Abstract
Ordered polarity alignment of cell populations plays vital roles in biology, such as in hair follicle alignment and asymmetric cell division. Although cell polarity is uniformly oriented along a tissue axis in many tissues, its mechanism is not well understood. In this paper, we propose a theoretical framework to understand the 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 the contacting surfaces between them. Using a perturbation method under the assumption of weak coupling between cells, we derive a reduced model in which polarity of each cell is described by only one variable. Essential dynamical properties including the effects of cell shape, coupling heterogeneity, external signal and noise can be clarified analytically. In particular, we show that the anisotropicity of the system, such as oriented cell elongation and axial asymmetry in the coupling strength, can serve as a global cue that drives the uniform orientation of cell polarity along a certain axis. Our study bridges the gap between detailed and phenomenological models, and it is expected to facilitate the study of polarity dynamics in various nonequilibrium systems.
Highlights
Ordered patterns are ubiquitous in nature and have been of central importance in various disciplines[1,2,3]
The generic dynamical properties of cell polarity alignment are examined by the derivation and analysis of a reduced model for coupled reaction–diffusion systems
As the model is manageable, essential dynamical properties including the effects of cell shape, external signal and noise can be clarified analytically, which have only been studied numerically in previous works using detailed models[6,8,9]
Summary
Our reaction–diffusion model given by equation (2) consists of N × M partial differential equations, where N and M are the numbers of cells and variables in each cell. For such a model, both analytical and numerical treatments are difficult. The phase φi(t) of Xi(θi, t) is defined such that Xi(θi, t) converges to XS(θi − φ) as t → ∞ in the unperturbed system. The eigenfunctions of zero eigenfunctions are and † are denoted by denoted by Y0 and Z0, Yi. e(.θ, ) aYn0d=Z (θ†Z) These eigenfunctions are assumed to form a complete orthonormal system and are normalised as
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