The influence of cooperativity on the determination of dissociation constants: examination of the Cheng–Prusoff equation, the Scatchard analysis, the Schild analysis and related power equations

Abstract

Use of the Cheng–Prusoff equation for determination of the equilibrium dissociation constant, K B, is based on an assumption that simple bi-molecular interaction kinetics are strictly followed. Under such circumstances, the slope parameters of the agonist concentration–response curve ( K) and that of the inhibition curve ( n) are unity. New equations are needed for calculating K B when slope parameters ( K and n) deviate from unity. In this article, the slope parameters K and n are used as indexes of cooperativity. Thus, the following new equations are derived: (1) For calculation of K B from IC 50, the new equation which incorporates both cooperativity indexes is described as K B=(IC 50) n /(1+ A K / K A)=(IC 50) n /[1+( A/EC 50) K ] where A is the concentration of the agonist against which the IC 50 is determined, and K A is the apparent equilibrium dissociation constant of the agonist. This new equation is applicable when the cooperativity indexes of K and n are less than, equal to, or greater than unity. This equation reduces to the Cheng–Prusoff equation when the cooperativity indexes K and n are unity. (2) For saturation binding assays, the enhanced Scatchard analysis is described by the equation: B/ F m =− B/ K D+ B max/ K D where B and F are the concentrations of the bound and free ligand, respectively, and m is the cooperativity index of the ligand. A plot of B/ F m versus B yields a straight line with a negative slope that equals 1/ K D, and an x-axis intercept that equals B max. When m equals unity, the above analysis reduces to the traditional Scatchard analysis. (3) The importance of the slope parameters ( K and n) on Schild analysis is illustrated by the equation: log (x K−1)= log B n− log K B , where x is the concentration ratio, and B is the concentration of the antagonist. The modified p A 2 is now defined as the −logarithm of the molar concentration of the antagonist ( B), power adjusted with the slope parameter ( B n ), that causes a two-fold shift of the agonist concentration–response curve ( x K =2), also power adjusted with the slope parameter K. When K and n equal unity, the above analysis reduces to the traditional Schild analysis. A total of six power equations are derived for estimating K B values covering situations with different cooperativity indexes of agonists and antagonists. These equations should yield more accurate estimations of K B values.