Minimal model for spontaneous cell polarization and edge activity in oscillating, rotating and migrating cells

Abstract

How cells break symmetry and organize activity at their edges to move directionally is a fundamental question in cell biology. Physical models of cell motility commonly incorporate gradients of regulatory proteins and/or feedback from the motion itself to describe the polarization of this edge activity. These approaches, however, fail to explain cell behaviour before the onset of polarization. We use polarizing and moving fish epidermal cells as a model system to bridge the gap between cell behaviours before and after polarization. Our analysis suggests a novel and simple principle of self-organizing cell activity, in which local cell-edge dynamics depends on the distance from the cell centre, but not on the orientation with respect to the front–back axis. We validate this principle with a stochastic model that faithfully reproduces a range of cell-migration behaviours. Our findings indicate that spontaneous polarization, persistent motion and cell shape are emergent properties of the local cell-edge dynamics controlled by the distance from the cell centre.

Main

The ability to break symmetry and move directionally is an essential property of most eukaryotic cells1,2,3. This happens in response to external stimuli, but also spontaneously4,5,6. Persistent motion requires segregation of cell-edge activities, so that protrusion happens predominantly at the front, and retraction at the back of the cell. In contrast, in cells exploring their environment, edge activity is on average spatially isotropic, but fluctuates in time between protrusion and retraction7,8,9,10,11. Thus both exploratory activity and the directional motion depend on the transitions between protrusion and retraction, but how the cell chooses between these two regimes to establish spatial and temporal patterns of edge activity remains unclear. It is believed that in migrating cells a directional mechanism at the scale of the whole cell—for example, a global gradient of cytoskeletal and/or signalling components—orchestrates cell-edge dynamics according to the overall motion direction1,5,12,13,14. This concept is limited in that external directional stimuli5,13 in combination with internal diffusible signals interacting through feedback loops15,16,17,18,19,20, feedback from the motion itself6,21,22,23,24, or very large and highly correlated perturbations to induce the polarized states25 have to be invoked to establish the polarity axis.

Fish epidermal keratocytes, thanks to their robust polarity, simple shape, and persistent motion, are a classic model system to study polarization and directional migration3,26. Analysis of the edge dynamics of these cells led to the concept of graded radial extension (GRE), which links local edge dynamics to the resulting overall cell motion: protrusion and retraction are directed normally to the cell edge with rates that are graded depending on the orientation with respect to the motion direction26. This model inspired several studies searching for the underlying mechanism in the form of a gradient of cytoskeletal and/or signalling components from the front to the back of the cell. Gradients of myosin II, actin, and small GTPases of the Rho family6,12,14,17,22,27,28,29 have been implicated, but it remains unclear how these gradients arise in the first place and whether they are cause or consequence of polarization and directed motion30. The concept of a directional gradient guiding the cell polarity axis cannot describe cell-edge activity in the presence of multiple leading edges31 or before the point when the direction of motion becomes defined. In this study, we start from quantification of experimentally observable cell-edge dynamics to identify features that are common to the migration and the polarization process. We find that the cell edge universally exhibits a distance-dependent switch from protrusion to retraction, and we develop a computational model to demonstrate that this property is sufficient for spontaneous polarization and directional motion.

Edge dynamics in migrating cells

Using a recently developed cell-edge segmentation and tracking method32, we have quantified protrusion velocities in keratocytes with high spatial and temporal resolution. During migration, the protrusion velocity in the regularly shaped (coherent) cells12,14 was nearly constant and stable in time along the central part of the leading edge, but declined and fluctuated towards the sides, becoming eventually negative (retraction; Fig. 1a, b). Whereas previous studies considered the edge velocity to be constant in time, and rather emphasized its gradation in space, we explore the outcome of temporal velocity fluctuations. We designed a method to identify the positions along the cell contour where the edge velocity switched in time from protrusion to retraction (PR) and from retraction to protrusion (RP) (Fig. 1c, Methods and Supplementary Fig. 1). Our mapping revealed that most of the switches localized to narrow zones at the lateral extremities of the cells. Plotting the switch probability as a function of the distance from the geometrical centre of the cell (see Methods) indicated that approximately 70% of switches occurred between 90 and 100% of the maximal cell extension (Fig. 1d). The majority of switches were from protrusion to retraction, but, interestingly, we observed that the cell extremities also exhibited RP switches (15% the total number of switches), suggesting metastable edge dynamics at these sites. Some switches marginally appeared (≤5%) at intermediate distances (Fig. 1d). They occurred at the back of the cell and were attributed to the dynamics of retraction fibres33.

