Ionic strength and counterion repulsion as factors in the behavior of polyions in orienting electric fields

The effect of counterion-counterion repulsion on the orientation of DNA, a polyion of high charge density is examined by electric-field orientation experiments. The charge species of the counterion and the ionic strength effect the orientation in a manner consistent with a theoretical treatment of the polarization of high charge density polyelectrolytes in terms of the effect of the applied field on the equilibrium distribution of condensed counterfoils on the polyion.     

Analysis of potentiometric titrations of heterogeneous natural polyelectrolytes in terms of counterion condensation theory: application to humic acid

A model, developed within the framework of the counterion condensation theory of linear polyelectrolytes, is presented in this paper to describe the acid-base properties of linear polyelectrolytes, consisting of several types of functional ionizable groups. This formalism has been successfully applied to Fluka humic acid under salt-free conditions, as well as in the presence of supporting simple 1:1 salt (KNO3) at three different concentrations. As part of this approach, the charge density of the humic acid is obtained from the activity coefficient measurements of potassium counterions at different humic acid concentrations at a constant degree of dissociation of the polyelectrolyte. The humic acid average charge density was found to be 0.80 +/- 0.05. Using the present model, we are able to satisfactorily describe the experimental data obtained from acid-base potentiometric titrations. Four main functional groups making up the polymer are determined through their fractional abundances (Xi) and intrinsic pK (pK0i) values. The fractional abundances remained constant and independent of the ionic strength, indicating that the humic acid constitution does not depend on the concentration of excess salts. The pK0i values show a small change with ionic strength, which can be explained by the polyelectrolytic behavior of the solution.     

Net proton charge of beta- and kappa-casein in concentrated aqueous electrolyte solutions

Titration experiments have been carried out in order to measure the net proton charge of beta- and kappa-casein in NaCl solutions at 0.1 M and 1 M salt concentrations, at 4 degrees C, in the pH range between 5.5 and 10.5. Experimental data are compared with model values calculated through pK(a)'s of titrable groups neglecting the electrostatic perturbation term (deltapK(a)) in order to evaluate the magnitude of the error caused by this approximation and to delimit its effectiveness. At both ionic strengths, the agreement is good for kappa-casein in the pH range [5.5, 9.5], while errors of up to 2 charges are observed for beta-casein in the same range. These deviations are likely to be caused by strong electrostatic effects induced by the high density of negative charges of beta-casein 1-21 peptide. In order to account for these electrostatic effects, the net proton charge on this peptide is evaluated through a model based on the counterion condensation theory developed for the titration of polyelectrolytes with different types of ionizable groups.     

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.     

Thermodynamics of the conformational transition of biopolyelectrolytes: the case of specific affinity of counterions.

A formal development of the Counterion Condensation theory (CC) of linear polyelectrolytes has been performed to include specific (chemical) affinity of condensed counterions, for polyelectrolyte charge density values larger than the critical value of condensation. It has been conventionally assumed that each condensed counterion exhibits an affinity free-energy difference for the polymer, (DeltaG(aff)). Moreover, the model assumes that the enthalpic and entropic contributions to DeltaG(aff), i.e., DeltaH(aff) and DeltaS(aff), are both independent of temperature, ionic strength and polymer concentration. Equations have been derived relative to the case of the thermally induced, ionic strength dependent, conformational transition of a biopolyelectrolyte between two conformations for which chemical affinity is supposed to take place. The experimental data of the intramolecular conformational transition of the ionic polysaccharide kappa-carrageenan in dimethylsulfoxide (DMSO) have been successfully compared with the theoretical predictions. This novel approach provides the enthalpic and entropic affinity values for both conformations, together with the corresponding thermodynamic functions of nonpolyelectrolytic origin pertaining to the biopolymer backbone change per se, i.e., DeltaH(n.pol) and DeltaS(n.pol), according to a treatment previously shown to be successful for lower values of the biopolyelectrolyte linear charge density. The ratio of DeltaH(n.pol) to DeltaS(n.pol) was found to be remarkably constant independent of the value of the dielectric constant of the solvent, from formamide to water to DMSO, pointing to the identity of the underlying conformational process.     

