Characterizing low affinity epibatidine binding to α4β2 nicotinic acetylcholine receptors with ligand depletion and nonspecific binding
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Along with high affinity binding of epibatidine (Kd1≈10 pM) to α4β2 nicotinic acetylcholine receptor (nAChR), low affinity binding of epibatidine (Kd2≈1-10 nM) to an independent binding site has been reported. Studying this low affinity binding is important because it might contribute understanding about the structure and synthesis of α4β2 nAChR. The binding behavior of epibatidine and α4β2 AChR raises a question about interpreting binding data from two independent sites with ligand depletion and nonspecific binding, both of which can affect equilibrium binding of [3H]epibatidine and α4β2 nAChR. If modeled incorrectly, ligand depletion and nonspecific binding lead to inaccurate estimates of binding constants. Fitting total equilibrium binding as a function of total ligand accurately characterizes a single site with ligand depletion and nonspecific binding. The goal of this study was to determine whether this approach is sufficient with two independent high and low affinity sites.
Computer simulations of binding revealed complexities beyond fitting total binding for characterizing the second, low affinity site of α4β2 nAChR. First, distinguishing low-affinity specific binding from nonspecific binding was a potential problem with saturation data. Varying the maximum concentration of [3H]epibatidine, simultaneously fitting independently measured nonspecific binding, and varying α4β2 nAChR concentration were effective remedies. Second, ligand depletion helped identify the low affinity site when nonspecific binding was significant in saturation or competition data, contrary to a common belief that ligand depletion always is detrimental. Third, measuring nonspecific binding without α4β2 nAChR distinguished better between nonspecific binding and low-affinity specific binding under some circumstances of competitive binding than did presuming nonspecific binding to be residual [3H]epibatidine binding after adding a large concentration of cold competitor. Fourth, nonspecific binding of a heterologous competitor changed estimates of high and low inhibition constants but did not change the ratio of those estimates.
Investigating the low affinity site of α4β2 nAChR with equilibrium binding when ligand depletion and nonspecific binding are present likely needs special attention to experimental design and data interpretation beyond fitting total binding data. Manipulation of maximum ligand and receptor concentrations and intentionally increasing ligand depletion are potentially helpful approaches.
KeywordsNicotine Binding Data Total Binding Equilibrium Binding Saturation Binding
95% confidence interval
nicotinic acetylcholine receptor
signal to noise ratio
Ligand depletion can significantly affect estimates for dissociation (Kd) or inhibition (Ki) constants from equilibrium binding data of epibatidine (EB) and α4β2 nicotinic acetylcholine receptor (nAChR) because of the high affinity of EB (Kd1≈10 pM). Errors from ligand depletion arise from inappropriately assuming that free ligand concentration equals total ligand concentration while using total ligand concentration as the independent variable for modeling the binding data. The assumption is attractive because total ligand concentration as the independent variable is suitable for least squares fitting of binding data [1, 2]. Ligand depletion can be minimized when designing binding experiments with EB and α4β2 nAChR. Radiolabeled EB with higher specific activity (for example, 125I instead of 3H) can lead to less ligand depletion by allowing a smaller concentration of α4β2 nAChR to produce useful data. A larger reaction volume at a fixed mole quantity of α4β2 nAChR reduces ligand depletion by reducing the difference between free and total concentration of radiolabeled EB. These avoidance strategies based on design of experiments, however, might be difficult to use in some situations. For example, a newly developed and 3H-labeled EB derivative might be available only with low specific activity. Large reaction volumes might be impractical for numerous samples associated with high throughput screening . When ligand depletion cannot easily be avoided, how can data with both ligand depletion and nonspecific binding (NSB) be correctly interpreted from EB and α4β2 nAChR?
Effects of ligand depletion on binding data have long been recognized, leading to models that correctly include ligand depletion with single and multiple specific binding sites [3, 4, 5, 6, 7, 8, 9]. For [3H]EB, a ligand with relatively low specific activity, and α4β2 nAChR, ligand depletion has been recognized and avoided as a potentially confounding factor for interpreting binding data [10, 11, 12, 13, 14, 15, 16, 17]. Alternatively, one site and two sites models for estimating binding constants have included ligand depletion with negligible NSB . Combining ligand depletion and NSB, however, imposes additional demands on binding models. For example, specific binding cannot be calculated simply by subtracting NSB from total binding. Instead, a binding model including both ligand depletion and NSB must fit total binding  as has been shown with one specific binding site . In addition to the high potency or high affinity site, functional data from electrophysiology and 86Rb+ flux [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41] and binding data [12, 18, 20, 42, 43] for α4β2 or α4β2-containing nAChR suggest a second, low potency or low affinity specific binding site. The difference in agonist potency at the two sites in functional assays has been attributed to α4β2 nAChR with different α:β stoichiometries [21, 25, 28, 30, 33]. (α4)2(β2)3 contributes high potency and (α4)3(β2)2 contributes low potency. Binding data from our laboratory suggest two independent sites and not two cooperative sites . The physical basis of low affinity equilibrium binding of [3H]EB detected under some conditions and the relationship between the low affinity site observed by equilibrium binding and the low potency site observed by functional methods are not known. On the other hand, a single binding site has been suggested for α4β2 nAChR [12, 14] and for extracellular domain α4β2 nAChR  from binding data. Photolabeling of α4β2 with [3H]EB also identified a single binding site . Models of α4β2 nAChR binding data, therefore, should not assume the presence of high and low affinity sites. Instead, an interpretation of binding data needs to test the hypothesis of one binding site versus more than one binding site.
