Continuum Model for Polymorphism of Bacterial Flagella

Bacterial flagellar filaments can abruptly change shape in response to mechanical load or changes in solution $p\mathrm{H}$ or ionic strength. These polymorphic transformations are an instance of a ubiquitous phenomenon, the spread of conformational change in large macromolecular assemblies. We propose a new theory for polymorphism, whose essential elements are two molecular switches and an elastic mismatch strain between the inner and outer cores of the filament. We calculate the phase diagram for helical and straight states, and the response of a helical filament to an external moment.

Figure 1

(a) Filament element with ${ϵ}_i=ϵ={ϵ}_0$. The gray lines represent the strands and the black disk represents a cross section of the central spring. (b) Filament element with $ϵ={ϵ}_0$, and a variation in the stretch ${ϵ}_i$ of the strands. (c) Gray curve: energy vs stretch for a protofilament; black curve: energy vs stretch for the central spring (dimensionless units). For the situation in (c), (b) has lower energy than (a).

Figure 2

Phase diagram for ground states. The vertical solid line separates left-handed and right-handed states. The vertical dash-dot lines define the limits of metastability for the twist potential. The diagonal line traces out the values of ${\overline{v}}_1$ and ${ϵ}_0$ used to calculate the curvature vs twist plot of Fig. 3.

Figure 3

Curvature vs twist, in dimensionless units. The solid line corresponds to minimum-energy states; the dashed lines correspond to metastable states. The parameters ${\overline{v}}_1$ and ${ϵ}_0$ vary along the diagonal line of Fig. 2.

Figure 4

Signed curvature ${\kappa }_2$ vs moment $m$, in dimensionless units, for $\tilde{u}=v$, ${\theta }_0=32°$. The solid line is the right-handed branch, and the dashed line is the left-handed branch (cf. Figure 5).

Figure 5

Twist ${\kappa }_3/\sigma$ vs moment $m$, in dimensionless units, for $\tilde{u}=v$ and ${\theta }_0=32°$.