Predicting the Activation of the Androgen Receptor by Mixtures of Ligands Using Generalized Concentration Addition

Abstract

Concentration/dose addition is widely used for compounds that act by similar mechanisms. But it cannot make predictions for mixtures of full and partial agonists for effect levels above that of the least efficacious component. As partial agonists are common, we developed generalized concentration addition, which has been successfully applied to systems in which ligands compete for a single binding site. Here, we applied a pharmacodynamic model for a homodimer receptor system with 2 binding sites, the androgen receptor, that acts according to the classic homodimer activation model: Each cytoplasmic monomer protein binds ligand, undergoes a conformational change that relieves inhibition of dimerization, and binds to DNA response elements as a dimer. We generated individual dose-response data for full (dihydroxytestosterone, BMS564929) and partial (TFM-4AS-1) agonists and a competitive antagonist (MDV3100) using reporter data generated in the MDA-kb2 cell line. We used the Schild method to estimate the binding affinity of MDV3100. Data for individual compounds fit the homodimer pharmacodynamic model well. In the presence of a full agonist, the partial agonist had agonistic effects at low effect levels and antagonistic effects at high levels, as predicted by pharmacological theory. The generalized concentration addition model fits the empirical mixtures data—full/full agonist, full/partial agonist, and full agonist/antagonist—as well or better than relative potency factors or effect summation. The ability of generalized concentration addition to predict the activity of mixtures of different types of androgen receptor ligands is important as a number of environmental compounds act as partial androgen receptor agonists or antagonists.

Nuclear receptors are ligand-activated transcription factors and are the mechanism by which lipophilic hormones and hormone-like molecules regulate gene expression. Thus, they are critical targets of endocrine disrupting chemicals. The androgen receptor is a nuclear receptor that binds ligands including hormones (eg, testosterone), pharmaceuticals (eg, flutamide), and environmental endocrine disrupting chemicals (eg, cyprodinil, vinclozolin, and procymidone). In females, androgen receptor signaling is necessary for efficient folliculogenesis (ie, androgen receptor knockout female mice have impaired folliculogenesis and display early onset infertility) (Matsumoto et al., 2008). Studies in androgen receptor knockout mice showed that in males androgen receptor signaling is necessary for the reproductive function (ie, development of reproductive organs, spermatogenesis, and brain masculinization). There is evidence for a decline in human sperm counts between 1973 and 2011 (Barratt et al., 2018), and an increased incidence of symptoms of testicular dysgenesis syndrome, including testicular cancer, disorders of sexual development, cryptorchidism, and hypospadias (Skakkebaek et al., 2016). A subset of these adverse outcomes is hypothesized to occur as a result of displacement of endogenous testosterone and dihydrotestosterone from the androgen receptor by androgen receptor antagonists, followed by changes in binding to DNA response elements and failure to initiate transcription of key androgen responsive genes. The involvement of these molecular events has been well documented using rodent models (Gray et al., 2004). Given the importance of androgen receptor signaling in physiology, increasing human exposure to environmental chemicals that are androgen receptor ligands and declining human reproductive health, it is necessary to be able to model how multiple ligands interact to activate the androgen receptor.

An approach commonly used to assess the effect of mixtures of endocrine disrupting chemicals is concentration addition, also called dose addition (Rajapakse et al., 2004). This model requires that all components of the mixture have the same efficacy or maximal response. However, ligands of nuclear receptors may act as full and partial agonists or competitive antagonists. Partial agonists are ligands that produce a lower maximal response than full agonists, and antagonists do not induce a response. We developed generalized concentration addition to resolve the inability of concentration addition to make predictions for mixtures including partial agonists and/or antagonists. Generalized concentration addition uses inverse mathematical functions allowing for some components to have submaximal efficacy (Howard and Webster, 2009). It successfully predicts such mixture effects for ligand-receptor systems with one binding site (eg, AhR, PPARγ) (Howard et al., 2010; Watt et al., 2016).

We recently developed a mathematical dose-response model for receptors that bind ligand at each monomer and then dimerize and showed that it fulfilled the requirements for generalized concentration addition (Webster and Schlezinger, 2019). Androgen receptor acts according to the classic homodimer activation model. Each cytoplasmic monomer protein binds ligand and undergoes a conformational change that relieves inhibition of dimerization and releases chaperone proteins such as heat shock protein 90 (Kuil et al., 1995; Wong et al., 1993). The androgen receptor translocates to the nuclear and binds to DNA response elements as a dimer (De Angelis et al., 2013; Wong et al., 1993). Transactivation is initiated by the recruitment of coactivators and transcriptional complex (Haelens et al., 2007).

