Receptor trafficking critically modulates both the potency and efficacy of ligand-induced signaling.

To investigate the effect of internalization and recycling on the production of cAMP, we first studied a model with two compartments, inspired from [11]. Model 1 (Fig 1) was developed to include reversible ligand-receptor binding, ligand-receptor internalization and receptor recycling, as well as catalytic cAMP production in both compartments and first-order cAMP degradation. Formulating model 1 (Fig 1) as a system of ordinary differential equations (ODEs) (Methods -Eq. (12)), we described the temporal dynamics of the system and derived its unique steady-state for the cAMP response which reads, as a function of the total quantity of ligand (L), receptors (R₀) and model parameters (Eq. (1)) [11]:

(1)

with . All calculations are detailed in Methods. From Eq. (1), we deduced that the efficacy is given by:

(2)

and the potency is determined by:

(3)

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Fig 1. Model 1.

The left panel illustrates receptor trafficking, whereas the right panel depicts second messenger production and degradation. R₁ denotes the amount of free receptors at the plasma membrane, LR₁ and LR₂ the amount of ligand-receptor complex at the plasma membrane and intracellular compartments. cAMP refers to the quantity of second messenger molecules. A parameter was associated to each reaction rate (represented by arrows).

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To investigate the effect of receptor trafficking, we compared expressions (1)–(3) with the classical ligand-receptor interaction model, when no internalization occurs (, Eq. (4)):

(4)

for which and .

We thus deduced that receptor trafficking () modulates both potency and efficacy of the signaling response, whereas second messenger production rates () influence efficacy alone.

We subsequently focused our analysis on scenarios mimicking receptor internalization inhibition. As illustrated in Fig 2, reducing the internalization rate can elicit diametrically opposite effects on the cellular response, depending on the relative magnitudes of signaling () and receptor () parameters. When the endocytic response is weaker than the plasma membrane response (), inhibiting receptor internalization increases efficacy (; Fig 2A and 2B). Conversely, when the endocytic response is stronger than the plasma membrane response (), inhibiting receptor internalization decreases efficacy (; Fig 2C and 2D).

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Fig 2. Impact of internalization inhibition on cAMP production using dose-response analysis.

In the different panels, the ligand (M) dose response relationship of model 1 (Fig 1) is plotted for different values of the internalization rate: (dotted line), (dash-dot line), (dashed line), (solid line). Circles represent the EC₅₀ value. Common parameters to all six panels are , R₀ = 1, k^– = 1, , . The remaining parameters values (, ) are given by: (A) , ; (B) , ; (C) , ; (D) , . Parameter values were arbitrarily chosen, for illustration purposes.

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Similarly, when ligand dissociation from the receptor is faster than receptor recycling (), inhibiting receptor internalization increases potency (EC₅₀; Fig 2B and 2D). In contrast, when ligand dissociation is slower than receptor recycling (), inhibiting receptor internalization decreases potency (EC₅₀; Fig 2A and 2C). Reduction in ligand efficacy (Fig 2C and 2D) is rather expected for receptors that have a strong intracellular activity, as inhibiting receptor internalization will prevent signaling from endosomal compartments. A decrease of ligand potency (Fig 2B and 2D; increase of EC₅₀) is more subtle, and shows that receptor trafficking may indeed affect the observed affinity from dose-response cellular assays. Going further, while inhibiting receptor internalization, the condition and (Fig 2D) leads to a globally less efficient response (i.e., decreases and EC₅₀ increases), and symmetrically the condition and (Fig 2A) leads to a globally more efficient response (i.e., increases and EC₅₀ decreases). However, as shown in Fig 2B and 2C, the effect of inhibiting receptor internalization may be more ambiguous. In these scenarios, efficacy and potency are affected in opposite directions. To resolve the complexities of cases (b) and (c) and characterize overall signaling efficiency under internalization blockade, we calculated the transducer ratio (Eq. (5)), which defines coupling efficiency as the ratio of efficacy to potency:

(5)

The transducer ratio (Eq. (5)) decreases when inhibiting receptor internalization if (Fig 2D), and increases in the opposite case (Fig 2A). However, the transducer ratio can increase or decrease in both cases (b) and (c). Given the widespread use of the transducer ratio to quantify ligand bias [21], these results further suggest that it is also a reliable proxy for assessing how ligand-dependent responses are altered when internalization is perturbed.

Analytic formula (Eqs. (2)-(3)) further reveal that the relative time scale of trafficking parameters () and ligand unbinding rates ( affects potency exclusively, leaving efficacy unchanged. Indeed, if the trafficking parameter time scale decreases, with a constant ratio , remains unchanged, while the EC₅₀ decreases. This indicates that, within this model, ligands inducing a slower receptor trafficking will have a tendency to have higher observed potency (Eq. (3)).

Spatial distribution of receptors modulates both the potency and efficacy of ligand-induced signaling.

