Comparison of polyelectrolyte theories of the binding of cations to DNA.

Predictions of the binding of counterions to DNA made using the counterion condensation theory developed by Manning are compared with those made using the Poisson-Boltzmann equation, solved numerically by the Runge-Kutta procedure. Ions are defined as territorially or atmospherically bound if they fall within a given distance, defined by counterion condensation theory, from the DNA surface. Two types of experimental situations are considered. The first is the delocalized binding of a single type of counterion to DNA. In this case the Poisson-Boltzmann treatment predicts somewhat lower extents of binding TO DNA, modeled as a 10-A radius cylinder, than does Manning theory. The two theories converge as the radius decreases. The second type of experiment is the competition of ions of different valence for binding to DNA. The theories are compared with literature values of binding constants of divalent ions in the presence of monovalent ions, and of spermidine 3+ in the presence of Na+ or Mg2+. Both predict with fair accuracy the salt dependence of the equilibrium constants.     

Monte Carlo and Poisson–Boltzmann calculations of the fraction of counterions bound to DNA

The counterion density and the condensation region around DNA have been examined as functions of both ion size and added-salt concentration using Metropolis Monte Carlo (MC) and Poisson–Boltzmann (PB) methods. Two different definitions of the “bound” and “free” components of the electrolyte ion atmosphere were used to compare these approaches. First, calculation of the ion density in different spatial regions around the polyelectrolyte molecule indicates, in agreement with previous work, that the PB equation does not predict an invariance of the surface concentration of counterions as electrolyte is added to the system. Further, the PB equation underestimates the counterion concentration at the DNA surface, compared to the MC results, the difference being greatest in the grooves, where ionic concentrations are highest. If counterions within a fixed radius of the helical axis are considered to be bound, then the fraction of polyelectrolyte charge neutralized by counterions would be predicted to increase as the bulk electrolyte concentration increases. A second categorization—one in which monovalent cations in regions where the average electrostatic potential is ledd than ?kT are considered to be bound—provides an informative basis for comparison of MC and PB with each other and with counterion-condensation theory. By this criterion, PB calculations on the B from of DNA indicate that the amount of bound counterion charge per phosphate group is about .67 and is independent of salt concentration. A particularly provocative observatiob is that when this binding criterion is used, MC calculations quantitatively reproduce the bound fraction predicated by counterion-condensation theory for all-atom models of B-DNA and A-DNA as well as for charged cylindera of varying lineat charge densities. For example, for B-DNA and A-DNA, the fractions of phosphate groups neutralized by 2 Å hard sphere counterions are 0.768 and .817, respectively. For theoretical studies, the rediys enclosing the region in which the electrostatic potential is calculated studies, the radius enclosing the region in which the electrostatic potential is calculated to be less than ?kT is advocated s a more suitable binding or condensation radius that enclosing the fraction of counterions given by (1 – ξ^?1). A comparsion of radii calculated using both of these definitions is presented. © 1994 John Wiley & Sons, Inc.     

The role of the ionic environment in the orientation of nucleic acids in electric fields

The relationship between polyelectrolyte theories based on linear charge density models and the electric-field induced orientation of the polyelectrolytes, poly(A), poly(C) and DNA is examined by varying their ionic environment with respect to ionic strength and acidity. The degree of counterion condensation on the polyelectrolytes predicted by the theories of Manning and Record is shown to be related linearly to the orientation as measured by their dichroism in the field. Micro-structural differences between poly(A) and poly(C) account for the differences in their dependence on the pH of the medium which affects the counterion condensation and thus the polarization in the orienting electric fields. The results consequently support recent treatments of linear polyelectrolytes having a high charge density which model them as smoothly charged linear polyions, but indicate that these models are insufficient to account for some of the effects of microstructural variations.     

