Generalized concentration addition for ligands that bind to homodimers

Abstract

Concentration addition/dose addition (CA) has proved to be a powerful tool for estimating the combined effect of mixtures that act by similar mechanisms. We earlier proposed generalized concentration addition (GCA) to deal with the inability of CA to estimate effects of mixtures above the level of the least efficacious component. GCA requires specifying mathematical concentration response functions for each mixture component that must be invertible, yielding real numbers. We construct concentration response functions using pharmacodynamic models of ligand-receptor interaction, an important molecular initiating event for adverse outcome pathways. Here, we extend our earlier work in two novel ways. First, we show how composite functions can be used to extend these predictions to downstream events. Second, we show that GCA can accommodate not only receptors with single binding sites but also receptors that bind ligand at each monomer and then dimerize. The derived concentration response functions for receptors that homodimerize meet the requirements for using GCA.

1. Introduction

Mixtures toxicologists have two general approaches: 1) whole mixtures methods examine mixtures such as commercial PCB formulations or water disinfection byproducts, and 2) component-based methods use information on mechanism and individual (marginal) concentration/dose response curves to estimate the effect of mixtures (In this paper we will focus on concentration at target tissues. Dose can also be used, taking into account pharmacokinetics). For component methods, toxicologists and pharmacologists generally employ two definitions of non-interaction: 1) concentration addition (CA), also known as dose addition or Loewe additivity, for compounds having the same or similar mechanisms of action and 2) independent action, also known as response addition, for compounds with disparate mechanisms. Mixtures with effect levels different from the chosen definition are said to interact, either synergize (for greater than non-interactive effects) or antagonize (for less than non-interactive effects). Not surprisingly, there is a lack of clarity about how similar or different mechanisms should be to apply these two criteria. A third criterion, effect summation, is generally rejected by mixtures toxicologists, although it is approximated by the other two in certain special cases [1].

This paper will discuss CA and its extension, generalized concentration addition (GCA), as well as the importance of the choice of the marginal concentration response functions. We extend our earlier work on this subject for receptors with a single binding site [1] to the case of ligands that bind receptors that homodimerize, i.e., receptors formed from two identical protein subunits each possessing a ligand binding site, e.g., the androgen receptor. Such biological systems are of importance for development, biological function, pharmacology and toxic effects such endocrine disruption [e.g., 2]. We also discuss methods for predicting effects downstream of ligand-receptor binding.

2. Background

2.1. Concentration-response functions, concentration addition and isoboles

Suppose we can write the marginal concentration response function for each component (indexed by i) acting alone (with other components equal to 0) as

$$\varphi =f_i[X_i]$$ (1)

We will assume the response ϕ and the concentrations X_i are non-negative real numbers (Later in this paper we will restrict response ϕ to a molecular initiating event of ligand activation of a receptor leading to an apical outcome such as reproductive toxicity). We will also assume that these functions are smooth and continuous, equal zero when X_i equals zero, increase monotonically, and asymptotically approach a maximum. In practice, these assumptions may require subtracting out background effects and avoiding high concentrations that cause general toxicity via mechanisms other than the one under study. Figure 1 shows some examples. In general, concentration-response functions may have different maximum effects (efficacy), require different amounts of compound to achieve the same effect level (potency), and different shapes. Note that the shape of graphs depends in part on scaling: e.g., toxicologists and pharmacologists typically plot concentrations on either arithmetic or logarithmic scales.

Figure 1.

Marginal concentration response curves for two compounds X₁ and X₂ with different maxima. Concentration addition can only make predictions for mixtures at effect levels (e.g., ϕ1) less than that of the least efficacious component (ϕ_crit), in this case X₁. Note that while the effect ϕ may be normalized (e.g., relative to a reference compound), the same normalization must be used for all mixture components.

For simplicity, we will discuss mixtures with two components (see appendix section 1 for higher dimensions). Assume we can write the biological response ϕ of exposure to the mixture as a joint function of the components

$$\varphi =f[X_1,X_2]$$ (2)

We can plot this response surface in 3 dimensions and project its contours (level sets) onto the X₁-X₂ plane. These contours are called isoboles and they play an important role in mixtures toxicology. Given the earlier assumptions about the individual concentration response functions, the isoboles do not intersect.

CA can be defined by the following equation:

$$\begin{gathered} \frac{X_1}{{\mathit{EC}}_{\varphi ,1}}+\frac{X_2}{{\mathit{EC}}_{\varphi ,2}}=1 \\ {\mathit{EC}}_{\varphi ,i}>0 \end{gathered}$$ (3)

where EC_ϕ,i is the concentration of compound i alone needed to achieve effect level ϕ [1,3]. For any choice of ϕ, equation (3) describes an isobole. As equation (3) is an equation for a line, mixtures that obey CA will have isoboles that are straight lines (planes or hyperplanes in higher dimensions). Since the EC_ϕ,i are positive, these isoboles are negatively sloped straight lines with slopes equal to the negative ratio of the EC_ϕ,i (Figure 2a). Under CA, the slopes of isoboles can vary, depending on the effect level ϕ [3]. In the special case where the slope is identical for all effect levels, yielding negatively sloped parallel isoboles, the slope is called the relative potency factor (RPF). A well known example of this application is used for dioxin-like compounds where they are called toxic equivalence factors (TEFs)[4]. A biological rationale for the definition of CA is that for some groups of compounds, one compound can substitute for another at fixed ratios without changing the effect level. Berenbaum provides an extensive discussion of CA as a definition of non-interaction [3].

