Counterion Condensation Theory Research Articles

Interaction of Heparin with Two Synthetic Peptides that Neutralize the Anticoagulant Activity of Heparin

Two synthetic analogues of the heparin-binding domain of heparin/heparan sulfate-interacting protein (Ac-SRGKAKVKAKVKDQTK-NH2) and the all-d-amino acid version of the same peptide (l-HIPAP and d-HIPAP, respectively) were synthesized, and their efficacy as agents for neutralization of the anticoagulant activity of heparin was assayed. The two analogue peptides were found to be equally effective for neutralization of the anticoagulant activity of heparin, as measured by restoration of the activity of serine protease factor Xa by the Coatest heparin method. The finding that l-HIPAP and d-HIPAP are equally effective suggests that d-amino acid peptides show promise as proteolytically stable therapeutic agents for neutralization of the anticoagulant activity of heparin. The interaction of l-HIPAP and d-HIPAP with heparin was characterized by 1H NMR, isothermal titration calorimetry (ITC), and heparin affinity chromatography. The two peptides were found to interact identically with heparin. Analysis of the dependence of heparin-peptide binding constants on Na+ concentration by counterion condensation theory indicates that, on average, 2.35 Na+ ions are displaced from heparin per peptide molecule bound and one peptide molecule binds per hexasaccharide segment of heparin. The analysis also indicates that both ionic and nonionic interactions contribute to the binding constant, with the ionic contribution decreasing as the Na+ concentration increases.

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 ∼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.

Specific Interactions versus Counterion Condensation. 2. Theoretical Treatment within the Counterion Condensation Theory

Polyuronates such as pectate and alginate are very well-known examples of biological polyelectrolytes undergoing, upon addition of divalent cations, an interchain association that acts as the junction of an eventually formed stable hydrogel. In the present paper, a thermodynamic model based on the counterion condensation theory has been developed to account for this cation-induced chain pairing of negatively charged polyelectrolytes. The strong interactions between cross-linking ions and uronate moieties in the specific binding site have been described in terms of chemical bonding, with complete charge annihilation between the two species. The chain-pairing process is depicted as progressively increasing with the concentration of cross-linking counterions and is thermodynamically defined by the fraction of each species. On these bases, the total Gibbs energy of the system has been expressed as the sum of the contributions of the Gibbs energy of the (single) chain stretches and of the (associated) dimers, weighted by their respective fractions 1 - theta and theta. In addition, the model assumes that the condensed divalent counterions exhibit an affinity free-energy for the chain, G(C)(aff,0), and the junction, G(D)(aff,0), respectively. Moreover, a specific Gibbs energy of chemical bonding, G(bond,0), has been introduced as the driving force for the formation of dimers. The model provides the mathematical formalism for calculating the fraction, theta, of chain dimers formed and the amount of ions condensed and bound onto the polyelectrolyte when two different types of counterions (of equal or different valence) are present. The effect of the parameter G(bond,0) has been investigated and, in particular, its difference from G(C,D)(aff,0) was found to be crucial in determining the distribution of the ions into territorial condensation and chemical bonding, respectively. Finally, the effect of the variation of the molar ratio between cross-linking ions and uronic groups in the specific binding sites, sigma0, was evaluated. In particular, a remarkable decrease in the amount of condensed counterions has been pointed out in the case of sigma0 = 1/3, with respect to the value of sigma0 = 1/4, characterizing the traditional "egg-box" structure, as a result of the drop of the charge density of the polyelectrolyte induced by complete charge annihilation.

Electric Polarizability of DNA in Aqueous Salt Solution

Monte Carlo simulations are performed to determine the anisotropy of the electric polarizability of a model DNA fragment in aqueous salt solution. By taking into consideration the participation of coions in the electroneutrality condition, at every simulation step, we obtain a list of counterions constituting the net charge arranged in increasing order of their distance from the DNA and calculate the contribution to the dipole moment from the first n counterions in the list. We define a partial polarizability tensor due to these n counterions to understand the origin of the polarizability in close relation to the solution structure. The ionic distributions are described by the counterion condensation theory. Characteristic features of the electric properties of polyelectrolytes are reproduced. The anisotropy of the electric polarizability Deltaalpha of DNA decreases with the addition of salt, yielding values comparable to experiment. The effect of electrophoretic motion of the polyion is examined by estimating its upper limit.

