The ability of SABRE, a new quantitative receptor function model, to quantify receptor binding from even challenging concentration-effect data with a single unified fit

The Signal Amplification, Binding affinity, and Receptor-activation Efficacy (SABRE) model is the most recent general and quantitative model of receptor function. A specific extension of the SABRE model enables the determination of K_d (the equilibrium dissociation constant of the agonist-receptor complex) and q (the fraction of receptors remaining operable after pretreatment with an irreversible receptor antagonist) from exclusively functional data. In the present investigation, we reevaluated the concentration-effect (E/c) data of our related recent study on the SABRE model to assess the properties of our newly developed multiline model, inspired by professional criticism of our previous study in question. We have found this multiline model, constructed within the framework of the SABRE model, to be capable of providing reliable results via one global fitting (i.e., with a single unified fit), even for our somewhat challenging data (containing some uncertainty). The multiline model that proved to be the most suitable for the current data was a relatively complex, six-model global fitting. These results further emphasize the significance of finding the best way to fit the equations of the SABRE model to the functional data to be evaluated.

1 Introduction

Modelling the function of receptors (in a pharmacological sense) can be useful in solving practical problems and may also provide valuable information about the underlying mechanisms of biological phenomena (Motulsky and Christopoulos, 2004; Kenakin, 2022). The most recent general and quantitative receptor function model is the Signal Amplification, Binding affinity, and Receptor-activation Efficacy (SABRE) model (Buchwald, 2017; Buchwald, 2019; Buchwald, 2020; Buchwald, 2022; Buchwald, 2023; Buchwald, 2025a). In addition to providing a tool to understand and simulate experimental phenomena, this model can be used to obtain estimates for K_d (the equilibrium dissociation constant of the agonist-receptor complex) and q (the fraction of the operable receptors after partial irreversible receptor inactivation) from purely functional data, specifically from appropriately constructed concentration-effect (E/c) curves (Buchwald, 2022).

In a previous study from our laboratory, we aimed to demonstrate the utility and main features of the SABRE model by evaluating E/c data generated with three A₁ adenosine receptor full agonists in isolated, paced guinea pig left atria, before and after a pretreatment with an irreversible A₁ adenosine receptor antagonist (Olah et al., 2025). In that study, we developed four fitting strategies of the SABRE model and compared them with one another. Finally, we recommended a two-step procedure (consisting of the third and fourth fitting strategies) as the best way to evaluate the presented sort of data (Olah et al., 2025).

Soon after, the results of our above-mentioned study (Olah et al., 2025) were commented on by Buchwald (2025b) and a single-step procedure was suggested as a more appropriate manner to achieve the same goal. We have found the core idea of this improvement noteworthy. Thus, we hereby propose our own new multiline model, designed for the current version of the GraphPad Prism software, that uses the same nomenclature and “fitting logic” as before (Olah et al., 2025), but incorporates the improvement suggested by Buchwald (2025b). Accordingly, we have named this multiline model the “fifth” fitting strategy, as a continuation of the four fitting strategies described in our previous work (Olah et al., 2025).

2 Methods

2.1 The reevaluated data

In the present study, we reevaluated the E/c data that were used in a recent study (Olah et al., 2025) and were originally generated in an earlier investigation from our laboratory (Gesztelyi et al., 2013). These E/c curves were constructed with NECA (5′-(N-ethylcarboxamido)adenosine), CPA (N⁶-cyclopentyladenosine) and CHA (N⁶-cyclohexyladenosine), generally considered to be A₁ adenosine receptor full agonists (Fredholm et al., 2001; Deb et al., 2019). The E/c curves were constructed in the absence (labelled “N”) and presence (labelled “X”) of a pretreatment with FSCPX (8-cyclopentyl-N³-[3-(4-(fluorosulfonyl)benzoyloxy)propyl]-N¹-propylxanthine), an irreversible A₁ adenosine receptor antagonist (Srinivas et al., 1996). Thus, we worked with “Furchgott-type” data, i.e., E/c data obtained at different levels of the operable receptors (Furchgott, 1966; Furchgott and Bursztyn, 1967).

For curve plotting and fitting, the independent variable (X value) was the common logarithm of the molar concentration of the agonists. To obtain the dependent variable (Y value), the percentage decrease in the initial contractile force of the isolated and paced guinea pig left atria was first determined (E), and then this effect value was expressed as the percentage of the maximal effect (E_max) achieved in the given experimental system during the given study (E/E_max %). Thus, in fact, we created E/E_max % vs. logc curves but, for simplicity, we called them E/c curves in this investigation as well.

