Cell intercalation and T1 cycle.

(a) A full cell intercalation, consists of an internal and four external T1 processes. The latter reconfigure the peripheral vertices of the primary cluster, thereby triggering new T1 processes across the neighboring cells. In the language of topological defects, the T1 translates to the (i) unbinding of a ±1/6 defect quadrupole and (ii) a further unbinding of the quadrupole into a pair of dipoles. These two processes are schematically presented in a specific temporal order, but, in practice, they occur simultaneously or nearly so. (b) In a T1 cycle, the primary cell cluster undergoes a T1, followed by an inverse T1, which restores its initial configuration. The process corresponds to (i) the unbinding of a defect quadrupole and (ii) its annihilation.

Cell intercalation and T1 cycle as defect unbinding and annihilation.

(a) Cell intercalaiton. (i) Backflow velocity field generated during the unbinding of an active, hexatic defect quadrupole. The three panels below show the orientation field associated with (ii) the quadruple in the initial configuration, (iii) as it unbinds in a pair of ±1/6 dipoles and (iv) after the dipoles have move outside of the region of interest, together with the corresponding configuration of the primary cluster. As the dipoles move away from each other, the cells surrounding the primary cluster rotate clockwise (blue) and counterclockwise (red). (b) T1 cycle. (i)-(iv) Analogous sequence as in panel (a), but associated with the annihilation of the defect quadrupole. Notice that, in panel (iii), the direction of the flow is reversed. The details of the finite difference simulations can be found in Methods.

Collective cell migration as defect unbinding in the multiphase field model.

(a-b) Color plots illustrating the longitudinal hexatic (a) and nematic (b) stresses in MPF simulations (refer to Methods). The color bar is normalized to the largest stress magnitude observed in the configuration. Notably, the stress is uniformly negative, reflecting the extensile characteristics of both hexatic and nematic stresses. (c) Example of a four-cell cluster as it undergoes a T1 process, together with (d) the reconstructed 6-fold orientation field. The 6-legged stars mark the local 6-fold orientation of the cells (see Methods), while the red and blue dots denotes the +1/6 and -1/6 defects. (e) Probability distribution of finding a T1 (red tones) and a random cell (yellow tones) at a given distance from a defect, for four different values of the rotational noise Dr. The data indicate a prominent correlation between T1 process and topological defects. (f) The mean square displacement (m.s.d) of cells versus defect density. We identify two distinct sub-populations of cells: “slow” (blue tones), with no correlation to the local density, and “fast” (yellow tones), located where the local defect density is higher. The former correspond to cells undergoing a T1 cycle and the latter participating to cell intercalation, hence to collective cell migration. (g) Temporal statistics of tissue remodelling events in multiphase field simulations. Average time between two intercalation events (orange) and average period of a T1 cycle (green) versus the rotational diffusion coefficient Dr. The box plot in the inset shows the statistics of events analyzed for the case at Dr = 4 × 10−5. (Pairwise comparisons was performed with the two-sided t-test: ***p < 10−3). In the main graph error bars are reported as the first (bottom bar) and third (upper bar) quartile of the dataset.

Shape function.

On the left, we see a graphical representation of the 6–fold shape function γ6 (see eq. (2) for more details) for a generic irregular polygon. On the right (black 6–legged star) the phase and magnitude of γ6 for the same cell.

Active hexatic defect quadrupole: convergent extension analytics

(a) Force field: Stream-density plot of the force field Eq. (C11). It exhibits a clear, local, convergent-extension pattern in the vicinity of the quadrupolar radius ℓ. (b) Velocity field: Stream density plot of the velocity field Eq. (C17). It exhibits a clear, local, convergent-extension flow pattern in the vicinity of the quadrupolar radius ℓ. (c) Velocity field approximated close to defect core: Stream density plot of the velocity field Eqs. (1). It exhibits a clear, local, convergent-extension flow pattern in the vicinity of the quadrupolar radius ℓ. In all plots, the black disk corresponds the the radius of the quadrupole. Our analytical solution is valid outside the disk.

Trajectories of annihilating defects in time

The red lines are the trajectories of positive and blue negative defects respectively. Defects are sped up by positive activity (α6 = 0.1), but they are slowed down instead by negative activity (α6 = -0.1).