  Understanding the mechanisms responsible for the formation and growth of FtsZ
polymers and their subsequent formation of the $Z$-ring is important for
gaining insight into the cell division in prokaryotic cells. In this work, we
present a minimal stochastic model that qualitatively reproduces {\it in vitro}
observations of polymerization, formation of dynamic contractile ring that is
stable for a long time and depolymerization shown by FtsZ polymer filaments. In
this stochastic model, we explore different mechanisms for ring breaking and
hydrolysis. In addition to hydrolysis, which is known to regulate the dynamics
of other tubulin polymers like microtubules, we find that the presence of the
ring allows for an additional mechanism for regulating the dynamics of FtsZ
polymers. Ring breaking dynamics in the presence of hydrolysis naturally induce
rescue and catastrophe events in this model irrespective of the mechanism of
hydrolysis.
