### The operational model of agonism

The pharmacological response to an agonist depends on the properties of the agonist and the system in which the response is measured. The operational model (OM) of agonism describes the system response using three objective parameters^{13}. According to OM the response of the system follows Eq. (1).

$$Response = frac{{left[ A right]tau_{A} E_{MAX} }}{{left[ A right]left( {tau_{A} + 1} right) + K_{A} }}$$

(1)

where [A] is the concentration of an agonist, E_{MAX} is the maximal possible response of the system, K_{A} is the equilibrium dissociation constant of the agonist-receptor complex and τ_{A} is the operational factor of efficacy. According to the OM, EC_{50} is related to K_{A} according to the following Eq. (2).

$$EC_{50} = frac{{K_{A} }}{{tau_{A} + 1}}$$

(2)

The apparent maximal response E’ _{MAX} observed as the upper asymptote of the functional response curve is given by Eq. (3).

$$E_{MAX}^{prime } = frac{{tau_{A} E_{MAX} }}{{tau_{A} + 1}}$$

(3)

The relationship between EC_{50} and the observed maximal response E’_{MAX} is given by Eq. (4).

$$E_{MAX}^{prime } = E_{MAX} – frac{{E_{MAX} EC_{50} }}{{K_{A} }}$$

(4)

For the derivation of equations, see Supplementary information, Eq. 1 to 5. From Eqs. (2) and (3) it is obvious that parameters E_{MAX}, τ_{A} and K_{A} are inter-dependent. The upper asymptote of functional response, E′_{MAX}, may be any combination of τ_{A} and E_{MAX}, provided their product equals E′_{MAX}. The same applies to EC_{50} value that may be any combination of τ_{A} and K_{A} provided that ratio K_{A} to 1 + τ_{A} equals EC_{50}. Therefore, for reliable determination of OM parameters of functional response to an agonist, we have proposed a two-step procedure^{15}. First, the apparent maximal response E’_{MAX} and half-efficient concentration EC_{50} are determined from a series of concentration–response curves, then the maximal response of the system E_{MAX} and equilibrium dissociation constant K_{A} are determined by fitting Eq. (4) to E’_{MAX} vs. EC_{50} values. Equation (1) is then fitted to the concentration–response curves with fixed E_{MAX} and K_{A} values to determine values of operational efficacy, τ_{A}.

### Allosteric modulation

An allosteric modulator is a ligand that binds to a site on the receptor that is spatially distinct from that of endogenous agonists and orthosteric ligands. Both agonist A and allosteric modulator B can bind to the receptor R simultaneously and form a ternary complex ARB (Figs. 1 and 2). The equilibrium dissociation constant K_{A} of an agonist A to the binary complex RB of the allosteric modulator and receptor differs from the equilibrium dissociation constant of agonist binding in the absence of allosteric modulator, K_{A}, by a factor of binding cooperativity α (K_{A}/α). The law of microscopic reversibility of thermodynamics dictates that the equilibrium dissociation constant of an allosteric modulator K_{B} to the binary complex AR of agonist and receptor differs from K_{B} by the same factor α (K_{B}/α). Values of the factor of binding cooperativity α greater than unity denote positive cooperativity, where binding of agonist and allosteric modulator mutually strengthens each other. Values of the factor of binding cooperativity α lower than 1 denote negative cooperativity, where binding of agonist and allosteric modulator mutually reduces the affinity of each other.

The thermodynamically complete description of allosteric modulation of receptor activation is described by the CTC model (Fig. 1)^{16}. Although the CTC model is simplified and omits improbable interactions of inactive-receptor complexes with G-proteins, besides modulation of binding affinity (equilibrium dissociation constant K_{A}), an allosteric ligand may affect the receptor activation constant (K_{ACT}). The CTC model of binding and activation (Fig. 1, left cube) therefore consists of three equilibrium constants and four factors of cooperativity. The allosteric modulator may also affect the affinity of the receptor complex for G-protein, K_{G}, and efficacy of G-protein activation, φ^{25,26,27}. Thus, it is obvious that such heuristic models are too complex to estimate any of their parameters.

The practical way to analyse allosteric modulation of pharmacological agonism is a parsimonious operational model where the effects of allosteric modulators on operational efficacy are quantified by the operational factor of cooperativity, β (Fig. 2). In this model the operational efficacy of the ternary complex of agonist, receptor and allosteric modulator, ARB, is β*τ_{A}. Values of operational cooperativity β greater than 1 denote positive cooperativity; the functional response to an agonist in the presence of allosteric modulator is greater than in its absence. Values of operational cooperativity β lower than 1 denote negative cooperativity, where the functional response to an agonist in the presence of an allosteric modulator is smaller than in its absence.

