A mathematical model of ParA filament-mediated chromosome movement in Caulobacter crescentus

Caulobacter crescentus uses the dynamic interactions between ParA and ParB proteins to segregate copies of its circular chromosome. In this paper, we develop two mathematical models of the movement of the circular chromosome of this bacterium during division. In the first model, posed as a set of stochastic differential equations (SDE), we propose that a simple biased diffusion mechanism for ParB/ParA interactions can reproduce the observed patterns of ParB and ParA localization in the cell. The second model, posed as a set of nonlinear partial differential equations, is a continuous treatment of the problem where we use results from the SDE model to describe ParB/ParA interactions and we also track ParA monomer dynamics in the cytoplasm. For both models, we show that if ParB complexes bind weakly and nonspecifically to ParA filaments, then they can closely track and move with the edge of a shrinking ParA filament bundle. Unidirectional chromosome movement occurs when ParB complexes have a passive role in depolymerizing ParA filaments. Finally, we show that tight control of ParA filament dynamics is essential for proper segregation.