A ligand binding model of counterion condensation to finite length polyelectrolytes

A ligand binding model of counterion association in finite length polyelectrolytes is presented. This model introduces counterion condensation features into a binding formalism. It agrees well with the predictions of other finite length models and is consistent with experimental data on helix–coil melting transitions for short nucleic acid oligomers. This model uses a discrete charge distribution for the polyelectrolyte. An expression for the electrostatic self-energy of finite length polyelectrolytes is derived using the Euler–Maclaurin sum formula. This sum is shown to be accurate over a wide range of salt concentrations. This electrostatic term is used in an energy minimization analysis. The energy minimization is solved analytically using a Lagrange inversion formula. This general procedure gives a rapidly convergent series and requires no assumptions with regard to “limiting law” behavior. However, when used in the Manning minimization formalism [(1977) Biophysical Chemistry, 24 , 2086], the volume of the condensed phase becomes unrealistically large at low ionic strength. The ligand binding model does not have a condensed phase volume as a parameter. It provides a single expression that agrees both with Manning's theory and with the theory of Ramanathan and Woodbury [(1982) Journal of Chemical Physics 77 , 4133] under the respective conditions of these theories.     

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.     

Flexible polymer-induced condensation and bundle formation of DNA and F-actin filaments

A simple semi-empirical theory is developed for the ionic strength dependence of the flexible polymer-induced condensation of semiflexible polyelectrolytes such as DNA and F-actin filaments. Critical concentrations of flexible polymer needed for condensation are calculated by comparing the free energies of inserting the semiflexible polyelectrolytes in a solution of flexible polymers, respectively, in their free state, and in their condensed state. Predictions of the theory are compared to experimental data on the condensation of DNA and F-actin filaments induced by the flexible polymer poly(ethylene oxide). The theory also predicts that reentrant decollapse is possible at low ionic strength and high concentrations of flexible polymer, as observed for DNA.     

The role of surface charge on the accelerating action of heparin on the antithrombin III-inhibited activity of alpha-thrombin

We have compared surface charge and the surface charge density on the polyanions heparin and potassium polyvinyl sulfate (KPVS), as well as on hydrolyzed heparin and KPVS, with their accelerating effect on the inhibitory action of antithrombin III on thrombin. Polyelectrolyte titration of thrombin with KPVS or heparin at pH 7.4 clearly indicates an electrostatic interaction. In contrast, at the same pH no electrostatic interaction is observed between polyanions and antithrombin III. KPVS accelerates the inhibitory action of antithrombin III to the same extent as heparin on the basis of charge equivalence. Heparin and KPVS with a mean distance between two charged centers of less than 0.75 and 0.95 nm, respectively, accelerate strongly whereas hydrolysates with lower charge densities are far less active. The following observations are indicated. Intramolecular neutralization of oppositely charged residues occurs within thrombin, antithrombin III, and partially hydrolyzed heparin. Heparin acts on the antithrombin III-thrombin reaction through cooperative electrostatic binding to thrombin and nonelectrostatic interaction with antithrombin III. This indicates a quasi-catalytic action of the polyelectrolyte. Hydrolysis of only a few N-sulfate residues within the heparin molecule decreases the linear surface charge density to such an extent that the accelerating action is drastically reduced. The loss of accelerating capacity agrees with the sudden loss of counterion condensation due to the decrease of the linear surface charge density beyond limits postulated by Manning in a theory of polyelectrolytes.     

