Equilibrium bundle size of rodlike polyelectrolytes with counterion-induced attractive interactions

Coalescence of liquid or gel-like DNA-encapsulating microdroplets

Liquid-liquid phase separation plays a prominent role in the physics of life, providing the cells with various membrane-less compartments. These structures exhibit a range of material properties that, in many cases, change over time. Inspired by this, we investigate here an aqueous two-phase system formed by mixing polyethylene glycol with dextran. We modulate the material properties of the resulting dextran droplets by adding DNA that readily enters the droplets. We find a non-monotonic dependence of the physical properties of the droplets under the imposed ionic conditions.

On Casimir and Helmholtz Fluctuation-Induced Forces in Micro- and Nano-Systems: Survey of Some Basic Results

Fluctuations are omnipresent; they exist in any matter, due either to its quantum nature or to its nonzero temperature. In the current review, we briefly cover the quantum electrodynamic Casimir (QED) force as well as the critical Casimir (CC) and Helmholtz (HF) forces. In the QED case, the medium is usually a vacuum and the massless excitations are photons, while in the CC and HF cases the medium is usually a critical or correlated fluid and the fluctuations of the order parameter are the cause of the force between the macroscopic or mesoscopic bodies immersed in it. We discuss the importance of the presented results for nanotechnology, especially for devising and assembling micro- or nano-scale systems. Several important problems for nanotechnology following from the currently available experimental findings are spelled out, and possible strategies for overcoming them are sketched. Regarding the example of HF, we explicitly demonstrate that when a given integral quantity characterizing the fluid is conserved, it has an essential influence on the behavior of the corresponding fluctuation-induced force.

Photoacid-macroion assemblies: how photo-excitation switches the size of nano-objects

Electrostatic self-assembly of photoacids with oppositely charged macroions yields supramolecular nano-objects in aqueous solutions, whose size is controlled through light irradiation. Nano-assemblies are formed due to electrostatic attractions and mutual hydrogen bonding of the photoacids. Irradiation with UV light leads to the deprotonation of the photoacid and, consequently, a change in particle size. Overall, the hydrodynamic radii of the well-defined photoacid-macroion nano-objects lie between 130 and 370 nm. For a set of photoacids, we determine the acidity constants in the ground and excited state, discuss the sizes of photoacid-macroion nano-objects (by dynamic and static light scattering), their composition and the particle shapes (by small-angle neutron scattering), and relate their charge characteristics to size, structure and shape. We investigate the association thermodynamics and relate nanoscale structures to thermodynamics and, in turn, thermodynamics to molecular features, particularly the ionization energy of the photoacid hydroxyl group proton. Structure-directing effects completely differ from those for previously investigated systems, with hydrogen bonding and entropic effects playing a major role herein. This combined approach allows developing a comprehensive understanding of assembly formation and photo-response, anchored in molecular parameters (pKa, ionization energy, substituent group location), charge characteristics, and the association enthalpy and entropy. This fundamental understanding again paves the way for tailoring application solutions with novel photoresponsive materials.

Coarse-grained explicit-solvent molecular dynamics simulations of semidilute unentangled polyelectrolyte solutions

We present results from explicit-solvent coarse-grained molecular dynamics (MD) simulations of fully charged, salt-free, and unentangled polyelectrolytes in semidilute solutions. The inclusion of a polar solvent in the model allows for a more physical representation of these solutions at concentrations, where the assumptions of a continuum dielectric medium and screened hydrodynamics break down. The collective dynamic structure factor of polyelectrolytes, S(q, t), showed that at [Formula: see text], where [Formula: see text] is the polyelectrolyte peak in the structure factor S(q) and [Formula: see text] is the correlation length, the relaxation time obtained from fits to stretched exponential was [Formula: see text], which describes unscreened Zimm-like dynamics. This is in contrast to implicit-solvent simulations using a Langevin thermostat where [Formula: see text]. At [Formula: see text], a crossover region was observed that eventually transitions to another inflection point [Formula: see text] at length scales larger than [Formula: see text] for both implicit- and explicit-solvent simulations. The simulation results were also compared to scaling predictions for correlation length, [Formula: see text], specific viscosity, [Formula: see text], and diffusion coefficient, [Formula: see text], where [Formula: see text] is the polyelectrolyte concentration. The scaling prediction for [Formula: see text] holds; however, deviations from the predictions for [Formula: see text] and D were observed for systems at higher [Formula: see text], which are in qualitative agreements with recent experimental results. This study highlights the importance of explicit-solvent effects in molecular dynamics simulations, particularly in semidilute solutions, for a better understanding of polyelectrolyte solution behavior.