Figure 2.

a). An isobole obeying concentration addition for a mixture of two components for effect level ϕ1: the isobole is linear and negatively sloped. b). An isobole obeying GCA with a positive slope. The inverse function for compound 1 at effect level ϕ2 yields an unobserved negative value.

Experimentalists often use one of several approaches to determine if a mixture obeys CA: 1) draw the isoboles of an empirical response surface and decide if they are consistent with being linear, 2) construct isoboles predicted by CA by connecting the empirical EC_ϕ,i and test whether experimental mixture results fall on it, 3) compare the observed response surface with that expected under CA [3]. The EC_ϕ,i can be graphically estimated from marginal concentration response curves as illustrated in Figure 1. If the concentration response curves are described by invertible mathematical functions, the EC_ϕ,i can be calculated

$${\mathit{EC}}_{\varphi ,i}=f_i^{-1}[\varphi ]$$ (4)

After inserting inverse functions into equation (3), one can sometimes, but not always, solve for the response surface function predicted by CA. Making predictions for CA does not require such a solution however, e.g., one can always use equation (3) to compute isoboles.

2.2. Generalized concentration addition (GCA)

While CA has received extensive use, it has a major limitation. It can only be used for effect levels that can be achieved by all components in the mixture individually. This is not an issue when all components have the same maximal effect. However, it is common in toxicology/pharmacology for some compounds to have lower maximal effect levels than others, i.e., lower efficacy. Such compounds are often referred to as partial agonists, as compared with full agonists. As illustrated in Figure 1, concentration addition can only be applied for effect levels less than that of the least efficacious component in the mixture. The reason is that for higher effect levels, the EC_ϕ,i does not exist, e.g., there is no positive concentration of compound 1 that can achieve this effect level [1,5].

As one solution to this problem, we defined generalized concentration addition (GCA) as

$$\begin{gathered} \frac{X_1}{f_1^{-1}[\varphi ]}+\frac{X_2}{f_2^{-1}[\varphi ]}=1 \\ {f}_i^{-1}[\varphi ]\in \Re \text{for all}\varphi \text{such that}0<\varphi <{\varphi }_{\max } \end{gathered}$$ (5)

where ℜrefers to real numbers. In other words the inverse functions must yield real numbers for the entire effect range, except possibly at asymptotes. ϕ_max is the maximum effect level for the most efficacious compound. Equation (5) is easily generalized to more than two compounds [1]. This definition differs from that for CA, equation (3), in three crucial ways: 1) One specifies mathematical functions f_i[X_i] for the concentration-response curves (not required for CA); 2) these functions must be invertible over the desired range of ϕ yielding real numbers, i.e., they are not restricted to positive numbers as in CA; 3) the latter requirement must hold for all compounds of interest. Let ϕ_crit be the lowest maximal value for an individual compound in a mixture (e.g., compound 1 in Figure 1). Then for ϕ <ϕ_crit, GCA yields the same results as CA (although CA can also use graphical methods). For ϕ >ϕ_crit, CA cannot be used but GCA may.

The generalization to real numbers allows denominators in equation (5) to be negative for ϕ >ϕ_crit (As discussed below, it is an open question whether equation (5) can be further expanded from real numbers to complex numbers). While the isoboles predicted by GCA are still linear for any choice of ϕ, their slopes can now be negative or positive. This is consistent with that empirically observed for mixtures of full agonists and partial agonists. At low effect levels, partial agonists add (via CA) to the effect of a full agonist. But at high effect levels, partial agonists antagonize full agonists: maintaining the same effect level ϕ (staying on the isobole) as one increases the concentration of the partial agonist requires increasing the concentration of the full agonist, i.e., the isobole has a positive slope [1]. As illustrated in Figure 2b, experimental data to test the predictions of GCA for mixtures exists for the first quadrant where concentrations are positive.

3. Results

3.1. Mixtures and downstream effects

Risk assessors typically want to make predictions for the effect of mixtures on adverse outcomes. However, increased attention is being given to the chain of events that lead to the outcome. Adverse outcome pathways (AOP) provide one method for organizing knowledge linking molecular initiating events (MIE) to adverse outcomes via a series of key events, [e.g., 6,7]. One important example of a MIE is receptor activation by a ligand.

This general concept can be of use in predicting the biological effect of mixtures (Figure 3). Suppose the following: i) we can predict the biological effect of the mixture on the MIE (measured by ϕ) using equation (2); ii) for a downstream outcome y the mixture components act via the same MIE and only via the MIE; iii) there exists some link function

$$y=h[\varphi ]$$ (6)

that connects the MIE to the outcome via the key event relationships, possibly empirically. We can then predict the effect of the mixture on the outcome y using the composite function

$$y=h[f[X_1,X_2]]$$ (7)

Figure 3.

Adverse outcome pathways are a method for organizing information on the linkage between molecular initiating events (MIEs) and adverse outcomes via a series of key events. Suppose that for outcome y, the mixture components act only via the same MIE. Then the shape of the isoboles at the MIE and the outcome will have the same shape.

Note that the link function can change the shape of the response surface and the marginal concentration-response functions. The operational model of pharmacology is one method used for examining this issue [8,9]. Importantly, the link function does not change the location and shape of isoboles: it changes the numerical value of an isobole but not its position in the X₁-X₂ plane [1]. This property may simplify the mixtures problem when it is easier to make biologically-based predictions for the MIE (ϕ) than for the outcome (y).

