Deciphering the interface laws of Turing foams

Protein pattern formation is central to the spatiotemporal self-organization of both prokaryotic and eukaryotic cells. Moreover, it is employed as a key spatial control system in the design of artificial cells. However, it remains unclear how the properties of the macroscopic, highly nonlinear reaction–diffusion patterns can be systematically linked to the underlying reaction network [1]. Here, we show that protein patterns are generically governed by an effective interfacial tension arising from cyclic steady-state currents of attachment and detachment at the interface. Angle laws for the non-equilibrium interface junctions follow that resemble but deviate systematically from the Neumann law in equilibrium phase separation. We furthermore recover generalized Plateau and von-Neumann laws for two-dimensional liquid foams in two-dimensional mesh patterns. This leads us to introduce “Turing foams,” which show generic behavior governed by the interplay of interfacial-tension-driven dynamics and interrupted coarsening, and that we observe experimentally in the in vitro Min protein system. Our theory offers a new ansatz to find principles of macroscopic self-organization in mass-conserving systems far from equilibrium.

[1] Halatek, J., Brauns, F. & Frey, E. Self-organization principles of intracellular pattern formation. Philos. Trans. R. Soc. B Biol. Sci. 373, 20170107 (2018).