Cell-alignment patterns in the collective migration of cells with polarized adhesion

Dictyostelium discoideum (Dd) utilizes inhomogeneities in the distribution of cell-cell adhesion molecules on cell membranes for collective cell migration. A simple example of an inhomogeneity is a front-side (leading-edge) polarization in the distribution at the early streaming stage. Experiments have shown that the polarized cell-cell adhesion induces side-by-side contact between cells [Beug et al., Nature (London) 274, 445 (1978)]. This result is counterintuitive, as one would expect cells to align front to front in contact with each other on the basis of front-side polarization. In this work, we theoretically examine whether front-side polarization induces side-by-side contact in collective cell migration. We construct a model for expressing cells with this polarization based on the two-dimensional cellular Potts model. By a numerical simulation with this model, we find cell-cell alignment wherein cells form lateral arrays with side-by-side contacts as observed in the experiments.

Figure 1

(a) Schematic image of a Dictyostelium discoideum cell in the early streaming stage. Darkened areas indicate regions where the concentration of the adhesion molecule gp24/DdCAD-1, $\rho \left(\mathit{r}\right)$, is high. Arrows represent the direction of cell polarity leading its motion $\mathbf{\mu }$ and the sawtooth shape at the front of the cell represents pseudopods. (b) Side-by-side contact between two cells. (c) Front-to-front contact between two cells.

Figure 2

(a) Schematic image of the correspondence between the configuration of cells and a Potts state. The left panel represents the configuration of two cells ($N_{\mathrm{Cell}}=2$) in an empty space. The right panel shows the corresponding Potts state. $L$ is the linear size of the system, and $\mathit{r}, {\mathit{r}}^{\prime }$, and ${\mathit{r}}^{\prime \prime }$ are neighboring sites occupied by cell 1, the empty space, and cell 2, respectively. (b) ${\mathit{e}}_m\left(\mathit{r}\right)$, representing the unit vector from the central position ${\mathit{R}}_m$ of cell $m$ to a peripheral site $\mathit{r}$. The dashed arrow represents the direction of ${\mathit{e}}_m\left(\mathit{r}\right)$. (c) A favorable pseudopod formation at $\mathit{r}$ given the configuration in (b) and the propulsion direction of cell $m$, indicated by the unit vector ${\mathbf{\mu }}_m$ (solid arrow). (d) An unfavorable pseudopod formation at $\mathit{r}$ given the configuration in (b) and the propulsion direction of cell $m$, indicated by the unit vector ${\mathbf{\mu }}_m$ (solid arrow).

Figure 3

Schematic images of adhesion molecule concentration $\rho \left(\mathit{r}\right)$ at cell periphery. Gray shades represent the adhesion molecule concentrations on the membrane: black indicates a high concentration, and white indicates a low concentration. (a) Adhesion contacts between the $m\mathrm{th}$ cell and the $m^{\prime }\mathrm{th}$ cell. $\mathit{r}$ represents a contacting site for the $m\mathrm{th}$ cell, and ${\mathit{r}}^{\prime }$ represents a contacting site for the $m$'th cell. (b) Excess energy of cell-cell adhesion, $J({\rho }_m\left(\mathit{r}\right),{\rho }_{m^{\prime }}\left({\mathit{r}}^{\prime }\right))$, as a function of the product of adhesion molecule concentrations, ${\rho }_m\left(\mathit{r}\right){\rho }_{m^{\prime }}\left(\mathit{r}\right)$. (c) Unpolarized component of adhesion molecule concentration. (d) Polarized component of adhesion molecule concentration. Vector ${\mathit{p}}_m$ represents the polarization direction of the adhesion molecule concentration. Vector ${\mathit{e}}_m\left(\mathit{r}\right)$ is a unit vector indicating a position $\mathit{r}$ on the periphery of the $m\mathrm{th}$ cell. ${\mathit{R}}_m$ and $\mathit{r}$ represent the center position of the $m\mathrm{th}$ cell and a position on the cell periphery, respectively.

