Chirality and equilibrium biopolymer bundles

Morphological control of bundled actin networks subject to fixed-mass depletion

Depletion interactions are thought to significantly contribute to the organization of intracellular structures in the crowded cytosol. The strength of depletion interactions depends on physical parameters such as the depletant number density and the depletant size ratio. Cells are known to dynamically regulate these two parameters by varying the copy number of proteins of a wide distribution of sizes. However, mammalian cells are also known to keep the total protein mass density remarkably constant, to within 0.5% throughout the cell cycle. We thus ask how the strength of depletion interactions varies when the total depletant mass is held fixed, a.k.a. fixed-mass depletion. We answer this question via scaling arguments, as well as by studying depletion effects on networks of reconstituted semiflexible actin in silico and in vitro. We examine the maximum strength of the depletion interaction potential U∗ as a function of q, the size ratio between the depletant and the matter being depleted. We uncover a scaling relation U∗ ∼ qζ for two cases: fixed volume fraction φ and fixed mass density ρ. For fixed volume fraction, we report ζ < 0. For the fixed mass density case, we report ζ > 0, which suggests that the depletion interaction strength increases as the depletant size ratio is increased. To test this prediction, we prepared our filament networks at fixed mass concentrations with varying sizes of the depletant molecule poly(ethylene glycol) (PEG). We characterize the depletion interaction strength in our simulations via the mesh size. In experiments, we observe two distinct actin network morphologies, which we call weakly bundled and strongly bundled. We identify a mass concentration where different PEG depletant sizes lead to weakly bundled or strongly bundled morphologies. For these conditions, we find that the mesh size and intra-bundle spacing between filaments across the different morphologies do not show significant differences, while the dynamic light scattering relaxation time and storage modulus between the two states do show significant differences. Our results demonstrate the ability to tune actin network morphology and mechanics by controlling depletant size and give insights into depletion interaction mechanisms under the fixed-depletant-mass constraint relevant to living cells.

Self-limiting stacks of curvature-frustrated colloidal plates: Roles of intraparticle versus interparticle deformations

In geometrically frustrated assemblies local intersubunit misfits propagate to intra-assembly strain gradients, giving rise to anomalous self-limiting assembly thermodynamics. Here we use theory and coarse-grained simulation to study a recently developed class of "curvamer" particles, flexible shell-like particles that exhibit self-limiting assembly due to the build up of curvature deformation in cohesive stacks. To address a generic, yet poorly understood aspect of frustrated assembly, we introduce a model of curvamer assembly that incorporates both intraparticle shape deformation as well as compliance of interparticle cohesive gaps, an effect we can attribute to a finite range of attraction between particles. We show that the ratio of intraparticle (bending elasticity) to interparticle stiffness not only controls the regimes of self-limitation but also the nature of frustration propagation through curvamer stacks. We find a transition from uniformly bound, curvature-focusing stacks at small size to gap opened, uniformly curved stacks at large size is controlled by a dimensionless measure of inter- versus intracurvamer stiffness. The finite range of interparticle attraction determines the range of cohesion in stacks that are self-limiting, a prediction which is in strong agreement with numerical studies of our coarse-grained colloidal model. These predictions provide critical guidance for experimental realizations of frustrated particle systems designed to exhibit self-limitation at especially large multiparticle scales.

Pulling on grafted flexible polymers can cause twisted bundles

Bundles of semiflexible polymers can twist at low temperatures to balance energy gain from attraction and energy cost from bending. This raises the question whether twisting can be also observed for bundles of rather flexible grafted polymers stretched out by pulling force. Here, we address this question using Monte Carlo computer simulations of small bundles. Our data show that for weak forces F < Fl, intertwined globular conformations are favored, whereas for strong forces F > Fu, rod-like bundles emerge. In the intermediate force window Fl < F < Fu, bundles with a helical twist can be clearly identified. Applying a field to all monomers yields qualitatively the same effect. This suggests the conclusion that rather flexible polymers under pulling force or field behave effectively like semiflexible polymers without external pull.

Understanding Self-Assembly and Molecular Packing in Methylcellulose Aqueous Solutions Using Multiscale Modeling and Simulations

We present a multiscale molecular dynamics (MD) simulation study on self-assembly in methylcellulose (MC) aqueous solutions. First, using MD simulations with a new coarse-grained (CG) model of MC chains in implicit water, we establish how the MC chains self-assemble to form fibrils and fibrillar networks and elucidate the MC chains' packing within the assembled fibrils. The CG model for MC is extended from a previously developed model for unsubstituted cellulose and captures the directionality of H-bonding interactions between the -OH groups. The choice and placement of the CG beads within each monomer facilitates explicit modeling of the exact degree and position of methoxy substitutions in the monomers along the MC chain. CG MD simulations show that with increasing hydrophobic effect and/or increasing H-bonding strength, the commercial MC chains (with degree of methoxy substitution, DS, ∼1.8) assemble from a random dispersed configuration into fibrils. The assembled fibrils exhibit consistent fibril diameters regardless of the molecular weight and concentration of MC chains, in agreement with past experiments. Most MC chains' axes are aligned with the fibril axis, and some MC chains exhibit twisted conformations in the fibril. To understand the molecular driving force for the twist, we conduct atomistic simulations of MC chains preassembled in fibrils (without any chain twists) in explicit water at 300 and 348 K. These atomistic simulations also show that at DS = 1.8, MC chains adopt twisted conformations, with these twists being more prominent at higher temperatures, likely as a result of shielding of hydrophobic methyl groups from water. For MC chains with varying DS, at 348 K, atomistic simulations show a nonmonotonic effect of DS on water-monomer contacts. For 0.0 < DS < 0.6, the MC monomers have more water contacts than at DS = 0.0 or DS > 0.6, suggesting that with few methoxy substitutions, the MC chains are effectively hydrophilic, letting the water molecules diffuse into the fibril to participate in H-bonds with the MC chains' remaining -OH groups. At DS > 0.6, the MC monomers become increasingly hydrophobic, as seen by decreasing water contacts around each monomer. We conclude based on the atomistic observations that MC chains with lower degrees of substitutions (DS ≤ 0.6) should exhibit solubility in water over broader temperature ranges than DS ∼ 1.8 chains.