Specific Interactions versus Counterion Condensation. 1. Nongelling Ions/Polyuronate Systems

The characteristics of the interaction between nongelling divalent cations (typically Mg(2+)) and polyuronates have been explored by means of isothermal calorimetry. In particular, three polyuronates mimicking separately guluronan (polyguluronate, polyG), mannuronan (polymannuronate, polyM), and polyalternating (polyMG), the three block-components of natural alginate samples, have been treated with divalent ions, and the enthalpy of mixing was determined for different values of the [M(2+)]/[Polym](rep.unit) ratio. Despite the absence of a site-specific chemical bonding between the two, as confirmed by circular dichroism spectroscopy, a substantial deviation of the experimental enthalpy of mixing from the theoretical behavior, as predicted by the classical counterion condensation (CC) theory, was observed. Such deviation has been interpreted in terms of a "generic" nonbonding affinity of the condensed divalent counterion for the polyelectrolytes. The mathematical formalism of the CC theory was extended to include a contribution to the (reduced) free energy and enthalpy arising from the counterion affinity, g(aff,0) and h(aff,0), and allowed the parametrical calculation of the fraction of divalent counterions condensed as function of the reduced thermodynamic quantity g(aff,0). A best fit procedure of the experimental enthalpy of mixing allowed the g(aff,0) and h(aff,0) pair to be estimated for each of the different polyuronates considered, revealing differences in the three samples. In qualitative terms, the results obtained seem to suggest a notable contribution of the desolvation process (i.e., release of structured water as a consequence of the interaction between the divalent counterion and the uronate group) to the enthalpy of affinity for polyM which is counterbalanced and overcome by an ion pairing term (i.e., partial formation of ion-ion and/or ion-dipole bonds) for polyG and polyMG, respectively.

Electrical conductivity of some cationic polysaccharides. I. Effects of polyelectrolyte concentration, charge density, substituent at the ionic group, and solvent polarity

AbstractElectrolytic conductivity behavior of some cationic polysaccharides in water, methanol, and the mixtures water/methanol is presented. The polyelectrolytes investigated contain quaternary ammonium salt groups, N‐alkyl‐N,N‐dimethyl‐2‐hydroxypropyleneammonium chloride, attached to a dextran backbone. This study considers the influences of polymer concentration (1 × 10−6 < C < 1 × 10−2 monomol L−1) and the charge density (ξ = 0.48–3.17) modified either by changing charge distance (b) or dielectric constant of the solvent (ε) on polyion–counterion interaction in salt‐free solutions. Above the critical value, ξc = 1, the variation of the equivalent conductivity (Λ) as a function of concentration is typical for a polyelectrolyte behavior. The conductometric data in water were analyzed in terms of the Manning's counterion condensation theory. The presence of longer alkyl chains at quaternary N atoms was found to have a negligible influence on the Λ values. The results show that the decrease of the medium polarity results in the decrease of the number of free ions and, consequently, of the equivalent conductivity values. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 3584–3590, 2005

New Insights into the Mechanism of Gelation of Alginate and Pectin: Charge Annihilation and Reversal Mechanism

