Evaluation of the counterion condensation theory of polyelectrolytes.

We compare free energies of counterion distributions in polyelectrolyte solutions predicted from the cylindrical Poisson-Boltzmann (PB) model and from the counterion condensation theories of Manning: CC1 (Manning, 1969a, b), which assumes an infinitely thin region of condensed counterions, and CC2 (Manning, 1977), which assumes a region of finite thickness. We consider rods of finite radius with the linear charge density of B-DNA in 1-1 valent and 2-2 valent salt solutions. We find that under all conditions considered here the free energy of the CC1 and the CC2 models is higher than that of the PB model. We argue that counterion condensation theory imposes nonphysical constraints and is, therefore, a poorer approximation to the underlying physics based on continuum dielectrics, point-charge small ions, Poisson electrostatics, and Boltzmann distributions. The errors in counterion condensation theory diminish with increasing distance from, or radius of, the polyion.     

Approach to the limit of counterion condensation

According to counterion condensation theory, one of the contributions to the polyelectrolyte free energy is a pairwise sum of Debye-Hückel potentials between polymer charges that are reduced by condensed counterions. When the polyion model is taken as an infinitely long and uniformly spaced line of charges, a simple closed expression for the summation, combined with entropy-derived mixing contributions, leads to the central result of the theory, a condensed fraction of counterions dependent only on the linear charge density of the polyion and the valence of the counterion, stable against increases of salt up to concentrations in excess of 0.1 M. Here we evaluate the sum numerically for B-DNA models other than the infinite line of B-DNA charges. For a finite-length line there are end effects at low salt. The condensation limit is reached as a flat plateau by increasing the salt concentration. At a fixed salt concentration the condensation limit is reached by increasing the length of the line. At moderate salt even very short B-DNA line-model oligomers have condensed fractions not far from the infinite polymer limit. For a long double-helical array with charge coordinates at the phosphates of B-DNA, the limiting condensed fraction appears to be approached at low salt. In contrast to the results for the line of charges, however, the computed condensed fraction varies strongly with salt in the range of experimentally typical concentrations. Salt invariance is restored, in agreement with both the line model and experimental data, when dielectric saturation is considered by means of a distance-dependent dielectric function. For sufficiently long B-DNA line and helical models, as typical salt concentrations, the counterion binding fraction approaches the polymer limit as a linear function of 1/P, where P is the number of phosphate groups of B-DNA.     

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.     

Trivalent counterion condensation on DNA measured by pulse gel electrophoresis

Pulse gel electrophoresis was used to measure the reduction of mobilities of λ-DNA-Hind III fragments ranging from 23.130 to 2.027 kilobase pairs in Tris borate buffer solutions mixed with either hexammine cobalt(III), or spermidine³⁺ trivalent counterions that competed with Tris⁺ and Na⁺ for binding onto polyion DNA. The normalized titration curves of mobility were well fit by the two-variable counterion condensation theory. The agreement between measured charge fraction neutralized and counterion condensation prediction was good over a relatively wide range of trivalent cation concentrations at several solution conditions (pH, ionic strength). The effect of ionic strength, trivalent cation concentration, counterion structure, and DNA length on the binding were discussed based on the experimental measurements and the counterion condensation theory. © 1996 John Wiley & Sons, Inc.     

Intrinsic binding effects in mixed counterionic polyelectrolyte systems: extension of condensation theory and comparison with voltammetric data

Voltammetric speciation data for the potassium/zinc/polymethacrylate system, recently obtained for various charge densities of the polyelectrolyte (Díaz-Cruz et al., Anal. Chim. Acta, 264 (1992) 163) and for different concentrations of monovalent counterions (van den Hoop and van Leeuwen, Anal. Chim. Acta, 273 (1993) 275), are compared with theoretical predictions computed according to a new thermodynamic model developed by Paoletti et al. (Biophys. Chem., 41 (1991) 73) and recently extended by Benegas and Paoletti (in preparation). The model allows: (i) the simultaneous condensation of both monovalent and divalent counterions and (ii) can account for a certain specific affinity of the polyelectrolyte for one type of the counterion over the other. For various charge densities of the polyelectrolyte, experimentally obtained speciation data for the K/Zn/PMA system agree well with theoretical predictions by considering an extra reduced molar affinity energy of -4RT for the Zn(2+) polyelectrolyte binding. The agreement between experimental and theoretical values for the distribution of Zn(2+) ions over the free and bound state becomes less perfect for relatively high concentrations of monovalent counterions.     

