Pattern Formation Beyond Turing: Physical Principles of Mass-Conserving Reaction--Diffusion Systems

Intracellular protein patterns govern essential cellular functions by dynamically redistributing proteins between membrane-bound and cytosolic states, conserving their total numbers. This review presents a theoretical framework for understanding such patterns based on mass-conserving reaction--diffusion systems. The emergence, selection, and evolution of patterns are analyzed in terms of mass redistribution and interface motion, resulting in mesoscale laws of coarsening and wavelength selection. A geometric phase-space perspective provides a conceptual tool to link local reactive equilibria with global pattern dynamics through conserved mass fluxes. The Min protein system of \emph{Escherichia coli} provides a paradigmatic example, enabling direct comparison between theory and experiment. Successive model refinements capture both the robustness of pattern formation and the diversity of dynamic regimes observed \emph{in vivo} and \emph{in vitro}. The Min system thus illustrates how to extract predictive, multiscale theory from biochemical detail, providing a foundation for understanding pattern formation in more complex and synthetic systems.