Evaluation of the profiles of CB₁ cannabinoid receptor signalling bias using joint kinetic modelling

Abstract

Background and Purpose

Biased agonism describes the ability of ligands to differentially regulate multiple signalling pathways when coupled to a single receptor. Signalling is affected by rapid agonist‐induced receptor internalisation. Hence, the conventional use of equilibrium models may not be optimal, because (i) receptor numbers vary with time and, in addition, (ii) some pathways may show non‐monotonic profiles over time.

Experimental Approach

Data were available from internalisation, cAMP inhibition and phosphorylation of ERK (pERK) of the cannabinoid‐1 (CB₁) receptor using a concentration series of six CB₁ ligands (CP55,940, WIN55,212‐2, anandamide, 2‐arachidonylglycerol, Δ⁹‐tetrahydrocannabinol and BAY59,3074). The joint kinetic model of CB₁ signalling was developed to simultaneously describe the time‐dependent activities in three signalling pathways. Based on the insights from the kinetic model, fingerprint profiles of CB₁ ligand bias were constructed and visualised.

Key Results

A joint kinetic model was able to capture the signalling profiles across all pathways for the CB₁ receptor simultaneously for a system that was not at equilibrium. WIN55,212‐2 had a similar pattern as 2‐arachidonylglycerol (reference). The other agonists displayed bias towards internalisation compared to cAMP inhibition. However, only Δ⁹‐tetrahydrocannabinol and BAY59,3074 demonstrated bias in the pERK–cAMP pathway comparison. Furthermore, all the agonists exhibited little preference between internalisation and pERK.

Conclusion and Implications

This is the first joint kinetic assessment of biased agonism at a GPCR (e.g. CB₁ receptor) under non‐equilibrium conditions. Kinetic modelling is a natural method to handle time‐varying data when traditional equilibria are not present and enables quantification of ligand bias.

Abbreviations

2‐AG

2‐arachidonoylglycerol

AEA

anandamide/N‐arachidonoylethanolamine

BAY

BAY59,3074

CAMYEL

cAMP sensor using YFP‐Epac‐RLuc

CB₁

cannabinoid‐1 receptor

CP

CP55,940

FOCE+I

first‐order conditional estimation with interaction method

FSK

forskolin

HA

haemagglutinin

HEK293, HEK

HEK cell

pERK

phosphorylation of ERK

PPP&D

population pharmacokinetic parameters and data

RSE

relative standard error

SR

SR141716A

THC

Δ⁹‐tetrahydrocannabinol

WIN

WIN55,212‐2

1.

What is already known

• Equilibrium models are often used to quantify ligand bias using non‐equilibrium bioassay data.

• The results produced by equilibrium methods when used to analyse non‐equilibrium data can be difficult to interpret.

What this study adds

• A kinetic analysis is applied for quantifying ligand bias at CB₁ receptor under non‐equilibrium conditions.

What is the clinical significance

• Under non‐equilibrium conditions, kinetic analysis can yield more robust conclusions about ligand bias compared to equilibrium analysis.

• Kinetic analysis may improve the success rate of translating lead compounds into innovative clinical therapies.

1. INTRODUCTION

The type one cannabinoid receptor (CB₁) is a GPCR. It is present at high levels throughout the CNS (Glass, Dragunow, & Faull, 1997). CB₁ is an attractive therapeutic target for numerous CNS diseases, including neurodegenerative disease (Scotter, Goodfellow, Graham, Dragunow, & Glass, 2010), multiple sclerosis (Correa et al., 2007) and pain (Sagar et al., 2009). However, despite many possible therapeutic targets, the clinical application of CB₁ ligands has been hampered due to their adverse on‐target effects (Volkow et al., 2016). Recently, interest has grown in the ability of ligands to differentially regulate multiple signalling pathways when coupled to a single receptor, termed biased agonism (Kenakin, 2007). Hence, a ligand will have a signature profile of signalling across the range of pathways of interest. It should be noted, however, that signalling profiles may not be quantitatively identical across different cells (including between heterologous cell lines and native expressers in vivo). One emerging therapeutic strategy is to improve the selectivity and safety of drugs through the development of ligands that are biased towards desired pathways (Kenakin & Christopoulos, 2013). Therefore, a comprehensive understanding of biased agonism at CB₁ signalling is crucial to lead compound optimisation.

However, as both ligand binding and signal transduction are dynamic processes, it is possible that stationary equilibria may not always be achieved at the time point of the assay (Lane, May, Parton, Sexton, & Christopoulos, 2017). An example of this includes a recent study of the dopamine D₂ receptor, in which non‐equilibrium conditions resulted in a reversal of bias quantification results and subsequent interpretation (Klein Herenbrink et al., 2016). Although signalling events can be measured in a way that does not account for internalisation of the receptor across the time course of the assay, sometimes the impact of internalisation on signalling cannot be ignored. For example, agonist‐induced internalisation of the CB₁ receptor leads to a rapid and remarkable decrease of the surface receptor following activation by some CB₁ agonists (Zhu, Finlay, Glass, & Duffull, 2019a). Similarly, a transient activation of phosphorylation of ERK (pERK) is observed in 3HA‐hCB₁ HEK cells leading to non‐monotonic signalling over time (Finlay et al., 2017). Stationary equilibria for these pathways are therefore not reached. Hence, non‐equilibrium analysis methods are warranted to quantify ligand bias under these conditions.

Although more and more investigators are using time‐resolved assays on signalling events, the quantification of biased agonism often utilises a selected single time point, either when the investigator believes a stationary equilibrium has occurred or at the time point with optimal signal to noise ratio (i.e. peak, in the case of transient responses). Then the operational model of agonism is applied separately to each pathway to quantify ligand bias based on the information collected at a timed snapshot of the whole process. The timed snapshot may either be at mixed times (i.e. a mixed time point ‘snapshot’ analysis) or the same time (e.g. a suitably long time into the experiment) (Kenakin, Watson, Muniz‐Medina, Christopoulos, & Novick, 2012; van der Westhuizen, Breton, Christopoulos, & Bouvier, 2014). An alternative approach is to examine signalling assays at multiple matched time points and then separately perform a series of snapshot biased agonism analyses to the data from each time point (a moving ‘cross‐sectional analysis’ [aka ‘movie’]) (Klein Herenbrink et al., 2016). Regardless of the technical difficulty in matching time points from different functional assays, repeated cross‐sectional analyses can illustrate the empirical relationship between time and apparent ligand bias, making it a good diagnostic tool to examine the robustness of an equilibrium assumption. For instance, when a moving cross‐sectional analysis was applied to the action of agonists at the dopamine D₂ receptor, it was demonstrated that the apparent ligand bias varied over time (in some cases, it even reversed) (Klein Herenbrink et al., 2016). However, even a moving cross‐sectional analysis lacks explanatory power and does not help understand the consequences of the various signalling pathways. In this sense, while it can describe the phenomenon of biased agonism, it may not be able to explain biased agonism. Hence, there is a need to view agonists' effects over the whole‐time course of their activities which can be achieved using a kinetic approach. Kinetic modelling is a promising technique to quantitatively assess the time‐dependent modulation of the receptor by ligands and provide novel mechanistic insights into the complex interplay among ligands, receptors and pathways (Bridge, Mead, Frattini, Winfield, & Ladds, 2018). In essence, this work intends to transform the standard mixed (and fixed) time snapshot analysis and the moving cross‐sectional analysis to a full longitudinal technique.

