Molecular basis of signaling specificity between GIRK channels and GPCRs

Stimulated muscarinic acetylcholine receptors (M2Rs) release Gβγ subunits, which slow heart rate by activating a G protein-gated K+ channel (GIRK). Stimulated β2 adrenergic receptors (β2ARs) also release Gβγ subunits, but GIRK is not activated. This study addresses the mechanism underlying this specificity of GIRK activation by M2Rs. K+ currents and bioluminescence resonance energy transfer between labelled G proteins and GIRK show that M2Rs catalyze Gβγ subunit release at higher rates than β2ARs, generating higher Gβγ concentrations that activate GIRK and regulate other targets of Gβγ. The higher rate of Gβγ release is attributable to a faster G protein coupled receptor – G protein trimer association rate in M2R compared to β2AR. Thus, a rate difference in a single kinetic step accounts for specificity.


Introduction
Heart rate is tightly regulated by the combined effects of the sympathetic and parasympathetic branches of the autonomic nervous system. These two branches control heart rate by stimulating different G protein-coupled receptors (GPCRs), which in turn activate ion channels that modify the electrical properties of cardiac pacemaker cells (DiFrancesco, 1993). Sympathetic stimulation accelerates heart rate through activation of beta-adrenergic receptors (bARs) and the stimulatory G protein (Ga s ) pathway, while parasympathetic stimulation slows heart rate through activation of the muscarinic acetylcholine receptor M 2 (M2Rs) of the inhibitory G protein (Ga i ) pathway (Brodde and Michel, 1999;Gordan et al., 2015).
G protein-activated inward rectifier K + (GIRK) channels are targeted by the parasympathetic nervous system (Loewi, 1921;Irisawa et al., 1993;Schmitt et al., 2014). Upon stimulation, acetylcholine (ACh) released from the vagus nerve binds to and activates M2Rs in sinoatrial node (SAN) pacemaker cells, promoting the engagement of the GDP-bound G protein trimer (Ga i (GDP)bg). The activated receptor catalyzes removal of GDP from the G protein alpha subunit (Ga i ), which allows intracellular GTP to bind. The GTP-bound Ga (Ga i (GTP)) and the G protein beta-gamma subunit (Gbg) then dissociate from the receptor and from each other ( Figure 1A) (Hilger et al., 2018). The Gbg subunit, now free to diffuse on the intracellular membrane surface (attached by a lipid anchor), binds to GIRK and causes it to open (Sakmann et al., 1983;Soejima and Noma, 1984;Logothetis et al., 1987;Wickman et al., 1994;Krapivinsky et al., 1995). Open GIRK channels hyperpolarize the cell membrane and thus lengthen the interval between cardiac action potentials (i.e. slow the heart rate) (DiFrancesco, 1993;Schmitt et al., 2014). The process is reversed by the alpha subunit, which hydrolyses GTP to GDP followed by reformation of the Ga i (GDP)bg complex.
Sympathetic stimulation of bAR speeds heart rate by opening excitatory ion channels through the Ga s pathway (DiFrancesco and Tortora, 1991;Simonds, 1999). Important to balanced opposing effects of sympathetic and parasympathetic input, bAR stimulation does not open GIRK channels even though Gbg subunits are released by this receptor (Hein et al., 2006;Digby et al., 2008). The reason why GIRK channel opening is specific to Ga i -coupled GPCR stimulation and not to Ga s -coupled GPCR stimulation has remained a long-standing unsolved puzzle, which we refer to as the Gbg Figure 1. Gbg specificity between GPCRs and GIRK channels. (A) A schematic representation of GPCR signal transduction and GIRK channel activation. Agonist binding promotes the formation of a GPCR-Ga(GDP)bg complex. The activated GPCR then triggers the exchange of GDP to GTP on the Ga subunit. Ga(GTP) and Gbg subunits subsequently dissociate from the GPCR. Dissociated Gbg directly binds to and activates GIRK channels. Dissociated Ga(GTP) hydrolyzes GTP to GDP, which then reassociates with Gbg to form Ga(GDP)bg. (B) A representative current-clamp recording of spontaneous action potentials from an acutely isolated murine sinoatrial node (SAN) cell. 1 mM isoproterenol (Iso) or acetylcholine (ACh) was applied as indicated. (C) A representative voltage-clamp recording from the same SAN cell in (B). The membrane potential was held at À80 mV, and 1 mM Iso or ACh was applied as indicated. (D) Representative voltage-clamp recordings of HEK-293T cells transiently co-transfected with GIRK channels, and either M2Rs, b2ARs or b1ARs. The membrane potential was held at À80 mV. 10 mM ACh or Iso was applied as indicated. (E) Validation of the function of bARs. HEK-293T cells expressing bARs or untransfected HEK-293T cells (Ctrl) were treated with 10 mM propranolol (Pro) or isoprennaline (Iso), and intracellular cAMP levels were quantified (N = 3, ±SD). See also Touhara and MacKinnon. eLife 2018;7:e42908. DOI: https://doi.org/10.7554/eLife.42908 specificity puzzle. One theory posited the existence of a macromolecular super-complex consisting of GIRK, G proteins and a Ga i -coupled GPCR, to endow specificity by proximity (Peleg et al., 2002;Ivanina et al., 2004;Clancy et al., 2005;Riven et al., 2006). Another theory suggested that stimulated Ga s -coupled receptors might generate insufficient quantities of free Gbg if Ga s (GTP) binds to Gbg with higher affinity (Digby et al., 2008). None of these studies provided sufficient data to strongly support a solution. Here we present data that support a simple biochemical solution to this puzzle.

