Model for polymorphic transitions in bacterial flagella

Many bacteria use rotating helical flagellar filaments to swim. The filaments undergo polymorphic transformations in which the helical pitch and radius change abruptly. These transformations arise in response to mechanical loading, changes in solution temperature and ionic strength, and point substitutions in the amino acid sequence of the protein subunits that make up the filament. To explain polymorphism, we propose a coarse-grained continuum rod theory based on the quaternary structure of the filament. The model has two molecular switches. The first is a double-well potential for the extension of a protofilament, which is one of the 11 almost longitudinal columns of subunits. Curved filament shapes occur in the model when there is a mismatch strain, i.e., when intersubunit bonds in the inner core of the filament prefer a subunit spacing which is intermediate between the two spacings favored by the double-well potential. The second switch is a double-well potential for twist, due to lateral interactions between neighboring protofilaments. Cooperative interactions between neighboring subunits within a protofilament are necessary to ensure the uniqueness of helical ground states. We calculate a phase diagram for filament shapes and the response of a filament to external moment and force.

Figure 1

Sketch of a short segment of the filament, showing the individual subunits that make up the protofilaments (the subunits of only three protofilaments are shown for clarity). The gray subunits are in the short conformation, and the white ones and all others not shown are in the other. To lower its energy, the filament will bend to form a helix (not shown).

Figure 2

The lattices of (a) $L$-type and (b) $R$-type filaments, at radius $a=4.5\mathrm{nm}$ and using data from Ref. 15. The vertical edges are identified. Note the 11-, 6-, and 5-start helices. It takes $n$ $n$-start helices to cover every point on the lattice of subunits. The protofilaments are the 11-start helices.

Figure 3

Sketch of a cross section of a filament of diameter $23\mathrm{nm}$, showing the inner core $\left(A\right)$, the outer core $\left(B\right)$, the outer domains $\left(C\right)$, and spokes $\left(S\right)$. The channel at the center has diameter $3\mathrm{nm}$.

Figure 4

Cartoon of a short segment of the $R$-type flagellar filament, showing the outer core $\left(A\right)$, connecting spokes $\left(B\right)$, inner core $\left(C\right)$, and lateral bonds $\left(D\right)$ between neighboring protofilaments $\left(E\right)$. The position of the spokes is not accurate, and is only meant to indicate that the inner and outer cores are connected. The outer domains are not shown. The lateral bonds lie along the five 5-start helices.

Figure 5

(a) The two preferred conformations of the subunit. (b) Sketch of double-well potential for stretching a protofilament, with minima corresponding to the two preferred states of (a). (c) The two preferred relative positions of neighboring subunits. (d) Sketch of double-well potential for twist ${\kappa }_3$.

Figure 6

Illustration of how twist causes neighboring subunits to slide past each other. The subunits rotate rigidly but the lattice of centers of mass shears. The angle of shear has been exaggerated for illustration.

Figure 7

Two rods with the same curvature and the same twist. The black line traces the path of the head of the vector ${\widehat{\mathbf{e}}}_1$. The circular rod is twisted through one turn to make ${\kappa }_3=1∕R_{\mathrm{c}}$, where $R_{\mathrm{c}}=1∕\kappa$ is the radius of the circle.

Figure 8

The basis vectors of the material and Serret-Frenet frames in the plane of the rod cross-section.

Figure 9

Protofilament extension ${ϵ}_i$ vs preferred extension ${ϵ}_0$ of the inner core for $i=1$ (a), $i=2$ and $i=11$ (b), $i=3$ and $i=10$ (c), $i=4$ and $i=9$ (d), $i=5$ and $i=8$ (e), $i=6$ and $i=7$ (f). The region between the horizontal dashed lines is the unstable region.

Figure 10

(a) Curvature vs ${ϵ}_0$ and (b) twist vs ${\overline{v}}_1$ for the simple model of Sec. 4. In (b), the solid lines correspond to the states of lowest energy, the dashed lines to metastable states, and the dotted line to unstable states.

Figure 11

(a) Applying twisting moments to the right-handed rod (top) causes it to snap through to a left-handed state (bottom). (b) The rod on top has ${\kappa }_2>0$; the rod on the bottom has ${\kappa }_2<0$. Applying appropriate moments to the top rod causes it to snap through.

Figure 12

Two helical filaments. The black line traces along the path of the protofilament $i=1$. Top: a filament with ${\kappa }_2<0$ and ${\kappa }_3<0$. Applying axial moments of sufficient magnitude causes the filament to snap through to a state with ${\kappa }_2>0$ and ${\kappa }_3>0$ (bottom).