A commentary on the screened-Oseen, counterion-condensation formalism of polyion electrophoresis

The use of linear theory, in particular, counterion condensation (CC) theory, in describing electrophoresis of polyelectrolyte chains, is criticized on several grounds. First, there are problems with CC theory in describing the equilibrium distribution of ions around polyelectrolytes. Second, CC theory is used to treat ion relaxation in a linear theory with respect to the polyion charge despite the fact that ion relaxation arises as a consequence of nonlinear charge effects. This nonlinearity has been well established by several investigators over the last 70 years for spherical, cylindrical, and arbitrarily shaped model polyions. Third, current use of CC theory ignores the electrophoretic hindrance as well as the ion relaxation for condensed counterions and only includes such interactions for uncondensed counterions. Because most of the condensed counterions lie outside the shear surface of the polyion (in the example of DNA), the assumption of ion condensation is artificial and unphysical. Fourth, the singular solution, based on a screened Oseen tensor, currently used in the above mentioned theories is simply wrong and fails to account for the incompressibility of the solvent. The actual singular solution, which has long been available, is discussed. In conclusion, it is pointed out that numerical alternatives based on classic electrophoresis theory (J.T.G. Overbeek, Kolloid-Beih, 1943, 54:287-364) are now available.     

Approach to the limit of counterion condensation

According to counterion condensation theory, one of the contributions to the polyelectrolyte free energy is a pairwise sum of Debye-Hückel potentials between polymer charges that are reduced by condensed counterions. When the polyion model is taken as an infinitely long and uniformly spaced line of charges, a simple closed expression for the summation, combined with entropy-derived mixing contributions, leads to the central result of the theory, a condensed fraction of counterions dependent only on the linear charge density of the polyion and the valence of the counterion, stable against increases of salt up to concentrations in excess of 0.1 M. Here we evaluate the sum numerically for B-DNA models other than the infinite line of B-DNA charges. For a finite-length line there are end effects at low salt. The condensation limit is reached as a flat plateau by increasing the salt concentration. At a fixed salt concentration the condensation limit is reached by increasing the length of the line. At moderate salt even very short B-DNA line-model oligomers have condensed fractions not far from the infinite polymer limit. For a long double-helical array with charge coordinates at the phosphates of B-DNA, the limiting condensed fraction appears to be approached at low salt. In contrast to the results for the line of charges, however, the computed condensed fraction varies strongly with salt in the range of experimentally typical concentrations. Salt invariance is restored, in agreement with both the line model and experimental data, when dielectric saturation is considered by means of a distance-dependent dielectric function. For sufficiently long B-DNA line and helical models, as typical salt concentrations, the counterion binding fraction approaches the polymer limit as a linear function of 1/P, where P is the number of phosphate groups of B-DNA.     

Evaluation of the counterion condensation theory of polyelectrolytes.

We compare free energies of counterion distributions in polyelectrolyte solutions predicted from the cylindrical Poisson-Boltzmann (PB) model and from the counterion condensation theories of Manning: CC1 (Manning, 1969a, b), which assumes an infinitely thin region of condensed counterions, and CC2 (Manning, 1977), which assumes a region of finite thickness. We consider rods of finite radius with the linear charge density of B-DNA in 1-1 valent and 2-2 valent salt solutions. We find that under all conditions considered here the free energy of the CC1 and the CC2 models is higher than that of the PB model. We argue that counterion condensation theory imposes nonphysical constraints and is, therefore, a poorer approximation to the underlying physics based on continuum dielectrics, point-charge small ions, Poisson electrostatics, and Boltzmann distributions. The errors in counterion condensation theory diminish with increasing distance from, or radius of, the polyion.     

Application of polyelectrolyte theories for analysis of DNA melting in the presence of Na+ and Mg2+ ions.