Here, we test the ability of generalized concentration addition to model the activation of androgen receptor by ligand mixtures and compare it to effect summation, a classic additivity model, as well as relative potency factors, a special case of concentration addition and generalized concentration addition. Using MDA-kb2 cells, which express a steroid response element-dependent luciferase reporter (Wilson et al., 2004), we generated dose-responses for individual synthetic androgen receptor ligands. We predicted mixture effects using the different additivity models and compared these predictions with empirical results. We show that the dissociation constant derived from the Schild method can be incorporated into generalized concentration addition in order to model mixtures containing antagonists. We establish generalized concentration addition as a model for predicting androgen receptor activation by mixtures containing diverse ligand types.

MATERIALS AND METHODS

 

 

Chemicals

BMS564929 (CAS No. 627530-84-1, Cat. No. 5274, purity ≥ 98%) and TFM-4AS-1 (CAS No. 188589-61-9, Cat. No. 3813, purity ≥ 98%) were purchased from Tocris Bioscience (Bristol, United Kingdom). MDV3100 (CAS No. 915087-33-1, Cat. No. sc-364354, purity ≥ 98%) was purchased from Santa Cruz Biotechnology (Dallas, Texas). Dihydroxytestosterone (DHT; CAS No. 521-18-6, Cat. No. 15874, purity ≥ 97%) was purchased from Cayman Chemical (Ann Arbor, Michigan). All other reagents were purchased from Thermo Fisher Scientific (Waltham, Massachusetts), unless indicated.

AR reporter assays

MDA-kb2 cells (RRID:CVCL_6421, CRL-2713; ATCC, Manassas, Virginia) were used for these studies; they were created from the MDA-MB-453 human breast carcinoma cell line (Cailleau et al., 1978). Cultures were maintained in Leibovitz’s L-15 medium (ATCC, Cat. No. 30-2008) supplemented with 10% fetal bovine serum (Gemini Bio-Products, West Sacramento, California) and antibiotic-antimycotic (100 U/ml penicillin, 100 μg/ml streptomycin, and 0.25 μg/ml amphotericin B) at 37°C without CO₂ supplementation. Seven days prior to plating for an experiment, the cultures were switched to Leibovitz’s L-15 medium supplemented with 10% charcoal/dextran-stripped fetal bovine serum and antibiotic-antimycotic. The medium was replaced once prior to experiment plating.

For experiments, MDA-kb2 cells were plated at 40 000 cells per well in 0.2-ml stripped serum medium in white-sided, 96-well plates. Following a 48-h incubation, cells received no treatment (medium only), Vh (dimethyl sulfoxide [DMSO], 0.25% for individual dose-responses or 0.5% for combination dose-responses), the positive control (DHT, 1 × 10⁻⁸ M) or androgen receptor ligands at a range of concentrations either alone or in combination (DHT, 1 × 10⁻¹³ to 1 × 10⁻⁶ M; BMS564929, 1 × 10⁻¹² to 1 × 10⁻⁶ M; TFM-4AS-1, 1 × 10⁻¹¹ to 4 × 10⁻⁴ M; and MDV3100, 1 × 10⁻¹⁰ to 2 × 10⁻⁵ M). BMS564929, TFM-4AS-1, and MDV3100 specifically bind to androgen receptor, without significant interaction with the glucocorticoid or progesterone receptors (Ostrowski et al., 2007; Schmidt et al., 2009; Tran et al., 2009). Vh and experimental chemicals were applied to duplicate wells. The positive control was applied to 6 wells of every experimental plate. Six wells were left untreated (Naïve) in every experimental plate. After 24 h of treatment, cells were analyzed using the Steady-Luc Firefly HTS Assay Kit (Biotium, Freemont, California). Luminescence was measured using a Synergy2 plate reader (Biotek Inc, Winooski, Vermont). In independent experiments to assess toxicity, plating and treatments were carried out as described above. MTS reagent (4-[5-[3-(carboxymethoxy)phenyl]-3-(4,5-dimethyl-1,3-thiazol2-yl)tetrazol-3-ium-2-yl]benze nesulfonate) was added 4 h prior to analysis of absorbance at 490 nm (Cat. No. ab197010, Abcam, Cambridge, United Kingdom).