Figure 1: Edge dynamics and distribution of switches between protrusion and retraction in migrating cells.
figure1

a, Normal edge velocity profiles as function of position along the cell contour averaged over consecutive 11 s intervals. Time is colour-coded from purple to yellow and profiles are spaced by 0.2 μm s−1 for clarity. The zero-contour position (middle of the leading edge) is illustrated on the cell image (inset). b, Mean normal velocity along the cell contour (top) and standard deviation of the velocity (bottom) over 199 s. Vertical lines mark the positions of zero mean velocity. c, Localization of PR (black) and RP (red) switches during 82 s of migration plotted in the co-moving frame (central panel). Detection of switching regions as intersections of protrusion (grey) and retraction (pink) in three consecutive contours (Methods) (left and right panels). d, Switch probability as function of the normalized distance from the cell centre. Distances were normalized by the maximum distance from the cell centre to aggregate data from cells of different size. Contours were extracted every 2 s from five different cells for a total of 236 contours. Scale bar is 6.45 μm.

We point out that the regions of most probable switching correspond at the same time to the maximal cell extension and to the parts of the cell edge displacing orthogonally to the direction of motion. To discriminate which of the two descriptors, orthogonal orientation with respect to the direction of motion or distance from the cell centre, defines the distribution of switches, we manipulated the cells to change the distance between the edge and the cell centre at different orientations. To increase the distance along the direction of motion, we placed a micropipette in front of the cell, allowing the leading edge to squeeze between the pipette and the substrate, but blocking the taller nucleus. The leading edge maintained its directional motion at constant velocity (right panel in Fig. 2a) until the cell extended to a distance similar to its original maximal lateral extension. At this point, the edge halted, fluctuated, and switched to retraction, resulting in a reversal of cell motion (Fig. 2a and Supplementary Video 1; similar responses were observed in 12 out of 12 cells). We also tested the effect of a reduction of the distance between the edge and the cell centre by cutting off a part of one lateral wing of a cell. The cut side exhibited persistent protrusion transverse to the direction of motion (inset in Fig. 2b) until the cell recovered its original width. The protrusion zone may even propagate from the site of the cut to the back of the cell, resulting in a turning of the cell in the direction of the cut (Fig. 2b and Supplementary Video 2; identical response observed in 29 out of 32 cells). Because pipette blocking induced switches at the front and cutting suppressed them at the side, both types of manipulations provide evidence that switches are regulated by distance from the cell centre rather than by angle with respect to the cell motion direction. If the angular hypothesis was true, the cell would continue protruding at the front in the pipette experiment and maintain switches at the site of the cut in the cutting experiment.

Figure 2: Protrusion–retraction switches depend on the distance from the cell centre, but not on the orientation.
figure2

a, Snapshots (left) and kymograph (right) of the cell with its body stalled by a pipette. Time is indicated in seconds after encounter with the pipette. b, Snapshots of the cell before, 1 s after, and 100 s after its right lateral part was cut off with a glass needle. Persistent protrusion at the cut site results in shape recovery and turning in the direction of cut. Inset in the right panel is a kymograph taken along the white dashed line shown in the middle panel. Kymographs were generated from the sequences taken at 2 s (a) and 3 s (b) frame intervals.

