On the analysis of ligand-directed signaling at G protein-coupled receptors

Abstract

The phenomenon of “ligand-directed signaling” is often considered to be inconsistent with the traditional receptor theory. In this review, I show how the mathematics of the receptor theory can be used to measure the observed affinity and relative efficacy of protean ligands at G protein-coupled receptors. The basis of this analysis rests on the assumption that the fraction of agonist bound in the form of the active receptor–G protein–guanine nucleotide complex is the biochemical equivalent of the pharmacological stimulus. Consequently, this stimulus function is analogous to the current response of a ligand-gated ion channel. Because guanosine triphosphate (GTP) greatly inhibits the formation of the active quaternary complex, even the most efficacious agonists probably only elicit partial receptor activation, and it seems likely that the ceiling of 100% receptor activation is not reached in the intact cell with high intracellular concentrations of GTP. Under these conditions, the maximum of the stimulus function is proportional to the ratio of microscopic affinity constants of the agonist for ground and active states. Ligand-directed signaling depends on the existence of different active states of the receptor with different selectivities for different G proteins or other effectors. This phenomenon can be characterized using classic pharmacological methods. Although not widely appreciated, it is possible to estimate the product of observed affinity and intrinsic efficacy expressed relative to that of another agonist (intrinsic relative activity) through the analysis of the concentration–response curves. No other information is required. This approach should be useful in quantifying agonist activity and in converting the two disparate parameters of potency and maximal response into a single parameter dependent only on the agonist–receptor–effector complex.

Appendix

Ternary complex model with guanine nucleotide and two G proteins

The ternary complex model can be described at three hierarchical levels of analysis. The first level is described in terms of the observed dissociation constant of the agonist for a given receptor complex. This dissociation constant has units of molarity, and it represents the concentration of agonist required for half-maximal formation of the receptor complex or state. The second level of analysis defines the relative abundances of the different types of receptor complexes (e.g., ARG ₁) in terms of the microscopic affinity constants of these complexes. The third level of analysis defines the relative abundances of the different states of the receptor for each receptor complex described at the second level. For the second and third levels of analysis, the microscopic affinity constants have units of inverse molarity.

In this section, the ternary complex model for two G proteins is derived at the level of receptor complexes (level 2 analysis), whereas the subsequent section describes the model at the level of receptor states (level 3 analysis). The model represents a composite of two models, each including a single G protein like that is shown in Fig. 5. In the composite model, the receptor (R) can potentially interact with two G proteins (G ₁ and G ₂), but only one G protein can bind to the receptor at a given time. Hence, the G proteins compete for the receptor. The equilibrium constants are denoted in the same manner as that described in Fig. 5 but with an additional subscript to indicate to which G protein the parameter is associated. The microscopic affinity constants describing the binding of agonist (A) to various receptor complexes (R, RG ₁, RG ₂, RG ₁ X, and RG ₂ X; X denotes guanine nucleotide) are:

$$K_1 = \frac{{\left[ {AR} \right]}}{{\left[ A \right]\left[ R \right]}}$$

(21)

$$\alpha _1 K_1 = \frac{{\left[ {ARG_1 } \right]}}{{\left[ A \right]\left[ {RG_1 } \right]}}$$

(22)

$$\alpha _2 K_1 = \frac{{\left[ {ARG_2 } \right]}}{{\left[ A \right]\left[ {RG_2 } \right]}}$$

(23)

$$\alpha _1 \gamma _1 K_1 = \frac{{\left[ {ARG_1 X} \right]}}{{\left[ A \right]\left[ {RG_1 X} \right]}}$$

(24)

$$\alpha _2 \gamma _2 K_1 = \frac{{\left[ {ARG_2 X} \right]}}{{\left[ A \right]\left[ {RG_2 X} \right]}}$$

(25)

The microscopic affinity constants of guanine nucleotide (X) for the various complexes of G protein are given by:

$$K_{31} = \frac{{\left[ {G_1 X} \right]}}{{\left[ X \right]\left[ {G_1 } \right]}}$$

(26)

$$K_{32} = \frac{{\left[ {G_2 X} \right]}}{{\left[ X \right]\left[ {G_2 } \right]}}$$

(27)

$$\beta _1 K_{31} = \frac{{\left[ {RG_1 X} \right]}}{{\left[ X \right]\left[ {RG_1 } \right]}}$$

(28)

$$\beta _2 K_{32} = \frac{{\left[ {RG_2 X} \right]}}{{\left[ X \right]\left[ {RG_2 } \right]}}$$

(29)

$$\beta _1 \gamma _1 K_{31} = \frac{{\left[ {ARG_1 X} \right]}}{{\left[ X \right]\left[ {ARG_1 } \right]}}$$

(30)

$$\beta _2 \gamma _2 K_{32} = \frac{{\left[ {ARG_2 X} \right]}}{{\left[ X \right]\left[ {ARG_2 } \right]}}$$

(31)

The interactions between the receptor and the two G proteins are described by the following equilibria. These have been normalized relative to the total receptor concentration ([R _T]) because the concentrations of the receptor and G protein are fixed at their local membrane concentrations.

$$K_{21} = \frac{{\left[ {RG_1 } \right]\left[ {R_{\text{T}} } \right]}}{{\left[ R \right]\left[ {G_1 } \right]}}$$

(32)

$$K_{22} = \frac{{\left[ {RG_2 } \right]\left[ {R_{\text{T}} } \right]}}{{\left[ R \right]\left[ {G_2 } \right]}}$$

(33)

$$\alpha _1 K_{21} = \frac{{\left[ {ARG_1 } \right]\left[ {R_{\text{T}} } \right]}}{{\left[ {AR} \right]\left[ {G_1 } \right]}}$$

(34)

$$\alpha _2 K_{22} = \frac{{\left[ {ARG_2 } \right]\left[ {R_{\text{T}} } \right]}}{{\left[ {AR} \right]\left[ {G_2 } \right]}}$$

(35)

$$\beta _1 K_{21} = \frac{{\left[ {RG_1 X} \right]\left[ {R_{\text{T}} } \right]}}{{\left[ R \right]\left[ {G_1 X} \right]}}$$

(36)

$$\beta _2 K_{22} = \frac{{\left[ {RG_2 X} \right]\left[ {R_{\text{T}} } \right]}}{{\left[ R \right]\left[ {G_2 X} \right]}}$$

(37)

$$\alpha _1 \beta _1 \gamma _1 K_{21} = \frac{{\left[ {ARG_1 X} \right]\left[ {R_{\text{T}} } \right]}}{{\left[ {AR} \right]\left[ {G_1 X} \right]}}$$

(38)

$$\alpha _2 \beta _2 \gamma _2 K_{22} = \frac{{\left[ {ARG_2 X} \right]\left[ {R_{\text{T}} } \right]}}{{\left[ {AR} \right]\left[ {G_2 X} \right]}}$$

(39)