Data from technical duplicates were averaged prior to analysis and to generate a single biological replicate per experiment. Biological replicates were normalized using a method to minimize intra- and inter-experimental variation (Rajapakse et al., 2004), as follows:

$$\text{Percent} \text{Maximum} \text{Activity}=\frac{\hspace{0.17em}\text{Exp}\hspace{0.17em} \text{Lum}-\text{Naïve} \text{Lum}}{\text{Positive} \text{Control} \text{Lum}-\text{Naïve} \text{Lum}},$$ (1)

where “Exp Lum” is the average luminescence in vehicle or treatment duplicate wells, “Naïve Lum” is the average luminescence in the 6 wells that contained untreated cells, and “Positive Control Lum” is the average luminescence in the 6 wells treated with the positive control (1 × 10⁻⁸ DHT). AR activation is reported as “% Maximum Activity.” Dose-response curves were fit with the “log(agonist) versus response – Variable Slope (4 parameters)” Hill function in Prism (version 6, GraphPad, San Diego, California); minimum, maximum, slope, and EC₅₀ values were determined from each fit.

Schild method

The Schild method was carried out as described (Kenakin, 2014). Using the MDA-kb2 experimental approach described above, dose-response data for the full agonist BMS564929 (1 × 10⁻¹¹ to 1 × 10⁻⁷ M) were generated in the presence of Vh or the competitive antagonist MDV3100 (4 × 10⁻⁸, 1 × 10⁻⁷, 4 × 10⁻⁷, 1 × 10⁻⁶, 2 × 10⁻⁶, or 4 × 10⁻⁶ M). Data from 6 experiments were averaged. The means for each dose-response curve were fit with the “log(agonist) versus response – Variable Slope (4 parameters)” Hill function in Prism; maximum, slope, and sum of squares values were determined from each fit. The mean of the maxima and of the slopes of each of the 7 dose-response curves were calculated. The dose-response curves then were refitted using the “log(agonist) versus response – Variable Slope (4 parameters)” function but were constrained by the means of the maxima and the slopes, and the sum of squares was determined.

The fits of the unconstrained and constrained models then were examined using Akaike’s information criterion for small sample sizes (AICc) (Portet, 2020), as follows:

$$\mathit{AICc}=n \mathrm{l}\mathrm{n}\left(\frac{\mathit{SSq}}{n}\right)+\left(\frac{2kn}{n-k-1}\right),$$ (2)

where n is the sample size, SSq in the sum of squared deviations between the model and the actual responses, and k is the number of parameters.

Using the data from the model (constrained) with the lower Akaike’s information criterion, we calculated the dose ratio for each dose-response curve by dividing the EC₅₀ of each curve by the EC₅₀ of the curve generated in the absence of MDV3100. The Schild plot was generated by plotting the log(dose ratio −1) versus the MDV3100 concentration. We assessed the suitability of the Schild plot by ensuring that the slope was > 0.8, that there were data near the log(dose ratio −1) = 0, and that a wide concentration range was represented (Kenakin, 2014).

Hill and pharmacodynamic models

Dose-response data for each individual compound except the competitive antagonist were modeled with a 4-parameter Hill function (equation 3):

$$y=a_0+\frac{a_{1i}{\left(\frac{x_i}{K_i}\right)}^{p_i}}{1+{\left(\frac{x_i}{K_i}\right)}^{p_i}},$$ (3)

where y is the response and x_i is the concentration of ligand i, and the 4 parameters are as follows: the background response in the absence of ligand (a₀); the span, the maximum response above background (a_1i); the EC₅₀ (K_i); and the slope parameter (p_i). The latter 3 may differ by compound.

Derivation of the pharmacodynamic model for homodimers such as the androgen receptor was described previously (Webster and Schlezinger, 2019). Briefly, we assume that receptor monomers reversibly bind ligand and dimerize leading to signal. We also assume equilibrium and constant total receptor concentration. The result can be written as a composite function:

$$\theta =\frac{{\alpha }_i\frac{x_i}{K_i}}{1+\frac{x_i}{K_i}},$$ (4a)

$$y=a_0+\lambda {\left(-\frac{1}{\theta }+\sqrt{\frac{1}{{\theta }^2}+1}\right)}^2,$$ (4b)

where equation 4a is a Hill function of slope one, but the compound-specific parameters (α_i, K_i) are interpreted somewhat differently. The second function contains the background response (a₀) and a scaling parameter (λ) that does not depend on the compound.