The results derived in Eqs. (1)-(3) can be readily generalized to an arbitrary number of receptor-containing compartments. Beyond receptor trafficking kinetics, the spatial distribution of receptors within the cell may also influence the cellular response. To investigate this, we extended the analysis to model 2 (Fig 3) with three compartments (e.g., very early and early endosomes in the case of gonadotropin receptors). Model 2 (Fig 3) was constructed to account for reversible ligand-receptor binding, ligand-receptor internalization to two different compartments (LR₂ and LR₃), and receptor recycling from both intracellular compartments. It further incorporates compartment maturation, whereby ligand-receptor complexes LR₂ transition to LR₃. As in model 1 (Fig 1), catalytic cAMP production occurs across all compartments, and cAMP degradation follows first-order kinetics. Interpreting again model 2 (Fig 3) as an ODE system (Methods -Eq. (14)), we obtained a unique steady-state for the cAMP response, derived in Methods -Eq. (15), which yields analytical expressions for the pharmacological quantities and EC₅₀ (Methods -Eqs. (16)-(17)). In order to investigate how the spatial distribution of ligand-receptor complexes influences the cellular response, we focused on conditions that promote their accumulation in the third compartment [24]. To this end, we introduced a parameter , representing the fraction of internalized receptors directed to LR₃ rather than LR₂. The internalization rates were thus defined as and where denotes the total internalization rate. From the analytical formula for the steady-state cAMP response (Methods -Eq. (15)), we obtained the efficacy, potency and transducer ratio values as follows:

(6)

(7)

and

(8)

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Fig 3. Model 2.

The left panel illustrates receptor trafficking, whereas the right panel depicts second messenger production and degradation. R₁ denotes the amount of free receptors at the plasma membrane, LR₁, LR₂ and LR₃ the amount of ligand-receptor complex at the plasma membrane and two types of intracellular compartments. cAMP refers to the quantity of second messenger molecules. A parameter was associated to each reaction rate (represented by arrows).

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As in Fig 2, Fig 4 illustrates that promoting ligand-receptor accumulation in the third compartment (e.g., increasing p₂₃) can exert opposing effects on potency, efficacy, and transducer ratio. This reflects the intricate interplay between receptor trafficking parameters and signaling parameters, as captured by Eqs. (6)-(8). Remarkably, inducing ligand-receptor accumulation in the third compartment (e.g., increasing p₂₃) always leads to a monotonic relation between , EC₅₀ and with respect to p₂₃. However, as shown on Fig 4, all qualitatively distinct cases are theoretically attainable. Characterizing the dynamics of each case could help identify the most plausible scenario based on current knowledge, and provide a framework for dissecting receptor trafficking and compartmentalized signaling. Notably, the compartment maturation rate from LR₂ to LR₃ () has no effect on the variations of the potency, efficacy or transduction coefficient. Potency is affected only by recycling rates, consistent with the result obtained for the first model. The interpretation of the first case (Fig 4A) is as follows: when the recycling rate is faster from LR₂ than from LR₃ (), and cAMP production in the third compartment is high enough (Condition 19), then the accumulation of receptor in the third compartment leads to globally more efficient response ( increases, EC₅₀ decreases, increases). The fourth case (Fig 4D) is symmetrically opposite to the first case, with a globally less efficient response.

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Fig 4. Impact of biasing receptor distribution on cAMP production using dose-response analysis.

In the different panels, the ligand (M) dose response relationship of model 2 (Fig 3) is plotted for different values of p₂₃. The values of p₂₃ represent the increasing of receptors in the third compartment. Common parameters are p₂₃ = 0 (dotted line), p₂₃ = 0.3 (dashdot line), p₂₃ = 0.7 (dashed line) and p₂₃ = 1 (solid line),; , R₀ = 1, k^– = 1, , , , , whereas the recycling ( and ) and production ( and ) parameters are specific to each panel: (A) , , , ; (B) , , , ; (C) , , , ; (D) , , , . Circles represent the value of EC₅₀. Parameter values were arbitrarily chosen, for illustration purposes.

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In contrast, the second (Fig 4B) and third cases (Fig 4C) are more subtle. In those later cases, the impact of receptor accumulation in the third compartment is dose-dependent, and the overall effect can be quantified by the transduction coefficient , whose variation depends on the relation between signaling efficacy and receptor recycling (Condition 20).

In conclusion, unlike the receptor internalization inhibition case studied in the first subsection, a change in induced by biasing receptor spatial distribution is harder to interpret in terms of parameters values and, therefore, receptor trafficking or cAMP production. However, a shift in the dose-response curve (EC₅₀) is indicative of differential recycling rates from distinct intracellular compartments, and the transduction coefficient may further provide insight into the signaling efficacy within each compartment.

Kinetic experiments and model can jointly reveal receptor trafficking and compartmentalized signaling parameters.

