Spatially compartmentalized phase regulation of a Ca2+-cAMP-PKA oscillatory circuit

Signaling networks are spatiotemporally organized to sense diverse inputs, process information, and carry out specific cellular tasks. In β cells, Ca2+, cyclic adenosine monophosphate (cAMP), and Protein Kinase A (PKA) exist in an oscillatory circuit characterized by a high degree of feedback. Here, we describe a mode of regulation within this circuit involving a spatial dependence of the relative phase between cAMP, PKA, and Ca2+. We show that in mouse MIN6 β cells, nanodomain clustering of Ca2+-sensitive adenylyl cyclases (ACs) drives oscillations of local cAMP levels to be precisely in-phase with Ca2+ oscillations, whereas Ca2+-sensitive phosphodiesterases maintain out-of-phase oscillations outside of the nanodomain. Disruption of this precise phase relationship perturbs Ca2+ oscillations, suggesting the relative phase within an oscillatory circuit can encode specific functional information. This work unveils a novel mechanism of cAMP compartmentation utilized for localized tuning of an oscillatory circuit and has broad implications for the spatiotemporal regulation of signaling networks.


Introduction
Cyclic adenosine monophosphate (cAMP) and Ca 2+ act as essential second messengers in almost every cell type and regulate many functional pathways within a cell, such as hormonal signal transduction, metabolism, and secretion (Clapham, 2007;Sassone-Corsi, 2012). In some cell types, including neurons, cardiomyocytes, and pancreatic b cells, these messengers' concentrations oscillate intracellularly (Dupont et al., 2011;Dyachok et al., 2006), and the oscillations encode critical signaling information (e.g. signal strength, duration, and target diversity) into parameters such as frequency and amplitude (Berridge et al., 1998;De Pittà et al., 2008;Parekh, 2011). This phenomenon is perhaps best exemplified in b cells, where oscillations of Ca 2+ drive pulsatile insulin secretion (Rorsman and Ashcroft, 2018) and oscillating cAMP levels (Nesher et al., 2002;Tengholm, 2012). Furthermore, Ca 2+ , cAMP, and the downstream cAMP-dependent kinase protein kinase (PKA) constitute a highly coordinated oscillatory circuit responsible for integrating metabolic and signaling information (Ni et al., 2011). In addition to temporal control, biochemical pathways are also spatially organized within the cell (Smith and Scott, 2002;White and Anderson, 2005). Both Ca 2+ and cAMP are highly spatially compartmentalized and form signaling microdomains or nanodomains (Calebiro and Maiellaro, 2014;Petersen, 2002). While Ca 2+ levels are locally controlled by channels, pumps, and intracellular buffering systems (Clapham, 2007;Stern, 1992), cAMP is thought to be regulated via controlled synthesis by adenylyl cyclases (ACs) and degradation by phosphodiesterases (PDEs) (Bender and Beavo, 2006;Defer et al., 2000). Despite extensive studies on cAMP compartmentation, the mechanisms that spatially constrain this mobile second messenger await to be fully elucidated (Saucerman et al., 2014;Lohse et al., 2017;Musheshe et al., 2018;Bock et al., 2020;Zhang et al., 2020). Furthermore, it is not clear how the spatial regulation of a second messenger influences its dynamic behaviors in the context of coordinated oscillations.
In this study, we investigated the spatiotemporal organization of the Ca 2+ -cAMP-PKA oscillatory circuit in MIN6 b cells and discovered that the relative oscillatory phase between cAMP/PKA and Ca 2+ is spatially regulated within signaling nanodomains. By combining dynamic live-cell imaging, super-resolution microscopy, and computational modeling, we further found that fine-scale, compartment-specific perturbations of this precise phase regulation impact Ca 2+ oscillations in b cells. These findings suggest that the relative phase in oscillatory signaling circuits, like the amplitude and frequency, represents yet another mode of informational encoding and processing, which is subjected to spatiotemporal regulation within the cell.