Generalized concentration addition for homodimers

Generalized concentration addition is an extension of concentration addition defined by the following equation:

$$\frac{x_1}{f_1^{-1}(y)}+\frac{x_2}{f_2^{-1}(y)}=1$$ (5)

for a binary mixture, easily generalized to more components (Howard and Webster, 2009). The denominators are the inverse dose-response functions for the individual compounds. To predict the combined effect of mixtures of compounds differing in efficacy, the inverse function must yield real numbers for partial agonists at effect levels above their maximal effect. As previously shown, this requirement is met by the homodimer pharmacodynamic model (equation 4) but not generally by Hill functions with slope parameters different from one and many other functions used for dose-response analysis (Webster and Schlezinger, 2019).

Application of generalized concentration addition replaces (4a) by the following equation for a binary mixture:

$$\theta =\frac{{\alpha }_1\frac{x_1}{K_1}+{\alpha }_2\frac{x_2}{K_2}}{1+\frac{x_1}{K_1}+\frac{x_2}{K_2}},$$ (6)

while the outer part of the composite function (4b) remains the same (Webster and Schlezinger, 2019). This result is easily generalized to more complicated mixtures. For application to a mixture of an agonist and a competitive antagonist, the alpha parameter for the latter is set to 0.

Effect summation

Effect summation is a mixtures model that sums the individual responses (rather than the concentrations) of the mixture components. The prediction for a binary mixture is

$$y=f_1(x_1)+f_2(x_2),$$ (7)

where f_i(x_i) is the dose-response for compound i alone. Note that effect summation is generally not considered to be a useful model in mixtures toxicology (Berenbaum, 1989).

Relative potency factor

The relative potency factor model for mixtures assumes that mixture components differ only in their potency. For Hill functions, this means that the maximal effects and slope parameters are the same. The relative potency factor is often based on the ratio of the EC₅₀ of a compound compared with a reference compound. The toxic equivalence factors used for dioxin-like compounds are a well-known example (Van den Berg et al., 2006). The relative potency factor model for a binary mixture is described by

$$y=f_1(x_1+{\gamma }_2{x}_2),$$ (8)

where f₁(.) is the dose-response function for the reference compound (x₁) and γ₂ is the relative potency factor for compound 2, typically estimated as the EC₅₀ for compound 1 divided by the EC₅₀ for compound 2. The relative potency factor model is a special case of concentration addition, which is in turn a special case of generalized concentration addition (Howard and Webster, 2009). Unlike concentration addition, the relative potency factor model can make predictions for mixtures of full and partial agonists at effect levels above the maximal effect of the partial agonist, but this violates the assumptions underlying the model.

Statistics and software

Dose-response data were generated from technical duplicates, which were averaged to generate a single biological replicate for each treatment in an experiment. No more than 2 independently plated experiments were conducted in a week, and each experiment was conducted at least 4 times. Dose-response data are presented as means of the biological replicates ± standard deviations. Four-parameter Hill functions were fit to individual compounds using R (R Core Team, 2017) as well as using Prism, yielding similar results (not shown). After subtracting background response, the homodimer pharmacodynamic model for each compound was fit using R for a range of values of λ yielding estimates of α_i and K_i. As λ is common across compounds (equation 4b), the value of λ was chosen that minimizes the mean squared error. Dose-response surfaces for mixtures data were generated using R. Predictions for the 3 mixtures models were made using the estimated parameters. For generalized concentration addition, the estimated values of a₀, α_i, K_i, and λ were substituted into equations 6 and 4b. Effect summation estimates (equation 7) and relative potency factors (equation 8) were based on the fits of the 4-parameter Hill function. The dose-response surfaces for each model were compared with empirical data using root-mean-square error (RMSE).

