Yeast GPCR signaling reflects the fraction of occupied receptors, not the number

Abstract According to receptor theory, the effect of a ligand depends on the amount of agonist–receptor complex. Therefore, changes in receptor abundance should have quantitative effects. However, the response to pheromone in Saccharomyces cerevisiae is robust (unaltered) to increases or reductions in the abundance of the G‐protein‐coupled receptor (GPCR), Ste2, responding instead to the fraction of occupied receptor. We found experimentally that this robustness originates during G‐protein activation. We developed a complete mathematical model of this step, which suggested the ability to compute fractional occupancy depends on the physical interaction between the inhibitory regulator of G‐protein signaling (RGS), Sst2, and the receptor. Accordingly, replacing Sst2 by the heterologous hsRGS4, incapable of interacting with the receptor, abolished robustness. Conversely, forcing hsRGS4:Ste2 interaction restored robustness. Taken together with other results of our work, we conclude that this GPCR pathway computes fractional occupancy because ligand‐bound GPCR–RGS complexes stimulate signaling while unoccupied complexes actively inhibit it. In eukaryotes, many RGSs bind to specific GPCRs, suggesting these complexes with opposing activities also detect fraction occupancy by a ratiometric measurement. Such complexes operate as push‐pull devices, which we have recently described.

To construct plasmid pMD45-K-hsRGS4-CFP we used PCR (primers 531 and 532) to amplify a DNA fragment from plasmid pRS406-K-hsRGS4-CFP which contained hsRGS4-CFP, the kanMX6 cassette [30], and 40 bp 5' and 3' regions homologous to the STE2 ORF and terminator regions, respectively. Amplified DNA was then transformed for integration in a yeast strain in which the only copy of the STE2 gene is a F204S mutant expressed from the centromeric plasmid pMD45 [14]. Transformants were selected on YPD agar plates supplemented with 350 µg/ml of antibiotic G418 (Geneticin; YPD-G418), after which modified plasmids were recovered by yeast plasmid rescue via transformation and amplification in E. coli, and verified by DNA sequencing.
Plasmid pBB13 was constructed by cloning a DNA fragment amplified by PCR from yeast genomic DNA (primers 223 and 224) containing the 403-916 nt region of the STE2 ORF and flanked with KpnI and RsrII sites, respectively, into a pRS406 plasmid containing CFP and the ADH1 terminator (tADH1). This resulted in the construct KpnI-STE2 403−916 -RsrII-CFP-tADH1.
We put STE2 under control of the GAL1 promoter (pGAL1) in strains YAB3930, YAB5302, and YAB5372 by transforming strains TCY3154, TCY394, and YPP3662, respectively, with a PCR product amplified from plasmid pFA6a-kanMX6-pGAL1 [30] (primers 416 and 417), carrying the KanMX6 resistance cassette, pGAL1, as well as 40 bp 5' and 3' sequence homology to the promoter region of STE2, such that integration replaced the 200 bp immediately upstream of the STE2 ORF. We selected for transformants on YPD-G418 plates and screened for positive clones based on their ability to respond to α-factor in SC-Gal, but not in SC-Glu medium.
Similarly, we put SST2 under control of pGAL1 in strain ACY5660 by transforming strain TCY394 with a PCR product amplified from plasmid pFA6a-kanMX6-pGAL1 [30] (primers 418 and 419) with 40 bp 5' and 3' sequence homology to the promoter region of SST2, such that integration replaced the 100 bp immediately upstream of the SST2 ORF. We selected for transformants on YPGal plates (2% galactose) supplemented with 350 µg/ml G418 (YPGal-G418), and identified positive clones in halo assays as those forming normal halos on YPGal, but large halos on YPD.
To construct strain YAB5313 constitutively expressing hsRGS4 from the ACT1 promoter (pACT1), we transformed strain YAB5302 with plasmid pRS406-hsRGS4-CFP digested with StuI to target it for integration at the URA3 locus. We selected for transformants on plates with synthetic medium lacking uracil (S -Ura) and identified positive clones by membrane associated CFP fluorescence.
To facilitate later PCR-mediated genomic modifications requiring selection with G418, we replaced the kanMX6 cassette in strains YAB5302 and YAB5372 by hphMX4 (which confers resistance to the antibiotic Hygromycin B), resulting in strains ACY5514 and ACY5588, respectively. To do this we digested plasmid pFA6a-hphMX4 [19] with BamHI and SacI and transformed it into said strains to promote homologous recombination between the shared TEF promoter (pTEF) and terminator (tTEF) sequences of hphMX4 and kanMX6 cassettes. We selected for transformants on YPD plates supplemented with 200 µg/ml of Hygromycin B (YPD-Hyg). Positive clones were then identified as those having gained resistance to hygromycin B to the detriment of G418 resistance, while retaining their ability to grow in medium lacking histidine (ACY5514) or supplemented with Nourseothricin (ACY5588), as the his3MX6 and natMX6 cassettes present in these strains also share homology with hphMX4 via pTEF and tTEF.
We constructed strains ACY5544, ACY5545, ACY5612, and ACY5613 expressing hsRGS4-CFP from the wild type SST2 promoter by transformation of strains TCY394, ACY5514, YPP3662, and ACY5588, respectively, with a PCR product amplified from plasmid pRS406-K-hsRGS4-CFP using primers 526 and 529. This fragment contains the hsRGS4-CFP ORF and the kanMX6 resistance cassette flanked by regions of homology to the promoter and terminator regions of SST2, thus replacing the entire SST2 ORF.
Similarly, we constructed strain ACY5549 expressing RGS hsRGS4 fused to STE2 by transforming strain ACY5514 with a PCR product amplified from plasmid pRS406-K-hsRGS4-CFP using primers primers 531 and 532. This fragment contains RGS hsRGS4 -CFP and the kanMX6 resistance cassette, flanked with 40 bp homology to the 3' end of the STE2 ORF and the STE2 terminator, thus fusing RGS hsRGS4 -CFP to the STE2 ORF.
