Competitive Binding by Multiple Ligands

Why Competition Changes Occupancy

A ligand’s affinity and free concentration determine how strongly it occupies a target when it binds alone. When several ligands compete for the same binding site, however, each ligand’s occupancy also depends on the concentrations and affinities of the others. An inhibitor can therefore reduce the fraction of target bound by a drug even if the drug’s own free concentration and affinity remain unchanged.

The key calculation is a set of equilibrium fractions: the proportion of total target that is unbound and the proportion bound to each ligand. These fractions apply to one homogeneous target population with a single binding site shared exclusively by the ligands. They use free ligand concentrations. If appreciable target binding depletes the ligands, total concentrations cannot simply be substituted for free concentrations; the binding mass balance must also be considered, as in the earlier material on ligand depletion and the binding mass balance.

Deriving the Competition Fractions

At equilibrium, each ligand follows the mass-action definition of its dissociation constant:

Rearranging gives \([RL_i]=[R]C_i/K_{D,i}\). Since the target is either unbound or bound to exactly one of the mutually exclusive ligands, total target is the sum of those states:

Dividing each state concentration by \(R_T\) gives the occupancy fractions:

Each ratio \(C_i/K_{D,i}\) is dimensionless and represents ligand \(i\)’s relative contribution to binding in this equilibrium model. The common denominator means that raising one ligand’s ratio reduces the fractions available to all other states. The fractions are nonnegative and sum to one:

For two ligands, A and B, the bound fractions are \(C_A/K_{D,A}\) and \(C_B/K_{D,B}\), respectively, divided by \(1+C_A/K_{D,A}+C_B/K_{D,B}\). If B is absent, its ratio is zero and the expression for A reduces to the familiar single-ligand equilibrium occupancy \(C_A/(K_{D,A}+C_A)\). This limiting case is a useful check on the algebra.

How Much Inhibitor Halves the Competing Ligand’s Occupancy?

For a tracer or drug A competing with inhibitor I, define \(x=C_A/K_{D,A}\) and \(y=C_I/K_{D,I}\). The fraction of target bound to A is \(x/(1+x+y)\). In the absence of inhibitor, that fraction is \(x/(1+x)\). The inhibitor concentration that halves A’s bound fraction can be derived by equating the occupied fraction in the presence of inhibitor to one-half its no-inhibitor value:

This result assumes that the free concentration of A is held fixed while inhibitor is added, and that both free concentrations are known. It describes a halving of A’s bound fraction relative to its own no-inhibitor value; it does not mean that half of all target is bound by inhibitor. The inhibitor’s own occupancy at that concentration is obtained from the full competition formula.

More Than Two Competing Ligands

The same expression applies when any number of ligands compete for the site: add each ligand’s free-concentration-to-\(K_D\) ratio to the denominator, then use the ratio for the ligand of interest as its numerator. This makes it possible to account for background endogenous ligands or several co-administered compounds without treating their effects as separate, independent occupancies.

Assumptions, Diagnostics, and Alternatives

The formula is appropriate when the target is represented by a homogeneous population of single binding sites, binding is reversible and at equilibrium, and ligands are mutually exclusive at that site. The \(K_D\) values must describe the same target state and conditions as the concentrations used. The calculation also presumes that each ligand’s free concentration is known. When target binding materially changes free ligand concentrations, concentrations in the formula must be obtained using the binding mass balance rather than assumed equal to total concentrations.

Several checks help reveal a misuse or a poor fit. First, the calculated fractions must lie between zero and one and sum to one. Second, increasing one ligand’s free concentration while holding the others fixed must increase its own fraction and decrease the unbound fraction; it must also decrease the fractions of its competitors. Third, predicted displacement should be compatible with the measured conditions and the stated \(K_D\) values. If the model cannot account for observed occupancy across conditions, consider whether free concentrations were misrepresented, equilibrium was not attained, target sites are not homogeneous, or the ligands do not share a single exclusive site. A mismatch does not by itself identify which assumption failed.

This competitive model differs from an independent-site model. For distinct, noncompeting sites, binding at one site does not exclude binding at another, and simultaneous occupancy can occur; the mutually exclusive state accounting used above would then be inappropriate. Likewise, a system with cooperative binding or multiple target populations requires a different model rather than simply adding more terms to this denominator. The current calculation’s strength is its transparent equilibrium interpretation; its limitation is that it is not a general model for every binding mechanism.

A frequent mistake is to calculate each ligand’s single-ligand fraction \(C_i/(K_{D,i}+C_i)\) separately and treat those values as simultaneous occupancies. This ignores competition: those separate calculations each allocate the target as though the other ligands were absent, so their fractions can sum to more than one. The correction is to use one shared denominator containing every competing ligand’s \(C_i/K_{D,i}\) term.