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. 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 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 , as follows: where r is the dose ratio is the concentration of the antagonist is the equilibrium constant of the binding of the antagonist to the receptor A Schild plot is a double logarithmic plot, typically as the ordinate and as the abscissa. This is done by taking the base-10 logarithm of both sides of the previous equation after subtracting 1: This equation is linear with respect to , allowing for easy construction of graphs without computations. This was particular valuable before the use of computers in pharmacology became widespread. The y-intercept of the equation represents the negative logarithm of and can be used to quantify the strength of the antagonist.