The following conservation of mass equations apply:

$$\begin{aligned}\left[ {R_{\text{T}} } \right] = \left[ R \right] + \left[ {AR} \right] + \left[ {RG_1 } \right] + \left[ {RG_2 } \right] + \left[ {ARG_1 } \right] + \left[ {ARG_2 } \right] \\+ \left[ {RG_1 X} \right] + \left[ {RG_2 X} \right] + \left[ {ARG_1 X} \right] + \left[ {ARG_2 X} \right] \\ \end{aligned} $$

(40)

$$\left[ {G_{1{\text{T}}} } \right] = \left[ {G_1 } \right] + \left[ {G_1 X} \right] + \left[ {RG_1 } \right] + \left[ {ARG_1 } \right] + \left[ {RG_1 X} \right] + \left[ {ARG_1 X} \right]$$

(41)

$$\left[ {G_{2T} } \right] = \left[ {G_2 } \right] + \left[ {G_2 X} \right] + \left[ {RG_2 } \right] + \left[ {ARG_2 } \right] + \left[ {RG_2 X} \right] + \left[ {ARG_2 X} \right]$$

(42)

In these equations, G _1T and G _2T denote the total concentrations of the two different G proteins. The constants described above completely define the model, and these parameters can be used to derive the equation for the binding of A to the receptor, which is:

$$\left[ {AR} \right] + \left[ {ARG_1 } \right] + \left[ {ARG_2 } \right] + \left[ {ARG_1 X} \right] + \left[ {ARG_2 X} \right] = \frac{{\left[ A \right]\left[ {R_{\text{T}} } \right]}}{{\left[ A \right] + K_{{\text{obs}}} }}$$

(43)

In this equation, K _obs denotes the observed dissociation constant of the agonist. The K _obs is defined in terms of the parameters of the ternary complex model as shown in the next equation. In solving for K _obs, depletion of the free concentration of G protein in the membrane as it binds to the receptor can be taken into account. This calculation, however, is unnecessary when there is an excess of G protein or when the concentration of guanine nucleotide is high. These are the conditions explored in this manuscript, so the requisite mathematics for G protein depletion are not described. Regardless, these mathematics are simple when there is one G protein; one has only to derive the roots to a quadratic equation with simple coefficients (see Ehlert 1985). When there are two G proteins; the analogous calculation requires solving the roots of a quartic equation with very complex coefficients, which is feasible. The resulting solution requires several pages of equations to describe, however. Because the condition of excess guanine nucleotide and G protein is accurately modeled without consideration for G protein depletion, these complex mathematics are not shown here.

When there is excess guanine nucleotide and G protein, K _obs is defined as:

$$K_{{\text{obs}}} = \frac{{1 + K_{21} \frac{{G_{1{\text{T}}} }}{{R_{\text{T}} }}\left( {1 + X\beta _1 K_{31} } \right) + K_{22} \frac{{G_{2{\text{T}}} }}{{R_{\text{T}} }}\left( {1 + \beta _2 K_{32} X} \right)}}{{K_1 \left( {1 + \alpha _1 K_{21} \frac{{G_{1{\text{T}}} }}{{R_{\text{T}} }}\left( {1 + X\beta _1 \gamma _1 K_{31} } \right)} \right) + \alpha _2 K_1 K_{22} \frac{{G_{2{\text{T}}} }}{{R_{\text{T}} }}\left( {1 + X\beta _2 \gamma _2 K_{32} } \right)}}$$

(44)

The equations describing the amount A bound in the form of quaternary complex (agonist–receptor–G protein–guanine nucleotide; ARGX) for the two different G proteins are:

$$\left[ {ARG_1 X} \right] = \frac{{\left[ A \right]R_{\text{T}} }}{{\left[ A \right] + K_{ARG_1 X} }}$$

(45)

$$\left[ {ARG_2 X} \right] = \frac{{\left[ A \right]R_{\text{T}} }}{{\left[ A \right] + K_{ARG_2 X} }}$$

(46)

In these equations, K _ARG1X and K _ARG2X denote the observed dissociation constants of the ARG ₁ X and ARG ₂ X complexes, respectively. For the reasons described above, these constants are defined without consideration for depletion of the free G protein concentrations in the membrane:

$$K_{ARG_1 X} = \frac{{1 + \left[ A \right]K_1 + K_{21} \frac{{G_{1{\text{T}}} }}{{R_{\text{T}} }}\left( {1 + \left[ A \right]\alpha _1 K_1 + X\beta _1 K_{31} } \right) + K_{22} \frac{{G_{2{\text{T}}} }}{{R_{\text{T}} }}\left( {1 + \left[ A \right]\alpha _2 K_1 + X\beta _2 K_{32} + \left[ A \right]X\alpha _2 \beta _2 \gamma _2 K_1 K_{32} } \right)}}{{\frac{{G_{1{\text{T}}} }}{{R_{\text{T}} }}\left( {X\alpha _1 \beta _1 \gamma _1 K_1 K_{21} K_{31} } \right)}}$$

(47)

$$K_{{ARG_{2} X}} = \frac{{1 + {\left[ A \right]}K_{1} + K_{{21}} \frac{{G_{{1{\text{T}}}} }}{{R_{{\text{T}}} }}{\left( {1 + {\left[ A \right]}\alpha _{1} K_{1} + X\beta _{1} K_{{31}} } \right)} + K_{{22}} \frac{{G_{{2{\text{T}}}} }}{{R_{{\text{T}}} }}{\left( {1 + {\left[ A \right]}\alpha _{2} K_{1} + X\beta _{2} K_{{32}} + {\left[ A \right]}X\alpha _{2} \beta _{2} \gamma _{2} K_{1} K_{{32}} } \right)}}}{{\frac{{G_{{1{\text{T}}}} }}{{R_{{\text{T}}} }}{\left( {X\alpha _{1} \beta _{1} \gamma _{1} K_{1} K_{{21}} K_{{31}} } \right)}}}$$

(48)

Conformational analysis

It is assumed that there are three states of the receptor—a ground state (R _s) and two active states ($R_{\text{s}}^*$ and $R_{\text{s}}^{**}$). In the model, $R_{\text{s}}^*$ interacts preferentially with G ₁, and $R_{\text{s}}^{**}$ interacts preferentially with G ₂. Each receptor complex (e.g., AR) described in the previous section can be divided into three components representing the ground (AR _s) and two active states of the receptor ($AR_{\text{s}}^*$ and $AR_{\text{s}}^{**}$). This section describes how to resolve each ARG ₁ X and ARG ₂ X complex into the two active states and how to define the various parameters of the level two analysis (α, β, γ, K ₁, K ₂, and K ₃) in terms of the microscopic affinity constants of the various states of the receptor.