DNA-protamine condensates under low salt conditions: molecular dynamics simulation with a simple coarse-grained model focusing on electrostatic interactions

Protamine, a small, strongly positively-charged protein, plays a key role in achieving chromatin condensation inside sperm cells and is also involved in the formulation of nanoparticles for gene therapy and packaging of mRNA-based vaccines against viral infection and cancer. The detailed mechanisms of such condensations are still poorly understood especially under low salt conditions where electrostatic interaction predominates. Our previous study, with a refined coarse-grained model in full consideration of the long-range electrostatic interactions, has demonstrated the crucial role of electrostatic interaction in protamine-controlled reversible DNA condensation. Therefore, we herein pay our attention only to the electrostatic interaction and devise a coarser-grained bead-spring model representing the right linear charge density on protamine and DNA chains but treating other short-range interactions as simply as possible, which would be suitable for real-scale simulations. Effective pair potential calculations and large-scale molecular dynamics simulations using this extremely simple model reproduce the phase behaviour of DNA in a wide range of protamine concentrations under low salt conditions, again revealing the importance of the electrostatic interaction in this process and providing a detailed nanoscale picture of bundle formation mediated by a charge disproportionation mechanism. Our simulations also show that protamine length alters DNA overcharging and in turn redissolution thresholds of DNA condensates, revealing the important role played by entropies and correlated fluctuations of condensing agents and thus offering an additional opportunity to design tailored nanoparticles for gene therapy. The control mechanism of DNA-protamine condensates will also provide a better microscopic picture of biomolecular condensates, i.e., membraneless organelles arising from liquid-liquid phase separation, that are emerging as key principles of intracellular organization. Such condensates controlled by post-translational modification of protamine, in particular phosphorylation, or by variations in protamine length from species to species may also be responsible for the chromatin-nucleoplasm patterning observed during spermatogenesis in several vertebrate and invertebrate species.

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.

Coacervation between two positively charged poly(ionic liquid)s

Complex coacervates are usually formed through electrostatic attraction between oppositely charged polyelectrolytes, with a few of exceptions such as coacervates of like-charge proteins and polyelectrolytes, both in vivo and in vitro. Understanding of the preparation and mechanism of these coacervates is limited. Here we design a positively charged poly(ionic liquid) poly(1-vinyl-3-benzylimidazolium chloride) (PILben) that bears benzene rings in repeating units. Fluidic coacervates were prepared by mixing the PILben aqueous solution with a like-charge poly(ionic liquid) named poly(dimethyl diallyl ammonium chloride) (PDDA). The effects of polymer concentration, temperature and ionic strength in the PILben-PDDA coacervate were studied. Raman spectroscopy and two-dimensional ¹ H-¹³ C heteronuclear single quantum coherence (¹ H-¹³ C HSQC) characterizations verify that the coacervate formation benefits from the cation-π interaction between PILben and PDDA. This work provides principles and understandings of designing coacervates derived from like-charge poly(ionic liquids) with high charge density. This article is protected by copyright. All rights reserved.

Protamine-Controlled Reversible DNA Packaging: A Molecular Glue

Packaging paternal genome into tiny sperm nuclei during spermatogenesis requires 10⁶-fold compaction of DNA, corresponding to a 10-20 times higher compaction than in somatic cells. While such a high level of compaction involves protamine, a small arginine-rich basic protein, the precise mechanism at play is still unclear. Effective pair potential calculations and large-scale molecular dynamics simulations using a simple idealized model incorporating solely electrostatic and steric interactions clearly demonstrate a reversible control on DNA condensates formation by varying the protamine-to-DNA ratio. Microscopic states and condensate structures occurring in semidilute solutions of short DNA fragments are in good agreement with experimental phase diagram and cryoTEM observations. The reversible microscopic mechanisms induced by protamination modulation should provide valuable information to improve a mechanistic understanding of early and intermediate stages of spermatogenesis where an interplay between condensation and liquid-liquid phase separation triggered by protamine expression and post-translational regulation might occur. Moreover, recent vaccines to prevent virus infections and cancers using protamine as a packaging and depackaging agent might be fine-tuned for improved efficiency using a protamination control.