RESULTS

We began by establishing the dose-responses of 3 androgen receptor ligands, DHT (full agonist), BMS564929 (full agonist), and TFM-4AS-1 (partial agonist). MDA-kb2 cells express 240 fmol androgen receptor/mg-protein and are stably transfected with a luciferase construct driven by a steroid response element (Hall et al., 1990; Ham et al., 1988; Wilson et al., 2004). The assay is included in Tox21 and ToxCast (Kleinstreuer et al., 2017). No toxicity was evident at the concentrations tested (Supplementary Figure S1). Fitted parameters and RMSEs are presented in Supplementary Tables S1 and S2, respectively. As shown in Figure 1, the dose-response data for BMS564929 and TFM-4AS-1 were very well fit by both the 4-parameter Hill function and the homodimer pharmacodynamic model; the Hill function fit slightly better for DHT.

Figure 1.

Dose-response curves for androgen receptor activation by full and partial agonists. MDA-kb2 cells were plated in medium with charcoal/dextran-stripped serum and dosed with Vh (0.5% DMSO), dihydroxytestosterone (DHT; 1 × 10⁻¹³ to 1 × 10⁻⁶ M), BMS564929 (1 × 10⁻¹² to 1 × 10⁻⁶ M), or TFM-4AS-1 (1 × 10⁻¹¹ to 4 × 10⁻⁴ M). Luminescence was measured after 24 h. Ligand-induced luminescence was normalized to the positive control-induced luminescence. Data from Vh-treated wells are reported as the lowest concentration in each plot. Dose-response data were fit with a Hill function or a homodimer pharmacodynamic model (PDM). A, Dose-response of full agonist DHT. B, Dose-response of full agonist BMS564929. C, Dose-response of partial agonist TFM-4AS-1. Data are presented as means ± SD of at least 4 independent experiments.

We hypothesized that mixtures of BMS564929 with DHT and with TFM-4AS-1 would show disparate effects on the overall efficacy of androgen receptor activation. For mixtures of BMS564929 with DHT, we expected that the efficacy of the mixture would reach 100% because each is a full androgen receptor agonist. This is evident, as shown in Figure 2A. For mixtures of BMS564929 with TFM-4AS-1, we expected that the mixture would cause a reduction in overall androgen receptor activation at effect levels above the efficacy of TFM-4AS-1, owing to the behavior of a partial agonist to act in the manner of a competitive antagonist in a mixture. Figure 2B shows that at high effect levels, the response to BMS564929 is reduced and the overall activation of androgen receptor by the mixture decreases.

Figure 2.

Efficacy of androgen receptor activation by a mixture of a full agonist with a full agonist (A) and with a partial agonist (B). MDA-kb2 cells were plated in medium with charcoal/dextran-stripped serum and dosed with Vh (0.5% DMSO, shown as lowest concentration) or BMS564929 (1 × 10⁻¹¹ to 1 × 10⁻⁸ M) with dihydroxytestosterone (DHT; 1 × 10⁻¹² to 1 × 10⁻⁶ M), or with TFM-4AS-1 (1 × 10⁻¹⁰ to 4 × 10⁻⁵ M). Luminescence was measured after 24 h. Ligand-induced luminescence was normalized to the positive control-induced luminescence. Data from Vh-treated wells are reported as the lowest concentration in each plot. Data are presented as means ± SD of at least 4 independent experiments.

To compare the empirical dose-response data with prediction of the different mixture models, three-dimensional response surfaces of androgen receptor activation by the full/full and full/partial agonist combinations were plotted. Figure 3 shows that for the full agonist/full agonist mixture, the generalized concentration addition model fits the experimental data the best, ie, it has the lowest RMSE, and the relative potency factor model fit almost as well. The latter is expected because relative potency factors are a special case of generalized concentration addition where compounds differ only in potency, as would be expected for full agonists. The small difference is due to the slightly different maximal effects and slopes for DHT and BMS as shown in Figure 1. As expected, the effect summation model fits poorly: It essentially double counts effects in the high-dose region where the dose-response curves flatten, a well-known problem with this model.

Figure 3.

Response surface modeling of androgen receptor activation by a mixture of full agonists. Androgen receptor reporter data were generated from mixtures of dihydroxytestosterone (DHT) and BMS564929 as described in Figure 2A. Vertical axis: % maximal androgen receptor (AR) activation. Marginal curves correspond to individual dose-response curves shown in Figure 1 (shown as lowest concentration on log scale) except for relative potency factor model. A, Experimental joint dose-response surface. B, Generalized concentration addition (GCA) model based on parameters for homodimer pharmacodynamic model. C, Relative potency factor (RPF) model using DHT as the reference compound and Hill functions for the margins. D, Effect summation (ES) model using Hill functions for the margins. RMSE, root-mean-square error comparing fit of model to experimental data.

