Research ArticleCell Migration

Spatial heterogeneities shape the collective behavior of signaling amoeboid cells

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Science Signaling  27 Oct 2020:
Vol. 13, Issue 655, eaaz3975
DOI: 10.1126/scisignal.aaz3975

Slime mold obstacle course

The slime mold Dictyostelium discoideum, which lives in the soil, is an experimental model of cell migration. Starving Dictyostelium cells secrete the chemoattractant cAMP, which stimulates the cells to migrate in head-to-tail streams, eventually leading to the formation of fruiting bodies required for cell survival. Eckstein et al. generated substrates containing pillars of different sizes and patterns and performed mathematical analysis of images of cAMP and the migrating cells to determine how the Dictyostelium cells interpreted waves of chemoattractant disrupted by obstacles. This analysis could be applied to understanding how collections of cells interact with spatial barriers in their environment.

Abstract

In its natural habitat in the forest soil, the cellular slime mold Dictyostelium discoideum is exposed to obstacles. Starving Dictyostelium cells secrete cAMP, which is the key extracellular signaling molecule that promotes the aggregation process required for their long-term survival. Here, we investigated the influence of environmental inhomogeneities on the signaling and pattern formation of Dictyostelium cells. We present experimental data and numerical simulations on the pattern formation of signaling Dictyostelium cells in the presence of periodic arrays of millimeter-sized pillars. We observed concentric cAMP waves that initiated almost synchronously at the pillars and propagated outward. In response to these circular waves, the Dictyostelium cells streamed toward the pillars, forming aggregates arranged in patterns that reflected the periodicity of the lattice of pillars. Our results suggest that, in nature, the excitability threshold and synchronization level of the cells are two key parameters that control the nature of the interaction between cells and spatial heterogeneities in their environment.

INTRODUCTION

An excitable medium is a spatially distributed dynamical system in which each constituting element has the property of excitability. Adjacent elements of an excitable system interact through diffusion-like local transport processes. A fundamental process that occurs in excitable reaction-diffusion systems is the propagation of nonlinear waves (14). Examples of such waves include chemical waves in the Belousov-Zhabotinsky (BZ) reaction (5), waves of CO oxidation on platinum catalytic surfaces (6), electrical waves in retinal and cortical nerve tissue (7), waves in heart muscle (8), and cyclic adenosine monophosphate (cAMP) waves in starved populations of Dictyostelium discoideum amoebae (9, 10). One of the important questions concerning nonlinear waves is how they propagate in the presence of obstacles. Various aspects of this question, such as the influence of one obstacle, a few large obstacles, or a large number of small obstacles on wave propagation in excitable media, have been the focus of several theoretical and experimental investigations (1115). Concentric target waves and outward rotating spiral waves are the basic spatiotemporal patterns in excitable media, and they have been observed in various biological and chemicals systems. Obstacles can stabilize rotating spiral waves by pinning the spiral core (8, 16), and understanding the interaction between spirals and obstacles is important in various physical, chemical, and biological systems (8, 1619). For example, in the heterogeneous heart tissue, once a free rotating spiral pins to an anatomical obstacle, physiologically life-threatening cardiac arrhythmias, such as fibrillation, can occur (15, 20, 21).

Signaling of D. discoideum cells is a classic example of an excitable medium and shows properties similar to those of heart tissue (20, 21) or chemical waves in the BZ reaction (5). This organism, naturally occurring in the forest soil, is an important model system for the study of chemotaxis, cell differentiation, and morphogenesis (22). Starvation of D. discoideum cells induces a developmental program in which cells align to form head-to-tail streams by signaling to each other with cAMP. Cells initiate the process by sending out pulsatile cAMP signals with a periodicity of several minutes, which propagate as waves. Over time, circular and spiral patterns of cAMP form, and cells respond chemotactically to cAMP waves that guide cell movement toward the signaling centers and form multicellular, centimeter-scale domains. The corresponding wave sources in each domain then act as aggregation centers, which eventually transform into millimeter-long slugs and lastly into fruiting bodies, bearing spores for long-term survival and long-range dispersal (23).

Chemotaxis assays in the laboratory are typically performed in uniform environments with a relatively deep and uniform layer of liquid over the cells. In their natural soil habitat, however, D. discoideum aggregation takes place in the presence of various sizes and shapes of impermeable particle barriers and what is likely a thin film of liquid, rather than a relatively deep fluid phase. These spatial barriers will substantially alter the processes of wave generation, propagation, and aggregation. As D. discoideum cells attempt to detect cAMP gradients and move directionally toward the source, obstacles may block the physical path of the cells or alter the flux of the chemical signal. It is speculated that chemotactic cells can avoid the obstacles by following the altered signal (24) or it is suggested that a self-assisted migration mechanism improves the navigation efficiency of the signaling cells in a complex environment (25). To understand cAMP wave propagation in these more complex, real-world environments, we have begun to investigate how barriers of various sizes, shapes, and positions influence the chemotactic signaling process. Here, we report experimental and numerical results on the spatiotemporal dynamics of populations of D. discoideum cells in the presence of nonexcitable obstacles. We showed that the chemotactic sensitivity and synchronization level of the D. discoideum cells are crucial parameters that determine the interaction between cells and the obstacles. Our experiments and numerical simulations indicate that in a synchronized population of starving cells with an increased level of chemotactic sensitivity, inert obstacles can act as aggregation centers if a small accumulation of either cells or cAMP occurs around them. However, unsynchronized cells with a reduced level of chemotactic sensitivity ignore the obstacles. These results suggest that possibly, in nature, an optimized level of synchrony and chemotactic sensitivity of the cells controls the ability of the cells to signal to each other and successfully navigate in the presence of physical constraints. These findings should help our understanding of aggregation in complex environments and, more broadly, how self-organizing living systems regulate their parameters to survive according to their particular reaction-diffusion characteristics and environmental restrictions.