For all strains constructed with PCR products from the pRS406-K-hsRGS4-CFP plasmid, colonies were first selected on YPD-G418 plates and then restreaked onto S -His plates and/or YPD plates supplemented with the antibiotics the target strain was expected to be resistant to. This procedure avoided the selection of clones in which homologous recombination from the introduced PCR product had resulted in replacing a selection cassette already present in the genome, rather than modifying the target locus. Finally, all colonies were verified by expression of CFP fluorescence and by formation of small halos in the α-factor halo assay.
We deleted the SST2 ORF in strains YAB5353 and ACY5609 by transforming strains YAB5313 and ACY5549, respectively, with a PCR product amplified from plasmid pFA6a-natMX6 [24] using primers 468 and 469. This DNA fragment contains the natMX6 resistance cassette flanked with 40 bp homology to the promoter and terminator regions of SST2, thus replacing the entire SST2 ORF. Transformants were selected on YPD plates supplemented with 100 µg/ml of the antibiotic Nourseothricin (YPD-Nat). We confirmed deletion of SST2 by colony PCR with primers 502 and 503. Note that we avoided passing through intermediate supersensitive strains with no functional RGS protein, as these tend to arrest spontaneously and accumulate sterile mutations.
We C-terminally tagged the genomic STE2 gene with CFP in strains YGV5564 and YGV5627 by transforming strains YAB5302 and ACY5545, respectively, with plasmid pRS406-STE2 W T -CFP digested with ClaI. This modification also introduced the ADH1 terminator (tADH1) after the STE2 -CFP ORF. Similarly, we constructed strains YGV5563, YGV5626, YGV5630, and YGV5580 that express STE2 20ST A−7KR -CFP (also referred to as STE2 RE in the main text) from the genomic locus by transforming strains TCY394, ACY5544, ACY5545, and YAB5302, respectively, with plasmid pRS406-STE2-20STA-7KR-CFP digested with ClaI. These chimeric constructs were also followed by tADH1. Likewise, we made strain YAB3724 that expresses a CFP-tagged truncated from of STE2 (STE2 1−305 ) by transforming strain ACL379 with plasmid pBB13 digested with ClaI. In all cases we selected for transformants on S -Ura plates and confirmed positive clones by expression of CFP fluorescence.
We introduced the GAL4-ER-VP16 (GEV) chimeric transcription factor [16] into strains YIP5370, YIP5581, ACY5620, and YGV5642 to control expression of STE2 from the GAL1 promoter by transforming strains YAB5353, YGV5564, ACY5609, and YGV5627, respectively, with plasmid pSZ111 digested with NdeI, targeting integration to the BMH2 locus. pSZ111 was kindly provided by Dr. Gustavo Pesce and contains GEV under control of the BMH2 promoter, which has low cell-to-cell variability (G. Pesce, personal communication). We selected for transformants in synthetic medium lacking leucine (S -Leu) and confirmed the clones phenotypically by their ability to respond to α-factor only in the presence of β-estradiol.

Cell treatments and microscopy methods
We prepared cells for microscopy as follows. We harvested exponential growing cells at an OD 600 around 0.02 (≈ 6 × 10 5 cells/ ml), briefly sonicated and placed them in a 384-well glass-bottom plate (BD Falcon, around 10 4 cells/well). To adhere the cells to the glass we pre-treated the wells with a solution of 1 mg/ml of concanavalin A type V (Sigma-Aldrich) for at least 20 minutes and washed twice with water. We prepared serial dilutions of chemically synthesized α-factor (Yale Small Scale Peptide Synthesis, New Haven, CT) in a factor 1 in √ 10 ≈ 3.16, or 1 in 3, using media supplemented with 40 µg/ml of casein (Roche Applied Science) at pH 5.5, or 0.1% w/v PEG (MW 3550, Sigma), to block unspecific binding to plastic material [12]. For fluorescent proteins (FP) reporter dose-response (DoR) experiments we stimulated the cells at the indicated concentrations of α-factor with 10 µm of the inhibitor of Cdc28-as2 1-NM-PP1, incubated at 30 o C for 2 hours, added cycloheximide to a final concentration of 100 µg/ml to inhibit translation and incubated at 30 o C for at least 2 more hours to allow complete maturation of the FP. When experiments involved induction of Ste2 by the GEV system, cells were incubated with different concentrations of β-estradiol for 3 hours at 30 o C followed by 2 hours α-factor treatment in the corresponding β-estradiol dose. When using supersensitive strains like P GAL1 -SST2 in SC-Glucose (where SST2 expression is repressed), to minimize the ligand depletion that may occur at pm concentration, cells were diluted 30-fold (OD 600 ≈ 0.0006 or 1.8 × 10 4 cells/ ml), incubated with α-factor and then concentrated by centrifugation.
Fluorescence microscopy-based cytometry was done as described elsewhere [12,20]. We acquired images using a fully motorized Olympus IX-81 inverted microscope, equipped with LED illumination (CoolLED), Zero Drift autofocus system (Olympus), CoolSnapHQ2 cooled CCD camera (Photometrix), adequate filter cubes (Chroma Technologies Corp. 41028 for YFP and Hilyte488, and 31044v2 for CFP) and an Olympus 60X PlanApo oil immersion objective (NA 1.4). We used our software Cell-ID [12] for image segmentation and our R package Rcell [8] for analysis and visualization of the dataset. Determination of total cell, membrane associated and membrane recruitment of fluorescence was done essentially as detailed elsewhere [8,20].
We used fluorescent-α-factor tagged with HiLyte488 (kindly provided by D.G. Drubin [42]) to measure receptor abundance over time. We incubated cells with 0 or 50 nm of unlabeled α-factor for 2 hours and then treated with inhibitor medium (SC medium containing 10 mm fluoride (NaF or KF) and 10 mm azide (NaN 3 )) for 30 min. Then, we washed cells by centrifugation 4 times in inhibitor medium to dissociate unlabeled α-factor, in order to reveal Ste2 binding sites by fluorescent-α-factor. Given the slow unbinding rate of unlabeled pheromone [27], we performed controls to verify that this protocol effectively removed the unlabeled α-factor. Due to the slow binding dynamics of the fluorescent pheromone ( Figure EV1C and [46]), we incubated for at least 3 hours at 30 o C to reach binding equilibrium. To correct for the autofluorescence of dead cells in the fluorescentpheromone channel, we subtracted the signal from cells treated identically but without addition of labeled-alpha-factor.
