A. The Expression of Amount of Drug: Concentration and Dose

1. Concentration. It is recommended that the molar concentration of substance X be denoted by either [X] or c_x, with the former preferred. Decimal multipliers should be indicated by the use of either Le Système International d'Unités (International System of Units) prefixes (e.g., μM, nM) or by powers of ten (e.g., 3 × 10^-8 M), with the former preferred.

2. Dose. In some circumstances (e.g., in therapeutics and clinical pharmacology, in in vivo experiments, and when tissues are perfused in vitro and exposed to a bolus application of drug), absolute drug concentrations are uncertain, and it becomes more appropriate to specify the quantity of drug administered. This may be done in terms of either mass or molar quantity. Units and routes of administration should be specified. In the case of in vivo experiments with animals, the quantity of drug is to be expressed per unit of animal mass (e.g., mol/kg, mg/kg). In therapeutics, milligrams per kilogram will normally be appropriate. Negative indices should be used where confusion otherwise arises (e.g., mg min^-1 kg^-1).

B. General Terms Used to Describe Drug Action

Table 1.

C. Experimental Measures of Drug Action

1. General Measures.Table 2.

2. Agonists.Table 3.

3. Antagonists.Table 4.

D. Terms and Procedures Used in the Analysis of Drug Action

1. The Quantification of Ligand-Receptor Interactions.Table 5.

2. Action of Agonists.Table 6.

3. Action of Antagonists.Table 7.

A. Microscopic and Macroscopic Equilibrium Constants

Microscopic and macroscopic equilibrium constants should be distinguished when describing complex equilibria, which occur with all agonists. The latter refers to a single constant describing the overall equilibrium (i.e., the value that would be obtained in a ligand binding experiment), whereas the former refers to each individual constant that describes each reaction step within the equilibrium. For the scheme the macroscopic equilibrium dissociation constant (denoted here as K_app for “Kapparent”) is given by Here K₁ and K₂ are the microscopic equilibrium constants for the first and second reactions, respectively. Note that in this scheme, saturation radioligand binding assays and Furchgott's (1966) irreversible antagonist method for determining the equilibrium dissociation constant for an agonist would each provide an estimate of K_app rather than K₁.

This distinction is also important when considering those receptors (e.g., ligand-gated ion channels) that have more than one binding site for the agonist.

B. Schild Equation and Plot—Further Detail

The Schild equation is based on the assumptions that (a) agonist and antagonist combine with the receptor macromolecule in a freely reversible but mutually exclusive manner, (b) equilibrium has been reached and that the law of mass action can be applied, (c) a particular level of response is associated with a unique degree of occupancy or activation of the receptors by the agonist, (d) the response observed is mediated by a uniform population of receptors, and (e) the antagonist has no other relevant actions, e.g., on the relationship between receptor and response. Under these circumstances, the slope of the Schild plot should be 1 and the resulting estimate of the pA₂ should be equal to the pK (negative logarithm of the antagonist equilibrium dissociation constant).

For an antagonist to be classified as reversible and competitive on the basis of experiments in which a biological response is measured, the following criteria must hold:

1. In the presence of the antagonist, the log agonist concentration-effect curve should be shifted to the right in a parallel fashion.

2. The relationship between the extent of the shift (as measured by the concentration ratio) and the concentration of the antagonist should follow the Schild equation over as wide a range of antagonist concentrations as practicable. Usually, the data are presented in the form of the Schild plot, and adherence to the Schild equation is judged by the finding of a linear plot with unit slope (see Note 2 below). Nonlinearity and slopes other than unity can result from many causes. For example, a slope greater than 1 may reflect incomplete equilibration with the antagonist or depletion of a potent antagonist from the medium, as a consequence either of binding to receptors or to other structures. A slope that is significantly less than 1 may indicate removal of agonist by a saturable uptake process, or it may arise because the agonist is acting at more than one receptor (the Schild plot may then be nonlinear). See Kenakin (1997) for a detailed account.

Note 1: The finding that the Schild equation is obeyed over a wide range of concentrations does not prove that the agonist and antagonist act at the same site. All that may be concluded is that the results are in keeping with the hypothesis of mutually exclusive binding, which may of course result from competition for the same site but can also arise in other ways (see Allosteric Modulators in Table 1 and Competitive Antagonism in Table 7).

Note 2: Traditional Schild analysis is based on the use of linear regression. Nowadays, with the almost ubiquitous availability of computers in most research environments, a more accurate approach to performing Schild analysis is to use computerized nonlinear regression to directly fit agonist/antagonist concentration-response data to the Gaddum/Schild equations. The advantages of this approach over traditional Schild analysis are described elsewhere (Waud, 1975; Black et al., 1985; Lew and Angus, 1995). One simple method is to fit agonist EC₅₀ data, determined in the absence and presence of antagonist, to the following equation: where pEC₅₀ and pA₂ are as defined previously in Tables 3 and 4, respectively, [B] denotes the antagonist concentration, S is a logistic slope factor analogous to the Schild slope and log c is a fitting constant (Motulsky and Christopoulos, 2003). This equation is based on a modification of the original Gaddum/Schild equations that results in more statistically reliable parameter estimates than those obtained using the original equations for nonlinear regression (Waud et al., 1978; Lazareno and Birdsall, 1993). If S is not significantly different from 1, then it should be constrained as such and the equation re-fitted to the data.

C. The Relationship between the Hill and Logistic Equation

The logistic function is defined by the equation where a and b are constants. If a is redefined as -log_e(K^b), and x as log_e z, then which has the same form as the Hill equation.