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.     

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.     

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.     

Analysis of potentiometric titrations of heterogeneous natural polyelectrolytes in terms of counterion condensation theory: application to humic acid

A model, developed within the framework of the counterion condensation theory of linear polyelectrolytes, is presented in this paper to describe the acid-base properties of linear polyelectrolytes, consisting of several types of functional ionizable groups. This formalism has been successfully applied to Fluka humic acid under salt-free conditions, as well as in the presence of supporting simple 1:1 salt (KNO3) at three different concentrations. As part of this approach, the charge density of the humic acid is obtained from the activity coefficient measurements of potassium counterions at different humic acid concentrations at a constant degree of dissociation of the polyelectrolyte. The humic acid average charge density was found to be 0.80 +/- 0.05. Using the present model, we are able to satisfactorily describe the experimental data obtained from acid-base potentiometric titrations. Four main functional groups making up the polymer are determined through their fractional abundances (Xi) and intrinsic pK (pK0i) values. The fractional abundances remained constant and independent of the ionic strength, indicating that the humic acid constitution does not depend on the concentration of excess salts. The pK0i values show a small change with ionic strength, which can be explained by the polyelectrolytic behavior of the solution.     

Limiting-laws of polyelectrolyte solutions. Ionic distribution in mixed-valency counterions systems. I: The model

An extension of the counterion-condensation (CC) theory of linear polyelectrolytes has been developed for the case of a system containing a mixture of counterions of different valency, i and j. The main assumption in the derivation of the model is that the relative amount of the condensed counterions of the type i and j is strongly correlated and it is determined by the overall physical bounds of the system. The results predicted by the model are consistent, in the limiting cases of single species component, with those of the original CC theory. The most striking results are obtained for the cases of low charge density and excess of counterion species: in particular, an apparent positive "binding" cooperativity of divalent ions is revealed for small, increasing additions of M2+ ions to a solution containing a swamping amount of monovalent salt and a polyelectrolyte of low charge density. Apparent "competitive binding" of mono- and divalent ions derives as a bare consequence of the electrostatic interactions. Theoretical calculations of experimentally accessible quantities, namely single-(counter) ion activity coefficients, confirm the surprising predictions at low charge density, which qualitatively agree with the measured quantities.     

Examination of the limiting laws of polyelectrolytes and counterion condensation II

The two phase model of polyelectrolyte solutions, which has been developed recently, is examined in a further detail. The binding free energy, which was introduced in the previous paper (J. Chem. Phys. 81 (1977) 1929), is replaced by the entropy of the condensed phase. This replacement leads to a detailed picture of the condensed phase. In a mixed system of mono- and divalent counterions, a couple of possibilities are examined in interpreting the condensation volume, which corresponds to the condensation entropy.     

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.     

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.     

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.     

Simple simulations of DNA condensation

Molecular dynamics simulations of a simple, bead-spring model of semiflexible polyelectrolytes such as DNA are performed. All charges are explicitly treated. Starting from extended, noncondensed conformations, condensed structures form in the simulations with tetravalent or trivalent counterions. No condensates form or are stable for divalent counterions. The mechanism by which condensates form is described. Briefly, condensation occurs because electrostatic interactions dominate entropy, and the favored coulombic structure is a charge-ordered state. Condensation is a generic phenomenon and occurs for a variety of polyelectrolyte parameters. Toroids and rods are the condensate structures. Toroids form preferentially when the molecular stiffness is sufficiently strong.     

A ligand binding model of counterion condensation to finite length polyelectrolytes

A ligand binding model of counterion association in finite length polyelectrolytes is presented. This model introduces counterion condensation features into a binding formalism. It agrees well with the predictions of other finite length models and is consistent with experimental data on helix–coil melting transitions for short nucleic acid oligomers. This model uses a discrete charge distribution for the polyelectrolyte. An expression for the electrostatic self-energy of finite length polyelectrolytes is derived using the Euler–Maclaurin sum formula. This sum is shown to be accurate over a wide range of salt concentrations. This electrostatic term is used in an energy minimization analysis. The energy minimization is solved analytically using a Lagrange inversion formula. This general procedure gives a rapidly convergent series and requires no assumptions with regard to “limiting law” behavior. However, when used in the Manning minimization formalism [(1977) Biophysical Chemistry, 24 , 2086], the volume of the condensed phase becomes unrealistically large at low ionic strength. The ligand binding model does not have a condensed phase volume as a parameter. It provides a single expression that agrees both with Manning's theory and with the theory of Ramanathan and Woodbury [(1982) Journal of Chemical Physics 77 , 4133] under the respective conditions of these theories.     

