Dielectric increment and dielectric dispersion of solutions containing simple charged linear macromolecules. II. Experimental results with synthetic polyelectrolytes

The electric permittivity of aqueous solutions of different synthetic polyelectrolytes have been measured as a function of frequency in the range 5 kHz up to 100 MHz in the absence of added salt. Solutions of polymethacrylic acid and polyacrylic acid of different degrees of polymerization, both partially neutralized with NaOH, were investigated as well as solutions of Na-polystyrenesulphonate at different concentrations.For all systems a dispersion profile with two separated dispersion regions was obtained with a molecular weight dependent value of the static electric permittivity. The low frequency dispersion region was found to be characterized by a molecular weight dependent mean relaxation time while for the high frequency dispersion region both the mean relaxation time and the dielectric increment are molecular weight independent. It is shown that the reciprocal values of the specific increments and of the relaxation times depend linearly on the macromolecular concentration. Extrapolation of the corresponding quantities to infinite dilution was found to be possible. A comparison of these extrapolated values with calculated ones according to the previously derived theory also applicable to flexible macromolecules establishes that this theory describes satisfactorily the dielectric behaviour of the systems investigated.The conclusion is reached that the high frequency dispersion and relaxation can be attributed to fluctuations in the distribution of bound counterions along limited parts of the macromolecule. The relaxation time of the low frequency dispersion region seems to be essentially determined by the rotation of the complete molecule and the static electric permittivity can he explained in terms of fluctuations in the counterion density extending over the whole macromolecule.     

Dielectric disperison of linear polyelectrolytes

A three-dimensional (gaussian) model is formulated that permits evaluation of the one-dimensional fourier cosine transform of the counterion-counterion coulombic interaction energy ?_k. It is shown that ?_kis much larger for small k than was assumed by Oosawa; consequently, the magnitude of his dielectric increment, and also his relaxation time, must be drastically revised to smaller values. Evidently, the self-field of the fluctuation is large enough to seriously depress the amplitude of the spontaneous fluctuations, and to force Ohmic conduction of the ions at a rate far in excess of the rate of translational diffusion. Finally, it is noted that in highly conducting bulk solvents, where both positive and negative counterfoils coexist in the ion-atmosphere, one might expect to observe a low-frequency component of the dielectric relaxation arising from unequal rates of diffusion of the positive and negative species, in analogy with the situation for spherical colloidal particles. In such a circumstance, the basic counterfoil fluctuation mechanism proposed by Oosawa would acquire a new validity.     

Dielectric behavior of linear polyelectrolytes

The dipolar correlation function for a system of counterions diffusing on the surface of a polyelectrolyte cylinder is computed. Repulsive coulombic interactions between the counterions are taken into account. Lateral dissociation and reassociation to the cylinder is treated microscopically. Numerical calculations needed to obtain quantitative results for the long time behavior are presented. The model dependence on its parameters is interpreted with special emphasis on parameter values typical of DNA.     

Dielectric properties of polyelectrolytes. II. A theory of dielectric increment due to ion fluctuation by a matrix method

The mean square of dipole moment of a linear macromolecule which is responsible for dielectric increment of aqueous polyelectrolyte solutions is calculated by means of a matrix method in which ion binding at discrete sites and the nearest-neighbor interaction are taken into account. On the basis of the relationship between polarization of poly-ion and fluctuation of bound counterions the present theory indicates that the loosely bound ions result in larger increment and otherwise smaller increment. Also, the theory predicts that the dielectric increment has a maximum at an intermediate monovalent–divalent ion ratio when both species coexist. These results are consistent with experiments on polyacrylic acid neutralized with NaOH and Ca(OH)₂. At large contents of divalent ions the effect of chelation is also discussed.     

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.     

The contribution of proton fluctuation to dielectric relaxation in protein solutions

Equations are derived which account for the effect of an applied electric field on the fluctuation of protons associated with a macromolecule. The contribution of this proton polarization to the complex permittivity of the macromolecules is evaluated in terms of its effect on both the dispersion due to proton fluctuation and the dispersion caused by dipolar rotation. An expression for the fluctuation correlation time τiδ is derived in terms of the mean lifetimes of the ionised and unionised state τ0 and τ+. If τiδ is very much less than the dipolar correlation time τi, the fluctuation dispersion will occur at much higher frequencies than the dispersion due to orientation polarization; hence the dispersions will be well separated. If τiδ » τi the two regions would overlap and would be indistinguishable as separate entities. At present insufficient data are available to test rigorously these conclusions but the potentialities of the theory in relation to small globular proteins is shown.     

