Effects of dielectric saturation on planar electric double layers in salt solutions

The effects of dielectric saturation on planar electric double layers in salt solutions are examined by solving the Poisson-Boltzmann equation analytically where the dielectric constant is given as a function of the electric displacement. The activity and the distribution of small ions, the surface potential and the Donnan potential are calculated. The salt exclusion parameter and the Donnan potential decrease while the surface potential increases as a result of the dielectric saturation. The electrostatic entropy is affected considerably by the dielectric saturation while the electrostatic energy is little influenced. Generally, the effects of dielectric saturation on the distribution of small ions and the thermodynamic properties are enhanced by the addition of salt.     

Glycocalyx electrostatic potential profile analysis: ion, pH, steric, and charge effects

The Poisson-Boltzmann equation is modified to consider charge ionogenicity, steric exclusion, and charge distribution in order to describe the perimembranous electrostatic potential profile in a manner consistent with the known morphology and biochemical composition of the cell's glycocalyx. Exact numerical and approximate analytical solutions are given for various charge distributions and for an extended form of the Donnan potential model. The interrelated effects of ionic conditions, bulk pH, ion binding, local dielectric, steric volume exclusion, and charge distribution on the local potential, pH, and charge density within the glycocalyx are examined. Local charge-induced, potential-mediated pH reductions cause glycocalyx charge neutralization. Under certain conditions, local potentials may be insensitive to ionic strength or may decrease in spite of increasing charge density. The volume exclusion of the glycocalyx reduces the local ion concentration, thereby increasing the local potential. With neutral lipid membranes, the Donnan and surface potential agree if the glycocalyx charge distribution is both uniform and several times thicker than the Debye length (approximately 20 A in thickness under physiological conditions). Model limitations in terms of application to microdomains or protein endo- and ectodomains are discussed.     

Site binding as a local equilibrium. Mn2+ binding to poly(acrylic acid) as a function of the degree of ionization

Mn2+ binding to poly(acrylic acid) at different degrees of ionization, alpha, has been studied from the frequency dependence of the water protons' relaxation rates T1(-1) and T2(-1). Site binding is treated as an equilibrium with the concentration of free ions at the immediate vicinity (CIV) of the polyion. The CIV is calculated as the solution of the Poisson-Boltzmann equation at the surface of the cylindrical polyion. A single value of K is shown to fit the results at all values of alpha. The amount of site binding is higher than the total amount of condensed divalent counterions predicted for a finite polyion concentration in the presence of monovalent counterions by Manning's theory.     

The theoretical distributions and diffusivities of small ions in chondroitin sulphate and hyaluronate

The electrostatic interactions between polyionic glycosaminoglycans and small mobile ions are investigated using the Poisson-Boltzmann equation and a rod-in-cell model of the polyelectrolyte. Calculations are made for the range of polyelectrolyte concentrations and buffer compositions for which measurements of ion distributions and diffusivities are reported in a companion paper (Maroudas et al., Biophys. Chem. 32 (1988) 257). We conclude that the distribution of mobile ions is largely determined by the 'far-field' potential and is adequately described by the Poisson-Boltzmann theory and also by more approximate theories such as ideal Donnan or 'condensation' theory. The measured variations in cation diffusivities, particularly the increase in diffusivity with increasing matrix concentration at low ionic strengths, are predicted qualitatively using an approximate diffusion theory together with the calculated potential fields. However, the same theory applied to anion diffusion gives qualitatively wrong results.     

Counterion diffusion in heparin solutions

The physiological importance of heparin is due to its strong interaction with bivalent counterions, especially Ca2+. A diffusional approach of this property is presented in this article: the observable is the self-diffusion coefficient of the counterions, as a function of the ratio of the polyelectrolyte over the added salt concentrations. All the results are in agreement with a simple "quasi-chemical model" in which two different states are assumed for the counterions: "free" or "bound." The proportions of these two types of ions are calculated according to the distribution function of the counterions around the polyion. We assume that those of counterions located at a distance closer than a, the characteristic distance, are bound; the others are free. The ionic distribution function is evaluated by a numerical integration of a cell model Poisson-Boltzmann equation. Finally, this model leads to a very good agreement with the experimental results, if the radius of heparin polyion is assumed to be 6 and 10 A.     

Electrostatic repulsion of ion penetrable charged membranes: role of Donnan potential

A model is proposed for the electrostatic repulsion between two ion-penetrable charged membranes in which fixed charges are uniformly distributed. This model assumes that the electric potential far inside the membrane is always equal to the Donnan potential, independent of the membrane separation. In this respect the present model completely differs from usual models for the electrostatic interaction of colloidal particles which assume that the surface potential or the surface charge density remains constant during interaction. It is shown that the rise in potential in the interacting membranes caused by their approach is greatly suppressed so that the potential in the membrane does not exceed the Donnan potential. Numerical results of the calculation of the repulsion by the non-linear Poisson-Boltzmann equation are displayed as a function of the membrane separation and an approximate formula is also derived.     