Figure 4

Snapshots of steady states for (a) $\gamma =0.25$ and $\beta =0.1$, (b) $\gamma =0$ and $\beta =0.1$, (c) $\gamma =-0.25$ and $\beta =0.1$, (d) $\gamma =0.25$ and $\beta =1.0$, (e) $\gamma =0$ and $\beta =1.0$, and (f) $\gamma =-0.25$ and $\beta =1.0$. Other parameters are given in the main text. Black regions represent empty spaces, and other colored regions represent cells, with different colors representing different cells. The white arrow at each cell's center indicates the direction of ${\mathbf{\mu }}_m$.

Figure 5

Time-averaged $n_{\mathrm{CC}}$ as a function of $J_{\mathrm{A}}$ for (a) $\gamma =0.25$, (b) $\gamma =0$, and (c) $\gamma =-0.25$. The data are shown by a solid line for $\beta =0.1$, a dashed line for $\beta =0.2$, a dotted line for $\beta =0.3$, a dashed-dotted line for $\beta =0.5$, and a dashed-double-dotted line for $\beta =1.0$. The other parameters are given in the main text. In (a), the data are not shown for $J_{\mathrm{A}}>3.0$ because the cells are unstable.

Figure 6

The time-averaged order parameter $M$ as a function of $J_{\mathrm{A}}$ for (a) $\gamma =0.25$, (b) $\gamma =0$, and (c) $\gamma =-0.25$. The data are shown by a solid line for $\beta =0.1$, a dashed line for $\beta =0.2$, a dotted line for $\beta =0.3$, a dashed-dotted line for $\beta =0.5$, and a dashed-double-dotted line for $\beta =1.0$. The other parameters are given in the main text. In (a), the data are not shown for $J_{\mathrm{A}}>3.0$ because the cells are unstable.

Figure 7

The correlation function $g\left(r\right)/g\left(0\right)$ for ${\mathbf{\mu }}_m$, for (a) $\gamma =0.25$ and (b) $\gamma =-0.25$. In both panels, the solid, dashed, dotted, and dashed-dotted lines represent $g\left(r\right)/g\left(0\right)$ for $\beta =1.0$ and $J_{\mathrm{A}}=0.0$, for $\beta =1.0$ and $J_{\mathrm{A}}=4.0$, for $\beta =0.2$ and $J_{\mathrm{A}}=0.0$, and for $\beta =0.2$ and $J_{\mathrm{A}}=4.0$, respectively. The other parameters are given in the main text. $d_{\mathrm{Cell}}$ represents the average cell diameter $\sqrt{V/\pi }\simeq 9.02a$.

Figure 8

The number of lateral contacts $n_{\mathrm{L}}$ (solid line and “$+$” markers) and the number of nonlateral contacts $n_{\mathrm{NL}}$ (dashed line and “$\times$” markers) as a function of $J_{\mathrm{A}}$. The data are calculated for $\beta =1.0$ and $\gamma =-0.25$. The other parameters are given in the main text. The inset shows the relative proportions of $n_{\mathrm{L}}$ and $n_{\mathrm{NL}}$.

Figure 9

Snapshots of steady states for (a) $J_{\mathrm{A}}=0$, (b) $J_{\mathrm{A}}=0.5$, (c) $J_{\mathrm{A}}=1.0$, and (d) $J_{\mathrm{A}}=4.0$. Other parameters are given in the main text. The black region represents $m=0$, and other colored regions represent cells, with different colors representing different cells. The white arrow at each cell's center indicates the direction of ${\mathbf{\mu }}_m$.

Figure 10

The number of lateral contacts $n_{\mathrm{L}}$ (solid line and “$+$” markers) and the number of nonlateral contacts $n_{\mathrm{NL}}$ (dashed line and “$\times$” markers) as a function of $E$. The data are calculated for $\beta =0.5$ and $\gamma =-0.25$. The other parameters are given in the main text. The inset shows the relative proportions of $n_{\mathrm{L}}$ and $n_{\mathrm{NL}}$.

Figure 11

Schematic images to explain side-by-side contact formation. The arrows in the cells represent cell polarity. Darkened areas indicate regions with high concentrations of cell-cell adhesion molecules. The sawtooth shape at the front of the cell indicates pseudopods. (a) Cells with front-to-front contact. (b) Cells whose front-to-front contact is destabilized by the motility of the cells. (c) Cells whose cell polarity and direction of movement are aligned in the same direction. (d) Cells that make side-by-side contact at their fronts.