Complex structures from patterned cell sheets

Abstract

The formation of three-dimensional structures from patterned epithelial sheets plays a key role in tissue morphogenesis. An important class of morphogenetic mechanisms relies on the spatio-temporal control of apical cell contractility, which can result in the localized bending of cell sheets and in-plane cell rearrangements. We have recently proposed a modified vertex model that can be used to systematically explore the connection between the two-dimensional patterns of cell properties and the emerging three-dimensional structures. Here we review the proposed modelling framework and illustrate it through the computational analysis of the vertex model that captures the salient features of the formation of the dorsal appendages during Drosophila oogenesis.This article is part of the themed issue 'Systems morphodynamics: understanding the development of tissue hardware'.

Keywords: cell mechanics; epithelial morphogenesis; vertex models.

Figure 1.

Examples of epithelial morphogenesis. (a) Schematic of a blastula-to-gastrula transition that involves localized bending of an epithelium. (b) Schematic of convergent extension during which cells intercalate, driving tissue elongation.

Figure 2.

Epithelial sheet and vertex model. (a) Schematic highlighting mechanical contributions of area elasticity and line tension terms in the vertex model (equation (2.1)). (b) Dependence of the cell side length l_e on the area elasticity coefficient μ_s for a uniform tissue at mechanical equilibrium. Solid (dashed) line: stable (unstable) equilibrium solutions. A representative flat homogeneous configuration is shown as an inset. (c) A representative configuration showing in-plane deformations due to spatial patterning of model parameters. (d) A representative configuration showing out-of-plane deformations due to patterning of model parameters. (e,f) Schematic of cell shape changes during bending. Each cell adopts a ‘wedge-like’ shape due to apical constriction, leading to the bending of an initially flat sheet.

Figure 3.

Two-dimensional model for three-dimensional epithelial deformations. (a) Schematic of an epithelial monolayer with thickness 2h. The two-dimensional model approximates the epithelial sheet by its midsurface (shown in black). (b) Two adjacent cells and their respective cell centres and unit normal in this model. The red line highlights a contractile segment that is offset by a distance h from the surface along the cell normals. Images are taken from [21]. (c) Schematic of a T1 transition event, in which a vanishing edge is resolved into a new edge reflecting new local cell connectivity (described in greater detail in electronic supplementary material, section S1).

Figure 4.

Dorsal appendage formation during Drosophila oogenesis. (a) Schematic of cell types in the developing Drosophila egg chamber and the fully formed eggshell. Blue highlights the roof cells, red the floor cells, yellow the midline cells and rest are main body cells. (bi–bv) Three-dimensional reconstructions of the apical surface of follicle cells at different time points during dorsal appendage formation in D. melanogaster. The flat primordium is transformed into a conical structure through a sequence of cell neighbour exchanges. The colour-coding of the cell contours in (b) is same as the colouring of the cell surface in (a). Images are taken from [39].

Figure 5.

Small appendage formation (the number of appendage-producing cells in the model is less than that present in the follicle cells in D. melanogaster). (a) Initial flat configuration highlighting different cell types present during appendage formation. Yellow cells indicate midline cells, grey main body cells, red floor cells and blue roof cells. (b) Schematic highlighting prepatterning of cell properties. The line tension coefficient for red edges σ_fm,0 (boundary of floor-midline cells) is larger than for the rest of the tissue. The contractile prepattern introduced to mimic apically constricting roof cells is shown in blue, for which Γ,h ≠ 0. (c) Plot of the system's energy as a function of time, relative to the energy of the initial homogeneous configuration. A magnified portion of the same plot and the equilibrium configuration are shown inset. (di–dix) Tissue configurations and associated floor cell connectivity diagram, which shows the cell centres with connectivity reflecting the actual configurations shown on top. Here, we are starting with the equilibrium solution for the prepatterning shown in figure 5b and introduce a position-dependent value of σ for red edges i.e. where θ denotes the angle that the corresponding edge midpoint makes with the reference line highlighted by θ = 0 (as shown in inset of figure 5e). Every image corresponds to the change in the connectivity of floor cells. (e) Plot of the system's energy as a function of time, relative to the energy of the initial homogeneous configuration. Parameter values used for the simulation: μ = 2.45, h =−0.5, Γ = 0.02. σ_fm,0 = 1.6, p = 6.0, G = 40.

Figure 6.

Model appendage formation (the number of appendage-producing cells in the model is similar to that present in the follicle cells in D. melanogaster). (a) Initial flat configuration highlighting different cell types present during appendage formation. (b) Schematic highlighting prepatterning of cell properties. For red edges σ = σ_fm,0 > 1; for grey edges σ = 0.1; otherwise σ = 1. For grey cells μ_s = 0.1; otherwise μ_s = 1. Offset contractile prepattern is shown in blue (Γ,h≠0). (ci–cv) Sequence of representative shapes that emerge during the formation of the appendages after introducing position-dependent value of σ for red edges (as described in figure 5e). Figures on the left in the sub-panel highlight the floor + roof + midline cells. Figures on the right in the same sub-panel highlight the floor cells only (above) or floor + roof cells (below). Viewpoints are chosen to highlight the increase in floor–floor cells' connectivity during the simulation. Parameter values used for the simulation: μ = 2.45, h = −0.5, Γ = 0.02, σ_fm,0 = 1.0, p = 8.0, G = 60.