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. 2023 Jan 3;122(1):130-142.
doi: 10.1016/j.bpj.2022.11.021. Epub 2022 Nov 17.

Coupling cell shape and velocity leads to oscillation and circling in keratocyte galvanotaxis

Ifunanya Nwogbaga  1 , Brian A Camley  2
Affiliations

Coupling cell shape and velocity leads to oscillation and circling in keratocyte galvanotaxis

Ifunanya Nwogbaga et al. Biophys J. .

Abstract

During wound healing, fish keratocyte cells undergo galvanotaxis where they follow a wound-induced electric field. In addition to their stereotypical persistent motion, keratocytes can develop circular motion without a field or oscillate while crawling in the field direction. We developed a coarse-grained phenomenological model that captures these keratocyte behaviors. We fit this model to experimental data on keratocyte response to an electric field being turned on. A critical element of our model is a tendency for cells to turn toward their long axis, arising from a coupling between cell shape and velocity, which gives rise to oscillatory and circular motion. Galvanotaxis is influenced not only by the field-dependent responses, but also cell speed and cell shape relaxation rate. When the cell reacts to an electric field being turned on, our model predicts that stiff, slow cells react slowly but follow the signal reliably. Cells that polarize and align to the field at a faster rate react more quickly and follow the signal more reliably. When cells are exposed to a field that switches direction rapidly, cells follow the average of field directions, while if the field is switched more slowly, cells follow a "staircase" pattern. Our study indicated that a simple phenomenological model coupling cell speed and shape is sufficient to reproduce a broad variety of different keratocyte behaviors, ranging from circling to oscillation to galvanotactic response, by only varying a few parameters.

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Conflict of interest statement

Declaration of interests The authors declare no competing interests.

Figures

Figure 1
Figure 1
Left: cells modeled as deformable ellipses, with a velocity v and an underlying biochemical polarity unit vector pˆ. Their orientation is characterized by θ, the shortest angle between the cell’s long axis and the xˆ axis. The unit vector nˆ points along the cell’s long axis. Cells begin as circular with radius R0. They can deform to ellipses with semimajor and semiminor axes lengths of R0±s/2. s controls eccentricity, with s=0 denoting no deformation from a circle. Right: schematic of the different terms in the equations of motion for velocity (Eq. 2), shape (Eq. 3), and polarity (Eq. 4). To see this figure in color, go online.
Figure 2
Figure 2
Schematic of the interaction between two competing processes: velocity alignment to the long axis and shape expansion perpendicular to the velocity. This competition leads to circular motion of the cell. To see this figure in color, go online.
Figure 3
Figure 3
(A) Linear stability phase diagram in γ−κ plane in the absence of electric field. Example trajectories for points i–iii are shown. (i) A 2.5-h trajectory at the “wild-type” (WT) values, (γ=γwt,κ=κwt). (ii) A 1-h trajectory when cell speed is increased, resulting in persistent circular motion, (γ=1.5γwt,κ=κwt). (iii) A 1.5-h trajectory with increased shape relaxation rate κ shows persistent random walk, (γ=1.5γwt,κ=2κwt). (B) Linear stability phase diagram in τb−χ plane. Transition lines in presence of field (solid line) and absence (dotted line) are shown, and examples iv–vi are shown with field. (iv) A 1.5-h WT trajectory following a field in +xˆ-direction. In this region of the plot, persistent motion is unstable, but the cell oscillates side to side while traveling in the field direction. (v) A 1.5-h trajectory following an field pointing in +xˆ-direction. In this region of the plot, persistent motion is stabilized in the presence of a field and there are no oscillations, (τb=τb,wt,χ=8χwt). (vi) A 1.5-h trajectory following an field pointing in +xˆ-direction, (τb=τb,wt,χ=50χwt). To see this figure in color, go online.
Figure 4
Figure 4
Cell response to a field being turned on. Here, ϕ is the angle of the cell’s velocity with respect to the field direction, and the field is turned on at 15 min. In blue are experimental data of the average response of 140 cells, extracted from (20). In red are our simulation data which we fit to experiment to calibrate our parameters, averaging over 1000 cells. Error bars are ±1 standard deviation of the average, computed by bootstrapping (Appendix B). To see this figure in color, go online.
Figure 5
Figure 5
Response to a field being turned on at 15 min, as in Fig. 4, with parameters varied from fit wild-type parameters. (A) Speed parameter γ. (B) Shape relaxation rate κ. (C) Velocity to polarity alignment rate χ. (D) Polarity to velocity alignment time τ. (E) Field alignment time τb. Each curve is the average response of 500 cells. Error bars are not shown to preserve clarity, but curve-to-curve variability is limited (see dotted red wild-type curve from panel to panel). To see this figure in color, go online.
Figure 6
Figure 6
(A) Schematic of simulation. We vary the electric field direction from +xˆ to +yˆ every tET and track the cell trajectory. Will the cell follow the field closely (dotted red arrows) or average the field directions (solid red arrow)? (B) Probability density plots. These are histogram heatmaps indicating the probability density of cell velocity orientation. Simulations are done for 100 simulation hours. The red lines correlate with the red arrows in (A). Center solid red line corresponds to a cell traveling around the average field direction. Dotted red lines corresponds to a cell following the field direction precisely. (C) A 10-h wild-type trajectory with slow switching time (2.5 h). (D) A 3-h wild-type trajectory with fast switching time (5 min). (E) A 4-h trajectory with high γ value where cell rotates faster than our averaging sample time. To see this figure in color, go online.
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