RESULTS

Formation of circular waves centered on the obstacles

In nature, the obstacles encountered by D. discoideum are randomly distributed in three dimensions, but as a first step toward understanding, we looked at a simpler system of cells in a two-dimensional (2D) geometry with a periodic arrangement of obstacles made from PDMS (polydimethylsiloxane). Our quasi-2D geometry consists of a regular array of millimeter-sized pillars that control the spatiotemporal dynamics of a population of uniformly distributed D. discoideum cells (fig. S1). Unlike other excitable systems (13, 17), initial experiments showed no interaction between obstacles and the appearing cAMP waves (movies S1 and S2). In our experiments, we observed spirals pinned to the unexcitable obstacles when they, by chance, appeared close to one (fig. S2 and movie S1), but they showed no systematic attraction or repulsion to them (fig. S3, A and B, and movie S2).

We then lowered the “apparent excitation threshold” of the cells by adding caffeine (which reduces the rate of cAMP production) to the system (26) and observed the resulting aggregation pattern (Fig. 1). Under these conditions, the spatial heterogeneities induced the formation of circular waves centered on the pillars, which stimulated chemotactic cell movement toward the pillars. This led to the formation of periodic domains that reflected the periodicity of the underlying macro-pillar array. Furthermore, we observed synchronized circular waves and regular domains only in the presence of caffeine (Fig. 1; fig. S3, A and B; and movie S2). In the absence of pillars but in the presence of caffeine, target patterns emerged at random locations on the PDMS substrate (fig. S3, C and D, and movie S3). This contrasted with the patterns formed in the presence of macro-pillars (movie S4) where waves originated at the pillars and propagated outward (Fig. 1). The four successive snapshots in (Fig. 1, A to D) revealed circular waves centered around the pillars, cells streaming toward the pillars, the formation of regular domains, and cell aggregation, respectively. In our experiments, we observed that concentric waves developed almost synchronously around the pillars and, because they had a slightly higher frequency, they dominated over the other firing centers, which also emitted periodic pulses of cAMP. We then analyzed the frequency of the firing centers (figs. S4 and S5). Waves propagated outward from the pillars and triggered chemotactic movement of the cells toward the pillars. As a result, periodic domains formed around the obstacles. The concentric waves around the pillars were visible in the spatial phase map (Fig. 1E) and in the processed video (movie S5). The phase map shows that (i) the circular waves were slightly off-center from the pillars, (ii) the territories that each circular wave propagated before annihilation had different sizes, and (iii) the frequency and phase of the circular waves varied between the pillars (movie S6). We then calculated the gradient vectors of the phase map around two neighboring pillars (Fig. 1F). The vector field changed direction where the concentric waves met and annihilate each other. Thus, the Laplacian of the phase map φ, defined as 2φ(x,y)/x2+2φ(x,y)/y2, takes on extreme values at the collision regions of two neighboring emitted waves and defines the boundaries of the domains (Fig. 1G). We also performed a 2D Fourier transform of these data and looked at the power spectrum (fig. S6, A, C, and E). This analysis showed dominant peaks corresponding to the 5-mm periodicity of the lattice. Moreover, presentation of power spectrum in polar coordinates showed a clear π/2-periodicity of the lattice in the presence of caffeine (fig. S6E). In contrast, in an experiment without caffeine (fig. S6, B, D, and F), the power spectrum showed a wider distribution, and the π/2-periodicity was absent.

Fig. 1 Formation of regular Voronoi domains.

(A to D) Top view of D. discoideum cells on a macro-pillar array. The experimental procedure and imaging technique are described in Materials and Methods. (A) Concentric waves initiated around the pillars and propagated outward. (B to D) Amoebae respond chemotactically to the circular waves and streamed toward the pillars. This led to the formation of regular domains around the pillars. Timestamps denote time from the start of starvation. Images are representative of at least 20 experiments. Dashed white line indicates the spatial position, which is used to make the space-time plot in Fig. 2. Scale bar, 5 mm. (E) Phase map of the observed spatiotemporal pattern showing the formation of circular waves of cAMP around the pillars at the same time point as shown in (A). (F) Gradient vectors of the phase map around two neighboring pillars. The white dashed line shows the location where two emitted waves collided and the vector field changed direction. The vectors are scaled 50 times larger than the original values for better visibility. Data are derived from the experiment presented in parts (A) to (D) at the same time point as shown in (A). (G) Laplacian of the phase map in units of mm−2 at the same time point as shown in (A). The gradient vector field switched direction at the boundary of the domains; thus, the Laplacian has higher values at these boundaries.

