Dynamics of Wound Closure in Living Nematic Epithelia

TL;DR

This work models wound closure in a living nematic epithelium as a two-dimensional incompressible active nematic with a circular hole, incorporating free-surface boundary conditions and an active stress $\alpha\mathbf{Q}$ that captures bulk tissue activity. A perturbative approach in the activity parameter $\epsilon_{\alpha}$ couples a passive axisymmetric base solution to nematic-driven corrections, predicting that contractile bulk stresses accelerate wound closure and elongate the wound along the far-field nematic axis under parallel anchoring, while extensile stresses slow closure. The model reproduces experimentally observed wound anisotropy and its correlation with bulk nematic order, and reveals the emergence and evolution of $n=0,2,4$ shape modes and $-\tfrac{1}{2}$ topological defects that annihilate as healing proceeds. Together with supplementary results, the study highlights the significant role of bulk active stresses and nematic length scales in epithelial re-epithelialisation, offering a continuum framework applicable to other epithelial wound-closure contexts.

Abstract

We study theoretically the closure of a wound in a layer of epithelial cells in a living tissue after damage. Our analysis is informed by our recent experiments observing re-epithelialisation in vivo of Drosophila pupae. On time and length-scales such that the evolution of the epithelial tissue near the wound is well captured by that of a 2D active fluid with local nematic order, we consider the free-surface problem of a hole in a bounded region of tissue, and study the role that active stresses far from the hole play in the closure of the hole. For parallel anchored nematic order at the wound boundary (as we observe in our experiments), we find that closure is accelerated when the active stresses are contractile and slowed down when the stresses are extensile. Parallel anchoring also leads to the appearance of topological defects which annihilate upon wound closure.

Figures (8)

• Figure 1: (a): Time-lapse movies of the developing Drosophila pupal epithelium (18 hrs APF) indicate that unwounded tissue has nematic order. Elongation and alignment of each epithelial cell is described by a traceless shape tensor, $q$, the magnitude of which $||q|| = \frac{1}{2} \mathrm{Tr} (q^2)$ describes anisotropy in shape and $\phi \in [-\pi/2, \pi/2)$ describes the direction of nematic alignment relative to the $x$-axis. (b) Epithelial cells align, on average, along the PD axis of the wing (roughly aligning with the $x$-axis of our imaging setup), the degree of elongation increasing as the tissue develops. (c): Snapshot of tissue $\sim 8$ minutes post wounding. Here, the cellular alignment, $\phi^\prime$ is measured relative to the radial direction in a polar basis centred on the wound as illustrated in the inset: red (blue) cells indicate alignment along $\pm \hat{\mathrm{e}}_\theta$ ($\pm \hat{\mathrm{e}}_r$). (d): Distribution of $|\phi^\prime|$'s for cells inside annulus illustrated in panel (c). We find that cells close to the wound initially tend to be aligned along the $\pm \hat{\mathrm{e}}_\theta$ direction. (e): Illustration of the model setup and (f): the active contractile and extensile force dipoles included in the model that coarse-grain to generate a stress $\sim \alpha \mathbf{{Q}}$.
• Figure 2: (a): Nematic texture (left) and active force (right) that appears in the Stokes equation and drives the flow. The blue squares mark positions of $-\frac{1}{2}$ topological defects. (b): Snapshots of nematic texture throughout closure show the motion of defects towards the origin, resulting in a topologically 'healed' final state. (c): Flow surrounding wound boundary for identical initial conditions but varying values of $\epsilon_\alpha$. (d): Wound areas throughout closure for different values of $\epsilon_\alpha$. Panels (a-d) plotted with $\Lambda=0.1$, $R_1^0(0)=0$, $R_2^0(0)=0$. (e): Experimental observation of correlation between wound anisotropy and degree of nematic order in the surrounding epithelium. $||\langle q_{...}\rangle ||$ corresponds to average magnitude of $q$ tensor for cells/wounds, averaging over all cells/wounds in first $\sim 10$ minutes (5 frames) of each time-lapse. Points are different wounds, with colour indicating value of $\theta_\mathrm{wound}$ -- illustrated in (f) -- the angle between the tissue axis and the major axis of the wound polygon averaged over initial $\sim 10$ minutes.
• Figure S1: Shape and orientation of each cell is described by traceless tensor $q_i$. (a) Examples of segmented cells with colour code indicating magnitude of $q^{xx}$ ($q^{xy}$) in the left (right) panel. Orange and cyan circles in left panel indicate cells that align strongly along the $x$ and $y$ axes (and so have large positive/negative values of $q^{xx}$) respectively. Orange and cyan circled cells in the right panel align strongly along the lines $y = x$ and $y=-x$ respectively. (b) Segmented cells surrounding a wound, colour coded by magnitude of $q^{rr}$. The orange circle highlights a cell elongated along the $\hat{\mathrm{e}}_r$ polar unit vector, and the cyan circle highlights a cell elongated along the $\hat{\mathrm{e}}_\theta$ polar unit vector.
• Figure S2: Example of shape mode dynamics for the inner boundary, $R_1$ (a-c) and outer boundary $R_2$ (d-f). Plotted using $\Lambda=0.1$ with initial conditions $R_1^0(0)=1, R_2^0(0)=20, \xi_i^k(0)=0 \forall i, k$ and $\eta_i^k(0)=0.05 \forall i,k$.
• Figure S3: Cartoon illustrating effect of many contractile force dipoles acting along the length of each nematogen/cell, representing the state of active stress in the tissue at the microscopic level.