Phase-field model for collective cell migration

We construct a phase-field model for collective cell migration based on a Ginzburg-Landau free-energy formulation. We model adhesion, surface tension, repulsion, coattraction, and polarization, enabling us to follow the cells' morphologies and the effect of their membranes fluctuations on collective motion. We were able to measure the tissue surface tension as a function of the individual cell cortical tension and adhesion and identify a density threshold for cell-sheet formation.

Figure 1

Fields ${\varphi }_i$ and ${\varphi }_j$ assigned to cells $i$ and $j$, respectively, together with their interaction ranges $C_i$ and $C_j$.

Figure 2

The scheme shows the repulsive forces exerted by the cells' membranes on each other.

Figure 3

The figure shows an aggregate of $N_c=16$ cells in yellow for $\frac{w}{\sigma }=0.5, b=0, r=30, \mathrm{\Gamma }=1, a=16$, and cell radii $R_c=4$.

Figure 4

The figure shows the polarization as a function of density. When $w/\sigma$ is increased, for each of the densities, the corresponding $P$ increases from the lower limit of the bar to its upper limit. The curve is given by $P={(n-n_0)}^{\gamma }$, where $\gamma =0.53\pm 0.05$ and $n_0=0.16\pm 0.02$.

Figure 5

The figure shows the polarization as a function the fraction of chemotactic cells.

Figure 6

The figure shows MSD as function of time for densities $n=0.16, n=0.36$, and $n=0.65$, respectively, in red (top line), blue (middle line), and black (bottom line) on a $log-log$ scale. The low-density results have asymptotic slopes consistent with diffusion.

Figure 7

We vary the tissue surface tension for $n=0.65$ with fixed initial conditions. The figure shows the dependence of the tissue surface tension on the ratio of adhesion to cortical tension. The red line is $T_{\mathrm{st}}/\sigma =2w/\sigma$, which equivalent to the differential adhesion hypothesis in two spatial dimensions.