Dielectric behavior of polyelectrolytes. III. The role of counterion interactions

The role of the Coulomb forces between the counterions on the surface of polyelectrolytes on the dielectric response is analyzed. An estimate of the maximum dielectric increment (as a function of the number of counterions) is found as a function of the molecular length. The minimum-energy configuration of the counterions on a cylinder is found to be a double helix, suggesting the fundamental importance of electrostatic interactions in determining structure. Solutions of the dynamical equations for a few counterions indicate that a single mode dominates the relaxation which is enhanced by the inter-ion repulsions. A lower bound is found for this mode based on analysis of the system response for short lengths. Sum rules for the rates and amplitudes of the dipolar correlation function are derived and lead to an upper bound for the rate of the dominant mode. These bounds approach one another for the parameters characteristic of restriction fragments of DNA. This permits a prediction of the magnitude and time scale of the dielectric response.     

Dielectric behavior of polyelectrolytes II. The cylinder

The dipolar correlation function for a system of coupterions diffusing on the surface of a polyelectrolyte cylinder is computed. The influence of screened coulombic repulsions on the dielectric increment is determined. Dissociation and reassociation of the counterions to the cylinder is treated microscopically and the coupled bulk diffusion is solved in the presence of the Poisson-Boltzmann potential. It is found that the correlation function contains a small, fast decaying, molecular weight independent part arising from diffusion around the cylinder and a large, slowly decaying, molecular weight dependent part arising from diffusion along the cylinder axis. The dissociation-reassociation kinetics can play a large, possible dominant, role in determining the relaxation rates.     

Monte Carlo calculations of ion distributions surrounding the oligonucleotide d(ATATATATAT)2 in the B, A, and wrinkled D conformations.

We calculated the uni-univalent ion distributions around the oligonucleotide d(AT)5.d(AT)5 in the A, B and wrinkled D conformation using the Metropolis Monte Carlo method. All atoms were included in the oligonucleotide model with partial charges and hard sphere radii assigned to each atom. The univalent counter- and coions were modeled as hard spheres with radius 0.3 nm. The solvent was assigned a dielectric constant of 80, corresponding to a temperature of 298K. The counterion distribution surrounding each of the conformers and the distribution surrounding an impenetrable cylinder, were calculated for four salt concentrations. We found significant counterion density in the major groove of the A DNA while fewer counterions occupied the grooves of B DNA. In the wrinkled D DNA, where groove occupancy is sterically hindered, the ion distributions were identical to the distributions surrounding the impenetrable, cylindrical model. This suggests that excluded volume effects significantly influence the details of the ion distributions near the oligomer, while the detailed charge distributions of the oligomer affects the ion distributions only minimally. Although substantial variation in counterion density was observed near the oligomers of differing conformations, the total number of counterions located within a cylinder surrounding the oligomer bounded radially by 2.4 nm was independent of the conformation of the oligomer. Therefore, for this model system, the local univalent counterion distributions are extremely sensitive to the geometry of the oligonucleotide whereas the extent of neutralization of the oligoanion is insensitive to the conformation of the oligomer.     

Polyelectrolyte theory. I. Counterion accumulation,site-binding,and their insensitivity to polyelectrolyte shape in solutions containing finite salt concentrations

We have computed the Poisson-Boltzmann distribution of counterions around polyelectrolytes in solutions containing finite salt concentrations. The polyelectrolytes considered here are highly charged in the sense that ξ > 1, ξ being the linear charge density parameter for cylinders, which is generalized by us to other shapes. Contrary to the situation at zero salt concentration, the counterion distribution is not strongly shape dependent, being similar for cylinders or spheres which have the same superficial charge density and radius of curvature R_c. The distribution resembles that in the neighborhood of a plane with the same charge density. Three regions are distinguished. (1) In the “inner region” which extends up to a distance R_c/2ξ from the surface, the counterion distribution is essentially salt independent. The counterion concentration in the immediate vicinity of the polyelectrolyte surface (CIV) is quite high, typically 1–10M, and proportional to the square of the surface charge density, which is its main determinant. (2) An intermediate region extends out to a distance where the electrostatic potential is equal to κT/e. This distance is comparable to λ for plane and cylinder, and smaller for the sphere. (3) In the outer region, the distribution is hardly influenced by the details of the inner region, on which it cannot, therefore, give much information. Colligative properties are dependent on the distribution in the outer region and are fairly well predicted even by a rudimentary theory. The large value of the CIV implies that site binding must often be significant. It can be computed by applying the mass-action law to site-bound counterions in equilibrium with the counterions in the neighborhood, whose concentration is the CIV, the relevant equilibrium constant being that for the binding of counterions to isolated monomer sites. Because the CIV is insensitive to salt concentration, this will also be the case for site binding. With the graphs provided, one can compute the extent of sitebinding within the Poisson-Boltzmann framework. The “condensation radius,” i.e., the radius encompassing a counterionic charge 1 ? ξ^?1 around a cylinder, is found to be large. It varies with salt concentration and tends to infinity as the salt is diluted. Neither this radius nor the charge fraction 1 ? ξ^?1 of condensation theory plays any special role in the counterion distribution. The “finite-salt” results apply to salt concentrations, typically as low as 1–10 mM. This encompasses, among others, all experiments on biological polyelectrolytes.     