How does interpreting binding data with ligand depletion with NSB and a single binding site [19, 45] need to be modified when a second, low-affinity specific binding site might be present from α4β2 nAChR? Detecting and accurately interpreting properties of the low affinity site is important because of the potential biological relevance of the low-affinity specific site. The low affinity binding site might reflect biologically important roles for α4β2 nAChR, reflect a variant structure of the agonist binding site, or give insight into the assembly of α4β2 nAChR. The goal of this study was to determine, using computational modeling, whether fitting total binding is sufficient for characterizing the low affinity binding site from α4β2 nAChR in the context of ligand depletion and NSB. The modeling simulated saturation binding, homologous competition, and heterologous competition. The experimental foundation for the modeling was reported previously with Kd1 = 13 pM for the high affinity site and Kd2 = 12 nM for the low affinity site . The findings are potentially relevant to other ligands and receptors when two or more specific binding sites are possible and when ligand depletion and NSB affect binding data.
Equations of the models
Analytical solutions of cubic equations are available that describe ligand depletion (with and without NSB) of two binding sites and one ligand or of two sites with homologous competition [3, 9, 18, 46, 47]. Analytical solutions of a quartic polynomial describing ligand depletion and NSB of three binding sites and one ligand or of two binding sites with homologous competition can be derived from the general solution of a quartic polynomial . Numerical solutions were used in this investigation because of the relative ease of implementation and the usefulness of numerical solutions when roots of quintic and higher order polynomials are needed to describe ligand depletion but for which analytical solutions are not available. For example, roots of a sixth order polynomial are needed to describe heterologous competition with two sites and ligand depletion and NSB, which precludes an analytical solution.
Data generation and model fitting
Data were generated with the following parameter values published by our laboratory  unless otherwise stated: Kd1 = 0.013 nM and Kd2 = 12 nM for [3H]EB; Ki1 = 0.84 and Ki2 = 775 nM for nicotine; fraction of R1T = 0.84; fraction of R2T = 0.16 (see Figure 1 for notation). When a R1T concentration is stated, the corresponding R2T concentration is implied.
The one site models for saturation binding and competition data were simpler cases of the two sites models, making these two types of models nested . Qualities of fit of the two types of models, therefore, were compared with the F-test . The level of significance for hypothesis testing was 0.05. The confidence level for a confidence interval (CI) was 95%. CIs for dissociation constants and average p values were based on logarithmic values.
Effects of ligand depletion and NSB on saturation binding to two specific sites
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.
Modeling specific binding and NSB as total binding
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 . 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?
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.
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.
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.
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.
Characterizing the low-affinity specific binding site with homologous competition when NSB is negligible
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 . 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
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.
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 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
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.
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.
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
A model that fits total binding data as a function of total ligand can correctly interpret those data when ligand depletion and NSB are significant . This approach is straightforward with one binding site. This study shows that the approach for [3H]EB, α4β2 nAChR, and two binding sites needs modifications for identifying binding to the low affinity site. In particular, identifying the low affinity site can be challenging because of phenomenological and computational similarities between low-affinity specific binding and NSB.
This study is novel because it shows that fitting total binding data from [3H]EB and α4β2 nAChR might be insufficient for characterizing the low affinity site when ligand depletion and NSB are significant. Moreover, this investigation develops four concepts for studying the low affinity binding site of α4β2 nAChR in the presence of ligand depletion and NSB that go beyond simply fitting total binding. First, binding of [3H]EB to the low affinity site in saturation data or homologous competition data can be misattributed to NSB. Low-affinity specific binding can be identified by using larger maximum concentrations of [3H]EB or cold competitor, simultaneously fitting apparent NSB, or obtaining data from multiple concentrations of α4β2 nAChR. Potential ambiguity between low-affinity specific binding and NSB arises because they share a similar appearance as long as R2≃R2T. Increasing [3H]EBmax for saturation binding or increasing the maximum concentration of cold competitor for competition binding breaks this similarity by creating conditions for which R2≪R2T, R2L≃R2T, and R2B≃R2T.
Second, when NSB is significant, ligand depletion can help characterize the low affinity site. Ligand depletion in binding studies is commonly believed to be only problematic. In contrast, increasing ligand depletion by increasing α4β2 nAChR concentration beneficially reduced NSB and significantly populated the low affinity site. The result was better detection of [3H]EB binding to the low affinity site.
