Modeling FtsZ ring formation in the bacterial cell-anisotropic aggregation via mutual interactions of polymer rods

The cytoskeletal protein FtsZ polymerizes to a ring structure (Z ring) at the inner cytoplasmic membrane that marks the future division site and scaffolds the division machinery in many bacterial species. FtsZ is known to polymerize in the presence of GTP into single-stranded protofilaments. In vivo, FtsZ polymers become associated with the cytoplasmic membrane via interaction with the membrane-binding proteins FtsA and ZipA. The FtsZ ring structure is highly dynamic and undergoes constantly polymerization and depolymerization processes and exchange with the cytoplasmic pool. In this theoretical study, we consider a scenario of Z ring self-organization via self-enhanced attachment of FtsZ polymers due to end-to-end interactions and lateral interactions of FtsZ polymers on the membrane. With the assumption of exclusively circumferential polymer orientations, we derive coarse-grained equations for the dynamics of the pool of cytoplasmic and membrane-bound FtsZ. To capture stochastic effects expected in the system due to low particle numbers, we simulate our computational model using a Gillespie-type algorithm. We obtain ring- and arc-shaped aggregations of FtsZ polymers on the membrane as a function of monomer numbers in the cell. In particular, our model predicts the number of FtsZ rings forming in the cell as a function of cell geometry and FtsZ concentration. We also calculate the time of FtsZ ring localization to the midplane in the presence of Min oscillations. Finally, we demonstrate that the assumptions and results of our model are confirmed by 3D reconstructions of fluorescently-labeled FtsZ structures in E. coli that we obtained.