Manipulation of Confined Polyelectrolyte Conformations through Dielectric Mismatch

We demonstrate that a highly charged polyelectrolyte confined in a spherical cavity undergoes reversible transformations between amorphous conformations and a four-fold symmetry morphology as a function of dielectric mismatch between the media inside and outside the cavity. Surface polarization due to dielectric mismatch exhibits an extra "confinement" effect, which is most pronounced within a certain range of the cavity radius and the electrostatic strength between the monomers and counterions and multivalent counterions. For cavities with a charged surface, surface polarization leads to an increased amount of counterions adsorbed in the outer side, further compressing the confined polyelectrolyte into a four-fold symmetry morphology. The equilibrium conformation of the chain is dependent upon several key factors including the relative permittivities of the media inside and outside the cavity, multivalent counterion concentration, cavity radius relative to the chain length, and interface charge density. Our findings offer insights into the effects of dielectric mismatch in packaging and delivery of polyelectrolytes across media with different relative permittivities. Moreover, the reversible transformation of the polyelectrolyte conformations in response to environmental permittivity allows for potential applications in biosensing and medical monitoring.

Cations Modulate Actin Bundle Mechanics, Assembly Dynamics, and Structure

Actin bundles are key factors in the mechanical support and dynamic reorganization of the cytoskeleton. High concentrations of multivalent counterions promote bundle formation through electrostatic attraction between actin filaments that are negatively charged polyelectrolytes. In this study, we evaluate how physiologically relevant divalent cations affect the mechanical, dynamic, and structural properties of actin bundles. Using a combination of total internal reflection fluorescence microscopy, transmission electron microscopy, and dynamic light scattering, we demonstrate that divalent cations modulate bundle stiffness, length distribution, and lateral growth. Molecular dynamics simulations of an all-atom model of the actin bundle reveal specific actin residues coordinate cation-binding sites that promote the bundle formation. Our work suggests that specific cation interactions may play a fundamental role in the assembly, structure, and mechanical properties of actin bundles.

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.

Self-assembled polyoxometalate-dendrimer structures for selective photocatalysis

We present a novel, self-assembled nanostructure with selective photocatalytic activity formed from anionic polyoxometalate clusters and cationic dendrimers by electrostatic self-assembly. The association of the components in aqueous solution is driven by ionic interaction and steric factors yielding stable aggregates of a defined size with a coil-like structure. The assemblies show high potential for the application in solar-energy conversion systems due to their enhanced and substrate specific photocatalytic activity.

Repulsion between oppositely charged rod-shaped macromolecules: Role of overcharging and ionic confinement

The interaction between two oppositely charged rod-shaped macro-ions in a micro-ion solution is investigated via Monte Carlo simulations of the primitive model. The focus is on the asymmetry in rod and/or ion charge, i.e., conditions where oppositely charged objects can repel one another. For equally and oppositely charged rods with asymmetric z:1 micro-ions, repulsion may be induced by overcharging one of the rods with the z valent ions. For asymmetrically charged rods in a symmetric z:z micro-ion solution, a repulsive interaction-at separation of the order of one ion diameter-can arise via an unbalanced osmotic pressure contribution from the ionic atmosphere in the inter-rod space, and an attractive interaction-at a smaller separation-may occur due to a "squeezing out" of the micro-ions from the space between the rods (with a consequent gain in entropy). The thermodynamics of each mechanism is investigated in terms of rod charge and size and micro-ion valence, size, and concentration. Our findings contribute to the understanding of the complex role of charge asymmetry on the interaction of, for example, oppositely charged polyelectrolytes, functionalized nanotubes, and rod-like biomolecules, e.g., viruses.

Aggregation of cyclic polypeptoids bearing zwitterionic end-groups with attractive dipole–dipole and solvophobic interactions: a study by small-angle neutron scattering and molecular dynamics simulation

The aggregation behavior of cyclic polypeptoids has been studied using experiments and simulations.

