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.     

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.     

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.     

Application of the Poisson Boltzmann polyelectrolyte model for analysis of thermal denaturation of DNA in the presence of Na+ and polyamine cations

The Poisson Boltzmann (PB) cell model of polyelectrolyte solution has been used for calculation of the electrostatic free energy difference, Delta G(el), between double- and single-stranded DNA. The calculations have been performed for conditions relevant to describe the DNA helix-coil transition in NaCl solution in the presence of the natural polyamines putrescine(2+), spermidine(3+), spermine(4+) and their synthetic homologs with different spacing between the charged amino groups, for which experimental values of the DNA 'melting' transition temperature (T(m)) are available. Using the PB theory and the polyamine ion radius as an adjusting parameter provides quantitative agreement between experimental and theoretical T(m)--salt concentration dependencies only by using physically unreasonable radii for the polyamine. Thus, modeling the linear and flexible polyamines as charged spheres within the PB cell model is an implausible oversimplification. We propose another explanation for the experimental observations, still within the frame of the 'primitive' PB polyelectrolyte theory. This explanation is based on an analysis of the Delta G(el) dependence on the stoichiometry of polyamine-polyanion binding to double- and single-stranded DNA.     

Cooperative interaction of the C-terminal domain of histone H1 with DNA

We have studied the interaction of the isolated C-terminal domain of histone H1 with linear DNA using precipitation curves and electron microscopy. The C-terminal domain shows a salt-dependent transition towards cooperative binding, which reaches completion at 60 mM NaCl. At this salt concentration, the C-terminal domain binds to some of the DNA molecules, leaving the rest free. A binding site of 22 base-pairs can be calculated from the stoichiometry of the precipitated fractions. The C-terminal domain condenses the DNA in toroidal particles. The average inner radius of the particles is of the order of 195 A. Consideration of the value of the inner radius of the toroids in the light of counterion condensation theory suggests that in these complexes the isolated C-terminal domain is capable of nearly full electrostatic neutralization of the DNA phosphate charge.     

Chromatin ionic atmosphere analyzed by a mesoscale electrostatic approach

Characterizing the ionic distribution around chromatin is important for understanding the electrostatic forces governing chromatin structure and function. Here we develop an electrostatic model to handle multivalent ions and compute the ionic distribution around a mesoscale chromatin model as a function of conformation, number of nucleosome cores, and ionic strength and species using Poisson-Boltzmann theory. This approach enables us to visualize and measure the complex patterns of counterion condensation around chromatin by examining ionic densities, free energies, shielding charges, and correlations of shielding charges around the nucleosome core and various oligonucleosome conformations. We show that: counterions, especially divalent cations, predominantly condense around the nucleosomal and linker DNA, unburied regions of histone tails, and exposed chromatin surfaces; ionic screening is sensitively influenced by local and global conformations, with a wide ranging net nucleosome core screening charge (56-100e); and screening charge correlations reveal conformational flexibility and interactions among chromatin subunits, especially between the histone tails and parental nucleosome cores. These results provide complementary and detailed views of ionic effects on chromatin structure for modest computational resources. The electrostatic model developed here is applicable to other coarse-grained macromolecular complexes.     

Limiting-laws of polyelectrolyte solutions. Ionic distribution in mixed-valency counterions systems. I: The model

An extension of the counterion-condensation (CC) theory of linear polyelectrolytes has been developed for the case of a system containing a mixture of counterions of different valency, i and j. The main assumption in the derivation of the model is that the relative amount of the condensed counterions of the type i and j is strongly correlated and it is determined by the overall physical bounds of the system. The results predicted by the model are consistent, in the limiting cases of single species component, with those of the original CC theory. The most striking results are obtained for the cases of low charge density and excess of counterion species: in particular, an apparent positive "binding" cooperativity of divalent ions is revealed for small, increasing additions of M2+ ions to a solution containing a swamping amount of monovalent salt and a polyelectrolyte of low charge density. Apparent "competitive binding" of mono- and divalent ions derives as a bare consequence of the electrostatic interactions. Theoretical calculations of experimentally accessible quantities, namely single-(counter) ion activity coefficients, confirm the surprising predictions at low charge density, which qualitatively agree with the measured quantities.     

Fluorescence energy transfer in one dimension: Frequency-domain fluorescence study of DNA–fluorophore complexes

