Cluster Formation in Solutions of Polyelectrolyte Rings

Abstract

We use molecular dynamics simulations to explore concentrated solutions of semiflexible polyelectrolyte ring polymers, akin to the DNA mini-circles, with counterions of different valences. We find that the assembly of rings into nanoscopic cylindrical stacks is a generic feature of the systems, but the morphology and dynamics of such a cluster can be steered by the counterion conditions. In general, a small addition of trivalent ions can stabilize the emergence of clusters due to the counterion condensation, which mitigates the repulsion between the like-charged rings. Stoichiometric addition of trivalent ions can even lead to phase separation of the polyelectrolyte ring phase due to the ion-bridging effects promoting otherwise entropically driven clustering. On the other hand, monovalent counterions cause the formation of stacks to be re-entrant with density. The clusters are stable within a certain window of concentration, while above the window the polyelectrolytes undergo an osmotic collapse, disfavoring ordering. The cluster phase exhibits characteristic cluster glass dynamics with arrest of collective degrees of freedom but not the self-ones. On the other hand, the collapsed phase shows arrest on both the collective and single level, suggesting an incipient glass-to-glass transition, from a cluster glass of ring clusters to a simple glass of rings.

Keywords: DNA mini-rings; clustering; molecular dynamics; polyelectrolytes; ring polymers; slow dynamics; threading.

Figure 1

Model. (a) Snapshot of a representative configuration of a system with both monovalent (red) and trivalent (yellow) ions with polyelectrolyte rings (shades of blue) at density ρσ³ = 0.40. (b) Typical configuration of a single ring (blue) from the system in the top left panel with the center of mass and the director, d⃗, shown in purple and the minimal surface of the ring as gray triangles. (c) Cylindrical stack (cluster) of rings with aggregation number N_s = 20 taken from the system in the top left panel, shown in both monomer-resolved (shades of blue) and coarse-grained representation (shades of purple) as defined in Methods. (d) Centers of mass and directors (shades of purple) of the system in the top left panel, showing the formation of stack clusters.

Figure 2

Structure factor. Static structure factor between centers of mass of the polyelectrolyte rings, for different densities and for the system with monovalent ions only in (a), both monovalent and trivalent ions in (b), and only trivalent ions in (c). The wavenumber is scaled by the radius of gyration of a single ring in infinite dilution. Analogous plots for additional values of the density are given in Section S1 in the Supporting Information.

Figure 3

Clustering. (a) Weight fraction, W, and distributions of aggregation numbers, N_s, of rings in the emerging cylindrical stacks for different densities for the system with monovalent ions (plots for other ion types are given in Section S1 in the Supporting Information). (b) Probability distributions of mutual orientations of directors, , or pairs of rings 1 and 2, separated by at most 0.5R_g,0 plotted for different densities for the system with monovalent ions only. (c) Weight-weighed mean aggregation number, , of rings in an average stack as a function of density for different ion types, with a purely neutral system with no ions for a reference. Four bold symbols in circles mark selected systems shown in the snapshots in the bottom panel. (d) Fraction of dangling rings, W(N_s = 1), i.e. rings not belonging to any cluster as a function of density. (e) Snapshots of selected systems using the color code of Figure 1.

Figure 4

Conformations. Joint probability distributions of the instantaneous radius of gyration and instantaneous prolateness (defined in eq 10) of individual polyelectrolyte rings in the systems. All panels correspond to the systems with only monovalent ions, at densities ρσ³ ∈ {0.10, 0.30, and 0.50} in (a–c) respectively. The three selected points and corresponding insets show representative conformations of the ring. The radius of gyration is scaled by the radius of gyration of a single ring at an infinite dilution. Dashed white lines mark and .

Figure 5

Counterions. (a) Pair correlation functions between monomeric units of rings and counterions of both types for the systems with a (+1)/(+3) mixture of ions. (b) Static structure factor between centers of mass of the polyelectrolyte rings, for the system with only monovalent ions and for the same system but with no electrostatics, thus turning ions into neutral crowders. The wavenumber is scaled by the radius of gyration of a single ring in infinite dilution. (c) Snapshots of the two systems from (b) showing recovery of the clusters, once the electrostatic is switched off. The color code is the same as in Figure 1.

Figure 6

Dynamics. Intermediate scattering functions (eqs 2 and 3) for centers of mass of the polyelectrolyte rings compared for different densities for the system with only monovalent ions in (a), both, monovalent and trivalent ions in (b), and only trivalent ions in (c). The incoherent (self) part is plotted with lines, while the coherent (collective) part is plotted with points and k_max is the wavenumber corresponding to the main clustering peak in Figure 2. Insets show the characteristic relaxation times, τ_R, defined as F(k_max, τ_R) = e^–1 for both functions, respectively. Analogous plots for more densities are shown in Section S4 in the Supporting Information.

Figure 7

Threading. (a) Joint probability distribution correlating the prolateness of a ring (defined in eq 10) with the number of passive threadings (how many other rings are threading this ring). Pinpointed is the data point for the blue oblate ring threaded by the gray rings. (b) Joint probability distribution correlating the prolateness of a ring with the number of active threadings (how many other rings are threaded by this ring). Pinpointed is the data point for the blue prolated ring threading the other gray rings. (c) Probability distributions of the number of passive partners (dashed line with points) and active partners (solid lines) per ring, plotted for different densities for the system with both monovalent and trivalent ions. Analogous plots for additional systems are given in Section S5 in the Supporting Information.