Results
The relative phase of b cell cAMP and Ca 2+ oscillations is compartmentalized In order to study the spatiotemporal relationship between key players of the Ca 2+ -cAMP-PKA circuit, we chose to focus our attention on an important class of molecular scaffolds, A-kinase anchoring proteins (AKAPs), which are responsible for recruiting PKA to specific substrates at distinct subcellular locations. In several excitable cell types, the plasma membrane (PM)-localized scaffold protein AKAP79 (rodent ortholog AKAP150) has been shown to organize a macromolecular complex with binding partners that include PKA, the voltage-gated Ca 2+ channel Ca V 1.2, Protein Kinase C (PKC), the Ca 2+ /calmodulin-dependent protein phosphatase calcineurin, Ca 2+ -sensitive ACs, AMPA receptors, and many others (Gold et al., 2011). Due to the extensive and multivalent nature of AKAP79/ 150 (as the scaffold is commonly referred) and a report describing the functional impairment of glucose-stimulated insulin secretion (GSIS) in pancreatic b cells upon its knock-out (Hinke et al., 2012), we hypothesized that the AKAP79/150 scaffold might play an important role in the spatiotemporal regulation of the Ca 2+ -cAMP-PKA oscillatory circuit. Specifically, we were interested in testing if AKAP79/150 is able to create a spatially-distinct compartment in which recruitment of signaling effectors can locally fine-tune and reshape signaling dynamics within the circuit (Beene and Scott, 2007;Greenwald and Saucerman, 2011).
To test this hypothesis, we monitored intracellular cAMP and Ca 2+ using the FRET-based cAMP biosensor (Ci/Ce)Epac2-camps (Everett and Cooper, 2013) and the red Ca 2+ indicator RCaMP (Akerboom et al., 2013). We measured cAMP concentration changes in the immediate vicinity of AKAP150 in mouse MIN6 b cells by using (Ci/Ce)Epac2-camps fused to the full-length AKAP79 scaffold (gene AKAP5) (Figure 1a). Although human AKAP79 and the rodent AKAP150 share only 53% sequence identity overall, the interaction motifs and association between the AKAP scaffold and other key signaling players such as PKA, the voltage-gated calcium channel, adenylyl cyclases, and calcineurin are highly conserved (Willoughby et al., 2010;Zhang et al., 2016). The primary difference between these two closely related scaffolds is the presence of an internal repetitive amino acid sequence of unknown function in AKAP150 (Robertson et al., 2009). The functional equivalence of AKAP79 and AKAP150 has been demonstrated in several reciprocal knock-out and recovery experiments in which AKAP79 or AKAP150 was knocked out and expression of the other was shown to rescue a measured phenotype (Hoshi et al., 2005;Zhang et al., 2011). As a control, we also targeted the cAMP probe to the general plasma membrane by adding a lipid modification domain . These targeted biosensors allowed us to compare the dynamics within the AKAP79/150-specific compartment versus the general plasma membrane compartment (Figure 1a).
Although both targeted sensors were trafficked to and distributed along the plasma membrane ( Figure 1-figure supplement 1a,b), we observed notable differences in their respective cAMP signals relative to Ca 2+ oscillations after triggering the circuit (Figure 1b) with tetraethylammonium chloride (TEA, 20 mM), a potent K + channel blocker. cAMP oscillations measured within the AKAP79/150 compartment were in-phase with oscillating Ca 2+ , such that each transient spike in intracellular Ca 2+ was closely associated with a transient increase in cAMP ( Figure 1c) (n = 60 cells). This was in sharp contrast to cAMP oscillations measured within the general plasma membrane compartment, where each local Ca 2+ peak corresponded to a local trough in cAMP (n = 24), followed by a slow reversal of both signals to a pre-stimulated baseline (Figure 1d). While these out-of-phase cAMP-Ca 2+ oscillations were consistent with those observed in the cytoplasm of b cells (Landa et al., 2005;Ni et al., 2011), in-phase cAMP-Ca 2+ oscillations had not previously been observed under these conditions. To quantify the cAMP-Ca 2+ phase relationship, we measured the lag time by calculating the cross-correlation between the two normalized, oscillatory signals and finding the shortest delay yielding the maximum correlation (see Appendix 1 for details) (Figure 1e). In-phase cAMP oscillations corresponded to short lag times (typically <20 s), while out-of-phase oscillations mostly possessed longer lag times. Within the AKAP79/150 compartment, cAMP lagged behind Ca 2+ by an average of only 13 ± 3 s (mean ± SEM, n = 60); however, cAMP within the general plasma membrane compartment oscillated with a lag time of 47 ± 4 s (n = 24) behind Ca 2+ (Figure 1e). In order to measure these different cAMP dynamics within the same cells, we transfected MIN6 cells with two cAMP sensors of different targeting sequences and colors, AKAP79-(Ci/ Figure 1. The phase of oscillating cAMP is shifted between the AKAP79/150 compartment and the general plasma membrane compartment, relative to Ca 2+ . (a) Depiction of the AKAP79/150 and plasma membrane compartments, including the targeted cAMP biosensor (Ci/Ce)Epac2-camps to measure the compartment-specific cAMP signaling. Schematics of the lyn-(Ci/Ce)Epac2-camps and AKAP79-(Ci/Ce)Epac2-camps sensors are shown above. (b) Network diagram describing the key players in the b cell Ca 2+ -cAMP-PKA oscillatory circuit. (c) Representative single-cell trace of in-phase oscillating responses of AKAP79-(Ci/Ce)Epac2-camps and RCaMP, whole-cell fluorescence measured. Red trace is cAMP (cyan direct channel divided by CY-FRET channel) and black trace is Ca 2+ (RFP). (d) Representative single-cell trace of out-of-phase oscillating responses of lyn-(Ci/Ce)Epac2-camps and RCaMP, whole-cell fluorescence measured. Blue trace is cAMP (cyan direct channel divided by CY-FRET channel) and black trace is Ca 2+ (RFP). (e) Crosscorrelation between the oscillatory Ca 2+ and cAMP signals from the representative in-phase AKAP79 (red) and out-of-phase plasma membrane (PM, blue) responses from c, d. Time lag (sec) between the cAMP and Ca 2+ signals for the two compartments (AKAP79/150, red, is 13 ± 3 sec n=60 and PM, blue, is 47 ± 4 s n=24). ****p<0.0001; unpaired two-tailed Student's t-test. The online version of this article includes the following figure supplement(s) for figure 1:  Ce)Epac2-camps and the red cytosolic cAMP sensor R-FlincA (Ohta et al., 2018). After stimulating the signaling circuit, we observed anti-correlated cAMP oscillations, indicating that cAMP oscillates with different phases between the cytosol/plasma membrane and the immediate vicinity of AKAP79/ 150 within the same cell (n = 25) (Figure 1-figure supplement 2). This stark difference in the cAMP-Ca 2+ phase relationship suggests that the relative phase of this oscillatory circuit is compartmentalized and hints at differential regulation of the circuit between the AKAP79/150 compartment and the general plasma membrane compartment.
Oscillatory phase is regulated by balanced activities of Ca 2+ -sensitive ACs and PDEs Given that in-phase cAMP oscillations were only observed within the AKAP79/150 compartment ( Figure 1c) and that out-of-phase cAMP oscillations were observed in the general plasma membrane compartment ( Figure 1d) and the cytoplasm (Ni et al., 2011), we hypothesized that Ca 2+ oscillations are coupled to cAMP oscillations by a ubiquitous mechanism throughout the cell, while additional mechanisms specifically regulate the phase relationship within the AKAP79/150 compartment. We first sought to identify the component that is responsible for coupling cAMP dynamics to Ca 2+ dynamics globally. Since TEA induces continuous Ca 2+ oscillations, we determined the temporal relationship between Ca 2+ and cAMP at the general plasma membrane more precisely by measuring the impulse response of the circuit following a transient membrane depolarization. After the addition of KCl (15 mM) followed by a subsequent washout to elicit a transient influx of Ca 2+ (Dou et al., 2015), we observed a synchronous cAMP decrease (n = 20) followed by a return to baseline ( Figure 2a). These data suggest that increasing cytosolic Ca 2+ was coupled to a decrease in cAMP at the plasma membrane through Ca 2+ -sensitive AC or PDE activities. Given that Ca 2+ -inhibited ACs (AC5, AC6) have low specific activity in both the presence and absence of physiological Ca 2+ , as well as relatively low expression in the pancreas (Defer et al., 2000), we instead focused on probing the role of PDEs. The Ca 2+ -dependent PDE1 family in b cells, specifically PDE1C, has been implicated in modulating GSIS (Han et al., 1999). Indeed, acute addition of 8MM-IBMX (100 mM), a relatively selective PDE1 inhibitor, effectively uncoupled cAMP dynamics from Ca 2+ oscillations ( Figure 2b, Figure 2-figure supplement 1a) (n = 18), indicating that Ca 2+ -triggered activation of PDE1 mediates the transient cAMP decreases. We also observed that the overall increase in cAMP led to an increase in the Ca 2+ oscillation frequency, consistent with the previously identified role of cAMP/ PKA in regulating the Ca 2+ oscillations (Ni et al., 2011). We tested the roles of two additional families of abundant PDEs in pancreatic b cells, PDE3 and PDE4, by acute pharmacologic inhibition. While treating cells with either milrinone (PDE3 inhibitor, 10 mM, n = 12) or rolipram (PDE4 inhibitor, 1 mM, n = 15) slightly increased cAMP levels, neither inhibitor had an effect on cAMP-Ca 2+ coupling or relative phase (Figure 2-figure supplement 1b,c). These data suggest that PDE1 is the key component that couples Ca 2+ and cAMP oscillations within this signaling circuit.
How is the phase relationship between Ca 2+and cAMP-regulated within distinct signaling compartments? In order to gain a more quantitative understanding of the regulation of the cAMP-Ca 2+ phase relationship, we created a simplified well-mixed mathematical model involving Ca 2+ , cAMP, and Ca 2+ -driven PDE and AC activity components (Cooper et al., 1995;Figure 2c, see Appendix 1 for details). This simple circuit represents the key aspects of the oscillatory cAMP-Ca 2+ circuit and is applicable to different signaling compartments. Opposite to the Ca 2+ -stimulated PDE1 (Ang and Antoni, 2002) is the Ca 2+ -stimulated AC8 (gene Adcy8) (Masada et al., 2009;Masada et al., 2012), an abundant Ca 2+ -sensitive transmembrane AC isoform in b cells that has been shown to mediate sustained insulin secretion and associate with the AKAP79/150 scaffold (Dou et al., 2015;Willoughby and Cooper, 2006;Willoughby et al., 2010). By computationally manipulating the activity of each arm, we found that cAMP can oscillate either out-of-phase or in-phase when a Ca 2+ pulse train is used as an input (Figure 2c). In particular, when the relative activity of PDE1 is greater than the activity of AC8, Ca 2+ -driven cAMP degradation dominates, resulting in an out-of-phase cAMP-Ca 2+ relationship. On the other hand, if the relative activity of AC8 is greater than that of PDE1, Ca 2+ -stimulated cAMP production is favored, and an in-phase relationship is observed, consistent with previous modeling studies (Fridlyand et al., 2007;Peercy et al., 2015).
Thus, our simplified model indicates that the phase relationship can be tuned by altering the relative strength between Ca 2+ -sensitive ACs and PDEs (Figure 2c). This model provided a blueprint for understanding the interplay between the Ca 2+ -stimulated AC/PDE balance and the cAMP-Ca 2+ Figure 2. The oscillation phase is regulated by a balance between Ca 2+ -sensitive AC and PDE activity. (a) Impulse response of plasma membrane cAMP (blue) to a spike in Ca 2+ entry (black), triggered by KCl-mediated membrane depolarization (wash in/out). The transient decrease in PM-cAMP is coupled to the transient increase in intracellular Ca 2+ . (b) Acute inhibition of Ca 2+ -sensitive PDE1 decouples the out-of-phase PM-cAMP oscillations from Ca 2+ oscillations, as observed in this representative cell trace (Ca 2+ -black, PM-cAMP -blue). (c) The oscillatory phase of cAMP can be manipulated by tuning the relative activity of Ca 2+ -sensitive PDE and AC, as demonstrated by a simplified mathematical model. The schematic shows the network architecture. The relative activities of AC and PDE, denoted as v(AC) and v(PDE), control the delay between Ca 2+ and cAMP. Point (i), with high AC activity and low PDE, shows in phase oscillations of Ca 2+ and cAMP, whereas point (ii), with high PDE and low AC activity shows out of phase oscillations. These dynamics are shown in the line graphs. (d) Knocking down AC8 is correlated with an increase in the time lag for oscillatory cAMP at the AKAP79/150 microdomain (37 ± 9 s, n = 11), indicating more cells exhibiting out-of-phase cAMP oscillations (representative cell trace, Ca 2+ -black, AKAP79/150-cAMP -red). (e) Over-expressing AC8 is sufficient to reverse the phase at the PM to in-phase (23 ± 2 s, n = 56) (representative cell trace, Ca 2+ -black, PM cAMP -blue). **p=0.0014, ****p<0.0001; unpaired two-tailed Student's t-test. The online version of this article includes the following figure supplement(s) for figure 2:   phase relationship within the AKAP79/150 compartment. Based on the findings from our model, we predicted that decreasing the relative contribution of AC8 will shift the cAMP-Ca 2+ phase relationship from in-phase to out-of-phase. To test this prediction, we knocked down endogenous AC8 in MIN6 cells as previously done  and observed that most cells exhibited out-ofphase cAMP oscillations within the AKAP79/150 compartment (average lag time 37 ± 9 s, n = 11) (Figure 2d), indicating an AC8-specific role in mediating the cAMP-Ca 2+ phase signature.