Using Bioluminescence Resonance Energy Transfer (BRET) measurements, we monitored the temporal dynamics of molecular species following FSH stimulation in a cell system overexpressing FSHR (Fig A in S1 Text) [30]. We considered four different BRET experiments. First, we quantified total cAMP levels using an intramolecular biosensor (EPAC) upon continuous stimulation with varying FSH concentrations (FSH = 0, 0.1, 0.3, 1, 3, 10 nM). At a single FSH concentration (1 nM), cAMP level was monitored in control conditions or in the presence of PitStop2, a chemical inhibitor of clathrin-mediated endocytosis (CME), hereafter referred to as ps₂. Localization of receptor in specific cellular compartments was assessed by monitoring interaction between FSHR fused to the luciferase RLuc8 (FSHR-RLuc8) and a fluorescent protein Ypet localized either at plasma membrane (Lyn-Ypet) or at early endosomes (Ypet-FYVE) (Fig B in S1 Text). Measuring the interaction between Lyn-Ypet and FSHR-RLuc8 enables to monitor FSHR localization at the plasma membrane, which decrease upon ligand-induced internalization. In contrast, monitoring interaction between FSHR-RLuc8 and Ypet-FYVE allowed to quantify the time-dependent accumulation of ligand-receptors complexes in early endosomes. Receptor internalization (PM_R) and trafficking to the EEs (EE_R) kinetics were assessed in basal conditions (FSH = 0) and with FSH = 100 nM.

Model 2 (Fig 3) represents the atypical trafficking a priori known for FSHR [6,8] in which LR₁, LR₂ and LR₃ represent ligand-receptor complexes at the plasma membrane (PM), in the very early endosomes (VEE) or early endosomes (EE). As cAMP data show moderate but persistent signal decreases at high doses (Fig A in S1 Text), we further incorporated the possibility of irreversible receptor desensitization from LR₁, at the plasma membrane, and from LR₃ at the early endosomes to obtain the more general model 3 (Fig 5). This complete model yielded an ODE system detailed in Methods -Eq. (21). We assumed the BRET signal is proportional to the quantity of measured species in the cell [31]. These signals were then linked to species evolution according to the following observable equations:

(9)

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Fig 5. Model 3.

The left panel illustrates receptor trafficking, whereas the right panel depicts second messenger production and degradation. R₁ denotes the amount of free receptors at the plasma membrane, LR₁, LR₂ and LR₃ the amount of ligand-receptor complex at the plasma membrane (PM) and two types of intracellular compartments. cAMP refers to the quantity of second messenger molecules. A parameter was associated to each reaction rate (represented by arrows).

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Importantly, prior to parameter estimation, it was essential to verify that the experimental design and model structure yielded a well-posed - i.e., identifiable - inverse problem. To this end, we first reparametrized the ODE system, together with the observable equations, and non-dimensionalized the parameters, so as to reduce the total number of parameters as much as possible. The resulting normalized model led to Eq. (22), comprising 19 unknown parameters instead of 21 (Tables 4 and 3) in which new parameters , and represent the cellular response produced by receptors (Eqs. (23)). Applying the methodology of [32], we demonstrated that the parameter estimation problem is locally identifiable given the observables (Eq. (9)) meaning that each set of observable outputs locally determines a single set of parameter values. Interestingly, when the internalization inhibition data with PitStop2 were excluded, the model became non identifiable for a subset of parameters, including several trafficking and cellular response parameters (, , , , , , , and ). This result demonstrates the value of the experimental design in jointly constraining - and thereby disentangling - receptor trafficking and compartmentalized signaling parameters.

Identifying a cluster of models.

We developed a model selection procedure aimed at identifying parsimonious models with fewer parameters than the complete model 3 (Fig 5), in order to discriminate among the following alternative questions, given the available data (Fig 6):

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Fig 6. Schematic model diagram highlighting the different questions (Q1-Q6) tested during the model selection.

The scheme was done on MatchaIO.

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1. Q1. Does the ligand-receptor complex internalize in one or two compartments (EE or VEE/EE) i.e., is ?
2. Q2. Is the ligand-receptor complex functionally active in two compartments (PM and VEE, i.e., is or PM and EE, i.e., is ) or three compartments (PM, VEE and EE)?
3. Q3. Is chemical internalization inhibition partially or fully efficient, i.e., is ps₂ = 0?
4. Q4. Are there irreversible receptor desensitization mechanisms at the plasma membrane, i.e., is ?
5. Q5. Are there irreversible receptor desensitization mechanisms at early endosomes, i.e., is ?
6. Q6. If there are three active signaling compartments, what is the trafficking pathway that the ligand-receptor complex may follow (internalization, traffic to an other compartment, recycling)– i.e., what is the most plausible trafficking model among 8 competitive models (Fig C in S1 Text)?