Figure 13

Two helices with the same curvature and twist, one with ${\kappa }_2<0$ (top), and one with ${\kappa }_2>0$ (bottom). In each case, the black line traces the path of the protofilament $i=1$.

Figure 14

(Color) (a) Signed curvature ${\kappa }_2$ vs $m$ for $\tilde{u}=v$ and ${\theta }_0=\pi ∕6$ (blue curves, I) and ${\theta }_0=\pi ∕3$ (red curves, II). The dashed lines correspond to left-handed states. (b) Twist ${\kappa }_3$ vs $m$ with parameters as in (a).

Figure 15

(a) Signed curvature ${\kappa }_2$ vs $m$ for $\tilde{u}=v$ and ${\theta }_0=43°\approx 0.2389\pi$. The dashed lines correspond to left-handed states. (b) Twist ${\kappa }_3$ vs $m$ with parameters as in (a).

Figure 16

(Color) (a) Signed curvature ${\kappa }_2$ vs $m$ for $\tilde{u}=v$ and ${\theta }_0=\pi ∕6$. The dashed lines correspond to left-handed states. The red curves correspond to helical solutions $\left(H\right)$; the green curves correspond to straight solutions $\left(S\right)$. (b) Twist ${\kappa }_3$ vs $m$ with parameters as in (a).

Figure 17

The line of action of the force coincides with the axis of the helix, leading to moments applied at each end.

Figure 18

(Color) (a) Signed curvature ${\kappa }_2$ vs $\tilde{F}$ for $\tilde{u}=v$, and ${\theta }_0=\pi ∕6$ (blue curves, I) and ${\theta }_0=\pi ∕4$ (red curves, II). The dashed lines correspond to left-handed states. (b) Twist ${\kappa }_3$ vs $\tilde{F}$ with parameters as in (a).

Figure 19

(Color) Phase diagram for $\gamma =1.8$, $\tilde{u}=v$, ${\overline{k}}_{\mathrm{s}}=8$, ${\overline{k}}_5=1$, and ${\overline{u}}_1=-0.5$; ${\overline{v}}_1$ and $c_2$ are dimensionless. The inclined dotted line defines the boundary where the energy of the left-handed state equals the energy of the right-handed state. The diagonal dashed-dotted line traces out the values of ${\overline{v}}_1$ and $c_2$ used to calculate the curvature vs twist plot of Fig. 20. The points $X$ and $Y$ in the metastable regions represent the corresponding parameters used for evaluating the response of a helical and straight filament to external loading respectively.

Figure 20

Curvature vs twist, in dimensionless units, for $\gamma =1.8$, $\tilde{u}∕v=1$, ${\overline{k}}_{\mathrm{s}}=8$, ${\overline{k}}_5=1$, and ${\overline{u}}_1=-0.5$. The parameters ${\overline{v}}_1$ and $c_2$ vary along the dashed-dotted line of Fig. 19. The dotted line corresponds to metastable states for which the lower energy states are of opposite handedness.

Figure 21

(a) Signed curvature ${\kappa }_2$ vs $\overline{M}$ for $\tilde{u}=v$, ${\overline{k}}_{\mathrm{s}}=8$, ${\overline{k}}_5=1$, $c_2=4$, ${\overline{v}}_1=-0.25$, and ${\overline{u}}_1=-0.5$. The dashed lines correspond to left-handed states. (b) Twist ${\kappa }_3$ vs $\overline{M}$ with parameters as in (a).

Figure 22

(Color) (a) Signed curvature ${\kappa }_2$ vs $\overline{M}$ for $\tilde{u}=v$, ${\overline{k}}_{\mathrm{s}}=8$, ${\overline{k}}_5=1$, $c_2=15$, ${\overline{v}}_1=-0.25$, and ${\overline{u}}_1=-0.5$. The red and green curves indicate the helical $\left(H\right)$ and straight $\left(S\right)$ filaments, respectively. The dashed lines correspond to left-handed states. (b) Twist ${\kappa }_3$ vs $\overline{M}$ with parameters as in (a).

Figure 23

(a) Signed curvature ${\kappa }_2$ vs $\overline{F}$ for $\tilde{u}=v$, ${\overline{k}}_{\mathrm{s}}=8$, ${\overline{k}}_5=1$, $c_2=15$, ${\overline{v}}_1=-0.25$, and ${\overline{u}}_1=-0.5$. The dashed lines correspond to left-handed states. (b) Twist ${\kappa }_3$ vs $\overline{F}$ with parameters as in (a).