We did halo assays in the following manner: 100 µl of exponential cell cultures (OD 600 =0.1 or ≈ 3 × 10 5 cells) were spread on agar plates of adequate medium. A sterile Whatman filter paper disk was placed on the surface of the agar and 5 µl of α-factor (0.1 nmol, 170 ng) was then placed on the disk. Plates were incubated at 30 o C for 2-4 days before imaging.

Statistical methods
In Figures 2G, 6A and 6B, to account for day-to-day differences in α-factor preparations, we normalized pheromone concentration as follows. For each strain and experiment, we fitted the data to a Hill function and then divided the α-factor concentrations by the correspondent EC 50 . The resulting DoR were therefore centered at 1. In order to recapture the sensitivity of each strain, we re-scaled the α-factor concentration using the average EC50 of each strain among the different experiments. Same measurements for Bmh2-YFP, a protein not related to the pheromone pathway. In both cases WT strain ACL379 was included as a autofluorescence control. Note the different y-scales.
As shown in Figure EV1B, the level of Ste2 GPCR is greatly reduced in the WT strain grown in medium with Galactose and Raffinose (SC-GalRaf), reaching levels 70% lower than the same strain grown in medium with Glucose (SC-Glu). Because of this unexpected result, we wondered if the abundance of other components of the pathway was also affected by the carbon source of the medium.
To this end, we tagged with YFP the endogenous copy of different components of the pheromone response pathway. Due to the low abundance of Ste5 [39], we used a strain in which STE5 was tagged with 3 YFP in tandem to increase the fluorescence signal. We grew these strains in either SC-Glu or SC-GalRaf overnight, and acquired epifluorescence images as described in the methods section.
Although there are significant differences between the abundances of some of the components of the pathway when grown in SC-Glu or SC-GalRaf (Table S5), the relative difference is much less than in the case of Ste2 GPCR . Interestingly Bmh2, a protein not related to the pheromone pathway, shows a sugar-dependent change in its abundance much greater than the observed effect for components of the pathway. It is also noteworthy that Sst2 RGS shows the strongest YFP signal among the pathway components, suggesting in particular that it is more abundant than Ste4 Gβ .

Strain
Genotype  Table S5. Corrected fluorescence values, ratios and significance for the data shown in Figure S1.

Model definition
The carousel model of G-protein activation results from the combination of the ternary complex model [13] and the G-protein activation cycle ( Figures 1D and 1E of main text).
The resulting model has many species and parameters, but can be conveniently represented in a 3D scheme as shown in Figure 3A of the main text, that evidences the symmetries in the model. The axial reactions (violet arrows in Figure 3A), represent binding and unbinding of the ligand L with the receptor R. The on-rate of these reactions are given by the k · on parameters, were the square superscripts denote the binding partners. For example k L·R on is the on-rate for free receptor with the ligand (not explicitly represented in the scheme), while k L·RG on 1 is the on-rate for the G-protein coupled receptor with the ligand. The offrates for the ligand-receptor complexes are written in the same notation. For example, k L·RGt of f is the off -rate of ligand from the receptor coupled to Gα GT P 2 . The radial reactions (blue arrows in Figure 3A) represent the coupling between the different states of the Gα subunit and the receptor. These reactions are also expressed in terms of on-rates and off -rates using the same notation as before. For example, the coupling between free receptor and Gα GDP is given by k R·Gd on .
The angular reactions represent progression through the G-protein activation cycle. The green arrows represent the exchange of GDP for GTP in the Gα subunit. The rates for these reactions are given by the k Ef (Exchange forward) parameters, where the square superscript indicates the molecular complex the Gα subunit is part of. Note that any Gα bound to GDP can undergo the exchange reaction, which includes the heterotrimeric Gαβγ protein and dissociated Gα GDP , either uncoupled from receptor, coupled to free receptor or coupled to ligand-occupied receptor.
Nucleotide exchange is not a simple reaction but the result of several elemental reactions; release of GDP from Gα GDP , binding of GTP to the free binding site of Gα, transition of Gα GTP to a conformation with low affinity for Gβγ, and dissociation of Gβγ from Gα GTP . Of these reactions, the dissociation of GDP from Gα has been reported to be the rate-limiting step in some systems [26,32]. All reactions are in principle reversible, so a reverse exchange were GTP is replaced by GDP is theoretically possible. We decided to neglect this reverse reaction as it is very unfavorable given the large [GTP]/[GDP] ratio in metabolically active cells [43]. Therefore, we assume that guanine nucleotide exchange in Gα is adequately modeled by a irreversible reaction.
The red arrows in Figure 3A represent the hydrolysis of the gamma phosphate of GTP bound to Gα. This results in the conversion of Gα GTP to Gα GDP and inorganic phosphate. The rates of these reactions are given by k Hf (Hydrolysis forward), where the square superscript indicates the complex Gα GTP is part of. For example, k LRGt Hf is the hydrolysis rate of Gα GTP coupled to ligand occupied receptor. We assume this reaction to be irreversible due to the large amount of free energy released during GTP hydrolysis. Note that in this model we assume that the concentrations of GTP, GDP and inorganic phosphate are not significantly influenced by the G-protein activation cycle and remain constant, therefore they are implicitly considered in the kinetic parameters.
The orange arrows in the carousel scheme represent the association between Gβγ and Gα GDP . These reactions are modeled as reversible associations with rates k Af and k Ar (Association forward and reverse), where the superindexes indicate the partners Gα GDP is bound to. For example k RGd Af is the rate for the association between Gβγ and Gα GDP coupled to unoccupied receptor.
