  Fine-grain patterns produced by juxtacrine signalling, have been studied
using static monolayers as cellular domains. Unfortunately, analytical results
are restricted to a few cells due to the algebraic complexity of nonlinear
dynamical systems. Motivated by concentric patterning of Notch expression
observed in the mammary gland, we combine concepts from graph and control
theory to represent cellular connectivity. The resulting theoretical framework
allows us to exploit the symmetry of multicellular bilayer structures in 2D and
3D, thereby deriving analytical conditions that drive the dynamical system to
form laminar patterns consistent with the formulation of cell polarity.
Critically, the conditions are independent of the precise dynamical details,
thus the framework allows for the utmost generality in understanding the
influence of cellular geometry on patterning in lateral-inhibition systems.
Applying the analytic conditions to mammary organoids suggests that intense
cell signalling polarity is required for the maintenance of stratified
cell-types within a static bilayer using a lateral-inhibition mechanism.
Furthermore, by employing 2D and 3D cell-based models, we highlight that the
cellular polarity conditions derived from static domains have the capacity to
generate laminar patterning in dynamic environments. However, they are
insufficient for the maintenance of patterning when subjected to substantial
morphological perturbations. In agreement with the mathematical implications of
strict signalling polarity induced on the cells, we propose an adhesion
dependent Notch-Delta biological process which has the potential to initiate
bilayer stratification in a developing mammary organoid.
