We study chiral symmetry-broken configurations of nematic liquid crystals (LCs) confined to cylindrical capillaries with homeotropic anchoring on the cylinder walls (i.e., perpendicular surface alignment). Interestingly, achiral nematic LCs with comparatively small twist elastic moduli relieve bend and splay deformations by introducing twist deformations. In the resulting twisted and escaped radial (TER) configuration, LC directors are parallel to the cylindrical axis near the center, but to attain radial orientation near the capillary wall, they escape along the radius through bend and twist distortions. Chiral symmetry-breaking experiments in polymercoated capillaries are carried out using Sunset Yellow FCF, a lyotropic chromonic LC with a small twist elastic constant. Its director configurations are investigated by polarized optical microscopy and explained theoretically with numerical calculations. A rich phenomenology of defects also arises from the degenerate bend/twist deformations of the TER configuration, including a nonsingular domain wall separating domains of opposite twist handedness but the same escape direction and singular point defects (hedgehogs) separating domains of opposite escape direction. We show the energetic preference for singular defects separating domains of opposite twist handedness compared with those of the same handedness, and we report remarkable chiral configurations with a double helix of disclination lines along the cylindrical axis. These findings show archetypally how simple boundary conditions and elastic anisotropy of confined materials lead to multiple symmetry breaking and how these broken symmetries combine to create a variety of defects.mirror symmetry | parity symmetry | topological defects | chiral defects T he emergence of chirality from achiral systems poses fundamental questions about which we have limited mechanistic understanding (1-11). When the chiral symmetry of an achiral system is broken, a handedness is established, and materials with different handedness commonly exhibit distinct and useful properties (10-14) relevant for applications ranging from chemical sensors (15, 16) to photonics (17-19). To date, considerable effort has been expended to control handedness in materials (for example, by chiral separation of racemic mixtures or chiral amplification of small enantiomeric imbalances) (1,8,(20)(21)(22). Recently and in a different vein, identification and elucidation of pathways by which achiral building blocks spontaneously organize to create chiral structures have become an area of active study. Examples of these pathways include packing with multiple competing length scales (8-10, 23, 24), reconfiguration through mechanical instabilities of periodic structures (20,25,26), and helix formation of flexible cylinders through inter-and intracylinder interactions (27,28). In addition, the system of a broken chiral symmetry often consists of domains of opposite handedness with defects separating the domains.Liquid crystals (LCs) are soft materials composed ...

An experimental and theoretical study of lyotropic chromonic liquid crystals (LCLCs) confined in cylinders with degenerate planar boundary conditions elucidates LCLC director configurations. When the Frank saddlesplay modulus is more than twice the twist modulus, the ground state adopts an inhomogeneous escapedtwisted configuration. Analysis of the configuration yields a large saddle-splay modulus, which violates Ericksen inequalities but not thermodynamic stability. Lastly, we observe point defects between opposite-handed domains, and we explain a preference for point defects over domain walls. The elastic properties of nematic liquid crystals (LCs) are crucial for liquid crystal display applications [1,2], and they continue to give rise to unanticipated fundamental phenomena [3][4][5][6][7][8][9]. Three of the bulk nematic LC deformation modes-splay, twist, and bend-are well known and have associated elastic moduli K 1 , K 2 , and K 3 , respectively. These moduli have been studied intensely because they are easy to visualize, and because it is possible to independently excite the modes via clever usage of sample geometry [10][11][12], LC boundary conditions [13,14], and external fields [15,16]. As a result, these moduli have been measured for a variety of thermotropic and lyotropic LCs [12,[16][17][18][19][20]. By contrast, a much less studied fourth independent mode [21-23] of elastic deformation in nematic LCs can exist; it is called saddle-splay. Saddle-splay is hard to visualize and to independently excite [23,24]. Moreover, the energy of this deformation class can be integrated to the boundary, so that the mode does not appear in the Euler-Lagrange equations, and with fixed boundary conditions the saddle-splay energy will have no effect on the LC director configuration. Even with free boundary conditions, the saddle-splay energy will not affect the bulk LC configuration unless the principal curvatures of the surface are different, i.e., saddle-splay effects are not expected for spherical or flat surfaces. Thus, although much progress in understanding saddle-splay has been made [25,26], especially with thermotropic nematic LCs, unambiguous determination of saddle-splay energy effects on liquid crystal configurations and measurement of the saddle-splay elastic modulus, K 24 , remain difficult [27].While the bulk elastic constants described above strongly influence LC director configurations, LC boundary conditions at material interfaces also influence bulk structure. Indeed, considerable effort has gone into development of surface preparation techniques to produce particular bulk director configurations [13,[28][29][30][31][32][33]. The saddle-splay term integrates to the boundary and effectively imposes boundary conditions at free surfaces favoring director alignment along the direction of highest surface curvature for positive K 24 [34] and outwardly * jjeong@unist.ac.kr pointing surface normals. For this effect to be present, the director cannot be held perpendicular to the surface, as was the case in ou...

