Effects of Ion Size and Valence on Ion Distribution in Mixed Counterion Systems of a Rodlike Polyelectrolyte Solution. 2. Mixed-Valence Counterion Systems

Research Progress of the Ion Activity Coefficient of Polyelectrolytes: A Review

Polyelectrolyte has wide applications in biomedicine, agriculture and soft robotics. However, it is among one of the least understood physical systems because of the complex interplay of electrostatics and polymer nature. In this review, a comprehensive description is presented on experimental and theoretical studies of the activity coefficient, one of the most important thermodynamic properties of polyelectrolyte. Experimental methods to measure the activity coefficient were introduced, including direct potentiometric measurement and indirect methods such as isopiestic measurement and solubility measurement. Next, progress on the various theoretical approaches was presented, ranging from analytical, empirical and simulation methods. Finally, challenges for future development are proposed on this field.

Direct Measurement of Interhelical DNA Repulsion and Attraction by Quantitative Cross-Linking

To better understand the forces that mediate nucleic acid compaction in biology, we developed the disulfide cross-linking approach xHEED (X-linking of Helices to measure Electrostatic Effects at Distance) to measure the distance-dependent encounter frequency of two DNA helices in solution. Using xHEED, we determined the distance that the electrostatic potential extends from DNA helices, the dependence of this distance on ionic conditions, and the magnitude of repulsion when two helices approach one another. Across all conditions tested, the potential falls to that of the bulk solution within 15 Å of the major groove surface. For separations of ∼30 Å, we measured a repulsion of 1.8 kcal/mol in low monovalent ion concentration (30 mM Na+), with higher Na+ concentrations ameliorating this repulsion, and 2 M Na+ or 100 mM Mg2+ eliminating it. Strikingly, we found full screening at very low Co3+ concentrations and net attraction at higher concentrations, without the higher-order DNA condensation that typically complicates studies of helical attraction. Our measurements define the relevant distances for electrostatic interactions of nucleic-acid helices in biology and introduce a new method to propel further understanding of how these forces impact biological processes.

Spatial segregation of mixed-sized counterions in dendritic polyelectrolytes

Langevin dynamics simulations are utilized to study the structure of a dendritic polyelectrolyte embedded in two component mixtures comprised of conventional (small) and bulky counterions. We vary two parameters that trigger conformational properties of the dendrimer: the reduced Bjerrum length, [Formula: see text], which controls the strength of electrostatic interactions and the number fraction of the bulky counterions, [Formula: see text], which impacts on their steric repulsion. We find that the interplay between the electrostatic and the counterion excluded volume interactions affects the swelling behavior of the molecule. As compared to its neutral counterpart, for weak electrostatic couplings the charged dendrimer exists in swollen conformations whose size remains unaffected by [Formula: see text]. For intermediate couplings, the absorption of counterions into the pervaded volume of the dendrimer starts to influence its conformation. Here, the swelling factor exhibits a maximum which can be shifted by increasing [Formula: see text]. For strong electrostatic couplings the dendrimer deswells correspondingly to [Formula: see text]. In this regime a spatial separation of the counterions into core-shell microstructures is observed. The core of the dendrimer cage is preferentially occupied by the conventional ions, whereas its periphery contains the bulky counterions.

Cation-Anion Interactions within the Nucleic Acid Ion Atmosphere Revealed by Ion Counting

