Monte-Carlo - Self Consistent Field Method in the Polyelectrolyte Theory

Abstract

A new time saving numerical method for calculation of equilibrium potential and density distribution of mobile ions around the polyion in a polyelectrolyte system is proposed: the region around the polyion is being divided into two zones - internal and external; in the internal zone all the ions are accounted explicity with the aid of Monte-Carlo procedure; in the external zone the combined Monte-Carlo - self consistent field method proposed earlier is applied, an exchange of ions between regions is being implied. For 1:1 electrolyte the optimal choice of the boundary between the zones has been demonstrated. As an example of a more complicated system calculation for 2:2:1:1 electrolyte was carried out.     

Monte Carlo-Self Consistent Field Study of the Symmetrical Models of Polyelectrolytes

Abstract

Time saving procedures unifying Monte Carlo and self consistent field approaches for the calculation of equilibrium potentials and density distributions of mobile ions around a polyion in a polyelectrolyte system are considered. In the final version of the method the region around the polyion is divided into two zones—internal and external; all the ions of the internal zone are accounted for explicitly in a Monte Carlo procedure, in the external zone the self consistent field approximation is applied with an exchange of ions between regions. Simulations are carried out for cylindrical and spherical polyions in solutions with mono-and divalent ions and their mixtures. The results are compared with Poisson—Boltzmann approximation and experimental data on intrinsic viscosity.     

Conductometric analysis of the competition between monovalent and divalent counterions in their interaction with polyelectrolytes

A procedure is described for the analysis of the conductivity of solutions of anionic polyelectrolytes in which both mono- and divalent counterions are present. The method is based on analysis of the relation between the overall conductivity of the system and the conductivity of the individual monovalent cations which are only electrostatically (non-specifically) bound. The system is described in terms of the two-state approach, implying that the counterions are considered to be either fully bound to the polyion or completely free. The potentialities of the proposed method are explored by studying solutions of alkali polyacrylates with and without added zinc nitrate at several alkali nitrate concentrations. The results give a picture of the composition of the counterionic atmosphere around the polyion in systems with both mono- and divalent counterions present. To a certain degree, the divalent ion Zn(II) was found to be bound quantitatively by the polyion. The composition of the counterionic atmosphere around the polyion was largely independent of alkali nitrate concentration when the latter was present in not too large an excess with respect to both Zn(II) and the charged monomers.     

Thermodynamics of polyelectrolyte solutions. An empirical extension of the manning theory to finite salt concentrations

The range of application of the Manning theory of polyelectrolyte solutions (J. Chem. Phys., 51 , 924 (1969)) is extended to finite concentrations of simple electrolyte by the empirical superposition of excess free energies arising from (i) interactions between mobile ions and polyions and (ii) mutual interactions between mobile ions. A comparison of published results with the modified theory shows excellent agreement over a wide range of concentrations. Results for a polyelectrolyte of low charge density suggest that the effective inter-charge spacing may be less than that of the fully extended polyion.     

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.     

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.     

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.     

Interaction of planar double layers in the modified Gouy-Chapman approximation

The force between two charged planar surfaces containing an electrolyte solution is calculated. The calculation is done for a 1-1 electrolyte with size-asymmetric ions using a Modified Gouy-Chapman theory. It is shown that at least part of the explanation for the sharp rise in the force between charged surfaces at small separations seen in experimental data may be related to finite-sized ion effects in the double layer. An interesting effect of size-asymmetric ions is the prediction of a force between uncharged surfaces.     

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.     

Donnan membrane equilibrium is not directly applicable to distributions of ions and water in gels or cells

