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The ability of cells to sense spatial gradients of chemoattractant factors governs the development of complex eukaryotic organisms. Cells exposed to shallow chemoattractant gradients respond with strong accumulation of the enzyme phosphatidylinositol 3-kinase (P13K) and its D3-phosphoinositide product (PIP3) on the plasma membrane side exposed to the highest chemoattractant concentration, whereas PIP3-degrading enzyme PTEN and its product PIP2 localize in a complementary pattern. Such an early symmetry-breaking event is a mandatory step for directed cell movement elicited by chemoattractants, but its physical origin is still mysterious. Here, we propose 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 reaction-diffusion lattice model that describes P13K and PTEN enzymatic activity, recruitment to the plasma membrane, and diffusion of their phosphoinositide products, we show that the effective enzyme-enzyme interaction induced by catalysis and diffusion introduces an instability of the system toward 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 naturally. The model is robust with respect to order- of-magnitude variations of the parameters. [ABSTRACT FROM AUTHOR]