Results
Gbg specificity in native and heterologously expressed GIRK channels Figure 1B shows spontaneous action potentials recorded from current-clamped murine SAN cells isolated from adult mice. With application of isoproterenol (Iso), action potential (AP) frequency increased, and with ACh firing altogether ceased. Somewhat surprisingly, we could not find in the literature a demonstration of both autonomic responses in the same cell. Here we observe that both Ga i -associated (via ACh to slow heart rate) and Ga s -associated (via Iso to speed heart rate) pathways are indeed activated within a single pacemaker cell. Figure 1C shows a voltage-clamp experiment performed on the very same cell shown in Figure 1B. ACh produces inward K + current through GIRK channels, which is the origin of action potential cessation in Figure 1B. Iso does not activate GIRK even though bAR stimulation is known to generate free Gbg subunits. Figure 1D shows voltage clamp experiments in human embryonic kidney 293T (HEK-293T) cells in which GIRK channels and GPCRs were heterologously expressed. M2R is a Ga i -coupled GPCR stimulated by ACh and beta 1-adrenergic receptor (b1AR) and beta 2-adrenergic receptor (b2AR) are both Ga s -coupled GPCRs stimulated by Iso. In each experiment, agonist (ACh or Iso) is applied to reveal the level of stimulated K + current. Only M2R receptor stimulation activates GIRK to a large extent. This expression is not due to endogenous M2Rs in HEK-293T cells, as ACh fails to stimulate GIRK channels unless M2R is expressed (Figure 1-figure supplement 1A). A difference in surface expression levels of the GPCRs does not explain this result, as Alexa Fluor 488-labeled M2Rs and b2ARs show similar fluorescence intensity at the plasma membrane (Figure 1-figure supplement 1B-1C). To ensure that expressed b1AR and b2AR are indeed functional in the cells and capable of initiating the Ga s pathway, the cAMP ELIZA assay was used to measure Iso-stimulated increases in cyclic adenosine monophosphate (cAMP) concentration, which is not observed in control cells and is thus dependent on the b1AR and b2AR expression ( Figure 1E). Similar experiments were carried out in chinese hamster ovary (CHO) cells (also mammal-derived) and Spodoptera frugiperda (Sf9) cells (insect-derived) (Figure 1-figure supplement 1D-1E). In each cell line only M2R receptor stimulation activates GIRK channels. These data demonstrate that specificity persists across mammalian and insect cells and is therefore a robust property of these signaling pathways. The results also imply that GIRK activation does not depend on Gbg subtypes, because different cell lines, particularly Sf9 cells, express subtypes of Gbg that are distinct from those in mammals (Leopoldt et al., 1997).

Effect of artificially enforced GPCR-GIRK co-localization
To test whether the macromolecular supercomplex hypothesis can account for Gbg specificity, we artificially enforced proximity by expressing GIRK linked to either M2R or b2AR within a single open reading frame, as shown ( Figure 2A). When expressed and analyzed using a western blot, the linked GIRK channel and GPCR run on SDS-PAGE gels as either full-length GIRK-GPCR units or as dimers, trimers and tetramers of those units ( Figure 2B). Therefore, when expressed, GIRK and the GPCR remain linked together. Because GIRK channels are tetramers under native conditions, expression of the GIRK-GPCR unit causes each channel to be surrounded by four GPCRs. Voltage-clamp experiments on HEK-293T cells transiently transfected with the M2R-GIRK construction showed GIRK activation in response to ACh stimulation ( Figure 2C). Iso stimulation with cells expressing the b2AR-GIRK construction did not activate GIRK ( Figure 2D), even though the b2AR is functional as evidenced by quantifying levels of stimulated cAMP ( Figure 2E). These experiments do not support the macromolecular supercomplex hypothesis as an explanation for Gbg specificity.

Influence of G protein levels on specificity
In the experiments described so far, activation of GIRK channels by GPCR stimulation was facilitated by endogenous levels of G proteins in the cells. We next ask what happens if the levels of G proteins available for mediating activation are altered? Using a cell line in which we established stable expression of GIRK channels and GPCRs, G protein levels were altered using transient transfection. In control experiments endogenous G protein levels support M2R stimulated GIRK channel activation ( Figure 3A), as was observed in Figure 1. Expression of additional Ga i1 subunits suppressed the level of M2R-stimulated GIRK current, presumably because excess Ga i1 subunits blunt the normal increase in Gbg concentration (i.e. Ga i1 can compete with the channel for available Gbg). Expression of additional Ga i1 and Gbg subunits, however, leads to M2R-stimulated GIRK current that exceeds levels mediated by endogenous G proteins alone ( Figure 3A and . This latter observation would seem to suggest that increased availability of Ga i (GDP)bg substrate (upon which stimulated M2R acts to generate free Gbg) leads to increased Gbg levels following M2R stimulation. The question then naturally arises, if sufficiently high levels of Ga s (GDP)bg substrate are provided, might the b2AR activate GIRK to a detectable extent? The answer is yes. Figure 2. Effect of artificially enforced GPCR-GIRK co-localization. (A) A schematic representation of GPCR-GIRK concatemer constructs. GIRK was directly fused to the C-terminus of GPCRs. A cleavable signal peptide and a Halo tag were added to the N-terminus of each concatemer. Additionally, a SNAP tag was added to the C-terminus of each concatemer. (B) Western-Blot analysis of GPCR-GIRK concatemer constructs. HEK-293T cells were transiently transfected with either M2R-GIRK or b2AR-GIRK concatemers. The expected size of these concatemers is~150 kDa. (C) (D) Representative voltage-clamp recordings of HEK-293T cells transiently transfected with M2R-GIRK concatemers or b2AR-GIRK concatemers. Membrane potential was held at À80 mV. 10 mM ACh or Iso was applied as indicated. (E) Validation of the function of b2AR-GIRK concatemers. HEK-293T cells expressing b2AR-GIRK concatemers were treated with 10 mM propranolol (Pro) or isoproterenol (Iso), and intracellular cAMP levels were quantified (N = 3, ±SD). DOI: https://doi.org/10.7554/eLife.42908.004 Touhara and MacKinnon. eLife 2018;7:e42908. DOI: https://doi.org/10.7554/eLife.42908 Figure 3. Influence of G protein levels on specificity. (A) GIRK currents induced by M2R agonist ACh. Cells from a stable HEK-293T cell line expressing M2Rs and GIRK channels were transiently transfected with a vector expressing either GFP (Ctrl), Ga i1 , or Ga i1 and Gbg. 10 mM ACh was applied, and the evoked inward current was normalized to the capacitance of the cell (±SEM). (B) GIRK currents induced by b2AR agonist Iso. Cells from a stable HEK-293T cell line expressing b2ARs and GIRK channels were transiently transfected with a control vector expressing either GFP, Ga s , or Ga s and Gbg. 10 mM Iso was applied, and the evoked inward current was normalized to the capacitance of the cell (±SEM). (C) A schematic representation of the BRET assay. Upon agonist stimulation of a GPCR, Gbg-Venus is released. Gbg-Venus then binds to GIRK-NLuc, which increases the BRET signal. (D) (E) Representative changes in BRET signal upon stimulation of GPCRs. In (D), HEK-293T cells were transfected with M2Rs, Gbg-Venus, GIRK-NLuc, and increasing amounts of Ga i1 . In (E), HEK-293T cells were transfected with b2ARs, Gbg-Venus, GIRK-NLuc, and increasing amounts of Ga s . Agonists were applied at t = 5 s. See also  Experiments using cells expressing GIRK channels and b2ARs show that excess Ga s and Gbg subunits give rise to b2AR-stimulated GIRK current ( Figure 3B and Figure 3-figure supplement 1B). This finding suggests that the specificity exhibited by Ga i -coupled GPCRs versus Ga s -coupled GPCRs is somehow related to differences in the levels of Gbg that they each are able to generate.

