Assuming that (fractional) occupancy and response are described by classic hyperbolic functions versus ligand concentration [L] (sigmoid on log-scale), they can be written in the general form of Eq. 2 as: f o c c u p = e 100 L L + K d (9) f r e s p = e max L L + K o b s (10)

Occupancy is assumed to always reach a maximum of 100% (hence, the e ₁₀₀ notation), while responses can plateau at smaller values for partial agonists (0 < e _max ≤ 100%). To connect f _resp and f _occup and obtain the general functional form illustrated in the bottom row of Figure 1 (essentially the general form connecting two different hyperbolic functions of ligand concentration characterized by e _max & K _obs and e ₁₀₀ & K _d, respectively), one can express [L] from Eq. 9 L = K d f o c c u p e 100 − f o c c u p (11) and plug this form shown in Eq. 11 in to Eq. 10, giving f r e s p = e max K d f o c c u p e 100 K o b s + f o c c u p K d − K o b s = e max K d K d − K o b s f o c c u p f o c c u p + e 100 K o b s K d − K o b s (12)

Thus, response (f _resp, characterized by e _max and K _obs = EC₅₀) is connected to occupancy, (f _occup, characterized by e ₁₀₀ = 1 and K _d) via a hyperbolic function as shown in Eq. 12 above. Introducing κ as a parameter quantifying the shift between response and occupancy (essentially a fold change in response vs. occupancy at the half-maximal values) κ = K d E C 50 = K d K obs (13) f _resp can be written as: f r e s p = e max κ κ − 1 f o c c u p f o c c u p + e 100 κ − 1 (14)

Notably, these general functional dependencies that rely only on the assumption of hyperbolic (sigmoid on log scale) functions (Eqs 9, 10, 14; Figure 1) fit well within the formalism of SABRE. The concentration-response form of SABRE (Eq. 4) links directly to the general sigmoid response function (Eq. 10, identical with Eq. 2) via Eqs. 6, 7 that define K _obs and e _max in terms of the ε and γ parameters of SABRE. Eqs. 6, 7 can also be used to connect f _resp to f _occup as in Eq. 14 above just using the parameters of the SABRE model. From Eq. 6 for K _obs, κ = K d K obs = ε γ − ε + 1 (15)

Thus f r e s p = e max κ κ − 1 f o c c u p f o c c u p + 1 κ − 1 = ε γ ε γ − ε + 1 ε γ − ε + 1 ε γ − ε + 1 − 1 f o c c u p f o c c u p + 1 ε γ − ε + 1 − 1 (16) giving the same form that has been deduced before after simplifying Eq. 16 above (Buchwald, 2019; Buchwald, 2020): f r e s p = γ γ − 1 f o c c u p f o c c u p + 1 ε γ − 1 (17)

Hence, SABRE can fit either the hyperbolic concentration-response functions directly using Eq. 4 or the corresponding (also hyperbolic) response versus occupancy functions using Eq. 17, and the results should be the same; an example is included below (Figure 2; Supplementary Table S1). When fitting with SABRE, there can be some ambiguity if there are not enough and sufficiently wide-spread data, especially as the efficacy values are dependent on the common gain parameter, and it is not always evident what is a true full agonist in the assay (except that the maximum efficacy is limited at ε = 1.0). Nevertheless, if there are enough good quality data covering an adequately large range, well-defined fit can be obtained.

An illustration of fit with SABRE using receptor binding (K _d) data assessed together with response is provided with the concentration-dependent contractions of isolated rat aorta induced by a series of imidazoline type α-adrenoceptor agonists such as phenylephrine, oxymetazoline, naphazoline, clonidine, tolazoline, and others (Ruffolo et al., 1979). Response data for seven compounds, which requires 14 parameters to fit with standard sigmoid curves (7 compounds × 2 parameters, EC₅₀ and e _max for each), can be fitted with SABRE without significant loss in the quality of fit (Figure 2) while using only 8 parameters (1γ + 7εs) and also integrating the experimental K _ds (Supplementary Table S1). Thus, SABRE, the more parsimonious model, gives a fit that is only slightly worse (as judged based on the sum of squared errors, SSE, which increased from 91.2 to 211.2, and the corrected Akaike Information Content, AICc, which increased from 81.2 to 93.9) while it also provides valuable additional insight, as it integrates the binding data, suggests a value of the amplification along the assayed pathway (γ = 11.6 ± 1.8), and quantifies the efficacies of the evaluated agonists (ε ranging from 1.0 for phenylephrine to 0.006 for tetrahydrozoline).