Studies have been undertaken on the binding of Mn2+ ions to two alginate samples of different mannuronate:guluronate ratios (M:G), a sample of low-ester amidated pectin and poly(acrylic acid) (PAA). The binding of Ca2+ ions has also been included for the latter for comparison. The binding curves showed an initial steep rise at low additions of Mn2+ or Ca2+ indicating that all of the ions were bound to the polymer chains with none remaining in solution. At higher additions, the binding curves showed a plateau region and the maximum amount bound, theta, was found to be 0.2, 0.2, 0.25, and 0.33 mol M(2+)/mol COO- for high M:G alginate, low M:G alginate, pectin, and PAA, respectively. The binding curves for Mn2+ and Ca2+ with PAA were superimposable. In all cases, theta was less than the stoichiometric equivalent and also less than predicted by Manning counterion condensation theory. The linear charge density, xi, for the polymers is 1.49, 1.55, 1.62, and 2.85, and it was found that at maximum binding the effective linear charge density, xi(effective), decreased to a value close to 1 in each case and not 0.5 as predicted from Manning's two-variable theory. The mobility of the PAA chains has been followed by electron spin resonance spectroscopy using nitroxide spin labels covalently attached to the polymer, and the gelation of the pectin and alginate samples has been monitored using small deformation oscillatory experiments. For PAA at maximum binding, it was noted that there was a loss of chain mobility and precipitation. For pectin and alginate, gelation occurred and the stoichiometric ratio for maximum binding corresponded to the stoichiometric ratio for the maximum in G'. Precipitation and gelation are attributed to the formation of polymer-metal complexes involving one or two carboxylate groups resulting in charge reversal or charge annihilation.

Electrostatic correlations and fluctuations for ion binding to a finite length polyelectrolyte

A statistical mechanical model is presented which explicitly accounts for the fluctuations, the electrostatic, and the excluded volume correlations for ions bound to a polyelectrolyte such as DNA. The method can be employed to treat a wide range of ionic conditions including multivalent ions. The microscopic framework of the theory permits the use of realistic finite length and grooved structural model for the polyelectrolyte and modeling of the finite size of the bound ions. Test against Monte Carlo simulations suggests that the theory can give accurate predictions for the ion distribution and the thermodynamic properties. For multivalent ions, the theory makes improved predictions as compared with the mean-field approach. Moreover, for long polyelectrolyte and dilute salt concentration, the theory predicts ion binding properties that agree with the counterion condensation theory.

Titration of poly(dA-dT) . poly(dA-dT) in solution at variable NaCl concentration.

CD and uv absorption data showed that high molecular weight poly(dA-dT) . poly(dA-dT), at 298 K, undergoes an acid-induced transition from B-double helix to random coil in NaCl solutions of different concentrations, ranging from 0.005 to 0.600M. Similarly, titration of the polynucleotide with a strong base causes duplex-to-single strands transition. The base- and acid-induced transitions were both reversible by back-titration (with an acid or, respectively, with a base): the apparent pKa were the same in both directions. However, the number of protons per titratable site (adenine N1) required to reach half-denaturation was in great excess over the stoichiometric value; to a much larger extent, the same effect was observed also for the deprotonation of the N3H sites of thymine. Moreover, in the basic denaturation experiments, at low salt concentrations ([NaCl]< or =0.300M) less acid than calculated was needed to back-titrate the base excess to half-denaturation. Both effects could be qualitatively justified on the basis of the counterion condensation theory of polyelectrolytes and considering the energy barrier created by the negatively charged phosphodiester groups to the penetration of the OH- ions inside the double helix and the screening effect of the Na+ ions on such charges, in the deprotonation experiments.

Interaction between Polyelectrolyte and Oppositely Charged Surfactant: Effect of Charge Density

The binding of dodecyltrimethylammonium bromide (DoTab) to poly(acrylic acid) (PAA) at different degrees of neutralization (α) was investigated using isothermal titration calorimetry (ITC) and laser light scattering (LLS) techniques. The surfactant binds to the polymer at all α; however, the mechanism varies. When α is lower than a critical value (αC), the hydrocarbon chains of DoTab cooperatively bind to the apolar segments of PAA driven by hydrophobic interaction at very low DoTab concentration (CDoTab ≤ 0.2 mM). In this binding region, the ITC profile exhibits a significant exothermic peak and the mixture precipitates, which is attributed to the interchain complexation via hydrogen bonding induced by the binding. The precipitation is soon resolubilized with further addition of surfactant as more DoTab micelles are bound to the polymer backbones with their ionic headgroups extending outward. When α > αC, the hydrophobic binding ceases as the polymer is progressively ionized and DoTab binds to the charged polymer chains driven by electrostatic attraction. The condensed counterions on the charged polymer chains are released via the ion-exchange process, resulting in an endothermic maximum on the ITC profile. The value of αC determined from ITC is approximately 0.3, which is reasonably close to the theoretical value derived from the Manning's counterion condensation theory (approximately 0.35).