We generated a space-time plot (Fig. 2A) by stacking up the measured light intensities (Fig. 1A, white dashed line, and movie S5). First, we observed synchronized bulk oscillations with a period of about 10 min. We attribute this finding to the initial starvation of the cells in a shaking suspension for 4 hours, which leads to cell synchronization (27). We also performed an experiment with caffeine but without initial cell starvation (movie S7). Note that the term “synchronization” here refers to the simultaneous generation of cAMP waves and not synchronization of the cell cycle, which is the most common use of the term. Cell cycle synchronization occurs by default when cells are starved and become mostly blocked in G2 phase. After about 1 hour, circular waves were initiated almost synchronously from the pillars, and they propagated outward (fig. S7 and movie S8). The waves annihilated as they collided with each other. These annihilation areas defined the boundaries of the regular domains. In an ideal experiment, in which the circular waves have the same frequency and phase, the size of the quadratic domains is the same as the pillar spacing (≃5 mm). However, in general, there is a frequency and phase difference between concentric waves originating from neighboring pillars consistent with the variation in target centers observed in other excitable systems (28). If two neighboring waves have almost the same frequency with a phase shift, due to this phase difference, the annihilation point of two neighboring circular waves is not located exactly at the middle of two pillars but is rather located closer to the pillar with a phase delay (Fig. 2A, dashed white line). Moreover, often in the experiments, there was also a small frequency difference between the waves initiating at the neighboring pillars. Consequently, the annihilation point of two neighboring concentric waves shifts toward the pillar with the smaller frequency (Fig. 2, A, red dashed line, and B, white dashed lines). The drift velocity of the annihilation point can be calculated to be vw(f2f1)/(f2 + f1), where vw is the wave propagation velocity, and f1 and f2 are the wave frequencies at the two neighboring pillars (fig. S8). In our experiments, vw was of the order of 0.4 mm/min, and the frequencies f1 and f2 were about 1/10 and 1/9 min−1, which resulted in vdrift being of the order of 0.02 mm/min. If the frequency difference persisted, eventually, the wavefront collided with the pillar and broke (Fig. 2C, white arrow). The waves recombined after passing through the obstacle and propagated further (Fig. 2D). In this case, the obstacle only broke the propagating wavefront (Fig. 2E), and there was no boundary formed between the middle and the right pillar.

Fig. 2 Space-time plots.

(A) Space-time plot of the experiment shown in Fig. 1. The light intensity from movie S5 was stacked up along the white dashed line shown in Fig. 1A. The black bars show the position of the pillars. The red arrows show a firing center other than the pillars that had a lower frequency and disappeared with time. The white dashed line traces the annihilation points of two waves that initiated on the neighboring post and had a similar frequency but a phase shift. The red dashed line shows the slow drift of the annihilation point toward the pillar with smaller frequency (left pillar). (B) Space-time plot in an experiment with pillars that were 50 μm in height. Only part of the kymograph with three pillars is shown. The red arrows point to a firing center not centered on a pillar, which had a slightly lower frequency than the waves initiated on the neighboring pillars and thereby disappeared with time. The white dashed lines show the movement of the annihilation points toward the pillars with lower wave frequency. The white arrows show the collision of the annihilation point with the right pillar. Images are representative of at least 20 experiments. (C to E) Right pillar in (B) failed to emit its own circular cAMP wave and only broke the wavefront initiated from the neighboring pillar. As the wavefront collided with the pillar on the right, it broke and recombined again. As a result, no boundary between two neighboring pillars formed.

Pillars with triangular and hexagonal arrangements

The phenomenon of initiation of synchronized circular waves around the pillars and formation of regular domains was robust with respect to the arrangement of the pillars. We repeated our experiments with the same pillar size and spacing, but triangular and hexagonal arrangements of the pillars, and observed hexagonal and triangular domains, respectively (Fig. 3 and movies S9 and S10). Furthermore, our experiments were robust with respect to pillar diameter (movie S11) and pillar height (movie S12) and even showed regular domains with holes (movie S13) and a planar obstacle (movie S14).

Fig. 3 Different arrangement of the obstacles.

(A and B) Triangular and hexagonal domains formed around the pillars with hexagonal and triangular arrangements, respectively. Timestamps denote time from the start of starvation. Images are representative of at least 10 experiments. Scale bar, 5 mm.

Cell distribution around the pillars

Next, we used bright-field microscopy to look closely at the wave propagation and cell streaming in the vicinity of the pillars (Fig. 4 and movie S15). A higher cell density around the pillars after plating the cells could explain the higher frequency of the waves initiating from the pillars. In separate experiments, we examined the effect of local cell density on the wave frequency (fig. S5 and movie S16). Our bright-field observations did not confirm a substantial cell accumulation around the pillars (Fig. 4E). However, on the basis of our numerical simulations, in the presence of caffeine, even a slight cAMP accumulation in the vicinity of the obstacles would be enough to trigger the formation of concentric cAMP waves around the pillars. Possible cell attachment to the side walls of the pillars, which we simulated as a small constant value of cAMP on the obstacles, is a plausible mechanism for the formation of circular waves around the pillars. Moreover, we emphasize that the initial starvation of the cells for 4 hours in a shaken suspension led to increased cell-cell adhesion; very small cell clusters were visible (Fig. 4A). As we mentioned earlier, concentric waves around the pillars were not always completely centered at the pillars but were slightly off-center. Therefore, the cells in most of our experiments streamed toward a point in the vicinity of the pillar (Fig. 4B, arrow, and movie S15) and then aggregated at the pillar itself. Note that in our typical experiments, which are performed with 20 ml of cell solution (see Materials and Methods), the depth of fluid was approximately 4.5 mm. Therefore, cells aggregated on the pillars but failed to form fruiting bodies because the air interface was relatively far away. To enable fruiting bodies to form, we performed another set of experiments with 2 ml of cell solution (but the same number of cells per area) to reduce the height of fluid to 400 μm. In this case (fig. S9 and movie S17), regular streaming domains emerged, and cells aggregated on the obstacles, eventually forming fruiting bodies on the pillars and in the area in close vicinity to the pillars. Note that in this experiment, pillars of height of 50 μm were used to avoid a fluid meniscus around the pillars, and the diameter of the pillars was 1 mm.