Electrostatic theory of the assembly of PAMAM dendrimers and DNA

The electrostatic interactions mediated by counterions between a cationic PAMAM dendrimer, modelized as a sphere of radius and cationic surface charge highly increasing with generation, and a DNA, modelized as an anionic elastic line, are analytically calculated in the framework of condensation theory. Under these interactions the DNA is wrapped around the sphere. For excess phosphates relative to dendrimer primary amines, the free energy of the DNA-dendrimer complex displays an absolute minimum when the complex is weakly negatively overcharged. This overcharging opposes gene delivery. For a highly positive dendrimer and a DNA fixed by experimental conditions to a number of phosphates less than the number of dendrimer primary amines, excess amine charges, the dendrimer may at the same time bind stably DNA and interact with negative cell membranes to activate cell transfection in fair agreement with molecular simulations and experiments.

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.

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.

A route to self-assemble suspended DNA nano-complexes

Highly charged polyelectrolytes can self-assemble in presence of condensing agents such as multivalent cations, amphiphilic molecules or proteins of opposite charge. Aside precipitation, the formation of soluble micro- and nano-particles has been reported in multiple systems. However a precise control of experimental conditions needed to achieve the desired structures has been so far hampered by the extreme sensitivity of the samples to formulation pathways. Herein we combine experiments and molecular modelling to investigate the detailed microscopic dynamics and the structure of self-assembled hexagonal bundles made of short dsDNA fragments complexed with small basic proteins. We suggest that inhomogeneous mixing conditions are required to form and stabilize charged self-assembled nano-aggregates in large excess of DNA. Our results should help re-interpreting puzzling behaviors reported for a large class of strongly charged polyelectrolyte systems.

Interaction modes between asymmetrically and oppositely charged rods

The interaction of oppositely and asymmetrically charged rods in salt-a simple model of (bio)macromolecular assembly-is observed via simulation to exhibit two free energy minima, separated by a repulsive barrier. In contrast to similar minima in the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, the governing mechanism includes electrostatic attraction at large separation, osmotic repulsion at close range, and depletion attraction near contact. A model accounting for ion condensation and excluded volume is shown to be superior to a mean-field treatment in predicting the effect of charge asymmetry on the free-energy profile.

Generic, phenomenological, on-the-fly renormalized repulsion model for self-limited organization of terminal supraparticle assemblies

Self-limited, or terminal, supraparticles have long received great interest because of their abundance in biological systems (DNA bundles and virus capsids) and their potential use in a host of applications ranging from photonics and catalysis to encapsulation for drug delivery. Moreover, soft, uniform colloidal aggregates are a promising candidate for quasicrystal and other hierarchical assemblies. In this work, we present a generic coarse-grained model that captures the formation of self-limited assemblies observed in various soft-matter systems including nanoparticles, colloids, and polyelectrolytes. Using molecular dynamics simulations, we demonstrate that the assembly process is self-limited when the repulsion between the particles is renormalized to balance their attraction during aggregation. The uniform finite-sized aggregates are further shown to be thermodynamically stable and tunable with a single dimensionless parameter. We find large aggregates self-organize internally into a core-shell morphology and exhibit anomalous uniformity when the constituent nanoparticles have a polydisperse size distribution.

Physical theory of ionomer aggregation in water

This article presents a physical theory for the aggregation of ionomer molecules in aqueous solution. To study this phenomenon, we consider a system of charged rigid rods with uniform surface charge immersed in water. The free-energy functional derived for this system consists of hydrophobic and direct electrostatic contributions as well as entropic terms. Energy minimization gives the stable aggregation number as a function of surface charge density, surface tension, geometric parameters, and density of rods in solution. We provide configuration diagrams of the system, which display the impact of the hydrophobic and electrostatic interaction strengths on the stabilization of finite-size bundles.

Supramolecular cellular filament systems: how and why do they form?

