Bifurcation analysis of a self-organizing signaling system for eukaryotic chemotaxis

Abstract

Phosphatidylinositol 3,4,5-trisphosphate ($\hbox {PIP}_3$) is a membrane lipid that works as a directional compass in migrating cells. Remarkably, $\hbox {PIP}_3$ shows both transient patterns and oscillatory patterns on the membrane, depending on experimental conditions (Arai et al. in Proc Natl Acad Sci 107:12399–12404, 2010). Here, we analyzed a reaction–diffusion model of the phosphatidylinositol signaling system that gives rise to transient pattern formation. Numerical bifurcation analysis showed that equilibrium solutions can be classified into uniform, unimodal and bimodal ones, among which the first and the second are stable for some parameter regions. We found that transient patterns of $\hbox {PIP}_3$ can be explained by the “ghost” after unimodal solutions disappear at a fold bifurcation. We further reduced the original PDEs to five-variable ODEs, considering only local and global concentrations. The bifurcation analysis of the reduced ODEs supports the above observation. Finally, we propose that trajectories of such transient patterns are determined by the phase space structure of the dynamical system.