  Organoids are in vitro cellular collectives from which brain-like, or
gut-like, or kidney-like structures emerge. To make quantitative predictions
regarding the morphology and rheology of a cellular collective in its initial
stages of development, we construct and study a three-dimensional vertex model.
In such a model, the cells are represented as deformable polyhedrons with cells
sharing faces such that there are no gaps between them, otherwise known as
confluent. In a bulk model with periodic boundary conditions, we find a
rigidity transition as a function of the target cell shape index $s_0$ with a
critical value $s_0^*=5.39\pm0.01$. For a confluent cellular collective with a
finite boundary, and in the presence of lateral extensile and in-plane, radial
extensile deformations, we find a significant boundary-bulk effect that is
one-cell layer thick. More specifically, for lateral extensile deformations,
the cells in the bulk are much less aligned with the direction of the lateral
deformation than the cells at the boundary. For in-plane, radial deformations,
the cells in the bulk exhibit much less reorientation perpendicular to the
radial direction than the cells at the boundary. In other words, for both
deformations, the bulk, interior cells are topologically-protected from the
deformations, at least over time scales much slower than the timescale for
cellular rearrangements and up to reasonable amounts of strain. Our results
provide an underlying mechanism for some observed cell shape patterning in
organoids. Finally, we discuss the use of a cellular-based approach to
designing organoids with new types of morphologies to study the intricate
relationship between structure and function at the multi-cellular scale for
example.