The ion atmosphere is a critical structural, dynamic, and energetic component of nucleic acids that profoundly affects their interactions with proteins and ligands. Experimental methods that "count" the number of ions thermodynamically associated with the ion atmosphere allow dissection of energetic properties of the ion atmosphere, and thus provide direct comparison to theoretical results. Previous experiments have focused primarily on the cations that are attracted to nucleic acid polyanions, but have also showed that anions are excluded from the ion atmosphere. Herein, we have systematically explored the properties of anion exclusion, testing the zeroth-order model that anions of different identity are equally excluded due to electrostatic repulsion. Using a series of monovalent salts, we find, surprisingly, that the extent of anion exclusion and cation inclusion significantly depends on salt identity. The differences are prominent at higher concentrations and mirror trends in mean activity coefficients of the electrolyte solutions. Salts with lower activity coefficients exhibit greater accumulation of both cations and anions within the ion atmosphere, strongly suggesting that cation-anion correlation effects are present in the ion atmosphere and need to be accounted for to understand electrostatic interactions of nucleic acids. To test whether the effects of cation-anion correlations extend to nucleic acid kinetics and thermodynamics, we followed the folding of P4-P6, a domain of the Tetrahymena group I ribozyme, via single-molecule fluorescence resonance energy transfer in solutions with different salts. Solutions of identical concentration but lower activity gave slower and less favorable folding. Our results reveal hitherto unknown properties of the ion atmosphere and suggest possible roles of oriented ion pairs or anion-bridged cations in the ion atmosphere for electrolyte solutions of salts with reduced activity. Consideration of these new results leads to a reevaluation of the strengths and limitations of Poisson-Boltzmann theory and highlights the need for next-generation atomic-level models of the ion atmosphere.

Competitive interaction of monovalent cations with DNA from 3D-RISM

The composition of the ion atmosphere surrounding nucleic acids affects their folding, condensation and binding to other molecules. It is thus of fundamental importance to gain predictive insight into the formation of the ion atmosphere and thermodynamic consequences when varying ionic conditions. An early step toward this goal is to benchmark computational models against quantitative experimental measurements. Herein, we test the ability of the three dimensional reference interaction site model (3D-RISM) to reproduce preferential interaction parameters determined from ion counting (IC) experiments for mixed alkali chlorides and dsDNA. Calculations agree well with experiment with slight deviations for salt concentrations >200 mM and capture the observed trend where the extent of cation accumulation around the DNA varies inversely with its ionic size. Ion distributions indicate that the smaller, more competitive cations accumulate to a greater extent near the phosphoryl groups, penetrating deeper into the grooves. In accord with experiment, calculated IC profiles do not vary with sequence, although the predicted ion distributions in the grooves are sequence and ion size dependent. Calculations on other nucleic acid conformations predict that the variation in linear charge density has a minor effect on the extent of cation competition.

Limiting law excess sum rule for polyelectrolytes

We revisit the mean-field limiting law screening excess sum rule that holds for rodlike polyelectrolytes. We present an efficient derivation of this law that clarifies its region of applicability: The law holds in the limit of small polymer radius, measured relative to the Debye screening length. From the limiting law, we determine the individual ion excess values for single-salt electrolytes. We also consider the mean-field excess sum away from the limiting region, and we relate this quantity to the osmotic pressure of a dilute polyelectrolyte solution. Finally, we consider numerical simulations of many-body polymer-electrolyte solutions. We conclude that the limiting law often accurately describes the screening of physical charged polymers of interest, such as extended DNA.

Brownian dynamics study of the association between the 70S ribosome and elongation factor G

Protein synthesis on the ribosome involves a number of external protein factors that bind at its functional sites. One key factor is the elongation factor G (EF-G) that facilitates the translocation of transfer RNAs between their binding sites, as well as advancement of the messenger RNA by one codon. The details of the EF-G/ribosome diffusional encounter and EF-G association pathway still remain unanswered. Here, we applied Brownian dynamics methodology to study bimolecular association in the bacterial EF-G/70S ribosome system. We estimated the EF-G association rate constants at 150 and 300 mM monovalent ionic strengths and obtained reasonable agreement with kinetic experiments. We have also elucidated the details of EF-G/ribosome association paths and found that positioning of the L11 protein of the large ribosomal subunit is likely crucial for EF-G entry to its binding site.

