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Fig 6. Spatial heterogeneity in stress magnitude as the mechanical cause of cell elimination.(A) The stress within a tissue was evaluated by two types of stress tensors that are discrete versions of Cauchy’s stress and defined by using the forces acting on the vertices that compose each polygonal cell. The forces acting on vertex i of cell α are composed of the pressure inside the cell, Pα, and tension at the two edges linked to the vertex, Tij,α, Tik,α. The edge tension involving cell α was assumed to be half of the tension acting on the focal edge, Tij, and the remaining half was allotted to the other cell β that shares the edge, i.e., Tij = Tij,α+Tij,β = 2Tij,α. The stress tensor σ(A)α given by Eq (25) was calculated using the force vector Fi, positional vector ri from cell center, and normal vector njk (right upper). The stress tensor σ(B)α given by Eq (27) was calculated using the pressure Pα, identity matrix I, tension Tij,α, and positional vector rij from focal vertex i to the adjacent one j (right lower; also see the Models section). (B) Stress magnitude, stress anisotropy and its orientation. The calculated tensors were characterized by the two scalars, stress magnitude σ1+σ2 and stress anisotropy σ1-σ2, where σ1 and σ2 (σ1>σ2) are the principal stresses. The left panel shows an example of stress distribution. The color indicates stress magnitude: blue for tensile states and red for compressed states. The line inside each cell indicates the degree of stress anisotropy and its orientation. Longer lines show more anisotropic states. The orientation is the direction of the maximum principal stress. (C) Correlations between the stress state and cellular geometry. Relationship between stress magnitude and cell area (top). Relationship between stress anisotropy and cell shape anisotropy (middle). Relationship between directions of stress anisotropy and cell shape anisotropy (bottom). Each graph is composed of 10,000 data points calculated from cells randomly selected at a specific time in one simulation run. Parameter values: the reference set, Λ = 0.14, Γ = 0.04, θT1 = 0.1, κ = 0 and μ = 3.47×10−3. Correlation coefficient: ρ≈0.94 (top), ρ≈0.97 (middle), ρ≈0.99 (bottom). (D) Good correlation between the mean change in local stress magnitude and stress heterogeneity over the tissue (left) or the elimination rate (right). All simulations were performed using a pure population. Symbols: blue crosses (Λ), cyan pluses (Γ), orange diamonds (κ), and red circles (μ). Image published in: Lee SW and Morishita Y (2017) © 2017 Lee, Morishita. This image is reproduced with permission of the journal and the copyright holder. This is an open-access article distributed under the terms of the Creative Commons Attribution license Larger Image Printer Friendly View |