Jamming and force distribution in growing epithelial tissue

We investigate morphologies of proliferating cellular tissues using a numerical simulation model for mechanical cell division and migration in two dimensions. The model is applied to a bimodal mixture consisting of stiff cells with a low growth potential and soft cells with a high growth potential; cancer cells are typically considered to be softer than healthy cells. In an even mixture, the soft cells develop into a tissue matrix and the stiff cells into a dendritelike network structure. When soft cells are placed inside a tissue consisting of stiff cells (to model cancer growth), the soft cells develop into a fast-growing tumorlike structure that gradually evacuates the stiff cell matrix. The model also demonstrates (1) how soft cells orient themselves in the direction of the largest effective stiffness as predicted by the theory of Bischofs and Schwarz [Proc. Natl. Acad. Sci. USA 100, 9274 (2003)] and (2) that the orientation and force generation continue a few cell rows behind the soft-stiff interface. With increasing intercell friction, tumor growth slows down, and cell death occurs. The contact force distribution between cells is demonstrated to be highly sensitive to cell type mixtures and cell-cell interactions, which indicates that local mechanical forces can be useful as a regulator of tissue formation. The results shed light on established experimental data.

Figure 1

Illustration of the model and the forces [Eq. (1)]. As the cells grow and deform, they reach the division threshold. At that time, a division plane is identified based on the long axis (Hertwig's) or some other selected rule. As the cell divides, new beads must be added to preserve the topology of the cells. Once division is completed, two child cells are produced. Colors indicate the different forces and the new beads that need to be created at the time of division.

Figure 2

Influence of stiffness differences in mixtures of stiff (red) and soft (blue) cells. The $X$ and $Y$ axes represent spatial coordinates in two dimensions (2D) in units of 10 $\mu \mathrm{m}$. (a) Initial configuration. Growth is simulated from this state onward until confluence with intermembrane friction coefficient $\mu =0.0\mu \mathrm{g}/\mathrm{s}$. (b) A confluent tissue of soft and stiff cells. Stiff cells form dendrite or veinlike structures in a matrix of soft cells. The regions marked with light purple and arrows are areas where cells interpenetrate and cell death occurs. The areas indicated with blue ovals show how the soft cells orient themselves at the boundary of stiff cells. The area surrounded by an oval with dashed lines shows how the stiff cells retain their shapes when away from the boundary. These issues are elaborated in the Discussion section. (c) Contact force distribution in the same tissue. Large contact forces are located at stiff cells and at boundaries between soft and stiff cells. White markers are the centers of masses of the stiff cells.

Figure 3

Influence of intermembrane friction on simulated cell morphologies. The $X$ and $Y$ axes represent spatial coordinates in two dimensions (2D) in units of 10 $\mu \mathrm{m}$. (a) Initial conditions used in both (b) and (c). Stiff cells are shown in red and soft ones in blue. The two are present in equal proportions. Morphologies after 10 division cycles in the cases of (b) zero intercellular friction and (c) high friction ($\mu =200.0\mu \mathrm{g}/\mathrm{s}$). Growth is faster at $\mu =0.0\mu \mathrm{g}/\mathrm{s}$, as indicated by the larger number of cells after the same number of division cycles. Morphologically, the high intermembrane friction system is more porous. The frictionless system corresponds to very early stages of development when junctions have not yet developed.

Figure 4

(a) Area and (b) cell-cell contact force (denoted by $F$) distributions at low ($\mu =0.0\mu \mathrm{g}/\mathrm{s}$) and high ($\mu =200.0\mu \mathrm{g}/\mathrm{s}$) intermembrane friction, corresponding to Figs. 3 and 3, respectively. (a) The peak at low areas ($A\approx 25 \mu {\mathrm{m}}^2$) corresponds to collapsed stiff cells. The dashed lines in (b) are fits with the green line being a fit to the $\mu =200.0\mu \mathrm{g}/\mathrm{s}$ case, ignoring the second peak. Solid lines are guide to the eye.

Figure 5

Configurations at confluence for the case of inclusion of soft (blue) cells in a matrix of stiff (red) cells when (a) $\mu =0.0\mu \mathrm{g}/\mathrm{s}$ and (b) $\mu =200.0\mu \mathrm{g}/\mathrm{s}$. The areas indicated with blue ovals show how the soft cells orient themselves at the boundary of stiff cells. Purple regions and arrows: areas were cells interpenetrate and death could occur. $X$ and $Y$ axes: spatial coordinates in units of 10 $\mu \mathrm{m}$.

Figure 6

Close-up showing (roughly) the circled area in Fig. 5. The color map shows the strain, that is, relative change in local membrane length as indicated by the color bar (1.0 equals to 100% strain). Soft cells orient themselves in the direction of the largest effective stiffness (the soft-stiff boundary). Since strain is given as stress/Young's modulus, it is evident that the soft cells at the boundary layer manifest force dipoles as predicted by Bischofs and Schwarz [85].

Figure 7

Distributions of cell-cell contact forces ($F$) when a soft cell is initially introduced into a tissue of stiff cells. (a) $\mu =0.0\mu \mathrm{g}/\mathrm{s}$ and (b) $\mu =200.0\mu \mathrm{g}/\mathrm{s}$. Configurations are shown in Figs. 5 and 5, respectively.