Ligand binding assays at equilibrium: validation and interpretation

Abstract

The focus of this review paper is factors affecting data interpretation in ligand binding assays under equilibrium conditions. Protocols for determining K(d) (the equilibrium dissociation constant) and K(dA) (the equilibrium inhibitor constant) for receptor ligands are discussed. The basic theory describing the interaction of a radiotracer and an unlabelled competitor ligand with a receptor is developed. Inappropriate experimental design may result in ligand depletion and non-attainment of equilibrium, distorting the calculation of K(d) and K(dA) . Strategies, both theoretical and practical, will be given to avoid and correct such errors, thus leading to the determination of reliable values for these constants. In determining K(dA) from competition binding studies, two additional concepts are discussed. First, the necessity to measure an adequate specific binding signal from the bound radiotracer ligand limits the range of affinity constants that can be measured: a particular set of assay conditions may lead to an upper limit on the apparent affinity of unlabelled ligands. Second, an extension of the basic assay methodology can indicate whether the interaction between the tracer and a test ligand is mediated by a competitive or an allosteric mechanism. Finally, the review ends with a discussion of two factors that are often overlooked: buffer composition and the temperature at which the assay is conducted, and the impact these can have on affinity measurements and the understanding of drug interactions.

Figure 1

Saturation binding curve for a radioligand. Specific binding of [³H]oxytropium to membranes from CHO cells expressing the human muscarinic M₃ receptor is shown. Radioligand depletion was 13% at the lowest concentration tested. The incubation time was 2 h at room temperature in 20 mM HEPES buffer, pH 7.4. Atropine was used to define non-specific binding. (A) Saturation binding curve plotted on a linear scale. The full line is the fit of equation 1b to the data. The estimated K_d was 0.05 nM, and the B_Max 1.4 pmol·mg⁻¹ protein; (B) data from (A) replotted against log concentration (x-axis) to reveal the characteristic sigmoid concentration–response curve.

Figure 2

Kinetic studies of radioligand association and dissociation. Kinetic studies of the binding of [³H]oxytropium to M₃ mAChRs: (A) association time-courses using several concentrations of oxytropium spanning the K_d; the full lines are fits to equation 2, giving values of k_obs, the apparent association rate constant; (B) determination of the association rate constant (k_on) from the fits shown in 2(A); k_obs was plotted against the concentration of oxytropium and the slope determined by linear regression, giving an estimate for k_on of 2.1 × 10⁹ M⁻¹·min⁻¹ (3.5 × 10⁷ M⁻¹·s⁻¹); (C) dissociation of oxytropium (0.3 nM) initiated with 10 µM atropine; the full line is the fit to a single exponential, with k_off 0.08 min⁻¹, corresponding to t_1/2= 8.7 min. The ratio of k_off/k_on gives 3.8 × 10⁻¹¹ M (0.038 nM) for the K_d, in good agreement with the value from the direct saturation study (Figure 1).

Figure 3

The effect of radioligand depletion on the apparent affinity. EC₅₀ values for binding of the high-affinity radioligand [³H]NMS to M₃ mAChRs are plotted against half of the concentration of receptor binding sites added to the assay. The data are taken from Carter et al. (2007). Assay volumes ranged from 50 to 1750 µL, and additions of receptor preparation (4 pmol·mg⁻¹ protein) from 5 to 50 µg. The straight line shows a linear regression with a slope of 1.3 and y-intercept corresponding to K_d= 1.07 × 10⁻¹⁰ M. The crosses show the radioligand depletion at the EC₅₀.

Figure 4

The allosteric ternary complex model of receptor–ligand interactions. Two ligands, L and A, bind to the receptor separately to give binary complexes RL and AR, governed by dissociation constants K_dL and K_dA respectively. The simultaneous binding of the two ligands to form a ternary complex, ARL, is subject to a cooperativity factor, α. The thicker arrows delineate a competitive interaction mechanism, when α= 0.

Figure 5

An affinity ratio plot distinguishes a competitive from a negatively cooperative binding interaction. The plot is analogous to a Schild plot of log₁₀(dose ratio − 1) against log₁₀[competitor]. The dose ratio is the ratio of the dissociation constant of the radioligand measured in the presence of the modulator to that in its absence. pK_dA is 8.0.

