Twisted protein aggregates and disease: the stability of sickle hemoglobin fibers

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.

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.

Crystal Structure and Twisted Aggregates of Oxcarbazepine Form III

Polymorphism and crystal habit play vital roles in dictating the properties of crystalline materials. Here, the structure and properties of oxcarbazepine (OXCBZ) form III are reported along with the occurrence of twisted crystalline aggregates of this metastable polymorph. OXCBZ III can be produced by crystallization from the vapor phase and by recrystallization from solution. The crystallization process used to obtain OXCBZ III is found to affect the pitch, with the most prominent effect observed from the sublimation-grown OXCBZ III material where the pitch increases as the length of aggregates increases. Sublimation-grown OXCBZ III follows an unconventional mechanism of formation with condensed droplet formation and coalescence preceding nucleation and growth of aggregates. A crystal structure determination of OXCBZ III from powder X-ray diffraction methods, assisted by crystal structure prediction (CSP), reveals that OXCBZ III, similar to carbamazepine form II, contains void channels in its structure with the channels, aligned along the c crystallographic axis, oriented parallel to the twist axis of the aggregates. The likely role of structural misalignment at the lattice or nanoscale is explored by considering the role of molecular and closely related structural impurities informed by crystal structure prediction.

Thermodynamic Size Control in Curvature-Frustrated Tubules: Self-Limitation with Open Boundaries

We use computational modeling to investigate the assembly thermodynamics of a particle-based model for geometrically frustrated assembly, in which the local packing geometry of subunits is incompatible with uniform, strain-free large-scale assembly. The model considers discrete triangular subunits that drive assembly toward a closed, hexagonal-ordered tubule, but have geometries that locally favor negative Gaussian curvature. We use dynamical Monte Carlo simulations and enhanced sampling methods to compute the free energy landscape and corresponding self-assembly behavior as a function of experimentally accessible parameters that control assembly driving forces and the magnitude of frustration. The results determine the parameter range where finite-temperature self-limiting assembly occurs, in which the equilibrium assembly size distribution is sharply peaked around a well-defined finite size. The simulations also identify two mechanisms by which the system can escape frustration and assemble to unlimited size, and determine the particle-scale properties of subunits that suppress unbounded growth.

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.

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.

Confined filaments in soft vesicles - the case of sickle red blood cells

Abnormal shapes of red blood cells (RBC) have been associated with various diseases. Diverse RBC shapes have also been intriguing for membrane biophysics. Here we focus on sickle shaped RBC which form due to abnormal growth of semi-rigid hemoglobin (HbS) fibers confined in RBC. Using the area difference elasticity (ADE) model for RBC and worm-like chain model for the confined HbS fibers, we explore shape deformations at equilibrium using Monte-Carlo simulations. We show that while a single HbS fiber is not rigid enough to produce sickle like deformation, a fiber bundle can do so. We also consider multiple disjoint filaments and find that confinement can generate multipolar RBC shapes and can even promote helical filament conformations which have not been discussed before. We show that the same model, when applied to microtubules confined in phospholipid vesicles, predicts vesicle tubulation. In addition we reproduce the tube collapse transition and tennis racket type vesicle shapes, as reported in experiments. We conclude that with a decrease in the surface area to volume ratio, and membrane rigidity, the vesicles prefer tubulation over sickling. The highlight of this work is several important non-axisymmetric RBC and vesicle shapes, which have never been explored in simulations.

Mobile obstacles accelerate and inhibit the bundle formation in two-patch colloidal particle

Aggregation of protein into bundles is responsible for many neurodegenerative diseases. In this work, we show how two-patch colloidal particles self-assemble into chains and a sudden transition to bundles takes place by tuning the patch size and solvent condition. We study the kinetics of formation of chains, bundles, and networklike structures using patchy Brownian cluster dynamics. We also analyze the ways to inhibit and accelerate the formation of these bundles. We show that in the presence of inert immobile obstacles, the kinetics of formation of bundles slows down. However, in the presence of mobile aggregating particles, which exhibit interspecies hard sphere repulsion and intraspecies attraction, the kinetics of bundle formation accelerates slightly. We also show that if we introduce mobile obstacles, which exhibit interspecies attraction and intraspecies hard sphere repulsion, the kinetics of formation of bundles is inhibited. This is similar to the inhibitory effect of peptide P4 on the formation of insulin fibers. We are providing a model of mobile obstacles undergoing directional interactions to inhibit the formation of bundles.

