Gβ Regulates Coupling between Actin Oscillators for Cell Polarity and Directional Migration

For directional movement, eukaryotic cells depend on the proper organization of their actin cytoskeleton. This engine of motility is made up of highly dynamic nonequilibrium actin structures such as flashes, oscillations, and traveling waves. In Dictyostelium, oscillatory actin foci interact with signals such as Ras and phosphatidylinositol 3,4,5-trisphosphate (PIP3) to form protrusions. However, how signaling cues tame actin dynamics to produce a pseudopod and guide cellular motility is a critical open question in eukaryotic chemotaxis. Here, we demonstrate that the strength of coupling between individual actin oscillators controls cell polarization and directional movement. We implement an inducible sequestration system to inactivate the heterotrimeric G protein subunit Gβ and find that this acute perturbation triggers persistent, high-amplitude cortical oscillations of F-actin. Actin oscillators that are normally weakly coupled to one another in wild-type cells become strongly synchronized following acute inactivation of Gβ. This global coupling impairs sensing of internal cues during spontaneous polarization and sensing of external cues during directional motility. A simple mathematical model of coupled actin oscillators reveals the importance of appropriate coupling strength for chemotaxis: moderate coupling can increase sensitivity to noisy inputs. Taken together, our data suggest that Gβ regulates the strength of coupling between actin oscillators for efficient polarity and directional migration. As these observations are only possible following acute inhibition of Gβ and are masked by slow compensation in genetic knockouts, our work also shows that acute loss-of-function approaches can complement and extend the reach of classical genetics in Dictyostelium and likely other systems as well.


Introduction
For cells to move, their cytoskeletal structures become spatially organized by internal polarity signals [1][2][3] as well as external chemoattractant [4][5][6]. How such signaling cues tame actin dynamics to produce a pseudopod and guide cellular motility remains a key question in eukaryotic chemotaxis.
By now, several key regulators of the actin cytoskeleton have been identified: in most cells, nucleation promoting factors (NPFs) of the Wiskott-Aldrich Syndrome Protein (WASP) and SCAR/WAVE family stimulate actin nucleation through the Arp2/3 complex and are essential for regulating polarity and motility for cells ranging from Dictyostelium [6,7] to metazoans [8][9][10]. NPFs themselves are regulated by self-association on the plasma membrane [1,11] and actin polymerization-based autoinhibition [1,12,13]; the actin polymer that they generate facilitates the removal of these NPFs from the plasma membrane. These positive and negative feedback interactions of the NPFs [1,14] and other actin regulators give rise to a range of highly dynamic, free-roaming, nonequilibrium actin structures such as flashes and traveling waves [1,2,5,6,[15][16][17][18][19][20][21], but how the actin machinery is coaxed to form these very different activity patterns is not well understood.
Particularly striking displays of NPF and actin dynamics are actin oscillations, which can be observed in many cell types and contexts [1,2,5,22,23]. Biological oscillations are typically generated through a combination of (1) fast positive feedback, which amplifies small signals into an all-or-none output; and (2) delayed inhibition, which turns the output off and resets the system for the next pulse. By spatially coupling oscillators, spreading or synchronization over long distances can be achieved [24][25][26].
Recently, small oscillating SCAR/WAVE foci have been discovered at the periphery of Dictyostelium cells [2]. These foci may constitute the basic cytoskeletal units from which pseudopods are formed. In the absence of signaling cues, these oscillators are present but lead to only small undulations of the cell boundary. In response to upstream signals, however, full-blown protrusions emerge [2,[27][28][29][30][31], likely from the coordination of these foci. Some intracellular signals (such as Ras and phosphatidylinositol 3,4,5-trisphosphate [PIP 3 ]) have been identified that affect this transition, but whether other signals link receptor activation with the SCAR/ WAVE foci, and, more generally, which properties of the foci are modulated to enable largescale coordination, are not known.
Here, we find that the heterotrimeric G-protein subunit Gβ sets the coupling range of an actin-based activator-inhibitor system. Specifically, acute sequestration of Gβ leads to strong global synchronization of normally weakly coupled cytoskeletal oscillators, and these effects are independent of known upstream regulators of these oscillators, such as Ras and PIP 3 . We show that this extended range of spatial coupling is detrimental for cell polarity, cell motility, and directional migration. To guide our intuition for how coupling between oscillators could affect the cell's ability to sense directional cues, we developed a simple mathematical model that represents its minimal features. Simulations show that the ability to sense a noisy input signal is facilitated by an intermediate strength of oscillator coupling, allowing different membrane regions to share information about the stimulus. We propose that in wild-type cells, Gβ sets the coupling strength of actin oscillators to an appropriate level to sense directional upstream cues.

