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Spatial organization of proteins in cells is important for many biologicalfunctions. In general, the nonlinear, spatially coupled models forprotein-pattern formation are only accessible to numerical simulations, whichhas limited insight into the general underlying principles. To overcome thislimitation, we adopt the setting of two diffusively coupled, well-mixedcompartments that represents the elementary feature of any pattern -- aninterface. For intracellular systems, the total numbers of proteins areconserved on the relevant timescale of pattern formation. Thus, the essentialdynamics is the redistribution of the globally conserved mass densities betweenthe two compartments. We present a phase-portrait analysis in the phase-spaceof the redistributed masses that provides insights on the physical mechanismsunderlying pattern formation. We demonstrate this approach for severalparadigmatic model systems. In particular, we show that the pole-to-pole Minoscillations in Escherichia coli are relaxation oscillations of the MinDpolarity orientation. This reveals a close relation between cell polarityoscillatory patterns in cells. Critically, our findings suggest that the designprinciples of intracellular pattern formation are found in characteristicfeatures in these phase portraits (nullclines and fixed points). These featuresare not uniquely determined by the topology of the protein-interaction networkbut depend on parameters (kinetic rates, diffusion constants) and distinctnetworks can give rise to equivalent phase portrait features.

Diffusive coupling of two well-mixed compartments elucidates elementary principles of protein-based pattern formation