2.2 The fifth fitting strategy, i.e., our newly proposed multiline model (constructed within the SABRE model)

Based on the suggestion of Buchwald (2025b), our new recommendation for fitting our “Furchgott-type” E/c data is the following six-model global fitting, presented as a multiline model designed for GraphPad Prism (RRID:SCR_002798):

$\text{NECA}=100*\left(\mathrm{\epsilon }\_\text{NECA}*\mathrm{\gamma }*10^\left(\mathrm{n}*\mathrm{X}\right)\right)/\left(\left(\mathrm{\epsilon }\_\text{NECA}*\mathrm{\gamma }-\mathrm{\epsilon }\_\text{NECA}+1\right)*10^\left(\mathrm{n}*\mathrm{X}\right)+10^\left(\mathrm{n}*\text{logKd}\_\text{NECA}\right))\right.$

$\text{NECAq}=100*\left(\mathrm{q}*\mathrm{\epsilon }\_\text{NECA}*\mathrm{\gamma }*10^\left(\mathrm{n}*\mathrm{X}\right)\right)/\left(\left(\mathrm{q}*\mathrm{\epsilon }\_\text{NECA}*\mathrm{\gamma }-\mathrm{q}*\mathrm{\epsilon }\_\text{NECA}+1\right)*10^\left(\mathrm{n}*\mathrm{X}\right)+10^\left(\mathrm{n}*\text{logKd}\_\text{NECA})\right)\right.$

$\text{CPA}=100*\left(\mathrm{\epsilon }\_\text{CPA}*\mathrm{\gamma }*10^\left(\mathrm{n}*\mathrm{X}\right)\right)/\left(\left(\mathrm{\epsilon }\_\text{CPA}*\mathrm{\gamma }-\mathrm{\epsilon }\_\text{CPA}+1\right)*10^\left(\mathrm{n}*\mathrm{X}\right)+10^\left(\mathrm{n}*\text{logKd}\_\text{CPA}\right)\right)$

$\text{CPAq}=100*\left(\mathrm{q}*\mathrm{\epsilon }\_\text{CPA}*\mathrm{\gamma }*10^\left(\mathrm{n}*\mathrm{X}\right)\right)/\left(\left(\mathrm{q}*\mathrm{\epsilon }\_\text{CPA}*\mathrm{\gamma }-\mathrm{q}*\mathrm{\epsilon }\_\text{CPA}+1\right)*10^\left(\mathrm{n}*\mathrm{X}\right)+10^\left(\mathrm{n}*\text{logKd}\_\text{CPA})\right)\right.$

$\text{CHA}=100*\left(\mathrm{\epsilon }\_\text{CHA}*\mathrm{\gamma }*10^\left(\mathrm{n}*\mathrm{X}\right)\right)/\left(\left(\mathrm{\epsilon }\_\text{CHA}*\mathrm{\gamma }-\mathrm{\epsilon }\_\text{CHA}+1\right)*10^\left(\mathrm{n}*\mathrm{X}\right)+10^\left(\mathrm{n}*\text{logKd}\_\text{CHA})\right)\right.$

$\text{CHAq}=100*\left(\mathrm{q}*\mathrm{\epsilon }\_\text{CHA}*\mathrm{\gamma }*10^\left(\mathrm{n}*\mathrm{X}\right)\right)/\left(\left(\mathrm{q}*\mathrm{\epsilon }\_\text{CHA}*\mathrm{\gamma }-\mathrm{q}*\mathrm{\epsilon }\_\text{CHA}+1\right)*10^\left(\mathrm{n}*\mathrm{X}\right)+10^\left(\mathrm{n}*\text{logKd}\_\text{CHA}\right)\right)$

$<\mathrm{A}>\mathrm{Y}=\text{NECA}$

$<\mathrm{D}>\mathrm{Y}=\text{NECAq}$

$<\mathrm{B}>\mathrm{Y}=\text{CPA}$

$<\mathrm{E}>\mathrm{Y}=\text{CPAq}$

$<\mathrm{C}>\mathrm{Y}=\text{CHA}$

$<\mathrm{F}>\mathrm{Y}=\text{CHAq}$

Here, the six-model global regression means that a separate equation should be fitted to each data set (six equations for the E/c curves of the six experimental groups of the original investigation: Gesztelyi et al., 2013). For this curve fitting, some parameters were shared among some data sets (Figure 1; Table 1). In the final regression, n and the three ε parameters were constrained to unity (to improve the reliability of the other estimates), as previously (Olah et al., 2025). Importantly, a single fitting can provide all estimates for the parameters.

