Statistical and Biological Physics

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Rethinking pattern formation in reaction–diffusion systems

Jacob Halatek and Erwin Frey

flat-field_normalComplex systems in biology, chemistry and physics are in general nonlinear and far from equilibrium, intricately coupled and heterogeneous. Why then do they not simply collapse, run out of control or become chaotic?

The canonical theory of pattern formation deals with nonlinear systems that are near a global equilibrium state and where nonlinear interactions have a stabilizing effect. However, in the real world this is rarely the case: collective effects in biological, ecological, or economic systems, can lead to sudden, dramatic changes in the dynamics (“catastrophes”) once the system reaches a “tipping point”, e.g. division and differentiation of cells in developing organisms, extinction of species, outbreaks of diseases, the emergence of migrating insect swarms, or crashes in financial markets. Despite the potential for such catastrophic events, most complex systems operate in a well-behaved manner despite being heterogeneous and far from equilibrium. Mechanisms for the maintenance and stabilization order in such systems are largely elusive.

In our work, we present a theoretical framework for self organized pattern formation in systems driven by mass conserving interactions (such as chemical reactions). The theory overcomes the limitations of previous approaches by decomposing complex systems into collections of small, coupled compartments. The dynamical state of the system is characterized by the local equilibria of all compartments which are determined by the amount of globally conserved mass in each compartment. We find that diffusive coupling generically leads to mass redistribution between compartments and thereby to dynamic shifting of local equilibria. Thus, moving local equilibria scaffold the dynamic evolution of patterns.

We apply this framework to study models for the regulation of cell polarity and cell division and find that moving local equilibria conceptually unify Turing instabilities and seemingly unrelated excitability phenomena that have been proposed as alternative mechanisms. Moreover, we find a new theoretical explanation for the emergence of chemical turbulence that is based on dynamic destabilization of local equilibria due to mass redistribution. Strikingly, our findings show that chemical turbulence is not the end of order, as suggested by the canonical theory. Rather, we find that order can emerge from turbulence by driving the system further from global equilibrium. The underlying mechanisms is based on the endogenous (self-organized) spatiotemporal control of local destabilization, which is conceptually similar to the action of a pacemaker that restores normal heart dynamics by inducing local perturbations.

These findings indicate that local equilibria and their dynamics constitute elementary concepts in a theory of complex dynamical systems driven by mass-conserving interactions.

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