Schild equation

In pharmacology, Schild regression analysis, based upon the Schild equation, both named for Heinz Otto Schild, are tools for studying the effects of agonists and antagonists on the response caused by the receptor or on ligand-receptor binding.

Concept Dose-response curves can be constructed to describe response or ligand-receptor complex formation as a function of the ligand concentration. Antagonists make it harder to form these complexes by inhibiting interactions of the ligand with its receptor. This is seen as a change in the dose response curve: typically a rightward shift or a lowered maximum. A reversible competitive antagonist should cause a rightward shift in the dose response curve, such that the new curve is parallel to the old one and the maximum is unchanged. This is because reversible competitive antagonists are surmountable antagonists. The magnitude of the rightward shift can be quantified with the dose ratio, r. The dose ratio r is the ratio of the dose of agonist required for half maximal response with the antagonist B {\displaystyle {\ce {B}}} present divided by the agonist required for half maximal response without antagonist ("control"). In other words, the ratio of the EC50s of the inhibited and un-inhibited curves. Thus, r represents both the strength of an antagonist and the concentration of the antagonist that was applied. An equation derived from the Gaddum equation can be used to relate r to [ B ] {\displaystyle [{\ce {B}}]} , as follows:

r = 1 + [ B ] K B {\displaystyle r=1+{\frac {[{\ce {B}}]}{K_{B}}}}

where

r is the dose ratio

[ B ] {\displaystyle [{\ce {B}}]} is the concentration of the antagonist

K B {\displaystyle K_{B}} is the equilibrium constant of the binding of the antagonist to the receptor A Schild plot is a double logarithmic plot, typically log 10 ⁡ ( r − 1 ) {\displaystyle \log _{10}(r-1)} as the ordinate and log 10 ⁡ [ B ] {\displaystyle \log _{10}[{\ce {B}}]} as the abscissa. This is done by taking the base-10 logarithm of both sides of the previous equation after subtracting 1:

log 10 ⁡ ( r − 1 ) = log 10 ⁡ [ B ] − log 10 ⁡ ( K B ) {\displaystyle \log _{10}(r-1)=\log _{10}[{\ce {B}}]-\log _{10}(K_{B})}

This equation is linear with respect to log 10 ⁡ [ B ] {\displaystyle \log _{10}[{\ce {B}}]} , allowing for easy construction of graphs without computations. This was particular valuable before the use of computers in pharmacology became widespread. The x-intercept (where log10(r-1)=0) of the graph represents the negative logarithm of K B {\displaystyle K_{B}} and can be used to quantify the affinity of the antagonist. 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. The fitting of the Schild plot to observed data points can be done with regression analysis.

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 = k − 1 k 1 {\displaystyle 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

[ A R ] = [ R ] t [ A ] [ A ] + K d {\displaystyle [AR]={\frac {[R]_{t}\,[A]}{[A]+K_{d}}}}

This is the Hill-Langmuir equation, which is practically the Hill equation 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).

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