From microscope to model: rotating signaling dynamics and cluster size in Dictyostelium discoideum aggregates

Abstract

Background

When starved, Dictyostelium discoideum cells form clusters that, when constrained in height, exhibit clear collective rotational motion. During this collective rotation, cells secrete and relay the chemoattractant cAMP, resulting in signal propagation in the form of spiral waves, which rotate in the opposite direction of the collective rotation. The quantification of this collective rotation with reference to cAMP signaling and the dependence on cluster size is currently unclear.

Results

In this study, we use experiments and modeling to investigate how cell motion and cAMP signaling dynamics depend on cluster sizes in Dictyostelium using aggregates that are confined to quasi-2D environments. For rotating clusters with a single cAMP spiral arm, we find that both cAMP wave and cell angular velocities decrease as cluster sizes increase, suggesting that larger clusters signal more slowly. We also show that the angular velocity of the cells () is linearly correlated with the angular velocity of the cAMP wave (). For clusters with multiple spiral arms, we observe a size dependence where larger clusters have a higher probability of having more spiral arms compared to smaller clusters. The qualitative experimental results are consistent with a computational model that couples a reaction diffusion model to cell motility and chemotaxis. Our simulation results suggest that a variable degradation rate of cAMP, which depends on cluster size, is a possible mechanism to explain the cluster size-dependent wave and cell angular velocities.

Conclusion

Together, these findings identify cluster size as a key regulator of both cAMP wave dynamics and collective cell motion and link slower signaling and enhanced multi-armed spiral formation to larger clusters. More broadly, our work suggests that size-dependent modulation of extracellular cAMP degradation can couple cell number to emergent spatiotemporal patterns in developing Dictyostelium aggregates.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12860-026-00593-9.

Background

Dictyostelium discoideum is a widely studied eukaryote and an NIH-recognized model organism. Some significant features include its robust growth and similarities with key cellular processes that are observed in major biological systems such as wound healing, cancer metastasis, and embryogenesis [1]. Dictyostelium cells undergo a life cycle after starvation that involves transitioning from single cells to multi-cellular aggregates. These aggregates then cluster together to form mounds, which transform into motile slugs before their final transition into a fruiting body. All these transitions are initiated by a chemoattractant, cyclic adenosine 3‘,5’-monophosphate (cAMP), that acts as a signal between cells and aggregates [2]. This chemical signaling regulates gene expression and furthers development through chemotactic aggregation and, eventually, cell differentiation [3]. When sufficient extracellular cAMP is bound to a cell’s receptors, it triggers an intracellular cascade of events that stimulates the synthesis and secretion of additional cAMP and phosphodiesterase (PDE), an enzyme that breaks down cAMP. The secreted cAMP molecules will then diffuse to the neighboring cells, triggering further cAMP secretion, and thus relaying the signal throughout the entire population. Additionally, as part of the intracellular cascade of events, cells also polarize in response to the external cAMP gradient by redistributing proteins to create a ‘front’ and ‘back’ of the cell, allowing for directed migration in the direction opposite to the signal propagation. This entire process creates a chemical wave that propagates through a population of cells and can be visualized using intracellular fluorescent reporters (Flamindo 2). These fluorescent reporters undergo conformational changes when bound to cAMP, resulting in a decrease in fluorescence intensity [4, 5]. This method allows for visualization of the signal in a single cell and any propagation through a population of cells.

Many experimental studies have sought to understand the cAMP signaling in Dictyostelium systems, both in a single cell [6–8] and at the population level [9–11]. There have also been attempts to understand these dynamics through theoretical studies [12–16]. Most of these studies, however, are limited to either the single cell, aggregation, or the slug stage. A detailed discussion on both the chemical signaling and the cell motility in the pre-mound formation stage is lacking. One key component of the pre-mound formation stage is the collective rotation of cells. As cells continue to signal to each other, they merge together to form small clusters, which subsequently coalesce into larger clusters. These clusters, containing anywhere from a few cells to a few thousands of cells depending on cluster size, collectively rotate in either clockwise or counterclockwise direction. As reported in earlier work, these aggregates also exhibit cAMP waves that rotate like spiral waves in the direction opposite to the cell rotation. The number of spiral arms within each cluster is not fixed and can vary [17–23]. A quantitative investigation of the rotational properties of cAMP waves and cells in rotating clusters remains unclear.

