Collective cell migration of epithelial cells driven by chiral torque generation

Various multicellular tissues show chiral morphology. Experimental studies have suggested that this can originate from the chirality of individual cells. However, no theory has been proposed to connect the cellular chiral torque and multicellular chiral morphogenesis. We propose a model of confluent tissue dynamics with cellular chiral torque. We found that cells migrate unidirectionally under a gradient of cellular chiral torque. While the migration speed varies depending on the tissue's mechanical parameters, it is determined solely by a structural order parameter for the liquid-to-solid transition in confluent tissues.

Figure 1

(a) Schematic of the disk-shaft model of a rotating cell on a frictional rigid substrate. The sense of a torque dipole driving the cellular rotation is represented by curved arrows. In the model, the apical and basal sides are modeled as red and black rotating disks, respectively. (b) Schematic of an epithelial tissue on a basal support substrate. (c) Schematic of the torque forces exerted on the $i\mathrm{th}$ vertex by the surrounding three cells generating chiral torque. The curved and solid arrows represent the chiral torque generation and torque forces, respectively, for positive ${\nu }_{\alpha }$. (d) 2D cellular sheet with the periodic boundary condition in the $x$ direction.

Figure 2

(a) Time evolution of a cellular configuration in a numerical simulation (${\nu }_{\alpha }=0.2,\sigma =0.3,N_y=6$). The time interval between each configuration is 40 time units. The Supplemental Material S1 [24] provides the movie. (b) Time-averaged flow profile for $N_y=40$ averaged over four samples (${\nu }_{\alpha }=1.0,\sigma =0.3$). The error bars represent the standard deviation. (c) Schematic of the torque forces exerted on the vertices on a boundary (${\nu }_{\alpha }>0$). The curved and solid arrows represent the chiral torque generation and the torque forces, respectively.

Figure 3

(a) Time evolution of a cellular configuration in a numerical simulation ($\lambda =0.1,\sigma =0.3,N_y=6$). The time interval between each configuration is 40 time units. Supplemental Material S2 [24] provides the movie. (b) Time-averaged flow profile for $N_y=40$ averaged over five samples ($\lambda =0.01,\sigma =0.3$). (c) Average velocity $V_{\mathrm{ave}}$ of cells is plotted against torque gradient $\lambda$ for different intensities $\sigma$ ($N_y=6$). The error bars represent the standard deviation.

Figure 4

(a) A cellular configuration for the theoretical analysis on the force exerted on a single vertex $O$. The curved arrows represent the direction of torque generations. (b) The normalized speed $V_{\mathrm{ave}}/v_{\mathrm{theory}}$ is plotted against the number $N_y$ of cells for different values of $\lambda$. The dashed line indicates $V_{\mathrm{ave}}/v_{\mathrm{theory}}=1$. (Inset) The dependence of $\langle q_{\alpha }\rangle$ on $N_y$ is shown. The dashed line indicates $\langle q_{\alpha }\rangle =q_{\mathrm{hex}}$.

Figure 5

The normalized bulk velocity $V_{\mathrm{bulk}}/v_{\mathrm{theory}}$ is plotted against the shape index ${\langle q_{\alpha }\rangle }_{\mathrm{BL}}$ in the boundary layers for different $\sigma$ and $P_0$ for a large system size $N_y=40$. $\lambda$ is fixed to 0.01. The different symbols are for different samples. (Inset) The normalized velocity $V_{\mathrm{ave}}/v_{\mathrm{theory}}$ is plotted against the target shape index $P_0$ for different values of $\sigma$. $V_{\mathrm{ave}}$ was averaged over three samples, and the error bars represent the standard deviation.

Figure 6

(a) The initial condition with regular hexagonal cells. The half regular hexagonal cells on the boundaries are shown with red dots. (b) A configuration after relaxation with $\sigma =0$ and ${\nu }_{\alpha }=0$ ($P_0=3.54$).

Figure 7

The tissue deformation observed without continuous cellular flow in the zero-noise limit $\sigma =0$ (a) under a homogeneous cellular chiral torque (${\nu }_{\alpha }=0.2$) and (b) under a gradient of cellular chiral torque ($\lambda =0.1$).

Figure 8

(a) Examples of the flow profiles are shown ($N_y=40,\lambda =0.01,p_0=3.33$). The $N_b\mathrm{th}$ and $N_t\mathrm{th}$ layers, determined by the above procedure, are also shown with the square markers. (b) Spatial profiles of shape index $\langle q_{\alpha }\rangle$ are shown. The $N_b\mathrm{th}$ and $N_t\mathrm{th}$ layers are also shown with the square markers. The data points for the top and bottom layers are marked with dashed ellipsoids as a guide for eye. In (a) and (b), each data point corresponds to each layer.

Figure 9

(a) ${\langle q_{\alpha }\rangle }_{\mathrm{BU}}$ is plotted against $P_0$ for different values of $\sigma$ for a large system size $N_y=40$. The torque gradient $\lambda$ is fixed to 0.01. The different symbols are for three different samples. (b) ${\langle q_{\alpha }\rangle }_{\mathrm{BU}}$ is plotted against ${\langle q_{\alpha }\rangle }_{\mathrm{BL}}$ for different $\sigma$ and $P_0$. The dashed line indicates ${\langle q_{\alpha }\rangle }_{\mathrm{BU}}={\langle q_{\alpha }\rangle }_{\mathrm{BL}}$. The different symbols are for three different samples.