASTILLO, J. & KATZ, B. (1957). Interation at end-plate reeptors between different holine derivatives. Pro. R. So. B., 146, DE LEAN, A., STADEL, J.M. & LEFKOWITZ, R.J. (1978). A ternary omplex model explains the agonist-speifi binding properties of the adenylate ylase oupled fl-adrenergi reeptor. J. Biol. Chem., 255, EGLEN, R.M., HUFF, M.M., MONTGOMERY, W.W. & WHITING, R.L. (1987). Differential effets of pertussis toxin and lithium on musarini responses in the atria and ileum: evidene for reeptor heterogeneity. Br. J. Pharmaol., 9, 6-8. arbahol produed very similar, and statistially indistinguishable, estimates of pka (4.95 and 5.3 respetively) and Em (97.3 and 96.7 respetively). Also the standard errors assoiated with the parameters, in partiular Em, were small, giving no indiation of the problems of goodness-of-fit assoiated with analysis of theoretial data. Therefore, in this pratial example there was no indiation of the errors whih the operation of the ternary omplex mehanism an, in theory, produe. Waud (1969) performed a similar study on piloarpine using the guinea-pig ileum preparation. He obtained a pka of 5.3 by the inativation method and one of 5.36 by the omparative method. Although that study was not onduted with the above issues in mind, it onfirms our findings. The absene of the predited theoretial errors in the present study does not rule out that the mehanism applies sine, under one ondition ([RO] > [To]), similar estimates of pka and Em are predited by theory. If this is the orret explanation, the results would imply that although Pbz must have altered the ratio between [RO] and [TO] the redution ould not have been suffiient for [RO] to then approximate to [TO]. That is, the ondition [RO] > [To] would have applied before and after reeptor inativation. Of ourse, this is not to say that the same initial ratio between [RO] and [To] would apply in other reeptor systems and in suh systems where [RO] < [To] the theoretially predited errors may be demonstrable. The results of the present study prompt the question as to whether the theoretial objetions raised about agonist affinity estimation by pharmaologial methods are justified sine one possible interpretation of these results is that the theories and, therefore, their preditions are inomplete or wrong, as disussed elsewhere (Leff et al., 199). Clearly, further appliation of experimental tests to detet errors in agonist quantifiation and inreased understanding of reeptor-effetor mehanisms will do muh to larify the issue. FURCHGOTT, R.F. (1966). The use of fi-haloalkylamines in the differentiation of dissoiation onstants of reeptor-agonist omplexes. Adv. Drug Res., 3, KENAKIN, T.P. (1989). Challenges for reeptor theory as a tool for drug and drug reeptor lassifiation. Trends Pharmaol. Si., 1, LEFF, P., DOUGALL, I.G., HARPER, D. & DAINTY, I.A. (199). Errors in agonist affinity estimation: do they and should they our in isolated tissue experiments? Trends Pharmaol. Si., 11, LEFF, P. & HARPER, D. (1989). Do pharmaologial methods for the quantifiation of agonists work when the ternary omplex mehanism operates? J. Theor. Biol., 14, LEFF, P., PRENTICE, D.J., GILES, H., MARTIN, G.R. & WOOD, J. (1989). Estimation of agonist affinity and effiay by diret, operational model-fitting. J. Pharmaol. Methods, 23, MACKAY, D. (1988). Continuous variation of agonist affinity onstants. Trends Pharmaol. Si., 9, STEPHENSON, R.P. (1956). A modifiation of reeptor theory. Br. J. Pharmaol. Chemother., 11, WAUD, D.R. (1969). On the measurement of the affinity of partial agonists for reeptors. J. Pharmaol. Exp. Ther., 17, (Reeived November 14, 1989 Revised Marh 21, 199 Aepted May 11, 199)