Radioligand Binding Assays: Kd, Bmax and Competition

A radioligand binding assay is asked for two numbers: how tightly a labeled molecule holds to a site, and how many such sites there are. The measurement is a subtraction between two counts, made on a system assumed to have settled, at a ligand concentration assumed to be the one written on the tube. Each of those three assumptions fails in a characteristic way, and each failure moves the answer in a predictable direction.

This guide covers the saturation and competition formats, what the specific-binding subtraction actually removes, the Cheng-Prusoff conversion and its assumptions, and why an incubation that looks long enough usually is not.

Two formats asking two different questions

Three basic radioligand protocols are in general use: saturation, indirect binding (competition, displacement or modulation), and kinetic binding. In one account written for G protein-coupled receptors, these three between them are used to assess receptor expression as Bmax, equilibrium dissociation constants for the radioligand (Kd) and for nonradioactive ligands (Ki), association and dissociation rates, and to distinguish competitive from allosteric mechanisms ^[1].

In a saturation experiment the labeled ligand concentration is varied and bound label is measured once the system has settled. Specific binding as a function of free ligand is fitted to a rectangular hyperbola, B = Bmax x [L] / (Kd + [L]). Kd is the free concentration at which half the sites are occupied; Bmax is the extrapolated plateau. One protocol built around an iodine-125-labeled ligand and cell saturation binding sets out the classic equation with a graphical method for extracting Kd and Bmax, and its authors characterize the format as a gold standard for quantifying ligand-receptor interactions ^[2].

The word extrapolated is doing real work there. Occupancy at 3 x Kd is 3 / (1 + 3) = 0.75, so a curve topping out at three times Kd has reached only 75% of the plateau it is being asked to define; reaching 90% requires 9 / (1 + 9), a top concentration of nine times Kd. A saturation curve often looks flat because the points are widely spaced, not because the sites are full.

In a competition experiment the labeled ligand is held at one fixed concentration and an unlabeled compound titrated against it. The output is an IC50: the competitor concentration that halves specific binding of the tracer. An IC50 is not an affinity. It is a property of that curve at that tracer concentration, and converting it into an affinity constant requires the arithmetic below.

Specific binding is a subtraction, and subtractions cost precision

Total binding is measured in one set of tubes, nonspecific binding in a parallel set containing a large excess of unlabeled ligand, and specific binding is the difference. The excess is conventionally a hundredfold to a thousandfold over the tracer, and the blocker is preferably structurally distinct from the tracer so that a shared off-target site is not silently subtracted away. Both are bench convention rather than findings from a primary study.

What the blank removes is not a single phenomenon. A two-component treatment of lipophilic radioligand binding to tissue homogenate describes the specific component as binding to a single uniform population of sites, and the nonspecific component as partitioning into the membrane together with entrapment of drug present in the aqueous phase before the particulate fraction is separated ^[3]. That analysis makes a subtler point too: capacity can be misestimated when access of a fraction of the radioligand to the receptor is blocked by the unlabeled drug, and that fraction then presents itself as nonspecific binding ^[3].

The cost of the subtraction is precision. Radioactive decay is a counting process, so the standard deviation on a count of N is approximately the square root of N; that is standard counting statistics, stated here without citation. Suppose total binding reads 4,800 counts and the nonspecific blank reads 3,600. Specific binding is 4,800 - 3,600 = 1,200 counts, which is 1,200 / 4,800 = 25% of total. The standard deviation of the difference is the square root of (4,800 + 3,600) = the square root of 8,400 = 91.7 counts, so the relative uncertainty on the specific signal from counting alone is 91.7 / 1,200 = 7.6%.

Now hold total binding at 4,800 and suppose the blank is only 480. Specific binding is 4,320, the standard deviation of the difference is the square root of 5,280 = 72.7, and the relative uncertainty is 72.7 / 4,320 = 1.7%. Same tracer, same counter, same count time, and the precision differs by a factor of roughly four and a half. A reported Kd should be read alongside the specific-to-total ratio at the concentrations used.

Depletion: when the x-axis is not the concentration at the receptor

The hyperbola is written in terms of free ligand. Experiments are set up in terms of added ligand. Those agree only when very little of the added tracer ends up bound.

