Fig 1.
Computational representations and analogous histological images.
Simulations can be developed at various scales, from a single cell to cell clusters and complex tissues. Cells and tissues are modelled using multiple particles (agents) which interact to each other; the macroscopic tissue behaviour emerges from these inter-particles interactions.
Fig 2.
Representations of cells and extracellular components in the proposed discrete element framework.
a) Example cell structure and possible interactions. b) Example tissue structure and possible interactions. The behaviour of each particle is influenced by its neighbouring particles through short range interaction forces.
Table 1.
Interaction matrix for the constituents of a cell.
Table 2.
Basic model parameters.
Fig 3.
Cell initialisation procedure.
a) Initial placement of particles defining the different parts of the cell. b) Cell after the initialisation procedure when no adhesion to the boundary is used. c) A spread cell geometry can be created by using attraction of membrane particles to the basement membrane (BM). The spread cell geometry depicted was achieved using a spring constant = 1×10−11 N/m and an influence radius δBM = 3μm for the BM. We have used a colour scale to indicate the net contact force experienced by each of the cell’s internal particles—the darker the colour the higher the force.
Fig 4.
Influence of BM parameters on cell spreading.
The attraction of membrane particles to the BM is characterised by both a linear attractive force (adhesion) spring constant and an influence radius. These quantities correspond to, for example, the extension distance of filopodia to a surface and the force generated by the filopodia. The influence radius δBM has a greater influence on the geometry of the cell as compared to the spring constant .
Table 3.
Nucleus calibration parameters used in the simulations.
Fig 5.
Comparison between experimental and simulation results for different values for Young’s modulus of the nucleus particles. Experimental data taken from Caille et al. [3]. The behaviour of the nucleus is well represented by the chosen force expressions and corresponding fitted parameters over a large range of deformation.
Table 4.
Upper and lower limits of cell calibration parameters used in the simulations.
Fig 6.
Comparison between experimental and simulation results for different values for Young’s modulus of the cytoplasm particles. a) Spread cell. b) Round cell. Experimental data taken from Caille et al. [3]. The behaviour of the cells is well represented by the chosen force expressions and corresponding fitted parameters over a large range of deformation.
Fig 7.
Spread cell compression based on EC = 2.5 kPa and EN = 25 kPa.
The computational simulation results for a) 10%, b) 20% and c) 30% compression are compared against the experimental results from [3]. The spread cell geometry depicted was achieved using a spring constant = 1×10−11 N/m and an influence radius δBM = 3μm for the BM. Very good match between the simulation and the experimental results can be observed.
Fig 8.
Round cell compression based on EC = 1.0 kPa and EN = 25 kPa.
The computational simulation results for a) 10%, b) 20% and c) 30% compression are compared against the experimental results from [3]. Very good match between the simulation and the experimental results can be observed.
Fig 9.
Influence of membrane and basement membrane spring constants on cell shapes in an epithelial layer.
In all cases the influence radii are δBM = 2μm and δM = 0.2μm. Increasing the adhesion to the basement membrane () leads to spread cells, while increasing adhesion between cells (
) leads to cells with a columnar appearance.
Fig 10.
Deformation of a single epithelial layer.
a) Control state ( = 1×10−11 N/m, δBM = 0.3 μm,
= 5x10-10 N/m, δM = 0.2 μm). At 40% compression a single buckle emerges. b) By increasing the BM adhesion (
= 1×10−10 N/m, δBM = 0.4 μm) and using the same cell-cell adhesion parameters the buckling is suppressed. c) Reducing BM and cell-cell adhesion (
= 1×10−12 N/m, δBM = 0.2 μm,
= 5x10-12 N/m, δM = 0.1 μm) causes the cell layer to cascade quickly and easily without any resistance. d) Increasing the BM adhesion and weakening the cell-cell adhesion (
= 5x10-10 N/m, δBM = 0.2 μm,
= 5x10-11 N/m, δM = 0.1 μm) results in the budding of a cell from the layer of cells.
Fig 11.
Influence of adhesion parameters on the shape of the buckle.
a) Weak BM adhesion (BMa) and weak cell-cell adhesion (CCa) ( = 1×10−11 N/m, δBM = 0.1 μm,
= 1x10-11 N/m, δM = 0.1 μm). b) Weak BM adhesion and strong cell-cell adhesion (
= 1×10−11 N/m, δBM = 0.1 μm,
= 1x10-9 N/m, δM = 0.3 μm). c) Strong BM adhesion and weak cell-cell adhesion (
= 1×10−9 N/m, δBM = 0.3 μm,
= 1x10-11 N/m, δM = 0.1 μm). d) Strong BM adhesion and strong cell-cell adhesion (
= 1×10−9 N/m, δBM = 0.3 μm,
= 1x10-9 N/m, δM = 0.3 μm). The adhesion parameters influence both the shape of the cells and the deformation of the epithelial layer.
Fig 12.
Influence of nuclei position on the behaviour of a cell layer under 40% compression.
a) Elevated nuclei position facilitates buckling. b) Nuclei positioned near the BM suppresses buckling. The BM and cell-cell adhesion parameters are given by = 1×10−11 N/m, δBM = 0.3 μm,
= 5x10-10 N/m, δM = 0.2 μm. The position of the nucleus at the base of the cell reduces the likelihood of buckling.