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

Abstract

Although previous studies suggested that the interfacial tension γ_cc acting along cell-cell boundaries and the effective viscosity μ 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 σ_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 μ. 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 μ of the cell cytoplasm, the tension γ_cc that acts along internal cell-cell interfaces and the surface tension σ_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 γ_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.

Recent experimental and theoretical investigations suggest that cell surface properties govern cell motions and tissue self-organization in settings that range from embryology to cancer to regenerative medicine (Alves et al., 2007; Brodland, 2004; Foty et al., 1996; Foty and Steinberg, 2004; Lecuit and Lenne, 2007; Liu et al., 2008; Muller et al., 2007; Napolitano et al., 2007; Preetha et al., 2005; Siniscalco et al., 2008; Steinberg, 2007; Velzenberger et al., 2009). As reviewed in Brodland (2004) and elsewhere, surface-associated proteins and other subcellular structural components give rise to net tensions along cell-cell and cell-medium interfaces. Three such tensions interact at each triple junction and, if the vector sum of the forces is not zero, there will be a tendency for the triple junction to move. Under suitable circumstances, these motions can accumulate and cause single cells or groups of cells to displace relative to their neighbors, thus, producing patterns of movement—such as sorting, engulfment, or dissociation—that are identifiable at a longer length scale (Brodland, 2004, 2002). Although the interfacial tensions γ_cc understood to act along cell-cell boundaries are crucial to many of the claims made, evidence for them has been almost entirely circumstantial. Information about these forces is essential for understanding the mechanisms by which self-arrangement of cells occurs during embryogenesis, the reasons that cells leave a primary tumor and attach at specific secondary sites during metastasis, why cells attach preferentially to certain biomaterials, and the mechanical conditions necessary for cells to form tight aggregates in engineered constructs.

The rate at which these motions occur is restrained by the ability of individual cells to deform, a characteristic captured by its effective viscosity μ. This property arises in part from the ability of molecules in the liquid constituents of the cell to move with respect to each other, as measured by fluorescence recovery and other methods (Verkman, 2002), but it is evidently determined to an even greater extent by the remodeling rates of filamentous networks coursing through the cell (Brodland, 2004; Chen and Brodland, 2000).

It has been known for more than a decade that if a spherical aggregate of embryonic cells is compressed between parallel plates (Beysens et al., 2000; Brodland, 2003; Davis et al., 1997; Forgacs et al., 1998; Foty et al., 1996, 1994; Phillips and Davis, 1978) the bulk (or macroscopic) surface tension σ_cm of the aggregate can be determined from the final geometry of the mass. This tension can be related to the tensions γ_cc and γ_cm acting, respectively, along the cell-cell boundaries and the cell-medium interface by the equation (Hutson et al., 2008)

$${\sigma }_{\text{cm}}={\gamma }_{\text{cm}}-\frac{{\gamma }_{\text{cc}}}{2}.$$ (1)

(A nomenclature summary is provided in Table S1 of the Supplementary Material.) A previous two-dimensional (2D) computational model of the compression test (Brodland, 2003) suggested that interfacial tension and effective cytoplasmic viscosity μ could also be determined from such tests since changes in cell shape, as described in the next section, occur with time. 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).

Here, that 3D finite element model is used to investigate how the platen forces in a compression test are related to the properties of the cells forming the aggregate. A detailed parametric study, involving hundreds of simulations, eventually led to an understanding of how the force-time curve is related to the geometry and properties of the cells in the aggregate. Insights gained from those simulations and from mechanical analyses they suggested lead to equations from which cell viscosity, interfacial tension, and surface tensions could be determined from experimental force-time curves and aggregate profiles. The equations were then used to determine the properties of 5-day embryonic chick mesencephalon, neural retina, liver, and heart cells.

THE PARALLEL PLATE COMPRESSION TEST

In suspension, a homotypic aggregate of embryonic cells will normally acquire a spherical shape (Steinberg, 1963), as suggested by Fig. 1a, and the polyhedral cells it contains will become statistically isotropic (Brodland and Veldhuis, 2003; Brodland, 2003; Phillips and Davis, 1978; Viens and Brodland, 2007). As discussed elsewhere (Brodland, 2004, 2002; Lecuit and Lenne, 2007), these characteristic geometries result from forces generated by cytoskeletal components, cell adhesions, and other mechanisms.