3.2. Choice of concentration-response functions for GCA: Hill functions

As discussed earlier, GCA requires specification of the mathematical concentration-response functions for each component. How should this be done? It is tempting to employ common descriptive models such as Hill functions

$$\varphi =f_i[X_i]=\frac{{\alpha }_i{\left(\frac{X_i}{K_i}\right)}^{p_i}}{1+{\left(\frac{X_i}{K_i}\right)}^{p_i}}$$ (8a)

where background effects have been subtracted out. The inverse function of equation (8a) is

$$X_i=f_i^{-1}[\varphi ]=K_i{\left(\frac{\varphi }{{\alpha }_i-\varphi }\right)}^{1∕p_i}$$ (8b)

Unfortunately, the invertible function requirement for GCA is not always met for equation (8b) for ϕ>ϕ_crit. For values of p greater than one (and 0<p<1 where 1/p is not an integer), the inverse will produce imaginary values of X_i for real values of ϕ>α_i, the maximum value for that compound. For example, suppose we are trying to estimate response under GCA for an effect level between the maxima of two compounds (0<α₁<ϕ<α₂) as in Figure 1 using two Hill functions with p=2. Application of equation (8b) for compound 1 yields an imaginary number (e.g., ia). Application of the inverse function for compound 2 produces a positive real number (e.g., b). Application of the GCA equation (5) and some algebra yields an isobole equation predicting the value of X₁ as a function of X₂:

$$\frac{X_1}{\mathit{ia}}+\frac{X_2}{b}=1\to X_1=i\left(a-\frac{a}{b}{X}_2\right)$$ (9)

On this isobole X₁ is purely imaginary for all real values of X₂. As a result, no predictions can be made for the first quadrant of the real X₁-X₂ plane that describes mixtures experiments. Note that taking the real part of equation (9) doesn’t help since then X₁=0 (a trivial solution since it is the marginal function). Hence, while GCA works mathematically in this situation, it cannot be used for predictions. This is why we restricted GCA in equation (5) to inverse functions producing real numbers. It is an open question whether other situations with complex values of ϕ and complex isoboles may be used.

3.3. Choice of concentration-response functions for GCA: pharmacodynamic models (PDM) for a receptor with a single binding site

The discussion above about MIEs provides another avenue for constructing concentration response functions using knowledge of mechanisms. Ligand-receptor interactions are enormously important in biology and form the MIE for many adverse outcome pathways in toxicology as well as playing a critical role in pharmacology [e.g., 6,7].

We previously examined the important case of receptor systems with a single binding site [1]. Briefly, assume a very simple two-step mechanism where i) a ligand (X) reversibly binds a receptor (R); ii) the ligand-receptor complex (XR) can transform into an active form (XR*) allowing biological signaling:

$$X+R\leftrightarrow \mathit{XR}\leftrightarrow {\mathit{XR}}^{\ast }$$ (10)

For example, transformation may involve conformational change, phosphorylation, shedding or addition of cofactors, etc. As is common in simple pharmacodynamic modeling [9], we assume mass action, equilibrium and effect ϕ equal (or proportional) to the concentration of activated ligand-receptor complexes XR*. We then derive the following concentration response function

$$f_i[X_i]=\frac{{\alpha }_i\frac{X_i}{K_i}}{1+\frac{X_i}{K_i}}$$ (11)

α_i describes the maximum effect level for compound i, K_i describes its EC₅₀ (Because of the activation step, the EC₅₀ is not the same as the binding affinity). While equation (11) is a Hill function, it has a Hill exponent (p) of one (implying linearity for X_i <<K_i and reaching a maximum for X_i >>K_i) and is invertible yielding real numbers. We can therefore apply GCA, producing the joint concentration-response function

$$\varphi =f[A,B]=\frac{{\alpha }_a\frac{A}{K_a}+{\alpha }_b\frac{B}{K_b}}{1+\frac{A}{K_a}+\frac{B}{K_b}}$$ (12)

where, to reduce notation, we let A stand for X₁ and B for X₂, and similarly for subscripts.

We have empirically shown that a set of mixtures of AhR ligands were fit by equation (12), consistent with GCA and possessing both negatively and positively sloped isoboles [10], and similarly for mixtures of PPARγ ligands [11]. Further support for equation (12) is provided by a mechanistic model for a mixture of ligands [1]. Extend the same reaction scheme to two ligands:

$$\begin{gathered} A+R\leftrightarrow \mathit{AR}\leftrightarrow {\mathit{AR}}^{\ast } \\ B+R\leftrightarrow \mathit{BR}\leftrightarrow {\mathit{BR}}^{\ast } \end{gathered}$$ (13)

Again assuming mass action, equilibrium and effect equal (or proportional) to the concentration of activated ligand-receptor complexes, one derives the same equation (12) as GCA [1]. (Equation 12) allows partial agonism via differences in the propensity for a ligand-receptor complex to take the active form. The partial agonist can then increase response at low effect levels and reduce response at high effect levels, switching at ϕ_crit (i.e., α_a if α_a <α_b). In the limiting case of the partial agonist having zero efficacy (e.g., α_a=0), equation (12) becomes the classical Gaddum equation for the effect of a competitive antagonist; it has only positively sloped isoboles [1].

Note that equation (5) is the definition of GCA, not (12). We would only expect equation (12) to apply for systems that meet the underlying mechanistic assumptions or, potentially, if equation (11) fits the marginal concentration-response data well. Receptors with multiple binding sites would not generally be expected to meet these assumptions. In particular, a Hill coefficient (exponent p) fit to the data is then unlikely to be one, changing the curvature of the concentration-response function. Use of equation (12) is then unlikely to fit data particularly well [e.g., 12].