The depletion interaction mediated by non-adsorbing polymers promotes condensation and assembly of repulsive colloidal particles into diverse higher-order structures and materials. One example, with particularly rich emergent behaviors, is the formation of two-dimensional colloidal membranes from a suspension of filamentous fd viruses, which act as rods with effective repulsive interactions, and dextran, which acts as a condensing, depletion-inducing agent. Colloidal membranes exhibit chiral twist even when the constituent virus mixture lacks macroscopic chirality, change from a circular shape to a striking starfish shape upon changing the chirality of constituent rods, and partially coalesce via domain walls through which the viruses twist by 180• . We formulate an entropicallymotivated theory that can quantitatively explain these experimental structures and measurements, both previously published and newly performed, over a wide range of experimental conditions. Our results elucidate how entropy alone, manifested through the viruses as Frank elastic energy and through the depletants as an effective surface tension, drives the formation and behavior of these diverse structures. Our generalizable principles propose the existence of analogous effects in molecular membranes and can be exploited in the design of reconfigurable colloidal structures.

Mitosis in the early syncytial Drosophila embryo is highly correlated in space and time, as manifested in mitotic wavefronts that propagate across the embryo. In this paper we investigate the idea that the embryo can be considered a mechanically-excitable medium, and that mitotic wavefronts can be understood as nonlinear wavefronts that propagate through this medium. We study the wavefronts via both image analysis of confocal microscopy videos and theoretical models. We find that the mitotic waves travel across the embryo at a well-defined speed that decreases with replication cycle. We find two markers of the wavefront in each cycle, corresponding to the onsets of metaphase and anaphase. Each of these onsets is followed by displacements of the nuclei that obey the same wavefront pattern. To understand the mitotic wavefronts theoretically we analyze wavefront propagation in excitable media. We study two classes of models, one with biochemical signaling and one with mechanical signaling. We find that the dependence of wavefront speed on cycle number is most naturally explained by mechanical signaling, and that the entire process suggests a scenario in which biochemical and mechanical signaling are coupled.