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.

In a typical parallel plate compression test (Beysens et al., 2000; Davis et al., 1997; Forgacs et al., 1998; Foty et al., 1996, 1994; Phillips and Davis, 1978), the cell mass is compressed rapidly between parallel plates, and individual cells flatten as the mass deforms [Fig. 1b]. The time rate of relative motion between the platens is called $\dot{H}$ and the force F required to compress the mass rises steeply (from A to B₁ 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, $\dot{H}$ 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). Here, the average aspect ratio κ^# of interior cells is defined as

$${\kappa }^{\#}=\text{avg}\left(\frac{d_1+d_2}{2d_3}\right),$$ (2)

where d₁⩾d₂⩾d₃ are the lengths of the principal axes of individual cells. Cell shape can be related to the compression ratio

$$\zeta =\frac{H}{D}$$ (3)

by the equation

$${\kappa }^{\#}=18.05{\zeta }^4-14.38{\zeta }^3+6.99{\zeta }^2-0.05\zeta +1$$ (4)

provided that the ratio

$$\Pi =-\frac{{\gamma }_{\text{cc}}\rho D}{\mu \dot{H}}$$ (5)

is less than or equal to 5. 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 γ_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). As soon as compression stops, the platen force drops by an amount F_μ from B₁ to B₂ because the viscous cytoplasm is no longer being deformed (Brodland, 2003).

Figure 2.

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

If the platen spacing then remains fixed, F decays from B₂ to C (Fig. 2), a further decrease in F_cc. The profile of the cell mass does not change appreciably (Davis et al., 1997), but the platen force decreases due to changes in cell shape associated with annealing (Brodland, 2003, 2004; Forgacs et al., 1998; Phillips and Davis, 1978; Phillips and Steinberg, 1978). The basis for this drop in force is explained in the Results and Discussion section.

When annealing is complete [Fig. 1c], the surface tension σ_cm along the cell-medium interface continues to act, producing an internal pressure in the aggregate. This pressure gives rise to the load F_cm associated with point C, and from it, an interfacial tension σ_cc can be determined (Foty et al., 1996, 1994; Hegedus et al., 2006; Jakab et al., 2008; Norotte et al., 2008). The interfacial tension σ_cp acting along the cell-platen boundary, which has not previously been reported, can then be determined by considering the angle β [Fig. 1b] formed where the edge of the aggregate meets the platen.

RESULTS AND DISCUSSION

In order to quantify how the platen force is related to specific cell properties, a detailed parametric study was carried out using the 3D Chen and Brodland model (Yang and Brodland, 2009). Various combinations of material properties μ, γ_cc, σ_cm, and σ_cp, compression rates $\dot{H}$, and maximum compression ratios ζ were used, and parameter choices and curve interpretations were informed by the earlier 2D model (Brodland, 2003).

That model suggested that F_μ, the contribution the effective viscosity of the cytoplasm makes to the platen force F, can be estimated by considering only the material between the aggregate-platen contact areas. In the 3D case, this material forms a cylindrical volume of radius R₃ and height H. Continuum mechanics can be used to derive formulas for the force that such a volume, if filled with material of viscosityμ, would produce if compressed at a rate $\dot{H}$. This approach assumes that partitioning of the cytoplasm in the aggregate by cell membranes does not affect the bulk force produced by the cytoplasm, an approach employed successfully elsewhere (Brodland et al., 2006; Brodland and Wiebe, 2004). It does not imply that the membranes and their associated proteins are insignificant, but only that separating the cytoplasmic masses associated with each cell from each other by a membrane does not affect the forces they generate. Analysis carried out in the “Materials and Methods” section leads to Eq. 16, which can be inverted to yield the relationship

$$\mu =\frac{F_{\mu }}{3\pi R_3^2\frac{\dot{H}}{H}},$$ (6)

from which the effective cell viscosity μ can be calculated.