3.4. Choice of concentration-response functions for GCA: pharmacodynamic models for receptors that homodimerize

Homodimers are an important class of receptors that participate in a vast array of biological processes from cell growth to organ system function. Receptors that act as homodimers include nuclear receptors (e.g., receptors for androgens, estrogens, progesterone, glucocorticoids, mineralocorticoids)[13], Toll-like receptors [14], receptor tyrosine kinases (e.g., epidermal growth factor receptors, fibroblast growth factor receptors [15,16], G protein-coupled receptors (i.e., receptors for light, odorants, peptides, neurotransmitters, hormones, lipids, and chemokines)[17]. These receptors are activated either by a single, bivalent ligand binding to each monomer or by ligands binding to each of the monomers, the focus of this paper. As a result of ligand-induced activation, transcription is activated either via homodimerization triggering signal transduction through a phosphorylation cascade (e.g., Toll-like receptors, receptor tyrosine kinases, G protein-coupled receptors) or via homodimerization triggering direct binding to DNA (e.g., nuclear receptors).

Suppose that a ligand-homodimer system obeys the following simple three-step reaction scheme [e.g., 18]

$$\begin{gathered} A+R\leftrightarrow \mathit{AR} \\ \mathit{AR}\leftrightarrow {\mathit{AR}}^{\ast } \\ {\mathit{AR}}^{\ast }+{\mathit{AR}}^{\ast }\leftrightarrow {\mathit{AR}}^{\ast \ast }\mathit{RA} \end{gathered}$$ (14)

This scheme assumes that a ligand-receptor complex (AR) transform into a state (AR*) that allows formation of the homodimer (AR**RA). Assume that only the active dimer (AR**RA) leads to signal transduction to the effect of interest. Let us again assume mass action, equilibrium, and effect equal to (or proportional) to the concentration of activated homodimers. As shown in the appendix section 2, the result can be written as a composite function

$$\varphi =f_a[A]=g[{\theta }_a[A]]$$ (15a)

$${\theta }_a[A]=\frac{{\alpha }_a\frac{A}{K_a}}{1+\frac{A}{K_a}}$$ (15b)

$$g[\theta ]=\lambda {\left(-\frac{1}{\theta }+\sqrt{\frac{1}{{\theta }^2}+1}\right)}^2\text{for}\theta >0,\varphi =0\text{for}\theta =0$$ (15c)

with K_a , α_a, λ>0 and where θ (and θ_a[A]) is the inner part of the composite function. The first two steps of the reaction scheme, ligand binding and formation of the active ligand-monomer complex, described by equation (15b), have the same form as the earlier result for a receptor with a single binding site. Homodimerization follows this step, described by the additional function (15c). The overall concentration response relationship is described by the composite function (15a). λ is a positive parameter that is independent of the ligand, including such factors as receptor concentration and potentially other aspects of the biology for assays used to measure ϕ. λ and α_i together determine the maximal effect for a compound (appendix section 3).

The homodimer function has a shape similar to, but different from a Hill function. To illustrate this, we generated synthetic data using equation (15) (with no random noise added) and fit a Hill function to them. Figure 4a shows a Hill plot of these data, plotting log[ϕ/(1-ϕ)] vs log[A], following normalization of ϕ [e.g., 19]. The points deviate from the straight line that would be produced by data exactly following a Hill function. In a Hill plot, the slope estimates the Hill coefficient p. Figure 4B displays the estimated Hill function and the homodimer function. The Hill coefficient estimated here, p=1.23 depends in part on the points used for the regression. Although similar, the homodimer function is steeper at low doses (slope of 2) and shallower at high doses (See appendix section 4).

Figure 4.

a) Hill plot of the homodimer function for one ligand. Dots are synthetic data (n=100) generated using the following parameters: α=1; K=10⁻⁶; background response of 10; λ=1000, and then normalized and transformed for the Hill plot. b) Plot of the homodimer function (solid) and the Hill function (dashed) fitted to these data using R package drc [20].

The inverse function for equation (15) can be described via the composite function

$$A={\theta }_a^{-1}[g^{-1}[\theta ]]$$ (16a)

$$\theta =g^{-1}[\varphi ]=\frac{2\sqrt{\frac{\varphi }{\lambda }}}{1-\frac{\varphi }{\lambda }}$$ (16b)

$$A={\theta }_a^{-1}[\theta ]=K_a\left(\frac{\theta }{{\alpha }_a-\theta }\right)$$ (16c)

Equation 16c is identical in form to our earlier result for a receptor with a single binding site. For ϕ and λ >0, equation (16b) generates real numbers (non-negative as shown in appendix section 5). Thus, as in the earlier single-binding site model, the sign of the overall inverse function is determined by whether θ is greater than or less than α_a. As the inverse function generates real numbers, it meets the requirements for GCA.

Applying GCA to equation (15) for two compounds differing in parameters α_i and K_i (with a common λ as discussed earlier), we obtain

$$\varphi =f[A,B]=g[\theta [A,B]]$$ (17a)

$$\theta [A,B]=\frac{{\alpha }_a\frac{A}{K_a}+{\alpha }_b\frac{B}{K_b}}{1+\frac{A}{K_a}+\frac{B}{K_b}}$$ (17b)

$$g[\theta ]=\lambda {\left(-\frac{1}{\theta }+\sqrt{\frac{1}{{\theta }^2}+1}\right)}^2$$ (17c)

again expressed as a composite function (see appendix section 6). The difference from the homodimer function for a single compound is that we have used equation (17b) instead of equation (15b). Equation (17b) has the same form as the solution we found for the single binding site model, equation (12). This correspondence occurs for several reasons: i) we can think of dimerization as an effect downstream of transformation of the ligand-receptor complex; this can be described mathematically using a composite function. ii) Applying a function to the outcome of a system that is GCA will remain GCA. As a result, application of equation (17c) to (17b) does not change the shape of the isoboles of equation (17b), leading to both negatively and positively sloped isoboles as in the single site receptor model applied to mixtures of full and partial agonists.

