Structural Transitions and Soft Modes in Frustrated DNA Crystals

Relying on symmetry considerations appropriate for helical biopolymers such as DNA and filamentous actin, we argue that crystalline packings of mutually repulsive helical macromolecules fall principally into two categories: unfrustrated (hexagonal) and frustrated (rhombohedral). For both cases, we construct the Landau-Ginzburg free energy for the 2D columnar-hexagonal to 3D crystalline phase transition, including the coupling between molecular displacements {\it along} biopolymer backbone to displacements in the plane of hexagonal order. We focus on the distinct elastic properties that emerge upon crystallization of helical arrays due to this coupling. Specifically, we demonstrate that frustrated states universally exhibit a highly anisotropic in-plane elastic response, characterized by an especially soft compliance to simple-shear deformations and a comparatively large resistance to those deformations that carry the array from the low- to high-density crystalline states of DNA.Comment: 7 pages, 3 figures (revised version

Shape selection of surface-bound helical filaments: biopolymers on curved membranes

Motivated to understand the behavior of biological filaments interacting with membranes of various types, we study a theoretical model for the shape and thermodynamics of intrinsically-helical filaments bound to curved membranes. We show filament-surface interactions lead to a host of non-uniform shape equilibria, in which filaments progressively unwind from their native twist with increasing surface interaction and surface curvature, ultimately adopting uniform-contact curved shapes. The latter effect is due to non-linear coupling between elastic twist and bending of filaments on anisotropically-curved surfaces, such as the cylindrical surfaces considered here. Via a combination of numerical solutions and asymptotic analysis of shape equilibria we show that filament conformations are critically sensitive to the surface curvature in both the strong- and weak-binding limits. These results suggest that local structure of membrane-bound chiral filaments is generically sensitive to the curvature-radius of the surface to which it is bound, even when that radius is much larger than the filament intrinsic pitch. Typical values of elastic parameters and interaction energies for several prokaryotic and eukaryotic filaments indicate that biopolymers are inherently very sensitive to the coupling between twist, interactions and geometry and that this could be exploited for regulation of a variety of processes such as the targeted exertion of forces, signaling and self-assembly in response to geometric cues including the local mean and Gaussian curvatures

Continuous Crystallization in Hexagonally-Ordered Materials

We demonstrate that the phase transition from columnar-hexagonal liquid crystal to hexagonal-crystalline solid falls into an unusual universality class, which in three-dimensional allows for both discontinuous transitions as well as continuous transitions, characterized by a single set of exponents. We show by a renormalization group calculation (to first order in $\epsilon = 4-d$) that the critical exponents of the continuous transition are precisely those of the XY model, which gives rise to a continuous evolution of elastic moduli. Although the fixed points of the present model are found to be identical to the XY model, the elastic compliance to deformations in the plane of hexagonal order, $\mu$, is nonetheless shown to critically influence the crystallization transition, with the continuous transition being driven to first order by fluctuations as the in plane response grows weaker, $\mu \to 0$.Comment: 4 pages, 2 figures (revised version

Topological Defects in Twisted Bundles of Two-Dimensionally Ordered Filaments

Twisted assemblies of filaments in ropes, cables and bundles are essential structural elements in wide use in macroscopic materials as well as within the cells and tissues of living organisms. We develop the unique, non-linear elastic properties of twisted filament bundles that derive from generic properties of two-dimensional line-ordered materials. Continuum elasticity reveals a formal equivalence between the elastic stresses induced by bundle twist and those induced by the positive curvature in thin, elastic sheets. These geometrically-induced stresses can be screened by 5-fold disclination defects in lattice packing, and we predict a discrete spectrum elastic energy groundstates associated with integer numbers of disclinations in cylindrical bundles. Finally, we show that elastic-energy groundstates are extremely sensitive to defect position in the cross-section, with off-center disclinations driving the entire bundle to buckle, adopting globally writhing configurations.Comment: 4.1 pages; 3 figure

Soft Spheres Make More Mesophases

We use both mean-field methods and numerical simulation to study the phase diagram of classical particles interacting with a hard-core and repulsive, soft shoulder. Despite the purely repulsive interaction, this system displays a remarkable array of aggregate phases arising from the competition between the hard-core and shoulder length scales. In the limit of large shoulder width to core size, we argue that this phase diagram has a number of universal features, and classify the set of repulsive shoulders that lead to aggregation at high density. Surprisingly, the phase sequence and aggregate size adjusts so as to keep almost constant inter-aggregate separation.Comment: 4 pages, 2 included figure

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 {\it 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 {\it 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, bacterial flagella -- which are universally chiral and are observed to form compact bundles under a broad range of conditions.Comment: 15 pages, 8 figure

Defects in Crystalline Packings of Twisted Filament Bundles: II. Dislocations and Grain Boundaries

Twisted and rope-like assemblies of filamentous molecules are common and vital structural elements in cells and tissue of living organisms. We study the intrinsic frustration occurring in these materials between the two-dimensional organization of filaments in cross section and out-of-plane interfilament twist in bundles. Using non-linear continuum elasticity theory of columnar materials, we study the favorable coupling of twist-induced stresses to the presence of edge dislocations in the lattice packing of bundles, which leads to a restructuring of the ground-state order of these materials at intermediate twist. The stability of dislocations increases as both the degree of twist and lateral bundle size grow. We show that in ground states of large bundles, multiple dislocations pile up into linear arrays, radial grain boundaries, whose number and length grows with bundle twist, giving rise to a rich class of "polycrystalline" packings.Comment: 10 pages, 7 figure