Effects of ligand depletion and NSB on saturation binding to two specific sites
The two sites modelfree generated errorless binding data using free [3H]EB as the independent variable to investigate how ligand depletion without NSB affected saturation binding behavior. Increasing the concentration of binding sites increased ligand depletion, shifted the total binding curve to the right, increased the steepness of the curve, and obscured the distinctive contour of the low affinity binding site (Figure 2A). The binding contour of the high affinity site began shifting noticeably to the right and showed an increasingly sharp bend at [3H]EB = R1T as R1T increased beyond Kd1 (0.013 nM) (Figure 2B). The binding contour of the low affinity site started shifting rightward as R1T approached Kd2 (12 nM). The rightward shift in the binding curves with ligand depletion means that relying on K0.5 as an estimate of Kd overestimates dissociation constants. Eq. (2)
correctly estimated Kd1 from the half-maximum for the high affinity site (K0.5, high) when K0.5, high was distinct [5, 50]. Eq. (2), however, became increasingly difficult to use as rightward shift of the binding curve from the high affinity site led to overlap with the binding curve from the low affinity site.
The two sites modelfree generated errorless binding data with both ligand depletion and NSB to investigate how combining NSB with ligand depletion affected binding behavior. The effect of NSB depended on the extent of ligand depletion. With negligible ligand depletion at R1T = 0.0001 nM (Figure 3A), NSB with α = 10-6 started obscuring the binding contour from the low affinity site. NSB with α = 10-3 obscured binding to the high affinity site. With significant ligand depletion at R1T = 0.3 nM (Figure 3B), NSB with α = 10-4 obscured the binding contour from the low affinity site. With extreme ligand depletion at R1T = 300 nM (Figure 3C), the contributions to total binding from the high affinity site and the low affinity site were not distinct even without NSB. NSB with α = 1 obscured specific binding to the high affinity site. Increasing ligand depletion also affected how NSB depended on total [3H]EB concentration. The leftward shift in NSB with each log unit increase in α was relatively uniform when ligand depletion was negligible at R1T = 10-4 nM (Figure 3D). The leftward shift in NSB with each log unit increase in α, however, became nonuniform when ligand depletion became large (Figure 3E; R1T = 0.3 nM) or extreme (Figure 3F; R1T = 300 nM).
Modeling specific binding and NSB as total binding
How can dissociation constants be estimated when both ligand depletion and NSB contribute significantly to [3H]EB binding? An effective approach when ligand depletion is negligible is to calculate specific binding as the difference between total binding and NSB measured without α4β2 nAChR (apparent NSB). In accord with a one binding site model including ligand depletion and NSB [19], this approach was incorrect when ligand depletion was significant (Figure 3G and 3H). NSB shifted rightward from the apparent NSB as R1T increased because increasing R1T decreased the free [3H]EB concentration for a given total concentration of [3H]EB. Subtracting apparent NSB from total binding led to calculated specific binding that was shifted rightward and downward compared to true specific binding (Figure 4A). This effect led to overestimating the values of Kd1 and Kd2 (Figure 4B and 4C), overestimating R1T (Figure 4D and 4E), and underestimating R2T (Figure 4D and 4E) as ligand depletion increased. The difference between total binding and NSB equals specific binding by definition. These results, however, showed NSB when α4β2 nAChR was present was not equal to NSB when α4β2 nAChR was absent (apparent NSB). Specific binding, therefore, did not equal the result of subtracting apparent NSB from total binding. This inequality arose because NSB with α4β2 nAChR present did not equal apparent NSB when ligand depletion was significant. This observation has been made previously for a one site model [19].
