The mean-field theory for attraction between like-charged macromolecules

A mean-field theory based on Gibbs-Bogoliubov inequality is constructed to study the interactions between two like-charged polyions. It is shown that contrary to the previously established paradigm, a properly constructed mean-field theory can quantitatively account for the attractive interactions between two like-charged rods. One of the most fascinating problems that has recently appeared in the field of condensed matter physics is the discovery of attraction between like-charged macromolecules[1]. This attraction plays a fundamental role in various biological processes such as the condensation of DNA [2,3] and the formation of fibers composing cellular cytosceleton [4]. The attraction between like-charged colloids has also been observed in various experiments and simulations [5–8]. It has been noted that the attraction appears only in the presence of multivalent counterions. A number of models have been proposed to try to explain the mechanism of these strange phenomena. It is now clear, from both simulations and experiments, that this effect is purely electrostatic and is produced by strong many body interactions present in polyelectrolyte solutions. In a beautiful set of experiments Tang et al[9] demonstrated how addition of simple monovalent salt produced dissociation of the actin bundles. The F-actin chains are highly charged polymers, which inspite of their large negative charge density, aggregate in well defined bundles in the presence of polyamines. However, this 1 Corresponding author [email protected] Preprint submitted to Elsevier Preprint 1 February 2008 bundling can be reversed by addition of simple monovalent salt which screens the electrostatic interactions between the polyions and the multivalent counterions. The first explanation of attraction between like charged surfaces in the presence of multivalent counterions was advanced by Kjellander and Marcelja [10] based on the integral equation formalism. From the numerical solution of the AHNC equation these authors came to conclude that for sufficiently high surface charge, an attraction can arise between like charged plates. A very simple physical picture to explain the mechanism of attraction was advanced by Rouzina and Bloomfield [11], and extended by Shklovskii [12]. These authors proposed that the condensed counterions around the two plates form strongly coupled Wigner crystals. In the case of rod-like polyions, a similar explanation has been advanced by Arenzon et al on the basis of an exactly solvable model[13,14]. A different mechanism, relying on correlated fluctuations, has been proposed by Ha and Liu [15], but has been criticized by Levin et al [16]. Since the beginning of the study of this interesting phenomenon there has been a general consensus that the attraction must arise as a result of correlations of condensed counterions [17]. It was, therefore, implicitly assumed that no mean-field theory would be able to account for this phnenomenon. This belief was further reinforced by the solutions of Poisson-Boltzmann equation (PB) which, of course, did not predict any attraction. Not all mean-fields, however, are equal. In this paper we shall present a mean-field theory, which quantitatively accounts for the attraction between like-charged rods in the presence of condensed multivalent counterion. We consider two parallel polyions modeled as rigid rods, each having Z charges of value −q, spaced uniformly with separation b along the length. The rods are separated by distance d = xb. The strong electrostatic interaction between the polyions and the multivalent counterions present in solution leads to counterion condensation [18–21]. The effect of n, α-valent condensed counterions, is approximated by the renormalization of local charge. Thus, if one of the charged sites of a polyion has an associated condensed counterion its effective charge becomes −q(1−α). Note that in this simple model the condensed counterions are assumed to reside only on top of the charged sites. The net charge of each polyion is (Z − αn)q. The Hamiltonian for the interactions between the two rods is [13], H = 1 2D Z