Association of two semiflexible polyelectrolytes by interchain linkers: Theory and simulations

Adsorption of semiflexible wormlike polymers to a bar and their double-chain complex formation

We theoretically study pairing (double-strand complexation) of semiflexible wormlike chains (WLC) due to their side-to-side attraction. Considering binding of two WLCs of high stiffness we start with the case of infinite stiffness of one chain which is replaced with a straight bar. A combination of the quantitative transfer matrix approach with scaling arguments in terms of trains, loops of different sizes, tails and supertrains allowed us to characterize all the regimes of semiflexible chain adsorption on a bar. In particular, we predict a self-similar monomer concentration profile c(r) ∝ r-10/3 near the bar (at distances r below the chain Kuhn length l) at the critical point for adsorption. Such localized critical profile leads to a sharp adsorption transition. Furthermore, we found that supertrains serve as the basic structural elements in WLC complexes leading to bridging, network formation and condensation of semiflexible polymers in dilute solutions. Polymer collapse (precipitation) and redissolution on increasing attraction strength are predicted in qualitative agreement with experiments on aqueous solutions of DNA and F-actin.

Alginate-metal cation interactions: Macromolecular approach

Alginates are a broad family of linear (unbranched) polysaccharides derived from brown seaweeds and some bacteria. Despite having only two monomers, i.e. β-d-mannuronate (M) and its C5 epimer α-l-guluronate (G), their blockwise arrangement in oligomannuronate (..MMM..), oligoguluronate (..GGG..), and polyalternating (..MGMG..) blocks endows it with a rather complex interaction pattern with specific counterions and salts. Classic polyelectrolyte theories well apply to alginate as polyanion in the interaction with monovalent and non-gelling divalent cations. The use of divalent gelling ions, such as Ca2+, Ba2+ or Sr2+, provides thermostable homogeneous or heterogeneous hydrogels where the block composition affects both macroscopic and microscopic properties. The mechanism of alginate gelation is still explained in terms of the original egg-box model, although over the years some novel insights have been proposed. In this review we summarize several decades of research related to structure-functionships in alginates in the presence of non-gelling and gelling cations and present some novel applications in the field of self-assembling nanoparticles and use of radionuclides.

Comparative Insights into the Fundamental Steps Underlying Gelation of Plant and Algal Ionic Polysaccharides: Pectate and Alginate

Pectate and alginate are among the most important biopolymers able to give rise to ionotropic gelation upon the addition of di- or multivalent counterions. The two ionic polysaccharides exhibit several common aspects of the gelation mechanism with calcium ions, the physiologically and commercially most relevant counterion type. The first one pertains to the role that specific Ca²⁺/polyion interactions play in the establishment of the ion-mediated chain/chain cross-links. Such interactions include both a specific affinity of the territorially condensed Ca²⁺ counterions for the polyuronate(s) and the formation of long-lasting chemical bonding (inner ion-sphere complex) of specific interchain sites accompanied by high conformational ordering. As to the first mechanism, it is dominated by the strong desolvation of the interacting ionic species, with concomitant positive variations in both enthalpy and entropy, the contribution of the latter prevailing over the former due to the favorable liberation of a very large number of water molecules of hydration. Both dilatometric and microcalorimetric data point to the higher affinity of Ca²⁺ for pectate than for alginate. The selective accumulation of calcium ions close to the polyanion(s) favors the onset of the second-chemical bonding-mode, which is associated with charge neutralization at the bonding site. This mode coincides with the largely accepted "egg-box" model for the calcium-mediated interchain junction of pectate and alginate. A new approach was devised for the calculation of the fraction of chemically bound divalent ions; it was based on the available circular dichroism data (further supported by scattering and viscosity results) and successfully tested by comparison with an independently determined fraction in the case of pectate. In detail, the strong bonding mode manifests in two sequential bonding modes. The first one (at low concentrations of added Ca²⁺ ions) entails a cross-link in which only one calcium ions is bracketed in a "twisted" egg-box between two chains; upon further counterion addition, a series of nearest-neighboring "perfect" egg-box structures develops. Both dilatometric and microcalorimetric changes associated with the latter chemical bonding modes are quantitatively larger for pectate than for alginate; clearly the latter polyuronate suffers from the relevant presence of the weakly calcium-binding mannuronic acid repeating units. Light-scattering experiments provided a clear-cut demonstration of the intermolecular bonding of calcium ions from the very beginning of the linker addition.