Density functional theory for the nonspecific binding of salt to polyelectrolytes: thermodynamic properties

The thermodynamics of the nonspecific binding of salt to a polyelectrolyte molecule is studied using a density functional approach. The polyelectrolyte molecule is modeled as an infinite, inflexible, and impenetrable charged cylinder and the counterions and co-ions are modeled as charged hard spheres of equal diameter. The density functional theory is based on a hybrid approach where the hard-sphere contribution to the one-particle correlation function is evaluated nonperturbatively and the ionic contribution to the one-particle correlation function is evaluated perturbatively. The advantage of the approach is that analytical expressions are available for all the correlation functions. The calculated single ion preferential interaction coefficients, excess free energy, and activity coefficients show a nonmonotonic variation as a function of polyion charge in the presence of divalent ions. These properties display considerable departure from the predictions of the nonlinear Poisson-Boltzmann (NLPB) equation, with qualitative differences in some cases, which may be attributed to correlation effects neglected in the NLPB theory.     

Dependence of DNA condensation on the correlation distance between condensed counterions

Based on the ground state of counterions condensed on a DNA molecule, a model has been developed to successfully detect the process of DNA condensation. Through further investigation, the process of DNA condensation strongly depends on the correlation distance between condensed counterions on DNA molecules. Generally, there are two routes. The process of DNA condensation with the correlation distance between condensed counterions being 2 nm or 4 nm is different from the one with the correlation distance between condensed counterions being 3 nm or 5 nm. Effects of ionic strength on the diameter of toroidal condensates originate from the increase of correlation distance between condensed counterions.     

Counterion diffusion in heparin solutions

The physiological importance of heparin is due to its strong interaction with bivalent counterions, especially Ca2+. A diffusional approach of this property is presented in this article: the observable is the self-diffusion coefficient of the counterions, as a function of the ratio of the polyelectrolyte over the added salt concentrations. All the results are in agreement with a simple "quasi-chemical model" in which two different states are assumed for the counterions: "free" or "bound." The proportions of these two types of ions are calculated according to the distribution function of the counterions around the polyion. We assume that those of counterions located at a distance closer than a, the characteristic distance, are bound; the others are free. The ionic distribution function is evaluated by a numerical integration of a cell model Poisson-Boltzmann equation. Finally, this model leads to a very good agreement with the experimental results, if the radius of heparin polyion is assumed to be 6 and 10 A.     

The contribution of transient counterion imbalances to DNA bending fluctuations

A two-sided model for DNA is employed to analyze fluctuations of the spatial distribution of condensed counterions and the effect of these fluctuations on transient bending. We analyze two classes of fluctuations. In the first, the number of condensed counterions on one side of the DNA remains at its average value, while on the other side, counterions are lost to bulk solution or gained from it. The second class of fluctuations is characterized by movement of some counterions from one side of the DNA to the other. The root-mean-square fluctuation for each class is calculated from counterion condensation theory. The amplitude of the root-mean-square fluctuation depends on the ionic strength as well as the length of the segment considered and is of the order 5-10%. Both classes of fluctuation result in transient bends toward the side of greater counterion density. The bending amplitudes are approximately 15% of the total root-mean-square bends associated with the persistence length of DNA. We are thus led to suggest that asymmetric fluctuations of counterion density contribute modestly but significantly toward the aggregate of thermalized solvent fluctuations that cause bending deformations of DNA free in solution. The calculations support the idea that counterions may exert some modulating influence on the fine structure of DNA.     

