Simulation of cell islands with more than two cells

Islands with more than two cells follow the same characteristics as the two-cell island case; however, the structure of the island and the organisation of the cells are different. In the hard regime, 𝛽𝛽 < 1, see Figure 3.7a, the cell boundaries are rigid and as such the cell island boundaries are also

rigid. In the soft regime, 𝛽𝛽 ≥ 1, see Figures 3.7b and 3.7c, the cell boundaries are malleable, even

forming tendrils in Figure 3.7c, and as a result the cell island boundary is much more regular. Aspects of the cell island morphology are best exhibited with a larger number of cells. Not only can the boundaries of a cell island be rigid in a hard regime, but the formation and optimal equilibrium configurations of the cell island can be difficult to achieve depending on the initial placement of the cells. Without an increase in fluctuation, from increased “noise”, on the cells’ boundaries, the cells become fixed and their configuration may not achieve the expected form. Softer, more malleable cells change their shape to geometrically pack themselves to achieve a minimum perimeter of the island, forming a “hexagonal” shape seen in Figure 3.7b. This is particularly evident on the edge of the island. In the soft regime we observe that there are physical differences between the outer layer of cells in contact with the empty space, see Figure 3.7b, compared to the cells surrounded by other cells. The outer layer of cells form “wedges” to fit the circular shape of the island, and the cell island is not restricted by this, forming any shape the parameters will allow.

A change in the number of cells in the simulations or geometric solutions has no significant effect on the average cell and cell island perimeter, in Figures 3.8a and 3.8d, respectively. Figure

3.8a plots the average cell perimeter against 𝛽𝛽, and shows that the number of cells in the island does

not affect the cell perimeter, as the value of the cell perimeter is similar for different numbers of cells in an island. Also, Figure 3.8d plots the average cell island perimeter divided by the square root of the number of cells in the islands against 𝛽𝛽, and, as with the average cell perimeters, the value of the rescaled cell island perimeters are similar for different numbers of cells in the islands. However, the difference between the simulation values and the geometric solutions seen in Figure 3.5d is also seen in Figure 3.8d.

There is a small difference in the cell and cell island areas as 𝛽𝛽 increases between the different numbers of cells in the cell islands, in Figures 3.8b and 3.8e. Figure 3.8b plots the average area of the cells in the cell islands and Figure 3.8e plots the rescaled value of the cell island area divided by the number of cells in the cell islands. The average area of the cells in the cell islands and the rescaled value of the cell island areas are equivalent. These figures suggest that systems with a smaller number of cells in the cell islands will have smaller cell areas than systems with cell islands containing a larger number of cells, especially for larger values of 𝛽𝛽 (i.e. soft cell regimes).

of the cell island. We note that in Figures 3.8b and 3.8e there is a larger difference between the

simulation and geometric solution values of the cell area when 𝛽𝛽 ≥ 4. This artefact is similar to the

divergence between the simulation and geometric solution values in Figures 3.6b and 3.6d, which were caused by the perimeter mechanisms influencing the cells more than the area constraint

mechanism for a “small” preferred area, 𝐴𝐴_𝑝𝑝 = 100. The divergence occurs here but at a larger value

of 𝛽𝛽 for a larger preferred area, 𝐴𝐴_𝑝𝑝 = 500. This suggests that the artefact occurs for any value of 𝐴𝐴𝑝𝑝 in the simulation of the cell island, but the value of 𝛽𝛽 the artefact occurs at depends on the value

of 𝐴𝐴_𝑝𝑝.

Given that the cell perimeters are similar and the differences in the cell areas is small for different numbers of cells in cell islands, the shape ratio of cells, 𝑔𝑔, plotted in Figure 3.8c against 𝛽𝛽, is similar for cells in cell islands containing different numbers of cells. Figure 3.8f plots the shape ratio of the cell islands, 𝑔𝑔_𝐼𝐼, against 𝛽𝛽 for cell islands containing different numbers of cells and shows that they are similar for any island size. The behaviour of both of these shape ratios is the same as the behaviour of the shape ratios of the islands with two cells in Section 3.4.2.