Each receptor state represents a unique structure with a unique affinity constant, independent of the nature other proteins in the complex. The various microscopic constants are defined as:

$$K_{\text{a}} = \frac{{\left[ {AR_{\text{s}} } \right]}}{{\left[ A \right]\left[ {R_{\text{s}} } \right]}} = \frac{{\left[ {AR_{\text{s}} G_1 } \right]}}{{\left[ A \right]\left[ {R_{\text{s}} G_1 } \right]}} = \frac{{\left[ {AR_{\text{s}} G_1 X} \right]}}{{\left[ A \right]\left[ {R_{\text{s}} G_1 X} \right]}} = \frac{{\left[ {AR_{\text{s}} G_2 } \right]}}{{\left[ A \right]\left[ {R_{\text{s}} G_2 } \right]}} = \frac{{\left[ {AR_{\text{s}} G_2 X} \right]}}{{\left[ A \right]\left[ {R_{\text{s}} G_2 X} \right]}}$$

(49)

$$K_{\text{b}} = \frac{{\left[ {AR_{\text{s}}^* } \right]}}{{\left[ A \right]\left[ {R_{\text{s}}^* } \right]}} = \frac{{\left[ {AR_{\text{s}}^* G_1 } \right]}}{{\left[ A \right]\left[ {R_{\text{s}}^* G_1 } \right]}} = \frac{{\left[ {AR_{\text{s}}^* G_1 X} \right]}}{{\left[ A \right]\left[ {R_{\text{s}}^* G_1 X} \right]}} = \frac{{\left[ {AR_{\text{s}}^* G_2 } \right]}}{{\left[ A \right]\left[ {R_{\text{s}}^* G_2 } \right]}} = \frac{{\left[ {AR_{\text{s}}^* G_2 X} \right]}}{{\left[ A \right]\left[ {R_{\text{s}}^* G_2 X} \right]}}$$

(50)

$$K_{\text{c}} = \frac{{\left[ {AR_{\text{s}}^{**} } \right]}}{{\left[ A \right]\left[ {R_{\text{s}}^{**} } \right]}} = \frac{{\left[ {AR_{\text{s}}^{**} G_1 } \right]}}{{\left[ A \right]\left[ {R_{\text{s}}^{**} G_1 } \right]}} = \frac{{\left[ {AR_{\text{s}}^{**} G_1 X} \right]}}{{\left[ A \right]\left[ {R_{\text{s}}^{**} G_1 X} \right]}} = \frac{{\left[ {AR_{\text{s}}^{**} G_2 } \right]}}{{\left[ A \right]\left[ {R_{\text{s}}^{**} G_2 } \right]}} = \frac{{\left[ {AR_{\text{s}}^{**} G_2 X} \right]}}{{\left[ A \right]\left[ {R_{\text{s}}^{**} G_2 X} \right]}}$$

(51)

$$K_{\text{e}} = \frac{{\left[ {R_{\text{s}} G_1 } \right]}}{{\left[ {R_{\text{s}} } \right]\left[ {G_1 } \right]}} = \frac{{\left[ {AR_{\text{s}} G_1 } \right]}}{{\left[ {AR_{\text{s}} } \right]\left[ {G_1 } \right]}} = \frac{{\left[ {AR_{\text{s}} G_1 X} \right]}}{{\left[ {AR_{\text{s}} } \right]\left[ {G_1 X} \right]}} = \frac{{\left[ {R_{\text{s}} G_1 X} \right]}}{{\left[ {R_{\text{s}} } \right]\left[ {G_1 X} \right]}}$$

(52)

$$K_{\text{f}} = \frac{{\left[ {R_{\text{s}} G_2 } \right]}}{{\left[ {R_{\text{s}} } \right]\left[ {G_2 } \right]}} = \frac{{\left[ {AR_{\text{s}} G_2 } \right]}}{{\left[ {AR_{\text{s}} } \right]\left[ {G_2 } \right]}} = \frac{{\left[ {AR_{\text{s}} G_2 X} \right]}}{{\left[ {AR_{\text{s}} } \right]\left[ {G_2 X} \right]}} = \frac{{\left[ {R_{\text{s}} G_2 X} \right]}}{{\left[ {R_{\text{s}} } \right]\left[ {G_2 X} \right]}}$$

(53)

$$K_{\text{g}} = \frac{{\left[ {R_{\text{s}}^* G_1 } \right]}}{{\left[ {R_{\text{s}}^* } \right]\left[ {G_1 } \right]}} = \frac{{\left[ {AR_{\text{s}}^* G_1 } \right]}}{{\left[ {AR_{\text{s}}^* } \right]\left[ {G_1 } \right]}} = \frac{{\left[ {AR_{\text{s}}^* G_1 X} \right]}}{{\left[ {AR_{\text{s}}^* } \right]\left[ {G_1 X} \right]}} = \frac{{\left[ {R_{\text{s}}^* G_1 X} \right]}}{{\left[ {R_{\text{s}}^* } \right]\left[ {G_1 X} \right]}}$$

(54)

$$K_{\text{h}} = \frac{{\left[ {R_{\text{s}}^* G_2 } \right]}}{{\left[ {R_{\text{s}}^* } \right]\left[ {G_2 } \right]}} = \frac{{\left[ {AR_{\text{s}}^* G_2 } \right]}}{{\left[ {A^* R_{\text{s}} } \right]\left[ {G_2 } \right]}} = \frac{{\left[ {AR_{\text{s}}^* G_2 X} \right]}}{{\left[ {A^* R_{\text{s}} } \right]\left[ {G_2 X} \right]}} = \frac{{\left[ {R_{\text{s}}^* G_2 X} \right]}}{{\left[ {R_{\text{s}}^* } \right]\left[ {G_2 X} \right]}}$$

(55)

$$K_{\text{i}} = \frac{{\left[ {R_{\text{s}}^{**} G_1 } \right]}}{{\left[ {R_{\text{s}}^{**} } \right]\left[ {G_1 } \right]}} = \frac{{\left[ {AR_{\text{s}}^{**} G_1 } \right]}}{{\left[ {AR_{\text{s}}^{**} } \right]\left[ {G_1 } \right]}} = \frac{{\left[ {AR_{\text{s}}^{**} G_1 X} \right]}}{{\left[ {AR_{\text{s}}^{**} } \right]\left[ {G_1 X} \right]}} = \frac{{\left[ {R_{\text{s}}^{**} G_1 X} \right]}}{{\left[ {R_{\text{s}}^{**} } \right]\left[ {G_1 X} \right]}}$$

(56)

$$K_{\text{j}} = \frac{{\left[ {R_{\text{s}}^{**} G_2 } \right]}}{{\left[ {R_{\text{s}}^{**} } \right]\left[ {G_2 } \right]}} = \frac{{\left[ {AR_{\text{s}}^{**} G_2 } \right]}}{{\left[ {AR_{\text{s}}^{**} } \right]\left[ {G_2 } \right]}} = \frac{{\left[ {AR_{\text{s}}^{**} G_2 X} \right]}}{{\left[ {AR_{\text{s}}^{**} } \right]\left[ {G_2 X} \right]}} = \frac{{\left[ {R_{\text{s}}^{**} G_2 X} \right]}}{{\left[ {R_{\text{s}}^{**} } \right]\left[ {G_2 X} \right]}}$$