Role of counterion condensation in folding of the Tetrahymena ribozyme. I. Equilibrium stabilization by cations

Folding of RNA into an ordered, compact structure requires substantial neutralization of the negatively charged backbone by positively charged counterions. Using a native gel electrophoresis assay, we have examined the effects of counterion condensation upon the equilibrium folding of the Tetrahymena ribozyme. Incubation of the ribozyme in the presence of mono-, di- and trivalent ions induces a conformational state that is capable of rapidly forming the native structure upon brief exposure to Mg2+. The cation concentration dependence of this transition is directly correlated with the charge of the counterion used to induce folding. Substrate cleavage assays confirm the rapid onset of catalytic activity under these conditions. These results are discussed in terms of classical counterion condensation theory. A model for folding is proposed which predicts effects of charge, ionic radius and temperature on counterion-induced RNA folding transitions.     

Application of polyelectrolyte theories for analysis of DNA melting in the presence of Na+ and Mg2+ ions.

Numerical calculations, using Poisson-Boltzmann (PB) and counterion condensation (CC) polyelectrolyte theories, of the electrostatic free energy difference, DeltaGel, between single-stranded (coil) and double-helical DNA have been performed for solutions of NaDNA + NaCl with and without added MgCl2. Calculations have been made for conditions relevant to systems where experimental values of helix coil transition temperature (Tm) and other thermodynamic quantities have been measured. Comparison with experimental data has been possible by invoking values of Tm for solutions containing NaCl salt only. Resulting theoretical values of enthalpy, entropy, and heat capacity (for NaCl salt-containing solutions) and of Tm as a function of NaCl concentration in NaCl + MgCl2 solutions have thus been obtained. Qualitative and, to a large extent, quantitative reproduction of the experimental Tm, DeltaHm, DeltaSm, and DeltaCp values have been found from the results of polyelectrolyte theories. However, the quantitative resemblance of experimental data is considerably better for PB theory as compared to the CC model. Furthermore, some rather implausible qualitative conclusions are obtained within the CC results for DNA melting in NaCl + MgCl2 solutions. Our results argue in favor of the Poisson-Boltzmann theory, as compared to the counterion condensation theory.     

Counterion exchange reactions on DNA: Monte Carlo and Poisson-Boltzmann analysis

In solutions containing DNA and cations of more than one type, the competitive interactions of these cations with DNA can be modeled as an ion exchange process that can be described quantitatively by means of the theoretical approach reported in this paper. Under conditions of experimental interest the radial distribution function of each type of counterion is calculated from the results of canonical Monte Carlo (MC) simulations using the primitive model for DNA (having a helical charge distribution) and for the electrolyte ions. These ions consist of monovalent coions, monovalent counterions intended to represent Na⁺, and counterions of a second type designated M^z+, having variable size and charge (z ≥ 1). The competitive association of these counterions with DNA is described in terms of D, a parameter analogous to an ion exchange equilibrium quotient. Values of D are calculated from the results of our MC simulations and compared with corresponding predictions of the Poisson–Boltzmann (PB) cell model and with results inferred from analyses of previously published nmr measurements. Over typical experimental concentration ranges (0.02M < [Na⁺] < 0.20M, 0.001 < [M^z+] < 0.160M), D^MC and D^PB both are predicted to be relatively independent of the bulk ion concentrations. For various specifications of the size and charge of the competing cation (M^z+), D^MC and D^PB exhibit similar trends, although the MC simulations consistently predict that the cations bearing a higher charge density than that of Na⁺ are somewhat stronger competitors than indicated by the PB calculations. For monovalent and divalent competitors of varying radii, theoretical predictions of D are compared with values obtained by fitting nmr measurements. If the hard-sphere radii specified in the simulations are the (hydrated) ionic radii determined from conductance measurements, then the MC predictions and the corresponding nmr results are in reasonable agreement for various monovalent competitors and for a divalent polyamine, but not for Ca²⁺ and Mg²⁺.     

Fluorescence energy transfer in one dimension: Frequency-domain fluorescence study of DNA–fluorophore complexes