This study aims to characterize the rotational properties of cell motion and cAMP signaling in the pre-mound stage clusters in more detail, using a combination of experiments and numerical simulations. We focus on cluster size as the key player in determining a number of signaling properties within a cluster, including the angular velocity of the cAMP wave (), the cell angular velocity (), and the number of spiral arms. We investigate this experimentally with GFP tagged Dictyostelium cell aggregates (AX4 Flamindo2) under agar confinement to analyze the rotations and cAMP signaling properties in clusters of various sizes. The agar confinement method is commonly used to improve visualization of cells and their cAMP signaling by restricting motion in the third dimension, allowing dynamics to be viewed in an effectively 2D slice [21]. Our experimental results show that in larger aggregates both and decrease. Furthermore, we demonstrate that and are linearly correlated. Finally, we also show that spirals with multiple arms are more probable within larger aggregates compared to smaller aggregates.

To gain more insight into the detailed mechanisms at play, we simulate the system using the Cellular Potts Model (CPM), a well-developed lattice-based computational model capable of simulating collective behaviors arising from interactions among many cells [24–28]. When incorporating cAMP signaling, cell motility and chemotaxis, and cAMP degradation through PDE, we show that this model is insufficient on its own to explain the experimental results. Specifically, it fails to replicate the cluster size dependence on cell and wave dynamics. However, when we systematically vary parameters that control extracellular cAMP levels, we find good agreement with experimental data, suggesting a cluster size dependent mechanism for extracellular cAMP modulation. Overall, this study provides a concise framework showing how cluster size influences signaling waves and cell motility in cluster rotations, supplying a foundation for subsequent studies to formalize a mechanistic understanding of what happens to these cells during different stages of development.

Materials and methods

Strains and cell culture

Dictyostelium discoideum cells (axenic strain AX2) expressing Flamindo2, a GFP based indicator for internal cAMP concentration, were grown in HL5 medium (including Glucose, 35.5 g HL5 powder per 1 L media [Formedium]) with Penicillin-Streptomycin (10,000 U/mL; Gibco, Thermo Fisher Scientific, USA) and Geneticin Selective Antibiotic (G418 Sulfate, 50 mg/mL, Gibco, Thermo Fisher Scientific, USA) in a shaking flask (165rpm) at room temperature (). For experiments, cells were prepared by first being starved by washing three times via centrifugation (3000rpm for 3 min) with buffer (14.6 mM , 5.4 mM , 100μ M, pH 6.4) to remove all nutrients. Cells were then plated for imaging.

Agar slice preparation for confinement

To prepare the agar slices used for confining the cells to 2D, 2% Agarose (w/v) was dissolved in DI water by stirring (20 g per 1 liter of water). The cloudy mixture was microwaved for 10-20s at a time until boiling, followed by stirring until transparent. To prepare the slices, three glass slides were arranged in an ‘H’ shape, where the two ‘walls’ on either side acted as spacers while the middle slide held the agar layer. 1 mL of hot agar mixture was pipetted onto the middle slide and another glass slide was used to quickly press down on top of the spacers to create the agar layer. This was held in place until the agar mixture solidified (within 30s). With a razor blade, thin agar slices were cut as needed [29]. This method is also outlined in dictybase as the Agar Overlay Method.

Plating cells for imaging

To plate the cells, 50–200μ L of starving cells (roughly cells/mL) were pipetted onto a Petri dish. After allowing the cells to settle to the surface, a thin agar slice was placed over the cells using forceps, restricting the cells in the third dimension [29]. The petri dish was covered during imaging to prevent the agar layer from drying out.

Imaging and analysis

Imaging began when cells formed rotating clusters. Starving cells were imaged every 30s using a spinning-disk confocal Zeiss Axio Observer inverted microscope and a Roper Cascade QuantEM 512SC camera. Differential Interference Contrast (DIC) and Fluorescent (GFP) images were taken using a 10X objective and viewed using Slidebook 6 (Intelligent Imaging Innovations). Since cells are tagged with the fluorescent reporter for cAMP, the fluorescent intensity decreases when the internal cAMP concentration increases, resulting in clearly visible cAMP propagation. Movie lengths varied between 2 hours and 14 hours. All images were viewed by ImageJ and analyzed using custom-made Matlab (R2024b) codes.