Numerical calculations, using Poisson-Boltzmann (PB) and counterion condensation (CC) polyelectrolyte theories, of the electrostatic free energy difference, DeltaGel, between single-stranded (coil) and double-helical DNA have been performed for solutions of NaDNA + NaCl with and without added MgCl2. Calculations have been made for conditions relevant to systems where experimental values of helix coil transition temperature (Tm) and other thermodynamic quantities have been measured. Comparison with experimental data has been possible by invoking values of Tm for solutions containing NaCl salt only. Resulting theoretical values of enthalpy, entropy, and heat capacity (for NaCl salt-containing solutions) and of Tm as a function of NaCl concentration in NaCl + MgCl2 solutions have thus been obtained. Qualitative and, to a large extent, quantitative reproduction of the experimental Tm, DeltaHm, DeltaSm, and DeltaCp values have been found from the results of polyelectrolyte theories. However, the quantitative resemblance of experimental data is considerably better for PB theory as compared to the CC model. Furthermore, some rather implausible qualitative conclusions are obtained within the CC results for DNA melting in NaCl + MgCl2 solutions. Our results argue in favor of the Poisson-Boltzmann theory, as compared to the counterion condensation theory.     

Role of counterion condensation in folding of the Tetrahymena ribozyme. I. Equilibrium stabilization by cations

Folding of RNA into an ordered, compact structure requires substantial neutralization of the negatively charged backbone by positively charged counterions. Using a native gel electrophoresis assay, we have examined the effects of counterion condensation upon the equilibrium folding of the Tetrahymena ribozyme. Incubation of the ribozyme in the presence of mono-, di- and trivalent ions induces a conformational state that is capable of rapidly forming the native structure upon brief exposure to Mg2+. The cation concentration dependence of this transition is directly correlated with the charge of the counterion used to induce folding. Substrate cleavage assays confirm the rapid onset of catalytic activity under these conditions. These results are discussed in terms of classical counterion condensation theory. A model for folding is proposed which predicts effects of charge, ionic radius and temperature on counterion-induced RNA folding transitions.     

Trivalent counterion condensation on DNA measured by pulse gel electrophoresis

Pulse gel electrophoresis was used to measure the reduction of mobilities of λ-DNA-Hind III fragments ranging from 23.130 to 2.027 kilobase pairs in Tris borate buffer solutions mixed with either hexammine cobalt(III), or spermidine³⁺ trivalent counterions that competed with Tris⁺ and Na⁺ for binding onto polyion DNA. The normalized titration curves of mobility were well fit by the two-variable counterion condensation theory. The agreement between measured charge fraction neutralized and counterion condensation prediction was good over a relatively wide range of trivalent cation concentrations at several solution conditions (pH, ionic strength). The effect of ionic strength, trivalent cation concentration, counterion structure, and DNA length on the binding were discussed based on the experimental measurements and the counterion condensation theory. © 1996 John Wiley & Sons, Inc.     

Novel chiral ionic liquids stationary phases for the enantiomer separation of chiral acid by high‐performance liquid chromatography

Novel chiral ionic liquid stationary phases based on chiral imidazolium were prepared. The ionic liquid chiral selector was synthesized by ring opening of cyclohexene oxide with imidazole or 5,6‐dimethylbenzimidazole, and then chemically modified by different substitute groups. Chiral stationary phases were prepared by bonding to the surface of silica sphere through thioene “click” reaction. Their enantioselective separations of chiral acids were evaluated by high‐performance liquid chromatography. The retention of acid sample was related to the counterion concentration and showed a typical ion exchange process. The chiral separation abilities of chiral stationary phases were greatly influenced by the substituent group on the chiral selector as well as the mobile phase, which indicated that, besides ion exchange, other interactions such as steric hindrance, π‐π interaction, and hydrogen bonding are important for the enantioselectivity. In this report, the influence of bulk solvent components, the effects of varying concentration, and the type of the counterion as well as the proportion of acid and basic additives were investigated in detail.     

Divalent cations and the electrostatic potential around DNA: Monte Carlo and Poisson-Boltzmann calculations

The predictions of counterion condensation theory for divalent ions were tested by comparison with the results of Monte Carlo calculations on an all-atom model of DNA. Monovalent-divalent competition at the polyelectrolyte surface was investigated by varying the partial molar volume of divalent ions. To assess the viability of using Poisson-Boltzmann (PB) calculations for determining divalent ion concentrations at DNA surfaces, Monte Carlo (MC) calculations were compared with PB calculations using different models of the dielectric continuum. It was determined that, while standard PB calculations of divalent ion surface densities are about 25-30% below those predicted by MC techniques, and somewhat larger than errors previously determined for monovalent ions, errors due to the use of the mean-field approximation of PB theory are smaller than those arising from common assumptions regarding the dielectric continuum.     

