Mathematical model for positioning the FtsZ contractile ring in Escherichia coli

During bacterial cytokinesis, a proteinaceous contractile ring assembles in the cell middle. The Z ring tethers to the membrane and contracts, when triggered, to form two identical daughter cells. One mechanism for positioning the ring involves the MinC, MinD and MinE proteins, which oscillate between cell poles to inhibit ring assembly. Averaged over time, the concentration of the inhibitor MinC is lowest at midcell, restricting ring assembly to this region. A second positioning mechanism, called Nucleoid Occlusion, acts through protein SlmA to inhibit ring polymerization in the location of the nucleoid. Here, a mathematical model was developed to explore the interactions between Min oscillations, nucleoid occlusion, Z ring assembly and positioning. One-dimensional advection-reaction-diffusion equations were built to simulate the spatio-temporal concentrations of Min proteins and their effect on various forms of FtsZ. The resulting partial differential equations were numerically solved using a finite volume method. The reduced chemical model assumed that the ring is composed of overlapping FtsZ filaments and that MinC disrupts lateral interactions between filaments. SlmA was presumed to break long FtsZ filaments into shorter units. A term was developed to account for the movement of FtsZ subunits in membrane-bound filaments as they touch and align with other filaments. This alignment was critical in forming sharp stable rings. Simulations qualitatively reproduced experimental results showing the incorrect positioning of rings when Min proteins were not expressed, and the formation of multiple rings when FtsZ was overexpressed.