Fiber Network Formation in Semi-Flexible Polymer Solutions: An Exploratory Computational Study

The formation of gels through the bundling of semi-flexible polymer chains into fiber networks is ubiquitous in diverse manufactured and natural materials, and, accordingly, we perform exploratory molecular dynamics simulations of a coarse-grained model of semi-flexible polymers in a solution with attractive lateral interchain interactions to understand essential features of this type of gel formation. After showing that our model gives rise to fibrous gels resembling real gels of this kind, we investigate how the extent of fiber bundling influences the "melting" temperature, T m , and the emergent rigidification of model bundled fibers having a fixed number of chains, N, within them. Based on our preliminary observations, we suggest the fiber size is kinetically selected by a reduced thermodynamic driving force and a slowing of the dynamics within the fibers associated with their progressive rigidification with the inclusion of an increasing number of chains in the bundle.

Mesoscopic Adaptive Resolution Scheme toward Understanding of Interactions between Sickle Cell Fibers

Understanding of intracellular polymerization of sickle hemoglobin (HbS) and subsequent interaction with the membrane of a red blood cell (RBC) is important to predict the altered morphologies and mechanical properties of sickle RBCs in sickle cell anemia. However, modeling the integrated processes of HbS nucleation, polymerization, HbS fiber interaction, and subsequent distortion of RBCs is challenging as they occur at multispatial scales, ranging from nanometers to micrometers. To make progress toward simulating the integrated processes, we propose a hybrid HbS fiber model, which couples fine-grained and coarse-grained HbS fiber models through a mesoscopic adaptive resolution scheme (MARS). To this end, we apply a microscopic model to capture the dynamic process of polymerization of HbS fibers, while maintaining the mechanical properties of polymerized HbS fibers by the mesoscopic model, thus providing a means of bridging the subcellular and cellular phenomena in sickle cell disease. At the subcellular level, this model can simulate HbS polymerization with preexisting HbS nuclei. At the cellular level, if combined with RBC models, the generated HbS fibers could be applied to study the morphologies and membrane stiffening of sickle RBCs. One important feature of the MARS is that it can be easily employed in other particle-based multiscale simulations where a dynamic coarse-graining and force-blending method is required. As demonstrations, we first apply the hybrid HbS fiber model to simulate the interactions of two growing fibers and find that their final configurations depend on the orientation and interaction distance between two fibers, in good agreement with experimental observations. We also model the formation of fiber bundles and domains so that we explore the mechanism that causes fiber branching.

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.

Biomechanics and biorheology of red blood cells in sickle cell anemia

Sickle cell anemia (SCA) is an inherited blood disorder that causes painful crises due to vaso-occlusion of small blood vessels. The primary cause of the clinical phenotype of SCA is the intracellular polymerization of sickle hemoglobin resulting in sickling of red blood cells (RBCs) in deoxygenated conditions. In this review, we discuss the biomechanical and biorheological characteristics of sickle RBCs and sickle blood as well as their implications toward a better understanding of the pathophysiology and pathogenesis of SCA. Additionally, we highlight the adhesive heterogeneity of RBCs in SCA and their specific contribution to vaso-occlusive crisis.

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.

Mechanical properties of normal and binormal double nanohelices

The normal double nanohelices made up of straight wires excel the binormal and rope-like double nanohelices in both load capacity and elasticity, and remain the mechanical stability.