Those six questions were associated to alternative model formulation that led to 200 submodels of the complete model 3 (Fig 5) (Table B in S1 Text). We ran the model selection algorithm on the 200 models and we ranked them according to the AIC criterion (Table B in S1 Text). Rather than a single best model, 5 models have a and are therefore considered as likely as the best model; moreover, up to 14 models have , and can thus be regarded as suitable alternatives. Importantly, all 19 best models share common features. Analysis below considers only the best 5 models, yet results extend to models from 3.6 to 3.19 (Table E in S1 Text). First, three compartments are required (Q1), with ligand-receptor complexes being functionally active within each compartment individually (Q2). Model selection rejects the scenario in which production arises solely from PM and VEE (LR₁ and LR₂) or from PM and EE (LR₁ and LR₃), even though biological model suggests that active receptors may be abundant in VEE (LR₂). The data are compatible with the hypothesis that the CME inhibitor is fully effective (Q3) (model 3.5 is the only model with a positive parameter ps₂, but it has very low value). The data also support the existence of an irreversible receptor desensitization mechanism at the plasma membrane (Q4), while we cannot conclude about the desensitization mechanism from early endosomes (Q5). Finally, we cannot currently identify a unique best trafficking model (Q6) (Table C in S1 Text), as three trafficking hypotheses (c, b and i) fall below the threshold (Table 1). These three trafficking models all exclude the internalization route from PM to EE (), but share the assumption that VEEs can mature into EEs (). In contrast, all models that omit the transition from VEE to EE (trafficking model g and h) have (Table B in S1 Text) and were thus clearly rejected.

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Table 1. Summary of results for the five best models selected based on the AIC criterion. The table below reports the outcomes of the selection procedure for models 3.1, 3.2, 3.3, 3.4 and 3.5, including the answers to 0–0 and the corresponding statistics for each model (negative log-likelihood, AIC, , number of estimated parameters, convergence status), revealing the existence of two distinct model clusters. Convergence was calculated as the number of runs with a negative log-likelihood value below the minimum negative log-likelihood value plus 0.1.

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The parameter estimation procedure yielded estimates for all parameters of each model (Table C in S1 Text), which were subsequently used for all predictions. We verified that the five best models yield comparable fits, as illustrated in Fig 7, which demonstrates strong agreement between model predictions and experimental data across all observables - cAMP production, receptor internalization, and receptor trafficking to early endosomes - both under control conditions and upon CME inhibition. Verifying practical parameter identifiability for the different models is essential to perform reliable model prediction. Hence, to further investigate the five best models (Table B in S1 Text, models 3.1-3.5), we performed a profile likelihood estimate for each parameter on each model (Fig 8). Some parameters are clearly identifiable with very small confidence intervals and common values across the eight models (, , , , , , , ), whereas some are consistent across models but have larger confidence intervals (, , , , ). The remaining parameters are consistent across some models but exhibit large confidence intervals in others (, , , ) (Fig 8).

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Fig 7. Model fitting for the five best models.

The five best fit of the ODE simulation (Table B in S1 Text, models 3.1-5) are superimposed while shading area indicates statistical uncertainty for the first model. Experimental data points are shown as crosses. For the 1 nM ligand condition, the bottom curve (shown in dark blue) corresponds to cAMP levels measured under PitStop2 treatment.

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Fig 8. Confidence intervals for all parameters in the five best models.

In each panel, we show the profile-based 95% confidence interval of the normalized parameter for the 5 selected best models (Table B in S1 Text). The optimum value is represented by the dark line.

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Parameters governing time scales, ligand-receptor interactions, and observables are well identified, while those related to receptor trafficking and localized cAMP production remain subject to greater uncertainty. Models 3.1, 3.4 and 3.5 share the common trafficking hypothesis c and lead to similar parameter values and practical identifiability (except for ). Similarly, models 3.2 and 3.3 also share similar parameter values and practical identifiability, but distinct from models 3.1, 3.4 and 3.5. This distinction is particularly evident for several parameters, including , and (Fig 8) Thus, within the five best models, two groups emerge (cluster 1 and cluster 2 -Table 1) that are compared in the subsequent study.

Uncertainties on internalized receptors do not affect predictions on localized cAMP production.

Despite our theoretical identifiability results, parameter uncertainty may arise from the inherent variability of the experimental data and/or from insufficient number of observables or experimental conditions. As apparent from Fig 7, experimental variability is high, which necessarily increases the variance of the statistical error model. In Fig 9, we plot the profile-based 95% confidence interval (shaded area), revealing that parameter uncertainty does not affect cAMP level predictions for model 3.1 (first column), but significantly impacts the predicted amount of internalized receptors. Similarly, Fig 9 (second column) shows that model predictions for each cAMP variable and for the amount of receptors at PM and in VEE (LR₁ ad LR₂) are consistent across the five best models. However, models differ substantially in their predictions of the amount of internalized receptor in the EE (LR₃). The prediction uncertainty in internalized receptor likely reflects variability in trafficking and localized production parameters, which may partially compensate for one another to reproduce a given cAMP output-at least within our dataset. A negative correlation can indeed be observed between the number of ligand-receptor complexes in EE (LR₃) and the cAMP production rate in this compartment (). Indeed, the amount of LR₃ is higher for models with trafficking hypotheses c (cluster 1, models 3.1, 3.4 and 3.5), while cAMP production in the third compartment is lower in these models (Fig 9 and Table C in S1 Text). In comparison, models 3.2 and 3.3 (cluster 2) have less internalized receptors and a highest cAMP production in the third compartment. A similar trend is observed within the 19 best models that are all consistent for cAMP dynamics but differs in internalized receptors (Fig H in S1 Text), possibly highlighting a third cluster (Table E in S1 Text). It is to be noted that the best models do not differ in their amount of internalized receptors (less than 1%, see next subsection) but differs in their relative proportion of LR₂ and LR₃ (Table 2).