Conversely, increasing the relative contribution of AC8, for example, by increasing the concentration of AC8 throughout the PM, should shift the cAMP-Ca 2+ phase relationship from out-of-phase to in-phase. To test this prediction, we overexpressed full-length AC8 and examined the effect in the general plasma membrane compartment. Interestingly, we found that AC8 overexpression reversed the out-of-phase cAMP-Ca 2+ phase relationship in a titratable manner, where the percentage of inphase oscillating cells correlated with increasing amounts of co-transfected AC8 (average lag time 23 ± 2 s, n = 56) (Figure 2e, Figure 2-figure supplement 2a-d). In order to examine the effects of the targeted cAMP biosensors on AC8 expression, we also measured endogenous AC8 levels in cells transfected with either lyn-(Ci/Ce)Epac2-camps or AKAP79-(Ci/Ce)Epac2-camps and observed no significant difference (Figure 2-figure supplement 3a,b). These data demonstrate that higher levels of AC8 are sufficient to reverse the cAMP phase at the plasma membrane. In summary, these phase-manipulation experiments suggest that the cAMP-Ca 2+ phase relationship is representative of a sensitive, compartmentalized balance between the Ca 2+ -stimulated activities of PDE1 and AC8.
Membrane-localized AKAP150:AC8 nanoclusters regulate cAMP-Ca 2+ oscillatory phase The close spatial juxtaposition between the AKAP79/150 and general plasma membrane compartments presents a significant challenge for cAMP compartmentation, in that cAMP oscillations must be distinctly regulated within these adjacent signaling domains. Indeed, how cAMP, a rapidly diffusing small molecule, is spatially compartmentalized in cells is not yet completely understood, especially given the relatively low catalytic efficiency of a single cAMP-producing AC and degrading PDE (Conti et al., 2014;Lohse et al., 2017;Bock et al., 2020;Zhang et al., 2020). Given that AKAP79/ 150 exists in nanoclusters at the plasma membrane in multiple cell types (Mo et al., 2017;Zhang et al., 2016) and associates with AC8 in b cells (Willoughby et al., 2010), we hypothesized that AC8 could form nanoclusters on the plasma membrane of MIN6 cells and compartmentalize cAMP dynamics. To test this hypothesis, we examined the spatial organization of AC8 and AKAP150 at the membrane using Stochastic Optical Reconstruction Microscopy (STORM). We found that AKAP150 molecules were organized in clusters with a mean radius of 127 ± 9 nm and an average nearest-neighbor spacing of 313 ± 20 nm between cluster centers (n = 20) (Figure 3a and Figure 3figure supplement 1), consistent with several recent reports demonstrating AKAP79/150's tendency to form nanoclusters in other cell types (Mo et al., 2017;Purkey et al., 2018;Tajada et al., 2017;Zhang et al., 2016). Thus, the AKAP79/150 compartment-specific cAMP phase is likely representative of the balanced cAMP generation and degradation within these AKAP clusters.
Due to the known interaction between AKAP79/150 and AC8 (Willoughby et al., 2010) and diffusion of membrane-localized signaling complexes, we next probed the spatial organization and mobility of AC8. We found AC8 also distributes non-uniformly at the plasma membrane and forms clusters with a mean radius of 88 ± 8 nm and an average nearest-neighbor spacing of 292 ± 16 nm between cluster centers (n = 16) ( Figure 3b; Figure 3-figure supplement 1). In order to measure the degree of colocalization between AKAP150 and AC8 nanoclusters, we also performed 2-color STORM and observed 72% of AKAP150 localizations were co-clustered with AC8 localizations, suggesting the presence of AKAP150:AC8 co-clustered nanodomains (Figure 3-figure supplement 2a-e). Next, to determine the mobility of AC8 molecules within the timescale of the oscillatory circuit's period, we performed Fluorescence Recovery After Photobleaching (FRAP) with EGFP-tagged AC8 and found AC8 diffuses slowly at the membrane, and that there exists a significant immobile fraction (D AC8 = 0.019 ± 0.002 mm 2 /s, avg. % immobile = 42.3%, n = 16) (Figure 3-figure supplement 3).
With the evidence of the nanoscale organization of AKAP150 and AC8 on the plasma membrane, we further hypothesized that the increased spatial density of Ca 2+ -driven cAMP sources within the AKAP150 clusters, in conjunction with dispersed PDE1 in the cytosol (Bender and Beavo, 2006;Goraya et al., 2008), is important in compartmentalizing cAMP production and mediating the in-phase cAMP signal. To test this idea, we sought to build a mathematical framework to describe the spatial compartmentalization of the in-and out-of-phase cAMP-Ca 2+ oscillations. Briefly, we expanded our network motif model (Figure 2c) by including a 3D spatial component with cAMP diffusion (D cAMP = 60 mm 2 /s; Agarwal et al., 2016) and incorporating our previously published wellmixed b cell model (Ni et al., 2011). We used the AKAP79/150:AC8 cluster pattern measurements from the STORM imaging to set model parameters in a hexagonal prism domain (200 nm edge, 600 nm depth), with one immobile AKAP79/150:AC8 cluster centered in the domain for simulation ( Figure 4a, see Appendix 1 for model development details). By localizing AC8 within the AKAP79/ 150:AC8 cluster on the plasma membrane face and leaving PDE1 well-mixed throughout the volume, we could simulate Ca 2+ -driven cAMP oscillations that were in-phase within the immediate vicinity of a cluster, but sharply transitioned out-of-phase outside the cluster. Specifically, during a Ca 2+ influx Ripley-K analysis measures the average radii of the nanoclusters and indicates that AKAP150 forms clusters of 127 ± 9 nm, n = 20 (uniform random distribution control in inset). The nearest-neighbor distance distribution describes the distance between nanoclusters (average distance for AKAP150 is 313 ± 20 nm). (b) Representative super-resolution STORM image of Ca 2+ -sensitive AC8 (scale 5 mm, inset 500 nm) depicts AC8 nanoclusters of average radius 88 ± 8 nm and average nearest-neighbor distance 292 ± 16 nm, n = 16. The online version of this article includes the following figure supplement(s) for figure 3:   event, Ca 2+ -triggered cAMP production dominated at the center of the AKAP79/150 cluster while Ca 2+ -triggered cAMP degradation was favored outside the cluster at the PM and in the center of the unit volume ( Figure 4a). Not surprisingly, the regime that recapitulates this phase relationship is sensitive to the spatially-restricted AC8/PDE1 balance and the diffusivity of cAMP. For example, by increasing the AC8 concentration within the cluster, the Ca 2+ -driven cAMP generative flux could exceed the rate of cAMP degradation by PDE1 and thus cAMP and Ca 2+ oscillations could exist inphase within both compartments (Figure 4-figure supplement 1). Alternatively, assuming that AC8 clustering is driven by AKAP150:AC8 interactions, weakening this interaction would then reduce the AC8 cluster stabilization and lead to a redistribution of AC8 away from the nanoclusters and a decrease in the local concentration of AC8 within the clusters (Figure 4b). Without the high local concentration of AC8 driving a net positive cAMP production within an AKAP79/150 cluster, the spatial domain where cAMP oscillates in-phase with Ca 2+ is predicted to shrink while the out-of-phase regime expands and can reverse the phase at the cluster center ( Figure 4c).
To test this prediction, we overexpressed the amino terminus of AC8 (AC8 1-106 ) required for interaction with AKAP79/150 (Willoughby et al., 2010) to compete with the binding of endogenous AC8 to the endogenous AKAP150 scaffold. The disruption of the AKAP150:AC8 interaction was validated using a proximity ligation assay (PLA) to visualize the AKAP150:AC8 interaction in situ. Compared to non-transfected cells, cells expressing the AC8 1-106 peptide had a 39 ± 4% reduction in the PLA signals, indicating a decrease in the AKAP150:AC8 interaction ( Figure 5-figure supplement 1a,b). Furthermore, STORM imaging showed that overexpression of the AC8 1-106 peptide led to a decrease in the percentage of AC8 single-molecule localizations within AC8 nanoclusters (n = 9) (Figure 5a), consistent with the predicted redistribution of AC8 molecules (Figure 4b). To test the impact of loss of AC8 molecules from the nanoclusters on the oscillation phase, we measured AKAP79/150-localized cAMP in the presence of AC8 1-106 and observed a significant increase in the average lag time (43 ± 6 s, n = 33) (Figure 5b). This is due to a higher proportion of cells exhibiting out-of-phase cAMP oscillations, indicating that the AKAP79/150:AC8 competitor peptide was Figure 5. Disruption of the AKAP79/150:AC8 interaction is associated with a redistribution of AC8 at the PM and a phase shift of cAMP at the AKAP79/150 nanodomain. (a) Over-expression of the N-terminus of AC8, which is necessary and sufficient for mediating the AKAP79/150:AC8 interaction, redistributes AC8 from within nanoclusters to the general PM, as seen in the STORM image (scale 5 mm, inset 500 nm) and measured by the percent of localizations that fall into nanoclusters. (b) Disruption of the AKAP79/150:AC8 interaction lengthens the time lag between the cAMP (red) and Ca 2+ (black) signals at the AKAP79/150 compartment (avg. time lag in absence of disruptor is 13 ± 3 s, n = 60, and in presence of disruptor 43 ± 6 s, n = 33) due to more cells displaying out-ofphase cAMP oscillations. *p=0.0341, ****p<0.0001; unpaired two-tailed Students t-test. The online version of this article includes the following figure supplement(s) for figure 5: sufficient in reversing the phase relationship in the AKAP79/150 compartment. Interestingly, we also observed many cells displaying irregular Ca 2+ oscillations, as indicated by a disruption in the periodic timing of individual cells' Ca 2+ peaks (Figure 5b, left). This nanoscale perturbation establishes the regulatory role of the AKAP79/150:AC8 interaction in mediating the compartmentalized cAMP-Ca 2+ phase relationship.
AKAP79/150-mediated phase relationship is critical for regulating oscillatory Ca 2+ Next we systematically examined the impact of perturbing the precisely regulated phase relationship within the AKPA79/150 compartment. Due to the modulatory role of PKA in the Ca 2+ -cAMP-PKA oscillatory circuit and the interaction between PKA and AKAP79/150, we wondered how the inphase cAMP oscillations with respect to Ca 2+ are translated into PKA activities and if spatial compartmentalization of the phase relationship is also maintained at the PKA activity level. Therefore, we extended our 3D model to include AKAP79/150-associated PKA (see Appendix 1 for model details). In this extended model, PKA activity oscillations exhibit distinct phase relationships with respect to Ca 2+ within and outside of the AKAP79/150 compartment ( Figure 6-figure supplement  1a). To test this prediction, we fused our FRET-based biosensor for PKA activity (AKAR4) (Depry et al., 2011) to either full-length AKAP79 or the PM-targeting motif and expressed the sensors in MIN6 cells. Upon TEA stimulation, PKA activity was observed to oscillate with a lag time of 25 ± 6 s (n = 15) within the AKAP79/150 compartment but with a lag time of 55 ± 8 s (n = 12) (Figure 6-figure supplement 1b-e) at the general plasma membrane, indicating that the compartmentalized phase relationship is preserved from cAMP to PKA.
Spatiotemporal organization of PKA signaling and its phosphorylation targets via AKAPs have been implicated in regulating several important pathways. For example, PKA has been shown to phosphorylate Ca V 1.2 in an AKAP79/150-dependent manner, and this modification can influence the open probability of the channel (Murphy et al., 2014), suggesting a mechanistic link between local cAMP/PKA activity and global oscillatory Ca 2+ . Thus, we sought to study the functional role of the spatially compartmentalized cAMP-Ca 2+ phase relationship in regulating intracellular Ca 2+ dynamics. We measured Ca 2+ oscillations by RCaMP in the presence of either the EGFP-tagged AKAP79/150: AC8 disruptor peptide, AC8 1-106 , or EGFP alone as a control. Population-wide differences in Ca 2+ dynamics, such as strength and timing, were observed in AC8 1-106 -transfected cells and visualized in heat maps depicting the normalized Ca 2+ signal per cell versus time (Figure 6a). Interestingly, we found that the expression of the disruptor peptide was correlated with a significant decrease in the peak ratio between the second Ca 2+ peak and the first Ca 2+ peak (control average À1.6%, n = 270; AC8 1-108 average À10.8%, n = 562) post TEA addition, indicative of less sustained oscillations (Figure 6b,c). In addition to intracellular Ca 2+ concentration, the precise timing of internal oscillatory events is critical for modulating b cell functions such as glucose homeostasis and pulsatile insulin secretion (Fridlyand et al., 2010). In the presence of the disruptor peptide, cells also exhibited a longer elapsed time between oscillatory Ca 2+ peaks (control average 3.9 ± 0.1 min, n = 270; AC8 1-108 average 4.6 ± 0.1 min, n = 562), suggesting that the timing of the signaling circuit was disturbed (Figure 6b,c). It is well established that besides the precise timing, the regularity of cytoplasmic Ca 2+ in b cells is crucial in mediating pulsatile insulin secretion from the pancreas (Gilon et al., 2002;Schmitz et al., 2002). By stratifying the disruptor peptide-expressing cell population into 'low, 'medium,' and 'high' expressers and performing a blinded classification of responding cells based on the regularity of the Ca 2+ oscillation (see Appendix 1 for details), we found a positive correlation between the percentage of cells exhibiting irregular oscillations and the expression level of the disruptor peptide (42% for low-expressing vs. 68% for high-expressing AC8 1-106 disruptor) (Figure 6d). Taken together, these data signify that the compartmentalized cAMP-Ca 2+ phase relationship regulates the oscillatory Ca 2+ signal and plays an important role in determining the pace, regularity, and sustainability of the Ca 2+ oscillations.

