Operator model

The various components of classical theory relating receptor occupancy to tissue response are shown schematically in Figure 3.5. It will be seen that this formally is identical to the equation for response derived in the operational model (see material following), where x = [Rt]e/p. 

Black and Leff [11] presented a model, termed the operational model, that avoids the inclusion of ad hoc terms for efficacy. This model is based on the experimental observation that the relationship between agonist concentration and tissue response is most often hyperbolic. This allows for response to be expressed in terms of... 

FIGURE 3.7 Principal components of the operational model. The 3D array defines processes of receptor occupation (plane 1), the transduction of the agonist occupancy into response (plane 2) in defining the relationship between agonist concentration, and tissue response (plane 3). The term a refers to the intrinsic activity of the agonist. 

FIGURE 3.8 Operational model of agonism. Ordinates response as a fraction of the system maximal response. Abscissae logarithms of molar concentrations of agonist, (a) Effect of changing i values, (b) Effect of changing KA. 

The operational model, as presented, shows dose-response curves with slopes of unity. This pertains specifically only to stimulus-response cascades where there is no cooperativity and the relationship between stimulus ([AR] complex) and overall response is controlled by a hyperbolic function with slope = 1. In practice, it is known that there are experimental dose-response curves with slopes that are not equal to unity and there is no a priori reason for there not to be cooperativity in the stimulus-response process. To accommodate the fitting of real data (with slopes not equal to unity) and the occurrence of stimulus-response cooperativity, a form of the operational model equation can be used with a variable slope (see Section 3.13.4) ... 

The operational model is used throughout this book for the determination of drug parameters in functional systems. 

A major modification to describe drug function is termed the operational model. This model is theoretically more sound than classical theory and is extremely versatile for the estimation of drug parameters in functional systems. 

Operational model forcing function for variable slope (3.13.4)... 

Operational Model Forcing Function for Variable Slope... 

The operational model allows simulation of cellular response from receptor activation. In some cases, there may be cooperative effects in the stimulus-response cascades translating activation of receptor to tissue response. This can cause the resulting concentration-response curve to have a Hill coefficient different from unity. In general, there is a standard method for doing this namely, reexpressing the receptor occupancy and/or activation expression (defined by the particular molecular model of receptor function) in terms of the operational model with Hill coefficient not equal to unity. The operational model utilizes the concentration of response-producing receptor as the substrate for a Michaelis-Menten type of reaction, given as... 

Therefore, the operational model for agonism can be rewritten for variable slope by passing the stimulus equation through the forcing function (Equation 3.51) to yield... 

Receptor density has disparate effects on the potency and maximal responses to agonists. The operational model predicts that the EC50 to an agonist will vary with receptor density according to the following relationship (see Section 3.13.3)... 

Dose-response curves to a full agonist [A] and a partial agonist [P] are obtained in the same receptor preparation. From these curves, reciprocals of equiactive concentrations of the full and partial agonist are used in the following linear equation (derived for the operational model see Section 5.9.2)... 

This method can also be employed with the operational model. Specifically, the operational model defines receptor response as... 

The Furchgott method can be effectively utilized by fitting the dose-response curves themselves to the operational model with fitted values of x (before and after alkylation) and a constant KA value. When fitting experimental data, the slopes of the dose-response curves may not be unity. This is a relevant factor in the operational model since the stimulus-transduction function of cells is an integral part of the modeling of responses. Under these circumstances, the data is fit to (see Section 3.13.3 and Equation 3.49)... 

In terms of the operational model, the EC50 of a partial agonist can also be shown to approximate the KA. The response to an agonist [A] in terms of the operational model is given as... 

In terms of the operational model, the response to a full [A] is given by... 

An identical equation results from utilizing the operational model. The counterpart to Equation 5.30 is... 

In systems of extremely poor receptor coupling, (3 will be a large value and e e /e (the relative maximal response approximates the relative efficacy of the agonists). In terms of the operational model, response is given by... 

The receptor occupancy curve can be converted to concentration-response curves by processing occupancy through the operational model for agonism (see Section 3.6). Under these circumstances, Equation 6.6 becomes... 

The effects of high values of constitutive activity can be determined for functional systems where function is defined by the operational model. Thus, it can be assumed in a simplified system that the receptor exists in an active (R ) and inactive (R) form and that agonists stabilize... 

Therefore, there are two efficacies for the agonism one for the full agonist (denoted 1) and one for the partial agonist (denoted r ). In terms of the operational model for functional response, this leads to the following expression for response to a full agonist [A] in the presence of a partial agonist [B] (see Section 6.8.6) ... 

Equation 6.8) into tissue response through the operational model ... 

Functional effects of an inverse agonist with the operational model (6.8.4)... 

Functional Effects of an Inverse Agonist with the Operational Model... 

Black, J. W., Feff, P, Shankley, N. P., and Wood, J. (1985). An operational model of pharmacological agonism The effect of E/[A] curve shape on agonist dissociation constant estimation. Br. J. Pharmacol. 84 561-571. 

Feff, P., Dougall, I. G., and Harper, D. (1993). Estimation of partial agonist affinity by interaction with a full agonist A direct operational model-fit approach. Br. J. Pharmacol. 110 239-244. 

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