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Table 2. Averaged outputs for the five best models. The table presents the proportion of internalized receptors () relative to total receptors, and the proportion of receptors in EE (LR₂) and VEE (LR₃) relative to internalized receptors, as well as endocytosis and recycling time and probability for a 5min incubation with 10nM of ligand. Values in brackets indicate uncertainty.

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Fig 9. Prediction and uncertainty quantification of cAMP levels and LR complexes in each compartment for the five best models.

Model 3.1 predictions are shown with 95% profile-based confidence intervals (left column), alongside single predictions for all best models (right column) in response to a single FSH dose (10 nM). Rounds show cluster 1 and crosses the cluster 2.

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To further discriminate among the best models, an interesting perspective would be to design experiments using sensors selectively localized in distinct endosomal compartments to monitor the quantities of LR₂ and LR₃ [6]. Such spatially resolved data would help discriminate between models and strengthen our conclusions regarding hypotheses 0–0. However, designing such selective sensors is currently challenging, as VEEs lack specific molecular markers. To illustrate this, Fig D in S1 Text shows model predictions for models 3.33, 3.60, 3.40, and 3.132, which all share the common c trafficking structure as their best model (Fig C in S1 Text), but differ in their assumptions regarding hypotheses: desensitization from EE (3.33) 0, no desensitization from PM (3.60) 0 and cAMP production at PM and VEE (3.40) or at PM and EE (3.132) 0, respectively (Table B in S1 Text). These alternative models deviate only marginally from the best model in terms of total cAMP - failing in particular to capture the long-time cAMP decrease - but diverge more substantially in their predictions of internalized receptor amounts and endosomal cAMP levels (Table D in S1 Text).

FSHR activation leads to a strong endosomal signaling response.

The five best models exhibit very few internalized receptors (less than 1%, Table 2) that yet elicit a strong cellular response ( and are three to four order of magnitude higher than , Table C in S1 Text). The first and transient wave of cellular response originates from signaling at the plasma membrane and in very early endosomes (LR₂), with a slightly higher response in VEE. A second and sustained wave of signaling corresponds to cAMP production in EEs (LR₃) (Fig 9). Further, receptor internalization rates (of order ) appear to be considerably slower than receptor recycling rates (of order ), the latter being comparable to ligand unbinding rate (of order ). All models predict an averaged endocytosis time of 58s with a rather sharp confidence interval, while the averaged receptor recycling time has much more uncertainty across models, ranging from 2 to 169s. Interestingly, for models with receptor recycling from VEE, the VEE recycling rate is much higher than the EE recycling rate (Table C in S1 Text), in agreement with previously reported dynamics for the LHR [6,8]. Finally, irreversible receptor desensitization occurs mostly at the plasma membrane (with a probability of 0.01 for receptors to be internalized in endosomes) while most receptors within the endosomes are recycled back to the plasma membrane. Results remain consistent across models 3.6 to 3.19 (Table F in S1 Text).

Given relatively comparable receptor recycling and ligand unbinding rates, but markedly contrasted signaling rates between the plasma membrane and endosomal compartments, we expect - in light of the Figs 2 and 4 - that internalization inhibition will primarily reduce signaling efficacy. Simulating dose-response experiments with decreasing internalization rates using our best models (3.1 and 3.2, representing each cluster), we show in Fig 10 that the model indeed predicts that internalization inhibition drastically reduces signaling efficacy, with no apparent effect on potency. Similarly, biasing the spatial distribution of receptors - by inducing ligand-receptor accumulation in the third compartment - has virtually no effect on potency but substantially impacts efficacy. However, the large parameter uncertainty and the lack of agreement between the two best models complicate interpretation: for model 3.1, efficacy decreases as the proportion of LR₃ increases, whereas in model 3.2, efficacy increases as the proportion of LR₃ increases. These contrasting predictions highlight that experimental approaches capable of biasing receptor spatial distribution will be extremely valuable, both for discriminating between models and for reducing parameter uncertainties within a given model.

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Fig 10. Dose response prediction of the total quantity of cAMP with internalization inhibition and with biased receptor spatial distribution for models 3.1 and 3.2 with 95% profile-based confidence intervals.