The angular velocity of the spiral cAMP wave () was analyzed from the GFP images by first splitting the cluster up into N = 72 wedges () to compute a per-wedge intensity time series. An autocorrelation analysis was performed to find the best period corresponding to the time between intensity peaks for each wedge. We then used the median period (T) per wedge to get the local period at each radius in the cluster. We report values from a narrow concentric ring around the cluster some fixed distance away from the center and the boundary. For the single arm spiral wave quantification, we analyzed 17 different clusters over 4 different experiments (days). Each data point represents the angular velocity corresponding to the median period for the ring (found from ).

The quantification of the angular velocity of the cells () was performed using Optical Flow Farneback in Matlab to track per-pixel radial and tangential velocities across frames. To plot this velocity as a function of r, we measured velocities in annular bins of fixed radial width dr. The data points for cell angular velocity were taken at the radius which corresponds to the maximum tangential velocity, roughly between and for all clusters. This maximum tangential velocity was used to calculate from . Error bars represent the median absolute deviation, defined as the median of the absolute differences between each data point and the median.

For each frame, the number of spiral arms was visually quantified and the probability distributions were determined.The number of spiral arms for a cluster was determined by the largest number of stable arms exhibited by a cluster for at least two full wave rotations (i.e. if the final stable state of the cluster has two arms for at least two full rotations, it is referenced here as a double-armed cluster). For n = 1 spiral arms, the size distribution of clusters was computed from 17 different clusters, while for n = 2 spiral arms we analyzed 7 different clusters from 3 different experiments (days).

Computational model

To quantitatively investigate cell movement within clusters, we developed a computational platform using the Cellular Potts Model (CPM), which has been utilized in diverse biophysical studies, including cell sorting and advection-diffusion properties in biological contexts [27, 30, 31]. In the CPM, each cell occupies a set of grid points and the evolution of a cell is determined by an energy minimization process. Our CPM incorporates cell deformation, cell polarity, chemotaxis, and cell-cell adhesion. In addition, our model simulates cAMP secretion, treating cells as excitable entities [32, 33], and it solves the resulting cAMP dynamics using a reaction-diffusion equation, using two different functional forms of the stimulus function. Full model details, equations, and numerical details are described in Supplementary Material 1. Furthermore, model parameter values are listed in Table S1.

Results

Experimental results

To investigate rotating signaling dynamics in Dictyostelium aggregates, we imaged different clusters in the pre-mound formation stage using an agar layer to confine the z-axis for better visualization, as described in the Methods section. With GFP expression for internal cAMP concentration, we simultaneously visualized the cell motility and the cAMP wave propagation within a cluster over time. Figure 1 shows representative examples of the cell and wave dynamics observed inside a cluster. We show two smaller clusters exhibiting single-armed spiral waves in the top panel (Fig. 1A) and one larger cluster with a double-armed spiral wave in the bottom panel (Fig. 1B). The first five images (starting from the left) in each panel show a time progression of the spiral arm from the GFP images where the white line represents the axis of the spiral arm. A single-armed wave extends from the center of the cluster to the outer perimeter and rotates around the cluster like a pinwheel with one arm. The double-armed waves appear like a pinwheel with two arms, where the arms are opposite each other and also extend from the center to the outer perimeter. The movies corresponding to these snapshots are shown in the Supplementary Material (2). We should note that we occasionally observed the formation of a core in aggregates, consistent with previous studies [12], but the experiments did not provide sufficient data to draw firm conclusions about this phenomenon.

Fig. 2.

cAMP wave and cell dynamics in Dictyostelium clusters confined in 2D. experimental snapshots showing the time progression of a Dictyostelium cluster with a single-armed spiral wave (A) and a double-armed spiral wave (B), visualized using Flamindo2-GFP. High/low fluorescent intensity corresponds to low/high intracelular cAMP concentration. White lines show axis of rotation to denote the location of the spiral arm(s). The t = 0 reference is arbitrarily chosen. The sixth image in both panels shows the cell velocity vectors with the size representing the magnitude of the velocities (scaled for visual purposes). Cell velocity was computed within the region indicated by the cyan line. Cells move in the opposite direction from the spiral wave. In all panels, the color of the arrows represent the direction with red/blue corresponding to counterclockwise/clockwise rotation

The last (rightmost) image in Fig. 1 shows the corresponding velocity flow vectors on the DIC image to quantify cell motility (obtained from Optical Flow using Matlab). The size, color and directions of the arrows represent the rotation magnitude (scaled for visual clarity) and direction (clockwise = blue or counterclockwise = red). These vectors were computed by excluding the outermost 8–12 pixels of the cluster, indicated by the cyan line. For each cluster, the spiral wave rotation is always opposite the direction of the cell rotation, as shown by the color and direction of the vector arrows.