Finally, the assumption of signalling pathway independency may preclude insights into biased agonism from observed phenomena, such as the natural correlation of a ligand's EC₅₀ values among different pathways (i.e. a drug's potency differences between pathways are constrained by the limits of occupancy) (Zhu, Finlay, Glass, & Duffull, 2019b). To reflect these biological linkages, we propose a joint kinetic model framework to share the kinetics of ligand binding across different pathways and to facilitate joint modelling of all the pathways. This approach is not only in line with the underlying theory of biased agonism but also helps improve the precision of parameter estimation by permitting the observations from each pathway to inform the common part of the model.

The overarching aim of this study is to evaluate biased agonism within the kinetic framework and the CB₁ cannabinoid receptor will be used as a model receptor. This encompasses two specific objectives: (i) to develop a joint kinetic model that quantitatively describes the time course data from internalisation, pERK and cAMP pathways coupled to CB₁ and (ii) to visualise the bias fingerprint profiles of CB₁ ligands.

2. METHODS

The methods are described in three parts. The first part provides the technical details of the experimental assays. The second part describes the kinetic modelling of joint CB₁ signalling. The third part describes the ligand bias analysis within a kinetic framework.

2.1. Part I

2.1.1. Drugs

Forskolin (FSK), CP55,940 (CP), WIN55,212‐2 (WIN) and BAY59‐3074 (BAY) were purchased from Tocris Bioscience (Bristol, UK); anandamide (AEA) and 2‐arachidonoylglycerol (2‐AG) were purchased from Cayman Chemical Company (Ann Arbor, MI); and Δ⁹‐tetrahydrocannabinol (THC) was purchased from THC Pharm (Frankfurt, Germany). SR141716A (SR, rimonabant) was a gift from Roche (Basel, Switzerland). All drugs were prepared as stocks in absolute ethanol (CP, WIN, AEA, 2‐AG, THC and BAY) or DMSO (FSK and SR). Drugs were stored as single‐use aliquots at −80°C prior to use. In order to assuage concerns about potential confounding of data due to the presence of endocannabinoids in serum, all drug stimulations were performed in the absence of FBS and following a short period of serum starvation (as described below). Drug vehicles (ethanol and DMSO) were controlled within each assay type, with a total load of 0.1% for each cannabinoid drug (ethanol) and 0.025% for 2.5‐μM FSK (DMSO). Note that the total ethanol load in the cAMP CAMYEL assay was somewhat higher than this, at 0.31%, as the BRET substrate (coelenterazine H) was also constituted in ethanol.

2.1.2. Cell culture

The cell culture medium was purchased from Hyclone Laboratories (GE Healthcare Life Sciences), all other culture reagents were purchased from Thermo Fisher Scientific and culture plasticware was purchased from Corning (Corning, NY). The 3HA‐hCB₁ HEK 293 (RRID:CVCL_0045) cell line (Cawston et al., 2013) was cultured in DMEM supplemented with 10% FBS and 250 μg·ml⁻¹ Zeocin, in a 5% CO₂, 37°C humidified incubator.

2.1.3. Receptor internalisation assays

Internalisation data analysed in the current study were first reported in Zhu et al. (2019a).

2.1.4. cAMP assays

All the cAMP assays were performed with multiple concentration levels (serial dilutions) of seven CB₁ ligands: CP, WIN, AEA, 2‐AG, THC, BAY and SR. The real‐time monitoring of cAMP was performed every 0.4–0.5 min for approximately 20 min. The measurement of cAMP was conducted using a commercially available kinetic BRET assay (CAMYEL), as previously described (Cawston et al., 2013; Finlay et al., 2017; Jiang et al., 2007). In brief, HEK 3HA‐hCB₁ cells were seeded in 10‐cm dishes. After 24 h of culture, 5 μg of pcDNA3L‐His‐CAMYEL was transfected into cells utilising linear polyethyleneimine (MW 25 kDa; Polysciences, Warrington, PA, USA). The day after transfection, cells were moved to poly‐d‐lysine (Sigma Aldrich, St. Louis, MO, USA) coated white CulturPlate™‐96 plates (PerkinElmer, Waltham, MA, USA) at a density of 60,000 cells per well. The cells were then cultured overnight prior to drug stimulation. Cells were washed with HBSS and HBSS (pH 7.4) containing 1 mg·ml⁻¹ BSA (ICPBio, Auckland, New Zealand) was dispensed. Cells were then returned to incubator for 30 min of serum starve. Coelenterazine H (5 μM, NanoLight Technologies, Pinetop, AZ, USA) was dispensed 5 min prior to addition of drugs. Immediately after addition of drugs, emission signals were detected simultaneously at 460 nm (RLuc) and 535 nm (YFP) with a LUMIstar® Omega luminometer (BMG Labtech, Ortenberg, Germany). Raw data were presented as an inverse BRET ratio of emission at 460/535 such that an increase in ratio correlated with an increase in cAMP production. All incubations and stimulations in cAMP assays were performed at 37°C.

2.1.5. Phosphorylated ERK assays

All assays for pERK were performed with multiple concentration levels (serial dilutions) of six CB₁ ligands: CP, WIN, AEA, 2‐AG, THC and BAY. Terminal sampling of the pERK was taken at 1, 3, 4, 5, 6, 8, 12, 20, 40 and 60 min. Quantification of pERK was performed using a commercially available AlphaLisa SureFire kit (Perkin Elmer), largely as previously described (Finlay et al., 2017). In brief, cells were plated in 96‐well culture plates. After approximately 24 h of culture, cells were serum starved for >16 h. A serial dilution of agonist was applied to cells and pERK time courses were performed with plates resting on a barely submerged stage in a 37°C water bath. After drug stimulation, plates were placed on ice, well contents were removed by aspiration and cells were immediately lysed in 50 μl of AlphaLisa lysis buffer. AlphaLisa detection was performed according to the manufacturer's instructions and plates were read in a CLARIOstar® plate reader (BMG Labtech, Ortenberg, Germany).