Edge dynamics in polarizing cells

Next, we investigated the dynamics of edge activity during the process of spontaneous cell polarization. In the course of polarization, most of the cells exhibited multiple irregular fluctuations and waves around the edge, with small regions of protrusion and retraction coalescing into larger zones32 (Fig. 3a and Supplementary Videos 3 and 4). A polarized cell with a single protruding and a single retracting region emerged as the result of these protrusion–retraction fluctuations that occurred during the polarization process. Interestingly, some cells were trapped during polarization in exotic two-fold or three-fold symmetric configurations, with several leading edges rotating for several minutes (Fig. 3d and Supplementary Video 4). Both fluctuating and rotating cells eventually polarized, forming a single protruding and a single retracting region. Mapping of switches in fluctuating cells showed that switch sites were distributed evenly along the cell edge (Fig. 3b) and both types of transition had similar probabilities to occur (55% of the total number were PR switches). The distribution of switching distances exhibited a peak of PR switches near the maximal cell extension and a peak of RP switches at intermediate distances (Fig. 3c). In the cells that exhibited multiple rotating edges, switches localized mostly at the tips and bases of the rotating segments (Fig. 3e), and the two types of switches were unbalanced (65% of PR switches). As in fluctuating and migrating cells, PR switches occurred near the maximal cell extension, whereas RP switches were more likely at short distances from the cell centre (Fig. 3f).

Figure 3: Distribution of switches between protrusion and retraction and shape properties during polarization.
figure3

a,d, Phase-contrast snapshots of fluctuating (a) and rotating (d) cells.b,e, Localization of PR (black) and RP (red) switches during 366 s of fluctuation (b) and 304 s of rotation (e). c,f, Switch probability as function of the normalized distance from the cell centre in polarizing (c) and rotating (f) cells. Contours of polarizing cells were extracted every 10 s (5 cells, 214 contours in total). Contours of rotating cells were extracted from two cells in two-fold and three-fold rotation every 6 s and 4 s, respectively, for a total number of 110 contours. Scale bar is 6.45 μm (polarizing) and 10.75 μm (rotating). g, Time course of maximal cell extension (red), average cell extension (green) and average PR switching distance (black) during the process of polarization. Snapshots of the cell at indicated times during polarization are shown in the insets. i, Kymograph (upper panel) of the polarizing cell taken along the white line shown in the insets in g. The transition time point (t = 550 s) is indicated in the kymograph (blue dashed line) when edge fluctuations stop and the cell starts moving. At this point the cell is not yet well polarized and the shape remains irregular (right inset in g). At the end of the sequence the cell is fully polarized (lower panel). In g and i the scale bar is 10 μm. h, Map of the time course of the angular distribution of protrusion–retraction switches (black dots). The angles are taken with respect to the direction of the instantaneous cell velocity. j, Protrusion–retraction switch probability as function of the angle before (upper panel) and after the transition point (lower panel).

Analysis of the dynamics of PR switches in time during the transition between isotropic and polarized states demonstrated that they remained near the maximal cell extension during the entire process of polarization, including the onset of persistent migration (Fig. 3g). Measuring the angle between the orientation of PR switches with respect to the cell centre and the direction of the instantaneous velocity showed that switches occurred at any angle up to the onset of the polarized state, then the range of angles became narrowed down to the two zones orthogonal to the velocity (Fig. 3h, j). These results indicate that, despite changes in cell morphology, the switches from protrusion to retraction always occur near maximal cell extension, suggesting that this is an important feature underlying cell-edge dynamics during the polarization process.

To gain insight into the mechanisms controlling this switching distance we tested if the cell width during migration and the response in body-blocking experiments depended on the cell volume, contractility, and the integrity of the microtubule system. Increasing the cell volume by hypo-osmotic treatment and inhibiting myosin-II-dependent contractility by blebbistatin both led to an increase in cell width (with eventual cell fragmentation in the case of myosin inhibition; Fig. 4a). Cell-body-blocking experiments with inflated cells showed a partial suppression of the distance sensing: in 50% of the cases (three out of six cells) cells kept extending their leading edge until it eventually detached from the cell body (Fig. 4c and Supplementary Video 1). The blebbistatin-treated cells extended their front lamellipodia away from the cell body, while remaining connected to the latter by a narrow stalk (in contrast to a wide bridge in the control case). However, in 75% of the cases (9 out of 12 cells) the lamellipodia did not extend continuously, instead turning to a motion direction parallel to the pipette (Fig. 4b and Supplementary Video 1). These results indicate that distance sensing is not a trivial consequence of finite membrane area or of cytoskeletal protein content, but depends on the cell volume and/or three-dimensional shape34 along with myosin-dependent contractility. In contrast, disassembly of microtubules with nocodazole affected neither the cell width (Fig. 4a), nor the cell response in the body-blocking experiments (Supplementary Video 5). This suggests that, unlike in other systems where microtubules were implicated in cell-size control35,36, they do not play such a role in keratocytes.

