Previous 2D and 3D models concluded that cell sorting requires cytoskeletal fluctuations and is stalled by high tension at heterotypic interfaces. New deterministic and stochastic models show that this is not true in 3D. Sorting in 3D involves both topological untangling and domain coalescence. Coalescence requires fluctuations and low tension, but untangling does not. It occurs by a Plateau-Rayleigh instability of cell threads-deterministically driven by high tension. At high minority-cell fractions, untangling dominates and significant partial sorting can occur without fluctuations.

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A 3D cell-based finite element (FE) model is used to show that the tensions postulated to act along cell-cell interfaces can be determined from tests in which an aggregate of cells is compressed between parallel plates. The aspect ratios kappa of individual cells are found to depend on whether they are in contact with the plates, on the surface of the aggregate or in the interior of the mass, a result confirmed experimentally. When the platen spacing is then held constant, interior cells anneal at a rate that depends only on the cell-cell interfacial tension gamma(cc), whereas the annealing rate of plate and surface cells depends on gamma(cc), the cell-plate interfacial tension gamma(cp) and the surface tension gamma(cm) acting along the cell-medium boundary. The model shows that the ratio of the interfacial tension gamma(cc) to the cytoplasmic viscosity mu can be determined from the shape history of the interior cells. Cell shape anisotropies make the relationship between cross-sectional shapes and 3D shapes nontrivial. Experiments by others are then used to determine, for the first time, and without compression force data, the interfacial tensions that act in aggregated cells. This property has implications for theories about the forces that drive tissue self-organization.