All cells, from simple bacteria to complex human tissues, rely on extensive networks of protein fibers to help maintain their proper form and function. These filament systems usually do not operate as single filaments, but form complex suprastructures, which are essential for specific cellular functions. Here, we describe the progress in determining the architectures of molecular filamentous suprastructures, the principles leading to their formation, and the mechanisms by which they may facilitate function. The complex eukaryotic cytoskeleton is tightly regulated by a large number of actin- or microtubule-associated proteins. In contrast, recently discovered bacterial actins and tubulins have few associated regulatory proteins. Hence, the quest to find basic principles that govern the formation of filamentous suprastructures is simplified in bacteria. Three common principles, which have been probed extensively during evolution, can be identified that lead to suprastructures formation: cationic counterion fluctuations; self-association into liquid crystals; and molecular crowding. The underlying physics of these processes will be discussed with respect to physiological circumstance.

Molecular structure encodes nanoscale assemblies: understanding driving forces in electrostatic self-assembly

Supramolecular nanoparticles represent a key field in recent research as their synthesis through self-assembly is straightforward and they often can respond to external triggers. A fundamental understanding of structure-directing factors is highly desirable for a targeted structure design. This contribution demonstrates a quantitative relation between the size of supramolecular self-assembled nanoparticles and the free energy of association. Nanoparticles are prepared by electrostatic self-assembly of cationic polyelectrolyte dendrimers as model macroions and oppositely charged di- and trivalent organic dye molecules relying on the combination of electrostatic and π-π-interactions. A systematic set of sulfonate-group carrying azo-dyes was synthesized. Light scattering and ζ-potential measurements on the resulting nanoparticles yield hydrodynamic radii between 20 nm < R(H) < 50 nm and positive ζ-potential values indicating a positive particle charge. Studies on dye self-aggregation and dendrimer-dye association by isothermal titration calorimetry (ITC) and UV-vis spectroscopy allow for the correlation of the thermodynamic parameters of dendrimer-dye association with the size of the particles, showing that at least a free energy gain of ΔG ≈ - 32 kJ mol(-1) is necessary to induce dendrimer interconnection. Structural features of the azo dyes causing these to favor or prevent nanoparticle formation have been identified. The dye-dye-interaction was found to be the key factor in particle size control. A simple model yields a quantitative relation between the free energy and the particle sizes, allowing for predicting the latter based on thermodynamic measurements. Hence, a set of different molecular "building bricks" can be defined where the choice of building block determines the resulting assembly size.

Modeling the formation of in vitro filopodia

Filopodia are bundles of actin filaments that extend out ahead of the leading edge of a crawling cell to probe its upcoming environment. In vitro experiments (Vignjevic et al. in J Cell Biol 160:951-962, 2003) have determined the minimal ingredients required for the formation of filopodia from the dendritic-like morphology of the leading edge. We model these experiments using kinetic aggregation equations for the density of growing bundle tips. In mean field, we determine the bundle size distribution to be broad for bundle sizes smaller than a characteristic bundle size above which the distribution decays exponentially. Two-dimensional simulations incorporating both bundling and cross-linking measure a bundle size distribution that agrees qualitatively with mean field. The simulations also demonstrate a nonmonotonicity in the radial extent of the dendritic region as a function of capping protein concentration, as was observed in experiments, due to the interplay between percolation and the ratcheting of growing filaments off a spherical obstacle.

Equilibrium polyelectrolyte bundles with different multivalent counterion concentrations

We present the results of molecular-dynamics simulations on the salt concentration dependence of the formation of polyelectrolyte bundles in thermodynamic equilibrium. Extending our results on salt-free systems we investigate here deficiency or excess of trivalent counterions in solution. Our results reveal that the trivalent counterion concentration significantly alters the bundle size and size distribution. The onset of bundle formation takes place at earlier Bjerrum length values with increasing trivalent counterion concentration. For the cases of 80%, 95%, and 100% charge compensation via trivalent counterions, the net charge of the bundles decreases with increasing size. We suggest that competition among two different mechanisms, counterion condensation and merger of bundles, leads to a nonmonotonic change in line-charge density with increasing Bjerrum length. The investigated case of having an abundance of trivalent counterions by 200% prohibits such a behavior. In this case, we find that the difference in effective line-charge density of different size bundles diminishes. In fact, the system displays an isoelectric point, where all bundles become charge neutral.

Hydrophilic macroionic solutions: what happens when soluble ions reach the size of nanometer scale?