(57)

$$K_{\text{k}} = \frac{{\left[ {G_1 X} \right]}}{{\left[ {G_1 } \right]\left[ X \right]}} = \frac{{\left[ {R_{\text{s}} G_1 X} \right]}}{{\left[ {R_{\text{s}} G_1 } \right]\left[ X \right]}} = \frac{{\left[ {AR_{\text{s}} G_1 X} \right]}}{{\left[ {AR_{\text{s}} G_1 } \right]\left[ X \right]}}$$

(58)

$$K_{\text{l}} = \frac{{\left[ {G_2 X} \right]}}{{\left[ {G_2 } \right]\left[ X \right]}} = \frac{{\left[ {R_{\text{s}} G_2 X} \right]}}{{\left[ {R_{\text{s}} G_2 } \right]\left[ X \right]}} = \frac{{\left[ {AR_{\text{s}} G_2 X} \right]}}{{\left[ {AR_{\text{s}} G_2 } \right]\left[ X \right]}}$$

(59)

$$K_{\text{m}} = \frac{{\left[ {R_{\text{s}}^* G_1 X} \right]}}{{\left[ {R_{\text{s}}^* G_1 } \right]\left[ X \right]}} = \frac{{\left[ {AR_{\text{s}}^* G_1 X} \right]}}{{\left[ {AR_{\text{s}}^* G_1 } \right]\left[ X \right]}}$$

(60)

$$K_{\text{n}} = \frac{{\left[ {R_{\text{s}}^* G_2 X} \right]}}{{\left[ {R_{\text{s}}^* G_2 } \right]\left[ X \right]}} = \frac{{\left[ {AR_{\text{s}}^* G_2 X} \right]}}{{\left[ {AR_{\text{s}}^* G_2 } \right]\left[ X \right]}}$$

(61)

$$K_{\text{o}} = \frac{{\left[ {R_{\text{s}}^{**} G_1 X} \right]}}{{\left[ {R_{\text{s}}^{**} G_1 } \right]\left[ X \right]}} = \frac{{\left[ {AR_{\text{s}}^{**} G_1 X} \right]}}{{\left[ {AR_{\text{s}}^{**} G_1 } \right]\left[ X \right]}}$$

(62)

$$K_{\text{p}} = \frac{{\left[ {R_{\text{s}}^{**} G_2 X} \right]}}{{\left[ {R_{\text{s}}^{**} G_2 } \right]\left[ X \right]}} = \frac{{\left[ {AR_{\text{s}}^{**} G_2 X} \right]}}{{\left[ {AR_{\text{s}}^{**} G_2 } \right]\left[ X \right]}}$$

(63)

$$K_{\text{q}} = \frac{{R_{\text{s}}^* }}{{R_{\text{s}} }}$$

(64)

$$K_{\text{r}} = \frac{{R_{\text{s}}^{**} }}{{R_{\text{s}} }}$$

(65)

It is possible to define the level 2 parameters of the ternary complex model for two G proteins in terms of the microscopic constants of the receptor states:

$$K_1 = \frac{{K_{\text{a}} + K_{\text{b}} K_{\text{q}} + K_{\text{c}} K_{\text{r}} }}{{1 + K_{\text{q}} + K_{\text{r}} }}$$

(66)

$$K_{21} = \frac{{K_{\text{e}} {\text{ + }}K_{\text{g}} K_{\text{q}} {\text{ + }}K_{\text{i}} K_{\text{r}} }}{{{\text{1 + }}K_{\text{q}} {\text{ + }}K_{\text{r}} }}$$

(67)

$$K_{22} = \frac{{K_{\text{f}} {\text{ + }}K_{\text{h}} K_{\text{q}} {\text{ + }}K_{\text{j}} K_{\text{r}} }}{{{\text{1 + }}K_{\text{q}} {\text{ + }}K_{\text{r}} }}$$

(68)

$$K_{31} = K_{\text{k}} $$

(69)

$$K_{32} = K_{\text{l}} $$

(70)

$$\alpha _1 = \frac{{\left( {K_{\text{a}} K_{\text{e}} + K_{\text{b}} K_{\text{g}} K_{\text{q}} + K_{\text{c}} K_{\text{i}} K_{\text{r}} } \right)\left( {1 + K_{\text{q}} + K_{\text{r}} } \right)}}{{\left( {K_{\text{e}} + K_{\text{g}} K_{\text{q}} + K_{\text{i}} K_{\text{r}} } \right)\left( {K_{\text{a}} + K_{\text{b}} K_{\text{q}} + K_{\text{c}} K_{\text{r}} } \right)}}$$

(71)

$$\alpha _2 = \frac{{\left( {K_{\text{a}} K_{\text{f}} + K_{\text{b}} K_{\text{h}} K_{\text{q}} + K_{\text{c}} K_{\text{j}} K_{\text{r}} } \right)\left( {1 + K_{\text{q}} + K_{\text{r}} } \right)}}{{\left( {K_{\text{f}} + K_{\text{h}} K_{\text{q}} + K_{\text{j}} K_{\text{r}} } \right)\left( {K_{\text{a}} + K_{\text{b}} K_{\text{q}} + K_{\text{c}} K_{\text{r}} } \right)}}$$

(72)

$$\beta _1 = \frac{{K_{\text{e}} K_{\text{k}} + K_{\text{g}} K_{\text{m}} K_{\text{q}} + K_{\text{i}} K_{\text{o}} K_{\text{r}} }}{{\left( {K_{\text{e}} + K_{\text{g}} K_{\text{q}} + K_{\text{i}} K_{\text{r}} } \right)K_{\text{k}} }}$$

(73)

$$\beta _2 = \frac{{K_{\text{f}} K_{\text{l}} + K_{\text{h}} K_{\text{n}} K_{\text{q}} + K_{\text{j}} K_{\text{p}} K_{\text{r}} }}{{\left( {K_{\text{f}} + K_{\text{h}} K_{\text{q}} + K_{\text{j}} K_{\text{r}} } \right)K_{\text{l}} }}$$

(74)

$$\gamma _1 = \frac{{\left( {K_{\text{a}} K_{\text{e}} K_{\text{k}} {\text{ + }}K_{\text{b}} K_{\text{g}} K_{\text{m}} K_{\text{q}} {\text{ + }}K_{\text{c}} K_{\text{i}} K_{\text{o}} K_{\text{r}} } \right)\left( {K_{\text{e}} {\text{ + }}K_{\text{g}} K_{\text{q}} {\text{ + }}K_{\text{i}} K_{\text{r}} } \right)}}{{\left( {K_{\text{e}} K_{\text{k}} {\text{ + }}K_{\text{g}} K_{\text{m}} K_{\text{q}} {\text{ + }}K_{\text{i}} K_{\text{o}} K_{\text{r}} } \right)\left( {K_{\text{a}} K_{\text{e}} {\text{ + }}K_{\text{b}} K_{\text{g}} K_{\text{q}} {\text{ + }}K_{\text{c}} K_{\text{i}} K_{\text{r}} } \right)}}$$