Engineering Rapamycin-Based Acute Inactivation of Gβ
Strong loss-of-function phenotypes in cell motility are rare [6,[32][33][34][35][36][37][38]. One reason may be that genetic perturbations are slow to act and may give cells time to compensate for gene loss [39][40][41][42]. Redundantly controlled processes like actin rearrangements during motility may be particularly susceptible to such compensation. To overcome this limitation, we developed a system that enables fast loss-of-function perturbations to cell signaling events involved in Dictyostelium cell motility. Here, we focus on its application to Gβ.
Heterotrimeric G-proteins consist of one α, β, and γ subunit and link receptor-mediated signals to directed migration and polarization in eukaryotic cells ranging from yeast to neutrophils to Dictyostelium [43][44][45][46]. Both intra-and extracellular signals can regulate the cytoskeleton, yet while knockout of the sole Gβ protein in Dictyostelium completely blocks chemotaxis, basal cytoskeletal dynamics and other directional responses such as shear-flowinduced motility and electrotaxis are still present, although somewhat reduced [2,3,44,47,48].
Gβ requires plasma membrane localization in order to signal; thus, removal from the plasma membrane should prevent it from activating downstream effectors. As Gβ is continually exchanged between membrane and cytoplasm with a half-life of 5 s [49], it should be possible to trap it by association with an internal anchor. We built a Gβ sequestration system using a chemical dimerization approach whereby the association of two protein domains (FKBP and FRB) is induced by the small molecule rapamycin [50][51][52][53][54]. Starting with Gβ-null cells [44], we expressed an FRB-Gβ fusion protein and an endoplasmic reticulum (ER)-localized FKBP (FKBP-calnex-inA [55]). Thus, addition of rapamycin should drive Gβ relocalization to the ER and suppress its signaling function, effectively rendering cells Gβ-null in an acute fashion (Fig 1A).
To test for rapamycin-induced sequestration, we measured the extent of ER-localized Gβ in single cells over time following rapamycin addition. We computed the correlation between each cell's fluorescence intensity in the ER anchor and Gβ channels to assess co-localization. FRB-RFP-Gβ was rapidly sequestered from the plasma membrane and increasingly co-localized in large clusters with FKBP-YFP-calnexinA (S1 Movie). Sequestration is fast: half-maximal correlation occurred 5.6 min after addition of the highest dose (5 μM) of rapamycin that was tolerated by cells (Fig 1B and 1C, S1 Data). Sequestration kinetics were similar for both 5 μM rapamycin and 1 μM rapamycin. Therefore, unless indicated otherwise, we used the lower concentration for subsequent experiments.
Gβ-null cells fail to transmit many signals triggered by G-protein-coupled receptors (GPCRs) [44,[56][57][58][59], and we should be able to recapitulate these defects with our sequestration approach. We thus assayed whether relocalization of Gβ to the ER inhibits transmission of signals from GPCRs to downstream effectors. Stimulating wild-type cells with chemoattractant Inducible protein sequestration as a method to acutely inactivate Gβ. (A) Inducible sequestration can be exploited to inactivate a protein of interest. Using the small molecule rapamycin (RAP), FRB-tagged Gβ can be recruited to an FKBP-tagged "anchor" at the endoplasmic reticulum (ER). Addition of RAP sequesters Gβ from its normal site of action at the plasma membrane and prevents it from activating downstream effectors. (B) Timecourse of Gβ sequestration. In cells lacking endogenous Gβ, but expressing FRB-RFP-Gβ and calnexinA-YFP-FKBP as an anchor at the ER, the speed and extent of sequestration were assayed by measuring the spatial correlation between YFP and RFP signals. For the highest dose of RAP, half-maximal heterodimerization is achieved within 5.6 min. To keep cells immobile, the experiment was performed in the presence of 10 μM latrunculinA. The spatial correlations between fluorescence signals from Gβ and anchor are plotted (n ! 20 cells per condition; mean +/-standard error of the mean [SEM]). Raw data can be found in S1 Data. (C) Representative images from a Gβ sequestration timecourse described in (B). Scale bar = 10 μm. (D) Gβ sequestration recapitulates Gβ-null phenotypes for receptor-stimulated signaling. Timecourses of chemoattractant stimulation (cyclic-AMP [cAMP]; 10 μM) are shown in four strains: wild-type (wt), Gβ-null, and cells expressing one or both components of the Gβ sequestration system. Each strain was stimulated in the presence and absence of rapamycin (5 μM; > 20 min incubation). Blot shows phosphorylation of PKBR1 (T309); Ras is used as a loading control. Schematic indicates the localization of Gβ (in orange) for each condition in test strains and the published localization for wt and Gβ-null cells. Further examples of signaling events blocked after Gβ sequestration can be found in S1 Fig. (cAMP) triggers transient responses, including phosphorylation of PKBR1, and this response is abolished in Gβ-null cells [33,56]. We found that introducing our FRB-Gβ construct in Gβnull cells rescued the PKBR1 response. Acute sequestration of FRB-Gβ to the ER anchor blocked PKBR1 phosphorylation, but only when all three components of our system-the ER anchor, FRB-Gβ, and rapamycin-are present (Fig 1D). Unfortunately experiments using the inducible sequestration system in developed cells were often problematic: Tagged Gβ and anchor components were frequently degraded during starvation and, likely as a consequence, cells failed to complete their developmental cycle. However, this problem was not observed in vegetative cells, in which the sequestration components remained intact. Gβ-dependent, chemoattractant-stimulated responses in vegetative cells, such as Ras activity and PIP 3 production [57,59,60], could also be blocked by Gβ-sequestration (S1 Fig). Taken together, these results demonstrate that in the absence of rapamycin, our inducible sequestration system sustains key Gβ-dependent signaling events. In the presence of rapamycin, Gβ is sequestered from its site of action, thereby blocking receptor-based signaling. In this respect, sequestration of Gβ recapitulates Gβ-null cells.

Gβ Sequestration Impairs Directional Migration
To probe for phenotypes that may only be apparent after rapid loss of Gβ, we turned to directional motility assays. We measured the behavior of Gβ-sequestered cells presented with two different directional cues-an attractive chemical (folate) or electric fields-and compared their responses with wild-type and Gβ-null cells. While chemotaxis is strictly dependent on Gβ, electrotaxis, the directed migration of Dictyostelium cells in response to electric fields, is not. While Gβ-null cells cannot move up a chemical gradient, they can move down electrical potential [44,47].
We took advantage of the heterogeneity in expression of components in Gβ-sequestered cells to internally control experiments. We can distinguish behavior of cells that, in the presence of rapamycin, are functionally wild-type (expressing RFP-FRB-Gβ, but no CFP-FKBPanchor), Gβ-null (with no detectable RFP-FRB-Gβ expressed), or Gβ-sequestered (expressing both RFP-FRB-Gβ and CFP-FKBP-anchor). For chemotaxis, we further compared these populations to true wild-type and true Gβ-null cells.

