Left–right asymmetric cell intercalation drives directional collective cell movement in epithelial morphogenesis

Morphogenetic epithelial movement occurs during embryogenesis and drives complex tissue formation. However, how epithelial cells coordinate their unidirectional movement while maintaining epithelial integrity is unclear. Here we propose a novel mechanism for collective epithelial cell movement based on Drosophila genitalia rotation, in which epithelial tissue rotates clockwise around the genitalia. We found that this cell movement occurs autonomously and requires myosin II. The moving cells exhibit repeated left–right-biased junction remodelling, while maintaining adhesion with their neighbours, in association with a polarized myosin II distribution. Reducing myosinID, known to cause counter-clockwise epithelial-tissue movement, reverses the myosin II distribution. Numerical simulations revealed that a left–right asymmetry in cell intercalation is sufficient to induce unidirectional cellular movement. The cellular movement direction is also associated with planar cell-shape chirality. These findings support a model in which left–right asymmetric cell intercalation within an epithelial sheet drives collective cellular movement in the same direction.

The model assumes that each vertex bears forces arising from various elements in the epithelial cells, such as adherens junctions. These forces are represented by a potential U (whose explicit form is given below). The vertex also bears a frictional force that is proportional to the velocity of the vertex, which expresses the difficulty of the motion of each vertex and hence junction remodeling progression. Time evolutions of the positions of the vertices are given by where ! = ! , ! is the position vector of the i -th vertex. Equation [1] expresses the balance of the frictional force − ! ! !" ! and potential forces −∇ ! ( ! , ). Here, µ i is the friction coefficient. In our model, µ i is allowed to depend on the vertex index i , which is based on the assumption that the rate of change of the length of cell boundaries may depend on the cell type. The potential energy U that is derived from the forces acting on the vertices except for the friction forces is given as a function of time and the positions of all the vertices, In our model, the potential U consists of four parts: Each part is explained in order below.
U area is the potential energy from the cell area change, given by where p 0 is the area elastic coefficient, which has the dimensions of pressure, n A is the area of the n -th cell, and 0 A is the preferred area. This potential energy represents the pressure acting on the cell boundaries. The symbol "cell n " under the summation indicates that the index n runs from 1 to the total cell number .
perimeter U is the potential energy from the cell perimeter change, given by where 0 k is the positive coefficient, n L is the perimeter of the n -th cell, and 0 L is the preferred perimeter. This term reflects the fact that the apical area of epithelial cells is surrounded by actomyosin cables at adherens junctions, and bears a tensile force from the cables.
This force tends to keep the cell shape round when 0 L is smaller than 4 ! . bond U is the potential energy from the change in edge length, given by where ij l is the length of the edge connecting the -and -th vertices, given by with random constant frequencies f ij ∈ [0,1] and random initial phases !" ∈ [0,2 ) for each cell boundary ij (Supplementary Figure 6d). Note that when 0 ij f = , ij γ does not fluctuate over time.
In our model, we must explicitly consider the boundaries of the epithelial sheet, because the tergite A8 we are concerned with is flanked by other segments, A7 and A9, whose properties are different from that of A8. The A7 segment does not move during genitalia rotation, so we consider A7 as a fixed object in space. Since A9 moves during the rotation, we regard the whole A9 as a solid disk that can move with a frictional force. Since the cells at the edges of A8 attach to these objects, the movement of these cells is constrained. This effect is expressed in terms of the potential ( ) i boundary U , which holds the -th vertex on the edge of the sheet. The explicit form of ( ) i boundary U reflects the shape of the sheet boundaries, so after the boundary shape is given, we can provide the explicit forms (see next section). The total potential representing the constraint of movement is formally written as

Supplementary Note 4 Two boundary cases in our model
(1) Flat boundary case To examine whether or not the whole epithelial sheet undergoes unidirectional motion, we consider the simplest case, a flat system shown in Figure 5a We examine the following two cases.   Figure 9 left).) The AP axis is assumed to be the direction specified by the ray from the center of the outer ring to the center of mass of the -th cell. The centers of both the outer and inner rings are set to be the origin of the x-y plane.