Theory of counter-ion condensation on flexible polyelectrolytes: adsorption mechanism.

A new model is presented for counterion distribution around flexible polyelectrolytes by considering (i) free energy of the polyelectrolyte chain, (ii) translational entropy of adsorbed counterions, (iii) adsorption energy, (iv) translational entropy of unadsorbed counterions, (v) fluctuations of dissociated ions, and (vi) correlation among ion-pairs formed by adsorbed counterions on the polymer. The effective charge and size of the polymer are calculated self-consistently. The degree of ionization f of the polymer decreases continuously with 1/epsilonT (epsilon and T are the dielectric constant of the solvent and temperature, respectively), depending sensitively on local dielectric heterogeneity. Further, f decreases with an increase in salt concentration, monomer concentration, or chain flexibility. The polymer size, accompanying the changes in f, depends nonmonotonically on 1/epsilonT. The predictions of the model are consistent with all trends observed previously in simulations and are distinctly different from the Manning argument for rodlike chains.

Effects of Sodium Ions on DNA Duplex Oligomers: Improved Predictions of Melting Temperatures

Melting temperatures, T(m), were systematically studied for a set of 92 DNA duplex oligomers in a variety of sodium ion concentrations ranging from 69 mM to 1.02 M. The relationship between T(m) and ln [Na(+)] was nonlinear over this range of sodium ion concentrations, and the observed melting temperatures were poorly predicted by existing algorithms. A new empirical relationship was derived from UV melting data that employs a quadratic function, which better models the melting temperatures of DNA duplex oligomers as sodium ion concentration is varied. Statistical analysis shows that this improved salt correction is significantly more accurate than previously suggested algorithms and predicts salt-corrected melting temperatures with an average error of only 1.6 degrees C when tested against an independent validation set of T(m) measurements obtained from the literature. Differential scanning calorimetry studies demonstrate that this T(m) salt correction is insensitive to DNA concentration. The T(m) salt correction function was found to be sequence-dependent and varied with the fraction of G.C base pairs, in agreement with previous studies of genomic and polymeric DNAs. The salt correction function is independent of oligomer length, suggesting that end-fraying and other end effects have little influence on the amount of sodium counterions released during duplex melting. The results are discussed in the context of counterion condensation theory.

Dynamic light scattering and 31 P NMR study of the self-assembly of deoxyguanosine 5?-monophosphate: the effect of added salt

Self-assembling and dynamical properties of deoxyguanosine 5'-monophosphate in isotropic aqueous solutions were studied by (31)P NMR spectroscopy and dynamic light scattering (DLS). All solutions had the same c = 4 wt% guanosine concentration, while the added KCl molarity ranged from 0 to 1.5 M. (31)P NMR measurements show that potassium ions strongly enhance the stacking process of guanosine tetramers until a saturation is reached at 0.1 M KCl with more than 70% of the molecules aggregated. Polarized light scattering reveals the presence of a fast relaxation mode that arises from the translational dynamics of the self-assembled stacks. The diffusion coefficient of this mode shows a strong dependence on molarity of added salt, which can be described in terms of the coupled mode and counterion condensation theories for polyelectrolyte solutions. Depolarized light scattering reveals the rotational dynamics of the self-assembled stacks which exhibits a pronounced slowing down with increasing the added salt content.

Proton–metal exchange processes in synthetic and natural polyelectrolyte solution systems

Proton–metal exchange processes that take place in polyelectrolyte solutions have been studied using previously reported potentiometric titration data for polyacrylic acid (PAA) and humic acid (HA) systems obtained at different degrees of polymer deprotonation, for various metals and at different added metal concentrations. It is shown that the extent of the exchange process, quantified by the parameter νexch, strongly depends on the way it has been determined, i.e. at constant polyelectrolyte characteristics or at constant pH, being significantly larger under constant pH conditions. In addition, it is found that the exchange process depends upon polyelectrolyte structural charge density, the degree of ionization, the type of metal and the total metal concentration in solution. For the experiments reported here, in general the larger exchange coefficients are found for PAA at the lowest reported degree of ionization (α=0.2) and low pH values, while the lower values correspond to HA at the highest studied degree of ionization (α=0.6) and high pH values. The ability of the heavy metal ions to induce H+ release increase following the order Ca≈Cd≈Ni<Pb≤Cu. The present analysis shows that counterion condensation theory of linear polyelectrolytes appropriately describes the experimental data, provided chemical bonding interactions are not too strong to disrupt the general polyelectrolytic behavior.

