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Rethinking pattern formation in reaction–diffusion systems


The present theoretical framework for the analysis of pattern formation in complex systems is mostly limited to the vicinity of fixed (global) equilibria. Here we present a new theoretical approach to characterize dynamical states arbitrarily far from (global) equilibrium. We show that reaction–diffusion systems that are driven by locally mass-conserving interactions can be understood in terms of local equilibria of diffusively coupled compartments. Diffusive coupling generically induces lateral redistribution of the globally conserved quantities, and the variable local amounts of these quantities determine the local equilibria in each compartment. We find that, even far from global equilibrium, the system is well characterized by its moving local equilibria. We apply this framework to in vitro Min protein pattern formation, a paradigmatic model for biological pattern formation. Within our framework we can predict and explain transitions between chemical turbulence and order arbitrarily far from global equilibrium. Our results reveal conceptually new principles of self-organized pattern formation that may well govern diverse dynamical systems.

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Fig. 1: Spatial compartmentalization of mass-conserving reaction–diffusion systems leads to the notion of moving local equilibria.
Fig. 2: In vitro Min protein patterns are slaved to stable local equilibria at onset.
Fig. 3: Local destabilization of Min protein patterns at onset leads to chemical turbulence.
Fig. 4: Control of local stability far from onset establishes spatiotemporal order.
Fig. 5: The in vitro Min protein system is an oscillatory medium.

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This research was supported by the German Excellence Initiative via the programme ‘NanoSystems Initiative Munich’ (NIM), and by the Deutsche Forschungsgemeinschaft (DFG) via project B02 within the SFB 1032 ‘Nanoagents for Spatio-Temporal Control of Molecular and Cellular Reactions’. The authors thank F. Brauns, C. Dekker and G. Pawlik for detailed feedback on the manuscript. The authors thank F. Brauns, J. Denk and D. Thalmeier for discussions and F. Brauns for his preliminary work on the CO–Pt system, which has significantly advanced our understanding of the mass-redistribution framework presented here.

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Authors and Affiliations



J.H. and E.F. conceived and performed the research. J.H. developed the analytical and numerical results. J.H. and E.F. discussed and analysed the results and wrote the manuscript.

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Correspondence to E. Frey.

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Supplementary information

Supplementary Information

Supplementary Notes, Supplementary Tables 1 and 2, Supplementary Figures 1–12, Supplementary References 1–18


Onset of chemical turbulence

Top: membrane profile of MinD during the transition from small amplitude standing waves to chemical turbulence for a control parameter (bulk height) h = 20µm. (Also, see main text Fig. 3b.) Bottom: temporal evolution of the system’s distribution in control space corresponding to the time step shown in the top wave profiles. (Also, see main text Fig. 3a.)

Emergence of standing wave order

Top: membrane profile of MinD at the onset of standing wave order for a control parameter (bulk height) h = 25µm. (Also, see main text Fig. 4c.) Bottom: temporal evolution of the system’s distribution in control space corresponding to the time step shown in the top wave profiles. (Also, see Supplementary Fig. 11b.)

Spiral wave patterns in a 3D box geometry

Spatiotemporal dynamics of membrane bound MinD in the simulation of the full 3D box geometry for a control parameter (bulk height) h = 100µm.

Chimera transition from standing to travelling waves

Top: Membrane profile of MinD during the chimera transition from from standing to travelling waves (light grey highlighted area) for a control parameter (bulk height) h = 33µm. (Also, see main text Fig. 5b.) Bottom: temporal evolution of the system’s distribution in control space corresponding to the time step shown in the top wave profiles. (Also, see main text Fig. 5e.) Data points corresponding to the spatial domain in the grey highlighted area (in the top wave profile) are plotted opaque to show the contraction of the distribution in control space to a well 67 defined cycle.

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Halatek, J., Frey, E. Rethinking pattern formation in reaction–diffusion systems. Nature Phys 14, 507–514 (2018).

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