We next quantified , obtained by analyzing the pixel intensities from the GFP images, as outlined in the Methods section. For 17 different clusters exhibiting single armed spiral waves, we observe the following: is found to decrease with cluster size, with values ranging between 1.8 rad/min for small clusters (4,000) and 0.8 rad/min for large clusters (70,000) (Fig. 2A). This cluster size range is equivalent to 50 to 1000 cells in a flattened pancake-shaped cluster (estimated using a circular cell with radius 5μ m). Fig. 2B shows as a function of normalized cluster radius (each radius divided by the maximum cluster radius) for three clusters of varying sizes (4,000 (blue), 35,000 (red), and 55,000 (yellow)). The large uncertainty at small radii for the blue curve in this panel arises from errors in determining the rotational center of the aggregate. Our results show that cells within smaller clusters generally have larger absolute angular velocities than cells that are part of bigger clusters. To further quantify the dependence of this angular velocity on cluster size, we computed at a radial distance corresponding to the maximum tangential velocity (see Methods). This clearly decreases as a function of cluster size (Fig. 2C), with values (0.1 - 0.5 rad/min) that are much lower than . Combining the data in Fig. 2A and C, we also observe a linear relationship between and with an of 0.824 (Fig. 2D). Together, the data presented so far shows that larger clusters with more cells exhibit slower cell rotations and slower angular cAMP wave velocities compared to smaller clusters with fewer cells and that these velocities are linearly correlated.

Fig. 1.

Cluster size and cAMP and cell dynamics. (A) Angular velocity of a single-armed cAMP spiral wave as a function of cluster size. (B) The angular velocity of the cell as a function of radius for three different cluster sizes. (C) Scatter plot of the angular velocity of the cells for different cluster sizes. (D) Angular velocity of a single-armed spiral wave vs. The angular velocity of the cell within a cluster. The straight line is a fit to the data with slope equal to 2.27. The error bars and shading represent the median-absolute-deviation

We evaluated the maximum number of spiral arms exhibited by a cluster and compared that to the size of the cluster to understand if size plays any role. In doing so, we found that larger clusters have a higher probability of having more spiral arms. Figure 3A shows the probability of observing single- and double- armed spiral waves for different cluster sizes, revealing that larger clusters are more likely to have more than one spiral arm compared to smaller clusters. Figure 3B presents the cluster-size distribution for single- and double- armed spirals, revealing a statistically significant difference between the two (p = 0.0008). These results show that double-armed spiral waves are more likely in larger clusters than in smaller clusters.

Fig. 3.

Spiral arm number and cluster size. (A) Probability of observing single- (blue) and double- armed (red) spiral waves for different cluster sizes. No clusters were present in the 80–100 cluster size bin. (B) Probability distribution of cluster area for single- and double-armed spiral waves. Error bars are bootstrapped 95% confidence intervals. Blue boxes represent the interquartile range (25th-75th percentiles) and the red horizontal line represents the median

Numerical results

We next turned to simulations, using a CPM consisting of motile, deformable cells that incorporates polarity, chemotactic motility, and cell-cell adhesion. The chemotactic motility is incorporated by assuming a higher preference of cell migration along the local direction of the cAMP gradient. In addition, to capture the persistent motion displayed by Dictyostelium cells, the cell polarity is assumed to align with the cell velocity such that cells have a tendency to migrate without changing direction. Furthermore, the model describes cAMP dynamics with reaction-diffusion equations which include cAMP secretion, diffusion, and degradation (see Methods and Supplementary Material 1 for full details). Example simulation snapshots are shown in Fig. 4, where we have used a similar color scheme as in the experimental figure (i.e., high/low intensity corresponds to low/high intracellular cAMP concentration). As in the experiments, the cAMP signaling takes on the form of a spiral wave, either single- (top row and Supplementary Material 3) or double- armed (lower row and Supplementary Material 3). Similar to Fig.1, the arrows in the last (rightmost) panel indicate the direction and magnitude of the cell’s velocity. Consistent with the experiments, cells move in the opposite direction from the wave direction.

Fig. 4.

