Continuum Theory for Planar Cell Polarity

Planar Cell Polarity (PCP), characterized by asymmetric localization of proteins at the cell membrane within the epithelial plane, plays essential roles in embryonic development and physiological functions. The significance of PCP can be appreciated by the outcomes of its failure in the form of defects in neural tube formation, tracheal malfunctions, organ shape misregulation, hair follicle misalignment etc. Extensive experimental works on PCP in fruit fly Drosophila melanogaster have classified the proteins involved in PCP into a core module, acting locally by inter-cellular protein interactions, and a global module, responsible for the alignment of cell polarities with that of the tissue axis. Despite the involvement of different molecular players the asymmetric localization of the proteins of the two modules on cell membrane primarily involves inter-cellular dimer formations. We have developed a continuum model of the localization of PCP proteins on the cell membrane and its regulation via intra- and inter-cellular protein-protein interactions. We have identified the conditions for the asymmetric protein localization, or PCP establishment, for uniform and graded protein expression levels in the tissue. We have found that in the absence of any tissue level expression gradient polarized state of the tissue does not arise. However, in the presence of tissue-level expression gradients of proteins the polarized state remains stable. We have also looked at the influence of the loss of PCP proteins from a select region of the tissue on the polarization of the cells outside of that region. This continuum theory of planar cell polarity can be coupled with active-matter hydrodynamics to study cell flows and their regulation by genetic machinery.