With these kinetic rates and following the laws of mass action, we defined a system of ordinary differential equations (ODEs) that describes the evolution of every specie in the model (Equations 1 to 12). For ease of reading, we grouped in parenthesis the forward and reverse terms of reversible reactions, and separated in lines terms that correspond to different reactions.
From these equations, we can see there are some conservations in the model, namely, the total amounts of R, Gα and Gβγ are conserved (equations 13 to 15). These conservations reduce the dimensionality of the ODE system from 12 to 9. The ligand concentration L is assumed to be constant, and equal to the applied ligand (i.e. no ligand depletion). Gβγ The complete carousel model has 38 parameters. Taking into account thermodynamic micro-reversibility considerations, we can impose a relation between parameters for every independent close loop of reversible reactions [51]. These conditions ensure that if there are two reversible routes from one specie to an other, both have the same equilibrium constant or, in other words, that there is no net tendency for the molecules to circulate in a preferential direction. Taking into account these relationships reduces the number of parameters to 33 and makes the model sounder and more biologically realistic.
Using a notation inspired by the Cubic Ternary Complex model [49], we encoded the micro-reversibility considerations using adimensional parameters (denoted by Greek letters) that relate the equilibrium constants for different reactions (equations 16 to 28) .

Symmetry assumptions: the simplified carousel model
As a staring point, and to simplify the analysis, we decided to study a simplified version of the carousel model, in which me make several symmetry assumptions: 1) Binding of ligand to receptor is independent on whether the receptor is coupled or not to Gα, and to the state of this subunit (Gα GTP , Gα GDP or Gαβγ).
Note that equation 29 defines equal equilibrium constants, while equation 30 defines equal kinetic rates.
This assumption is supported by experiments in yeast, in which deletion of GPA1 Gα shows very small changes in affinity, from K αF·ST E2 d = 3.4 nm in WT cells to 4.8 nm in ∆gpa1 Gα cells [3]. On the other hand, experiments with membrane preparations do show a effect on K αF·Ste2 d , depending on the presence of GTP-γ-S (supposed to lock Gα in the active state), but only in non-physiological conditions (high ionic strength and pH) [6].
In other GPCR systems, there is strong evidence for an effect of G-protein coupling to the receptor on the receptor-ligand affinity. In fact, this was the original motivation for the Ternary Complex Model [13]. We decided to do the current assumption because it appears to be the case in the yeast pheromone pathway [3], and because it greatly simplifies the analysis of the model.
2) Coupling between Gα and the receptor does not depend on the state of Gα.
Inmuno-precipitation experiments show that somewhat less Gpa1 Gα co-precipitates with Ste2 GPCR after treatment with saturating α-factor than with no pheromone [50]. This suggests some degree of reduction in affinity between the receptor and Gα when the pathway is active, and therefore more molecules are in the Gα GTP state. Nevertheless, this loss of affinity is partial and hard to estimate from the available data. The co-precipitation experiment gives no information on the kinetic rates of these interactions. In order to reduce the number of parameters of the model, we made the assumption described in equations 31 and 32, even though these might not be exact in the case of the pheromone pathway.
3) Association between Gα GDP and Gβγ does not depend on whether Gα is coupled or not to the receptor, or the state of the receptor. ρ = 1 and λ = λ d (33) k Gd Ar = k RGd Ar = k LRGd Ar (34) Note that the assumption for the equilibrium constants (equation 33) is implied by the previous two assumptions (equations 29 and 31) and the micro-reversibility conditions.
4) The exchange rate of GDP for GTP on Gα GDP is not affected by Gβγ binding.
Dissociation of GDP from rabbit Gsα and Goα (the rate limiting step in the exchange reaction), is slower in the presence of Gβγ [7], but this effect disappears as Mg 2+ concentrations increase [26]. Given the [Mg 2+ ] of yeast cytoplasm [21], this suggests that Gβγ should have only a small effect on GDP exchange in physiological conditions (see discussion in [41] and YeastPheromoneModel.org).
5) The hydrolysis rate of GTP by Gα GTP coupled to receptor, does not depend on whether the receptor is bound to ligand or not.
Note that this does not imply that the rate of GTP hydrolysis by free (receptor uncoupled) Gα GTP should be the same as for receptor-coupled Gα GTP (see below). To our knowledge there is no mechanism that could explain a change in GTP hydrolysis rate due to the ligand-occupation state of the receptor.
6) There are equimolar amounts of Gα and Gβγ.
Measurements of pathway components abundance by quantitative western blot resulted in an estimation of 2390 ± 262 molecules cell −1 for Gpa1 Gα and 2045 ± 107 molecules cell −1 for Ste4 Gβ [39], which are consistent with equimolar amounts of these components.
These assumptions have different degrees of experimental support for the yeast pheromone pathway. Regardless if they hold exactly for this system, we believe they constitute a reasonable simplification of the carousel model that facilitates the analysis, while allowing for interesting behaviors. Further relaxations of these assumptions could result in new and interesting features, but their analysis goes beyond the scope of this work.

Parameters of the simplified carousel model
These symmetry assumptions result in a simplified carousel model, with 13 free parameters to determine. Seven of these parameters have been determined for the yeast's pheromone response. These are the abundances of the components of the system, the on and off rates of α-factor with the receptor, the basal exchange rate of GDP for GTP in Gα GDP , the basal hydrolysis rate of GTP in Gα GTP and the increase in this rate induced by Sst2 RGS (see Table 1 in main text).
Other parameters have been measured in other GPCR signaling systems and can be considered as a good estimation of the correspondent parameters in the pheromone response pathway. These are the coupling and uncoupling rate of GPCRs with G proteins [1,2,23], and the exchange rate of GDP for GTP in Gα GDP stimulated by the GEF activity of occupied receptors [5,32]. A thorough review of the evidence for these rates can be found in Ty Thomson's PhD thesis [41] and in the YeastPheromoneModel.org [40] site.