Helical motors and formins synergize to compact chiral filopodial bundles: A theoretical perspective

Chiral actin bundles have been shown to play an important role in cell dynamics, but our understanding of the molecular mechanisms which combine to generate chirality remains incomplete. To address this, we numerically simulate a crosslinked filopodial bundle under the actions of helical myosin motors and/or formins and examine the collective buckling and twisting of the actin bundle. We first show that a number of proposed mechanisms to buckle polymerizing actin bundles without motor activity fail under biologically-realistic parameters. We then demonstrate that a simplified model of myosin spinning action at the bundle base effectively "braids" the bundle, but cannot control compaction at the fiber tips. Finally, we show that formin-mediated polymerization and motor activity can act synergitically to compact filopodium bundles, as motor activity bends filaments into shapes that activate twist forces induced by formins. Stochastic fluctuations of actin polymerization rates and slower cross linking dynamics both increase buckling and decrease compaction. We discuss implications of our findings for mechanisms of cytoskeletal chirality.

How a Soft Rod Wraps around a Rotating Cylinder

The unique characteristics of helical coils are utilized in nature, manufacturing processes, and daily life. These coils are also pivotal in the development of soft machines, such as artificial muscles and soft grippers. The stability of these helical coils is generally dependent on the mechanical properties of the rods and geometry of the supporting objects. In this Letter, the shapes formed by a flexible, heavy rod wrapping around a slowly rotating rigid cylinder are investigated through a combination of experimental and theoretical approaches. Three distinct morphologies-tight coiling, helical wrapping, and no wrapping-are identified experimentally. These findings are rationalized by numerical simulations and a geometrically nonlinear Kirchhoff rod theory. Despite the frictional contact present, the local shape of the rod is explained by the interplay between bending elasticity, gravity, and the geometry of the system. Our Letter provides a comprehensive physical understanding of the ordered morphology of soft threads and rods. Implications of this understanding are significant for a wide range of phenomena, from the recently discovered wrapping motility mode of bacterial flagella to the design of an octopus-inspired soft gripper.

Thermal stability and secondary aggregation of self-limiting, geometrically frustrated assemblies: Chain assembly of incommensurate polybricks

In geometrically frustrated assemblies, equilibrium self-limitation manifests in the form of a minimum in the free energy per subunit at a finite, multisubunit size which results from the competition between the elastic costs of frustration within an assembly and the surface energy at its boundaries. Physical realizations-from ill-fitting particle assemblies to self-twisting protein superstructures-are capable of multiple mechanisms of escaping the cumulative costs of frustration, resulting in unlimited equilibrium assembly, including elastic modes of "shape flattening" and the formation of weak, defective bonds that screen intra-assembly stresses. Here we study a model of one-dimensional chain assembly of incommensurate "polybricks" and determine its equilibrium assembly as a function of temperature, concentration, degree of shape frustration, elasticity, and interparticle binding, notably focusing on how weakly cohesive, defective bonds give rise to strongly temperature-dependent assembly. Complex assembly behavior derives from the competition between multiple distinct local minima in the free-energy landscape, including self-limiting chains, weakly bound aggregates of self-limiting chains, and strongly bound, elastically defrustrated assemblies. We show that this scenario, in general, gives rise to anomalous multiple aggregation behavior, in which disperse subunits (stable at low concentration and high temperature) first exhibit a primary aggregation transition to self-limiting chains (at intermediate concentration and temperature) which are ultimately unstable to condensation into unlimited assembly of finite-chains through weak binding beyond a secondary aggregation transition (at low temperature and high concentration). We show that window of stable self-limitation is determined both by the elastic costs of frustration in the assembly as well as energetic and entropic features of intersubunit binding.

Ionic environment effects on collagen type II persistence length and assembly

Collagen type II is a main structural protein in cartilage and forms fibrils. The radius of the fibrils ranges from 50 nm to a few hundred nm, and previous theoretical studies point to electrostatics and collagen elasticity (measured as the persistence length, l_p) as the main origin for the self-limiting size scales. In this study, we have investigated the collagen triple helical structure and fibril size scales in pH 2 solutions with varying NaCl concentrations from 10^-4 to 100 mM, at which collagen is positively charged, and in pH 7.4 solutions, with varying ionic strengths from 100 to 250 mM, at which collagen is both positively and negatively charged. Using static and dynamic light scattering, the radius of gyration (R_g), hydrodynamic radius (R_h), and second virial coefficient (A₂) of collagen triple helices are determined, and l_p is calculated. With increasing ionic strength, triple helical l_p decreases in pH 2 solutions and increases in pH 7.4 solutions. The value ranges from 60 to 100 nm depending on the ionic environment, but at the salt concentration at which A₂ is near zero, there are no net backbone interactions in solution, and the intrinsic collagen triple helix l_p is determined to be 90-95 nm. Electron microscopy is used to determine the diameter of fibrils assembled in pH 7.4 conditions, and we compare l_p of the collagen triple helices and fibril diameter using recent theory on fibril assembly. By better understanding collagen l_p and fibril assembly, we can further understand mechanisms of biomacromolecule self-assembly.