Gβ Sequestration Drives Large-Scale Oscillations of Cortical F-actin
Closer examination of Gβ-sequestered cells by confocal microscopy revealed a striking change in the organization of the actin cytoskeleton. While wild-type cells have fairly stable levels of cortical and cytoplasmic actin, sequestration of Gβ induces striking oscillations of LimE-GFP, a reporter for dynamic F-actin (Fig 3A and 3B) [61]. Periodic loss of cytoplasmic LimE-GFP intensity is accompanied by a corresponding accumulation of F-actin around the entire periphery of the cell (S2 and S3 Figs). The cytoskeletal oscillations induced by Gβ sequestration are present in the majority of cells and have well-defined characteristics. By automatically tracking cells over time and measuring their cytoplasmic LimE-GFP intensity, we identified oscillating cells from the characteristic peak induced in their Fourier spectrum (S4 Fig). After rapamycin addition, the fraction of oscillating cells rises from 6% to 52%, but only when the ER anchor is co-expressed ( Fig 3C and S1 Data). The period of oscillation (measured as the peak frequency of the Fourier-transformed signal) is tightly controlled across all oscillating Gβ-sequestered cells (12.9 +/-3.2 s, n = 83) (S4 Fig). We also observed a second F-actin phenotype upon acute loss of Gβ. In~10% of cells, waves of F-actin polymerization travel around the cell perimeter with a similar period as the whole field oscillations, taking 10-20 s for a full cycle (S5 Fig and  S3 Movie).
Two lines of evidence confirm that acute Gβ loss of function through sequestration is required to initiate this actin oscillation phenotype. First, oscillations are not observed when the ER is forced into proximity of the plasma membrane, arguing against an ER-specific recruitment phenotype (S3 Fig). Most importantly, when Gβ is overexpressed and sequestered Gβ sequestration impairs directional migration. (A) Cells of the Gβ sequestration strain were incubated with rapamycin and exposed to a gradient of folate. Based on the expression of sequestration components, different subpopulations were identified, and directionality was measured after 30 min of migration. Plotted are the means (+/-S.E.M) of wild-type (wt): Gβ+/anchor-cells (n = 30, red); Gβ-null (Gβ-): Gβ-/anchor-cells (n = 48, green); and Gβsequestered: Gβ+/anchor+ cells (n = 31, yellow). ** indicates a highly significant p-value of < 0.02; n.s. indicates a not-significant p-value of > 0.05 (Student's two tailed t test). Data are derived from five videos in two independent experiments. For comparison, directedness of wt (DH1) and Gβ-cells (n = 97 and n = 98, data from two videos in single experiments, respectively) is shown in light and dark grey bars. Raw data can be found in S1 Data. (B) Cells of the Gβ sequestration strain were incubated with rapamycin and exposed to an electrical field. Based on the expression of sequestration components, different subpopulations were identified, and directionality was measured after 30 min of migration. Plotted are the means (+/-stdev) of wt: Gβ+/anchor-cells (n = 33, red), Gβ-null: Gβ-/anchor-cells (n = 34, green); and Gβ-sequestered: Gβ+/anchor+ cells (n = 34, yellow). ** indicates a highly significant p-value of < 0.01; n. s. indicates a not-significant p-value of > 0.05 (Student's two tailed t test). Data are combined from several fields of view of movies recorded on two separate days. A movie corresponding to the stills in Fig 2B is included as S2 Movie. Raw data can be found in S1 Data.  in wild-type cells (which harbor endogenous Gβ that cannot be recruited), no actin oscillations are induced (Fig 3D and S1 Data).
Individual cells transition abruptly into the oscillatory mode. Oscillations become apparent as soon as rapamycin-induced sequestration of Gβ can be observed ( S6 Fig and S3 Movie) and can continue for days (see later; Fig 4C and S1 Data). By treating cells with both rapamycin (the FKBP-FRB heterodimerizer) and a competitive inhibitor of heterodimerization (the small molecule FK506, an FKBP-FKBP homodimerizer), we titrated Gβ levels over the full dynamic range of the sequestration system (S7 Fig). As the amount of sequestered Gβ is increased, the properties of the oscillating state such as its period and amplitude did not change (S8 Fig). The oscillations have characteristics of an all-or-none behavior: only the percentage of oscillating cells increased (Fig 4A and 4B, S1 Data).
These phenotypes-whole-cell oscillations and traveling waves of actin polymerizationare reminiscent of previously observed actin-based activator-inhibitor systems [1,2,5,6,[16][17][18][19][20]. However, the oscillations we observe here are triggered, persistent, and have an unusually large spatial range and high amplitude. This suggests that acute loss of Gβ pushes the cytoskeleton into an unusual state.

Acute Inactivation of Gβ Differs from Gβ-Null Cells
Our observation that cortical F-actin oscillations follow acute sequestration of Gβ raises a key question: why did previous Gβ-null analyses fail to uncover this striking cytoskeletal phenotype? Consistent with published work [2,3], we find that very few Gβ-null cells display LimE-GFP oscillations cells (Fig 4C and S1 Data). We reasoned that if cells compensate for the loss of Gβ function over time, the phenotype induced by acute sequestration of Gβ should approach the Gβ-null phenotype after sufficient time has passed. Consistent with this hypothesis, the fraction of oscillating cells decreases over days of continuous Gβ sequestration and eventually approaches the small fraction seen in Gβ nulls ( Fig 4C and S1 Data). Similar compensatory phenomena have been previously observed in other Dictyostelium signaling contexts. For example, the effect of LY294002, a PI3K inhibitor, on Dictyostelium cell migration fades during prolonged treatment [2], likely due to compensation by redundant signaling pathways [35]. In another case, the actin nucleator WASP relocalizes to the leading edge and compensates for SCAR/ WAVE function when SCAR/WAVE is deleted [6]. Our findings suggest that a compensatory mechanism is also at work here: the globally oscillating state is suppressed in Gβ-null cells.
Our results highlight the value of using acute inhibition to uncover protein function. We have used rapamycin-induced Gβ sequestration to interrogate loss-of-function phenotypes along two "axes" (Fig 4D). By titrating the amount of sequestration while retaining its fast timescale (axis 1), it is possible to interrogate how a phenotype emerges, distinguishing between an all-or-none or gradual transition. Conversely, varying the timescale of perturbation (axis 2) reveals whether phenomena such as cellular compensation can mask an acutely induced phenotype. Applied to Gβ sequestration, we find that a new phenotype-a globally oscillating Factin cytoskeleton-can be uncovered at points in this "phenotypic space" that are not accessible to standard genetic perturbations.

Whole-Field Oscillations Emerge by Synchronizing Preexisting Oscillators
Multiple oscillating actin foci localize around the cell periphery on the basal surface of chemotactic cells. These foci often originate from previously aborted pseudopods that remain attached to the substrate. Internal and external signaling inputs are thought to entrain these foci, but how their dynamics are controlled for this to happen remains unknown (e.g., Raw data can be found in S1 Data. (C) The percentage of oscillating cells decreases over time during Gβ sequestration and approaches the terminal Gβnull state. Cells were incubated with 1 μM rapamycin, and the fraction of oscillating cells was determined at the timepoints indicated (n > 25 Gβ-sequestered cells per condition). Raw data can be found in S1 Data. (D) Acute inhibition via rapamycin mediated protein sequestration can reveal phenotypes that are not accessible through classic genetic perturbations. First, it can reveal consequences of protein depletion to intermediate levels, such as the gradual or all-or-none emergence of phenotypes (axis 1). Second, rapid inactivation can reveal immediate phenotypes that are not accessible to slower methods of gene inactivation (axis 2). oscillation dynamics are unchanged in Ras, PI3K, and Gβ nulls) [2]. The large-scale cortical actin oscillations we observe here are similar in period to the previously described oscillating foci (13 +/-3 s versus 9 +/-2 s, respectively), suggesting that these two forms of cytoskeletal dynamics may be closely related. Thus, we tested whether our acute sequestration of Gβ would reveal signaling control over these oscillatory actin foci.
To analyze individual actin foci, we collected confocal movies imaged in the plane where cells contact the coverslip. We developed a computational approach to comprehensively track and quantify the dynamics of actin foci by automatically identifying each cell's periphery, subdividing it into ten degree sectors (thereby generating 36 tracked regions per cell), and measuring the mean intensity in each sector over time ( Fig 5A). Consistent with previous results [2], we found large-amplitude oscillations in LimE-GFP intensity in some sectors ( We next addressed how the dynamics of actin foci compare between wild-type (Gβ-unsequestered) cells and Gβ-sequestered cells that exhibit whole-field oscillation. In both cases, individual sectors oscillate. However, the mean LimE intensity across all sectors in Gβ unsequestered cells does not show a marked oscillatory behavior (Fig 5C), whereas the mean intensity of sectors in Gβ-sequestered cells clearly oscillates ( Fig 5D). Thus, the whole-field oscillations we observe upon Gβ-sequestration in the middle plane of cells ( Fig 3A) are also reflected in the behavior of membrane-plane actin foci.
What properties of these individual oscillators change as cells transition to whole-field oscillation? We reasoned that changes in the amplitude, period, or the synchronization in phase between individual oscillating sectors could be responsible. We developed an automated approach using the Hilbert transform [63,64], which has been used extensively to analyze neuronal activity [65,66], to quantify the amplitude, period, and phase of individual oscillators over time (S11 Fig). Using this algorithm, we extracted the oscillation phase (i.e., whether currently at a peak or trough) as well as the instantaneous period (i.e., how fast the phase is changing) at each timepoint. Strikingly, only the phase synchrony differs in Gβ-sequestered cells (