The functional response to an agonist in the presence of an allosteric modulator is given by Eq. (5)^{17}.

$$Response = frac{{E_{MAX} tau_{A} left[ A right]left( {K_{B} + alpha beta left[ B right]} right)}}{{left[ A right]K_{B} + K_{A} K_{B} + left[ B right]K_{A} + alpha left[ A right]left[ B right] + tau_{A} left[ A right]left( {K_{B} + alpha beta left[ B right]} right)}}$$

(5)

where [A] and [B] are the concentrations of an agonist and allosteric modulator, respectively, E_{MAX} is the maximal response of the system, K_{A} and K_{B} are the equilibrium dissociation constants of the agonist-receptor and allosteric modulator-receptor complex, respectively, and τ_{A} is the operational factor of efficacy of an agonist. As can be seen, even Eq. (5) is difficult to fit the functional response data directly. As we have shown previously^{15}, all three parameters of OM (Eq. (1)), E_{MAX}, K_{A} and τ_{A} are inter-dependent. Therefore, they cannot be reliably determined by fitting of Eq. (1) to functional response data. These parameters are also inter-dependent in Eq. (5). Moreover, this equation is more complex than Eq. (1). Below we analyse the operational model of allosterically-modulated agonism (OMAM) by the same approach used previously for the OM^{15}.

From Eq. (5) apparent half-efficient concentration of an agonist, EC’_{50}, is given by Eq. (6) and the apparent maximal response induced by an agonist, E’_{MAX}, is given by Eq. (7). Alternative expressions of EC′_{50} and E′_{MAX} can be found in Supplementary Information.

$$EC_{50}^{prime } = frac{{K_{A} left( {left[ B right] + K_{B} } right)}}{{alpha left[ B right] + left( {alpha beta left[ B right] + K_{B} } right)tau_{A} + K_{B} }}$$

(6)

$$E_{MAX}^{prime } = frac{{left( {alpha beta left[ B right] + K_{B} } right)tau_{A} E_{MAX} }}{{alpha left[ B right] + left( {alpha beta left[ B right] + K_{B} } right)tau_{A} + K_{B} }}$$

(7)

From Eq. (6) it is obvious that both factors α and β affect EC′_{50}. Thus, α and β are the fourth and fifth inter-dependent parameters with τ_{A}, E_{MAX} and K_{A}. Equation (7) indicates that the factor of operational cooperativity β affects observed maximal response E′_{MAX}. For saturation concentrations of an allosteric modulator B Eq. (7) becomes Eq. (8).

$$E_{MAX}^{prime } = frac{{beta tau_{A} E_{MAX} }}{{beta tau_{A} + 1}}$$

(8)

The factor of binding cooperativity, α, can be determined from the dependence of the dose ratio of EC_{50} values on the concentration of allosteric modulator. The dose ratio of EC_{50} in the absence of allosteric modulator to EC’_{50} in its presence at the concentration [B] is given by Eq. (8).

$$frac{{EC_{50} }}{{EC_{50}^{prime } }} = frac{{alpha left[ B right] + left( {K_{B} + alpha beta left[ B right]} right)tau_{A} + K_{B} }}{{left( {tau_{A} + 1} right)left( {K_{B} + left[ B right]} right)}}$$

(9)

Values of the ratio greater than 1 where EC’_{50} is lower than EC_{50}, denote an increase in potency mediated by positive cooperativity. Ratio values smaller than 1 denote negative cooperativity and a decrease in potency.

In case the allosteric modulator does not affect operational efficacy τ_{A} (β = 1), Eq. (9) simplifies to Eq. (10).

$$frac{{EC_{50} }}{{EC_{50}^{prime } }} = frac{{K_{B} + alpha left[ B right]}}{{K_{B} + left[ B right]}}$$

(10)

The factor of binding cooperativity α affects only the apparent half-efficient concentration, EC’_{50}, of an agonist (Fig. 3). In the case of negative cooperativity (Fig. 3, left), the allosteric modulator concentration-dependently increases the value of EC’_{50} without a change in the apparent maximal response, E’_{MAX}. In the case of positive cooperativity (Fig. 3, right), the allosteric modulator decreases EC’_{50} without a change in E’_{MAX}. The maximal dose ratio is equal to α as for [B] much greater than K_{B}, the right side of Eq. (10) becomes equal to α (Fig. 3, bottom).

