Experimental evaluation of apparent tissue surface tension based on the exact solution of the Laplace equation

Abstract

The notion of tissue surface tension has provided a physical understanding of morphogenetic phenomena such as tissue spreading or cell sorting. The measurement of tissue surface tension so far relied on strong approximations on the geometric profile of a spherical droplet compressed between parallel plates. We solved the Laplace equation for this geometry and tested its solution on true liquids and embryonic tissue fragments as well as multicellular aggregates. The analytic solution provides the surface tension in terms of easily and accurately measurable geometric parameters. Experimental results show that the various tissues and multicellular aggregates studied here are incompressible and, similarly to true liquids, possess effective surface tensions that are independent of the magnitude of the compressive force and the volume of the droplet.PACS numbers: 68.03. Cd, 68.05.Gh, 87.18.La In the absence of external forces, an embryonic tissue fragment (typically less than 1 mm in size) rounds up similarly to a liquid drop, to minimize its surface energy. Cells of two distinct tissues randomly intermixed within a single multicellular aggregate sort into separate regions similarly to coalescing immiscible liquids. To account for these observations Steinberg formulated the Differential Adhesion Hypothesis (DAH), which states that embryonic tissues or, more generally, tissues composed of motile cells with different homotypic adhesive strengths should behave analogously to immiscible liquids [1,2]. DAH implies that such tissues, similarly to ordinary liquids, possess measurable surface and interfacial tensions generated by adhesive and cohesive interactions between the component subunits (molecules in one case, cells in the other). Predictions of DAH have been confirmed both in vitro [3] and in vivo [4,5,6], and the surface tensions of different embryonic tissues were measured and the values accounted for the observed mutual sorting behavior [3,7]. Currently, the only available method to measure the surface or interfacial tension σ of submillimeter size droplets of tissue aggregates is by compression plate tensiometry [7,8]. The method, based on the Laplace equation, so far relied on various approximations of the geometrical profile of an equilibrated spherical droplet compressed between two parallel plates and yielded σ values that can at best only be considered relative and their independence on droplet size and compressive force questionable. In this Letter, by analytically solving the corresponding Laplace equation, we determine the exact profile of a compressed droplet, which allows, for the first time, to accurately and reliably determine the absolute value of tissue surface tensions in terms of easily and accurately measurable geometric parameters. Furthermore, we show that the method can readily be extended to the simultaneous compression of several droplets of different sizes. Finally, we apply our new method to a large number of compression plate measurements for calculating σ of true liquids (for validati...

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

“…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 A RT I C L E associated with the surface tension cm remains sensibly constant throughout the test.…”

“…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 R 2 to R 1 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.…”

“…1(c)], the surface tension 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 cc can be determined (Foty et al, 1996(Foty et al, , 1994Hegedus et al, 2006;Jakab et al, 2008;Norotte et al, 2008). The interfacial tension cp acting along the cell-platen boundary, which has not previously been reported, can then be determined by considering the angle ␤ [ Fig.…”

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

“…Such cell sorting can be interpreted through cell adhesion differences: Motile cells rearrange to minimize the overall energy of the aggregate (6). By compressing an aggregate between two parallel plates (7), measuring its pressure and curvatures determines its surface tension (assuming that Laplace's law holds) (3,7,8). Comparing the surface tension values obtained for different cell types enables one to predict their sorting behavior (9).…”

Coaction of intercellular adhesion and cortical tension specifies tissue surface tension

“…Cell tracking in vivo and in vitro highlights (i) largescale flows, (ii) exchange of nearest neighbors in a cellular aggregate, and (iii) rounding-up and fusion of aggregates (1). Macroscopic rheological properties such as surface tension can be measured using a tissue surface tensiometer (TST) (1)(2)(3)(4)(5)(6)(7)(8) or micropipette aspiration (9), and surface tension can be used to explain tissue self-organization in embryogenesis (8,(10)(11)(12) or cancer (13,14). In particular, cell sorting and tissue spreading can be explained in terms of tissue surface tensions that differ among cell types (1, 3-5, 8, 15, 16).…”