Actin and PIP3 waves in giant cells reveal the inherent length scale of an excited state

Actin-based force generation is the principal mechanism of motility in eukaryotic cells. A signature of directional locomotion in many cell types is the polarization into well-defined front and tail regions that differ in terms of the composition and dynamics of the actin cytoskeleton and in the phosphoinositide signaling system of the membrane. In the front region, actin is polymerizing to drive the formation of membrane protrusions, whereas in the tail region, the cell body retracts owing to the contractile action of myosin II. The leading edge corresponds to a phosphatidylinositol (3,4,5)-trisphosphate (PIP3)-enriched membrane at the cell front. The front is distinguished from the back of the cell, which is dominated by the PIP3-degrading phosphatase PTEN (Funamoto et al., 2002; Iijima and Devreotes, 2002). Front- and tail-type regions can also self-organize into dynamic patterns on the planar lower surface of a cell, where transition zones between the two states travel along the substrate-attached membrane (Arai et al., 2010; Schroth-Diez et al., 2009; Shibata et al., 2012; Taniguchi et al., 2013). These zones are decorated with an actin-rich band, called an actin wave.

An overview of various types of actin-based waves has been provided by Allard and Mogilner (Allard and Mogilner, 2013). The actin waves studied by us in Dictyostelium discoideum cells differ in principle from waves that are produced in an in vitro motility assay by instabilities in the myosin-driven translocation of pre-established actin filaments (Schaller et al., 2010). As revealed by fluorescence recovery after photobleaching (FRAP), the actin waves in Dictyostelium travel by the net polymerization of actin at their front and net depolymerization at their back (Bretschneider et al., 2009). Similarly, waves of Hem-1 (also known as NCKAP1L), a subunit of the WAVE complex that promotes Arp2/3-mediated actin polymerization, propagate by the recruitment of Hem-1 at their front and its release at their back (Weiner et al., 2007). In human osteosarcoma cells, ventral actin waves can generate transmembrane signals by integrin activation (Case and Waterman, 2011). These adhesive waves are associated with the Arp2/3 complex, as are the actin waves in Dictyostelium.

No membrane folds have been detected beneath the actin waves on the substrate-attached surface of Dictyostelium cells (Gerisch et al., 2004). The planar membrane on which these waves are traveling distinguishes them from circular dorsal ruffles, which are shaped by curved proteins that bind to convex or concave portions of the membrane, where they activate the polymerization of actin (Peleg et al., 2011).

Actin waves on a planar substrate-attached cell surface are not necessarily associated with cell movement or changes in cell shape, and thus differ from waves that reflect spatiotemporal patterns in the protrusive activity of a motile cell. The latter class of waves comprises periodic actin-based protrusions in various mouse and Drosophila cells (Döbereiner et al., 2006), and curvature waves in Dictyostelium that travel from the leading edge to the tail of a cell (Driscoll et al., 2012).

The actin waves propagating on the substrate-attached surface of Dictyostelium cells are supposed to be instrumental in the search for bacteria to be taken up by phagocytosis (Gerisch et al., 2009). These waves typically display a closed, ring-shaped structure and circumscribe a PIP3-rich inner territory that differs from an external area decorated with PTEN, a PIP3-degrading phosphoinositide 3 (PI3) phosphatase. Several criteria relate the inner territory to the front and the external area to the tail region of a polarized cell. The membrane of the inner territory is distinguished not only by the enrichment in PIP3 and the lack of PTEN, but also by the activation of Ras (Gerisch et al., 2011). Correlated with differences in the membrane are differences in the structure and composition of the underlying actin network – the inner territory of densely packed actin filaments is enriched in the Arp2/3 complex, whereas the external area consists of wide-meshed actin bundles that are associated with myosin-II and the actin-bundling protein cortexillin (Schroth-Diez et al., 2009). Thus, the wave patterns on the substrate-attached cell surface provide an opportunity to image, within one plane of focus, transitions between two states of the cell membrane in line with changes in the underlying actin cortex (Gerisch et al., 2012).

