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.     

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.     

Limiting laws and counterion condensation in polyelectrolyte solutions: IV. The approach to the limit and the extraordinary stability of the charge fraction

The limiting laws for polyelectrolyte solutions developed in previous papers of this series have been amply confirmed by measurement. A surprising result of the accumulated data is that the limiting polyelectrolyte charge fraction (fraction of fixed charges uncompensated by condensed counterions in the limit of zero concentration), persists up to concentrations of 0.1 M or even higher. Here the theory is extended in a simple manner to finite concentrations, and the stability of the charge fraction is found to be firmly based on consequences of the long-range polyelectrolyte field. The associated counterions are assumed to translate freely in a region centered on the contour axis of the polyion. The numerical value of the free volume is determined self-consistently from the axial charge density of the polyelectrolyte and is used as the general framework within which specific binding effects are treated.     

Monte Carlo and Poisson–Boltzmann calculations of the fraction of counterions bound to DNA

The counterion density and the condensation region around DNA have been examined as functions of both ion size and added-salt concentration using Metropolis Monte Carlo (MC) and Poisson–Boltzmann (PB) methods. Two different definitions of the “bound” and “free” components of the electrolyte ion atmosphere were used to compare these approaches. First, calculation of the ion density in different spatial regions around the polyelectrolyte molecule indicates, in agreement with previous work, that the PB equation does not predict an invariance of the surface concentration of counterions as electrolyte is added to the system. Further, the PB equation underestimates the counterion concentration at the DNA surface, compared to the MC results, the difference being greatest in the grooves, where ionic concentrations are highest. If counterions within a fixed radius of the helical axis are considered to be bound, then the fraction of polyelectrolyte charge neutralized by counterions would be predicted to increase as the bulk electrolyte concentration increases. A second categorization—one in which monovalent cations in regions where the average electrostatic potential is ledd than ?kT are considered to be bound—provides an informative basis for comparison of MC and PB with each other and with counterion-condensation theory. By this criterion, PB calculations on the B from of DNA indicate that the amount of bound counterion charge per phosphate group is about .67 and is independent of salt concentration. A particularly provocative observatiob is that when this binding criterion is used, MC calculations quantitatively reproduce the bound fraction predicated by counterion-condensation theory for all-atom models of B-DNA and A-DNA as well as for charged cylindera of varying lineat charge densities. For example, for B-DNA and A-DNA, the fractions of phosphate groups neutralized by 2 Å hard sphere counterions are 0.768 and .817, respectively. For theoretical studies, the rediys enclosing the region in which the electrostatic potential is calculated studies, the radius enclosing the region in which the electrostatic potential is calculated to be less than ?kT is advocated s a more suitable binding or condensation radius that enclosing the fraction of counterions given by (1 – ξ^?1). A comparsion of radii calculated using both of these definitions is presented. © 1994 John Wiley & Sons, Inc.     

A commentary on the screened-Oseen, counterion-condensation formalism of polyion electrophoresis

The use of linear theory, in particular, counterion condensation (CC) theory, in describing electrophoresis of polyelectrolyte chains, is criticized on several grounds. First, there are problems with CC theory in describing the equilibrium distribution of ions around polyelectrolytes. Second, CC theory is used to treat ion relaxation in a linear theory with respect to the polyion charge despite the fact that ion relaxation arises as a consequence of nonlinear charge effects. This nonlinearity has been well established by several investigators over the last 70 years for spherical, cylindrical, and arbitrarily shaped model polyions. Third, current use of CC theory ignores the electrophoretic hindrance as well as the ion relaxation for condensed counterions and only includes such interactions for uncondensed counterions. Because most of the condensed counterions lie outside the shear surface of the polyion (in the example of DNA), the assumption of ion condensation is artificial and unphysical. Fourth, the singular solution, based on a screened Oseen tensor, currently used in the above mentioned theories is simply wrong and fails to account for the incompressibility of the solvent. The actual singular solution, which has long been available, is discussed. In conclusion, it is pointed out that numerical alternatives based on classic electrophoresis theory (J.T.G. Overbeek, Kolloid-Beih, 1943, 54:287-364) are now available.     

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.     

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.     

On the mechanism of dielectric relaxation in aqueous DNA solutions.

The complex dielectric response of calf thymus DNA in aqueous saline solutions has been measured from 1 MHz to 1 GHz. The results are presented in terms of the relaxation of the incremental contributions to the permittivity and conductivity from the condensed counterions surrounding the DNA molecules. Measurements of the low-frequency conductivity of the samples also lends support to the condensed counterion interpretation.     

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.     

Counterion fluctuation and dielectric dispersion in linear polyelectrolytes

The thermal fluctuation in the concentration of counterions bound to a rodlike polyion was analyzed by expanding the fluctuation in a Fourier series along the rod. The amplitude and the relaxation time of fluctuations of various wave lengths were obtained as functions of the charge density and the length of the polyion. From these results the real and imaginary parts of the dielectric constant of the polyelectrolyte solution were derived as the sum of contributions of fluctuations of different modes. The dielectric dispersion curve or the Cole-Cole plot obtained was found to be in good agreement with experimental data.     

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.     

