- 1An 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].
- 2Irreversible 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.
- 35-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.
- 4Fitting 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.
- 5The 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.
Theory: The operational model of agonism
Hyperbolic El/A] curves
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.
Non-hyperbolic E/[A] curves
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.
E/[A] curve gradients
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.
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.
KA estimation by [Ro] reduction
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.
Rabbit thoracic aorta
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 118.41, NaHCO3 25.00, KCI 4.75, KH2PO4 1.19, MgSO4 1.19, glucose 11.10, CaCl2 2.50, aerated continuously with 95:5; O2:CO2.
Tissue responses were measured as increases in isometric force using Grass FT03C strain gauges and displayed on Bryans 2000 single-channel flat-bed, recorders.
The protocol adopted for these experiments is shown in Figure 3.
The second force application was made to reinstate a total load of 3 g. Tissues achieved steady tension before each drug intervention. All washes (indicated by W in the protocol) were by exchange of bath contents for fresh Krebs solution. Benextramine tetrahydrochloride (BHC) was included in order to avoid possible confounding of 5-HT responses due to direct or indirect α-adrenoreceptor stimulation by the agonist (Innes, 1962; Fozard & Mwaluko, 1976; Apperley et al., 1976; Marin et al., 1981). The sighting dose of 10 µM 5-HT was used only to establish tissue viability. Two cumulative 5-HT curves were constructed in each tissue according to Van Rossum (1963) using 0.5log10M increments. 5-HT responses after Pbz treatment were measured as fractions of the maximum of the corresponding control curve.
Paired curve analysis
Each aorta provided six ring segments, three of which were used for Pbz treatment and the other three to provide replicate 5-HT curves. For both control and Pbz treatment groups a total of nine pairs of curves were obtained, the control pairs providing information on time-dependent differences between first and second curves.
Drugs and solutions
Drugs used were: pentobarbitone sodium (Sagatal, May and Baker), 5-HT creatinine sulphate complex (Sigma), benextramine tetrahydrochloride monohydrate (BHC) (Aldrich Chemical Company), phenoxybenzamine (Pbz) (Dibenyline, SK & F). 5-HT and BHC solutions were aqueous. Pbz was prepared in acidic solution which, at the concentrations used in the experiment, produced no alteration in the Krebs solution pH. 5-HT and Pbz solutions were freshly prepared and BHC solutions prepared from a frozen 1 mM stock. The maximum volume of drug solution administered to the organ bath was 300 µl.
All the following fitting procedures were unweighted iterative least squares minimization computer programmes. They were locally written with the exception of BMDP Module AR (BMDP Statistical Software, 1981) which was the programme used to fit data to the model of agonism.
Individual sets of 5-HT E/[A] curve data were fitted to a logistic function of the form
in which a, b and p are the asymptote, location ([A50]) and slope parameters respectively. The location parameters were actually estimated as logarithms by making the subsitution b = 101gb.
Pairs of control and Pbz-treatment 5-HT curves were fitted to model equation (8) providing direct estimates of Em, n and KA, and two estimates of τ for each curve in a pair. Goodness-of-fit was assessed by examining fitted and experimental data points for systematic deviations. A more comprehensive goodness-of-fit analysis will be the subject of a subsequent paper.
Estimation of KA by Furchgott's method
Equi-effective 5-HT concentrations were interpolated from logistic curves fitted to each pair of control and Pbz treatment 5-HT curves. Five interpolations were made for each pair using the upper part of the Pbz treatment curve as recommended by Thron (1970). KA values were estimated by fitting the interpolated values, [A] and [A]′ (the equi-effective concentrations of 5-HT before and after Pbz treatment respectively) to the equation
in which M =KA(l − y1)/y1 and K=KA/y1 where y1 is the fraction of receptors irreversibly occluded. KA is estimated as K − M (see Parker & Waud, 1971).
5-Hydroxytryptamine E/[A] curves
5-HT produced sustained, concentration-dependent, contractions of the aortic rings. This was the case for both curves in control and Pbz-treatment pairs. Typical tracings are reproduced in Figure 4. Analysis of the paired curves was initially performed by logistic curve fitting using equation (13) and the estimated asymptote, location and slope parameters were compared by paired t test.
Analysis of paired control 5-HT curves indicated a significant fractional increase in asymptote between first and second curves of 0.094 (P < 0.001, n= 9). This factor was used to correct 5-HT responses after Pbz treatment. No significant slope parameter changes were observed (P= 0.402, n= 9) but a small but significant (P= 0.044, n= 9) left shift in the location parameter was noted. This increment in log10 units was only 0.05 and was considered to be too small to be meaningful since it was of the same order as the fitting error on individual curve location parameters.
