Mathematical Modeling of Sub-Cellular Asymmetry of Fat-Dachsous Heterodimer for Generation of Planar Cell Polarity

Planar Cell Polarity (PCP) is an evolutionarily conserved characteristic of animal tissues marked by coordinated polarization of cells or structures in the plane of a tissue. In insect wing epithelium, for instance, PCP is characterized by en masse orientation of hairs orthogonal to its apical-basal axis and pointing along the proximal-distal axis of the organ. Directional cue for PCP has been proposed to be generated by complex sets of interactions amongst three proteins - Fat (Ft), Dachsous (Ds) and Four-jointed (Fj). Ft and Ds are two atypical cadherins, which are phosphorylated by Fj, a Golgi kinase. Ft and Ds from adjacent cells bind heterophilically via their tandem cadherin repeats, and their binding affinities are regulated by Fj. Further, in the wing epithelium, sub-cellular levels of Ft-Ds heterodimers are seen to be elevated at the distal edges of individual cells, prefiguring their PCP. Mechanisms generating this sub-cellular asymmetry of Ft-Ds heterodimer in proximal and distal edges of cells, however, have not been resolved yet. Using a mathematical modeling approach, here we provide a framework for generation of this sub-cellular asymmetry of Ft-Ds heterodimer. First, we explain how the known interactions within Ft-Ds-Fj system translate into sub-cellular asymmetry of Ft-Ds heterodimer. Second, we show that this asymmetric localization of Ft-Ds heterodimer is lost when tissue-level gradient of Fj is flattened, or when phosphorylation of Ft by Fj is abolished, but not when tissue-level gradient of Ds is flattened or when phosphorylation of Ds is abrogated. Finally, we show that distal enrichment of Ds also amplifies Ft-Ds asymmetry. These observations reveal that gradient of Fj expression, phosphorylation of Ft by Fj and sub-cellular distal accumulation of Ds are three critical elements required for generating sub-cellular asymmetry of Ft-Ds heterodimer. Our model integrates the known experimental data and presents testable predictions for future studies.