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# Schild regression

Schild regression analysis, named for Heinz Otto Schild, is a useful tool for studying the effects of agonist 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] / KB. 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 commonly experiments use cellular response as a measure of the effect, the effect is basically a result of the binding kinetics, so in order to illustrate the mechanism ligand binding is used. A ligand A with bind to a receptor R according to an equilibrium constant :

Kd=k-1/k1

Although the equilibrium constant is more meaningful, often its inverse called affinity constant Kaff = K1/k-1 is mentioned in text as it is easier to grasp: a better binding means an increase of binding affinity. The equation for simple ligand binding to a single homogenous receptor is [AR] = [R]t[A] / ([A] + Kd) In chemistry it is called the Langmuir equation that describes the adsorption of molecules onto sites of a surface. (see adsorbtion) [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 to, meaning that the more ligand there is the more binding sites are occupied 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, 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 toghter or even more bizarre effects when the system studying is more intact (tissue absoption, densitization, and other non equililibrium stedy-state can be a problem). A surmountable drug changes the binding affinty:

• competitive ligand K'd=Kd (1+[B]/Kb)
• cooperative allosteric ligand K'd=Kd (KB+[B]/(KB+[B]/α))

A nonsurmountable drug changes the maximum binding:

• noncompetitive binding [R]t'=[R]t /(1+[B]/Kb)
• 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 was done in 1970 by Lefkowitz et al, using a radiolabeled hormone to determine the binding affinity for its receptor. A radio-receptor assay requires the separation of the bound from the free ligand. This is done by filtration, centrifugation or dialysis. A method that does not require separation is the scintillation proximity assay that relies on the fact that β-particles from 3H travel extremely short distances so the receptors are bound to beads coated with a polyhydroxy scintillant, resulting in only the bound ligands to be detected. Today, the fluorescent 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. The 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, length of the linker, and its position must be carefully selected. An example is by using FRET, where the fluorophore of the ligand transfers its energy to the fluorophore of an antibody raised against the receptor. Other methods such as surface plasmon resonance do not even require fluorophores.

## References

Ligand receptor binding: Kenakin T, 1993. Pharmacological analysis of drug-receptor interaction New York: Raven Press

"Radioligand binding assays" Lutea A et al, 2005. Receptor–ligand binding assays: Technologies and Applications journal of chromatography 829 1–25

Lefkowitz RJ et al, 1970. Radioreceptor Assay of Adrenocorticotropic Hormone: New Approach to Assay of Polypeptide Hormones in Plasma Science 170 633