Fig 1.
(A,B) Schematic of 2D Voronoi model (green, degrees of freedom are cell center) and 2D vertex model (purple, degrees of freedom are cell vertices). (C,D) Snapshot of the 3D Voronoi model (green) and vertex (purple) simulations with a heterotypic interface between light and dark cells. (E) Demixing behavior of the 3D Vertex model initialized in a mixed state with DP ≈ 0. The color represents different values of the heterotypic interfacial tension: σ = 0.1 (light pink), σ = 0.2 (medium purple), σ = 0.5 (dark magenta). (F) Demixing as a functional of initial conditions between 3D Vertex and Voronoi models. The dark green (Voronoi) and dark magenta (vertex) are initialized in a sorted state and remain sorted. The bright green (Voronoi) and light pink (vertex) are initialized in a mixed state and rapidly sort.
Fig 2.
A) Zoomed simulation snapshot of the 3D Vertex model with heterotypic interfacial tension between cells of different types. B) Cell shape s as a function of the magnitude of heterotypic interfacial tension σ for cells that are adjacent to the boundary (solid lines) and cells that are in the bulk and do not share a heterotypic interface (dashed lines). In both models, the shape index of the boundary cells increases, while the shape index of the bulk cells decreases as a function of increasing interfacial tension. Error bars represent the standard error with respect to each ensemble. C) The probability distribution (pdf(z)) of the heights of cell centers (z) reported in natural length units from the bottom of the box. Different colors correspond to different magnitudes of the heterotypic interfacial tensions σ ∈ 0.04, 0.08, 0.16, 0.32, 0.64 from light pink to dark magenta. D) The registration of cell centers on either side of the interface (defined by Eq 4) increases for Voronoi (green) and vertex (magenta) models as a function of increasing interfacial tension. Error bars represent the standard error.
Fig 3.
(A-C) Simulation snapshots of tissues with different values of average orientation 〈O〉 defined by Eq 6 (0.24, 0.38, 0.18 for A,B,C respectively) for cells adjacent to the heterotypic boundary (D) Average orientation 〈O〉 as a function of the magnitude of the heterotypic interfacial tension σ for the Voronoi (green) and vertex (magenta) models. Error bars represent the standard error with respect to each ensemble.
Fig 4.
(A) The average cusp-like restoring force for cells, 〈Fr〉, being perturbed a distance ϵ perpendicular to the heterotypic boundary. The solid green lines are from 3D Voronoi model simulations with σ ∈ 0.04, 0.08, 0.16, 0.32, 0.64, with the colorscale ranging from light green at the lowest tensions to dark green at the highest, and the dashed purple lines are from the 3D vertex model with σ = 0.04, 0.08, 0.16, 0.32, 0.64 with the color shade ranging from light pink to dark magenta. (B) The plateau values at low tensions extracted from panel (A). (C) The distribution of restoring forces with ϵ = 10−4 for each set of parameters over an ensemble of N = 100 systems. In both systems, the forces are Gaussian-distributed around a central peak (Shapiro-Wilk test with P-values between 1e–5 and 1e–15), with colorscale the same as in panel (A). (D) Numerical simulations of the restoring force generated by perturbing a single Voronoi cell in 9 cell configuration in 2 dimensions. (E) Schematic diagram illustrating cell boundaries after a perturbation of ϵ = 10−1 to the cell center of a Voronoi (E, green) cell. (F) Schematic diagram illustrating cell boundaries after adding a random displacement of magnitude ϵ = 10−1 to each vertex of the center cell. In both (E,F), the blue lines represent the heterotypic interface with increased interfacial tension, and the red dots highlight positions of the vertices along the interface.
Fig 5.
(A) Schematic of perturbations to the x–component of the initial position xi of a single vertex relative to a putative four-fold coordinated vertex on the boundary. The green line represents a perfect four-fold vertex where xi is zero, the blue represents a coordinate that has a positive xi towards the interface and the red represents a vertex that is further from the interface xi negative. (B) Analytic restoring force (Eq 7) due to the interfacial tension alone due to the perturbations of vertices on the interface. The solid lines represent intial perturbations of the moving cell’s vertex a distance xi = {−10−4, 0, 10−5, 10−4, 10−3} colored {red, green, dark blue, blue, cyan} respectively with yi = 1. The dashed blue line represents a perturbation parallel to the interface such that {xi, yi} = {10−4, 1.1}. (C-E) Schematic diagrams of how the larger-scale cellular structure changes when a nearly four-fold coordinated vertex is perturbed as shown in (A).
Fig 6.
(A) Representative image from a confocal z-stack from one Xenopus gastrula across the ectoderm-mesoderm boundary. Scale bar is 50 μm. (B) Example of manual tracing of projected cell shapes from this image. (C) Image illustrating calculation of center of mass (black dot) and best-fit ellipse (red ellipses) for cell shapes across the heterotypic interface. (D-E) Histograms of the average registration R for pairs of cells (D), and orientation O for all cells (E) in images from seven Xenopus gastrula. (F) Bar plot illustrating the average O and R in experiments (box with black and white lines), with an error bar representing the standard deviation. For comparison, the average O and R for the vertex (light magenta) and Voronoi (dark green) simulations across a broad physiological range of heterotypic interfacial tensions (between 0.1 and 1) is also shown, with the error bar representing the max and min values of those observed parameters in simulations for that range of tensions.