There are three parameters for which (to our knowledge) there are no experimental estimations available in any GPCR system. We estimated the values of these parameters as follows.
We estimated K Gd·Gβγ d based on our experimental measurement of the affinity between Ste5 and Gβγ [8]. To do this, we first considered the closed cycle of reversible reactions between Gα GDP , Gβγ and Ste5, shown in equation 38.
Here, K 1 to K 4 represent the correspondent association constants. Because of microreversibility considerations [51], these constants have to satisfy the following equation.
K 1 is the reciprocal of the dissociation constant between Ste5 and free Gβγ, a parameter for which we have experimentally determined an upper bound of 1 K 1 = K Gβγ·Ste5 d < 0.65 nm in a previous work [8]. K 3 is the reciprocal of K Gd·Gβγ d , the value we attempt to estimate. We expect K 3 to be relatively large, because Gα GDP tightly binds Gβγ [11]. On the other hand, Ste5 cannot bind the heterotrimeric G protein (only free Gβγ), therefore we assume K 4 ≈ 0. In consequence, as follows from equation 39, the affinity constant between Gα GDP and the complex between Gβγ and Ste5 is negligible (K 2 ≈ 0). In other words, Ste5 and Gα GDP bind to Gβγ in a mutually exclusive manner. In unstimulated cells, where almost all Gα is in the GDP form, Gα GDP has to outcompete Ste5 for Gβγ. Otherwise the pathway would be active in the absence of pheromone. This implies that On the other hand, the strongest protein-protein interactions reported in heterodimeric complexes have dissociation constants in the pm range [9]. Therefore, we decided to use a estimated value of K Gd·Gβγ d = 0.01 nm, which is much lower (higher affinity) than our upper bound for the Ste5 and Gβγ dissociation constant, but well within the physiological range.
We chose the association rate k Gd Af such that the hydrolysis of GTP, and not the association of Gα with Gβγ, is the limiting step in the deactivation of G-protein.
Finally, we set the value of the exchange rate for G protein coupled to unoccupied receptor to the same value as for uncoupled G protein (k RG Ef = k G Ef ).
All the components of the carousel model are permanently associated to the plasma membrane. The ODEs that define the model (Equations 1 to 12) are expressed in terms of concentrations of the different species, therefore we need to define a reaction volume. We considered a perimembrane space spanning ∆h = 50 nm across the plasma membrane, that contains all the components of the model. We defined the perimembrane volume as V perimembrane = 4πR 2 cell ∆h = 3.92 fl, where R cell = 2.5 µm. ODE models of reaction networks assume well-mixed compartments, an assumption unlikely to hold due to the relatively slow diffusion of membrane associated components and the geometry of the perimembrane volume. Another complication arises from the fact that at high doses of α-factor the signaling proteins concentrate at the shmoo tip, in a much smaller volume, within the first hour of stimulation (see Figure S4 in [46]). We decided to assume a well-mixed compartment of constant size as a starting point in our work, as this greatly simplifies the analysis of the model. The qualitative conclusions obtained do not depend strongly on the reaction volume used in the model, therefore one can interpret the model as the interactions that take place in a membrane microdomain, which can be considered a well-mixed compartment.
Calibrated western-blot estimations of the abundance of Ste2 GPCR receptors result in 6622 ± 361 molecules cell −1 [39]. Ste2 GPCR receptors have been shown to form dimmers in vivo [33], therefore we estimated around 3300 dimmers cell −1 . Although we assumed monomeric receptors in our model, we used the abundance of dimmers to estimate R tot . The rationale for this is that the model mainly focuses in the coupling between receptors and G proteins, and we believe that the number of coupling partners is better estimated in this manner. As described in the main text, the model shows robustness to receptor abundance, so the exact number used for R tot has no influence on the conclusions of this work.
The regulator of G-protein signaling (RGS) is not explicitly represented in the carousel model. Nevertheless, its effect can be captured by the parameters of the model, specifically in the GTP hydrolysis rates. In the pheromone response pathway of yeast, Sst2 RGS physically interacts with the cytoplasmic C-termial domain of Ste2 GPCR , and this interaction is required for Sst2's activity on Gα [4]. The effect and localization of the RGS can be captured in the parameters of the model, if the GTP hydrolysis rate is much greater for Gα coupled to receptor than for uncoupled Gα (equation 40). Note that this equation is satisfied by the parameter in Table 1

Extended carousel model: Gβγ-Ste5 interaction
To asses the effect of downstream components on the dynamic of the G-protein cycle, we extended the carousel model to include the interaction between free Gβγ and the scaffold protein Ste5. This interaction mediates the recruitment of Ste5 to the membrane and is required for signal transmission in the yeast pheromone system [34].
This extension adds four new parameter to the model; the total amount of scaffold protein (Ste5 tot ), the off -rate between Gβγ and Ste5 (k Note that, as discussed in section 3.3, we are assuming that Ste5 and Gα bind in a mutually exclusive manner to Gβγ.
• Like most components of the pathway, the abundance of Ste5 has been measured by quantitative western blot, resulting in Ste5 tot = 480 ± 60 molecules/cell [39]. Note that this is one of the components in lowest abundance.
• We estimated k Gβγ·Ste5 of f based on a FRAP experiment [45]: cells expressing Ste5-GFP were stimulated with α-factor 3 µm for 90 minutes. In these conditions Ste5 localized to the shmoo tip, where it was photobleached. The fluorescence at the shmoo tip recovered in a half time of τ 1/2 = 8.2 ± 1.3 s [45]. Because cytoplasmic diffusion is much faster than the binding reaction, this recovery time is mainly determined by the off -rate between Ste5 and Gβγ, which we can estimate as k Gβγ·Ste5 • In a previous work we have experimentally measured the affinity between Ste5 and Gβγ [8]. We found that the affinity is modulated during the response, starting with a high "peak" affinity (low dissociation constant) of K Gβγ·Ste5 d,peak < 0.65nM , which then goes to a "steady-state" value of K Gβγ·Ste5 d,ss = 17 ± 9nM , within the first five minutes of the response. This modulation of the affinity results in a peak and decline   • As Ste5 resides in the cytoplasm before it is recruited to the membrane, we had to defined the volume of the cytoplasmic compartment V cyt . Note that the relevant volume is the one accessible to Ste5. We estimated this volume to be 36.4 fl based on the soft X-ray tomography data presented in [44].
