Geometry of T1 transitions in epithelia

Abstract

The flows of tissues of epithelial cells often involve T1 transitions. These neighbour exchanges are irreversible rearrangements crossing an energy barrier. Here, by an exact geometric construction, I determine this energy barrier for general, isolated T1 transitions dominated by line tensions. I~show how deviations from regular cell packing reduce this energy barrier, but find that line tension fluctuations increase it on average. By another exact construction, I prove that the nonlinear tensions in vertex models of tissues also resist T1 transitions. My results thus form the basis for coarse-grained understanding of cell neighbour exchanges for continuum descriptions of epithelia.

Figures (4)

• Figure 1: T1 transitions. (a) Example of a two-dimensional epithelial tissue, the Drosophila wing disc epithelium, taken from Ref. dye21. Lines are cell boundaries. Scale bar: $10\,\text{\textmu m}$. (b) Schematic of a T1 transition: before the T1 transition, the "top" and "bottom" cells are neighbours; after the transition, the "left" and "right" cells are neighbours. (c) Generic energy landscape of a T1 transition, plotted against the T1 transition coordinate $L$, the signed length of the central edge. The energy landscape has a cusp at the T1 transition point $L=0$. The energy minimum for $L<0$ is at $L=-\mathcal{L}$, where $\mathcal{L}>0$. This and the energy $\mathcal{E}$ at the T1 transition define the energy barrier $\mathcal{E}_\text{b}$. The energy landscape for $L>0$ is similar. Shape insets: T1 transition and definition of $L$.
• Figure 2: Geometry of an isolated T1 transition dominated by (homogeneous) line tensions. (a) Plot of an isolated T1 transition: The outer vertices (red) are fixed as the central edge formed by the inner vertices (blue) undergoes a T1 transition. These outer vertices define a quadrilateral with diagonals $\mathcal{D}_1,\mathcal{D}_2$ that are at an angle $\theta$ to each other. (b) Ensembles of isolated T1 transitions: (i) In the equal-area ensemble, the area $\mathcal{A}$ of the quadrilateral is fixed. (ii) In the equal-energy ensemble, the energy $\mathcal{E}$ of the transition configuration is held constant. (c) Geometric construction of the positions $F_1,F_2$ of the inner vertices minimising the line-tension energy for fixed outer vertices $A,B,C,D$. See text for further explanation. (d) Calculation of the energy barrier $\mathcal{E}_\text{b}$ and the distance $\mathcal{L}$ from the T1 transition based on this geometric construction. See text for further explanation. (e) Contour plot of $\mathcal{L}$ against $\theta$ and the asymmetry $\Delta=|\mathcal{D}_1-\mathcal{D}_2|$ in the equal-area ensemble (i). The red mark corresponds to a regular hexagonal lattice $\theta=\theta_6$, $\Delta=0$. On the (blue) line $\theta=2\pi/3$, $\mathcal{L}=0$. (f) Analogous plot for the equal-energy ensemble (ii). (g) Plot of the energy barrier $\mathcal{E}_\text{b}$ against $\mathcal{L}^2$, for the equal-area ensemble (i) and for different values of $\Delta$. The dotted line shows the asymptotic relation $\mathcal{E}_\text{b}/\mathcal{L}^2\sim 3(\mathcal{D}_1+\mathcal{D}_2)/(8\mathcal{D}_1\mathcal{D}_2)$. (h) Analogous plot for the equal-energy ensemble (ii).
• Figure 3: Geometry of an isolated T1 transition dominated by (homogeneous) nonlinear line tensions. (a) Construction of the positions $G_1,G_2$ minimising the nonlinear line-tension energy for fixed outer vertices $A,B,C,D$. See text for further explanation. (b) Contour plot of the length $\mathcal{L}$ of the central edge against $\theta$ and the asymmetry $\Delta=|\mathcal{D}_2-\mathcal{D}_1|$ [Fig. \ref{['fig2']}] in the equal-area ensemble [Fig. \ref{['fig2']}, case (i)]. The red mark corresponds to a regular hexagonal lattice $\theta=\theta_6$, $\Delta=0$. The blue mark is the single point $\Delta=0$, $\theta=\pi$ at which ${\mathcal{L}=0}$. (c) Analogous contour plot in the equal-energy ensemble [Fig. \ref{['fig2']}, case (ii)]. See text for further explanation.
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Theorems & Definitions (2)

• Lemma
• Lemma