Iterated Moiré maps and braiding of chiral polymer crystals

Mechanics of Metric Frustration in Contorted Filament Bundles: From Local Symmetry to Columnar Elasticity

Bundles of filaments are subject to geometric frustration: certain deformations (e.g., bending while twisted) require longitudinal variations in spacing between filaments. While bundles are common-from protein fibers to yarns-the mechanical consequences of longitudinal frustration are unknown. We derive a geometrically nonlinear formalism for bundle mechanics, using a gaugelike symmetry under reptations along filament backbones. We relate force balance to orientational geometry and assess the elastic cost of frustration in twisted-toroidal bundles.

Fast dynamics and emergent topological defects in long-range interacting particle systems

Long-range interacting systems exhibit unusual physical properties not shared by systems with short-range interactions. Understanding the dynamical and statistical effects of long-range interactions yields insights into a host of physical systems in nature and industry. In this work, we investigate the classical microscopic dynamics of screened Coulomb interacting particles confined in the disk, and reveal the featured dynamics and emergent statistical regularities created by the long-range interaction. We highlight the long-range interaction driven fast single-particle and collective dynamics, and the emergent topological defect structure. This work suggests the rich physics arising from the interplay of long-range interaction, topology and dynamics.

Gnomonious projections for bend-free textures: thoughts on the splay-twist phase

The Hopf fibration has inspired any number of geometric structures in physical systems, in particular, in chiral liquid crystalline materials. Because the Hopf fibration lives on the three sphere,S 3 , some method of projection or distortion must be employed to realize textures in flat space. Here, we explore the geodesic-preserving gnomonic projection of the Hopf fibration, and show that this could be the basis for a new liquid crystalline texture with only splay and twist. We outline the structure and show that it is defined by the tangent vectors along the straight line rulings on a series of hyperboloids. The phase is defined by a lack of bend deformations in the texture, and is reminiscent of the splay-bend and twist-bend nematic phases. We show that domains of this phase may be stabilized through anchoring and saddle-splay.

Topological vacancies in spherical crystals

Understanding geometric frustration of ordered phases in two-dimensional condensed matter on curved surfaces is closely related to a host of scientific problems in condensed matter physics and materials science. Here, we show how two-dimensional Lennard-Jones crystal clusters confined on a sphere resolve geometric frustration and form pentagonal vacancy structures. These vacancies, originating from the combination of curvature and physical interaction, are found to be topological defects and they can be further classified into dislocational and disclinational types. We analyze the dual role of these crystallographic defects as both vacancies and topological defects, illustrate their formation mechanism, and present the phase diagram. The revealed dual role of the topological vacancies may find applications in the fabrication of robust nanopores. This work also shows the promising potential of exploiting richness in both physical interactions and substrate geometries to create new types of crystallographic defects, which have strong connections with the design of crystalline materials.

Observation of the Chiral and Achiral Hexatic Phases of Self-assembled Micellar polymers

We report the discovery of a thermodynamically stable line hexatic (N + 6) phase in a three-dimensional (3D) system made up of self-assembled polymer-like micelles of amphiphilic molecules. The experimentally observed phase transition sequence nematic (N)  N + 6  two-dimensional hexagonal (2D-H) is in good agreement with the theoretical predictions. Further, the present study also brings to light the effect of chirality on the N + 6 phase. In the chiral N + 6 phase the bond-orientational order within each "polymer" bundle is found to be twisted about an axis parallel to the average polymer direction. This structure is consistent with the theoretically envisaged Moiré state, thereby providing the first experimental demonstration of the Moiré structure. In addition to confirming the predictions of fundamental theories of two-dimensional melting, these results are relevant in a variety of situations in chemistry, physics and biology, where parallel packing of polymer-like objects are encountered.

Making the cut: lattice kirigami rules

In this Letter we explore and develop a simple set of rules that apply to cutting, pasting, and folding honeycomb lattices. We consider origami-like structures that are extrinsically flat away from zero-dimensional sources of Gaussian curvature and one-dimensional sources of mean curvature, and our cutting and pasting rules maintain the intrinsic bond lengths on both the lattice and its dual lattice. We find that a small set of rules is allowed providing a framework for exploring and building kirigami—folding, cutting, and pasting the edges of paper.