Excitability of the membrane and cortical actin layer has been implicated in the periodicity of actin polymerization at the leading edge (Ryan et al., 2012) and in the chemotaxis of eukaryotic cells (Shi and Iglesias, 2013; Nishikawa et al., 2014). The wave dynamics in Dictyostelium provide a system to study the propagation of an excited state within a single cell (Gerisch et al., 2012). The wave patterns that have been previously analyzed were formed within the boundaries of a single cell where the lateral borders confined the space for wave expansion to 10–20 µm, allowing only sections of the pattern to develop. To eliminate these restrictions of pattern development, we used cells in which the total membrane area is greatly enlarged such that the cell borders are far off.

Giant cells were either produced by cultivating myosin-II-deficient cells in suspension where the mutant cells are unable to divide (DeLozanne and Spudich, 1987; Knecht and Loomis, 1987) or they were generated in a wild-type background by electric-pulse-induced cell fusion (Gerisch et al., 2013). The giant cells enabled us to investigate the unpinned and undampened propagation of actin waves over distances that exceeded the radius of a normal cell by an order of magnitude. Here, we demonstrate that PIP3 patterns self-organize into dynamic zones with an intrinsic length scale. Independent of cell size, this length scale determines the distance between leading and trailing actin waves that are coupled to the borders of the PIP3-rich territories.

To monitor actin waves on the substrate-attached surface of giant cells, we fused cells that expressed markers for both the front and tail region of a normal motile cell – mRFP–LimEΔ, a label for filamentous actin that is enriched at the front of the cell (Fischer et al., 2004) and GFP–myosin-II heavy chain, a marker for the tail region (Moores et al., 1996). The electrofused cell shown in Fig. 1 exemplifies three features that are typical of the wave patterns in giant cells: (1) the waves can propagate with undiminished velocity over long distances, largely undisturbed by the cell boundary; (2) when waves collide, they tend to extinguish each other; and (3) there is no persistent pacemaker that serves as the origin of a periodic wave pattern. In this large cell, a zone enriched in myosin II was formed at an average distance of 6 µm behind the peak of the actin wave, independently of the cell border.

A variety of wave patterns in giant cells is illustrated in Fig. 2, showing a large myosin-II-null cell that expressed LimEΔ–GFP as a label for filamentous actin. Dominant features of the patterns are actin-enriched bands with peaks of actin accumulation at their border (Fig. 2A). These bands are polarized and mobile. They propagate across the substrate-attached surface of the cell with one of their broadsides ahead with an average velocity of 0.12 µm×s⁻¹±0.06 (±s.d.). The actin-rich border of the band corresponds to the actin waves previously studied in normal-sized cells (Gerisch et al., 2011). In the giant cells, these waves consist of a leading and a trailing segment (Fig. 2B; supplementary material Movie 1). In accordance with the terms employed for normal-sized cells, we designate the band-shaped regions surrounded by the actin waves as ‘inner territory’ and the regions outside the wave as ‘external area’ (Gerisch et al., 2011). In normal-sized cells, the inner territory has been shown to be enriched in the Arp2/3 complex, the external area to be associated with myosin II (Schroth-Diez et al., 2009).

Typically, pattern formation begins with the local clustering of actin, followed by the circular spreading of a wave from the initiation site (Fig. 2C,D). The concentric pattern eventually breaks and gives rise to waves that continue to propagate as actin-rich bands in radial direction (Fig. 2E, 553 and 561 s frames). When the bands collide, they extinguish each other locally followed by their fusion, as shown in Fig. 2E (582 to 618 s frames) and in supplementary material Movie 1.

The overview of shape dynamics shown in Fig. 2 and supplementary material Movie 1 reveals a preference for band-shaped waves with a preserved width. To substantiate this observation, we have developed an image-processing tool to represent the wave width and its changes over time. If a pattern is dominated by a specific width this will produce a peak in the distribution plot. The evolution of a dominant width is reflected in changing positions of the most prominent peak as a function of time. This analysis is illustrated in Fig. 3A–D for two scenarios, the expansion of a circular wave and the propagation of a band-shaped wave.

