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.
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Barbara Niethammer, Matthias Röger, Juan J.L. Velázquez, A bulk-surface reaction-diffusion system for cell polarization. Interfaces Free Bound. 22 (2020), no. 1, pp. 85–117DOI 10.4171/IFB/433