The iso-competition point for counterion competition binding to DNA: calculated multivalent versus monovalent cation binding equivalence.

In this paper we introduce an important parameter called the iso-competition point (ICP), to characterize the competition binding to DNA in a two-cation-species system. By imposing the condition of charge neutralization fraction equivalence theta1 = ZthetaZ upon the two simultaneous equations in Manning's counterion condensation theory, the ICPs can be calculated. Each ICP, which refers to a particular multivalent concentration where the charge fraction on DNA neutralized from monovalent cations equals that from the multivalent cations, corresponds to a specific ionic strength condition. At fixed ionic strength, the total DNA charge neutralization fractions thetaICP are equal, no matter whether the higher valence cation is divalent, trivalent, or tetravalent. The ionic strength effect on ICP can be expressed by a semiquantitative equation as ICPZa/ICPZb = (Ia/Ib)Z, where Ia, Ib refers to the instance of ionic strengths and Z indicates the valence. The ICP can be used to interpret and characterize the ionic strength, valence, and DNA length effects on the counterion competition binding in a two-species system. Data from our previous investigations involving binding of Mg2+, Ca2+, and Co(NH3)63+ to lambda-DNA-HindIII fragments ranging from 2.0 to 23.1 kbp was used to investigate the applicability of ICP to describe counterion binding. It will be shown that the ICP parameter presents a prospective picture of the counterion competition binding to polyelectrolyte DNA under a specific ion environment condition.     

Polyelectrolyte theory. I. Counterion accumulation,site-binding,and their insensitivity to polyelectrolyte shape in solutions containing finite salt concentrations

We have computed the Poisson-Boltzmann distribution of counterions around polyelectrolytes in solutions containing finite salt concentrations. The polyelectrolytes considered here are highly charged in the sense that ξ > 1, ξ being the linear charge density parameter for cylinders, which is generalized by us to other shapes. Contrary to the situation at zero salt concentration, the counterion distribution is not strongly shape dependent, being similar for cylinders or spheres which have the same superficial charge density and radius of curvature R_c. The distribution resembles that in the neighborhood of a plane with the same charge density. Three regions are distinguished. (1) In the “inner region” which extends up to a distance R_c/2ξ from the surface, the counterion distribution is essentially salt independent. The counterion concentration in the immediate vicinity of the polyelectrolyte surface (CIV) is quite high, typically 1–10M, and proportional to the square of the surface charge density, which is its main determinant. (2) An intermediate region extends out to a distance where the electrostatic potential is equal to κT/e. This distance is comparable to λ for plane and cylinder, and smaller for the sphere. (3) In the outer region, the distribution is hardly influenced by the details of the inner region, on which it cannot, therefore, give much information. Colligative properties are dependent on the distribution in the outer region and are fairly well predicted even by a rudimentary theory. The large value of the CIV implies that site binding must often be significant. It can be computed by applying the mass-action law to site-bound counterions in equilibrium with the counterions in the neighborhood, whose concentration is the CIV, the relevant equilibrium constant being that for the binding of counterions to isolated monomer sites. Because the CIV is insensitive to salt concentration, this will also be the case for site binding. With the graphs provided, one can compute the extent of sitebinding within the Poisson-Boltzmann framework. The “condensation radius,” i.e., the radius encompassing a counterionic charge 1 ? ξ^?1 around a cylinder, is found to be large. It varies with salt concentration and tends to infinity as the salt is diluted. Neither this radius nor the charge fraction 1 ? ξ^?1 of condensation theory plays any special role in the counterion distribution. The “finite-salt” results apply to salt concentrations, typically as low as 1–10 mM. This encompasses, among others, all experiments on biological polyelectrolytes.     

