Fig 1.
The cellular automaton simulations reproduce the biological cell segregation experiments of Méhes et al. [20].
The segregation indices γi(t) for the two experiments PFK (dark red) with EPC (dark green) and HaCaT (dark blue) with EPC, within the observed time interval 50–1000 min, match with the segregation indices predicted by the cellular automaton (lines with corresponding brighter colors, dashed lines for PFK with EPC and dotted lines for HaCaT with EPC). Within the given time interval (grey box in top panel displayed again in bottom panel), the segregation indices seem to decay algebraically with exponent 1/3 (black dashed line) as expected asymptotically for fluid segregation. For the simulation of the segregation indices γi(t) of PFK (i = PFK) mixed with EPC (i = EPC), we obtain a cell type ratio of NPFK/NEPC = 41.2/58.8 and fit the adhesion parameters (βPFK-PFK, βEPC-PFK, βEPC-EPC) = (−8.06, −6.56, −0.06) and the time scale of migration τPFK-EPC ≈ 4.2 min. For the simulation of the segregation indices γi(t) of HaCaT (i = HaCaT) mixed with EPC (i = EPC) we obtain a cell type ratio of NHaCaT/NEPC = 35.2/64.8 and fit the parameters (βHaCaT-HaCaT, βEPC-HaCaT, βEPC-EPC) = (−7.93, −5.44, 0.06) and τEPC-HaCaT ≈ 35.1 min. In both cases 1402 cells are simulated, comparable to the cells visible in the experiments, starting from a random mixture. Snapshots of the cell mixtures at the points marked with crosses labeled A, C and B, D are displayed in Fig 3.
Fig 2.
The Cahn-Hilliard simulations reproduce the biological cell segregation experiments of Méhes et al. [20].
The segregation indices γi(t) for the two experiments PFK (dark red) with EPC (dark green) and HaCaT (dark blue) with EPC within the observed time interval 50–1000 min match the segregation indices predicted by the Cahn-Hilliard simulation (lines with corresponding brighter colors, dashed lines for PFK with EPC and dotted lines for HaCaT with EPC). Within the given time interval (grey box in top panel displayed again in bottom panel), the segregation indices of the Cahn-Hilliard simulation decay algebraically with exponent 1/4 (black dotted line) rather than 1/3 (black dashed line), which implies that the segregation process is in an intermittent regime of fluid segregation, see text for details. By using a mapping from the cellular automaton model to the Cahn-Hilliard model, see Materials and methods, parameters are set analogous to the parameters used in Fig 1 except for the mobility constant D, which is fitted to for the mixture of PFK with EPC and
for the mixture of HaCaT with EPC. Snapshots of the cell mixtures at the points marked with crosses labeled C, E and D, F are displayed in Fig 3. Note, that the Cahn-Hilliard model is shown after the settling process took place, see Fig E in S1 Text.
Fig 3.
The cellular automaton reproduces morphology and size distribution of the cell clusters in the experiments of Méhes et al. [20] of EPC (green) with PFK (red) closer than the Cahn-Hilliard model.
The snapshots of the cell mixtures A, C, E of the first row are taken at a segregation index of EPC γEPC = 0.25, at the start of the experimental recording, while the snapshots B, D, F in the second row are at a segregation index of EPC γEPC = 0.1, at the end of the recording. A and B show the cellular automaton, C and D show the experiments and are taken from video S5 in Méhes et al. [20], and E and F show the Cahn-Hilliard model. The snapshots A, B, E and F show a detail from the simulations, such that approximately 1002 cells are visible, to match the spatial scale of the snapshots C and D of the experiments. The time points corresponding to the images are marked by black crosses in Figs 1 and 2.
Fig 4.
The cellular automaton reproduces the cluster size distribution ρ of the experiments of Méhes et al. [20] for EPC with PFK closer than the Cahn-Hilliard model.
Shown is the reverse cumulative probability that a randomly drawn cell belongs to a cluster of respective size. For both models and the video S5 from Ref. [20], two separate cluster size distributions are shown, one at an early stage (t ≈ 55min) and one at a later stage (t ≈ 800min). The cluster size distributions represent exclusively PFK clusters, since EPC as the more abundant cell type forms one large connected sea, which we ignore in the distributions. Note that clusters below 2 cells are neglected as they can not be resolved in the video, see S1 Text.
Fig 5.