Effects of Ion Size and Valence on Ion Distribution in Mixed Counterion Systems of a Rodlike Polyelectrolyte Solution. 2. Mixed-Valence Counterion Systems

The characteristics of the ion distributions are investigated in the systems of salt-free polyelectrolyte solution with mixed-valence counterions using a cylindrical cell model. The mixtures of two species of the counterion having different valences are simulated systematically by means of the Monte Carlo method in a primitive model of the rodlike polyelectrolyte solution. The comparisons of the free fractions and the selectivity coefficients by this method are performed with the numerical solutions of the Poisson−Boltzmann equation. A significant accumulation of the multivalent counterions found in the Monte Carlo simulations is examined under various conditions. The differences between both methods and additional effects of ion size are discussed in terms of the relation to the simple counterion condensation theory and the effect of ion−ion correlations.

Salt Separation by Charged Gel-Filled Microporous Membranes

A series of membranes composed of a microporous polypropylene substrate filled with a polyelectrolyte gel of poly(N-benzyl-4-vinylpyridinium chloride) of different gel polymer volume fractions has been used to study a pressure-driven separation of mono-monovalent salts from aqueous solutions. The experimental results are compared with that obtained from theoretical calculations based on the Teorell−Meyer−Sievers model of charged membranes combined with the Manning−Oosawa counterion condensation theory being used to estimate the membranes effective charge density. For counterions of low specific binding to the polyelectrolyte such as chloride and fluoride, the theoretical predictions were found to be in a good agreement with the experiment over a wide range of the gel polymer volume fractions. For strong binding counterions such as iodide, the separation is substantially lower than that predicted and is accompanied by an almost 3-fold increase in flux suggesting a microphase separation in the gel. The implications of this study on the design of gel-filled membranes are discussed.

Extensions of Counterion Condensation Theory. 2. Cell Model and Osmotic Pressure of DNA

In a previous paper, counterion condensation theory was extended by invoking a new alternative auxiliary assumption in order to treat isolated polyions with alternative geometries (beyond a line charge) at finite salt concentrations. Now this theory is further extended to treat a cell model and the osmotic pressures of relatively concentrated DNA solutions in equilibrium with bathing solutions of different salt concentration. Results of the present theory are compared with recently published experimental data of Raspaud et al. and with theoretical calculations of Hansen et al., who combined zero-salt solutions of the nonlinear Poisson−Boltzmann equation for a uniformly charged cylinder with the Donnan approximation. Surprisingly good agreement with the experimental data is observed, especially at the higher DNA concentrations. Agreement with the calculations of Hansen et al. is also rather good at the higher DNA concentrations, but at lower DNA concentrations the results of Hansen et al. fall significantl...

Net proton charge of β- and κ-casein in concentrated aqueous electrolyte solutions

Titration experiments have been carried out in order to measure the net proton charge of β- and κ-casein in NaCl solutions at 0.1 M and 1 M salt concentrations, at 4 °C, in the pH range between 5.5 and 10.5. Experimental data are compared with model values calculated through p K a's of titrable groups neglecting the electrostatic perturbation term (Δp K a) in order to evaluate the magnitude of the error caused by this approximation and to delimit its effectiveness. At both ionic strengths, the agreement is good for κ-casein in the pH range [5.5, 9.5], while errors of up to 2 charges are observed for β-casein in the same range. These deviations are likely to be caused by strong electrostatic effects induced by the high density of negative charges of β-casein 1–21 peptide. In order to account for these electrostatic effects, the net proton charge on this peptide is evaluated through a model based on the counterion condensation theory developed for the titration of polyelectrolytes with different types of ionizable groups.