The theory for the salt dependence of the free energy, entropy, and enthalpy of a polyelectrolyte in the PB (PB) model is extended to treat the nonspecific salt dependence of polyelectrolyte–ligand binding reactions. The salt dependence of the binding constant (K) is given by the difference in osmotic pressure terms between the react ants and the products. For simple 1-1 salts it is shown that this treatment is equivalent to the general preferential interaction model for the salt dependence of binding [C. Anderson and M. Record (1993) Journal of Physical Chemistry, Vol. 97, pp. 7116–7126]. The salt dependence, entropy, and enthalpy are compared for the PB model and one specific form of the preferential interaction coefficient model that uses counterion condensation/limiting law (LL) behavior. The PB and LL models are applied to three ligand–polyelectrolyte systems with the same net ligand charge: a model sphere–cylinder binding reaction, a drug–DNA binding reaction, and a protein–DNA binding reaction. For the small ligands both the PB and limiting law models give (ln K vs. In [salt]) slopes close in magnitude to the net ligand charge. However, the enthalpy/entropy breakdown of the salt dependence is quite different. In the PB model there are considerable contributions from electrostatic enthalpy and dielectric (water reorientation) entropy, compared to the predominant ion cratic (release) entropy in the limiting law model. The relative contributions of these three terms in the PB model depends on the ligand: for the protein, ion release entropy is the smallest contribution to the salt dependence of binding. The effect of three approximations made in the LL model is examined: These approximations are (1) the ligand behaves ideally, (2) the preferential interaction coefficient of the polyelectrolyte is unchanged upon ligand binding, and (3) the polyelectrolyte preferential interaction coefficient is given by the limiting law/counterion-condensation value. Analysis of the PB model shows that assumptions 2 and 3 break down at finite salt concentrations. For the small ligands the effects on the slope cancel, however, giving net slopes that are similar in the PB and LL models, but with a different entropy/enthalpy breakdown. For the protein ligand the errors from assumptions 2 and 3 in the LL model do not cancel. In addition, the ligand no longer behaves ideally due to its complex structure and charge distribution. Thus for the protein the slope is no longer related simply to the net ligand charge, and the PB model gives a much larger slope than the LL model. Additionally, in the PB model most of the salt dependence of the protein binding comes from the change in ligand activity, i.e. from nonspecific anion effects, in contrast to the small ligand case. While the absolute binding is sensitive to polyelectrolyte length, little length effect is seen on the salt dependence for the small ligands at 0.1M salt, and for lengths > 60 Å. Almost no DNA length dependenceis seen in the salt dependence of the protein binding, since this is determined primarily by the protein, not the DNA. © 1995 John Wiley & Sons, Inc.     

Binding Sites of the Polyamines Putrescine,Cadaverine, Spermidine and Spermine on A- and B-DNA Located by Simulated Annealing

Abstract

Molecular dynamics simulations with simulated annealing are performed on polyamine-DNA systems in order to determine the binding sites of putrescine, cadaverine, spermidine and spermine on A- and B-DNA. The simulations either contain no additional counterions or sufficient Na⁺ ions, together with the charge on the polyamine, to provide 73% neutralisation of the charges on the DNA phosphates. The stabilisation energies of the complexes indicate that all four polyamines should stabilise A-DNA in preference to B-DNA, which is in agreement with experiment in the case of spermine and spermidine, but not in the case of putrescine or cadaverine. The major groove is the preferred binding site on A-DNA of all the polyamines. Putrescine and cadaverine tend to bind to the sugar-phosphate backbone of B-DNA, whereas spermidine and spermine occupy more varied sites, including binding along the backbone and bridging both the major and minor grooves.     

Divalent cations and the electrostatic potential around DNA: Monte Carlo and Poisson-Boltzmann calculations

The predictions of counterion condensation theory for divalent ions were tested by comparison with the results of Monte Carlo calculations on an all-atom model of DNA. Monovalent-divalent competition at the polyelectrolyte surface was investigated by varying the partial molar volume of divalent ions. To assess the viability of using Poisson-Boltzmann (PB) calculations for determining divalent ion concentrations at DNA surfaces, Monte Carlo (MC) calculations were compared with PB calculations using different models of the dielectric continuum. It was determined that, while standard PB calculations of divalent ion surface densities are about 25-30% below those predicted by MC techniques, and somewhat larger than errors previously determined for monovalent ions, errors due to the use of the mean-field approximation of PB theory are smaller than those arising from common assumptions regarding the dielectric continuum.     

Catalytic influence of heparin on auramine O hydrolysis: A basis for differentiating heparin from other glycosaminoglycans based on its properties as a polyelectrolyte