The usual working threshold is that bound radioactivity above roughly 10% of total added radioligand constitutes depletion and compromises the accuracy of Kd and Ki, and that it can be limited by adjusting receptor concentration and by radioligand choice ^[1]. The underlying design rule is to keep the receptor concentration substantially, at least tenfold, below the Kd of the radioligand and the Ki of the test compound. One methods paper calls this a golden rule that is difficult to adhere to with ultra-high-affinity radioligands: with low-specific-activity tritiated tracers routinely applied at or below their Kd, the very small amounts of receptor protein the rule demands give low levels of bound radioactivity, while the alternative of larger assay volumes makes 96-well filtration devices difficult to apply ^[4].

Work the bias through. If 12% of the added tracer is bound, free ligand is 0.88 of total. At a point on the curve where 12% of the added tracer is bound, plotting against total rather than free places that point at a concentration (1 - 0.88) / 0.88 = 0.12 / 0.88 = 13.6% higher than the concentration the sites actually experienced. The offset is not the same at every point, because the bound fraction is largest where ligand is scarce relative to sites and falls as tracer is raised, so depletion bends the curve as well as displacing it. The fitted Kd inherits that error in the same direction: it comes out too high, so affinity is understated. Note which value is the denominator. The error is expressed relative to the free concentration, because free is the quantity the model refers to.

The artifact has been shown directly for a lipophilic tracer: both the apparent capacity and the apparent affinity of dopamine D2 receptors labeled by tritiated spiperone were found to depend on the concentration of tissue used, with a reported dissociation constant of about 50 pM, a membrane-to-buffer partition coefficient of 410, and artifacts negligible below 0.1 mg of protein per mL of incubate in that specific example ^[3]. The authors also note that with filtration, removal of an indeterminate amount of radioligand during washing precludes correction based on the partition coefficient, and that about 30% of total protein, with a comparable percentage of receptors, was lost on washing ^[3].

Bmax carries a second dependency unrelated to binding. It is reported per milligram of protein, so its denominator comes from a separate protein assay. If that assay overstates protein by 15%, the reported Bmax is B / 1.15 = 0.870 of the true value, understated by 13.0%. Two laboratories using different protein assays can disagree about Bmax while agreeing about binding.

Fitting: what the linear plots cost

Scatchard and other linearizations turn a hyperbola into a straight line, and in doing so transform the error structure into one that least squares no longer suits. Comparing six calculation techniques, four linear and two nonlinear, on a constructed data set into which randomized errors had been introduced to varying extents, one evaluation concluded that nonlinear evaluation gives more reliable results, particularly when the errors are large ^[5]. Linearizations now serve mainly to display data, not to estimate parameters.

That evaluation raises a point which survives the choice of method entirely: an inevitable correlation between Bmax and Kd exists regardless of technique, which makes simultaneous changes in both parameters open to misinterpretation ^[5]. A report in which Kd and Bmax rose together has not necessarily observed two things. It may have observed one poorly determined plateau. Confidence intervals on both, and the correlation between them, belong in the report.

Cheng-Prusoff and the conditions it assumes

The relationship between an inhibition constant and the inhibitor concentration producing 50% inhibition was set out by Cheng and Prusoff for enzymatic reactions ^[6]. Applied to competitive radioligand displacement, the familiar form is Ki = IC50 / (1 + [L] / Kd), where [L] is the free concentration of the tracer and Kd is the tracer's own dissociation constant measured under the same conditions.

A worked example. Suppose an IC50 of 12 nM is measured with tracer at 0.5 nM and the tracer Kd is taken as 0.25 nM. Then 1 + 0.5 / 0.25 = 3, and Ki = 12 / 3 = 4 nM. Now suppose the true tracer Kd was 0.20 nM, so the value used was (0.25 - 0.20) / 0.20 = 0.05 / 0.20 = 25% high. The correct denominator is 1 + 0.5 / 0.20 = 3.5 and the correct Ki is 12 / 3.5 = 3.43 nM. The reported 4 nM is then high by (4 - 3.4286) / 3.4286 = 0.5714 / 3.4286 = 16.7%, that is exactly one sixth: a 25% error in the tracer Kd has propagated into a 16.7% error in Ki. A Ki is never better characterized than the Kd used to correct it, and that Kd must come from the same buffer, temperature and preparation.

The conversion assumes:

• competitive interaction at a single class of sites;
• attainment of equilibrium;
• negligible depletion of both tracer and competitor;
• knowledge of the free tracer concentration rather than the added one.

It is also, contrary to common belief, not exact. Munson and Rodbard showed that the correction is an approximation, that under many commonly used experimental conditions it can produce a 3- to 6-fold or larger error in Ki, and that a relatively simple exact rule is available in its place ^[7].