Mechanically tough, elastic and stable rope-like double nanohelices

Double helix nanostructures have been the object of intense theoretical and experimental investigations in recent years due to their various types of available materials and unique morphology. Among these structures, rope-like double nanohelices of two strands in contact along a line can be obtained using any one-dimensional nanostructure. In this work, we establish a novel theory for quantitatively exploring the statics and dynamics of rope-like double nanohelices by employing the concept of the extensible Cosserat curve. The rope-like double nanohelices are tough, relatively elastic, and mechanically stable, which agrees well with the experiments. The characteristics of the interaction between the two strands, the tensile modulus and the torque are precisely described and explained across the entire stretching region. The proposed model offers in depth quantitative insight into the mechanics of double helix nanostructures, and supplies a reliable reference for further experimental research.

Methods for modeling cytoskeletal and DNA filaments

This review summarizes the models that researchers use to represent the conformations and dynamics of cytoskeletal and DNA filaments. It focuses on models that address individual filaments in continuous space. Conformation models include the freely jointed, Gaussian, angle-biased chain (ABC), and wormlike chain (WLC) models, of which the first three bend at discrete joints and the last bends continuously. Predictions from the WLC model generally agree well with experiment. Dynamics models include the Rouse, Zimm, stiff rod, dynamic WLC, and reptation models, of which the first four apply to isolated filaments and the last to entangled filaments. Experiments show that the dynamic WLC and reptation models are most accurate. They also show that biological filaments typically experience strong hydrodynamic coupling and/or constrained motion. Computer simulation methods that address filament dynamics typically compute filament segment velocities from local forces using the Langevin equation and then integrate these velocities with explicit or implicit methods; the former are more versatile and the latter are more efficient. Much remains to be discovered in biological filament modeling. In particular, filament dynamics in living cells are not well understood, and current computational methods are too slow and not sufficiently versatile. Although primarily a review, this paper also presents new statistical calculations for the ABC and WLC models. Additionally, it corrects several discrepancies in the literature about bending and torsional persistence length definitions, and their relations to flexural and torsional rigidities.

A clear view of polymorphism, twist, and chirality in amyloid fibril formation

The self-assembly of protein molecules into highly ordered linear aggregates, known as amyloid fibrils, is a phenomenon receiving increasing attention because of its biological roles in health and disease and the potential of these structures to form artificial proteinaceous scaffolds for biomaterials applications. A particularly powerful approach to probe the key physical properties of fibrillar structures is atomic force microscopy, which was used by Usov et al. in this issue of ACS Nano to reveal the polymorphic transitions and chirality inversions of amyloid fibrils in unprecedented detail. Starting from this study, this Perspective highlights recent progress in understanding the dynamic polymorphism, twisting behavior, and handedness of amyloid fibrils and discusses the promising future of these self-assembling structures as advanced functional materials with applications in nanotechnology and related fields.

Micellization model for the polymerization of clathrin baskets

A thermodynamic model is used to investigate the conditions under which clathrin triskelions form polyhedral baskets. The analysis, which is similar to classical methods used to study micelle formation, relates clathrin basket energetics to system parameters linked to triskelial rigidity, the natural curvature of an isolated triskelion, and interactions between triskelial legs in the assembled polyhedra. Mathematical theory predicts that a minimal ("critical") clathrin concentration, C(C), needs to be surpassed in order for basket polymerization to occur, and indicates how C(C), and the amount of polymerized material, depend on the chosen parameters. Analytical expressions are obtained to indicate how changes in the parameters affect the sizes of the polyhedra which arise when the total clathrin concentration exceeds C(C). A continuum analytic approximation then is used to produce numerical results that illustrate the derived dependences.

Effect of chain chirality on the self-assembly of sickle hemoglobin

We present simulation results on the self-assembly behavior of sickle hemoglobin (HbS). A coarse-grained HbS model, which contains hydrophilic and hydrophobic particles explicitly, is constructed to match the structural properties and physical description of HbS. The hydrophobic interactions are shown to be necessary with chirality being the main driver for the formation of HbS fibers. In the absence of chain chirality, only small self-assembled aggregates are observed whereas self-assembled elongated steplike bundle microstructures appear when we include chain chirality. We also investigate the effect of confinement on self-assembly, and find that elongated fibers-similar to open-space ones-can be obtained in hard confinement domains but cannot be formed within compliant red blood cell (RBC) domains under the same assumptions. We show, however, that by placing explicitly HbS fibers inside the RBCs and subjecting them to linear elongation and bending, we obtain different types of sickle-shaped RBCs as observed in sickle cell anemia.

