Collective epithelial migration mediated by the unbinding of hexatic defects
- Instituut-Lorentz, Universiteit Leiden, Netherlands
- Physics Department, College of Sciences, Koç University, Turkiye
Figures
Figure 1
Cell intercalation and T1 cycle.
(a) A full cell intercalation consists of an internal and four external T1 processes. The latter reconfigures 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 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.
Figure 2
Cell intercalation and T1 cycle as defect unbinding and annihilation.
(a) Cell intercalation. (i) Backflow velocity field generated during the unbinding of an active, hexatic defect quadrupole (Video 1). 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 dipoles, and (iv) after the dipoles have moved outside of the region of interest, together with the corresponding configuration of the primary cluster (Video 2). 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’.
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Figure 2—source data 1
Values of the hexatic orientation field and velocity field used to produce the panels in this figure, obtained from finite-difference numerical solutions of Equation 3a, Equation 3b, Equation 3c.
- https://cdn.elifesciences.org/articles/105397/elife-105397-fig2-data1-v1.zip
Figure 3
Collective cell migration as defect unbinding in the multiphase field model (MPF).
(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 sixfold orientation field. The six-legged stars mark the local sixfold orientation of the cells (see ‘Methods’), while the red and blue dots denote the and defects. For such a four-cell cluster in real epithelial cell monolayer, please see Armengol-Collado et al., 2023a. (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 . 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 computed over a time window of iterations, chosen to match the typical duration of T1 events and defect lifetimes. We identify two distinct subpopulations 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 corresponds to cells undergoing a T1 cycle and the latter participating in cell intercalation, hence to collective cell migration. (g) Temporal statistics of tissue remodeling 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 . The box plot in the inset shows the statistics of events analyzed for the case at . (Pairwise comparisons were performed with the two-sided t-test: ). In the main graph, error bars are reported as the first (bottom bar) and third (upper bar) quartile of the dataset.
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Figure 3—source data 1
Numerical data displayed in panels (e), (f) and (g).
- https://cdn.elifesciences.org/articles/105397/elife-105397-fig3-data1-v1.zip
Figure 4
Shape function.
On the left, we see a graphical representation of the sixfold shape function (see Equation 2 for more details) for a generic irregular polygon. On the right (black six-legged star) the phase and magnitude of for the same cell.
Figure 5
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 . (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 . (c) Velocity field approximated close to defect core: Stream density plot of the velocity field Equation 1. It exhibits a clear, local, convergent-extension flow pattern in the vicinity of the quadrupolar radius . In all plots, the black disk corresponds to the radius of the quadrupole. Our analytical solution is valid outside the disk.
Figure 6
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 (), but they are slowed down instead by negative activity ().
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Figure 6—source data 1
All positions of topological defects for all iterations needed to reproduce the defect annihilation trajectory plots in this figure.
- https://cdn.elifesciences.org/articles/105397/elife-105397-fig6-data1-v1.zip
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Figure 6—source data 2
All positions of topological defects for all iterations needed to reproduce the defect annihilation trajectory plots in this figure.
- https://cdn.elifesciences.org/articles/105397/elife-105397-fig6-data2-v1.zip
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Figure 6—source data 3
All positions of topological defects for all iterations needed to reproduce the defect annihilation trajectory plots in this figure.
- https://cdn.elifesciences.org/articles/105397/elife-105397-fig6-data3-v1.zip
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Figure 6—source data 4
All positions of topological defects for all iterations needed to reproduce the defect annihilation trajectory plots in this figure.
- https://cdn.elifesciences.org/articles/105397/elife-105397-fig6-data4-v1.zip
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Figure 6—source data 5
All positions of topological defects for all iterations needed to reproduce the defect annihilation trajectory plots in this figure.
- https://cdn.elifesciences.org/articles/105397/elife-105397-fig6-data5-v1.zip
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Figure 6—source data 6
All positions of topological defects for all iterations needed to reproduce the defect annihilation trajectory plots in this figure.
- https://cdn.elifesciences.org/articles/105397/elife-105397-fig6-data6-v1.zip
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Figure 6—source data 7
All positions of topological defects for all iterations needed to reproduce the defect annihilation trajectory plots in this figure.
- https://cdn.elifesciences.org/articles/105397/elife-105397-fig6-data7-v1.zip
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Figure 6—source data 8
All positions of topological defects for all iterations needed to reproduce the defect annihilation trajectory plots in this figure.
- https://cdn.elifesciences.org/articles/105397/elife-105397-fig6-data8-v1.zip
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Figure 6—source data 9
All positions of topological defects for all iterations needed to reproduce the defect annihilation trajectory plots in this figure.
- https://cdn.elifesciences.org/articles/105397/elife-105397-fig6-data9-v1.zip
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Figure 6—source data 10
All positions of topological defects for all iterations needed to reproduce the defect annihilation trajectory plots in this figure.
- https://cdn.elifesciences.org/articles/105397/elife-105397-fig6-data10-v1.zip
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Figure 6—source data 11
All positions of topological defects for all iterations needed to reproduce the defect annihilation trajectory plots in this figure.
- https://cdn.elifesciences.org/articles/105397/elife-105397-fig6-data11-v1.zip
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Figure 6—source data 12
All positions of topological defects for all iterations needed to reproduce the defect annihilation trajectory plots in this figure.
- https://cdn.elifesciences.org/articles/105397/elife-105397-fig6-data12-v1.zip
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Figure 6—source data 13
All positions of topological defects for all iterations needed to reproduce the defect annihilation trajectory plots in this figure.
- https://cdn.elifesciences.org/articles/105397/elife-105397-fig6-data13-v1.zip
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Figure 6—source data 14
All positions of topological defects for all iterations needed to reproduce the defect annihilation trajectory plots in this figure.
- https://cdn.elifesciences.org/articles/105397/elife-105397-fig6-data14-v1.zip
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Figure 6—source data 15
All positions of topological defects for all iterations needed to reproduce the defect annihilation trajectory plots in this figure.
- https://cdn.elifesciences.org/articles/105397/elife-105397-fig6-data15-v1.zip
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Figure 6—source data 16
All positions of topological defects for all iterations needed to reproduce the defect annihilation trajectory plots in this figure.
- https://cdn.elifesciences.org/articles/105397/elife-105397-fig6-data16-v1.zip
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Figure 6—source data 17
All positions of topological defects for all iterations needed to reproduce the defect annihilation trajectory plots in this figure.
- https://cdn.elifesciences.org/articles/105397/elife-105397-fig6-data17-v1.zip
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Figure 6—source data 18
All positions of topological defects for all iterations needed to reproduce the defect annihilation trajectory plots in this figure.
- https://cdn.elifesciences.org/articles/105397/elife-105397-fig6-data18-v1.zip
Videos
Video 1
Backflow velocity field (vector plot) generated during the unbinding of an active, hexatic defect quadrupole (color plot, magnitude of order parameter).
Video 2
Imaginaty part of hexatic order parameter (color plot) during the unbinding of an active, hexatic defect quadrupole.
Additional files
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MDAR checklist
- https://cdn.elifesciences.org/articles/105397/elife-105397-mdarchecklist1-v1.pdf
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Source code 1
C code used for the multiphase field model simulations.
- https://cdn.elifesciences.org/articles/105397/elife-105397-code1-v1.zip
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Collective epithelial migration mediated by the unbinding of hexatic defects
eLife 14:RP105397.
https://doi.org/10.7554/eLife.105397.3
