  The plane of bacterial cell division must be precisely positioned. In the
bacterium Myxococcus xanthus, the proteins PomX and PomY form a large cluster,
which is tethered to the nucleoid by the ATPase PomZ and moves in a stochastic,
but biased manner towards midcell, where it initiates cell division.
Previously, a positioning mechanism based on the fluxes of PomZ on the nucleoid
was proposed. However, the cluster dynamics was analyzed in a reduced,
one-dimensional geometry. Here we introduce a mathematical model that accounts
for the three-dimensional shape of the nucleoid, such that nucleoid-bound PomZ
dimers can diffuse past the cluster without interacting with it. Using
stochastic simulations, we find that the cluster still moves to and localizes
at midcell. Redistribution of PomZ by diffusion in the cytosol is essential for
this cluster dynamics. Our mechanism also positions two clusters equidistantly
on the nucleoid. We conclude that a flux-based mechanism allows for cluster
positioning in a biologically realistic three-dimensional cell geometry.
