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

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Abstract

Use of the Cheng–Prusoff equation for determination of the equilibrium dissociation constant, KB, 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 KB 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 KB from IC50, the new equation which incorporates both cooperativity indexes is described as KB=(IC50)n/(1+AK/KA)=(IC50)n/[1+(A/EC50)K] where A is the concentration of the agonist against which the IC50 is determined, and KA 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/KD+Bmax/KD 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/Fm versus B yields a straight line with a negative slope that equals 1/KD, and an x-axis intercept that equals Bmax. 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(xK−1)=logBnlogKB, where x is the concentration ratio, and B is the concentration of the antagonist. The modified pA2 is now defined as the −logarithm of the molar concentration of the antagonist (B), power adjusted with the slope parameter (Bn), 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 KB values covering situations with different cooperativity indexes of agonists and antagonists. These equations should yield more accurate estimations of KB values.

Introduction

The equilibrium dissociation constant (KB) of a competitive antagonist is often obtained by analyzing concentration ratios of agonist concentration–response curves in the absence and presence of a series of concentrations of the competitive antagonist [1], [2]. However, under some special circumstances, KB has also been determined in the absence and presence of only a single concentration of the antagonist [3]. In other experimental conditions, such as in enzyme inhibition studies, an inhibition study is first performed to obtain an IC50 which is then used for calculating KB or Ki by using the Cheng–Prusoff equation [4] which is shown in Eq. (1):Ki=IC501+(S/Km)where IC50 is the concentration of the competitive inhibitor producing a 50% inhibition, S is the substrate concentration used, and Km is the Michaelis constant of the substrate for the enzyme.

Eq. (1) has also been widely used in radioligand binding assays for calculating KB (or Ki) values. When drug–receptor interactions obey bi-molecular interaction kinetics, Eq. (1) can also be applied to functional studies. In those circumstances, the term Km is replaced by the EC50 of the agonist [5]. Based on the proportionality approach and the mass action law, the author has recently derived a power equation which can accommodate different slope parameters (K) of agonist concentration–response curves [6]. The equation is shown as follows:KB=IC501+(AK/KA)=IC501+(A/EC50)Kwhere KA is the apparent equilibrium dissociation constant of the agonist based on the proportionality theory (previously designated as KP [7]), and is a composite of the rate constants of association, dissociation, product formation, and any cooperativity between the agonist and the receptor. EC50 is also incorporated into the equation for functional studies. Eq. (2) can be used to calculate KB values when slope parameters of agonist concentration–response curves are greater than, equal to, or less than unity. One interesting finding from that study [6] is that the inhibition curve has a slope parameter of unity regardless of the slope parameter of the agonist concentration–response curves. In radioligand binding assays and functional assays, it is very common to have inhibition curves whose slope parameters deviate from unity. Although the Hill equation is originally derived for describing the kinetics of enzyme–substrate interaction for allosteric enzymes with multiple subunits, the Hill coefficient is now believed to signify a cooperativity of the drug–receptor interaction [8], [9]. Cooperativity is observed in ion-channel-coupled receptors and in G-protein-coupled receptors. Similarly, antagonists may also bind to antagonist allosteric binding sites which are different from the agonist binding site. For incorporating the influence of cooperativity of both the agonist and the antagonist on the determination of KB, Eq. (3) is thus derived (Appendix A):KB=(IC50)n1+(AK/KA)=(IC50)n1+(A/EC50)Kwhere n is the cooperativity index of the antagonist (or the slope parameter of the inhibition concentration–response curve). EC50 is also incorporated into the equation for functional studies.

In this article, the derivation of Eq. (3) is described. Simulation data are generated and analyzed. The enhanced Scatchard analysis, the enhanced Schild analysis, and additional related power equations and their use are also illustrated. Literature data are also analyzed by using the equations described. A preliminary form of this article has been presented [10]. In this article, the slope parameter K and n are also referred to as the cooperativity index of the agonist and antagonist, respectively.

Section snippets

Derivation of the power equations

Derivation of the power equations incorporating slope parameters of the agonist and the antagonist are described in Appendix A. It is based on the proportionality approach [7] and the mass action law.

The occupancy theory, the kinetics of the drug–receptor interaction, and substrate–enzyme interactions are then applied to the derivation of the new power equations [11], [12], [13], [14].

Generation of the simulation data

Concentration–response curves are generated according to Eq. (A.12) (Appendix A) in which the agonist

Effects of K and n on concentration–response curves

The cooperativity index of the agonist (K) has three profound effects on the concentration–response curves: (1) As shown in column 1 of Fig. 1, Fig. 2, Fig. 3, the greater the K value, the smaller the response; in other words, the greater the K value, the farther the curve is located to the right. (2) The greater the K value, the steeper the slope of the agonist concentration–response curve. (3) The greater the K value, the smaller the spacing of the shifts produced by various concentrations of

Discussion

This study has demonstrated the importance of the slope parameters (K and n) in the determination of dissociation constants of antagonists. Eq. (A.4), which is derived according to a proportionality approach, is the same as the Hill equation or the logistic equation [7]. Although the Hill equation was originally used for describing the kinetics of enzyme–substrate interaction for allosteric enzymes with multiple subunits, the Hill coefficient is now believed to signify a degree of cooperativity

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