To calculate the contribution F_cc that the interfacial tension γ_cc makes to the plate force is more involved. In brief, this force arises because cells in the mass are not isotropic in geometry. As a result, cell edges are, on average, more nearly parallel to the platens than normal to them. The tensions γ_cc acting along these edges thus produce net tension-induced stresses ${\sigma }_x^{\gamma }\cong {\sigma }_y^{\gamma }$ in the directions parallel to the platens that are greater than the stress ${\sigma }_z^{\gamma }$ they produce normal to it. This imbalance causes the aggregate to push against the platens with a pressure

$$P_{\text{cc}}={\sigma }_x^{\gamma }-{\sigma }_z^{\gamma }.$$ (7)

This pressure acts over the aggregate-platen contact area, a circle of radius R₃, producing a force F_cc, which can be directly related to γ_cc and κ^#, as shown in the Materials and Methods section. During annealing, the cells become increasingly isotropic, the stresses ${\sigma }_x^{\gamma }$ and ${\sigma }_z^{\gamma }$ become more similar to each other, the pressure P_cc decreases, and so does the platen force F_cc. This relationship forms the basis of Eq. 20 in the Materials and Methods section, which can be inverted to yield the relationship

$${\gamma }_{\text{cc}}=\frac{F_{\text{cc}}}{\pi R_3^2\rho [-0.0368{\kappa }^2+0.7746\kappa -0.7404]},$$ (8)

from which γ_cc can be determined.

Others (Foty et al., 1996, 1994) have noted that the bulk surface tension σ_cm is related to the geometry of a compressed aggregate by the formula

$${\sigma }_{\text{cm}}=\frac{F_{\text{cm}}}{\pi R_3^2\left(\frac{1}{R_1}+\frac{1}{R_2}\right)}.$$ (9)

An exact solution of the Laplace equation has recently become available (Norotte et al., 2008) and, as the authors of that work point out, some of the assumptions made previously about the relationship of R₂ to R₁ or to H do not lead to accurate results. Here the aggregate profile was assumed to be a circular arc because a circular arc could be measured easily and the experimental data did not indicate the need to use a more complex profile. By considering the interface angles formed where the edge of the aggregate contacts the platens, the surface tension σ_cp along the aggregate-platen interface can also be determined. The governing formula, derived in the Materials and Methods section, is

$${\sigma }_{\text{cp}}=\frac{{\sigma }_{\text{cm}}H}{2R_2}.$$ (10)

To evaluate the efficacy of Eqs. 6, 7, 8, 9, 10, a representative model aggregate was compressed, and the descriptors shown in Fig. 1b were determined from the aggregate profile while κ^# was calculated from the model cells using Eq. 2. Equations 16, 20, 22 were then used to estimate the contributions made by F_μ, F_cc, and F_cm to the total platen loads (dashed curve in Fig. 2). The aggregate profile at C was not meaningfully different than that at B₂ (or B₁). The formulas provide reasonable approximations to the behavior of the model aggregate, even without any adjustment factors. To verify the accuracy of the custom computer-controlled aggregate compression instrument (Fig. 3) and the curve-fitting algorithms used to characterize the aggregate profile (Fig. 4), small air bubbles were formed and the interfacial tension of the associated air-water interfaces measured (0.081±0.021 N∕m) and compared with precise known values (0.072 N∕m at 25 °C).

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).

The interfacial and surface tensions and viscosities of 5-day embryonic chick mesencephalon, neural retina, liver, and heart cells were then measured. Standard methods, described in the Materials and Methods section, were used to form 100–500 μm-diameter spherical aggregates of each cell type, which were then compressed. Force-time curves were obtained and images of the lateral aspect of the aggregates were collected (Fig. 4). The experimental force-time profile shown is consistent with the theoretical prediction. Key points on the force-time curves were determined manually, and descriptors of the aggregate profile were extracted from collected images using manually chosen edge points and custom curve-fitting algorithms. Tensions and viscosities were calculated using a spreadsheet (Supplementary Material, Table 3) and summaries of the resulting material properties are given in Table 1.

Table 1.

Summary of measured properties. Means and their standard errors are shown.

