Mathematical modeling of planar cell polarity: principles, approaches, and open questions.

Planar cell polarity represents a fundamental mechanism by which cells within epithelial sheets align their orientation, enabling coordinated tissue morphogenesis and function. Disruption of PCP leads to developmental defects and disease, highlighting the importance of understanding its establishment and maintenance. While experimental studies have identified key protein molecules that drive PCP, mathematical and computational modeling have become indispensable in connecting molecular interactions to tissue-level outcomes. Over the last couple of decades, diverse approaches, such as agent-based models, Cellular Potts frameworks, Petri nets, continuum theories, and phenomenological models, have been developed to capture distinct aspects of PCP dynamics. These frameworks allow systematic exploration of nonlinear feedback, intracellular and intercellular signaling, and the influence of geometry and mechanics, and noise on polarization. This review summarizes these mathematical and computational developments in PCP modeling, emphasizing methodological assumptions, insights gained, and open challenges. By bridging experiment and theory, PCP modeling advances both mechanistic understanding and predictive capacity for tissue-scale organization.