Comparison of polyelectrolyte theories of the binding of cations to DNA.

Predictions of the binding of counterions to DNA made using the counterion condensation theory developed by Manning are compared with those made using the Poisson-Boltzmann equation, solved numerically by the Runge-Kutta procedure. Ions are defined as territorially or atmospherically bound if they fall within a given distance, defined by counterion condensation theory, from the DNA surface. Two types of experimental situations are considered. The first is the delocalized binding of a single type of counterion to DNA. In this case the Poisson-Boltzmann treatment predicts somewhat lower extents of binding TO DNA, modeled as a 10-A radius cylinder, than does Manning theory. The two theories converge as the radius decreases. The second type of experiment is the competition of ions of different valence for binding to DNA. The theories are compared with literature values of binding constants of divalent ions in the presence of monovalent ions, and of spermidine 3+ in the presence of Na+ or Mg2+. Both predict with fair accuracy the salt dependence of the equilibrium constants.     

Simulations of the solvent structure for macromolecules. III. Determination of the Na+ counter ion structure

We report on a computer experiment in which, using Monte Carlo techniques, we considered a three-turn (30-base-pairs) B-DNA fragment as a solute and a set of 1200 water molecules and 60 sodium counterions (at a temperature of 300 K) as a solvent. From a statistical analysis of the Monte Carlo simulation (applied to the water molecules and counterions in the B-DNA field), we determined that the counterions themselves conform to two helical structures intertwined with the two strands. The strutures of the water molecules solvating both counterion helices and the two B-DNA strands are fully analyzed and described in detail. A model for base-pair recognition based on the above findings is proposed. Aspects of the unwinding mechanism are discussed.     

Effect of finite ionic size on the solution of the Poisson-Boltzmann equation: Application to the binding of divalent metal ions to DNA

The nonlinear Poisson-Boltzmann equation is solved for a cylindrical polyelectrolyte solution containing mono- and divalent counterions and monovalent coions. The finite size of the ions is taken into account by the introduction of the distances of closest approach between the ionic charges and the surface of the polyelectrolyte. The choice of these distances is based on the physicochemical properties of the polyelectrolyte and ions in solution. The effects of the finite ionic size on the distribution of the counterions around the polyelectrolyte and on the local ion concentration and the integrated charge fraction of the divalent cations in the vicinity of the polyelectrolyte are discussed. Theoretical predictions regarding the overall extent of binding and the extent of inner-sphere binding of divalent counterions to rodlike polyions are compared with the results of nmr studies of the binding of divalent metal ions to DNA.     

Counterion charge density determines the position and plasticity of RNA folding transition states

The self-assembly of RNA structure depends on the interactions of counterions with the RNA and with each other. Comparison of various polyamines showed that the tertiary structure of the Tetrahymena ribozyme is more stable when the counterions are small and highly charged. By monitoring the folding kinetics of the ribozyme as a function of polyamine concentration, we now find that the charge density of the counterions determines the positions of the folding transition states. The transition state ensemble (TSE) between U and N moves away from the native state as the counterion valence and charge density increase, as predicted by the Hammond postulate. The TSE is broader and less structured when the RNA is refolded in polyamines rather than Mg2+. That the charge density of the counterions determines the plasticity of the TSE demonstrates the importance of interactions among condensed counterions for the self-assembly of RNA structures. We propose that the major barrier to RNA folding is dominated by entropy changes when counterion charge density is low and enthalpy differences when it is high.     

Dielectric increment and dielectric dispersion of solutions containing simple charged linear macromolecules. I. Theory

A theory is derived for the static and frequency dependent value of the electric permittivity for model systems representing a solution of a macromolecule bearing a large number of identical charges. The polyion is represented either as a charged rigid rod (A) or as a sequence of charged rodlike subunits in an arbitrary but fixed configuration (B) and it is assumed that a certain fraction of the counterions is closely associated to the macromolecule. The dielectric properties are described in terms of fluctuations in the distribution of the associated counterions along the polyion. These fluctuations can occur locally between potential barriers marking the ends of the subunits (if considered) but can also extend over the whole molecule. Neglecting correlations between different associated counterions expressions for the static value of the dielectric increment are obtained which reveal its dependence on the fraction of bound ions, on the charge of the counterions and on the length of the molecule for model A or the radius of gyration for model B. The dynamic behaviour of A is distinguishable from that of B as the former will present one single dispersion curve of the frequency dependent electric permittivity while the latter may give rise to two different dispersion regions. This will be the case if both the exchange between bound and free ions and the rotation of the complete molecule are relatively slow in comparison to the local bound counterion density fluctuations and if these fluctuations occur on a much shorter time scale than the ion density fluctuations extending over the complete macromolecule.     

