John Warren Williams

The osmotic, sedimentation, and diffusion coefficients of sonicated sodium DNA in aqueous solutions which were rendered rigorously free of salts have been obtained in order to study the polyelectrolyte effect on these parameters. Both native and heat denatured DNA's were studied. All three parameters were found to be independent of concentration between 2 and 10 mmole/1. The osmotic coefficient gives the fraction of counterions not immobilized by the polyelectrolyte potential, under conditions of equilibrium. The value (ca. 0.18) obtained for this fraction agrees very well with that predicted by a theory of rodlike polyelectrolytes. A simplified model that permits the computation of fractions of mobile ions from sedimentation and diffusion coefficients, when applied to the experimental data, showed that under these conditions of flow between 0.5 to 0.9 of counterions are not immobilized. The experimental data also showed that for native DNA the translation frictional coefficient seems to have different values in sedimentation and diffusion, which agrees with a finding reported for another saltfree polyelectrolyte. For denatured DNA, however, the values were found to be equal: no satisfactory explanation could be found for this difference of behavior.     

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.     

Activity coefficients of counterions in solutions of diethylaminoethyl dextran hydrochloride

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

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

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

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.     

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.     

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.     

Ion binding or condensation to polyelectrolytes?: A detailed comparison,on the example of sodium heparinate

A modified Poisson-Boltzmann treatment is applied to aqueous solutions of sodium heparinate, in the presence of added salts (NaCl, LiCl, KCl, CaCl2). The results show that 23Na-NMR line widths for the counterions are determined nearly exclusively by Na+ in the immediate vicinity of the linear polyelectrolyte. Hence, two-state treatments, despite their crude character, can reproduce such experimental data rather well.     

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 diffusion in heparin solutions

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

Magnetic resonance distinction between site bound and atmospherically bound paramagnetic counterions in polyelectrolyte solutions.

A combination of the water protons NMR chemical shifts, longitudinal and transversal relaxation rates and of the paramagnetic counterion EPR signal is shown to provide a clear distinction between site binding, atmospheric trapping and free counterions in solutions of polyelectrolyte TMA salts with increasing concentrations of the divalent counterions Co++ and Mn++. Site binding is defined by the loss of water in the counterion first hydration shell while atmospheric binding results in a change in the counterion correlation time as compared to a free ion.     

Divalent paramagnetic counterions site binding in polyelectrolyte solutions. Analysis of the frequency dependence of the water protons magnetic relaxation and the characteristic parameters of "site binding"

Chemical shift and relaxation time measurements on the water protons in polyelectrolyte solutions containing divalent paramagnetic counterions have shown the existence of three types of counterions: - site bound with loss of water molecules and partial or complete release of the electrostriction in the first hydration sphere, - atmospherically trapped with no change in hydration, - free. The overall stoichiometry of the two former is in agreement with Manning's fraction of condensed counterions. A complete analysis of the frequency dependent contribution of site bound counterions to the water protons relaxation times leads us to interesting conclusions on the modifications of the first hydration shell and on the life time of site binding.     

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.     

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.     

Electroviscosity of polyelectrolyte solutions

A theoretical expression for the electroviscous effect in polyelectrolyte solutions, caused by the distortion of counterion-distribution and counterion flow around a polyion under a velocity gradient of solvent flow, was obtained to elucidate the characteristic behaviour of the viscosity of highly charged polyelectrolyte solutions observed at low salt concentration. The derivation of the theory was performed on the basis of the Navier-Stokes-Onsager equation, Poisson equation, and diffusion equations for low molecular ions by the use of a cell model (free-volume model) for a polyion. Energy dissipation was obtained without directly solving these equations. It was found that the derived expression of viscosity explained the experimental results satisfactorily, and that the streaming potential effect caused by the counterion flow played an essential role in the increase in viscosity of polyelectrolyte solutions at finite polymer concentration and low salt concentration ranges.     

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.     

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.     

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.     

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.