Fluid model of epithelial morphogenesis: Oscillations and structuring

We have revised a mathematical model of epithelial morphogenesis by Belintsev et al. (1987) (BB model) taking into account the oscillatory nature of morphogenesis and stability analysis (Cherdantsev, 2014). Following the BB model in considering the feedback control of cell shape changes by mechanical forces, we modify it to represent epithelial surface movements observed in different types of Metazoan gastrulation. Basing on these observations, we argue that the epithelial surface movement is that of an incompressible fluid supplemented by a positive feedback between the movement and spreading of the surface flow. Dipole interactions between sources and sinks of surface energy provide a single mechanism both of short-ranged and long-ranged regulation of collective cell and surface movements whose basic variables are the space averaged epithelial surface curvature and lateral pressure within the epithelial surface flux negatively related to its velocity. The short-ranged activation means a movement of the surface up to the lateral pressure gradient under non-linear feedback control of the surface flexure. The break of this feedback with equalization of the surface curvature is sufficient for a self-restriction of the movement spreading. Owing to bistable interdependence between the lateral pressure and epithelial surface curvature, we get a generic oscillatory contour in which the same region oscillates being alternately a sink and source of the surface flow. The opposite phase oscillations of the lateral pressure and curvature allow for both directional propulsion of the surface through the same region and spatial differentiation based on parametric differences between the large-scaled regions that correspond to sources and sinks of the surface.