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 fixed concentration 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 is drug 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, but concentration 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 the receptor. (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 of receptor 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 as concentration, 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 linear regression 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.

An alternative to Schild plots

Analyzing a Schild plot with linear regression of dose-ratios is not the best way to determine the Kd of the antagonist from functional data. The problem is that the EC50 of the control curve is used to compute dose ratios for all other curves. Any error in that control value shows up in all the data points. The Schild plot was developed in an era when nonlinear regression was unavailable, so it was necessary to transform data to a linear form. This is no longer an advantage, and Schild plots can be thought of in the same category as Scatchard plots. See Avoid Scatchard, Lineweaver-Burke and similar transforms.

Lew and Angus (Trends Pharmacol. Sci., 16:328-337, 1995) have presented an alternative method for analyzing Schild experiments using nonlinear regression instead of the linear regression method of standard Schild analysis. This alternative method also avoids the need to calculate dose ratios.Start with the Gaddum equation for occupancy as a function of agonist and antagonist concentrations:

Simple algebra expresses the equation this way:

Thus you can obtain any particular occupancy f, with any concentration of antagonist ([B]) so long as you adjust A to keep the quantity in the parentheses constant (C).

Rearrange to show how you must change the agonist concentration to have the same response in the presence of an antagonist.

The EC50 is the concentration needed to obtain 50% of the maximal response. You don't know the fraction of receptors occupied at that concentration of agonist, but can assume that the same fractional occupancy by agonist leads to the same response, regardless of the presence of antagonist. So you can express the equation above to define EC50 as a function of the antagonist concentration [B].

You determined the EC50 at several concentrations of antagonist (including 0), so you could fit this equation to your data to determine a best-fit value of Kb (and C, which you don't really care about). But it is better to write the equation in terms of the logarithm of EC50, because the uncertainty is more symmetrical on a log scale. See Why Prism fits the logEC50 rather than the EC50.  By tradition, we use the negative logarithm of EC50, called the pEC50.  For similar reason, you want to determine the best-fit value of log(Kb) (logarithm of the dissociation constant of the antagonist) rather than Kb itself.

Define Y to be the pEC50, X to be the antagonist concentration [B], and a new constant P to be log(c). Now you have an equation you can use to fit data:

To determine the Kb using Prism:

Determine the EC50 of the antagonist in the presence of several concentrations of antagonist, including zero concentration. Enter these values into a Prism data table. Into the X column, enter the antagonist concentrations in micromolar. Into the Y columns, enter the negative log of the EC50. Use nonlinear regression to fit this equation.

Y=-1*log((X*1e-6)+(10^logKb))-P

Note that the parameter logKb in the first equation has been replaced with logK in the above equation. This is because, strictly speaking, if the value for the slope is significantly different than one, then the antagonist fitting parameter is not the log of the Kb!

Enter the following user-defined equation into Prism:

Y=-1*log(((X*1e-6)^slope)+(10^logK))-P

When performing this analysis, it is a good idea to use Prism to fit the data to both equations at the same time and allow the program to decide, via the F-test, which one is the more appropriate equation.  If the simpler equation is the better equation, then the logKb estimate may be quoted.  Otherwise, you must conclude that your data are not consistent with a model of simple competition.