Many eukaryotic cell types share the ability to migrate directionally in response to external chemoattractant gradients. This ability is central in the development of complex organisms, and is the result of billion years of evolution. Cells exposed to shallow gradients in chemoattractant concentration respond with strongly asymmetric accumulation of several signaling factors, such as phosphoinositides and enzymes. This early symmetry-breaking stage is believed to trigger effector pathways leading to cell movement. Although many factors implied in directional sensing have been recently discovered, the physical mechanism of signal amplification is not yet well understood. We have proposed that directional sensing is the consequence of a phase ordering process mediated by phosphoinositide diffusion and driven by the distribution of chemotactic signal. By studying a realistic computational model that describes enzymatic activity, recruitment to the plasmamembrane, and diffusion of phosphoinositide products we have shown that the effective enzyme-enzyme interaction induced by catalysis and diffusion introduces an instability of the system towards phase separation for realistic values of physical parameters. In this framework, large reversible amplification of shallow chemotactic gradients, selective localization of chemical factors, macroscopic response timescales, and spontaneous polarization arise.