How do we study and quantify the effect a drug has on the body? In order for a drug to exert its effects, it needs to reach its site of action. The study of how it gets to its target site and, subsequently, how it gets eliminated from the body, is the subject of pharmacokinetics. What the drug does at its site of action (its receptor), and the quantification of its interaction with the receptor, is precisely what basic pharmacodynamics is concerned with.

This article aims to give an overview of basic pharmacodynamics, introducing some of the fundamental mathematical descriptions of drug receptor interactions and the use of these principles in drug research and development.

The really basic pharmacodynamics:  Affinity, Potency and Efficacy

The basic terms that will be used in this article have already been introduced in ‘How drugs work?; however, it is important to refresh these before we go any further:

Affinity – this refers to the ability of a given drug to bind to its receptor by direct chemical interactions (electrostatic, hydrophobic etc.) with the binding site on the receptor. It is therefore an intrinsic property of the drug stemming from its chemical structure and how well that structure complements the binding site.

Potency – this can be defined as the ability of a given drug to produce a tissue response. The lower the concentration of that drug needed to give a specified response (mostly a 50% response – the EC50) the more potent the drug is. However, it is important to emphasise that binding to a receptor does not mean activation of the receptor and thus a tissue response.

Early pharmacologists assumed that receptor occupancy was equal to tissue response; however, this idea was later found to be too simplistic, especially when studying the neuromuscular junction where roughly only 1% of total receptors need to be occupied to give a maximal tissue response (muscle contraction). Therefore, a new concept was introduced and termed efficacy, and the drug-receptor interaction scheme was revised to its current version (sometimes referred to as del Castillo-Katz mechanism, shown in Figure 1).

Del Castillo - Katz mechanism for a drug binding to a single site on its receptor

Figure 1 del Castillo – Katz mechanism for a drug binding to a single site on its receptor. Drug A binds to its receptor binding site, R, forming drug-receptor complex AR, which is, in turn, in an equilibrium with the activated receptor complex AR*.

Efficacy – the ability of a given drug, once bound to its receptor, to activate that receptor and thus initiate cellular signalling pathways that lead to tissue response. By definition, the efficacy of an antagonist is 0 and the efficacy of a partial agonist will always be less than that of a full agonist in a given tissue preparation. Unlike partial agonists, a full agonist has a large enough efficacy to cause a maximal response in a given tissue.

Efficacy is more difficult to quantify but it has been possible to do so with the invention of the patch clamp technique, where the probability of opening of a single ligand-gated ion channel may be directly assessed but this is beyond the scope of this article. Affinity of a drug is relatively accurately quantified by in vitro radio-ligand binding studies, from which the binding parameters for a given drug acting on a specific receptor may be determined. The next section aims to demonstrate this process.

Basic Pharmacodynamics: Quantifying drug-receptor interactions

The del Castillo-Katz scheme shown in Figure 1 is the basis for the mathematical description of drug-receptor interactions. As this article is mainly focussing on affinity, we will zoom in on the first half of the scheme to explain how the binding parameters for a given drug are derived from radio-ligand binding experiments (Figure 2).

Drug-receptor binding site equilibrium

Figure 2 Drug-receptor binding site equilibrium. This shows the binding step of the del Castillo – Katz mechanism that is used to determine the drug’s affinity at a particular receptor. Drug A binds to binding site R create a drug-receptor complex AR, which is in a dynamic equilibrium with the free drug (A) and free receptor (R) that is governed by association rate constant k+1, and dissociation rate constant k-1.

The key to deriving the expression that describes the proportion of occupied receptors by drug A is to assume that this reaction is in dynamic equilibrium. Second assumption we need to make is that the Law of Mass Action applies, which states that the rate of any chemical reaction is proportional to the product of the masses of the reactants raised to the power equal to the coefficients that occur in the chemical reaction. In this case, all coefficients are equal to 1 and, if we assume a constant volume, the rate of each reaction in the equilibrium will therefore be proportional to the product of the concentrations of its reactants. As the rate constant of each reaction is the proportionality constant, we can derive the expression shown in Figure 3 and Figure 4.

First steps in the derivation of an expression for the proportion of occupied receptors

Figure 3: First steps in the derivation of an expression for the proportion of occupied receptors. This Figure shows that the derivation is based on the fact that the rates of backward and forward reactions are equal when the reaction reaches equilibrium, as assumed. From equation (4) onwards, [AR] and [R] are expressed as pAR and pR, respectively, meaning that the concentrations of either the drug-receptor complex, AR, or the free receptor, R, are now expressed as proportions of the total receptor population, so that pAR = [AR]/[R]total and pR = [R]/[R]total.

