Abstract
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.
Highlights
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
We show that nonlinear traveling waves inevitably change into reflection-induced standing waves in sufficiently short, confined systems. Since this generic phenomenon relies on basic universal principles of pattern formation, we explore it at first within a minimal model for nonlinear traveling waves
As figures 4(a) and (b) show, the qualitative behavior of nonlinear waves in both of these models is very similar to the generic complex Swift– Hohenberg (CSH) model: in sufficiently strong confinement, traveling wave patterns inevitably change into reflection-induced standing waves
Summary
A variety of fascinating patterns emerges spontaneously in a wealth of living or inanimate driven systems [1–13]. They explore, for instance, the important functions patterns fulfill: self-organized patterns in biology guide size sensing [6], positioning of protein clusters [7], self-driven morphogenesis [8] and communication between species [10] They enhance heat transport in fluid systems [3, 11] and are the basis of successful survival strategies for vegetation in water-limited systems [12–14]. The resulting system-spanning properties can be transferred to related phenomena in nature: in the Min system, e.g., traveling waves form by coordinated attachment and detachment of Min proteins from the membrane This protein system originates from E. coli bacteria where it plays an important role in the cell division process [31–33]: inside the rod-shaped E. coli bacteria, oscillating proteins shuttle between the two cell poles. A deeper understanding of generic properties of nonlinear waves in confinement will help to reconcile these seemingly contradictory observations
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