Mathematical modeling of spatiotemporal protein localization patterns in C. crescentus bacteria: A mechanism for asymmetric FtsZ ring positioning.

We examine the localization patterns of ParA, ParB, PopZ, and MipZ, which are key division proteins in C. crescentus bacteria. While Par and PopZ proteins have been implicated in the physical segregation of the replicated chromosome, MipZ dimers control the placement of the cell division plane by preventing FtsZ proteins from assembling into a Z-ring. MipZ proteins generate bipolar gradients that are sensitive to Par protein localization, however, it is not understood how the MipZ gradient is shaped so as to allow for the correct Z-ring placement during asymmetric cell division in C. crescentus. In this paper, we develop and analyze a mathematical model that incorporates the known interactions between Par, PopZ, and MipZ proteins and use it to test mechanisms for MipZ gradient formation. Using our model, we show that gradient-dependent ParB advection velocities in conjunction with a ParA polar recycling mechanism are sufficient to maintain a robust new pole-directed ParA dimer gradient during segregation. A "saturation of binding site" hypothesis limiting access of ParA and MipZ to the ParB complex is then necessary and sufficient to generate time-averaged bipolar MipZ protein gradients with minima that are skewed toward ParA gradient peaks at the new pole, in agreement with data. By analyzing reduced versions of the model, we show the existence of oscillatory ParA localization regimes provided that cytoplasmic PopZ oligomers interact with ParA and ParA is over-expressed. We use our model to study mechanisms by which these protein patterns may simultaneously direct proper chromosome segregation and division site placement in C. crescentus.