Fig 1.
Numerical simulation using a 2D vertex model.
(a) Diagram of forces around vertex in 2D vertex model. Blue arrows represent intracellular pressure toward the vertex and red arrows represent tension at cell-cell boundaries. (b) Illustration of T1 transition implemented in 2D vertex model. The left and right cells approach each other. When the length of the edge shown in red becomes shorter than a certain threshold, the cells acquire a common edge and top and bottom cells separate. (c) Flow of 2D vertex model. In the period 0 ≤ t ≤ tf the system is relaxed by adding a fluctuation term to the line tension. In the period tf ≤ t ≤ ttotal, the system is transformed to a steady state to minimize energy by removing the fluctuation term.
Table 1.
List of constants and variables used in 2D vertex model.
Table 2.
Typical ranges of parameters with “Control” as set value in Table 1.
Fig 2.
Parameter dependence of estimation accuracy in systems with homogeneous cells.
(a) Heatmap of RMSE in 2D parameter space of parameter elasticity and line tension (Λ, Γ). Gray cells indicate parameter sets for which the simulation stopped due to a large distortion of cell morphology. (b1, b2) Dependence of estimation accuracy on Λ and Γ. Dashed line shows the threshold of high-accuracy estimation. (c1-c4) Scatter plots of estimated values and true values at four representative points (1–4) indicated in (a), namely (Λ, Γ) = (−0.8, 0.11), (Λ, Γ) = (−0.3, 0.11), (Λ, Γ) = (0.3, 0.16), and (Λ, Γ) = (0.1, 0.04). The parameter set (Λ, Γ) = (0.1, 0.04) was used by Ishihara et al. (2012) for verifying their technique. All edge tensions (red) and cell pressures (blue) are plotted. (d1-d4) Cell morphology for four conditions simulated using 2D vertex model.
Fig 3.
Parameter dependence of cell shape characteristics and correlation with estimation accuracy.
(a1-a3) Heatmap of cell shape characteristics (circularity, perimeter, and polygonal number) in 2D parameter space of parameter elasticity and line tension (Λ, Γ). Solid line is the threshold (RMSE = 0.2) and dashed lines divide the parameter region defined in a previous study [17]. Points 1–4 correspond to representative points in Fig 2(a). (b1-b3) Scatter plots of RMSE versus cell shape characteristics obtained by comparing (a1-a3) and Fig 2(a).
Fig 4.
Cell morphology and estimation accuracy in systems with heterogeneous cells.
(a1-a4) Cell morphology calculated using 2D vertex model with perimeter elasticity having Gaussian distribution (expressed by color contour). The results were obtained under the conditions where the standard deviation of the perimeter elasticity is set as σΓ = 0.01, 0.02, 0.03, and 0.04. (b1-b4) Scatter plots of true and estimated values for each cell morphology.
Fig 5.
Dependence of estimation accuracy and circularity on cell heterogeneity.
(a) Dependence of estimation accuracy on the standard deviation of perimeter elasticity σΓ for system with heterogeneous cells. Dashed line is the RMSE threshold (0.2). (b) Circularity for each analysis condition. Points represent the mean circularity value and error bars represent its standard deviation. Dashed line is the circularity threshold (0.82). SD: standard deviation.