Active Curved Polymers form Vortex Patterns on Membranes
A critical component of the bacterial cell division machinery is a contractile polymer structure, called Z-ring, which performs cytokinesis. Rather than being a single, closed polymer ring, the Z-ring is composed of multiple, overlapping FtsZ filaments. However, how self-organization into this structure occurs, remains unknown and is subject to extensive research. Recent experiments  of FtsZ on a supported lipid membrane have shown that FtsZ polymerizes into curved polymer filaments, which effectively move via treadmilling. These filaments collect into dynamic rings which resemble the Z-ring in size and structure. The effect of chirality and directed motion on the collective dynamics remains, however, poorly understood. In our study, we provide a quantitative analysis how these systems self-organize into different patterns.
In our paper we model systems of active FtsZ polymers on the membrane, by studying self-propelled particles that move along circular, chiral paths. Our work predicts self-organization into vortex structures, which recall the ring patterns observed by Loose and Mitchison. The results are consolidated by the phenomenological agreement of two different approaches: we employ Brownian dynamics simulations and a kinetic Boltzmann approach to study these systems on microscopic and mesoscopic length scales, respectively. We obtain a phase diagram featuring a confined parameter region of steady, dense swirls. In the Boltzmann framework, we determine the nature and stability of patterns and quantify all corresponding phase transitions. Notably, the onset of pattern formation is described by an extended complex Ginzburg-Landau equation that represents an interesting mathematical problem in its own right.
Applying our theory to treadmilling FtsZ polymers - the main motivation of our study - shows that steric interactions already suffice to obtain a vortex phase. The generic and robust existence of such a vortex phase is important, as it might facilitate Z-ring formation.
In a broader perspective our study extends previous work on systems of self-propelled particles where soliton waves accompany the onset of collective motion. We find that curved motion inhibits these wave patterns in favor of vortices. Interestingly, our phase analysis is independent of curvature and therefore also valid for straight-moving particles.