Self-organization of the MinE ring in subcellular Min oscillations

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.

💡 Research Summary

The paper presents a minimal theoretical framework to understand how the MinE ring self‑organizes during the subcellular oscillations of MinD and MinE in rod‑shaped Escherichia coli. By focusing exclusively on the dynamics at the tip of a depolymerizing MinD filament, the authors derive a steady‑state description that isolates the essential physics of MinE recruitment, independent of the full oscillatory cycle. The model assumes that MinE molecules detach from the filament tip, diffuse either along the long axis of the cell (1‑dimensional diffusion) or in the three‑dimensional space surrounding the filament tip, and then rebind to the tip with a rate k_rebind. The key dimensionless parameter is the scaled cell shape (\tilde{r}=R/L) (cell radius divided by cell length). Small (\tilde{r}) values correspond to a geometry where axial (1D) diffusion dominates, whereas large (\tilde{r}) values give prominence to radial (3D) diffusion.

Two qualitatively distinct MinE‑ring profiles emerge from the analysis. In the “strong” or plateau‑like regime (small (\tilde{r})), MinE diffuses efficiently along the cell length, leading to a broad, nearly uniform occupancy of the filament tip. The tip occupation fraction θ saturates near unity, making the depolymerization speed v_dep essentially independent of (\tilde{r}) and only weakly dependent on the total MinE/MinD stoichiometry. Consequently, the filament’s shrinkage rate is robust against changes in cell shape or modest fluctuations in protein concentrations.

In the “weak” or cusp‑like regime (large (\tilde{r})), radial diffusion limits the supply of MinE to the tip. The occupancy θ remains low and varies strongly with both (\tilde{r}) and the MinE‑to‑MinD ratio. Here, v_dep is highly sensitive to these parameters: a modest increase in cell radius or a decrease in MinE concentration can dramatically slow filament depolymerization. The model predicts that the width of the weak ring scales with (\tilde{r}), while the strong ring’s width is set primarily by the diffusion constant along the cell axis.

When the authors map realistic E. coli dimensions onto the model, they find that the bacterium operates near the transition between the two regimes ( (\tilde{r}) ≈ 0.1–0.2). This positioning implies that natural variations in cell shape, growth stage, or protein expression can shift the system from a robust to a highly tunable depolymerization regime, providing a plausible mechanism for the observed sensitivity of division site positioning to cell geometry.

Beyond the in‑vivo context, the framework explains a puzzling experimental observation from in‑vitro ATPase assays. In open‑geometry setups (e.g., MinD filaments anchored on a surface), the time required for a MinE ring to form can be tens of seconds to minutes, especially at low MinE concentrations. The model shows that this delay arises solely from the limited rebinding flux of MinE to the filament tip, without invoking any cooperative activation of MinE. Hence, the long lag in ATPase activity reported in earlier studies can be accounted for by simple diffusion‑limited rebinding.

Overall, the study provides a concise, analytically tractable description of MinE‑ring formation that captures the essential dependence on cell geometry, protein stoichiometry, and rebinding kinetics. It bridges the gap between detailed stochastic simulations and experimental phenomenology, offering clear predictions that can be tested by manipulating cell shape (e.g., using filamentous mutants), altering MinE/MinD expression ratios, or varying the diffusion environment in vitro. The work deepens our mechanistic understanding of how spatial self‑organization emerges from minimal physical principles in bacterial cell division.