Monte Carlo determination of the distribution of ions about a cylindrical polyelectrolyte

We report a calculation of the distribution of small ions around a charged cylinder representing a polyelectrolyte molecule in solution. The Monte Carlo method of Metropolis, Rosenbluth, and Teller was used to avoid the inaccuracies known to be associated with the Poisson-Boltzmann equation. The systems examined contained a long polyelectrolyte cylinder with charge parameter, χ, equal to 4.2, corresponding approximately to a DNA molecule. In one model, the cylinder had charges on its axis and an exclusion radius to the center of the small ions equal to 10 Å, while the small ions had various radii in the range from 1 to 10 Å and one or two protonic charges. Various systems were studied; some had one species of small ion alone, others had mixtures of different types. The results showed good agreement with the solution of the Poisson-Boltzmann equation when only the species with 1-Å radius was present, but considerable discrepancies appeared with larger ions as a result of excluded volume interactions between the latter. Deviations from the Poisson-Boltzmann equation also appeared when both positive and negative small ions were present; the deviations were in the direction of a higher concentration of both counter- and co-ions, but particularly co-ions, close to the polyelectrolyte. In another model, the charges were arranged along two helices on the surface of the cylinder; the resulting radial distribution of small ions was not much different from that found when the charges were situated on the axis. In all cases there was a striking accumulation of counterions in a layer of concentration exceeding 1 mol/L at the surface of the polyion.     

Calculations of the spatial distribution of charge density in the environment of DNA

A technique for modeling the structured environmental charge distribution about isolated polyions of arbitrary geometry is presented and applied to B-DNA. It describes the three-dimensional variation of the continuous space charge and allows estimation of local electrostatic potentials and fields that the electrolytic environment induces at nuclei of the polyion. Calculations involve an iterative solution to the set of equations coupling electrostatic potential and average charge density in space. By dividing the region around a DNA segment into finite volume elements, sets of numerically stable atmospheric charge densities have been obtained over a range of concentrations of added monovalent salt. Results are in good agreement with those of Poisson-Boltzmann calculations on comparable systems and are consistent with findings from Monte Carlo simulations of DNA.     

Electrostatic contribution to the bending of DNA

A model is derived that accounts for the short-range electrostatic contribution to the bending of DNA molecule in solution and in complexes with proteins in terms of the non-linear Poisson-Boltzmann equation. We defined that the short-range electrostatic interactions depend on the changes of the polyion surface charge density under deformation, while the long-range interactions depend on the bending-induced changes in distances between each two points along the polyion axis. After an appropriate simplification of the Poisson-Boltzmann equation, the short-range term is calculated separately giving the lower limit for the electrostatic contribution to the DNA persistence length. The result is compared with the theoretical approaches developed earlier [M. Fixman, J. Chem. Phys. 76 (1982) 6346; M. Le Bret, J. Chem. Phys. 76 (1982) 6243] and with the experimental data. The conclusion is made that the results of Fixman-Le Bret, which took into account both types of the electrostatic interactions for a uniformly bent polyion, give the upper limit for the electrostatic persistence length at low ionic strength, and the actual behavior of the DNA persistence length lies between two theoretical limits. Only the short-range term is significant at moderate-to-high ionic strength where our results coincide with the predictions of Fixman-Le Bret. The bending of DNA on the protein surface that is accompanied by an asymmetric neutralization of the DNA charge is also analyzed. In this case, the electrostatic bending energy gives a significant favorite contribution to the total bending energy of DNA. Important implications to the mechanisms of DNA-protein interactions, particularly in the nucleosome particle, are discussed.     

Dielectric increment and dielectric dispersion of solutions containing simple charged linear macromolecules. I. Theory

A theory is derived for the static and frequency dependent value of the electric permittivity for model systems representing a solution of a macromolecule bearing a large number of identical charges. The polyion is represented either as a charged rigid rod (A) or as a sequence of charged rodlike subunits in an arbitrary but fixed configuration (B) and it is assumed that a certain fraction of the counterions is closely associated to the macromolecule. The dielectric properties are described in terms of fluctuations in the distribution of the associated counterions along the polyion. These fluctuations can occur locally between potential barriers marking the ends of the subunits (if considered) but can also extend over the whole molecule. Neglecting correlations between different associated counterions expressions for the static value of the dielectric increment are obtained which reveal its dependence on the fraction of bound ions, on the charge of the counterions and on the length of the molecule for model A or the radius of gyration for model B. The dynamic behaviour of A is distinguishable from that of B as the former will present one single dispersion curve of the frequency dependent electric permittivity while the latter may give rise to two different dispersion regions. This will be the case if both the exchange between bound and free ions and the rotation of the complete molecule are relatively slow in comparison to the local bound counterion density fluctuations and if these fluctuations occur on a much shorter time scale than the ion density fluctuations extending over the complete macromolecule.     

