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 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 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 r, the occupation fraction of MinE at the MinD filament tip is saturated and hence the depolymerization speed do not depend strongly on r. Conversely, for weak E-rings both r and the MinE to MinD stoichiometry strongly control the tip occupation and hence the depolymerization speed. MinE rings 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 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.