Dynamic Self-Organization of Idealized Migrating Cells by Contact Communication

This Letter investigates what forms of cellular dynamic self-organization are caused through intercellular contact communication based on a theoretical model in which migrating cells perform contact following and contact inhibition and attraction of locomotion. Tuning those strengths causes varieties of dynamic patterns. This further includes a novel form of collective migration, snakelike dynamic assembly. Scrutinizing this pattern reveals that cells in this state can accurately respond to an external directional cue but have no spontaneous global polar order.

Figure 1

Schematics of the types of intercellular communication in the model. (a) Contact following. The strength is denoted by ${\alpha }_{\mathrm{CF}}$. (b) Contact inhibition and attraction of locomotion (left and right, respectively). The strength is denoted by ${\alpha }_{\mathrm{CIL}}$. In each, the top and bottom subfigures describe the cases for cell migration and idealized situations, respectively. Green arrows indicate intrinsic polarities of cells, ${\mathit{q}}_j$ (for the $j$th cell).

Figure 2

Phase diagram for $\beta =1.0$, $D=0.1$, $A=0.0$, $\rho =1.0$, and $N=10000$. (a) Fraction of space covered by high-density regions [51], against ${\alpha }_{\mathrm{CF}}$ and ${\alpha }_{\mathrm{CIL}}$. (See Fig. S2 for the heat maps of other quantities [51].) Curves indicate the phase boundaries [51]. Broken curves specifically indicate the apparent phase boundaries expected to disappear for the large $N$ and long-time limits. (b)–(d),(f)–(j) Snapshots. Each small colored arrow represents the location and polarity direction of each cell, and the color indicates the polarity direction corresponding to the color wheel for better visibility. Large black arrows indicate the apparent motile directions of clusters. The observed patterns include (b) ring and spiral, (c) ring-formation failure, (d) SDA, (f) polar traveling band, (g) homogeneous polar flock, (h) static aggregate, (i) motile aggregate, and (j) rotating aggregate. The values of ${\alpha }_{\mathrm{CF}}$ and ${\alpha }_{\mathrm{CIL}}$ for each snapshot are indicated in (a) by the marks. Figure S3 displays more snapshots [51]. (e) Schematic showing directions of each cell polarity and stripe-shaped cluster’s motion (green and large black arrows, respectively) in (b)–(d). Orange line describes the stripe’s center line.

Figure 3

SDA, for $\beta =1.0$, $D=0.1$, and $\rho =1.0$. $N=80000$ except for (d). (a) Snapshots for ${\alpha }_{\mathrm{CF}}=0.5$ and ${\alpha }_{\mathrm{CIL}}=0.0$. Small colored arrows are defined in the Fig. 2 legend. Black arrows indicate the velocity vectors of 80 representative cells. See also Movie S7 [51]. (b) $\text{tMSD}(\mathrm{\Delta }t)$ for ${\alpha }_{\mathrm{CF}}=0.5$ and various ${\alpha }_{\mathrm{CIL}}$. Main: $\widetilde{\text{tMSD}}\equiv \text{tMSD}(\mathrm{\Delta }t)/\text{tMSD}(1)$ divided by $\mathrm{\Delta }t$. Inset: $\text{tMSD}$ itself. For (b), simulations were performed up to $t=9600$. (c) Global PO $R$ against ${\alpha }_{\mathrm{CIL}}$ and ${\alpha }_{\mathrm{CF}}$. Figure S4 displays the corresponding snapshots [51]. (d) $R$ (marks) and median cell speed $V$ (curves) for ${\alpha }_{\mathrm{CIL}}=0.0$ for various ${\alpha }_{\mathrm{CF}}$ and $N$. (e) ROI linear size $l$ dependence of local PO $R_l$ (i.e., PO parameter calculated in rectangular ROIs with size $l\times l$). Inset: $\log (l)$ dependence of $R_l$ for the SDA state. Thick line segments are eye guides for power-law slopes. Error bars indicate standard deviations over time for (d) and time and ROIs for (e).

Figure 4

Directional migration under external directional bias $A=0.01$ for $\beta =1.0$, $D=0.1$, and $\rho =1.0$ with $N=10000$ for (a)–(d). (a) CI for various ${\alpha }_{\mathrm{CF}}$ and ${\alpha }_{\mathrm{CIL}}$. Curves indicate the phase boundaries determined for $A=0.0$ [Fig. 2]. (b),(c) Snapshots. The parameter values are indicated in (a). Small colored and black arrows are defined in the Fig. 2 and 3 legends, respectively. Huge red arrows indicate the bias direction. Figure S5 displays more snapshots [51]. (d) CI calculated in each frame for various ${\alpha }_{\mathrm{CF}}$ and ${\alpha }_{\mathrm{CIL}}$. (e) CI for ${\alpha }_{\mathrm{CIL}}=0.0$ for various ${\alpha }_{\mathrm{CF}}$ and $N$. Marks and error bars indicate means and standard deviations over time, respectively.