Pearling Instability in Mass-Conserving Pattern-Forming Systems: from Reaction-Diffusion to Active-Matter Models

The Min system is a protein complex involved in cell division in Escherichia coli. Its dynamics arise from the interplay between biochemical reactions and diffusion, which can drive the system out of its homogeneous state and lead to self-organized pattern formation via a diffusion-driven instability mechanism. While in vivo the Min proteins typically exhibit pole-to-pole oscillations that ensure symmetric cell division, in vitro reconstituted systems display a rich variety of stationary patterns, including stripes, dots, and foam-like structures. These patterns have been successfully and quantitatively reproduced using theoretical models based on reaction-diffusion equations. We study a minimal three-component mass-conserving reaction-diffusion model inspired by the Min system. The model consists of an inactive cytosolic species corresponding to MinD-ADP, an active cytosolic species corresponding to MinD-ATP, and a membrane-bound density representing all membrane associated protein, including both bound MinD and transient MinD-MinE complex involved in detachment. The dynamics conserve total mass and incorporate nucleotide exchange together with membrane attachment and detachment processes. The resulting phase diagram, controlled by the total protein mass and the nucleotide exchange rate, reproduces the stationary patterns observed in vitro. In particular, variations of the total mass can induce a transition from stripe-like to dot-like structures. The main goal of the project is to characterize the stability of localized stripe solutions and to understand the pearling instability, through which an extended stripe becomes unstable and breaks up into a sequence of aligned dots. This mechanism provides insight into the selection and robustness of localized patterns in mass-conserving reaction-diffusion systems. Finally, we discuss how the theoretical framework developed can be extended to other pattern-forming systems with conserved dynamics exhibiting localized structures, in particular Active Model B-.