Direct measurement of the Gbg-GIRK interaction
We explored the influence of G protein levels further using a more direct measurement to estimate the Gbg-GIRK interaction. After fusing the modified yellow fluorescent protein Venus to Gbg and the bioluminescent protein Nano-Luciferase (NLuc) to GIRK (GIRK-NLuc) we monitored their proximity by measuring the bioluminescent resonance energy transfer (BRET) ratio (Masuho et al., 2015). The idea is, following GPCR stimulation Gbg-Venus separates from the GPCR-G protein complex and binds to GIRK, bringing Venus close to NLuc on the channel and thus increasing the BRET ratio ( Figure 3C).
Two initial controls were carried out. First, we examined the binding of Gbg-Venus to the membrane anchored C-terminal PH domain of GRK3 fused to NLuc (masGRK3ct-NLuc), which is known to bind to Gbg with~20 nM affinity (Pitcher et al., 1992). This experiment produced a robust increase in the BRET signal following M2R stimulation ( Figure 3-figure supplement 1C). Second, we examined the binding of Gbg-Venus to Kir2.2 fused to NLuc. Kir2.2 is structurally similar to GIRK but does not bind to Gbg. No change in BRET signal occurred following M2R stimulation (Figure 3-figure supplement 1D). These positive and negative controls imply that the BRET assay may be suitable for monitoring a specific interaction between GIRK and Gbg subunits released following GPCR stimulation.
HEK-293T cells were transiently transfected with M2Rs, Gbg-Venus, GIRK-NLuc, and varying concentrations of Ga i1 . The BRET signal was then monitored over time following ACh stimulation ( Figure 3D and Figure 3-figure supplement 1E). Even in the absence of additional Ga i1 , the BRET signal showed a time-dependent increase, consistent with Gbg-Venus being released from M2Rs and then binding to the GIRK channel. As the amount of Ga i1 expression was increased the BRET signal increased further, consistent with more Gbg-Venus being generated as a result of greater Ga i1 (GDP)bg-Venus substrate availability. Note that this result is not inconsistent with the reduced current generated in Figure 3A upon excess Ga i1 expression because in the BRET experiment ( Figure 3D) Gbg-Venus is also over-expressed. As the level of Ga i1 expression is increased a maximum BRET signal is reached, suggesting that an aspect of this signaling pathway other than Ga i1 availability eventually becomes limiting. When the same experiment was carried out with the b2AR only a very small change in the BRET signal was observed in the absence of Ga s transfection ( Figure 3E and Figure 3-figure supplement 1E), consistent with the failure of b2AR stimulation (in the absence of Ga s transfection) to activate GIRK channels ( Figure 3B). In accord with the ability of Ga s and Gbg over-expression to over-ride specificity and permit b2AR-stimulated GIRK current ( Figure 3B), the BRET ratio increased with increased expression of Ga s (and Gbg-Venus). The electrophysiological and BRET assays are in complete agreement with each other and suggest that specificity in Ga i -coupled GPCR signaling results from higher Gbg concentrations achieved when Ga i -coupled receptors are stimulated compared to Ga s -coupled receptors.

Generalization of Ga i -coupled GPCR target specificity
If specificity results from higher levels of Gbg generated when Ga i -coupled receptors are stimulated rather than from a specific protein-protein interaction and localization of the receptor with GIRK, then other targets upon which Gbg acts might also exhibit similar specificity. To test this idea, we carried out experiments using the transient receptor potential melastatin 3 (TRPM3) channel, which is inhibited by Gbg ( Figure 4A) (Badheka et al., 2017;Quallo et al., 2017;Dembla et al., 2017). TRPM3 channels and M2Rs were transiently transfected into HEK-293T cells and whole-cell voltageclamp recordings were performed. TRPM3 channels were first activated by a chemical ligand, pregnenolone sulphate (PS), and then inhibited (85 ± 10%) by stimulating M2R with Ach ( Figure 4B and C). Similar experiments with Iso-stimulated b2ARs showed only modest inhibition (17 ± 10%), consistent with some degree of specificity as a result of there being insufficient concentrations of Gbg generated by the Ga s -coupled pathway ( Figure 4B and D). As in the GIRK experiments, specificity is lost when Ga s and Gbg are over-expressed (inhibition 73 ± 14%) ( Figure 4B-4F). These observations further strengthen the idea that Ga i -coupled receptors generate higher concentrations of Gbg in the setting of endogenous G protein concentrations and that these higher Gbg levels account for Gbg specificity. These observations also further reject the macromolecular supercomplex hypothesis as a tenable explanation, because similar Gbg specificity is observed with a completely different protein target of the Gbg pathway.