Dielectric properties of polyelectrolytes. V. Dielectric dispersion of heavy meromyosin

Dielectric constant and dielectric loss of heavy meromyosin (HMM) were measured with varying pH. HMM showed a broader dispersion pattern than that with a single relaxation time especially on the high-frequencey side. The dielectric increment increased sharply with pH, above p^{H 6}, whereas the mean relaxation time and whole dispersion pattern were unchanged in the same region. The values of the increment and the mean relaxation time were much larger than those of usual globular proteins. The dispersion profile, pH dependence, and values of the increment are well explained by Oosawa's counterion fluctuation theory. Other mechanisms are more or less inadequate to our results. In the low pH region below the isoelectric precipitation region, both the increment and the mean relaxation time decreased; this is probably due to partial denaturation and suppression of the dissociation of carboxyl groups. An experiment on a urea-denatured sample supports this assumption. The biological significance of the pH dependence is discussed.     

Self-association of hemoglobin: a dielectric dispersion study

The concentration dependence of the dielectric dispersion between 1 kHz and 0.5 MHz was investigated for dense hemoglobin solutions. The concentration was varied between 4 and 26 mM/l. per heme. From the characteristic changes in dielectric parameters at some threshold concentration, we observe that the association of hemoglobin macromolecules occurs. This statement is also confirmed by viscosity measurement data.     

Temperature dependence of DNA dielectric dispersion at radiofrequency

We have studied the dielectric behavior of DNA aqueous solutions at various ionic strengths and in the presence of the specific DNA ligand ethidium bromide, in the frequency range 1 MHz-1 GHz, at different temperatures ranging from 5 to 40 degrees C. The activation enthalpies of the dielectric relaxations studied were obtained by Arrhenius plots of In(tau T)-1 vs. T-1. The results are consistent with a counterion fluctuation model as previously developed by Mandel and colleagues.     

Effect of ions on counterion fluctuation in low-molecular weight DNA dielectric dispersions

The dielectric permittivity of aqueous solutions of low-molecular weight DNA (Mr = 3.2 X 10(5) ) in the presence of MgCl2 and AgNO3 has been measured in the frequency range from 5 kHz to 30 MHz, at a temperature of 25 degrees C. The DNA concentration was 3.5 X 10(-4) M in terms of phosphate and the salt concentration was varied from 1 X 10(-5) to 2 X 10(-4) M. The dielectric results have been analyzed in terms of two contiguous dielectric dispersions, and characteristic parameters have been discussed on the basis of polyelectrolyte theories which deal with counterion fluctuation. Some molecular parameters of the DNA molecule in electrolyte solutions are estimated.     

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.     

Counterion condensation in polyelectrolyte theory

Condensation of counterions to a single, highly charged polyelectrolyte is formulated in terms of the two-dimensional coulomb gas theory of Kosterlitz and Thouless. The critical condensation criterion is calculated and is shown to vary as a function of the added salt concentration.     

Dielectric properties of polyelectrolytes. III. Effect of divalent cations on dielectric increment of polyacids

Dielectric dispersion of polyacrylic acid (PAA) and polystyrene sulfonic acid (PSS) was measured in the presence of divalent cations. Effects of divalent ions were studied by neutralization with varying ratios of sodium hydroxide and divalent base concentration, addition of salts of divalent cations, and neutralization with divalent bases only. Two dispersion regions were observed in all cases, i.e., low-frequency dispersion (10²–10⁴ Hz) and high-frequency dispersion (10⁵–10⁶ Hz). The dielectric increment increases in the presence of sodium and alkaline earth metal ions together, but not with sodium and transition metal ions. This is due to the increment of low-frequency dispersion and is attributable to the fluctuation of bound counterions which is explained by our theory previously reported.¹ In the case of PAA neutralized with large fractions of divalent ions, or with divalent ions only, the increment is very small because of reduction of the fluctuation by interaction between bound ions at the neighboring sites and reduction of the effective length of polyion probably due to chelation by divalent ions. There are some differences among the effects of Mg⁺⁺, Ca⁺⁺, and Ba⁺⁺ on dielectric increment which may result from affinity or chelating ability of these ions.     

Allowance for spatial dispersion of dielectric permittivity in polyelectrolyte model of DNA.

In order to allow for real dielectric properties of a solvent in calculating of electrostatic characteristics of strongly charged polyions such as DNA in salt solution we consider a simple model of linear dielectric response of a medium. The interactions between charged particles are treated in the framework of self-consistent-field approximation. The basic characteristic of the problem, electrostatic potential, can be found from the solution of non-linear integro-differential equation. Specifically we consider so-called quasimacroscopic model where dielectric response of a medium depends only on the distance from the polyion. Application of the approach for calculating of the B-to-Z free energy qualitatively retains the main conclusion obtained previously within the model with fixed dielectric constant: non-monotonous behavior of the free energy differences as a function of ionic strength. At the same time, essential sensitivity of the results to specific values of dielectric parameters is observed.