Spatial Coupling Bypasses Established Cytoskeletal Signaling Pathways
Downstream of Gβ, three signaling pathways, defined by PI3K, TORC2, and PLA2, are known to instruct actin-based motility in Dictyostelium (Fig 6D). Ras activity can feed into both PI3K and TORC2, and downstream, Rac activation is thought to connect these signaling modules to the actin cytoskeleton [31,33,38,56]. Enhanced activity of these pathways leads to wider, more stable zones of actin polymerization compared to the isolated oscillating foci.
We investigated whether Gβ uses any of these signaling pathways to regulate spatial coupling of actin foci. First, we analyzed the dynamics of Ras activity, PIP 3 levels, and Rac activity in single cells. Gβ sequestration neither induced oscillations nor caused any other apparent changes to these signaling currencies on a timescale of minutes ( Fig 6A). Second, we perturbed the activities of members of these pathways in wild-type cells to determine whether global LimE oscillations would emerge. Neither inducing Rac activity (Tet-On: GFP-Rac1A[V12]), blocking all three pathways (using a pharmacological cocktail: BEL|LY294002|pp242), nor raising the levels of intracellular Ca 2+ (a messenger commonly oscillating in other systems [22,67]) led to global oscillations of F-actin ( Fig 6B and S1 Data). Third, we interfered with these pathways in Gβ-sequestered oscillatory cells to determine whether their activity was required for synchrony. Acute inhibition of all three pathways caused only a very small decrease in the number of oscillating cells, while unbalancing Ca 2+ levels did not inhibit global oscillations at all (Fig 6C and S1 Data). We conclude that Gβ's control over the coupling range of actin oscillators likely involves a different, currently unidentified mediator.

Increased Spatial Coupling of Oscillators Impairs the Establishment of Cell Polarity
How can hypercoupling between cytoskeletal oscillators lead to a defect in directed cell migration? The coupling state among the oscillators might be an important parameter for upstream cues to polarize the cytoskeleton-a prerequisite for cell motility. To investigate this question, we tracked individual Gβ-sequestration cells over time, simultaneously monitoring cytosolic actin dynamics and cell migration in both the presence and absence of rapamycin.
For this analysis, we returned to confocal imaging in the midplane of the cell. Here, polarization events are distinguished by a relatively stable actin patch that coincides with a substantial drop in cytoplasmic LimE-GFP reporter levels (Fig7A and 7B and S14 Fig). In both control and Gβ-sequestered cells, polarized patches are of similar intensity (S15 Fig), and phases of polarity alternate with apolar phases, which can easily be visualized in t-stack kymographs ( Fig  7A and 7B; left panels). In this representation, the y-axis represents time, and the lateral surface of the cell is shown for each timepoint along the x-axis.
We found that Gβ-sequestered as well as Gβ unsequestered cells were capable of cycling between polarized and apolar states (S4 and S5 Movies). Consistent with our prior results, acute sequestration of Gβ induced large-amplitude oscillations of F-actin. However, long-term imaging revealed that these oscillations are largely restricted to apolar phases-times when the cell is not undergoing protrusion or migration (Fig 7B and S5 Movie). Thus, phases of polarization appear to be incompatible with whole cell oscillations. While increased coupling in Gβsequestered cells did not affect the lifetime of poles once they successfully formed (Fig 7C and S1 Data), Gβ sequestration significantly (p < 10 -4 , Student's two-tailed t test) impaired the establishment of new poles (Fig 7D and S1 Data). Consistent with a reduced number of cell polarization events, sequestered cells translocate at a significantly reduced speed (p < 0.003, Student's two-tailed t test, Fig 7E and S1 Data).
Taken together, our data show that appropriate control of coupling between localized cytoskeletal oscillators is essential for efficient polarization and motility as well as directional sensing. Increasing the strength of coupling-through acute loss of Gβ-synchronizes actin dynamics, which hampers the entrainment of the actin cytoskeleton by both internal polarity cues as well as entrainment by the external cues that are necessary to direct motility (Fig 8).