Relative rates of Gbg release by Ga i versus Ga s -coupled receptors
By what mechanisms do M2Rs generate higher Gbg concentrations than b2ARs? If Ga i subunits were more abundant in cells than Ga s subunits then higher rates of Gbg generation would be expected. This explanation seems unlikely though, because the endogenous levels of Ga s in HEK-293T cells are actually higher than Ga i when we measure levels directly using a Western blot assay in the same cells TRPM3 and b2ARs (E) TRPM3, b2ARs, and Ga s , or (F) TRPM3, b2ARs, Ga s , and Gbg. A ramp protocol from À100 mV to +100 mV was applied to the cells every second. The currents at +100 mV were plotted. TRPM3 currents were evoked by 10 mM pregnenolone sulfate (PS). M2Rs and b2ARs were stimulated by 10 mM ACh and Iso, respectively. DOI: https://doi.org/10.7554/eLife.42908.008 (Figure 3-figure supplement 2). Higher levels of Ga s in HEK-293T cells were also reported previously on the basis of RNA levels (Atwood et al., 2011).
Alternatively, differences in the affinity of Gbg for Ga s -GTP versus Ga i -GTP could potentially account for differences in the levels of free Gbg generated during b2AR versus M2R stimulation. To test this possibility, we assessed the relative ability of Ga s -GTP versus Ga i1 -GTP to bind to Gbg. Because the affinity of GTP-bound forms of Ga for Gbg are so low we contrived the experiment shown in Figure 5A. GIRK channels and Gbg were reconstituted into planar lipid bilayers at a mass ratio of~1:0.1. In the presence of 8 mM Na + and 32 mM C8-PIP 2 a fraction of GIRK channels are activated in the context of limiting Gbg concentration ( Figure 5-figure supplement 1A). Under this condition, sufficiently high concentrations of Ga(GTP-gS) can inhibit GIRK activation through competition by binding to Gbg. Thus, known amounts of Ga i1 (GTP-gS) or Ga s (GTP-gS) were added by replacing the lipid tail with a His 10 tag and including in the bilayer 3% Ni-NTA lipids. After saturation of Ni-NTA lipids these conditions should yield a Ga(GTP-gS) concentration adjacent to the membrane~3 mM Touhara et al., 2016). Inhibition of GIRK current was observed, but with no significant difference between Ga i1 (GTP-gS) and Ga s (GTP-gS), suggesting that their affinities for Gbg are similar ( Figure 5B-5D). Thus, lower Gbg concentrations following Ga s -coupled receptor stimulation cannot be attributed to sequestration by Ga s (GTP).
Next, we tested the possibility that Ga i -coupled receptors catalyze intrinsically faster Gbg release. We developed an assay by attaching Venus to Ga, NLuc to Gbg, and measured the BRET ratio Figure 5. Ga s (GTP-gS) and Ga i1 (GTP-gS) do not differentially compete with GIRK channels for Gbg. (A) A schematic representation of the competition assay between His 10 -Ga(GTP-gS) and GIRK for Gbg in a reconstituted planar lipid bilayer system. In these experiments, we controlled the amount of lipid-associated Ga(GTP-gS) to evaluate the competition quantitatively. We first incorporated a fixed amount of Ni-NTA-lipids into the lipid bilayer and applied enough His 10 -Ga(GTP-gS) to saturate all the available Ni-NTA binding sites ( Figure 5-figure supplement 1B-1D). Tethered His 10 -Ga(GTP-gS) competes with GIRK for Gbg and therefore inhibits GIRK. (B) Current inhibition by His 10 -Ga(GTP-gS) was normalized to the initial current levels (N = 3, ±SD). (C) (D) Representative inward GIRK currents from lipid bilayers. GIRK was partially activated by PIP 2 , Na + , and a low concentration of Gbg. Dashed lines represent the baseline current (0 pA). (C) His 10 -Ga i1 (GTP-gS) or (D) His 10 -Ga s (GTP-gS) was directly perfused to the bilayer membrane several times followed by mixing the solutions in the bilayer chamber. The transient decrease in the current upon Ga(GTP-gS) application is an artifact due to the absence of Na + in His 10 -Ga(GTP-gS) solution. See also  change to monitor GPCR-mediated dissociation of Gbg-NLuc from Ga-Venus ( Figure 6A). We also expressed masGRK3ct in the same cells to sequester Gbg-NLuc once it is released, thus reducing the extent to which Gbg-NLuc will rebind to Ga-Venus. Two different Ga-Venus insertion constructs were made -into the aa-ab loop or into the ab-ac loop of Ga -to ensure that the observed behavior does not depend on the site of insertion ( Figure 6-figure supplement 1A). Prior to GPCR stimulation, N-Luc intensity and BRET ratio were nearly constant and approximately similar in magnitude in all experiments ( Figure 6B-6C, Figure 6-figure supplement 1, and Table 1). Following GPCR stimulation the BRET ratio change was small for the b2AR and comparatively large for M2R. Similar experiments were also carried out with the Ga i -coupled dopamine receptor (D2R), which activates GIRK ( Figure 6-figure supplement 2A), and the Ga s -coupled b1AR, which does not. Again, we observe that Gbg-dissociation from Ga is much greater for the Ga i -coupled receptor ( Figure 6D-6E, Figure 6-figure supplement 2C-2D, and Table 1). The small signal associated with bAR stimulation is not due to malfunctioning of the Venus-inserted Ga s constructs because Gbg-NLuc dissociation from Ga s -Venus is observable in controls in which G proteins were over-expressed to higher levels (i.e. when BRET ratio prior to bAR stimulation was higher) (Figure 6-figure supplement 2B). We conclude from these experiments that the Ga i -coupled receptors M2R and D2R generate more rapid Gbg release than the Ga s -coupled receptors b1AR and b2AR due to a higher intrinsic turnover rate.

Kinetic model of Gbg specificity
We developed a kinetic model for GIRK activation to test whether we could replicate Gbg-specificity on the basis of differences in Ga i versus Ga s -coupled receptor turnover rates. The model consists of a G protein turnover reaction cycle and a GIRK-Gbg binding reaction that leads to channel activation ( Figure 7A). Numerous studies have provided estimates for rate constants in the reaction cycle ( Table 2) (Breitwieser and Szabo, 1988;Sarvazyan et al., 1998;Sungkaworn et al., 2017), and the GIRK-Gbg binding reaction has been studied in detail, providing estimates for k 56 and k 65 as well as a cooperativity factor m (Shea et al., 1997;Wang et al., 2016;Touhara et al., 2016).
The G protein reaction cycle models the conversion of Ga(GDP)bg (the G protein trimer) into Ga (GTP) and Gbg in two kinetic transitions. The first transition (k 12 ) describes the formation of a productive complex between the G protein trimer and an active (ligand-bound) GPCR (R*). The second (k 23 ) combines multiple reactions, including GDP/GTP exchange and Ga(GTP) and bg dissociation. In our experiments, the observation that G protein over-expression increases levels of stimulated Gbg in cells (Figure 3 and Figure 4B) implies that the k 12 transition is to some extent rate-limiting under physiological G protein conditions. A single molecule study of the a2 adrenergic receptor (a2AR; a Ga i -coupled GPCR) also concluded that complex formation between activated receptor and G protein trimer (i.e. the k 12 transition) was rate-limiting (Sungkaworn et al., 2017). Furthermore, the same study found that k 12 for the b2AR was ten times smaller than for the a2AR.
GPCR density over the entire membrane of atrial cardiac myocytes and in CHO cells is approximately 5 mm À2 (Nenasheva et al., 2013). However, G protein signaling occurs within 'hotspots' that Table 1. Quantitative-BRET measurements of Gbg release from different Ga constructs. Averaged Nano-Luc intensity, basal BRET ratio, and DBRET ratio were summarized (N = 3-4, ±SD).