The actin wave shown in Fig. 3A undergoes expansion until, at a critical size, a circular trailing wave is inserted that leads to a ‘doughnut’ pattern. In the evolution of the width distribution, the circular expansion is reflected in a gradual shift of the dominant peak to larger values, until at 48 s the preferred width of ∼12 µm is restored by the emerging doughnut shape (Fig. 3E).

The width of the wave displayed in Fig. 3B and supplementary material Movie 2 stays constant at about 15 µm for >5 min until, after 400 s, a segment of the wave widens, resulting in a second peak in the distribution that gradually shifts to larger values (Fig. 3F). After 470 s, the widened area is converted into an arc-shaped structure and the distribution returns to a single peak between 10 and 15 µm. The overview of supplementary material Movie 3 adds further examples for relaxation of the system into that preferred width between the outer borders of the actin wave. In conclusion, a key principle of the wave pattern is an inherent length scale that does not vary with cell size.

The territory enclosed by the leading and trailing segments of an actin wave propagates as a coherent PIP3-enriched band (Fig. 4A,B; supplementary material Movie 4). To explore the PIP3 dynamics during wave propagation, we measured the fluorescence intensities of the marker GFP–PHcrac (Parent and Devreotes, 1999) at multiple points on the substrate-attached surface of giant cells. For an unbiased sampling of temporal changes in a field of PIP3 waves, points of measurement were distributed on a rectangular grid with a distance of 10 µm between the points (Fig. 4C). This is about the diameter of a non-fused normal-sized cell. At each site, the passage of a wave produced a transient increase in PIP3. As an example, the time-series obtained at the point encircled in Fig. 4C is displayed in Fig. 4D. The distribution of peak-to-peak intervals extends over a wide range, suggesting that the system did not oscillate at a specific frequency (Fig. 4E).

Among the transient PIP3 increases, we selected those corresponding to waves that were not influenced by the cell border or by interference with another wave. Averaging the scans shown in Fig. 4F resulted in a curve with a width at half-maximum of 44 s. A rise time of 17 s from half-maximum to maximum and a slower decay of 27 s from maximum down to half-maximum results in an asymmetric shape of the temporal PIP3 profile. No extended plateau of PIP3 accumulation was observed. The spatial profile of PIP3 normal to the direction of wave propagation is shown in Fig. 4G. This profile is characterized by a distance from half-maximum to maximum of 2.1 µm, and a distance from maximum back to half-maximum of 3.0 µm. The mean velocity of propagation was 0.13 µm×s⁻¹±0.03 (±s.d.).

As shown in Fig. 4F,G, the PIP3 increase during the passage of a band-shaped wave across a point of the cell surface has a limited lifetime of ∼44 s and a width in space of 5 µm at half-maximum. This pattern evolves from circular waves that radially expand from their site of origin, as shown for an early stage of pattern formation in Fig. 1. If the lifetime of the PIP3-rich state of the membrane is limited, there should be a critical radius, beyond which the territory within a circular wave becomes unstable such that its center will turn from the PIP3-rich into a PIP3-depleted state. Fig. 5 shows three examples of how the conversion of a circular PIP3 pattern into a propagating band pattern takes place. In the case of Fig. 5A, also shown in supplementary material Movie 5, the circular area is almost symmetrically divided into two bands by the lateral ingression of external area from two opposite sides. The measurement of PIP3 dynamics at different points in the field indicates a change from longer to shorter persistence times during evolution of the circular area into propagating bands (Fig. 5B). It appears, therefore, that during expansion of a circular wave, PIP3 is in a metastable state, before it relaxes into the steady state of synthesis and degradation that dominates the band pattern of PIP3 in giant cells. Often the external area ingresses only from one side, converting the circular PIP3-rich area into a horseshoe-shaped band with two open ends.