Figure 4: Effect of blebbistatin, hypo-osmotic treatment, and nocodazole on cell width and behaviour in body-blocking experiments.
figure4

a, Dynamics of the average normalized cell width after treatment with blebbistatin (black, 11 cells), nocodazole (green, 10 cells), and hypo-osmotic shock (red, 9 cells). The width of each cell was normalized with its average before treatment. b, A blebbistatin-treated cell extends its front on a narrow stalk after body blocking and then turns parallel to the pipette. c, The cell after hypo-osmotic shock keeps extending its front.

Computational model

We developed a stochastic model incorporating the features of cell-edge dynamics uncovered in our experiments. The model is deliberately reductionist and minimal, to test whether switches induced by a threshold distance are indeed sufficient to explain the transition to a polarized state, directional motion and cell shape. We describe the cell edge by a set of points (that is, computational particles) Xi in two possible states: protrusion (P) or retraction (R). PR switches occur at a threshold distance rmax from the cell centre, whereas RP switches occur at a smaller threshold distance rmin. The state of a point also changes if most of its neighbours within a defined interaction range are in the opposite state, providing coupling that enables local ordering of the edge activity. Specifically, each point i, at a distance r from the cell centre has a probability PPR (resp. PRP) to switch from protrusion to retraction (resp. from retraction to protrusion), given by:

where τ is an overall rate of transition (Δt−1) that takes into account the self-persistence of the point state. It effectively allows increasing the displacement of a point between transitions while preventing numerical instabilities. is a Gaussian random number with mean rmax and variance σ2 (corresponding to a noisy threshold distance), ni is the number of neighbours (within the range of interaction N) of i that are in the opposite state; ni/N thus accounts for coupling. The position of each point i is then updated as:

with protruding nodes moving outwards normally to the edge at constant velocity Vp (Fig. 1a) and retracting nodes moving towards the cell centre (xc, yc) with a distance-dependent velocity Vr(r) (Methods and Supplementary Fig. 2). At each time step, the position of the cell centre is calculated from the coordinates of the edge points (xi, yi), as described in Methods.

For a wide range of parameters (Supplementary Figs 3–5), this combination of isotropic rules of local edge dynamics led to self-organization with spontaneous transition from disordered initial states to a polarized motile state with emergent shapes reproducing experimental observations (Fig. 5). The simulations were started from a circular outline with a random distribution of protruding and retracting points; first, the edge exhibited strong shape fluctuations comparable to those observed in experiments with small protruding and retracting regions travelling and fusing into larger zones (Fig. 5a and Supplementary Video 6). Then, the persistently migrating state was reached with stable typical keratocyte shapes elongated perpendicular to the direction of motion (Fig. 5b and Supplementary Videos 6 and 7). The analysis of the resulting cell outlines with the same switch mapping protocol that was used for real cells yielded distributions of switches similar to the experimental ones both for polarizing and migrating cells (Fig. 5d). The model was also able to sustain the rotating state of a specific initial three-fold configuration with three leading and trailing edges (Fig. 5c and Supplementary Video 8). We also simulated cell-blocking experiments with a virtual pipette in the form of a line that could not be passed by the cell centre. In this numerical experiment, the outline backed up in a way similar to real cells: by extending at a constant speed, halting, fluctuating in place, and eventually retracting and reversing its direction (Fig. 5e, and Supplementary Video 9). Finally, removing the lateral part of the cell outline and replacing it with a straight line segment resulted in persistent protrusion and turning of the cell towards the removed part of the edge (Supplementary Video 10), in agreement with the cutting experiments.

