The law of mass action in pharmacodynamics

This chapter answers parts from Section C(v) of the 2023 CICM Primary Syllabus, which encourages the exam candidate to *"explain the Law of Mass Action and describe affinity and dissociation constants". *At this stage, no sadistic SAQ or viva question has asked about this specific topic, even though the question of affinity and dissociation constants had emerged in Question 12 from the second paper of 2007. Because the law of mass action is fundamental to the understanding of receptor-effector coupling, this topic receives more attention here than it does in the exam.

The official college pharmacology texts either omit or skim over this topic. *Basic and Clinical Pharmacology *(13th ed) mentions it briefly in Chapter 2 (*Drug Receptors & Pharmacodynamics*) where the hyperbolic relation of concentration and effect is discussed. Fortunately there is published peer-reviewed literature. The best articles on this topic are probably the 2015 paper by Terry Kenakin or the 1979 paper by Norman, both of which are available for free.

In summary:

- The Law of Mass Action dictates that the rate of a reaction is dependent on the concentration of reagents.
- At equilibrium, the ratio of the reagents to products of a reaction is constant.
- Therefore, at equilibrium the rate of association and the rate of dissociation are constant (k
_{on}= k_{off}).- The ratio (k
_{off }/ k_{on}) describes K_{d}, the apparent dissociation constant- When [D] = K
_{d}, the concentration of drug D is enough to bind 50% of the total number of receptors- When plotted on a semilogarithmic scale of receptor occupancy and drug concentration, this relationship presents as a sigmoid curve

The equation offered here is a simple one.

A + B ⇌

a+b

Consider two "masses", A and B. These masses react and produce population of product molecules, *a* and *b*. Given an infinite timeline, the concentrations of parent molecules and daughter molecules will achieve an equilibrium. At equilibrium, the product of A and B on one side of the equation divided by the product of *a* and *b* on the other side of the equation is a constant. This constant (k) is independent of the absolute amount of substances at the start of the reaction.

To borrow nomenclature used in Question 12 from the second paper of 2007, the two masses acting on each other in pharmacology are the drug [D] and the receptor it binds [R]. If these associate with each other with the association constant k_{on}, the rate of association can be described as the product k_{on}[D][R]. The rate of dissociation of the drug from the receptor can be described as k_{off} [DR], where k_{off} is the rate constant for dissociation. At equilibrium, the rates of association and dissociation are equal, so the amount of drug bound to the receptor remains stable:

k

_{on}[D][R] = k_{off}[DR]

One can then define K_{d}, the apparent dissociation constant at which equilibrium occurs:

K

_{d}= k_{off }/ k_{on}

Thus, when [D] = K_{d} the reaction is at equilibrium and the concentration of drug D is just enough to bring the reaction to half-maximal completion, i.e. half of all receptors are occupied.

The mass action equation for drug/receptor interaction is therefore:

[DR] = [D][R

_{T}] / ([D] + K_{d})

where R_{T} is the total number of receptors and [DR] is the concentration of the complex of drug and receptor.

We can look at the proportion of the receptors occupied by the drug (which would be [DR]/[Rt]) if we rearrange the equation like so:

[DR] / [Rt] = [D] / ([D] + K

_{d})

This equation describes a sigmoid curve on a semi-logarithmic scale, which will be instantly familiar to all people who had to sit through a semester of undergraduate pharmacology:

This relationship makes several assumptions:

- All ligands and all receptors are equally available to each other
- The binding of drug and receptor is reversible (it frequently is not, as in the case of phenoxybenzamine)
- The binding of drug and receptor does not alter either the drug or the receptor (that's not the case when the drug is a substrate for a receptor which is a metabolic enzyme, for one example)
- The receptor and drug are either bound to each other, or are not bound to each other (i.e. there are no ambiguous partial states).

Moreover, there are problems when the drug and receptor exist in a complex system of interrelated and mixed mass action reactions.

Kenakin, Terry. "The mass action equation in pharmacology." *British journal of clinical pharmacology* 81.1 (2016): 41-51.

Érdi, Péter, and János Tóth. *Mathematical models of chemical reactions: theory and applications of deterministic and stochastic models*. Manchester University Press, 1989.

Waage, Peter, and Cato Maximilian Gulberg. "Studies concerning affinity." *J. Chem. Educ* 63.12 (1986): 1044.

Norman, J. "Drug-receptor reactions." *BJA: British Journal of Anaesthesia* 51.7 (1979): 595-601.