NLuc intensity
Basal BRET ratio DBRET ratio  we estimate to cover about 10% of the membrane surface (Sungkaworn et al., 2017). Thus, we assume the receptor density to be 50 mm À2 within a hotspot and assume an initial Ga(GDP)bg density of 100 mm À2 . When the reaction is switched on (i.e. ligand stimulation) at t = 0 by changing k 12 from 0 to 0.2 mm 2 molecule À1 sec À1 (Sungkaworn et al., 2017), Gbg concentration increases (along with time-dependent concentration changes of other components) and GIRK channels activate to a steady state value within a few seconds following a time course similar to M2R stimulated GIRK currents in SAN cells ( Figure 7B and Figure 7-figure supplement 1A). We note that time courses vary from cell to cell, but that the modeled time course falls within the experimental range.
To model the b2AR receptor we reduced k 12 ten times, consistent with Sungkaworn et al, leaving all other quantities the same. Lower concentrations of Gbg are predicted and along with significantly less GIRK activation ( Figure 7B). Figure 7C displays in greater detail calculated GIRK-(Gbg) 4 concentration (i.e. channel activation) as a function of k 12 magnitude. A steep dependence occurs right around the experimentally determined value for the Ga i -coupled receptor turnover rate constant (Sungkaworn et al., 2017). Thus, the model predicts that higher rates of G protein turnover catalyzed by Ga i -coupled compared to Ga s -coupled GPCRs can account for Gbg specificity.
Partial agonists by definition activate GPCRs with reduced efficacy compared to full agonists (McKinney et al., 1991). The effects of two partial agonists, oxotremorine and pilocarpine, on M2R activation of GIRK are shown (Figure 7-figure supplement 1B-1C). A study recently concluded that for the b2AR, the distinction between partial and full agonist action lies in the magnitude of k 12 , its value being smaller for partial agonists (Gregorio et al., 2017). We think this conclusion likely applies to M2R as well, based on the following observations. When the partial agonists oxotremorine (Oxo) and pilocarpine (Pilo) are used to stimulate M2R, reduced GIRK currents are associated with reduced BRET signals for Gbg-Venus binding to GIRK-NLuc (blue symbols in  Table 2. Parameters used for simulation of GPCR-activation of GIRK. k 12 : The rate of formation of the productive GPCR-G protein complex (Sungkaworn et al., 2017). k 21 : The rate of dissociation of the productive GPCR-G protein complex (Sungkaworn et al., 2017). k 23 : The rate of nucleotide exchange and subsequent dissociation of GPCRs, Ga(GTP), and Gbg (Sungkaworn et al., 2017). k 32 : The rate of the reverse reaction of nucleotide exchange and dissociation of GPCRs and G proteins. k 34 : The rate of GTP hydrolysis, based on Breitwieser and Szabo, 1988. k 43 : The rate of the reverse reaction of GTP hydrolysis. k 45 : The on-rate between Ga(GDP) and Gbg, adapted from Sarvazyan et al., 1998. k 54 : The off-rate between Ga (GDP) and Gbg, calculated based on k 45 and K d = 3 nM (Sarvazyan et al., 1998). k 56 : The on-rate between the GIRK and Gbg is diffusion limited . k 65 : The off-rate between the GIRK and Gbg were calculated based on k 56 and our previous K d measurement (Shea et al., 1997).

Reaction
Forward-rate Backward-rate Note Ga(GTP) * ) Ga(GDP) + P i 2 sec À1 (k 34 ) 0 M À1 sec À1 (k 43 ) Breitwieser and Szabo, 1988 Ga(GDP) + Gbg * ) Ga(GDP)bg 0.7 Â 10 6 M À1 sec À1 (k 45 ) 0.002 sec À1 (k 54 ) Sarvazyan et al., 1998 GIRK-bg n-1 + (5 -n)Gbg * ) GIRK-bg n + (4 -n)Gbg (5 -n) Â 1 Â 10 7 M À1 sec À1 ((5 -n) Â k 56 ) n Â m n-1 Â 600 sec À1 (n Â m n-1 Â k 65 ) supplement 1D). Furthermore, when amounts of available Ga i1 are increased (so that more Ga i1 (GDP)bg-Venus can form), the partial agonist Oxo gives rise to a BRET signal as strong as that of ACh (orange symbols in Figure 7-figure supplement 1D). A similar effect was also observed with Pilo, although to a lesser extent. These results are explicable on the basis of the G protein trimer-GPCR on-rate determining the efficacy of different agonists. Thus, k 12 can explain the difference in agonists versus partial agonists as well as the fundamental difference between M2R and bARs with respect to their ability to activate GIRK channels. In the model we present, k 12 , is rate limiting under physiological G protein concentrations, and its magnitude determines differential rates of Gbg generation.