Multiscale hierarchical structures from a nanocluster mesophase

Spontaneous hierarchical self-organization of nanometre-scale subunits into higher-level complex structures is ubiquitous in nature. The creation of synthetic nanomaterials that mimic the self-organization of complex superstructures commonly seen in biomolecules has proved challenging due to the lack of biomolecule-like building blocks that feature versatile, programmable interactions to render structural complexity. In this study, highly aligned structures are obtained from an organic-inorganic mesophase composed of monodisperse Cd₃₇S₁₈ magic-size cluster building blocks. Impressively, structural alignment spans over six orders of magnitude in length scale: nanoscale magic-size clusters arrange into a hexagonal geometry organized inside micrometre-sized filaments; self-assembly of these filaments leads to fibres that then organize into uniform arrays of centimetre-scale bands with well-defined surface periodicity. Enhanced patterning can be achieved by controlling processing conditions, resulting in bullseye and 'zigzag' stacking patterns with periodicity in two directions. Overall, we demonstrate that colloidal nanomaterials can exhibit a high level of self-organization behaviour at macroscopic-length scales.

A chiral smectic phase induced by an alternating external field

Using simple achiral building blocks modulated by an external field to achieve chiral liquid crystal phases remains a challenge. In this study, a chiral helix liquid crystal phase is obtained for a simple Gay-Berne ellipsoid model under an alternating external field by using molecular dynamics simulations. Our results show that the chiral helix liquid crystal phase can be observed in a wide range of external field strengths when the oscillation period is smaller than the rotational characteristic diffusion timescale of ellipsoids. In addition, we find that the pitch and tilt angle of the helix structure can also be adjusted by changing the strength and oscillation period of the applied alternating external field. This may provide a feasible route for the regulation of chiral liquid crystal phases by an alternating external field.

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.

Topological defects produce kinks in biopolymer filament bundles

Bundles of stiff filaments are ubiquitous in the living world, found both in the cytoskeleton and in the extracellular medium. These bundles are typically held together by smaller cross-linking molecules. We demonstrate, analytically, numerically, and experimentally, that such bundles can be kinked, that is, have localized regions of high curvature that are long-lived metastable states. We propose three possible mechanisms of kink stabilization: a difference in trapped length of the filament segments between two cross-links, a dislocation where the endpoint of a filament occurs within the bundle, and the braiding of the filaments in the bundle. At a high concentration of cross-links, the last two effects lead to the topologically protected kinked states. Finally, we explore, numerically and analytically, the transition of the metastable kinked state to the stable straight bundle.

Equilibrium mechanisms of self-limiting assembly

Self-assembly is a ubiquitous process in synthetic and biological systems, broadly defined as the spontaneous organization of multiple subunits (e.g. macromolecules, particles) into ordered multi-unit structures. The vast majority of equilibrium assembly processes give rise to two states: one consisting of dispersed disassociated subunits, and the other, a bulk-condensed state of unlimited size. This review focuses on the more specialized class of self-limiting assembly, which describes equilibrium assembly processes resulting in finite-size structures. These systems pose a generic and basic question, how do thermodynamic processes involving non-covalent interactions between identical subunits "measure" and select the size of assembled structures? In this review, we begin with an introduction to the basic statistical mechanical framework for assembly thermodynamics, and use this to highlight the key physical ingredients that ensure equilibrium assembly will terminate at finite dimensions. Then, we introduce examples of self-limiting assembly systems, and classify them within this framework based on two broad categories: self-closing assemblies and open-boundary assemblies. These include well-known cases in biology and synthetic soft matter - micellization of amphiphiles and shell/tubule formation of tapered subunits - as well as less widely known classes of assemblies, such as short-range attractive/long-range repulsive systems and geometrically-frustrated assemblies. For each of these self-limiting mechanisms, we describe the physical mechanisms that select equilibrium assembly size, as well as potential limitations of finite-size selection. Finally, we discuss alternative mechanisms for finite-size assemblies, and draw contrasts with the size-control that these can achieve relative to self-limitation in equilibrium, single-species assemblies.

