Equilibrium bundle size of rodlike polyelectrolytes with counterion-induced attractive interactions

Abstract

Multivalent counterions can induce an effective attraction between like-charged rodlike polyelectrolytes, leading to the formation of polelectrolyte bundles. In this paper, we calculate the equilibrium bundle size using a simple model in which the attraction between polyelectrolytes (assumed to be pairwise additive) is treated phenomenologically. If the counterions are point-like, they almost completely neutralize the charge of the bundle, and the equilibrium bundle size diverges. When the counterions are large, however, steric and short-range electrostatic interactions prevent charge neutralization of the bundle, thus forcing the equilibrium bundle size to be finite. We also consider the possibility that increasing the number of nearest neighbors for each rod in the bundle frustrates the attractive interaction between the rods. Such a frustration leads to the formation of finite size bundles as well, even when the counterions are small. The mean-field Poisson-Boltzmann (PB) theory of electrostatic interactions predicts that two identical macromolecules in any salt solution will repel each other [1]. However, the presence of multivalent counterions can actually induce an attraction between like-charged polyelectrolytes (PEs). This has been experimentally observed for several different PEs, including doublestranded DNA [2, 3], F-actin [4,5], microtubules [4,6], and the fd, M13, and tobacco mosaic viruses [4,7]. Computer simulations of both homogeneously charged rods [8,9,10,11] and realistic DNA molecules [11,12,13] unambiguously show that attractive interactions can arise solely from counterion correlations not included in PB theory. Several theories that take these correlations into account -including perturbative expansions of PB theory [14,15], structural-correlation theory [16,17,18], and strong-coupling theory [19] -obtain an attractive interaction between two rods. It is still a matter of discussion, however, as to which of these theories is the most appropriate description of the correlation-induced attraction seen in experiments and simulations. Furthermore, it is unknown whether the interactions between multiple rods is pairwise additive or not [17,20,21].Under experimental conditions in which the interaction between PEs is attractive, the PEs typically form dense, ordered bundles of a well-defined size, rather than precipitating into a PE-rich phase [2,3,4,5,6,7]. In this paper, we theoretically investigate the thermodynamic stability of these bundles (if bundle growth is not limited thermodynamically, then it must be limited by kinetic barriers [21,22,23]). We assume that the attractive interactions are pairwise additive, but do not specify the precise nature of the counterion correlations. Rather, we simply introduce a phenomenological parameter γ to characterize the attractive energy between two PEs in a bundle.Consider, then, an aqueous solution of volume V with N identical rodlike PEs of length L, radius a 0 , and a uniform linear charge density −eλ 0 (the aggregation of flexible PEs has been cons...

Braided bundles and compact coils: The structure and thermodynamics of hexagonally packed chiral filament assemblies

“…According to the thermodynamic description of such an assembly process, the classical nucleation model, there can be no equilibrium limitation to the growth in the presence of a net free energy gain per aggregated filament. To explain this apparent contradiction, a number of theoretical mechanisms for the self-limited growth have emerged, focusing variously on the specialized nature of forces between filaments condensed in the presence of multi-valent counterions [41,42] as well as the elastic cost of defects forced into bundles by rapid quenching [43] or by the toroidal topology of bundles form by long strands of DNA [44]. In conflict with assumptions of "electrostatic" mechanisms for limited growth, finite-sized bundles are observed even when bundles are condensed in the absence of multi-valent ions, for example, by depletion forces [37,38] or through the incorporation of specialized cross-linking proteins [39,40].…”

Aggregation kinetics of stiff polyelectrolytes in the presence of multivalent salt

“…an infinitely large bundle [6], experiments always find finite-sized bundles [2,3]. To explain this, it has been suggested that the theoretically expected phase separation may be hindered by kinetic barriers [7], steric effects [8], or frustration of the local structure with energy penalty [3,9]. The phenomenon of bundle formation has also been studied using computer simulation, which indicated a tendency towards a well defined finite size [10,11].…”

Helical twist controls the thickness of F-actin bundles

“…The stabilization mechanism of counterioninduced bundles is proposed to be similar to that of colloidal clusters (17,18); steric and short-range electrostatic interactions or frustration within the bundles prevent charge neutralization and limit the bundle size (19). Alternatively, the finite size of chiral biopolymers has been suggested to result from a buildup of in-plane shear elastic stresses (20), which can result in braided structures (21).…”

“…The bundle diameter could, in principle, be either kinetically (25)(26)(27) or thermodynamically (19,20) constrained. However, the independence of the bundle diameter on the preparation method and system used strongly suggests an equilibrium mechanism.…”