The operational model of allosteric modulation of pharmacological agonism

Proper determination of agonist efficacy is indispensable in the evaluation of agonist selectivity and bias to activation of specific signalling pathways. The operational model (OM) of pharmacological agonism is a useful means for achieving this goal. Allosteric ligands bind to receptors at sites that are distinct from those of endogenous agonists that interact with the orthosteric domain on the receptor. An allosteric modulator and an orthosteric agonist bind simultaneously to the receptor to form a ternary complex, where the allosteric modulator affects the binding affinity and operational efficacy of the agonist. Allosteric modulators are an intensively studied group of receptor ligands because of their selectivity and preservation of physiological space–time pattern of the signals they modulate. We analysed the operational model of allosterically-modulated agonism (OMAM) including modulation by allosteric agonists. Similar to OM, several parameters of OMAM are inter-dependent. We derived equations describing mutual relationships among parameters of the functional response and OMAM. We present a workflow for the robust fitting of OMAM to experimental data using derived equations.

Scientific RepoRtS | (2020) 10:14421 | https://doi.org/10.1038/s41598-020-71228-y www.nature.com/scientificreports/ better. In case of agonist binding to the receptor that is allosterically modulated, the parsimonious OM needs to be extended by the operational factor of cooperativity (β) and equilibrium dissociation constant of the allosteric modulator (K B ) at the active state of the receptor (Fig. 2) 17 . The operational factor of cooperativity, β, quantifies the overall effect of an allosteric modulator on operational efficacy of an orthosteric agonist, τ A , and thus brings inter-dependence of three OM parameters with K B . The resulting operational model of allosterically-modulated agonism (OMAM) is thus very complex and has five inter-dependent parameters: E MAX, K A , τ A , K B and β. Several allosteric ligands of various receptors activate the receptor in the absence of an agonist [18][19][20][21][22][23][24] . These ligands are termed allosteric agonists. Their intrinsic activity, τ B , can be ranked according to the OM. The OMAM that describes the functional response to an agonist in the presence of an allosteric agonist is even more complex than the OMAM for pure allosteric modulators that lack agonistic activity. The number of inter-dependent parameters rises to six.
In this paper, we analyse the OMAM and derive equations describing mutual relations among parameters of functional response and OMAM, both for pure allosteric modulators and allosteric agonists. These equations would be useful in the analysis of experimental data of such complex systems. For this purpose, we also present a workflow for the reliable fitting of OMAM to experimental data using the derived equations to avoid fitting equations with inter-dependent parameters.
Description and analysis of models 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).
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).
The apparent maximal response E' MAX observed as the upper asymptote of the functional response curve is given by Eq. (3). The cubic ternary complex model of allosteric modulation of receptor activation. Receptor exists in two states; inactive, R, and active, R*. The equilibrium between these two states is given by the activation constant K ACT . An orthosteric agonist A binds to the inactive receptor with an equilibrium dissociation constant K A . An allosteric modulator binds to the inactive receptor with an equilibrium dissociation constant K B . α, the factor of binding cooperativity between the orthosteric agonist A and allosteric modulator B. β, the factor of cooperativity between binding of agonist A and receptor activation. γ, the factor of cooperativity between the binding of allosteric modulator B and receptor activation. δ, the factor of cooperativity between the binding of an allosteric modulator and agonist-induced receptor activation. Further, equilibrium dissociation constant K G of G-protein, G, may differ among individual complexes of R* by cooperativity factors ε and η. Also signalling efficacy φ of G-protein may differ among complexes. Adopted from .
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 inactivereceptor 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 Scheme of the ternary complex model of allosteric interaction. Representation of allosteric interaction between an orthosteric agonist A (blue circle) and an allosteric modulator B (green diamond) at the receptor R (red U-shape). An orthosteric agonist A binds to the receptor R with an equilibrium dissociation constant K A . An allosteric modulator B binds to the receptor R with an equilibrium dissociation constant K B . The orthosteric agonist A and the allosteric modulator B can bind concurrently to the receptor R to form a ternary complex ARB. The factor of binding cooperativity α is the ratio of the equilibrium dissociation constant to empty receptor to the equilibrium dissociation to binary complex AR or RB. The complex of receptor and orthosteric agonist (AR) has operational efficacy τ A . The complex of receptor and allosteric agonist (RB) has operational efficacy τ B . The factor of operational cooperativity β is the ratio of operational efficacy of the ternary complex ARB to operational efficacy of binary complex AR.
Scientific RepoRtS | (2020) 10:14421 | https://doi.org/10.1038/s41598-020-71228-y www.nature.com/scientificreports/ 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 .
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. From Eq. (6) it is obvious that both factors α and β affect EC′ 50 . Thus, α and β are the fourth and fifth interdependent 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).
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).
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.
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).
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 Scientific RepoRtS | (2020) 10:14421 | https://doi.org/10.1038/s41598-020-71228-y www.nature.com/scientificreports/ 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).
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 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). 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).
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. www.nature.com/scientificreports/ 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).
To separate individual factors of cooperativity, the dose ratio may be expressed by Eq. (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).
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).
For a pure allosteric agonist the maximal dose ratio is given by Eq. (21) (Fig. 6, bottom).
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 concentrationdependently 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).
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 Scientific RepoRtS | (2020) 10:14421 | https://doi.org/10.1038/s41598-020-71228-y www.nature.com/scientificreports/ 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).
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

