Making waves

The surfaces of many biological systems exhibit intricate structures and patterns. Think of the stripes of zebras and tigers, the geometric patterns on sea shells, the coloring of reptiles and fish, or fingerprints. Since the beginning of mankind, people have sought explanations for these patterns and physics offers appropriate concepts for providing them.

Whereas the patterns mentioned above form during development and remain essentially static, others are continuously maintained by dynamic processes, for example, the stripes of a zebra fish [1]. This becomes apparent when these structures are perturbed and recover with time. Alan Turing proposed a generic mechanism based on reactions and diffusion to explain their appearance [2]. It involves at least two chemical species with distinct diffusion constants, which is notably realized in systems with local excitation and global inhibition (LEGI) [3].

Reaction-diffusion systems are not limited to generate static patterns but can also cause traveling waves as was discussed already by Turing in his original work [2]. Whereas his example, namely, the beating of spermatozoan flagella is now attributed to mechanical feedback between bending of the flagellum and molecular motors [4], there are biological patterns that result from bona fide Turing-like mechanisms. The most prominent example is arguably the Min system in Escherichia coli: the two proteins MinD and MinE shuttle between the two poles of the rod-shaped bacterium, where they inhibit cell division, which is thus directed to the cell center [5]. As an interesting twist to common reaction-diffusion systems, however, the molecular reactions of the proteins are essentially restricted to conformational changes that are accompanied by translocations from the cytoplasm, i.e., the cell interior, to the cytoplasmic membrane, i.e., the inner surface of the bacterial hull. In short, MinD first bind the cellular fuel Adenosine-Triphosphate and then adhere to the cytoplasmic membrane. There they recruit MinE to the membrane, which in turn trigger the detachment of MinD into the cytoplasm after which also MinE are released from the membrane. In addition to theoretical analysis [6], in vitro studies [7, 8] as well as patterns in bacteria [9] of unusual shape or size provide ample evidence that this core mechanism underlies the formation of the Min-protein patterns in E. coli.

Typically, the emergence of traveling wave patterns in Turing systems requires some special conditions and is studied for specific systems. As Levine and Kessler now show, however, all systems with two globally conserved molecular species and sufficiently different corresponding diffusion constants are prone to present a traveling-wave instability [10]. In case of the Min system these conditions are satisfied as the numbers of MinD and MinE molecules are separately conserved [11] and because the mobility of membrane-bound and cytoplasmic molecules differ by orders of magnitude [12]. In general, the presence of two globally conserved quantities implies the existence of two marginally stable modes with wave-number k = 0. Under appropriate conditions, there is then a band of unstable oscillating modes extending up to a finite critical wave number. For system lengths L smaller than the wavelength corresponding to the critical wave number, the homogenous state is stable. As the system size is increased, the homogenous state undergoes a (supercritical) bifurcation towards a traveling wave with one peak. Upon further increase of the system length, the number of peaks increases.

Living cells make ample use of exchanging molecules between the cytoplasm and the plasma membrane, often as a part of signaling cascades. As a consequence of the instability reported by Levine and Kessler one may thus expect further discoveries of intracellular protein waves. It could also be interesting to go beyond pure reaction-diffusion systems for studying the impact of globally conserved quantities on the spontaneous formation of traveling waves. This applies notably to active matter [13]. As an example, the actin cytoskeleton, which is a network of linear polymers and molecular motors that is involved in cell motility and division, is known to readily self-organize into waves. Whether the principle described in [10] is relevant for this case remains to be seen. In any case, it seems safe to say that it will be helpful for understanding and possibly making (nonlinear) waves.