  We model the self-organization of the MinE ring that is observed during
subcellular oscillations of the proteins MinD and MinE within the rod-shaped
bacterium {\it Escherichia coli}. With a steady-state approximation, we can
study the MinE-ring generically -- apart from the other details of the Min
oscillation. Rebinding of MinE to depolymerizing MinD filament tips controls
MinE ring formation through a scaled cell shape parameter $\tilde{r}$. We find
two types of E-ring profiles near the filament tip: a strong plateau-like
E-ring controlled by 1D diffusion of MinE along the bacterial length, or a weak
cusp-like E-ring controlled by 3D diffusion near the filament tip. While the
width of a strong E-ring depends on $\tilde{r}$, the occupation fraction of
MinE at the MinD filament tip is saturated and hence the depolymerization speed
do not depend strongly on $\tilde{r}$. Conversely, for weak E-rings both
$\tilde{r}$ and the MinE to MinD stoichiometry strongly control the tip
occupation and hence the depolymerization speed. MinE rings {\em in vivo} are
close to the threshold between weak and strong, and so MinD-filament
depolymerization speed should be sensitive to cell shape, stoichiometry, and
the MinE-rebinding rate. We also find that the transient to MinE-ring formation
is quite long in the appropriate open geometry for assays of ATPase activity
{\it in vitro}, explaining the long delays of ATPase activity observed for
smaller MinE concentrations in those assays without the need to invoke
cooperative MinE activity.