Equilibration of ions and water with a charged gel does not follow the simple equations of the classical Gibbs-Donnan membrane equilibrium. Partition of ions between the gel and the external solution show specific effects, which require that activity coefficients are different in the two compartments. Highly hydrated ions, such as Na⁺ and H⁺ are accumulated into the gel water, whereas less highly hydrated ions, such as K⁺ and NH₄⁺ accumulate in the external water. This selectivity is the obverse of that found for gels containing low-density, expanded water. Water in a charged gel equilibrated with solutions of MgCl₂ was found to be more dense than bulk water at the same temperature. It is proposed that gels imbibe water to maximize the entropy of the system. Ions and water then equilibrate under those constraints. The chemical potential of water in the two compartments equalizes by an increase in density in the compartment of higher osmolality (the charged gel) and a decrease in density in the compartment of lower osmolality (the external solution). Electrolytes equilibrate so that macroscopic electroneutrality is conserved, and the chemical potential of an electrolyte is the same in each compartment. Because activity coefficients are different in the two compartments this results in asymmetric distributions of ions.Because real gels usually contain both charged and hydrophobic regions of surface, populations of water molecules of different density coexist even in very small pores. This accounts for the common failure to detect this phenomenon experimentally.     

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.     

Electrostatic Potential of B-DNA: Effect of Interionic Correlations

Modified Poisson-Boltzmann (MPB) equations have been numerically solved to study ionic distributions and mean electrostatic potentials around a macromolecule of arbitrarily complex shape and charge distribution. Results for DNA are compared with those obtained by classical Poisson-Boltzmann (PB) calculations. The comparisons were made for 1:1 and 2:1 electrolytes at ionic strengths up to 1 M. It is found that ion-image charge interactions and interionic correlations, which are neglected by the PB equation, have relatively weak effects on the electrostatic potential at charged groups of the DNA. The PB equation predicts errors in the long-range electrostatic part of the free energy that are only ∼1.5 kJ/mol per nucleotide even in the case of an asymmetrical electrolyte. In contrast, the spatial correlations between ions drastically affect the electrostatic potential at significant separations from the macromolecule leading to a clearly predicted effect of charge overneutralization.     

Translocation of polysialic acid across model membranes: kinetic analysis and dynamic studies.

Transmembrane translocation of polyion homopolymers takes place in the case of polyanionic polysialic acid (polySia), polyanionic polynucleotides and polycationic polypeptides. The purpose of this work was to determine the role of membrane electrical parameters on the kinetics of polyion translocation, the influence of polysialic acid on ion adsorption on positively charged membrane surface and the dynamics of the phospholipid hydrocarbon chains and choline group by using 1H-NMR. The analysis of polyion translocation was performed by using the electrical equivalent circuit of the membrane for the initial membrane potential equal to zero. The changes in polysialic acid flux was up to 75% after 1 ms in comparison with the zero-time flux. Both a decrease of membrane conductance and an increase of polyion chain length resulted in the diminution of this effect. An increase of praseodymium ions adsorption to positively charged liposomes and an increase of the rate of segmental movement of the -CH2 and -CH3 groups, and the choline headgrup of lipid molecules, was observed in the presence of polySia. The results show that the direction of the vectorial polyion translocation depends both on the membrane electrical properties and the degree of polymerization of the polymer, and that polysialic acid can modulate the degree of ion adsorption and the dynamics of membrane lipids.     

Polyelectrolyte properties of biopolymers: conductivity and secondary structure of polyriboadenylic acid and its salts in solutions

Polyriboadenylates of alkali metals were obtained from (1) K(+)-poly(A) (salts 1) and (2) H(+)-poly(A) (salts II) by the ion-exchange method. The conductivity of these salts as well as of H(+)-poly(A) were studied. Salts I and II of the same counterion were shown to have significantly different conductivity coefficients (f) and polyion conductances (lambda 0p). the charge density parameter (xi) was 1.3 and 2.5, respectively, with lambda 0p equal to 44 and 83 ohm-1 cm2 mole-1 for poly(A)-I and poly(A)-II salts, respectively. This is credited to the difference in the conformations of corresponding polyions. The linear dependence of equivalent conductivity on the square root of polymer concentration (Kohlrausch coordinates), earlier obtained for DNA, is also satisfied for the studied polynucleotides. A comparison of the slopes of straight lines in Kohlrausch coordinates for poly(A), simple electrolytes, and for earlier studied polyribouridylic acid salts lends credence to the concepts, developed by a number of authors, that DNA can act as a "buffer" against the ion-ion interaction in concentrated electrolyte solutions. Using the approximation that the polyion conductance is independent of the counterion nature, parameter f (agreeing in this case with Eisenberg parameter phi) has been shown to decrease as the polynucleotide concentration is increased; the decrease is caused by the relaxation effect. The transference numbers of counterions, which have negative values in poly (A)-II solutions, grow with the increase in polymer concentration; the higher the xi, the more apparent is this increase. This is explained by the increase in the fraction of conductivity along the polyion chains ("surface" conductivity) with the growth of polyelectrolyte concentration.     