The two cases shown in Fig. 5C,D are distinguished by a prolonged preservation of symmetry. Here, the coherent PIP3-rich area is transformed into a ‘doughnut’ pattern consisting of a PIP3-rich annulus and a PIP3-depleted area in the center of the doughnut (Fig. 5C; supplementary material Movie 6). During the decline of PIP3 in this area, a circular actin wave is inserted. To provide evidence that the PIP3-depleted area changes its specification, we have co-labeled cells with GFP–myosin-II heavy chains. This label indicates that filamentous myosin II strongly assembles in the central area of a doughnut pattern, thus underscoring the transition from a front-like into a tail-like state (Fig. 5D; supplementary material Movie 7). The preference for band patterns with a defined width is also maintained during splitting, curling and budding of PIP3-rich territories. In particular, lateral expansion contributes to the elongated shape of the bands (supplementary material Fig. S1).

In an excitable system, a period of stimulation is typically followed by a refractory phase during which a stimulus cannot evoke a second response. The unidirectional propagation of PIP3 waves suggests that, after a period of activation, the PI3 kinases pass a state of refractoriness. Moreover, when two waves collide, they most often extinguish each other, as shown in Fig. 2E and supplementary material Movies 4, 8. This behavior is typical of waves in an excitable system that are followed by a refractory phase. Nevertheless, a detailed analysis of pattern dynamics in giant cells indicated that the refractory phase can be short, and an absolute refractoriness is sometimes lacking. In the giant cells, we investigated the variability of refractory phases using waves that collide with the cell border.

Frequently, waves disappeared from the substrate-attached cell surface after collision with the cell border, as expected if excitation is followed by a refractory phase (Fig. 6A,B). In other cases, the waves were not completely extinguished at the border of the cell; remnants persisted and propagated backward along the substrate-attached cell membrane. In these remnants of a wave, PIP3 remained at a high level for a longer period of time than during unperturbed wave propagation (Fig. 6C,D as compared to Fig. 4F).

The subject of the present paper is the self-organization of PIP3 and actin waves on the inner face of the substrate-attached membrane in giant cells of D. discoideum. We produced these cells by electric-pulse-induced cell fusion or by myosin II knockout in order to investigate pattern dynamics not limited by the narrow borders of normal-sized cells, which have been studied previously (Gerisch et al., 2012). As with the wave patterns in normal cells, those in giant cells are generated spontaneously. The patterns develop independently of external chemoattractant gradients or any other structured impact from the environment, indicating that they evolve by self-organization.

The principal pattern elements in giant cells are PIP3-rich bands of finite width that propagate along the membrane. These bands are flanked at their front by the leading segment of an actin wave and at their back by a weaker (sometimes missing) trailing segment. This means that each segment of the actin wave separates two states of the membrane, a PIP3-rich and a PIP3-depleted state. The pattern-generating system can be considered as an excitable medium, the excited state of which consists of a composite wave embracing the PIP3-rich territory together with the flanking segments of an actin wave (Fig. 7). This implies that the actin pattern is coupled to the on and off of PIP3 synthesis, and that segments of an actin wave are formed at two transition states – one from low to high, the other from high to low PIP3.

In previous experiments, we have enhanced the wave formation by pre-treating cells with latrunculin A to block actin polymerization (Gerisch et al., 2004). During recovery from the drug, the cells consistently undergo a stage of profuse wave formation. However, actin waves are also formed on the substrate-attached surface of untreated cells, in particular during the first few hours of starvation (Bretschneider et al., 2004; Taniguchi et al., 2013), and the giant cells maintain this behavior.

In normal-sized cells, actin waves have been shown to alternately expand and retract (Gerisch et al., 2011), resulting in the presence of actin waves at the front of an expanding and at the back of a shrinking PIP3-rich area (Asano et al., 2008). The pattern in giant cells shows that actin waves at the front and at the back of a PIP3-rich area are in fact sectors of a wave that are coupled to each other through the finite width of the PIP3-rich zone. The expanding waves correspond to the leading segments of an actin wave in giant cells, and the retracting waves to its trailing segments. This means that the wave patterns previously observed in normal-sized cells are confined sectors of the multitude of configurations that, due to the larger space available, can freely evolve in giant cells (Fig. 8). Notably, the leading segment of an actin wave in giant cells propagates from a PIP3-rich territory towards the PIP3-depleted external area, whereas the trailing segment moves the opposite way towards increasing concentrations of PIP3.