(75)

$$\gamma _2 = \frac{{\left( {K_{\text{a}} K_{\text{f}} K_{\text{l}} {\text{ + }}K_{\text{b}} K_{\text{h}} K_{\text{n}} K_{\text{q}} {\text{ + }}K_{\text{c}} K_{\text{j}} K_{\text{p}} K_{\text{r}} } \right)\left( {K_{\text{f}} {\text{ + }}K_{\text{h}} K_{\text{q}} {\text{ + }}K_{\text{j}} K_{\text{r}} } \right)}}{{\left( {K_{\text{f}} K_{\text{l}} {\text{ + }}K_{\text{h}} K_{\text{n}} K_{\text{q}} {\text{ + }}K_{\text{j}} K_{\text{p}} K_{\text{r}} } \right)\left( {K_{\text{a}} K_{\text{f}} {\text{ + }}K_{\text{b}} K_{\text{h}} K_{\text{q}} {\text{ + }}K_{\text{c}} K_{\text{j}} K_{\text{r}} } \right)}}$$

(76)

Using the microscopic affinity constants for the different receptor states, it is possible to define the equilibrium between the inactive from of the quaternary complex (AR _s GX) and the two active forms ($AR_{\text{s}}^* GX$ and $AR_{\text{s}}^{**} GX$):

$$\frac{{\left[ {AR_{\text{s}}^* G_1 X} \right]}}{{\left[ {AR_{\text{s}} G_1 X} \right]}} = \frac{{K_{\text{b}} K_{\text{g}} K_{\text{m}} K_{\text{q}} }}{{K_{\text{a}} K_{\text{e}} K_{\text{k}} }}$$

(77)

$$\frac{{\left[ {AR_{\text{s}}^{**} G_2 X} \right]}}{{\left[ {AR_{\text{s}} G_2 X} \right]}} = \frac{{K_{\text{c}} K_{\text{j}} K_{\text{p}} K_{\text{r}} }}{{K_{\text{a}} K_{\text{f}} K_{\text{l}} }}$$

(78)

Using these relationships, it is possible to estimate the proportion of bound drug in the form of the two active quaternary complexes as:

$$AR_{\text{s}}^* G_1 X = \frac{1}{{1 + \frac{{K_{\text{a}} K_{\text{e}} K_{\text{k}} }}{{K_{\text{b}} K_{\text{g}} K_{\text{m}} K_{\text{q}} }}}} \times \frac{{\left[ A \right]R_{\text{T}} }}{{\left[ A \right] + K_{ARG_1 X} }}$$

(79)

$$AR_{\text{s}}^{**} G_{21} X = \frac{1}{{1 + \frac{{K_{\text{a}} K_{\text{f}} K_{\text{l}} }}{{K_{\text{c}} K_{\text{j}} K_{\text{p}} K_{\text{r}} }}}} \times \frac{{\left[ A \right]R_{\text{T}} }}{{\left[ A \right] + K_{ARG_2 X} }}$$

(80)

in which K _ARG1X and K _ARG2X are defined above in Eq. 47 and 48.

Estimation of agonist RA_i values

As described previously (Griffin et al. 2007), the estimation of RA_i is simple when the E _max values of the agonists are the same (i.e., RA_i value equals the potency ratio) or when the Hill slopes of the agonist concentration–response curves are equal to 1 (see Eq. 17). When the E _max values of the agonists are different and the Hill slopes of the agonist–concentration response curves differ from 1, global nonlinear regression analysis is required (Griffin et al. 2007). In this section, specific instructions are given for this analysis. If the concentration–response curves are consistent with a logistic equation, then the operational model can be used to estimate RA_i; otherwise, a null method can be used (see below).

Use of the operational model to estimate RA_i

The use of the operational model will be explained using the data in Fig. 11, which shows the activity of muscarinic agonists for eliciting contraction of the guinea pig ileum. The analysis requires the concentration–response curve of the standard agonist and one or more test agonists. The standard agonist is defined as the agonist with the largest E _max value (carbachol). The RA_i values of pilocarpine and McN-A-343 are estimated relative to carbachol. The first step in the analysis involves estimating the E _max and EC₅₀ values of the concentration–response curves using nonlinear regression analysis with a logistic equation. This step can be easily done with Prism (GraphPad Software, San Diego, CA) using the variable-slope dose–response curve equation. For the data shown in Fig. 11, the log EC₅₀ and E _max values are: carbachol, −7.01 and 100%, pilocarpine, −6.05 and 69%, and McN-A-343, −5.23 and 62%, respectively. The log EC₅₀ and E _max values are then used to calculate initial parameter estimates for global nonlinear regression analysis. The parameters for global nonlinear regression analysis are (1) the logarithm of the observed dissociation constant (LOGK1) of the standard agonist, (2) the logarithm of the τ _B K _B value of the standard agonist (LOGR), (3) the logarithm of the observed dissociation constant of the test agonist (LOGK2), (4) the logarithm of the RA_i value of the test agonist (LOGRA), (5) the maximum response of the system (M), and (6) the transducer slope factor (N) in the operational model. The theoretical basis for these parameters is described below. The equations for calculating the initial parameter estimates are given next. In these equations, the E _max and log EC₅₀ values of the different agonists are denoted by subscripts. If the standard agonist is a full agonist, then its dissociation constant is set to an arbitrarily high value (e.g., 10⁻¹; see comment at the end of this section for situations in which the standard agonist is not a full agonist). The initial parameter estimates (parameter′) for carbachol are:

$${\text{LOGK}}1' = - 1$$

(81)

$${\text{LOGR'}} = - \log \left( {{\text{EC}}_{50 - {\text{carb}}} } \right)$$

(82)

For the first test agonist, pilocarpine, the initial parameter estimates are:

$${\text{LOGK}}2' = \log {\text{EC}}_{50 - {\text{pilocarpine}}} $$

(83)

$${\text{LOGRA'}} = \log \left( {\frac{{E_{\max - {\text{pilo}}} }}{{E_{\max - {\text{carb}}} }}} \right) + \log \left( {{\text{EC}}_{50 - {\text{carb}}} } \right) - \log \left( {{\text{EC}}_{50 - {\text{pilo}}} } \right)$$

(84)

The parameter estimates for the second test agonist, McN-A-343, are estimated similarly:

$$LOGK2' = \log {\text{EC}}_{50 - {\text{McN}}} $$

(85)

$${\text{LOGRA'}} = \log \left( {\frac{{E_{\max - {\text{McN}}} }}{{E_{\max - {\text{carb}}} }}} \right) + \log \left( {{\text{EC}}_{50 - {\text{carb}}} } \right) - \log \left( {{\text{EC}}_{50 - {\text{McN}}} } \right)$$

(86)

The initial estimates of the shared system parameters of the operational model are estimated as:

$$M' = E_{\max - {\text{carb}}} $$

(87)

$$N' = 1$$

(88)

