Polarization wave at the onset of collective cell migration

Collective cell migration underlies morphogenesis, tissue regeneration, and cancer progression. How the biomechanical coupling between epithelial cells triggers and coordinates the collective migration is an open question. Here, we develop a one-dimensional model for an epithelial monolayer which predicts that after the onset of migration at an open boundary, cells in the bulk of the epithelium are gradually recruited into outward-directed motility, exhibiting traveling-wave-like behavior. We find an exact formula for the speed of this motility wave proportional to the square root of the cells' contractility, which accounts for cortex tension and adhesion between adjacent cells.

Figure 1

One-dimensional particle model. (a) Top: initially, cells are uniformly distributed and spaced at equilibrium distance at positions $x_i\left(0\right)$ and only the leading cell $N$ is polarized. Bottom: elastic forces applied on a cell at time $t$. For example, cell $N-1$ feels force $F_{N-1,+}$ toward the cell-free surface ($+$, green arrow); and force $F_{N-1,-}$ in the backward ($-$, green arrow) direction. (b) Steady-state polarities of a single cell visualized as intersections of the active velocity $a^2/(a^2+{\alpha }^2)$ (solid curves) and polarity $a$ (gray dotted line). For half-saturation polarity $\alpha =0.3$ (black) three steady-state polarities are $a_1^{*}=0$ (stable, solid arrow), $a_2^{*}=0.1$ (unstable, dashed arrow), and $a_3^{*}=0.9$ (stable, solid arrow). Thus, starting with an initial polarity greater than $a_2^{*}$, the cell will ultimately move with constant velocity and polarity. If $\alpha$ is too large, e.g., 0.6 (red dashed), there is no nonzero steady-state polarity.

Figure 2

Migratory dynamics of a group of $N=20$ cells obtained from the particle model with half-saturation polarity $\alpha =0.3$ and Hill coefficient $n=2$. The cells' displacement (top row) and polarity (bottom row) are shown versus time. Initially, only the leading cell (solid arrow, in contrast to dashed arrow showing last cell) is polarized, $a_{20}\left(0\right)=0.8$, $a_1\left(0\right)=...=a_{19}\left(0\right)=0$. The cells' contractility is either weak [(a), (b) $\kappa =0.1]$ or strong [(c), (d) $\kappa =1]$ in which case polarization propagates significantly faster [compare (a) and (c)].

Figure 3

Position of $N=50$ cells as a function of time derived from the particle model, when white noise ${\xi }_i$ with magnitude $\zeta$ is added to Eq. (2): $d{a}_i/dt=-\beta a_i+\gamma d{x}_i/dt+\zeta {\xi }_i$, for $N$ cells with indices $i=1,...,N$. Simulation parameters are contractility $\kappa =1$, half-saturation polarity $\alpha =0.45$, Hill coefficient $n=10$, and $\zeta =0.45$ (a) and 0 (b), where the initial polarity of 10 cells close to the free edge is given by 0.8. The solid and dashed arrows point to the position of cells 50 (leading) and 1 (last). In the presence of random polarization direction, shown in (a), cells occasionally move backward. In both (a) and (b) the polarization wave propagates through the layer of cells with similar speed (dashed lines).

Figure 4

Traveling wave analysis. (a), (b) Qualitative shapes of polarity $A$ and density $R$ wave profiles versus wave variable $z$. Rest state: $(R_{-\infty },A_{-\infty })$; motile state: $(R_{\infty },A_{\infty })$. Arrow shows the direction of the polarization wave speed $c$. (c) Phase diagram of the system (6) for a set of parameters ($\alpha =0.2$, $n=10$, $\kappa =1$, ${\rho }_0=1$) and the approximately corresponding wave speed $c=2$ obtained from Eq. (9). The system exhibits a heteroclinic orbit (green solid curves) corresponding to the traveling wave profiles. Red closed circles: saddle points; red open circle: the focus; purple square: $(R_{\infty },\alpha )$. Additional trajectories (blue dashed) are depicted.

Figure 5

Inverse function $g^{-1}\left(Q\right)$ for $g\left(y\right)=1/y+log\left(1/y-1\right)$.

Figure 6

Polarization wave speed $c$ versus cellular contractility $\kappa$. Speed $c$ is derived from the particle model for $N=2000$ cells in the contractility range $0.1\le \kappa \le 100$ at two different values for half-saturation polarity $\alpha =0.05$ and 0.3, each repeated for three different Hill coefficients $n=2,6$, and 10 (circle, square, and triangle symbols, respectively). Speed values correspond to mean $\pm$ standard deviation (SD). The dashed lines are the fits to the mean velocities. For $\alpha =0.05$: $c=4.71\sqrt{\kappa }$ (dashed blue), $c=4.25\sqrt{\kappa }$ (dashed green), and $c=4.23\sqrt{\kappa }$ (dashed magenta). For $\alpha =0.3$: $c=1.58\sqrt{\kappa }$ (dashed blue), $c=1.48\sqrt{\kappa }$ (dashed green), and $c=1.49\sqrt{\kappa }$ (dashed magenta).

Figure 7

Comparison of the particle and continuum models. (a) Polarization wave speed $c$ versus cells' contractility $0.1\le \kappa \le 100$. Velocities (mean $\pm$ SD) are calculated by the particle model for $N=2000$ cells at half-saturation polarities $\alpha =0.05$ and 0.3, each repeated for Hill coefficients $n=2,6$, and 10 (circle, square, and triangle symbols, respectively). The output of the continuum model is derived from Eq. (9) (black curves). Both models show that $c$ is proportional to $\sqrt{\kappa }$, independently of $n$; as shown in Fig. 6. (b) Polarization wave speed $c$ versus $\alpha$. Particle and continuum models are run for two different $\kappa$, shown with symbols (mean $\pm$ SD) and solid curves, respectively. At $\alpha \ge 0.5$, the motility does not propagate into the interior of the cell layer in the particle model simulations; see Fig. 8. (c), (d) Quantitative shapes of polarity and density wave profiles in the instantaneous polarization limit $n\to \infty$ [for formulas, see Eqs. (11) and (12)].

Figure 8

Position of cells versus time derived from the particle model for $N=50$ cells with contractility $\kappa =0.1$, half-saturation polarity $\alpha =0.6$, and Hill coefficient $n=10$, where the initial polarity of 10 cells from the leading edge was assigned as 0.8. The solid arrow points to the position of the leading cell 50. The motility wave does not propagate into the monolayer, as $\alpha >0.5$. Instead, the trajectories exhibit a depolarization wave.