Ion Condensation onto Ribozyme Is Site Specific and Fold Dependent

The highly charged RNA molecules, with each phosphate carrying a single negative charge, cannot fold into well-defined architectures with tertiary interactions in the absence of ions. For ribozymes, divalent cations are known to be more efficient than monovalent ions in driving them to a compact state, although Mg²⁺ ions are needed for catalytic activities. Therefore, how ions interact with RNA is relevant in understanding RNA folding. It is often thought that most of the ions are territorially and nonspecifically bound to the RNA, as predicted by the counterion condensation theory. Here, we show using simulations of Azoarcus ribozyme, based on an accurate coarse-grained three-site interaction model with explicit divalent and monovalent cations, that ion condensation is highly specific and depends on the nucleotide position. The regions with high coordination between the phosphate groups and the divalent cations are discernible even at very low Mg²⁺ concentrations when the ribozyme does not form tertiary interactions. Surprisingly, these regions also contain the secondary structural elements that nucleate subsequently in the self-assembly of RNA, implying that ion condensation is determined by the architecture of the folded state. These results are in sharp contrast to interactions of ions (monovalent and divalent) with rigid charged rods, in which ion condensation is uniform and position independent. The differences are explained in terms of the dramatic nonmonotonic shape fluctuations in the ribozyme as it folds with increasing Mg²⁺ or Ca²⁺ concentration.     

Salt dependence of DNA structural stabilities in solution. Theoretical predictions versus experiments

The predictions of currently available theories for treating DNA-diffuse ionic cloud free energy contributions to conformational stability have been tested against experimental data for salt induced B-Z and B-A transitions. The theories considered are (i) Manning's counterion condensation approach (CC), (ii) the idealized Poisson-Boltzmann approximation (PB), and (iii) the potentials of mean force (PMF) approach proposed by Soumpasis. As far as we can judge from comparison with the set of experimental data currently available, it is found that only the latter theory yields satisfactory quantitative results for the dependence of the B-Z and B-A relative stabilities on monovalent salt concentration. The correct application of the PB and CC theories does not yield very low salt Z-B transitions, in contradiction to earlier assertions. At low salt concentrations the PB theory is qualitatively correct in predicting that the B form is electrostatically more favorable than both the A and Z forms, whereas the CC theory is qualitatively wrong predicting that Z-DNA is more stable than both B and A DNA.     

Roles of Boundary Conditions in DNA Simulations: Analysis of Ion Distributions with the Finite-Difference Poisson-Boltzmann Method

The wide use of lattice-sum strategies in biomolecular simulations has raised many questions on potential artifacts in these strategies. One interesting question is the artifacts in the counterion distributions of highly charged systems. As one would anticipate, Coulombic interactions under the periodic boundary condition may deviate noticeably from those under the free boundary condition in the highly charged systems, significantly influencing their counterion distributions. On the other hand, the electrostatic screening due to water molecules and mobile ions may effectively damp the possible periodic distortions in Coulombic interactions. Therefore, the magnitude of periodicity-induced artifacts in counterion distributions is not straightforward to dissect without detailed analyses. In this study, we have developed a hybrid explicit counterion/implicit salt representation of mobile ions to address this question. We have chosen a well-studied DNA for easy validation of the minimal hybrid ion representation. Our detailed analysis of continuum ion distributions, explicit ion distributions, radial counterion distribution functions, and sequence-dependent counterion distributions, however, indicates that periodicity artifacts are not apparent at the surface of the tested DNA. Nevertheless, influence of boundary conditions does show up starting at the second solvation shell and becomes apparent at the cell boundary.     

Linearity, saturation and blocking in a large multiionic channel: divalent cation modulation of the OmpF porin conductance

Measurement of unitary conductance is a fundamental step in the characterization of a protein ion channel permeabilizing a membrane. We study here the effect of salts of divalent cations on the OmpF channel conductance with a particular emphasis in dissecting the role of the electrolyte itself, the role of the counterion accumulation induced by the protein channel charges and other effects not found in salts of monovalent cations. We show that current saturation and blocking are not exclusive properties of narrow (single-file) ion channels but may be observed in large, multiionic channels like bacterial porins. Single-channel conductance measurements performed over a wide range of salt concentrations (up to 3 M) combined with continuum electrodiffusion calculations demonstrate that current saturation cannot be simply ascribed to ion interaction with protein channel residues.     

