Soft yet Sharp Interfaces in a Vertex Model of Confluent Tissue

How can dense biological tissue maintain sharp boundaries between coexisting cell populations? We explore this question within a simple vertex model for cells, focusing on the role of topology and tissue surface tension. We show that the ability of cells to independently regulate adhesivity and tension, together with neighbor-based interaction rules, lets them support strikingly unusual interfaces. In particular, we show that mechanical- and fluctuation-based measurements of the effective surface tension of a cellular aggregate yield different results, leading to mechanically soft interfaces that are nevertheless extremely sharp.

Figure 1

The effective interfacial tension measured by parallel plate compression matches the imposed microscopic tension. (a) Snapshot of a droplet of cells at the beginning of a parallel plate compression simulation. (b) Geometric quantities used to compute the effective surface tension, as described in the main text or via the Young-Laplace law. (c) Measured surface tension normalized by ${\gamma }_0$ as a function of $p_0$. Circles, squares, diamonds, and triangles correspond, respectively, to ${\gamma }_0=0.05$, 0.1, 0.2, and 0.4, with $v_0=0.1$ and $D_r=10$. These data are representative of our results for all simulations performed. Inset: Example of the mean force in the $y$ direction on lower plate, shown on a logarithmic scale, as a function of time.

Figure 2

Interfaces are much sharper than expected based on the imposed microscopic tension. Normalized growth of the interfacial width $w$ as a function of the size of the periodic simulation domain $L_y$. Solid green squares refer to an equilibrium system at the same $T_{\text{eff}}$ as the orange diamonds, which are the self-propelled cell model with $v_0=0.1$, $D_r=1$. Blue circles refer to an active system with $v_0=0.1$, $D_r=0.1$. Open green squares refer to 2D vertex model simulations with the same parameters as the Voronoi simulations represented by solid green squares. The dashed red line is the expected scaling based on the standard capillary wave argument using $\gamma ={\gamma }_0=0.01$ as the interfacial tension and $w_0=0$. Inset: Sample image of two cell types in a strip geometry.

Figure 3

Short interfacial edges become more probable with increasing tension. Probability distribution of interfacial edge lengths, $P(l_i)$, for $p_0=3.95$, $v_0=0.1$, $D_r=1$ and ${\gamma }_0=0.005–0.64$ (light red to dark blue curves). The distribution of cell edge lengths in the bulk is given by the dashed black curve. The $l_i\ll 1$ distributions are approximately exponential, with characteristic decay length $l_p$. Inset: The length $l_p$, characterizing the short edge distribution for $D_r=1$ and $v_0=0.1$, 0.05, 0.025. The dashed red line is a guide to the eye with slope -1, corresponding to a Boltzmann expectation $P(l_i)\sim \exp [-l_i/(T_{\text{eff}}{l}_p)]$.

Figure 4

The restoring force changes discontinuously when interfacial cells are displaced. The distribution of restoring forces, $P(f_r)$, on cells displaced by ${\epsilon }_x={10}^{-4}$ in the inherent state of a system prepared in a strip geometry with $p_0=3.95$, $D_r=1$, $v_0=0.1$. From left to right ${\gamma }_0=0.04$, 0.08, 0.16, 0.32, 0.64. The dashed vertical lines show the magnitude of the restoring force predicted from the analytical calculation in the Supplemental Material [39] based on breaking fourfold vertices in a square-lattice geometry for each value of ${\gamma }_0$ (see inset). For each value of ${\gamma }_0$ the mean of the restoring force is nearly independent of the displacement for small displacements, demonstrating the non-Hookean behavior of the interface. The dashed line is the Hookean fit to the ${\gamma }_0=0.04$ data at large displacements.