We can also derive a homodimer function for a mixture by expanding the reaction mechanism equation (14) to include two ligands, allowing for formation of the mixed ligand homodimer AR**RB:

$$A+R\leftrightarrow \mathit{AR}\leftrightarrow {\mathit{AR}}^{\ast }$$ (18a)

$${\mathit{AR}}^{\ast }+{\mathit{AR}}^{\ast }\leftrightarrow {\mathit{AR}}^{\ast \ast }\mathit{RA}$$ (18b)

$$B+R\leftrightarrow \mathit{BR}\leftrightarrow {\mathit{BR}}^{\ast }$$ (18c)

$${\mathit{BR}}^{\ast }+{\mathit{BR}}^{\ast }\leftrightarrow {\mathit{BR}}^{\ast \ast }\mathit{RB}$$ (18d)

$${\mathit{AR}}^{\ast }+{\mathit{BR}}^{\ast }\leftrightarrow {\mathit{AR}}^{\ast \ast }\mathit{RB}$$ (18e)

Suppose the K₃, K₄, K₅ are the equilibrium constants describing the formation of the activated homodimers in equations (18b), (18d) and (18e), respectively. As shown in the appendix section 7, the system yields the same solution as equation (17), i.e., it is GCA, provided that formation of the mixed ligand homodimer (AR**RB) occurs with propensity equal to the geometric mean of the propensities for the single ligand dimers AR**RA and BR**RB:

$$K_5=\sqrt{K_3{K}_4}$$ (19)

This condition can be thought of as a type of non-cooperativity. Equation (19) must be true if formation of a homodimer doesn’t depend on the type of activated monomer, i.e., K₃=K₄. Cooperativity, or non-cooperativity could be empirically tested using assays that specifically examine the dimerization step or, more generally, by the fit of data to equation (17). Homodimerization could still depend on ligands through differential ability to form activated monomers.

4. Discussion

This paper extends our earlier work on mixtures that act on receptors with single binding sites [1] to one mechanism for receptors that homodimerize. We show conditions under which the latter meet the requirements for GCA, unlike general Hill functions. The homodimer concentration response function is derived from pharmacodynamic models and the GCA solution matches the mechanistically-based solution for mixtures under certain conditions. We focus on ligand-receptor interactions because this type of molecular initiating event (MIE) is crucial to many adverse outcome pathways, e.g., androgen receptor ligands and male reproductive effects, e.g. [21]. We also show how and when composite functions can be used to link these MIEs to downstream outcomes, potentially altering the shape of the concentration response curves but retaining the shape of isoboles.

The androgen, glucocorticoid, progesterone and mineralocorticoid receptors appear to follow the homodimer mechanism we have analyzed, equation (14), at least approximately [e.g., 18]. Our mechanistic model is of course simplified in many respects. The estrogen receptor is evolutionarily distinct from the other steroid receptors [22], and some important details of the mechanism for the estrogen receptor, another homodimer system, appear to differ. These will be discussed in another paper. Real world mixtures may include different types of ligands for these receptors—endogenous ligands, pharmaceuticals, plant-based compounds and synthetic chemicals—that may differ in their pharmacological properties such as potency and efficacy. It is thus important to be able to deal with mixtures containing full and partial agonists.

The pharmacodynamic concentration-response function we derived for homodimers has the property that it is concave-up at low concentration, approximating a quadratic at very low concentrations. This contrasts with the linearity at low concentration of the model for activation of receptors with a single binding site. This property makes application of GCA more appropriate for ligands of homodimers where non-linearity at low concentration is typically expected.

A few other approaches have been proposed for analyzing mixtures of full and partial agonists [12,23,24]. An advantage of the GCA approach is that it has a pharmacological basis and is equivalent to CA for effect levels below the maximum of the least efficacious mixture component. A weakness of GCA is that the concentration–response functions must meet certain mathematical requirements. For example, the homodimer function we derived meets the requirements of GCA, but the commonly used general Hill function does not. We have focused on ligand-receptor interactions for the MIE as these are both biologically very important and many aspects are understood allowing modeling. This potential limitation may be partly mitigated by the use of link functions to extend mixtures predictions from MIEs to downstream events. More research is needed on a number of fronts, e.g.: on the possibility of extending GCA to complex isoboles, ligand-receptor mechanisms different than those discussed here, extension from MIEs to events downstream on AOPs, mixtures where components bind different sites of heterodimers, and situations where mixture components act at different points of the AOP [e.g., 25]. Empirical data for homodimer systems following the modeled mechanism (e.g., androgen receptor ligands) are needed to test whether they obey GCA.

An interesting aspect of the pharmacodynamic models discussed here for mixtures of ligands that bind to single site-receptors and homodimers is that they are GCA, yielding linear isoboles with slopes that can vary in sign. The extension from negatively sloped isoboles (CA) to negatively or positively sloped isoboles (GCA) has been theoretically and empirically examined for receptors with single binding sites [1,10,11] and is consistent with the Gaddum equation for the effect of a competitive antagonist as discussed earlier. The results presented here suggest the same pattern of isoboles may occur for homodimer systems.

An important aspect of these pharmacodynamic models is that ligands compete for the same site, partly in keeping with one of the rationales for CA—one compound substituting for another at some fixed ratio without changing the effect level—but extending this idea to full and partial agonists. This raises the hypothesis that systems where ligands compete for the same site(s) will obey GCA. As the pharmacodynamic model for homodimers shows, exceptions are possible where there is cooperativity (or anti-cooperativity) between ligands in the formation of the homodimer.