Waves in giant cells can propagate with undiminished velocity and amplitude over distances almost one order of magnitude larger than the radius of a normal cell (Fig. 1). The average velocity of 0.11 µm×s⁻¹ for the propagation of PIP3 bands in electrofused cells is similar to that in normal-sized cells, where an average of 0.14 µm×s⁻¹ was found (Gerisch et al., 2012). In the cell shown in Fig. 1, a velocity of 0.043 µm×s⁻¹ was found for an actin wave (Fig. 1B). This exceptionally low value is probably due to the fact that in this case a circular wave was addressed at an early stage of pattern development.

The actin waves are independent of myosin II (Bretschneider et al., 2009). Cells lacking myosin II heavy chains profusely form waves (Fig. 2; supplementary material Movie 1), which propagate at a velocity of 0.12 µm×s⁻¹ – similar to those in wild-type cells – indicating that the filamentous myosin has no substantial influence on wave propagation. As in wild-type cells, the actin waves in myosin-II-null cells enclose a dense fabric of actin filaments and separate this inner territory from an area with a loose filamentous network (Fig. 2B). These data indicate that myosin II, although being enriched in the external area, is not required for the switch from inner to external area nor for differentiation of the cortical actin structure.

The evolution of spatiotemporal patterns in giant cells starts with expanding circular waves. When their radius exceeds a critical length, the PIP3-rich membrane area becomes unstable. Depletion of PIP3 and assembly of filamentous myosin II indicates a state transition that corresponds to a switch from the ‘front’ to the ‘tail’ state of membrane and cortex organization (Fig. 5D). As a consequence of the limited width, curved bands are generated that are rich in PIP3 and surrounded by an actin wave. These PIP3-rich bands have an average width at half-maximum of 5.1 µm (Fig. 4G). This width is determined by the temporal sequence of PIP3 synthesis and degradation, which limits the persistence of the high-PIP3 state to 44.5 s at half-maximum (Fig. 4F). This characteristic lifetime of the PIP3-rich state in giant cells is similar to 46 s±7 (±s.d.), the lifetime in normal non-fused cells (Gerisch et al., 2012).

The giant cells show that the width of the PIP3-rich bands is not determined by the cell border or any other compartmentalizing membrane. The defined width appears to be unique to PIP3-rich regions of the membrane; we did not find a fixed length scale for PIP3-depleted areas. As a consequence, the PIP3 bands are not generated by a stable pacemaker that sets a specific frequency. Instead, the peak-to-peak intervals of PIP3 waves measured at various points on the cell surface show a broad distribution (Fig. 4E).

The mechanism that determines the intrinsic width of a PIP3 band might be relevant to the polar organization of a normal-sized cell with a front-to-tail distance of 10–20 µm. This mechanism seems to limit not only the front region in a motile cell but also the length of the PIP3-rich zone in a phagocytic cup. In phagocytic cups that enclose a long cylindrical particle, the PIP3-rich section of the membrane tube that encloses the particle has a limited length of ∼8 µm, even when the entire phagocytic cup is longer (Gerisch et al., 2009).

The wave-generating cell membrane together with the associated cortical region behaves as an excitable system consisting of multiple components that are coupled to each other. Two features unveiled in giant cells illustrate specific properties of the underlying excitable system. First, sites of wave initiation are generated stochastically. The wave-generating system in Dictyostelium cells thus behaves as an excitable medium with random fluctuations that occasionally cross the excitation threshold so that a wave is initiated at a random location. This is reminiscent of an ‘extreme event’ that might occur in an excitable system modeled by diffusively coupled FitzHugh–Nagumo units (Ansmann et al., 2013).

