Self-organization of the MinE protein 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 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: either a strong plateaulike E ring controlled by one-dimensional diffusion of MinE along the bacterial length or a weak cusplike E ring controlled by three-dimensional 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 does 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 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 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.

Figure 1

(Color online) Fractional occupancy of MinE on the MinD filament $\rho$ vs distance along the bacterial axis $z$ for the 3D stochastic model (continuous lines) and the 1D model (dashed lines) for parameters typical of E. coli: $L=2\mu \text{m}$, $R=0.5\mu \text{m}$, and ${\rho }_0=0.35$. The MinE binding parameter was ${\sigma }_3=0.3\mu {\text{m}}^3/\text{s}$, while for the 1D model $f=0.06$ was used. One MinD filament supports either (a) a strong plateaulike E ring for pitch $p=0.45\mu \text{m}$ or (b) a weak cusplike E ring for pitch $p=\infty$ (straight filament). The width $W$ of the strong E ring, given by $\rho \left(W\right)=\left(1+{\rho }_0\right)/2$, is indicated. The inset illustrates the cylindrical geometry of the 3D model, showing the underlying helical MinD filament with its depolymerizing tip at $z=0$. The helical pitch $p$ is indicated.

Figure 2

(Color online) MinE profile, characterized by $W/L$ and ${\rho }_{tip}$, as a function of the scaled aspect ratio $\tilde{r}$. (a) and (b) Straight filaments ($p=\infty$, ${\rho }_0=0.35$). 1D model results are shown with solid lines (using $f=0.06$); 3D stochastic results are indicated by symbols. Single filaments [$n=1$, green (light gray) data] for $L=1\mu \text{m}$ (◻), $L=2\mu \text{m}$ (○), and $L=3\mu \text{m}$ (△); multiple filaments [$n=2$ or 5, blue (dark gray ▽) data] for $L=1\mu \text{m}$. (c) and (d) Helical filaments ($n=1$, ${\rho }_0=0.35$) with $L/p=20$ (◻), 10 (○), 4 (△), and 0 (◇). For all these data, ${\sigma }_3=0.3\mu {\text{m}}^3/\text{s}$, and $R$ is varied to explore $\tilde{r}$. Similar results are obtained when ${\sigma }_3$ is varied.

Figure 3

(Color online) (a) and (b) MinE profile, as characterized by $W/L$ and ${\rho }_{tip}$ obtained by the 3D model (points) and the 1D model (lines, using $f=0.06$) as a function of $\tilde{r}$ for different values of stoichiometry; ${\rho }_0=0.2$ (◻, continuous lines), 0.35 (○, dashed lines), and 0.50 (△, dotted lines). For each stoichiometry, the same collapse as Figs. 2a, 2b is obtained: data are compiled for $n=1$, 2, 3, 4, and 5, $L=1$, 2, and $3\mu \text{m}$, $p=\infty$, ${\sigma }_3=0.3\mu {\text{m}}^3/\text{s}$, and $R$ varies to explore $\tilde{r}$. Similar results are obtained when ${\sigma }_3$ is varied.

Figure 4

(Color online) Transients and E-ring structure for an “inside-out” open geometry appropriate for in vitro experiments, where a MinD filament is tightly wound on the outside of a cylinder of small radius $\left(R=50\text{nm}\right)$ with open boundaries at $R=\infty$. (a) Evolution of ${\rho }_{tip}$ as a function of the number of depolymerization steps $N$ (measured in thousands) after the uniform initial conditions for ${\rho }_0$ equal to 0.8 (solid), 0.4 (long dash), 0.3 (short dash), and 0.2 (dotted); (b) steady state $\rho \left(z\right)$ as a function of axial distance $z$ along the helical axis for the same ${\rho }_0$.

Figure 5

(Color online) For the same inside-out in vitro geometry described in the previous figure. Cumulative ATPase activity $N\left(t\right)$ (measured in thousands of depolymerization steps) versus time $t$ for various ${\rho }_0$. Asymptotic behavior is plotted as thin dotted lines.