A Parabolic Free Boundary Problem Arising in a Model of Cell Polarization
Anna Logioti, Barbara Niethammer, Matthias Röger, Juan J. L. Velázquez · SIAM Journal on Mathematical Analysis · 2021
The amplification of an external signal is a key step in direction sensing of biological cells. We consider a simple model for the response to a time-depending signal, which was previously proposed by the last three authors. The model consists of a bulk-surface reaction-diffusion model. We prove that in a suitable asymptotic limit the system converges to a bulk-surface parabolic obstacle--type problem. For this model and a reduction to a nonlocal surface equation we show an $L^1$-contraction property and, in the case of time-constant signals, the stability of stationary states.