Figure 4 shows that for the full agonist/partial agonist mixture, generalized concentration addition fits the experimental data quite well. The relative potency factor model falls short due to the difference in efficacy between the 2 compounds. Effect summation fits the data poorly.

Figure 4.

Response surface modeling of androgen receptor activation by a mixture of a full and a partial agonist. Androgen receptor reporter data were generated from mixtures of BMS564929 and TFM-4AS-1 as described in Figure 2B. Vertical axis: % maximal androgen receptor (AR) activation. Marginal curves correspond to individual dose-response curves shown in Figure 1 (shown as lowest concentration on log scale) except for relative potency factor model. A, Experimental joint dose-response surface. B, Generalized concentration addition (GCA) model based on parameters for homodimer pharmacodynamic model. C, Relative potency factor (RPF) model using BMS as the reference compound and Hill functions for the margins. D, Effect summation (ES) model using Hill functions for the margins. RMSE, root-mean-square error comparing fit of model to experimental data.

Next, we tested the applicability of generalized concentration addition in modeling androgen receptor activation in the presence of a competitive antagonist. We began by establishing the dose-response of MDV3100; it fully antagonized androgen receptor activation by BMS564929 (Figure 5). MDV3100 is a competitive inhibitor of DHT binding to androgen receptor (Tran et al., 2009). Therefore, we used the Schild method to estimate the equilibration dissociation constant for MDV3100 binding to androgen receptor in MDA-kb2 cells (Figure 6). We fit BMS564929 dose-response curves generated in the presence of increasing concentrations of MDV3100 with a 4-parameter Hill model (Figure 6A) and a 2-parameter Hill model constrained to the mean slope and maxima (Figure 6B). We determined that the fit of the curves with the means generated a lower Akaike’s information criterion (AICc, Figure 6C); therefore, this model is statistically preferable. Using the EC₅₀s in the dose-response curves in Figure 6B, we calculated the dose ratios for each curve relative to the curve without MDV3100 and used these values to generate a Schild plot (Figure 6D). Because the 95% confidence interval of the slope (0.8663–1.038) included unity, the data were refit to a slope of 1, and the pK_b was estimated from the x intercept when y = 0, yielding an estimate of 69 nM. In generalized concentration addition, a competitive antagonist can be considered a partial agonist with zero efficacy. We generated three-dimensional response surfaces of androgen receptor activation by the full agonist and antagonist mixture, to compare the empirical dose-response data with those predicted by different additive models. As shown in Figure 7, the generalized concentration addition model predicts the experimental data much better than effect summation. The relative potency factor model is not relevant for this case.

Figure 5.

Dose-response for androgen receptor activation by an antagonist in the absence and presence of a full agonist. MDA-kb2 cells were plated in medium with charcoal/dextran-stripped serum and dosed with Vh (0.5% DMSO), BMS564929 (1 × 10⁻⁷ M), and/or MDV3100 (1 × 10⁻¹⁰ to 4 × 10⁻⁶ M). The cultures were then dosed with Vh (0.5% DMSO) or BMS564929 (1 × 10⁻⁹ M). Luminescence was measured after 24 h. Ligand-induced luminescence was normalized to the positive control-induced luminescence. Data from Vh-treated wells are reported as the lowest concentration in each plot. Data are presented as means ± SD of at least 4 independent experiments.

Figure 6.

Schild analysis of MDV3100. MDA-kb2 cells were plated in medium with charcoal/dextran-stripped serum and dosed with Vh (0.5% DMSO), BMS564929 (1 × 10⁻¹¹ to 1 × 10⁻⁷ M), and/or MDV3100 (1 × 10⁻¹⁰ to 2 × 10⁻⁵ M). Luminescence was measured after 24 h. Ligand-induced luminescence was normalized to the positive control-induced luminescence. Data from Vh-treated wells are reported as the lowest concentration in each plot. Data are presented as means ± SD (n= 6). A, Individual dose-response curves fitted with a 4-parameter Hill function. B, Individual dose-response curves fitted a 4-parameter Hill function constrained to the mean slope and maximum. C, Akaike’s information criterion (AIC) for the individual and constrained fits. D, Schild plot generated from the experimental data fit with the constrained Hill function and analyzed using linear regression. pK_b is the estimate of the log of the dissociation constant of the competitive antagonist.

