Mathematical Modeling of the Invagination of Epithelial Layers in Embryogenesis
Invagination of epithelial sheets is an important type of morphogenetic deformation. Primary
invagination during gastrulation in the sea urchin provides one of the simplest and best-studied examples.
The specific mechanisms of invagination remain unclear in spite of numerous observations. The problem of
plane-stress deformation of an initially circular layer exposed to a constant internal pressure is considered.
Active forces developed by cells are characterized by an active moment. The rheology of a layer is described
by a Maxwell-type viscoelasticity equation, which links the passive bending moment with the curvature of the
layer. The presence of a passive moment threshold below which bending is purely elastic is taken into account.
The active moment is defined as a function of coordinates and time that is nonzero in a certain limited region.
The function is assumed to gradually increase, reach a steady state, and then decline gradually. Both constant-
and alternating-sign spatial distributions of the active moment are considered. Numerical simulation
showed that among all of the considered variants a realistic sequence of shapes can only be obtained if the
layer is viscoelastic, there is a finite threshold for the passive bending moment, and the distribution of the
active moment is of an alternating-sign type. The sign of the active moment differs between the inner and
outer areas of the active region, tending to bend the sheet inward in the inner area and outward in the outer
area. This study made it possible to reach several conclusions on the nature of the macroscopic organization
of invagination and to outline avenues of research into the cellular mechanisms that are capable of developing
the corresponding forces.