Monte Carlo determination of the distribution of ions about a cylindrical polyelectrolyte

We report a calculation of the distribution of small ions around a charged cylinder representing a polyelectrolyte molecule in solution. The Monte Carlo method of Metropolis, Rosenbluth, and Teller was used to avoid the inaccuracies known to be associated with the Poisson-Boltzmann equation. The systems examined contained a long polyelectrolyte cylinder with charge parameter, χ, equal to 4.2, corresponding approximately to a DNA molecule. In one model, the cylinder had charges on its axis and an exclusion radius to the center of the small ions equal to 10 Å, while the small ions had various radii in the range from 1 to 10 Å and one or two protonic charges. Various systems were studied; some had one species of small ion alone, others had mixtures of different types. The results showed good agreement with the solution of the Poisson-Boltzmann equation when only the species with 1-Å radius was present, but considerable discrepancies appeared with larger ions as a result of excluded volume interactions between the latter. Deviations from the Poisson-Boltzmann equation also appeared when both positive and negative small ions were present; the deviations were in the direction of a higher concentration of both counter- and co-ions, but particularly co-ions, close to the polyelectrolyte. In another model, the charges were arranged along two helices on the surface of the cylinder; the resulting radial distribution of small ions was not much different from that found when the charges were situated on the axis. In all cases there was a striking accumulation of counterions in a layer of concentration exceeding 1 mol/L at the surface of the polyion.     

Binding of positive charged ligands to DNA. The effect of ionic strength, charge and size of the ligand

The electrostatic interaction of extended cationic ligands with DNA has been considered on the basis of the analytical solution of a simplified Poisson--Boltzmann equation for the charged polyion cylinder. The control numerical solution of rigorous Poisson--Boltzmann equation shows that the assumption about the absence of coions in the vicinity of the highly charged polyion cylinder does not significantly influence the accuracy of solution and DNA electrostatic free energy evaluation. It was found that the basic contribution to the free energy of electrostatic ligand-DNA interaction is the mixing entropy change due to release of counterions from the vicinity of DNA. The equation for the dependence of the ligand to DNA binding constant K upon ionic strength c has been derived without introduction of any empirical parameters. This equation is consistent with the experimental data and can be used for the determination of a number of ligand--DNA ionic contacts in a wide range of salt concentrations. The main consequences of Manning and Record et al. theories can be considered as limiting cases of the theory presented. In particular the equation d(lnK)/d(lnc) = -0.88 N by Record et al. has a restricted range of application and it can be used only for a relative approximate estimation of the number of electrostatic bonds in ligand-DNA complexes. The analysis of electrostatic interaction of DNA with ligands which neutralize only part of phosphate groups in the binding site of DNA was also performed.(ABSTRACT TRUNCATED AT 250 WORDS)     

Importance of explicit salt ions for protein stability in molecular dynamics simulation.

The accurate and efficient treatment of electrostatic interactions is one of the challenging problems of molecular dynamics simulation. Truncation procedures such as switching or shifting energies or forces lead to artifacts and significantly reduced accuracy. The particle mesh Ewald (PME) method is one approach to overcome these problems by providing a computationally efficient means of calculating all long-range electrostatic interactions in a periodic simulation box by use of fast Fourier transformation techniques. For the application of the PME method to the simulation of a protein with a net charge in aqueous solution, counterions are added to neutralize the system. The usual procedure is to add charge-balancing counterions close to charged residues to neutralize the protein surface. In the present article, we show that for MD simulation of a small protein of marginal stability, the YAP-WW domain, explicit modeling of 0.2 M ionic strength (in addition to the charge-balancing counterions) is necessary to maintain a stable protein structure. Without explicit ions throughout the periodic simulation box, the charge-balancing counterions on the protein surface diffuse away from the protein, resulting in destruction of the beta-sheet secondary structure of the WW domain.     