Fig. 4 Cell distribution around the pillars.

(A to D) Magnified view of the pillars with a bright-field microscope. The experimental procedure used is described in Materials and Methods. (A) Initial distribution of the cells shows small clumps. (B to D) Cells chemotactically moved toward the center of the concentric waves, which were sometimes slightly off-center from the pillars. During the streaming process, the cells joined small clusters to make larger ones, and eventually, large clusters aggregate on the post (see also movie S15). Timestamps denote time from the start of starvation. Images are representative of at least 15 experiments. Scale bar, 5 mm. (E and F) Systematic measurements of cell density around the pillars of 1 mm in diameter [6 min after the start of experiment as in (A)] showed no initial substantial cell accumulation in the vicinity of the obstacles. The distance was measured from the center of the pillar, and SDs were calculated from the measurements for ≃11 pillars. Data are representative of at least 15 experiments. Note that the area around each pillar is divided into circular bands, and the occupied area fraction was measured for each band, as shown in (F).

Pillars of smaller diameter with a random distribution

In nature, obstacles encountered by D. discoideum cells are smaller in size and have a random distribution. Therefore, to better approximate the natural habitat of the cells, we performed multiple sets of experiments with (i) pillars of random diameter (0.1 ≤ d ≤ 1 mm) distributed randomly on a PDMS substrate, (ii) pillars of 1 mm in diameter distributed randomly, and (iii) pillars of 100 μm in diameter in a periodic lattice (fig. S10 and movies S18 to S20). We found that the pillars of the smallest diameter of 100 μm mostly became wave centers, and the cells streamed toward the obstacles (fig. S10, E and F). However, as we further reduced the pillar diameter to 50 μm, the obstacles were mostly ignored by the cells, suggesting a minimum obstacle size for pillars to act as a wave center. Moreover, in the case of random pillar distribution, we observed that two pillars located in close vicinity to each other effectively acted as one wave center (fig. S10D and movie S19), which suggests that an optimal distance between obstacles is needed such that each pillar successfully emits its own circular wave and becomes an aggregation center.

Numerical simulations

We performed numerical simulations of the model proposed by Martiel and Goldbeter (MG) for the creation and relay of cAMP in D. discoideum (29, 30). Many approaches have been used to create spirals and target centers in this model, most of which define the position of the localized structures through perturbations or diversity of developmental stage among the cells (31, 32). We instead used a dynamical approach in which centers appear naturally in areas of higher local density (33) (see Materials and Methods and movie S21). Here, cells are distributed inside a grid; thus, a grid containing a cell is an occupied one and can produce and relay cAMP, whereas empty grids without cells can only degrade the signal through the action of an external phosphodiesterase (PDE). This mixture of occupied and unoccupied areas on the system breaks the system homogeneity and enables clusters of higher local cell density to become target centers. The lower-density areas are still capable of sustaining waves; thus, the waves generated by these clusters get relayed by the rest of the system. We measured the dispersion relation of such waves and showed that they had the behavior of trigger waves (Fig. 5) (34). These dispersion relations at different cell densities also showed that the wave velocity increased with cell density, which is necessary to produce aggregation streams (35). Note that trigger waves are nonlinear excitation waves that propagate in excitable media when a perturbation above a threshold is applied. In these systems, small perturbations become dampened, but suprathreshold perturbations are amplified and excite the neighboring area, enabling wave propagation. A trigger wave has a velocity that is nonlinearly selected by the system, and both the propagation speed and its profile are uniquely determined by the properties of the medium. Another important characteristic is that a new trigger wave cannot enter the system until some recovery time has elapsed. Last, we emphasize that trigger waves should be distinguished from phase waves that exist due to phase gradients between coupled oscillators, and unlike phase waves, trigger waves present actual chemical transport. Phase waves are largely independent of diffusion and have a variable speed, which is not intrinsic to the system and can be arbitrarily large (36).

Fig. 5 Characterization of the MG model with caffeine.

(A) Phase diagram of the MG model used for simulating the experimental setup. In the area marked as stable, the system has one solution, which is stable. In the oscillatory region, the system shows one unstable steady state surrounded by a limit circle. In the excitable regime, the system has three steady states, two unstable ones and a stable one, which is excitable. In the regime marked as E(1), the system shows one steady state, which is excitable (fig. S17). Stability was calculated through linear analysis, whereas excitability was calculated through no-space simulations. (B to D) The red dashed line in (A) shows the path that km is changed in parts (B) to (D) at a fixed value of ke = 5.0 min−1. Note that higher Km values correspond to higher caffeine concentrations. (B) Dispersion relations of the supported wave trains for 40% surface coverage, where T is the wave period, and c the wave velocity. Waves with periods below those shown do not get relayed by the system. (C) Effect of a boundary condition γ0: For higher values of km, the minimum amount of γ0 required to produce a target center decreases. (D) Minimum number N of consecutive cells (in one dimension) needed to produce a cluster with self-sustained oscillations.