Defects in crystalline packings of twisted filament bundles. I. Continuum theory of disclinations

We develop the theory of the coupling between in-plane order and out-of-plane geometry in twisted, two-dimensionally ordered filament bundles based on the nonlinear continuum elasticity theory of columnar materials. We show that twisted textures of filament backbones necessarily introduce stresses into the cross-sectional packing of bundles and that these stresses are formally equivalent to the geometrically induced stresses generated in thin elastic sheets that are forced to adopt spherical curvature. As in the case of crystalline order on curved membranes, geometrically induced stresses couple elastically to the presence of topological defects in the in-plane order. We derive the effective theory of multiple disclination defects in the cross section of bundle with a fixed twist and show that above a critical degree of twist, one or more fivefold disclinations is favored in the elastic energy ground state. We study the structure and energetics of multidisclination packings based on models of equilibrium and nonequilibrium cross-sectional order.

Formation of impeller-like helical DNA-silica complexes by polyamines induced chiral packing

The helicity of DNA and its long-range chiral packing are widespread phenomena; however, the packing mechanism remains poorly understood both in vivo and in vitro. Here, we report the extraordinary DNA chiral self-assembly by silica mineralization, together with circular dichroism measurements and electron microscopy studies on the structure and morphology of the products. Mg(2+) ion and diethylenetriamine were found to induce right- and left-handed chiral DNA packing with two-dimensional-square p4mm mesostructures, respectively, to give corresponding enantiomeric impeller-like helical DNA-silica complexes. Moreover, formation of macroscopic impeller-like helical architectures depends on the types of polyamines and co-structure-directing agents and pH values of reaction solution. It has been suggested that interaction strength between negatively charged DNA phosphate strands and positively charged counterions may be the key factor for the induction of DNA packing handedness.

Structure of toroidal DNA collapsed inside the phage capsid

The structure of DNA toroids made of individual DNA molecules of various lengths (3,000 to 55,000 bp) was studied, by using partially filled bacteriophage capsids in conjunction with cryoelectron microscopy. The tetravalent cation spermine was diffused through the capsid to condense the DNA under conditions that were chosen to produce a hexagonal packing. Our results demonstrate that the frustration arising between chirality and hexagonal packing leads to the formation of twist walls; the correlation between helices combined with their strong curvature impose variations of the DNA helical pitch.

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.

Helical twist controls the thickness of F-actin bundles

In the presence of condensing agents such as nonadsorbing polymer, multivalent counter ions, and specific bundling proteins, chiral biopolymers typically form bundles with a finite thickness, rather than phase-separating into a polymer-rich phase. Although short-range repulsive interactions or geometrical frustrations are thought to force the equilibrium bundle size to be limited, the precise mechanism is yet to be resolved. The importance of the tight control of biopolymer bundle size is illustrated by the ubiquitous cytoskeletal actin filament bundles that are crucial for the proper functioning of cells. Using an in vitro model system, we show that size control relies on a mismatch between the helical structure of individual actin filaments and the geometric packing constraints within bundles. Small rigid actin-binding proteins change the twist of filamentous actin (F-actin) in a concentration-dependent manner, resulting in small, well defined bundle thickness up to approximately 20 filaments, comparable to those found in filopodia. Other F-actin cross-linking proteins can subsequently link these small, well organized bundles into larger structures of several hundred filaments, comparable to those found in, for example, Drosophila bristles. The energetic tradeoff between filament twisting and cross-linker binding within a bundle is suggested as a fundamental mechanism by which cells can precisely adjust bundle size and strength.

Hexagonal order in crystalline and columnar phases of hard rods

We report a study of colloidal suspensions of highly monodisperse semiflexible chiral rodlike viruses, denoted fd, in the range of high concentrations. Small angle x-ray scattering experiments reveal the existence of two hexagonal phases: the first one is crystalline and the second one is hexatic columnar, as shown by its short-range positional order. The suspension of rodlike viruses is the first experimental system showing the whole phase sequence with increasing particle concentration theoretically predicted for systems of hard rods, ranging from the chiral nematic via the smectic to columnar and crystalline phases.