A second characteristic feature is that refractory phases are variable and sometimes indistinct. Refractory phases are crucial when waves interact with each other or with the cell border. If a refractory phase does exist, waves annihilate each other at the site of collision (Fig. 2E; supplementary material Movie 4). First, the leading segments of the actin waves fuse and disappear while the trailing segments continue to propagate until both types of segments are extinguished at the site of collision (Fig. 2E, 582 and 612 s frames). To the left and right of this site, the waves will fuse such that two new waves are formed that propagate in opposite directions, each of them normal to the previous axis of wave propagation (Fig. 2E, 618 s frame; supplementary material Movie 8).

For waves interacting with the cell border we observed two different scenarios attributable to variability in refractoriness (Fig. 6). Consistent with full refractoriness, a wave can be annihilated upon collision with the cell border (Fig. 6A,B). However, an excited state can also persist at the border until the PIP3-rich territory expands in the reverse direction, so that the wave appears to be reflected, implying the absence of a refractory phase (Fig. 6C,D). Lack of a refractory phase is also evident when a wave changes direction, as shown in Fig. 3B and diagrammed in Fig. 8A. In this case, the trailing segment of an actin wave is converted into a leading one and the PIP3-rich territory turns from shrinkage to expansion. From this flexibility in behaviour, a broad range of dynamical regimes of the underlying pattern generating system can be deduced.

Wave reflection at non-flux boundaries has been reported for various excitable systems (Petrov et al., 1994; Argentina et al., 1997; Hayase and Ohta, 1998). Among them are different versions of the Oregonator model of the Belousov–Zhabotinsky reaction (Kosek and Marek, 1995; Bordyugov and Engel, 2008) and of the Hodgkin–Huxley model (Aslanidi and Mornev, 1997). In a system with three coexisting steady states discussed by Petrov et al. (Petrov et al., 1994), the waves reverse direction without annihilation of their maxima, similar to the waves reflected at the border of a Dictyostelium cell (Fig. 6C,D). The reversal of direction in the model system has been attributed to the effect of a boundary on the diffusive fluxes of reactants into and out of the wave.

Recent models of actin waves focus on the interactions of small GTPases with the machinery of actin polymerization, consistent with the finding that actin waves in Dictyostelium are coupled to patterns of PIP3 (Asano et al., 2008; Gerisch et al., 2009) and of Ras signaling (Gerisch et al., 2011) on the underlying membrane. In the model analyzed by Mata et al. (Mata et al., 2013), the GTPase enhances its own activation by positive feedback, and is inactivated by negative feedback from F-actin. It is essential for this model that the concentration of a GTPase, i.e. the sum of its concentrations in the active and inactive state, is constant. This is a reasonable assumption for the closed volume of a living cell that continues to generate waves for more than an hour. The model allows wave reflection on a diffusion barrier, a behavior that we have obtained. In a certain parameter space, the model proposes pinning of the actin waves, which we did not observe for waves that travel along the extended membrane area of the giant cells.

The model of Khamviwath et al. (Khamviwath et al., 2013) proposes a positive-feedback circuit between GTPase, PIP3 and F-actin, based on experimental data by Sasaki et al. (Sasaki et al., 2007), and a second positive-feedback loop due to Arp2/3-mediated branching of actin filaments, as suggested by Carlsson (Carlsson, 2010). The decay on the back of the waves is thought to be caused by local exhaustion of one of the constituents of the actin network. This model is consistent with the observed changes of direction in the propagation of actin waves and with their annihilation at the sites of collision.

The patterns obtained in giant cells provide a basis for explicit models of traveling actin waves as they are generated spontaneously in Dictyostelium cells. A characteristic of these waves is their division into two segments that are separated by a finite width of a PIP3-enriched membrane space.