Fig. 11

Concentration–response curves of selected agonists for eliciting contraction of the isolated guinea pig ileum. The data are from Ehlert et al. (1999). Mean values ± SEM are shown. The theoretical curves represents the least squares fit of Eqs. 89 and 90 to the data

The next step involves estimating the RA_i values of the two test agonists (pilocarpine and McN-A-343) by global nonlinear regression analysis of the concentration–response curves. In this analysis, Eq. 89 is fitted to the data for the standard agonist carbachol, and Eq. 90 is fitted to the data for the test agonists.

$${\text{response}} = \frac{{MX^N }}{{X^N + \frac{{\left( {X + 10^{{\text{LOGK}}1} } \right)^N }}{{\left( {10^{\left( {{\text{LOGK}}1 + {\text{LOGR}}} \right)} } \right)^N }}}}$$

(89)

$${\text{response}} = \frac{{MX^N }}{{X^N + \frac{{\left( {X + 10^{{\text{LOGK2}}} } \right)^N }}{{\left( {10^{\left( {{\text{LOGK2 + LOGR + LOGRA}}} \right)} } \right)^N }}}}$$

(90)

in which X denotes the concentration of the agonist. The theoretical basis for Eqs. 89 and 90 is described in Griffin et al. (2007). Specifically, these equations can be derived from Eq. 11 in Griffin et al. (2007) with the following substitutions for the standard agonist:

$${\text{LOGK1}} = \log \left( {K_A } \right)$$

(91)

$${\text{LOGR}} = \log \left( {\frac{{\tau _A }}{{K_A }}} \right)$$

(92)

The following substitutions for the test agonists:

$${\text{LOGK}}2 = \log \left( {K_B } \right)$$

(93)

$${\text{LOGRA}} = \frac{{\tau _B K_A }}{{\tau _A K_B }}$$

(94)

and the following substitutions for the operational model:

$$N = m$$

(95)

$$M = M_{sys} $$

(96)

In situations where the agonist concentration–response curves are decreasing, such as in agonist-mediated inhibition of forskolin-stimulated cAMP accumulation, the following equations are used for the standard agonist and test agonists, respectively, instead of Eqs. 89 and 90:

$${\text{response}} = B - \left( {\frac{{MX^N }}{{X^N + \frac{{\left( {X + 10^{{\text{LOGK1}}} } \right)^N }}{{\left( {10^{\left( {{\text{LOGK1}} + {\text{LOGR}}} \right)} } \right)^N }}}}} \right)$$

(97)

$${\text{response}} = B - \left( {\frac{{MX^N }}{{X^N + \frac{{\left( {X + 10^{{\text{LOGK1}}} } \right)^N }}{{\left( {10^{\left( {{\text{LOGK1}} + {\text{LOGR}} + {\text{LOGRA}}} \right)} } \right)^N }}}}} \right)$$

(98)

in which B denotes the estimate of the response measured in the absence of agonist. In the case of agonist-mediated inhibition of forskolin-stimulated cAMP accumulation, B represents the estimate of cAMP accumulation in the presence of forskolin and absence of agonist. For the remainder of this discussion, the data in Fig. 11 will be analyzed, which requires the use of an increasing equation for the concentration–response curve (i.e., Eqs. 89 and 90). For the RA_i analysis, Eqs. 89 and 90 are fitted simultaneously to the concentration–response curves in Fig. 11, sharing the estimate of M and N among the curves and obtaining a unique estimate of LOGR for the standard agonist and unique estimates of LOGK2 and LOGRA for each test agonist. To explain how this process works, I describe how one can use a spreadsheet to do the analysis (see Fig. 12a):

• 1. The data are entered into a spreadsheet, with the log agonist concentrations entered in column A (cells A11–A24). The response values for each agonist are entered in column B (carbachol, cells B11–B18), E (pilocarpine, cells E13–E21), and H (McN-A-343, cells H16–H24).

• 2. The value (−1) to which the parameter LOGK1 is constrained as a constant is entered into cell C4.

• 3. The initial estimates of the shared parameters M (100) and N (1) are entered into cells C2 and C3, respectively. The values of these initial estimates are designated in Eqs. 87 and 88, respectively.

• 4. The initial estimate of LOGR for the standard agonist carbachol is entered into cell C5. The estimate (7) is calculated from Eq. 82.

• 5. The initial estimates for parameters LOGK2 and LOGRA for pilocarpine are entered into cells F4 and F5, respectively. These estimates (−6.05 and −1.11) are calculated from Eqs. 83 and 84, respectively. The corresponding estimates for McN-A-343 (−5.23 and −1.98) are entered into cells I4 and I5, respectively.

• 6. A modification of Eq. 89 is entered into column C, cells C11–C18 to generate the predicted response values for the standard agonist carbachol. The equation is modified so that the logarithm of the agonist concentration is the independent variable instead of the agonist concentration. The entry in cell C11 is:

  $$ = \$ {\text{C}}\$ 2*{{\left( {\left( {10^ \wedge \$ {\text{A}}11} \right)^ \wedge \$ {\text{C}}\$ 3} \right)} \mathord{\left/ {\vphantom {{\left( {\left( {10^ \wedge \$ {\text{A}}11} \right)^ \wedge \$ {\text{C}}\$ 3} \right)} {\left( {\left( {\left( {10^ \wedge \$ {\text{A}}11} \right)^ \wedge \$ {\text{C}}\$ 3} \right) + {{\left( {\left( {10^ \wedge \$ {\text{A}}11 + \left( {10^ \wedge \$ {\text{C}}\$ 4} \right)} \right)^ \wedge \$ {\text{C}}\$ 3} \right)} \mathord{\left/ {\vphantom {{\left( {\left( {10^ \wedge \$ {\text{A}}11 + \left( {10^ \wedge \$ {\text{C}}\$ 4} \right)} \right)^ \wedge \$ {\text{C}}\$ 3} \right)} {\left( {\left( {10^ \wedge \left( {\$ {\text{C}}\$ 4 + \$ {\text{C}}\$ 5} \right)} \right)^ \wedge \$ {\text{C}}\$ 3} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\left( {10^ \wedge \left( {\$ {\text{C}}\$ 4 + \$ {\text{C}}\$ 5} \right)} \right)^ \wedge \$ {\text{C}}\$ 3} \right)}}} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\left( {\left( {10^ \wedge \$ {\text{A}}11} \right)^ \wedge \$ {\text{C}}\$ 3} \right) + {{\left( {\left( {10^ \wedge \$ {\text{A}}11 + \left( {10^ \wedge \$ {\text{C}}\$ 4} \right)} \right)^ \wedge \$ {\text{C}}\$ 3} \right)} \mathord{\left/ {\vphantom {{\left( {\left( {10^ \wedge \$ {\text{A}}11 + \left( {10^ \wedge \$ {\text{C}}\$ 4} \right)} \right)^ \wedge \$ {\text{C}}\$ 3} \right)} {\left( {\left( {10^ \wedge \left( {\$ {\text{C}}\$ 4 + \$ {\text{C}}\$ 5} \right)} \right)^ \wedge \$ {\text{C}}\$ 3} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\left( {10^ \wedge \left( {\$ {\text{C}}\$ 4 + \$ {\text{C}}\$ 5} \right)} \right)^ \wedge \$ {\text{C}}\$ 3} \right)}}} \right)}}$$

  The symbol $ is used to restrict the address of a cell so that the function entered in C11 can be copied and pasted into cells C12–C18 without disturbing the addresses of the parameters.