Computational results for cAMP wave and cell rotations. snapshots of simulations exhibiting a single-armed spiral wave (A) and double-armed spiral wave (B) at four successive time points for N=200 cells. All model parameters were chosen to be size-independent. White lines represent the axis of the spiral arm, blue arrows indicate clockwise rotational direction, green regions represent low cAMP concentration, and t=0 is chosen arbitrarily. The final panel shows the corresponding velocity vector field of the cells at the last simulation time step. The size of the red arrows (denoting counterclockwise rotation) are proportional to the magnitude of velocity of cells. For visual purposes, only a few velocity vectors are plotted

To investigate the effect of cell size on wave dynamics and cell velocity, we carried out simulations in which we varied the number of cells from 100 to 300 cells in steps of 50, corresponding to a range of cluster sizes from . For each size, we performed 10 simulations and measured . Interestingly, this quantification reveals that in these simulations, this velocity remains roughly constant and is independent of cluster size (red symbols; Fig. 5A), in contrast to the experimental result (Fig. 2A). This result is perhaps not surprising since the model parameters (listed in Table S1) are size independent. This indicates that the propagation speed of a cAMP wave, which is largely determined by its diffusive properties and by cell-to-cell relay, will not depend on the size of the cluster.

Fig. 5.

Quantification of cAMP and cell dynamics in simulations. (A) as a function of cluster size. The red symbols represent the angular velocity when the degradation rate is constant and size-independent () while the blue symbols correspond to the model with size-dependent degradation (). (B) as a function of radius (normalized by the cluster radius) for different cluster sizes. (C) as a function of cluster size. (D) plotted as a function of . The dashed line shows the linear fit. The error bars represent the standard deviation

Motivated by this observation, we asked ourselves how could become size dependent. Since it is unlikely that the properties of the cells change as a function of the cluster size, we did not modify model parameters related to intracellular signaling, cell area, secretion, or polarization. Instead, we focused on how extracellular cAMP influences wave propagation and, in particular, on the role of PDE, the enzyme responsible for the degradation of cAMP. The regulation of extracellular cAMP by PDE has been extensively investigated in earlier theoretical and experimental studies, which identified cAMP degradation as a key control parameter for oscillatory signaling and for deviations from scale-invariant behavior [15, 34, 35]. In contrast to these studies, which emphasized cell density, our experiments focus on aggregates with different cell numbers and show that cluster size is associated with systematic changes in both wave and cell rotational dynamics. We therefore propose that the size dependence of the wave angular velocity reflects changes in PDE levels. Since PDE is diffusible [36], a constant production rate per cell would lead to higher extracellular PDE concentrations in larger clusters and consequently lower extracellular cAMP levels. This reduction in external cAMP concentration will result in a weaker stimulus, slowing cells’ intracellular signaling rate and consequently reducing the wave propagation velocity.

To verify this mechanism, we modeled the cAMP degradation rate, β, to be proportional to the number of cells in a cluster, such that this rate is equal to the original rate for our smallest cluster (N=100), but becomes smaller when we increase the number of cells in the simulation: . In Supplementary Material 1, we show, by explicitly solving for the PDE concentration, that this linear dependence on the cluster size is an excellent approximation (Fig. S1). The simulation results of this modified model are shown in Fig. 5A as a blue line and demonstrate that this simple assumption reproduces the wave speed as observed in the experiment for single-armed clusters. In particular, the simulations show that now decreases with cluster size, with values ranging from 1.25 - 1 rad/min, representative of the experimental results (Fig. 2A).

We also quantified as a function of cluster size. Fig. 5B shows as a function of the radius (normalized by the maximum radius) for different cluster sizes. For each cluster size, the angular velocity is not constant but decreases with increasing radius. In other words, the cluster does not rotate as a rigid body; instead, cells continuously slide past one another, with those farther from the center exhibiting greater linear velocities. Since the linear velocities of cells in the simulations are bounded by an upper limit, the angular velocity at the perimeter of a cluster (Fig. 5C) must diminish as the cluster grows. Consistent with this interpretation, a similarly strong decline in angular velocity is found even in the model assuming size-independent degradation, demonstrating that size-dependent degradation is not essential to account for this behavior.

We can also examine the correlation between the two angular velocities. Consistent with the experiments, our simulations also show a linear relationship between and (Fig. 5D, ). Clearly, this linear correlation is only present in the model version with size-dependent degradation, indicating the importance of the extracellular cAMP concentration throughout the development process.