(a) 𝑆𝑆 = 1000 (b) 𝑆𝑆 = 2000

(c) 𝑆𝑆 = 5000

Figure 3.7. Example of cell islands a) in the hard regime (Γ = 6, 𝑆𝑆 = 1000, 𝛽𝛽 < 1) b) the

soft regime (Γ = 6, 𝑆𝑆 = 2000, 𝛽𝛽 > 1), and c) very soft regime (Γ = 6, 𝑆𝑆 = 5000, 𝛽𝛽 > 1).

(a) (d)

(b) (e)

(c) (f)

Figure 3.8. Plots, with the preferred area 𝐴𝐴_𝑝𝑝 = 500, for cell islands simulated on a

hexagonal pixel grid with the CPM showing the (a) mean cell perimeter with the geometric solutions from Equation (3.8), (b) mean cell area with the geometric solutions from

Equation (3.10) and (c) ratio 𝑔𝑔 = 𝐴𝐴 𝐿𝐿⁄ against parameter β, for different number of cells in 2

the cell islands, 𝑁𝑁; and the (d) mean cell island perimeter with the geometric solutions from

Equation (3.9), (e) mean cell island area with the sum of the geometric solutions of cell area and (f) ratio 𝑔𝑔_𝐼𝐼 = 𝐴𝐴_𝐼𝐼⁄ against parameter 𝛽𝛽, for different number of cells in the cell 𝐿𝐿_𝐼𝐼2

islands, 𝑁𝑁. Each measurement of the cell is averaged over the cells for the last 50 of 1000

iterations and the island measurements are averaged over the last 50 of 1000 iterations. The

3.5 DISCUSSION

The cell and cell island size are strongly influenced by the interacting mechanisms of the cell area constraint and perimeter contraction, and the cell and cell island shapes are influenced by the interacting mechanisms of the cell perimeter contraction and adhesion.

In Chapter 2 a geometric representation was developed from the simulated system; however, in this case we consider an optimisation approximation of the energy function. These geometric solutions are comparable to the simulation results, with a few differences.

The cell island can still be separated into a hard and soft regime as described for the

epithelial layer in Chapter 2; however, with the addition of “empty space” cells can become smaller

and hence the perimeters can become smaller than the calculated minimum perimeter, 𝐿𝐿_{𝑚𝑚𝑖𝑖𝑚𝑚}. This

behaviour is the same as the wound case in Chapter 2. Although there can be a decrease of the cell perimeter, the perimeter is still larger than the preferred perimeter, 𝐿𝐿_𝑝𝑝, and the cells follow the same behaviour, with stiff polygon shapes in the hard regime and malleable cell perimeters in the soft regime.

The different shapes of cells in the hard and soft regimes affect the perimeter of the free edge of the cell island and that of epithelial cell aggregates. The hard systems create stiff cells with a small number of fluctuations compared to the soft systems. This results in a cell island boundary with irregular edges created by the tiling of the cells. In addition, the cells have difficulty attaining the preferred perimeter due to the small number of fluctuations. These outcomes can be countered by increasing the noise of the system. Comparatively, softer systems create smoother boundaries due to the malleable nature of the cell boundaries. This difference between cells and the system edges in the simulation of the hard and soft regimes is the reason for the difference in results between the simulations and the geometric solutions, since the optimisation assumes specific cell and cell island shapes.

An unexpected result from the investigation of the cell island case is the behaviour of the cell island in a soft regime. Both the simulations and geometric solutions show the cell island will become smaller in softer regimes. However, it was found that the cells themselves follow this pattern as well. In addition, the cell area never reaches the preferred area of 𝐴𝐴_𝑝𝑝, and the larger the contraction or the number of cells, the larger the decrease of the relative area. This result is due to only one mechanism acting on the perimeter of island, namely the perimeter contraction mechanism.

Chapter 4: The Three-Dimensional CPM and