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.