Figure 5: Computer model simulations of cell-edge dynamics.
figure5

Successive edge configurations during polarization (a), migration (b, snapshot every 10,000 time steps) and rotation (c) with the corresponding maps of protrusion (right panels). Protruding (retracting) points are shown in black (red), and the axes are in units of rmax. In the protrusion maps, the fraction of time spent in protrusion (colour scale) is computed for each point in a sliding window of 100 time steps. The zero curvilinear abscissa corresponds to the intersection of the cell edge with the line emerging from the cell centre in the average direction of motion. Positive (negative) coordinates stand for clockwise (anticlockwise) directions along the contour. The simulations were run with a range of interaction N = 4 and a noise σ = 0.15rmax. The points had random initial states and were initially equi-distributed on a circle. The simulation for the rotation was run with the following parameters: Vmax = 1.2Vp, Vmin = 0.8Vp, N = 48, and the coupling term (n/N)5. d, Switch probability as function of the normalized distance from the cell centre computed from the sequences shown in b and c over ten independently generated simulations of polarization. Contours were recorded every 500 time steps. e, Numerical experiment with a polarized cell facing an obstacle (grey line). Left panel: The cell is in contact with the obstacle (top), then it is blocked (middle), and finally it backs up from the obstacle and regains its polarized shape in the opposite direction (bottom). Right panel: Kymograph extracted from the sequences along the yellow line. Scale bars are shown in the number of time steps (vertical) and units of rmax (horizontal). f, Instantaneous aspect ratio and velocity during migration for experiments (black) and simulations (red). Experimental points were obtained from 21 cells (total migration time 15 h, frame rate 1/30 s). To simulate cells with different aspect ratios, we modify the threshold distance rmax while keeping all other parameters constant. Simulations lasted for 106 time steps for each rmax; velocity and aspect ratio are reported every 5,000 time steps.

The simulation model also allowed us to test different mechanisms of shape feedback. If the distance limit was replaced by linking the switching probability to either cell area or to the overall ratio of protruding versus retracting points, the system developed neither stable shape nor directional motion (Supplementary Video 7). Any feedback coupling local edge dynamics to a mean cell property that affects all points would prevent local order from propagating. In contrast, the distance limit, although still encoding global information through the position of the cell centre, affects only specific points. A distance threshold coupled to the motion of the cell creates localized zones of frustration at the cell edge, where PR switch is favoured, maintaining phase segregation of cell-edge activity and hence stabilizing the system. Experimentally, only the lateral extremities of migrating cells reach the critical distance and switch from protrusion to retraction, thus stabilizing the width of the cell, whereas the points at the front and back move in concert and maintain their state as long as they do not pass the critical distance threshold. This switch distribution pattern is possible only if the cell has an elongated shape, explaining why such anisotropic shapes, minimizing the edge fluctuations, emerge from the isotropic distance-threshold mechanism. This is consistent with keratocytes being one of the most efficient types of migrating cells. If, for any reason, the aspect ratio of the cell decreases and approaches unity, our model predicts that the distance threshold would induce switches to retraction at the front and fluctuations of the velocity of the leading edge, resulting in a decrease of the net velocity of motion. Indeed, it has been experimentally observed that keratocyte velocity depends on the cell aspect ratio, with cells of smaller aspect ratio having variable and lower velocities14. We have confirmed this effect experimentally and also reproduced it in the simulations (Fig. 5f).

