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Two types of critical cell density for mechanical elimination of abnormal cell clusters from epithelial tissue
Fig 2
Numerical simulations showed that there are two types of phase diagrams for elimination success/failure depending on the physical property of tissues.
(A) Typical time course of the number of abnormal cells in cases of elimination failure (top), growth suspension (middle), and elimination success (bottom) under lower tissue fluidity. (B, E) The dependence of elimination success on interfacial contractility μΛ for tissues with lower fluidity (B) or higher fluidity (E); the frequencies of elimination failure (red), growth suspension (G.S.; black), and elimination success (green). Note that the growth suspension phase appeared when the tissue fluidity was lower. The frequencies at different time points are represented as lines with different transparencies; the curves for the different time points overlap in (E). Each point (i.e., circle, triangle, or cross) represents the average result at t = 100τA for 20 simulation runs. The thick lines were obtained by fitting the Hill functions (i.e., f(x) = Kh/(xh+Kh) for “Failure”, g(x) = xh/(xh+Kh) for “Success”, and 1−f(x)−g(x) for “G.S.”). (C) The growth suspension phase showing that the size of the abnormal cell cluster remains nearly constant for a long period, with the size depending on the initial cluster size Nθ (top), but the cell density remains nearly constant independently of Nθ (bottom). Orange lines represent Nθ = 100 and blue lines Nθ = 200. (D, F) The time averages of the cell area (
) for each simulation run in the case of Nθ = 100. For each μ value, the results from 20 simulation runs were plotted. For the tissue with lower fluidity (D), a plateau density (
= ρ*) appeared for intermediate values of μ, suggesting that ρ* is the critical density associated with mechanical elimination of abnormal cell clusters. As shown later, ρ* is almost equal to the mechanical homeostatic cell density ρ1 (see also Figs 3C and 4B). On the other hand, for the tissue with higher fluidity (F), 
is an upwardly convex function at smaller μ values, without a plateau density, before jumping to zero at a higher value of μ. The cell density just before the jump, denoted as ρ**, provides another critical density for mechanical elimination different from ρ1 (see Fig 4C and the text for details). (G) The dependence of elimination success on interfacial contractility μΛ for different sets of mechanical parameters; the frequencies of elimination failure (red), growth suspension (black), and elimination success (green). Each curve was obtained using the same Hill function fitting explained above. It should be noted that the phase diagrams for elimination success/failure were similar for the sets of mechanical parameters yielding the same bulk modulus value K(Λ,Γ). Each vertical broken line in the phase diagram shows the critical contractility μ2Λ (black) or μ1Λ (red) obtained from analytical solutions (see the text and Fig 4D). The parameter set (Λ,Γ = 0.12, 0.04) was used in (A–D) and (Λ,Γ = 0.01, 0.025) in (E and F). In (G), the nine parameter sets shown in Fig 1B were used. All results were obtained under Nθ = 100 and Scenario 1, except for those in (C), which were obtained under Nθ = 100, 200.
doi: https://doi.org/10.1371/journal.pcbi.1010178.g002