The different values of internalization are obtained by varying the parameters ps₂: ps₂ = 1, (dotted line), (dash-dot line), (dashed line), no internalization: ps₂ = 0 (solid line). The different values for p₂₃ are 0 (dotted line), 0.3 (dash-dot line), 0.7 (dashed line) and 1 (solid line), while the internalization rate remains constant for each model (, and ). Circles represent the value of EC₅₀. The remaining parameters used for the prediction correspond to the best estimates obtained during model selection (Table C in S1 Text).

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The model accurately predicts pulse-chase ligand stimulation data.

To strengthen our model predictions, we performed additional experiments for model validation, using both continuous and pulse-chase ligand stimulation. In the later design, the ligand is washed out after one to several minutes of exposure (Fig A in S1 Text-validation data). Pulse-chase experiments are particularly well suited for dissecting the molecular events that occur after ligand binding to its receptor at the plasma membrane, while preventing new ligand binding. Validation data include an additional ligand concentration (30 nM), that was not included in the model fitting data. Three BRET assays were used to monitor total cAMP levels, receptor internalization, and ligand-receptor complexes located in early endosomes (EE). In receptor internalization assays, the CAAX sequence was used instead of LYN to adress the Ypet at PM, but the assay similarly monitored the observable PM_R.

Fig 11 shows the predictive performance of these models in representing pulse-chase data for a single ligand concentration, across all observables (cAMP, PM_R, and EE_R) (Fig E in S1 Text for all ligand concentrations). Nearly all validation data points ( for all the concentrations of ligand) fall within the 95% confidence interval of the statistical error model. Although a systematic model bias is apparent, this may be attributable to variability between independent experiments, as observed in Fig 7. Some differences between cAMP dynamics have to be noticed. Firstly, a notable difference in the timing of cAMP decrease can be observed between conditions. In both cases, cAMP production decreases faster than suggested by the validation data points. One possible explanation is that our model relies on a minimal representation of the cAMP signaling cascade, whereas it is well established that additional molecular actors are involved, as well as non mass-action kinetics phenomena (e.g., phase-separation [33]). Furthermore, for the two highest ligand concentrations (L = 10 nM and L = 30 nM), model predictions exceed the validation data points during the first 30 minutes, before decreasing faster than observed experimentally. The plateau level () is subject to systematic variability across replicate (Fig 7), potentially due to variability in receptor and/or sensor expression (the validation data seems to have reached saturation phase starting from L = 3 nM, which is not the case of the experimental data used for fitting). Finally, one notable model limitation emerges in the pulse-chase condition: following ligand washout, the number of membrane receptors first decreases rapidly due to internalization, then rebounds after approximately 20 minutes, presumably reflecting the fraction of receptors recycled back to the plasma membrane (Fig 11 - right column). This biphasic dynamic is not captured by our model, possibly due to the large experimental variability that obscures the underlying trend - most data points nonetheless remain within the confidence interval derived from the statistical observation model. The observed decrease in plasma membrane receptors can be attributed to both internalization and desensitization reactions. When we simulated the model 3.1 with its best fit parameters, but with no desensitization () and with a faster internalization (), we were able to reproduce the bounce effect (Fig 12). Similarly, this phenomenon appears when we simulated a model without desensitization at the plasma membrane (Fig 12, model 3.60/best fit), with a faster internalization rate (). Overall, this result demonstrates that our set of models can ultimately predict this behavior when considering an alternative set of parameters.

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Fig 11. Prediction of observables with data validation for the five best models.

The first column shows results under continuous ligand stimulation, and the second column under pulse-chase stimulation, both at a single FSH concentration of 30 nM. The first row displays cAMP levels, followed by LR₃-FYVE (second row) and plasma membrane receptors-CAAX (third row). Scatter points represent validation data under both conditions (continuous and pulse-chase stimulation). Shaded areas indicate the 95% confidence interval of the statistical observation model across the five best models. The parameter values used for these predictions correspond to the best estimates obtained during model selection (Table C in S1 Text).

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Fig 12. Predictions of plasma membrane receptors.

Simulations were performed for both the best-selected model (3.1) and a similar model lacking desensitization from the plasma membrane (3.60). Parameters used for the prediction are: 1) Model 3.1: best estimates (dashed line) and and (solid line) and 2) Model 3.60: best estimates (dashed line) and (solid line).

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The model accurately predicts binding affinity.

We note that the binding affinity of FSH is predicted at nM, (95% confidence interval: nM), consistently with previously reported values [34]. We additionally designed an experiment to further validate our binding affinity prediction. We characterized FSH binding to its cognate receptor using a first classical binding assay with a recombinant FSH fused to mNeonGreen as a BRET acceptor (FSH^mNG) and a FSHR fused to a N-terminal NanoLuciferase tag (Nluc-FSHR) as the BRET donor. We then performed a competitive binding experiment where the concentration of FSH^mNG was held fixed while unlabelled FSH was added in excess (Fig A in S1 Text, last row). This approach allowed indirect quantification of reference FSH binding through the dose-dependent decrease in FSH^mNG-induced BRET signal. The first binding assay corresponds to the traditional equilibrium function with f the amount of ligand-receptor complexes:

(10)

where is the fluorescent FSH^mNG concentration, and its binding affinity to FSHR, while the competitive assay corresponds to:

(11)

We jointly optimized Eqs. (10)-(11) in , with fixed to its model-prediction ( nM in the best model) with a least squares fitting. Given the good data-fitting in Fig 13 and the fact that the fitted ( nM) is very close to the EC₅₀ measured in the experiment, we conclude that the model-predicted value is coherent with our competitive binding assays.