Discussion
The essential conclusion of this study is that M2R catalyzes the generation of Gbg at a higher rate than b2AR, thus achieving higher concentrations of Gbg to activate GIRK. The concentrations of Ga s (GTP) generated by b2AR are obviously sufficient to stimulate the downstream-amplified Ga s pathway and speed heart rate, but the lower Gbg levels generated are insufficient to activate GIRK to a great extent. The higher rate of Gbg generation by M2R likely stems from an intrinsically higher rate of association with G protein trimer. This conclusion is most easily appreciated through careful inspection of Figure 3D and E and Figure 3-figure supplement 1E, where it is shown that endogenous levels of Ga (in the presence of expressed Gbg-Venus to detect Gbg binding to GIRK) permit Gbg generation by M2R, but not by b2AR. Furthermore, over-expression of Ga and Gbg increases the rate of Gbg generation in both cases, but higher levels of Ga expression are needed for the b2AR to reach its maximum rate. Thus, Gbg specificity is explicable on the basis of a difference in the rate at which M2R and b2AR associate with G protein trimer, M2R being faster. The forward rate in the first step of the reaction cycle ( Figure 7A) is k 12 [G protein trimer] [GPCR]. Therefore, a difference in the rate constant k 12 or either reactant concentration will change the forward rate. We have four reasons to conclude that the difference lies primarily in a difference in k 12 . First, we have shown in HEK-293T cells that the endogenous concentration of Ga i does not exceed Ga s (Figure 3-figure supplement 2) and therefore it is unlikely that Ga i trimer exceeds Ga s trimer. Moreover, we estimate the levels of GPCR density expressed in HEK-293T cells to be similar for M2R and b2AR (Figure 1-figure supplement 1B and C). Second, given that endogenous levels of Ga i are not greater than Ga s in HEK-293T cells, the difference in the rate of Gbg generation (mediated by M2R versus b2AR) as reported by the BRET assay (Figure 3-figure supplement 1E) is explicable on the basis of a difference in k 12 . Third, the higher rate of Gbg dissociation from Ga, mediated by M2R versus b2AR (and also D2R versus b1AR), is most simply explained by a difference in k 12 ( Figure 6). Fourth, the recent single molecule study showing an intrinsically larger k 12 for a2AR (a Ga i -coupled GPCR) compared to b2AR is completely consistent with the three reasons enumerated above (Sungkaworn et al., 2017). Therefore, we conclude that the basis of specificity we are describing here is explained by a difference in k 12 . It is possible that differences in Ga i versus Ga s concentrations in certain cell types could further contribute to specificity. However, a difference in k 12 alone can explain specificity.
When G protein trimer associates with a GPCR, both Ga and receptor undergo a series of conformational changes (Rasmussen et al., 2011). A chimera Ga subunit containing mostly Ga i1 amino acids and only 13 C-terminal Ga s amino acids -that engage the receptor -is known to permit bAR activation of GIRK (Leaney et al., 2000). This observation suggests that the Ga conformational change, which involves the main body of the Ga subunit, might be more important in determining the rate of G protein trimer-GPCR association.
The M2R-GIRK signaling pathway is characterized by four key features: M2R density over the entire cell membrane is relatively low (~5 mm À2 ), M2R turnover rate (i.e. Gbg generation rate) is slow (maximum rate k 23~1 sec À1 ), Gbg lifetime is short (~1 s), and the affinity of Gbg for GIRK is not very high (1.9 mM for the first Gbg and a cooperativity factor of 0.3 for each successive Gbg). These features have important consequences. The expected steady state concentration profile of Gbg surrounding an isolated M2R catalyzing even at its maximum rate (k 23 ) shows that Gbg never reaches sufficient levels to activate GIRK ( Figure 7D and F). This is because as Gbg is generated it both diffuses away (diffusion coefficient, D~0.2 mm 2 sec À1 ) and is re-sequestered by Ga(GDP) in approximately one second (k~1 sec À1 ), causing the Gbg concentration to decay over a characteristic distance (D/k) 1/2 . This circumstance explains why b2AR does not activate GIRK even when it is tethered to the channel (Figure 2). And it implies that the macromolecular super-complex hypothesis can not work very well to activate GIRK or to explain Gbg specificity. At a density of 5 mm À2 , M2Rs are too far apart from each other to build up the Gbg concentration. But at a density of 50 M2Rs mm À2 , sufficiently high Gbg concentrations can be reached: the expected concentration profile surrounding a disk-shaped 'hotspot' of radius 0.3 mm is shown ( Figure 7E and F). In the middle of the hotspot, which contains about 14 M2Rs, Gbg concentration reaches 12.5 mm À2 (2.5 mM in a layer 80 Å thick beneath the membrane), which is enough to activate GIRK channels that happen to be located within the disk. At a fixed GPCR density (50 mm À2 ), the steady state Gbg concentration depends on the size of the disk ( Figure 7G). It is notable that the predicted disk size -several hundred nm to 1 mm -required to achieve sufficiently high concentrations of Gbg to activate GIRK matches well with G protein signaling hotspots observed in cells (Sungkaworn et al., 2017).
In summary, Gbg specificity is determined by more rapid Ga i (GDP)Gbg association with M2R compared to Ga s (GDP)Gbg association with b2AR. A sufficient density of GPCRs is required to achieve GIRK-activating concentrations of Gbg. This is apparently achieved through the formation of hotspots of higher GPCR and G protein density (Sungkaworn et al., 2017). But specificity is explained by the magnitude of a rate constant. Experimental model and subject details Animals C57BL/6J (Jackson Labs) male and female adult mice (!10 weeks old) were used. Animals were kept in cages with a 12:12 hr light/dark cycle and unrestricted access to food and water. All experimental procedures were carried out according to a protocol approved by the Institutional Animal Care and Use Committee (IACUC) of The Rockefeller University (Protocol #16864).

Cell lines
HEK-293T tsA201 cells were obtained from Sigma and maintained in DMEM (Thermo Fisher) supplemented with 10% Fetal Bovine Serum (FBS, Thermo Fisher) and 1% L-glutamine (Thermo Fisher). Chinese Hamster Ovary cells were obtained from Sigma and maintained in DMEM/F12 (Thermo Fisher) supplemented with 10% FBS and 1% L-glutamine. Sf9 cells were obtained from Sigma and maintained in Grace's Insect medium (Thermo Fisher) supplemented with 10% FBS and Pluronic TM F-68 (Thermo Fisher).

