Modeling Radioligand Displacement Assays

What a Displacement Curve Measures

A radioligand displacement assay measures how the amount of bound labeled ligand changes as increasing concentrations of an unlabeled competitor are added. The competitor’s effect is interpreted through its competition with the tracer for target binding sites. The measured signal, however, is not automatically equal to tracer bound at the target: it can also include nonspecific association and background. The model must connect these components before a displacement curve can support an affinity estimate.

Here, $L$ denotes the free concentration of radioligand, $I$ the free concentration of competing ligand, $K_{d,L}$ the equilibrium dissociation constant of the radioligand, and $K_i$ the equilibrium inhibition constant of the competitor at the same binding site. $B_{\max}$ is the concentration of available binding sites expressed in the assay’s binding units. Concentrations and affinity constants must use compatible units.

Definition: In a one-site equilibrium displacement model, the specifically bound radioligand is $$ B_{\mathrm{spec}}(I)=B_{\max}\frac{L/K_{d,L}}{1+L/K_{d,L}+I/K_i} =B_{\max}\frac{L}{K_{d,L}(1+I/K_i)+L}. $$ This model assumes that tracer and competitor bind reversibly and mutually exclusively to one class of sites, the assay has reached equilibrium, and the concentrations used in the equation are free concentrations. Nonspecific binding and background are additional signal components, not part of $B_{\mathrm{spec}}$.

The model predicts a decreasing curve: at $I=0$, tracer binding is $B_{\max}L/(K_{d,L}+L)$; as $I$ increases, the competitor occupies a larger share of the available sites and less tracer is specifically bound. This describes competition for target binding. It does not mean that the competitor necessarily reduces the total signal by the same fraction, because some signal may remain even when target-specific tracer binding is nearly eliminated.

From Bound Tracer to Measured Signal

A useful observation model separates target-specific signal from nonspecific binding and background. If the instrument response is proportional to bound radioligand, a simple form is

Y(I)=gB_{\mathrm{spec}}(I)+N(I)+C,

where $Y(I)$ is the measured signal, $g$ converts bound tracer into signal units, $N(I)$ is nonspecific signal, and $C$ is instrument or assay background. In an assay where nonspecific binding is approximately unchanged across competitor concentrations, $N(I)$ can be treated as a constant. That approximation should be checked rather than assumed automatically. For example, a competitor might alter tracer adsorption to the assay vessel or affect the measurement process.

Nonspecific binding is commonly estimated in a parallel condition containing a sufficiently high concentration of an unlabeled ligand that displaces tracer from the target. Signal remaining in that condition is treated as nonspecific binding plus background. The estimate is then subtracted from each total signal to obtain specific binding. This procedure relies on the blocking condition actually displacing target-bound tracer and on the nonspecific component being comparable between conditions.

For a displacement curve, define $B_0$ as specific tracer binding without competitor and $B(I)$ as specific binding at competitor concentration $I$. The fraction of control binding remaining is $B(I)/B_0$. The concentration $IC_{50}$ is the competitor concentration at which this fraction is one-half under the particular assay conditions. It is not generally equal to $K_i$, and it should be defined using nonspecific-corrected binding rather than the uncorrected total signal.

Deriving the IC50–Ki Relationship

Under the one-site equilibrium model, the fraction of control specific binding remaining is found by dividing the expression for $B_{\mathrm{spec}}(I)$ by that for $B_{\mathrm{spec}}(0)$. The common factors $B_{\max}$ and $L$ cancel:

\frac{B_{\mathrm{spec}}(I)}{B_{\mathrm{spec}}(0)} =\frac{K_{d,L}+L}{K_{d,L}(1+I/K_i)+L}.

At $I=IC_{50}$, this ratio is one-half. Equating the denominator to twice the numerator gives

K_{d,L}\left(1+\frac{IC_{50}}{K_i}\right)+L =2(K_{d,L}+L).

Subtracting $K_{d,L}+L$ from both sides yields $K_{d,L}IC_{50}/K_i=K_{d,L}+L$. Rearranging gives the Cheng–Prusoff relationship:

K_i=\frac{IC_{50}}{1+L/K_{d,L}}, \qquad IC_{50}=K_i\left(1+\frac{L}{K_{d,L}}\right).