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Fig 13. Validation of the FSH binding affinity.

Binding assay of FSH^mNG to FSHR (left panel) and competitive binding of FSH to FSHR in presence of FSH^mNG (right panel, FSH^mNG = 3 nM). Data points are shown in red crosses, the solid red line represents the mean of the data, and the dotted blue line corresponds to the prediction. The fitted FSH^mNG binding affinity constant is shown by the circle on the left panel.

https://doi.org/10.1371/journal.pcbi.1014790.g013

Model 1 (Fig 1).

Model 1 (Fig 1) leads to the following ODE system:

(12)

In short, through binding (resp. unbinding) reactions at rate (resp. ), lead to the formation of a ligand-receptor complex (LR₁) at the plasma membrane. This complex is then internalized into the second compartment at rate , while the receptor is recycled back to the membrane at rate . The second messenger (cAMP) is produced in each compartment, in proportion to the amount of ligand-receptor complex , with i = 1,2 with compartment-specific production rates at the plasma membrane and in the intracellular compartment and is degraded according to first-order kinetics, at a rate k^–. Table A in S1 Text summarizes all variables and parameters of this model.

By mass conservation, the receptor trafficking subsystem () can be reduced to a two-dimensional affine system whose system matrix is stable (by simple calculations). Thus, the receptor variables are globally exponentially stable, and the same holds for the second messenger cAMP, due to its first-order degradation kinetics. To compute the steady state, consider the system:

we deduced thanks to simple algebraic calculations the expressions:

and then using:

we obtained Eq. (1). The later Eq. (1) has a similar expression as the standard phenomenological dose-response curve which we recall:

(13)

In Eq. (13), quantifies the ligand efficacy (maximum of the cellular response) and EC₅₀ is the ligand potency (quantity of ligand needed to obtain half of the maximum of the dose response) [20]. Identifying and EC₅₀ in Eq. (1) leads to Eqs. (2)-(3).

Model 2 (Fig 3).

Model 2 (Fig 3) leads to the following ODE system (Eq. (14)):

(14)

In short, binding and unbinding reactions, occurring at rates and respectively, lead to the formation of a ligand-receptor complex (LR₁) at the plasma membrane. The ligand-receptor complex LR₁ is then internalized in two intracellular compartments, called LR₂ or LR₃, at rates and , respectively. LR₂ is recycled at rate (respectively LR₃ at rate ) and receptors in the second compartment, called LR₂, can transfer to the third compartment, called LR₃, at rate . The second messenger (cAMP) is produced in each compartment proportionally to the ligand-receptor complex with compartment-specific production rates for i = 1,2,3, and is degraded according to first-order kinetics, at a rate k^–. Table A in S1 Text summarizes all variables and parameters of this model. Using the same methodology as the previous section, we derived the analytical formula for the steady-state cAMP response, which reads as follows. By simple algebraic calculations, we obtained the following expressions:

The steady-state for cAMP is the following:

(15)

with the values of potency and efficacy:

(16)

(17)

Now using and , we obtained efficacy, potency and transducer ratio values given in Eqs. (6)-(8). We studied the effect of increasing the proportion of receptors in the third compartment (p₂₃) on the efficacy , potency (EC₅₀) and (Eqs. (6)-(8)), to obtain the following necessary and sufficient conditions:

(18)

(19)

and,

(20)

Model 3 (Fig 5).

The general model 3 (Fig 5) for the parameter estimation problem is the following ODE system (Eq. (21) and Table A in S1 Text):

(21)

together with the observable-equation system (Eq. (9) in the main text),

The ODE model is exactly the same model as Eq. (14) but includes the desensitization/degradation of LR₁ at the plasma membrane () and of LR₃ in the early endosomes (). Table A in S1 Text summarizes all variables and parameters of this model.

General formula.

Endocytosis and recycling events are not almost certain events from a probability point of view given that the model includes irreversible receptor desensitization. Therefore, we calculated conditional expectation times using the following method [61]. Given a continuous-time Markov chain (CTMC) of infinitesimal generator L on a state-space where A and B are two disjoint absorbing ensembles, we calculated the solution of

(28)

In Eq. (28), h(x) is the absorbing probability in A, starting from , and f is the conditional expectation time:

with and the absorbing times in A et B respectively.

Endocytosis time and probability.