Large, hydrophilic inorganic ions (mostly polyoxometalate macroions and cationic metal-organic hybrid nanocages) with high solubility in water and/or other polar solvents demonstrate unique solution behaviors. In dilute solutions, they behave significantly different from small simple ions (as described by the Debye-Hückel theory) because the macroions cannot be treated as point charges or large, insoluble colloidal suspensions (usually described by the DLVO theory) because the macroions form homogeneous, stable "real solutions". The size disparity between the macroions and their counterions results in complex macroion-counterion interaction and leads to the self-assembly of macroions into single-layered, hollow, spherical "blackberry" structures. The blackberries, with robust and very stable structures mimicking biological membranes, can adjust their size accurately and reversibly in response to the change of solvent content, charge density on the macroions, or in some cases merely solution pH. The blackberry membrane is permeable to small cations. The inorganic macroions with well-defined size, shape, mass, charge density (even accurately tunable within certain range), and no intramolecular interaction can be treated as simple model systems to understand the intermolecular interaction in polyelectrolyte solutions. The blackberry structures show certain similarities to the spherical virus capsids, from the overall structure to the kinetic properties of formation.

Suprastructures and dynamic properties of Mycobacterium tuberculosis FtsZ

Tuberculosis causes the most death in humans by any bacterium. Drug targeting of bacterial cytoskeletal proteins requires detailed knowledge of the various filamentous suprastructures and dynamic properties. Here, we have investigated by high resolution electron microscopy the assembly of cell division protein and microtubule homolog FtsZ from Mycobacterium tuberculosis (MtbFtsZ) in vitro in the presence of various monovalent salts, crowding agents and polycations. Supramolecular structures, including two-dimensional rings, three-dimensional toroids, and multistranded helices formed in the presence of molecular crowding, were similar to those observed by fluorescence microscopy in bacteria in vivo. Dynamic properties of MtbFtsZ filaments were visualized by light scattering and real time total internal reflection fluorescence microscopy. Interestingly, MtbFtsZ revealed a form of dynamic instability at steady state. Cation-induced condensation phenomena of bacterial cytomotive polymers have not been investigated in any detail, although it is known that many bacteria can contain high amounts of polycations, which may modulate the prokaryotic cytoskeleton. We find that above a threshold concentration of polycations which varied with the valence of the cation, ionic strength, and pH, MtbFtsZ mainly formed sheets. The general features of these cation-induced condensation phenomena could be explained in the framework of the Manning condensation theory. Chirality and packing defects limited the dimensions of sheets and toroids at steady state as predicted by theoretical models. In first approximation simple physical principles seem to govern the formation of MtbFtsZ suprastructures.

Molecular mechanism of bundle formation by the bacterial actin ParM

The actin homolog ParM plays a microtubule-like role in segregating DNA prior to bacterial cell division. Fluorescence and cryo-electron microscopy have shown that ParM forms filament bundles between separating DNA plasmids in vivo. Given the lack of ParM bundling proteins it remains unknown how ParM bundles form at the molecular level. Here we show using time-lapse TIRF microscopy, under in vitro molecular crowding conditions, that ParM-bundle formation consists of two distinct phases. At the onset of polymerization bundle thickness and shape are determined in the form of nuclei of short helically disordered filaments arranged in a liquid-like lattice. These nuclei then undergo an elongation phase whereby they rapidly increase in length. At steady state, ParM bundles fuse into one single large aggregate. This behavior had been predicted by theory but has not been observed for any other cytomotive biopolymer, including F-actin. We employed electron micrographs of ParM rafts, which are 2-D analogs of 3-D bundles, to identify the main molecular interfilament contacts within these suprastructures. The interface between filaments is similar for both parallel and anti-parallel orientations and the distribution of filament polarity is random within a bundle. We suggest that the interfilament interactions are not due to the interactions of specific residues but rather to long-range, counter ion mediated, electrostatic attractive forces. A randomly oriented bundle ensures that the assembly is rigid and that DNA may be captured with equal efficiency at both ends of the bundle via the ParR binding protein.