Application of models to experimental data
In this section, we demonstrate the application of OMAM equations described above to experimental data. Binding parameters. The binding parameters were determined in competition experiments with the radiolabelled orthosteric antagonist N-methylscopolamine ([ 3 H]NMS). Both agonists, carbachol and iperoxo, displayed binding to two populations of binding sites (Fig. 11). Their high-affinity binding of 390 nM and 200 pM, respectively, can be taken as candidate values for equilibrium dissociation constants K A in OM. It should be noted that not always high-affinity binding site corresponds to the conformation of the receptor that initiates the signalling 15 . Allosteric modulator BQCA caused incomplete inhibition of [ 3 H]NMS indicating the allosteric mode of interaction (Fig. 12, left). BQCA decreased affinity of [ 3 H]NMS more than threefold. BQCA increased affinity of carbachol 33-times and iperoxo 4.8-times. BQCA apparent K B was about 50 μM. In contrast to BQCA,

Parameters of OM.
The parameters of functional response to sole orthosteric agonists carbachol or iperoxo (K A , τ A ) and allosteric ligands BQCA and TBPB (K B and τ B ) were determined from changes in intracellular IP X levels in the presence of these ligands (Fig. 13). The basal level of IP X in the absence of agonist was about 0.88% of incorporated radioactivity. Both agonists produced immense response increasing the IP X level more than 50-times. Both allosteric ligands increased level of IP X . The E′ MAX of response to TBPB was close to E′ MAX of full agonist carbachol. First, the logistic Eq. (31) was fitted to the data. The slope of response curves was equal to unity in all cases (Supplementary information Table S3). Maximal system response E MAX determined by the procedure described earlier using a batch of agonists with a full spectrum of efficacies 15 was 98-fold over basal   (Fig. 14, red) and iperoxo (Fig. 15, red) indicating positive binding cooperativity (α > 1) as well as operational cooperativity (β > 1). BQCA in concentrations 30, 100 and 300 μM had the same effect on both EC′ 50 and E′ MAX values indicating saturation of its effect that is the sign of allosteric interaction. TBPB decreased E′ MAX values of both carbachol (Fig. 14, blue) and iperoxo (Fig. 15, blue) indicating negative operational cooperativity (β < 1). TBPB increased EC′ 50 values of carbachol and iperoxo. The increase is in part due to TBPB operational efficacy (τ B > 0). Negative binding cooperativity (α < 1) may also contribute to the observed increase in EC′ 50 . The effects of TBPB on E′ MAX and EC′ 50 were the same at 3 and 10 μM concentration indicating saturation of its effect that is the sign of allosteric interaction. From E′ MAX /E MAX ratios and τ A value (predetermined in Fig. 13) factors of operational cooperativity β were calculated according to Eq. (8). Subsequently, the factor of binding cooperativity α was calculated according to Eq. (15). Calculated cooperativity factors α and β were used as initial guesses in fitting Eq. (13) with E MAX , K A , K B , τ A and τ B fixed to values predetermined in functional experiments (Fig. 13). Global fitting of Eq. (13) to experimental data resulted in small SDs of estimated parameters indicating reliable results. In contrast, global   27) and (28)  In the case of BQCA, the factors of binding cooperativity α were the same in binding experiments (Fig. 12) as in functional experiments (Figs. 14 and 15). In the case of TBPB, however, observed negative binding cooperativity was about twice stronger in functional experiments (Figs. 14 and 15) than in binding experiments (Fig. 12).

