Polarity-driven laminar pattern formation by lateral-inhibition in 2D and 3D bilayer geometries

Fine-grain patterns produced by juxtacrine signalling have previously been studied using static monolayers as cellular domains. However, analytic results are usually restricted to a few cells due to the algebraic complexity of non-linear 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 and behaviour. 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 formation of cell polarity by activator localization. Critically, the patterning conditions are independent of the precise dynamical details, thus the framework allows for generality in understanding the influence of cellular geometry and signal polarity on patterning using 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 can 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 that has the potential to initiate bilayer stratification in a developing mammary organoid.