  We propose a model for cell polarization as a response to an external signal
which results in a system of PDEs for different variants of a protein on the
cell surface and interior respectively. We study stationary states of this
model in certain parameter regimes in which several reaction rates on the
membrane as well as the diffusion coefficient within the cell are large. It
turns out that in suitable scaling limits steady states converge to solutions
of some obstacle type problems. For these limiting problems we prove the onset
of polarization if the total mass of protein is sufficiently small. For some
variants we can even characterize precisely the critical mass for which
polarization occurs.
