Electrostatic Persistence Length Research Articles

Dimension and Flexibility of Polystyrenesulfonate Chains in Methanol-Water.

The influence of the relative permittivity of the solvent medium on the single-chain dimension and flexibility of sodium polystyrenesulfonate chains has been investigated in mixed solvent media of methanol and water using viscosity experiments. Particular attention has been paid to explore the effect of the added low-molar-mass electrolyte. The root-mean-square (rms) radii of gyration of the chains in the unperturbed state have been calculated by applying the Flory model, while the intrinsic persistence lengths by the Benoit-Doty equation on the basis of the Kratky-Porod worm-like chain model. Estimation of the expansion factors for the rms radius of gyration, and the electrostatic persistence length helps evaluate the rms radii of gyration and the total persistence length of polystyrenesulfonate chains in the presence of varying amount of the supporting electrolyte. The polyion chains are highly extended at low ionic strengths but exhibit coil-like behavior with small persistence lengths when an excess of the supporting electrolyte is added in all the methanol-water mixtures investigated. Specifically, in the investigated solvent media, the polystyrenesulfonate chains have been found to shrink by ∼63-65% in the θ-state from their expanded conformation in the presence of 0.0001 mol L-1 NaCl. The chain dimensions pass through a maximum as the medium becomes richer in methanol, which could be explained by the formation and breakup of internal rings involving the polyion chain and water and/or methanol molecules. The intrinsic persistence length of sodium polystyrenesulfonate in a methanol-water mixture containing 0.1 mole fraction of methanol is ca. 1.3 times that in a medium with 0.3 mole fraction of methanol, indicating that flexibility of the polyion depends appreciably on the relative permittivity of the medium.

Polysaccharide-based polyampholyte complex formation: Investigating the role of intra-chain interactions

The difference in inter-chain and intra-chain electrostatic attraction was investigated in polyelectrolyte and polyampholyte electrostatic complex formation. Three polymers with similar backbone molecular structures including chitosan (Ch) polycation, carboxymethyl cellulose (CMCe) polyanion, and carboxymethyl chitosan (CMCh) polyampholyte were used for this purpose. The turbidimetric, water content, and rheological measurements for polyampholyte self-complex showed more dependence on the ionic strength rather than the polyelectrolyte one. The degree of dissociation (α), dissociation constant (pKa), and intrinsic persistence length were calculated by applying the Katchalsky-Lifson model to potentiometric data. We studied the gyration radii as a function of Debye length and observed the polyampholyte chain contractions due to the intra-chain electrostatic attractions, which minimize the entropic gain of the inter-chain complex formation. This is in accordance with the decrease in pKa by αc for CMCh which is the opposite of that for the Ch and CMCe samples. We also found that the polyampholyte has less intrinsic and electrostatic persistence length compared with both polyanion and polycation with similar chain structures indicating the impact of the inter-chain electrostatic interaction on the complex properties. This study deepens our insight about the behavior of CMCh and the nature of difference between CMCh and Ch/CMCe electrostatic complexes.

Flexible, charged biopolymers in monovalent and mixed-valence salt: Regimes of anomalous electrostatic stiffening and of salt insensitivity.

The conformations of biological polyelectrolytes (PEs), such as polysaccharides, proteins, and nucleic acids, affect how they behave and interact with other biomolecules. Relative to neutral polymers, PEs in solution are more locally rigid due to intrachain electrostatic repulsion, the magnitude of which depends on the concentration of added salt. This is typically quantified using the Odijk-Skolnick-Fixman (OSF) electrostatic-stiffening model, in which salt-dependent Debye-Hückel (DH) screening modulates intrachain repulsion. However, the applicability of this approach to flexible PEs has long been questioned. To investigate this, we use high-precision single-molecule elasticity measurements to infer the scaling with salt of the local stiffness of three flexible biopolymers (hyaluronic acid, single-stranded RNA, and single-stranded DNA) in both monovalent and mixed-valence salt solutions. In monovalent salt, we collapse the data across all three polymers by accounting for charge spacing, and find a common power-law scaling of the electrostatic persistence length with ionic strength with an exponent of 0.66±0.02. This result rules out simple OSF pictures of electrostatic stiffening. It is roughly compatible with a modified OSF picture developed by Netz and Orland; alternatively, we posit the exponent can be explained if the relevant electrostatic screening length is the interion spacing rather than the DH length. In mixed salt solutions, we find a regime where adding monovalent salt, in the presence of multivalent salt, does not affect PE stiffness. Using coarse-grained simulations, and a three-state model of condensed, chain-proximate, and bulk ions, we attribute this regime to a "jacket" of ions surrounding the PE that regulates the chain's effective charge density as ionic strength varies. The size of this jacket in simulations is again consistent with a screening length controlled by interion spacing rather than the DH length. Taken together, our results describe a unified picture of the electrostatic stiffness of polyelectrolytes in the mixed-valence salt conditions of direct relevance to cellular and intercellular biological systems.

