Estimating Interfacial Tension from the Shape Histories of Cells in Compressed Aggregates: A Computational Study

Abstract

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.

Cellular interfacial and surface tensions determined from aggregate compression tests using a finite element model

“…During the compression phase of the test, Ḣ is negative because the spacing of H between the plates reduces with time. Reshaping of cells inside the aggregate is quite uniform, but different than for cells in contact with the medium or the platens, a result confirmed experimentally (Yang and Brodland, 2009 provided that the ratio…”

“…The time rate of relative motion between the platens is called Ḣ and the force F required to compress the mass rises steeply (from A to B 1 in Fig. 2), as interior cells are carried by the bulk deformation of the aggregate (Yang, 2008;Yang and Brodland, 2009). During the compression phase of the test, Ḣ is negative because the spacing of H between the plates reduces with time.…”

“…Equation (4) is a best-fit polynomial to data (Fig. S1) from a previous geometric study (Yang and Brodland, 2009), and it is applicably provided that the platens move together sufficiently quickly that the interfacial tensions ␥ cc do not have time to meaningfully reshape (anneal) the cells during the compression process (see Supplementary Material). Annealing, a spontaneous process in anisotropic close-packed cells, occurs because force imbalances at triple junctions cause them to move, reshaping and rearranging the cells, so as to reduce the total interface area (Yang and Brodland, 2009).…”

“…Because aggregates in two dimensions can behave quite differently from those in three dimensions (Hutson et al, 2008), quantitative relationships between cell properties and aggregate behavior could not be obtained without a three-dimensional (3D) model. A 3D model with the capacity to explicitly model realistic cell geometries, cell rearrangements, and the mechanical effects of interfacial tensions and cytoplasm viscosity based the geometries of individual cells was built (Viens and Brodland, 2007) and used to carry out a detailed, experimentally confirmed study of the cell geometries produced during a compression test (Yang and Brodland, 2009). …”

The role of fluctuations and stress on the effective viscosity of cell aggregates

“…For instance, Leghorn embryo cell aggregates subjected to brief centrifugation periods (8-16 min) flatten quickly (cell shapes flatten too), and round up quickly (in Ͻ2 min) once centrifugal forces are removed, reaching a shape very similar to the initial one, like an elastic solid (16). In contrast, during prolonged centrifugation (1 day), the same aggregates continue to flatten more slowly (while cells gradually reassume their original shapes) (17,18); once centrifugal forces are removed, these aggregates round up slowly (in Ϸ1 d), like a viscous liquid.…”