Oscillator Coupling Is Sufficient to Increase Sensitivity to Noisy Inputs
One of the most remarkable features of chemotaxis is the ability of migrating cells to accurately sense extraordinarily shallow chemical gradients [68]. Previous work has suggested that the signaling network downstream of Gβ plays a crucial role in this input sensing [5,29,69,70]. Here, we have uncovered a separate link between Gβ and the cytoskeleton in tuning coupling between actin oscillators. Might oscillator coupling also play a role in input sensitivity?
We reasoned that oscillator-to-oscillator coupling might represent a means of sharing information between nearby regions of the cell periphery. By comparing noisy receptor-ligand interactions at multiple locations, cells might improve their ability to discriminate signal from noise when choosing a migration direction. To test this hypothesis in a simple context, we built a mathematical model representing input sensing at the cell's periphery (Fig 9). It should be emphasized that this model is not meant to capture the full complexity of the cell's gradient sensing and chemotaxis pathways, but rather represents a minimal model to quantitatively interrogate the essential elements of oscillator-to-oscillator coupling and entrainment to an input. Our model incorporates a circular lattice of actin oscillators representing the cell's cortex. Oscillators are coupled to one another by a term that increases sinusoidally with their difference in phase [71] and can also be coupled to an oscillating input signal using the same mechanism. Although the chemoattractant signals presented to a real cell are unlikely to oscillate in this fashion, the exact mechanism for input coupling is unknown, and our simplifying Similarly, increasing the intracellular concentration of Ca 2+ has no effect. Data represent the means of more than 35 cells from at least 2 d for each condition (+/-stdev). Raw data can be found in S1 Data. (C) Inhibition of core chemotactic regulatory pathways does not abolish Gβ mediated global oscillations of LimE-GFP. Gβ-sequestered, oscillating cells were treated with various drugs to determine their effect on oscillatory behavior. Neither a triple-drug cocktail (BEL|LY294002|pp242) that simultaneously blocks PLA2-, PI3K-, and TORC2-mediated signaling, nor unbalancing Ca 2+ levels (blocking PLC with U73122, supplying Ca 2+ or chelating any Ca 2+ present in the buffer with EGTA) had a significant effect on the presence of global LimE-GFP oscillations. Data represent the means of more than 25 cells from at least 2 d for each condition (+/-stdev). Raw data can be found in S1 Data. (D) Gβ appears to bypass established signaling pathways to regulate the spatial range of coupling. S19 Fig shows data to validate the use of pp242 as an inhibitor for TORC2 mediated signaling in Dictyostelium cells.  assumption allowed us to model oscillator-to-input and oscillator-to-oscillator coupling in a single unified framework. Our model includes three parameters that define the coupling between an external input and the nearby membrane (k IN ) and the coupling between membrane oscillators (parameters k 1 and k 2 for input-coupled and non-input-coupled membrane oscillators). We also include a term (σ) to represent noise in input-to-oscillator coupling.
Our model reproduced well-known features of coupled oscillator systems. Increasing oscillator-to-oscillator coupling showed an abrupt transition to global synchrony, consistent with prior work modeling the synchronization of weakly coupled oscillators as a phase transition (S16 Fig) [68,69]. This is analogous to the effect observed after Gβ sequestration, in which the transition to global oscillation appears to be all-or-none in individual cells (Fig 4 and S1 Data).
To test how coupled oscillators are affected by features of the input signal, we set out to determine how oscillator-to-oscillator coupling affected sensing of weak inputs (low values of k IN ) or noisy inputs (high values of σ). We found that increasing coupling could not improve sensing of weak noise-free inputs but rather led to spontaneous synchronization as coupling strength is increased (S21 Fig). In contrast, oscillator-to-oscillator coupling markedly improved sensing of noisy inputs (Fig 9). For simulations with little or no coupling, the effect of noise was dominant, and membrane oscillators were unable to accurately couple to inputs (Fig 9; k 1 = 0.1). Conversely, for very strong coupling, oscillators became synchronized to one another so Fig 8. Acute loss of Gβ induces a hypercoupled cytoskeleton. By synchronizing weakly coupled peripheral oscillators, a hypercoupled state is induced that is apparent as whole-field oscillations. This pathologic state is less permissive to the establishment of cell polarity and continuous realignment of polarity in a gradient. We suggest that this hypercoupled state prevents oscillators from becoming patterned by upstream signaling cues from inside or outside the cell. strongly that they were completely input-insensitive (Fig 9; k 1 = 3.5) [25,71]. Between these two extremes, our model revealed an optimum of input sensitivity at an intermediate coupling strength (Fig 9; k 1 = 2.5).
If weak oscillator-to-oscillator coupling was indeed beneficial for input sensing, one would expect wild-type cells to exhibit some coupling between oscillating foci. Indeed, we find experimentally that in wild-type cells the relative phases of oscillators are not random but loosely correlated (Fig 5E, asynchrony; phase distribution width Θ 50 < 90 [deg.]; S12 Fig and S1 Data). Thus, we propose that upstream signaling cues optimally entrain the cytoskeleton when the coupling strength between its dynamic units is of intermediate strength.
Electrotaxis revealed this new role for Gβ in directed cell migration. However, we expect the link between Gβ and cytoskeletal dynamics to be essential for interpreting other cues as well. During chemotaxis, when the activity of the Gαβγ heterotrimer is proportional to the amount of the chemical signal the cell experiences [75], fine control of the magnitude and intracellular distribution [76] of oscillator coupling may be possible.
In future work, it will be important to learn more about how Gβ exerts this control. Common signaling pathways involving PI3K, TORC2, and PLA2 appear to be not essential. Similarly, perturbing levels of Ca 2+ , a messenger known to oscillate in many systems [22,67], including chemotaxing Physarum polycephalum cells [77], shows no obvious effect on coupling between actin oscillators. P. polycephalum, however, is a beautiful, conceptual precedent for the idea that cell movement may be governed by the coupling between independent oscillators: in this organism, periodic streams of small pieces of cytoplasm can become entrained to each other, which, through further modulation by attractants or repellants, supports directional movement [78].
Recent evidence in Dictyostelium shows that Gβ interacts with Elmo, which suggests a possible direct link to the cytoskeleton bypassing the other signaling pathways [79]. We observed oscillation of HSPC-300, a member of the actin-nucleating SCAR/WAVE complex. This may be the most upstream oscillator, with F-actin reporters and disassembly factors (e.g., Coronin) following its dynamics. In this case, SCAR/WAVE's relevant regulators will need to be identified [80]. Mechanistically, how could loss of Gβ increase the strength of coupling? Based on the mechanisms through which oscillators are coupled in other systems, possible explanations include (1) increasing the density of oscillators at the periphery while keeping the coupling range of each constant [81], and/or (2) directly increasing the range of a diffusible or mechanical signal that is generated by the oscillators [26]. Our experimental data support the first hypothesis. In strongly coupled Gβ-sequestered cells, a larger fraction of sectors contain actin oscillators compared to Gβ-unsequestered cells or Gβ-null cells (S17A Fig). Moreover, upon Gβ sequestration, the number of membrane sectors that contain an actin oscillator increases, while the amplitude of the oscillators remains constant (S17B Fig). Additionally, we find that during cell polarization, oscillators largely disappear from the sides and back of the cell (S18 Fig). Taken together, our data suggest that the number or density of oscillators is regulated, and this may be used as a mechanism to control coupling strength. Additional mechanisms could affect the firing threshold or the refractory period of the oscillators. Our simple mathematical model helps to guide intuition on why coupling between oscillators could be advantageous for polarity and directional movement. For both cases, the signals that need to be interpreted can be noisy, and in these scenarios moderate coupling between oscillators can provide an advantage-input-coupled oscillators can "share" information to filter noise and better entrain to an input signal. Our results are consistent with recent predictions in bacterial chemotaxis, in which an optimal membrane distribution of receptors balances sensitivity to spatially correlated external noise and spatially uncorrelated intrinsic noise (which can be filtered out by a similar mechanism of local information sharing) [82].

