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The operational model of agonist action Limitations of dose-response curves Fitting a sigmoidal (logistic) equation to a dose-response curve to determine EC50 (and perhaps slope factor) doesn't tell you everything you want to know about an agonist. The EC50 reflects both the ability of the drug to bind to its receptor (the agonist's affinity) and the ability of the drug to cause a response once it is bound (the agonist's efficacy). Thus, the EC50 you observe could have different causes. The agonist could bind with very high affinity, but have low efficacy once bound. Or it could bind weakly with low affinity, but have very high efficacy. Two very different drugs could have the same EC50s and maximal responses (in the same tissue). One drug could have high affinity and low efficacy, while the other has low affinity and high efficacy. Since efficacy reflects properties of both agonist and tissue, a single drug acting on one kind of receptor could have different EC50 values in different tissues. Derivation of the operational model Black and Leff (Proc. R. Soc. Lond. B, 220:141-162, 1983) developed the operational model of agonism to help understand the action of agonists and partial agonists, and to develop experimental methods to determine the affinity of agonist binding and a systematic way to measure relative agonist efficacy based on an examination of the dose-response curves. Start with a simple assumption: Agonists bind to receptors according to the law of mass action. At equilibrium, the relationship between agonist concentration ([A]) and agonist-occupied receptor ([AR]) is described by the following hyperbolic equation:
[RT] represents total receptor concentration and KA represents the agonist-receptor equilibrium dissociation constant. What is the relationship between agonist occupied receptor (AR) and receptor action? We know biochemical details in some cases, but not in others. This lack of knowledge about all the steps between binding and final response prevents the formulation of explicit, mechanistic equations that completely describe a dose-response curve. However, Black and Leff derived a "practical" or "operational" equation that encompasses the behavior of all of these unknown biochemical cascades. They began with the observation that dose-response curves often have a sigmoidal shape with a Hill Slope of 1.0, (the curves are hyperbolas when response is plotted against agonist concentration, sigmoidal when response is plotted against the log of concentration). They then proved mathematically that if agonist binding is hyperbolic and the dose-response curve has a Hill slope of 1.0, the equation linking the concentration of agonist occupied receptors to response must also be hyperbolic. This second equation, shown below, has been termed the "transducer function", because it is a mathematical representation of the transduction of receptor occupation into a response:
The parameter, Effectmax, is the maximum response possible in the system. This may not be the same as the maximum response that a particular agonist actually produces. The parameter KE is the concentration of [AR] that elicits half the maximal tissue response. The efficacy of an agonist is determined by both KE and the total receptor density of the tissue ([RT]). Black and Leff combined those two parameters into a ratio ([RT]/KE) and called this parameter tau(t), the "transducer constant". Combining the hyperbolic occupancy equation with the hyperbolic transducer function yields an explicit equation describing the effect at any concentration of agonist:
This equation can be rewritten as follows, to make it easier to compare the operational model with the equation of agonist occupancy of receptors.
This form of the equation makes it clear that the maximum effect seen with a particular agonist is not Effectmax, but rather is Effectmax multiplied by tau/(tau+1). Only a full agonist in a tissue with plenty of receptors (high values of t) will yield a maximum response that approaches Effectmax. The EC50 does not equal KA (the equilibrium dissociation constant for agonist binding to the receptors) but rather KA/(1+t). With a strong agonist, you'll get half-maximal response by binding fewer than half the receptors, so the EC50 will be much less than KA. This figure shows a dose-response curve for a partial agonist, and shows the relationship between EC50 and maximum response to terms in the operational model.
The parameter, tau, is a practical measure of efficacy. It equals the total concentration of receptors in the system divided by the concentration of receptors that need to be occupied by agonist to provoke a half-maximal tissue response. The tau value is the inverse of the fraction of receptors that must be occupied to obtain 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 with a partial agonist or with a full agonist in a tissue where the receptors had been significantly 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. The equations here show agonist stimulated response, so the curves all begin at zero. It is easy to add a basal term to model observed response, so the response with no agonist equals basal rather than zero. Shallower and steeper dose-response curves Some dose-response curves are steeper or shallower than a sigmoid curve with standard slope. The operational model can be extended to analyze these curves. If you assume the initial binding of the agonist to the receptor follows the law of mass-action (hill slope equals 1 for the binding step), then transduction step(s) between occupancy and final response must follow an equation that allows for variable slope. If the dose-response curve is still symmetrical and sigmoid, then the operational model can be extended fairly simply, by including a slope parameter, n. The extended form of the operational model is:
The relationship between this operational model and the variable slope sigmoid equation are as follows:
When n equals 1, the equation is the same as those shown earlier, describing dose-response curves with Hill slopes of 1.0. However, n is not always the same as the Hill Slope (but the two values will be close for full agonists). Designing experiments to fit to the operational model If you try to fit the operational model equation indiscriminately to dose-response data, you'll run into a problem. Either the curve-fitting program will report an error message, or it will report best-fit values with enormously wide confidence intervals. Any symmetrical dose-response curve is defined by four parameters: Bottom (response with no agonist), Top (response at very high concentrations), EC50 (concentration of agonist needed to provoke a response halfway between Bottom and Top) and the Hill Slope. However, the operational model equation has five parameters: Basal (response with no agonist), KA (dissociation constant of agonist binding), Effectmax (maximum possible effect with a full agonist and plenty of receptors), t (a measure of agonist efficacy) and n. Since the operational model has more parameters than are needed to describe a sigmoid dose-response curve, any curve can be defined by an infinite combination of operational model parameters. Even if a curve fitting program could find best-fit values (rather than report an error message), the best-fit parameter estimates may not be correct. To fit the operational model to data, you must compare curves. The most common approach has been to reduce the receptor number, usually with an irreversible alkylating agent, to such an extent that a full agonist can no longer produce the maximal tissue response, no matter how high a concentration is used. The agonist curve before alkylation is then compared to the curve after alkylation. Alkylation is not required with partial agonists. Instead, the dose-response curve of the partial agonist curve is compared to the dose-response curve of the full agonist. All the dose-response curves should be obtained in the same tissue, in order to reduce the variability in Effectmax estimates that can occur between tissues. Fitting the operational model with Prism To fit the operational model, the analysis must simultaneously account for the dose-response curve of a full agonist as well as one or more curves from a partial agonist or a full agonist in receptor-depleted tissue. Some investigators have fit the two (or more) curves at once, using a nonlinear regression program that can fit several data sets at once (Leff et al., J. Pharmacol. Meth., 23: 225-237, 1990). However, Prism can only fit one curve at a time (it will fit a whole family of curves as part of one analysis, but each fit is mathematically independent). Use the following approach (devised by A. Christopoulos). The first step is to fit the dose-response curve for the full agonist. Use the sigmoidal dose-response (variable slope) equation using nonlinear regression. Record the best-fit values for Top, Bottom and HillSlope. (The EC50 is not relevant.) The second step is to fit the dose-response curve for each partial agonist (or full agonist in receptor depleted tissue) to the operational model written as a user-defined Prism equation. When writing the equation for Prism, consider these three points.
Since you fix the first three parameters to constant values, nonlinear regression will find values for logKA and logTau. |
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