A separate criticism concerns the transfer of the relationship out of biochemistry. Writing about functional experiments rather than binding assays, Craig argued that the widespread perception that antagonist Kb values can only be obtained from inhibition curves via Cheng-Prusoff is mistaken, and that a related equation requiring no knowledge of agonist affinity is the theoretically valid one for that setting ^[8]. Leff and Dougall extended those arguments, describing further ways in which applying the method to pharmacological experiments is likely to produce misinterpretation and erroneous affinity estimates ^[9]. Both papers concern functional inhibition curves and neither overturns the binding-assay use of the equation, but they are a reminder that an equation carries its derivation with it.

Equilibrium, and why the incubation is usually too short

Every equation above assumes the system has stopped changing. Inappropriate experimental design may result in ligand depletion and non-attainment of equilibrium, distorting the calculation of both the dissociation constant and the inhibitor constant ^[10]. Equilibration time for high-affinity ligands with slow dissociation rates may require much extended incubation times or increased incubation temperature ^[1].

The kinetics of competitive binding were solved from the law of mass action by Motulsky and Mahan, whose analysis addressed how long such an incubation takes to equilibrate and how the IC50 varies over time ^[11]. Their answers depend largely on the relative dissociation rate constants of tracer and competitor. When the competitor dissociates from the receptors more rapidly than the radioligand, the IC50 first decreases and then increases, though it never falls below the Ki; when the competitor dissociates more slowly, the time required to equilibrate depends only on the dissociation rate constant of the competitor, and the IC50 decreases over time ^[11]. At low radioligand concentrations, equilibrium is reached in the same amount of time the radioligand requires to dissociate completely as determined in an off-rate experiment, and at higher radioligand concentrations that time is halved ^[11].

So a competition experiment can produce a smooth, reproducible, well-fitted curve whose midpoint is the wrong number, with nothing in the curve announcing it. The tell is not the fit. It is running the same experiment at two incubation times and finding the IC50 has moved.

Arithmetic makes the scale concrete, for the low-tracer case in which equilibration tracks the tracer's own dissociation ^[11]. If a tracer had a dissociation rate constant of 0.01 per minute, its dissociation half-life would be ln(2) / 0.01 = 0.693 / 0.01 = 69.3 minutes. Reaching 95% of the way to equilibrium takes ln(20) / ln(2) = 2.996 / 0.693 = 4.32 half-lives, that is 4.32 x 69.3 = 299 minutes, close to five hours; a 60-minute incubation would be under one half-life. The rate constant is a stated hypothetical, and the remedy is to measure the off-rate rather than assume it.

One assumption inside that framework is invisible in practice. The Motulsky and Mahan model is a one-state model, assuming the labeled ligand has no preference among distinct states or conformations of the target; where a radioligand displays biphasic association kinetics the one-state model is not applicable, which is what motivated an extension to a two-state model, exemplified for ligands at the adenosine A1 receptor ^[12]. A monophasic-looking association curve is weak evidence for a single state, not proof.

What a reported Kd does and does not tell you

A dissociation constant is a conditional number: it belongs to a buffer, an ionic composition, a temperature, a preparation, and a separation method. Buffer composition and assay temperature are described as two often-overlooked factors that affect affinity measurements and the understanding of ligand interactions ^[10]. A Kd quoted without them is incomplete, in the way a purity percentage quoted without its chromatographic method is incomplete.

There is also a ceiling on what the format can resolve. The need to measure an adequate specific binding signal from the bound tracer limits the range of affinity constants that can be measured, so a particular set of assay conditions may impose an upper limit on the apparent affinity of unlabeled ligands ^[10]. A table of Ki values clustered near the tracer concentration should be read as the assay reporting its own floor, not as a set of compounds that happen to share an affinity.

Finally, competition experiments do not by themselves establish mechanism. An extension of the basic assay methodology can indicate whether the interaction between tracer and test ligand is competitive or allosteric ^[10], and distinguishing the two is listed among the purposes of the three protocols taken together ^[1]. A displacement curve alone is consistent with both. What is unsettled in routine practice is less the theory than its uptake: corrections for depletion and for non-equilibrium have existed for decades ^{[7, 11]}, yet the approximate forms remain in routine use, and a paper that reports a Ki without saying which form it used, and why the approximation holds in that system, has left the question open.