Thus, for a fixed competitor, the observed $IC_{50}$ depends on both competitor affinity and the radioligand concentration relative to its $K_{d,L}$. The conversion is justified only when the equilibrium competition assumptions hold and the stated $L$ and $K_{d,L}$ describe the assay. In particular, $L$ must be the free tracer concentration, not merely the amount added when depletion makes those concentrations differ.

Worked Example: Predicting Tracer Binding at a Competitor Concentration

Hypothetical assay. Suppose $B_{\max}=120$ fmol/mg, the free radioligand concentration is $L=2$ nM, $K_{d,L}=4$ nM, and the free competitor concentration is $I=12$ nM with $K_i=3$ nM. Assume one class of mutually exclusive sites, equilibrium, and negligible ligand depletion.

With no competitor, predicted specific binding is

B_{\mathrm{spec}}(0)=120\frac{2}{4+2}=40.0\ \text{fmol/mg}.

At $I=12$ nM, the competitor-to-affinity ratio is $I/K_i=12/3=4$. Therefore,

B_{\mathrm{spec}}(12)=120\frac{2}{4(1+4)+2} =120\frac{2}{22} =10.91\ \text{fmol/mg}.

The predicted fraction of control binding remaining is $10.91/40.0=0.2728$, or about 27.3%. This agrees with the ratio formula: $(4+2)/[4(1+4)+2]=6/22=0.2727$, with the small difference due to rounding.

Worked Example: Correcting a Displacement Curve for Nonspecific Signal

Hypothetical measurements. An assay produces the following total counts per minute (cpm). A parallel condition with excess unlabeled ligand estimates nonspecific signal plus background as 120 cpm. Assume this 120-cpm component is constant across competitor concentrations.

Competitor $I$ (nM)	Total signal (cpm)	Specific signal after subtraction (cpm)
0	920	800
2.5	760	640
10	520	400
40	280	160

For example, at 10 nM competitor the specific signal is $520-120=400$ cpm. Relative to the no-competitor specific signal, the fraction remaining is $400/800=0.500$. Thus, the hypothetical curve has $IC_{50}=10$ nM. Notice that the total signal at this point is 520 cpm, not half the total control signal: half of 920 cpm is 460 cpm. Using total signal without accounting for the 120-cpm floor would therefore misidentify the midpoint.

Suppose the assay used free tracer concentration $L=16$ nM and its dissociation constant is $K_{d,L}=4$ nM. The competition correction is $1+L/K_{d,L}=1+16/4=5$. The inferred competitor affinity is

K_i=\frac{10\ \text{nM}}{5}=2.00\ \text{nM}.

This estimate is conditional on the equilibrium, one-site and free-concentration assumptions. The arithmetic is exact for the hypothetical values given; real measurements would generally require estimating the curve from all observed concentrations rather than relying on a single midpoint observation.

Estimating Affinity Across Assay Conditions

The dependence of $IC_{50}$ on tracer conditions is important when comparing results. If the same competitor is tested with different tracer concentrations or tracers with different $K_d$ values, the raw $IC_{50}$ values can differ even when the underlying competitor $K_i$ is unchanged. Applying the correction separately can reveal whether the results are consistent with a common affinity under the model.

Worked Example: Comparing Two Tracer Conditions

Hypothetical results. The same competitor is tested in two equilibrium assays, with free concentrations and tracer affinities known. Assay A has $L=2$ nM and $K_{d,L}=2$ nM, and gives $IC_{50}=10$ nM. Assay B has $L=6$ nM and $K_{d,L}=3$ nM, and gives $IC_{50}=15$ nM. Assume each assay meets the one-site competition assumptions and has adequate nonspecific correction.

For Assay A, the correction factor is $1+2/2=2$, so $K_i=10/2=5.00$ nM. For Assay B, the correction factor is $1+6/3=3$, so $K_i=15/3=5.00$ nM. The two raw $IC_{50}$ values differ, but both imply the same $K_i$ under the model. The difference is explained by the different tracer conditions rather than by a different competitor affinity.