Ionic correlations in the inhomogeneous atmosphere surrounding cylindrical polyions: catalytic effects of polyions

The structural properties of linear polyelectrolyte solutions in the presence of a salt as evidenced through ionic correlations in the inhomogeneous atmosphere around a polyion and their consequence such as the catalytic potential are studied by using Monte Carlo simulation techniques. The simulations are performed on the cylindrical cell model where a uniformly charged hard cylinder mimics the linear polyion, which is caged in its own cylindrical cell containing counterions and salt. The cell (volume) average of the interionic correlations is presented as a function of the polyion and salt concentrations and ion radius. These results are utilized to study the catalytic effects of polyions as manifested through the changes in the collision frequency between ions in the double layer surrounding the polyion relative to that in the pure electrolyte solution. The reported results suggest a strong influence of the added salt/polyelectrolyte concentration ratio on the structural properties of the solution and hence on ion-ion collision frequency. The machine simulations are supplemented by nonlinear Poisson-Boltzmann results. Fair agreement between two different theoretical methods of calculating the collision frequency is obtained.

Polyelectrolyte hydration: Theory and experiment

A short review of recent theoretical and experimental advances in studies of polyelectrolyte solutions is presented. The focus is on ion-specific effects as revealed in measurements of osmotic pressure and enthalpy of dilution. We review the experimental results for two different polyelectrolyte systems: (i) salts of polyanetholesulfonic acid, and (ii) aliphatic ionenes (polycations) in aqueous solution with various counterions. A theoretical approach based on the extension of Wertheim's integral equation theory [J. Stat. Phys.35, 19 (1984)] is used to analyze the experimental data. Preliminary results, based on the all-atom simulation of model 3,3 ionene oligomers, are discussed in the light of polyelectrolyte hydration.

Quantitative and comprehensive decomposition of the ion atmosphere around nucleic acids

The ion atmosphere around nucleic acids critically affects biological and physical processes such as chromosome packing, RNA folding, and molecular recognition. However, the dynamic nature of the ion atmosphere renders it difficult to characterize. The basic thermodynamic description of this atmosphere, a full accounting of the type and number of associated ions, has remained elusive. Here we provide the first complete accounting of the ion atmosphere, using buffer equilibration and atomic emission spectroscopy (BE-AES) to accurately quantitate the cation association and anion depletion. We have examined the influence of ion size and charge on ion occupancy around simple, well-defined DNA molecules. The relative affinity of monovalent and divalent cations correlates inversely with their size. Divalent cations associate preferentially over monovalent cations; e.g., with Na+ in 4-fold excess of Mg2+ (20 vs 5 mM), the ion atmosphere nevertheless has 3-fold more Mg2+ than Na+. Further, the dicationic polyamine putrescine2+ does not compete effectively for association relative to divalent metal ions, presumably because of its lower charge density. These and other BE-AES results can be used to evaluate and guide the improvement of electrostatic treatments. As a first step, we compare the BE-AES results to predictions from the widely used nonlinear Poisson Boltzmann (NLPB) theory and assess the applicability and precision of this theory. In the future, BE-AES in conjunction with improved theoretical models, can be applied to complex binding and folding equilibria of nucleic acids and their complexes, to parse the electrostatic contribution from the overall thermodynamics of important biological processes.

Counterion-counterion correlation in the double layer around cylindrical polyions: counterion size and valency effects

Monte Carlo simulation and Poisson-Boltzmann results on some aspects of structure and thermodynamics of aqueous polyelectrolyte solutions are presented. The polyelectrolyte solution is described by an infinitely long cylindrical polyion surrounded by counterions modeled as rigid ions moving in a continuum dielectric. Ion-ion correlations in the form of volume average of the counterion-counterion distribution function in the double layer surrounding the polyion are reported for mono- and divalent counterions and for a range of polyion concentrations and charge density parameters in each case. These results confirm again strong influence of the charge density parameter of polyions on properties of polyelectrolyte solutions. The structural information is supplemented by the calculated thermodynamic properties such as osmotic coefficients and heats of dilutions; the latter quantity has not been examined yet in detail by computer simulations. The results are discussed in view of the existing experimental data from the literature for these properties.