Self-Limiting Polymerization of DNA Origami Subunits with Strain Accumulation

Biology demonstrates how a near infinite array of complex systems and structures at many scales can originate from the self-assembly of component parts on the nanoscale. But to fully exploit the benefits of self-assembly for nanotechnology, a crucial challenge remains: How do we rationally encode well-defined global architectures in subunits that are much smaller than their assemblies? Strain accumulation via geometric frustration is one mechanism that has been used to explain the self-assembly of global architectures in diverse and complex systems a posteriori. Here we take the next step and use strain accumulation as a rational design principle to control the length distributions of self-assembling polymers. We use the DNA origami method to design and synthesize a molecular subunit known as the PolyBrick, which perturbs its shape in response to local interactions via flexible allosteric blocking domains. These perturbations accumulate at the ends of polymers during growth, until the deformation becomes incompatible with further extension. We demonstrate that the key thermodynamic factors for controlling length distributions are the intersubunit binding free energy and the fundamental strain free energy, both which can be rationally encoded in a PolyBrick subunit. While passive polymerization yields geometrical distributions, which have the highest statistical length uncertainty for a given mean, the PolyBrick yields polymers that approach Gaussian length distributions whose variance is entirely determined by the strain free energy. We also show how strain accumulation can in principle yield length distributions that become tighter with increasing subunit affinity and approach distributions with uniform polymer lengths. Finally, coarse-grained molecular dynamics and Monte Carlo simulations delineate and quantify the dominant forces influencing strain accumulation in a molecular system. This study constitutes a fundamental investigation of the use of strain accumulation as a rational design principle in molecular self-assembly.

Mechanical basis for fibrillar bundle morphology

Understanding the morphology of self-assembled fibrillar bundles and aggregates is relevant to a range of problems in molecular biology, supramolecular chemistry and materials science. Here, we propose a coarse-grained approach that averages over specific molecular details and yields an effective mechanical theory for the spatial complexity of self-assembling fibrillar structures that arises due to the competing effects of (the bending and twisting) elasticity of individual filaments and the adhesive interactions between them. We show that our theoretical framework accounting for this allows us to capture a number of diverse fibril morphologies observed in natural and synthetic systems, ranging from Filopodia to multi-walled carbon nanotubes, and leads to a phase diagram of possible fibril shapes. We also show how the extreme sensitivity of these morphologies can lead to spatially chaotic structures. Together, these results suggest a common mechanical basis for mesoscale fibril morphology as a function of the nanoscale mechanical properties of its filamentous constituents.

Phase diagram of two-patch colloids with competing anisotropic and isotropic interactions

Patchy particles are considered to be a good model for protein aggregation. We propose a novel method to generate different structures of glucose isomerase protein such as chains, crystals and bundles by utilising aggregation of two-patch colloidal particles in presence of competing isotropic and anisotropic potential. We calculate the equilibrium phase diagram of two-patch colloidal particles and demonstrates the coexistence of different phases like disordered clusters, chains, crystals and bundles depending on the relative strength of isotropic and anisotropic potential. We also show that the formation of network of bundles is metastable against the formation of thermodynamically favored finite sized bundles along with thermodynamically stable crystals. These bundles appear to be helical in structure similar to that observed in sickle cell hemoglobin. The simulation results show that the method can characterize phase behaviour of glucose isomerase protein, which provides a novel tool to unveil self-assembly mechanism of protein under different conditions.

Polymer-Peptide Conjugates Convert Amyloid into Protein Nanobundles through Fragmentation and Lateral Association

The assembly of proteins into amyloid fibrils has become linked not only with the progression of myriad human diseases, but also important biological functions. Understanding and controlling the formation, structure, and stability of amyloid fibrils is therefore a major scientific goal. Here we utilize electron microscopy-based approaches combined with quantitative statistical analysis to show how recently developed kind of amyloid modulators-multivalent polymer-peptide conjugates (mPPCs)-can be applied to control the structure and stability of amyloid fibrils. In doing so, we demonstrate that mPPCs are able to convert 40-residue amyloid beta fibrils into ordered nanostructures through a combination of fragmentation and bundling. Fragmentation is shown to be consistent with a model where the rate constant of fibril breakage is independent of the fibril length, suggesting a local and specific interaction between fibrils and mPPCs. Subsequent bundling, which was previously not observed, leads to the formation of sheet-like nanostructures which are surprisingly much more uniform than the starting fibrils. These nanostructures have dimensions independent of the molecular weight of the mPPC and retain the molecular-level ordering of the starting amyloid fibrils. Collectively, we reveal quantitative and nanoscopic understanding of how mPPCs can be applied to control amyloid structure and stability, and demonstrate approaches to elucidate nanoscale amyloid phase behavior in the presence of functional macromolecules and other modulators.

Chiral and achiral mechanisms of self-limiting assembly of twisted bundles

A generalized theory of the self-limiting assembly of twisted bundles of filaments and columns is presented. Bundles and fibers form in a broad variety of supramolecular systems, from biological to synthetic materials. A widely-invoked mechanism to explain their finite diameter relies on chirality transfer from the molecular constituents to collective twist of the assembly, the effect of which frustrates the lateral assembly and can select equilibrium, finite diameters of bundles. In this article, the thermodynamics of twisted-bundle assembly is analyzed to understand if chirality transfer is necessary for self-limitation, or instead, if spontaneously-twisting, achiral bundles also exhibit self-limited assembly. A generalized description is invoked for the elastic costs imposed by twist for bundles of various states of intra-bundle order from nematic to crystalline, as well as a generic mechanism for generating twist, classified both by chirality but also the twist susceptibility of inter-filament alignment. The theory provides a comprehensive set of predictions for the equilibrium twist and size of bundles as a function of surface energy as well as chirality, twist susceptibility, and elasticity of bundles. Moreover, it shows that while spontaneous twist can lead to self-limitation, assembly of twisted achiral bundles can be distinguished qualitatively in terms of their range of equilibrium sizes and thermodynamic stability relative to bulk (untwisted) states.