Sequence-Dependent Persistence Length of Long DNA.

Using a high-throughput genome-mapping approach, we obtained circa 50million measurements of the extension of internal human DNA segments in a 41 nm×41 nm nanochannel. The underlying DNA sequences, obtained by mapping to the reference human genome, are 2.5-393kilobase pairs long and contain percent GC contents between 32.5% and 60%. Using Odijk's theory for a channel-confined wormlike chain, these data reveal that the DNA persistence length increases by almost 20% as the percent GC content increases. The increased persistence length is rationalized by a model, containing no adjustable parameters, that treats the DNA as a statistical terpolymer with a sequence-dependent intrinsic persistence length and a sequence-independent electrostatic persistence length.

Bending stiff charged polymers: The electrostatic persistence length

Many charged polymers, including nucleic acids, are locally stiff. Their bending rigidity —quantified by the persistence length— depends crucially on Coulombic features, such as the ionic strength of the solution which offers a convenient experimental route for tuning the rigidity. While the classic Odijk-Skolnick-Fixman treatment fails for realistic parameter values, we derive a simple analytical formula for the electrostatic persistence length. It is shown to be in remarkable agreement with numerically obtained Poisson-Boltzmann theory results, thereby fully accounting for non-linearities, among which counter-ion condensation effects. Specified to double-stranded DNA, our work reveals that the widely used bare persistence length of 500 Å is overestimated by some 20%.

Role of the ratio of biopolyelectrolyte persistence length to nanoparticle size in the structural tuning of electrostatic complexes.

Aggregation of nanoparticles of given size R induced by addition of a polymer strongly depends on its degree of rigidity. This is shown here on a large variety of silica nanoparticle self-assemblies obtained by electrostatic complexation with carefully selected oppositely charged biopolyelectrolytes of different rigidity. The effective rigidity is quantified by the total persistence length L_{T} representing the sum of the intrinsic (L_{p}) and electrostatic (L_{e}) polyelectrolyte persistence length, which depends on the screening, i.e., on ionic strength due to counterions and external salt concentrations. We experimentally show that the ratio L_{T}/R is the main tuning parameter that controls the fractal dimension D_{f} of the nanoparticles' self-assemblies, which is determined using small-angle neutron scattering: (i) For L_{T}/R<0.3 (obtained with flexible poly-l-lysine in the presence of an excess of salt), chain flexibility promotes easy wrapping around nanoparticles in excess, hence ramified structures with D_{f}∼2. (ii) For 0.3<L_{T}/R≤1 (semiflexible chitosan or hyaluronan complexes), chain stiffness promotes the formation of one-dimensional nanorods (in excess of nanoparticles), in good agreement with computer simulations. (iii) For L_{T}/R>1,L_{e} is strongly increased due to the absence of salt and repulsions between nanoparticles cannot be compensated for by the polyelectrolyte wrapping, which allows a spacing between nanoparticles and the formation of one-dimensional pearl necklace complexes. (iv) Finally, electrostatic screening, i.e., ionic strength, turned out to be a reliable way of controlling D_{f} and the phase diagram behavior. It finely tunes the short-range interparticle potential, resulting in larger fractal dimensions at higher ionic strength.

Multi-region to single region shear thinning transitions in drying PEDOT:PSS dispersions: contributions from charge density fluctuations.

