  Ordered polarity alignment of a cell population plays a vital role in
biology, such as in hair follicle alignment and asymmetric cell division. Here,
we propose a theoretical framework for the understanding of generic dynamical
properties of polarity alignment in interacting cellular units, where each cell
is described by a reaction-diffusion system and the cells further interact with
one another through their proximal surfaces. The system behavior is shown to be
strongly dependent on geometric properties such as cell alignment and cell
shape. Using a perturbative method under the assumption of weak coupling
between cells, we derive a reduced model in which each cell is described by
just one variable, the phase. The reduced model resembles an XY model but
contains novel terms that possesses geometric information, which enables the
understanding of the geometric dependencies as well as the effects of external
signal and noise. The model is simple, generic, and analytically and
numerically tractable, and is therefore expected to facilitates studies on
cellular polarity alignment in various nonequilibrium systems.