The Benefits of Acute Perturbations: A Novel Cytoskeletal Role for Gβ
Our work highlights limitations in classical genetic approaches. Genetic nulls are the most common means of assaying gene function in Dictyostelium. However, many genetic mutants give no or mild phenotypes, and they often require combined hits in multiple signaling pathways to significantly inhibit chemotaxis [6,[32][33][34][35][36]. In theory, two mechanisms can account for this: a selection on the population levels can favor a subset of cells (potentially carrying suppressor mutations) that best cope with the genetic change. Alternatively, intrinsic redundancy with parallel pathways or slow compensation via negative feedback can obscure the true role of a gene in cell behavior [42]. Such compensation enables robust function and is a widely employed characteristic of adaptive/homeostatic systems. For example, pharmacological inhibition of synapses transiently inhibits signal transmission, but homeostatic mechanisms restore function within minutes [39][40][41]. The motor of bacteria is another example. It compensates for persistent changes in the level of internal signaling components to maintain the robustness of chemotaxis [41].
Which mechanism is at play in our case? Both Gβ-null knockout and wild-type cells lack excessive coupling of actin oscillators, albeit likely for different reasons. In wild-type cells, Gβ suppresses the coupling, while Gβ-null knockout cells have, over time, arrived at a Gβ independent steady state that does not support oscillations. Transformation of Gβ-null knockout cells with the sequesterable Gβ construct restores wild-type physiology, which can become transiently unbalanced upon acute Gβ sequestration. This imbalance remains for days-long enough for us to observe its effect on oscillator coupling-but eventually the steady state of Gβnull knockout cells is assumed again, potentially due to compensation from parallel pathways. We favor this possibility over genetic suppression based on the speed with which oscillations disappear again after induction.
As a consequence of compensation, different modes of gene inactivation can result in strikingly different phenotypes. In zebrafish, gene knockdowns can produce strong phenotypes that are masked by compensation in genetic knockouts [83]. Our data suggest that additional phenotypes appear when proteins become inactivated even more rapidly. Gβ-knockout and knockdown cells have been extensively studied in Dictyostelium and other systems [43,44,84]. Although defects have been reported for a wide range of chemoattractant-stimulated responses, including directed migration [44], these cells display normal basal polarity and actin dynamics [2,3]. Acute sequestration was essential to uncover the role of Gβ in tuning cytoskeletal dynamics and initiating cell polarity. In this light, our work suggests that much can be learned by revisiting classical mutants with acute perturbation approaches, and not only in instances in which a loss-of-function mutation is lethal.

Dictyostelium Cell Culture and Sequestration Experiments
Dictyostelium strains were grown at 22°C in HL5 medium (ForMedium) in Nunclon tissue culture dishes or in suspension in flasks shaken at 180 rpm. Cells were routinely used from nonaxenic cultures. In this case, cells were grown in association with Klebsiella aerogenes (K.a.) on SM agar plates and used for assays when bacteria began to get cleared [85]. Growth under these conditions gave the strongest responses to stimulation with folate, so this condition was used for most subsequent rapamycin-mediated sequestration experiments. However, sequestration of Gβ also induced oscillations in F-actin when cells were grown in HL-5 instead. For imaging experiments, a scrap of cells was seeded in 200 μl HL5 in a Lab-Tek II 8 well chamber (Nunc), allowed to settle, and washed one to two times in KK2 (16.5 mM KH 2 PO 4 , 3.9 mM K 2 HPO 4 , 2 mM MgSO 4 ) immediately before the assay. Rapamycin (SIGMA) was freshly prepared at 2 μM in KK2 and added 1:1 in sequestration experiments. To render cells responsive to cAMP (Fig 1C), aggregation-competent amoebae were prepared by resuspending washed cells at 2 x 10 7 cells/ml in KK2, starving them for 1 h while shaking at 180 rpm, followed by pulsing the cells with 70-90 nM cAMP (final conc.) for another 4 h. Before stimulation with 1 μM cAMP, cells were basalated (shaking at 180 rpm in the presence of 5 mM caffeine for 20 min) with or without 5 μM rapamycin, washed in ice-cold KK2, and kept on ice until stimulation. For analysis by western blotting, samples were resolved on 4%-15% SDS-PAGE gels. After electrophoresis, proteins were transferred to PVDF membrane and probed with antiphospho-PKC (pan) antibody from Cell Signaling Technology (190D10), which was used to detect the activation loop (T309) phosphorylation of PKBR1, and anti-pan-Ras antibody from EMD (Ab-3). Sequestration of Gβ was confirmed by fluorescence microscopy.

Microscopy
A spinning disc Nikon Eclipse Ti fitted with a spinning disc head, 405 nm, 488 nm, and a 561 nm laser line and appropriate emission filters were used to record CFP, RFP, and GFP (or YFP) double-or triple-labeled cells at room temperature. Images were routinely recorded using a 60x (1.45 NA) objective, a Clara Interline CCD camera (Andor Technologies), and NIS Elements software. After analysis, when necessary for presentation, contrast was adjusted uniformly using ImageJ or Photoshop, and to image sets of some experiments a uniform Gaussian Blur was applied. To quantify oscillations, a single two-or three-channel image was taken to assess Gβ sequestration, followed by a 2-min movie (1 frame/second) to record behavior in the reporter channel at the lowest laser intensity necessary for reasonable signal-to-noise. Longer imaging periods (10 min) and/or adjustment of the focal plane close to the coverslip were used when necessary (e.g., to record individual oscillating foci or alternating polar and apolar states).

Electrotaxis Experiments
The electric fields were applied as previously described for vegetative Dictyostelium cells [88] by using μ-Slides (Ibidi). These tissue-culture-treated slides with small cross-sectional area provide high resistance to current flow and minimized Joule heating during experiments. To eliminate toxic products from the electrodes that might be harmful to cells, agar salt bridges made with 1% agar gel in Steinberg's salt solution were used to connect silver/silver chloride electrodes in beakers of Steinberg's salt solution to pools of excess developing buffer (5 mM Na 2 HPO 4 , 5 mM KH 2 PO 4 , 1 mM CaCl 2 , and 2 mM MgCl 2 , pH 6.5) [89] at either side of the chamber slide. EF strength is empirically chosen (~10V/cm) based on our previous experience [90] and measured by a voltmeter before and after each experiment. Fields of HO547 cells were chosen based on the presence of Gβ and anchor expressing cells, which were distinguished by fluorescence imaging (see Microscopy section for details). High-definition DIC movies (1 frame/30 s) were recorded at room temperature for at least 30 min after the electric field was switched on. To quantify directionality and speed, time-lapse images were imported into Ima-geJ (http://rsbweb.nih.gov/ij). Tracks were marked by using the MtrackJ tool and plotted by using the Chemotaxis tool described [91]. All experiments were repeated and produced similar results. Data are combined and presented as means +/-SEM (standard error). To compare group differences, unpaired, two-tailed Student's t test was used. A p-value of less than 0.05 is considered significant.

Folic Acid Chemotaxis Experiments
HO543, DH1, or LW6 (Gβ null) cells were grown in HL5 medium containing 20 μg/ml G418 and 50 μg/ml hygromycin. Two days before the experiment, 2x10 5 cells were mixed with an overnight culture of K.a. in 250 μl streptomycin-free HL-5 medium and plated on an SM agar plate. On the day of the experiment, cells were washed off the SM plate with DB buffer, washed once, and resuspended in DB at 2x10 7 cells/ml. Suitable amount of cells were transferred to LabTek II chambered coverglass (Nalge Nunc) containing DB with 5 μM rapamycin and 0.05% DMSO. For folic acid chemotaxis, Femtotips microcapillary pipettes (Eppendorf) filled with 1 mM folic acid were used. Microscopy for this set of experiments was carried out with a Nikon Eclipse TiE microscope illuminated by an Ar laser (YFP) and a diode laser (RFP). Time-lapse images in bright field, YFP, and RFP channels were acquired by a Photometrics Evolve EMCCD camera controlled by Nikon NIS-Elements. Tracks of cell migration were analyzed in ImageJ to obtain directedness and speed of cells.

