Cato Guldberg and Peter Waage, the history of the Law of Mass Action, and its relevance to clinical pharmacology

Abstract

We have traced the historical link between the Law of Mass Action and clinical pharmacology. The Law evolved from the work of the French chemist Claude Louis Berthollet, was first formulated by Cato Guldberg and Peter Waage in 1864 and later clarified by the Dutch chemist Jacobus van ‘t Hoff in 1877. It has profoundly influenced our qualitative and quantitative understanding of a number of physiological and pharmacological phenomena. According to the Law of Mass Action, the velocity of a chemical reaction depends on the concentrations of the reactants. At equilibrium the concentrations of the chemicals involved bear a constant relation to each other, described by the equilibrium constant, K. The Law of Mass Action is relevant to various physiological and pharmacological concepts, including concentration–effect curves, dose–response curves, and ligand–receptor binding curves, all of which are important in describing the pharmacological actions of medications, the Langmuir adsorption isotherm, which describes the binding of medications to proteins, activation curves for transmembrane ion transport, enzyme inhibition and the Henderson–Hasselbalch equation, which describes the relation between pH, as a measure of acidity and the concentrations of the contributory acids and bases. Guldberg and Waage recognized the importance of dynamic equilibrium, while others failed to do so. Their ideas, over 150 years old, are embedded in and still relevant to clinical pharmacology. Here we explain the ideas and in a subsequent paper show how they are relevant to understanding adverse drug reactions.

Introduction

Just over 150 years ago, on 15 March 1864, Peter Waage and Cato Guldberg (Figure 1) published a paper in which they propounded what has come to be known as the Law of Mass Action 1. In this article we review the history of its discovery and early applications in pharmacology. As we explain, the Law of Mass Action is the foundation on which the interpretation of the relationship between dose and response is based. We illustrate the importance of the Law in interpreting adverse drug reactions in a companion review 2.

Figure 1.

Cato Guldberg and Peter Waage

The origins of the Law of Mass Action

The concept of ‘affinity’ as the chemical force that holds together dissimilar substances is ascribed to Herman Boerhaave (1668–1738), the influential Leiden physician and the author of Elementa chemiae 3. As he wrote: ‘…the dissolving and dissolved particles […] unite, by the affinity of their own nature, into homogeneous bodies’ 4.

The French chemist Claude Louis Berthollet (1748–1822) then considered the driving force behind chemical reactions and established the relation between the mass of a substance and the rate at which it undergoes a chemical reaction. He set this idea down in his 1801 treatise Recherches sur les lois de l’affinité, in which he described ‘the laws of the affinity by which bodies tend to join together or combine’ 5. The relevance of chemical affinity to therapeutics was quickly recognized. For example, Jonathan Pereira in The Elements of Materia Medica wrote that ‘The action of a medicine on one organ rather than on another is accounted for on the chemical hypothesis, by assuming the existence of unequal affinities of the medicinal agent for different tissues’ 6.

It was not until 1864, however, that the Norwegian mathematician Cato Guldberg (1836–1902) and the chemist Peter Waage (1833–1900), Guldberg's brother‐in‐law, clearly propounded the Law of Mass Action 1. That their paper was neglected was perhaps foreseeable, since it was written in Norwegian. The population of Norway was at that time around 1.7 million 7 and few outside the country would have spoken the language in which the paper was written. Guldberg and Waage therefore extended the reach of their contribution, by publishing a further paper in French in 1867, entitled Etudes sur les affinités chimiques 8. Their theory was subsequently supported by the work of the Danish chemist Julius Thomsen, using thermochemical techniques 9, 10, but the idea was still largely neglected until it was rediscovered in 1877 by the Dutch chemist Jacobus van ’t Hoff (1852–1911), who classified chemical reactions and defined their orders 11 and showed how the Law of Mass Action could be deduced directly from thermodynamic principles. Based on van ’t Hoff's work, the Swedish chemist Svante August Arrhenius (1859–1927) deduced a formula for the effect of temperature on the rate constant of a reaction.

The problem that had occupied Guldberg and Waage was how to give a mathematical description of the affinities that drive chemical reactions. This involved both the rates of the reactions, which they described as chemical dynamics, and, for reversible reactions, the final equilibrium between the forward reaction and the backward reaction, which they designated chemical statics. The two separate parts of the problem have sometimes been forgotten, as Mysels 12 and Guggenheim 13 pointed out.

Taking first the question of the rate of reaction, the Norwegians proposed that for a reversible reaction taking place in one phase (their data were derived from reactions in solution), the rate was proportional to the ‘active mass’ of the reactants (‘La force est porportionelle au produit des masses actives des deux corps A et B’ 8), where ‘active mass’ is the mass per unit volume ‐ that is, the concentration.

$$\mathrm{A}+\mathrm{B}\rightleftharpoons \mathrm{A}\text{'}+\mathrm{B}\text{'}$$

The rate is therefore related to the number of species involved in the reaction, and so depends on the precise reaction path. This means that if there is a series of intermediate reactions whose form is not known, it may not be possible to calculate the rate of reaction, even when the overall reaction is known. As Guggenheim later pointed out, ‘The point… that the kinetic behaviour of a particular reaction must be determined experimentally and cannot be predicted from the stoichiometric formula was better appreciated by… van ’t Hoff than by many subsequent writers, especially writers of textbooks’ 12. Nevertheless, the simple analysis of the rate of change of reactant and the pharmacokinetic treatment of the disappearance of a drug from the sampling compartment are identical. In particular, if the disappearance of a reactant R obeys first‐order kinetics it follows that:

$$–\mathrm{d}\left[\mathrm{R}\right]/\mathrm{d}\mathrm{t}=\mathit{k}\left[\mathrm{R}\right]\text{.}$$

This is precisely the form of rate equation used to describe the first‐order disappearance of drug from a sampling compartment. The theoretical treatments of reaction kinetics have evolved, and now take into account the energy required for a species to react, the time for a suitably energized species to react, and the spatial arrangement of the reactants 14.

