Colligative Properties of Helical Polyelectrolytes

In the context of a continuum model, the excess electrostatic free energy, F,,, of helical charge distributions has been obtained. .More specifically, we calculate F,,, of cy, DNA-like, and infinite pitch helical lines of charge embedded on the surface of a low dielectric cylinder. The deviation of F,,, from the uniformly charged cylinder value is evaluated as a function of screening length. The deviation is small for DNA, smaller for the a helix, and significant for the helix of infinite pitch. Moreover, employing the ideas of Manning, the colligative properties of such polyelectrolyte solutions have been obtained. The deviation from a line of charge in bulk solvent model is small at low salt concentration but becomes appreciable at higher ionic strengths. In a recent paper,’ hereafter designated paper 1, the Debye-Huckel equation was solved for the electrostatic interaction energy of point charges on the surface of a dielectric cylinder immersed in salt water; major deviations from the interaction in bulk solvent were found. We then applied the analytical results to calculate that portion of the potential, A s , of helical charge distributions which depends on salt concentration. For a and DNA-like helical charge distributions, the dominant contribution to A s is the potential characteristic of a uniformly charged cylinder. In toto, the treatment of paper 1 indicated the large and a t times surprising effect a low dielectric, salt-excluding cylinder can have on the electrostatic potential. The present work deals with the calculation of A@ and various colligative properties of a, DNA-like, and infinite pitch helical charge distributions. Our approach is in the spirit of Manning,2-4 and by analogy to simple electrolyte theory, the derived quantities have been designated extending Manning colligative properties. The reader is referred to paper 1 for a brief review of the 1iterature.l As a first approximation, it seems reasonable to view the helical polyion as a thin helical line of charge embedded on the surface of a low-dielectric, mobile, ion-free cylinder. a helices are represented by a single helical line of charge; DNA-type double helices are modeled by two helical lines of charge 180° out of phase with each other. When the pitch of the helix is infinite, the helix becomes a line of charge embedded on the surface of a low dielectric cylinder. Such a model is perhaps appropriate for linear, nonhelical polyelectrolytes3 and is therefore examined. A quantity of primary interest is the difference in reversible work required to charge up a given charge distribution in the presence and absence of salt, F,,,,,,. The difference between the electrostatic free energy of a uniform charge distribution and the helical line is found to be rather small for either the a helix or for DNA but increases with increasing pitch and increasing salt concentration. Possibly, the difference might be significant for DNA a t high salt concentrations. Although the dielectric cylinder exerts a drastic influence on the interaction between point charges, the effect has a different sign depending on whether the charges reside on the same or opposite sides of the cylinder, and there appears to be a significant cancellation for a helical charge distribution. However, over a considerable range of ionic strengths, major deviations in FeXCeSS of the infinite pitch helix from both a line and uniform charge model are observed. As one would expect, the deviations become greater with increasing salt concentration. * Address correspondence to this author at Bell Laboratories, Murray Hill, New Jersey 07974. Since the deviations from ideality of polyelectrolyte colligative properties can be related to (aFexcess /aK)T,V, we determine this quantity for the a helix, line of charge, and DNA double helix. In each case, we delineate the conditions under which the line of charge in bulk solvent is the appropriate limiting case and when corrections become necessary. Finally, we formulate extended Manning colligative properties for the osmotic and activity coefficients of a helical, DNA-like helical, and infinite pitch helical polyelectrolytes. The Excess Electrostatic Free Energy and Colligative Properties of Helical Charge Distributions A. Description of the Model. In view of the quite large effect of a cylinder on the interaction between point charges on its surface,l especially at high salt concentration, we have examined the self-energy of helical distributions of charges. Specifically, one of the questions put in this section is whether the effect of varying salt concentration on the self energy differs between the helical array and a uniform charge distribution. The self energy is independent of the internal dielectric constant for the uniform distribution. Having isolated that portion of the self energy which depends on salt concentration, Fexcess, we then consider the influence of the helical charge distribution on the colligative properties of the polyelectrolyte solution. For a and DNA-like helical lines, our numerical work indicates that the difference in electrostatic free energy between the helical and uniform distributions is rather small. As Manning‘s limiting laws are easy to apply and represent a well-defined reference ~ t a t e , ~ ~ it seems worthwhile to examine the predictions of Manning theory a t finite salt concentration. Otherwise stated, we postulate the following at finite salt concentration: (I) If .$ > 1 counterions condense on the polyion to reduce E to one. Here, E = q 2 / D 2 k B T b , where q is the protonic charge, k B is Boltzmann’s constant, T i s the absolute temperature, b is the linear distance between charges in the configuration of maximum extension, and D2 is the bulk solution dielectric constant. A 1:l supporting electrolyte is assumed to be present. (This is Manning’s assumption D.)2 (11) We may treat the uncondensed ions in the Debye-Huckel approximation. (This is Manning’s assumption E.)2 (111) We neglect interactions between different polyions in the calculation of colligative properties. (IV) The helical polyelectrolyte is assumed to be infinitely long. (V) The helical polyelectrolyte is a salt-excluding, low-dielectric cylinder immersed in bulk solvent; the helical lines of charge reside on the surface. Plausible rationalizations for assumptions I and I1 are as follows: First and foremost, we wish to define a simple 0024-9297/79/2212-0515$01.00/