“Additivity” and “interaction” are only meaningful in pharmacology and toxicology (as well as epidemiology and biostatistics) with respect to a definition of non-interaction [1]. GCA provides one such definition that extends previous ideas, allowing predictions to be made for mixtures of full and partial agonists.

Highlights.

• Effect of chemical mixtures on outcomes downstream of a molecular initiating event

• Mixtures of full and partial agonists

• Pharmacodynamic model for ligands of homodimers such as androgens

• The homodimer function follows generalized concentration addition

Abbreviations:

CA

concentration addition

GCA

generalized concentration addition

RPF

relative potency factor

MIE

molecular initiating event

AOP

adverse outcome pathway

Appendix

1. Definition of CA and GCA for higher dimensions

$$\begin{gathered} \sum _{i=1}^N\frac{X_i}{{\mathit{EC}}_{\varphi ,i}}=1,{\mathit{EC}}_{i,\varphi }>0 \\ \sum _{i=1}^N\frac{X_i}{f_i^{-1}[\varphi ]}=1,f_i^{-1}[\varphi ]\in \Re \mathrm{for}\mathrm{all}\varphi \mathrm{such}\mathrm{that}0<\varphi <{\varphi }_{\max } \end{gathered}$$ (A.1)

except at asymptotes.

2. Homodimer dose-response function

assumptions:

• Ligand A reversibly binds to receptor R

• The ligand-receptor complex AR can transform to an active form AR*

• The activated complex can dimerize, forming the homodimer AR**RA

• The response of interest ϕ is equal (or proportional) to the concentration of active homodimers AR**RA

• Equilibrium

• Mass balance (molar) of receptors with constant total concentration R₀

• Receptor binding does not appreciably affect the concentration of ligand A, for example if there is a large extracellular volume [9]

The goal is to derive ϕ as a function of A. To simply notation we will not use the bracket notation often employed for concentrations, i.e., A is equivalent to [A].

Assuming equilibrium, we can, from the reaction scheme, directly define the microscopic equilibrium constants, K_A, K₁, K₃, assumed constant and >0

$$\begin{gathered} A+R\leftrightarrow \mathit{AR}{K}_A=\frac{\mathit{AR}}{\mathit{AR}} \\ \mathit{AR}\leftrightarrow {\mathit{AR}}^{\ast }{K}_1=\frac{\mathit{AR}}{{\mathit{AR}}^{\ast }} \\ {\mathit{AR}}^{\ast }+{\mathit{AR}}^{\ast }\leftrightarrow {\mathit{AR}}^{\ast \ast }\mathit{AR}{K}_3=\frac{{\left({\mathit{AR}}^{\ast }\right)}^2}{2{\mathit{AR}}^{\ast \ast }\mathit{RA}} \end{gathered}$$ (A.2)

where the ½ in K₃ is a statistical factor [26] that will become important when we come to mixtures.

Express response ϕ in terms of A and R using (A.2)

$$\varphi ={\mathit{AR}}^{\ast \ast }\mathit{RA}=\left(\frac{A^2}{2K_3{K}_1^2{K}_A^2}\right)R^2$$ (A.3)

where we’ve omitted any proportionality constant for simplicity.

Mass balance on R; solve quadratic for R, using the positive root.

$$\begin{gathered} R_0=R+\mathit{AR}+{\mathit{AR}}^{\ast }+2{\mathit{AR}}^{\ast \ast }\mathit{RA} \\ =R\left(1+\frac{A}{K_A}+\frac{A}{K_1{K}_A}\right)+2R^2\left(\frac{A^2}{2K_3{K}_1^2{K}_A^2}\right) \end{gathered}$$ (A.4a)

$$\begin{gathered} R=\frac{-\left(1+\frac{A}{K_A}+\frac{A}{K_1{K}_A}\right)+\sqrt{{\left(1+\frac{A}{K_A}+\frac{A}{K_1{K}_A}\right)}^2+4\left(\frac{A^2}{K_3{K}_1^2{K}_A^2}\right)R_0}}{2\left(\frac{A^2}{K_3{K}_1^2{K}_A^2}\right)}\mathit{for}A\ne 0 \\ =R_0\mathit{for}A=0 \end{gathered}$$ (A.4b)

Derive ϕ = f_a[A] by substituting (A.4b) into (A.3)

$$\varphi =\frac{A^2}{2K_3{K}_1^2{K}_A^2}{\left(\frac{-\left(1+\frac{A}{K_A}+\frac{A}{K_1{K}_A}\right)+\sqrt{{\left(1+\frac{A}{K_A}+\frac{A}{K_1{K}_A}\right)}^2+4\left(\frac{A^2}{K_3{K}_1^2{K}_A^2}\right)R_0}}{2\left(\frac{A^2}{K_3{K}_1^2{K}_A^2}\right)}\right)}^2$$ (A.5)

Rewrite (A.5) with some algebra

$$\begin{gathered} \varphi =\frac{R_0}{2}{\left(\frac{-\left(1+\frac{A}{K_A}+\frac{A}{K_1{K}_A}\right)}{2\sqrt{\frac{R_0{A}^2}{K_3{K}_1^2{K}_A^2}}}+\sqrt{\frac{{\left(1+\frac{A}{K_A}+\frac{A}{K_1{K}_A}\right)}^2}{4\left(\frac{R_0{A}^2}{K_3{K}_1^2{K}_A^2}\right)}+1}\right)}^2\mathit{for}A\ne 0 \\ =0\mathit{for}A=0 \end{gathered}$$ (A.6)

Express (A.6) as a composite function

$$\begin{gathered} \varphi =g[\theta ]=\lambda {\left(-\frac{1}{\theta }+\sqrt{\frac{1}{{\theta }^2}+1}\right)}^2\text{for}\theta \ne 0 \\ =0\text{for}\theta =0 \end{gathered}$$ (A.7)

$$\theta [A]=\frac{\frac{2\sqrt{R_0}}{\sqrt{K_3}{K}_1}\frac{A}{K_A}}{1+\frac{A}{K_A}\left(\frac{1+K_1}{K_1}\right)}$$ (A.8)

Note: λ does not depend on the ligand and exceeds zero. In practice, it will typically contain a proportionality constant.