Azimuthal frustration and bundling in columnar DNA aggregates

The interaction between two stiff parallel DNA molecules is discussed using linear Debye-Hückel screening theory with and without inclusion of the dielectric discontinuity at the DNA surface, taking into account the helical symmetry of DNA. The pair potential furthermore includes the amount and distribution of counterions adsorbed on the DNA surface. The interaction does not only depend on the interaxial separation of two DNA molecules, but also on their azimuthal orientation. The optimal mutual azimuthal angle is a function of the DNA-DNA interaxial separation, which leads to azimuthal frustrations in an aggregate. On the basis of the pair potential, the positional and orientational order in columnar B-DNA assemblies in solution is investigated. Phase diagrams are calculated using lattice sums supplemented with the entropic contributions of the counterions in solution. A variety of positionally and azimuthally ordered phases and bundling transitions is predicted, which strongly depend on the counterion adsorption patterns.     

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.     

Molecular dynamics simulation study of oriented polyamine- and Na-DNA: sequence specific interactions and effects on DNA structure

Molecular dynamics (MD) computer simulations have been carried out on four systems that correspond to an infinite array of parallel ordered B-DNA, mimicking the state in oriented DNA fibers and also being relevant for crystals of B-DNA oligonucleotides. The systems were all comprised of a periodical hexagonal cell with three identical DNA decamers, 15 water molecules per nucleotide, and counterions balancing the DNA charges. The sequence of the double helical DNA decamer was d(5'-ATGCAGTCAG)xd(5'-TGACTGCATC). The counterions were the two natural polyamines spermidine(3+) (Spd(3+)) and putrescine(2+) (Put(2+)), the synthetic polyamine diaminopropane(2+) (DAP(2+)), and the simple monovalent cation Na(+). This work compares the specific structures of the polyamine- and Na-DNA systems and how they are affected by counterion interactions. It also describes sequence-specific hydration and interaction of the cations with DNA. The local DNA structure is dependent on the nature of the counterion. Even the very similar polyamines, Put(2+) and DAP(2+), show clear differences in binding to DNA and in effect on hydration and local structure. Generally, the polyamines disorder the hydration of the DNA around their binding sites whereas Na(+) being bound to DNA attracts and organizes water in its vicinity. Cation binding at the selected sites in the minor and in the major groove is compared for the different polyamines and Na(+). We conclude that the synthetic polyamine (DAP(2+)) binds specifically to several structural and sequence-specific motifs on B-DNA, unlike the natural polyamines, Spd(3+) and Put(2+). This specificity of DAP(2+) compared to the more dynamic behavior of Spd(3+) and Put(2+) may explain why the latter polyamines are naturally occurring in cells.     

Influence of tetraalkyl ammonium ions on the structure of poly (rG-dC).poly (rG-dC): unexpected transitions among the Z, A and B conformations.

The conformation of the double-stranded, mixed ribodeoxyribo polynucleotide, poly (rG-dC).poly (rG-dC), has been examined in the presence of tetraalkyl ammonium ions. Tetramethyl ammonium ion stabilizes the "low salt" Z conformation (1) of the polymer from submillimolar to molar concentrations of the counterion. In the presence of tetraethyl and tetrapropyl ammonium ions the polymer exists in the low salt Z form up to 2 mM concentration of the counterions and then flips to the right hand helical A form. With tetrabutyl ammonium counterions the polymer is in an A conformation at low ion concentrations and converts to a B form at concentrations greater than thirty millimolar. These results are interpreted in terms of electrostatic and solvent interactions of the polynucleotide.     

Thermodynamics of the conformational transition of biopolyelectrolytes: the case of specific affinity of counterions.

A formal development of the Counterion Condensation theory (CC) of linear polyelectrolytes has been performed to include specific (chemical) affinity of condensed counterions, for polyelectrolyte charge density values larger than the critical value of condensation. It has been conventionally assumed that each condensed counterion exhibits an affinity free-energy difference for the polymer, (DeltaG(aff)). Moreover, the model assumes that the enthalpic and entropic contributions to DeltaG(aff), i.e., DeltaH(aff) and DeltaS(aff), are both independent of temperature, ionic strength and polymer concentration. Equations have been derived relative to the case of the thermally induced, ionic strength dependent, conformational transition of a biopolyelectrolyte between two conformations for which chemical affinity is supposed to take place. The experimental data of the intramolecular conformational transition of the ionic polysaccharide kappa-carrageenan in dimethylsulfoxide (DMSO) have been successfully compared with the theoretical predictions. This novel approach provides the enthalpic and entropic affinity values for both conformations, together with the corresponding thermodynamic functions of nonpolyelectrolytic origin pertaining to the biopolymer backbone change per se, i.e., DeltaH(n.pol) and DeltaS(n.pol), according to a treatment previously shown to be successful for lower values of the biopolyelectrolyte linear charge density. The ratio of DeltaH(n.pol) to DeltaS(n.pol) was found to be remarkably constant independent of the value of the dielectric constant of the solvent, from formamide to water to DMSO, pointing to the identity of the underlying conformational process.     

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.     

Limiting laws and counterion condensation in polyelectrolyte solutions: V. Further development of the chemical model

Counterion binding to polyelectrolyte chains is formulated as a chemical reaction M^z(free) → M^z(bound). Expressions for the chemical potentials of free and bound counterions are set equal to obtain the reaction equilibrium. The results are equivalent to those in the previous paper of this series. An additional result obtained here is that a polyion holds its bound counterion layer with a strength on the order of 100 $\text{kcal}\text{(mole cooperative unit)}$. The method is then applied to the calculation of the polarizability along the chain due to the bound (condensed) counterions.     

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.     

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.