The law governing the equilibrium of reversible chemical reactions, in contrast to the rate of the reaction, does not depend on any intermediate steps, only on the concentrations of the reactants and products. For a simple one‐step reaction, in which A + B ⇌ AB, the ratio of the concentrations of reactants to product is a constant, the equilibrium constant, K _A, given by:

$$\frac{\left[\text{AB}\right]}{\left[\mathrm{A}\right]\left[\mathrm{B}\right]}={\mathit{K}}_{\mathrm{A}}$$

where K _A is the ratio of the backward and forward rate constants. [By convention, k is used to designate a rate constant, and K is used to designate an equilibrium constant].

More complex interactions lead to correspondingly more complex equilibrium conditions.

Later developments—defining dose–responsiveness

At the end of the 19th century, Krönig & Paul, acknowledging their debt to studies in plants published by Kahlenberg & True in the Botanical Gazette in 1896, applied chemical principles to disinfection 15. Harriette Chick at the Lister Institute extended their results, demonstrating that the rate of bacterial death was proportional to the number of surviving bacteria, n:

$$–\mathrm{d}\mathit{n}/\mathrm{d}\mathrm{t}=\mathit{k}\times \mathit{n}$$

‘in accord with the Law of Guldberg & Waage’ 16. For the number of anthrax spores remaining after the addition of 5% phenol, the value of k [K in the original] calculated by Chick was approximately 0.047 h^–1.

Paul Ehrlich first set out his view of the specificity of reactions with cellular components in his doctoral thesis on the staining of tissues with aniline dyes, ‘Beiträge zur Theorie und Praxis der histologischen Färbung’, which he presented at Leipzig University on 18 June 1878 17. He argued that the interaction was purely chemical, that the specificity was explained by preferential reactions, and that the structure of the aniline dye determined what cellular components it would stain. He later more explicitly stated that there is a direct relation between chemical structure and function, and that the binding of molecules by receptors mediates most physiological functions 18, 19. In his 1908 Nobel Lecture, Ehrlich explained why the action of a toxin ‘can only be caused by the adhesion of the toxic substance to quite definite cell complexes’ which he called ‘receptors’ 20. This was the foundation of receptor theory as a chemical phenomenon. A corollary was that that the interactions of drugs with the body could be analyzed in terms of chemical theory.

The concept of receptors and the consequences of receptor theory were elaborated during the first half of the 20th century by Langley and others 21. Hill, working with Langley, examined the temperature coefficient of the effect of nicotine on muscle contraction. Citing Arrhenius's work, he concluded in 1909 that the effect could not be accounted for by physical diffusion. The 'combination between nicotine or curare and the combining constituent of the muscle is of an ordinary chemical nature. This combination is a reversible one between two molecules…' 22. Hill also derived the hyperbolic form of the dose–response curve from the premise of a chemical combination of drug and receptor. The first example we have found of a plot of response against the logarithm of the dose (i.e. the classical log dose–response curve) was published in 1926, when Clark used it to demonstrate the action of acetylcholine on frog rectus abdominis 23. He noted that ‘the relation between the concentration and action of acetylcholine in most cases follows the formula K × x = y/(100 – y) and the simplest explanation of this fact is to suppose that a reversible monomolecular reaction occurs between the drug and some receptor in the cells’.

Clark was also aware of the work of Galton 24 and MacAlister 25 on the log‐normal distribution, which could be used to interpret many data on human attributes, including those attributes explained by Weber's law, later extended by Fechner, that the least discernible increment in stimulus was a constant proportion of the initial stimulus.

Guldberg and Waage's work on chemical reactions was the foundation for our quantitative treatment of pharmacological phenomena. Hill postulated a simple reversible combination between two molecules, and this formulation has served pharmacologists well. Modern analysis, which allows for more complex interactions and incorporating more recent ideas of chemical kinetics, is dealt with by Kenakin 26.

Conclusion

The fundamental ideas behind Guldberg and Waage's Law of Mass Action, namely that in an elementary reaction the rate depends on the concentrations of reactants and the stoichiometry of the reaction and that at equilibrium the products and reactants are in fixed ratio, have profoundly influenced pharmacology. Ehrlich and others proposed that interactions between drugs and the body took place at specific receptor sites, and that in turn allowed Langley and Hill to apply chemical theory to the processes. The theory is still the foundation of our quantitative understanding of drug–receptor interactions.

Competing Interests

Both authors have completed the Unified Competing Interest form at www.icmje.org/coi_disclosure.pdf (available on request from the corresponding author) and declare no support from any organization for the submitted work. JKA is a member of a Technology Appraisal Committee of NICE, Editor of Meyler's Side Effects of Drugs: The International Encyclopedia of Adverse Drug Reactions and Interactions, Editor of the Side Effects of Drugs Annuals, and a member of the Editorial Advisory Board of the Adverse Drug Reactions Bulletin. REF is a member of the Appeals Panel of NICE, Editor of the Adverse Drug Reactions Bulletin, and Director of the MHRA Yellow Card Centre West Midlands. There are no other relationships or activities that could appear to have influenced the submitted work. The views expressed in this article are those of the authors and are not necessarily shared by these institutions or others associated with them.

Ferner, R. E. , and Aronson, J. K. (2016) Cato Guldberg and Peter Waage, the history of the Law of Mass Action, and its relevance to clinical pharmacology. Br J Clin Pharmacol, 81: 52–55. doi: 10.1111/bcp.12721.