Twisting transition between crystalline and fibrillar phases of aggregated peptides

We study two distinctly ordered condensed phases of polypeptide molecules, amyloid fibrils and amyloidlike microcrystals, and the first-order twisting phase transition between these two states. We derive a single free-energy form which connects both phases. Our model identifies relevant degrees of freedom for describing the collective behavior of supramolecular polypeptide structures, reproduces accurately the results from molecular dynamics simulations as well as from experiments, and sheds light on the uniform nature of the dimensions of different peptide fibrils.

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.

Defects in crystalline packings of twisted filament bundles. II. Dislocations and grain boundaries

Twisted and ropelike assemblies of filamentous molecules are common and vital structural elements in cells and tissues 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 nonlinear 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.

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.

NMR characterization of the formation kinetics and structure of di-O-benzylidene sorbitol gels self-assembled in organic solvents

The molecule 1,3:2,4-di-O-benzylidene sorbitol (DBS) is a common "gelator" that forms thermally reversible gels in diverse organic solvents. Solid-state (13)C and (1)H NMR techniques, along with electron microscopy, are utilized in an exploratory study of DBS in the gelled state where we consider both in situ and dried gels. The gels were formed in either acetone or benzene, with the former being a better solvent for DBS. We find the in situ or dried DBS gels to be composed of rigid twisted nanofibrils (∼15 to 21 nm in diameter). The fibrils show local molecular ordering, but not crystalline order, and they contain no trapped solvent. The molecular mobility at the fibril surface is modestly enhanced, and all the free hydroxyl groups of the sorbitol moiety are involved in strong hydrogen bonding. We also attempted to find a truly crystalline form of DBS whose structure, as judged by the similarity of (13)C spectra, is close to that of the fibrils. We partially succeeded in this quest, employing melt crystallization followed by slow cooling. However, this sample was a mixed crystal having small domains, where only one type of domain was structurally similar to the fibrils. We also investigated the long-time evolution of the in situ DBS gel network. Specifically, high-resolution NMR kinetic studies were performed over periods of days where the residual concentration of DBS in acetone solution was monitored during and after gel formation. The DBS concentration on these long timescales evolved slowly, and we introduce a simple mathematical model and equation to describe this phenomenon.

Structural reorganization of parallel actin bundles by crosslinking proteins: incommensurate states of twist

We construct a coarse-grained model of parallel actin bundles crosslinked by compact globular bundling proteins, such as fascin and espin, necessary components of filopodial and mechanosensory bundles. Consistent with structural observations of bundles, we find that the optimal geometry for crosslinking is overtwisted, requiring a coherent structural change of the helical geometry of the filaments. We study the linker-dependent thermodynamic transition of bundled actin filaments from their native state to the overtwisted state and map out the "twist-state" phase diagram in terms of the availability as well as the flexibility of crosslinker proteins. We predict that the transition from the uncrosslinked to fully crosslinked state is highly sensitive to linker flexibility: flexible crosslinking smoothly distorts the twist state of bundled filaments, while rigidly crosslinked bundles undergo a phase transition, rapidly overtwisting filaments over a narrow range of free crosslinker concentrations. Additionally, we predict a rich spectrum of intermediate structures, composed of alternating domains of sparsely bound (untwisted) and strongly bound (overtwisted) filaments. This model reveals that subtle differences in crosslinking agents themselves modify not only the detailed structure of parallel actin bundles, but also the thermodynamic pathway by which they form.