Finally, we quantified the number of spiral wave arms as a function of cluster size. For this, we initialized simulations with either single- or double- armed spiral configurations for both activator and inhibitor concentrations.For each size, 10 ensemble simulations were performed and the number of spiral arms in the final configurations, obtained after iterations, was recorded. Since, in some simulations, the spiral structure disappeared during evolution, we determined the prevalence of the single- and double- armed spirals and the conditional surviving probability as

where E(n) denotes the number of simulation events in which the final configurations exhibit n spiral arms. With this definition, a higher value of P(n) indicates a greater survival probability of a given spiral configuration within clusters of a specified size.

Fig. 6 shows the surviving probability of the spiral arm number obtained for different cluster sizes using the model with size-dependent degradation. Consistent with the experiments (Fig. 3A), double-armed spirals tend to be more prevalent in larger clusters, while smaller clusters only support single-armed spiral waves. Moreover, the average cluster area increases with the number of spiral arms (Fig. 6B). This trend largely agrees with our experimental observations (Fig. 3A & B), although in our largest simulated clusters, the probability of forming a double-armed spiral is only about 50%, whereas the largest experimental clusters always exhibit double armed spirals. This discrepancy is likely attributable to computational limitations, which required us to restrict the simulated cluster-size range to values smaller than those probed experimentally.

Fig. 6.

Quantification of the number of spiral arms in simulations. (A) Surviving probability distribution of single- and double- armed spiral waves across different cluster sizes. The degradation rate β is set to . (B) Distribution of the average cluster size for different spiral arm numbers (center red line shows the median cluster size, where the bottom and top of the box correspond to the 25th and 75th percentiles, and whiskers represent the maximum and minimum cluster area)

Discussion

In this work, we combined quantitative experiments and simulations to examine how aggregate size controls rotating cAMP waves and collective cell motion in Dictyostelium discoideum under quasi-2D confinement [21]. We found that both and decrease with cluster area, and that these two velocities are linearly correlated over nearly an order of magnitude in cluster size (Figs. 2 and 5). In addition, larger aggregates are more likely to host multi-armed spiral waves than smaller aggregates (Figs. 3, and 6). A Cellular Potts Model (CPM), which models deformable cells as a collection of grid points [24–28], coupled to an excitable reaction–diffusion description of cAMP signaling, qualitatively reproduces several key features of the experiments, including spiral cAMP waves, counter-rotation of waves and cells, and a higher likelihood of multi-armed spirals in larger clusters. However, the model requires a size-dependent modulation of extracellular cAMP degradation to capture the observed cluster-size dependence of angular velocities.

The direction of wave propagation is always opposite to the direction of cell rotation (Fig. 1). The wave propagation is unidirectional because of the refractory property of the cell - cells that have just secreted cAMP are in a refractory state and are unable to respond to a new stimulus. At the same time, cells at the wave front sense the cAMP gradient and chemotax towards the direction opposite to the wave propagation. When cells move into the wave back (where the direction of the cAMP gradient changes), they exhibit a memory effect, which has been extensively discussed in other works [37–41], that prevents them from suddenly switching direction and allows for persistent motion along the direction opposite to the wave. This asymmetry between refractoriness, excitation, and cell memory makes counter-rotation of waves and cells a generic outcome of chemotaxis in an excitable medium.

Our experiments show that there is a linear relationship between the angular velocity of the wave and the cell (Fig. 2D). The extrapolated intercept in this figure, where the cell velocity would vanish, corresponding to a wave period of approximately 9 min. This is shorter than the timescale reported in experiments in which single cells were exposed to waves with a well-defined period [38, 39]. In those studies, cells were able to fully or partially ignore the back of the wave, and thus exhibit memory, when the wave period was below ≈20 min. This difference may reflect both the later developmental stage of the cells studied here and the collective signaling dynamics within aggregates. In addition, the extrapolation to zero cell speed should be interpreted with caution, since a state with stalled cells but sustained coherent wave rotation is unlikely to be physically realizable.

Our experimental measurements further show that larger clusters rotate more slowly and exhibit slower cAMP waves than smaller clusters. For single-armed spirals, decreases by more than a factor of two across the observed cluster-size range, and shows a similar decline (Fig. 2). Within a given cluster, the angular velocity is not spatially uniform but decreases with radius: cells near the center rotate more rapidly than those near the periphery. This indicates that the aggregates do not behave as rigidly rotating bodies and that cells continuously slide past each other along circular trajectories. Since cell speeds are bounded by a maximum linear velocity, the circular geometry of the clusters implies that the angular velocity at the cluster edge must decrease as the radius grows. Thus, the size dependence of already arises from purely kinematic constraints and is also present in our model when the cAMP degradation rate is size-independent.