The shift between response and occupancy, as quantified by κ (Eq. 13) from the experimental data for the full antagonist phenylephrine, provides context for the gain parameter γ of SABRE (and its interplay with ε). As shown in Eq. 15, κ can be written as εγ–ε +1 in terms of SABRE parameters, so that for a full agonist (ε = 1), κ = γ. Here, γ estimated from fitting the whole dataset (11.63 ± 1.83) indeed agrees very well with κ of the full agonist phenylephrine (12.3) (Supplementary Table S1). For the other compounds, which are all partial agonists, the shifts are smaller, but they are also well reproduced by the SABRE estimates (Supplementary Table S1) with the note that the EC₅₀ and, hence, κ estimates for the weak partial agonists (e.g., those with e _max < 30%) cannot be considered reliable as well-defined values could not be obtained for them due to the limited range of responses.

As the hyperbolic concentration-response shapes seem to be maintained for the right-shifted cases, the derived equations linking response to occupancy can still be used, just with κ < 1 for Eq. 14 (i.e., fold decrease and not increase in response vs. occupancy at the half-maximal values). Similarly, right-shifted responses can very nicely be accommodated within the formalism of SABRE by allowing the gain parameter to be less than unity (γ < 1). With this, the corresponding equations such as Eqs 4, 17 can be used without needing any further modifications. As γ > 1 is an indication of signal amplification, this suggests the opposite, i.e., an apparent signal attenuation (dampening) or loss.

Data from three different experimental works by three different groups studying possible biased agonism at the MOPr (a class A GPCR) are used here for a SABRE-based quantitative analysis of right-shifted responses. MOPr signaling has been in particular focus recently because of the purported possibility of achieving improved analgesia by reducing the unwanted side-effects of opiate therapeutics via biased signaling. Data used here are from three different published works that, however, all (i) included both DAMGO (D-Ala², N-MePhe⁴, Gly-ol⁵–enkephalin) and morphine as agonist ligands, (ii) quantified both G-protein activation and β-arrestin recruitment in cell-based assays, and (iii) in addition to responses also measured MOPr binding affinities: works by McPherson and coworkers at the University of Bristol (UK; 2010) (McPherson et al., 2010), Hothersall and coworkers at Pfizer (Cambridge, UK; 2017) (Hothersall et al., 2017), and Pedersen and coworkers at the University of Copenhagen (Denmark; 2019) (Pedersen et al., 2019) (see Methods for relevant experimental details). As shown in Table 1, there were some differences in the measured binding (log K _d) and activity (log EC₅₀) data among these works; nevertheless, in all cases, the G protein activation responses were left-shifted compared to occupancy (κ _Gprt = K _d/EC_50,Gprt > 1.0), whereas β-arrestin recruitment responses were right-shifted (κ _βArr = K _d/EC_50,βArr < 1.0) despite originating from the same receptors.

Plotting of the data either as classic concentration-response curves (in parallel with occupancies) or as response vs. occupancy (Figure 3, top and bottom rows, respectively), clearly shows that the β-arrestin responses lag behind the occupancy even for the full agonist DAMGO (light blue versus dashed curves) while the G-protein responses (dark blue) are well ahead. At the lower end of fractional occupancy where, e.g., only a quarter of the receptors is occupied, f _occup = 25%, β-arrestin responses are minimal, f _resp,βArr < 5%, whereas corresponding G-protein responses are already approaching their maximum, f _resp,Gprot > 80% (Figures 3B, C). Thus, there has to be something limiting the β-arrestin response as compared to receptor occupancy.