Specific binding and NSB of [3H]EB and α4β2 nAChR needed to be modeled together as total binding using the two sites modeltotal. This conclusion was consistent with the findings from a general one site model [19]. Accuracy of the two sites modeltotal for calculating saturation binding data was tested by comparing predicted [3H]EB binding to [3H]EB binding calculated with two sites modelfree. The concentration of bound [3H]EB calculated by the two methods agreed to at least fourteen significant digits across this range of parameters: 10-6 nM ≤ R1T ≤ 104 nM and 0 ≤ α ≤ 102 with 10-6 nM ≤ [3H]EBfree ≤ 106 nM. These results confirmed the accuracy of the binding calculations using the two sites modeltotal.
Potential for failing to identify low-affinity specific binding when modeling only total saturation binding
An important role for the two sites modeltotal is to estimate dissociation constants and binding site concentrations from noisy binding data. These estimates, however, are valid only when the two sites modeltotal fits data better than does the one site modeltotal according to statistical testing. Under what circumstances are binding data from the two sites of α4β2 nAChR adequately explained by the one site modeltotal? In these situations, specific binding to the low affinity site is indistinguishable from high-affinity specific binding, NSB, or noise. On the other hand, what circumstances favor identifying the low-affinity specific binding site?
Deriving a computational expression for NSB from the general expression for binding to a single site suggested potential confusion between low-affinity specific binding and NSB as defined in Figure 1 (symbols similar to Figure 1):
where α = (RTNSB/Kd, NSB) and RNSB = RTNSB by the definition of homogeneous NSB. If NSB arises from a collection of heterogeneous sites, then
By analogy with these derivations, binding to the low affinity site also can be modeled as constant*Lf, similar to NSB, when R2≃R2T. On the other hand, low-affinity specific binding behaves differently from NSB when the approximation R2≃R2T fails. This approximation most likely fails as total [3H]EB approaches its maximum concentration ([3H]EBmax) in a saturation binding experiment. In contrast and by definition, RNSB≃RTNSB is valid for NSB; and NSB equals α*Lf at any [3H]EBmax. When [3H]EBmax is sufficiently small that R2≃R2T is valid for the low-affinity specific binding site, the two sites modeltotal does not fit significantly better than one sites modeltotal. This outcome supports the incorrect conclusion that a second low affinity site is not present. These observations led to the hypothesis that modeling total saturation binding data with ligand depletion and NSB can blur the important biological distinction between low-affinity specific binding and NSB for [3H]EB and α4β2 nAChR.
Three approaches to characterizing the low-affinity specific binding site with saturation binding
To test this hypothesis, the one site modeltotal was compared to the two sites modeltotal by fitting noisy total binding data from the two sites modelfree with zero NSB (α = 0). The data (60 data points and 20 total concentrations of [3H]EB) with R1T = 0.13 nM and [3H]EBmax = 2 nM were generated with the two sites modelfree and an unrealistically large maximum signal-to-noise ratio (S/N) of 13,300 (SD = 1 × 10-5 nM). The two sites modeltotal fitted the data significantly better than the one site modeltotal (p values of 1.5 × 10-24, 2.2 × 10-22, and 1.3 × 10-20 for three trials). This result showed that fitting high precision total binding data with the two models identified low-affinity specific binding.
Reducing the precision of the data was expected to make detection of binding to the low affinity site more difficult. To test this expectation, binding data with the same R1T and [3H]EBmax were generated with a tenfold smaller but still unrealistically large maximum S/N of 1,330 (SD = 1 × 10-4 nM). Under these conditions, the two sites modeltotal did not fit the data significantly better than the one site modeltotal with five of five data sets (p = 0.33, 0.13, 0.24, 0.73, and 1.0). Fitting noisier data led to the misleading conclusion that only one specific binding site plus NSB satisfactorily accounted for the total binding data.
