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Review
. 2016 Jan;81(1):41-51.
doi: 10.1111/bcp.12810. Epub 2015 Dec 21.

The Mass Action Equation in Pharmacology

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Free PMC article
Review

The Mass Action Equation in Pharmacology

Terry Kenakin. Br J Clin Pharmacol. .
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Abstract

The mass action equation is the building block from which all models of drug-receptor interaction are built. In the simplest case, the equation predicts a sigmoidal relationship between the amount of drug-receptor complex and the logarithm of the concentration of drug. The form of this function is also the same as most dose-response relationships in pharmacology (such as enzyme inhibition and the protein binding of drugs) but the potency term in dose-response relationships very often differs in meaning from the similar term in the simple mass action relationship. This is because (i) most pharmacological systems are collections of mass action reactions in series and/or in parallel and (ii) the important assumptions in the mass action reaction are violated in complex pharmacological systems. In some systems, the affinity of the receptor R for some ligand A is modified by interaction of the receptor with the allosteric ligand B and concomitantly the affinity of the receptor for ligand B is modified to the same degree. When this occurs, the observed affinity of the ligand A for the receptor will depend on both the concentration of the co-binding allosteric ligand and its nature. The relationships between drug potency in pharmacological models and the equilibrium dissociation constants defined in single mass action reactions are discussed. More detailed knowledge of efficacy has led to new models of drug action that depend on the relative probabilities of different states, and these have taken knowledge of drug-receptor interactions beyond Guldberg and Waage.

Keywords: dose-response; mass action; pharmacology; receptor.

Figures

Figure 1
Figure 1
The mass action equation as applied to the binding of a drug A to receptor R, to form complex AR. The amount of AR complex plotted as a logarithmic function of the concentration yields a characteristic sigmoidal curve. The sensitivity of the system to A is given by the KA term (apparent dissociation constant) and gives the location parameter of the curve along the x axis. The maximal ordinate value is given by [RT], the maximal amount of receptor in the system. This curve closely resembles the dose–response relationships for many drugs
Figure 2
Figure 2
Pharmacological systems as combinations of the simple mass action reaction scheme (A). (B) A series mass action system where the product of the first reaction becomes the reactant for the second. (C) A system where two series mass action reactions are aligned in parallel with each other, with a thermodynamic link between them. In this case, a protein (ion channel) can exist in an open and closed conformation and drug A can interact with both of them
Figure 3
Figure 3
Series and parallel mass action reactions specified as distinct pharmacological models. (A) Mass action binding of a substrate to an enzyme begins the process of enzyme catalysis in the Michaelis–Menten model 13 for enzyme function. Once the substrate is bound, the enzyme catalyses the reaction to production of product through k2 (also referred to as kcat). (B) Two common settings for series mass action reactions are the agonist‐mediated production of ternary complexes for G protein‐coupled receptors and agonism as described by the Black–Leff operational model 14. (C) Two common parallel mass action reactions involve drug activation of ion channels and the allosteric model for seven transmembrane receptors 27, 28
Figure 4
Figure 4
The sinistral displacement of functional activity curves for agonists from the initial binding curve. The property of efficacy causes the concentration of agonist producing 50% maximal response (EC50) to be of a lower magnitude than the binding constant KA. In the Black–Leff operational model, it is assumed that the initial agonist–receptor complex AR interacts with response elements in the cell (denoted as E) to produce a further complex that causes cellular response. The sensitivity factor for this second reaction is denoted KE and represents numerous saturable processes within the cell cytosol. Efficacy in this model is given as τ, where τ = [RT]/KE, ([RT] is the total number of receptors
Figure 5
Figure 5
Receptors are often pleiotropically linked to numerous signalling systems in the cell, and the activation of each of these can generate a separate concentration–response curve for drug effect. The key to observing these is the availability of separate assays for each of the responses. Panel A. Schematic showing the interacting species of ligand (A), receptor® and various response coupling elements labeled E1 to E5. Panel B. Simulated concentration‐response curves for interaction of the ligand‐bound receptor with the various response element
Figure 6
Figure 6
The stabilization of unique receptor states can lead to the selective activation of some of the pleiotropic signalling mechanisms linked to the receptor. The activation of angiotensin receptors by angiotensin II normally leads to the activation of the Gq protein and interaction of the receptor with β‐arrestin. The structural analogue TRV120027 stabilizes a conformation of the receptor that causes interaction of the receptor with β‐arrestin, not Gq. This phenomenon is generally referred to as biased receptor signalling. Data for TRV120027 are redrawn from 36. Panel A: Concentration response curves for angiotensin activation of two signaling pathways (G‐protein and β‐arrestin‐ see left panel). Panel B: Biased signaling for TRV120027 showing activation of β‐arrestin but no activation of G protein
Figure 7
Figure 7
Receptor conformational ensembles as Boltzmann distributions of tertiary states. Shown are natural ensembles that lead to various physiological responses such as activation of Gαs, Gαi protein, β‐arrestin interaction and receptor internalization. As different ligands bind to the ensemble, certain conformations are stabilized through selective high affinity at the expense of other conformations. If the stabilized conformation coincides with a physiologically active conformation within the ensemble, this confers that efficacy upon the ligand. Shown are two different ligands. Ligand 1 activates Gαs, Gαi and β‐arrestin (but does not internalize receptors) while ligand 2 activates β‐arrestin and causes receptor internalization

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