An operational model of pharmacological agonism: the effect of E/[A] curve shape on agonist dissociation constant estimation

Abstract

An operational model of pharmacological agonism has been analysed to predict the behaviour of rectangular hyperbolic and non-hyperbolic agonist-concentration effect, E/[A], curves with variation in receptor concentration, [Ro]. Irreversible antagonism is predicted to cause E/[A] curve gradient changes in non-hyperbolic cases but not in hyperbolic cases; in both cases estimation of agonist dissociation constants (KAs) is theoretically valid. 5-Hydroxytryptamine (5-HT) produced ‘steep’ E/[A] curves in contracting the rabbit isolated aorta preparation. Irreversible antagonism by phenoxybenzamine (Pbz) produced a flattened E/[A] curve, consistent with theoretical predictions. Fitting 5-HT E/[A] curves in the presence and absence of Pbz to the model provided an estimate of KA for 5-HT which was not significantly different from the estimate obtained using Furchgott's null method. The operational model of agonism appears to account qualitatively and quantitatively for the effects of [Ro] changes on hyperbolic and non-hyperbolic E/[A] curves. Under conditions where irreversible antagonism may be used to estimate KAs, fitting the operational model directly to E/[A] data represents a valid, economical and analytically simple alternative to the conventional null method. In a previous paper, Black & Leff (1983) undertook an analysis of pharmacological agonism, aiming to develop an operational model which explicitly described agonist-concentration effect, E/[A], curves and their behaviour under various experimental conditions. The approach to the problem began with the assumption that the initial event in the generation of a pharmacological effect by an agonist, hormone or drug, is a simple bimolecular interaction with a receptor obeying the Law of Mass Action. An attempt was made to define the ‘transducer function’ linking agonist occupied receptors, AR, to pharmacological effect, E, which would account for experimentally observed E/[A] curves. For rectangular hyperbolic E/[A] curves this transducer function is necessarily hyperbolic (Paton & Rothschild, 1965; Black & Leff, 1983). For E/[A] curves which deviate from hyperbolicity an alternative transducer function is necessary and alogistic-type function was sufficient to account for such cases. Here, the theoretical analysis is extended to predict the behaviour of non-hyperbolic E/[A] curves, particularly in circumstances where total receptor concentration, [Ro], is varied. [Ro] variation may occur physiologically between different tissues for the same agonist-receptor combination and [Ro] variation is introduced using irreversible antagonists to allow estimation of agonist dissociation constants (KAs) by the method of Furchgott (1966). In the latter circumstances particularly, theoretical expectation of E/[A] curve shape behaviour with variable [Ro] is relevant in order to establish conditions under which estimation of KAs are valid. In a recent paper (McPherson et al., 1983) the absence of such a framework for expectation led to the assertion that ‘to calculate accurate KA values it is essential that the pre- and post-inactivation concentration-response curves [in the method of Furchgott (1966)] be rectangular hyperbolas’, a claim which contravenes the principle of the null methods (for reviews see Colquhoun, 1973; Jenkinson, 1979). In this paper we use the operational model of agonism (Black & Leff, 1983) to demonstrate that KA estimation using irreversible receptor antagonism should be valid, in principle, regardless of the shape of the E/[A] curve so long as only one receptor-transducer system is involved and that the agonist-receptor interaction is a simple, non-cooperative, one. Both assumptions are implicit in Furchgott's method. The validity of the operational model as a basis for analysis is assessed by comparing its theoretical predictions with experimental data obtained using 5-hydroxytryptamine (5-HT) as agonist and phenoxybenzamine (Pbz) as irreversible antagonist. In the rabbit isolated thoracic aorta preparation, 5-HT produces E/[A] curves which are ‘steep’ compared to rectangular hyperbolae. Therefore this system offered the opportunity to challenge the ability of the operational model to account quantitatively for nonhyperbolic E/[A] curves and, in addition, to allow the assertion of McPherson et al., to be examined experimentally. In analysing the interaction between 5-HT and Pbz we compare the KA value estimated using Furchgott's method with that obtained by fitting the operational model directly to E/[A] curve data. In deriving the operational model ofagonism (Black & Leff, 1983) it was assumed that the initial event in the production of a pharmacological effect by an agonist, A, is a simple bimolecular interaction between the agonist and a receptor obeying the law of mass action. Therefore, at equilibrium, the concentration of occupied