Microvillar ion channels: cytoskeletal modulation of ion fluxes

The recently presented theory of microvillar Ca(2+)signaling [Lange, K. (1999) J. Cell. Physiol.180, 19-35], combined with Manning's theory of "condensed counterions" in linear polyelectrolytes [Manning, G. S. (1969). J. Chem. Phys.51, 924-931] and the finding of cable-like ion conductance in actin filaments [Lin, E. C. & Cantiello, H. F. (1993). Biophys. J.65, 1371-1378], allows a systematic interpretation of the role of the actin cytoskeleton in ion channel regulation.Ion conduction through actin filament bundles of microvilli exhibits unique nonlinear transmission properties some of which closely resemble that of electronic semiconductors: (1) bundles of microfilaments display significant resistance to cation conduction and (2) this resistance is decreased by supply of additional energy either as thermal, mechanical or electromagnetic field energy. Other transmission properties, however, are unique for ionic conduction in polyelectrolytes. (1) Current pulses injected into the filaments were transformed into oscillating currents or even into several discrete charge pulses closely resembling that of single-channel recordings. Discontinuous transmission is due to the existence of counterion clouds along the fixed anionic charge centers of the polymer, each acting as an "ionic capacitor". (2) The conductivity of linear polyelectrolytes strongly decreases with the charge number of the counterions; thus, Ca(2+)and Mg(2+)are effective modulator of charge transfer through linear polyelectrolytes. Field-dependent formation of divalent cation plugs on either side of the microvillar conduction line may generate the characteristic gating behavior of cation channels. (3) Mechanical movement of actin filament bundles, e.g. bending of hair cell microvilli, generates charge translocations along the filament structure (mechano-electrical coupling). (4) Energy of external fields, by inducing molecular dipoles within the polyelectrolyte matrix, can be transformed into mechanical movement of the system (electro-mechanical coupling). Because ionic transmission through linear polyelectrolytes is very slow compared with electronic conduction, only low-frequency electromagnetic fields can interact with the condensed counterion systems of linear polyelectrolytes.The delineated characteristics of microvillar ion conduction are strongly supported by the phenomenon of electro-mechanical coupling (reverse transduction) in microvilli of the audioreceptor (hair) cells and the recently reported dynamics of Ca(2+)signaling in microvilli of audio- and photoreceptor cells. Due to the cell-specific expression of different types and combinations of ion channels and transporters in the microvillar tip membrane of differentiated cells, the functional properties of this cell surface organelle are highly variable serving a multitude of different cellular functions including receptor-mediated effects such as Ca(2+)signaling, regulation of glucose and amino acid transport, as well as modulation of membrane potential. Even mechanical channel activation involved in cell volume regulation can be deduced from the systematic properties of the microvillar channel concept. In addition, the specific ion conduction properties of microfilaments combined with their proposed role in Ca(2+)signaling make microvilli the most likely cellular site for the interaction with external electric and magnetic fields.     

Polyelectrolytic effects in semi-flexible carboxylate polysaccharides. Part 2

A statistical model is presented in which an ionic polymer is taken as a linear arrangement of flexible segments (similar to the spring-bead model). This model is used to represent semi-flexible linear ionic polysaccharides in solution. The distribution of end-to-end distances is taken from Monte Carlo calculations of the amylosic chain conformation and combined with Manning's counterion condensation theory of linear polyelectrolytes. The excess thermodynamic properties are then calculated as a function of the degree of ionization and of a number of physical variables. The results show that the statistical average of thermodynamic functions taken over the conformational states is not equivalent to the thermodynamic function of the average conformation. This has important implications for the correct comparison of theoretical prediction with the experimental results.     

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.     

Calculations on the effect of methylation on the electrostatic stability of the B- and Z-conformers of DNA

The three-dimensional Poisson–Boltzmann equation for the distribution of counterion charge density around double-helical DNA has been solved for solutions of .01M, .10M, and .20M monovalent salt. The polymers, poly[d(CpGp)] and poly[d(m⁵CpGp)], were studied in the B- and the Z-conformations. The effect of methylation on the relative stabilities of these conformers in solutions of different ionic strengths is known to favor the Z-form. Accumulation of charge density around the B- and the Z-conformers is compared in detail. The relative electrostatic stabilities of the B- and Z-conformers in .01M, .10M, and .20M solutions are compared and discussed in terms of the ion–DNA interactions and the self-energy of the structured ionic environment. The ion–DNA interaction energies, termed “phosphate screening,” monotonically decrease with ionic strength and are consistent with a B-to-Z conformation change induced in either polymer by increased electrolyte concentration. However, these calculated energies alone do not account for the fact that the ionic strength at the midpoint of the transition of the methylated polymer is substantially lower than that of its unmethylated analogues. The phosphate screening effect is counterbalanced by changes in the self-energy required for the creation of the structured counterion environment. This self-energy of the electrolyte environment monotonically increases with ionic strength. Methylation-induced shifts in the overall conformational equilibria depend on the relative changes of these competing effects. Increasing salt concentration is calcualted to favor the Z-conformer. The effect of methylation, lowering the ionic strength of the transition midpoint, is proposed to originate in minor structural changes in the Z-form of the polymer, making the groove more accessible to counterions in the G(3′ – 5′)C region. This allows a redistribution of counterion density and a lowering of the self-energy of the ionic environment, conferring added stability to the Z-conformation, as indicated by calculations of relative entropies. The experimentally observed temperature dependence of the B-to-Z transition, however, cannot be explained without assuming the release of bound water. Maps of the calculated three-dimensional structure at the counterion distribution near the surface of these molecules in both the B- and the Z-forms are also presented.     

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.     