We wanted to study the extent to which Ste5 can sequester Gβγ form Gα in an unstimulated system, based on our estimation of the model's parameters. To do this we simulated the basal levels of active G protein (i.e. Gβγ dissociated from Gα, either free or bound to Ste5) and membrane recruited Ste5, for different total abundances of Ste5.
As can be observed in Figure S2 there are around 4 active G proteins (0.2% of the total 2,042 G proteins simulated) in the absence of Ste5. When the simulation was run using the steady-state affinity K Gβγ·Ste5 d,ss = 17nM , we found a very small increase in the number of active G proteins, reaching 8 molecules (0.4%) for a tenfold over-expression of Ste5. On the other hand, when using the peak affinity K Gβγ·Ste5 d,peak = 0.6nM , the effect is somewhat stronger; 12 G-protein molecules (0.6%) are active with wt abundance of Ste5 and 34 G-proteins (1.7%) are activated with a tenfold increase in the amount of Ste5.
For steady-state simulations as the ones used in this work, the most reasonable affinity value to use is K Gβγ·Ste5 d,ss = 17nM . In any case, based on the value used for the model's parameters, the sequestration effect of Ste5 on Gβγ appears to affect a very small percentage of the total G-protein. Therefore, for the rest of this work we neglected the effect of Ste5 on the activation of the G protein.
We do not have the quantitative data required to do a similar analysis with other binding partners of Gβγ, like Far1 or Ste20. We therefore make the reasonable assumption that they also have a negligible effect on the activation of the G protein, as the model suggest for Ste5. It is interesting to note that there is four times more G protein than Ste5 molecules [39]. Therefore Ste5 cannot titrate free Gβγ at high pheromone doses, leaving sites available for the interaction with other partners as Far1.

Extended carousel model: Receptor-RGS interaction
Due to the importance of the receptor-RGS interaction suggested by our previous results, we decided to explicitly include the RGS component in our model. To express all the possible states of the receptor in a convenient way, we introduce the following notation, where x can be either nothing or L, y can be nothing, G, Gd or Gt (representing Gαβγ, Gα GDP or Gα GTP , respectively) and z can be nothing or RGS. Therefore, this extended model has 2 × 4 × 2 = 16 species representing different states of the receptor, 4 species for the free G protein (Gαβγ, Gα GDP , Gα GTP and Gβγ) and one specie representing free cytoplasmic RGS. As can be seen, adding explicitly the RGS protein and its interaction with the receptor greatly increases the complexity of the model, going from 12 to 21 species, from 35 to 81 reactions and from 38 to 85 total parameters ( Figure EV3A). This is an example of combinatorial explosion in the complexity of a model.
In order to keep the model tractable, we did additional symmetry or independence assumptions (the numeration is correlative with the assumptions in section 3.2).

7) Binding affinity and kinetics between ligand and receptor is not influenced by the
binding of the RGS to the receptor.
Several studies show that STE2 GPCR alleles with no C-terminal domain have the same affinity for α-factor than wt receptors [15,28,35]. As Sst2 RGS binds to the Cterminal domain of Ste2 GPCR , this suggests there is no effect of the RGS on ligand binding affinity. Note that combining this assumption with symmetry assumption 1 results in the same affinity and kinetics of ligand binding for all receptor species (i.e. all violet "axial" arrows in Figure EV3A).

8) Coupling between Gα and the receptor is not influenced by the receptor-RGS interaction.
k xR·y of f = k Two studies propose that Sst2 RGS increases the affinity between Ste2 GPCR and Gpa1 Gα [29,36]. In the first study the authors found that the pheromone induced cell-cycle arrest is robust to receptor abundance in wt cells, but not in ∆sst2 RGS cells [36]. The second study shows that Sst2 RGS is required for the dominant-negative effect of loss-of-function mutations in STE2 GPCR [29]. Their interpretation of these observations was that the receptor pre-couples with the G protein (forming a stable complex), and that Sst2 RGS is required for this interaction. However, Weiner et al. specifically looked for, but were unable to find any evidence that Sst2 RGS modulates the affinity between receptor and G protein [48]. In this work we propose an alternative explanation for the mentioned observations, namely, that unoccupied receptors inactivate G proteins due to their interaction with Sst2 RGS . In this view it is not necessary to postulate any modulation of the receptor-G protein affinity due to Sst2 RGS .
This assumption combined with assumptions 2 results in the same rates for coupling between Gα (in any state) to the receptor (ligand-occupied or not). In other words, all blue "radial" arrows in Figure EV3A are equivalent.
9) Association between the receptor-Gα GDP complex and Gβγ, does not depend on whether the RGS is bound to the receptor or not.
k xRGd·Gβγ To our knowledge there is no evidence for a change in the rate of re-association of the G protein due to the interaction between the Sst2 RGS and Ste2 GPCR . Taken together with symmetry assumption 3, this implies that all orange "angular" reactions in Figure EV3A are equivalent.
10) The receptor induced exchange rate of GDP for GTP in Gα is not influenced by binding of the RGS to the receptor.
This assumption indicates that the correspondent green "angular" arrows in the left and right carousel schemes of Figure EV3A are equivalent.
11) The interaction between the receptor and the RGS is not influenced by the sate of the receptor.