receptors, [AR], is given by in which [Ro] is the total receptor concentration and KA is the dissociation constant for AR. Equation (1) describes a rectangular hyperbolic, [AR]/[A], function. For E/[A] curves which are rectangular hyperbolic we proved that the relation between pharmacological effect, E, and [AR] must also be rectangular hyperbolic, that is in which KE is the value of [AR] for half the maximum possible effect, Em. This ‘transducer function’ could also be linear, which is the limiting case of the rectangular hyperbola with KE >> [AR], but in order to account for receptor reserve the relation is necessarily saturable. The resulting E/[A] relation, found by substitution of (1) into (2), is: τ, which we called the ‘transducer ratio’, defines the operational efficacy, for an agonist. This can be appreciated by considering the location, [A50], and asymptote, α, parameters of the E/[A] function: When τ is large, α is approximately Em and [A50] is much less than KA, meaning full agonism. When τ is small, α is less than Em and [A50] is approximately equal to KA, meaning partial agonism or, for very small τ, competitive antagonism. For non-rectangular hyperbolic E/[A] curves, if receptor occupancy is still assumed to be non-cooperative, and to obey equation (1), then an alternative transducer function to equation (2) is required. In our previous analysis we chose a transducer function ofthe logistic form Combination of equations (1) and (7) retains the definition and meaning of the transducer ratio, τ, and leads to an E/[A] function of the form The location and asymptote parameters of this function are: While a flexible transducer function is required to account for non-hyperbolic E/[A] curves, the particular function used, equation (7), cannot be regarded as necessary. In the present study, the sufficiency of this choice of function is all that can be examined. Figure 1 illustrates the model in the three-dimensional display adopted previously (Black & Leff, 1983). The operational model of pharmacological agonism: the model was simulated with n = 1 and [Ro]/KE= 10. So α= 0.91 Em and [A50]=0.09 KA. The model predictions for EI[A] curve gradients are conveniently made by finding the derivative of equation (8) at the mid-point of the curve. Pharmacological E/[A] curves are usually displayed in semi-logarithmic (base 10) mode, so equation (8) must be differentiated with respect to lg[A], then [A50], as defined by equation (9), is substituted for [A]. Also, in order to making meaningful comparisons of curves with different asymptotes, the gradient must be ‘normalized’ by dividing the derivative by Emτn/(l +τn) (equation 10). The resulting definition of the mid-point gradient is Note that when n = 1, corresponding to a rectangular hyperbolic E/[A] curve, the gradient G has the value 0.576 which is the mid-point gradient for normalized semi-logarithmic rectangular hyperbolae. This value is constant regardless of the value of τ. When n is not equal to 1 equation (11) does not simplify and G is dependent on τ. When τ is very large, that is for substantial receptor reserves, G→0.576n which is the expected value of a logistic function of the form, This reflects the fact that for large τ the transducer function (equation (7)) determines the form of the E/[A] relation. This is because occupancy, [AR], is proportional to [A] in the range that [AR] is saturating the E/[AR] function (Paton & Rothschild, 1965). When τ is very small, G→0.576n (2(1 − 1/21/n)). In cases when n > 1, G decreases with decreasing τ; in cases when n < 1, G increases with decreasing τ. The dependence of G on τ and n is illustrated in Figure 2 which shows computer simulated E/[A] curves and accompanying mid-point gradient values. E/[A] curve simulations for varying n and t: the operational model was simulated using equation (8) with KA fixed at 10−6. In a(i), (ii) and (iii), n is set at 0.5, 1.0 and 2.0 respectively. In each panel t is varied as shown. Mid-point gradients, calculated from equation (11), are given by each curve together with the corresponding τ value. The dotted lines drawn through the curves indicate the trend in p[A50]s as τ→0. In panels b (i), (ii) and (iii), double reciprocal plots are constructed, according to the method of Furchgott (1966). Equi-effective agonist concentrations [A] and [A]′ were interpolated from curves at τ= 100 and τ= 1 respectively. Interpolations were made in the effect range 0.3 to 0.999 of the maximum of the curve at τ= 1. The three graphs are actually identical although the range of [A] and [A]′ values differ due to the different relative gradients of the paired E/[A] curves in each case. The value of (slope − 1)/intercept was 10−6 in each instance corresponding to the true KA for the simulated curves. The important point from this analysis is that the gradients of the non-hyperbolic E/[A] curves are predicted by the model to vary with changes in the receptor reserve τ, whereas the gradient of hyperbolic E/[A] curves remain