Exemplary comparison of the average cluster diameter in the segregation of PFK and EPC for the cellular automaton model (red line), the Cahn-Hilliard model (orange line) and the experimental data (blue line, based analysis of video S5 of Méhes et al. [20]) computed with two-point correlation method, see Materials and methods.
Note that average cluster diameter in both models, cellular automaton and Cahn-Hilliard, are inverse proportional to their segregation indices. In contrast, the average cluster diameter obtained for the experimental data displays a steeper power law than expected from the corresponding segregation indices.
Fig 6.
Example representation of the metrics segregation indices γi, average cluster diameter and cluster size distribution ρ with γ-ρ-fitted parameters for the cellular automaton for the PFK and EPC experiment of Méhes et al. [20].
Panels A and B are analogous to Fig 1, Panel C is analogous to Fig 5 and Panel D is analogous to Fig 4. The simulation used 1402 cells with a cell type ratio of NPFK/NEPC = 41.2/58.8, the adhesion parameter (βPFK-PFK, βEPC-PFK, βEPC-EPC) = (−8.0, −5.5, 0.0) and a time scale of migration τPFK-EPC ≈ 20.0 min.
Table 1.
Summary of the averaged mean squared deviation Δγ and the Kolmogorow-Smirnow-Distance (KSD) between each model and the corresponding experiment, see Materials and methods for details.
Fig 7.
The cellular automaton with γ-ρ-fitted parameters reproduces the morphology and size distribution of the cell clusters ρ in the experiments of Méhes et al. [20] of EPC (green) with PFK (red) closer than the cellular automaton with the γ-fitted parameters.
The snapshots of the cell mixtures A, C, E of the first row are taken at a segregation index of EPC γEPC = 0.25, at the start of the experimental recording, while the snapshots B, D, F in the second row are at a segregation index of EPC γEPC = 0.1, at the end of the recording. A and B show the cellular automaton with optimised parameters, C and D show the experiments and are taken from video S5 in Méhes et al. [20], and E and F show the cellular automaton with the Δγ fitted parameters. The snapshots A, B, E and F show a detail from the simulations, such that approximately 1002 cells are visible, to match the spatial scale of the snapshots C and D of the experiments.
Fig 8.
A 1/3 exponent forms an upper bound for the pseudo-algebraic scaling in cellular automaton.
Segregation indices obtained from the simulation are shown for a range of the effective parameters db and β*, but only for the last two orders of magnitudes in time before the segregation index reaches γ = 0.1 in each simulation. For comparability, the time scale of each simulation is set such that all simulations reach γ = 0.1 at
. For each simulation we use 1002 cells, a cell type ratio of 50/50, periodic boundary conditions, and a random mixture γ = 0.5 as initial configuration.
Fig 9.
Segregation indices obtained from the simulation shown for a range of cell type ratios.
For each simulation we use 1002 cells, periodic boundary conditions, db = 0, β* = 3, and a random mixture as initial configuration. For comparability, the time scale of migration τ of each simulation is set such that all simulations reach segregation indices γ0 and γ1 with γ0N0 = γ1N1 = 500 at dimensionless time . Every color represents a specific cell type ratio, while each cell type ratio was simulated five times. Panel A shows the raw data of the simulations. The black lines correspond to an even cell type ratio, for which both segregation indices match, while for uneven ratios the segregation index of the more abundant cell type is below the black line and the other above. Panel B shows the same data where each segregation index γi is rescaled to a segregation index
at an even ratio according to
.
Fig 10.
The influence on the time scale is trival for the parameter d, and non trival for the parameter db.
Shown is, for a constant β* = 3 and random initial conditions ξ with a 50/50 cell type ratio, the color coded sum λ0 of all heterotypic transitions rates. In direction of (1, 1, 1)T, the value of λ decreases and therefore the simulation time Δtswap for two neighboring heterotypic cells to change positions increases. In direction of (−1, 0, 1)T, the time dependency is nontrivial, but symmetric to db = 0.
Fig 11.
The cell type ratio for the simulation can be obtained from the experiments of Méhes et al. [20].
Shown is a comparison of the cell type ratio r = γ0(t)/γ1(t) from the experiments of Méhes et al. [20] (dotted in color) and from the cellular automaton (dashed in corresponding color). The cell type ratio in the cellular automaton is set to the approximate mean of the ratios observed in the experiment (dashed black lines). The cell type ratio was calculated by the ratio of the type specific segregation indices for each time t per experiment EPC with PFK (red) and HaCaT with EPC (blue).