Flexible Polymers Also Counterattract

In their article (Physics Today, September 2000, page 38), William M. Gelbart and coauthors Robijn F. Bruinsma, Philip A. Pincus, and V. Adrian Parsegian describe the condensation of double-stranded DNA by multivalent ions. They discuss many models that feature counterion-mediated rigid DNA attractions. Electrostatically driven precipitation, however, is not unique to rigid DNA fragments. It also occurs in flexible charged polymers,¹ including single-stranded DNA and polysterene sulfonate; results in these polymers can be used to determine the correct physical mechanism of the precipitation. The precipitation can be explained by a simple mean-field theory of counterion condensation along the chains if correlations between the condensed counterions and the monomers are included in the analysis.² The correlations must be computed with solid-state physics techniques,³ rather than liquid-state theory, given that the precipitate is a dense system of charges. An ionic glass structure is formed,^3,4 which is reminiscent of the Wigner crystal arrangement of counterions found in simpler morphologies, such as between aligned charged plates (Gelbart et al., ref. 6) or rods (see papers cited in Gelbart et al., ref. 3). The cohesive energy of the ionic glass precipitate increases as the size of the multivalent ions decreases, as in conventional ionic glasses or crystals.

The physical size of the chain, which is determined by its conformation, is another important scale to describe the monomolecular collapse of DNA into toroidal conformations. Electrostatic interactions in low ionic strength solutions lead to two possible polyelectrolyte conformations: stretched with a reduced charge and compacted with nearly zero charge.² The renormalized or effective charge of chains with N charged monomers due to counterion condensation strongly depends on the chain conformation. Mean-field models, including Poisson-Boltzmann of ion penetrable spheres, give a reduced effective charge proportional to the size of a charged sphere R. If the stretched chain (R ∼ N) collapses into a dense sphere (R ∼ N^1/3), its reduced charge is much less than the effective charge of the stretched conformation, N ln N. If the correlations between the condensed ions and the monomers are included, the dense sphere is nearly neutral.^2,3

Chain precipitation occurs if the entropy decrease of the counterions neutralizing the sphere's charge is overcompensated by the gain in short-range electrostatic attractions per monomer in the sphere.³ These attractions cause the chains to compact: to a toroid in semiflexible chains or a sphere in flexible chains. Consequently, monomolecular DNA condensation occurs because finite, stretched, rodlike chains of charged monomers have higher effective charge, and therefore higher energy, than chains collapsed to their smallest possible size. Gelbart and coauthors mention that isolated spheres have higher effective charge than isolated rods. But this is only true if the finite rods have no short-range cutoffs, as in a continuous zero-width line of charge with zero-size counterions,² and if the concentration of chains is identically zero, which is physically impossible.

The size of the multivalent ions is also important in determining the possibility of charge inversion (see Gelbart et al., ref. 11), and the re-dissolution of the precipitate at large salt concentrations.³ These phenomena are determined by the relation of the chemical potential of the multivalent ions in the solution and the inverse screening length. Large multivalent particles do not contribute to screening; they readily overcharge a polyelectrolyte such as DNA wrapped around histones. Small multivalent salts, however, have complex thermodynamics in concentrated ionic solutions, and do not necessarily lead to polyelectrolyte charge inversion.

In summary, contrary to our intuition, short length scales strongly influence the physics of large charged systems.

Gelbart, Bruinsma, Pincus, and Parsegian reply: We thank Francisco Solis and Monica Olvera de la Cruz for calling attention to the novel statistical mechanical properties that arise with flexible polyelectrolytes. We also agree that the line- and point-charge condensation behaviors we describe do not occur except in the ideal limit of infinite dilution (as we did specify in our text). At the levels of both mean-field and correlation approaches, physical issues arise with flexible chains that are related to--but different from--those treated in our article. Because of space constraints we focused primarily on rigid polyelectrolytes, and omitted many other interesting aspects of recent studies on molecular and colloidal electrostatics.

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