# Schild regression

Schild regression analysis, named for Heinz Otto Schild, is a useful tool for studying the effects of agonists and antagonists on the cellular response caused by the receptor or on ligand-receptor binding.

Using a dose-response curve or an equivalent curve with concentration and binding %, it is possible to determine the dose ratio, this is a measure of the potency of a drug; it is obtained by dividing the increased equilibrium constant due to drug inhibition by the equilibrium constant without the drug. A Schild plot is a double logarithmic plot, typically Log(dr-1) as the ordinate and Log[B] as the abscissa. This is because a competitive drug B will have a linear plot with the $dr=1+[B]/K_B$. These experiments must be carried out on a very wide range (therefore the logarithmic scale) as the mechanisms differ over a large scale, such as at high concentration of drug.

## Schild regression for ligand binding

Although most experiments use cellular response as a measure of the effect, the effect is, in essence, a result of the binding kinetics; so, in order to illustrate the mechanism, ligand binding is used. A ligand A will bind to a receptor R according to an equilibrium constant :

$K_d = \frac{ k_{-1} }{k_1}$

Although the equilibrium constant is more meaningful, texts often mention its inverse, the affinity constant (Kaff = k1/k-1): A better binding means an increase of binding affinity.

The equation for simple ligand binding to a single homogeneous receptor is

$[AR]=\frac{[R] \, t \, [A]}{[A]+K_d}$

This is the Hill-Langmuir equation, which is practically the Hill equation (biochemistry) described for the agonist binding. In chemistry, this relationship is called the Langmuir equation, which describes the adsorption of molecules onto sites of a surface (see adsorption).

[R]total is the total number of binding sites, and when the equation is plotted it is the horizontal asymptote to which the plot tends; more binding sites will be occupied as the ligand concentration increases, but there will never be 100% occupancy. The binding affinity is the concentration needed to occupy 50% of the sites; the lower this value is the easier it is for the ligand to occupy the binding site.

The binding of the ligand to the receptor at equilibrium follows the same kinetics as an enzyme at steady-state (Michaelis-Menten equation) without the conversion of the bound substrate to product.

Agonists and antagonists can have various effects on ligand binding. They can change the maximum number of binding sites, the affinity of the ligand to the receptor, both effects together or even more bizarre effects when the system being studied is more intact, such as in tissue samples. (Tissue absorption, densitization, and other non equilibrium steady-state can be a problem.)

A surmountable drug changes the binding affinity:

• competitive ligand: $K_d'= K_d \frac{1+[B]}{K_b}$
• cooperative allosteric ligand: $K_d'= K_d \frac{K_B + [B]}{K_B + \frac{[B]}{\alpha} }$

A nonsurmountable drug changes the maximum binding:

• noncompetitive binding: $[R] \, t' = \frac{[R] \, t}{1 + \frac{[B]}{K_b} }$
• irreversible binding

The Schild regression also can reveal if there are more than one type of receptor and it can show if the experiment was done wrong as the system has not reached equilibrium.

The first radio-receptor assay (RRA) was done in 1970 by Lefkowitz et al., using a radiolabeled hormone to determine the binding affinity for its receptor.[1]

A radio-receptor assay requires the separation of the bound from the free ligand. This is done by filtration, centrifugation or dialysis.[2]

A method that does not require separation is the scintillation proximity assay that relies on the fact that β-rays from 3H travel extremely short distances. The receptors are bound to beads coated with a polyhydroxy scintillator. Only the bound ligands to be detected.

Today, the fluorescence method is preferred to radioactive materials due to a much lower cost, lower hazard, and the possibility of multiplexing the reactions in a high-throughput manner. One problem is that fluorescent-labeled ligands have to bear a bulky fluorophore that may cause it to hinder the ligand binding. Therefore, the fluorophore used, the length of the linker, and its position must be carefully selected.

An example is by using FRET, where the ligand's fluorophore transfers its energy to the fluorophore of an antibody raised against the receptor.

Other detection methods such as surface plasmon resonance do not even require fluorophores.