• 7. A modification of Eq. 90 is entered into column F, cells F13–F21, to generate the predicted response values for the test agonist pilocarpine. The equation is modified so that the logarithm of the agonist concentration is the independent variable instead of the agonist concentration. The entry in cell F13 is:

  $${\text{ = \$ C\$ 2*}}{{\left( {\left( {{\text{10}}^ \wedge {\text{\$ A13}}} \right)^ \wedge {\text{\$ C\$ 3}}} \right)} \mathord{\left/ {\vphantom {{\left( {\left( {{\text{10}}^ \wedge {\text{\$ A13}}} \right)^ \wedge {\text{\$ C\$ 3}}} \right)} {\left( {\left( {\left( {{\text{10}}^ \wedge {\text{\$ A13}}} \right)^ \wedge {\text{\$ C\$ 3}}} \right){\text{ + }}{{\left( {\left( {{\text{10}}^ \wedge {\text{\$ A13 + }}\left( {{\text{10 $ \hat{} $ F\$ 4}}} \right)} \right)} \right)^ \wedge {\text{\$ C\$ 3)}}} \mathord{\left/ {\vphantom {{\left( {\left( {{\text{10}}^ \wedge {\text{\$ A13 + }}\left( {{\text{10 $ \hat{} $ F\$ 4}}} \right)} \right)} \right)^ \wedge {\text{\$ C\$ 3)}}} {\left( {\left( {{\text{10}}^ \wedge \left( {{\text{F\$ 4 + \$ C\$ 5 + F\$ 5}}} \right)} \right)^ \wedge {\text{\$ C\$ 3}}} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\left( {{\text{10}}^ \wedge \left( {{\text{F\$ 4 + \$ C\$ 5 + F\$ 5}}} \right)} \right)^ \wedge {\text{\$ C\$ 3}}} \right)}}} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\left( {\left( {{\text{10}}^ \wedge {\text{\$ A13}}} \right)^ \wedge {\text{\$ C\$ 3}}} \right){\text{ + }}{{\left( {\left( {{\text{10}}^ \wedge {\text{\$ A13 + }}\left( {{\text{10 $ \hat{} $ F\$ 4}}} \right)} \right)} \right)^ \wedge {\text{\$ C\$ 3)}}} \mathord{\left/ {\vphantom {{\left( {\left( {{\text{10}}^ \wedge {\text{\$ A13 + }}\left( {{\text{10 $ \hat{} $ F\$ 4}}} \right)} \right)} \right)^ \wedge {\text{\$ C\$ 3)}}} {\left( {\left( {{\text{10}}^ \wedge \left( {{\text{F\$ 4 + \$ C\$ 5 + F\$ 5}}} \right)} \right)^ \wedge {\text{\$ C\$ 3}}} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\left( {{\text{10}}^ \wedge \left( {{\text{F\$ 4 + \$ C\$ 5 + F\$ 5}}} \right)} \right)^ \wedge {\text{\$ C\$ 3}}} \right)}}} \right)}}$$

  This entry is also pasted into cells F14–F21. The entry in cell F13 can also be pasted into cells I16–I24 to calculate the predicted response values for McN-A-343.

• 8. The squared difference between the observed and predicted response of the standard agonist is calculated with the following entry in cell D11:

  $$ = \left( {{\text{B}}11 - {\text{C}}11} \right)^ \wedge 2$$

Fig. 12

Spreadsheet for the estimation of agonist RA_i values. a The spreadsheet has been set up to estimate the RA_i values for the data shown in Fig. 11. b The spreadsheet as it appears after the Solver function has been used to minimize the entry in cell D28, the total sum of squared residuals. The estimates of the log RA_i values for pilocarpine and McN-A-343 are shown in cells F5 and I5, respectively. Further details are described under the Appendix

This entry is copied and pasted into cells D12–D18 to estimate the squared differences between the observed and predicted responses at each concentration of the standard agonist. The entry is also copied and pasted into cells G13–G21 and cells J16–J24 to calculate the corresponding values for pilocarpine and McN-A-343, respectively.

• 9. The sum of the squared deviations between the observed and predicted response values for the standard agonist is calculated with the following entry in cell D26:

  $$ = {\text{SUM}}\left( {{\text{D11:D24}}} \right)$$

This entry is copied and pasted into cells G26 and J26 to calculate the corresponding values for pilocarpine and McN-A-343, respectively.

• 10. The total sum of squared deviation between the observed and predicted response values for the three agonists is calculated by taking the sum of the values in cells D26, G26, and J26. This sum is calculated in cell D28 with the following entry:

  $$ = {\text{D26 + G26 + J26}}$$

At this point in the process, the spreadsheet should look like that shown in Fig. 12a. To obtain the best fitting parameter estimates, nonlinear regression analysis is accomplished using the Solver function in Excel.

• 11. Cell D28 is selected, and the Solver function is selected from the Tools menu. A dialog box (Solve Parameters) should appear. More information about the use of the dialog box is given by Christopoulos and Mitchelson (1998). The following steps are completed in the Solve Parameters dialog box:

• a. Insure that cell D28 is entered in the Set Target Cell box.

• b. Click on the Min button to insure that a least squares fit is calculated.

• c. In the By Changing Cells box, enter the cells containing the parameter estimates with a comma between each cell. The entry should look like:

  $${\text{\$ C\$ 2,\$ C\$ 3,\$ C\$ 5,\$ F\$ 4,\$ F\$ 5,\$ I\$ 4,\$ I\$ 5}}$$

• d. Click on the Options button and a Solver Options dialog box should appear in which the following stringent conditions are set:

• i. Leave the Max Time on the default value of 100 s.

• ii. Set Iterations to 2,000.

• iii. Set Precision to 1e-7.

• iv. Set Tolerance to 1e-7.

• v. Set Convergence to 1e-7.

• vi. Select the Use Automatic Scaling button.

• vii. Click on the OK button to return to the Solver Options dialog box.

• e. Upon returning to the Solver Parameters dialog box, click the Solve button to complete the regression analysis and obtain the best fitting parameters. After a few seconds, the iteration process is complete, and a Solver Results dialog box appears. If it appears that solution has been found click OK; otherwise, click Cancel. After clicking OK, the spreadsheet for this example should look like Fig. 12b. The parameter estimates are given in their respective cells. These are: the maximum response of the system (M, cell C2): 97.71, the transducer slope factor of the operational model (N, cell C3): 1.75, the logarithm of the ratio of τ/K for the standard agonist (LOGR, cell C5): 7.01, the logarithm of dissociation constant of pilocarpine (LOGK2, cell F4): −5.81, the logarithm of the RA_i value of pilocarpine (LOGRA, cell F5): −0.91, the logarithm of the dissociation constant of McN-A-343 (LOGK2, cell I4): −5.13, and the log RA_i value of McN-A-343 (LOGRA, cell I5): −1.71.

