The Relationship between Ligand Concentration and Receptor Occupancy

1 The Relationship between Ligand Concentration and Receptor Occupancy 

We begin with the simplest possible representation of the combination of a ligand, A, with its binding site on a receptor, R  

binding is regarded as a bimolecular reaction and k+l and are, respectively, the association rate constant (M 1 s-1) and the dissociation rate constant (s-1). 

The law of mass action states that the rate of a reaction is proportional to the product of the concentrations of the reactants. We will apply it to this simple scheme, making the assumption that equilibrium has been reached so that the rate at which AR is formed from A and R is equal to the rate at which AR dissociates. This gives  

It may seem odd to refer to receptor concentrations in this context when receptors can often move only in the plane of the membrane (and even then perhaps to no more than a limited extent, as many kinds of receptors are anchored). However, the model can be formulated equally well in terms of the proportions of a population of binding sites that are either free or occupied by a ligand. If we define pR as the proportion free, equal to [R]/[R]T, where [R]T represents the total concentration of receptors, and pAR as [AR]/[R]T, we have  

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This is the important Hill-Langmuir equation. A. V. Hill was the first (in 1909) to apply the law of mass action to the relationship between ligand concentration and receptor occupancy at equilibrium and to the rate at which this equilibrium is approached. The physical chemist I. Langmuir showed a few years later that a similar equation (the Langmuir adsorption isotherm) applies to the adsorption of gases at a surface (e g., of a metal or of charcoal). 

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