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.     

Role of counterion condensation in folding of the Tetrahymena ribozyme. II. Counterion-dependence of folding kinetics

Condensed counterions contribute to the stability of compact structures in RNA, largely by reducing electrostatic repulsion among phosphate groups. Varieties of cations induce a collapsed state in the Tetrahymena ribozyme that is readily transformed to the catalytically active structure in the presence of Mg2+. Native gel electrophoresis was used to compare the effects of the valence and size of the counterion on the kinetics of this transition. The rate of folding was found to decrease with the charge of the counterion. Transitions in monovalent ions occur 20- to 40-fold faster than transitions induced by multivalent metal ions. These results suggest that multivalent cations yield stable compact structures, which are slower to reorganize to the native conformation than those induced by monovalent ions. The folding kinetics are 12-fold faster in the presence of spermidine3+ than [Co(NH3)6]3+, consistent with less effective stabilization of long-range RNA interactions by polyamines. Under most conditions, the observed folding rate decreases with increasing counterion concentration. In saturating amounts of counterion, folding is accelerated by addition of urea. These observations indicate that reorganization of compact intermediates involves partial unfolding of the RNA. We find that folding of the ribozyme is most efficient in a mixture of monovalent salt and Mg2+. This is attributed to competition among counterions for binding to the RNA. The counterion dependence of the folding kinetics is discussed in terms of the ability of condensed ions to stabilize compact structures in RNA.     

A new model of counterion condensation in polyelectrolyte solutions. III. Theoretical predictions of competitive condensation between counterions of different valences

Our model has been extended for theoretical estimation of competitive condensation of counterions of different valences onto polyelectrolytes in solution. The estimations are compared with those obtained from Manning theory and with experimental data on counterion activity coefficients. The agreement with the data for sodium polystyrenesulfonate/MgCl2, CaCl2 is satisfactory.     

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.     

The polyelectrolyte behavior of actin filaments: a 25Mg NMR study.

Under physiological conditions, filamentous actin (F-actin) is a polyanionic protein filament. Key features of the behavior of F-actin are shared with other well-characterized polyelectrolytes, in particular, duplex DNA. For example, the bundle formation of F-actin by polyvalent cations, including divalent metal ions such as Mg2+, has been proposed to be a natural consequence of the polyelectrolyte nature of actin filaments [Tang and Janmey (1996) J. Biol. Chem. 271, 8556-8563]. This recently proposed model also suggests that weak interactions between F-actin and Mg2+ ions reflect a nonspecific trapping of counterions in the electric field surrounding F-actin due to its polyelectrolyte nature. To test this hypothesis, we have performed 25Mg NMR measurements in F-actin solutions. Based on the NMR data, we estimate that the rotational correlation times of Mg2+ are independent of the overall rotational dynamics of the actin filaments. Moreover, competitive binding experiments demonstrate a facile displacement of F-actin-bound Mg2+ by Co(NH3)63+. At higher Co(NH3)63+ concentrations, a fraction of the magnesium ions are trapped as actin filaments aggregate. ATP also competes effectively with actin filaments for binding to Mg2+. These results support the hypothesis that magnesium ions bind loosely and nonspecifically to actin filaments, and thus show a behavior typical of counterions in polyelectrolyte solutions. The observed features mimic to some extent the well-documented behavior of counterions in DNA solutions.     

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.     

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.     

Gel electrophoresis measurement of counterion condensation on DNA

We used agarose gel electrophoresis to measure the effective charge neutralization of DNA by counterions of different structure and valence, including Na⁺, Mg²⁺, Co(NH₃), and sperinidine³⁺, which competed for binding with an excess of Tris acetate buffer. Linear DNA molecules ranged in size from 1 to 5 kilobases, and supercoiled plasmid pUC18 was also measured. In all cases, the results were in good agreement with theoretical predict ions from counterion condensation theory for two-counterion mixtures. © 1995 John Wiley & Sons, Inc.