Analysis of spontaneous emergence of cell polarity with delayed negative feedback

Cell polarity refers to spatial differences in the shape and structure of cells, which leads to the generation of diverse cell types playing different roles in biological processes. Cell polarization usually involves the localization of some specific signaling molecules to a proper location of the cell membrane. Recent studies proposed that delayed negative feedback may be important for maintaining the robustness of cell polarization and the observed oscillating behavior of signaling cluster. However, the fundamental mechanisms for achieving cell polarization under negative feedback remain controversial. In this paper, we formulate the cell polarization system as a non-local reaction diffusion equation with positive and delayed negative feedback loops. Through the Turing stability analysis, we identify the parameter conditions, including the range of the time delay constant, for achieving cell polarization without any inhomogeneous spatial cues. Also, our numerical results support that by controlling the length of the time delay in negative feedback and the magnitude of positive feedback, the oscillating behavior of signaling cluster can be observed in our simulations.