Discussion
Proper determination of agonist efficacy is a cornerstone in the assessment of possible agonist selectivity and signalling bias. Apparent agonist efficacy is dependent on the system in which it is determined. The operational model of agonism (OM) 13 can reliably rank agonist efficacies at any receptor effector system 14 . However, the inherent glitch in OM is that the objective parameters (agonist equilibrium dissociation constant K A , its operational efficacy τ A and maximal possible response of the system E MAX ) describing it are inter-dependent 15 . To circumvent this pitfall, we proposed a two-step procedure of fitting of OM to experimental data. First, E MAX and K A are determined by fitting Eq. (4) to the observed E' MAX and EC 50 values. Then Eq. (1) is fitted to the concentration-response curves with E MAX and K A fixed to predetermined values. This two-step procedure yields robust fits. Allosteric modulators are intensively studied for their selectivity and preservation of space-time pattern of signalization they modulate [8][9][10] . They bind to a receptor concurrently with an orthosteric agonist and change the equilibrium dissociation constant K A of the agonist by a factor of binding cooperativity α ( Figs. 1 and 2). Values of binding cooperativity α greater than unity denote positive cooperativity; an increase in binding affinity that is reflected in a decrease in EC' 50 values (Fig. 3, right). Values of α lower than one denote negative cooperativity; a decrease in binding affinity that is manifested as an increase in EC' 50 values (Fig. 3, left).
Besides modulation of ligand binding affinity, an allosteric ligand can also affect receptor activation, the affinity of the receptor for G-proteins and efficacy of G-protein activation (Fig. 1) 16,25,26 . Such a multitude of possibilities makes it inconceivable to estimate any of the parameters of heuristic models due to their complexity. In the parsimonious operational model of allosterically-modulated agonism (OMAM) the operational factor of cooperativity β quantifies the overall effect of an allosteric modulator on the operational efficacy of an orthosteric agonist τ A (Fig. 2) 17 . Values of operational cooperativity β greater than one denote positive cooperativity, leading to an increase in the observed maximal response E' MAX (Fig. 4, right). Values of β lower than one denote negative cooperativity, leading to a decrease in E' MAX values (Fig. 4, left).
From Eqs. (6) and (7) it is obvious that both factors α and β are inter-dependent with τ A , E MAX and K A . Following the logic of the two-step procedure of fitting OM to experimental data described above, the factor of binding cooperativity α should be determined from dose ratios according to Eq. (8), alongside with determination of parameters τ A , K A and E MAX before fitting Eq. (5) to experimental data to yield reliable results (Fig. 10).
In case of allosteric agonists, the observed half-efficient concentration of an orthosteric agonist EC' 50 is affected not only by factors of cooperativity α and β but also by the operational efficacy of an allosteric agonist τ B (Fig. 6). Thus, the operational efficacy of allosteric agonist τ B becomes the sixth inter-dependent parameter with parameters α, β, τ A , K A and E MAX . Moreover, as a factor of operational efficacy of the orthosteric agonist τ A is inter-dependent with agonist K A , a factor of operational efficacy of the allosteric agonist τ B is inter-dependent with its K B . To yield reliable results, operational efficacies τ A , τ B , equilibrium dissociation constants K A and K B ,  www.nature.com/scientificreports/ the factor of binding cooperativity α and the maximal response of the system E MAX should be determined before fitting Eq. (14) to experimental data (Fig. 10). Similar to the case of an orthosteric agonist, parameters of an allosteric agonist can be determined by fitting Eq. (4) to the observed E' MAX and EC 50 values of functional response to the allosteric agonist. The factor of operational cooperativity β can be calculated (no regression necessary) according to Eq. (8). Subsequently, the factor of binding cooperativity α can be calculated (no regression necessary) according to Eqs. (15) or (17). Equations (5) and (14) describing the OMAM are markedly more complex than Eq. (1) describing the OM. While OM has 3 inter-dependent parameters, τ A , K A and E MAX , the OMAM has two additional inter-dependent parameters, α and β. In the case of an allosteric agonist, a sixth inter-dependent parameter, the operational efficacy of allosteric agonist τ B , comes into play. Therefore, it is necessary to experimentally determine as many parameters as possible before fitting Eqs. (5) or (14) to the data (Fig. 10). Values of K A and τ A of an orthosteric agonist and values of K B and τ B of an allosteric agonist should be determined in functional experiments as described above. Values of K B and α of an allosteric ligand can be determined in binding experiments 28 . However, it should be noted that both values of binding cooperativity α between an orthosteric agonist and an allosteric modulator and equilibrium dissociation constant of an allosteric modulator K B differ for the low-affinity binding site (inactive receptor) and high-affinity binding site (active receptor) 29 . Thus, values of binding parameters of allosteric modulators measured indirectly, e.g. using a radiolabelled antagonist as a tracer 30 , need not be suitable for the fitting of the OMAM. In case values of K B and α cannot be measured directly in the binding experiment, they can be inferred from dose ratios of functional response to an agonist in the presence of an allosteric modulator as described above.
In practice, the analysis of functional responses may be further complicated by response curves not following rectangular hyperbola (n H ≠ 1). Flat curves may indicate negative cooperativity between two sites or non-equilibrium conditions 31 . Steep curves may indicate positive cooperativity between two sites or assay clipping. Such situations deserve further analysis. However, as n H of logistic Eq. (31) does not affect inflexion point (EC′ 50 ) or upper asymptote (E′ MAX ) derived equation describing relations of EC′ 50 and E′ MAX are valid also for flat or steep response curves. This represents another advantage over the direct fitting of Eqs. (5) and (14) as the introduction of n H brings an additional degree of freedom to them.
As a case study, we present the application of derived equations of OMAM on allosteric modulation of M 1 receptors. We followed the workflow outlined in Fig. 10 to avoid fitting equations with inter-dependent parameters. Instead, we analysed the effects of ligands on apparent half-efficient concentration EC′ 50 and maximal response E′ MAX . First, binding parameters (K I , K A , K B and α) were determined in binding experiments (Figs. 11 and 12). Both agonists displayed two binding sites. The high to low-affinity ratio was greater for iperoxo than for carbachol indicating iperoxo has greater efficacy than carbachol 32 .
Parameters of functional response to sole orthosteric agonists (K A , τ A ) and allosteric agonists (K B and τ B ) were determined in functional experiments (Fig. 13). To determine values of K A and τ A or K B and τ B , logistic Eq. (31) was fitted to the data and the system maximal response E MAX was determined by the procedure described earlier 15 . Then Eqs. (1) or (13) with E MAX fixed to predetermined value was fitted to the response curves. The functional experiments confirmed that iperoxo has greater efficacy than carbachol. Obtained K A values corresponded to high-affinity K I s indicating that observed high-affinity sites correspond to receptor conformation initiating the signalling. In contrast, K B values determined in functional experiments (Fig. 13) were higher than K B values determined in binding experiments (Fig. 12) indicating that K B determined in the binding experiments is not K B of the receptor in the conformation that initiates the signalling. Rather it is an inactive conformation induced by the antagonists [ 3 H]NMS used as a tracer.
Subsequently, functional response to agonist in the presence of allosteric agonists was measured (Figs. 14 and 15). First logistic Eq. (31) was fitted to the experimental data. The factor of operational cooperativity β was calculated (no regression necessary) from E′ MAX /E MAX ratio according to Eq. (8). Then factor of binding cooperativity α was calculated (again no regression necessary) from EC 50 / EC′ 50 ratio according to Eq. (15). Finally, Eq. (14) with K A , K B , τ A and τ B fixed to values predetermined in functional experiments (Fig. 13) was fitted to the concentration-response curves (global fit) to determine confidence intervals of cooperativity factors α and β. In the case of BQCA, the factors of binding cooperativity α determined in binding experiments (Fig. 12) were the same as those determined in functional experiments (Figs. 14 and 15). In the case of TBPB, they differed. It should be noted that the estimates of TBPB α values in binding experiments were associated with high SDs as result of complete inhibition of [ 3 H]NMS binding by TBPB making estimation of the binding cooperativity between [ 3 H]NMS and TBPB unreliable. Low SDs obtained by the presented procedure indicate that estimates of α and β are reliable. In contrast, the fitting Eq. (1) (OM) with three inter-dependent parameters is problematic (Supplementary information Table S3B) 15 . OMAM Eq. (5) possesses five and Eq. (14) possesses six interdependent parameters making their direct fitting to the experimental data impossible.