We observed in our simulations that a no-flux boundary condition at the obstacles was not sufficient to produce centers. Similarly, when this boundary condition was applied, waves only broke after colliding with the obstacles, and then they recombined (fig. S11 and movie S22). This finding is consistent with our initial experiments without caffeine. Another boundary condition that produces traveling waves in numerical simulations is the Dirichlet boundary condition, where the boundary is held to a fixed value. In our system, a fixed value of cAMP at the obstacles generated wave trains emitted from the obstacles if the fixed value was larger than a threshold (Fig. 6, A and B, and movie S23). This minimum amount of cAMP needed for the obstacles to act as a wave source depends on the system parameters. The minimum accumulation of cAMP or cells necessary for an unexcitable obstacle to become a center was great enough so that the cells did not aggregate around the obstacles naturally. To obtain aggregation domains around the obstacles, we varied different parameters, which can be controlled in our experiments. To achieve a lower excitation threshold, we varied the parameter Km, which is the Michaelis constant of the reaction in which adenosine 5′-triphosphate (ATP) is used to produce intracellular cAMP. Increasing this parameter reduced the affinity between ATP and the enzyme adenylate cyclase (ACA), thus reducing the production rate of intracellular cAMP and accounting for the effects of adding caffeine to the experimental system.

Fig. 6 Numerical simulations of the reaction-diffusion model.

(A and B) Numerical simulations of cAMP waves with four pillars as obstacles with fixed boundary condition γ0 = 10 nM, km = 0.508 mM, ke = 5.0 min−1, 50% of cell surface coverage. (A) Concentric waves of cAMP emanating from the pillars. (B) Cell distribution after t = 100 min, showing regular domains around the pillars. Gray squares show grid points containing a cell, whereas black squares show the empty grid points. (C and D) Numerical simulation showing the effect of a higher cell accumulation around the pillars. km = 0.48 mM, ke = 5.0 min−1, 60% of cell coverage around the pillar, 40% in the rest of the system. (C) Waves of cAMP at t = 15 min. (D) Cell distribution after t = 150 min. Gray squares show grid points containing a cell, whereas black squares show the empty grid points (movie S26). (E) cAMP waves emanating from pillars with a triangular arrangement. Fixed boundary condition γ0 = 10 nM, km = 0.508 mM, ke = 5.0 min−1, 40% of cell surface coverage. (F) Cell distribution after t = 150 min showing regular hexagonal domains. (G) cAMP waves emanating from pillars with a hexagonal arrangement. Fixed boundary condition γ0 = 10 nM, km = 0.508 mM, ke = 5.0 min−1, 40% of cell surface coverage. (H) Cell distribution after t = 150 min showing regular triangular domains (see movies S29 and S30).

We performed linear stability analysis of the MG model with modifications in Km and characterized its different regimes (Fig. 5A). The other parameter that we varied is the degradation rate of external PDE ke. We chose the parameters such that the system was in the oscillatory regime, that is, a stable limit cycle existed, and the cell coverage (ratio of the number of occupied grids to the total number of grids) was high enough for the waves to get relayed. At higher values of Km, which is equivalent to a higher concentration of caffeine in our experiments, the number of firing centers in the system was decreased (fig. S12 and movie S24). These simulations were consistent with our control experiments with caffeine, which showed a decrease in the number of firing centers and an increase in the wave period (see figs. S13 and S14 and movie S25). The reason is that the minimum cluster size (measured as consecutive cells in a 1D setting) required to produce a self-sustained oscillatory center was increased (Fig. 5D). Increasing the amount of caffeine (higher Km values) also decreased the wave velocity and increased the minimum wave period that the system could sustain. We plotted different dispersion relations that showed this effect (Fig. 5B). We studied the boundary effect in this system and found that at a higher caffeine concentration, the trigger waves required a smaller amount of cAMP to be triggered; thus, the obstacles acted as a wave source at smaller values of the fixed boundary γ0 (Fig. 5C and movie S23).

Another mechanism for wave creation observed in our simulations was a higher local cell density around the pillars. This can be achieved either by inhomogeneous initial cell distribution or by adhesion of the cells to the pillars after colliding due to random movement. These locally high-density cell clusters triggered the formation of wave centers and acted as aggregation centers (Fig. 6, C and D, and movie S26). To investigate the effect of PDE accumulation or depletion in the vicinity of the obstacles, we also performed numerical simulations assuming slightly higher or lower PDE amounts around the obstacles. We found that obstacles acted as a wave center in the case of PDE depletion but that they were ignored if PDE accumulated around them (movies S27 and S28). Note that we also performed numerical simulations with triangular and hexagonal arrangements of the pillars (Dirichlet boundary condition) and observed hexagonal and triangular patterns, respectively (Fig. 6, E to H, and movies S29 and S30). Last, it was previously suggested that the effect of caffeine on ACA inhibition is possibly mediated by different targets, one of which inhibits the activation of the G protein G2, which is part of the signal transduction cascade that normally activates ACA (37). To investigate the effect of having less activated ACA, we also performed simulations by modifying the parameter ε, which controls the ratio of the active to inactive forms of ACA. These simulations showed similar results to those obtained by modifying Km (figs. S15 and S16 and movie S31). Therefore, regardless of the particular way in which caffeine affects ACA, our numerical results provide explanations for the observed behavior.