Electrostatic free energy of the DNA double helix in counterion condensation theory

Polyelectrolyte theory based on counterion condensation is extended from the standard line charge model to helical and double helical charge arrays. The number of condensed counterions turns out to be the same as for a line charge with charge density equal to the axial charge density of the helix. Also, the logarithmic salt dependence of the electrostatic free energy is the same in the range of lower salt concentration, so that the limiting laws remain unchanged. However, the internal free energy of the condensed layer of counterions and the overall electrostatic free energy depend on the helical parameters. At higher salt, the free energies of both single and double helix are negative, indicating electrostatic stabilization of the helical charge lattices due to the mixing entropy of the condensed counterions. Except at very low salt, the free energy of a single helix is higher than the free energy of a double helix with twice the charge density. With B-DNA parameters and single strands modeled as single helices, the predicted salt dependence of the free energy of transition from double helix to separated single strands has a maximum at approximately 0.2 M salt, close to the location in the laboratory of this well-known feature of the DNA strand separation transition. We also calculate the electrostatic free energy for the transition of the DNA double helix from the B to the A conformation. The B form is electrostatically stable over most of the salt range, but there is a spontaneous electrostatic transition to A near 1 M salt. The electrostatic free energy values are close to the experimental values of the overall (electrostatic plus non-electrostatic) transition free energies for A-philic base pair sequences. We are led to suggest that the experimentally observed B-to- A transition for A-philic sequences near 1 M salt in water is governed by the polyelectrolyte properties of these two conformations of the DNA double helix. The effect of ethanol, however, cannot be attributed to lowering of the bulk dielectric constant.

Extensions of counterion condensation theory. I. Alternative geometries and finite salt concentration

The counterion condensation theory originally proposed by Manning is extended to take account of both finite counterion concentration ( m C) and the actual structure of the array of discrete changes. Counterion condensation is treated here as a binding isotherm problem, in which the unknown free volume is replaced by an unknown local binding constant β′, which is expected to vary with m C and polyion structure. The relation between the condensed fraction of counterion charge, r, β′ and m C is obtained from the relevant grand partition function via the maximum term method. In the case of the single polyion in a large salt reservoir, the result is practically identical to Manning's equation. In order to determine the values of β′ and r at arbitrary m C, a second relation between r, β′ and m C is required. We propose an alternative auxiliary relation that is equivalent to previous assumptions near m C=0, but which yields qualitatively correct and quantitatively useful results at finite m C. Simple expressions for r vs. m C and β′ vs. m C are obtained by simultaneously solving the binding isotherm and auxiliary equations. Then r and β′ are evaluated for five different linear arrays of infinite extent with different geometries: (1) a straight line of charges with uniform axial spacing; (2) two parallel lines of in-phase uniformly spaced charges; (3) a single-helix of discrete charges with uniform axial spacing; (4) a double-helix of discrete charges with uniform axial spacing of pairs of charges; (5) a cylindrical array of many parallel charged lines, chosen to simulate a uniformly charged cylinder. In all cases, the computed binding isotherms exhibit qualitatively correct behavior. As m C approaches zero, r approaches the Manning limit, r=1−1/( L B/ b) where b is the average axial spacing of electronic charges in the array and L B is the Bjerrum length. However, β′ varies with polyion geometry, even in the zero salt limit, and matches the Manning value only in the case of a single straight charged line. With increasing m C, r declines significantly below its limiting value whenever λ b≳0.3, where λ is the Debye screening parameter. In the case of cylindrical arrays containing either 2 or 100 parallel charged lines, r also decreases, whenever λ d≳2.0, where d is the diameter of the array. In the case of two parallel charged lines, each with axial charge spacing b=3.4 Å, which are separated by d=200 Å, r exhibits a plateau value, 0.76, characteristic of the two combined lines, when λ d≪2.0, and declines with increasing m C to a shelf value, 0.52, characteristic of either single line, when λ d≳2.0 and the lines become effectively screened from one another. β′ behaves in a roughly similar fashion. In the case of a cylindrical array of charged lines with the diameter and linear charge density of DNA, the r-values predicted by the present theory agree fairly well with those predicted by non-linear Poisson–Boltzmann theory up to 0.15 M uni-univalent salt.