Conformational switching of chiral colloidal rafts regulates raft-raft attractions and repulsions

Membrane-mediated particle interactions depend both on the properties of the particles themselves and the membrane environment in which they are suspended. Experiments have shown that chiral rod-like inclusions dissolved in a colloidal membrane of opposite handedness assemble into colloidal rafts, which are finite-sized reconfigurable droplets consisting of a large but precisely defined number of rods. We systematically tune the chirality of the background membrane and find that, in the achiral limit, colloidal rafts acquire complex structural properties and interactions. In particular, rafts can switch between 2 chiral states of opposite handedness, which alters the nature of the membrane-mediated raft-raft interactions. Rafts with the same chirality have long-ranged repulsions, while those with opposite chirality acquire attractions with a well-defined minimum. Both attractive and repulsive interactions are qualitatively explained by a continuum model that accounts for the coupling between the membrane thickness and the local tilt of the constituent rods. These switchable interactions enable assembly of colloidal rafts into intricate higher-order architectures, including stable tetrameric clusters and "ionic crystallites" of counter-twisting domains organized on a binary square lattice. Furthermore, the properties of individual rafts, such as their sizes, are controlled by their complexation with other rafts. The emergence of these complex behaviors can be rationalized purely in terms of generic couplings between compositional and orientational order of fluids of rod-like elements. Thus, the uncovered principles might have relevance for conventional lipid bilayers, in which the assembly of higher-order structures is also mediated by complex membrane-mediated interactions.

Helical Superstructure of Intermediate Filaments

Intermediate filaments are the least explored among the large cytoskeletal elements. We show here that they display conformational anomalies in narrow microfluidic channels. Their unusual behavior can be understood as the consequence of a previously undetected, large-scale helically curved superstructure. Confinement in a channel orders the otherwise soft, strongly fluctuating helical filaments and enhances their structural correlations, giving rise to experimentally detectable, strongly oscillating tangent correlation functions. We propose an explanation for the detected intrinsic curving phenomenon-an elastic shape instability that we call autocoiling. The mechanism involves self-induced filament buckling via a surface stress located at the outside of the cross section. The results agree with ultrastructural findings and rationalize for the commonly observed looped intermediate filament shapes. Beyond curvature, explaining the molecular origin of the detected helical torsion remains an interesting challenge.

Modulation of toll-like receptor signaling by antimicrobial peptides

Antimicrobial peptides (AMPs) are typically thought of as molecular hole punchers that directly kill pathogens by membrane permeation. However, recent work has shown that AMPs are pleiotropic, multifunctional molecules that can strongly modulate immune responses. In this review, we provide a historical overview of the immunomodulatory properties of natural and synthetic antimicrobial peptides, with a special focus on human cathelicidin and defensins. We also summarize the various mechanisms of AMP immune modulation and outline key structural rules underlying the recently-discovered phenomenon of AMP-mediated Toll-like receptor (TLR) signaling. In particular, we describe several complementary studies demonstrating how AMPs self-assemble with nucleic acids to form nanocrystalline complexes that amplify TLR-mediated inflammation. In a broader scope, we discuss how this new conceptual framework allows for the prediction of immunomodulatory behavior in AMPs, how the discovery of hidden antimicrobial activity in known immune signaling proteins can inform these predictions, and how these findings reshape our understanding of AMPs in normal host defense and autoimmune disease.

Theory of microphase separation in bidisperse chiral membranes

We present a Ginzburg-Landau theory of microphase separation in a bidisperse chiral membrane consisting of rods of opposite handedness. This model system undergoes a phase transition from an equilibrium state where the two components are completely phase separated to a state composed of microdomains of a finite size comparable to the twist penetration depth. Characterizing the phenomenology using linear stability analysis and numerical studies, we trace the origin of the discontinuous change in microdomain size that occurs during this phase transition to a competition between the cost of creating an interface and the gain in twist energy for small microdomains in which the twist penetrates deep into the center of the domain.

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 employ a theoretical model for the shape and thermodynamics of intrinsically helical filaments bound to curved membranes. We show that filament-surface interactions lead to a host of nonuniform 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 nonlinear 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's 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.

How geometric frustration shapes twisted fibres, inside and out: competing morphologies of chiral filament assembly

Chirality frustrates and shapes the assembly of flexible filaments in rope-like, twisted bundles and fibres by introducing gradients of both filament shape (i.e. curvature) and packing throughout the structure. Previous models of chiral filament bundle formation have shown that this frustration gives rise to several distinct morphological responses, including self-limiting bundle widths, anisotropic domain (tape-like) formation and topological defects in the lateral inter-filament order. In this paper, we employ a combination of continuum elasticity theory and discrete filament bundle simulations to explore how these distinct morphological responses compete in the broader phase diagram of chiral filament assembly. We show that the most generic model of bundle formation exhibits at least four classes of equilibrium structure-finite-width, twisted bundles with isotropic and anisotropic shapes, with and without topological defects, as well as bulk phases of untwisted, columnar assembly (i.e. 'frustration escape'). These competing equilibrium morphologies are selected by only a relatively small number of parameters describing filament assembly: bundle surface energy, preferred chiral twist and stiffness of chiral filament interactions, and mechanical stiffness of filaments and their lateral interactions. Discrete filament bundle simulations test and verify continuum theory predictions for dependence of bundle structure (shape, size and packing defects of two-dimensional cross section) on these key parameters.