Tissue	No. of specimens n	Cell-medium surface tension σcm (N∕m)	Cell-platen surface tension σcp (N∕m)	Cell-cell interfacial tension γcc (N∕m)	Cytoplasm viscosity μ (N s∕m2)
Mesencephalon	4	12.9±0.9	12.0±1.0	1.40±0.23	27.1±3.0
Neural retina	5	14.1±2.2	13.2±1.9	1.54±0.29	20.4±3.4
Liver	6	23.3±2.4	20.9±2.7	1.52±0.36	39.5±3.3
Heart	6	72.4±11.0	62.3±10.2	1.59±0.29	60.7±7.2

All of the values measured were found to be tissue-specific, except for the cell-cell interfacial tension γ_cc, which was remarkably uniform. The cell-medium surface tension σ_cm values found here are higher than those reported elsewhere for these tissues (Forgacs et al., 1998; Foty et al., 1996), but tissue-specific tensions varied over a range of more than five times in those studies and the present one, and they fell in the same order. It is difficult to account for the nearly constant factor of approximately 8 between the previous and present studies, but forces in some of the earlier studies were reported in units of “mg.” The values reported here are closer to more recent measurements made, though in different tissues (Hegedus et al., 2006; Jakab et al., 2008), and the method used here correctly predicts the tension of air-water interfaces. The cell-platen interfacial tension σ_cp of each tissue is very similar to its cell-medium tension σ_cm, which is not surprising since the platens were coated with poly(2-hydroxymethylmethacrylate) to minimize cell-platen interactions. The cell-platen tensions are slightly smaller than the cell-medium tensions, suggesting that there was slight adhesion between the cells and platens because any such adhesion would reduce the net interfacial tension (Brodland, 2002; Lecuit and Lenne, 2007). The cell viscosities μ reported here fall within the broad range reported elsewhere (Valberg and Feldman, 1987) and are found to vary substantially with embryonic cell type, a finding that is strengthened here by the use of identical protocols for all tissues. One might expect heart cells to have an even higher relative viscosity compared to mesencephalon or neural retina cells due to the dense myofilament arrays they contain, but the dynamics of cell-specific intracellular filamentous networks may be just as important in determining effective viscosity μ, as are their morphologies. The interfacial tensions γ_cc, for which no comparable data exist, are small compared to the surface tension values. The experimental results also validate the computational model and the theory on which the formulas are based. Whether heterotypic interfacial tensions could also be determined from aggregate tests is an open question that might be answered by additional simulations and theory.

In a previous study, the authors showed that the interfacial tension γ_cc could be determined from the shape history of the cells in the interior of a compressed aggregate, an approach independent of the data used here. In that approach, cell shape data, like those that could be obtained using a confocal microscope, are required, rather than force-time data and aggregate geometry. Unfortunately, both sets of data have not been obtained for any one cell type so that γ_cc values obtained by the two approaches could be compared. Either method could be used to study aggregates that are compressed multiple times, provided that they are unloaded for a sufficient period of time that the cells completely anneal. If force-time and cell shape data were obtained simultaneously for a particular aggregate, the two approaches could be combined and the model challenged in more detail. For example, the actual time course of cell shape might be monitored or accurate cell shape data might be used in recompression experiments with incomplete preannealing. In time, when suitable instrumentation becomes available, cell pressure data might also be used to validate the model (Viens and Brodland, 2007).

The new kinds of measurements made possible by the present study provide crucial information for evaluating theories about the forces that drive tissue self-organization. For example, the differential interfacial tension hypothesis states that cell sorting, a phenomenon that has been studied extensively (Armstrong, 1989; Beysens et al., 2000;Brodland, 2002; Graner and Glazier, 1992; Hutson et al., 2008; Krieg et al., 2008; Mombach et al., 1995), requires the tensions along heterotypic interfaces to be higher than those along homotypic interfaces. However, the homotypic tensions in the two sorting populations can have the same value. In contrast, the differential adhesion hypothesis requires type-specific differences in how homotypic pairs of cells interact with each other, a demand inconsistent with the present data. One might argue that the observed constant interfacial tensions could result if the various membrane-associated systems and cell-cell adhesion forces (Brodland, 2002; Lecuit and Lenne, 2007) increased in exactly the same way, but such a scenario seems highly improbable given that the values found here are statistically indistinguishable across four cell types.

