﻿ Equation: Allosteric EC50 shift

# Equation: Allosteric EC50 shift

## Introduction

An allosteric modulator can reduce or enhance agonist binding. This model fits entire dose-response curves determined in the absence and presence of a modulator. The goal is to learn the affinity of the modulator for binding to its site, and also determine the value of alpha, the ternary complex constant that quantifies the degree to which binding of the modulator alters the affinity of the radioligand for the receptor site.

## Step by step

1. Create an XY data table.

2.Enter the logarithm of the concentration of the agonist ligand into X. If the concentration is 1nM, enter -9.

3.Enter response into Y in any convenient units. Enter data with no modulator into column A. Enter data collected with a constant concentration of modulator into column B. Repeat, if you have data, for column C, D, E, ..., each with a different concentration of modulator.

4.Enter the modulator concentration (in molar so 1nM is entered as '1e-9') into the column titles. Don't forget to enter '0' as the column title for data set A. These column titles are not just labels. The values you enter become part of the analysis.

5.From the data table, click Analyze, choose nonlinear regression, and choose the panel of equations:  Dose-Response -- Special. Then choose Allosteric EC50 shift.

You do not need to constrain any parameters to constant values

## Model

EC50=10^LogEC50

KB=10^LogKB

alpha=10^Logalpha

Antag=(1+B/KB)/(1+alpha*B/KB)

LogEC=Log(EC50*Antag)

Y=Bottom+(Top-Bottom)/(1+10^((LogEC-X)*HillSlope))

EC50 is the concentration of agonist that gives half maximal response in the absence of modulator.

Kb is the equilibrium dissociation constant (Molar) of modulator binding to its allosteric site. It is in the same molar units used to enter the modulator concentration into column titles on the data table.

Alpha is the ternary complex constant. When alpha=1.0, the modulator won't alter binding. If alpha is less than 1.0, then the modulator reduces ligand binding. If alpha is greater than 1.0, then the modulator increases binding. In the example shown about, alpha equals 0.01 so the modulator greatly decreases binding.

Top and Bottom are plateaus in the units of the Y axis.

## Notes

This model is designed to analyze data when the modulator works via an allosteric site. Since the agonist and modulator are acting via different sites, it is incorrect to refer to the modulator as a competitor.

The model is written to fit the logarithm of alpha, rather than alpha itself. This is because alpha is asymmetrical: All values from 0 to 1 mean that the modulator decreases binding, while all values from 1 to infinity mean that the modulator enhances binding. On a log scale, its values are more symmetrical, so the confidence interval computed on a log scale (as Prism does) are more accurate. Prism reports both alpha and log(alpha).

This model assumes that the allosteric modulator is present in excess, so the concentration you added is very close to its free concentration. This model won't work when the concentration of allosteric modulator is limiting (as it is when G proteins alter agonist binding to many receptors). No explicit model can handle this situation. You need to define the model with an implicit equation (Y on both sides of the equals sign) and Prism cannot handle such equations.

Note two important points about the progression of curves from left to right with increasing concentrations of the allosteric modulator. First, note that the maximum response doesn't change. Second, note that the effect of the modulator to right-shift the dose-response curve reaches a maximum as the modulator saturates its binding site.

## Reference

A. Christopoulos and T. Kenakin, Pharmacol Rev, 54: 323-374, 2002