Competitive Ligands at a Shared Binding Site

Why shared-site competition changes occupancy

When two ligands bind mutually exclusively to the same receptor site, a receptor bound to one ligand cannot be bound to the other at the same time. The occupancy of either ligand therefore depends not only on its own free concentration and affinity, but also on the competing ligand. This matters when interpreting target engagement: an increase in one ligand’s concentration can reduce the fraction of receptors available to the other, even if the other ligand’s concentration is unchanged.

The equations below describe equilibrium occupancy in terms of free ligand concentrations. This distinction is important when ligand depletion is appreciable: total concentration cannot simply be substituted for free concentration. Equilibrium occupancy is also appropriate only when binding has had time to approach equilibrium under conditions where the free concentrations are effectively constant.

Deriving the two-ligand occupancy equations

At equilibrium, each ligand’s bound-to-unbound receptor ratio is its free concentration divided by its dissociation constant. In particular, \([RA]/[R]=[A]/K_{D,A}\) and \([RB]/[R]=[B]/K_{D,B}\). These relationships use the usual one-site equilibrium definition of \(K_D\); they assume that each receptor has one relevant binding site and that binding of one ligand excludes the other.

Define the dimensionless concentration-to-affinity ratios \(x=[A]/K_{D,A}\) and \(y=[B]/K_{D,B}\). Substituting \([RA]=x[R]\) and \([RB]=y[R]\) into the receptor total gives \(R_{\mathrm{tot}}=[R](1+x+y)\). Dividing each state amount by this total yields all three state fractions:

Here \(\theta_0\) is the unoccupied fraction. The denominator includes the unbound state, represented by 1, as well as both bound states. The fractions are nonnegative and sum to one: \(\theta_0+\theta_A+\theta_B=1\). Total occupancy by either ligand is \(\theta_A+\theta_B\), which is at most one.

A short sensitivity result makes the competition explicit. Holding \([B]\), \(K_{D,A}\), and \(K_{D,B}\) fixed, \(y\) is constant and \(x=[A]/K_{D,A}\). Differentiating \(\theta_B=y/(1+x+y)\) with respect to \([A]\) gives:

Thus increasing free \([A]\) cannot increase B’s equilibrium occupancy in this model; it lowers that occupancy whenever \(y>0\). The effect is an occupancy shift, not evidence that ligand A changes ligand B’s affinity. With B’s free concentration held fixed, \(\theta_A\) increases as \([A]\) increases, while the unoccupied fraction decreases. The size of the change depends on both ligands’ concentration-to-affinity ratios, not on concentration alone.

Predicting the effect of a concentration change

To predict a change, recalculate the shared denominator after changing the free concentration and hold other stated quantities fixed. Comparing the new and old occupancies helps distinguish a change in the competing ligand’s occupancy from a change in total receptor occupancy.

Sometimes the question is how much A is needed to achieve a specified occupancy despite B’s competition. Set \(\theta_A=q\), where \(q\) is the desired fractional occupancy, and solve the occupancy equation for \(x\). Provided \(0\leq q<1\):

The required free concentration increases with B’s dimensionless ratio \(y\). If \(q=1\), no finite concentration gives exactly complete occupancy in this equilibrium model; the occupancy approaches one as \([A]\) increases without bound.

Assumptions, checks, and alternatives

The formula is useful when the model assumptions match the question. It describes equilibrium binding of two ligands to a homogeneous receptor population with one shared site per receptor, mutually exclusive binding, and no additional binding states. Each ligand’s affinity is represented by a fixed equilibrium dissociation constant. The free concentrations must be known or adequately maintained while the receptor state fractions are evaluated.

For a proposed calculation, check that the two ligands really compete for the same site and that equilibrium is a defensible approximation. Check units as well: \([A]\) and \(K_{D,A}\) must use the same concentration units, as must \([B]\) and \(K_{D,B}\). The ratios \(x\) and \(y\) should be dimensionless. Finally, confirm that each calculated fraction lies between zero and one and that all three fractions sum to one.

The main alternatives address different conditions. Applying a one-ligand occupancy equation separately to A and B ignores competition: it can assign incompatible fractions whose sum exceeds one. If the relevant input is total ligand and depletion is appreciable, the free concentrations must instead be found using ligand mass balances together with receptor binding. The earlier material on ligand depletion establishes why free and total concentrations can differ; the shared-site occupancy equations still apply once the free concentrations are known. If concentrations change before equilibrium is reached, an equilibrium calculation does not describe the transient response; a time-dependent binding model is needed.

Common mistakes and interpretation

• Using total concentration as if it were free. This can give an incorrect occupancy when appreciable ligand is bound. Use free concentrations or solve the relevant mass balances.
• Calculating each ligand independently. Separate one-ligand calculations omit the shared denominator. The two occupancy fractions must be calculated together and cannot sum to more than one.
• Comparing concentrations without accounting for affinity. Competition is governed by \([A]/K_{D,A}\) and \([B]/K_{D,B}\), not by the raw concentrations alone.
• Calling an occupancy change an affinity change. Under this model, increasing A changes the fraction of receptors bound to B through competition; it does not change \(K_{D,B}\).
• Confusing total occupancy with occupancy by a particular ligand. Total occupancy can increase while B occupancy decreases, as in the worked concentration-change example.

This model is intentionally limited: it does not represent receptor subtypes, cooperativity, allosteric states, or non-equilibrium binding. Its value is that it makes the consequences of simple shared-site competition explicit and calculable. When a diagnostic check fails—for example, fractions do not sum to one, a required free concentration is unavailable, or the binding process is not near equilibrium—the model’s assumptions should be reconsidered before interpreting the result.