Étude,par diffusion centrale des Rayons X,des polyelectrolytes rigides en solution. Cas des sels de Li,Na et Cs du DNA

The study of polyelectrolytes in solution by small-angle x-ray scattering techniques involves special problems, raised by the presence of several species: solvent, macroions, counterions, and possibly the ions of an additional electrolyte. A theoretical treatment of this problem is developed, based upon geometric concepts; the treatment applies to globular and to rodlike particles for the case of x-ray experiments carried out on an absolute scale. The equation obtained involves several parameters: mass and radius of gyration for globular particles; mass per unit length and axial radius of gyration for rodlike particles; partial specific volumes of the neutral macromolecular component of the counterions and of the added electrolyte; solvation of the macromolecular species; fraction of the counterions osmotically free. The equation is used to interpret a series of experiments performed with the Li, Na, and Cs salts of DNA in solution in water containing variable amounts of the chlorides of each cation. The effects of concentration are first eliminated by extrapolating to infinite dilution the experiments carried out at different concentrations; then the effects of the solvation are eliminated by extrapolating to pure water the results obtained at different electrolyte concentration. The parameters still involved at this stage are the mass per unit length, the partial specific volumes of the DNA and of the counterions, and the fraction of the counterions osmotically free. If the partial specific volumes are chosen in agreement with other data of the literature, and if the fraction of the counterions osmotically free is assumed to be 0.30 for the three salts, as suggested by other workers, the structure of the DNA molecules turns out to be the same for three cations, and to agree with the Watson-Crick model. These results are confirmed by the study of the liquid–crystalline gels, obtained at higher concentration, that lead to a direct determination of the mass per unit length of the rods. Moreover the solvation of the DNA molecules is determined as is shown to be different for each of the three cations. These results are in excellent agreement with those obtained by other techniques.     

Limiting-laws of polyelectrolyte solutions. Ionic distribution in mixed-valency counterions systems. II. A comparison of conductometric data and theoretical predictions

The competitive binding of monovalent and divalent counterions (M+ and M2+, respectively) has been studied by a conductometric procedure as described by De Jong et al. (Biophysical Chemistry 27 (1987) 173) for aqueous solutions of alkali metal polymethacrylates in the presence of Ca (NO3)2 or Mg(NO3)2. The experimentally obtained fractions of conductometrically free counterions are compared with theoretical values computed according to a new thermodynamic model recently developed by Paoletti et al. (Biophysical Chemistry, 41 (1991) 73). For the systems studied, the fractions of free monovalent and divalent counterions can be fairly well described by the theory. In fact, the results support the assumption that under the present conditions the conductometrically obtained distribution parameters (l) and (2) approximate the equilibrium fractions of free monovalent and divalent counterions. For a degree of neutralization of 0.8 and a molar concentration ratio of divalent counterions and charged groups on the polyion up to 0.25, the mean M+/M2+, exchange ratio nu has been found to be 1.39 +/- 0.03 and 1.33 +/- 0.03 for the alkali metal/Ca/PMA and alkali metal/Mg/PMA systems, respectively. These values agree well with the theoretical value, which for this particular case is 1.38.     

Clustering of macroions in solutions of highly asymmetric electrolytes

In this paper, we present results of computer simulations for a primitive model of asymmetric electrolyte solutions containing macroions, counterions and in a few cases, also co-ions. The results show that the valency of counterions plays an important role in shaping the net interaction between the macroions. For solutions with monovalent counterions, the macroions are distributed at larger distances, and in solutions with divalent counterions, the macroions come closer to each other and share a layer of counterions, whereas, in solutions with trivalent counterions, the macroions form clusters. These clusters dissolve upon dilution or addition of a simple electrolyte. These findings suggest a mechanism whereby the nonuniform distribution of macroions observed experimentally in charged systems may occur.     

Activity coefficients of counterions in solutions of diethylaminoethyl dextran hydrochloride

Activity coefficients of counterions in solutions of diethylaminoethyl dextran hydrochloride have been determined. It has been observed that they increase with decreasing concentration of the polyelectrolyte. The experimental values have been compared with those calculated using Oosawa's theory of activity coefficients. The calculated values are higher than those observed, which suggests that the rodlike model on which Oosawa's theory is based is inadequate for the present case. Activity coefficients of counterions of some solutions containing NaCl and KC1, respectively, have also been determined. It has been found that the additivity rule for activity of counterions applies for these solutions.