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.
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) = (IC50)n/(1 + A(K)/K(A)) = (IC50)n/[1 + (A/EC50)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/Fm = -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 Bn - log K(B), where x is the concentration ratio, and B is the concentration of the antagonist. The modified pA(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 (xK = 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.