Bubbly vertex dynamics: A dynamical and geometrical model for epithelial tissues with curved cell shapes

In order to describe two-dimensionally packed cells in epithelial tissues both mathematically and physically, there have been developed several sorts of geometrical models, such as the vertex model, the finite element model, the cell-centered model, and the cellular Potts model. So far, in any case, pressures have not neatly been dealt with and the curvatures of the cell boundaries have been even omitted through their approximations. We focus on these quantities and formulate them in the vertex model. Thus, a model with the curvatures is constructed, and its algorithm for simulation is provided. The possible extensions and applications of this model are also discussed.

Figure 1

Examples of epithelial tissue with curved cell shapes. (a) Epithelial tissue in the abdomen of Drosophila pupa, 19hr15min APF, DECadGFP knock-in [14], courtesy of K. Sugimura (WPI-iCeMS, Kyoto University). Scale bar: 10 $\mu \mathrm{m}$. (b) Apical surface view of the olfactory epithelium. Olfactory epithelium consists of two different types of cells, olfactory cells (small, rounded shape) and supporting cells. Olfactory epithelium of mouse was stained with ZO-1. Courtesy of S. Katsunuma and H. Togashi (Graduate School of Medicine, Kobe University). Scale bar: 5 $\mu \mathrm{m}$.

Figure 2

An illustration for the Young-Laplace law in two dimensions. A uniform tension $T$ along the one-dimensional interface with the curvature radius $R$ is balanced by the pressure difference $p_{\alpha }-p_{\beta }$.

Figure 3

A simple illustration of a wedge part of the cell area.

Figure 4

A variation of the area energy of the single cell $\alpha$ with respect to $V_{\alpha }$. The area is normalized by the least energy area, and the coefficients are given so that the behavior around the minimum coincides with that of the “area constraint” with ${\kappa }_{\alpha }^{\left(2\right)}=1$. These fix two parameters of three, and the variation can be given by the value of $V_{\alpha }$.

Figure 5

A schematic picture of the modified Young-Laplace function $G\left(\rho \right)$ of the central edge with its normalized curvature $\rho$ in three typical cases. The orientations of the central edges are shown in the associated diagrams. The two saturated cases are modified at $\rho =-1,1$, respectively.

Figure 6

An illustration of the T1 topological change.

Figure 7

An illustration of the T2 topological change.

Figure 8

An illustration of the ${\text{T2}}^{\prime }$ topological change.

Figure 9

The initial configurations of the vertices and the curvatures used for the examples. The initial positions of the vertices are given by the Voronoi diagram (left). The initial configuration of the curvatures (right) is determined by the simple collection of the bisection methods and the method of steepest descent for the parameters (47). The bar at the bottom represents the unit length. In epithelial tissues, the unit length is approximately $O\left(10\mu \mathrm{m}\right)$. The colors (tones) represent relative values of cell areas within an image: The darker the color is, the smaller is the cell area.

Figure 10

A series of images for a time course of the BVD generated by simulation. The images were acquired at $t=0.000$, $t=2.006$, and $t=100.006$ (from top left to bottom right). We ran the simulation up to $t\simeq 10000$, and no significant change was observed in the tissue shape after $t\simeq 100$. The colors (tones) represent relative values of cell areas within an image: The darker the color is, the smaller is the cell area.

Figure 11

The clustered histograms show a time course of the frequency distribution of the cells with respect to the number of edges per cell and the experimental observation in [11]. To compare with the experiment, only the nonperipheral cells are counted. From left to right in each cluster, they are of $t=0.000$, $t=2.006$, $t=100.006$, and the mean with SD of the experimental data of Drosophila imaginal discs in the late third instar stage. The line plots show a time course of the average cell areas for different numbers of edges. At $t=100.006$ (solid line with circles), the polygonal distribution approximately matches the experimental data, and the areas are roughly proportional to the number of edges per cell.

Figure 12

An example of the mitotic cell rounding during cell division obtained by simulation. The snapshots were acquired at $t=0.000,6.510,11.610,13.214$, and $30.004$. The first three images show the extracted part of the initial configuration, the cell volume growth with the mitotic cell rounding in the $I$ phase, and the beginning of the $M$ phase. The last three images show the $M$ phase and the finalization of cytokinesis. A peanut shape appears after the cytoplasmic division at $t=11.610$.

Figure 13

The dark cell at the center indicates the emerging two-vertex cell, and the other small cells share the same target area ${\tilde{A}}_{\mathrm{minor}}^{\left(0\right)}=0.2$ and the coefficient ${\Gamma }_{\mathrm{minor}}^{\left(L\right)}=0.09$. The bright large cells are of the ordinary dominant type with the target area ${\tilde{A}}^{\left(0\right)}=1.0$ and ${\Gamma }^{\left(L\right)}=0.03$. The two-vertex cell is surrounded by the dominant type of cells. To produce a stable two-vertex cell, we used a designed set of the parameters described in the paper and the initial configuration different from that in Fig. 9.

Figure 14

Successive images showing the creation of a two-vertex cell. The small, dark cell of the minor type is effectively being pushed by the light-colored, large cell of the dominant type through the repulsive interaction ${\Lambda }_{\mathrm{repulsion}}=0.30$. The images were acquired at $t=0.00,2.00$, and $23.60$, respectively.

Figure 15

The $\delta {\mathbf{x}}_i$ contribution in $\delta A_{\alpha }^{\left(\mathrm{poly}\right)}$.