Bidirectional molecular transport shapes cell polarization in a two-dimensional model of eukaryotic chemotaxis

Chemotacting eukaryotic cells can sense shallow gradients of chemoattractants and respond by assuming an asymmetric shape with well-defined front and back regions. Such a striking polarization phenomenon is produced largely through the interconversions and interactions between several cellular components, including Rac GTPase (Rac), phosphoinositide 3-kinase (PI3K), tensin homology protein (PTEN), phosphatidylinositol 3,4,5-trisphosphate (PI(3,4,5)P3) and phosphatidylinositol (4,5)-bisphosphate (PI(4,5)P2). Here, we developed a mathematical model of cell polarization by exploring bidirectional molecular transport that arose from phosphoinositides (PIs) and Rac-mediated feedback loops. We assumed a static gradient of activated Rac derived from an external signal field as the internal trigger signal. The evolution of PI(3,4,5)P3 and PI(4,5)P2 along with PI3K and PTEN that act as activator and inhibitor, respectively, were described by a pair of coupled transient reaction-diffusion equations. The entire system was solved using a Lattice-Boltzmann method with an embedded Monte-Carlo method to track the stochastic translocation behaviors of discrete PI3K/PTEN molecules. We first showed that, upon a graded external stimulus, the Rac to PI(3,4,5)P3 cascade exhibited a short range positive-feedback loop, while the PTEN to PI(4,5)P2 cascade contributed another long range negative-feedback loop, which dominated the "forward" and "backward" molecular transport, respectively. Second, polarization was governed by the ratio of [PI3K] to [PTEN], and manifested a switch-like behavior. Third, with a uniform stimulus, spontaneous polarization could occur in PTEN-deficient cells.