Single-Mode Turbulence in Pattern-Forming Protein Systems

Protein pattern formation often relies on proteins that cycle between a cyctosolic bulk and a membrane at which they undergo molecular interactions. On a flat membrane, this cycling can lead to intriguing protein patterns including spiral waves as well as more irregular dynamics such as chemical turbulence. While theoretical approaches have been able to reproduce various experimentally observed protein patterns, the underlying mechanisms for pattern selection remain poorly understood. Motivated by the bacterial Min protein system, we present a spatially reduced reaction-diffusion model to study pattern selection in protein systems with bulk-membrane coupling. Remarkably, we find that already a single-mode instability can lead to turbulent dynamics at the onset of pattern formation. Further away from this onset, we observe a transition from turbulent to coherent patterns, which can be explained on the basis of diffusively coupled local equilibria. Our study yields insights into a novel route to chaos for a widespread class of mass-conserving reaction-diffusion systems with bulk-boundary coupling.