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.
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 validation) and several tissue and multicellular aggregates. Our results show that the studied systems are incompressible and the obtained surface ten-sions are independent of the magnitude of the compressive force and the volume of the droplets. Our results provide strong evidence for the concept of embryonic tissue liquidity and support its usefulness for the interpretation of early morphogenetic processes. More importantly, since surface tension is a measure of the liquid's cohesivity, in the case of tissues it must be related to molecular parameters. Indeed, it was shown on theoretical grounds that σ ∝ JNτ, where J, N and τ are respectively the bond energy between two homotypic cell adhesion molecules (CAMs), the surface density of CAMs and the effective life time of the adhesive bond [8]. The linear dependence of σ on N has recently been confirmed experimentally [9]. Thus our method of determining the absolute value of σ has important biological implications as it quantitatively relates a macroscopic tissue property to biomolecular entities.
The surface tension σ of a small liquid droplet compressed between two pressure plates can be determined from its geometric shape (Fig. 1A-B). Here we consider droplets with radius R 0 much smaller than the corresponding capillary length R c ≈ (σ /ρg) 1/2 , for which the effect of gravity can be neglected (e.g., for water, with σ = 0.07 N/m and density ρ = 10 3 kg/m 3 , R c ≈ 2.7 mm). Thus, the shape of a submillimeter liquid drop placed on a horizontal plate (Fig. 1A) is a spherical cap of radius R 10 and height H 0 . With these parameters, simple geometric considerations provide: (i) the (complementary) contact angle θ = cos -1 (H 0 /R 10 -1), and, assuming incompressibility, (ii) the radius of the suspended drop, R 0 = R 10 [(2cos θ ) cos 4 (θ /2)]. While H 0 and R 10 can be measured with high accuracy (e.g., < 1%), the relative error ∆θ /θ ≈ [(1 + cosθ )/θ sin θ ](∆H 0 /H 0 + ∆R 10 /R 10 ) can still be very large (e.g., ∼ 30% for θ = 20 • and ≫ 130% for θ < 10 • ). Thus, for determining the surface tension it is desirable to reduce the adherence between the drop and plates and avoid using quantities that explicitly contain the contact angle.
The compressed drop (Fig. 1B) has rotational symmetry about the z-axis and reflection symmetry with respect to the equatorial plane z = H/2. In this plane the surface of the drop has two principal radii of curvatures R 1 and R 2 (Fig. 1B). R 3 is the radius of the droplet’s circular area of contact with either compression plates. The degree of compression depends on the magnitude of the compression force F applied to the upper (or lower) plate. In terms of R 1 and R 2 the excess pressure inside the drop due to the surface tension is given by the Laplace formula ∆p = σ (1/R 1 + 1/R 2 ). Thus, at mechanical eq
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