PSS), in water during the drying process is explored. PSS dispersions in water can be non-stoichiometric polyelectrolyte complexes (PECs) with free polyanion (PSS) chains resulting in scrambled egg conformations. They also behave as charged colloidal particles. The behaviour of PSS dispersions as PECs and charged colloidal systems during drying is analyzed using rheology and TEM, zeta potential, conductivity, pH, dynamic light scattering (DLS) and rheo-small angle light scattering (rheo-SALS) information. The rheological behaviour is found to have significant contributions from conformational changes in the PSS PEC due to charge density fluctuations affecting the electrostatic persistence length, le. A low concentration regime with multiple shear thinning behaviour and a high concentration regime with single shear thinning behaviour were observed. The PEC behavior of PSS is further explored by charge neutralizing the dispersions with salts.

Single-Stranded DNA is not a Worm-Like Chain

The structure of unfolded, flexible polyelectrolytes in solution, while an important subject for understanding various biological and technological phenomena such as RNA and protein folding and the self-assembly of DNA nanostructures, is poorly understood. While it is well-known and tested that semi-flexible polyelectrolytes, such as dsDNA, behave as Worm-Like Chains (WLC) with a scale-dependent electrostatic persistence length, validity of this model for flexible polyelectrolytes is doubtful. However, throughout the literature, data on flexible polyelectrolytes are naively fit to WLC models due to the simplicity of the analytical expressions. Here, we examine the validity of the WLC model for flexible polyelectrolytes using single-molecule force spectroscopy on ssDNA. We find that the force-extension data cannot be adequately fit to the Marko-Siggia prediction for the WLC even after accounting for electrostatic effects. Rather, our data reveal a self-avoiding walk regime consistent with scaling predictions at low force followed by a regime where the extension scales as a logarithm with force over a broad range of moderate forces and monovalent salt concentrations. Further, we run molecular dynamics simulations on a bead-spring model polyelectrolyte under tension and reproduce this logarithmic behavior in monovalent salt, indicating that it is indeed a general behavior for flexible polyelectrolytes. Examining the structure factor of the simulated polymer reveals a highly wrinkled, ion-stabilized structure at length scales smaller than a Debye length which defies characterization as a WLC. Addition of divalent salt to either the simulation or experiment results in enhanced flexibility indicating increased wrinkling or polymer “wrapping” around the divalent ions.

Persistence length of polyelectrolytes with precisely located charges

The conformation of polyelectrolytes in aqueous salt solutions is closely related to their self-assembly properties. In particular, the persistence length has a large impact on how the chain can arrange itself. In this work, biomimetic poly N-substituted glycines (polypeptoids) have been designed to position charged side chains at precise distances from each other to elucidate the relationship between the spacing of the charges along the backbone, the ionic strength, and the persistence length. Using small angle neutron scattering (SANS), it is shown that at low ionic strength, polypeptoids with charged groups located closer to each other along the polymer backbone are stiffer than those with the charged groups spaced further apart. At high ionic strength, the total persistence length decreases for both macromolecules because the electrostatic repulsions between ionized groups are screened. The measured persistence lengths were compared to those calculated using a discrete chain model with bending rigidity, and it is shown that the electrostatic persistence length scales quadratically with the Debye screening length. It is also shown that the bare persistence length of a molecule with alternating ionizable and hydrophilic groups is larger than that of a molecule containing 100% ionizable groups. This difference can be attributed to the longer hydrophilic side chains that may induce local chain stiffening.

Variation of Weak Polyelectrolyte Persistence Length through an Electrostatic Contour Length

There is currently no consensus that the classical perspective of electrostatic persistence length (the so-called Odijk–Skolnick–Fixman, or OSF, formulation), derived by equating changes in electrostatic energy with bending rigidity, is applicable to weak polyelectrolytes. Here, with a focus on polyelectrolyte chains of finite length at a single scale, we formulate a simple and general theoretical model featuring the electrostatic persistence length, Pe, through the introduction of an electrostatic contour length, Le, such that the electrostatic energy balance is applicable to highly flexible charged polyelectrolytes (where the intrinsic stiffness, P0 is on the order of Pe). To isolate the effect of ionization, only the salt-free regime is considered. At the upper limit (relatively rigid molecules), the new formulation converges to the classical OSF form, while the lower bound (highly flexible molecules) approaches proportionality to the Debye screening length, κ–1. In general, the electrostatic persistence can be described retroactively by the Debye screening, κ–1, and the ratio of electrostatic to intrinsic contour lengths, A/a, complementary to predictive theoretical formulations. The theory is validated via full atomistic molecular dynamics simulations of single, isolated model weak polyelectrolyte chains in explicit solvent—specifically poly(acrylic acid), PAA, and poly(allylamine hydrochloride), PAH, both of which undergo significant increases in persistence length under ionization. An ensemble of equilibrated polymer states is obtained via temperature assisted sampling, implementing molecular dynamics to drive the polymers into physically accessible conformations via cyclical temperature fluctuations and equilibration.