2.2. Part II

2.2.1. Data and analysis

The data and statistical analysis comply with the recommendations of the British Journal of Pharmacology on experimental design and analysis in pharmacology (Curtis et al., 2018). This study was designed to generate groups of equal size (n = 3). However, the analysts need to be unblinded in order to apply the models and analysis techniques that are used here. This study relies heavily on mechanistic receptor theory which could not otherwise be applied. The time course data of receptor internalisation and cAMP inhibition were replicated in three independent experiments. The time course data of ERK phosphorylation were obtained from a representative experiment, with a qualitatively typical profile and time course (detailed comparisons among replicates in Supporting Information; replicates could not be combined due to different nominal signal magnitudes in AlphaLisa units). This may appear to conflict with the BJP minimum experimental replication requirement (n = 5). However, as the data are analysed by simultaneous global fitting of responses from three signalling pathways at multiple time points and multiple concentration levels, the effective sample size is much larger than five. We add, however, that no statistical comparison was performed on these data; rather, they were used for the development of a kinetic model of CB₁ signalling. Plots show representative data (mean ± SEM) of technical replicates, unless otherwise specified.

2.2.2. Overview of joint CB₁ signalling model

The joint CB₁ signalling model consists of three parts (Figure 1): (i) internalisation, (ii) cAMP signalling and (iii) pERK signalling. The internalisation model describes the interplay between ligand and receptor and the dynamic change of receptor occupancy (Zhu et al., 2019a). The stimulus response model converts stimulus (S, resulting from receptor occupancy) into a final response via a transducer function (f), which allows the response to represent a potential cascade of cellular signalling (Visser et al., 2002). The cAMP model describes how the capacity‐limited stimulatory effect of FSK and the inhibitory effect of CB₁ ligands act on the natural turnover of cAMP. The pERK model describes the natural turnover of pERK and the time‐dependent stimulatory effect of CB₁ ligands on the synthesis of pERK. The detailed description of the joint CB₁ signalling model is provided in Appendix A.

FIGURE 1.

Schematic overview of the structure of the joint CB₁ signalling model. Here, A denotes the agonist, R denotes the free CB₁ receptor and AR denotes the agonist–receptor complex. The joint CB₁ signalling model contains three main components: (i) receptor internalisation, (ii) cAMP signalling and (iii) pERK signalling. The receptor internalisation model combines the ligand binding kinetics, receptor turnover and agonist‐induced receptor internalisation. The cAMP signalling model describes the natural turnover of cAMP, the capacity limited stimulation of signal amplifier forskolin (FSK) and the inhibitory effect of CB₁ ligands on the synthesis. The pERK signalling model describes the natural turnover of pERK and the time‐dependent stimulatory effect of CB₁ ligands on the synthesis of pERK. The details of the model are given in Appendix A

2.2.3. Model development software and criteria

The mechanism‐based stimulus response model for CB₁ signalling in multiple pathways was developed sequentially following a PPP&D modelling approach (the internalisation model was fixed before developing the models for the other pathways) (Zhang, Beal, & Sheiner, 2003) in the non‐linear mixed effect modelling software NONMEM (version 7.3, RRID:SCR_016986) with subroutine ADVAN13 (ICON Development Solutions, Ellicott City, Maryland). The first‐order conditional estimation with interaction method (FOCE+I) was used for model estimation (Beal, Sheiner, Boeckmann, & Bauer, 1992). The inter‐individual variability was described by an exponential distribution (Equations A3 and A13) and the residual error was described by combined error model. Due to the intensive sampling in cAMP measurement (every 0.4–0.5 min for approximately 20 min), an AR(1) model (detailed in Supporting Information) was used to account for correlation in the residual errors (Karlsson, Beal, & Sheiner, 1995). The code of the final model is provided in Supporting Information. The model was evaluated and determined based on successful convergence, objective function value, parameter precision and the visual inspection of the goodness‐of‐fit plots. The R package Xpose 4.5.3 was used for creating goodness‐of‐fit plots (Keizer, Karlsson, & Hooker, 2013).

2.3. Part III

2.3.1. Visualisation of bias fingerprint

Agonist‐induced internalisation occurs during the time course of the kinetic processes of other signalling pathways. Consequently, this rapid change of surface receptor leads to a time‐varying transducer ratio τ ( $\epsilon ·\frac{R_T}{K_E}$; ε: ligand intrinsic efficacy, R_T: total receptor number, K_E: coupling efficiency of signal transduction), which affects the suitability of the current ligand bias metric $\left(\mathrm{\Delta \Delta }\mathit{log}\left(\frac{\tau }{K_A}\right)\right)$ for quantifying functional selectivity within a kinetic framework. Because of this, it is necessary to go back to the fundamental parameters of the ligand which do not vary over time (e.g. ε). Hence, we used the time‐invariant intrinsic efficacy (ε) derived from kinetic modelling of CB₁ signalling to quantify ligand bias (Equations A2, A9 and A15). In our study, 2‐AG was selected as the reference ligand because (i) it is an endogenous agonist of CB₁, (ii) it is a full agonist in all the pathways in this study and (iii) it enabled us to compare our results with a related previous study (Khajehali et al., 2015). To determine the relative effectiveness of the ligands to activate the signalling pathway, the logε value of each test ligand was normalised to that of reference ligand 2‐AG at the same pathway (Δlogε, Equation 1). This produces a bias metric that has the same conceptual use as the $\mathit{log}\left(\frac{\tau }{K_A}\right)$ but without issues of non‐stationarity.

$$\Delta \mathit{\text{logε}}=\mathit{\text{logε}}-{\mathit{\text{logε}}}^{2\mathit{AG}}.$$ (1)

Then the relative effectiveness of the ligand was compared across different signalling pathways to give a ΔΔlogε value (Equation 2). Here, we referred to this metric as ‘ligand bias’ within a kinetic framework because ΔΔlogε has a similar fundamental interpretation as the standard ligand bias metric ΔΔlogR under equilibrium condition. In the following, the ligand bias, relative to 2‐AG, was calculated as the inverse logarithm of the ΔΔlogε using Equation 3.

$$\mathrm{\Delta \Delta }\mathit{log}{\epsilon }_{1-2}=\Delta \mathit{log}{\epsilon }_1-\Delta \mathit{log}{\epsilon }_2,$$ (2)

$$\mathit{\text{Bias}}={10}^{\mathrm{\Delta \Delta }\mathit{log}{\epsilon }_{1-2}}.$$ (3)

Here, a subscript number ‘1’ indicates pathway 1 and ‘2’ indicates pathway 2.

Finally, the ligand bias values were graphically presented in a web plot using Microsoft Excel 2016 (RRID:SCR_016137), to illustrate a specific bias ‘fingerprint’ for each ligand.