The data suggest that the tensions γ_cc acting along typical homotypic interfaces may be minimal. That these tensions are relatively small compared to surface tensions is reasonable if their primary purpose is simply to keep the geometry of individual cells reasonably compact. Whether γ_cc is small or large would make no meaningful difference in the final geometries of cells inside an aggregate, provided that rates of bulk deformation, mitoses, and other events that otherwise disrupt isotropic cells (Brodland et al., 2006) are sufficiently modest. Generation of larger forces would presumably involve higher rates of energy consumption and, in general, would be disadvantageous to the cell. The present analysis assumes that these tensions are a local property, independent of cell shape, and that they are insensitive to strain rates of the scale involved in these tests. It also assumes that the cytoskeleton inside the cells of the aggregate and the cell-cell connections are not significantly affected by the steps required to create the aggregates. These are substantial assumptions, but they seem to be consistent with the present experimental evidence. In time, as further experiments are carried out, they may need to be modified or qualified.

The amount of tension along heterotypic interfaces is relatively large and cell type-specific and is also reasonable if the purpose of these tensions is to move single cells or groups of cells relative to each other and to do so over reasonably short periods of time. The significance of this idea has been recognized for many years in the context of embryogenesis, where these forces are considered important to tissue ordering and morphogenetic movements (Beysens et al., 2000; Chen and Brodland, 2008; Foty et al., 1996). It has also been recognized in the context of biomaterials, where surface properties affect cell-implant interactions (Alves et al., 2007; Bren et al., 2004; Velzenberger et al., 2009). More recently, cell motions relevant to cancer metastases and tissue engineering have been explained in terms of cell surface tensions (Foty and Steinberg, 2004; Muller et al., 2007; Napolitano et al., 2007; Preetha et al., 2005; Siniscalco et al., 2008).

Unfortunately, the importance of interfacial tensions in these settings has not received the attention it deserves because not only do the relative values of interfacial and surface tensions govern whether an aggregate dissociates (Brodland, 2004, 2002), but the values of these forces are crucial for working out force balances (Fig. 5) that determine whether cells leave an aggregate and whether they invade a target site, issues of fundamental relevance to cancer metastases and tissue engineering.

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.

MATERIALS AND METHODS

Some of the calculations necessary for the present study are quite involved, and details are shown in the Supplementary Material. This section presents the main steps in the calculations along with useful intermediate results and approximations. Biological protocols associated with the experiments are also presented here.

Geometric relationships

As an initially spherical aggregate of cells is compressed, its barrel-shaped geometry can, in general, be described by any three of the four dimensions shown in Fig. 1b. In the present analysis, R₁, R₂, and H will be considered the primary measurements, and the radius R₃ will be derived from these using the geometric relationship

$$R_3=R_1-R_2+\sqrt{R_2^2-\frac{H^2}{4}}.$$ (11)

As the mass is compressed, its volume

$$V=2\pi {\int }_0^{H∕2}{\left[\left(R_1-R_2\right)+\sqrt{R_2^2-y^2}\right]}^{2}dy=\pi H\left[2R_2^2+R_1^2-2R_1{R}_2-\frac{H^2}{12}+\left(R_1-R_2\right)\sqrt{R_2^2-\frac{H^2}{4}}\right]+2\pi R_2^2\left(R_1-R_2\right){\text{sin}}^{-1}\left(\frac{H}{2R_2}\right)$$ (12)

remains constant, making it possible to estimate its original diameter

$$D=\sqrt[3]{\frac{V}{4\pi }},$$ (13)

even if images of that state are not available. The exterior angle β can be determined from H and R₂ using the equation

$$\beta =\text{arccos}\left(\frac{H}{2R_2}\right).$$ (14)

If the profiles of Norotte (Norotte et al., 2008) were used, a somewhat smaller angle could result.