In case the allosteric modulator does not affect the equilibrium dissociation constant of an agonist K_{A} (α = 1), Eq. (9) simplifies to Eq. (11).

$$frac{{EC_{50} }}{{EC_{50}^{prime } }} = frac{{tau_{A} left( {beta left[ B right] + K_{B} } right) + left[ B right] + K_{B} }}{{tau_{A} left( {left[ B right] + K_{B} } right) + left[ B right] + K_{B} }}$$

(11)

In contrast to the factor of binding cooperativity α, the factor of operational cooperativity β affects both the observed maximal response E’_{MAX} and the observed half-efficient concentration of an agonist EC’_{50} (Fig. 4). In the case of negative operational cooperativity (Fig. 4, left), the allosteric modulator concentration-dependently increases the value of EC’_{50} and decreases the observed maximal response E’_{MAX}. In the case of positive operational cooperativity (Fig. 4, right), the allosteric modulator decreases EC’_{50} and increases E’_{MAX}. The maximal dose ratio is given by Eq. (12) (Fig. 4, bottom).

$$For left[ B right] gg K_{B} ;, frac{{EC_{50} }}{{EC_{50}^{prime } }} = frac{{beta tau_{A} + 1}}{{tau_{A} + 1}}$$

(12)

Additional combinations of types of cooperativity of binding, α, and operational efficacy, β, between an orthosteric agonist and allosteric modulator are illustrated in Fig. 5. The meta-analysis of concentration–response curves is in Supplementary information Figure S1 and S2. For the derivation of equations, see Supplementary information, Eqs. 6 to 29.

### Allosteric agonists

The allosteric ligand may possess its own intrinsic activity, i.e., being able to activate the receptor in the absence of an agonist. Such allosteric ligand is termed allosteric agonist. According to the OM, the response to an allosteric agonist is given by Eq. (13).

$$Response = frac{{left[ B right]tau_{B} E_{MAX} }}{{left[ B right]left( {tau_{B} + 1} right) + K_{B} }}$$

(13)

where [B] is the concentration of an allosteric agonist, E_{MAX} is the maximal response of the system, K_{B} is the equilibrium dissociation constant of the complex of allosteric agonist and receptor and τ_{B} is the operational factor of efficacy of the allosteric modulator. In the presence of an allosteric modulator, the response to an orthosteric agonist is given by Eq. (14).

$$Resp = frac{{E_{MAX} left( {tau_{A} left[ A right]left( {K_{B} + alpha beta left[ B right]} right) + tau_{B} left[ B right]K_{A} } right)}}{{left[ A right]K_{B} + K_{A} K_{B} + left[ B right]K_{A} + alpha left[ A right]left[ B right] + tau_{A} left[ A right]left( {K_{B} + alpha beta left[ B right]} right) + tau_{B} left[ B right]K_{A} }}$$

(14)

Equation (14) is even more complex than Eq. (5). From Eq. (14) the apparent half-efficient concentration of an agonist, EC’_{50}, is given by Eq. (15) and apparent maximal response induced by an agonist, E’_{MAX}, is given by Eq. (16). Alternative expressions of EC′_{50} and E′_{MAX} can be found in Supplementary Information, Eq. 38, 41 and 42.

$$EC_{50}^{prime } = frac{{K_{A} left( {left[ B right] + K_{B} } right) + tau_{B} left[ B right]K_{B} }}{{alpha left[ B right] + left( {K_{B} + alpha beta left[ B right]} right)tau_{A} + K_{B} }}$$

(15)

$$E_{MAX}^{prime } = frac{{left( {K_{B} + alpha beta left[ B right]} right)tau_{A} E_{MAX} }}{{alpha left[ B right] + left( {K_{B} + alpha beta left[ B right]} right)tau_{A} + K_{B} }}$$

(16)

Apparent maximal response E′_{MAX} to an agonist in the presence of allosteric agonist at saturation concentration is independent of operational efficacy of allosteric agonist τ_{B} and thus is given by Eq. (8). The dose ratio of EC_{50} in the absence of allosteric modulator to EC’_{50} in its presence at concentration [B] is given by Eq. (17).

$$frac{{EC_{50} }}{{EC_{50}^{prime } }} = frac{{alpha beta left[ B right]tau_{A} + alpha left[ B right] + K_{B} tau_{A} + K_{B} }}{{left( {tau_{A} + 1} right)left( {left[ B right]tau_{B} + left[ B right] + K_{B} } right)}}$$

(17)

To separate individual factors of cooperativity, the dose ratio may be expressed by Eq. (18).

$$frac{{EC_{50} }}{{EC_{50}^{prime } }} = frac{{E_{MAX} left( {alpha left[ B right] + K_{B} } right)}}{{left( {tau_{A} + 1} right)left( {E_{MAX} – E_{MAX}^{prime } } right)left( {left[ B right]tau_{B} + left[ B right] + K_{B} } right)}}$$

(18)

The principal difference between an allosteric agonist and allosteric modulator is that the former increases the basal level of functional response on its own. Even if an allosteric agonist exerts neutral binding cooperativity (α = 1) and does not affect the operational efficacy of the orthosteric agonist (β = 1), it increases the half-efficient concentration, EC′_{50}, of the orthosteric agonist regardless the ratio of operational efficacies τ_{A} and τ_{B}. (Fig. 6). Figure 6 illustrates pure allosteric interaction with τ_{B} lower (left) and greater (right) than τ_{A}. As it can be seen in Fig. 6, the observed maximal response, E′_{MAX}, is given by the factor of operational efficacy of the orthosteric agonist, τ_{A}, according to Eq. (18).