Lipid rafts are hypothesized to facilitate protein interaction, tension regulation, and trafficking in biological membranes, but the mechanisms responsible for their formation and maintenance are not clear. Insights into many other condensed matter phenomena have come from colloidal systems, whose micron-scale particles mimic basic properties of atoms and molecules but permit dynamic visualization with single-particle resolution. Recently, experiments showed that bidisperse mixtures of filamentous viruses can self-assemble into colloidal monolayers with thermodynamically stable rafts exhibiting chiral structure and repulsive interactions. We quantitatively explain these observations by modeling the membrane particles as chiral liquid crystals. Chiral twist promotes the formation of finite-sized rafts and mediates a repulsion that distributes them evenly throughout the membrane. Although this system is composed of filamentous viruses whose aggregation is entropically driven by dextran depletants instead of phospholipids and cholesterol with prominent electrostatic interactions, colloidal and biological membranes share many of the same physical symmetries. Chiral twist can contribute to the behavior of both systems and may account for certain stereospecific effects observed in molecular membranes. membrane rafts | liquid crystals | chirality | self-assembly | colloids F ilamentous viruses have proved to be a fruitful colloidal system (1-19). They serve as monodisperse, rigid, and chiral rods that are ∼1 µm in length and interact effectively through hard-core repulsion (2, 7). When suspended in an aqueous solution at increasing concentrations, they transition from a disordered isotropic phase to a cholesteric (chiral nematic) phase characterized by alignment along a director field that twists with a preferred handedness and wavelength (1, 6). The addition of a nonadsorbing polymer such as dextran induces lateral virusvirus attraction via the depletion interaction (10,12,20,21). The viruses self-assemble into monolayers that exhibit fluid-like dynamics internally (10) and sediment to the bottom of glass containers, which are coated with a polyacrylamide brush to suppress depletion-induced virus-wall attractions (22). The rich physics and phenomenology of membranes formed from singlevirus species have been thoroughly studied (8)(9)(10)(11)(12)(13)(14)(15)(16)(17)19). However, two-species membranes demonstrate a novel set of behaviors that are not adequately understood (18). We review these behaviors now before describing a theory that can explain them.fd-Y21M and M13KO7, which we shorten to fd and M13 for convenience, are two species of filamentous virus that have slightly different lengths and form cholesteric phases of opposite handednesses (Table 1 and Fig. 1A). Membranes composed of both fd and M13 viruses are circular with interior particles aligned largely perpendicularly to the membrane plane and edge particles tilted azimuthally, as in single-species membranes (13). At low dextran concentrations, the two spec...

Grid cells in the medial entorhinal cortex (MEC) respond when an animal occupies a periodic lattice of ‘grid fields’ in the environment. The grids are organized in modules with spatial periods, or scales, clustered around discrete values separated on average by ratios in the range 1.4–1.7. We propose a mechanism that produces this modular structure through dynamical self-organization in the MEC. In attractor network models of grid formation, the grid scale of a single module is set by the distance of recurrent inhibition between neurons. We show that the MEC forms a hierarchy of discrete modules if a smooth increase in inhibition distance along its dorso-ventral axis is accompanied by excitatory interactions along this axis. Moreover, constant scale ratios between successive modules arise through geometric relationships between triangular grids and have values that fall within the observed range. We discuss how interactions required by our model might be tested experimentally.

Grid cells in the medial entorhinal cortex (MEC) respond when an animal occupies a periodic lattice of "grid fields" in the environment. The grids are organized in modules with spatial periods, or scales, clustered around discrete values separated by ratios in the range 1.2-2.0. We propose a mechanism that produces this modular structure through dynamical self-organization in the MEC. In attractor network models of grid formation, the grid scale of a single module is set by the distance of recurrent inhibition between neurons. We show that the MEC forms a hierarchy of discrete modules if a smooth increase in inhibition distance along its dorso-ventral axis is accompanied by excitatory interactions along this axis. Moreover, constant scale ratios between successive modules arise through geometric relationships between triangular grids and have values that fall within the observed range. We discuss how interactions required by our model might be tested experimentally.

Persistent cohomology is a powerful technique for discovering topological structure in data. Strategies for its use in neuroscience are still undergoing development. We comprehensively and rigorously assess its performance in simulated neural recordings of the brain's spatial representation system. Grid, head direction, and conjunctive cell populations each span low-dimensional topological structures embedded in high-dimensional neural activity space. We evaluate the ability for persistent cohomology to discover these structures for different dataset dimensions, variations in spatial tuning, and forms of noise. We quantify its ability to decode simulated animal trajectories contained within these topological structures. We also identify regimes under which mixtures of populations form product topologies that can be detected. Our results reveal how dataset parameters affect the success of topological discovery and suggest principles for applying persistent cohomology, as well as persistent homology, to experimental neural recordings.