Ultrasonic study of the kinetics of counterion site binding in aqueous solutions of polyelectrolytes

The excess ultrasonic absorption due to counterion binding has been studied as a function of frequency for a series of polysalts in the range 1–150 MHz. All the relaxation spectra can be represented by a relaxation equation with two relaxation terms. The relaxation frequencies appear concentration independent and the relaxation amplitudes seem proportional to concentration. The low frequency relaxation process appears to depend mainly on the nature of the counterion while the high frequency relaxation process seems to be mostly dependent on the nature of the polyion. These results are quite similar to those obtained in ultrasonic studies of ion-pairing in solutions of divalent sulfates. The kinetic model used for the quantitative analysis of these results has been modified for polysalts through introducing the concept of“counterion condensation”. In this modified model the excess absorption is assigned to the perturbation by the ultrasonic waves of the equilibria between the three states of hydration of ths complex formed by a counterion and that part of the polyion where it is bound. Analytical expressions of the relaxation amplitudes have been derived using classical procedures for this modified kinetic model. In the case of cobalt-polyphosphate (Co-PP), the ultrasonic data together with the results of NMR measurements on either Co²⁺ or Co-PP have been used for the evaluation of the volume changes, the rate constants and the fractions of counterions in the three states of hydration involved in the binding equilibria. The volume changes obtained in this manner depend only slightly on the method of calculation and appear to be consistent with volume changes for outer-sphere and inner-sphere complex formation. These results are discussed.     

Salt Dependence of DNA Structural Stabilities in Solution. Theoretical Predictions Versus Experiments

Abstract

The predictions of currently available theories for treating DNA-diffuse ionic cloud free energy contributions to conformational stability have been tested against experimental data for salt induced B-Z and B-A transitions. The theories considered are (i) Manning's counterion condensation approach (CC), (ii) the idealized Poisson-Boltzmann approximation (PB), and (iii) the potentials of mean force (PMF) approach proposed by Soumpasis. As far as we can judge from comparison with the set of experimental data currently available, it is found that only the latter theory yields satisfactory quantitative results for the dependence of the B-Z and B-A relative stabilities on monovalent salt concentration. The correct application of the PB and CC theories does not yield very low salt Z-B transitions, in contradiction to earlier assertions. At low salt concentrations the PB theory is qualitatively correct in predicting that the B form is electrostatically more favorable than both the A and B forms, whereas the CC theory is qualitatively wrong predicting that Z-DNA is more stable than both B and A DNA.     

Ion condensation and signal transduction

Many abiotic and other signals are transduced in eukaryotic cells by changes in the level of free calcium via pumps, channels and stores. We suggest here that ion condensation should also be taken into account. Calcium, like other counterions, is condensed onto linear polymers at a critical value of the charge density. Such condensation resembles a phase transition and has a topological basis in that it is promoted by linear as opposed to spherical assemblies of charges. Condensed counterions are delocalised and can diffuse in the so-called near region along the polymers. It is generally admitted that cytoskeletal filaments, proteins colocalised with these filaments, protein filaments distinct from cytoskeletal filaments, and filamentous assemblies of other macromolecules, constitute an intracellular macromolecular network. Here we draw attention to the fact that this network has physicochemical characteristics that enable counterion condensation. We then propose a model in which the feedback relationships between the condensation/decondensation of calcium and the activation of calcium-dependent kinases and phosphatases control the charge density of the filaments of the intracellular macromolecular network. We show how condensation might help mediate free levels of calcium both locally and globally. In this model, calcium condensation/decondensation on the macromolecular network creates coherent patterns of protein phosphorylation that integrate signals. This leads us to hypothesize that the process of ion condensation operates in signal transduction, that it can have an integrative role and that the macromolecular network serves as an integrative receptor.