A minimal vertex model explains how the amnioserosa avoids fluidization during Drosophila dorsal closure

Abstract

Dorsal closure is a process that occurs during embryogenesis of Drosophila melanogaster. During dorsal closure, the amnioserosa (AS), a one-cell thick epithelial tissue that fills the dorsal opening, shrinks as the lateral epidermis sheets converge and eventually merge. During this process, both shape index and aspect ratio of amnioserosa cells increase markedly. The standard 2-dimensional vertex model, which successfully describes tissue sheet mechanics in multiple contexts, would in this case predict that the tissue should fluidize via cell neighbor changes. Surprisingly, however, the amnioserosa remains an elastic solid with no such events. We here present a minimal extension to the vertex model that explains how the amnioserosa can achieve this unexpected behavior. We show that continuous shrinkage of the preferred cell perimeter and cell perimeter polydispersity lead to the retention of the solid state of the amnioserosa. Our model accurately captures measured cell shape and orientation changes and predicts nonmonotonic junction tension that we confirm with laser ablation experiments.

Keywords: Drosophila dorsal closure; amnioserosa; epithelial tissue; morphogenesis; vertex model.

Fig. 1.

Experiment and vertex model for the amnioserosa during dorsal closure. (A) The geometry of the dorsal hole during early (Left), Middle (center), and late (Right) dorsal closure. Enlargements show tissue with selected cells, several of which ingress (highlighted by triangles). (B) We model the dorsal closure process as a quasistatic uniaxial deformation. The geometry of the model is shown at the beginning (Left), in the Middle (center, at 45% closure), and toward the end (Right, 80% closure) of the process. $\mathrm{\Delta }A(t)=\frac{A_0-A(t)}{A_0}$ is the fractional change in total AS area of the closure process, where $A_0$ is the AS area at the onset of dorsal closure. (C) An initial normal distribution of the preferred shape index of the model tissue (dashed red) with the standard deviation adjusted to be 0.45, leads to a distribution of the actual shape index after minimization (solid blue) that is in excellent agreement with the distribution of the experimentally observed shape index (solid black) at the beginning of dorsal closure. (D) Sketch of AS tissue regions included in model comparison (white center), with edge regions excluded (gray regions). (E) In the model, we reduce the preferred cell perimeter at a linear rate (blue) to capture the experimentally observed decrease of junction lengths (black). For comparison, we normalize the average perimeter by its value at the onset of the process. Inset: schematic representation of the reduction of cellular junction length and apical area during dorsal closure.

Fig. 2.

Results from experiment (black solid) and model (blue dashed). (A) A comparison of measured mean shape index $\overline{q}$ (blue curve) as a function of $\mathrm{\Delta }A(t)=\frac{A_0-A(t)}{A_0}$. Here $A_0$ is the AS area at the onset of dorsal closure and $A(t)$ is the area as it shrinks during dorsal closure, so that $\mathrm{\Delta }A(t)=0$ at onset. The red curve shows the model target shape index $q_0$. The shaded regions indicate the SD among 12 embryos (experiment) or 10 different initial configurations (model). (B) Comparison of cell to cell SD of the shape index (${\sigma }_q$) during dorsal closure. (C) Comparison of cell aspect ratio during dorsal closure. (D) Orientational order parameter ($\overline{Q}$) of the cells during dorsal closure. (E) Experimental initial junction recoil velocity (Left y-axis) of the vertices after performing laser ablation of the junction and predicted average cellular cortical tension ($\overline{{\tau }_J}$)(Right y-axis) of the model during dorsal closure. The boxplots represent data across three intervals of $\mathrm{\Delta }A$ ($\mathrm{\Delta }A<0.4$, $0.4\le \mathrm{\Delta }A<0.7$, $\mathrm{\Delta }A\ge 0.7$). Whiskers extend to the 5th and 95th percentiles, while the boxes delineate the interquartile range, and the horizontal lines within the boxes indicate the median values. An ANOVA followed by a post hoc Tukey’s HSD test was conducted to assess statistical significance (*$P<0.1$, **$P<0.05$). We performed and evaluated cuts of $N=97$ junctions. (F) Average initial recoil velocity of vertices after laser cutting as a function of junction straightness (ratio of the intervertex distance ($d_v$) to the junction length ($L$), see Inset) immediately before cutting. Junction recoil velocity is independent of junction straightness (fitted with the red dashed line) until $S=d_v/L\gtrsim 0.93$. The cross-over point at $d_v/L\approx 0.93$ marks the intersection of the red and blue dashed lines; the latter fits the data points in the gray-shaded region, indicating that the recoil velocity increases strongly and approximately linearly with junction straightness in this regime. (G) Comparison of experimental junction straightness (Left y-axis) and model cellular junction tension (Right y-axis) during dorsal closure.

Fig. 3.

Percolation analysis, shear modulus, and phase diagram of AS during dorsal closure. (A) Percolation analysis of tissue rigidity, showing the fraction of junctions with nonzero tension in the model (blue line) and experiment (black line). Additional lines represent isotropic deformation (red dashed), model without shrinking preferred cell perimeters (blue dashed), and percolation thresholds in x-direction (green solid) and y-direction (green dashed). (B) Shear modulus of the minimal model throughout dorsal closure. The blue line shows that the shear modulus is nonzero for the model with shrinking preferred cell perimeter, while the red line shows that the system fluidizes (the shear modulus vanishes) when the preferred perimeters are held constant. (C) Phase diagram in $\overline{q}$ vs. $\mathrm{\Delta }A$ space, illustrating solid (open black circles) and fluid (open gray squares) states. The solid-fluid transition is denoted by the black dashed line. Trajectories of both the experiment (green dashed line) and model (blue dashed line) remain within the solid phase throughout dorsal closure.