In contrast, the experimentally observed decrease in with cluster size does not emerge in simulations in which all parameters are size-independent. In that case, the cAMP wave speed in the simulations is independent of cluster area (Fig. 5A). Because wave propagation in the FitzHugh–Nagumo description of intracellular signaling is largely controlled by the excitability of the cell, which is set by the balance between cAMP production, diffusion, and degradation, we asked whether extracellular degradation could be size dependent. Extracellular cAMP diffuses with a diffusion constant determined by its molecular properties, while extracellular PDE is secreted by cells and itself diffuses in the medium. If each cell secretes PDE at a roughly constant rate, the average extracellular PDE concentration, and thus the effective cAMP degradation rate, should increase with the number of cells in an aggregate. One might expect the effects of PDE and cAMP to negate each other since larger clusters have more cells, which synthesize more cAMP and PDE. However, since cAMP and PDE have physiologically different diffusion and degradation values, the two do not cancel each other out. Incorporating this mechanism in the model by making the degradation rate proportional to cluster size is sufficient to transform a size-independent wave speed into a decreasing function of cluster area that matches the experimental trend and restores the linear correlation between and (Fig. 5A & D).

This provides a potential mechanism by which aggregate size can influence excitability. Extracellular PDE is required to reduce external cAMP to levels that allow cells to respond to successive waves [42, 43] and is involved in regulating cluster sizes [44]. In this interpretation, increased effective degradation in larger clusters lowers extracellular cAMP, thereby reducing the stimulus experienced by cells and slowing wave propagation. Changes in PDE activity can therefore be translated into changes in wave angular velocity. Thus, PDE activity, and more generally mechanisms that regulate extracellular cAMP availability, may help tune excitability during the transition from small aggregating groups to larger rotating pre-mound structures. We emphasize, however, that this size-dependent degradation mechanism should be viewed as a plausible hypothesis rather than a unique explanation. Our current data do not directly measure extracellular PDE concentrations or degradation rates as a function of cluster size or developmental stage.

This size-dependent mechanism may act alongside, but appears distinct from, known density-dependent mechanisms of aggregation. Previous work has shown that cell density and extracellular factors can regulate responsiveness to cAMP [3, 45]. However, in the loose mound stage studied here, cells are already locally packed, implying that cell density alone is unlikely to explain the observed size-dependent changes in wave frequency and rotational motion. Instead, our findings point to mechanisms that depend on total cell number or aggregate size, potentially providing an additional layer of regulation during aggregation.

Together, these findings place the rotating clusters studied here within the broader framework of Dictyostelium aggregation. During starvation, aggregation is not simply a process in which cells move toward a fixed center; rather, it is an excitable, self-organized signaling process in which cAMP waves coordinate signal relay and directed cell motion. Early aggregation requires cells to repeatedly sense, amplify, relay, and reset their response to extracellular cAMP, while later pre-mound and mound stages use these same signaling dynamics to organize increasingly large multicellular structures. In this context, our observation that larger clusters exhibit slower cAMP wave and cell angular velocities suggests that aggregation dynamics are not scale-neutral: as more cells join an aggregate, the collective signaling environment changes, altering both the timescale of wave propagation and the associated cell motion.

Other factors could in principle contribute to the observed size dependence. For example, even though we think this is unlikely, it is possible that hypoxia plays a role in the size-dependent observations. Previous studies have shown that Dictyostelium behavior changes below a critical oxygen threshold [24, 46], with aggregates dispersing at 2% oxygen. Since the aggregates in our experiments remain stable, including under agar confinement, our results argue against hypoxia being a significant factor. Also, it is possible that the wave speed depends on wave curvature, as described by the eikonal formulation for excitable media developed by Keener [47]. However, because of the stochastic motion of cells, we are not able to reliably quantify the relationship between wave speed and curvature in our system. Other mechanisms, including size-dependent changes in cell packing, local confinement, receptor adaptation, cell–substrate interactions, or intracellular signaling or secretion rates, could also modulate excitability and thereby alter wave speed. Nonetheless, the PDE-based mechanism is biophysically plausible, relies only on the known production and diffusion of extracellular PDE, and naturally couples cell number to the local cAMP landscape. It therefore provides a concrete, experimentally testable route for future perturbation studies, for instance by altering PDE production or activity and quantifying the resulting changes in wave and rotation dynamics.