Discussion
Biological oscillations represent a rich way of encoding information using the amplitude and frequency. Here, we show the phase in an oscillatory signaling circuit represents a novel mode of informational encoding for which the phase itself can be spatiotemporally regulated. In the case of the Ca 2+ -cAMP-PKA circuit, the oscillatory cAMP/PKA phase relative to a widespread Ca 2+ signal is distinctly regulated within two adjacent membrane compartments through intracellular organization of scaffolds and signaling effectors. Localized perturbation of this spatial phase signature disrupts global Ca 2+ oscillations and thus has far-reaching consequences on the functional landscape of the b cell.
Compartmentalization of cAMP/PKA signaling is instrumental in processing a diverse set of inputs and mediating specific cellular functions; however, the mechanistic details of compartmentalization await to be fully elucidated (Musheshe et al., 2018). Given the measured kinetic rates of most ACs and PDEs, the generation of local cAMP gradients around single enzymes appears unfeasible (Conti et al., 2014). Context-dependent discrepancies in some of the kinetics (i.e. differences between in vitro and in vivo measurements) and slower cAMP diffusion due to buffering have been shown to be important for cAMP compartmentalization (Bock et al., 2020;Zhang et al., 2020). Here, we propose the nanoscale organization of key cAMP effectors and regulators as a novel mechanism for cAMP compartmentation. Despite the slow rates measured for individual ACs, we computationally and experimentally describe conditions in which the generation of compartmentalized Figure 6. Ca 2+ oscillatory dynamics are affected by expression of the disruptor peptide in b cells. (a) Heatmaps depicting Ca 2+ oscillations for 220 randomly selected cells co-expressing EGFP alone (control, left) or EGFP-tagged AC8 1-106 (AKAP79/150:AC8 disruptor, right), ordered by a mixed parameter describing the time lag between the first two Ca 2+ peaks and the avg. timelag between all Ca 2+ peaks. (b) Schematic describing two Ca 2+ oscillatory parameters: the ratio between the first two Ca 2+ peaks (R = P 2 / P 1 ) and the interpeak timing (t). (c) The peak ratio R is decreased in the presence of the AKAP79/150:AC8 disruptor (left), indicating less of a sustained Ca 2+ oscillatory response. Over-expression of the disruptor also lengthens the timing between peaks (right). (d) The level of disruptor peptide expression is correlated with an increase in the percentage of cells exhibiting irregular oscillations (total n = 562). p=0.0054, ordinary one-way ANOVA. The online version of this article includes the following figure supplement(s) for figure 6:  cAMP can emerge from the clustering of many relatively immobile AC8 enzymes at the membrane and bulk distribution of PDE1 in the cytoplasm. Alterations to this nanoscale organization lead to dysregulated Ca 2+ oscillations, demonstrating the functional importance of maintaining this organization. This system also serves as a nanoscale demonstration of how a cell can translate a global signal (Ca 2+ ) into a compartmentalized signal (cAMP/PKA activity) by local activation and global inhibition, a strategy that is likely utilized in many other cellular contexts (Levchenko et al., 2000;Purvis and Lahav, 2013).
In order to understand the functional impact of the nanoscale architecture, it is important to also consider the discrete scale of such signaling organization. According to our rough estimates, approximately 15-150 individual AC8 molecules comprise AKAP150:AC8 nanoclusters (see Materials and methods for STORM-based estimation). To investigate the role of small molecule numbers, we recast our previous network motif model as a stochastic system and simulated the impact of changing initial conditions of PDE1 and AC8. We found that even in these discrete simulations, the previously noted phase relationships are preserved. As the balance between PDE and AC is perturbed, the system moves from an in-phase to an out-of-phase cAMP response with respect to Ca 2+ . Interestingly, we also noted that increasing the number of AC8 molecules leads to a reduction of the noise in the cAMP response. Thus, we predict that tuning AC8 cluster size might be a biological design strategy of nanoclusters to maintain accurate responses to perturbations ( Figure 6-figure supplement 2ad).
The interplay between Ca 2+ and cAMP is very intricate, and multiple mechanisms could contribute to the observed dysregulated signaling effects upon perturbation of the AKAP79/150-compartmentalized cAMP-Ca 2+ phase (Figure 6a-d). AKAPs can recruit PKA to regulate channel activities (Dell'Acqua et al., 2006;Mo et al., 2017;Torres-Quesada et al., 2017), such as in the regulation of voltage-mediated Ca 2+ entry via PKA-dependent phosphorylation of Ca V 1.2 (Murphy et al., 2014) or the modulation of store-operated Ca 2+ entry by both PKA-dependent STIM1 and Orai1 phosphorylation (Thompson and Shuttleworth, 2015;Zhang et al., 2019). Additional levels of regulatory feedback within the Ca 2+ -cAMP-PKA oscillatory circuit have also been identified, such as a negative feedback loop involving PKA phosphorylation of AC8, thereby fine-tuning the circuit dynamics (Willoughby et al., 2012).
The precisely regulated Ca 2+ and cAMP oscillations are likely functionally important as insulin secretion requires coordination between Ca 2+ , cAMP, and PKA. Localized cAMP/PKA signaling at the AKAP79/150 scaffold and the in-phase oscillations of cAMP and Ca 2+ within this compartment might play a critical role in regulating insulin secretion due to close interactions between AKAP79/ 150 and insulin secretory granules via Ca V 1.2 (Barg et al., 2001). Several important processes and components of the secretory machinery have been identified as targets of PKA signaling here, such as PKA-dependent mobilization of granules (Renströ m et al., 1997) and modulation of the synaptosomal protein SNAP25 (Gao et al., 2016). These PKA phosphorylation events may be highly coordinated with oscillations of Ca 2+ , which drive the exocytosis process for pulsatile insulin secretion. In addition to PKA-dependent secretory control, cAMP has recently been implicated to play a direct role in fusion pore formation via the cAMP-regulated guanine exchange factor Epac (Gucˇek et al., 2019). Compartmentalized phase regulation of the Ca 2+ -cAMP-PKA oscillatory circuit at the AKAP79/150 macromolecular complex is likely involved in the regulation of many b cell processes, in addition to insulin secretion, and more work will be needed to further establish the mechanisms involved in decoding the information embedded in the local phase relationship.
The Ca 2+ -cAMP-PKA oscillatory circuit in pancreatic b cells integrates many important regulators of cellular function, and the precise coordination of each is required for proper signaling control. Here, we have uncovered a spatiotemporal organization of the circuit where the oscillatory phase between cAMP/PKA and Ca 2+ depends on the spatial proximity of the AKAP79/150 scaffold protein and AC8. The construction principles of this signaling nanodomain, including the spatial distributions of sinks and sources, likely represent a generalized strategy for the generation of other compartmentalized signals and provide a unique modality in which cells embed, process, and produce signaling information.