Automated Identification of Cells and Subcellular Regions
For all other analyses, cells were identified, tracked, and processed to extract various properties (e.g., cytoplasmic fluorescence, membrane fluorescence, extent of polarization, angle of polarization) using custom code written in Matlab. First, initial locations for each cell were provided by hand-drawn masks such that each mask contains a single cell at the first timepoint. At each subsequent timepoint, each cell was tracked by extracting a 100x100 pixel box centered at that cell's prior location in the LimE-GFP fluorescent channel. To identify the cell within this box region, interior pixels were separated from background intensity using a fixed intensity threshold, followed by binary erosion with a single-pixel structuring element (to remove isolated noncell pixels) and a hole-filling operation (to fill all pixels within the cell). The largest connected component within this image was assumed to be the cell.
For each cell and at each timepoint, we extracted the following features: • Centroid: The "middle" of the cell • Center of mass: The intensity-weighted center of mass of the LimE-GFP channel (e.g., cells with a bright actin pole would have a center of mass biased toward the pole).
• Cytoplasmic intensity: The mean LimE-GFP intensity was extracted from a disk with a radius of 10 pixels, centered at the cell's centroid.
• Cell membrane: From a cell's mask at each timepoint, we subtract a mask that has been eroded by a disk of radius 5 to identify a 10-pixel-wide "rim" around the cell.
• Membrane sector intensity: By extending lines from the cell's centroid in 10-degree increments, we subdivided the cell into 36 equal-angle regions. The intensity was then measured in a region formed by their intersection with the previously identified membrane region. The sector size was chosen because it was sufficiently small to be unlikely to contain multiple foci; doubling the number of sectors did not qualitatively change our results.
• Gβ-anchor correlation: To measure the extent of sequestration of Gβ to the ER at each timepoint, we computed the correlation of all cellular pixels (including both membrane and cytoplasm) between the Gβ and ER channels using each cell's mask as described above.

Identifying Cytoplasmic Oscillation
To identify which cells in a population were oscillating and characterize the timescale of oscillation, we turned to a Fourier approach (for the analyses of Figs 3 and 4). We found that the cytoplasmic LimE-GFP levels undergo strong, regular periodic fluctuations. From each cytoplasmic intensity timecourse, we subtracted a 30 s moving average to center cytoplasmic fluctuations on a mean value of zero (eliminating intensity fluctuations during cell movement or photobleaching) and computed the discrete Fourier transform of this mean-centered signal. Cells were then marked as "oscillating" if any sampling frequency between 0.05 and 0.2 Hz contained at least 10% of the cytoplasmic signal's total power (see S4 Fig for oscillating and nonoscillating representative cells). These frequencies correspond to periods ranging from 5 to 20 s, which covered the range of frequencies we observed in a preliminary analysis across more than 50 oscillating cells. Each cell's oscillation frequency was then taken to be the sampling frequency at which the power was maximal.

Analysis of Dual Reporter Movies
To understand how cortical LimE dynamics relate to those of other cytoskeletal factors, we sought to correlate LimE-RFP with other reporters (GFP fusions to HSPC300, Coronin, the ABD actin binding domain of ABP120, and Arp2). To identify cells expressing both LimE and a second reporter, we thresholded cells using both GFP and RFP fluorescence. The cell's cortex was identified as a 5-pixel-wide shell of this thresholded image for each cell. To compute the intensity of cytoskeletal foci around the cell's cortex, we then subdivided the cortex into 36 equal-angle segments (sweeping out 10 degrees each) and measured the fluorescence intensity in both the GFP and RFP channels. We then sought to compare the temporal dynamics of GFP and RFP in each spatial region from each cell. To do so, we calculated the cross-correlation between these two channels. For uncorrelated cytoskeletal factors (e.g., myosin, paxillin), we found that dynamics in GFP and RFP were uncorrelated, leading to a low-magnitude, flat cross-correlation. For correlated cytoskeletal factors (e.g., HSPC300, Coronin, Arp2, and the actin binding domain ABD), the crosscorrelation peaked at the characteristic delay time between LimE and that particular cytoskeletal factor. We estimated this delay time by fitting a Gaussian distribution to the cross-correlation to identify the location of this peak-the resulting delay times are shown in Fig 5B.

Measuring Polarization and Identifying Polarized and Unpolarized Time Periods
From the centroid and center of mass measurements described above, the direction and extent of polarization was determined by computing the vector between the center of mass (c) and centroid (ñ).p

¼c Àñ
The magnitude ofp describes the extent of polarization, while its direction reflects the pole's orientation.
We were also interested in identifying periods of time in which cells exhibit long-term, stable polarization (for the analyses of Fig 7). By inspecting many cell trajectories, we found that stable polarization was associated with a consistent direction of polarity-cells would retain a pole with a similar directional orientation, and changes in direction were associated with the formation of a new pole. Conversely, during unpolarized phases, fluctuations of actin around the membrane would lead to frequent changes in the direction ofp (S14 Fig; lower panels). Thus, we implemented a greedy search algorithm to find continuous periods of time when the angle of polarization was contained in a 1-radian window and lasted at least 25 s, and measured the number and duration of these polar regions for each cell (S14 Fig shows two representative cells).

Computing Hilbert Transform; Instantaneous Phase and Period
To assess the synchrony of oscillation between different membrane regions of a cell, we set out to measure each region's oscillation phase at each timepoint. The phase of oscillation describes the current position of an oscillating signal on a sinusoidal curve (i.e., the rising or falling edge), and periodically rises from 0 to 2π. Thus, by comparing the phases between different regions of the membrane, we could assess whether they were oscillating in synchrony, with the phase rising and falling together, or whether at a single timepoint different membrane regions were at different points in their oscillating trajectories.
The analytic representation of a signal provided by the Hilbert transform is an ideal way to measure instantaneous properties of a signal containing periodic fluctuations such as the oscillation phase. For the time-varying LimE-GFP intensity in the n th membrane sector x n (t), the analytic signalx Phase measurement can be improved by first applying a low-pass filter to avoid noisy fluctuations from being interpreted as oscillation. Thus, we first applied a low-pass filter (an 8 th order Butterworth filter with a cutoff of 0.2 Hz) to each membrane trajectory before calculating its Hilbert transform, using custom Matlab code. We found this procedure to yield highly robust measurements of oscillation phase (S11 Fig) in both Gβ-sequestered and Gβ-functional cells. The instantaneous frequencies we measure from this approach are closely centered at~10 s (S9 Fig) and are strikingly similar to those measured by Fourier analysis of cytoplasmic oscillation (Fig 3).

Computing Synchrony
To assess synchrony between different membrane regions, we measured the breadth of spread in oscillation phase between them, at all timepoints during oscillation. We first computed the "group phase"-the vector sum of all regions' individual phases, weighted identically.