Figure 1

Figure 1. The fifth fitting strategy of the SABRE receptor function model applied to six data sets, consisting of E/c curves of three synthetic A₁ adenosine receptor full (or close to full) agonists (NECA, CPA, CHA), constructed in isolated, paced guinea pig left atria, in the absence (“N”) or presence (“X”) of a pretreatment with FSCPX, an irreversible A₁ adenosine receptor antagonist. The SABRE parameters, which were shared among the data sets (at least before their constraint to unity), are marked in blue. E/c, concentration-effect; SABRE, Signal Amplification, Binding affinity, and Receptor-activation Efficacy; NECA, 5′-(N-ethylcarboxamido)adenosine; CPA, N⁶-cyclopentyladenosine; CHA, N⁶-cyclohexyladenosine; FSCPX: 8-cyclopentyl-N³-[3-(4-(fluorosulfonyl)benzoyloxy)propyl]-N¹-propylxanthine.

Table 1

Table 1. Results provided by the SABRE model using the fifth fitting strategy (i.e., by fitting the multiline model presented).

The parameters characterizing the postreceptorial signal handling (γ and n) apply to all data sets, while the parameters describing the particular agonist-receptor interactions (ε_NECA, ε_CPA, ε_CHA, logK_d_NECA, logK_d_CPA and logK_d_CHA) refer only to the data set pairs generated with the same agonist. Furthermore, the parameter characterizing the efficiency of the pretreatment with FSCPX (q) applies only to the three data sets that underwent this pretreatment. For the classic appearance of the multiline model, see: Supplementary Appendix.

2.3 Data processing and presentation

Curve plotting and fitting were implemented with GraphPad Prism 10.6.1 for Windows (GraphPad Software Inc., La Jolla, CA, USA).

The precision of regression was characterized by the width of the 95% confidence interval (CI) of the best-fit values. For computing 95% CIs, the “asymmetrical” option was always chosen. The precision of the curve fitting and the precision of the E/c curve data were characterized by the distance of the best-fit curve from the corresponding 95% confidence bands and by the 95% prediction bands, respectively.

When setting the way in which the software checks how well the experimental data define the model, the option “Identify ambiguous fits” was chosen.

The degree to which each variable parameter was intertwined with all the others was indicated by dependency, the value of which could range from 0 (independent parameter) to 1 (redundant parameter). Dependency values greater than 0.9 and 0.99 could be considered high and unacceptably high, respectively.

The goodness of fit of the model was quantified by the “individual” and global values of the coefficient of determination (R²) and the adjusted value of the global R². This adjusted global R² is much lower than the global R² if the model contains redundant parameters (Graphpad, 2025).

3 Results

The logK_d and q values provided by the fifth fitting strategy were similar to the corresponding values obtained previously using the third fitting strategy (Olah et al., 2025). Moreover, the related reliability measures (95% confidence limits and dependency) showed some improvement owing to the fifth fitting strategy (Table 1). The γ and (especially) q values (plus the related reliability measures) determined with the fifth fitting strategy were also close to the corresponding data obtained with the fourth fitting strategy (Olah et al., 2025) (Table 1). The measures characterizing the fitting as a whole (the different sorts of R²) and the appearance of the best-fit curves and their confidence and prediction bands were also similar to those presented in our original work (Olah et al., 2025) (Table 1; Figure 2).

Figure 2

Figure 2. The E/c curves of three A₁ adenosine receptor full agonists, NECA (panel (A)), CPA (panel (B)) and CHA (panel (C)), generated in isolated, paced guinea pig left atria in the absence (filled symbols) and presence (open symbols) of a pretreatment with FSCPX, an irreversible A₁ adenosine receptor antagonist. The x-axis shows the common logarithm of the molar concentration of the given agonist, while the y-axis denotes the direct negative inotropic effect expressed as a percentage of the maximal effect achieved in this system (±SEM). The continuous lines show the best-fit curves of the SABRE model, fitted according to the fifth fitting strategy. For more details (sharing, constraints), see: Table 1. The thick dotted lines indicate the 95% confidence bands, while the thin dotted lines represent the 95% prediction bands. E/c, concentration-effect; SABRE, Signal Amplification, Binding affinity, and Receptor-activation Efficacy; NECA, 5′-(N-ethylcarboxamido)adenosine; CPA, N⁶-cyclopentyladenosine; CHA, N⁶-cyclohexyladenosine; FSCPX, 8-cyclopentyl-N³-[3-(4-(fluorosulfonyl)benzoyloxy)propyl]-N¹-propylxanthine.