Regardless of this, fitting with SABRE produces good results for both responses in all three datasets (Figure 3) and with consistent sets of parameters (Table 1). In all cases, the G-protein activation responses are amplified with the gain parameter γ > 1 with values ranging somewhere between 10 and 30. Meanwhile, the β-arrestin responses gave γ < 1 with values ranging from ∼1/15 to 1/1.5. Allowing different efficacies for the two divergent pathways [as possible sign of biased agonism (Buchwald, 2019; Buchwald, 2020)—see also below], did not result in significantly different values in any of these cases; therefore, all fittings shown here (Figure 3; Table 1) were obtained with a single efficacy for each ligand (i.e., εs restricted to the same value for both pathways, ε _Gprt = ε _βArr), which also made the fittings much better defined. Results indicated DAMGO to be a full or very close to full agonist (ε ≈ 1.0) and morphine a weaker partial agonist in all three data sets (Table 1).

For pleiotropically linked receptors such as MOPr, responses along different pathways could be different even with the same activation signal, often for obvious physiological reasons. For example, G protein activations often assessed through second messenger assays (e.g., measurement of cAMP) tend to be highly amplified, whereas β-arrestin complementation ones do not, resulting in different potencies (e.g., EC_50,Gprt << EC_50,βArr). In addition to this phenomenon, which is termed system bias, ligands might show what has been designated as biased agonism (functional selectivity), i.e., activate these pathways to different degrees even if they originate from the same receptor (Kenakin, 1995; Urban et al., 2007; Rajagopal et al., 2011; Kenakin and Christopoulos, 2013; Luttrell, 2014; Shonberg et al., 2014; Stahl et al., 2015; Ehlert, 2018; Michel and Charlton, 2018; Smith et al., 2018; Wootten et al., 2018; Kenakin, 2019; Karl et al., 2023). Biased agonism is an intriguing option for improved therapeutic action, and G-protein biased ligands at MOPr have been particularly pursued following the suggestion that they might be less likely to induce unwanted side effects, such as constipation and respiratory depression, than commonly used opioids due to differential engagement of G proteins versus β-arrestins. However, quantifying bias is challenging and might not even be achievable in most cases (Onaran et al., 2017; Kenakin, 2018b; Michel and Charlton, 2018). Visual evaluations can be done best with bias plots that show the response produced in one signaling pathway as a function of the response produced in the other at the same ligand concentration as shown, for example, in Figure 4D.

For quantitative assessment, current methods typically rely on calculating ΔΔlog (τ/K _D) or ΔΔlog (E _max/EC₅₀) versus a selected reference compound (Onaran et al., 2017; Michel and Charlton, 2018). SABRE allows a conceptually different approach as long as there is sufficient data and adequate fit can be achieved (Buchwald, 2019; Buchwald, 2020). This is done by allowing each ligand to have different, pathway-specific efficacies (not just a single receptor-specific efficacy) and then comparing the obtained fitted values for indication of bias. Efficacy values that are significantly different (ε _Pk ≠ ε _Pl ) can be considered as indication of biased agonism (Buchwald, 2019; Buchwald, 2020). If γ and ε values cannot be obtained in sufficiently well-defined manner, εγ products can be compared (with a designated reference compound).

For illustration, a set of data generated within the framework of SABRE for two divergent pathways (P₁, P₂) originating from the same receptor but with different amplifications (γ _P1, γ _P2) and assuming three compounds having different affinities (CpdTst1, 2, and 3 with log K _ds of −7.0, −6.5, and −8.0, respectively) is shown in Figure 4. CpdTst1 (blue) was assumed to be a balanced and full agonist for both pathways (ε _1,P1 = ε _1,P2 = 1.0), CpdTst3 (green) a weak balanced agonist (ε _3,P1 = ε _3,P2 = 0.25), and CpdTst2 (red) a biased agonist with higher efficacy for pathway 1 than for 2 (ε _2,P1 = 0.8, ε _2,P2 = 0.2). Corresponding data are shown as typical concentration-response curves (f _resp vs. log C; Figures 4B1, B2), response versus occupancy graphs (f _resp vs. f _occup; C1, C2), and classic bias plot used in such cases [f _resp1 vs. f _resp2 (Gregory et al., 2010; Kenakin and Christopoulos, 2013); D]. As the pathways have different amplifications (γ _P1 ≠ γ _P2; here, γ _P1 = 15 and γ _P2 = 5), response plots are curvilinear to different degrees even for balanced (non-biased) ligands (system bias) making it challenging to identify biased agonists using these plots (e.g., Figures 4C1 vs C2). Thus, it is difficult to notice that CpdTst2 is biased (red vs. blue and green) except in the bias plot (Figure 4D) directly depicting f _resp1 versus f _resp2. Systems that are strongly biased and have widely different amplifications resulting in highly curved plots can further mask this.