Third, directly measuring NSB without α4β2 nAChR can more reliably interpret NSB than does modeling NSB as a component of total binding in competition binding. Whether [3H]EB binding at a particular large concentration of competitor arises solely from NSB depends on Ki2 and concentration of the low affinity site. Removing α4β2 nAChR from the assay, when feasible, is a more rigorous way than is using a large concentration of competitor to ensure that [3H]EB binding arises from NSB and does not involve the low affinity site of α4β2 nAChR.
Fourth, αcompetitor needs to be considered when interpreting heterologous competition data with [3H]EB and α4β2 nAChR because it increases Ki1, app and Ki2, app. The true values of Ki1 and Ki2, therefore, can be determined only when αcompetitor is known. Regardless of αcompetitor, however, Ki2, app/Ki1, app is invariant and equals Ki2/Ki1. This ratio can help compare structural features of the two binding sites of α4β2 nAChR. For example, variations in the ratio for a series of competitors with systematic structural variations might correlate with structural features of the two binding sites.
The findings presented in this study have limitations. First, modeling explored conditions suitable for characterizing low affinity binding that might not match conditions readily available in a laboratory. One such condition is nanomolar concentrations of α4β2 nAChR. This high range of α4β2 nAChR concentration might be more available in the future with high level heterologous expression of α4β2 nAChR. Quantitative results, such as concentration ranges that identify the low affinity site, are a reasonable but not definitive guide to conditions for studying the low affinity site of α4β2 nAChR with [3H]EB. For example, values of α might be substantially smaller than the values illustrating NSB in this study. With membrane homogenates from stably transfected HEK 293 cells, α was on the order of 0.001 . In addition, changes in the fraction of low affinity site, as might occur with different expression conditions, will change the appearance of data. A larger fraction of low affinity site would make detection and analysis of this site easier. Second, the simulations included large numbers of data points with the goal of reliably describing binding data. Fewer data points would need higher precision in the data to identify the low affinity site and would lead to reduced precision of binding parameter estimates. Third, the properties of noise imposed on errorless data in this study do not necessarily reflect properties of real noise and uncertainties in experiments. Fourth, based on binding data from our laboratory , this study assumes two independent binding sites in α4β2 nAChR. Other descriptions of binding sites (for example, two cooperative binding sites, a combination of cooperative and independent binding sites, or more than two independent sites) might better describe binding data from α4β2 nAChR under other conditions. Fifth, the linear relationship between free [3H]EB and NSB led to the phenomenological and computational similarity between low affinity binding and NSB expressed in Eqs. (3)-(5). This linear relationship usually describes the behavior of NSB. This linear relationship might be unsuitable for some situations. For example, if NSB in the absence of specific binding is observed to be saturable [53, 54, 55], the linear relationship would need to be modified. Sixth, statistical comparisons using the F-test and p values between the one site modeltotal and two sites modeltotal were suitable because of the nested nature of the two models. In other words, the two sites modeltotal contained all the features of the one site modeltotal and extended those features by a second specific binding site. Other statistical methods for comparing models do not need nested models, such as Akaike's information criterion [7, 56, 57]. Seventh, the independent variable for the models in this study is the concentration of total ligand ([3H]EB for saturation binding or a cold ligand for competition). This variable usually is accurately known and was presumed to be free of uncertainty. Using the measured concentration of free ligand as the independent variable simplifies the model equations. The measured free ligand concentrations, however, will have nonnegligible uncertainty. The method of least squares might not reliably estimate parameter values when the values of the independent variable are uncertain [1, 2, 58].
Characterizing the low affinity site potentially will contribute understanding of structure, function, and synthesis of α4β2 nAChR in native and heterologous expression systems. For example, the low affinity site might arise from an immature form of α4β2 nAChR or be involved in ligand-induced upregulation [32, 59, 60, 61]. Heterologous competition data similar to Figure 12B were found with cytisine, nicotine, and acetylcholine as competitors of [3H]EB binding with α4β2 nAChR immunoisolated with monoclonal antibody (mAb) 295 but not with other mAbs . This similarity suggests that mAb 295 might isolate a distinctive form of low affinity α4β2 nAChR. Homologous competition data might help further characterize this form of α4β2 nAChR. An intriguing possibility is that this low affinity form contributes to the biological roles of α4β2 nAChR. This study should help investigators design experiments and develop computational approaches for interpreting data from [3H]EB and α4β2 nAChR when ligand depletion and NSB are significant. Manipulation of maximum ligand and receptor concentrations and intentionally increasing ligand depletion are potentially helpful approaches. Extending the modeling and numerical solution method to three or more binding sites and to cooperative binding with ligand depletion and NSB is straightforward. Although applied specifically to [3H]EB and α4β2 nAChR, the methods should be relevant to other contexts of multiple binding sites, ligand depletion, and NSB.
This study was financially supported by the Texas A&M University System Health Science Center.
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