In the work by Ballon et al. where the DEP domains of Sst2 RGS were first described, the authors showed that the affinity between Sst2 RGS and Ste2 GPCR is reduced when the C-terminal domain of the receptor is phosphorylated by the kinases Yck1 and Yck2 [4]. Because phosphorylation of the receptor by Yck1/2 is promoted by pheromone binding to the receptor [25], this suggests that the affinity of Sst2 RGS for pheromone bound receptors should be lower than for unoccupied receptors (K LRy·RGS d > K Ry·RGS d for y ∈ {∅, G, Gd, Gt}). This can have interesting consequences, as ligand-occupied receptors could have reduced GAP activity besides their increased GEF activity, therefore enhancing the functional antagonism with unoccupied receptors. Furthermore, this could explain different behaviors at different receptor occupancies that correspond to high and low pheromone doses.
Despite this, as we are not modeling the receptor's phosphorylation and turnover explicitly, and in order to keep the extended carousel model as simple as possible, we decided to do the current symmetry assumption.
In terms of the scheme of Figure EV3A this means that the horizontal black arrows for all receptor species are equivalent.
The fundamental difference between RGS bound receptors and receptors with no RGS is in the GTP hydrolysis rates of the Gα subunits coupled to them. The GAP activity, and therefore the enhanced hydrolysis rate, is only present on the complexes with RGS (compare red "angular" arrows in Figure EV3A). This condition is mathematically defined by equation 52.
Taking into account the additional symmetry assumptions (7 to 11) the extended carousel model has 5 new parameters to estimate: • RGS tot , the total amount RGS protein. The experimental estimations for the abundance of Sst2 RGS in unstimulated cells ranges from 2000 molecules/cell in [22], to 6000 molecules/cell in [17]. Furthermore, this protein is induced during the response [22], something that our model can not capture in its current form. We decided to use the value of 6000 molecules/cell, roughly 3 times the value used for G tot (2045 ± 107 molecules/cell [39]), as this is consistent with our estimation based on YFP tagged proteins for Sst2 RGS and Ste4 Gβ (table S5 and Figure S1).
• V cyt , the cytoplasmic volume accessible to free RGS. As before (section 3.4) we estimated this volume to be 36.4 fl [44].
• K R·RGS d , the dissociation constant between the RGS and the receptor. To estimate this affinity we noted that in unstimulated cells Sst2 RGS appears to be mainly in the cytoplasm [4]. To assign a numeric value to K R·RGS d , we solved the amount of RGS in complex with receptors in steady state, which is given by equation 53.
Here xRy ss RGS represents the sum of all species of receptors with RGS at steady state.
Using the estimated values for R tot , RGS tot and V cyt , and assuming that only 20% of Sst2 RGS is bound to Ste2 GPCR in unstimulated cells (xRy ss RGS = 0.2 × RGS tot ), we estimated a value of K R·RGS d = 383 nm.
• k R·RGS of f , the off -rate between the RGS and the receptor. To our knowledge there is no experimental estimation of this value. We reasoned that if the off -rate between receptor and RGS is much faster than the off -rate between receptor and G protein (k R·RGS of f k R·G of f ), then the time averaged effect of the RGS would be homogeneous among receptors. Therefore, we used the value k R·RGS of f = 3 s −1 (roughly 30 × k R·G of f ).
• k xRGt RGS Hf , the hydrolysis rate of Gα GTP bound to receptor with RGS. Based on equation 53 and the estimated values of R tot , RGS tot , V cyt and K R·RGS d , we estimate that only 37% of the receptors are bound to RGS at any given instant. Equivalently, by ergodicity, a given receptor is bound to RGS only 37% of the time. Therefore, to maintain the same "effective" time-averaged GAP activity used in the simplified carousel model for wt abundances of the components (section 3.

Restriction analysis of the simplified carousel model
To draw conclusions that do not depend on the precise value of the reference parameters ( Table 1 in main text), we analyzed the behavior of the carousel model in a large region of parameter space. We varied each parameter 4 orders of magnitude to each side of its reference value, in logarithmic scale. To explore parameter space efficiently we used Latin Hypercube Sampling (LHS) [31]. For each point p in the LHS sample, we numerically simulated a steady-state DoR curve for different receptor abundances, ranging from 30 to 3 × 10 5 receptors per cell. For each DoR curve we fitted a Hill function (54), and saved the parameters of these fitted curves, namely Basal, Amplitude, EC 50 and n (the Hill coefficient).
Next, we defined different sets, based on the behavior of the DoR curves. The criteria for each set are defined in equations 55 to 59, where the wt super index denotes the value of the parameters at WT receptor abundance (R tot = 3300 molecules cell −1 ) [39], and max and min are taken over the different receptor abundances. The amount of total G protein was kept constant in the simulations to G tot = 2040 molecules cell −1 [39] and the ligand-receptor dissociation constant was set to one (K L·R d = 1), which is equivalent to adimensionalize the extracellular ligand concentration.
The set S wa (55) corresponds to points in parameter space that have wt amplitude; DoR curves with low Basal and high Amplitude at WT receptor abundance, and therefore are capable of responding to the ligand L. S re (56) are the points with robust EC 50 , defined as the intersection between S wa and the points that maintain a fairly constant EC 50 at all tested receptor abundances. S ra (57) are the points with robust Amplitude, defined as the intersection between S wa and the points that maintain a fairly constant Amplitude at all tested receptor abundances. S rr (58) is the intersection between S re and S ra , that is, the points with a robust response. These points show robust sensitivity and robust amplitude, and therefore the overall response is independent on variations in receptor abundance. Finally, S rd (59) are the points that show robust DoR alignment, and is defined as the intersection between S rr and the points that show a EC 50 almost equal to the dissociation constant of the receptor K R·L D . These points show both robustness to receptor abundance and dose-response alignment (DoRA).
To visualize the distribution of the points in each set, we plotted a matrix of 2D histograms, as shown in Figure S3 for the points in S wa . The panels on the diagonal show the histogram, for the parameters indicated in the top and right margins. Each panel in the upper left triangle shows the 2D histograms for the points in S wa , projected on the plane defined by the parameters labeled on the top (x) and right (y) margins. The frequency of points in each bin is represented by the yellow-red color scale.
