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Conceived and designed the experiments: MKD JTF CP WL. Performed the experiments: MKD CM RK. Analyzed the data: MKD TH. Contributed reagents/materials/analysis tools: MKD CM RK TH. Wrote the paper: MKD JTF CP WL.

The authors have declared that no competing interests exist.

We observe and quantify wave-like characteristics of amoeboid migration. Using the amoeba

Migration of cells on surfaces and through tissues is an important part of life, from the amazingly coordinated migration of cells during development to the uncontrollable migration of metastatic cancer cells. Here we investigate the physics of cell migration with the goal of gaining new insights into how cells move and how they respond to obstacles. Through detailed quantitative analysis of time-dependent cell shapes, we demonstrate the existence of wave-like dynamic shape changes during the migration of

During chemotaxis, a chemical gradient directs cell migration. Chemotaxis occurs in phenomena as diverse as wound healing

The social amoeba ^{−} mutation do not produce cAMP and, in its absence, are mostly round and immotile ^{−} cells polarize and begin to migrate. If stimulated with a uniform concentration of cAMP, ^{−} cells exhibit chemokinesis (chemically induced random migration). In response to a cAMP gradient, these cells chemotax effectively

The regulatory processes behind

Quantitative studies of cell shape and motion that follow protrusions are now emerging, and indicate that protrusion dynamics are richer than anticipated

How are such complex protrusion dynamics relevant to chemotaxis and directed cell migration? For the models of chemotaxis reviewed above, and for fast migration, a local protrusion phenotype is usually assumed in which protrusions are a (noisy) output in response to a chemotactic signal. This view appears to be supported by experiments in which a strong chemotactic signal is placed in close proximity to individual cells, causing a pseudopod to form in the direction of the signal

However, recent experiments and modeling efforts point to a more complex protrusive machinery. Indeed, given signal strengths at physiological levels (and even in the absence of chemotactic gradients), protrusions tend to form on alternating sides of the leading edge of the cell, resulting in a zig-zag migration pattern

The conclusion that alternating pseudopods are prominent in fast migrating cells relies on thresholds to separate individual pseudopods in a consistent way. However, it is unclear whether the underlying biology of protrusions justifies such thresholding. Instead, zig-zagging and alternating of pseudopods may be the result of wave-like behavior of the protrusive machinery. In developing tissues, actin waves can be seen to propagate across groups of cells

To study the character of protrusions during fast cell migration and chemotaxis, we present new methods for the quantification of the dynamic shape of migrating cells. Using these methods, we demonstrate that protrusions in

To study the changing shape of migrating

(A) The overlapped light gray boundaries show the shape every 4 seconds, while the alternating dark gray and colored boundaries show the shape every 2 minutes. Colored boundaries represent curvature. (Scale bar, 20 µm.) (B) The boundary curvature overlaid on the original fluorescence images (^{th} boundary point (

To study the motion and evolution of these propagating curvature peaks, we labeled the boundary in order to follow individual boundary points as they move. We defined a 1∶1 mapping between the boundary points in successive frames, choosing the mapping that minimized the total boundary point displacement. To create such a 1∶1 mapping, we described the boundary with the same number of points in every frame (400), even though the length of the perimeter of the cell varies from frame to frame.

With the boundary points labeled in this way, we are able to visualize and analyze how local properties of the boundary, such as curvature, vary as a function of both space and time.

We have previously found that alternative tracking mappings yield similar results, but generally do not track the entire local boundary

Migration requires not only pseudopods but also adhesion to a surface. Thus, surface contact can indicate whether bumps on the side of the cell reflect pseudopods that have successfully adhered (and thus can contribute to motion) or unsuccessful pseudopods that failed to adhere. We imaged fluorescently-labeled, developed, WT cells, while simultaneously using internal reflection microscopy (IRM) to image the region of cell-surface contact. We extracted both the boundary of the entire cell and the boundary of the surface contact region(s) from the images.