The estimated standard error for test ligand was calculated using Equation 4 (assuming independence—which for the standard post hoc analysis is implicit).

$${\mathit{SE}}_{\left(\mathrm{\Delta \Delta }\mathit{log}{\epsilon }_{1-2}\right)}=\sqrt{{\mathit{SE}}_{\left(\mathit{log}{\epsilon }_1\right)}^2+{\mathit{SE}}_{\left(\mathit{log}{\epsilon }_1^{2\mathit{AG}}\right)}^2+{\mathit{SE}}_{\left(\mathit{log}{\epsilon }_2\right)}^2+{\mathit{SE}}_{\left(\mathit{log}{\epsilon }_2^{2\mathit{AG}}\right)}^2}.$$ (4)

Note, other ways to describe pathway preference can be considered but are outside the scope of the work presented here.

2.4. Nomenclature of targets and ligands

Key protein targets and ligands in this article are hyperlinked to corresponding entries in http://www.guidetopharmacology.org, the common portal for data from the IUPHAR/BPS Guide to PHARMACOLOGY (Harding et al., 2018) and are permanently archived in the Concise Guide to PHARMACOLOGY 2019/20 (Alexander et al., 2019).

3. RESULTS

All pathways were modelled simultaneously in order to ensure system information and correlations are preserved appropriately across pathways. The results are, however, presented for each pathway separately in order for the influence of the ligand effect on each pathway to be explored.

3.1. Agonist‐induced internalisation of CB₁ receptor

As shown in Figures 2 and S1, the quasi‐steady state internalisation model provided an acceptable description of the kinetic profile of agonist‐induced internalisation of CB₁ receptor over a large concentration range of six investigated CB₁ agonists. The parameter estimates were precisely estimated (Table 1). Large variability in the potency parameter (logK_SS) was observed among six investigated CB₁ agonists, with the estimated value ranging from −7.88 to −4.34. Among them, CP was the most potent and 2‐AG the least. All investigated CB₁ agonists exhibited substantial effects on internalisation, with the estimated logarithmic intrinsic efficacy (logε_int) ranging from 1.23 to 2.83 (Table 1). The order in efficacies proceeded 2‐AG (most efficacious) > WIN > AEA ≈ CP > THC ≈ BAY (Table 1). As agonist‐induced internalisation was much faster than constitutive internalisation of the CB₁ receptor, it was essential to consider this rapid agonist‐induced receptor internalisation when modelling other signalling pathways coupled to CB₁ receptor. Thus we have assumed that receptor signalling ceases on internalisation.

FIGURE 2.

Individual fitting results of the agonist‐induced internalisation in 3HA‐hCB₁ HEK cells on stimulation with six CB₁ agonists: CP55,940 (a), WIN55,212‐2 (b), AEA (c), 2‐AG (d), THC (e) or BAY59,3074 (f). The line here represents the prediction from quasi‐steady state model of agonist‐induced internalisation for ‘live‐at‐start’ assay and the symbols (mean ± SEM) here are the representative data from the ‘live‐at‐start’ internalisation assay

TABLE 1.

Parameter estimation results for the agonist‐induced internalisation of CB₁ receptor from joint CB₁ signalling model

Parameter	Definition	Parameter estimate [RSE%]
kcon (min−1)	Constitutive internalisation rate constant	0.0016 [16%]
α (105 AU)	Coefficient to link the unit of receptor density and fluorescence intensity	2.6 [2%]
TVR0	Typical value of initial receptor density	1 FIX
logεint	Intrinsic efficacy of ligand on internalisation in logarithm	CP	1.9 [4%]
		WIN	2.27 [3%]
		AEA	1.92 [8%]
		2‐AG	2.83 [4%]
		THC	1.36 [5%]
		BAY	1.23 [6%]
logKSS	Quasi‐steady state equilibrium constant in logarithm that represents the dissociation constant for the ligand–receptor complex	CP	−7.88 [1%]
		WIN	−6.17 [1%]
		AEA	−5.81 [4%]
		2‐AG	−4.34 [2%]
		THC	−7.38 [2%]
		BAY	−6.06 [1%]
IIVR0	Inter‐individual variability on R0	12.7% [12%]
${{\sigma }_{\mathit{add}}^2}_{\mathit{int}}$	Additive error for internalisation	0.0167 [34%]
${{\sigma }_{\mathit{\text{prop}}}^2}_{\mathit{int}}$	Proportional error for internalisation	0.00607 [30%]

Note: ‘1 FIX’ means that TVR₀ was fixed to 1 (arbitrary units) and not estimated.

3.2. Kinetic modelling of cAMP signalling

The joint CB₁ signalling model adequately described the data from the CB₁ agonists‐mediated inhibition offorskolin‐induced cAMP production (Figures 3 and S2–S6). All model parameters were estimated precisely (Table 2). The estimated maximal stimulatory effect of FSK demonstrated a 10.8‐fold (18% relative standard error [RSE]) increase on the synthesis rate of cAMP. The estimated system maximal inhibitory effect of CB₁ agonists caused a 76% (5% RSE) reduction on the synthesis rate of cAMP. Our modelling results revealed large variability (equating to approximately 400‐fold in non‐log units) in the logarithmic intrinsic efficacies of the tested ligands on cAMP signalling (logε_c). Among the six agonists tested, 2‐AG was the most efficacious. Meanwhile, the reported partial agonists (De Vry et al., 2004; Pertwee, 2008), THC and BAY, demonstrated lower efficacies on the inhibition of cAMP production than the other agonists.

FIGURE 3.

Individual fitting results of the temporal cAMP BRET signalling in 3HA‐hCB₁ HEK cells on stimulation with 7.9‐μM forskolin (‘F’; FSK) and CP55,940 (a), WIN55,212‐2 (b), AEA (c), 2‐AG (d), THC (e) or BAY59,3074 (f). The line here represents the prediction from joint CB₁ signalling model for cAMP signalling and the symbols (mean ± SEM) here are the representative data from the kinetic CAMYEL biosensor data (The fitting results for the stimulation with 2.5‐ or 25‐μM forskolin are presented in Figure S5 and S6. The individual kinetic curve for each concentration level was presented separately in Figure S11–S28.)

TABLE 2.

