Domain coarsening in eukaryotic directional sensing

The cells of multicellular organisms are endowed with a chemical compass of amazing sensitivity, formed as a result of billions of years of evolution. Concentration differences of the order of a few percent in attractant chemicals from side to side are sufficient to induce a chemical polarization of the membrane leading to cell migration towards the signal source. It has been realized recently that this early polarization process is the result of a phase-separation instability in a well characterized network of diffusion-controlled chemical reactions. We give a universal description of this early symmetry breaking process. Our description implies the existence of two clearly separated polarization regimes depending on the presence or absence of an anisotropic component in the activation pattern, and the existence of a sensitivity threshold for the anisotropic component. In particular, we find that the polarization time t_ε in the presence of an anisotropic extracellular signal depends on the anisotropy degree ε through the power law t_ε α ε², and that in a cell of radius R there should exist a threshold value ε_th α R^-1 for the smallest detectable anisotropy. Our results are in agreement with existing experimental data and explain the recent observation of a threshold in the degree of detectable anisotropy.