Chirality and equilibrium biopolymer bundles

We use continuum theory to show that chirality is a key thermodynamic control parameter for the aggregation of biopolymers: chirality produces a stable disperse phase of hexagonal bundles under moderately poor solvent conditions, as has been observed in in vitro studies of F actin [O. Pelletier et al., Phys. Rev. Lett. 91, 148102 (2003)]. The large characteristic radius of these chiral bundles is not determined by a mysterious long-range molecular interaction but by in-plane shear elastic stresses generated by the interplay between a chiral torque and an unusual, but universal, nonlinear gauge term in the strain tensor of ordered chains that is imposed by rotational invariance.

Linear Solvation Energy Correlation of the Solvatochromic Response of Amino Acid Functionalized Chromophores

Solvatochromism of the long wavelength band in the electronic absorption spectra of N-(4-nitrophenyl)-L-proline, NLP, N-(4-nitrophenyl)-D-proline, NDP, and N-(4-nitro-phenyl)-trans-4-hydroxy-L-proline, NHP, was studied and quantitatively described with Kamlet-Taft solvent polarity parameters. To evaluate the environmental effects for NLP, NDP, and NHP, the UV-vis spectroscopic behavior of these compounds was also investigated as pure crystals, as a guest in 2,6-O-dimethyl-beta-cyclodextrin, and also when adsorbed on Aerosil 300 silica particles from nonhydrogen-bond accepting solvents. Excellent Kamlet-Taft solvatochromic correlations were established for the three compounds in most solvents. Multiparameter correlations show the existence of specific solute-solvent interactions. A strong positive solvatochromic behavior was found for these compounds, indicating that their dipole moments were higher in the excited singlet state than in the ground state.

Triply periodic smectic liquid crystals

Twist-grain-boundary phases in smectics are the geometrical analogs of the Abrikosov flux lattice in superconductors. At large twist angles, the nonlinear elasticity is important in evaluating their energetics. We analytically construct the height function of a pi2 twist-grain-boundary phase in smectic-A liquid crystals, known as Schnerk's first surface. This construction, utilizing elliptic functions, allows us to compute the energy of the structure analytically. By identifying a set of heretofore unknown defects along the pitch axis of the structure, we study the necessary topological structure of grain boundaries at other angles, concluding that there exist a set of privileged angles and that the pi2 and pi3 grain boundary structures are particularly simple.

Molecular scale imaging of F-actin assemblies immobilized on a photopolymer surface

A photo-immobilization based process is presented for direct imaging of hierarchical assemblies of biopolymers using atomic force microscopy (AFM). The technique was used to investigate the phase behavior of F-actin aggregates as a function of concentration of the divalent cation Mg2+. The data provided direct experimental evidence of a coil-on-coil (braided) structure of F-actin bundles formed at high Mg2+ concentrations. At intermediate Mg2+ concentrations, the data showed the first images of the two-dimensional nematic rafts discovered by recent x-ray studies and theoretical treatments.

Entropy-driven formation of a chiral liquid-crystalline phase of helical filaments

We study the liquid-crystalline phase behavior of a concentrated suspension of helical flagella isolated from Salmonella typhimurium. Flagella are prepared with different polymorphic states, some of which have a pronounced helical character while others assume a rodlike shape. We show that the static phase behavior and dynamics of chiral helices are very different when compared to simpler achiral hard rods. With increasing concentration, helical flagella undergo an entropy-driven first order phase transition to a liquid-crystalline state having a novel chiral symmetry.

Liquid crystals of polyelectrolyte networks

The Onsager theory of nematic liquid crystals is extended to rigid polyelectrolytes cross-linked by polyvalent ions. Recent synchrotron x-ray diffraction experiments showed that dilute, birefringent networks are formed under these conditions. The application of Onsager theory to this system leads to the prediction of the existence of a range of exotic mesophases such as the "cubatic," the "tetratic," and the "trigatic." The exotic network phases appear on the border of regions of phase coexistence of network phase with isotropic material (at low polyvalent ion concentration) and with dense bundles (at high polyvalent ion concentration).

Chiral discotic columnar germs of nucleosome core particles

In concentrated solution and in the presence of high concentrations of monovalent cations, nucleosome core particles order into a discotic columnar mesophase. This phase is limited to finite-sized hexagonal germs that further divide into six coiled branches, following an iterative process. We show how the structure of the germs comes from the competition between hexagonal packing and chirality with a combination of dendritic facetting and double-twist configurations. Geometrical considerations lead us to suspect that the chirality of the eukaryotic chromosomes may originate from the same competition.