Physics

Dynamical systems theory

Instability

Mechanics

Diffusion

Non-equilibrium thermodynamics

Biology

Quantum mechanics

Pattern formation

Reaction–diffusion system

Statistical physics

Classical mechanics

Genetics

Autocatalytic reaction

Thermodynamics

Nature Physics

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. Canonical pattern formation relies on a system being close to an instability and stabilized by nonlinearities — but real systems seldom conform to these conditions. A new theory solves the problem by recasting it in terms of moving local equilibria.

Rethinking pattern formation in reaction–diffusion systems