(A) An IRM image sequence overlaid with the boundary of the surface contact region, shown in purple, and the boundary of the entire cell, shown colored by curvature (^{−} (non cAMP releasing) cell that is electrostatically repulsed from the substrate, and so is not adhered to the surface. The curvature peak that travels from the cell front to the cell back is moving with respect to the substrate. The centroid positions were aligned to account for drift. (4 sec. apart.) (C) The boundary curvature kymograph of this non-adherent cell (

At the sides of migrating cells, regions of high curvature are in contact with the surface and remain stationary. However, in analyzing the shapes of migrating cells it is difficult to determine if regions of high curvature are stationary at the front of cells. To analyze the behavior of high curvature regions in the absence of surface contact, we analyzed the shape of ^{−}

We also analyzed the shape of ^{−}^{−}^{−}^{−}

Using surfaces with-three dimensional topography, we also analyzed the shapes of cells that adhere only at their back, even in standard salt concentrations. To guide a cell to move away from the surface, we placed a point source of cAMP above and over the edge of a microfabricated ramp that terminated with a 75 µm tall cliff, such that the surface closest to the point source was the cliff edge. A schematic of our set-up is shown in ^{−}

(A) A schematic of the 3-D surface on which cells are guided over a cliff edge. The surface closest to a cAMP-releasing needle is the cliff edge. (B) The overlaid boundaries show a cell extended over the edge of a cliff and wiggling rapidly (boundaries are 1.6 seconds apart). (C) An image sequence of a propagating curvature wave. (

To explore the onset of curvature peaks at the sides of cells, we analyzed the dynamic shape of polarizing ^{−}^{−}^{−}^{−}

(A) During polarization, non-circularity, the normalized ratio of perimeter to the perimeter of a circle with the same area, increases in an oscillatory fashion. (B) The speed of the cell centroid. (C) Boundary curvature, which prior to polarization is mostly static, begins exhibiting organized curvature waves (

When a cell polarizes, its shape elongates. To quantify the degree of polarization, we define the non-circularity as the ratio of the cell perimeter to its area, normalized so that the non-circularity of a circle is 1. The non-circularity, centroid velocity, and boundary curvature of a polarizing ^{−}

Curvature peaks are suggestive of protrusions, because a localized protrusion is necessarily associated with a localized region of high curvature. To compare boundary curvature to motion, we developed a measure of local boundary motion. We calculated the motion of each boundary point by measuring the distance to the closest boundary point in a later frame and then smoothing over the list of mapped to boundary points. Protrusive motion was defined to be positive, while retractive motion was defined to be negative. ^{−}

(A) The boundary curvature kymograph of the cell shown in

We can compare boundary curvature to boundary motion by comparing the curvature and local motion kymographs. Curvature peaks (shown as dashed black lines) are overlaid on

Protrusive motion has often been discretized into pseudopod extension and retraction events. Here we analyze protrusive motion both with and without discretization and show that discretization may hide the wave-like nature of the protrusive process. We first analyze boundary motion under the assumption that protrusions and retractions are discrete events. The times and locations of individual protrusions and retractions along the boundary were defined as the peaks and valleys of the local motion measure, respectively.

(A) In this plot of the evolution of local boundary motion, protrusive motion appears red, while retractive motion appears blue (

While analyzing protrusions as discrete events yields results consistent with prior work

To analyze protrusive and retractive motion as continuous boundary movement, we define the location of greatest protrusion and retraction activity for each frame as the location of the weighted average of the protrusive or retractive motion.

We measured the mean squared displacement (MSD) of the average protrusion location along the boundary (

Together, these findings indicate that for adherent cells, protrusive activity is continuous and constantly shifts along the leading edge of the cell in a wave-like manner similar to the dynamics of the wave-like high curvature regions observed both in suspended cells and in cells extended over cliff edges.