Effects of the Chiral Interface and Orientation-Dependent Segmental Interactions on Twisting of Self-Assembled Block Copolymers

The chirality effect on the self-assembly of block copolymers (BCPs) composed of chiral entities, denoted as chiral BCPs (BCPs*), gives rise to the formation of the helical phase (H*) due to intermolecular chiral interactions. The corresponding twisting of self-assembled H* might be initiated from the microphase-separated chiral interface and/or attributed to chiral orientation-dependent segmental interactions. To examine the origins of the twisting mechanisms, a series of polylactide-containing BCPs* with different sequences and molecular weights of chiral block, poly(l-lactide) (PLLA), and achiral block, poly(d,l-lactide) (PLA), were designed for self-assembling. Accordingly, the impact of sequence and length of the blocks on self-assembled BCPs* can be scrutinized. For the ones with the PLLA as a middle block, polystyrene-b-poly(l-lactide)-b-poly(d,l-lactide), the formation of H* can be achieved even with a short PLLA segment. By contrast, the self-assembled morphology of polystyrene-b-poly(d,l-lactide)-b-poly(l-lactide) with the PLA as a middle block is dependent upon the length of the PLA block. On the basis of the self-assembled results, the chirality effect on the self-assembly of BCP* is originated from both the chiral interface and the chiral orientation-dependent segmental interactions. The present work on the study of the formation mechanisms of the H* thus provides insights into the intermolecular chiral interactions from the self-assembly of BCP*.

Genetic defects in β-spectrin and tau sensitize C. elegans axons to movement-induced damage via torque-tension coupling

Our bodies are in constant motion and so are the neurons that invade each tissue. Motion-induced neuron deformation and damage are associated with several neurodegenerative conditions. Here, we investigated the question of how the neuronal cytoskeleton protects axons and dendrites from mechanical stress, exploiting mutations in UNC-70 β-spectrin, PTL-1 tau/MAP2-like and MEC-7 β-tubulin proteins in Caenorhabditis elegans. We found that mechanical stress induces supercoils and plectonemes in the sensory axons of spectrin and tau double mutants. Biophysical measurements, super-resolution, and electron microscopy, as well as numerical simulations of neurons as discrete, elastic rods provide evidence that a balance of torque, tension, and elasticity stabilizes neurons against mechanical deformation. We conclude that the spectrin and microtubule cytoskeletons work in combination to protect axons and dendrites from mechanical stress and propose that defects in β-spectrin and tau may sensitize neurons to damage.

DOI:http://dx.doi.org/10.7554/eLife.20172.001

Faceted particles formed by the frustrated packing of anisotropic colloids on curved surfaces

We use computer simulations and simple theoretical models to analyze the morphologies that result when rod-like particles end-attach onto a curved surface, creating a finite-thickness monolayer aligned with the surface normal. This geometry leads to two forms of frustration, one associated with the incompatibility of hexagonal order on surfaces with Gaussian curvature, and the second reflecting the deformation of a layer with finite thickness on a surface with non-zero mean curvature. We show that the latter effect leads to a faceting mechanism. Above threshold values of inter-particle attraction strength and surface mean curvature, the adsorbed layer undergoes a transition from orientational disorder to an ordered state that is demarcated by reproducible patterns of line defects. The number of facets is controlled by the competition between line defect energy and intra-facet strain. Tuning control parameters thus leads to a rich variety of morphologies, including icosahedral particles and irregular polyhedra. In addition to suggesting a new strategy for the synthesis of aspherical particles with tunable symmetries, our results may shed light on recent experiments in which rod-like HIV GAG proteins assemble around nanoscale particles.

Dilute Semiflexible Polymers with Attraction: Collapse, Folding and Aggregation

We review the current state on the thermodynamic behavior and structural phases of self- and mutually-attractive dilute semiflexible polymers that undergo temperature-driven transitions. In extreme dilution, polymers may be considered isolated, and this single polymer undergoes a collapse or folding transition depending on the internal structure. This may go as far as to stable knot phases. Adding polymers results in aggregation, where structural motifs again depend on the internal structure. We discuss in detail the effect of semiflexibility on the collapse and aggregation transition and provide perspectives for interesting future investigations.

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.

Stability of actin-lysozyme complexes formed in cystic fibrosis disease

Finding the conditions for destabilizing actin-lysozyme complexes is of biomedical importance in preventing infections in cystic fibrosis. In this manuscript, the effects of different charge-mutants of lysozyme and salt concentration on the stability of actin-lysozyme complexes are studied using Langevin dynamics simulation. A coarse-grained model of F-actin is used in which both its twist and bending rigidities are considered. We observe that the attraction between F-actins is stronger in the presence of wild-type lysozymes relative to the mutated lysozymes of lower charges. By calculating the potential of mean force between F-actins, we conclude that the stability of actin-lysozyme complexes is decreased by reducing the charge of lysozyme mutants. The distributions of different lysozyme charge-mutants show that wild-type (+9e) lysozymes are mostly accumulated in the center of triangles formed by three adjacent F-actins, while lysozyme mutants of charges +7e and +5e occupy the bridging regions between F-actins. Low-charge mutants of lysozyme (+3e) distribute uniformly around F-actins. A rough estimate of the electrostatic energy for these different distributions proves that the distribution in which lysozymes reside in the center of triangles leads to more stable complexes. Also our results in the presence of a salt suggest that at physiological salt concentration of airway, F-actin complexes are not formed by charge-reduced mutants of lysozyme. The findings are interesting because if we can design charge-reduced lysozyme mutants with considerable antibacterial activity, they are not sequestered inside F-actin aggregates and can play their role as antibacterial agents against airway infection.