How can low-affinity specific binding be distinguished more reliably from NSB as S/N values decrease to realistic levels? Eqs. (3)-(5) suggested increasing [3H]EBmax so the approximation R2≃R2T no longer would be valid near [3H]EBmax. The approximation would break down because increased binding of [3H]EB to R2 at large values of [3H]EB would cause a significant decrease in R2 as [3H]EB approaches the increased value of [3H]EBmax. To determine whether increasing [3H]EBmax helped distinguish the low affinity binding site from NSB in the presence of ligand depletion, the one site modeltotal and the two sites modeltotal were fitted to noisy data with zero NSB and with [3H]EBmax increased from 2 nM (60 data points) to 5 nM (63 data points). The maximum S/N of the data again was 1,330 (SD = 1 × 10-4 nM). With [3H]EBmax = 5 nM, the two sites modeltotal fit better than the one site modeltotal in five of five data sets (p = 4.6 × 10-11, 1.8 × 10-9, 2.8 × 10-9, 2.5 × 10-9, and 1.7 × 10-12). Increasing the data points from 60 to 63 did not account for this improved detection of low-affinity specific binding. Instead, this result was consistent with a breakdown of the approximation R2≃R2T as [3H]EBmax increased, leading to better discernment of binding at the low affinity site at [3H]EBmax = 5 nM compared to 2 nM.
To explore whether larger values of [3H]EBmax could distinguish low-affinity specific binding from NSB in data with more realistic precision, the one site modeltotal and two sites modeltotal were fitted to noisy data (maximum S/N = 36; SD = 0.0041 nM) and zero NSB (α = 0) (Figure 5A). When [3H]EBmax was 10 nM, the two sites modeltotal usually did not fit the data better than the one site modeltotal. As [3H]EBmax increased, the likelihood of better fitting by the two sites modeltotal and the likelihood of support for the presence of the low affinity site also increased. At [3H]EBmax = 100 nM with fitting total binding data only, the two sites modeltotal fitted the data better than the one site modeltotal for all trials. The increase in data points with increasing [3H]EBmax did not account for this improved detection of low-affinity specific binding.
As a second potential approach, fitting apparent NSB while simultaneously fitting total binding data might help distinguish low-affinity specific binding from NSB by directly evaluating NSB. To test this hypothesis, total binding data (40 data points) and apparent NSB binding data (20 data points) were generated with the same conditions (maximum S/N = 1,300; SD = 1 × 10-4 nM) that failed to distinguish the low affinity binding site with total binding data only. Simultaneously fitting total binding data (Figure 5B) and apparent NSB (Figure 5C) led to the two sites modeltotal fitting the data significantly better than the one site modeltotal in five of five data sets. The p values were vanishingly small (p = 6.5 × 10-31, 7.3 × 10-34, 3.2 × 10-33, 1.3 × 10-28, and 2.1 × 10-33). Figure 5C shows how fitting apparent NSB led to better detection of low-affinity specific binding. The one site modeltotal could not fit total binding and simultaneously accurately fit the apparent NSB. In contrast, the two sites modeltotal accurately fit the contribution from the low affinity site to total binding and simultaneously accurately fit the apparent NSB. With more realistic precision (maximum S/N = 36; SD = 0.0041 nM), the two sites modeltotal usually fit the data better than did the one site modeltotal for [3H]EBmax ≥ 22 nM (Figure 5A). In addition, simultaneously fitting both total binding and apparent NSB data more reliably identified low-affinity specific binding than did fitting only total binding. These results suggested that simultaneously fitting both total binding and apparent NSB could be superior to fitting only total binding for detecting low-affinity specific binding when NSB was negligible.
