Coordinated motion of epithelial layers on curved surfaces

TL;DR

It is demonstrated that extrinsic curvature effects can explain the alignment of cell elongation with the principal directions of curvature and postulate its importance for tissue morphogenesis.

Abstract

Coordinated cellular movements are key processes in tissue morphogenesis. Using a cell-based modeling approach we study the dynamics of epithelial layers lining surfaces with constant and varying curvature. We demonstrate that extrinsic curvature effects can explain the alignment of cell elongation with the principal directions of curvature. Together with specific self-propulsion mechanisms and cell-cell interactions this effect gets enhanced and can explain observed large-scale, persistent and circumferential rotation on cylindrical surfaces. On toroidal surfaces the resulting curvature coupling is an interplay of intrinsic and extrinsic curvature effects. These findings unveil the role of curvature and postulate its importance for tissue morphogenesis.

Figures (14)

• Figure 1: Geometries and cell shapes. Red and blue lines mark periodic boundaries, which are glued together in b) and c) (not to scale). Three individual cells are shown in their equilibrium configuration in a) and b). The colors correspond to the parameter $Ec$ which models extrinsic curvature effects, see eq. \ref{['eq:FEC']}. $Ec = 0$ (purple) leads to a (geodesic) circle on both geometries. $Ec > 0$ (green) favours an alignment in direction of maximum absolute curvature and $Ec < 0$ (yellow) an alignment in direction of minimal absolute curvature. The elongation is marked and enhanced for visibility. On toroidal surfaces cell shapes depend on position. In c) trajectories and final positions and shapes of the cells are shown. The effect of extrinsic curvature is not visible. All shapes are obtained by solving eq. \ref{['eq:evol']} with $v_0 = 0$.
• Figure 2: Evolution on a cylinder. a) Time instance of the evolution together with overlayed cell shapes ($\phi_i = 0$) and cells at previous time steps for three cells. For corresponding movie see SI. b) and c) Kymographs and graphs displaying the average velocities of the cells from a) in azimuthal and longitudinal directions as function of time.
• Figure 3: Distribution of direction of motion and elongation direction on cylindrical surfaces ($r_{Cyl}, h_{Cyl}$). The angle between longitudinal direction and direction of movement or elongation direction is used as angular coordinate and the ratio of cells with this property as radial coordinate. $a)$-$i)$ Direction of movement color coded by mean velocity, $j)$-$r)$ direction of elongation. The data are averaged over time and three simulations for each configuration.
• Figure 4: Distribution of direction of motion and elongation direction on toroidal surfaces ($R_T, r_T$). The angle between poloidal direction and direction of movement or elongation direction is used as angular coordinate and the ratio of cells with this property as radial coordinate. $a)$-$f)$ Direction of movement color coded by mean velocity, $g)$-$l)$ direction of elongation. The data are averaged over time and three simulations for each configuration. $m)$-$o)$ Angle of elongation direction as function of Gaussian curvature averaged over the area of the cell $K_{cell}$.
• Figure S1: Principal directions of curvature. a) for cylindrical and b) for toroidal surfaces.