Figure 3a introduces a new variable into the expression, KA. KA is the dissociation equilibrium constant for the binding of drug A to its binding site, and represents the ratio of the dissociation rate constant, k-1, to the association rate constant, k+1, so that:

Dissociation equilibrium constant

KA is the key parameter of drug binding, because it describes the affinity of the drug to its binding site, and is therefore the key parameter in experiments that determine the basic pharmacodynamics of a given drug. The smaller the KA, the greater the affinity, as the dissociation rate constant of the drug from the receptor is much smaller than the association rate. The KA has the units of molar (M; moles per litre).

The next step in the derivation of the final expression for the proportion of occupied receptors by drug A, pAR, is the assumption of receptor conservation. One therefore assumes that the total population of receptors in the binding preparation remains constant, and therefore, the sum of the proportion of free receptors, pR, and those occupied by drug A, pAR, has to equal 1. The next steps of the derivation are shown in Figure 4.

Final steps for the derivation of Hill-Langmuir equation

Figure 4: Final steps for the derivation of Hill-Langmuir equation. The final expression for the proportion of occupied receptors by drug A (sometimes this expression is called the Hill-Langmuir equation) is derived by substituting the terms that were obtained from the initial assumption of equilibrium (Figure 3) into the expression that describes receptor conservation (Equation 1).

Now that we have derived the Hill-Langmuir equation, it is crucial to look at its application in pharmacological research, specifically, in radio-ligand saturation binding experiments that allow a precise determination of both the affinity of the drug (expressed as the drug’s KA), but also the total amount of receptors in the tissue preparation.

Radio-ligand binding studies have developed their own nomenclature. The idea of these studies is to measure the amount of specific binding i.e. measure the radioactivity given out by the radio-ligand only when it is bound to its binding site on the receptor, thus when it formed drug-receptor complex, AR. Specific binding is denoted as B. The dissociation equilibrium constant for radio-ligand L is called KL. The amount of maximum specific binding in a given preparation, i.e. when all the binding sites are saturated (hence the name saturation experiments), is denoted Bmax, and thus represents the amount of receptor in that preparation. Using this nomenclature gives an expression for specific binding shown in Figure 5.

Equation for the specific binding of a radio-ligand L to its receptor

Figure 5: Equation for the specific binding of a radio-ligand L to its receptor. Since pAR in the Hill-Langmuir equation is in fact [AR]/[R]total, then it must follow that B/Bmax since the maximum amount of specific binding will be achieved if all specific binding sites for radio-ligand L are occupied.

At this stage, it is important to note that this equation, which describes the binding of a radio-labelled ligand to its receptor is only applicable for the specific binding. Non-specific binding, such as binding to the cell membranes and other irrelevant sites needs to be subtracted before the analysis can be performed. In the classical saturation experiments, this is performed by saturating the cellular preparation with a very high concentration of the radioligand first and then repeat the experiment so that the non-specific binding fraction can be determined (see Figure 6). The KL and Bmax parameters are frequently determined graphically by rearranging the equation above to a form, for example by the means of a Scatchard plot (Figure 7).

Curves showing the total, specific and non-specific binding obtained in saturation assays

Figure 6: Curves showing the total, specific and non-specific binding obtained in saturation assays. The total drug bound to the preparation consists of two components: the specific binding fraction and the non-specific binding fraction. The specific binding is the desired function (hyperbola) that is then further used for analysis. Non-specific binding represents binding to for example cell membranes, other receptors in cell preparations etc., and increases linearly with concentration of radio-ligand (x-axis).

An example of a Scatchard plot for radioligand L

An example of a Scatchard plot for saturation binding assay

Figure 7: An example of a Scatchard plot for saturation binding assay. The Scatchard plot is used to obtain the dissociation equilibrium constant of the radioligand, KL, from the gradient, and the Bmax (total number of receptors in the preparation) from the x-intercept.

To summarise, this article reviewed the drug-receptor binding scheme, terms such as affinity and efficacy and described the approach to the derivation of basic pharmacodynamic parameters that are commonly used to quantify drug binding to its receptor. Specifically, it is the estimation of the dissociation equilibrium constant that is key to quantifying the affinity of a drug for its binding site.

Michal Barabas is currently a medical student at the University of Cambridge, where he also teaches pharmacology to undergraduates in small group seminars. Michal obtained a BSc. degree in Pharmacology from UCL in 2013 and an M. Phil degree in Translational Medicine from the University of Cambridge in 2014.

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