Figure 7.

Androgen receptor activation by a mixture of a full agonist and a competitive antagonist. Androgen receptor reporter data were generated from mixtures of BMS564929 and MDV3100 as described in Figure 6. Vertical axis: % maximal androgen receptor (AR) activation. Marginal curves correspond to individual dose-response curves shown in Figure 1 for BMS and Figure 5 for MDV3100 (shown as lowest concentration on log scale). A, Experimental joint dose-response surface. B, Generalized concentration addition (GCA) model based on parameters for homodimer pharmacodynamic model and Schild analysis. C, Effect summation (ES). The relative potency factor model is not relevant for this case.

DISCUSSION

Toxicologists typically study one compound at a time, but humans are exposed to a large number simultaneously. Furthermore, the National Institute of Environmental Health Science has made “unraveling the effects of environmental mixtures” a priority (Carlin et al., 2013). The significance of nuclear receptor-based mixture effects was underlined by research demonstrating that mixtures of environmental estrogens, each at its no observable effect level, elicited significant biological responses (Silva et al., 2002). Here, we focused on the androgen receptor not only because of its physiological importance but also because it allowed us to test the efficacy of an expansion of our novel model of additivity, generalized concentration addition, to a homodimer receptor system with 2 binding sites. We show that the model can accommodate the pharmacodynamics of homodimers and accurately predict androgen receptor activation by mixtures of full agonists, full and partial agonists, and a full agonist and an antagonist.

We initially developed generalized concentration addition in order to address a limitation of concentration addition: It cannot model the effect of a mixture that contains ligands of differing efficacies at effect levels above that of the least efficacious mixture component (Howard and Webster, 2009; Silva et al., 2002). We previously showed that generalized concentration addition can accurately model the activation of AhR and PPARγ by mixtures of ligands containing full and partial agonists (Howard et al., 2010; Watt et al., 2016). Although these receptors have distinct biological functions, they both can be modeled using a pharmacodynamic model for receptors with one binding site, a Hill function with a slope parameter of 1. To expand the utility of generalized concentration addition, we derived a pharmacodynamic model for homodimer receptors in which each monomer binds ligand and then dimerizes. The resulting function meets the mathematical requirements for application of generalized concentration addition: The function is invertible yield real numbers. This property is not true for many commonly used empirical dose-response functions based on curve fitting, eg, Hill functions with slope parameter > 1 (Webster and Schlezinger, 2019).

We tested the extension of generalized concentration addition to modeling homodimers using empirical data generated with binary mixtures of ligands of the androgen receptor. The combination of 2 full agonists (DHT and BMS564929) generated mixture responses that were positively additive until the maximum activation level was reached; once the maximum was reached the mixture continued to fully activate the androgen receptor. Generalized concentration addition accurately modeled androgen receptor activation at all dose combinations. However, effect summation did not; as expected, it overestimated the combined effect of the ligands at high concentrations. The relative potency factor model, a special case of concentration addition and generalized concentration addition, performed nearly as well in this situation. The combination of a full and a partial agonist (BMS564929 and TFM-4AS-1) generated responses that were positively additive at low concentrations. At high combined concentrations, androgen receptor activation was reduced to the maximum activation induced by TFM-4AS-1 alone (ie, ≈ 60% activation). This was predicted by generalized concentration addition. Again, effect summation inadequately modeled the combined effect. The relative potency factor model performed significantly worse than generalized concentration addition as the assumption of equal efficacy is violated. These results indicate that generalized concentration addition is a viable model in the case of 2 chemicals acting as either full or partial androgen receptor agonists.

Our ultimate goal is to use generalized concentration addition to model androgen receptor activation by mixtures of natural ligands, pharmaceuticals, and environmental ligands, where many of the latter act as competitive antagonists (Luccio-Camelo and Prins, 2011). Thus, we next tested the ability of generalized concentration addition to accommodate a mixture containing a competitive antagonist. Because the apparent IC₅₀ of an antagonist depends upon the concentration of the agonist in the mixture, we used the Schild method to estimate the equilibrium constant for binding of a synthetic, competitive androgen receptor antagonist (Wyllie and Chen, 2007). The Schild method has been previously used to estimate the apparent binding affinity of a mixture of 8 androgen receptor antagonists (Orton et al., 2012). The linear shape of the Schild plot and the slope of ≈ 1 indicated that MDV3100 is a competitive androgen receptor antagonist. When the Schild method-derived equilibrium dissociation constant is incorporated into generalized concentration addition, it accurately models the activation of androgen receptor by a mixture of a full agonist and an antagonist.