A third approach for potentially distinguishing low-affinity specific binding from NSB is based on how NSB varies with α4β2 nAChR concentration. Suppose, in an idealized case, that NSB arises solely from sources (e. g., walls of a test tube, surface of a glass filter, or a constant volume of cell membranes) that are independent of α4β2 nAChR. The independence of NSB from α4β2 nAChR suggests the hypothesis that varying α4β2 nAChR concentration helps distinguish low-affinity specific binding from NSB when ligand depletion is significant. Variation in α4β2 nAChR concentration could arise by injecting different amounts of cRNA into oocytes or by transfecting different amounts of cDNA into cells. To test this hypothesis, the one site modeltotal and two sites modeltotal with implicit fitting of NSB were fitted to noisy [3H]EB binding data (maximum S/N = 36) generated at three different concentrations of α4β2 nAChR and with zero NSB (Figure 6A). The two sites modeltotal consistently fit the data better than the one site modeltotal for [3H]EBmax ≥ 22 nM (Figure 6B). In contrast, [3H]EBmax in the range of 100 nM was needed when the same numbers of data points were generated under similar conditions from a single α4β2 nAChR concentration (Figure 5A). These results suggested that simultaneous fitting of data from various α4β2 nAChR concentrations, when NSB is independent of α4β2 nAChR concentration, could help distinguish binding to the low affinity binding site better than fitting data from a single α4β2 nAChR concentration.
Potentially, both sources independent of α4β2 nAChR concentration and sources correlated with α4β2 nAChR concentration might contribute significantly to NSB. The equation describing NSB in this case needs to include a component independent of (RLNSB, indep) and a component dependent on α4β2 nAChR concentration (RLNSB, dep). Based on Eqs. (3)-(5) and if RTNSB, dep is directly proportional to α4β2 nAChR, the relationship between NSB and free ligand becomes:
Eq. (12) for NSBtotal or other expressions for RTNSB, dep can be incorporated into binding equations (Figure 1) when the low affinity binding site is investigated with various α4β2 nAChR concentrations and binding models.
Characterizing the low-affinity specific binding site by ligand depletion
How does combining NSB with ligand depletion affect the interpretation of saturation binding with ligand depletion? Without ligand depletion, large NSB tended to overwhelm the signal from the low affinity site when total and free [3H]EB were high enough to populate the low affinity binding site (Figure 3A). Conditions leading to ligand depletion, however, would increase the concentration of the low affinity site, reduce free [3H]EB and NSB, and lead to relatively more binding to the low affinity site than to NSB. With α = 0.1 and R1T = 0.00013 nM (negligible depletion), the ratio R2L/NSB was 1.1 × 10-5 at [3H]EB = 12 nM and 4.4 × 10-6 at [3H]EB = 50 nM. As expected, NSB overwhelmed the signal from the low affinity site at and above [3H]EB = Kd2, which was the minimal concentration range needed to significantly populate the low affinity site. In contrast, with R1T = 20 nM (substantial depletion) and the low affinity site starting to participate in ligand depletion, the ratio R2L/NSB was much larger: 3.2 at [3H]EB = 12 nM and 1.0 at [3H]EB = 50 nM.
To test this promising usefulness for ligand depletion, noisy data (maximum S/N = 50 at each R1T) with α = 0.1 and significant ligand depletion at three values of R1T (0.13, 3, and 20 nM; [3H]EBmax = 0.15, 3.6, and 24 nM) were fitted by the one site modeltotal and the two sites modeltotal. The two sites modeltotal fit the data better in ten of ten trials and produced CIs that included the true values for the parameters (Kd1 = 0.0133 nM, CI = 0.0120-0.0149 nM; Kd2 = 11.9 nM, CI = 9.0-15.8 nM; fraction of low affinity site = 0.180, CI = 0.156-0.204; α = 0.098, CI = 0.092-0.103). To test the effect of simultaneously fitting apparent NSB, noisy data (maximum S/N = 50) with α = 0.1 at three values of R1T (0 nM for apparent NSB alone, 0.13, and 20 nM) were fitted by the one site modeltotal and the two sites modeltotal. The two sites modeltotal fit the data better in ten of ten trials and produced CIs including the true values for the parameters (Kd1 = 0.0123 nM, CI = 0.0097-0.0156 nM; Kd2 = 31.8 nM, CI = 6.5-155 nM; fraction of low affinity site = 0.291, CI = 0.133-0.450; α = 0.0997, CI = 0.0987-0.101). These results suggested that increasing ligand depletion might be useful for detecting and characterizing the low affinity site when NSB is significant in saturation binding data.