It is also possible to estimate agonist RA_i values using Prism (GraphPad Software). It is assumed that the reader is familiar with Prism, and specific instructions are only given for analyzing the data by global nonlinear regression analysis. The data are entered in a datasheet for the XY graph. The replicate experiments for the standard agonist are entered in subcolumns of column A, and the replicate experiments for the test agonists (e.g., pilocarpine and McN-A-343) are entered in subcolumns of columns B and C. Because the dissociation constant is fixed as a constant for the standard agonist and allowed to vary for each test agonist, it is necessary to define variables in the regression equation that change depending upon whether the data for the standard agonist or test agonist is being analyzed. This can be accomplished by entering the following five lines in as the equation for nonlinear regression analysis:

$$\begin{array}{*{20}l} { < A > P = LOGK1} \hfill \\ { < A > Q = LOGK1 + LOGR} \hfill \\ { < \sim A > P = LOGK2} \hfill \\ { < \sim A > Q = LOGK2 + LOGR1 + LOGRA} \hfill \\ {Y = {{\left\{ {\left[ {\left( {10^ \wedge X} \right)^ \wedge N} \right]*M} \right\}} \mathord{\left/ {\vphantom {{\left\{ {\left[ {\left( {10^ \wedge X} \right)^ \wedge N} \right]*M} \right\}} {\left\{ {{{\left[ {\left( {10^ \wedge X} \right)^ \wedge N} \right] + \left\{ {\left[ {\left( {10^ \wedge X} \right) + \left( {10^ \wedge P} \right)} \right]^ \wedge N} \right\}} \mathord{\left/ {\vphantom {{\left[ {\left( {10^ \wedge X} \right)^ \wedge N} \right] + \left\{ {\left[ {\left( {10^ \wedge X} \right) + \left( {10^ \wedge P} \right)} \right]^ \wedge N} \right\}} {\left[ {\left( {10^ \wedge Q} \right)^ \wedge N} \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {\left( {10^ \wedge Q} \right)^ \wedge N} \right]}}} \right\}}}} \right. \kern-\nulldelimiterspace} {\left\{ {{{\left[ {\left( {10^ \wedge X} \right)^ \wedge N} \right] + \left\{ {\left[ {\left( {10^ \wedge X} \right) + \left( {10^ \wedge P} \right)} \right]^ \wedge N} \right\}} \mathord{\left/ {\vphantom {{\left[ {\left( {10^ \wedge X} \right)^ \wedge N} \right] + \left\{ {\left[ {\left( {10^ \wedge X} \right) + \left( {10^ \wedge P} \right)} \right]^ \wedge N} \right\}} {\left[ {\left( {10^ \wedge Q} \right)^ \wedge N} \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {\left( {10^ \wedge Q} \right)^ \wedge N} \right]}}} \right\}}}} \hfill \\ \end{array} $$

The first set of two lines define the variables P (LOGK1) and Q (LOGK1 + LOGR) in the regression equation for data set A (standard agonist), and the second set of two lines define the same variables (P = LOGK2, Q= LOGK2 + LOGR + LOGRA) for each test agonist. The regression equation is shown in the last line, which is more easily recognized in the following form:

$$Y = \frac{{\left( {10^X } \right)^N M}}{{\left( {10^X } \right)^N + \frac{{\left( {10^X + 10^P } \right)^N }}{{\left( {10^Q } \right)^N }}}}$$

(99)

The independent variable X represents the logarithm of the agonist concentration. Global nonlinear regression analysis is done setting LOGK1 to a constant (−1), sharing M and N among all data sets, and obtaining a unique estimate of LOGR for the standard agonist and unique estimates of LOGRA and LOGK2 for each test agonist. The initial parameter estimates are calculated as described above in Eqs. 81 to 88.

If the standard agonist (i.e., the agonist with the largest E _max) is not a full agonist, it may be possible to obtain an estimate of its dissociation constant by doing the regression analysis without setting LOGK1 to a constant but by estimating a unique estimate for this parameter. In such instances (i.e., when the standard agonist is not a full agonist), we have found a very small improvement in the residual sum of squares and a very minor, insignificant change in the estimate of LOGRA for the test agonist. If the standard agonist is a full agonist, there is no improvement in residual error at all, and it may be impossible to obtain a fit unless LOK1 is fixed at an arbitrarily high value.

Use of a null method to estimate RA_i

Concentrations of the standard agonist (A _i) that elicit responses equivalent to those of the concentration–response of the test agonist (B _i) are estimated as described previously (Griffin et al. 2007). These pairs of equiactive agonist concentrations are converted to logarithms (Y and X, respectively), and the following equation is fitted to the data by nonlinear regression analysis:

$$Y = \frac{{10^{X + {\text{LOGP + LOGRA + LOGK1}}} }}{{10^X \left( {1 - 10^{{\text{LOGP + LOGRA}}} } \right) + 10^{{\text{LOGP + LOGK1}}} }}$$

(100)

Regression analysis is done with the parameter LOGK1 set as a constant to an arbitrary value (−1), and estimates of LOGP and LOGRA are obtained. The regression analysis only involves fitting pairs of equiactive concentrations of the standard and a single test agonist at a time—no global regression analysis is needed. Equation 100 can be derived from Eq. 8 in Griffin et al. (2007) with the following substitution:

$${\text{LOGP}} = \log \left( {\frac{{K_B }}{{K_A }}} \right)$$

(101)

The other parameters are defined as described above (see Eqs. 91 and 94). The initial parameter estimates (parameter′) are either set as a constant (LOGK1′) or calculated (LOGP′ and LOGRA′) using the following equations with the subscripts A and B referring to the standard and test agonists, respectively:

$${\text{LOGK1'}} = - 1$$

(102)

$${\text{LOGP'}} = \log \left( {{\text{EC}}_{50 - B} } \right) - {\text{LOGK1'}}$$

(103)

$${\text{LOGRA'}} = \log \left( {\frac{{E_{\max - B} }}{{E_{\max - A} }}} \right) + \log \left( {{\text{EC}}_{50 - A} } \right) - \log \left( {{\text{EC}}_{50 - B} } \right)$$

(104)

The regression analysis can be done using a spreadsheet or Prism in a manner analogous to that described above for the operational model. The process is simpler because standard regression analysis is used, not global regression analysis. The estimate of the logarithm of the dissociation constant of the test agonist can be calculated from the constant LOGK1 and the estimate of LOGP:

$$\log \left( {K_B } \right) = {\text{LOGK1}} + {\text{LOGP}}$$

(105)