On the Molecular Mechanism of the Calcium-Induced Gelation of Pectate. Different Steps in the Binding of Calcium Ions by Pectate

Pectic acid/sodium pectate is one of the most widespread hydrocolloid used in the food industry. It is able to form strong ionotropic gels by the addition of ions, in particular, calcium ions. The initial steps of binding Ca²⁺ ions to a sample of sodium pectate with a composition close to 90% of ideal Na⁺-poly(galacturonate) were investigated by means of circular dichroism (CD), microcalorimetry, dilatometry, viscosity, and membrane osmometry, as a function of increasing R_j, R_j being the ratio of the molar concentrations of Ca²⁺ and pectate repeating units. Data were collected in aqueous NaClO₄ at 25 °C. The key instrument of interpretation has been the counterion condensation theory (CCT) of linear polyelectrolytes, modified to include the presence of both specific affinity of the divalent counterion for the polysaccharide ("territorial binding"), and, very importantly, strong chemical bonding (not a covalent bonding, though) of Ca²⁺ on conformationally well-defined sites on the polymer, with local charge annihilation. Intrinsic viscosity and number-average molar mass data as a function of R_j showed that calcium bonding brings about chain association right from the beginning of addition to pectate. The analysis of the microcalorimetric curve using the modified CCT revealed two types of bonding. In the order of development as a function of R_j, the first mode (type 1) could be reconciled with the "tilted egg-box" type, recently proposed for Ca²⁺ binding to alginate and the second mode (type 2) with the "shifted egg-box" proposed for calcium pectate on the basis of conformational analysis investigation. Likewise, the two types of bonding turned out to be superimposable with similar bonding categories proposed for alginate and low-methoxyl pectin (LMP), on the one side, and for the association of semiflexible polyelectrolytes, on the other. The analysis allowed us to obtain standard Gibbs free energy, enthalpy, entropy, and volume molar values both for the affinity and the chemical bonding processes. Interestingly, the analysis of the dependence of the gelation temperatures, T_g, of LMP upon increasing additions of calcium ions provided the values of T_g and standard Gibbs free-energy of calcium-to-pectate association coinciding with those obtained from calorimetry for the type-2 bonding process. This finding corroborated previously reported evidence on the enthalpic nature of the elasticity of Ca²⁺-pectate gels. Finally, comparative analysis of different techniques, but of CD in particular, enabled proposing a "loose-2₁-helix" as the starting conformation of sodium pectate in aqueous solution.

Discontinuous bundling transition in semiflexible polymer networks induced by Casimir interactions

Fluctuation-induced interactions are an important organizing principle in a variety of soft matter systems. We investigate the role of fluctuation-based or thermal Casimir interactions between cross linkers in a semiflexible network. One finds that, by integrating out the polymer degrees of freedom, there is an attractive logarithmic potential between nearest-neighbor cross linkers in a bundle, with a significantly weaker next-nearest-neighbor interaction. Here we show that a one-dimensional gas of these strongly interacting linkers in equilibrium with a source of unbound ones admits a discontinuous phase transition between a sparsely and a densely bound bundle. This discontinuous transition induced by the long-ranged nature of the Casimir interaction allows for a similarly abrupt structural transition in semiflexible filament networks between a low cross linker density isotropic phase and a higher cross link density bundle network. We support these calculations with the results of finite element Brownian dynamics simulations of semiflexible filaments and transient cross linkers.

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.

Structure-driven homology pairing of chromatin fibers: the role of electrostatics and protein-induced bridging

Chromatin domains formed in vivo are characterized by different types of 3D organization of interconnected nucleosomes and architectural proteins. Here, we quantitatively test a hypothesis that the similarities in the structure of chromatin fibers (which we call "structural homology") can affect their mutual electrostatic and protein-mediated bridging interactions. For example, highly repetitive DNA sequences in heterochromatic regions can position nucleosomes so that preferred inter-nucleosomal distances are preserved on the surfaces of neighboring fibers. On the contrary, the segments of chromatin fiber formed on unrelated DNA sequences have different geometrical parameters and lack structural complementarity pivotal for stable association and cohesion. Furthermore, specific functional elements such as insulator regions, transcription start and termination sites, and replication origins are characterized by strong nucleosome ordering that might induce structure-driven iterations of chromatin fibers. We propose that shape-specific protein-bridging interactions facilitate long-range pairing of chromatin fragments, while for closely-juxtaposed fibers electrostatic forces can in addition yield fine-tuned structure-specific recognition and pairing. These pairing effects can account for some features observed for mitotic and inter-phase chromatins.