The basis for heparin's ability to accelerate the conversion of auramine O (AuO) to Michlers ketone [P. Band & A. Lukton (1982) Anal. Biochem. 120 , 19–24] is here explained in terms of well-known polyelectrolyte phenomena. Acceleration of this acidcatalyzed hydrolysis appears to be due to colocalization of the cationic dye and hydronium ion within the microenvironment of heparin's electrostatic domain. The dependence of reaction rate on polymer, dye, and hydronium ion concentration, ionic strength, and various ratios of these parameters is consistent with what others have observed for polyelectrolyte catalysts. Dextran sulfate, polyvinyl sulfate, and polyanethole sulfonate likewise accelerate this reaction, thus precluding any explanation of the catalysis in terms of a specific catalytic site. A striking aspect about the polyanion–AuO system is the inability of all other natural glycosaminoglycans tested to catalyze this reaction, despite their analogous polyanionic nature. Manning's limiting laws describing counterion condensation in polyelectrolyte solutions provide a simple context within which to interpret these results. Calculation of the structural linear charge density parameter, ξ, based on analytically determined sulfate-to-hexosamine ratios, reveals that heparin is unique among the glycosaminoglycans in possessing an ξ value greater than unity under the conditions described. Thus, heparin's differential ability to catalyze AuO hydrolysis is a reflection of the fact that only heparin is sulfated to an extent great enough to maintain ξ > 1, even when the negative charge of carboxylate groups is neutralized. It is proposed that this distinction may be important to the unique biochemical attributes of heparin and that such considerations may prove useful in establishing structure–function relationships when comparing different glycosaminoglycan classes.     

Binding sites of the polyamines putrescine, cadaverine, spermidine and spermine on A- and B-DNA located by simulated annealing

Molecular dynamics simulations with simulated annealing are performed on polyamine-DNA systems in order to determine the binding sites of putrescine, cadaverine, spermidine and spermine on A- and B-DNA. The simulations either contain no additional counterions or sufficient Na+ ions, together with the charge on the polyamine, to provide 73% neutralisation of the charges on the DNA phosphates. The stabilisation energies of the complexes indicate that all four polyamines should stabilise A-DNA in preference to B-DNA, which is in agreement with experiment in the case of spermine and spermidine, but not in the case of putrescine or cadaverine. The major groove is the preferred binding site on A-DNA of all the polyamines. Putrescine and cadaverine tend to bind to the sugar-phosphate backbone of B-DNA, whereas spermidine and spermine occupy more varied sites, including binding along the backbone and bridging both the major and minor grooves.     

Polyion-counterion interaction behavior for sodium carboxymethylcellulose in methanol-water mixed solvent media

Precise measurements on the electrical conductivities have been reported for solutions of sodium carboxymethylcellulose in methanol-water mixed solvent media. The conductivity vs. concentration data have been analyzed on the basis of the scaling theory approach for semidilute polyelectrolyte conductivity. The effects of the temperature, the medium, and the polymer concentration on the fractions of uncondensed counterions, the polyion conductivities, the standard state free energies of counterion association, and the coefficients of friction between the polyion and the solvent have also been investigated and the results have been interpreted from the viewpoints of polyion-countreion interactions, effective charge on the polyion, solvation of counterions and the polyionic sites, and counterion dissociation.     

Coarse-grained simulations of the salt dependence of the radius of gyration of polyelectrolytes as models for biomolecules in aqueous solution

The salt dependent radius of gyration of a polyelectrolyte in aqueous solution is calculated in an environment where the polyelectrolyte is surrounded by a permeable membrane that exchanges only solvent particles with the bulk. We obtain additionally the scaling exponent of the gyration radius as a function of the polymerization degree, and find that the polyelectrolyte retains a stretched conformation during the condensation and re-expansion process, indicating that these effects are of an electrostatic nature. The solvent quality is also shown to affect the polyelectrolyte conformation, especially for the poor solvent case. These results are obtained using a hybridized Monte Carlo technique with the coarse-grained, dissipative particle dynamics method with fluctuating number of solvent particles. The full range of the electrostatic interactions is included in the simulations, using the Ewald sum method, and the counterions and solvent molecules are included explicitly. In the complex systems mentioned above, the electrostatic interactions and the solvent quality play a key role in understanding phenomena that do not occur in uncharged systems. Our results are compared and validated with the behavior of some biomolecules under similar environments.     

Calculations on the effect of methylation on the electrostatic stability of the B- and Z-conformers of DNA