Protrusion/retraction transition is a ubiquitous cellular phenomenon that has been linked with various molecular processes7,8,9,10 and with mechanical force generation11. However, previous studies did not provide a unified model to relate local cell-edge fluctuations to either cell symmetry breaking during polarization or overall cell shape and motion. In this paper, we used high-resolution edge segmentation and cell-shape modification experiments, to show that the dynamics of switches from protrusion to retraction is controlled by the distance from the cell centre. We developed a top–down stochastic model to demonstrate that this isotropic property of the distance dependence leads to spontaneous symmetry breaking. Our model thus relates overall cell behaviour to local protrusion–retraction dynamics. The model simulations have shown that the hypothesized mechanism is indeed sufficient to reproduce the experimentally observed cell behaviour, also under perturbations. Such distance sensing leads to a genuine self-organization, with emerging shape and motion being the macroscopic manifestations of rotational symmetry breaking in the localization of switches. Our findings suggest that cell volume and contractility, but not the microtubule system, may play a role in the mechanism of distance sensing. One possibility related to contractile properties is that traction stresses generated by the acto-myosin network scale with the size of the network and induce detachment from the substrate and collapse at a critical distance. Another possibility is that the distance and volume can influence the force balance at the cell edge through three-dimensional geometry: extension of the cell is expected to flatten its apical surface and therefore increase the components of membrane34 and cortical tension37 in the substrate plane, whereas increasing the cell volume would have the opposite effect on the orientation of forces and thus suppress distance sensitivity. These force-related mechanisms do not exclude a contribution of intra-cellular gradients of structural or regulatory components, which may depend on cell volume and contractility. A unique feature of our model, however, is that gradients do not necessarily have to be oriented in the direction of motion. Instead, we propose a centre-to-periphery radial gradient of the probability of protrusion–retraction switches. This probability may depend on regulatory factors, for example, small GTPases, that could develop a radial gradient by a reaction–diffusion mechanism16,18, marking the cell margins in a way that is similar to how the Min-protein system in bacteria marks the middle of the cell38. After motion onset, front-to-back gradients may develop as well, but their role would be to reinforce motion rather than to initiate it. Another important feature of our model is that cell-edge activity is not imposed deterministically. Instead, the position in the putative radial gradient encodes only disjoint zones of increased switching probability. The behaviour between these zones may rely on cytoskeletal reactions favouring persistence, for example, autocatalytic actin network branching39 and contraction that may be self-sustained owing to its ability to concentrate myosin motors6,25,33. Future studies will uncover the interplay of cell contractility, cell volume, and other factors involved in radial organization to reveal the physical and molecular underpinnings of distance sensing and to determine how this mechanism is involved in different cell shapes and behaviours.

Methods

Experimental procedures.

Black tetra (Gymnocorymbus ternetzi) epidermal keratocytes were cultured and imaged as described32,33,34. To obtain isolated keratocytes, the cell colonies were treated either with 2.5 mM ethylenediaminetetraacetic acid (EDTA) or 0.2% trypsin and 0.02% EDTA in phosphate-buffered saline. This disrupted cell–cell and cell–substrate adhesions and rendered the cells isotropic. The specimens were then replenished with fresh culture medium to allow for cell polarization.

Mapping protrusion–retraction switches.

Cell outlines were extracted from the phase-contrast image sequences as described32. From segmented cell contours, protruding (retracting ) regions were defined as intersections of the parts of the image outside (inside Ω(n − 1)) the cell contour in frame n − 1 with the regions inside Ω(n) (outside ) the contour in frame n.

Consequently, switches from protrusion PR(n) (retraction RP(n)) to retraction (protrusion) were determined as intersections of protruding (retracting) regions between frames n and n − 1 with the retracting (protruding) regions between frames n and n + 1 (Fig. 1c). This defines switching sites in a substrate-fixed coordinate system. For migrating cells, switching sites defined for each three consecutive frames were subsequently aligned with the cell position throughout the image sequence.

Each pixel located in PR(n) and RP(n) defines a switching site in a substrate-fixed coordinate system.

The frame rate was chosen to keep edge displacement between frames significantly smaller than the cell size to resolve small regions of protrusion and retraction, but at the same time large enough to reliably detect them with the segmentation algorithm. This algorithm detected the small features of the cell edge with high accuracy, except small edge irregularities (for example, small and thin retraction fibres) which were regularized by the torsion stiffness of the active contour model. We tested how frame rate influenced switch distance distribution and the number of switches detected per time unit (Supplementary Fig. 1). We observed similar distributions in an interval of frame rates between 2 and 10 s/frame for migrating cells and between 6 and 14 s/frame for polarizing cells. There was no significant difference in number of switches between polarizing and migrating cells, except at a low frame rate, which produces an artificially high switch frequency in moving cells due to a large edge displacement. Thus, switch frequency and switch probability distribution were largely independent of frame interval. Most of the analysis was performed with the frame interval of 2 s for migrating cells (where edge displacement was faster allowing for accurate segmentation with a shorter time interval) and of 10 s for polarizing cells.

For migrating cells, switching sites defined for each three consecutive frames in substrate-fixed coordinates were subsequently aligned with the cell position throughout the image sequence.

Model parameters.