The results of logistic fitting of paired 5-HT and corrected Pbz-treatment curves are shown in Table 1.
Analysis showed that there was significant differences between their slope parameters (P < 0.001, n= 9). The control 5-HT curves were steeper than the Pbz treatment curves in all cases, although both groups were steeper than rectangular hyperbolae. Pbz treatment reduced asymptotes significantly (P < 0.001, n= 9) as expected and also produced significant right shift (P <0.001, n= 9) of the location parameters.
Model-fitting of E/[A] curves
Each pair of control and corrected Pbz-treatment curves were fitted to the operational model of agonism (equation 8). A typical set of data with the fitted lines superimposed is shown in Figure 5. Comparisons of observed data points and corresponding fitted values revealed systematic deviations for the Pbz-treatment curves. However, even the largest residual difference between observed and fitted points represented only 5% of the amplitude of the corresponding curve and generally the residuals were much smaller. Figure 5 gives an indication of the extent of departure of model-fitted lines from the data points.
For each pair of curves estimates were made of Em,n, KA and two τ values, τ1 and τ2, corresponding to the control and Pbz-treatment curves respectively. The ratio between τ2 and τ1 for each pair reflects the reduction in [Ro] due to Pbz-treatment. In addition to the slope parameter, n, estimated for each pair the mid-point gradients G1 and G2 of each curve in a pair were calculated from model equation (11). Table 2 summarizes the parameter estimates. Comparison of G1 and G2 illustrates the flattening effect of irreversible antagonism on the steep 5-HT E/[A] curves, seen previously by logistic curve fitting.
Estimated Em values were invariably greater than the asymptote of the control 5-HT curve, the latter always being set at unity. This indicates that 5-HT was not behaving as a full agonist but was on average generating only 85% of the maximum possible response.
Comparison of the control τ values, τ1, with the ratio τ2/τ1, implied that although the initial operational efficacy of 5-HT varied between tissues, presumably due to [Ro] differences, the effect of Pbz in each tissue was similar, reducing [Ro] and, therefore T, by almost the same fraction in each case. Agonist affinity estimates, expressed as pKAs, were evidently consistent across the nine preparations with an average value of 6.95.
Estimation of agonist affinity by Furchgott's method
Analysis of paired control Pbz treatment curves by the method of Furchgott, gave the results shown in Table 3. Agonist affinities as pKAs are given for each pair and also the estimated fraction of receptors remaining after Pbz treatment. The average pKA value estimated by this method was 6.81.
|1 − y1||0.23||0.28||0.58||0.34||0.37||0.31||0.20||0.21||0.18|
Comparison of parameter estimation by model-fitting and Furchgott's method
The average pKA values estimated by operational model fitting and Furchgott's method were evidently similar, the difference being only 0.14 logl0 units.
There appears to be no obvious way of comparing these two values statistically due to the different approaches used in their estimation and in particular the somewhat arbitrariness of certain steps involved in the Furchgott method of calculation. The similarity between the two estimates seems to make complex statistical analysis inappropriate.
The ratio τ2/τ1 provided by operational modelfitting is equivalent to (1 − y1) in Furchgott's method, both corresponding to the fractional concentration of receptors remaining after irreversible antagonism. The average value of τ2/τ1 was 0.35 compared with an average value of 1 − y1 of 0.30. Once again a similarity between the two methods of estimation is evident.
In this paper we have extended our previous operational analysis of pharmacological agonism in order to make quantitative predictions about nonrectangular hyperbolic E/[A] curves, particularly under conditions where receptor concentration, [Ro], is varied. In order to test the utility of the operational model in this respect, we have examined an agonistreceptor combination which clearly demonstrates ‘non-hyperbolic’ agonism.
Pragmatic, logistic, curve fitting of 5-HT E/[A] data in the rabbit aortic ring preparation showed that the curves were steeper than rectangular hyperbolae. It was also evident that, after irreversible antagonism, the curves became inherently flattened, an effect which is emphatically not merely due to depression of the asymptote of the curves. Logistic slope parameters and mid-point gradients have been calculated for selfnormalized E/[A] curves in order to avoid such confusion. This result is entirely consistent with model predictions when a ‘steep’ transducer E/[AR] function operates, and fitting the E/[A] curve data to the model showed that the model quantitatively accommodates this flattening effect.
By simple inspection of observed and model-fitted curves and examination of residuals for each of the nine replicate experiments we conclude that the operational model accounts quantitatively and qualititatively for the data presented here.
