Size matters for nonlinear (protein) wave patterns

Pattern formation and selection are fundamental, omnipresent principles in nature-from small cells up to geological scales. In E. coli bacteria, for example, self-organized pole-to-pole oscillations of Min proteins-resembling a short standing wave-ensure correct positioning of the cell division site. The same biochemical reaction leads to traveling protein waves on extended membranes in in vitro experiments. Are these seemingly contradictory observations of system-spanning importance? We show that a transition of nonlinear traveling wave patterns to reflection-induced standing waves in short systems is a generic and robust phenomenon. It results from a competition between two basic phenomena in pattern formation theory. We confirm the generic findings for the cell-biological Min reaction and for a chemical reaction-diffusion system. These standing waves show bistability and adapt to varying system lengths similar as pole-to-pole oscillations in growing E. coli. Our generic results highlight key functions of universal principles for pattern formation in nature.