Imaging
Cells were washed twice with Hank's balanced salt solution buffer and maintained in the dark at room temperature. Cells were imaged on a Zeiss Axiovert 200M microscope with a cooled chargecoupled device camera (MicroMAX BFT512, Roper Scientific, Trenton, NJ) controlled by META-FLUOR 6.2 software (Universal Imaging, Downingtown, PA). Dual cyan/yellow emission from FRET used a 420DF20 excitation filter, a 450DRLP dichroic mirror, and two emission filters [475DF40 for CFP and 535DF25 for YFP]. RFP fluorescence was imaged using a 568D55 excitation filter, a 600DRLP dichroic mirror, and a 653DF95 emission filter. GFP fluorescence was imaged using a 480DF30 excitation filter, a 505DRLP dichroic mirror, and a 535DF45 emission filter. YFP fluorescence was imaged using a 495DF40 excitation filter, a 515DRLP dichroic mirror, and a 535DF25 emission filter. These filters were alternated by a filter-changer Lambda 10-2 (Sutter Instruments, Novato, CA). Exposure time was 50-500 ms, and images were taken every 10-30 s. For confocal imaging, images were collected with a C2 plus on a Nikon Ti2 inverted microscope equipped with a Plan Apo lambda 60x oil immersion objective NA 1.4. YFP fluorescence was excited with the 488 nm line from a LU-N4 laser. Images were acquired with a DUVB detector collecting emission from 495 nm to 600 nm with a virtual spectral GaAsP detector controlled by NIS Elements software. The pinhole was set at 30 mm. Frame size was 1024 Â 1024 pix.
For FRAP acquisition, cells were imaged using a C2 plus mounted on a Nikon Ti2 with a 488 nm laser (LU-N4 laser engine, Nikon Instruments) and sensitivity PMT detector (C2-DU3, Nikon) with 525/50 nm bandpass filter. The excitation light is reflected by a dichroic mirror 405/488/561/640 and focused through an Apo 100 Â 1.49 NA objective. Image stacks were acquired in Galvano mode with unidirectional scanning with a 488 nm laser at 1.5% laser power with frame size 512 Â 512 at scan zoom 2, one frame per second and 97.1 mm pinhole size. Small stimulation ROIs were drawn at the plasma membrane. The total FRAP series contained three images before bleaching (obtained with 2 s intervals), two cycles of ROI bleaching with the 488 nm laser at 100% laser power (5 frames at one frame per second), and two minutes of continuous acquisition to monitor fluorescence recovery.