Speed Determination for Fig 7E
To characterize the migration of Gβ-sequestered and Gβ-unsequestered cells, we tracked individual cells during 10 min movies, where fluorescent images were acquired once per second. Cells were automatically segmented by thresholding the fluorescent channel, and the centroid of each cell was automatically determined at each timepoint. At least 28 cells were tracked in each condition. From each cell's centroid data, we calculated the root-mean-squared displacement x rms over time for each cell, choosing 300 distinct 5-min intervals for each cell during the where d = 2 is the dimensionality, D is the diffusion constant, and t is the current time. From this model, we estimated the diffusion coefficient for each cell, and computed the p-value for a difference in diffusion coefficients between Gβ-sequestered and Gβ-unsequestered cells (Fig 7E).

Modeling
Constructing a simple model of coupled oscillators and input sensing. To get some insight into how oscillator coupling can affect input sensing, we built a simple model incorporating the essential elements of this process. We reiterate that the goal of this model is not to provide a detailed account of the full biochemical network of either chemoattractant sensing or cell polarization, and a number of excellent models have already been published for both [5,69,92]. Rather, we are interested in whether we could construct a simple system to understand if coupling between individual oscillators can improve the system's ability to entrain to an external input, and under what circumstances this may play an important role.
Modeling the cell membrane as a set of weakly coupled oscillators. Our model consists of N membrane domains arranged in a circle, each representing a single cytoskeletal oscillator. We assume each oscillator has a defined phase θ i , which progresses from 0 to 2π at a constant rate over one period. Each oscillator's frequency ω i is drawn randomly from a uniform distribution on the interval [ω 0 −δω, ω 0 +δω] (to account for this random sampling, each simulation was run at least 20 times, starting at different random initial conditions). Thus, the phase over time can be represented by the following expression: The following subsections will describe how we implement terms to account for oscillatorto-oscillator coupling and input-to-oscillator coupling.
Incorporating oscillator coupling. To implement coupling between oscillators, we assume that each oscillator θ j has an effect on an oscillator θ i , speeding it up or slowing it down in proportion to the difference in their phase. We based this relationship on the well-known Kuramoto model of coupling in populations of oscillators [25,71].
We also modeled an input source coupling to the first M oscillators. This input represents a localized source of activation, such as a spatially restricted source of chemoattractant.
We are interested in the effects of Gβ tuning the coupling between input-coupled oscillators (G protein-coupled receptors are activated locally upon chemoattractant binding, suggesting that Gβ could locally influence oscillator coupling). We therefore focused on one model in which the input sensing increases oscillator coupling: in this model, the coupling strength parameter is increased for all oscillators that receive an input stimulus. Thus, Incorporating input coupling. We sought a simple way to implement oscillator coupling to an input, such that this coupling could be easily measured and is compatible with the modeling framework described above. We chose to describe the input as another oscillator that autonomously runs at a frequency o IN ¼ 1 2 o 0 . The input frequency is therefore easily distinguishable from the natural frequencies of all membrane oscillators, and we can implement input coupling using the same mathematical term as for oscillator-to-oscillator coupling within the membrane.
Finally, we hypothesized that oscillator-to-oscillator synchrony might improve coupling in the case that input sensing is noisy-thus, sharing information between input-coupled oscillators may improve their ability to detect the input signal. To model input noise, we incorporated a single noise term η(t) in input-to-oscillator coupling. The noise function η(t) is drawn from a normal distribution at discrete sampling times (the sampling rate is chosen to be many times faster than the oscillation period). Taking into account all of these interactions, our final model can be represented as: Z½t k $ Nð0; sÞ Measuring synchrony. As a metric of synchrony, we reported what proportion of the total simulation time we observed input-coupled oscillators (membrane locations 1 to M) oscillating at a similar frequency to our input. Two oscillators were said to have similar frequencies when their frequencies differed by less than Δω = 0.1 rad/s, thus satisfying the inequality As a control for our methodology, we also measured the input entrainment of oscillators without direct input-coupling terms (i.e., membrane locations M+1 to N). Broadly, we never observed substantial input coupling by these nonmembrane coupled regions, with coupling reported <10% of the time.

Modeling Results
Strong oscillator-to-oscillator coupling leads to global, synchronized oscillation (S16 Fig). We first tested whether increasing oscillator-to-oscillator coupling (represented by parameter k 2 ) led to the expected increase in cell-cell coupling. This experiment is input-independent (the number of oscillators coupled to input, M, is set to 0), which renders other parameters (k 1 , k IN , σ) completely dispensable. Consistent with prior results describing the synchronization of weakly coupled oscillators as an abrupt phase transition, we found spontaneous large-scale synchrony emerge at a critical coupling strength of approximately k 2 = 0.13 (S16 Fig). The parameter set used is shown in Table 1. Oscillator-to-oscillator coupling does not increase sensitivity to weak, noise-free inputs (S21 Fig). We next implemented input coupling to our membrane oscillators, with a coupling strength of k IN . We set out to understand how oscillator-to-input coupling and oscillator-tooscillator coupling interact with one another, by varying the parameters (k 1 and k IN , respectively) that determine their strength (see S21A Fig). We found that strong oscillator-to-input coupling (high k IN ) drove complete frequency synchronization of membrane oscillators to the input, although with a constant phase lag (S21A and S21B Fig; upper panel).
Importantly, increased oscillator-to-oscillator coupling only served to weaken input sensing in this model (see  Table 2. Oscillator-to-oscillator coupling can increase sensitivity to noisy inputs (Fig 9). Are there any circumstances under which oscillator-to-oscillator coupling can improve input sensing? We hypothesized that coupling could provide a means of "sharing information" between membrane regions to better filter signal from noise in a noisy input (such as in the case of stochastic ligand-receptor binding within each local region of the plasma membrane). To test this hypothesis, we implemented a noise term in the model's input coupling and varied oscillatorto-oscillator coupling to test for an increase or decrease in synchrony. The parameter set used is shown in Table 3.
In contrast to the case of deterministic input sensing, noisy input sensing showed a clear benefit to intermediate levels of oscillator-to-oscillator coupling (Fig 9). For weak coupling, noise destroyed the ability for the membrane oscillators to lock onto the input (Fig 9, k 1 = 0.1). For strong coupling, oscillators synchronized to each other, as seen for high coupling strength in our noise-free models (Fig 9, k 1 = 3.5). However, for intermediate coupling strength, an enhancement of input sensing was clearly apparent, both from trajectories and from a metric assessing the fraction of time spent coupled to the input, which rose from approximately 50% to 75% (Fig 9,    Schematic of data processing steps to assess cytoplasmic LimE-GFP oscillations. Slow fluctuations in mean intensity were removed from each single-cell cytoplasmic trajectory by subtracting a 30 s moving average. The Fourier transform for each trajectory was then computed and normalized to the same total signal power to account for differences in reporter expression level and oscillation amplitude. When a single frequency peak contained more than 10% of the total signal power, a trajectory was considered oscillating. The peak frequency was also measured.