The other criterion by which the model can be judged quantitatively is in its ability to determine accurately the agonist dissociation constant, KA, and also the fractional reduction in receptor concentration produced by irreversible antagonism. To this end the data were analysed by the method of Furchgott (1966) following the optimizing recommendations of Thron (1970) and Parker & Waud (1971). This null approach, in principle, should provide the most assumption-free, pragmatic, estimate of KA and, in addition, fractional receptor reduction. Comparison of the estimates of these two quantities by Furchgott's method and by model-fitting, showed negligible differences. Therefore, on the basis of the present data, we conclude that the operational model, despite containing an explicit assumption regarding the transducer function (an assumption which the null method of Furchgott explicitly avoids), can be used to provide a reliable estimate of the dissociation constant by a direct model-fitting approach.
For the purposes of data analysis such a modelfitting approach has a number of attractions. In Furchgott's method a number of choices have to be made regarding data to be analysed and the means by which to analyse them. Equi-effective agonist concentrations in the presence and absence of irreversible antagonism are interpolated between curves, either drawn by eye or produced by logistic fitting. In neither case do the interpolated concentrations necessarily correspond to ‘real’ data points and differences in the choice of interpolations introduces differences in precision of KA estimation (Thron, 1970). The optimal way of analysing data by the Furchgott method appears to be that described by McPherson et al. (1983) in which both pre- and post-inactivation E/[A] curves are fitted by a logistic function. However, the validity of using the logistic function may be questioned as it can be proven (see Appendix) that there is no single definable transducer function which can translate a set of rectangular hyperbolic occupancy functions with varying [Ro]s into a corresponding set of logistic E/[A] relations. As the transducer function, is in principle, fixed for a particular system (assuming that the irreversible antagonist does not affect it), the corollary is that the resulting E/[A] function cannot be logistic. It is to be emphasized that the E/[A] curve as defined by the operational model (equation (8)) is not of the logistic form (see equation (12) for contrast). Having calculated equi-effective agonist concentrations, by whatever means, the choice must then be made regarding which graphical method to use to relate them in order to estimate KA (Parker & Waud, 1971). At this stage questions arise as to whether regression is an appropriate means of analysis as both sets of concentration data are estimated with error (Thron, 1970).
The direct-fitting approach using the operational model not only eliminates the need for all these considerations but also uses the raw E/[A] curve data without transformation. Therefore, from the point of view of analytical simplicity, this approach appears to be favourable.
While the operational model accounts for the present data, its general acceptability depends on its application to a variety of agonist-receptor combinations. Essentially, the feature of the model under test is the transducer function, as it is conventional in the analysis of pharmacological agonism (Stephenson, 1956; Mackay, 1966; Furchgott, 1966; Barlow et al., 1967), to avoid explicit assumptions about the form of the E/[AR] relation. Several important points should be made about the nature of this relation. Firstly, for rectangular hyperbolic E/[A] curves a rectangular hyperbolic transducer function can be deduced (Black & Leff, 1983). Therefore, in these cases the operational model (with n = 1, see equation (4)) must, in principle, serve as a quantitative basis for analysis. When E/[A] curves are non-rectangular hyperbolic, if hyperbolic occupancy is still assumed, it can be inferred that the non-hyperbolicity must reside at the transducer level. While the precise choice of logistic function at this level is not necessary it can be concluded to be sufficient from the present study. We have also performed similar analyses of experimental data in the literature (not shown). Model-fitting of non-hyperbolic E/[A] data from Van Rossum (1966) and Besse & Furchgott (1976), involving irreversible antagonism at histamine (H1) and α-adrenoceptors respectively, produced the same estimates of KA as those obtained by the null method.
In the case of E/[A] curves, which have steeper midpoint gradients than rectangular hyperbolae, accepting that the steepness resides in the transducer function, means that a logistic transducer relation with n > 1 provides at least a qualitative general guide to expectations, even if our quantitative analyses were not to hold in general. Under these circumstances, irreversible antagonism is expected to produce a progressive reduction of E/[A] curve mid-point gradients, as shown in Figure 2. This result should hold generally and indeed was a feature of the data presented here.
McPherson et al. (1983), in their analysis of β-adrenoceptor irreversible antagonism by Ro 03-7894, assume that the pre- and post-inactivation curves should have the same gradient even although they are inherently steep. Having shown by computer simulation that Furchgott's method does not hold in these circumstances, they deduced that the irreversible antagonism method is not applicable for non-hyperbolic E/[A] curves. Our analysis implies that gradients for pre- and post-inactivation curves should only remain unchanged by [Ro] variation in the case when the control, pre-inactivation curve, is rectangular hyperbolic. As shown in Figure 2, Furchgott's method is valid regardless of the gradients of the control E/[A] curve so long as the non-hyperbolicity of the curve resides at the transducer level, an assumption which is explicit in the operational model and implicit in Furchgott's method.
