Two types of critical cell density for mechanical elimination of abnormal cell clusters from epithelial tissue
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.