Specific interactions versus counterion condensation. 2. Theoretical treatment within the counterion condensation theory

Polyuronates such as pectate and alginate are very well-known examples of biological polyelectrolytes undergoing, upon addition of divalent cations, an interchain association that acts as the junction of an eventually formed stable hydrogel. In the present paper, a thermodynamic model based on the counterion condensation theory has been developed to account for this cation-induced chain pairing of negatively charged polyelectrolytes. The strong interactions between cross-linking ions and uronate moieties in the specific binding site have been described in terms of chemical bonding, with complete charge annihilation between the two species. The chain-pairing process is depicted as progressively increasing with the concentration of cross-linking counterions and is thermodynamically defined by the fraction of each species. On these bases, the total Gibbs energy of the system has been expressed as the sum of the contributions of the Gibbs energy of the (single) chain stretches and of the (associated) dimers, weighted by their respective fractions 1 - theta and theta. In addition, the model assumes that the condensed divalent counterions exhibit an affinity free-energy for the chain, G(C)(aff,0), and the junction, G(D)(aff,0), respectively. Moreover, a specific Gibbs energy of chemical bonding, G(bond,0), has been introduced as the driving force for the formation of dimers. The model provides the mathematical formalism for calculating the fraction, theta, of chain dimers formed and the amount of ions condensed and bound onto the polyelectrolyte when two different types of counterions (of equal or different valence) are present. The effect of the parameter G(bond,0) has been investigated and, in particular, its difference from G(C,D)(aff,0) was found to be crucial in determining the distribution of the ions into territorial condensation and chemical bonding, respectively. Finally, the effect of the variation of the molar ratio between cross-linking ions and uronic groups in the specific binding sites, sigma0, was evaluated. In particular, a remarkable decrease in the amount of condensed counterions has been pointed out in the case of sigma0 = 1/3, with respect to the value of sigma0 = 1/4, characterizing the traditional "egg-box" structure, as a result of the drop of the charge density of the polyelectrolyte induced by complete charge annihilation.     

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.     

Cation radius effects on the helix-coil transition of DNA. Cryptates and other large cations.

Most polyelectrolyte theories of the effect of ions on the thermal melting of DNA assume that the predominant influence of the cations comes through their charge. Ion size and structure are treated, for analytic convenience, as negligible variables. We have examined the validity of this assumption by measuring the melting temperature of calf thymus DNA as a function of salt concentration with four univalent cations of different hydrated radii. These are K+ (3.3 A), (n-Pr)4N+ (4.5 A), (EtOH)4N+ (4.5 A), and C222-K+ (5 A). C222-K+ is a complex of cryptand C222 with K+. With K+ as the sole cation, Tm varies linearly with the log of ionic strength over the range 0.001-0.1 M. With all the K+ sequestered by an equimolar amount of C222, Tm is depressed by 10-20 degrees C and the slope of Tm vs. ionic strength is lower. At low ionic strength, an even greater reduction in Tm is achieved with (n-Pr)4N+; but the similar-sized (EtOH)4N+ gives a curve more similar to K+. Theoretical modeling, taking into account cation size through the Poisson-Boltzmann equation for cylindrical polyelectrolytes, predicts that larger cations should be less effective in stabilizing the double helix; but the calculated effect is less than observed experimentally. These results show that valence, cation size, and specific solvation effects are all important in determining the stability of the double-helical form of DNA.     

Charge patch attraction and reentrant condensation in DNA-liposome complexes

We investigated the formation of complexes between cationic liposomes built up by DOTAP and three linear anionic polyions, with different charge density and flexibility, such as a single-stranded ssDNA, a double-stranded dsDNA and the polyacrylate sodium salt [NaPAA] of three different molecular weights. Our aim is to gain further insight into the formation mechanism of polyion-liposome aggregates of different sizes (lipoplexes), by comparing the behavior of DNA with a model polyelectrolyte, such as NaPAA, with approximately the same charge density but with a higher flexibility. We employed dynamic light scattering (DLS) and transmission electron microscopy (TEM) measurements, in order to explore both the hydrodynamic and structural properties of the aggregates resulting from polyion-liposome interaction and to present a comprehensive picture of the complexation process. The phenomenology can be summarized in a charge ratio-dependent scenario, where the main feature is the formation of large equilibrium clusters due to the aggregation of intact polyion-coated vesicles. At increasing polyion-liposome ratio, the size of the clusters continuously increases, reaching a maximum at a well-defined value of this ratio, and then decreases (“reentrant” condensation). The aggregation mechanism and the role of the polyion charge density in the complex formation are discussed in the light of the recent theories on the correlated adsorption of polyelectrolytes at charged interfaces. Within this framework, the phenomena of charge inversion and the reentrant condensation, peaked at the isoelectric point, finds a simple explanation.     

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.     

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.