Topological defects in twisted bundles of two-dimensionally ordered filaments

Twisted assemblies of filaments in ropes, cables, and bundles are essential structural elements in both macroscopic materials and living organisms. We develop the unique, nonlinear 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 are screened by fivefold disclination defects in the lattice packing, and we predict a discrete spectrum of elastic-energy ground states associated with integer numbers of disclinations in cylindrical bundles. Finally, we show that elastic-energy ground states are extremely sensitive to the defect position in the cross section, with off-center disclinations driving the entire bundle to buckle and writhe.

Self-limited self-assembly of chiral filaments

We discuss dynamical simulations and free energy calculations on patchy spheres with chiral pair interactions that spontaneously assemble into filamentous bundles. The chirality frustrates long-range crystal order by introducing twist between interacting subunits. For some ranges of system parameters this constraint leads to bundles with a finite diameter, and in other cases frustration is relieved by the formation of defects. While some self-limited structures can be modeled as twisted filaments arranged with local hexagonal symmetry, other structures are surprisingly complex.

Isotropic-nematic phase transition in amyloid fibrilization

We carry out a theoretical study on the isotropic-nematic phase transition and phase separation in amyloid fibril solutions. Borrowing the thermodynamic model employed in the study of cylindrical micelles, we investigate the variations in the fibril length distribution and phase behavior with respect to changes in the protein concentration, fibril's rigidity, and binding energy. We then relate our theoretical findings to the nematic ordering experimentally observed in Hen Lysozyme fibril solution.

Theoretical issues relating to thermally reversible gelation by supermolecular fiber formation

Existing models of the thermodynamics and dynamics of self-assembly are summarized to provide a context for discussing the difficulties that arise in modeling supermolecular fiber assembly and the formation of thermally reversible gels through fiber growth and branching. Challenging problems in this field, such as the physical origin of fibers of uniform diameter and fiber twisting, the kinetics of fiber growth, the hierarchical bundling of fibers into "superfibers", fiber branching, gelation through fiber impingement and the associated phenomenon of fractal fiber network and spherulite formation, and the origin and control of structural polymorphism in the fiber and superfiber geometry, are discussed from a personal perspective. Suggestions are made for integrating current research efforts into a more coherent multiscale description of fiber formation and gelation on molecular, mesoscopic, and macroscopic scales.

A generic mechanism of emergence of amyloid protofilaments from disordered oligomeric aggregates

The presence of oligomeric aggregates, which is often observed during the process of amyloid formation, has recently attracted much attention because it has been associated with a range of neurodegenerative conditions including Alzheimer's and Parkinson's diseases. We provide a description of a sequence-indepedent mechanism by which polypeptide chains aggregate by forming metastable oligomeric intermediate states prior to converting into fibrillar structures. Our results illustrate that the formation of ordered arrays of hydrogen bonds drives the formation of beta-sheets within the disordered oligomeric aggregates that form early under the effect of hydrophobic forces. Individual beta-sheets initially form with random orientations and subsequently tend to align into protofilaments as their lengths increase. Our results suggest that amyloid aggregation represents an example of the Ostwald step rule of first-order phase transitions by showing that ordered cross-beta structures emerge preferentially from disordered compact dynamical intermediate assemblies.

Hofmeister effect and the phase diagram of lysozyme

The phase diagrams for lysozyme are calculated for two different precipitant salts, NaCl and NaSCN, using a potential of mean force that takes into account contributions from ion-dispersion forces [M. Boström, J. Phys. Chem. B 110, 24757 (2006)]. Our results are consistent with a recent perturbation theory calculation (referenced above) in that the phase diagram for lysozyme with NaCl is quite different than for lysozyme with NaSCN for the same molar concentration (0.2M) . However, in contrast to the perturbation theory calculation, we find that the lysozyme phase diagram with NaCl has a metastable fluid-fluid coexistence curve and that the metastability gap in the case of NaSCN is much larger than predicted by perturbation theory.

Polymer science and biology: structure and dynamics at multiple scales

I give a brief, biased survey of some recent problems in molecular and cell biology from the perspective of physical science, with a few answers, but a great many questions, challenges and opportunities.