fixed and independent of τ. Therefore, in experiments employing irreversible receptor antagonism the variation in τ produced by [Ro] reduction is expected to be accompanied by E/[A] curve gradient changes if the initial, control, curve is non-hyperbolic. Model equations (9) and (10) establish the predicted behaviour of the location and asymptote curve parameters with variations in τ. The relationship between [A50] and KA as τ is reduced is of particular relevance to the estimation of KA by irreversible antagonism. Equation (9) predicts that as [Ro], and therefore τ, is reduced [A50] approaches a limiting value of KA/(21/n− 1). Under this condition the asymptote α is reduced towards zero and A behaves as a progressively less efficacious partial agonist. When n = 1, the limiting value of[A50] is KA and therefore KA can be estimated ‘by eye’ by extrapolating the trend in the [A50] values. However, when n is not equal to 1 the extrapolated [A50] value no longer corresponds to KA. For example, when n = 2, the [A50] approaches a limiting value of KA/0.41 which therefore, is an underestimate of agonist affinity. This case, which implies positive cooperativity at the transducer level, was considered by Furchgott (1966). Conversely, when n is less than 1, the limiting value of [A50] is an overestimate of agonist affinity. Figure 2 exemplifies these predictions. Importantly, regardless of n, and therefore regardless of the gradients of the E/[A] curves, the method of Furchgott (1966) to estimate KA can be shown to be valid. This is also shown in Figure 2, where double reciprocal plots of equi-effective agonist concentrations have been constructed from the simulated curves according to Furchgott's method. In each instance a control curve corresponding to τ= 100 and a postantagonism curve corresponding to τ= 1 were used. Straight lines are evident in all three cases corresponding to the same true KA. Male NZW rabbits (2.0–2.5 kg) were killed by intravenous injection of pentobarbitone sodium (60 mg kg−1). Isolated thoracic aortic ring segments were prepared according to the method of Stollak & Furchgott (1983) and suspended in 20 ml organ baths, thermostatically controlled at 37°C, containing modified Krebs solution, pH 7.4, of the following (mM) composition: NaCl with were as increases in using and displayed on The adopted for these experiments is shown in Figure = Pbz = The was made to a total of each by in the were by of for Krebs was in order to possible of 5-HT due to or by the agonist & et al., et al., The of 5-HT was used only to establish 5-HT curves were constructed in each according to using 5-HT Pbz were as of the maximum of the corresponding control curve. aorta provided ring three of which were used for Pbz and the three to 5-HT curves. For both control and Pbz a total of of curves were the control on between and curves. used pentobarbitone sodium and 5-HT phenoxybenzamine (Pbz) & 5-HT and were Pbz was prepared in at the concentrations used in the produced no in the Krebs 5-HT and Pbz were prepared and prepared from a 1 The maximum of to the organ was the following fitting were computer were with the of which was the used to data to the model of agonism. of 5-HT E/[A] curve data were to a logistic function of the form in which b and are the asymptote, location and parameters respectively. The location parameters were actually estimated as by making the b = of control and 5-HT curves were to model equation (8) of n and KA, and of τ for each curve in a was assessed by and experimental data for A analysis be the of a Equi-effective 5-HT concentrations were interpolated from logistic curves to each of control and Pbz 5-HT curves. were made for each using the of the Pbz curve as by KA values were estimated by fitting the interpolated [A] and [A]′ equi-effective concentrations of 5-HT and Pbz to the equation in which − and where is the of KA is estimated as − & 5-HT produced of the aortic This was the case for both curves in control and are in Figure of the paired curves was by logistic curve fitting using equation and the estimated asymptote, location and parameters were compared by paired t 5-hydroxytryptamine (5-HT) E/[A] curve a of control and phenoxybenzamine curves are shown. concentrations of 5-HT are given in of paired control 5-HT curves a in asymptote between and curves of < This was used to 5-HT Pbz changes were observed but a small but in the location was This in was only and was considered to be small to be meaningful it was of the same order as the fitting on curve location The of logistic fitting of paired 5-HT and curves are shown in 1. that was between their parameters < The control 5-HT curves were than the Pbz curves in all although both were than rectangular hyperbolae. Pbz reduced significantly < as expected and also produced of the location of control and curves were to the operational model of agonism (equation A set of data with the lines is shown in Figure of observed data and corresponding values for the