Contribution F_μ from cytoplasm deformation

Previous models of 2D aggregates (Brodland, 2003) suggest that, for ζ⩽0.6, the cytoplasmic contribution to the platen force can be adequately estimated by assuming a state of uniform strain in the cytoplasm between the platen contact zones and ignoring the material outside this zone. In three dimensions, the contact areas are circles of radius R₃ and the volume between them has height H. If this cylinder-shaped region is shortened at a rate of $-\dot{H}$ (a positive value for $\dot{H}$ corresponds to separation of the platens), and allowed to expand isochorically in the radial direction, the axial strain rate will be $\dot{H}∕H$ and the radial strain rate will be $-\dot{H}∕\left(2H\right)$, where these calculations must be based on the current value of H. The resulting tensile axial stress (Malvern, 1969) will be

$${\sigma }_z^{\mu }=3\mu \frac{\dot{H}}{H}.$$ (15)

If this stress acts over the circular area of contact between the aggregate and the platen, the contribution of the cytoplasm viscosity to the platen loading will be

$$F_{\mu }=3\pi \mu R_3^2\frac{\dot{H}}{H}.$$ (16)

Contribution F_cc from cell-cell interfacial tensions

To calculate the stresses generated by tensions γ_cc that act along each cell-cell interface, consider a cylindrical volume V (Fig. 2) having a cell interface density

$$\rho =\frac{Q}{V},$$ (17)

where Q is the total interface area per unit volume. Interface density can be related to the number of cells m in volume V by (Yang and Brodland, 2009)

$$\rho =g\sqrt[3]{\frac{m}{V}}=g\frac{1}{\sqrt[3]{{\overline{v}}_0}}=\frac{g}{{\overline{r}}_0\sqrt[3]{\frac{4}{3}\pi }},$$ (18)

where g is a form factor, ${\overline{v}}_0$ is the average cell volume, and ${\overline{r}}_0$ is the radius of a typical cell. For a regular cubic mesh g=3, while for a typical polyhedral mesh g=2.81.

The Supplementary Material shows how statistical mechanics, like that used elsewhere to estimate forces in cell sheets, can be used to calculate the stresses and pressures produced in a 3D aggregate when tensions γ_cc act along the edges of cells that have an anisotropic fabric. A key outcome of that analysis is the intracellular pressure P_cc, which can be approximated by the relationship

$$P_{\text{cc}}\cong \left[-0.0368{\kappa }^2+0.7746\kappa -0.7404\right]\rho {\gamma }_{\text{cc}},$$ (19)

which when it acts over the circular aggregate-platen contact area contributes a force of

$$F_{\text{cc}}=\pi R_3^2{P}_{\text{cc}}.$$ (20)

The calculations in this section were validated numerically, as reported in the Supplementary Material.

Contribution to F_cm from surface tension

If a surface tension σ_cm exists between a cell mass and the medium that surrounds it, an internal pressure will develop in the mass. This pressure is given by (Flugge, 1973)

$$P_{\text{cm}}={\gamma }_{\text{cm}}\left(\frac{1}{R_1}+\frac{1}{R_2}\right).$$ (21)

The pressure P_cm will act over the circle of contact between the aggregate and the platen and will contribute a force of

$$F_{\text{cm}}=\pi {\sigma }_{\text{cm}}{R}_3^2\left(\frac{1}{R_1}+\frac{1}{R_2}\right)$$ (22)

to the platen load. Although these radii are an approximation to the true solution of the Laplace equation (Norotte et al., 2008), our numerical solutions of the exact differential equations for geometries like those observed in our experiments show the differences to be largely inconsequential. Once the maximum degree of compression is reached, the profile of the mass does not change and the pressure and platen forces associated with the surface tension σ_cm remains sensibly constant throughout the test.

TOTAL FORCES

The total platen force will be the sum of the forces from cytoplasm deformation, surface tension, and interfacial tension, i.e.,

$$F_{\text{total}}=F_{\text{cyto}}+F_{\text{cm}}+F_{\text{cc}}.$$ (23)

For equilibrium to exist in the radial direction at the edge of the cell mass, where it contacts the platen at angle β [Fig. 1b], the following equation must be satisfied:

$${\sigma }_{\text{cp}}={\sigma }_{\text{cm}}\text{\hspace{0.17em}cos\hspace{0.17em}}\beta =\frac{{\sigma }_{\text{cm}}H}{2R_2}.$$ (24)

