Pulling Helices inside Bacteria: Imperfect Helices and Rings

We study steady-state configurations of intrinsically-straight elastic filaments constrained within rod-shaped bacteria that have applied forces distributed along their length. Perfect steady-state helices result from axial or azimuthal forces applied at filament ends, however azimuthal forces are required for the small pitches observed for MreB filaments within bacteria. Helix-like configurations can result from distributed forces, including coexistence between rings and imperfect helices. Levels of expression and/or bundling of the polymeric protein could mediate this coexistence.

Figure 1

Bifurcation diagram showing perfect helices with $Z\equiv z^{\prime }$ vs scaled axial force $F$ at filament tips. Stable branches are solid lines, unstable branches are dashed. Red squares are at the critical points $\pm (F^{*},Z^{*})=\pm [4/(3\sqrt{3}),1/\sqrt{3}]$. The experimental range of $Z$ observed for MreB homologs is indicated by the shaded region. The inset shows the bifurcation diagram for $Z$ vs scaled azimuthal force $N$.

Figure 2

Equilibrium configurations $Z\equiv z^{\prime }$ vs $s/L=\tilde{s}/\alpha$ for uniformly distributed azimuthal forces $\tau (s)$ following Eq. (8). (a) Fixed scaled length $\alpha =20$ and scaled azimuthal forces $N_0=0.75$, 1.5, 4, 6, and 12 [blue (dark gray) curves]. The real-space helical configuration for $N_0=3$ [medium-weight blue (dark gray) curve] is shown in red (light gray). The $\alpha =\infty$ limits for the same range of azimuthal forces are shown by black dashed lines. Significant $\alpha$ dependence is seen both at the free tip and at the transition to $Z=0$. (b) Fixed azimuthal force $N_0=4$ and various scaled lengths $\alpha =5$, 10, 20, 40, 80 [blue (dark gray) curves], and $\infty$ [thick red (light gray) curve]. The left inset shows the pitch distributions of the configurations with finite $\alpha$. The right inset, plotting $Z_{\alpha }-Z_{\infty }$ vs $s/R$, shows that the boundary layer, near $s=0$, has a width of order $R$.

Figure 3

In the limit of long filaments, with $\alpha =\infty$, the range of filament length $s/L$ and scaled azimuthal force $N$ that have rings ($Z=0$) are shown above the thick red line, while open helical configurations are seen below the line. No rings are seen below $N^{*}=2$. For shorter filaments, this phase diagram still applies approximately. Shown are the lines separating $Z<0.01$ and $Z>0.01$ for $\alpha =5$, 10, 20, 40, and 80, using Eq. (8). For shorter filaments (as shown here for $\alpha =5$), the entire short filament is ring-shaped at larger forces.