Effects of ligand depletion and NSB on homologous competition
To investigate effects of ligand depletion and NSB on homologous competition, a two sites modelfree and a two sites modeltotal were developed using concentration of free or total cold EB as the independent variable (Figure 1B). Calculations of total binding using the two sites modeltotal agreed with calculations with two sites modelfree to at least fourteen significant digits. The ranges of parameters tested were 1 × 10-6 nM ≤ R1T ≤ 1 × 104 nM and 0 ≤ α ≤ 20 with 1 × 10-6 nM ≤ [3H]EBtotal ≤ 1 × 106 nM. These results confirmed the accuracy of modeling homologous competition using total cold EB concentration as the independent variable.
Increasing ligand depletion by increasing R1T changed the appearance of homologous competition data using 0.013 nM [3H]EB, which equaled the Kd for the high affinity binding site (Figure 7A). At R1T = 0.00013 nM, ligand depletion was negligible. The binding curve was symmetric about IC50 = 0.02612 nM with a Hill coefficient of -0.9995. The Kd calculated from a modified Cheng-Prusoff equation for homologous competition [51], which ignores ligand depletion:
was 0.01316 nM, close to the value of Kd for the high affinity site. Increasing ligand depletion distorted the competition curve away from a sigmoidal shape and shifted the curve rightward. The curve at R1T = 130 nM was asymmetric about IC50 = 306 nM and did not follow Eq. (13). When [3H]EB was increased to 13 nM, [3H]EB concentration controlled IC50 when ligand depletion was negligible, agreeing with Eq. (13) (Figure 7B). IC50, therefore, remained about 13 nM for R1T < 13 nM. Increasing ligand depletion shifted IC50 rightward when R1T ≥ 13 nM and made the homologous competition curves asymmetric around IC50. These results showed that increasing ligand depletion in homologous competition data shifted IC50 rightward and caused asymmetric curves around IC50.
As ligand depletion increased, its effect on binding to the high affinity site became qualitatively different from its effect on binding to the low affinity site. With negligible ligand depletion at R1T = 0.00013 nM and [3H]EB = 0.013 nM, homologous competition of [3H]EB binding to the high and low affinity sites produced similarly shaped sigmoidal competition curves (Figure 8A and 8B). With substantial ligand depletion at R1T = 130 nM and [3H]EB = 0.013 nM, [3H]EB binding to the high affinity site acquired a sharp shoulder but continued to decrease monotonically with increasing cold EB (Figure 8C). At the low affinity site, substantial ligand depletion produced an asymmetric peak of [3H]EB binding (Figure 8D).
Characterizing the low-affinity specific binding site with homologous competition when NSB is negligible
How well can homologous competition data with ligand depletion identify the low affinity binding site? Comparing fits from the one site modeltotal and two sites modeltotal to noisy data from a single [3H]EB concentration reliably achieved this goal only with highly precise data (maximum S/N = 1000) (Figure 9A). With 20 nM [3H]EB, [3H]EB occupied a large fraction (62%) of the low affinity binding site when cold EB was absent. The result with 20 nM [3H]EB and 0.13 nM R1T suggested that occupying both high and low affinity sites using one high [3H]EB concentration was insufficient to identify the low affinity site when S/N values were realistic. Figure 8C and 8D, however, suggested that combining concentrations of [3H]EB and binding sites on the order of Kd2 might lead to a distinctive concentration dependence of [3H]EB binding that would identify the low affinity binding site with less precise data. Indeed, concentrations of 20 nM [3H]EB and 20 nM R1T reliably achieved this goal with less precise data (maximum S/N = 50) (Figure 9A). These results suggested that homologous competition data from a single [3H]EB concentration could identify the low affinity binding site with realistically precise data using large concentrations of [3H]EB and α4β2 nAChR binding sites. This approach, however, consumed large amounts of [3H]EB and α4β2 nAChR.