Parameter estimation results for cAMP signalling from joint CB₁ signalling model

Parameter	Definition	Parameter estimate [RSE%]
Csyn	Synthesis rate of cAMP	0.043 [22%]
kc (min−1)	cAMP elimination rate constant	0.49 [3%]
EmaxFSK	Maximum response of FSK stimulatory effect on cAMP synthesis	10.8 [18%]
logC50FSK	The log concentration of FSK that leads to half of maximum response	−5.23 [1%]
Im	System maximum response of inhibitory effect on cAMP synthesis	0.76 [5%]
KI	Coupling efficiency in cAMP	0.44 [27%]
logεc	Intrinsic efficacy on cAMP in logarithm	CP	1.66 [6%]
		WIN	2.63 [8%]
		AEA	1.49 [7%]
		2‐AG	3.42 [5%]
		THC	0.85 [12%]
		BAY	0.87 [11%]
TVsignal0	Typical value of signal at time 0	0.80 [0.5%]
slope	The change rate of background signal	0.0026 [6%]
IIV − R0	Inter‐individual variability on R0	12.7% FIX
IIV − signal0	Inter‐individual variability on signal0	5.8% [7%]
${{\sigma }_{\mathit{\text{prop}}}^2}_c$	Proportional error for cAMP	0.00067 [7%]

3.3. Kinetic modelling of pERK signalling

The joint CB₁ signalling model well described the time course of pERK activation (Figures 4 and S7). All model parameters were precisely estimated (Table 3). The estimated system maximal stimulation was 56‐fold over baseline (24% RSE). The estimated duration of stimulation was 3.76 min (3% RSE), which was consistent with observed peak time (from 3 to 5 min). This time‐dependent stimulation might be explained by the reported negative feedback in pERK signalling network (Cirit, Wang, & Haugh, 2010). Compared to the other two signalling pathways, the CB₁ agonists tested demonstrated the smallest variation on the logarithmic intrinsic efficacy in pERK signalling (logε_p) (Table 3), with around 20‐fold difference in the intrinsic efficacies. The order of efficacy from most to least efficacious was 2‐AG > WIN > CP ≈ AEA > BAY > THC.

FIGURE 4.

Individual fitting results of the time course of pERK activation in 3HA‐hCB₁ HEK cells on stimulation with CP55,940 (a), WIN55,212‐2 (b), AEA (c), 2‐AG (d), THC (e) or BAY59,3074 (f). The line here represents the prediction from joint CB₁ signalling model for pERK activation and the symbols here are the representative data from the pERK assay

TABLE 3.

Parameter estimation results for pERK signalling from joint CB₁ signalling model

Parameter	Definition	Parameter estimate [RSE%]
Psyn (105AU)	Synthesis rate of pERK	0.16 [8%]
kp (min−1)	pERK elimination rate constant	0.58 [5%]
ktr (min−1)	Stimulus transition rate constant	0.1 FIX
Em	System maximum response of stimulatory effect on pERK synthesis	56 [24%]
KE	Coupling efficiency in pERK	11 [40%]
sTime (min)	After this time of stimulation, the system will be tolerant to the given stimulus	3.76 [3%]
logεp	Intrinsic efficacy on pERK in logarithm	CP	1.77 [12%]
		WIN	2.16 [11%]
		AEA	1.8 [12%]
		2‐AG	2.78 [9%]
		THC	1.46 [15%]
		BAY	1.66 [23%]
IIV − R0	Inter‐individual variability on R0	12.7% FIX
tcorr (min)	Constant determining how fast the correlation decreases with time	0.70 [11%]
${{\sigma }_{\mathit{add}}^2}_p$	Additive error for pERK	0.052 [48%]
${{\sigma }_{\mathit{\text{prop}}}^2}_p$	Proportional error for pERK	0.026 [93%]

3.4. Bias fingerprint of CB₁ agonists

Figure 5 illustrates the bias fingerprint of CB₁ agonists relative to the reference ligand 2‐AG. Each vertex represents the relative ligand activity relative to the reference compound and compared to the specified pathway. The reference ligand (2‐AG) is shown as the purple line and its bias values are normalised to unity. ‘pERK–cAMP’ indicates the bias value between pERK and cAMP pathways. A value greater than 1 means that relative to 2‐AG, the test ligand is biased towards pERK over cAMP and vice versa. This applies similarly to ‘pERK–internalisation’ and ‘internalisation–cAMP’ as well.

FIGURE 5.

Visualisation of bias fingerprint relative to 2‐AG for CB₁ agonists. Bias values ( ${10}^{\mathrm{\Delta \Delta }\mathit{log}{\epsilon }_{1-2}}$) obtained for the agonists between the agonist‐induced internalisation, inhibition of forskolin‐induced cAMP production and ERK1/2 phosphorylation presented in Table 4 are shown in a web of bias

From the visualisation of the bias fingerprint profiles, the panel of six CB₁ agonists could be classified into two groups. The agonist WIN had a similar pattern of bias to that of 2‐AG. The remaining four agonists (i.e., CP, AEA, THC and BAY) had a bias fingerprint distinct from that of 2‐AG and WIN predominantly due to lower relative effectiveness in the inhibition of cAMP production compared to other signalling assays. All of these four agonists displayed bias towards agonist‐induced internalisation when compared to the inhibition of cAMP production (Table 4). However, only THC and BAY demonstrated bias in the pERK–cAMP pathway comparison (Table 4). CP and AEA displayed moderate bias towards the ERK1/2 phosphorylation over inhibition of cAMP production. Furthermore, it was also noted that the selectivity of investigated agonists towards the agonist‐induced internalisation and ERK1/2 phosphorylation were similar and none of them exhibited bias between these two signalling pathways (Table 4).

TABLE 4.

Ligand bias of CB₁ receptor agonists on internalisation, cAMP and pERK signalling in 3HA‐hCB₁ HEK cells with 2‐AG as the reference (bias is a unitless ratio and is given as 10^ΔΔlogε)

Ligand	pERK–internalisation		Internalisation–cAMP		pERK–cAMP
	ΔΔlogε	Bias	ΔΔlogε	Bias	ΔΔlogε	Bias
2‐AG	−0.00 ± 0.37	1.00	0.00 ± 0.29	1.00	0.00 ± 0.41	1.00
CP	−0.08 ± 0.35	0.83	0.83 ± 0.24	6.76	0.75 ± 0.37	5.62
WIN	−0.06 ± 0.36	0.87	0.23 ± 0.30	1.70	0.17 ± 0.43	1.48
AEA	−0.07 ± 0.37	0.85	1.02 ± 0.28	10.5	0.95 ± 0.38	8.91
THC	0.15 ± 0.35	1.41	1.10 ± 0.24	12.6	1.25 ± 0.38	17.8
BAY	0.48 ± 0.46	3.02	0.95 ± 0.24	8.91	1.43 ± 0.48	26.9

4. DISCUSSION

Equilibrium pharmacological models including the E_max model and operational model are widely used to describe ligand–receptor interactions, whether at the level of binding or a functional assay of a response variable. However, these models are so ingrained in analytical pharmacology that the equilibrium assumptions attached to them are often overlooked. Recent studies highlight that equilibrium methods can be problematic for time‐varying systems (Bdioui et al., 2018; Klein Herenbrink et al., 2016) and a violation of the equilibrium assumption would lead to misinterpretation of experimental data and erroneous descriptions of ligand effects.