Salt-induced aggregation of stiff polyelectrolytes

Molecular dynamics simulation techniques are used to study the process of aggregation of highly charged stiff polyelectrolytes due to the presence of multivalent salt. The dominant kinetic mode of aggregation is found to be the case of one end of one polyelectrolyte meeting others at right angles, and the kinetic pathway to bundle formation is found to be similar to that of flocculation dynamics of colloids as described by Smoluchowski. The aggregation process is found to favor the formation of finite bundles of 10-11 filaments at long times. Comparing the distribution of the cluster sizes with the Smoluchowski formula suggests that the energy barrier for the aggregation process is negligible. Also, the formation of long-lived metastable structures with similarities to the raft-like structures of actin filaments is observed within a range of salt concentration.

Braided bundles and compact coils: the structure and thermodynamics of hexagonally packed chiral filament assemblies

Molecular chirality frustrates the two-dimensional assembly of filamentous molecules, a fact that reflects the generic impossibility of imposing a global twisting of layered materials. We explore the consequences of this frustration for hexagonally ordered assemblies of chiral filaments that are finite in lateral dimension. Specifically, we employ a continuum-elastic description of cylindrical bundles of filaments, allowing us to consider the most general resistance to and preference for chiral ordering of the assembly. We explore two distinct mechanisms by which chirality at the molecular scale of the filament frustrates the assembly into aggregates. In the first, chiral interactions between filaments impart an overall twisting of filaments around the central axis of the bundle. In the second, we consider filaments that are inherently helical in structure, imparting a writhing geometry to the central axis. For both mechanisms, we find that a thermodynamically stable state of dispersed bundles of finite width appears close to but below the point of bulk filament condensation. The range of thermodynamic stability of dispersed bundles is sensitive only to the elastic cost and preference for chiral filament packing. The self-limited assembly of chiral filaments has particular implications for a large class of biological molecules--DNA, filamentous proteins, viruses, and bacterial flagella--which are universally chiral and are observed to form compact bundles under a broad range of conditions.

Effects of osmotic force and torque on microtubule bundling and pattern formation

We report effects of polyethylene glycol (PEG, molecular weight of 35 kDa ) on microtubule (MT) bundling and pattern formation. Without PEG, polymerizing tubulin solutions of a few mg/ml that are initially subjected to a field that aligns MTs can spontaneously form striated birefringence patterns. These patterns form through MT alignment, bundling, and coordinated bundle buckling. With increasing PEG concentrations, solutions form progressively weaker patterns. At a sufficiently high PEG concentration ( approximately 0.5% by weight), the samples maintain a nearly uniform birefringence (i.e., no pattern) and laterally contract at a later stage. Concomitantly, on a microscopic level, the network of dispersed MTs that accompany the bundles in pure solutions disappear and the bundles become more distinct. We attribute the weakening of the pattern to the loss of the dispersed MT network, which is required to mediate the coordination of bundle buckling. We propose that the loss of the dispersed network and the enhanced bundling result from PEG associated osmotic forces that drive MTs together and osmotic torques that facilitate their bundling. Similarly, we attribute the lateral contraction of the samples to osmotic torques that tend to align crossing bundles in the network.

Packing defects and the width of biopolymer bundles

The formation of bundles composed of actin filaments and cross-linking proteins is an essential process in the maintenance of the cells' cytoskeleton. It has also been recreated by in-vitro experiments, where actin networks are routinely produced to mimic and study the cellular structures. It has been observed that these bundles seem to have a well-defined width distribution, which has not been adequately described theoretically. We propose here that packing defects of the filaments, quenched and random, contribute an effective repulsion that counters the cross-linking adhesion energy and leads to a well-defined bundle width. This is a two-dimensional strain-field version of the classic Rayleigh instability of charged droplets.

Helical twist controls the thickness of F-actin bundles

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

Strain induced clustering in polyelectrolyte hydrogels

Systematic large strain compression measurements have been performed on polyelectrolyte hydrogels based on modified PAA crosslinked by bifunctional thiols. For compressive strains larger than a critical value depending on polymer concentration, we observed a significant hysteresis, strain-hardening and a stress plateau during unloading. This was attributed to strain-induced ionic clustering due to electrostatic interactions that can become attractive if chains are close enough to each other. This phenomenon is dynamic and reversible but a long lifetime for the clusters has been identified. Although clustering between like-charge chains has been reported for hydrogels, it is the first time that this phenomenon is caused by deformation. This effect is potentially important as we strive to understand the behaviour of all polyelectrolyte hydrogels at large strains which are highly relevant for fracture properties.