curves. However, the between observed and only of the of the corresponding curve and the were much Figure an of the of of lines from the data of E/[A] curve the shows the of 5-hydroxytryptamine (5-HT) data for a using equation values of KA and n were estimated and values of τ to the pre- and post-inactivation curves respectively. For each of curves were made of KA and τ and corresponding to the control and curves respectively. The between and for each reflects the reduction in [Ro] due to In to the n, estimated for each the mid-point gradients and of each curve in a were calculated from model equation 2 the of and illustrates the effect of irreversible antagonism on the 5-HT E/[A] curves, previously by logistic curve Em values were than the asymptote of the control 5-HT curve, the latter set at This that 5-HT was not as a full agonist but was on only of the maximum possible of the control τ with the that although the initial operational of 5-HT varied between due to [Ro] the effect of Pbz in each was [Ro] and, therefore by the same in each case. as were consistent the with an value of of paired control Pbz curves by the method of the shown in as are given for each and also the estimated of Pbz The value estimated by this method was The values estimated by operational model fitting and Furchgott's method were the only appears to be no of comparing these values due to the different approaches used in their estimation and in particular the of involved in the Furchgott method of The between the to analysis The provided by operational is to − in Furchgott's both corresponding to the concentration of irreversible antagonism. The value of was compared with an value of 1 − of a between the methods of estimation is In this paper we have extended our previous operational analysis of pharmacological agonism in order to predictions hyperbolic E/[A] curves, particularly under conditions where receptor concentration, [Ro], is varied. In order to the of the operational model in this we have examined an combination which agonism. curve fitting of 5-HT E/[A] data in the rabbit aortic ring that the curves were than rectangular hyperbolae. was also evident irreversible the curves an effect which is not due to of the asymptote of the curves. parameters and mid-point gradients have been calculated for E/[A] curves in order to such This is consistent with model predictions when a ‘steep’ transducer E/[AR] function and fitting the E/[A] curve data to the model that the model quantitatively this simple of observed and curves and of for each of the experiments we that the operational model quantitatively and for the data The by which the model can be quantitatively is in its ability to the agonist dissociation KA, and also the reduction in receptor concentration produced by irreversible antagonism. this the data were analysed by the method of Furchgott (1966) following the of and & This null in principle, should the estimate of KA and, in addition, receptor of the of these by Furchgott's method and by Therefore, on the basis of the present we that the operational containing an assumption the transducer function assumption which the null method of Furchgott explicitly can be used to a estimate of the dissociation constant by a For the of data analysis such a approach has a of In Furchgott's method a of have to be made data to be analysed and the by which to Equi-effective agonist concentrations in the presence and absence of irreversible antagonism are interpolated between curves, drawn by or produced by logistic In case the interpolated concentrations necessarily to data and in the choice of in of KA estimation The of analysing data by the Furchgott method appears to be that described by McPherson et (1983) in which both pre- and post-inactivation E/[A] curves are by a logistic function. However, the validity of using the logistic function may be as it can be that is no transducer function which can a set of rectangular hyperbolic occupancy with varying into a corresponding set of logistic E/[A] the transducer is in principle, fixed for a particular system that the irreversible does not the is that the resulting E/[A] function cannot be is to be that the E/[A] curve as defined by the operational model (equation is not of the logistic form equation for calculated equi-effective agonist by the choice must then be made which method to use to in order to estimate KA & this as to is an of analysis as both of concentration data are estimated with The approach using the operational model not only the for all these but also the E/[A] curve data Therefore, from the point of of this approach appears to be While the operational model for the present its on its to a of agonist-receptor the of the model under is the transducer as it is conventional in the analysis of pharmacological agonism et al., to assumptions the form of the E/[AR] relation. important should be made the of this relation. for rectangular hyperbolic E/[A] curves a rectangular hyperbolic transducer