Semiflexible Biopolymers in Bundled Arrangements

Bundles and networks of semiflexible biopolymers are key elements in cells, lending them mechanical integrity while also enabling dynamic functions. Networks have been the subject of many studies, revealing a variety of fundamental characteristics often determined via bulk measurements. Although bundles are equally important in biological systems, they have garnered much less scientific attention since they have to be probed on the mesoscopic scale. Here, we review theoretical as well as experimental approaches, which mainly employ the naturally occurring biopolymer actin, to highlight the principles behind these structures on the single bundle level.

Morphology selection via geometric frustration in chiral filament bundles

In assemblies, the geometric frustration of a locally preferred packing motif leads to anomalous behaviours, from self-limiting growth to defects in the ground state. Here, we demonstrate that geometric frustration selects the equilibrium morphology of cohesive bundles of chiral filaments, an assembly motif critical to a broad range of biological and synthetic nanomaterials. Frustration of inter-filament spacing leads to optimal shapes of self-twisting bundles that break the symmetries of packing and of the underlying inter-filament forces, paralleling a morphological instability in spherical two-dimensional crystals. Equilibrium bundle morphology is controlled by a parameter that characterizes the relative costs of filament bending and the straining of cohesive bonds between filaments. This parameter delineates the boundaries between stable, isotropic cylindrical bundles and anisotropic, twisted-tape bundles. We also show how the mechanical and interaction properties of constituent amyloid fibrils may be extracted from the mesoscale dimensions of the anisotropic bundles that they form.

Bundling in semiflexible polymers: A theoretical overview

Supramolecular assemblies of polymers are key modules to sustain the structure of cells and their function. The main elements of these assemblies are charged semiflexible polymers (polyelectrolytes) generally interacting via a long(er)-range repulsion and a short(er)-range attraction. The most common supramolecular structure formed by these polymers is the bundle. In the present paper, we critically review some recent theoretical and computational advances on the problem of bundle formation, and point a few promising directions for future work.

A review of immune amplification via ligand clustering by self-assembled liquid-crystalline DNA complexes

We examine how the interferon production of plasmacytoid dendritic cells is amplified by the self-assembly of liquid-crystalline antimicrobial peptide/DNA complexes. These specialized dendritic cells are important for host defense because they quickly release large quantities of type I interferons in response to infection. However, their aberrant activation is also correlated with autoimmune diseases such as psoriasis and lupus. In this review, we will describe how polyelectrolyte self-assembly and the statistical mechanics of multivalent interactions contribute to this process. In a more general compass, we provide an interesting conceptual corrective to the common notion in molecular biology of a dichotomy between specific interactions and non-specific interactions, and show examples where one can construct exquisitely specific interactions using non-specific interactions.

Multi-scale strain-stiffening of semiflexible bundle networks

Bundles of polymer filaments are responsible for the rich and unique mechanical behaviors of many biomaterials, including cells and extracellular matrices. In fibrin biopolymers, whose nonlinear elastic properties are crucial for normal blood clotting, protofibrils self-assemble and bundle to form networks of semiflexible fibers. Here we show that the extraordinary strain-stiffening response of fibrin networks is a direct reflection of the hierarchical architecture of the fibrin fibers. We measure the rheology of networks of unbundled protofibrils and find excellent agreement with an affine model of extensible wormlike polymers. By direct comparison with these data, we show that physiological fibrin networks composed of thick fibers can be modeled as networks of tight protofibril bundles. We demonstrate that the tightness of coupling between protofibrils in the fibers can be tuned by the degree of enzymatic intermolecular crosslinking by the coagulation factor XIII. Furthermore, at high stress, the protofibrils contribute independently to the network elasticity, which may reflect a decoupling of the tight bundle structure. The hierarchical architecture of fibrin fibers can thus account for the nonlinearity and enormous elastic resilience characteristic of blood clots.

Physical model for the geometry of actin-based cellular protrusions

Actin-based cellular protrusions are a ubiquitous feature of cell morphology, e.g., filopodia and microvilli, serving a huge variety of functions. Despite this, there is still no comprehensive model for the mechanisms that determine the geometry of these protrusions. We present here a detailed computational model that addresses a combination of multiple biochemical and physical processes involved in the dynamic regulation of the shape of these protrusions. We specifically explore the role of actin polymerization in determining both the height and width of the protrusions. Furthermore, we show that our generalized model can explain multiple morphological features of these systems, and account for the effects of specific proteins and mutations.

Geometry of flexible filament cohesion: better contact through twist?