Predicting nonspecific ion binding using DelPhi

Ions are an important component of the cell and affect the corresponding biological macromolecules either via direct binding or as a screening ion cloud. Although some ion binding is highly specific and frequently associated with the function of the macromolecule, other ions bind to the protein surface nonspecifically, presumably because the electrostatic attraction is strong enough to immobilize them. Here, we test such a scenario and demonstrate that experimentally identified surface-bound ions are located at a potential that facilitates binding, which indicates that the major driving force is the electrostatics. Without taking into consideration geometrical factors and structural fluctuations, we show that ions tend to be bound onto the protein surface at positions with strong potential but with polarity opposite to that of the ion. This observation is used to develop a method that uses a DelPhi-calculated potential map in conjunction with an in-house-developed clustering algorithm to predict nonspecific ion-binding sites. Although this approach distinguishes only the polarity of the ions, and not their chemical nature, it can predict nonspecific binding of positively or negatively charged ions with acceptable accuracy. One can use the predictions in the Poisson-Boltzmann approach by placing explicit ions in the predicted positions, which in turn will reduce the magnitude of the local potential and extend the limits of the Poisson-Boltzmann equation. In addition, one can use this approach to place the desired number of ions before conducting molecular-dynamics simulations to neutralize the net charge of the protein, because it was shown to perform better than standard screened Coulomb canned routines, or to predict ion-binding sites in proteins. This latter is especially true for proteins that are involved in ion transport, because such ions are loosely bound and very difficult to detect experimentally.     

Density functional theory for the nonspecific binding of salt to polyelectrolytes: thermodynamic properties

The thermodynamics of the nonspecific binding of salt to a polyelectrolyte molecule is studied using a density functional approach. The polyelectrolyte molecule is modeled as an infinite, inflexible, and impenetrable charged cylinder and the counterions and co-ions are modeled as charged hard spheres of equal diameter. The density functional theory is based on a hybrid approach where the hard-sphere contribution to the one-particle correlation function is evaluated nonperturbatively and the ionic contribution to the one-particle correlation function is evaluated perturbatively. The advantage of the approach is that analytical expressions are available for all the correlation functions. The calculated single ion preferential interaction coefficients, excess free energy, and activity coefficients show a nonmonotonic variation as a function of polyion charge in the presence of divalent ions. These properties display considerable departure from the predictions of the nonlinear Poisson-Boltzmann (NLPB) equation, with qualitative differences in some cases, which may be attributed to correlation effects neglected in the NLPB theory.     

Average electrostatic potential between the filaments in striated muscle and its relation to a simple Donnan potential.

An approximate analytical solution to the Poisson-Boltzmann equation for a cylindrical particle was used to calculate the relationship between the charge on the filaments and the average electrostatic potential. Both thick and then filaments were considered in the muscle lattice with a filament charge ratio of 4 to 1. Comparing this with a similar relationship obtained using simple Donnan theory showed a discrepancy at high charge where the Poisson-Boltzmann equation leads to saturation of the average potential. However, using two separate experiments from the literature, we have shown that at pH 7.0 muscle must not be close to saturation and thus is in a region of the curve where the two approaches agree.     

Donnan potential and surface potential of a charged membrane.

A model is presented for the electrical potential distribution across a charged biological membrane that is in equilibrium with an electrolyte solution. We assume that a membrane has charged surface layers of thickness d on both surfaces of the membrane, where the fixed charges are distributed at a uniform density N within the layers, and that these charged layers are permeable to electrolyte ions. This model unites two different concepts, that is, the Donnan potential and the surface potential (or the Gouy-Chapman double-layer potential). Namely, the present model leads to the Donnan potential when d much greater than 1/k' (k' is the Debye-Hückel parameter of the surface charge layer) and to the surface potential as d----0, keeping the product Nd constant. The potential distribution depends significantly on the thickness d of the surface charge layer when d less than or approximately equal to 1/k'.     

A self-consistent calculation of the free energy and electrostatic potential for a cylindrical polyion

The linearized Poisson-Boltzmann equation is solved for a cylindrical polyion immersed in an ionic solution of specified pH and ionic strength. The boundary condition at the surface of the cylinder is determined self-consistently, so that the only input required is the density of ionizable groups on the cylinder surface and their dissociation characteristics. An expression is also derived for the free energy of the system and it is shown that the degree of dissociation calculated via the self-consistent boundary condition yields the minimum value of the free energy. Calculations are presented for parameters that are relevant to several systems of biological interest and the response of the system to changes in pH and ionic strength is discussed in detail.     