Figure 6

Opposite time dependences of the apparent inhibition constants of a ‘fast’ and a ‘slow’ competitor. pIC₅₀ values for atropine (a fast competitor) and tiotropium (a slow competitor) were determined at room temperature by competition with [³H]NMS (3.5 × 10⁻¹⁰ M) for binding to M₃ mAChRs expressed in CHO cells. Incubation times ranged from 5 to 1200 min. The data are replotted from Dowling and Charlton (2006).

Figure 7

Choice of unlabelled ligand to define non-specific binding for a lipophilic radioligand. Binding of (−)[³H]quinuclidinyl benzilate (3 × 10⁻¹⁰ M), a tertiary amine muscarinic antagonist, was measured over 24 h using a recombinant M₁ muscarinic receptor construct (50 fmol·mL⁻¹) expressed in Escherichia coli spheroplast membranes suspended in 50 mM sodium phosphate, pH 8.0: (A) Inhibition of [³H]QNB binding by a quaternary amine antagonist, (−)N-methylscopolamine, could not be fitted by a single-site inhibition curve, but required the addition of about 10% of sites with ca. 50-fold lower affinity; (B) inhibition of [³H]QNB binding by scopolamine, the tertiary analogue of NMS, was adequately described by a single site model for a homogeneous set of binding sites.

Figure 8

Quality control checks on the radioligand: bindability. The equilibrium binding of (−)[³H]NMS to recombinant M₁ mAChRs was measured after 60 min at 30°C in 50 mM sodium phosphate, pH 8.0. Assay volume was 1.0 mL. (A) Determination of the fractional bindability, FrB, of a batch of [³H]NMS. Saturation binding assays (0.0078–2.4 nM [³H]NMS).were performed using a range of dilutions of an M₁ mAChR preparation initially containing 47.4 pmol·mL⁻¹ binding sites (13 pmol·mg⁻¹ protein). Non-specific binding was measured using atropine (10 µM). The data were globally fitted to equations 3b and 3c with the total ligand concentration calculated as [L_TB]=FrB*(LTdpm −Bgd)/(2220*SPact*V). The full lines show the set of fitted curves. FrB was estimated to be 0.75 ± 0.05. Restriction of FrB to 1.0 gave a poor fit to the data at the highest mAChR concentration (grey line). An F-test on the sum of squares of the weighted residuals gave P < 10⁻⁵ with respect to the unrestricted fit. pK_d was 10.10 ± 0.05. The coefficients of non-specific binding were: N_S, 3.7 × 10⁻⁴; N_R, 5.2 × 10⁻⁴ nM⁻¹. (B) Plot of 1/dpm bound versus 1/[R_T] for the lowest concentration of [³H]NMS. Comparison of the y-intercept with the inverse of LTdpm, the total radioactivity added to the assay (arrow), gave an estimated FrB of 0.83.

Figure 9

Quality control checks on the radioligand: specific radioactivity. Estimation of the specific radioactivity of a batch of [³H]NMS. Assay conditions were as in Figure 8. A direct saturation curve (A) was compared to a homologous competition curve (B) using a receptor concentration of 50 fmol·mL⁻¹. The full data set from (A) and (B) was analysed simultaneously, with FrB fixed at 0.745, giving an estimated specific radioactivity (SPact) of 67.4 ± 2.7 Ci·mmol⁻¹. The pK_d of NMS was 10.06 ± 0.04.

Figure 10

Probing the mechanism of interaction between a test ligand and the radiotracer using several concentrations of radioligand. Binding of [³H]NMS to wild-type and mutant M₁ mAChRs expressed in membranes from COS-7 cells was measured at 30°C for 2 h in 20 mM Na HEPES, 100 mM NaCl, 1 mM MgCl₂, pH 7.5. Concentrations of [³H]NMS equivalent to 0.2×, 2× and 10×K_d were used. Serial dilutions of the agonist 77-LH-28-1 in dimethyl sulphoxide were added to the assays (final DMSO concentration 1%). Non-specific binding was measured with 10⁻⁵ M scopolamine. Assays were performed in quadruplicate. Maximum radioligand depletion was 10%. Data were globally fitted to the allosteric ternary complex model equation 5a embedded in equation 3b, and modified to allow non-unitary slopes for the competition curves. (A) Wild-type receptor showing competitive inhibition; pK_dL ([³H]NMS), 9.74; pK_dA (77-LH-28-1), 6.54; nH 0.84; log(α) was fixed at −6. (B) Tyr82 Ala mutant showing allosteric inhibition: pK_dL 9.45; pK_dA 6.67; nH 0.50; log(α), −1.33.