| Abstract | The ability of eukaryotic cells to polarize is essential for their division, differentiation into distinct tissues, and migration. During polarization various polarity proteins segregate to form a distinct front and rear. To understand a mechanism for polarization we consider a simplified PDE model describing the interchange of a polarity protein, such as Rho or Rap family GTPase, between an active membrane-bound form and an inactive cytosolic form. An initial transient signal results in a traveling front of activation that stops at some point in the domain, representing segregation of the cell into front and back. Using phase plane methods and numerical continuation we analyze the transition from a spatially heterogeneous (pinned wave) to a spatially homogeneous steady state as the ratio of the diffusion coefficients of the two forms and the total amount of material in the domain is varied. We discover a second spatially heterogeneous solution that acts as a threshold for polarity establishment, and give biological interpretation for this phenomenon. |
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