Hierarchical self-assembly of actin in micro-confinements using microfluidics

We present a straightforward microfluidics system to achieve step-by-step reaction sequences in a diffusion-controlled manner in quasi two-dimensional micro-confinements. We demonstrate the hierarchical self-organization of actin (actin monomers-entangled networks of filaments-networks of bundles) in a reversible fashion by tuning the [Formula: see text] ion concentration in the system. We show that actin can form networks of bundles in the presence of [Formula: see text] without any cross-linking proteins. The properties of these networks are influenced by the confinement geometry. In square microchambers we predominantly find rectangular networks, whereas triangular meshes are predominantly found in circular chambers.

Electrostatics of strongly charged biological polymers: ion-mediated interactions and self-organization in nucleic acids and proteins

Charges on biological polymers in physiologically relevant solution conditions are strongly screened by water and salt solutions containing counter-ions. However, the entropy of these counterions can result in surprisingly strong interactions between charged objects in water despite short screening lengths, via coupling between osmotic and electrostatic interactions. Widespread work in theory, experiment, and computation has been carried out to gain a fundamental understanding of the rich, yet sometimes counterintuitive, behavior of these polyelectrolyte systems. Examples of polyelectrolyte association in biology include DNA packaging and RNA folding, as well as aggregation and self-organization phenomena in different disease states.

Polyamine-induced bundling of F-actin

To better understand the mechanism of actin filament (F-actin) bundling by polyamines, we have measured the onset of bundling as a function of polyamine concentration. Samples were centrifuged at low speeds to separate bundles from unbundled actin, and the relative amounts of actin in the pellet and supernatant were determined via gel electrophoresis, yielding a description of the bundling transition as a function of actin and polyamine concentrations. These experiments were carried out for two different polyamines, spermine (tetravalent) and spermidine (trivalent). We found that the threshold concentration of polyamine needed to bundle actin is independent of both actin concentration and Mg2+ concentration over a wide range in Mg2+ concentration. We also find that spermine in F-actin bundles is essentially invisible in solution-phase proton NMR, suggesting that it is bound so tightly to F-actin that it is immobilized.

Simulation and theory of self-assembly and network formation in reversibly cross-linked equilibrium polymers

A simulation model of hard spheres capable of reversible assembly into chains, which then may reversibly cross-link into networks, has been studied through grand canonical Monte Carlo simulation. Effects of varying intra- and interchain bond strengths, chain flexibilities, and restrictions on cross-linking angle were investigated. Observations including chain-length distributions and phase separation could be captured in most cases using a simple model theory. The coupling of chain growth to cross-linking was shown to be highly sensitive to the treatment of cross-linking by chain ends. In some systems, ladderlike domains of several cross-links joining two chains were common, resulting from cooperativity in the cross-linking. Extended to account for this phenomenon, the model theory predicts that such cooperativity will suppress phase separation in weakly polymerizing chains and at high cross-link concentration. In the present model, cross-linking stabilizes the isotropic phase with respect to the nematic phase, causing a shift in the isotropic-nematic transition to higher monomer concentration than in simple equilibrium polymers.

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.

Structural polymorphism of the actin-espin system: a prototypical system of filaments and linkers in stereocilia

We examine the interaction between cytoskeletal F-actin and espin 3A, a prototypical actin bundling protein found in sensory cell microvilli, including ear cell stereocilia. Espin induces twist distortions in F-actin as well as facilitates bundle formation. Mutations in one of the two F-actin binding sites of espin, which have been implicated in deafness, can tune espin-actin interactions and radically transform the system's phase behavior. These results are compared to recent theoretical work on the general phase behavior linker-rod systems.

A semiflexible chain model of local denaturation in double-stranded DNA

Double-stranded DNA (dsDNA) is known to form a locally denatured structure ("bubble") below its denaturation temperature (T(c)). We have investigated the physical behavior of the bubbles using a model of dsDNA as two wormlike chains containing interacting complementary base pairs. The model incorporates two biologically relevant features, i.e., semiflexibility of the strand and overdamping nature of aqueous background. Computer simulations using the Langevin equation are performed to examine the size distribution and dynamics of bubbles. The results show that the entropy associated with semiflexibility of DNA sensitively affects the size distribution and lifetime of bubble. In particular, the lifetime grows with bubble size m as m(2.7) at temperature close to T(c), which is consistent with our analysis based on a stochastic model of bubble size dynamics.