For estimation, one can fit the mechanistic equation directly to nonspecific-corrected binding measurements, estimating $K_i$ while supplying or estimating appropriate tracer parameters. Alternatively, one can estimate an $IC_{50}$ from the displacement curve and apply the correction when its assumptions are defensible. Direct fitting preserves the model’s concentration dependence, whereas a curve summarized only by its midpoint can discard information and conceal whether the curve shape is compatible with the assumed model.

Assumptions, Diagnostics, and Alternatives

Before interpreting a fitted curve, check the conditions that give the equation its meaning. Equilibrium must be reached at the time of measurement; tracer and competitor must compete for the same mutually exclusive site; and a single affinity class must be an adequate representation for the assay. The free concentrations should be known or reasonably approximated by the added concentrations. When binding appreciably depletes either ligand, free and total concentrations differ, and a mass-balance treatment is needed rather than substitution of added concentration into the simple equation.

The nonspecific estimate also needs scrutiny. A high concentration of unlabeled ligand should be sufficient to suppress specific tracer binding, and the resulting residual signal should represent the nonspecific and background components relevant to the displacement wells. A residual signal that changes with competitor concentration, or a poorly defined no-competitor plateau, makes the corrected midpoint unreliable. Replicate measurements and adequate concentrations on both sides of the midpoint help distinguish a well-supported curve from one whose $IC_{50}$ is largely extrapolated.

Useful warning signs include a non-flat lower plateau after nonspecific correction, a curve that does not decline across the tested range, or systematic differences between observations and the fitted one-site curve. These can arise from insufficient competitor range, incomplete equilibration, concentration or signal errors, ligand depletion, or a binding system more complex than the model. A steep or shallow curve alone does not establish a particular mechanism; it indicates that the one-site model should be checked rather than treated as automatically correct.

The main alternatives answer somewhat different questions. A saturation-binding assay varies radioligand concentration to characterize tracer binding, while a displacement assay holds tracer conditions fixed and varies competitor concentration. An empirical sigmoidal curve can summarize a response and provide a midpoint without claiming the one-site equilibrium mechanism, but its midpoint is not thereby a mechanistic $K_i$. If the assay has not reached equilibrium, a kinetic binding model is more appropriate than an equilibrium conversion. The appropriate choice depends on the assay design and the scientific quantity sought.

A frequent mistake is to call the measured $IC_{50}$ the competitor’s $K_i$, or to take half of the uncorrected total signal as the displacement midpoint. The correction follows from competition in the presence of tracer, and the midpoint applies to specific binding after accounting for nonspecific signal. These are separate issues: subtracting nonspecific binding gives the relevant curve, while the Cheng–Prusoff relationship adjusts its midpoint for tracer concentration and affinity.

Key takeaway: A displacement curve estimates competitor affinity only through a stated binding model. Correct nonspecific signal first, define $IC_{50}$ on specific binding, and apply the tracer-dependent correction only when equilibrium competition and free-concentration assumptions are credible.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A curve has $IC_{50}=18$ nM, with free tracer $L=3$ nM and $K_{d,L}=3$ nM. Under the equilibrium competition model, what is $K_i$?
Why should the midpoint be calculated from nonspecific-corrected binding rather than total signal?
If tracer concentration is increased while competitor affinity stays constant, what happens to $IC_{50}$ under the model?
Name two conditions that would make it inappropriate to use the simple Cheng–Prusoff correction.

Solutions. 1. The correction factor is $1+3/3=2$, so $K_i=18/2=9.00$ nM. 2. Total signal includes nonspecific binding and background, which can remain when specific binding is half-displaced; using total signal can therefore locate the wrong midpoint. 3. The model gives $IC_{50}=K_i(1+L/K_{d,L})$, so increasing $L$ while $K_{d,L}$ and $K_i$ remain fixed increases $IC_{50}$. 4. Examples include failure to reach equilibrium, appreciable ligand depletion when free concentrations are not used, and competition that does not follow a single mutually exclusive binding-site model.