To illustrate the effect of right-shifted responses on this, responses for the same three hypothetical compounds (CpdTsts1–3) but with a second pathways that now has right- and not left-shifted response (γ _P2 < 1) are depicted in Figure 5. Accordingly, the curvature in the corresponding response 2 vs. occupancy plot (Figures 5C2 vs 4C2) is different, and because of the signal attenuation, partial agonists produce only weak responses here regardless of whether they are biased or not. With the values used here (γ _P1 = 15 and γ _P2 = 0.15), both compounds 2 and 3 produce only very weak responses in this second pathway; thus, not just biased (CpdTst2, red) but also weak balanced agonists produce no detectable responses (CpdTst3, ε ₃ = 0.25, green; Figure 5). Notably, even in the bias plot (Figure 5D) that typically allows the best discrimination, the weak balanced agonist CpdTsts3 groups more with the biased agonist (CpdTsts2) than the balanced full agonists (CpdTst1).

Such rapid deterioration of responses produced by partial agonists in pathways with right-shifted responses (i.e., with apparent signal attenuation/loss, γ < 1) as efficacy decreases is well-illustrated by the β-arrestin recruitment data at MOPr as assayed by Pedersen and co-workers (Pedersen et al., 2019), part of which was used earlier for DAMGO and morphine (Figure 3C). As for the β-arrestin pathway here γ << 1, weak agonists such as buprenorphine and oliceridine cause very little response (Figure 6). These data can still be fitted well with SABRE using single efficacies for both pathways (Supplementary Table S2); thus, without having to assume biased agonism as discussed before. Note that oliceridine (R-TRV130, included here together with its S isomer, S-TRV130) was developed as a MOPr biased agonist, and it was approved by the FDA for clinical use in 2020 (Olinvyk) as one of the first possible products showing the clinical promise in developing biased agonists (Wadman, 2017). However, it has been suggested that the improved safety profiles of such compounds are due not to the relative reduction in β-arrestin mediated signaling because of biased agonism, but to the low intrinsic efficacy in all signaling pathways (Gillis et al., 2020a; Gillis et al., 2020b; Gillis et al., 2020c). Fit with SABRE here (Figure 6; Supplementary Table S2) seems to suggest the same, i.e., that the weak β-arrestin recruitment is due to the combination of weak ligand agonism (low ε) and signal attenuation/right-shifted response in that pathway (γ < 1) without a need for biased agonism.

Such right-shifted responses mean that responses are still far from the maximum when occupancy is already approaching saturation (e.g., at [L] = 10×K _d where f _occup = 91%) and then catch up abruptly (Figure 1C, bottom). As mentioned, right-shifted responses are usually considered indications of an occupancy threshold issue where receptor concentrations are not negligible compared to ligand concentrations. However, for cases such as those discussed here involving responses generated by the same ligands at the same receptors with one response being left- and one right-shifted (e.g., Figures 3, 6), it is unlikely that the right-shift results from an occupancy threshold issue. Producing part of the effect at a different target that binds the ligand with lower affinity could cause a right-shift; however, this should result either in a noticeable two-step response, if the two affinities are significantly (>100-fold) different, or in a Hill slope that is less then unity (n < 1), if the affinities are closer to each other [see, e.g., Figure 3 in Buchwald (2017)]. This is unlikely here as a unity Hill slope (n = 1) fits all three data sets well for both DAMGO and morphine (Figure 3) (quality of fit is more difficult to confirm for morphine as its β-arrestin response has a limited range in all cases, <30%). Some pathway-specific threshold issue such as one along the β-arrestin pathway that does not affect the G protein activation pathway might be a possibility. For example, a need to reach some minimum threshold of activation, including a threshold of sufficiently prolonged activation, or a need for reenforced signals that have to come from multiple occupied receptors within the same system. Loss of weak overall signal or some other issue could also be involved.