Cells of D. discoideum strain AX2–214 were transfected with integrating vectors to express GFP and/or mRFP fusion proteins, as listed in supplementary material Table S1. Cells were cultivated in HL5 or modified maltose-containing medium with selection reagents blasticidin, hygromycin and/or G418. The PIP3 marker GFP–PHcrac was used in two versions – for Fig. 4A,B it was expressed on an extrachromosomal vector in AX3 cells (Parent and Devreotes, 1999); otherwise it was expressed as a superfolding GFP construct in AX2 cells (Müller-Taubenberger and Ishikawa-Ankerhold, 2013). Cells of the AX2-derived myosin-II-heavy-chain-null mutant HS2205 (Manstein et al., 1989) were cultivated for 2 days in shaken suspension to produce large cells by the prevention of cytokinesis. Temperature was within 20 to 23°C throughout the experiments.

Transformed cells were fused by electric pulses using one of two different protocols.

Cells were harvested from non-confluent Petri dishes, washed twice in 17 mM K⁺/Na⁺-phosphate buffer pH 6.0, adjusted in the buffer to 1.5×10⁷ cells/ml, and gently shaken for 3 h in roller tubes, allowing the cells to cluster. Using a cut-off pipet tip to prevent dissociation of the clusters, aliquots of the suspension were transferred to electroporation cuvettes with an electrode distance of 4 mm and fused in a BioRad Gene Pulser Model 1652077 (Bio-Rad Laboratories, Hercules, CA94547) by applying three pulses of 1 kV and 1 or 3 µF at 1-s intervals. A 20-µl aliquot of the fused-cell suspension was transferred into an open chamber on a glass coverslip. After 5 min, 1 ml of the phosphate buffer supplemented with 2 mM CaCl₂ and 2 mM MgCl₂ was added and, after settling, the cells were subjected to imaging.

Cells were cultivated in suspension up to the end of the exponential growth phase. Then the cells were washed in an aqueous solution of 340 mM glucose and resuspended at 1×10⁷ cells/ml in this dielectric medium. After shaking for 12 h, the cells were washed twice and adjusted to 5×10⁶ cells/ml in the same solution. Cells were fused within glass-bottom culture dishes equipped with aluminium electrodes spaced by 4 mm. An ECM 2001 Electro Cell Manipulator (Harvard Apparatus, Holliston, MA 01746-1388) was set to loop nine times over a dielectrophoresis step of 70 V for 8 s, followed by a 1 kV pulse for 50 µs and another dielectrophoresis step for 1 s. The entire procedure was repeated and the cells were gently transferred to 35-mm glass-bottom culture dishes where they were stepwise equilibrated with 17 mM K⁺/Na⁺-phosphate buffer pH 6.0, supplemented with 2 mM CaCl₂ and 2 mM MgCl₂.

Confocal images were acquired using a Zeiss LSM 780 equipped with a Plan Apo 63×/NA 1.46 or with a Plan Apo 40×/NA 1.4 oil-immersion objective (Carl Zeiss Microscopy, 07745; Jena, Germany). An algorithm was designed to analyze the shape of waves as follows. At a pixel size of 0.181 µm for Fig. 3A and 0.191 µm for Fig. 3B, the actin wave was defined manually by pointing on its perimeter and was rendered automatically by a polygon with a point-to-point distance of 10 pixels. A subroutine calculated the length of all secants directed from each point of the polygon perpendicularly across the wave (for examples, see Fig. 3C,D). The probability of secant lengths was color-coded and the temporal evolution of the length distribution presented as a kymograph (Fig. 3E,F).

Fluorescence intensities were processed using the image processing package Fiji developed by Schindelin et al. (Schindelin et al., 2012) on the basis of ImageJ. Plug-ins of Fiji were used for point scans with an approximately circular area of 12 pixels and for line scans with a width of 30 pixels. Data were copied into a Microsoft Excel spreadsheet for calculation and chart plotting.

We thank Kirsten Krüger (Institute of Physics and Astronomy, University of Potsdam, Germany), Jana Prassler (Max Planck Institute of Biochemistry, Martinsried, Germany) for providing cell cultures and Annette Müller-Taubenberger (Ludwig Maximilian University of Munich, Germany) for superfolding GFP-PHcrac.