Comparing individual-based approaches to modelling the self-organization of multicellular tissues

Abstract

The coordinated behaviour of populations of cells plays a central role in tissue growth and renewal. Cells react to their microenvironment by modulating processes such as movement, growth and proliferation, and signalling. Alongside experimental studies, computational models offer a useful means by which to investigate these processes. To this end a variety of cell-based modelling approaches have been developed, ranging from lattice-based cellular automata to lattice-free models that treat cells as point-like particles or extended shapes. However, it remains unclear how these approaches compare when applied to the same biological problem, and what differences in behaviour are due to different model assumptions and abstractions. Here, we exploit the availability of an implementation of five popular cell-based modelling approaches within a consistent computational framework, Chaste (http://www.cs.ox.ac.uk/chaste). This framework allows one to easily change constitutive assumptions within these models. In each case we provide full details of all technical aspects of our model implementations. We compare model implementations using four case studies, chosen to reflect the key cellular processes of proliferation, adhesion, and short- and long-range signalling. These case studies demonstrate the applicability of each model and provide a guide for model usage.

Fig 1. Schematics of the cell-based models considered in this study.

(a) Cellular automaton (CA). (b) Cellular Potts (CP) model. (c) Overlapping spheres (OS) model. (d) Voronoi tessellation (VT) model. (e) Vertex model (VM). (f) Flow chart of cell-based simulation algorithm. See text for full details.

Fig 2. Simulations of cell sorting due to differential adhesion.

Snapshots are shown at selected times for each model. Cells of type A and B are shown in purple and green, respectively. Engulfment of type-B cells occurs most readily in the CA, CP and VM models. Parameter values are given in Tables 1 and 2, with k_pert = 1. A video of these simulations is given in S1 Movie.

Fig 3. Comparison of cell sorting dynamics across differential adhesion simulations.

Left, as a measure of sorting, the fractional length is computed as a function of time for each model. Results are shown for varying multiples of the baseline level of noise, T or ξ (whose value is defined for each model in Table 2), by multiplying by k_pert. Each line is the mean value of 10 simulations. Right, the magnitude in the fluctuation of the fractional length curves (calculated as the mean squared error between the original curves and smoothed versions of the same curves, using a 10 hour smoothing range). The simulations from Fig 2 (with k_pert = 1) are denoted, on the left and right, by black lines and black crosses, respectively. Parameter values are given in Tables 1 and 2.

Fig 4. Effect of perturbations on cell sorting.

Snapshots at t = 100 (unless otherwise indicated) for the simulations presented in Fig 2 for varying k_pert = 10⁻², 10⁻¹, 1, 10, 10². Parameter values are given in Tables 1 and 2. For CP simulations with k_pert = 10² a snapshot is given at t = 3 as by this time in the simulation cells are already dissociated and have left the viewing window. For VT simulations with k_pert = 10² a snapshot is given at t = 3 as by this time in the simulation a cell has left the tissue domain and caused an infinite Voronoi region and the simulation is halted. For VM simulations with k_pert = 10¹ a snapshot is given at t = 1 as by this time in the simulation the perturbations cause cells to become inverted and the simulation is halted. There is no plot for VM simulations with k_pert = 10² as for this level of perturbation cells become inverted at the first time step so the simulations are not run. Note the central column corresponds to the t = 100 snapshots in Fig 2.

Fig 5. Simulations of monoclonal conversion in the colonic crypt.

Snapshots are shown at selected times for each model. In each simulation at time t = 0, every cell is regarded as a clonal population and given a different colour, which is inherited by its progeny. These populations evolve in time due to cell proliferation and sloughing from the crypt orifice, resulting in a single clone eventually taking over the entire crypt. Parameter values are given in Tables 1 and 3, with r_CI = 0.8. A video of these simulations is given in S2 Movie.

Fig 6. Comparison of clonal population dynamics and cell velocity across crypt simulations for varying levels of contact inhibition.

Left: the number of clones remaining in the crypt is computed as a function of time for each model: CA; CP; OS; VT; and VM. Right: the vertical component of cell velocity is computed for each model. For each model, the mean and standard error (not shown on clonal plots for clarity) from 10 simulations are shown for three levels of contact inhibition, quantified by the parameter r_CI. The vertical dotted line corresponds to the height of the proliferative compartment, y_prolif. Parameter values are given in Tables 1 and 3.

Fig 7. Number of cells in the crypt for varying contact inhibition.

For each model, the mean number of cells from 10 simulations are shown for three levels of contact inhibition, quantified by the parameter r_CI. Parameter values are given in Tables 1 and 3.

Fig 8. Simulations of lateral inhibition in a proliferating tissue.

For each model, snapshots are shown for three levels of cell proliferation, quantified by the parameter p_div. Parameter values are given in Tables 1 and 4. A video of these simulations, for p_div = 0.1, is given in S3 Movie.

Fig 9. Comparison of cell fate patterning, and cell compression across lateral inhibition simulations.

Left: as a measure of patterning, the ratio of cells in the heterogeneous steady state to those not in this state is computed, by averaging across times t = t_steady to t = t_end, as a radial distribution across the tissue (calculated using a bin size of 1 CD) for each model: CA; CP; OS; VT; and VM. Right: as a measure of compression, the number of cells per unit target area, for each model, is averaged as for the patterning ratios. For each model, the mean and standard error from 10 simulations are shown for three levels of cell proliferation, quantified by the parameter p_div. The vertical dotted line corresponds to the radius of the proliferative compartment, R_P. Parameter values are given in Tables 1 and 4.

Fig 10. Simulations of morphogen-dependent proliferation.

Snapshots of the tissue and associated morphogen distribution are shown at selected times for each model. Parameter values are given in Tables 1 and 5. A video of these simulations is given in S4 Movie.

Fig 11. Comparison of spatio-temporal morphogen and tissue shape dynamics across simulations.

Left: the morphogen distribution is plotted at selected times as an average over the x direction and over 20 simulations. Results are shown for each model: CA; CP; OS; VT; and VM. Right: as a measure of tissue anisotropy, the ratio of the widths of the tissue in the x and y directions is computed as a function of time for each model. For each model, we plot the mean and standard error of this ratio across 20 simulations. Parameter values are given in Tables 1 and 5.

Fig 12. Illustration of edge artefact in VT simulations on growing domains.

Closeup of bottom right of the t = 100 snapshot from the VT model in Fig 10.