Notably, response of the β-arrestin pathway for receptors other than MOPr does not appear to be right-shifted (κ _βArr < 1.0), at least, for the few cases where occupancy data were also assessed in the same work. For example, for epinephrin at β₂ adrenergic receptors: log K _d = −6.54, log EC_{50,Gprt, cAMP} = −9.01, and log EC_50,βArr2 = −7.26 corresponding to κ _βArr = 5.2 in (Rajagopal et al., 2011) and log K _d = −5.70, log EC_50,Gprt,Gs = −7.00, and log EC_50,βArr2 = −6.75 corresponding to κ _βArr = 11.2 in (Onaran et al., 2017). For angiotensin II at the angiotensin II type 1 receptor (AT1R): log K _d = −7.90, log EC_50,Gprt,IP1 = −8.84, and log EC_50,βArr2 = −7.90 corresponding to κ _βArr = 1.0 in (Rajagopal et al., 2011) and log K _d = −7.61, log EC_50,Gprt,Gq = −8.62, and log EC_50,βArr2 = −8.43 corresponding to κ _βArr = 6.6 in (Wingler et al., 2020). Note that several of these left-shifted β-arrestin responses [e.g., (Rajagopal et al., 2011; Wingler et al., 2020)] were obtained with the same assay as the one used in the MOPr studies shown in Figure 3 (PathHunter) making it unlikely that the right-shift at MOPr is due to an assay-related issue even if fusing the assay-specific fragments to different receptors might have different effects on the readouts. One difference for MOPr is that contrary to most GPCRs, where bound ligands and especially agonist ligands are deeply buried within the receptor (Buchwald, 2019), the ligands within the binding pocket of MOPr are more exposed to the extracellular surface (Manglik et al., 2012; Zhuang et al., 2022)—see Figures 7 versus Figure 8. This is a likely reason why even potent opioids are rapidly dissociating from their receptor with half-lives of only minutes (Manglik et al., 2012), e.g., 0.5 and 0.7 min for morphine and DAMGO (Pedersen et al., 2019), the compounds discussed here, compared to, for example, tiotropium, which has a dissociation half-life (t _1/2 = ln2/k _off = ln2×t _res) of >30 h at the muscarinic M3 receptor (Tummino and Copeland, 2008; Guo et al., 2014). The resulting short life of the agonist bound MOPr complex could be a possible reason why the concentration-response of arrestin recruitment lags behind the occupancy (is right-shifted) for this pathway.

As their name implies, a main function of these proteins is to terminate (“arrest”) signaling through GPCRs (Wess et al., 2023). For the two known β-arrestins (β-arrestin-1 and 2, βarr1 and βarr2—also known as arrestin-2 and -3, respectively), activation involves two steps: phosphorylation of the activated receptor by specialized GRKs followed by binding of the arrestin(s) to the active phosphorylated receptor to interfere with receptor/G protein coupling (Wess et al., 2023). It is conceivable that an “arresting” response is only triggered when a minimum threshold of activation or sufficiently prolonged activation is achieved, and this might require higher agonists concentrations than just binding (occupancy), especially if activated receptors are short-lived. Along similar lines, it is also possible that termination signaling is not triggered unless multiple receptors are activated within the same system, so that low occupancy does not cause response in this pathway. However, a putative mechanism for this is unclear and such activation threshold assumptions are likely to cause responses that are more abrupt than typical hyperbolic ones, which does not seem to be the case here (Figure 3). For example, if two independent receptors within the same system need to be occupied to trigger the response, then f _resp ∝ (f _occup) ^ν with ν = 2, and this produces a response corresponding to γ ≈ 0.4 that is just a bit more abrupt than hyperbolic (Supplementary Figure S1); however, larger νs needed for larger shifts result in gradually more abrupt responses.