Using quantitative analysis of cell shape dynamics, we have demonstrated the existence of wave-like characteristics in the local shape of

In order to track the local boundary from frame to frame, we maintained a constant number of boundary points per frame. This approach allows for a robust, 1∶1 mapping between points in subsequent frames (see SI). Other recent approaches in which the lengths of boundary segments are maintained

Kymographs of boundary curvature reveal regions of high curvature that start at the leading edge of the cell and travel backward along alternating sides of the cell at an approximately constant speed of ∼35 µm/min. Such curvature waves are prominent in cells that are not adherent to the surface (^{−}

Using a measure of local motion, we determined that the locations of protrusive activity are associated with curvature peaks at the fronts of cells (

Local protrusive motion at the cell front transitions from ballistic to caged at about 20 seconds. This 20 second characteristic time is consistent with the finding of Meili

Indeed, we find the same results both for self-aggregating WT cells and for chemokinesing ^{−}^{−}

The similarities in protrusive activity and shape dynamics of chemotaxing, chemokinesing, and suspended cells is hard to reconcile with models in which pseudopods are triggered directly by directional chemical signals. Our results are consistent with recent models that treat the cellular migratory apparatus as an excitable system

Wave-like protrusions provide a simple and robust mechanism for directed migration. Chemotactic signals are not needed to trigger migration, since protrusive activity is self-sustaining. This view is consistent with observations that cells continue to migrate for hours after a temporary migratory signal

Wave-like protrusions may allow cells to migrate in a viscous environment. Recent reports indicate that both

Wave-like protrusions may allow cells to search for surfaces in 3D environments. Away from a surface, a protrusion that advects backwards along the edge of the cell seems to lead to a wiggling motion (

^{−}^{−}^{−}^{−}^{−}^{−3} diluted phosphate buffer (see

Image sequences that show the migration of individual cells were pre-processed using ImageJ to enhance contrast and to remove other cells. The shape of the cell was then extracted using a snake algorithm. We adapted sample code

With a constant number of boundary points per frame,

At each boundary point, we calculate the boundary curvature by fitting a circle to that boundary point and the two points that are 10 boundary points away from it. The magnitude of the boundary curvature is then defined as the reciprocal of the radius of that circle. If the midpoint of the two points 10 boundary points away is inside the cell, the curvature is defined as positive, otherwise it is defined as negative. For visualization, the curvature is smoothed over 3 boundary points and 3 frames, and the color scale is cut off at a maximum curvature magnitude.

To calculate local motion, we first mapped each boundary point to the closest boundary point in the frame 12 seconds later. This time interval was chosen so that boundary motion dominates over image noise. Next, we used an averaging window to smooth the mapping twice. (The first smoothing had a window size of 19 boundary points, while the second had a size of 15 boundary points.)

The (A) boundary curvature and (B) local motion kymographs of a self-aggregating, wild-type cell (

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The polarization of a fluorescently dyed ^{−}

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Distributions of the magnitude of mean protrusion displacements along the boundary. The durations of the displacements vary from 4 to 160 seconds and are 4 seconds apart.

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Curvature waves are visible in the cellular footprint (

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The local motion mapping. (A) First, each boundary point in a frame is mapped to the closest boundary point in the frame 12 seconds later. A representative frame's boundary is shown as a solid line, the boundary in the frame 12 seconds later by a dashed line, and the mapping between boundary points by blue lines. (B) Next, we smooth over target boundary points, pulling mapping vectors into protrusions, and more evenly distributing vectors in retractions. The magnitude of these blue mapping vectors is then defined as the magnitude of our local motion measure.

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Supplemental materials and methods.

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A migrating, wild-type

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A migrating, wild-type cell. Colored markers indicate the position of every 50^{th} boundary point. (There are 400 total boundary points.) The red dot represents boundary point 0, the orange dot boundary point 50, etc. (see

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An additional, migrating, wild-type cell. The boundary is colored by curvature (see

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Curvature waves in the boundary, shown colored by curvature, and footprint, shown in purple, of a migrating cell (see

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^{−}

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Curvature waves in a cell extended over the edge of a cliff (see

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The polarization of an ^{−}

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The polarization of an additional ^{−}

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The local motion measure. Each frame's boundary is shown as a solid line, the boundary in the frame 12 seconds later as a dashed line, and every 6^{th} tracking mapping vector as a line colored by the value of the local motion measure (see

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Tracking the local boundary of a migrating, wild-type cell. Each frame's boundary is shown as a solid line, the boundary in the next frame as a dashed line, and every 20^{th} tracking mapping vector as a red line (see

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We thank Linjie Li for assistance with cliff fabrication, as well as Simon Freedman, Abby Goldman and John Watts for experimental and analytical contributions.