Differences in efficacy between receptor ligands have important consequences for the design of mixtures experiments, which typically begin by characterizing the concentration-response functions of individual compounds where this property may not be known. Most obviously, the effect of a competitive antagonist will only be observed in the presence of an agonist. Schild analysis provides one method for determining the binding affinity of competitive antagonists that are needed for mixtures predictions. Partial agonists are determined by comparing maximal effect versus a suitably chosen full agonist. When the system under analysis (eg, androgen receptor) has a natural ligand (or pharmaceutical) that is a full agonist, the design of mixtures experiments should consider its normal physiologic (or therapeutic) concentration.

“Interaction” and “noninteraction” are ambiguous concepts in the absence of a definition of the latter. Our results suggest that ligands of the androgen receptor behave in a “noninteractive” manner using the definition of generalized concentration addition that encompasses full and partial agonists as well as competitive antagonists. For example, we do not see evidence of cooperativity in the dimerization of androgen receptor when there are different ligands (Webster and Schlezinger, 2019). We note that generalized concentration addition may not apply where compounds act by different mechanisms (Webster, 2013), although there is some evidence that concentration addition performs better than the independent action model for the androgen system (eg, NRC, 2008).

MDA-kb2 cells are one of the assays included in Tox21 and ToxCast to characterize androgen receptor ligands (Kleinstreuer et al., 2017). A limitation of MDA-kb2 cells is that they are stably transfected with a luciferase construct driven by a steroid response element (Hall et al., 1990; Ham et al., 1988; Wilson et al., 2004). The steroid response element is not specific to androgen receptor, but also can be transactivated by glucocorticoid receptor (Ham et al., 1988). Thus, we chose to use highly specific androgen receptor ligands to test the efficacy of generalized concentration addition in predicting androgen receptor activation by mixtures. Such specificity is unlikely with many environmental androgen receptor ligands. Thus, in order to use MDA-kb2 cells in future studies of mixtures of environmental ligands, an additional experimental step will be necessary. Each ligand should be tested for its ability to induce reporter activity in MDA-kb2 cells in the presence of a glucocorticoid receptor antagonist. If there is a significant contribution of the glucocorticoid receptor, the MDA-kb2 cells could still be used, but expression of an endogenous androgen receptor target gene should be used for androgen receptor dose-response and generalized concentration addition modeling. A potential benefit to the steroid receptor driven reporter gene is that MDA-kb2 cells could be used in future studies of more complex mixtures containing both androgen receptor and glucocorticoid receptor ligands.

There are many chemicals in our indoor and outdoor environments, as well as in consumer products, that are androgen receptor ligands. They have the potential to disrupt both male and female reproductive health. We show that generalized concentration addition can accommodate dose-response functions for ligands (full agonists, partial agonists, and competitive antagonists) of homodimers, which significantly expands its utility. Future research will investigate the ability of generalized concentration addition to model androgen receptor activation by complex (3+ components) environmental ligand mixtures and compare predictions with other approaches used to model androgen receptor activation (Birkhoj et al., 2004; Blake et al., 2010; Ermler et al., 2011; Ezechiáš and Cajthaml, 2018; Kjeldsen et al., 2013). Additionally, we are exploring the use of generalized concentration addition to model other homodimer systems (de la Rosa et al., manuscript). Hazardous site risk assessments in the United States typically examine mixtures using a hazard index and do not generally consider issues related to efficacy (USEPA, 1986). Generalized concentration addition may be useful in future risk assessment by examining mixtures at molecular initiating events (using new approach methods) in conjunction with adverse outcomes pathways linking such events to downstream adverse outcomes (USEPA, 2020).

SUPPLEMENTARY DATA

Supplementary data are available at Toxicological Sciences online.

DECLARATION OF CONFLICTING INTERESTS

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

FUNDING

National Institute of Environmental Health Sciences (R01 ES027813).