Double-exponential decay of orientational correlations in semiflexible polyelectrolytes

In this paper we revisited the problem of persistence length of polyelectrolytes. We performed a series of Molecular Dynamics simulations using the Debye-Hückel approximation for electrostatics to test several equations which go beyond the classical description of Odijk, Skolnick and Fixman (OSF). The data confirm earlier observations that in the limit of large contour separations the decay of orientational correlations can be described by a single-exponential function and the decay length can be described by the OSF relation. However, at short countour separations the behaviour is more complex. Recent equations which introduce more complicated expressions and an additional length scale could describe the results very well on both the short and the long length scale. The equation of Manghi and Netz when used without adjustable parameters could capture the qualitative trend but deviated in a quantitative comparison. Better quantitative agreement within the estimated error could be obtained using three equations with one adjustable parameter: 1) the equation of Manghi and Netz; 2) the equation proposed by us in this paper; 3) the equation proposed by Cannavacciuolo and Pedersen. Two characteristic length scales can be identified in the data: the intrinsic or bare persistence length and the electrostatic persistence length. All three equations use a single parameter to describe a smooth crossover from the short-range behaviour dominated by the intrinsic stiffness of the chain to the long-range OSF-like behaviour.

Polyelectrolytes in Salt Solutions: Molecular Dynamics Simulations

We present results of the molecular dynamics simulations of salt solutions of polyelectrolyte chains with number of monomers N = 300. Polyelectrolyte solutions are modeled as an ensemble of bead–spring chains of charged Lennard-Jones particles with explicit counterions and salt ions. Our simulations show that in dilute and semidilute polyelectrolyte solutions the electrostatic induced chain persistence length scales with the solution ionic strength as I–1/2. This dependence of the chain persistence length is due to counterion condensation on the polymer backbone. In dilute polyelectrolyte solutions the chain size decreases with increasing the salt concentration as R ∝ I–1/5. This is in agreement with the scaling of the chain persistence length on the solution ionic strength, lp ∝ I–1/2. In semidilute solution regime at low salt concentrations the chain size decreases with increasing polymer concentration, R ∝ cp–1/4, while at high salt concentrations we observed a weaker dependence of the chain size on th...

Conformational analysis and estimation of the persistence length of DNA using atomic force microscopy in solution

The worm-like chain model describes the mechanical properties of semi-flexible polymers by introducing a certain correlation length along the contour. This correlation length is called the persistence length. Using atomic force microscopy in solution, we performed measurements of the persistence length of DNA molecules. We found good agreement between the theoretical model and experimental data. However, the measured persistence length values in solution differ from those found by several authors using the same technique but in dry air. In order to determine the contribution of the electrostatic persistence length to the total persistence length, we varied the salt concentration. We found a large discrepancy between the Odijk, Skolnick and Fixman theory and our measurements. The effect of divalent ions and the omission in the theory of the dependence of non-electrostatic persistence length on salt concentration are qualitatively invoked.

Structure and dynamics of hyaluronic acid semidilute solutions: A dielectric spectroscopy study