Helices

Helices are among the simplest shapes that are observed in the filamentary and molecular structures of nature. The local mechanical properties of such structures are often modeled by a uniform elastic potential energy dependent on bending and twist, which is what we term a rod model. Our first result is to complete the semi-inverse classification, initiated by Kirchhoff, of all infinite, helical equilibria of inextensible, unshearable uniform rods with elastic energies that are a general quadratic function of the flexures and twist. Specifically, we demonstrate that all uniform helical equilibria can be found by means of an explicit planar construction in terms of the intersections of certain circles and hyperbolas. Second, we demonstrate that the same helical centerlines persist as equilibria in the presence of realistic distributed forces modeling nonlocal interactions as those that arise, for example, for charged linear molecules and for filaments of finite thickness exhibiting self-contact. Third, in the absence of any external loading, we demonstrate how to construct explicitly two helical equilibria, precisely one of each handedness, that are the only local energy minimizers subject to a nonconvex constraint of self-avoidance.

Anisotropy in Sickle Hemoglobin Fibers from Variations in Bending and Twist

We have studied the variations of twist and bend in sickle hemoglobin fibers. We find that these variations are consistent with an origin in equilibrium thermal fluctuations, which allows us to estimate the bending and torsional rigidities and effective corresponding material moduli. We measure bending by electron microscopy of frozen hydrated fibers and find that the bending persistence length, a measure of the length of fiber required before it starts to be significantly bent due to thermal fluctuations, is 130microm, somewhat shorter than that previously reported using light microscopy. The torsional persistence length, obtained by re-analysis of previously published experiments, is found to be only 2.5microm. Strikingly this means that the corresponding torsional rigidity of the fibers is only 6x10(-27)Jm, much less than their bending rigidity of 5x10(-25)Jm. For (normal) isotropic materials, one would instead expect these to be similar. Thus, we present the first quantitative evidence of a very significant material anisotropy in sickle hemoglobin fibers, as might arise from the difference between axial and lateral contacts within the fiber. We suggest that the relative softness of the fiber with respect to twist deformation contributes to the metastability of HbS fibers: HbS double strands are twisted in the fiber but not in the equilibrium crystalline state. Our measurements inform a theoretical model of the thermodynamic stability of fibers that takes account of both bending and extension/compression of hemoglobin (double) strands within the fiber.

Structural biology of cellular machines

Multi-component macromolecular machines contribute to all essential biological processes, from cell motility and signal transduction to information storage and processing. Structural analysis of assemblies at atomic resolution is emerging as the field of structural cell biology. Several recent studies, including those focused on the ribosome, the acrosomal bundle and bacterial flagella, have demonstrated the ability of a hybrid approach that combines imaging, crystallography and computational tools to generate testable atomic models of fundamental biological machines. A complete understanding of cellular and systems biology will require the detailed structural understanding of hundreds of biological machines. The realization of this goal demands a concerted effort to develop and apply new strategies for the systematic identification, isolation, structural characterization and mechanistic analysis of multi-component assemblies at all resolution ranges. The establishment of a database describing the structural and dynamic properties of protein assemblies will provide novel opportunities to define the molecular and atomic mechanisms controlling overall cell physiology.

Triggers of full-length tau aggregation: a role for partially folded intermediates

Alzheimer's disease is characterized in part by the accumulation of full-length tau proteins into intracellular filamentous inclusions. To clarify the events that trigger lesion formation, the aggregation of recombinant full-length four-repeat tau (htau40) was examined in vitro under near-physiological conditions using transmission electron microscopy and spectroscopy methods. In the absence of exogenous inducers, tau protein behaved as an assembly-incompetent monomer with little tertiary structure. The addition of anionic inducers led to fibrillization with nucleation-dependent kinetics. On the basis of circular dichroism spectroscopy and reactivity with thioflavin S and 8-anilino-1-naphthalenesulfonic acid fluorescent probes, the inducer stabilized a monomeric species with the folding characteristics of a premolten globule state. Planar aromatic dyes capable of binding the intermediate state with high affinity were also capable of triggering fibrillization in the absence of other inducers. Dye-mediated aggregation was characterized by concentration-dependent decreases in lag time, indicating increased nucleation rates, and submicromolar critical concentrations, indicating a final equilibrium that favored the filamentous state. The data suggest that the rate-limiting barrier for filament formation from full-length tau is conformational and that the aggregation reaction is triggered by environmental conditions that stabilize assembly-competent conformations.