This work provides the first report for the evaluation of biased agonism at a GPCR (e.g. CB₁ receptor) using a kinetic framework. To do this, we first performed time course characterisation of a panel of representative CB₁ ligands in three signalling pathways (internalisation, cAMP and pERK pathways). Second, based on receptor theory and experimental data, a joint kinetic model of biased agonism at CB₁ receptor was developed to delineate the time‐dependent complex interplays among ligands, receptor and pathways. Finally, based on the insights from the kinetic model, fingerprint profiles of the CB₁ ligand bias were constructed and visualised. As demonstrated in the current study, non‐equilibrium bioassays can be used to assess the ligand characteristics (e.g. signalling bias profiles) when a full kinetic model is used to analyse data.

Ideally, experimentation would be able to extract the individual components and allow these to be investigated in isolation. Unfortunately, this cannot be easily done with the biased agonism system due to the concurrence of pathways (e.g. internalisation and other signalling pathways). Hence, one possible way to gain mechanistic insight is to simultaneously assess all signalling pathways via a joint modelling approach. Furthermore, this approach helps improve the precision of parameter estimation by permitting the observations from each pathway to inform the common part of the model. In the current study, the data from the internalisation pathway provide solid information of ligand affinity, which help to disentangle the effects of affinity and efficacy. In another example, application of joint modelling with the intact operational model to an equilibrium experiment (i.e. a model that shared functional affinity across all the pathways) rendered precise estimation of functional affinity even when the ligand displayed full agonism in one pathway (Zhu et al., 2019b).

A previous snapshot bias analysis has investigated biased agonism between cAMP inhibition and pERK activation at the CB₁ receptor (Khajehali et al., 2015). Relative to the reference ligand 2‐AG, WIN, CP, AEA and THC were 1.7, 3.9, 6.8 and 5.6 times more biased towards the inhibition of cAMP inhibition over pERK activation, although none of them reached statistical significance. However, the current kinetic analysis produced the opposite result. Relative to 2‐AG, the other agonists WIN, CP, AEA and in particular THC were 1.5, 5.6, 8.9 and 17.8 times more biased (respectively) towards pERK activation over cAMP inhibition (Table 4). This discrepancy in the result of bias analysis may have arisen from choosing a snapshot in a non‐stationary system: accumulation for the inhibition of cAMP production at 30 min and transient response for pERK activation at 2.5–5 min. In the current study, however, among all of the tested ligands, the reference ligand 2‐AG had the highest efficacy for agonist‐induced internalisation (Table 1). Thus, rather than the prima facie interpretation that ligands which internalise receptors slowly (e.g. THC) have a bias towards pERK, our bias findings for the pERK–cAMP pathway comparison (Figure 5) are likely a manifestation of ‘anti‐cAMP’ bias (i.e. bias away from cAMP rather than towards pERK). This is because 2‐AG reduces CB₁ receptor number by internalisation so quickly that the high efficacy cAMP signal at later time points occurs even with negligible receptor number (implying 2‐AG cAMP‐bias). In contrast, the relative abundance of receptors remaining on the cell surface for ligands that internalise more slowly (THC) produce a similar efficacy cAMP response. This higher receptor number implies lower THC stimulus per receptor—an anti‐cAMP bias. Hence, during internalisation the apparent bias towards pERK for slowly internalising ligands is most likely due to a bias away from cAMP (relative to 2‐AG) rather than towards pERK.

Similarly, a previous study has investigated the action of dopamine D₂ receptor ligands between pERK activation at 5 min and the inhibition of cAMP production over time (Klein Herenbrink et al., 2016). It was shown that the slow dissociating ligands became more biased towards the inhibition of cAMP production over time, because of the relative increase in receptor occupancy over time compared to the rapid dissociation of the reference ligand. A full kinetic evaluation, however, captures the whole system over time and hence provides a stationary reference comparison method.

It is important to note that the metric for the estimate of signalling bias is limited by the available assay data. With only equilibrium functional assay data (and no binding data), the transduction ratio(τ/K_A) is the minimal element that can be directly estimated from the operational model (Zhu, Finlay, Glass, & Duffull, 2018). However, for a time‐resolved assay as reported in the current study, τ ( $\epsilon ·\frac{R_T}{K_E}$) is time varying due to rapid internalisation of total receptor (R_T) which occurs over the same timeframe as other signalling pathways. Therefore, τ cannot be used to quantify ligand bias (other than by a snapshot analysis). For this reason, a ligand‐specific, efficacy‐based constant (ε) was used for bias comparisons which is naturally included as a component of τ. It is therefore possible to separate efficacy and affinity, particularly when the ligand affinity information can be obtained from an independent assay (e.g., binding and internalisation assay). It has been shown that functional affinity (K_A) has an overbearing influence on the transduction ratio (τ/K_A) (Kenakin, 2015; Kenakin, 2018). The consequence of this is that ligand bias is often dominated by affinity at the cost of learning about efficacy. Overall, the efficacy‐based metric, along with functional affinity (K_A), may therefore be better suited for drug discovery (Kenakin, 2018; Onaran et al., 2017).

It is evident that the kinetic context plays a critical role in apparent biased agonism at GPCRs (Klein Herenbrink et al., 2016). However, due to computational requirements, it is often considered impractical to apply a kinetic analysis for large‐scale screening in drug discovery (Lane et al., 2017). In general, we suggest this can be overcome by matching experimental pharmacologists with biologically based data modellers (e.g. pharmacometricians) for analysis of data. In the absence of this symbiotic relationship, it is possible to construct a potential compromise in reducing the number of tested ligands in the full kinetic investigation. Instead of testing all compounds, only the ligands with unique pharmacological action patterns may be selected (i.e. maximise representativeness). One possible approach is to conduct binding and functional assays for screening, followed by compound clustering via traditional equilibrium methods (e.g. operational model of agonism). Then a kinetic investigation for a panel of representative ligands from each cluster can be done to gain deeper mechanistic insights for biased agonism (as demonstrated in the current work). This piece of knowledge could help refine the clustering criteria in the subsequent testing and bridge the gap between in vitro pharmacological responses and in vivo therapeutic effects.

In the current work, a joint kinetic model of biased agonism at CB₁ receptors was developed that incorporated essential (but not comprehensive) mechanistic elements. Due to limitations in our understanding of the mechanism of activation and inactivation of the signalling pathways of interest (especially the pERK pathway), an empirical time‐dependent switch function (Equation A18) was implemented to delineate the rapid signal decay in pERK signalling. This function serves as an approximation of the negative feedback in pERK signalling network with a steep concentration–response curve (Cirit et al., 2010). This would be an area for further exploration to establish a mechanistic framework for this feedback process. Our joint kinetic model is semi‐mechanistic in nature and while it cannot explain the deeper mechanisms of the signalling pathways (e.g. potential crosstalk between pathways), it is able to accurately characterise the observed pathway responses (Figures 2, 3, 4 and S1–S7), indicating that it is able to represent the complex time course data. Therefore, we can conclude that the available data align with the biological processes implemented in the developed kinetic model. The current kinetic model of biased agonism is expected to provide a good starting point to develop a more mechanistic model, including phosphorylation, desensitisation and feedback.