4 Discussion

In the present study, we have improved our previous work (Olah et al., 2025) by elaborating a fifth fitting strategy incorporating Buchwald’ suggestion, the core idea of which is that each agonist-related parameter should be individualized (for the specific agonist used: NECA, CPA and CHA) (Buchwald, 2025b). Thus, instead of ε and logK_d, ε_agonist and logK_d_agonist (e.g., ε_NECA and logK_d_NECA) should be fitted. The curve fitting software, used here, allows the E/c curves generated with a particular agonist to be fitted only to the equation that contains the corresponding agonist-related parameters. This maneuver is similar to the one that allows the E/c curves without and with an FSCPX pretreatment to be fitted separately (and adequately). In this way, six equations can be obtained (as a combination of the three agonists and the two outcomes of the FSCPX pretreatment). The global nature of regression can be ensured by sharing the agonist-related parameters (ε_agonist and logK_d_agonist), γ, n and q.

In fact, the six equations in question were almost ready as the fourth fitting strategy (Olah et al., 2025). To finalize these equations, we replaced the nonspecific parameter ε and the specific logK_d values of the three agonists with the appropriate individualized parameters (ε_NECA, ε_CPA, ε_CHA, logK_d_NECA, logK_d_CPA and logK_d_CHA). To ensure a single unified fit, all these individualized agonist-related parameters were shared among the data sets (in addition to γ, n and q). The fitting constraints, which assigned each of the six equations to the corresponding one of the six data sets, could be left the same for the fifth strategy as for the fourth strategy.

5 Conclusion

Our present work has demonstrated that the most recent method for estimating K_d values of agonist-receptor complexes from E/c data (Buchwald, 2022), an application of the SABRE receptor function model (Buchwald, 2017; 2019), is capable of providing reliable results through one global fitting (i.e., with a single unified fit), even for our data with some challenge. This challenge stemmed from the fact that our E/c curves, constructed in the presence of an irreversible antagonist, did not show a decrease in their maximal effect (as compared to the corresponding control E/c curves), rendering these “Furchgott-type” data somewhat uncertain. Our results further emphasize the importance of finding the best way to fit the equations of the SABRE model to this type of functional data. In the present case, this “best way” was a properly constructed multiline model (i.e., a six-model global fitting).

Data availability statement

The datasets presented in this study can be found in online repositories. The names of the repository/repositories and accession number(s) can be found in the article/Supplementary Material.

Ethics statement

The animal study was approved by Committee of Animal Research, University of Debrecen, Hungary (DE MAB 35/2007). The study was conducted in accordance with the local legislation and institutional requirements.

Author contributions

BO: Investigation, Writing – original draft. VT: Data curation, Resources, Writing – review and editing. GV: Data curation, Resources, Writing – review and editing. IO: Data curation, Resources, Writing – review and editing. AC: Data curation, Resources, Writing – review and editing. ZS: Funding acquisition, Writing – review and editing. BJ: Data curation, Resources, Writing – review and editing. JZ: Data curation, Resources, Writing – review and editing. RG: Conceptualization, Formal Analysis, Writing – review and editing. TE: Conceptualization, Formal Analysis, Writing – review and editing.

Funding

The author(s) declared that financial support was received for this work and/or its publication. This research was supported by the European Union and the State of Hungary under the grant number TKP2021-EGA-19 (TKP2021-EGA-19 has been implemented with the support provided by the Ministry of Culture and Innovation of Hungary from the National Research, Development and Innovation Fund, financed under the TKP2021-EGA funding scheme). Furthermore, this research was supported by the University of Debrecen Program for Scientific Publication.

Conflict of interest

The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphar.2026.1715771/full#supplementary-material

References

Buchwald, P. (2017). A three-parameter two-state model of receptor function that incorporates affinity, efficacy, and signal amplification. Pharmacol. Res. Perspect. 5, e00311. doi:10.1002/prp2.311

PubMed Abstract | CrossRef Full Text | Google Scholar

Buchwald, P. (2019). A receptor model with binding affinity, activation efficacy, and signal amplification parameters for complex fractional response Versus occupancy data. Front. Pharmacol. 10, 605. doi:10.3389/fphar.2019.00605

PubMed Abstract | CrossRef Full Text | Google Scholar

Buchwald, P. (2020). A single unified model for fitting simple to complex receptor response data. Sci. Rep. 10, 13386. doi:10.1038/s41598-020-70220-w

