A receptor is a molecule, often a protein, that can bind a particular ligand. Binding creates a receptor-ligand complex and can change receptor activity, but binding itself is first a reversible chemical equilibrium.

For a single binding site,

$$\mathrm{R+L\rightleftharpoons RL}.$$

A useful equilibrium measure is the dissociation constant

$$\boxed{K_d=\frac{[R][L]}{[RL]}}.$$

A smaller $K_d$ corresponds to tighter binding: less free ligand is needed to favor the bound state.

Fractional occupancy

Let the total receptor concentration be

$$[R]_T=[R]+[RL].$$

The fractional occupancy is the fraction of receptors carrying ligand:

$$\theta=\frac{[RL]}{[R]_T}.$$

For one independent binding site, when free ligand is much more abundant than receptor so that $[L]$ is effectively fixed,

$$\boxed{\theta=\frac{[L]}{K_d+[L]}}.$$

This equation has an important interpretation:

• if $[L]\ll K_d$, few receptors are occupied;
• if $[L]=K_d$, $\theta=1/2$;
• if $[L]\gg K_d$, occupancy approaches 1.

Worked example

If a receptor has

$$K_d=10\ \mathrm{nM}$$

and free ligand concentration is

$$[L]=30\ \mathrm{nM},$$

then

$$\theta=\frac{30}{10+30}=0.75.$$

About 75% of receptor sites are occupied.

Occupancy is not the same as response

A cell does not necessarily produce 50% of its maximum response at 50% receptor occupancy. Downstream amplification and saturation can change the relation between receptor binding and biological output.

The reusable distinction is therefore

ligand concentration → receptor occupancy → receptor activity → downstream response

$K_d$ characterizes the binding equilibrium. It does not, by itself, specify the magnitude or time course of the cellular response.