Image analysis
Fluorescence images were background-corrected by subtracting the fluorescence intensity of background with no cells from the emission intensities of cells expressing fluorescent reporters. Cells for experiments using the targeted biosensors were manually segmented from microscopy images based on emission intensity (YFP) and RCaMP (RFP) using custom-written Java and MATLAB scripts and whole-cell fluorescence emission was measured. The ratio of yellow/cyan emission (AKAR4) or cyan/yellow emission (Epac2-camps) and RFP intensity were then calculated at different time points using MATLAB scripts. The values of all-time courses were normalized by dividing each by the average basal value before drug addition. FRAP measurements were background-subtracted, bleachcorrected, and normalized, and fit to a previously described diffusion-dominant model (Ellenberg et al., 1997;Phair et al., 2004). For the AKAP150:AC8 disruption experiments, cells were segmented with custom CellProfiler (Broad Institute) pipelines based on RCaMP (RFP) fluorescence emission, and both RFP and GFP fluorescence intensities (EGFP-AC8 1-106 ) per cell were measured every 30 s. Graphpad Prism eight was used for visualization and statistical analysis. Customwritten MATLAB scripts were used for correlation analysis and regularity classification.

Super-resolution imaging (STORM)
For fixed-cell stochastic optical reconstruction microscopy (STORM) imaging, cells were fixed with 4% paraformaldehyde (PFA) and 0.2% glutaraldehyde (GA) for 20 min and then washed with 100 mM glycine in HBSS to quench the free PFA. Cells were permeabilized and blocked in a permeabilization solution with 0.1% Triton X-100, 0.2% bovine serum albumin, 5% goat serum, and 0.01% sodium azide in HBSS. The cells were then incubated overnight at 4˚C with an anti-AC8 antibody (Abcam, ab196686) at a 1:2000 dilution or an anti-AKAP150 (Millipore Sigma, 07-210) antibody at a 1:500 dilution, followed by 1 to 2 hr with goat anti-rabbit Alexa 647-conjugated antibody (Thermo-Fisher Scientific, A21245) at 1:1000 dilution. For the 2-color STORM experiments, cells were incubated with both the anti-AC8 antibody (1:2000 dilution) and an anti-AKAP79 antibody that recognizes a conserved epitope in AKAP150 (BD, 610314; raised against the C-terminal 248 residues of AKAP79 which shares 70% similarity with a region in AKAP150; 1:500 dilution), followed by a 2 hr incubation with a goat anti-rabbit Alexa 647-conjugated antibody (1:1000 dilution) and a goat antimouse Alexa 568-conjugated antibody (ThermoFisher Scientific, A11031; 1:1000 dilution). The cells were then post-fixed again in 4% PFA and 0.2% GA, quenched with 100 mM glycine in HBSS, and washed with HBSS to prepare for imaging. Immediately before imaging, the medium was changed to STORM-compatible buffer (50 mM Tris-HCl, pH 8.0, 10 mM NaCl, and 10% glucose) with glucose oxidase (560 mg/ml), catalase (170 mg/ml), and mercapto-ethylamide (7.7 mg/ml). STORM images were obtained using a Nikon Ti total internal reflection fluorescence (TIRF) microscope with N-STORM, an Andor IXON3 Ultra DU897 EMCCD, and a 100 Â oil immersion TIRF objective. Photoactivation was driven by a Coherent 405 nm laser, while excitation was driven with a Coherent 647 nm laser or Coherent 561 nm laser.
All image analysis and image reconstruction were performed using both Nikon Elements analysis software and custom-written MATLAB scripts. Blinking correction was performed by implementing the pairwise Distance Distribution Correction (DDC) algorithm (Bohrer et al., 2019). Cluster property measurements were performed using Ripley-K analysis and custom mean-shift MATLAB code for segmentation, as described before (Mo et al., 2017). Co-clustering measurements were made using custom MATLAB scripts implementing Getis-Franklin pattern analysis, as described previously (Getis and Franklin, 1987;Mo et al., 2017). Briefly, we first calculated the degree of local clustering of AKAP150 molecules using the L(r) spatial statistic at r = 200 nm which counts the number of AKAP150 localizations within 200 nm (length scale of nanoclusters) of each AKAP150 localization, normalized appropriately. We then calculated Lcross(r) at r = 200 nm which counts the number of AC8 localizations within 200 nm of each AKAP150 localization, normalized appropriately. Plots of Lcross(200) vs. L(200) were generated and an L threshold of 150 was chosen based on the proportion of localizations found within clusters from the mean-shift segmentation analysis, and used to divide the plot into quadrants to calculate the proportion of AKAP150 localizations co-clustered with AC8 as previously described (Mo et al., 2017).
We obtained a very rough estimate of AC8 molecules within nanoclusters from our blink-corrected STORM datasets by assuming background localizations outside the cell are due to single secondary antibodies with conjugated dyes. We measured an average of 2 localizations per secondary antibody outside the cell and approximately 50 localizations per AC8 nanocluster within the cell. Estimating a labeling stoichiometry of 0.4-1.5 secondary antibodies per primary antibody and 0.4-1.5 primary antibodies per AC8 molecule (Zanacchi et al., 2017;Nieuwenhuizen et al., 2015), we can roughly approximate the number of AC8 molecules per nanocluster to be between 15 and 150 molecules/cluster (N = 3 cells).