Ultimately, the goal of kinetic and mechanistic developments of quantitative analysis methods is to improve the efficiency of identifying lead compounds (e.g. in applications such as high‐throughput screening of ligands for receptors like CB₁) and pathways of therapeutic value, for further investigation in translational research contexts. Despite these useful developments, it is worth remembering that there is an underlying disconnect between in vitro observations and the complexity of in vivo native functional contexts.

In conclusion, a joint kinetic model approach provides a reasonable description of the biased agonism at the CB₁ receptor over time, leading to a better understanding of CB₁ system. It also allows for a visual assessment of the ability of ligands to selectively and differentially activate certain signalling pathways. In so doing, it could facilitate the discovery of novel therapeutics with improved selectivity and safety.

CONFLICT OF INTEREST

The authors declare no conflicts of interest.

AUTHOR CONTRIBUTIONS

X.Z. and S.B.D. designed and performed model‐based analysis and wrote the paper. D.B.F. designed and performed experiments, analysed the data and wrote the paper. M.G. designed experiments and wrote the paper.

DECLARATION OF TRANSPARENCY AND SCIENTIFIC RIGOUR

This Declaration acknowledges that this paper adheres to the principles for transparent reporting and scientific rigour of preclinical research as stated in the BJP guidelines for Design & Analysis and as recommended by funding agencies, publishers and other organisations engaged with supporting research.

Supporting information

Data S1. Supporting information

APPENDIX A.

A.1. Kinetic model for internalisation assay

In previous study (Zhu et al., 2019a), a quasi‐steady state internalisation model (Equation A1, the same as eq. A11 in Zhu et al., 2019a) was developed to describe the data from ‘live‐at‐start’ internalisation assay measured using fluorescence intensity and the structural form of the model as

$$\mathit{FI}=\alpha ·R_0·e^{-\left(k_{\mathit{con}}+\frac{\left(k_{\mathit{int}}-k_{\mathit{con}}\right)·A}{K_{\mathit{SS}}+A}\right)·t},$$ (A1)

where k_con is the constitutive internalisation rate constant, k_int is the ligand‐mediated internalisation rate, K_SS is the quasi‐steady state equilibrium constant, R₀ represents the initial receptor density and α is a coefficient to link the unit of receptor density and fluorescence intensity (FI).

In current study, this model is reused after two modifications. First, ε_int is introduced as a ligand‐specific intrinsic efficacy for the internalisation pathway. The classic definition of efficacy uses the system maximum response as the reference. However, in this study, it was impossible to precisely estimate the system maximum internalisation rate constant with the available data. Only the constitutive internalisation rate constant of the system (k_con) could be precisely estimated. Hence, the ligand‐specific internalisation rate constant (k_int) was normalised to the system constitutive internalisation rate constant (k_con) to get a unit‐free metric for indicating the relative efficacy of ligand (ε_int). Note that ε_int is a ligand‐specific parameter and constant over time.

$$\mathit{FI}=\alpha ·R_0·e^{-k_{\mathit{con}}·\left(1+\frac{\left({\epsilon }_{\mathit{int}}-1\right)·A}{K_{\mathit{SS}}+A}\right)·t}.$$ (A2)

Second, the inter‐individual variability of initial receptor density (R₀) is introduced to reflect the situation that different assay batches have slightly different initial receptor densities (i.e., the initial fluorescence intensity is not the same among different ligand concentrations). This inter‐individual variability is described by an exponential equation (Equation A3). Due to a model identifiability issue, it is not possible to precisely estimate both TVR₀ (the typical value of R₀) and α. In order to circumvent this problem, TVR₀ is fixed to 1 (arbitrary units) and α is defined as the fluorescence intensity relative to 1 unit of labelled receptors, accordingly.

$$R_0={\mathit{TVR}}_0·\exp \left({\eta }_1\right).$$ (A3)

Here, η₁ represents the inter‐individual variability on R₀.

A.2. Kinetic internalisation model for other signalling pathways

The receptor internalisation occurs over the same time course as other signalling pathways. Therefore, it is critical to take receptor internalisation into account when modelling other signalling pathways. However, the previous internalisation model cannot be directly applied here. The reason for this is because in that model the receptor synthesis rate is assumed to be 0, as only the receptor labelled by the primary antibody at the beginning of the internalisation assay can be detected; however, in this experiment, the newly synthesised receptors contribute to observed pharmacological responses in other signalling assays (e.g., cAMP and pERK assays). Hence, the receptor synthesis process needs to be incorporated into the internalisation model when describing cAMP and pERK assays.

Here, integrating receptor synthesis process and substituting the intrinsic efficacy for internalisation (ε_int) into the kinetic model for the internalisation assay (eq. A7 in Zhu et al., 2019a) yield the kinetic internalisation model for other signalling pathways (Equation A4).

$$\begin{array}{l}\frac{{\mathit{dR}}_T}{\mathit{dt}}=R_{\mathit{syn}}-k_{\mathit{con}}\cdot \left(1+\frac{\left({\epsilon }_{\mathrm{int}}-1\right)\cdot A}{K_{\mathit{SS}}+A}\right)\cdot R_T\\ R_T\left(t=0\right)=R_0,R_{\mathit{syn}}=k_{\mathit{con}}\cdot R_0.\end{array}$$ (A4)

Similar to the kinetic model for the internalisation assay, the inter‐individual variability of initial receptor density (R₀) is modelled by an exponential equation (Equation A3).

Furthermore, based on the quasi‐steady state assumption, the expressions for free receptor (R) and occupied receptor (AR) are derived:

$$R\left(t\right)=\frac{K_{\mathit{SS}}·R_T\left(t\right)}{K_{\mathit{SS}}+A},$$ (A5)

$$\mathit{AR}\left(t\right)=\frac{A·R_T\left(t\right)}{K_{\mathit{SS}}+A}.$$ (A6)

Note here that R (the free receptor) and AR (the ligand–receptor complex) are time varying (in relation to R_T) and are the drivers for both cAMP and pERK systems. Since they are time varying, from here, they are defined as R(t) and AR(t).