Multiple concentrations of [3H]EB that explored a wide range of fractional occupancies of the two binding sites might identify the low affinity binding site while consuming less [3H]EB and α4β2 nAChR. Improving the interpretation of homologous competition data from two binding sites by using several concentrations of radioligand has been described for a general case [7]. To test this method with [3H]EB and α4β2 nAChR, homologous competition data sets from [3H]EB concentrations of 0.013, 0.3, and 20 nM and R1T = 0.13 nM were generated (Figure 9B-E). Multiple concentrations of [3H]EB required less precise data and consumed less [3H]EB and α4β2 nAChR to identify the low affinity site than did a single large [3H]EB concentration (Figure 9A).
Characterizing the low-affinity specific binding site with homologous competition when NSB is significant
In practice, NSB is not zero and needs to be included in a model of homologous competition data. NSB, as expected, moved the baseline above zero at large concentrations of cold EB. Increasing ligand depletion shifted IC50 rightward and distorted the monotonically decreasing sigmoidal shape of the competition curve (Figure 10A). As expected from modeling of one specific binding site [19], the contribution of NSB to total [3H]EB binding across the range of cold EB concentration depended on the extent of ligand depletion (Figure 10B). The dependence of NSB on ligand depletion showed that simply subtracting the baseline of bound [3H]EB at large cold EB concentration from total bound [3H]EB did not accurately calculate specifically bound [3H]EB. Instead, and similar to saturation binding with ligand depletion and NSB, interpreting properties of specific binding of homologous competition data with NSB needed fitting of total binding.
Homologous competition without NSB suggested simultaneously fitting data from several [3H]EB concentrations at a constant concentration of α4β2 nAChR better identified the low affinity site than did fitting data from a single [3H]EB concentration (Figure 9A). Applying this approach at 0.013, 0.3, and 20 nM [3H]EB to homologous competition with R1T = 0.13 nM and α = 0.1, however, revealed that NSB overwhelmed specific binding at 20 nM [3H]EB. 92% of total [3H]EB binding was NSB, 7% was bound to the high affinity site, and only 1% was bound to the low affinity site in the absence of cold EB.
As suggested by Figures 8 and 9A, concentrations of both [3H]EB and R1T on the order of Kd2 might help identify binding to the low affinity site. This method populates the low affinity site relative to the high affinity site and to NSB (Figure 11A). This method with [3H]EB and R1T at 20 nM identified binding to the low affinity site with five of five data sets at S/N = 50 and three of five data sets at S/N = 25 (Figure 11B). The consumption of a large concentration of [3H]EB and α4β2 nAChR at all data points, however, was an undesirable outcome.
To reduce [3H]EB and α4β2 nAChR consumption, both binding sites and [3H]EB were varied. This method could sample a wide range of fractional occupancies of the two binding sites, which suggested a potential advantage for interpreting binding to the specific sites (Figure 11A). The maximum fractional occupancies (R1L/R1T) of the high affinity site by [3H]EB were 0.089, 0.29, and 0.97 at [3]EB = 0.013, 0.3, and 20 nM and at R1T = 0.13, 1, and 20 nM. For the low affinity site, the maximum fractional occupancies (R2L/R2T) were 0.00081, 0.014, and 0.29. NSB made a greater fractional contribution to total binding than the low affinity site for all concentrations of cold EB when [3H]EB = 0.013 nM and R1T = 0.13 nM. With [3H]EB and R1T at 20 nM, however, [3H]EB binding by the low affinity site was greater than NSB up to 24 nM cold EB (Figure 11A). These results suggested this method might adequately sample the contribution by the low affinity site to total binding during fitting of noisy data when NSB was significant.
The method was tested by comparing one site modeltotal and two sites modeltotal fits to noisy data from three pairs of [3H]EB concentrations and binding site concentrations. The low affinity site was identified with five of five data sets with S/N = 50 and four of five data sets with S/N = 25 (Figure 11B). These results suggested that simultaneous fitting of homologous competition data from several concentrations of [3H]EB and binding sites has the potential to identify low-affinity specific binding in the presence of NSB.