Proximity ligation assay
Antibodies for AC8 and AKAP150, mentioned in STORM section, were buffer exchanged into DPBS and conjugated with MINUS or PLUS oligos, following the Sigma DuoLink in situ Probemaker kits. PLA experiments were performed using the Duolink in situ red kit for PLAs according to the provided protocol. The only protocol modification was to extend the amplification time by 50 min. Briefly, cells were fixed and permeabilized as in the STORM experiments before incubation with PLUS and MINUS oligo-conjugated primary antibodies for 30 min at 37˚C each, with washes after each step. Ligation of the nucleotides and amplification of the strand occurred sequentially by incubating cells with first ligase then polymerase and detection solution. PLA experiments with AKAP95 antibodies from different species were used as positive controls in HEK293T cells, and experiments with just one oligo-labeled primary antibody or the other were our negative control. Images were acquired on a Nikon Ti Eclipse epifluorescence scope with z-control and maximum intensity projections were created. A cross section of the DAPI-stained nucleus (3.6-5 mm) was also acquired and the number of dots per cell was counted using the nucleus as reference.

Statistical analysis
Statistics, such as means, SEMs, and comparisons were calculated using Graphpad Prism 8 software. Experiments comparing multiple conditions were analyzed using one-way ANOVA or unpaired t-tests. Statistical significance was assessed using p<0.05 as a cutoff. Experiments were repeated >3 times. Individual cells with high expression of the AKAP79/150-fused biosensors were removed from analysis (see Appendix 1 for details). The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication. Appendix 1 cAMP analysis and quantification AKAP79-(Ci/Ce)Epac2-camps transfected cells displayed cAMP oscillations that were either in-phase or out-of-phase with their respective Ca 2+ signal. This was in sharp contrast to responsive lyn-(Ci/Ce) Epac2-camps transfected cells where all cells yielded only out-of-phase oscillations. We found a strong correlation between the AKAP79-fused sensor expression level and the observed cAMP-Ca 2+ phase relationship, with cells having lower levels of sensor present displaying predominantly inphase cAMP oscillations and cells with higher levels of the AKAP79/150-fused biosensor exhibiting out-of-phase oscillations (Appendix 1-figure 1). Overexpression of the AKAP79 scaffold likely changed the stoichiometry of the signaling complexes and resulted in unsuccessful targeting of the biosensor to functional AKAP79/150 domains, and so in this manuscript we considered only TEAresponsive cells below an AKAP79 expression threshold determined by the YFP fluorescence (Appendix 1- figure 1).

Time lag calculation
Due to the heterogeneity of cellular Ca 2+ and cAMP/PKA activity oscillatory responses in each cell (i.e. variations in frequency, amplitude, and regularity), we sought an applicable metric to describe the phase relationship. Here we measure the lag time (sec) between the Ca 2+ signal trace and the cAMP/PKA activity signal trace. Specifically, we high-pass filtered both the Ca 2+ and cAMP/PKA activity traces (approximately 20 min) to subtract out slowly varying baseline changes, normalized the traces so that the maximum intensity/FRET ratio was set to 1, and then computed the cross-correlation to measure the signal overlap for different lag times. To calculate the lag time, we identified peaks in the cross-correlation passing a peak prominence cutoff and found the absolute value of the shortest lag time corresponding to a peak maximum. For in-phase oscillations, the lag time was typically small (t 20 s) due to the two signal traces oscillating in synchrony. On the other hand, out-ofphase oscillations typically corresponded to longer lag times (t >20 s) due to the anti-phasic relationship seen in the peak timing and peak shape. Analysis was performed with custom scripts in MAT-LAB and Java, and pipelines in CellProfiler.