Comparison of Poisson-Boltzmann and condensation model expression for the colligative properties of cylindrical polyions

Closed-form expression have been derived for the polyelectrolyte contribution to the colligative properties of solutions containing rodlike polyions in the presence of excess added salt. The derivations are based on: the conventional Poisson-Boltzmann equation for cylindrical symmetry; the thermodynamics of the cell model developed by Marcus [J. Chem. Phys. 23 , 1057–1068 (1955)]; and an equation derived from the cylindrical Poisson-Boltzmann cell model by Anderson and Record [Biophys. Chem. 11 , 353–360 (1980)]. Subject to the inherent limitations of the Poisson-Boltzmann approximation [Fixman (1979) J. Chem. Phys. 70 , 4995–5005], the resulting expressions are nevertheless applicable outside the “limit of infinite dilution.” They conform over a range of salt concentrations to the limiting laws deduced by Manning from the hypothesis of counterion condensation [J. Chem. Phys. 51 , 924–933 (1969)]. This hypothesis is found to be compatible with the Poisson-Boltzmann cell model but is not required in the derivation of the thermodynamic coefficients presented here. It is demonstrated that the magnitude of the polyion axial charge density plays a critical role in determining the low-salt limiting forms of the colligative properties obtained from the Poisson-Boltzmann equation, in close analogy with Manning's model.     

A new radiochemical method to investigate ion binding with polyelectrolytes

A new method for investigating the binding of ions with polyelectrolytes has been developed. This method, based on Donnan equilibrium and an isotope exchange between the electrolyte and polyelectrolyte, can distinguish territorial from specific binding of ions and can determine fractions of ions bound with the polyion. This method can determine ion binding with polyelectrolytes in a wide range of polyelectrolyte concentrations in multicomponent solutions. The method was tested with radioactive tracers 22Na+, 36Cl- and heparin sodium salt. The influence of the ionic strength on the Na+ binding with heparin was investigated at 310 K. In the limit of zero ionic strength, all Na+ ions are bound to heparin, but only 45% of them are exchangeable. Thus Na+ ions can be bound both territorially and specifically. The fraction of bound ions decreases rapidly with increasing ionic strength. The fraction of the specifically bound ions becomes negligible when the ionic strength exceeds 0.01 M, whereas the fraction of territorially bound ions can be neglected at ionic strengths higher than 0.45 M.     

Contact pairs of RNA with magnesium ions-electrostatics beyond the Poisson-Boltzmann equation

The electrostatic interaction of RNA with its aqueous environment is most relevant for defining macromolecular structure and biological function. The attractive interaction of phosphate groups in the RNA backbone with ions in the water environment leads to the accumulation of positively charged ions in the first few hydration layers around RNA. Electrostatics of this ion atmosphere and the resulting ion concentration profiles have been described by solutions of the nonlinear Poisson-Boltzmann equation and atomistic molecular dynamics (MD) simulations. Much less is known on contact pairs of RNA phosphate groups with ions at the RNA surface, regarding their abundance, molecular geometry, and role in defining RNA structure. Here, we present a combined theoretical and experimental study of interactions of a short RNA duplex with magnesium (Mg²⁺) ions. MD simulations covering a microsecond time range give detailed hydration geometries as well as electrostatics and spatial arrangements of phosphate-Mg²⁺ pairs, including both pairs in direct contact and separated by a single water layer. The theoretical predictions are benchmarked by linear infrared absorption and nonlinear two-dimensional infrared spectra of the asymmetric phosphate stretch vibration which probes both local interaction geometries and electric fields. Contact pairs of phosphate groups and Mg²⁺ ions are identified via their impact on the vibrational frequency position and line shape. A quantitative analysis of infrared spectra for a range of Mg²⁺-excess concentrations and comparison with fluorescence titration measurements shows that on average 20–30% of the Mg²⁺ ions interacting with the RNA duplex form contact pairs. The experimental and MD results are in good agreement. In contrast, calculations based on the nonlinear Poisson-Boltzmann equation fail in describing the ion arrangement, molecular electrostatic potential, and local electric field strengths correctly. Our results underline the importance of local electric field mapping and molecular-level simulations to correctly account for the electrostatics at the RNA-water interface.     

Dielectric increment in polyion solutions due to the distortion of counterion distribution

A theory of DC dielectric increment, delta, has been constructed based on the Fokker- Planck and Poisson equations on the mechanism of the distortion of spatial counterion distribution around a rodlike polyion under the applied electric field, E, perpendicular to the polyion lod. The DC dielectric constant of polyion solutions can be obtained analytically in the both cases of absence and presence of counterion-flow. The nonlinearity of the Poisson-Boltzmann equation in the absence of E has given the interesting behaviors of delta. The calculation in the absence of counterion-flow has shown the delta-values having the same order of magnitude as the experimental data at the higher frequency range (around 100 kHz), and the importance of this mechanism has been pointed out.     

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.