Condensation of DNA-actin polyelectrolyte mixtures driven by ions of different valences

Multivalent ions can induce condensation of like-charged polyelectrolytes into compact states, a process that requires different ion valences for different polyelectrolyte species. In this work we examine the condensation behavior in binary anionic polyelectrolyte mixtures consisting of DNA coils and F-actin rods in the presence of monovalent, divalent, and trivalent ions. As expected, monovalent ions do not condense either component and divalent ions selectively condense F-actin rods out of the polyelectrolyte mixture. For trivalent ions, however, we observe a microphase separation between the two polyelectrolytes into coexisting finite-sized F-actin bundles and DNA toroids. Further, by increasing the DNA volume fraction in the mixture, condensed F-actin bundles can be completely destabilized, leading to only DNA condensation within the mixture. We examine a number of possible causes and propose a model based on polyelectrolyte competition for ions.

Linear polyelectrolytes in tetravalent salt solutions

The effect of adding tetravalent salt of different sizes to a solution of linear and flexible polyelectrolytes is investigated by molecular dynamics simulations. Upon the addition of salt, a chain reexpansion takes place, following a well-known collapsed conformation. The degrees of collapse and reexpansion increase with ion size. In the solution, tetravalent counterions replace monovalent ones and condense onto the chains. The condensation for small ions displays a profile different from that for large ones. In a high-salt region, ions can form layering orders around a polyelectrolyte and locally overcompensate the charge inside. Consequently, the integrated charge distribution reveals an oscillatory behavior away from a chain. By studying the radial distribution function between monomers on different polyelectrolytes, like-charge attraction between chains is demonstrated. This attraction is a prerequisite to chain aggregation or precipitation. The results show a strong dependence of salt concentration and ion size on the properties of polyelectrolyte solutions.

Salt effects on aggregation of O-carboxymethylchitosan in aqueous solution

The effects of salt with different valences (NaCl, CaCl2 and CrCl3) on the aggregation of O-carboxylmethylchitosan (OCMCS) in dilute aqueous solution were investigated using viscometry, dynamic laser light scattering (DLS) and atomic force microscopy (AFM). With increasing OCMCS concentration beyond a critical aggregation concentration (cac) of approximately 0.045 g/l, the aggregation of OCMCS appears in solution. The driving forces of the OCMCS aggregation are intermolecular hydrogen bond, hydrophobic interaction and electrostatic repulsion. The OCMCS aggregation behavior strongly depends on the valence of salt. When NaCl is added, the aggregate size increases with NaCl concentration. When CaCl2 or CrCl3 is added to a given OCMCS concentration, there exists a critical concentration each of Ca2+ and Cr3+. Before the critical concentration, the aggregates decrease in size with increasing salt concentration due to the intra-aggregate complexation; while after the critical concentration, the size of the aggregates increases with salt concentration due to the inter-aggregate complexation. Moreover, the effect of Cr3+ on the OCMCS aggregation is greater than that of Ca2+. The formation of the intra-aggregate complexation is found to be a kinetic process and the aggregate size decreases with time; the formation of the inter-aggregate complexation is also kinetic where the aggregate size increases with time. The aggregates dominated by the intra-aggregate complexation are small, compact and spherical, while the aggregates dominated by the inter-aggregate complexation show the big, compact and spherical morphology.

Intermolecular adhesion in conducting polymers

We analyze the interaction of two conducting, charged polymer chains in solution using a minimal model for their electronic degrees of freedom. We show that a crossing of the two chains in which the polymers pass within angstroms of each other leads to a decrease of the electronic energy of the combined system that is significantly larger than the thermal energy and thus promotes interchain aggregation. We consider the competition of this attractive interaction with the screened electrostatic repulsion and thereby propose a phase diagram for such polymers in solution; depending on the charge density and persistence length of the chains, the polymers may be unbound, bound in loose, braidlike structures, or tightly bound in a parallel configuration.