Pathways of tau fibrillization

New methods for analyzing tau fibrillization have yielded insights into the biochemical transitions involved in the process. Here we review the parallels between the sequential progression of tau fibrillization observed macroscopically in Alzheimer's disease (AD) lesions and the pathway of tau aggregation observed in vitro with purified tau preparations. In addition, pharmacological agents for further dissection of fibrillization mechanism and lesion formation are discussed.

Site-specific pseudophosphorylation modulates the rate of tau filament dissociation

Hyperphosphorylation of tau is of fundamental importance for neurofibrillary lesion development in Alzheimer's disease, but the mechanisms through which it acts are not clear. Experiments with pseudophosphorylation mutants of full-length tau protein indicate that incorporation of negative charge into specific sites can modulate the aggregation reaction, and that this occurs by altering the critical concentration of assembly. Here, the kinetic origin of this effect was determined using quantitative electron microscopy methods and pseudophosphorylation mutant T212E in a full-length four-repeat tau background. On the basis of disaggregation rates, decreases in critical concentration resulted primarily from decreases in the dissociation rate constant. The results suggest a mechanism through which site-specific posttranslational modifications can modulate filament accumulation at low free intracellular tau concentrations.

Measuring forces between protein fibers by microscopy

We propose a general scheme for measuring the attraction between mechanically frustrated semiflexible fibers by measuring their thermal fluctuations and shape. We apply this analysis to a system of sickle hemoglobin (HbS) fibers that laterally attract one another. These fibers appear to "zip" together before reaching mechanical equilibrium due to the existence of cross-links into a dilute fiber network. We are also able to estimate the rigidities of the fibers. These rigidities are found to be consistent with sickle hemoglobin "single" fibers 20 nm in diameter, despite recent experiments indicating that fiber bundling sometimes occurs. Our estimate of the magnitude of the interfiber attraction for HbS fibers is in the range 8 +/- 7 kBT/microm, or 4 +/- 3 k(B)T/microm if the fibers are assumed, a priori to be single fibers (such an assumption is fully consistent with the data). This value is sufficient to bind the fibers, overcoming entropic effects, although extremely chemically weak. Our results are compared to models for the interfiber attraction that include depletion and van der Waals forces. This technique should also facilitate a similar analysis of other filamentous protein assembles in the future, including beta-amyloid, actin, and tubulin.

Evidence for an intermediate in tau filament formation

Alzheimer's disease is defined in part by the intraneuronal accumulation of filaments comprised of the microtubule-associated protein tau. In vitro, fibrillization of full-length, unphosphorylated recombinant tau can be induced under near-physiological conditions by treatment with various agents, including anionic surfactants. Here we examine the pathway through which anionic surfactants promote tau fibrillization using a combination of electron microscopy and fluorescence spectroscopy. Protein and surfactant first interacted in solution to form micelles, which then provided negatively charged surfaces that accumulated tau aggregates. Surface aggregation of tau protein was followed by the time-dependent appearance of a thioflavin S reactive intermediate that accumulated over a period of hours. The intermediate was unstable in the absence of anionic surfaces, suggesting it was not filamentous. Fibrillization proceeded after intermediate formation with classic nucleation-dependent kinetics, consisting of lag phase followed by the exponential increase in filament lengths, followed by an equilibrium phase reached in approximately 24 h. The pathway did not require protein insertion into the micelle hydrophobic core or conformational change arising from mixed micelle formation, because anionic microspheres constructed from impermeable polystyrene were capable of qualitatively reproducing all aspects of the fibrillization reaction. It is proposed that the progression from amorphous aggregation through intermediate formation and fibrillization may underlie the activity of other inducers such as hyperphosphorylation and may be operative in vivo.