Potential misinterpretation of low-affinity specific binding as NSB in homologous competition binding
Even with highly precise data, Eqs. (3) to (5) suggested a possibility of misinterpreting low-affinity specific binding as NSB in homologous competition data when only fitting total binding data. A low affinity, second specific binding site with a large relative concentration could mimic NSB as long as R2≈R2T over the range of cold EB concentration. Although a large relative concentration of the second binding site was not observed from expression of α4β2 nAChR in oocytes [18], such a condition potentially could arise in a different heterologous expression system. The potential for confusing low-affinity specific binding and NSB was explored by comparing homologous competition data from a one site modelfree with α = 0.2 with data from a two sites modelfree with α = 0 and Kd2 = R2T/0.2. As R2T and Kd2 increased, the upper limit of cold EB concentration for which R2 R2T remained valid also increased. The data from the two sites modelfree with zero NSB, therefore, displayed increasingly long plateaus mimicking NSB at large concentrations of cold EB. The long plateaus, however, arose from specific binding to the low affinity α4β2 nAChR binding site and not from NSB. Figure 12A suggested that homologous competition data at a single [3H]EB concentration might not distinguish binding to a low affinity site from NSB unless either the maximum concentration of cold EB exceeded Kd2 or NSB was measured without α4β2 nAChR.
Heterologous competition with ligand depletion and NSB
Homologous competition is a specific case of the more general case of heterologous competition, for which the dissociation constants of the radioligand and the heterologous competitor differ. For heterologous competition, identification of a low affinity site and estimates for dissociation constants for [3H]EB to high and low affinity sites typically are determined from saturation binding. In this case, inhibition constants (Ki1 and Ki2 in Figure 1) for the competitor and the concentration of binding sites are the only unknowns when fitting heterologous displacement data. This study focuses on how ligand depletion and NSB affects heterologous competition with high and low affinity binding sites of [3H]EB. In addition, this study investigates concentrations of [3H]EB and α4β2 nAChR that might facilitate studying the low affinity site.
To determine how ligand depletion without NSB affects heterologous competition with [3H]EB and α4β2 nAChR, competition data at increasing concentrations of binding sites were generated with nicotine as the competitor. The dissociation constants for nicotine were 0.84 nM for the high affinity site [18] and 775 nM for the low affinity site. The inhibition constant for nicotine at the low affinity site was assigned so that Ki2/Ki1 for nicotine = Kd2/Kd1 for EB. When ligand depletion was negligible, IC50 values varied only slightly with binding site concentration (Figure 13A-F). The Ki values derived from these IC50 values and the Cheng-Prusoff equation (Eq. (14)),
which assumes a single binding site without ligand depletion, were close to Ki1 for nicotine (0.90, 0.87, and 0.96 nM at 0.013, 0.3, and 20 nM [3H]EB and R1T = 0.00013 nM). As increasing ligand depletion shifted IC50 rightward (Figure 13A-F), the estimate of Ki from the Cheng-Prusoff equation no longer closely matched Ki1 for nicotine. The shape of the competition curve remained approximately sigmoidal with a Hill coefficient consistently near -1 at all levels of ligand depletion.
Although nicotine binds more weakly than [3H]EB to α4β2 nAChR, other ligands developed in the future, especially derivatives of EB, conceivably might bind more tightly than [3H]EB. To determine how ligand depletion affects heterologous competition with a superhigh affinity competitor, heterologous competition data were generated with two dissociation constants 100-fold tighter (1.3 × 10-4 and 0.12 nM) than the two dissociation constants for [3H]EB. When ligand depletion of [3H]EB was negligible, IC50 values were independent of binding site concentration and led to slightly high estimates of Ki1 (1.4 × 10-4 nM) using Eq. (14); Hill coefficients were about -1 (Figure 13G-L). Increasing ligand depletion shifted IC50 rightward and, in contrast to nicotine, shifted Hill coefficients to strongly negative values (for example, -35 with [3H]EB = 0.013 nM and R1T = 130 nM). These results showed the effect of ligand depletion on the Hill coefficient depended markedly on whether the competitor bound more tightly or less tightly than [3H]EB.