Quantification of nanodomain perturbation effects on global Ca 2+
In order to measure the effects of AKAP79/150:AC8 disruption on Ca 2+ dynamics, we transiently transfected and expressed AC8 1-106 in MIN6 cells and measured Ca 2+ with RCaMP. For quantification of the interpeak timing and peak ratio, we first selected cells that responded to the TEA treatment, identified Ca 2+ peaks passing a peak prominence cutoff, and finally calculated the avg. time between peak maxima and RFP intensity ratio between the second and first Ca 2+ peak maxima. To find the percentage of cells with regular vs. irregular Ca 2+ oscillations, we randomized all Ca 2+ traces from the experimental and control samples and performed a blinded classification to sort the single cell traces as regular, irregular, or nonresponsive. Analysis was performed with custom scripts in MATLAB and Java, and pipelines in CellProfiler.
Computational modeling Well-mixed system Assumptions . Signaling components are present in large enough quantities that concentration changes are smooth and move in a deterministic fashion. . Well-mixed kinetic rate constants contain conversion factors between compartments, that is membrane to cytosol. . Binding interactions occur rapidly enough such that any kinetic parameter, k, remains constant on surfaces (Berry, 2002). . AKAP does not alter the activity of the catalytic subunits of Protein kinase A (PKA) instead it only affects localization.
. Ca 2+ independent activity of Adenylyl cyclase (AC) is of the same strength as inactive Ca 2+ dependent Adenylyl cyclase (AC8) and stays at a constant value.

Mass-action kinetics
We generated an ordinary differential equation (ODE) for every species using mass-action kinetics for each binding interaction. The law of mass-action states that the rate of a chemical reaction is proportional to the product of the concentration of the reactants raised to the power of their stoichiometric coefficient (Guldberg and Waage, 1879). Mass-action kinetics rely on the assumption that the rate constant, k, is constant over time. For example, consider the one-reaction system: where the forward and backward rates are k 1 and k 2 . The differential equations describing the dynamics of species A; B, and C under mass-action kinetics are:

Michaelis-Menten kinetics
We used Michaelis-Menten kinetics to model the kinetics of enzyme-catalyzed reactions. When a reaction is catalyzed by an enzyme with kinetic properties k cat and K M , S E ! P then the reaction rate is given by For Michaelis-Menten kinetics, concentrations of reactants and products must be in large enough quantities, and one of the following conditions must apply: the concentration of the substrate is much larger than the concentration of products, ½S ) ½p, and/or the energy released in the reaction is very large, DG ( 0.

Model development
We constructed a biochemical network to represent interactions between Ca 2+ and cAMP in b92cells, Figure 1b, (Fridlyand and Philipson, 2016;Ni et al., 2011) after a depolarization event. The computational model considered the dynamics of calcium, potassium, leaky, and calcium-sensitive potassium channels (Appendix 1-table 1). Importantly, we included feedback of PKA with KATP channels and the inclusion of Ca 2+ -sensitive ACs and PDEs. The model contains a total of 92 parameters with 11 free parameters. The values of the parameters are constrained through both previously peer-reviewed publication results (Ang and Antoni, 2002;Boras et al., 2014;Lai et al., 2015;Masada et al., 2009;Ni et al., 2011) and with new experimental results using obtained FRET measurements. To constrain source and sink activation rates, previously published literature of AC and PDE stimulus-response curves (of related isoforms) was utilized (Ang and Antoni, 2002;Masada et al., 2009). COPASI was used to calculate initial guesses for kinetic activation parameters. FRET measurements took precedence over binding curves, especially for the spatial model where further fitting routines were performed to refine the model. Model variations were performed to attain both semi-physiological concentrations (within the ranges of the sensor) and phase (period and relation) information. Predictions are made on qualitative behavior as opposed to quantitative as proper parameter fitting would require much more data than available for this system.
Values and reaction sets used in the well-mixed model can be found in Appendix 1-tables 1-5. The network of interactions was constructed using COPASI (version 4.23, build 184) (http://www.nrcam.uchc.edu, http://copasi.org/) (Hoops et al., 2006). The model was built in COPASI to leverage the inbuilt fitting techniques for initial parameter guesses pre-FRET.

Well-mixed computational results
The network shown in Figure 1b has been shown to exhibit oscillations through cAMP variation due to the action of PKA on IP3 receptors and KATP plasma membrane channels. This network has been studied in many labs previous work (Fridlyand et al., 2007;Fridlyand et al., 2003;Han et al., 1999;Ni et al., 2011) and has been explored to show: . The system can be moved in and out-of-phase through tuning of AC (Fridlyand and Philipson, 2016). . The oscillation rate can be tuned through PKA feedback to KATP channels (Ni et al., 2011). . Both Ca 2+ sensitive PDE and AC is necessary for oscillations to occur (Ni et al., 2011).
Our model results agree with the above findings. Variations in the connection strength of sources and sinks also introduced another findingchanging the component's (source or sink) variability in the regime of Ca 2+ spiking (0.1-1.2 mM) a switch in-phase can also be observed, Appendix 1-figure 2. Compared to well-mixed results, Appendix 1-figure 2a, by decreasing the activation rate of CaMAC from 59.5 to 0.6 s À1 , a switch to out-of-phase can occur, Appendix 1-figure 2c. Yet, we notice how increasing basal PDE activity does not switch the phase Appendix 1-figure 2b, only after subsequently decreasing PDE and CaM association does the phase change Appendix 1-figure 2d. This is due to both changes being required for the activity variation of PDE to outperform that of AC. These findings lead to more questions about how source and sink activities relate in a variable Ca 2+ regime.

Well-mixed reaction tables
Appendix Continued on next page Since we consider the sign we can then characterize the solution by This model was then computationally run to confirm the results with arbitrary parameters (Figure 2c). The system was simulated with all parameters set to 1 except v 1 and v 2 at values of 4 and 2, respectively and vice-versa. The pulse train is ON for 1 sec (S h = 10) and OFF for 3 sec (S 0 = 1), and repeats until the simulation total time is reached.

Numerical implementation of the minimal model
We performed numerical simulations in MATLAB R2018b and checked against analytic solutions provided in the previous section. Minimal model solutions are shown in Figure 2c.

Stochastic implementation of the minimal model
To further explore the possible effects of noise and small numbers of molecules on the simplified four component model, we created a stochastic version of the model. Due to the nature of stochastic simulations, all reactions were rewritten to multistep interactions. In order to allow for oscillatory kinetics at the membrane, a Lotka-Volterra model (in a larger volume, for numerical stability) was utilized to simulate depolarization events shown in Appendix 1-table 8. The system was solved using the Gibson method in the Virtual Cell modeling platform. All Ca 2+ , AC, and PDE reactions are shown in Appendix 1-tables 6-9.
Appendix 1-table 6. Initial conditions for the stochastic model.

Species
Number of molecules in the system