Figure 3. Vortex formation in expanding tissues.

(A) Vortical flows seen from 10 hr traces of cell trajectories in small (left) and large (right) tissues. We color each trajectory according to its local orientation. (B) Growth rate of perturbations of wave vector modulus q around the unpolarized state of the tissue bulk, Equation 3. Perturbations with wavelength longer than $2\pi /q_c$ grow ($\mathrm{\Omega }>0$), leading to large-scale spontaneous flows in the tissue bulk. We show curves for the following values of the polarity-velocity coupling parameter: ${\nu }_s=0,1,2,3,4$ mm⁻¹. For the remaining parameters, we took $T_a=100$ Pa/μm, $\xi =100$ Pa⋅s/μm², $\eta =25$ MPa⋅s, $\gamma =10$ kPa⋅s, $a=20$ Pa, $K=10$ nN, as estimated in Pérez-González et al., 2019. (C) Average kymographs of vorticity show that the vortex in small tissues appears in the center and expands to >1 mm (n = 16), while vorticity in large tissues is present only during the early stages of tissue expansion (n = 16). The black bars indicate a characteristic vortex size. (D) Characteristic vortex size (marker size), time (horizontal axis), and intensity (vertical axis) of each tissue’s maximal vortex intensity. Small tissue vortices are generally more intense, with $p<0.0001$. (E) For small tissues, the time of maximal vortex intensity decreases with the initial cell density.

Figure 3—figure supplement 1. Trajectory displacement quantification of vortex.

Radial and tangential displacements of 18 h cell trajectories for a representative small tissue as a function of the initial radial position of each cell trajectory. Trajectories were selected from the vortex-dominated period of 24–42 hr. The tangential displacement (blue markers) most clearly reveals the presence of the vortex, with 350 μm tangential displacement throughout the central 750 μm of the tissue. At the outer zone, tangential displacements drop to 0 μm. In contrast, radial displacement (red markers) increases roughly linearly from the center to the outer edge of the tissue, with no sharp change when moving from the central to the outer region. The fact that radial displacements are largely insensitive to the vortical flows explains why the presence of a vortex has no noticeable impact on the overall expansion of the tissue. Together, the radial and tangential displacements of the tissue reveal a spiraling vortical flow that combines tangential shear with radial expansion.

Figure 3—figure supplement 2. Representative kymographs and heatmaps of vorticity.

Representative kymograph and heatmap of vorticity for (A) small and (B) large tissues.

Figure 3—figure supplement 3. Kymographs of enstrophy for small tissues of varying density and large tissues.

Averaged kymographs of enstrophy, which partition the vorticity power in modes of different wavelengths for each timepoint. (A) From left to right, enstrophy kymographs of small tissues for decreasing starting density. Initial densities were grouped as in Figure 1D. Decreasing starting density clearly delays the onset of high power, long wavelength (small wavevector) vorticity. Small tissue enstrophy peaks at a wavevector of $\sim 3.14m{m}^{-1}$, which corresponds to a wavelength of 2 mm. Data from n = 16 tissues for $\rho ={\rho }_1$ (left); n = 13 tissues for $\rho ={\rho }_2$ (middle); and n = 11 tissues for $\rho ={\rho }_3$ (right). (B) Average kymograph of enstrophy for large tissues ($\rho ={\rho }_1$, n = 16). The peak at large wavelength is not evident since the vortex is not as prevalent in large tissues. The initial cell density ranges ${\rho }_1$, ${\rho }_2$, and ${\rho }_3$ are the same as in Figure 1D.

Figure 3—video 1. Vortex formation in a sample small expanding tissue from t = 22 hr to 40 hr of expansion.

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Left panel shows phase-contrast microscopy of an expanding monolayer. Right panel draws the trajectories of individual cells as they evolve through time.

Figure 3—video 2. Vortex formation in a sample large expanding tissue from t = 12 hr to 30 hr of expansion.

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Left panel shows phase-contrast microscopy of an expanding monolayer. Right panel draws the trajectories of individual cells as they evolve through time.