The three-dimensional Poisson–Boltzmann equation for the distribution of counterion charge density around double-helical DNA has been solved for solutions of .01M, .10M, and .20M monovalent salt. The polymers, poly[d(CpGp)] and poly[d(m⁵CpGp)], were studied in the B- and the Z-conformations. The effect of methylation on the relative stabilities of these conformers in solutions of different ionic strengths is known to favor the Z-form. Accumulation of charge density around the B- and the Z-conformers is compared in detail. The relative electrostatic stabilities of the B- and Z-conformers in .01M, .10M, and .20M solutions are compared and discussed in terms of the ion–DNA interactions and the self-energy of the structured ionic environment. The ion–DNA interaction energies, termed “phosphate screening,” monotonically decrease with ionic strength and are consistent with a B-to-Z conformation change induced in either polymer by increased electrolyte concentration. However, these calculated energies alone do not account for the fact that the ionic strength at the midpoint of the transition of the methylated polymer is substantially lower than that of its unmethylated analogues. The phosphate screening effect is counterbalanced by changes in the self-energy required for the creation of the structured counterion environment. This self-energy of the electrolyte environment monotonically increases with ionic strength. Methylation-induced shifts in the overall conformational equilibria depend on the relative changes of these competing effects. Increasing salt concentration is calcualted to favor the Z-conformer. The effect of methylation, lowering the ionic strength of the transition midpoint, is proposed to originate in minor structural changes in the Z-form of the polymer, making the groove more accessible to counterions in the G(3′ – 5′)C region. This allows a redistribution of counterion density and a lowering of the self-energy of the ionic environment, conferring added stability to the Z-conformation, as indicated by calculations of relative entropies. The experimentally observed temperature dependence of the B-to-Z transition, however, cannot be explained without assuming the release of bound water. Maps of the calculated three-dimensional structure at the counterion distribution near the surface of these molecules in both the B- and the Z-forms are also presented.     

Counterion-induced condesation of deoxyribonucleic acid. a light-scattering study.

The addition of the trivalent or tetravalent cations spermidine or spermine to a solution of T7 DNA in aqueous solution causes an alteration of the DNA from its extended coil form to a condensed form. If performed at low DNA concentration and at low ionic strengths, this transformation results in a monomolecular collapse to form a particle with a hydrodynamic radius of about 500 A. We have monitored this change using quasielastic and total intensity light scattering. In a solution of 50% methanol in water, the divalent cations Mg2+ and putrescine also can cause the condensation of DNA. Using Manning's (1978) counterion condensation theory, we calculate a striking unity among these disparate ions: the collapse occurs in each case when from 89 to 90% of the DNA phosphate charges are neutralized by condensed counterions.     

Condensation of Co++ ions on chondroitin sulfate from transport coefficients and proton NMR measurements in aqueous solutions

The “condensed” counterions which characterize high-charge-density polyelectrolyte solutions can be analyzed into two subpopulations: (1) site-bound counterions and (2) atmospherically entrapped counterions. The distinction is achieved experimentally by combining the data from self-diffusion coefficient or electrical mobility measurements, which give the amount of “condensed” ions, and those from nmr, chemical shift measurements, which indicate the amount of site-bound ions. In the case of a solution of chondroitin sulfate with excess Co⁺⁺ counterions, it can be estimated that 20% of the structural charge of the polyion is neutralized by site-bound, dehydrated, condensed counterions, while a further 30% is neutralized by atmospherically entrapped, hydrated counterions.     

Structure of hydrated Na+ ions around a region of A- or B-DNA helix

A molecular-dynamics simulation was used to carry out an introductory study of the hydration of a section of a rigid single A- or B-DNA helix with one Na⁺ counterion per nucleotide. Four Na⁺ ions and four nucleotides and periodic boundary conditions were used to mimic an infinite helix. The atoms of the helix and the Na⁺ ions were assumed to be Lennard-Jones spheres that also carried charges. Stillinger four-point charge model water molecules were used. We carried out five calculations, for 26 and 46 water molecules in B-DNA and 20, 32, and 46 in A-DNA fragments. The arrangements of the Na⁺ ions are found to have some similarities to those obtained by Clementi and Corongiu. In the calculations with 46 water molecules, we found that two Na⁺ ions can be bridged by about two water molecules and form a hydrated bound pair, which in turn forms a bridge between the guanine N7 and a near phosphate group. These bound pairs may be important in stabilizing the helix structure of DNA molecules.     

Counter-Ion Condensation and System Dimensionality

Abstract

We review some of the characteristic properties of the structure of polyelectrolyte solutions: the condensed layer of counterions that forms abruptly at a critical threshold charge density on the polymer chain; the more diffuse Debye-Hückel cloud, which is spatially distinct from the condensed layer; and the entropie release of counterions from the condensed layer as a driving force for the binding of oppositely charged ligands. We present a reminder of the basis of our current understanding in a variety of experiments, simulations, and theories; and we attempt as well to clarify some misunderstandings. We present a new analysis of a lattice model that suggests why the limiting laws for polyelectrolyte thermodynamics have proved to be accurate despite the neglect of polymer-polymer interactions in their original derivation. We sketch recent progress in constructing a potential between counterion and polyion. A counterion located in the interface between condensed layer and Debye cloud is repelled from the polyion, creating a sharp boundary between the two counterion populations.