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This equation is used when X values are logarithms of doses or concentrations. Use a related equation when X values are concentrations or doses.

Introduction to the operational model

The EC50 is determined by two properties of the agonist:

How well it binds to the receptor, quantified by the affinity of the drug for binding to its receptor.

How well it causes a response once bound. This property is known as the agonist’s efficacy. Since efficacy depends on both agonist and tissue, a single drug acting on a single kind of receptor can have different efficacies, and thus different EC50 values, in different tissues.

A single dose-response experiment cannot determine affinity and efficacy. A drug that binds with high affinity but has low efficacy will produce exactly the same dose-response curve as a drug with low affinity and high efficacy.

To untangle affinity from efficacy, globally fit a dose-response curve of a full agonist and a second dose-response curve determined after treating the cells or tissue with an alkylating agent (or some other irreversible treatment) that reduces the number of accessible receptors. With fewer receptors, the dose response curve is shifted down and usually to the right.

The operational model assumes that the affinity of the drug for the receptors is not altered by reducing the number of available receptors. It also assumes that the maximum possible response in the tissue remains unchanged (the treatment was specific for the receptors you are studying). Accepting these assumptions, fitting the operational model globally will determine the affinity of the agonist for the receptors.

Step by step

Create an XY data table. Enter the logarithm of the concentration of the agonist ligand into X. Enter response into Y in any convenient units. Enter data with a full agonist and no receptor depletion into column A. Enter data collected after receptor depletion into column B. Repeat, if you have data with different levels of receptor depletion for column C, D, E, ... You don't have to know the degree to which the receptors are depleted, and don't have to enter any values in the column titles (although they are useful as labels).

From the data table, click Analyze, choose nonlinear regression, and choose the panel of equations:  Dose-Response -- Special, X is log(concentration). Then choose Operational Model - Depletion, X is log(concentration).

If you have subtracted off any basal response, consider constraining the parameter Basal to a constant value of zero.

Also consider constraining the transducer slope n to a constant value of 1.0. When set to 1.0, all dose-response curves are constrained to have Hill slopes of 1.0, which is observed commonly.


operate= (((10^logKA)+(10^X))/(10^(logtau+X)))^n

Y=Basal + (Effectmax-Basal)/(1+operate) 


Interpret the parameters

Effectmax is the maximum possible system response, in units of the Y axis. It is the top plateau of the dose-response curve obtained with a full agonist without receptor depletion. If your agonist isn't a full agonist, the EffectMax might be higher than the top plateau of the no depletion curve.

Basal is the response in absence of agonist, in same units as Y. If you have subtracted off any basal response, constrain basal to a constant value of zero.

KA is the agonist-receptor dissociation constant, in same units as X (usually molar). It measures the affinity of the full agonist for the receptors, which is the main goal of this kind of experiment. Prism reports both KA and its logarithm. It is not the same as the EC50.

tau is the transducer constant, a practical measure of efficacy. It is the inverse of the fraction of receptors that must be occupied by agonist to obtain the half-maximal response. If tau equals 10, that means that occupation of only 10% of the receptors leads to a half-maximal response. If tau equals 1.0, that means that it requires occupation of all the receptors to give a half-maximal response. This would happen in a tissue where the receptors had been substantially depleted. Because tau is a property of both the tissue and receptor system, it is not a direct measure of intrinsic efficacy, which is commonly defined as a property belonging only to an agonist-receptor pair, irrespective of the assay system in which it is measured. Prism reports both tau and its logarithm for each data set.

n is the Unitless transducer slope. It is similar to, but not identical to, the Hill slope. In most cases, n is constrained to a constant value of 1.0, in which case all the dose-response curves will have Hill slopes of 1.0. If n does not equal 1.0, the Hill Slope does not equal either 1.0 or n.


Since Tau measures efficacy, Prism fits a different value of tau for each data set. Receptor depletion reduce the value of tau. The other parameters are fit globally, to find one best-fit value for all the data sets.


Black and Leff (Proc. R. Soc. Lond. B, 220: 141-162, 1983

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