A.3. Kinetic model for cAMP signalling

The red block in the schematic plot of joint CB₁ signalling model (Figure 1) illustrates the kinetic model for cAMP. The natural turnover of cAMP is controlled by the synthesis rate of cAMP (C_syn) and the elimination rate constant (k_c).

$$\begin{array}{l}\frac{\mathit{\text{dcAMP}}}{\mathit{dt}}=C_{\mathit{syn}}-k_c\cdot \mathit{\text{cAMP}}\\ \mathit{\text{cAMP}}\left(t=0\right)=0.\end{array}$$ (A7)

The basal cAMP is set to zero at the beginning of the assay because of serum starvation.

In the cAMP assay, the signal amplifier FSK is added to enhance the synthesis of cAMP. Hence, an E_max model (Equation A8) is introduced to describe this capacity‐limited stimulation by FSK on the synthesis rate of cAMP:

$$E_{\mathit{FSK}}=\frac{\mathit{FSK}·{E_{\max }}_{\mathit{FSK}}}{{C_{50}}_{\mathit{FSK}}+\mathit{FSK}},$$ (A8)

where E_maxFSK is the maximal stimulatory effect of FSK and C_50FSK is the concentration of FSK that reaches half maximal stimulatory effect.

The effect of CB₁ ligands is added via a stimulus response model with inhibition on the cAMP synthesis. The stimulus for cAMP signalling (S_c) results from receptor occupancy (Equation A9):

$$S_c=R\left(t\right)+{\epsilon }_c·\mathit{AR}\left(t\right),$$ (A9)

where ε_c is the intrinsic efficacy in cAMP signalling pathway. This is a ligand‐specific parameter and constant over time. Note here that S_c is not constant because the amount of R(t) and AR(t) is changing due to receptor internalisation that occurs simultaneously to all other pathways. S_c can be related through a series of steps to final inhibitory effect (I) giving a rectangular hyperbolic function:

$$I=f\left(S_c\right)=\frac{S_c·I_m}{K_I+S_c},$$ (A10)

where K_I represents the efficiency of coupling between S_c and I and I_m is the system maximal inhibitory response.

Integrating FSK stimulatory effect and CB₁ ligand inhibitory effect into the natural turnover of cAMP yields the kinetic model for cAMP signalling (Equation A11):

$$\begin{array}{l}\frac{\mathit{\text{dcAMP}}}{\mathit{dt}}=C_{\mathit{syn}}\cdot \left(1+E_{\mathit{FSK}}\right)\cdot \left(1-I\right)-k_c\cdot \mathit{\text{cAMP}}\\ \mathit{\text{cAMP}}\left(t=0\right)=0.\end{array}$$ (A11)

There is evidence of a background signal at the start of the experiment, probably due to the technique nature of the measurement method for cAMP. A linear model is implemented to describe the time‐varying background signal (Equation A12).

$$\mathit{\text{BRET}}\left(t\right)=\mathit{\text{cAMP}}\left(t\right)+{\mathit{\text{signal}}}_0+\mathit{\text{slope}}·t,$$ (A12)

where signal₀ represents background signal at time 0 and slope is the increased rate of the background signal. As different assay batches show slightly different initial background signal, the inter‐individual variability of signal₀ is implemented and described by an exponential equation (Equation A13) (i.e., Y‐cut is not shared among different ligand concentrations).

$${\mathit{\text{signal}}}_0={\mathit{\text{TVsignal}}}_0·\mathit{exp}\left({\eta }_2\right).$$ (A13)

Here, η₂ represents the inter‐individual variability on signal₀.

A.4. Kinetic model for pERK signalling

The green block in the schematic plot of joint CB₁ signalling model (Figure 1) illustrates the kinetic model for pERK. The natural turnover of pERK (Equation A14) is controlled by the pERK synthesis rate (P_syn) and the elimination rate constant (k_p).

$$\begin{array}{l}\frac{\mathit{\text{dpERK}}}{\mathit{dt}}=P_{\mathit{syn}}-k_p\cdot \mathit{\text{pERK}}\\ \mathit{\text{pERK}}\left(t=0\right)=0.\end{array}$$ (A14)

The basal pERK is set to zero at the beginning of the assay because of serum starvation.

The action of CB₁ ligands on pERK signalling could be conceptually divided into three steps. First, the ligand binds to the receptor and the stimulus for pERK signalling (S_p) is generated (Equation A15).

$$S_p=R\left(t\right)+{\epsilon }_p·\mathit{AR}\left(t\right),$$ (A15)

where ε_p is the intrinsic efficacy in pERK signalling pathway. This is a ligand‐specific parameter and constant over time. S_p is the sum of the stimulus generated by free receptor R(t) and the stimulus generated by ligand–receptor complex ε_p · AR(t). Here, the R(t) component represents the contribution of constitutive activity and the ε_p · AR(t) component represents the contribution of ligand‐mediated activity.

Then the generated stimulus transits through signal transduction process to the effect site with a time course governed by the transition rate constant (k_tr).

$$\begin{array}{l}\frac{{\mathit{dS}}_{\mathit{eff}}}{\mathit{dt}}=k_{\mathit{tr}}\cdot \left(S_p-S_{\mathit{eff}}\right)\\ S_{\mathit{eff}}\left(t=0\right)=0.\end{array}$$ (A16)

At the final step, stimulus at the effect site (S_eff) exhibits a stimulatory effect on the synthesis of pERK via a rectangular hyperbolic function (Equation A17).

$$E=f\left(S_{\mathit{eff}}\right)=\frac{E_m·S_{\mathit{eff}}}{K_E+S_{\mathit{eff}}},$$ (A17)

where K_E represents the efficiency of coupling between S_eff and E.

In addition, there is a negative feedback in pERK signalling (Cirit et al., 2010). Hence, an empirical time‐dependent switch function (Equation A18) is implemented to delineate the rapid signal decay. This function indicates that after certain time of stimulation, the system will become tolerant to further stimulus for the remaining time of experiment.

$$\mathit{\text{Feedback}}=\left\{\begin{array}{c}1t\le \mathit{\text{stopTime}}\\ 0t>\mathit{\text{stopTime}}\end{array}\right..$$ (A18)

Incorporating the stimulatory effect of CB₁ ligands and the negative feedback into the natural turnover of pERK yields the kinetic model for pERK signalling (Equation A19):

$$\begin{array}{l}\frac{\mathit{\text{dpERK}}}{\mathit{dt}}=P_{\mathit{syn}}\cdot \left(1+E\cdot \mathit{\text{Feedback}}\right)-k_p\cdot \mathit{\text{pERK}}\\ \mathit{\text{pERK}}\left(t=0\right)=0.\end{array}$$ (A19)

Zhu X, Finlay DB, Glass M, Duffull SB. Evaluation of the profiles of CB₁ cannabinoid receptor signalling bias using joint kinetic modelling. Br J Pharmacol. 2020;177:3449–3463. 10.1111/bph.15066