  Spatial organization of proteins in cells is important for many biological
functions. In general, the nonlinear, spatially coupled models for
protein-pattern formation are only accessible to numerical simulations, which
has limited insight into the general underlying principles. To overcome this
limitation, we adopt the setting of two diffusively coupled, well-mixed
compartments that represents the elementary feature of any pattern -- an
interface. For intracellular systems, the total numbers of proteins are
conserved on the relevant timescale of pattern formation. Thus, the essential
dynamics is the redistribution of the globally conserved mass densities between
the two compartments. We present a phase-portrait analysis in the phase-space
of the redistributed masses that provides insights on the physical mechanisms
underlying pattern formation. We demonstrate this approach for several
paradigmatic model systems. In particular, we show that the pole-to-pole Min
oscillations in Escherichia coli are relaxation oscillations of the MinD
polarity orientation. This reveals a close relation between cell polarity
oscillatory patterns in cells. Critically, our findings suggest that the design
principles of intracellular pattern formation are found in characteristic
features in these phase portraits (nullclines and fixed points). These features
are not uniquely determined by the topology of the protein-interaction network
but depend on parameters (kinetic rates, diffusion constants) and distinct
networks can give rise to equivalent phase portrait features.
