Braided bundles and compact coils: The structure and thermodynamics of hexagonally packed chiral filament assemblies

Molecular chirality frustrates the two-dimensional assembly of filamentous molecules, a fact that reflects the generic impossibility of imposing a global twisting of layered materials. We explore the consequences of this frustration for hexagonally ordered assemblies of chiral filaments that are finite in lateral dimension. Specifically, we employ a continuum-elastic description of cylindrical bundles of filaments, allowing us to consider the most general resistance to and preference for chiral ordering of the assembly. We explore two distinct mechanisms by which chirality at the molecular scale of the filament frustrates the assembly into aggregates. In the first, chiral interactions between filaments impart an overall twisting of filaments around the central axis of the bundle. In the second, we consider filaments that are inherently helical in structure, imparting a writhing geometry to the central axis. For both mechanisms, we find that a thermodynamically stable state of dispersed bundles of finite width appears close to but below the point of bulk filament condensation. The range of thermodynamic stability of dispersed bundles is sensitive only to the elastic cost and preference for chiral filament packing. The self-limited assembly of chiral filaments has particular implications for a large class of biological molecules—DNA, filamentous proteins, viruses, and bacterial flagella—which are universally chiral and are observed to form compact bundles under a broad range of conditions.

Figure 1

(Color online) Two packing motifs through which molecular structure of constituent filaments imparts chirality on the organization of densely packed bundles. In (a), intermolecular forces of chiral filaments favor relative tilting of neighboring filaments, leading to an overall twisted bundle. In (b), the unstressed shape of the filament itself is helical, imparting a writhe to the central axis of the bundle.

Figure 2

(Color online) (a) shows a cartoon of the torsional deformation of a vertical section of the bundle near $z=0$, demonstrating that the hexagonal packing is largely preserved. The filled (open) circles show the original (deformed) position of filaments. (b) depicts cylindrical sections from a twisted bundle at various radii. Starting from the center and moving outward, according to Eq. (10), the tilt of filaments relative to the central axis grows approximately linearly. The tilt angle of outermost filaments is then $\theta =\Omega R$, which leads to a reduction in the spacing between adjacent filaments, $d$, below the preferred spacing, $d_0$.

Figure 3

(Color online) Curves predicting the radial-dependence equilibrium twist angle for fixed elastic constants—according to the minima of Eq. (17)—are shown in (a) and (b). Each solid curve depicts a different value of the preference from twist ${\gamma }_{\text{Tw}}$. Depending on the relative sizes of elastic parameters ${\lambda }_{3\perp }$ and ${\lambda }_{2\parallel }$, the variation in $\theta$ with $R$ may be summarized by one of two scenarios, which are shown in (a) and (b) and described by four possible scaling regimes. Note, in particular, that $\theta \left(R\right)$ has only a single maximum.

Figure 4

(Color online) The diagram of state for the assembly of chiral filaments into straight bundles of twisted filaments. When the net free energy gain per unit aggregated filament length, $\Delta ϵ<0$, filaments are dispersed in solution in one of two states. Above a critical value of surface energy, ${\Sigma }_c$, filaments do not aggregate, and for $\Sigma <{\Sigma }_c$ bundles of finite radius are stable. The (dark) blue curve shows the boundary between these states for columnar-liquid-crystalline bundles and the (bright) red curve shows the same boundary for a strongly cross-linked solid bundle.

Figure 5

(Color online) A plot of equilibrium radius of straight bundles of twisted filaments along the line of bulk condensation, where, $\Delta ϵ=0$. For a bundle that is strictly liquid crystalline, ${\mu }_{\parallel }=0$, the growth of the bundle is shown by the black curve. The equilibrium curves spanning the range of weakly solid, ${\theta }_{\parallel \perp }^2={10}^{-4}$, to strongly solid, ${\theta }_{\parallel \perp }^2={10}^3$, shown in color. The increased resistance to out-of-plane shear reduces the bundle size, hence, $R_0$ monotonically decreases with ${\theta }_{\parallel \perp }^2$. This figure highlights the ${\theta }_{\parallel \perp }^2\ne 0$ crossover for the divergence of bundle size from columnar response $\left(R_0\sim {\Sigma }^{-3}\right)$ to solid response $\left(R_0\sim {\Sigma }^{-1}\right)$.

Figure 6

(Color online) Three helical bundles described by Eqs. (32, 33). An isometric bundle—with a constant separation between all nearest-neighbor filaments—can only be obtained if there is no net twist, $\Omega +\mathcal{T}=0$ (i.e., the coordinate frame compensates for the natural rotation of the Frenet frame).

Figure 7

(Color online) Plots of equilibrium structure of bundles of helical filaments for easy-twist model (bright red curves) and easy bend model (dark blue curves). The dependence of the curvature of the central axis of the bundle, ${\mathcal{K}}_0$, on the cohesive energy gain per unit filament length is shown in (a). The dependence equilibrium size of bundles on cohesive energy is shown in (b), with the inset highlighting the singular growth at the onset of bundling: $R_0\sim {\left(\Delta ϵ\right)}^{1/2}$. The ratio of bundle torsion to bundle curvature, ${\mathcal{T}}_0/{\mathcal{K}}_0$, is shown in (c). This ratio corresponds to the aspect ratio of the helical bundle and helical pitch relative to helical radius of bundle axis.

Figure 8

(Color online) The predicted diagram of state for the assembly of helical filaments in solution. If the net cohesive free-energy gain per bundled filament is below a critical value both dispersed filament and dispersed filaments may be stable, depending on the size of the surface energy of the bundles. The boundaries between these to states are shown for the easy-twist (bright red curve) and easy bend (dark blue curve) models.