Electroviscosity of polyelectrolyte solutions

A theoretical expression for the electroviscous effect in polyelectrolyte solutions, caused by the distortion of counterion-distribution and counterion flow around a polyion under a velocity gradient of solvent flow, was obtained to elucidate the characteristic behaviour of the viscosity of highly charged polyelectrolyte solutions observed at low salt concentration. The derivation of the theory was performed on the basis of the Navier-Stokes-Onsager equation, Poisson equation, and diffusion equations for low molecular ions by the use of a cell model (free-volume model) for a polyion. Energy dissipation was obtained without directly solving these equations. It was found that the derived expression of viscosity explained the experimental results satisfactorily, and that the streaming potential effect caused by the counterion flow played an essential role in the increase in viscosity of polyelectrolyte solutions at finite polymer concentration and low salt concentration ranges.     

The relationship between the poisson-boltzmann model and the condensation hypothesis: an analysis based on the low salt form of the Donnan coefficient

Two common models for the interaction of counterions with cylindrical polyions are considered in the context of the Donnan membrane equilibrium. General analytic expressions are obtained from the Poisson-Boltzmann equation for the Donnan coefficient in terms of the potential at the surface of the polyion or the local concentration of unbound ions at the surface. Analysis based on these expressions shows that if, and only if, the polyion charge density exceeds a certain critical value a large local concentration of ions will persist near the polyion surface at low ionic strengths. We therefore conclude that this principal hypothesis of the condensation model is consistent with the characteristics of the Poisson-Boltzmann potential at the surface of the polyion.     

Quasielastic light scattering by biopolymers. VI. Diffusion of mononucleosomes and oligonucleosomes in the presence of static and sinusoidal electric fields

Quasielastic light scattering and electrophoretic light scattering studies were carried out on mononucleosome and oligonucleosome systems. The electrophoretic light scattering experiments employed static and sinusoidal electric fields. Data are presented that suggest at least two relaxation modes. It is proposed that the small amplitude sinusoidal field effectively polarizes the ion atmosphere about the polyion, thus leading to an induced dipole moment that varies sinusoidally in time. This model is, in essence, an extension of the current interpretation of low-frequency dielectric dispersion data on DNA as being due to fluctuations of counterions along the polyion.     

Monte Carlo simulation of the double layer at an electrode including the effect of a dielectric boundary

Monte Carlo values of the density profiles and related properties of the double layer formed by an electrolyte near a charged electrode are reported for the cases where the electrode has a dielectric coefficient greater, equal, and smaller than that of the electrolyte that causes a surface polarization that can be represented by electrostatic images. As expected, compared to the case where there is no dielectric boundary the ions near the electrode are attracted or repelled by the electrode if the dielectric coefficient is greater or smaller, respectively, than that of the electrolyte. This effect is most pronounced near the electrode and is stronger for 2:2 electrolytes than for 1:1 electrolytes. For both monovalent and divalent ions the effect of the dielectric boundary is stronger at low concentrations.     

Condensation of Co++ ions on chondroitin sulfate from transport coefficients and proton NMR measurements in aqueous solutions

The “condensed” counterions which characterize high-charge-density polyelectrolyte solutions can be analyzed into two subpopulations: (1) site-bound counterions and (2) atmospherically entrapped counterions. The distinction is achieved experimentally by combining the data from self-diffusion coefficient or electrical mobility measurements, which give the amount of “condensed” ions, and those from nmr, chemical shift measurements, which indicate the amount of site-bound ions. In the case of a solution of chondroitin sulfate with excess Co⁺⁺ counterions, it can be estimated that 20% of the structural charge of the polyion is neutralized by site-bound, dehydrated, condensed counterions, while a further 30% is neutralized by atmospherically entrapped, hydrated counterions.