(Un)buckling mechanics of epithelial monolayers under compression

Abstract

When cell sheets fold during development, their apical or basal surfaces constrict and cell shapes approach the geometric singularity in which these surfaces vanish. Here, we reveal the mechanical consequences of this geometric singularity for tissue folding in a minimal vertex model of an epithelial monolayer. In simulations of the buckling of the epithelium under compression and numerical solutions of the corresponding continuum model, we discover an "unbuckling" bifurcation: At large compression, the buckling amplitude can decrease with increasing compression. By asymptotic solution of the continuum equations, we reveal that this bifurcation comes with a large stiffening of the epithelium. Our results thus provide the mechanical basis for absorption of compressive stresses by tissue folds such as the cephalic furrow during germband extension in Drosophila.

Figures (4)

• Figure 1: Mechanics of tissue folds. (a) Cephalic furrow in Drosophilavellutini23 with constricted triangular cell shapes. Arrows show the effective tissue compression from germband extension after the active formation of the furrow. Dotted lines outline one constricted cell. Image provided by Bruno Vellutini. Scale bar: 20 µ m. (b) Cartoon of buckling of a tissue under external compression: As the tissue folds, the initially rectangular cells become trapezoidal. The apical and basal sides of the tissue, and the tangent angle $\psi$ of the tissue midline are also shown. (c) Geometric singularity of constriction in a minimal vertex model: Cell shapes are isosceles trapezoids of respective apical, basal, and lateral sides $L_\text{a},L_\text{b},L_\ell$ and fixed area $a$, with equal apical and basal tensions $\Gamma$ and lateral tension $\Gamma_\ell$. As the sides bend, the cells become triangular, at which point their lateral sides cannot bend further.
• Figure 2: (Un)buckling of a compressed epithelial monolayer as its relative compression $D$ is increased, in a minimal vertex model (top row) and in the corresponding continuum limit (bottom row). (a) Flat configuration of the discrete monolayer before buckling. Red dots mark cell centres. Inset: definition of the aspect ratio $r$ from the undeformed shape of the monolayer. (b) At $D=D_\ast$, the monolayer buckles. As $D>D_\ast$ increases, so does the buckling amplitude, and the cell sides bend. (c) At $D=D_\triangle$, cells in the crest and the trough of the buckled shape become triangular, and, as $D>D_\triangle$ increases, fans of triangular cells (darker shading) expand from the crest and the trough. For still larger compressions $D>D_\text{bif}$, all cells are triangular and unbuckling is possible depending on the aspect ratio of the monolayer: (d) If $r>r_\text{bif}$, the buckling amplitude continues to increase with $D$. (e) If $r<r_\text{bif}$ however, at the same compression, the amplitude decreases with increasing $D$, and the triangular cells become taller and thinner. (f)--(j) Corresponding shapes in the continuum limit. The red line marks the midline of the cells, and the blue lines their apical and basal surfaces. Kinks in the apical or basal surfaces for $D>D_\triangle$ herald a fan of triangular cells (darker shading) at their crests and troughs.
• Figure 3: Unbuckling bifurcation. (a) Plot of the buckling amplitude $A$ against relative compression $D$ in the vertex model and for different cell numbers $N$, showing sharp unbuckling for small enough $N$. Parameter value for numerical calculations: $\ell_0=3$. (b) Corresponding plot for the continuum model, showing the unbuckling transition as the aspect ratio $r$ of the monolayer increases. A critical branch (black lines) at $r=r_\text{bif}$ separates buckling and unbuckling behaviour, bifurcating at $D=D_\text{bif}$. The upper branch is dashed to indicate steric self-intersection for $D>D_\text{bif}$ on this branch. The critical values $D_\ast,D_\triangle$ are highlighted on the $D$-axis for the critical branch $r=r_\text{bif}$. Triangles on branches mark $D=D_\triangle$. Parameter value: ${\ell_0=\sqrt{10}}$. Inset: behaviour of the critical branch $r=r_\text{bif}$ as $N$ and $\ell_0$ increase, showing that $D_\text{bif}\to D_0$; the dashed line shows the prediction $D_0\approx 0.21$ from asymptotic calculations. (c) Convergence $D_\text{bif}\to D_0$, $r_\text{bif}\to r_0$ as $N$ and $\ell_0$ increase: Numerical results (dot markers) match the asymptotic prediction (dashed line). (d) Plot of the fraction $f$ of triangular cells against $D$ for the critical branch $r=r_\text{bif}$ in panel (b), rising from $0$ for $D>D_\triangle$ and approaching $1$ as $D$ approaches $D_\text{bif}$. Inset: plot of $1-f$ against $D_\text{bif}-D>0$, confirming that the numerical results (dot markers) match the asymptotic calculations (dashed line). (e) Plot of the compressive force $\mu$ against relative compression $D$ for the critical branches $r=r_\text{bif}$ in panel (b), showing a large increase near the bifurcation at $D=D_\text{bif}$. The vertical arrow indicates the asymptotic orders of magnitude over which the monolayer stiffens near $D=D_\text{bif}$. Inset: same plot, zooming in on smaller $\mu$ and $D$ and showing the decrease of the apparent stiffness $\partial\mu/\partial D$ of the monolayer associated with the buckling at $D=D_\ast$, and its increase for $D>D_\triangle$ (triangular shape inset); the dotted line shows the slower increase of $\mu$ if the geometric singularity is ignored (self-intersecting shape inset).
• Figure 4: Mechanism of unbuckling. (a) Buckled shapes with all cells triangular: Each half of the buckled shape consists of two matched fans of $N/4$ triangular cells each. Inset: isosceles triangular cell with base angle $2\phi$, lateral side $\mathit{\Lambda}\ell_0$. (b) Plot of the relative compression $D$ against the semi-angle $\phi$, with minimum $D_\text{min}=D_\text{bif}$ at $\phi=\phi_\text{bif}$. For $D\gtrsim D_\text{bif}$, there are two solutions, $\phi_1<\phi_\text{bif}$, $\phi_2>\phi_\text{bif}$. (c) Plots of the energy $E$ against $\phi$: If $\partial E/\partial\phi(\phi_\text{bif})<0$, then $\phi_2$ is selected (buckling); if $\partial E/\partial\phi(\phi_\text{bif})>0$, then $\phi_1$ is selected (unbuckling).