Sinoatrial node (SAN) isolation
Adult mice (!10 weeks old) were anesthetized with 90-150 mg/kg ketamine and 7.5-16 mg/kg xylazine IP (Sigma-Aldrich). After 5-10 min, when mice stopped responding to tail/toe pinches they were secured in the supine position by gently fixing their forepaws and hindpaws to a pinnable work surface on an animal surgery tray. SAN isolation was performed according to a published procedure (Sharpe et al., 2016). A midline skin incision was made from the mid abdomen to the diaphragm with a surgical scissor. The heart was exposed after cutting the diaphragm and holding the sternum with curved serrated forceps. The heart was lifted and dissected out of the thoracic cavity as near as possible to the dorsal thoracic wall. The isolated heart was transferred to a petri dish containing Tyrode's solution (140 mM NaCl, 5.4 mM KCl, 1.2 mM KH 2 PO 4 , 1 mM MgCl 2 , 2 mM CaCl 2 , 5.5 mM D-glucose, 1 mg/mL BSA, 5 mM HEPES-NaOH [pH 7.4]), and quickly washed several times to remove residual blood. The heart was dissected and the ventricles were removed. The atria were transferred to a silicone dissection dish and pinned through the inferior and superior vena cavae and the right and left atrial appendages. The interatrial septum was exposed by opening the anterior wall. Next, the right atrial appendage was removed and the SAN was isolated by cutting along the cristae terminals. The isolated SAN was transferred to low-Ca 2+ /Mg 2+ Tyrode's solution (140 mM NaCl, 5.4 mM KCl, 1.2 mM KH 2 PO 4 , 0.5 mM MgCl 2 , 0.2 mM CaCl 2 , 5.5 mM D-glucose, 50 mM Taurine, 1 mg/mL BSA, 5 mM HEPES-NaOH [pH 7.4]) and incubated for 5 min at 37˚C. Next the SAN was washed with low-Ca 2+ /Mg 2+ Tyrode's solution twice, transferred to low-Ca 2+ /Mg 2+ Tyrode's solution with enzymes (0.5 mg/mL Elastase [Worthington], 1.0 mg/mL Type II Collagenase [Worthington], and 0.5 mg/mL Protease xiv [Sigma-Aldrich]), and incubated for 15-20 min at 37˚C. Digested tissue was transferred to Kraftbrü he (KB) medium (100 mM K-glutamate, 10 mM K-aspartate, 25 mM KCl, 10 mM KH 2 PO 4 , 2 mM MgSO 4 , 20 mM Taurine, 5 mM Creatine, 0.5 mM EGTA, 20 mM D-glucose, 1 mg/mL BSA, 5 mM HEPES-KOH [pH 7.2]), and gently washed. The tissue was washed two more times with KB medium and cells were dissociated by constant trituration at approximately 0.5-1 Hz for 5-10 min. CaCl 2 solution was added stepwise (200 mM, 400 mM, 600 mM, and 1 mM) every 5 min to reach to a final concentration of 1 mM. Subsequently an equal volume of Tyrode's solution was gradually added to the KB solution with dissociated cells. Finally, dissociated cells were centrifuged for 3 min at 150 g, resuspended in Tyrode's solution, and plated onto PDL/Laminin pre-coated glass bottom dishes for~1 hr prior to electrophysiological recordings.

Establishment of the stable HEK-293T cell lines
A SNAP tag was fused to the C-terminus of the full-length GIRK4 channel. A serotonin 5-HT signal peptide and a Halo tag were fused to the N-terminus of human full-length M2R or b2AR. Both GIRK4-SNAP and Halo-M2R or Halo-b2AR were cloned into the pcDNA5/FRT/TO vector. An internal ribosome entry site (IRES) sequence was inserted between SNAP-GIRK4 and Halo-GPCR to allow for their simultaneous expression under the same promoter. Stable HEK-293T cell lines were produced using the Flp-In T-REx-293 System according to the manufacturer's protocol (ThermoFisher).

Whole-cell voltage clamp recordings on HEK-293T cells expressing GPCR-GIRK concatemers
Full-length human GIRK4 was fused to the C-terminus of full-length human M2R or b2AR. A serotonin 5-HT cleavable signal peptide and a Halo tag were fused to the N-terminus of each concatemer. Additionally, a SNAP tag was fused to the C-terminus of each concatemer. Concatemers were transiently transfected and cells were incubated at 37˚C for 20-24 hr. Cells were then dissociated and plated on PDL/Laminin-pre-coated glass coverslips for electrophysiological recordings. Whole-cell voltage clamp recordings were performed with the same system, pipettes, perfusion system, and solutions as described above.
Whole-cell voltage clamp recordings on HEK-293T cells expressing TRPM3 channels Mouse TRPM3a2 was cloned into a pEG BacMam vector. A PreScission protease cleavage site, an enhanced green fluorescent protein (eGFP), and 1D4 peptide tag were placed at the C-terminus of the TRPM3 construct. TRPM3-eGFP, Sero-SNAP-GPCR, and G proteins were transiently transfected to HEK-293T cells and cells were incubated at 30˚C for 48-72 hr. Cells were then dissociated and plated on PDL/Laminin-pre-coated glass coverslips for electrophysiological recordings. Whole-cell voltage clamp recordings were performed as described above. The currents were recorded using a ramp protocol from À100 mV to +100 mV, applied every second, and the currents at +100 mV were plotted. TRPM3 currents were evoked by 10 mM pregnenolone sulfate (PS) (Tocris).

cAMP quantification assay
Untransfected HEK-293T cells or HEK-293T cells transfected with bARs were cultured in 12-well plates for 20-24 hr. Sf9 cells infected with P3 baculovirus of bARs were cultured in 12-well plates for 40-48 hr. Cells were treated with either 10 mM isoprenaline or propranolol for 10 min and washed twice with PBS + 500 mM isobutylmethylxanthine (IBMX). Cells were collected in 200 mL PBS + IBMX, exposed to four freeze-thaw cycles, and centrifuged (14,000 rpm) for 10 min at 4˚C. The supernatant was analyzed for cAMP content according to the manufacturer's protocol (cAMP ELISA Detection Kit, GeneScript).
For BRET measurements between Ga-Venus and Gbg-NLuc, HEK-293T cells were transfected with Sero-SNAP-GPCR (90 ng), Gbg-NLuc (90 ng), masGRK3ct (90 ng), and Ga-Venus (90-450 ng), and incubated for 20-24 hr at 30˚C or 37˚C. The measured light emitted by Gbg-NLuc is proportional to the amount of Gbg-NLuc in the sample, and the measured light emitted by Ga-Venus is proportional to the amount of G protein trimers in the sample. By having equal intensities for Gbg-NLuc and Ga-Venus (i.e. NLuc intensity and basal BRET ratio), the rate of Gbg release can be compared and contrasted for different GPCRs (Table 1). Therefore, samples of each GPCR were prepared with different transfected Ga-Venus-DNA amounts (90-450 ng) to carry out these experiments.

BRET measurements
After 20-24 hr incubation, transfected HEK-293T cells were washed with PBS twice and detached by incubation in PBS + 5 mM EDTA for 5 min at room temperature. Cells were harvested by centrifugation at 300 g for 3 min and resuspended into 350 mL BRET buffer (PBS supplemented with 0.5 mM MgCl 2 and 0.1% D-glucose). 25 mL of the suspension containing~70,000 cells was transferred to each well in a 96-well flat-bottom white microplate (Greiner CELLSTAR). The NLuc substrate (Promega) was diluted into the BRET buffer according to the manufacturer's protocol, and 25 mL of diluted NLuc substrate were added to the cells in 96-well plates. BRET measurements were made with a microplate reader (Synergy Neo, BioTek) equipped with two emission photomultiplier tubes. The BRET signal was determined by calculating the ratio of the light emitted by Venus (535 nm with a 30 nm band width) to the light emitted by NLuc (475 nm with a 30 nm bandwidth).