conclusions
The described workflow analysis of functional response represents a robust way of fitting the operational model of allosterically-modulated agonism (OMAM) to experimental data. We believe that the workflow and derived equations describing relations among functional response to agonists and parameters of OMAM will be helpful to many for proper analysis of experimental data of allosteric modulation of receptors.
Methods cell culture and membrane preparation. CHO cells were grown to confluence in 75 cm 2 flasks in Dulbecco's modified Eagle's medium (DMEM) supplemented with 10% fetal bovine serum. Two million cells were subcultured in 100 mm Petri dishes. The medium was supplemented with 5 mM sodium butyrate for the Analysis of experimental data. Data from experiments were processed in Libre Office and then analysed and plotted using program Grace (https ://plasm a-gate.weizm ann.ac.il/Grace ). The following equations were used for non-linear regression analysis: where y is specific binding at free concentration x, B MAX is the maximum binding capacity, and K D is the equilibrium dissociation constant.
where y is specific radioligand binding at concentration x of competitor expressed as per cent of binding in the absence of a competitor, IC 50 is the concentration causing 50% inhibition of radioligand binding at high (IC 50high ) and low (IC 50low ) affinity binding sites, f low is the fraction of low-affinity binding sites expressed in per cent. Inhibition constant K I was calculated as: where [D] is the concentration of radioligand used and K D is its equilibrium dissociation constant.

Allosteric interaction. Interaction between tracer ([ 3 H]NMS) and allosteric modulator:
where y is specific radioligand binding at concentration x of the allosteric modulator as per cent of binding in the absence of allosteric modulator. Where Competition binding where y is response normalized to basal activity in the absence of allosteric ligand at concentration x, E' MAX is the apparent maximal response, EC 50 is concentration causing half-maximal effect, and n H is Hill coefficient.
Concentration − response curve