Reparameterize (A.8), reducing the number of parameters

$$\theta [A]=\frac{{\alpha }_a\frac{A}{K_a}}{1+\frac{A}{K_a}};{\alpha }_a=\frac{2\sqrt{R_0}}{\sqrt{K_3}(1+K_1)};K_a=K_A\left(\frac{K_1}{1+K_1}\right)$$ (A.9)

Note that α_a,K_a > 0

3. Maximum value of ϕ

$$\begin{gathered} \mathit{As}A\to \infty ,\theta \to {\alpha }_a \\ \mathit{As}\theta \to {\alpha }_a,\varphi \to \lambda {\left(-\frac{1}{{\alpha }_a}+\sqrt{\frac{1}{{\alpha }_a^2}+1}\right)}^2 \end{gathered}$$ (A.10)

4. Low dose behavior of the homodimer function

$$\begin{gathered} \varphi =g[\theta ]=\lambda {\left(-\frac{1}{\theta }+\sqrt{\frac{1}{{\theta }^2}+1}\right)}^2~\lambda {\left(-\frac{1}{\theta }+\left(\frac{1}{\theta }+\frac{\theta }{2}+O[\theta {]}^3\right)\right)}^2\mathrm{using}\mathrm{Taylor}\mathrm{Series}\mathrm{on}\theta \\ ~\frac{\lambda }{4}{\theta }^2 \\ ~\left(\frac{\lambda {\alpha }_a^2}{4K_a^2}\right)A^2\mathit{since}\theta [A]~{\alpha }_a\frac{A}{K_a}\mathrm{at}\mathrm{low}\mathrm{doses}A<<K_A \end{gathered}$$ (A.11)

5. Inverse homodimer function

The inverse functions for (A.7) and (A.9) are

$$\theta =g^{-1}[\varphi ]=\frac{2\sqrt{\frac{\varphi }{\lambda }}}{1-\frac{\varphi }{\lambda }}$$ (A.12)

$$A={\theta }^{-1}[\theta ]=K_A\left(\frac{\theta }{{\alpha }_a-\theta }\right)$$ (A.13)

Assume ϕ, λ, α>0. From (A.12), θ is real. Using (A.10), one can show that ϕ_max < λ.

Hence, θ is non-negative.

6. GCA solution for two ligands obeying the homodimer function

Substituting the inverse function into the GCA equation we obtain

$$\begin{gathered} \frac{A}{{\theta }_a^{-1}[g^{-1}[\varphi ]]}+\frac{B}{{\theta }_b^{-1}[g^{-1}[\varphi ]]}=1 \\ \frac{A}{K_A\left(\frac{g^{-1}[\varphi ]}{{\alpha }_a-g^{-1}[\varphi ]}\right)}+\frac{B}{K_B\left(\frac{g^{-1}[\varphi ]}{{\alpha }_b-g^{-1}[\varphi ]}\right)}=1 \end{gathered}$$

Simplifying, we obtain

$$\begin{gathered} g^{-1}[\varphi ]=\frac{{\alpha }_a\frac{A}{K_a}+{\alpha }_b\frac{B}{K_b}}{1+\frac{A}{K_a}+\frac{B}{K_b}} \\ \varphi =g\left(\frac{{\alpha }_a\frac{A}{K_a}+{\alpha }_b\frac{B}{K_b}}{1+\frac{A}{K_a}+\frac{B}{K_b}}\right) \end{gathered}$$ (A.14)

7. Derivation of the homodimer function for a mixture of two ligands

assumptions:

• Ligands A and B reversibly bind to receptor R

• The ligand-receptor complexes AR, BR can transform to active forms AR*, BR*

• The activated complex can dimerize, forming homodimers AR**RA, BR**RB, AR**RB

• The response of interest ϕ is equal (or proportional) to the sum of the concentration of active homodimers AR**RA + BR**RB + AR**RB

• Equilibrium

• Mass balance (molar) of receptors with constant total concentration R₀

• Receptor binding does not appreciably affect the concentration of ligands, for example if there is a large extracellular volume

Definition of microscopic equilibrium constants, assumed constant and >0

$$\begin{gathered} A+R\leftrightarrow \mathit{AR}{K}_A=\frac{\mathit{AR}}{\mathit{AR}} \\ \mathit{AR}\leftrightarrow {\mathit{AR}}^{\ast }{K}_1=\frac{\mathit{AR}}{{\mathit{AR}}^{\ast }} \\ {\mathit{AR}}^{\ast }+{\mathit{AR}}^{\ast }\leftrightarrow {\mathit{AR}}^{\ast \ast }\mathit{RA}{K}_3=\frac{{\left({\mathit{AR}}^{\ast }\right)}^2}{2{\mathit{AR}}^{\ast \ast }\mathit{RA}} \\ B+R\leftrightarrow \mathit{BR}{K}_B=\frac{BR}{\mathit{BR}} \\ \mathit{BR}\leftrightarrow {\mathit{BR}}^{\ast }{K}_2=\frac{\mathit{BR}}{{\mathit{BR}}^{\ast }} \\ {\mathit{BR}}^{\ast }+{\mathit{BR}}^{\ast }\leftrightarrow {\mathit{BR}}^{\ast \ast }\mathit{RB}{K}_4=\frac{{\left({\mathit{BR}}^{\ast }\right)}^2}{2{\mathit{BR}}^{\ast \ast }\mathit{RB}} \\ {\mathit{AR}}^{\ast }+{\mathit{BR}}^{\ast }\leftrightarrow {\mathit{AR}}^{\ast \ast }\mathit{RB}{K}_5=\frac{\left({\mathit{AR}}^{\ast }\right)\left({\mathit{BR}}^{\ast }\right)}{{\mathit{AR}}^{\ast \ast }\mathit{RB}} \end{gathered}$$ (A.15)