PubMed Abstract | CrossRef Full Text | Google Scholar

Buchwald, P. (2022). Quantification of receptor binding from response data obtained at different receptor levels: a simple individual sigmoid fitting and a unified SABRE approach. Sci. Rep. 12, 18833. doi:10.1038/s41598-022-23588-w

PubMed Abstract | CrossRef Full Text | Google Scholar

Buchwald, P. (2023). Quantitative receptor model for responses that are left- or right-shifted versus occupancy (are more or less concentration sensitive): the SABRE approach. Front. Pharmacol. 14, 1274065. doi:10.3389/fphar.2023.1274065

PubMed Abstract | CrossRef Full Text | Google Scholar

Buchwald, P. (2025a). Quantification of signal amplification for receptors: the Kd/EC50 ratio of full agonists as a gain parameter. Front. Pharmacol. 16, 1541872. doi:10.3389/fphar.2025.1541872

PubMed Abstract | CrossRef Full Text | Google Scholar

Buchwald, P. (2025b). Commentary: the flexibility of SABRE, a new quantitative receptor function model, in fitting challenging concentration-effect data. Front. Pharmacol. 16, 1675039. doi:10.3389/fphar.2025.1675039

PubMed Abstract | CrossRef Full Text | Google Scholar

Deb, P. K., Deka, S., Borah, P., Abed, S. N., and Klotz, K. N. (2019). Medicinal chemistry and therapeutic potential of agonists, antagonists and allosteric modulators of A1 adenosine receptor: current status and perspectives. Curr. Pharm. Des. 25, 2697–2715. doi:10.2174/1381612825666190716100509

PubMed Abstract | CrossRef Full Text | Google Scholar

Fredholm, B. B., IJzerman, A. P., Jacobson, K. A., Klotz, K. N., and Linden, J. (2001). International Union of pharmacology. XXV. Nomenclature and classification of adenosine receptors. Pharmacol. Rev. 53, 527–552.

PubMed Abstract | CrossRef Full Text | Google Scholar

Furchgott, R. F. (1966). The use of b-haloalkylamines in the differentiation of receptors and in the determination of dissociation constants of receptor-agonist complexes. Adv. Drug Res. 3, 21–55.

Google Scholar

Furchgott, R. F., and Bursztyn, P. (1967). Comparison of dissociation constants and of relative efficacies of selected agonists acting on parasympathetic receptors. Ann. N. Y. Acad. Sci. 144, 882–899. doi:10.1111/j.1749-6632.1967.tb53817.x

CrossRef Full Text | Google Scholar

Gesztelyi, R., Kiss, Z., Wachal, Z., Juhasz, B., Bombicz, M., Csepanyi, E., et al. (2013). The surmountable effect of FSCPX, an irreversible A(1) adenosine receptor antagonist, on the negative inotropic action of A(1) adenosine receptor full agonists in isolated Guinea pig left atria. Arch. Pharm. Res. 36, 293–305. doi:10.1007/s12272-013-0056-z

PubMed Abstract | CrossRef Full Text | Google Scholar

Graphpad (2025). Curve fitting guide. Available online at: https://www.graphpad.com/guides/prism/latest/curve-fitting/index.htm (accessed on June 22, 2025).

Google Scholar

Kenakin, T. (2022). A pharmacology primer. Techniques for more effective and strategic drug discovery. Academic Press. ISBN: 9780323992893.

Google Scholar

Motulsky, H. J., and Christopoulos, A. (2004). “Fitting models to biological data using linear and nonlinear regression,” in A practical guide to curve fitting. San Diego: GraphPad Software Inc. Available online at: https://cdn.graphpad.com/faq/2/file/Prism_v4_Fitting_Models_to_Biological_Data.pdf (accessed on June 22, 2025).

Google Scholar

Olah, B., Tarjanyi, V., Takacs, B., Pluzsnyik, E., Viczjan, G., Ovari, I., et al. (2025). The flexibility of SABRE, a new quantitative receptor function model, when fitting challenging concentration-effect data. Front. Pharmacol. 16, 1591761. doi:10.3389/fphar.2025.1591761

PubMed Abstract | CrossRef Full Text | Google Scholar

Srinivas, M., Shryock, J. C., Scammells, P. J., Ruble, J., Baker, S. P., and Belardinelli, L. (1996). A novel irreversible antagonist of the A1-adenosine receptor. Mol. Pharmacol. 50, 196–205.

PubMed Abstract | CrossRef Full Text | Google Scholar