Direct observation of counterion organization in F-actin polyelectrolyte bundles

Attractions between like-charged polyelectrolytes have been observed in a variety of systems (W.M. Gelbart, R.F. Bruinsma, P.A. Pincus, V.A. Parsegian, Phys. Today 53, September issue, 38 (2000)). Recent biological examples include DNA, filamentous viruses, and F-actin. Theoretical investigations on idealized systems indicate that counterion correlations play a central role, but no experiments that specifically probe such correlations have been performed. Using synchrotron X-ray diffraction, we have directly observed the organization of multivalent ions on cytoskeletal filamentous actin (a well-defined biological polyelectrolyte) and found an unanticipated symmetry-breaking collective counterion mechanism for generating attractions. Surprisingly, the counterions do not form a lattice that simply follows actin's helical symmetry; rather, the counterions organize into "frozen" ripples parallel to the actin filaments and form structures reminiscent of charge density waves. Moreover, these 1D counterion charge density waves form a coupled mode with twist deformations of the oppositely charged actin filaments. This counterion organization is not sensitive to thermal fluctuations in temperature range accessible to protein-based polyelectrolyte systems. Moreover, the counterion density waves are "pinned" to the spatial periodicity of charges on the actin filament even if the global filament charge density is varied, indicating the importance of charge periodicity on the polyelectrolyte substrate.

Multiscale study of counterion-induced attraction and bundle formation of F-actin using an Ising-like mean-field model

An Ising-like counterion-binding model is developed and solved by a mean-field method. For G-actin, the calculated affinity constants of all the binding sites ranging from loose to tight binding match the experimental data. The model is used to calculate the interaction energy between two F-actin filaments. Within a certain counterion concentration range, a rapidly decaying attractive force between two parallel filaments is produced not only by the correlation of the counterion distributions on the two filaments, but also by the correlation of the configurations of the two filaments with fixed counterion positions, which has been ignored in previous calculations. The bundling energy depends strongly on the configuration of the filaments. Upon bundling, the tightly bound counterion site is not affected, but the medium and loosely bound ones are. The model reproduces the observed minimal divalent counterion concentration for bundling, and naturally predicts the resolubilization of bundles which is seen in recent experiments. At the optimal counterion concentration, we obtain a bundling energy of approximately -0.01 eV per monomer along the filament. The counterion valence strongly affects the optimal counterion concentration, but has only minor effects on the optimal bundling energy. We show that the attractive potential between filaments can be simplified as the sum of interactions between their monomers. This simplification makes it possible to calculate the exact free energy of a two-F-actin-filament system. We are thus able to probe the effects of filament length on F-actin bundling and obtain a critical length for bundling of 59 monomers at 1 microM monomer concentration and pH=7.2.

Thermoreversible crosslinking of polyelectrolyte chains

Thermoreversible crosslinking of polyelectrolyte chains via short-range attractions such as hydrogen bonding induced by uncharged or charged particles is studied within the Flory model of ideal association. Electrostatic interactions between the charges at different linking fractions are taken into account by using a generalized random phase approximation approach which includes the network connectivity. We find that at certain concentration of linking agents an infinitely large polymer network is formed. We calculate the structural gelation lines for linkers of different charges and functionalities.

Onset of DNA aggregation in presence of monovalent and multivalent counterions

We address theoretically aggregation of DNA segments by multivalent polyamines such as spermine and spermidine. In experiments, the aggregation occurs above a certain threshold concentration of multivalent ions. We demonstrate that the dependence of this threshold on the concentration of DNA has a simple form. When the DNA concentration c(DNA) is smaller than the monovalent salt concentration, the threshold multivalent ion concentration depends linearly on c(DNA), having the form alphac(DNA) + beta. The coefficients alpha and beta are related to the density profile of multivalent counterions around isolated DNA chains, at the onset of their aggregation. This analysis agrees extremely well with recent detailed measurements on DNA aggregation in the presence of spermine. From the fit to the experimental data, the number of condensed multivalent counterions per DNA chain can be deduced. A few other conclusions can then be reached: 1), the number of condensed spermine ions at the onset of aggregation decreases with the addition of monovalent salt; 2), the Poisson-Boltzmann theory overestimates the number of condensed multivalent ions at high monovalent salt concentrations; and 3), our analysis of the data indicates that the DNA charge is not overcompensated by spermine at the onset of aggregation.

Polyelectrolytes in the presence of multivalent ions: gelation versus segregation

We analyze solutions of strongly charged chains bridged by linkers such as multivalent ions. The gelation induced by the strong short range electrostatic attractions is dramatically suppressed by the long range electrostatic correlations due to the charge along the non-cross-linked monomers and ions. A modified Debye-Hückel approach of cross-linked clusters of charged chains is used to determine the mean field gelation transition self-consistently. Highly dilute polyelectrolyte solutions tend to segregate macroscopically. Semidilute solutions can form gels if the Bjerrum length l(B) and the distance between neighboring charged monomers along the chain b are both greater than the ion size a.