K₄ has a statistical factor of ½, parallel to K₃, but K₅ does not. The statistical factors become important because we are actually examining an overall equilibrium.

$${\mathit{AR}}^{\ast \ast }\mathit{RA}+{\mathit{BR}}^{\ast \ast }\mathit{RB}\leftrightarrow 2A+2B+4R\leftrightarrow 2{\mathit{AR}}^{\ast \ast }\mathit{RB}$$ (A.16)

Use (A.15) to express response ϕ in terms of A, B and R:

$$\varphi ={\mathit{AR}}^{\ast \ast }\mathit{RA}+{\mathit{BR}}^{\ast \ast }\mathit{RB}+{\mathit{AR}}^{\ast \ast }\mathit{RB}=\left(\frac{A_2}{2K_3{K}_1^2{K}_A^2}+\frac{B^2}{2K_4{K}_2^2{K}_B^2}+\frac{AB}{K_5{K}_1{K}_2{K}_A{K}_B}\right)R_2$$ (A.17)

Mass balance on R

$$\begin{gathered} R_0=R+\mathit{AR}+{\mathit{AR}}^{\ast }+\mathit{BR}+{\mathit{BR}}^{\ast }+2{\mathit{AR}}^{\ast \ast }\mathit{RA}+2{\mathit{BR}}^{\ast \ast }\mathit{RB}+2{\mathit{AR}}^{\ast \ast }\mathit{RB} \\ =R\left(1+\frac{A}{K_A}+\frac{A}{K_1{K}_4}+\frac{B}{K_B}+\frac{B}{K_2{K}_B}\right)+2R^2\left(\frac{A^2}{2K_3{K}_1^2{K}_A^2}+\frac{B^2}{2K_4{K}_2^2{K}_B^2}+\frac{AB}{K_5{K}_1{K}_2{K}_A{K}_B}\right) \end{gathered}$$ (A.18)

The steps for solving the quadratic, deriving the function and rewriting, are parallel with (A.4b), (A.5), (A.6) for A and B ≠0

$$\varphi =\frac{R_0}{2}{\left(\frac{-\left(1+\frac{A}{K_A}+\frac{A}{K_1{K}_A}+\frac{B}{K_B}+\frac{B}{K_2{K}_B}\right)}{2\sqrt{\frac{R_0{A}^2}{K_3{K}_1^2{K}_A^2}+\frac{R_0{B}^2}{K_4{K}_2^2{K}_B^2}+\frac{2R_0\mathit{AB}}{K_5{K}_1{K}_2{K}_A{K}_B}}}+\sqrt{\frac{{\left(1+\frac{A}{K_A}+\frac{A}{K_1{K}_A}+\frac{B}{K_B}+\frac{B}{K_2{K}_B}\right)}^2}{4\left(\frac{R_0{A}^2}{K_3{K}_1^2{K}_A^2}+\frac{R_0{B}^2}{K_4{K}_2^2{K}_B^2}+\frac{2R_{0}AB}{K_5{K}_1{K}_2{K}_A{K}_B}\right)}+1}\right)}^2$$ (A.19)

which can be rewritten as a composite function using the same outer function (A.7) but substituting the following for the inner function (A.8)

$$\theta [A,B]=\frac{2\sqrt{\frac{R_0{A}^2}{K_3{K}_1^2{K}_A^2}+\frac{R_0{B}^2}{K_4{K}_2^2{K}_B^2}+\frac{2R_0\mathit{AB}}{K_5{K}_1{K}_2{K}_A{K}_B}}}{1+\frac{A}{K_A}\left(\frac{1+K_1}{K_1}\right)+\frac{B}{K_B}\left(\frac{1+K_2}{K_3}\right)}$$ (A.20)

Reparameterize (A.20) using (A.9) for A and (A.21) for B, we obtain (A.22)

$${\alpha }_b=\frac{2{\sqrt{R}}_0}{K_4(1+K_2)};K_b=K_B\left(\frac{K_2}{1+K_2}\right)$$ (A.21)

$$\theta [A,B]=\frac{\sqrt{{\left({\alpha }_a\frac{A}{K_a}\right)}^2+{\left({\alpha }_b\frac{B}{K_b}\right)}^2+2{\alpha }_a{\alpha }_b\frac{A}{K_a}\frac{B}{K_b}\left(\frac{\sqrt{K_3{K}_4}}{K_5}\right)}}{1+\frac{A}{K_a}+\frac{B}{K_b}}$$ (A.22)

$$\theta [A,B]=\frac{{\alpha }_a\frac{A}{K_a}+{\alpha }_b\frac{B}{K_b}}{1+\frac{A}{K_a}+\frac{B}{K_b}}\mathit{if}{K}_5=\sqrt{K_3{K}_4}$$ (A.23)

Hence, the pharmacodynamic model for the mixture of ligands for the homodimer system gives us the GCA solution (A.14) if $K_5=\sqrt{K_3{K}_4}$