Modeling epithelial tissue and cell deformation dynamics using a viscoelastic slab sculpted by surface forces

During morphogenesis, epithelial monolayers actively alter their shape to create future body parts of the animal; this makes the epithelium one of the most active and critical structures in early animal development. Even though epithelial cells exist and move in three dimensions, mathematical models frequently describe them as merely two-dimensional. However, recent imaging technology has begun to reveal pertinent dynamics in the third dimension of the tissue. With the importance of the third dimension in mind, we have developed a self-sculpting, three-dimensional model of epithelia whose dynamics are driven by active forces on its surface. We present a first, fundamental study for a reduced version of epithelia that investigates how surface forces affect its internal dynamics. Our model captures the 3D slab-like geometry of epithelia, viscoelasticity of tissue response, fluid surroundings, and driving from active surface forces. We represent epithelial tissue as a thick slab, a 3D continuum comprised of a Stokes fluid with an extra viscoelastic stress. Employing this model, we present both analytical and numerical solutions of the system and make quantitative predictions about cell shapes, cell dynamics, and the tissue's response to surface force in a three-dimensional setting. In particular, we elucidate the initiation of ventral furrow invagination and T1 transitions in Drosophila embryogenesis. In the former, we demonstrate the importance of fluid and geometric surroundings to drive invagination. In the latter, we show the limitations of surface forces alone to drive T1 transitions.