DISCUSSION

The results presented here show that external obstacles can substantially affect the generation of cAMP waves in starving populations of D. discoideum cells when the excitability threshold has been lowered. We observed circular waves that initiated almost synchronously at the pillars and propagated outward. Chemotactically competent cells detected the cAMP gradient and crawled toward the pillars, forming a periodic array of domains. This phenomenon was also observed for triangular and hexagonal arrangements of the pillars, leading to the formation of hexagonal and triangular domains, respectively (movies S9 and S10). Most of our experiments were performed with pillars of 1 mm in diameter, but the phenomenon was fairly robust to different types of obstacles, because we observed a similar phenomenon with pillars of 1.5 mm in diameter (movie S11), random pillar diameters (movie S18), random distributions of the pillars (movie S19), pillars of 100 μm in diameter (movie S20), shallow holes of depth of 100 μm in PDMS (movie S13), smaller center-to-center spacing of the pillars (3.75 mm instead of 5 mm), and even with a PDMS wall (movie S14). The phenomenon was also robust with respect to the pillar height because we observed regular domains with pillar heights of as low as 50 μm (Fig. 2, B to E, and movie S12).

The concentric waves emitted from pillars had a slightly higher frequency than those of other firing centers, thereby dominating the system dynamics. Cells attached to the side walls of the PDMS can trigger higher-frequency waves. In our simulations, we included this effect by assuming either a fixed value of cAMP around the pillars or slightly higher cell accumulation in the vicinity of the pillars. Future experiments using fluorescent indicators of extracellular cAMP will be valuable to visualize any possible cAMP accumulation around the pillars (38, 39). Note that several factors may influence the dependency of wave frequency on cell density. Whereas experiments with well-stirred cell suspensions (40) showed weak to no density dependence of oscillation frequency for early waves and the speed of wave propagation increased as the wave moved outward from the center, in other experiments (39), wave frequency increased with density, and waves propagated at uniform velocity. Our experimental assay and simulations show results consistent with those of Gregor et al. (39) where we measured a higher frequency at higher local cell densities.

Another scenario that requires further biochemical analysis is the possibility of accumulation or depletion of any chemical (such as PDE) in the vicinity of the pillars. Our numerical simulations showed that in the case of PDE depletion, pillars could act as a wave source in the system, which might drive oscillations in the vicinity of the obstacles (movie S27) (4143). In the opposite case of PDE accumulation, pillars were “ignored” and only broke the colliding wavefronts (movie S28). However, note that to prevent the adsorption of chemicals to the PDMS substrate, we repeated our experiments with substrates treated with bovine serum albumin (BSA) and still observed concentric waves and periodic domains around the pillars (movie S32). Last, although PDMS is useful because of its biocompatibility, deformability, and permeability to gas, it has certain properties that might complicate our results. To rule this out, we used pillars made of polymethyl methacrylate (PMMA) and again observed the regular streaming domains (movie S33).

In our system, we observed periodic domains within a range of caffeine concentration between 1 and 5 mM, when cells were initially synchronized in an agitated starving period. Caffeine, which is a highly specific inhibitor of cAMP relay (26, 4446), reduced both the cAMP production rate and the wave frequency in a dose-dependent manner (fig. S13). We did not observe ordered aggregation territories without caffeine (fig. S3, A and B, and movie S2). Our experiments and earlier experiments by Siegert and Weijer (47) with caffeine showed that the aggregation territories were much larger in the presence of caffeine compared to those in the control experiments (fig. S14 and movie S25). This suggests that the number of firing centers was decreased, which is also consistent with our numerical simulations in the presence of caffeine with higher values of Km (fig. S12 and movie S24). We believe that in our experiments, a lower number of firing centers was crucial for the circular waves emitted from pillars to successfully dominate the system dynamics (41, 48, 49). We also observed that in the presence of obstacles, caffeine substantially reduced the number of spirals that appeared in the system. Because spirals had a higher frequency than target patterns, they dominated the dynamics once they appeared. Moreover, we emphasize that in our experiments, without initial starvation of the cells in an agitated suspension, mostly spirals formed, confirming that the synchronization level of cells plays an important role in shaping the overall wave pattern (movie S7). In other excitable systems, such as the BZ reaction, spirals can be pinned to obstacles with lower excitability (13). In that reaction-diffusion system, it has also been shown numerically that the spiral tip interacts with the (no-flux) obstacle boundary through attraction and repulsion (17). We observed neither attraction nor repulsion of the spiral cores to the obstacles in both the experimental setup and the numerical simulations. We believe that this crucial difference is due to the lack of meandering of the spiral tip in D. discoideum. Furthermore, in the presence of caffeine, spiral core size increases in a dose-dependent manner (45). In those occasions when a spiral did appear next to a pillar, it got pinned to the pillar and remained rotating around it (fig. S2 and movie S1).

Comparing our numerical simulations with modified Km to the unmodified parameters, the system showed fewer centers due to the higher minimum cluster size necessary to produce pacemakers; it also showed smaller frequency, consistent with the experimental measurements (fig. S13). The minimum amount of cAMP necessary to produce cell activation was reduced, enabling easier wave relay. This is consistent with the findings of Brenner and Thoms (26), showing that caffeine increases the apparent chemotactic sensitivity of the cells. Thus, in the presence of caffeine, the transition boundary between “source” and ignored obstacles occurs at lower amounts of cAMP accumulation around the obstacles. For this reason, the presence of caffeine was necessary in our experiments to trigger the formation of concentric waves around the pillars. Note that there is a subtle difference between our experiments and those of Brenner and Thoms (26). In that study, the authors exposed caffeine-treated cells and untreated cells to small droplets of cAMP and observed that caffeine-treated cells respond to shallower gradients of cAMP than did untreated cells. The authors hypothesized that the reason that treated cells can sense diffusional gradients of lower initial concentration than those sensed by untreated cells is because cAMP relay is reduced in the presence of caffeine, so there is less locally produced cAMP to compete with exogenous diffusing cAMP to bind to receptors. Thereby, in caffeine-treated cells, exogenous cAMP dominates signaling. The local cAMP concentration is a complex combination of the synthesis rate of cAMP, the diffusion of distantly produced cAMP, and the local concentration of membrane-bound and cytosolic PDE. In the presence of caffeine, cells are in an environment in which they can sense lower concentrations of cAMP most probably because the production of endogenous cAMP is reduced. In our experiments, however, cAMP was produced by the cells themselves, and no exogenous cAMP was added, and so caffeine should, in theory, result in waves of exogenous cAMP that are lower in concentration due to the damped production of cAMP. We believe that further detailed experiments (for example, with labeled extracellular cAMP) are necessary to measure the cAMP concentration in the vicinity of cells with and without caffeine to examine whether the hypothesis of Brenner and Thoms (26) is valid in our system.

