Affinity, association constant and dissociation constant

This chapter answers parts from Section C(v)  of the 2017 CICM Primary Syllabus, which asks the exam candidate to "describe affinity and dissociation constants".  This has been asked about in Question 12 from the second paper of 2007. Specifically, the college wanted to know what the term "affinity" meant, and expected the candidates to be able to recognise that "k_off/k_on" is the relationship which describes the dissociation constant. The question was passed by 14% of the exam candidates, and has never appeared in the exam again. 

In summary:

• k_off  is the rate constant of dissociation of the drug from the receptor
• k_on is the rate constant of association of the drug to the receptor
• Dissociation constant (K_d ) is the rate constant of dissociation at equilibrium, defined as the ratio k_off / k_on
• Association constant (k_a or K_a ) is the opposite of K_d
  • When K_a is high, K_d is low, and the drug has a high affinity for the receptor (fewer molecules are required to bind 50% of the receptors)
• Affinity in chemistry is the tendency of dissimilar chemical species to form chemical compounds.
• The major factors which affect affinity and dissociation constant in clinical pharmacology are temperature and the presence of a catalyst.

Drug and receptor binding

The relationship of drugs combining with their receptors can be described as this:

In human language, the population of drug molecules and receptor molecules combine at a certain rate k_on,  and then separate again at another (possibly different) rate k_off. When the system is allowed to rest for an infinitely long time these reactions will run to an equilibrium, where there will be some constant concentration of free drug, unbound receptor and drug/receptor complexes. At this point, the relationship between k_on and k_off will stabilise. 

A moment of silence for nomenclature. Who decided we were going to call things k_something, and why? Well. The exact origins of "k" are lost in antiquity, and one can only assume that "k" stands for konstant, from German. By agreed-upon convention of pharmacology, lower case "k" is used to denote rate constants such as k_on and k_off, whereas uppercase K is used for equilibrium constants (such as K_d). To be precise, k_on and k_off are actually not the official terms - the International Union of Pharmacology Committee on Receptor Nomenclature and Drug Classification would prefer you to use k₊₁ for the association reaction and k_-1 for the dissociation of drug from receptor. The "on" and "off" terminology has been retained here because it is used by the college examiners in  Question 12 from the second paper of 2007.

Rate constants (k_on and k_off) and the dissociation constant (K_d)

k_on is the rate constant of association. Units for k_on are units of [D] multiplied by the units of [R] over time, i.e. it is usually expressed in moles per second (or picomoles per hour, depending on how lazy your reaction rate and how scarce the reagents). 

k_off  is the rate constant of dissociation. Units for k_off are units of [DR] over time.

K_d is the dissociation constant. This is the constant which describes the drug / receptor interactions at equilibrium.

K_d = k_off / k_on 

The units for K_d are mol/L , i.e. units of concentration. When 50% of the receptors are occupied,  K_d = [D] (i.e. [R] = [DR]). So, when K_d is high, it means that a large concentration of the drug is required to occupy 50% of the receptors, i.e. the drug and the receptor have a low affinity for one another. Conversely, a low K_d value means that the drug has high affinity for the receptor and fewer molecules of the drug are required to occupy 50% of the receptors.

K_a is the association constant.  It is the opposite of K_d; i.e. when a drug has a low K_d it has a high K_a (i.e. it binds avidly to the receptor). The colloquial term "affinity" is often used interchangeably with K_a; however affinity as a chemical definition is actually something slightly different. 

Affinity

Affinity is usually used to describe how avidly a drug binds to its receptor. This concept is borrowed from chemical physics and physical chemistry, where affinity is defined as the quantifiable representation of the tendency of dissimilar chemical species to form chemical compounds. Eddy (2004) presents an essay review of a book by Kim, describing the crazy trajectory we as a species have taken to arrive at the modern IUPAC Green and Gold Book definition, which holds that:

"...affinity A is the negative partial derivative of Gibbs free energy G with respect to extent of reaction ξ at constant pressure and temperature."

To borrow the formula from Wikipedia, 

In this context, one can see that a positive affinity value A would result in a reaction which takes place spontaneously, and a negative affinity would require the reagents to be heated or pressurised until A becomes positive. Which segues nicely into a discussion of the factors which determine affinity and dissociation constant.

Determinants of affinity (A) and dissociation constant (K_d)

The major factors which affect affinity and dissociation constant are temperature and the presence of a catalyst. In general, the rate of any reaction is  determined by the Arrhenius equation:

So, it is composed of several immutable constants, eg. the gas constant and temperature Within the confines of the fragile reaction vessel which is the human body, temperature is fairly constant and probably close to 37°C. The only things which will vary for any given chemical reaction will be E_a, which can be reduced in the presence of a catalyst. Additionally, there is a wild card in the shape of A, an experimentally derived pre-exponential factor which is different for every chemical reaction and which is basically a reflection of the number of molecular collisions which occur per second, specifically ones which put the reagents in an orientation which is just right for the reaction to take place. This factor is usually derived from experiments, and as far as one can tell there is no clever way to predict what A is going to be for any given molecular interaction.