Dose-response curves in the presence of antagonists
Competitive antagonists

The term antagonist refers to any drug that will block, or partially block, a response. When investigating an antagonist the first thing to check is whether the antagonism is surmountable by increasing the concentration of agonist. The next thing to ask is whether the antagonism is reversible. After washing away antagonist, does agonist regain response? If an antagonist is surmountable and reversible, it is likely to be competitive (see next paragraph). Investigations of antagonists that are not surmountable or reversible are beyond the scope of this manual.

A competitive antagonist binds reversibly to the same receptor as the agonist. A dose-response curve performed in the presence of a fixedconcentration of antagonist will be shifted to the right, with the same maximum response and (generally) the same shape.

Gaddum derived the equation that describes receptor occupancy by agonist in the presence of a competitive antagonist. The agonist isdrug A. Its concentration is [A] and its dissociation constant is Ka. The antagonist is called drug B, so its concentration is [B] and dissociation constant is Kb. If the two drugs compete for the same receptors, fractional occupancy by agonist (f) equals:


The presence of antagonist increases the EC50 by a factor equal to 1+[B]/Kb. This is called the dose-ratio. You don't have to know the relationship between agonist occupancy and response for the equation above to be useful in analyzing dose response curves.

You don't have to know what fraction of the receptors is occupied at the EC50 (and it doesn't have to be 50%). Whatever that occupancy, you'll get the same occupancy (and thus the same response) in the presence of antagonist when the agonist concentration is multiplied by the dose-ratio.

The graph below illustrates this point. If concentration A of agonist gives a certain response in the absence of antagonist, butconcentration A' is needed to achieve the same response in the presence of a certain concentration of antagonist, then the dose-ratio equals A'/A. You'll get a different dose ratio if you use a different concentration of antagonist.

If the two curves are parallel, you can assess the dose-ratio at any point. However, you'll get the most accurate results by calculating the dose-ratio as the EC50 in the presence of antagonist divided by the EC50 in the absence of antagonist. The figure below shows the calculation of dose ratio.


Schild plot

If the antagonist is competitive, the dose ratio equals one plus the ratio of the concentration of antagonist divided by its Kd for thereceptor. (The dissociation constant of the antagonist is sometimes called Kb and sometimes called Kd)


A simple rearrangement gives:


If you perform experiments with several concentrations of antagonist, you can create a graph with log(antagonist) on the X-axis and log(dose ratio -1 ) on the Y-axis. If the antagonist is competitive, you expect a slope of 1.0 and the X-intercept and Y-intercept will both equal the Kd of the antagonist.


If the agonist and antagonist are competitive, the Schild plot will have a slope of 1.0 and the X intercept will equal the logarithm of the Kd of the antagonist. If the X-axis of a Schild plot is plotted as log(molar), then minus one times the intercept is called the pA2 (p for logarithm, like pH; A for antagonist; 2 for the dose ratio when the concentration of antagonist equals the pA2). The pA2 (derived from functional experiments) will equal the Kd from binding experiments if antagonist and agonist compete for binding to a single class ofreceptor sites.

Creating and analyzing Schild plots with Prism

Enter your dose-response data with X as log of the agonist concentration, and Y as response. (If you enter your data with X asconcentration, do a transform to create a table where X is log of agonist concentration). Label each Y column with a heading (title) that is the log of antagonist concentration. The first column should be the control, with agonist only (no antagonist). Label this column "control".

Use nonlinear regression to fit a sigmoid dose-response curve. Choose a standard slope or variable slope, depending on your data. From the nonlinear regression dialog, check the option to calculate dose-ratios for Schild plots.

The values of the dose ratio can only be interpreted if all the dose-response curves are parallel. If you selected the sigmoid curve with a standard slope, this will be true by definition. If you let Prism determine the slope factor for each curve, look at these (and their standard errors) to see if they differ significantly. If the slope factors differ, then the interaction is probably not strictly competitive, and Schild analysis won't be useful. If the slope factors are indistinguishable, consider holding all the slope factors constant to a single value.

The curve fit results include a results view called Summary table which tabulates the log(DR-1) for each data set (except the first, which is the control). To graph these data, go to the graph section and click the button New graph. Choose a new graph from the summary table of the nonlinear regression results.

First fit to linear regression to determine slope and intercept. If the antagonist is competitive, the Schild plot ought to have a slope that is indistinguishable from 1.0. You can check this assumption by seeing whether the confidence interval for the slope includes 1.0.

If the confidence interval for the slope does not include 1.0, your antagonist is probably not a simple competitive antagonist. For suggestions of further analyses, see T. Kenakin, Pharmacologic Analysis of Drug-Receptor Interaction, 3rd Ed. Lippincott-Raven Press, 1997.

If the confidence interval does include 1.0, refit the line constraining the slope to equal 1.0. You cannot do this with Prism's linearregression analysis. However, you can use Prism's nonlinear regression to fit a line with a constant slope. Use this equation:
Y = X - pA2

When X=pA2, Y=0. As X increases above pA2, Y increases as well the same amount. Fit this equation to determine the pA2 of the antagonist.