Lateral inhibition-induced pattern formation controlled by the size and geometry of the cell.

Pattern formation in development biology is one of the fundamental processes by which cells change their functions. It is based on the communication of cells via intra- and intercellular dynamics of biochemicals. Thus, the cell is directly involved in biochemical interactions. However, many theoretical approaches describing biochemical pattern formation have usually neglected the cell's role or have simplified the subcellular process without considering cellular aspects despite the cell being the environment where biochemicals interact. On the other hand, recent experimental observations suggest that a change in the physical conditions of cell-to-cell contact can result in a change in cell fate and tissue patterning in a lateral inhibition system. Here we develop a mathematical model by which biochemical dynamics can be directly observed with explicitly expressed cell structure and geometry in higher dimensions, and reconsider pattern formation by lateral inhibition of the Notch-Delta signaling pathway. We explore how the physical characteristic of cell, such as cell geometry or size, influences the biochemical pattern formation in a multi-cellular system. Our results suggest that a property based on cell geometry can be a novel mechanism for symmetry breaking inducing cell asymmetry. We show that cell volume can critically influence cell fate determination and pattern formation at the tissue level, and the surface area of the cell-to-cell contact can directly affect the spatial range of patterning.