Last, the natural life cycle of D. discoideum cells is expected to involve the upward migration of slug through the soil. Our experiments suggest that the attraction of cells toward obstacles followed by upward migration and formation of fruiting body on the obstacles may commonly occur in nature. In contrast, the attraction of cells toward “holes” as spatial heterogeneities in our experiments is less understandable in terms of the survival strategy of D. discoideum cells because it may complicate the generation of a mound and the consequent phases of the life cycle. Because in nature an inhomogeneous distribution of cells and physical obstacles is expected, we believe that a combination of two factors, namely the “apparent chemotactic sensitivity” and the synchronization level in a population of signaling cells, may strongly influence the interaction between cells and obstacles. However, it is also plausible to think that cells in nature are very likely not to be synchronized and that aggregation occurs in the absence of optimized synchrony or chemotactic sensitivity. We hope that future experiments examining the aggregation of asynchronous cells (for example, in mixtures of cells with various starvation times) may shed some light on the effect of asynchrony and chemotactic sensitivity on the interaction between cells and obstacles. From a general perspective, D. discoideum presents itself as a very suitable model system with which to further elucidate how cells interact with spatial barriers in their natural habitat, and investigations in this direction are underway in our laboratory.

MATERIALS AND METHODS

Experimental methods

The D. discoideum cells (strain AX2-214) were grown at 22°C in HL5 medium, harvested in the exponential growth phase, and starved for 4 hours in 10 ml of phosphate buffer supplemented with 2 mM caffeine in a shaking suspension. After 4 hours of starvation, they were centrifuged and diluted to a density of 2 × 106 cells/ml in fresh phosphate buffer containing 2 mM caffeine. Next, 20 ml of cell solution (≃0.9 × 106 cells/cm2 ≃0.9 monolayer) was transferred to a modified petri dish with a plasma-treated PDMS substrate (50). The PDMS had a periodic array of macro-pillars characterized by pillar dimensions and spacing (fig. S1). If not stated otherwise, pillars of 1 mm in diameter and a height of 3 mm were arranged on square, triangular, or hexagonal lattices with a lattice size of 5 mm. Patterns of cAMP waves are indirectly visualized by dark-field microscopy (10, 51, 52). Note that under dark-field illumination, the light scattered by the cells is a measure of changes in the cell shape and indirectly reflects the concentration of cAMP. Direct visualization of cAMP waves using isotope dilution-fluorographic technique (53) has shown that cell shape change and concentration of cAMP are correlated. Whereas at low cAMP concentrations, cells are more or less round and scatter a little light, at high concentrations of cAMP, chemotactic cells are elongated and scatter more light. Pairs of dark-field images separated by 1 min were subtracted from one another and then bandpass filtered to reduce spatial noise. These filtered images were used to obtain phase maps for visualizing wave propagation in oscillatory systems. They were obtained using a mathematical operation called the Hilbert transform, which converts a real signal to an analytical signal with real and imaginary parts. The phase angle of this analytical signal is then between [ − π, π] and represents the phase of the oscillatory system. Waves propagate in regions where the phase field is continuous and, in standard convention, in the direction from positive to negative phase. Note that in the areas where the waves collide, the phase field is no longer continuous.

Numerical methods

The reaction-diffusion equations used for modeling this system arek11t ρi=f1(γ(xi,yi))ρi+f2(γ(xi,yi))(1ρi)t βi=sΦ(ρi,γ(xi,yi))(ki+kt)βit γ=D2γkeγ+ΣiNH(i,x,y)kt βi/hwith f1(γ)=1+κ γ1+γ, f2(γ)=L1+κ L2 c γ1+c γ, Φ(ρ,γ)=λ1+Y2λ2+Y2, Y(γ,ρ)=ρ γ1+γ, and s = qσα/(1 + α), where γ(x, y) and βi are the amounts of extracellular and intracellular cAMP, respectively. ρi corresponds to the percentage of active cAMP receptors on the cell surface and acts effectively as the slow variable that gives the system its excitable capabilities. ke corresponds to the extracellular PDE, and s controls the amount of cAMP produced inside the cells. H(i, x, y) is an index variable with values 1, if the ith cell is located in (x, y), and 0, if there is no cell at this position. Therefore, on the grid points containing amoebas, the cAMP is transported from the intra- to the extracellular medium, whereas on the empty cell spaces, the wave is degraded by the free diffusing PDE. The system was simulated using discrete differences, a five-point Laplacian, and a time-adaptive Runge-Kutta scheme. The used parameters were as follows: σ = 0.55 min−1, k1 = 0.09 min−1, κ = 18.5, ℒ1 = 10, ℒ2 = 0.005, c = 10, q = 4000, α = 1.2/Km, λ1 = 10−4/ε, λ2 = 0.2575/ε, ki = 1.7 min−1, kt = 0.9 min−1, D = 0.024 mm2/min, and h = 5. For simulating the effects of caffeine, either Km = 0.4 − 0.6 mM and ε = 1.0 or Km = 0.4 mM and ε = 0.6 − 1.0 were used. Note that higher Km values correspond to higher concentrations of caffeine, whereas higher values of ε are a measure of lower caffeine concentrations (see movie S31 for a simulation of different values of ε). With these parameters, the system is in an oscillatory state, meaning that a limit cycle exists. If a cluster of cells of large enough size exists, it acts as a pacemaker, producing trigger (chemical) waves that are relayed by the system, as long as a minimum percentage (measured as surface coverage) of cells exists. How this model produces target centers was previously described (33). Emulating the experimental observations, after some simulation time, when the waves have been established, we allowed the cells to be chemotactically competent. The movement rules were as follows. If a cell detected a cAMP gradient larger than a threshold (∇γ > gth) and it was in the excitable state (ρ > ρth), it moved against the gradient with a velocity vc as long as the two previous conditions continued to be fulfilled. If the new position fell in a different grid space, the movement occurred only if the new grid point did not already contain a cell. The parameters used were gth = 25.98 nM/mm, ρth = 0.6, and vc = 20 μm/min, and we allowed cell movement after t = 50 min.

