Figure 6. An example of an intrinsically 3d event, budding, simulated using our version of the vertex model.

A. Initial configuration of cells in simulation; a small perturbation in the z direction is given to allow out-of-plane motion; the domain of cells is fixed at its equilibrium size at the boundaries. Color-coding denotes vertex model parameter patterning: blue cells have dimensionless parameter values a_α = a = 1, b_α = b = 0.1, and σ_ij = σ = 1; gray cells have a_α = fa = f, b_α = fb = f0.1, and σ_ij = fσ = f; pink line indicates edges with σ_ij ≈ T ≥ 1 with some noise (see discussion in the main text). The ring of high tension at the boundary between patch and non-patch cells eventually causes the patch to “pinch off”. B. Plotted are the critical tension T_crit as a function of the bending parameter c_αβ required to cause the patch of cells to pinch off. The value T_crit is explored for f = 1 (blue), f = 0.5 (green), f = 0.2 (orange), and f = 0.1 (red). It is clear that T_crit is lower for smaller values of f, and T_crit is larger for larger values of c_αβ since tissues averse to bending will require larger tensions to pinch off. C. Frames from the simulation with parameters in part A, setting f = 0.5, T = 5.5 and c_αβ = 0.1, viewed from beneath the dome of cells (Movie S10); the row of patch cells initially contacting the non-patch cells are colored pink for easy identification. It is clear that several intercalations occur between initial and final configurations at the patch edge due to the presence high tension there. D. Final steady states of the system viewed from the side with parameters in part C except for c_αβ which is varied as c_αβ = 0, 0.1 and 0.2. While for c_αβ = 0, 0.1, the patch of cells pinches off (Movies S9–S10), for c_αβ = 0.2, this does not occur (Movies S11–S12). E. Total energy E plotted as a function of time for cells time-evolved from an initial configuration, for c_αβ = 0, 0.1, 0.2 and for a system confined to 2d (z component is 0). While for c_αβ = 0, 0.1 the energy drops sharply when the patch of cells pinch off, for c_αβ = 0.2 and for the cells confined to 2d, the energy equilibrates to a stable value. Note that the final stable energy E is lower for the cases where cells are allowed to move in 3d than for the case where the cells are confined to 2d. Note also that the final pinched-off configuration for the c_αβ = 0 and c_αβ = 0.1 cases are not equilibrium states.