The theory for the salt dependence of the free energy, entropy, and enthalpy of a polyelectrolyte in the PB (PB) model is extended to treat the nonspecific salt dependence of polyelectrolyte–ligand binding reactions. The salt dependence of the binding constant (K) is given by the difference in osmotic pressure terms between the react ants and the products. For simple 1-1 salts it is shown that this treatment is equivalent to the general preferential interaction model for the salt dependence of binding [C. Anderson and M. Record (1993) Journal of Physical Chemistry, Vol. 97, pp. 7116–7126]. The salt dependence, entropy, and enthalpy are compared for the PB model and one specific form of the preferential interaction coefficient model that uses counterion condensation/limiting law (LL) behavior. The PB and LL models are applied to three ligand–polyelectrolyte systems with the same net ligand charge: a model sphere–cylinder binding reaction, a drug–DNA binding reaction, and a protein–DNA binding reaction. For the small ligands both the PB and limiting law models give (ln K vs. In [salt]) slopes close in magnitude to the net ligand charge. However, the enthalpy/entropy breakdown of the salt dependence is quite different. In the PB model there are considerable contributions from electrostatic enthalpy and dielectric (water reorientation) entropy, compared to the predominant ion cratic (release) entropy in the limiting law model. The relative contributions of these three terms in the PB model depends on the ligand: for the protein, ion release entropy is the smallest contribution to the salt dependence of binding. The effect of three approximations made in the LL model is examined: These approximations are (1) the ligand behaves ideally, (2) the preferential interaction coefficient of the polyelectrolyte is unchanged upon ligand binding, and (3) the polyelectrolyte preferential interaction coefficient is given by the limiting law/counterion-condensation value. Analysis of the PB model shows that assumptions 2 and 3 break down at finite salt concentrations. For the small ligands the effects on the slope cancel, however, giving net slopes that are similar in the PB and LL models, but with a different entropy/enthalpy breakdown. For the protein ligand the errors from assumptions 2 and 3 in the LL model do not cancel. In addition, the ligand no longer behaves ideally due to its complex structure and charge distribution. Thus for the protein the slope is no longer related simply to the net ligand charge, and the PB model gives a much larger slope than the LL model. Additionally, in the PB model most of the salt dependence of the protein binding comes from the change in ligand activity, i.e. from nonspecific anion effects, in contrast to the small ligand case. While the absolute binding is sensitive to polyelectrolyte length, little length effect is seen on the salt dependence for the small ligands at 0.1M salt, and for lengths > 60 Å. Almost no DNA length dependenceis seen in the salt dependence of the protein binding, since this is determined primarily by the protein, not the DNA. © 1995 John Wiley & Sons, Inc.     

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.     

Atomic-scale structure and electrostatics of anionic palmitoyloleoylphosphatidylglycerol lipid bilayers with Na+ counterions

Anionic palmitoyloleoylphosphatidylglycerol (POPG) is one of the most abundant lipids in nature, yet its atomic-scale properties have not received significant attention. Here we report extensive 150-ns molecular dynamics simulations of a pure POPG lipid membrane with sodium counterions. It turns out that the average area per lipid of the POPG bilayer under physiological conditions is approximately 19% smaller than that of a bilayer built from its zwitterionic phosphatidylcholine analog, palmitoyloleoylphosphatidylcholine. This suggests that there are strong attractive interactions between anionic POPG lipids, which overcome the electrostatic repulsion between negative charges of PG headgroups. We demonstrate that interlipid counterion bridges and strong intra- and intermolecular hydrogen bonding play a key role in this seemingly counterintuitive behavior. In particular, the substantial strength and stability of ion-mediated binding between anionic lipid headgroups leads to complexation of PG molecules and ions and formation of large PG-ion clusters that act in a concerted manner. The ion-mediated binding seems to provide a possible molecular-level explanation for the low permeability of PG-containing bacterial membranes to organic solvents: highly polar interactions at the water/membrane interface are able to create a high free energy barrier for hydrophobic molecules such as benzene.     

Interactions of mono- and divalent counterions with alkali- and enzyme-deesterified pectins in salt-free solutions

The free fractions of monovalent and divalent counterions were determined on salt-free solutions of pectins. The effects of charge density, distribution of the carboxyl groups, polymer concentration, and the nature of the counterion were investigated by determinating the calcium and sodium activity coefficients (with specific electrodes) and by measuring the transport parameters (by conductimetry). Poor agreement for calcium ions was found with the Manning theory. The strong binding of these ions to highly charged polymers, which is ascribed to a dimerization process was demonstrated in very dilute solutions.     

Effective water model for Monte Carlo simulations of proteins

We present an effective theory for water. Our goal is to formulate on accurate model for the effects of solvation on protein dynamics, without incurring the huge computational cost and the slow temporal evolution typical of molecular dynamics simulations of liquids. We replace the individual water molecules in an all-atom potential with a local dielectric density field, with self interactions given by the Landau-Ginzburg free energy and external interactions by Lennard-Jones forces at the surface of the protein atoms. We explore conformational space with finite temperature Monte Carlo dynamics, using parallel Langevin and Fourier acceleration algorithms well suited to data-parallel computer architectures such as the Connection Machine. To establish the validity of our approximations, we compare our electrostatic contribution to the solvalion energy with the results of Lim, Bashford, and Karplus using a conventional static continuum dielectric cavity model, and the non electrostatic contributions with estimates of hydrophohic surface free energy. Our model can also accommodate ionic charges and temperature fluctuations, We propose future investigations extending our effective theory of solvation to include explicit orientational entropy and hydroxen-bonding terms. © 1995 John Wiley & Sons, Inc.     

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.