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

Abstract

Although previous studies suggested that the interfacial tension gamma(cc) acting along cell-cell boundaries and the effective viscosity mu of the cell cytoplasm could be measured by compressing a spherical aggregate of cells between parallel plates, the mechanical understanding necessary to extract this information from these tests-tests that have provided the surface tension sigma(cm) acting along cell-medium interfaces-has been lacking. These tensions can produce net forces at the subcellular level and give rise to cell motions and tissue reorganization, the rates of which are regulated by mu. Here, a three-dimensional (3D) cell-based finite element model provides insight into the mechanics of the compression test, where these same forces are at work, and leads to quantitative relationships from which the effective viscosity mu of the cell cytoplasm, the tension gamma(cc) that acts along internal cell-cell interfaces and the surface tension sigma(cp) along the cell-platen boundaries can be determined from force-time curves and aggregate profiles. Tests on 5-day embryonic chick mesencephalon, neural retina, liver, and heart aggregates show that all of these properties vary significantly with cell type, except gamma(cc), which is remarkably constant. These properties are crucial for understanding cell rearrangement and tissue self-organization in contexts that include embryogenesis, cancer metastases, and tissue engineering.

Figure 1. A 3D finite element model of aggregate compression.

(a) Initial configuration of 454 cells for which μ=2000, γ_cc=10 000, σ_cm=15 000, and σ_cp=15 000. (b) Immediately following compression, the cells are visibly deformed. (c) Following annealing, the cells are nearly isotropic in shape.

Figure 2

Force-time curves associated with the simulation shown in Fig. 1.

Figure 3. The aggregate compression apparatus.

The beam is moved downward by the linear actuator under computer control until the aggregate is sufficiently compressed. A laser rangefinder determines the displacement of the end of the beam, and the difference between the motions of the actuator and the beam end indicates the degree of beam bending, a quantity used to calculate the applied compression force. As the load in the beam decreases during the annealing phase of the test, rangefinder data are used to adjust the actuator so that a constant degree of aggregate compression is maintained. Images of the aggregate profile are provided by the camera and its associated optics.

Figure 4. Force-time curve from a typical experiment.

A 275-μm-diameter aggregate of chick heart cells (the one reported as Trial 3 in Table S3) was compressed to ζ=0.33. The step undulations in the experimental curve (shown solid) result from the limited resolution of the rangefinder (Fig. 3). The insets show the aggregate at T=300 s, with the image on the left indicating the clarity of the aggregate profile and that on the right the manually traced edges of the aggregate with the corresponding best-fit arcs to the points along its left and right edges. The radius R₂ was set equal to the average of the two arc radii. The theoretical curve (shown dashed) was obtained by estimating K^# at the end of the compression phase using Eq. 4, using Eq. 10, and then Eq. 9 from a previous paper by the authors (Yang and Brodland, 2009) to estimate K^#≅K₁₃ as a function of τ, and then using Eqs. 19, 20 of this paper to estimate the contribution F_cc to F. No adjustments were made to the equations, as might be done acknowledge that K^# evidently approaches 1 in real aggregates, not 1.4, as in the model studies (Yang and Brodland, 2009).

Figure 5. Cell ingression and egression.

As noted elsewhere (Brodland, 2004, 2002), motion of cell A relative to the cells that surround it, whether of the same phenotype or not, is governed by the relationship between the tension γ_AM along the cell A-medium interface, the tension γ_BM along the cell B-medium interface, and the tension γ_AB along the cell A-cell B interface. Until one knows the value of the interfacial tension γ_AB, one cannot know whether cell A leaves the mass, attaches to its surface, or ultimately moves into the interior of the mass.