Thickness distribution of actin bundles in vitro

Bundles of filamentous actin form the primary building blocks of a broad range of cytoskeletal structures, including filopodia, stereocilia and microvilli. In each case, the cell uses specific associated proteins to tailor the dynamics, dimensions and mechanical properties of the bundles to suit a specific cellular function. While the length distribution of actin bundles was extensively studied, almost nothing is known about the thickness distribution. Here, we use high-resolution cryo-TEM to measure the thickness distribution of actin/fascin bundles, in vitro. We find that the thickness distribution has a prominent peak, with an exponential tail, supporting a scenario of an initial fast formation of a disc-like nucleus of short actin filaments, which only later elongates. The bundle thicknesses at steady state are found to follow the distribution of the initial nuclei indicating that no lateral coalescence occurs. Our results show that the distribution of bundles thicknesses can be controlled by monitoring the initial nucleation process. In vivo, this is done by using specific regulatory proteins complexes.

Aggregation kinetics of stiff polyelectrolytes in the presence of multivalent salt

Using molecular dynamics simulations, the kinetics of bundle formation for stiff polyelectrolytes such as actin is studied in the solution of multivalent salt. The dominant kinetic mode of aggregation is found to be the case of one end of one rod meeting others at a right angle due to electrostatic interactions. The kinetic pathway to bundle formation involves a hierarchical structure of small clusters forming initially and then feeding into larger clusters, which is reminiscent of the flocculation dynamics of colloids. For the first few cluster sizes, the Smoluchowski formula for the time evolution of the cluster size gives a reasonable account of the results of our simulation without a single fitting parameter. The description using the Smoluchowski formula provides evidence for the aggregation time scale to be controlled by diffusion, with no appreciable energy barrier to overcome.

Chirality and equilibrium biopolymer bundles

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

Evolution of growth modes for polyelectrolyte bundles

Multivalent ions induce attractions between polyelectrolytes, but lead to finite-sized bundles rather than macroscopic phase separation. The kinetics of aggregation and bundle formation of actin is tracked using two different fluorescently labeled populations of F-actin. It is found that the growth mode of these bundles evolves with time and salt concentration, varying from an initial lateral growth to a longitudinal one at later stages. The results suggest that kinetics play a role in bundle growth, but not in the lateral size of bundles, which is constant for linear and branched topologies.

Deprotonations and charges of well-defined {Mo72Fe30} nanoacids simply stepwise tuned by pH allow control/variation of related self-assembly processes

The solution behavior of the largest inorganic acid known thus far, the neutral, spherical iron/molybdenum/oxide nanocluster {Mo72Fe30} ([triple bond{(MoVI) MoVI5}12FeIII30 1a), including the pH-controlled deprotonation, is reported. The acidic properties are due to the 30 peripheral, weakly acidic FeIII(H2O) groups that form a unique Archimedean solid with all edges and dihedral angles being equal, the icosidodecahedron, and therefore an "isotropic" surface. Interestingly, the aqueous solutions are stable even for months because of the inertness of the spherical solutes and the presence of the hard FeIII and MoVI centers. The stability can be nicely proven by the very characteristic Raman spectrum showing, because of the (approximately) icosahedral symmetry, only a few lines. Whereas the {Mo72Fe30} clusters exist as discrete, almost neutral, molecules in aqueous solution at pH < 2.9, they get deprotonated and self-associate into single-layer blackberry-type structures at higher pH while the assembly process (i.e., the size of the final species) can be controlled by the pH values; this allows the deliberate generation of differently sized nanoparticles, a long-term goal in nanoscience. The average hydrodynamic radius (Rh) of the self-assembled structures decreases monotonically with increasing number of charges on the {Mo72Fe30} macroanions (from approximately 45 nm at pH approximately 3.0 to approximately 15 nm at pH approximately 6.6), as studied by laser light scattering and TEM techniques. The {Mo72Fe30} macroions with high-stability tunable charges/surfaces, equal shape, and masses provide models for the understanding of more complex polyelectrolyte solutions while the controllable association and dissociation reported here of the assembled soft magnetic materials with tuneable sizes could be interesting for practical applications.