Fig 1.
Schematic diagrams for vertex dynamics model and simulation settings.
(A) A scheme for mechanical elimination of abnormal cell clusters by interfacial contraction in epithelial tissues. (B) (left) Contours of bulk modulus K (black lines) and shear modulus G (gray broken lines) in cell mechanical parameter (Λ-Γ) space and (middle) a scheme for the apical dynamics of epithelial cells with Λ and Γ. For smaller Λ and Γ values, the tissue fluidity is higher. (right) The relationship between K and G, where the values calculated at 100 points in the gray region of (Λ-Γ) space are plotted. Red closed circles in the left and right panels (labeled as (i)–(ix)) show the parameter sets used for the simulations. (C) Two types of scenarios for the simulations. Scenario 1 is a artificial/simplified situation, in which an abnormal cell cluster appears at a certain moment within a normal cell population, and immediately after its appearance, the interfacial contraction μΛ comes into play (top). Scenario 2 is a more realistic situation, in which a single abnormal cell appears in a large population of normal cells and grows via cell division. Once the number of abnormal cells reach a certain size (Nθ), interfacial contraction μΛ occurs (bottom). See the text for details. (D) The relationship between the average stress state of the normal cells at the interface and the number of abnormal cells N (black lines). The gray regions show the standard deviation. σ1 and σ2 are the maximum (tensile, tangential to the interface) and minimum (compressive, perpendicular to the interface) principal stresses, respectively. In particular, σ2 and σ1 − σ2 show a clear dependence on N. See [37] for the calculation of cell stress tensor.
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.
Fig 3.
Mechanical homeostatic cell density and scaling of phase diagrams for elimination success/failure.
(A) The simulation settings in which cells continue proliferating within a restricted space with a fixed boundary (gray cells; left). The dependency of the number of cells in the domain (black) and mean cell area (red) on the domain size R (right). Each open circle indicates the temporal average of a single simulation run. The error bars indicate the standard deviation over time. (B) The distribution of the number of cell sides within an abnormal cell cluster in the growth suspension phase (black) and in a growing tissue composed of a single cell type under free boundary (white) and fixed boundary (gray) conditions. These histograms were obtained by the temporal average over a single simulation run. (C) The reciprocal values of mechanical homeostatic cell density ρ1 (red) and the upper limit of surrounding cell density below which a regular polygon with i edges can exist (ρMCE(i), i = 3, blue; i = 4, green; i = 5, black) for different sets of mechanical parameters (Λ,Γ). (D) Pairwise plots of 1/ρ1 and 1/ρMCE(i) (i = 3, left; i = 4, middle; i = 5, right) showing their positive correlations. (E) Comparison of the values of ρ1 and the multiple linear regression results for the nine different mechanical parameter sets (see also Eq (1)). The broken line represents
= ρ1. (F) A schematic diagram for the 2D Laplace’s law. At the interface, the energy is higher by ΛΔμ = Λ(μ−1) per unit edge length. PN and PA are the pressure within normal and abnormal cell populations, respectively, the difference in which is denoted by ΔP = PA − PN. Req is the radius of the abnormal cell cluster (denoted by R) at mechanical equilibrium. (G, H) The frequencies of elimination failure (red), growth suspension (black), and elimination success (green) against the rescaled contractility
for the tissues with lower fluidity (G) or higher fluidity (H). The thin curves show the results (fitted by the Hill functions) for Nθ = 100, 150, 200, and 250, and the thick curves show the approximations using the Hill functions for all of the simulation data with four different values of Nθ. In (A), (B), and (G), the parameter set (Λ,Γ = 0.12, 0.04) was used. In (C), (D), and (E), the nine parameter sets in Fig 1B were used. In (H), the parameter set (Λ,Γ = 0.01, 0.025) was used.
Fig 4.
Mathematical analysis for deriving analytical solutions.
(A) A simplified model for deriving approximate analytical solutions for elimination conditions (see main text for details). (B, C) Dependency of the cell area a (= 1/ρ) at equilibrium and its local stability (thick black curve) on interfacial contractility μΛ obtained by our approximate analytical solutions for tissues with lower fluidity (B) or higher fluidity (C) under Scenario 1. There are two types of characteristic cell density that can be critical for the mechanical elimination of abnormal cell clusters; one is mechanical homeostatic density (ρ1, red horizontal broken line), and the other is related to mechanical stability as a population (ρ2, gray). For tissues with lower fluidity (B), the inequality ρ2 > ρ1 holds. Thus ρ1 is reached first when μ increases and functions as a critical density for mechanical elimination, where the corresponding contractility is denoted by μ1Λ. On the other hand, for tissues with higher fluidity (C), ρ2 < ρ1 holds, and ρ2 functions as the critical density, for which the corresponding contractility is μ2Λ. (D) Dependency of the cell area at equilibrium and its local stability on interfacial contractility for different sets of mechanical parameters under Scenario 1 (thick solid/dotted curves). The red and black vertical dotted lines represent μ1Λ and μ2Λ, respectively, and are the same as those in Fig 2G, showing that the derived analytical solution explains the simulation results well. In theory, when ρ2 > ρ1, a growth suspension phase is expected. (E) The results of a similar analysis under Scenario 2; (bottom) phase diagram from the simulations and (top) analytical solutions. When the tissue fluidity is lower, the differences between ρ1 and ρ2 and between μ1Λ and μ2Λ are more marked. In (B) and (C), the parameter sets (Λ,Γ) = (0.12, 0.04) and (Λ,Γ) = (0.01, 0.025) were used, respectively. In (D), the nine parameter sets shown in Fig 1B were used. In (E), the three parameter sets (Λ,Γ) = (0.01, 0.025), (0.06, 0.035), and (0.12, 0.04) were used. All results were calculated for Nθ = 100.
Fig 5.
Effects of the relative proliferation rate and the difference in physical properties between normal and abnormal cells.
(A) The dependence of the relative growth rate of abnormal cells (r) on the phase diagram for elimination success. Simulations were performed under Scenario 2 and Nθ = 100. (B) When a cell divides, its daughter cells have fewer sides on average compared with the mother cell, while the number of sides of the two cells adjacent to the divided cell increases. As there is a positive correlation between the number of cell sides and the average cell size (Lewis’s law), the areas of the adjacent cells increase. Thus, division of normal cells at the interface could be a mechanism for increasing the likelihood of outward expansion of abnormal cell populations. (C, D) The effects of different physical properties between normal and abnormal cells on the phase diagram. When the abnormal cells have a fluidity higher than that of the surrounding normal cells, the critical contractility required for elimination of an abnormal cell cluster becomes smaller than that when normal cells have the same physical property as abnormal cells, and vice versa. The broken lines show the case in which both normal and abnormal cells are fluid (C) or solid (D).
Table 1.
Descriptions of the parameter values used in our simulations.