$$E_{MAX}^{prime } = frac{{tau_{A} E_{MAX} }}{{tau_{A} + 1}}$$

(18)

The EC′_{50} value depends on the concentration of pure allosteric agonist (α = 1, β = 1) according to Eq. (19).

$$EC_{50}^{prime } = frac{{K_{A} left( {left[ B right]tau_{B} + left[ B right] + K_{B} } right)}}{{left( {tau_{A} + 1} right)left( {left[ B right] + K_{B} } right)}}$$

(19)

It is evident from Fig. 6 and Eq. (19) that the factor of operational efficacy of an allosteric modulator, τ_{B}, affects the EC′_{50} value of the orthosteric agonist. Thus, τ_{B} is the sixth inter-dependent parameter in addition to α, β, τ_{A}, E_{MAX} and K_{A}. For a pure allosteric agonist (α = 1, β = 1) Eq. (16) simplifies to Eq. (20).

$$frac{{EC_{50} }}{{EC_{50}^{prime } }} = frac{{left[ B right] + K_{B} }}{{left[ B right]tau_{B} + left[ B right] + K_{B} }}$$

(20)

For a pure allosteric agonist the maximal dose ratio is given by Eq. (21) (Fig. 6, bottom).

$$Forleft[ B right] gg K_{B}; frac{{EC_{50} }}{{EC_{50}^{prime } }} = frac{1}{{tau_{B} + 1}}$$

(21)

In case an allosteric agonist does not affect the operational efficacy τ_{A} (β = 1) but allosterically modulates the affinity of the orthosteric agonist (α ≠ 1), Eq. (16) simplifies to Eq. (22).

$$frac{{EC_{50} }}{{EC_{50}^{prime } }} = frac{{alpha left[ B right] + K_{B} }}{{left[ B right]tau_{B} + left[ B right] + K_{B} }}$$

(22)

The factor of binding cooperativity α affects only the apparent half-efficient concentration, EC’_{50}, of an orthosteric agonist (Fig. 7). In the case of negative cooperativity (Fig. 7, left), an allosteric agonist concentration-dependently increases the value of EC’_{50} without a change in apparent maximal response E’_{MAX}. In the case of positive cooperativity (Fig. 7, right), an allosteric agonist decreases EC’_{50} without a change in E’_{MAX}. For an allosteric agonist with neutral operational cooperativity (β = 1), the maximal dose ratio is given by Eq. (23) (Fig. 7, bottom).

$$For left[ B right] gg K_{B} ; frac{{EC_{50} }}{{EC_{50}^{prime } }} = frac{alpha }{{tau_{B} + 1}}$$

(23)

In case allosteric agonist does not affect equilibrium dissociation constant of an orthosteric agonist K_{A} (α = 1) but allosterically modulates the operational efficacy of the orthosteric agonist (β ≠ 1) Eq. (16) simplifies to Eq. (24).

$$frac{{EC_{50} }}{{EC_{50}^{prime } }} = frac{{left[ B right] + beta left[ B right]tau_{A} + K_{B} tau_{A} + K_{B} }}{{left( {tau_{A} + 1} right)left( {left[ B right] + left[ B right]tau_{B} + K_{B} } right)}}$$

(24)

In contrast to the factor of binding cooperativity, α, the factor of operational cooperativity, β, affects both the observed maximal response E’_{MAX} and observed half-efficient concentration of an agonist EC’_{50} (Fig. 8). In the case of negative operational cooperativity (Fig. 8, left), the allosteric agonist concentration-dependently decreases the observed maximal response E’_{MAX} and increases the value of EC’_{50}. In the case of positive operational cooperativity (Fig. 8, right), the allosteric agonist increases E’_{MAX} but, paradoxically, may increase EC’_{50} as shown in Fig. 8, right. This increase in EC’_{50} value happens when a decrease in the EC’_{50} value due to positive operational cooperativity β is smaller than an increase in the EC’_{50} value due to operational activity of allosteric agonist τ_{B}. Maximal dose ratio is given by Eq. (25) (Fig. 8, bottom).

$$For left[ B right] gg K_{B} ; frac{{EC_{50} }}{{EC_{50}^{prime } }} = frac{{beta tau_{A} + 1}}{{left( {tau_{A} + 1} right)left( {tau_{B} + 1} right)}}$$

(25)

Additional combinations of types of cooperativity of binding α and operational efficacy β between orthosteric and allosteric agonists are illustrated in Fig. 9. The meta-analysis of concentration–response curves is in Supplementary information Figure S3 and S4. For the derivation of equations, see Supplementary information, Eqs. 30 to 56.