function can be (Black & Leff, 1983). Therefore, in these cases the operational model n = 1, see equation in principle, as a basis for When E/[A] curves are non-rectangular hyperbolic, if hyperbolic occupancy is still it can be that the must at the transducer While the choice of logistic function at this is not necessary it can be to be sufficient from the present have also of experimental data in the of non-hyperbolic E/[A] data from (1966) and & Furchgott irreversible antagonism at and produced the same of KA as obtained by the null method. In the case of E/[A] curves, which have gradients than rectangular that the in the transducer that a logistic transducer relation with n > 1 at a to if our were not to in Under these irreversible antagonism is expected to a reduction of E/[A] curve mid-point gradients, as shown in Figure This should and was a of the data McPherson et in their analysis of irreversible antagonism by that the pre- and post-inactivation curves should have the same gradient although are shown by computer that Furchgott's method does not in these that the irreversible antagonism method is not for non-hyperbolic E/[A] curves. analysis implies that gradients for pre- and post-inactivation curves should only remain by [Ro] variation in the case when the control, curve, is rectangular shown in Figure 2, Furchgott's method is regardless of the gradients of the control E/[A] curve so long as the of the curve at the transducer level, an assumption which is in the operational model and implicit in Furchgott's method. The illustrates the of the of agonism that the operational model In this instance the model the basis for E/[A] curve gradient changes with varying [Ro], the conditions under which KA estimation is valid. in a particular experimental the pre- and post-inactivation E/[A] curves were actually observed to be and of the same then the model predictions to the to an which may of pharmacological in the agonist or in the irreversible antagonist. The estimation of KA by Furchgott's method would be as McPherson et (1983) point but the should be with a particular system than with a principle that the method is than for hyperbolic curves. their initial using the same have the of this as an et al., of the model predictions in the present experimental analysis is for E/[A] curves, the location [A50], approaches a concentration value which is than the KA as [Ro] is the extrapolated value of the [A50] agonist affinity. In for the 5-HT E/[A] curves the control [A50] was very to the estimated 1 and Figure The curves obtained Pbz were on at This is important as it is when E/[A] curves to to occupancy curves and, therefore, to their location parameters to KA by such can be In the operational in to a framework for the of pharmacological agonism, appears to quantitatively in for the effects of irreversible antagonists on non-rectangular hyperbolic E/[A] curves. The of our analysis that the model may be directly to E/[A] curve data to agonist dissociation constants and that this approach is a valid, and economical to such data. The are to for and to and for the In this it is shown that a set of rectangular hyperbolic occupancy curves, with the same location, KA, but with different asymptotes, [Ro] cannot be into an set of logistic E/[A] curves using a transducer function. The occupancy relation is of the form A logistic E/[A] curve can be as [Ro], KA, b and are as defined in the and (ii) then the transducer can be found by to define [A] in of [AR] and into This in which Equation describes the form of the transducer function. When = 1, that for rectangular hyperbolic E/[A] curves, to In this case, a and can be in order to the E/[AR] relation independent of [Ro]. a = and to be This is the hyperbolic transducer function which is independent of [Ro]. When 1, the parameters in equation cannot be in such a as to the transducer function independent of [Ro]. in this case, [Ro] necessarily determines the form of the transducer for different values of [Ro] a different transducer function Therefore, if a set of logistic E/[A] curves is each of the set of occupancy curves a different transducer function. An of the problem is to use of the relation between equi-effective agonist concentrations which is the basis for Furchgott's null method (1966). This relation can be as that a in which [A] and [A]′ the equi-effective agonist concentrations in the absence and presence of irreversible antagonist. The problem is to E/[A] which their form when [Ro] is that when equation in of reciprocal we a function so that = is of the same form as the logistic E/[A] relation (equation in reciprocal form when is substituted by equation is of a different form from equation when is no longer the reciprocal form of a When = 1, that for rectangular hyperbolic E/[A] curves, both and are of the same Therefore, it is that a set E/[A] curves cannot be as a of [Ro] that the occupancy relation is rectangular hyperbolic and that the transducer is fixed and independent of [Ro].