The panels on the bottom right triangle show the difference between the frequency of points in S wa that fall in a bin, and the frequency expected assuming independent distributions for the parameters. This null hypothesis distributions (H 0 ), is constructed by combining the marginal distributions represented by the histograms on the diagonal. The difference between the frequency in S wa and H 0 is represented by the blue-red color scale. To determine which panels show a distributions significantly different from H 0 , we performed Monte Carlo simulations (N = 10 5 ) using the Cramér-von Mises statistic (ω 2 = (Freq−H 0 ) 2 ), and determined significance using a Bonferroni corrected α. Panels with significant structure are indicated with an asterisk. Note that correspondent panels in the upper an lower triangles of the matrix are labeled with the same number on the top left corner of the panel, and that all parameters are plotted in log 10 scale. Many panels of Figure S3 show significant structure (see Table S7 for details). In some of these panels, there are regions almost or totally devoid of points, as in panels R45, R47, R48, R57 and R89. In these cases we say there is a restriction between the two parameters that define the panel. For instance, all points in S wa fall on top of the line defined by log 10 (k LRG Ef ) = 1.7 + log 10 (k RG Ef ), as shown in panel R45 of Figure S3. This means that points in S wa satisfy the restriction k RG Ef R45 k LRG Ef 3 . In other words, DoR curves with high amplitude and low basal are obtained only when the GEF activity is much higher in ligand-occupied receptors than in unoccupied receptors, which makes intuitive sense. Another restriction can be observed in panel R89 of Figure S3, which involves parameters k LRGd Af 4 and k Gd Ar . These two parameters have different units, s −1 nM −1 and s −1 respectively. In order to make them comparable, we express the restriction in terms of G tot , the total number of G proteins, kept fixed in the simulations. Doing this we obtain

R89
G tot , which implies complete re-association of Gβγ with Gα, when Gα is in the GDP state.
Because all points in S wa satisfy these two restrictions, we call them necessary restrictions and represent them with solid lines in Figure S3. On the other hand, there are panels in which we can define soft restrictions. In these panels there are regions almost, but not totally, devoid of points (e.g. R47, R48 and R57 in Figure S3). We define soft restrictions such that 95% of the points satisfy them, and represent them with dashed lines.
There are two soft restrictions on k LRGt Hf 5 shown in panels R47 and R57 of Figure S3; In other words, the GTP hydrolysis rate has to be slower than the GEF activity of occupied receptor, but higher than the GEF rate of unoccupied receptor. Combining R47 and R57 we get k RG Ef < k LRG Ef , which is in line with R45 (k RG Ef R45 k LRG Ef ). Consistent with this, for p ∈ LHS, P (R45|R47∧R57) = 0.99 , which means that of the points that satisfy both R47 and R57, 99% also satisfy R45. This kind of relationships between restrictions are shown in Table S8. k LRGt Hf is the only parameter that has upper and lower restrictions for points in S wa , and this is reflected in the 1D histogram for this parameter, which is the only one that shows a maximum at intermediate values. In Figure S4 we show the matrix of 2D histograms for the points in S re , that is, the points that show robust EC 50 and respond to ligand at wt receptor abundance (belong to S wa ). The restrictions defined in Figure S3 are satisfied for points in S re (blue lines in Figure S4), as expected by the fact that S re ⊂ S wa . There are also new restrictions for S re , shown by orange lines in Figure S4, Figure S5 we show the matrix of 2D histograms for the points with robust amplitude, that is for S ra . As before, the restrictions for S wa are satisfied, as expected based on the relation S ra ⊂ S wa . There are more restrictions for S ra than for S re , which is consistent with fact that there are four times more points in S re than in S ra (Table S7). The new restrictions are shown in violet in Figure S5. These restrictions involve several parameters, but can be summarized as low k G Ef (R13, R23, R35, R37, Table S7) and low k Gt Hf (R16, R26, R56, Table S7). In line with this, the only new necessary restriction is k R·G of f R13 k G Ef , i.e. that the nucleotide exchange of uncoupled Gα has to take much longer than the life time of the R · G complex. In other words, to maintain a robust amplitude, the probability of uncoupled Gα to change state has to be low.  Figure S6 we show the points in S rr = S re ∩ S ra , that is, the points that have robust amplitude and robust EC 50 , and therefore show a robust overall response. We include in Figure S6 the restrictions defined for S wa , S re , S ra and the new restrictions that appear for S rr (red lines). It is interesting to note that the restrictions for S wa and S ra are satisfied, but the restrictions for S re are not (see R18, R17, R15 and R12 in Figure S6 and in Table  S7). These were soft restrictions, therefore the 5% that did not satisfied them in S re are over-represented in S rr . This means that low k R·G of f (which results in pre-coupling) is not necessarily required to have a robust response.
There are only two new restrictions for S rr that do not appear in the previous sets. One is that k LRGt Hf R78 < G tot k LRGd Af , which means that the hydrolysis has to be the rate limiting step in the reassociation of the G protein.
The other new restriction (k Gt Hf R67 k LRGt Hf ) is a necessary restriction and requires that the hydrolysis rate of uncoupled Gα should be much slower than the hydrolysis of receptor coupled Gα. Note that this is exactly what we expect in systems in which the RGS physically interacts with the receptor, as this will increase the hydrolysis only when Gα is coupled to the receptor. Due to this, we call R67 the localized RGS restriction. In Figure S7 we show the points in S rd , that show both a robust response and Dose Response Alignment (DoRA). Interestingly, almost all of the previous restrictions are satisfied for this set, including restrictions for S re like R17. See also Figure 8 and main text for more details.  Table S6 and G tot = 2042 molecules cell −1 . Columns LHS, S wa , S re , S ra , S rr , S rd show the fraction of points in each of these sets that satisfy the restrictions. The asterisk indicates significant difference from the null hypothesis obtained assuming independent distributions of the X and Y parameters, calculated using Monte Carlo simulations on the Cramér-von Mises statistic, and a family-wise error rate of α = 0.05 (Bonferroni correction).  Table S8. Relationships between different restrictions. (1) Restriction label as shown in Table S7. (2) Conditional probability of the restriction, for points in LHS.