SUPPLEMENTARY MATERIALS

stke.sciencemag.org/cgi/content/full/13/655/eaaz3975/DC1

Methods

Fig. S1. Setup.

Fig. S2. Spiral pinning.

Fig. S3. Effects of caffeine and pillars.

Fig. S4. Frequency of firing centers.

Fig. S5. Dependency of frequency on local cell density.

Fig. S6. Comparison of experiments with and without caffeine in the Fourier domain.

Fig. S7. Analysis to quantify the degree of synchronization.

Fig. S8. Drift of annihilation point.

Fig. S9. Formation of fruiting bodies with a thin layer of fluid.

Fig. S10. Experiments with pillars of random diameter and spacing.

Fig. S11. Numerical simulations with and without pillars.

Fig. S12. Numerical simulations showing the effects in amount of target centers when adding caffeine to D. discoideum.

Fig. S13. Period measurements in the simulations.

Fig. S14. Comparison of aggregation territories in two experiments.

Fig. S15. Phase diagram.

Fig. S16. Numerical simulations with pillars and a fixed boundary condition.

Fig. S17. The number of fixed points at different regimes of the MG model.

Movie S1. Spiral pinning.

Movie S2. Experiment with pillars without caffeine.

Movie S3. Experiment without pillars but with caffeine.

Movie S4. Experiment showing regular patterns and streaming.

Movie S5. Image processing of an experiment showing regular patterns and streaming.

Movie S6. Phase map.

Movie S7. Experiment with no initial starvation but with caffeine.

Movie S8. Degree of synchronization.

Movie S9. Experiment with hexagonal arrangement of obstacles.

Movie S10. Experiment with triangular arrangement of obstacles.

Movie S11. Experiment with pillars 1.5 mm in diameter.

Movie S12. Experiment with pillars 50 μm in height.

Movie S13. Experiment with holes.

Movie S14. Experiment with a wall.

Movie S15. Experiment showing a bright-field view of one pillar.

Movie S16. Experiment showing wave initiation at clusters with higher local cell density.

Movie S17. Experiment showing the formation of fruiting bodies.

Movie S18. Experiment with pillars of random diameter.

Movie S19. Experiment with a random distribution of pillars.

Movie S20. Experiment with pillars 100 μm in diameter.

Movie S21. Simulations showing cell streaming.

Movie S22. Simulations with pillars without caffeine.

Movie S23. Simulations with pillars with caffeine.

Movie S24. Simulation showing a reduced number of firing centers in the presence of caffeine.

Movie S25. Experiment without and with caffeine.

Movie S26. Simulations with higher cell density around pillars.

Movie S27. Simulations with a reduced cAMP degradation rate around the pillars.

Movie S28. Simulation with an increased cAMP degradation rate around the pillars.

Movie S29. Simulation with a triangular arrangement of pillars.

Movie S30. Simulation with a hexagonal arrangement of pillars.

Movie S31. Simulation with ε.

Movie S32. Experiments with a BSA coating.

Movie S33. Experiments with pillars made from PMMA.

Reference (54)

REFERENCES AND NOTES

Acknowledgments: We are grateful to E. Bodenschatz, E. Frey, V. Zykov, O. Steinbock, F. Mohammad-Rafiee, L. Turco, I. Guido, H. Nobach, and B. Eltzner for fruitful discussions and to the unknown referees for helpful comments and suggestions. Funding: T.F.E. acknowledges Deutsche Forschungsgemeinschaft (DFG), project Nr. GH184/1-1. E.V.-H. thanks the Deutsche Akademische Austauschdienst (DAAD), Research Grants–Doctoral Programs in Germany. A.G. and A.J.B. acknowledge the MaxSynBio Consortium, which is jointly funded by the Federal Ministry of Education and Research of Germany and the Max Planck Society. Author contributions: T.F.E. performed experiments; E.V.-H. designed and performed numerical simulations; T.F.E., E.V.-H., A.J.B., and A.G. analyzed data; A.J.B. and A.G. designed research; T.F.E., E.V.-H., A.J.B., and A.G. wrote the paper. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper or the Supplementary Materials.
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