We used the CTMC whose graph is the following:

with , and (the absorbing desensitization state). We obtained for the conditional expectation time for endocytosis, with :

Recycling time and probability.

We used the CTMC whose graph is the following:

with , A={R₁}, and (the absorbing desensitization state). We obtained for the endocytosis, with and :

Finally, we averaged out these quantities according to the initial point, which is LR₂ with probability , and LR₃ with probability . We thus obtained:

Receptors proportion.

The proportion of receptors is:

Ligands and drugs.

Recombinant FSH (GONAL-f^R) was kindly provided by Merck (Darmstadt, Germany) and resuspended in mQ H₂O. FSH-mNeonGreen (FSH-mNG) was designed in our group and produced in ExpiCHO-S cells expression system. ExpiCHO-S cells at 6.10⁶ living cells/mL density were transfected with 20 g of pcDNA3.1 encoding the FSH-mNG, according to ExpiCHO-S Expression System (Gibco) manufacturer’s instructions. Cells were cultured for one day at C/8% CO₂ on a shaker platform (120 rpm) before supplementation with feed and enhancer solution (provided in the ExpiCHO-S expression system transfection kit). Cells were then cultured for 12 days at C/5% CO₂ at 120 rpm. The culture was centrifuged at 500 g/30 min/C, and the supernatant was collected. The supernatant was then centrifuged at 5000g/30min/C. The newly collected supernatant was dialyzed in regenerated cellulose membrane tubing with a 6–8 kDa molecular weight cut-off (MWCO) (Spectra/Por^M) against a pH 8.0 buffer (50 mM Tris HCl, 100 mM NaCl) overnight at C under continuous stirring. The dialyzed supernatant was centrifuged (5000g/30min/C) to remove insoluble debris, and passed through a buffer equilibrated (pH 8.0, 50mM TrisHCl, 100mM NaCl) Protino Ni-IDA packed column (Macherey-Nagel). The column was washed with three times its volume of the following buffers: i) pH 8.0, 50mM TrisHCl, 100mM NaCl, ii) pH 8.0, 50mM TrisHCl, 1M NaCl, iii) pH 8.0, 50mM TrisHCl, 100mM NaCl. The FSH-mNG elution was performed with a pH 8.0, 50mM TrisHCl, 100mM NaCl, 500mM imidazole buffer. The FSH-mNG was buffer exchanged into a pH 8.0, 50mM TrisHCl, 100mM NaCl buffer, using 10DG Desalting Prepacked Gravity Flow Columns (BioRad Laboratories). The FSH-mNG was then concentrated using 30kDa MWCO Amicon Ultra centrifugation unit (Sigma-Aldrich) according to the manufacturer’s instructions. After analysis by Coomassie blue SDS-PAGE gel staining, purified FSH-mNG was stored at -C. PitStop2 was purchased at Sigma-Aldrich, and used at a final concentration of 30M.

Plasmids.

The FSHR-RLuc8 plasmid was kindly provided by Pr. Aylin Hanyaloglu (Imperial College London, United Kingdom). All other plasmids were designed in our group and synthesized by Twist Bioscience. The cAMP BRET sensor NLuc-EpacD602A-VV-NES was designed from the cAMP BRET sensor NLuc-Epac-VV [62] by adding a nuclear exclusion signal (NES) sequence and the D602A mutation position to improve the sensor’s dynamic range [63].

Cell culture.

Human Embryonic Kidney 293 (HEK293A) (Thermo Fisher Scientific) cells were cultured in DMEM (Eurobio) medium containing Glutabio and NaHCO₃ and supplemented with 10% heat inactivated fetal bovine serum (FBS) (Eurobio), 100 IU/mL penicillin and 0.1 mg/mL streptomycin (Eurobio). Cells were kept at C in a humidified 5% CO₂ incubator.

Bioluminescence Resonance Energy Transfer (BRET).

Forty thousand HEK293A cells per well were seeded in previously 0.01% poly-lysine coated 96-well plates, and transiently transfected in suspension using Metafectene Pro transfection reagent (Biontex Laboratories) according to the manufacturer’s instructions. DNA quantities used in each type of experiment are detailed in the following table:

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https://doi.org/10.1371/journal.pcbi.1014790.t005

Of note, Lyn and CAAX motifs were used to address Ypet to the lipid rafts and negative charges of PM, respectively, whereas FYVE motif allowed Ypet addressing to the EEs.

48 hours after transfection, BRET measurements were performed upon addition of 5 M coelenterazine-H (Interchim) diluted in Ca²⁺/Mg²⁺free PBS, containing no or different concentrations of FSH. For experiments performed in presence of PitStop2, cells were pre-incubated 35 minutes in presence of drug-containing buffer before measurements and ligand stimulation. Signals were recorded for at least 60 minutes with a Mithras LB 943 plate reader (Berthold Technologies GmbH & Co.). BRET ratios were calculated as follows: 480nm/540nm for cAMP experiments; 540nm/480nm for receptor internalization and FSH-mNG binding experiments.