The preceding argument illustrates the utility of the explicit description of agonism that the operational model provides. In this instance the model provides the basis for predicting E/[A] curve gradient changes with varying [Ro], thereby clarifying the conditions under which KA estimation is valid. If, in a particular experimental system, the pre- and post-inactivation E/[A] curves were actually observed to be steep and of the same gradient, then the model predictions serve to alert the experimenter to an inconsistency which may reflect expression of other pharmacological properties in the agonist or in the irreversible antagonist. The estimation of KA by Furchgott's method would indeed be invalid as McPherson et al. (1983) point out, but the result should be associated with a particular system rather than with a general principle that the method is invalid other than for hyperbolic curves. Indeed, since their initial study using Ro 03-7894, the same workers have questioned the classification of this compound as an irreversible, β-adrenoceptor antagonist (Krstew et al., 1984).
Another of the model predictions borne out in the present experimental analysis is that, for steep E/[A] curves, the location parameter [A50], approaches a concentration value which is higher than the KA as [Ro] is reduced; the extrapolated value of the [A50] underestimates agonist affinity. In fact, for the 5-HT E/[A] curves the average control [A50] (6.98) was very similar to the pKA estimated (6.95) (see Table 1 and Figure 5). The curves obtained after Pbz treatment were located on average at 6.48. This result is important as it is intuitively tempting when inspecting ‘partial agonist’ E/[A] curves to approximate them to occupancy curves and, therefore, to approximate their location parameters to affinity constants. Rough KA estimates by such ‘eye-fitting’ evidently can be substantially inaccurate.
In conclusion, the operational model, in addition to providing a general qualitative framework for the understanding of pharmacological agonism, appears to succeed quantitatively in accounting for the effects of irreversible antagonists on non-rectangular hyperbolic E/[A] curves. The results of our analysis suggest that the model may be fitted directly to E/[A] curve data to obtain agonist dissociation constants and that this model-fitting approach is a valid, efficient and economical way to treat such data.
The authors are indebted to Miss J.M. Morse for excellent technical assistance, and to Mrs K. Egan and Ms L. Joyce for preparing the manuscript.
In this appendix it is shown that a set of rectangular hyperbolic occupancy [AR]/[A] curves, with the same location, KA, but with different asymptotes, [Ro] cannot be transformed into an equivalent set of logistic E/[A] curves using a single, fixed, transducer function.
The occupancy relation is of the form
A logistic E/[A] curve can be represented as follows:
[Ro], KA, a, b and p are as defined in the text.
If (i) and (ii) hold, then the transducer function, E/[AR], can be found by rearranging (i) to define [A] in terms of [AR] and substituting into (ii). This gives
in which K= b/KA.
Equation (iii) describes the form of the transducer function. When p = 1, that is, for rectangular hyperbolic E/[A] curves, (iii) simplifies to
In this case, a and K can be redefined in order to make the E/[AR] relation independent of [Ro]. Defining a = Em[Ro]/([Ro]+KE) and K=KE/([Ro]+KE) allows (iv) to be rewritten
This is the familiar hyperbolic transducer function (see text), which is clearly independent of [Ro]. When p ≠ 1, the parameters in equation (iii) cannot be redefined in such a way as to make the transducer function independent of [Ro]. Consequently in this case, [Ro] necessarily determines the form of the transducer function; for different values of [Ro] a different transducer function pertains.
Therefore, if a set of logistic E/[A] curves is assumed, each individual member of the equivalent set of occupancy curves requires a different transducer function.
An alternative, more general, way of approaching the problem is to make use of the fundamental relation between equi-effective agonist concentrations which is the basis for Furchgott's null method (1966). This relation can be written as
that is, a straight line in which [A] and [A]′ are, respectively, the equi-effective agonist concentrations in the absence and presence of irreversible antagonist. The problem is to identify E/[A] functions which preserve their form when [Ro] is varied, that is, when equation (vi) applies. Considered in terms of reciprocal concentrations, we require a function f so that E = f (1/[A]) is of the same form as E =f (α+β/[A]′). Writing the logistic E/[A] relation (equation (ii)) in reciprocal form
when 1/[A] is substituted by (α+β/[A]′)
By inspection, equation (viii) is of a different form from equation (vii) when p ≠ 1; (viii) is no longer the reciprocal form of a logistic. When p = 1, that is, for rectangular hyperbolic E/[A] curves, both (vii) and (viii) are of the same form.
Therefore, it is proven that a set oflogistic E/[A] curves cannot be generated as a result of [Ro] variation, assuming that the occupancy relation is rectangular hyperbolic and that the transducer is fixed and independent of [Ro].