BIOLOGICAL PROTOCOLS

Fertile white leghorn chicken eggs (Shaver Poultry Breeding Farms Ltd. Ontario, Canada) were incubated at 37 °C and 50% humidity for 5 days. Neural retina heart, mesencephalon, and liver tissues were obtained from 5-day embryos. Dissections and dissociations were adapted from previously published methods (Freshney, 2000). Briefly, dissociations of dissected tissue fragments were conducted in calcium and magnesium-free Hanks’ Balanced Salt Solution (HyQ). Tissues were dissected and placed into a solution of ice cold 0.25% trypsin (Gibco). Dissected tissues were kept at 4 °C for 3 h. Excess trypsin was removed and tissues were incubated in residual trypsin at 37 °C until tissue came apart with gentle pipetting. To stop trypsin action 1 ml of a solution containing 50% Horse Serum (Gibco), 50% complete media was added. Complete media consisted of MEM∕EBSS (HyQ) with 50 mg∕ml DNase I (Roche Diagnostics), 26 mM bicarbonate, and the following antibiotics: 70 mg∕ml genetamycin (Gibco), 50 mg∕ml penicillin∕streptomycin∕neomycin (Gibco), and 50 mg∕ml kanamycin (Gibco). After addition cell solutions were centrifuged at 1000 g for 4 min to settle the cells. Supernatant solution was removed, and cells were resuspended in 1 ml complete media. Cells were then passed through nitex membrane to yield single cells and counted using a hemocytometer. Cell concentrations were diluted to 10⁶ cells∕ml and 3 ml were transferred to 5 ml round bottom flasks. Flasks were placed in a water bath∕shaker at 37 °C, 5% CO2 for 1–2 days at 120 rpm until cells formed spherical aggregates. Tests were performed in CO2 independent media heated to 37 °C (Gibco).

Aggregates were compressed using the instrument shown schematically in Fig. 3. For the compression phase of the test, the linear actuator (T-LA-28A, Zaber Technologies Inc., Vancouver) was manually operated and used to move the base of the cantilever beam until its tip just contacted the aggregate. Then, it was typically advanced by 200 μm at a rate of 2,000 μm∕s under computer control. Part of this motion produced compression of the aggregate while the balance produced bending of the beam. The beam, which was cut from a thermanox plastic cover slip, had a Young’s modulus of E=7,060 MPa and was typically L=36.2 mm long by h=180 μm thick and b=2 mm wide, was intentionally flexible so that the load it carried could be related to its degree of bending by the end-loaded cantilever beam formula (Hibbeler, 2005)

$$\delta =\frac{P{L}^3}{3EI},$$ (25)

where I is its moment of inertia (I=bh³∕12), and δ is its end deflection. The end deflection was defined as the difference between the linear actuator displacement and the beam tip displacement, as determined by a laser rangefinder (Opto NCDT1700, Micro-epsilon, Ortenburg, Germany). During the hold phase of the test, the beam load decreased and rangefinder output was used to adjust the position of the beam base under computer control so that the tip did not move. A camera (SensicamEM, Cooke Corp., Romulus MI) and lens (10X MPlanApo, Mitutoyo, Japan) were used to capture side views of the aggregate every second.

Figure 4 shows a typical force-time curve and the inset shows the aggregate in profile at the end of the compression phase. Software was written to fit circular arcs to the manually selected data points at the ends of the aggregate and to calculate the platen spacing from the chosen points along its top and bottom. The software automatically calculated R₁, R₂, R₃, and H [Fig. 1b]. Equation 11 was used to check the verity of the input numbers. This information was then entered into a spread sheet (Table S3, Supplementary Material), along with force values corresponding to points B₁, B₂, and C. The force F_μ was set equal to the difference in the forces between points B₁ and B₂, F_cc was set equal to the difference between the forces at B₂ and C, and F_cm was set equal to the force at C. Equation 12 was used to estimate the original volume V of the mass and Eq. 13 was used to estimate its original diameter. Compression ratio ζ was then determined using Eq. 3 and κ^# was estimated using Eq. 4. Equations 6, 7, 8, 9, 10 were then used to calculate μ, γ_cc, σ_cm and σ_cp.