Dielectric spectroscopy is used to investigate fundamental length scales describing the structure of hyaluronic acid sodium salt (Na-HA) semidilute aqueous solutions. In salt-free regime, the length scale of the relaxation mode detected in MHz range scales with HA concentration as c(HA)(-0.5) and corresponds to the de Gennes-Pfeuty-Dobrynin correlation length of polyelectrolytes in semidilute solution. The same scaling was observed for the case of long, genomic DNA. Conversely, the length scale of the mode detected in kilohertz range also varies with HA concentration as c(HA)(-0.5) which differs from the case of DNA (c(DNA)(-0.25)). The observed behavior suggests that the relaxation in the kilohertz range reveals the de Gennes-Dobrynin renormalized Debye screening length, and not the average size of the chain, as the pertinent length scale. Similarly, with increasing added salt the electrostatic contribution to the HA persistence length is observed to scale as the Debye length, contrary to scaling pertinent to the Odijk-Skolnick-Fixman electrostatic persistence length observed in the case of DNA. We argue that the observed features of the kilohertz range relaxation are due to much weaker electrostatic interactions that lead to the absence of Manning condensation as well as a rather high flexibility of HA as compared to DNA.

Conformations of End-Tethered DNA Molecules on Gold Surfaces: Influences of Applied Electric Potential, Electrolyte Screening, and Temperature

We describe the behavior of 72mer oligonucleotides that are end-tethered to gold surfaces under the influence of applied electric fields. The DNA extension is measured by fluorescence energy transfer as a function of the DNA hybridization state (single- and double-stranded), the concentration of monovalent salt in solution (100 microM to 1 M NaCl), the applied electrode potential (-0.6 to +0.1 V vs Pt), and the temperature (1 to 50 degrees C). At high ionic strength, the DNA conformations are very robust and independent of the applied electrode potential and temperature variations. In solutions of medium ionic strength, the DNA conformation can be manipulated efficiently by applying bias potentials to the Au electrodes. The molecules are repelled at negative potentials and attracted to the surface at positive potentials. The conformation transition occurs abruptly when the electrode bias is swept by merely 0.1 V across the transition potential, which shifts negatively when the salinity is decreased. The behavior can be understood by electrostatic screening arguments and, in the case of single-stranded DNA, when secondary structures are taken into account. At low ionic strength, the experiments reveal an intriguing temperature-dependent stiffening of single-stranded DNA, which can be rationalized by combining counterion condensation theory with the Odjik-Skolnick-Fixman description of the electrostatic persistence length and the unstacking of bases at elevated temperatures.

Electric potential and bending rigidity of a wormlike particle in electrolyte solution

Using the linearized Poisson-Boltzmann equation (LPB) we derive an asymptotic expansion for the electrostatic potential of charged torus immersed in solution of an electrolyte in the limit of high salinity and large major radius of the torus. The small parameter of this expansion is the ratio of the Debye length to the minor radius of the torus. We derive asymptotic expressions for the electrostatic free energy and for the electrostatic persistence length of a polyion of a finite thickness. We propose a simple interpolation formula, xi(el)=l(B)(sigma(0)/e)(2)bkappa(D)[1+kappa(D)/(4b)], that gives the electrostatic persistence length in terms of the Debye length kappa(D), the linear charge density (sigma(0)/e), and the thickness of the polyion, 2b. This formula reproduces the exact results from the LPB theory in the limits of high and low salt concentrations. For the entire range of salinities, our formula is in excellent agreement with the numerical LPB results for wormlike particles of varying thickness. For particles of vanishing thickness, this formula reduces to the classical Odijk-Skolnick-Fixman expression.

DNA on a tube: electrostatic contribution to stiffness.

Two simple models are used to estimate the electrostatic contributions to the stiffness of short DNA fragments. The first model views DNA as two strands that are appropriately parametrized and are wrapped helically around a straight cylinder radius equal to the radius of the DNA molecule. The potential energy of the DNA due to phosphate-phosphate electrostatic interactions is evaluated assuming that the charges interact through Debye-Hückel potentials. This potential energy is compared with the potential energy as computed using our second model in which DNA is viewed as two helical strands wrapping around a curved tube whose cross-section is a disk of radius equal to the radius of the DNA. We find that the electrostatic persistence length for B-DNA molecules in the range of 105-130 bp is 125.64 angstroms (37 bp) and 76.05 angstroms (23 bp) at 5 and 10 mM monovalent salt concentration, respectively. If the condensed fraction theta is taken to be 0.715 at 10 mM, then the electrostatic persistence length is 108.28 angstroms (32 bp), while that based on taking into account end effects is 72.87 angstroms (21 bp). At 5 mM monovalent salt, the total persistence length for DNA fragments in this length range is approximately 575.64 angstroms (171 bp), using the best estimate for nonelectrostatic contribution to persistence length. Electrostatic effects thus contribute 21.8% to DNA stiffness at 5 mM for fragments between 105- to 130-bp. In contrast, electrostatics are calculated to make a negligible contribution to the DNA persistence length at physiological monovalent cation concentration. The results are compared with counterion condensation models and experimental data.

