Figure 4. Phases of collective motion.

(${\displaystyle 4}$-cell systems; confinement radius $r_0=30.6$; area stiffness ${\kappa }_A=0.18$; average polarization field ${ϵ}_0=225$; signaling radius $R=5$; cytoskeletal update rate $\mu =0.1$; cell-cell adhesion $B=0$; cell-cell dissipation $\mathrm{\Delta }B=12$; cell-substrate dissipation $D=0$; cell-substrate adhesion penalty $\phi =0$ ($r<r_0$), ${\displaystyle \phi \to -\mathrm{\infty }}$ ($r>r_0$); 100 independent simulations for each set of parameters). (A) Characteristic observables of collective cell rotation at different values of the cell perimeter stiffness parameter ${\kappa }_P$: mean (${\displaystyle ⟨|\omega |⟩}$) and standard deviation (${\sigma }_{\omega }$) of the magnitude of the cell cluster's angular velocity, and the standard deviation of the cell perimeter (${\sigma }_P$). The statistics of collective cell motion depends only on the ratio of maximum cell polarity, $\mathrm{\Delta }ϵ$, to cell contractility, ${\kappa }_P$ (specific polarizability). (B) Representative angular trajectories and (C) cell shapes (color code represents cell polarization; cf. Figure 1C) for the different parameter regimes as described in the main text. The cellular dynamics in the different parameter regimes are shown in Figure 4—video 1, Figure 4—video 2 and Figure 4—video 3.

Figure 4—figure supplement 1. Collective motion for varying number of cells at low polarizability.

($N$-cell systems; confinement radius $r_0=\sqrt{234N}$; stiffness parameters ${\kappa }_P=0.060$, ${\kappa }_A=0.18$; average polarization field ${ϵ}_0=225$; maximum cell polarity $\mathrm{\Delta }ϵ=28$; signaling radius $R=5$; cytoskeletal update rate $\mu =0.1$; cell-cell adhesion $B=0$; cell-cell dissipation $\mathrm{\Delta }B=12$; cell-substrate dissipation $D=0$; cell-substrate adhesion penalty $\phi =0$ ($r<r_0$), $\phi \to -\mathrm{\infty }$ ($r>r_0$); ${\displaystyle 100}$ independent simulations for each set of parameters). For this choice of parameters ${\displaystyle 4}$-cell populations rotate in the ${ℛ}_1$-phase. We observe a similar behavior here: the cell clusters rotate slowly and reorient frequently. (A) Characteristic observables of collective cell rotation at different values of the cell perimeter stiffness parameter ${\kappa }_P$: mean (${\displaystyle ⟨|\omega |⟩}$) and standard deviation (${\sigma }_{\omega }$) of the angular velocity magnitude of cell motion, and the standard deviation of the cell perimeter (${\sigma }_P$). The black line corresponds to a power-law fit of the form $⟨|\omega |⟩\propto N^{-k/2}\propto r_0^{-k}$ with the fitted exponent $k\approx 8/3$. (B) Representative angular trajectories and (C) cell shapes (color code represents cell polarization; cf. Figure 1) for the different parameter regimes as described in the main text. .

Figure 4—figure supplement 2. Collective motion for varying number of cells at intermediate polarizability.

($N$-cell systems; confinement radius $r_0=\sqrt{234N}$; stiffness parameters ${\kappa }_P=0.060$, ${\kappa }_A=0.18$; average polarization field ${ϵ}_0=225$; maximum cell polarity $\mathrm{\Delta }ϵ=50$; signaling radius $R=5$; cytoskeletal update rate $\mu =0.1$; cell-cell adhesion $B=0$; cell-cell dissipation $\mathrm{\Delta }B=12$; cell-substrate dissipation $D=0$; cell-substrate adhesion penalty $\phi =0$ ($r<r_0$), $\phi \to -\mathrm{\infty }$ ($r>r_0$); ${\displaystyle 100}$ independent simulations for each set of parameters). For this choice of parameters ${\displaystyle 4}$-cell populations rotate in the ${ℛ}_2$-phase. We observe a similar behavior here: highly correlated rotations with no changes in rotational direction. (A) Characteristic observables of collective cell rotation at different values of the cell perimeter stiffness parameter ${\kappa }_P$: mean (${\displaystyle ⟨|\omega |⟩}$) and standard deviation (${\sigma }_{\omega }$) of the angular velocity magnitude of cell motion, and the standard deviation of the cell perimeter (${\sigma }_P$). The black line corresponds to a power-law fit of the form $⟨|\omega |⟩\propto N^{-1/2}\propto r_0^{-1}$. (B) Representative angular trajectories and (C) cell shapes (color code represents cell polarization; cf. Figure 1) for the different parameter regimes as described in the main text. .

Figure 4—figure supplement 3. Collective motion for varying number of cells at high polarizability.

($N$-cell systems; confinement radius $r_0=\sqrt{234N}$; stiffness parameters ${\kappa }_P=0.060$, ${\kappa }_A=0.18$; average polarization field ${ϵ}_0=225$; maximum cell polarity $\mathrm{\Delta }ϵ=70$; signaling radius $R=5$; cytoskeletal update rate $\mu =0.1$; cell-cell adhesion $B=0$; cell-cell dissipation $\mathrm{\Delta }B=12$; cell-substrate dissipation $D=0$; cell-substrate adhesion penalty $\phi =0$ ($r<r_0$), $\phi \to -\mathrm{\infty }$ ($r>r_0$); ${\displaystyle 100}$ independent simulations for each set of parameters). For this choice of parameters ${\displaystyle 4}$-cell populations rotate in the ${ℛ}_3$-phase. We observe a similar behavior here: highly correlated rotations. (A) Characteristic observables of collective cell rotation at different values of the cell perimeter stiffness parameter ${\kappa }_P$: mean (${\displaystyle ⟨|\omega |⟩}$) and standard deviation (${\sigma }_{\omega }$) of the angular velocity magnitude of cell motion, and the standard deviation of the cell perimeter (${\sigma }_P$). The black line corresponds to a power-law fit of the form $⟨|\omega |⟩\propto N^{-1/2}\propto r_0^{-1}$. (B) Representative angular trajectories and (C) cell shapes (color code represents cell polarization; cf. Figure 1) for the different parameter regimes as described in the main text.

Figure 4—video 1. Collective rotations of ${\displaystyle 4}$ cells in the ${ℛ}_1$-phase (${\displaystyle \mathrm{\Delta }ϵ=28}$; ${\displaystyle \mathrm{\Delta }ϵ/{\kappa }_P=467}$).

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Figure 4—video 2. Collective rotations of ${\displaystyle 4}$ cells in the ${ℛ}_2$-phase (${\displaystyle \mathrm{\Delta }ϵ=50}$; ${\displaystyle \mathrm{\Delta }ϵ/{\kappa }_P=833}$).

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Figure 4—video 3. Collective rotations of ${\displaystyle 4}$ cells in the ${ℛ}_3$-phase (${\displaystyle \mathrm{\Delta }ϵ=70}$; ${\displaystyle \mathrm{\Delta }ϵ/{\kappa }_P=1167}$).

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