Ki2 for a competitor potentially can be estimated with procedures analogous to procedures investigated for homologous competition. To test the approach described in Figures 9 and 11 for homologous competition, noisy heterologous competition data for nicotine and [3H]EB with ligand depletion and NSB were fit with the two sitestotal model (Figure 14). A single 0.013 nM concentration of [3H]EB with R1T = 0.13 nM did not significantly populate the low affinity site (Figure 14A). That concentration combination produced reliable estimates of Ki2 only with highly precise data (maximum S/N ≥ 1000) (Figure 14C). At maximum S/N = 100, fits with competition by nicotine at the high and low affinity sites generally were not significantly better than fits with competition by nicotine at only the high affinity site (p > 0.05 for six of six trials). Similar to the findings in Figures 9 and 11, increasing ligand depletion and populating both the high and low affinity sites with larger concentrations of [3H]EB and α4β2 nAChR (Figure 14B) allowed more reliable estimates of Ki2 with less precise data (Figure 14C). At maximum S/N = 15 with this approach, fits with competition by nicotine at the two [3H]EB binding sites generally were significantly better than fits with competition by nicotine at only the high affinity site (0.007 < p < 5 × 10-10 for six of six trials). These results suggest that fitting data with large ligand depletion might identify the presence of nicotine competition at the low affinity site even if those data have a low S/N and an estimate of Ki2 has low precision.
Similar to homologous competition data (Figure 12A), low-affinity specific binding might be misinterpreted as NSB when fitting heterologous competition data with a model of total binding. To investigate this possibility with a nicotine-like inhibitor (Ki1 = 0.84 nM), heterologous depletion data from the one site modelfree with NSB (α = 0.2) were compared to data from the two sites modeltotal without NSB. With R2T = 2.4 nM and various values of Ki2, the two sites modeltotal produced a long plateau mimicking NSB (Figure 12B). The value of Ki2 at this constant value of R2T determined the length of the plateau along the x-axis. One log unit increase of the value of Ki2 lengthened the plateau of binding to the low affinity site by one log unit. A competitor binding more tightly than [3H]EB to the high affinity binding site produced similar results (Figure 12C). These results suggested that binding to the low affinity site might be identified as NSB at a single [3H]EB concentration unless either the maximum competitor concentration was greater than Ki2 or NSB was measured without α4β2 nAChR.
Characterizing high and low affinity binding sites when NSB of a heterologous competitor is unknown
The NSB of an unlabeled competitor is not measured by heterologous competition measurements and often is assumed to be zero. The true value of αcompetitor, therefore, presents a source of uncertainty about values of inhibition constants. This uncertainty was investigated by increasing values of αcompetitor while nicotine (Figure 15A) or a superhigh affinity competitor (Figure 15C) inhibited binding of [3H]EB to α4β2 nAChR. As the true value of αcompetitor for nicotine increased, apparent values of Ki1 (Ki1, app) and Ki2 (Ki2, app) also increased (Figure 15B). The contours of competition curves with the superhigh affinity competitor changed as αcompetitor increased (Figure 15C), in contrast to the constant contours with nicotine. The ratio Ki2, app/Ki1, app for the superhigh affinity competitor, however, was invariant as αcompetitor increased (Figure 15D). The invariance of Ki2, app/Ki1, app at the two binding sites of α4β2 nAChR is important because the ratio represents the difference in free energy of binding at the two binding sites. This difference reflects differences in the interactions between the competitor and binding sites and structural differences between the high and low affinity binding sites. This measured free energy difference is independent of αcompetitor.