To represent a smooth cell contour, simulations must be run with a sufficient number of points. We have used 4,096 points to describe the cell edge. rmax is the most probable distance from the cell centre for switches from protrusion to retraction; it defines the maximal cell extension (2rmax).

The other parameter values were defined in units of rmax (the velocities of protrusion and retraction were chosen to be small with respect to rmax to ensure a smooth evolution of the cell edge). From typical cell sizes and velocities, the real time equivalent of the simulation time step could be estimated. The simulation protrusion velocity was rmax × 10−4, taking rmax = 20 μm gives a protrusion velocity of 2 nm/time step. Comparing it to a typical experimental cell velocity of 250 nm s−1, we obtain time step equivalent of 8 ms.

Table 1 Parameters of the model.

Cell centre, loop elimination and contour re-sampling.

In both experiments and simulations, we calculate at each time step the coordinates (xc, yc) of the cell centre as the centroid of a non-intersecting polygon:

where (xi, yi) are the coordinates of the cell edge points defining the vertices of the polygon, and is the signed area of the polygon.

Because the previous equations are valid only for non-intersecting polygons, in the simulations we use at each time step a sweep-line algorithm40 to check if loops have formed. In every detected loop, the inner points are deleted and these points are re-inserted in the sparsest region of the contour. These re-inserted points are in protrusion (resp. in retraction) states if the two surrounding neighbours are in protrusion (resp. retraction) states; otherwise their states are chosen randomly.

To ensure a homogeneous distribution of points along the evolving cell edge, the simulated contour is re-sampled (re-meshed) at each time step according to the following procedure: When the maximal distance between two adjacent points is larger than twice the minimal distance, we add a new point in the middle of the largest segment and replace the two points bordering the smallest segment by a single point in their middle. The state of the new points is defined as in the loop elimination procedure.

Retraction velocity.

We choose the following form for the retraction velocity

This is motivated by the results of tracking actin motion using fluorescence speckle microscopy of phalloidin-injected cells11. Actin motion is relevant for the retraction velocity, because, unlike protrusion, which is due to the formation of new actin filaments, retraction is largely due to motion of the existing network32. Consistent with previous findings28,32, actin structures at the back of the lateral cell sides moved nearly parallel to the rear edge in the direction from the sides to the centre, whereas structures at the central part of the rear edge moved nearly normally to the edge in the forward direction (Supplementary Fig. 2 a, b). Overall, the direction of retraction was neither normal to the edge, nor centripetal, and the retraction velocity was higher at the lateral wings than in the centre (Supplementary Fig. 2 c, d).

For simplicity, we choose the centripetal direction in the model. In the case of outline concavities, the direction towards the cell centre points outside the cell. In this case, we set Vr = 0.

Change history

  • 02 March 2016

    In the version of this Article originally published, in the Methods section 'Mapping protrusion–retraction switches', a minus sign was mistakenly omitted from a sentence, which should have read: 'Consequently, switches from protrusion PR(n) (retraction RP(n)) to retraction (protrusion) were determined as intersections of protruding (retracting) regions between frames n and n − 1 with the retracting (protruding) regions between frames n and n + 1 (Fig. 1c).' This has been corrected in all versions of the Article.

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Acknowledgements

We would like to thank F. Nédélec and M. Balland for useful discussions, S. Bohnet for the first observation of cell rotation, and H. Troyon for experimental assistance. This work is supported by Swiss National Science Foundation Grant 31003A-135661. M.E.A. was funded by a PhD fellowship from the Swiss National Competence Center for Biomedical Imaging, to A.B.V. and I.F.S.

Author information

F.R. and A.B.V. designed the study. M.E.A., C.G., A.B. and A.B.V. performed the experiments. F.R. and M.E.A. analysed the data. F.R. developed the numerical model. F.R., I.F.S. and A.B.V. wrote the paper. All authors have discussed the results and the interpretation. F.R. and M.E.A. have contributed equally to the study.

Correspondence to Franck Raynaud.

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Raynaud, F., Ambühl, M., Gabella, C. et al. Minimal model for spontaneous cell polarization and edge activity in oscillating, rotating and migrating cells. Nature Phys 12, 367–373 (2016). https://doi.org/10.1038/nphys3615

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