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.     

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

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

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

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

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

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

A new model of counterion condensation in polyelectrolyte solutions. III. Theoretical predictions of competitive condensation between counterions of different valences

Our model has been extended for theoretical estimation of competitive condensation of counterions of different valences onto polyelectrolytes in solution. The estimations are compared with those obtained from Manning theory and with experimental data on counterion activity coefficients. The agreement with the data for sodium polystyrenesulfonate/MgCl2, CaCl2 is satisfactory.     

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

Abstract

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

The release of monovalent counterions by addition of divalent countemons to aqueous solutions of maleic acid copolymer

Divalent cation binding and the release of monovalent cations accompanying the cation binding were experimentally studied by ion-selective electrode methods in aqueous solutions of copolymer of maleic acid and ethyl vinyl ether. It was found that in the process of Ca2+ addition, all the Ca2+ added was bound to polyions and the initially condensed Na+ was released in proportion to the concentration of the added Ca2+ up to the critical concentration of added Ca2+ at which the condensation of Ca2+ ceases. Values of the structural charge density parameter xi(s), were determined from the end-points of condensation of Ca2+. The process of Na+ release by adding Ca2+ was analyzed on the basis of the counterion condensation theory by using these xi(s) values. In addition, the relationship between the activity coefficient gamma-- of Ca2+ and degree of neutralization alpha in salt-free solutions was obtained from the Manning theory. Agreement between the calculated and experimental values was excellent in both cases.     

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.     

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.     

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

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

The osmotic coefficients of the sodium form of some biopolymers

The osmotic coefficients phi p,Na of dilute solutions of the sodium form of some weakly acidic polymers are theoretically predicted in this work. Based on the measured value 0.73 of gamma Na, the activity coefficient of free Na+, of the completely ionized humic acid (sodium salt) in a salt-free solution, the effective interligand distance b is calculated to be 11.34 A by using Manning's counterion condensation theory [Manning, G. S. (1969) J. Chem. Phys. 51(3), 924]. The corresponding values of gamma Na (measured experimentally) and b for the completely ionized exopolymer of Pseudomonas atlantica are 0.624 and 7.57 A when cultivated at a dilution rate D = 0.015 h-1, 0.647 and 8.19 A at D = 0.025 h-1, and 0.613 and 7.29 Aat D = 0.06 h-1. For alginic acid (in the completely ionized sodium form), gamma Na = 0.40 and b = 4.71 A. The osmotic coefficients phi p,Na for the partially and the completely ionized polymers are then predicted with Manning's theory as well.     

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.     

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.     

Interpreting protein/DNA interactions: distinguishing specific from non-specific and electrostatic from non-electrostatic components

We discuss the effectiveness of existing methods for understanding the forces driving the formation of specific protein-DNA complexes. Theoretical approaches using the Poisson-Boltzmann (PB) equation to analyse interactions between these highly charged macromolecules to form known structures are contrasted with an empirical approach that analyses the effects of salt on the stability of these complexes and assumes that release of counter-ions associated with the free DNA plays the dominant role in their formation. According to this counter-ion condensation (CC) concept, the salt-dependent part of the Gibbs energy of binding, which is defined as the electrostatic component, is fully entropic and its dependence on the salt concentration represents the number of ionic contacts present in the complex. It is shown that although this electrostatic component provides the majority of the Gibbs energy of complex formation and does not depend on the DNA sequence, the salt-independent part of the Gibbs energy--usually regarded as non-electrostatic--is sequence specific. The CC approach thus has considerable practical value for studying protein/DNA complexes, while practical applications of PB analysis have yet to demonstrate their merit.     

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.     

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 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.     

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

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

A relationship between 13C-chemical-shift displacements and counterion-condensation theory, in the binding of calcium ion by heparin

Characteristics of the interaction between heparin and calcium ion in the presence of sodium ion have been examined by monitoring the 13C-chemical shift changes as a function of the calcium ion concentration and the total ionic strength. The results indicated that the association between the polyanion and the divalent cation is a delocalized process, as opposed to one involving specific binding. The correspondence found between chemical shift and the number of Ca2+ ions bound per charged group, as derived from the Manning counterion-condensation model, showed that the stoichiometry is not a constant quantity but, rather, varies throughout the titration, and approaches a limiting value of 2 at high dilution. Additional measurements of T1 and line-width were consistent with an intramolecular order-disorder conformational process induced by the binding of calcium ion. Moreover, binding does not occur or is relatively weak with N-desulfated heparin, or chondroitin 4-sulfate and 6-sulfate, each of which possesses fewer sulfate groups than heparin. These differences serve to emphasize the importance of the charge-density parameter in the control of counterion condensation according to the Manning model, and suggest that the spacing between the negatively charged groups is an associated factor.