In the present model, extracellular cAMP influences the excitable dynamics mainly through the input term, so that cluster-size-dependent changes in cAMP degradation mainly affect wave propagation and excitability rather than the refractory period. Previous experimental and theoretical studies, however, indicate that cAMP turnover can also shape recovery dynamics and refractoriness [48, 49]. Incuding such effects in a more detailed model would likely reinforce the observed cluster-size dependence of wave angular velocity. Extending our modeling approach to a more complete description of degradation-dependent signaling dynamics should shed further light on this.

Our data and simulations also indicate that larger clusters are more likely to support multi-armed spiral waves. Experimentally, double-armed spirals are more likely in larger aggregates (Fig. 3), and the CPM with size-dependent degradation reproduces this qualitative trend (Fig. 6). A plausible explanation is that the formation and persistence of multiple spiral arms impose both temporal and spatial constraints. Temporally, each wave arm is accompanied by a refractory period; in small clusters, most cells remain close to at least one passing arm and experience frequent excitation, limiting their ability to recover and respond to additional stimuli. As a result, secondary arms are more likely to be extinguished by refractoriness or by collisions, favoring single-armed patterns. In larger clusters, a substantial fraction of cells resides farther away from any given arm and can fully recover between wave passages, providing a reservoir of excitable tissue that can sustain additional arms. Spatially, larger perimeters can accommodate several wavelengths along the boundary, allowing distinct arms to remain separated and reducing destructive interference when they rotate around the center. Together, these geometrical and dynamical considerations make multi-armed spirals more stable in large aggregates and inherently unstable in small ones, consistent with both experimental and numerical observations. We should note that sometimes we observed a transition from single to multi-armed spiral waves, as reported in earlier studies [17, 19, 22]. However, a detailed investigation of the dynamics of spiral arm selection will require a larger-scale dataset.

Our approach has several experimental and numerical limitations. Experimentally, were inferred from optical flow applied to DIC images, which assumes approximately constant pixel intensity over time. However, since intensity can change due to vertical motion or subtle imaging artifacts, the extracted velocities need to be interpreted as relative measures rather than absolute cell speeds. We also focused on clusters that exhibit clear single- or double-armed spirals and did not consider cluster merging events. In addition, our experiments were carried out in a quasi-2D environment and the wave and cell dynamics in 3D clusters remains to be explored. More extensive datasets, combined with controlled perturbations of PDE or other signaling components, will be essential to validate the proposed mechanisms.

On the modeling side, we adopted several simplifying assumptions. Cells were treated as excitable throughout the simulations, and extracellular degradation was taken to be spatially homogeneous, neglecting the contribution of membrane-bound PDE and potentially heterogeneous secretion. Furthermore, to speed up simulations, we seeded spiral structures through prescribed initial conditions rather than allowing them to arise solely via spontaneous symmetry breaking. Relaxing these assumptions and systematically exploring parameter space could reveal additional routes to size-dependent dynamics, including changes in aggregate shape, mechanical feedback, or coupling to tissue-scale deformations.

Conclusion

The combined experimental and numerical results establish that cluster size is a key control parameter for rotating cAMP waves and collective motility in Dictyostelium aggregates. Larger clusters rotate more slowly, harbor slower cAMP spirals, and are more likely to exhibit multi-armed spiral waves. A modeling framework that incorporates size-dependent extracellular cAMP degradation captures these trends and suggests a direct link between the number of cells in an aggregate, the extracellular signaling environment, and the emergent spatiotemporal patterns. This framework lays the groundwork for future work on how chemical cues, cell motion, and overall cluster shape combine to guide the growth from early cell clusters to fully formed multicellular structures.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Author contributions

SS, MHT, JJ performed experiments. YCL carried out simulations. SS, YCL, and MR analyzed data. WJR conceived and directed the study, and obtained funding. SS, YCL, and WJR created the first draft. All authors read and approved the final manuscript.

Funding

This work was supported by NSF MCB 2,426,002 and NSF PHY 2,310,496.

Data availability

Code used to generate the simulation data can be found at https://github.com/Rappel-lab/Dicty-Cluster-CPM.

Declarations

Ethics approval

Not applicable.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.