Cohesive interactions between filamentous molecules have broad implications for a range of biological and synthetic materials. While long-standing theoretical approaches have addressed the problem of inter-filament forces from the limit of infinitely rigid rods, the ability of flexible filaments to deform intra-filament shape in response to changes in inter-filament geometry has a profound affect on the nature of cohesive interactions. In this paper, we study two theoretical models of inter-filament cohesion in the opposite limit, in which filaments are sufficiently flexible to maintain cohesive contact along their contours, and address, in particular, the role played by helical-interfilament geometry in defining interactions. Specifically, we study models of featureless, tubular filaments interacting via: (1) pair-wise Lennard-Jones (LJ) interactions between surface elements and (2) depletion-induced filament binding stabilized by electrostatic surface repulsion. Analysis of these models reveals a universal preference for cohesive filament interactions for non-zero helical skew, and further, that in the asymptotic limit of vanishing interaction range relative to filament diameter, the skew-dependence of cohesion approaches a geometrically defined limit described purely by the close-packing geometry of twisted tubular filaments. We further analyze non-universal features of the skew-dependence of cohesion at small-twist for both potentials, and argue that in the LJ model the pair-wise surface attraction generically destabilizes parallel filaments, while in the second model, pair-wise electrostatic repulsion in combination with non-pairwise additivity of depletion leads to a meta-stable parallel state.

Chirality Transfer in Block Copolymer Melts: Emerging Concepts

Chirality transfer from molecule to assembly is a ubiquitous process, occurring in every class of self-assembling materials, from liquid crystals to biological matter. Yet, a basic understanding of the influence of molecular chirality on the mesoscopic assembly of block copolymers lags decades behind nearly all other aspects of their structure (e.g., chain composition, topology, stiffness, interactions). This Viewpoint highlights recent experimental and theoretical studies of mesochiral assemblies of chiral block copolymers that are beginning to shed light onto the necessary conditions for and principle outcomes of chirality transfer in block copolymer melts.

Helical buckling in columnar assemblies of soft discotic mesogens

We investigate the emergence of chiral meso-structures in one-dimensional fluids consisting of stacked discotic particles and demonstrate that helical undulations are generated spontaneously from internal elastic stresses. The stability of these helical conformations arises from an interplay between long-ranged soft repulsions and nanopore confinement which is naturally present in columnar liquid crystals. Using a simple mean-field theory based on microscopic considerations we identify generic scaling expressions for the typical buckling radius and helical pitch as a function of the density and interaction potential of the constituent particles.

Designing highly tunable semiflexible filament networks

Semiflexible polymers can generate a range of filamentous networks significantly different in structure from those seen in conventional polymer solutions. Our coarse-grained simulations with an implicit cross-linker potential show that networks of branching bundles, knotted morphologies, and structural chirality can be generated by a generalized approach independent of specific cross-linkers. Network structure depends primarily on filament flexibility and separation, with significant connectivity increase after percolation. Results should guide the design of engineered semiflexible polymers.

Chirality in block copolymer melts: mesoscopic helicity from intersegment twist

We study the effects of chirality at the segment scale on the thermodynamics of block copolymer melts using self-consistent field theory. In linear diblock melts where segments of one block prefer a twisted, or cholesteric, texture, we show that melt assembly is critically sensitive to the ratio of random coil size to the preferred pitch of cholesteric twist. For weakly chiral melts (large pitch), mesophases remain achiral, while below a critical value of pitch, two mesoscopically chiral phases are stable: an undulated lamellar phase and a phase of hexagonally ordered helices. We show that the nonlinear sensitivity of mesoscale chiral order to preferred pitch derives specifically from the geometric and thermodynamic coupling of the helical mesodomain shape to the twisted packing of chiral segments within the core, giving rise to a second-order cylinder-to-helix transition.

Non-euclidean geometry of twisted filament bundle packing

Densely packed and twisted assemblies of filaments are crucial structural motifs in macroscopic materials (cables, ropes, and textiles) as well as synthetic and biological nanomaterials (fibrous proteins). We study the unique and nontrivial packing geometry of this universal material design from two perspectives. First, we show that the problem of twisted bundle packing can be mapped exactly onto the problem of disc packing on a curved surface, the geometry of which has a positive, spherical curvature close to the center of rotation and approaches the intrinsically flat geometry of a cylinder far from the bundle center. From this mapping, we find the packing of any twisted bundle is geometrically frustrated, as it makes the sixfold geometry of filament close packing impossible at the core of the fiber. This geometrical equivalence leads to a spectrum of close-packed fiber geometries, whose low symmetry (five-, four-, three-, and twofold) reflect non-euclidean packing constraints at the bundle core. Second, we explore the ground-state structure of twisted filament assemblies formed under the influence of adhesive interactions by a computational model. Here, we find that the underlying non-euclidean geometry of twisted fiber packing disrupts the regular lattice packing of filaments above a critical radius, proportional to the helical pitch. Above this critical radius, the ground-state packing includes the presence of between one and six excess fivefold disclinations in the cross-sectional order.

Hierarchical self-assembly of chiral fibres from achiral particles

We investigate, by molecular dynamics simulation, the behaviour of discotic particles in a solvent of Lennard-Jones spheres. When chromonic disc-sphere interactions are imposed on these systems, three regimes of self-assembly are observed. At moderate temperatures, numerous short threads of discs develop, but these threads remain isolated from one another. Quenching to low temperatures, alternatively, causes all of the discs to floc into a single extended aggregate which typically comprises several distinct sections and contains numerous packing defects. For a narrow temperature range between these regimes, however, defect-free chiral fibres are found to freely self-assemble. The spontaneous chirality of these fibres results from frustration between the hexagonal packing and interdigitation of neighbouring threads, the pitch being set by the particle shape. This demonstration of aggregate-wide chirality emerging owing to packing alone is pertinent to many biological and synthetic hierarchically self-assembling systems.