Monitoring the Synthesis and Properties of Copolymeric Polycations

The kinetics; evolution of molar mass; solution conductivity, sigma; intrinsic viscosity; and average composition drift; and distribution were determined by monitoring the synthesis of copolymeric polycations of acrylamide (Am) and [2-(acryloyloxy)ethyl]-trimethylammonium chloride (Q9). The quantitative relationship between diminishing sigma and charged co-monomers incorporation was monitored for the first time and provided novel data on counterion condensation, which occurs gradually over a broad composition regime. This new capability allows predictions concerning the relationship between copolymer composition and linear charge density, xi, to be tested and models of trivariate mass, composition, and xi distributions to be built. This approach, hence, brings together the previously disparate fields of synthetic chemistry of copolymers and physical chemical properties of polyelectrolytes. Monitoring was achieved with a new implementation of the ACOMP (automatic continuous online monitoring of polymerization reactions) platform. Reactivity ratios determined by ACOMP were rQ9 = 0.47 and rAm = 1.10. Opposite trends in composition drift and final molar mass were found; low starting percentage of Q9 led to low composition drift and high molar mass, whereas the opposite was found at high starting percentage of Q9. Complementary end-product analysis by multidetector gel permeation chromatography supported the ACOMP results. End-product polyelectrolyte properties were characterized by automatic continuous mixing, revealing that combined electrostatic persistence length and excluded volume effects led to the expected large changes in polyelectrolyte conformation and interactions. These results set the groundwork for semibatch control of molar mass, composition, and xi, and eventually for monitoring and control for inverse emulsion-based reactions of this type.

Adsorption of Highly Charged Polyelectrolytes onto an Oppositely Charged Porous Substrate

The adsorption behavior of highly charged cationic polyelectrolytes onto porous substrates is electrostatic in nature and has been shown to be highly dependent on the polyelectrolyte properties. Copolymers of acrylamide (AM) and diallyldimethylammonium chloride (DADMAC) were synthesized to have a range of macromolecular properties (i.e., charge density and molecular mass). Traditional titration methods have been complemented by fluorescence labeling techniques that were developed to directly observe the extent that fluorescently labeled poly(AM- co-DADMAC) adsorbs into the pore structure of a cellulosic substrate. Although contributing to the electrostatic driving force, the charge density acts to limit adsorption to the outermost surface under electrolyte-free conditions. However, adsorption into the pores can occur if both the molecular mass and charge density of poly(AM- co-DADMAC) are sufficiently low. Adsorption initially increases as the electrolyte concentration is increased. However, the electrostatic persistence length of poly(AM- co-DADMAC) restricts the polyelectrolyte from entering the pores. Therefore, changes in the adsorption behavior at moderate electrolyte concentrations have been attributed to swelling of the polyelectrolyte layer at the fiber exterior. The adsorption behavior changes again at high electrolyte concentrations such that poly(AM- co-DADMAC) could adsorb into the pore structure. This occurred when the electrolyte concentration was sufficient to screen the electrostatic persistence length of poly(AM- co-DADMAC), provided that the entropic driving force for adsorption still existed. It is suggested that adsorption into the pore structure is a kinetic process that is governed by localized electrostatic interactions between poly(AM- co-DADMAC) and the charges located within the pores.

Effect of a Low-Dielectric Interior on DNA Electrostatic Response to Twisting and Bending

We study the effects of a low-dielectric core of rod-like macromolecules on their electrostatic persistence lengths. We use the exact solution of the linear Poisson-Boltzmann equation for the potential of a charge on the surface of a low-dielectric cylinder. We apply the results to the B-DNA molecule, modeled as a double helical array of discrete charges wound on the surface of a low-dielectric rod. For this charge geometry, we calculate the change in the electrostatic twist persistence as compared to DNA with a water-permeable core. We also discuss possible effects of the low-dielectric molecular core on DNA bending persistence.