Expression and purification
Human full-length GIRK4 was cloned into a pEG BacMam vector (Goehring et al., 2014). A PreScission protease cleavage site, an enhanced green fluorescent protein (eGFP) and a 1D4 peptide tag were placed for purification at the C-terminus of the GIRK4 construct. For overexpression and protein purification, HEK-293S GnTlcells were grown in suspension, infected with P3 BacMam virus of the GIRK4-1D4 and incubated at 37˚C. At 8-12 hr post-infection, 10 mM sodium butyrate was added to the culture, and cells were harvested 60 hr post-transduction. Cells were harvested by centrifugation, frozen in liquid N 2 , and stored at À80˚C until needed. Frozen cells were solubilized in 50 mM HEPES (pH 7.35), 150 mM KCl, 4% (w/v) n-decyl-b-D-maltopyranoside (DM), and the protease inhibitor cocktail (0.1 mg/mL pepstatin, 1 mg/mL leupeptin, 1 mg/mL aprotinin, 0.1 mg/mL soy trypsin inhibitor, 1 mM benzamidine, and 1 mM phenylmethylsulfonyl fluoride). After 2 hr of solubilization, lysed cells were centrifuged at 36,000 g for 30 min and the supernatant was incubated with 1D4 affinity resin for 1 hr at 4˚C with gentle mixing. The resin was loaded onto a column and washed with buffer A (50 mM HEPES [pH 7.0], 150 mM KCl, 0.4% [w/v] DM). 5 mM DTT and 1 mM EDTA were added, and eGFP and affinity tags were cut with PreScission protease overnight at 4˚C. The cleaved protein was then concentrated and run on a Superose 6 10/300 GL gel filtration column in 20 mM Tris-HCl (pH 7.5), 150 mM KCl, 0.2% (w/v) DM, 20 mM DTT, and 1 mM EDTA.
Human lipid-anchored Gb 1 g 2 , and soluble Gb 1 g 2 were purified as described previously .
Human full-length Ga i1 Ga i2 , Ga i3 , Ga o , and Ga s were cloned into a pET28a vector. A PreScission protease cleavage (PPX) site followed by a deca-histidine tag was fused to the N-terminus of Ga. The His 10 -PPX-Ga-pET28a vector was transformed into BL21(DE3) E. coli cells and transformants were cultured in LB medium containing 50 mg/L of kanamycin at 37˚C for 4 hr. Isopropyl-thio-b-Dgalactopyranoside was added to a final concentration of 0.5 mM to induce protein expression. Following an additional incubation at 25˚C for 12 hr, the cells were harvested by centrifugation and resuspended in buffer B (200 mM HEPES-NaOH [pH 7.5], 300 mM NaCl, 2 mM MgCl 2 , and 10 mM MgGDP) and a protease inhibitor cocktail. Cell extracts were obtained by sonication followed by centrifugation at 36,000 g for 30 min. The supernatant was incubated with Talon metal affinity resin (Clontech) for 1 hr at 4˚C with gentle mixing. The resin was washed in batch with five column volumes of buffer B, then loaded onto a column and further washed with 10 column volumes of buffer B + 20 mM imidazole. The column was then eluted with buffer B + 200 mM imidazole.
For Western Blotting analysis, the eluted protein was concentrated and run on a Superdex 200 10/300 GL gel filtration column in 10 mM potassium phosphate (pH 7.4), 150 mM KCl, 2 mM MgCl 2 , and 10 mM GDP.
For the planar lipid bilayer experiment, the eluted protein was concentrated and run on a Superdex 200 10/300 GL gel filtration column in 10 mM potassium phosphate (pH 7.4), 150 mM KCl, and 2 mM MgCl 2 . 1 mM GTP-gS was then added to~1 mg/mL purified proteins and incubated at 37˚C for 30 min to produce His 10 -PPX-Ga(GTP-gS). Residual amounts of His 10 -PPX-Ga(GDP) affect the results of the subsequent bilayer experiment described below. Therefore purified His 10 -PPX-Ga (GTP-gS) was mixed with soluble Gbg at a ratio of 4:1 (molar:molar) to chelate all the possibly contaminating His 10 -PPX-Ga(GDP). This low concentration of Gbg does not affect GIRK activity. acquired. The fluorophore was excited with a white light laser of 488 nm. The fluorescence intensity at the edge of the GUVs was measured using the Zeiss ZEN two software.

Kinetic simulation
Mass balance equations were derived based on the model ( Figure 7A). Rate constants used in the simulation are presented and referenced in Table 2. The set of first order differential equations was solved using the NDSolve function in Mathematica (Wolfram).

Diffusion model
The graphs in Figure 7D and F (dashed) were generated by solving for the concentration of Gbg (C r ð Þ) analytically, using DSolve in Mathematica, the equation r Á DrC r ð Þ À kC r ð Þ ¼ 0 in polar coordinates with the near boundary condition set at the perimeter of an 0.01 mm radius circle corresponding to a flux of 15.9 Gbg mm À1 sec À1 (which corresponds to a single GPCR inside the circle with a turnover rate of 1 Gbg sec À1 ) and far boundary condition zero. D is the diffusion coefficient (0.2 mm 2 sec À1 ) and k is the Gbg decay constant (1 sec À1 ). To model a GPCR density of 5 mm À2 , on average the small circle with a GPCR resides within a circle of radius~0.25 mm. At 0.25 mm the Gbg concentration has not decayed to zero and therefore neighboring GPCRs increase the Gbg concentration slightly above what is shown for a lone GPCR in an essentially infinite membrane, but not enough to activate GIRK. The graphs in Figure 7E and F (solid curve) were generated numerically, using NDSolve in Mathematica, the equation r Á DrC r ð Þ þ s hotspot ð Þ À kC r ð Þ ¼ 0, applying a circular finite element mesh with a zero concentration boundary condition at the perimeter, far from a smaller, central 'hotspot' circle. s refers to Gbg generation (molecules mm À2 sec À1 ) and is applied over the hotspot. The magnitude of s was selected to be near the steady state value of dc dt in the kinetic equations. At a radius 3.16 times the hotspot radius (i.e. corresponding to a circular area 10 times the hotspot area) the Gbg concentration is not zero ( Figure 7F) and therefore in a cell with hotspots covering 10% of the membrane the Gbg concentration will be slightly higher than shown for the case of a lone hotspot. Figure  7G graphs the C r ¼ 0 ð Þ solution to equation r Á DrC r ð Þ þ s hotspot ð Þ À kC r ð Þ ¼ 0, solved as described.