Analysis of cooperativity and ion effects in the interaction of quinacrine with DNA

The interaction of quinacrine with calf thymus DNA was monitored at several different ionic strengths using spectrophotometric and equilibrium dialysis techniques. The binding results can be explained, assuming each base pair is a potential binding site, using a model containing two negative cooperative effects: (1) ligand exclusion at binding sites adjacent to a filled binding site and (2) ligand–ligand negative cooperativity at adjacent filled binding sites. The logarithm of the observed equilibrium constant (K_obs) determined by this model varies linearily with log[Na⁺], as predicted by the ion condensation theory for polyelectrolytes. When the log K_obs plot is correlated for sodium release by DNA in the intercalation conformational change, the predicted number of ion pairs between the ligand and DNA is approximately two, as expected for the quinacrine dication. Even though K_obs depends strongly on ionic strength, the ligand negative cooperativity parameter ω was found to be indpendent of ionic strength within experimental error. This finding is also in agreement with the ion condensation theory, which predicts a relatively constant amount of condensed counterion on the DNA double helix over this ionic strength range. Drugs would, therefore, experience a relatively constant ionic environment when complexed to DNA even though the ionic conditions of the solvent could change considerably.     

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.     

Approach to the limit of counterion condensation

According to counterion condensation theory, one of the contributions to the polyelectrolyte free energy is a pairwise sum of Debye-Hückel potentials between polymer charges that are reduced by condensed counterions. When the polyion model is taken as an infinitely long and uniformly spaced line of charges, a simple closed expression for the summation, combined with entropy-derived mixing contributions, leads to the central result of the theory, a condensed fraction of counterions dependent only on the linear charge density of the polyion and the valence of the counterion, stable against increases of salt up to concentrations in excess of 0.1 M. Here we evaluate the sum numerically for B-DNA models other than the infinite line of B-DNA charges. For a finite-length line there are end effects at low salt. The condensation limit is reached as a flat plateau by increasing the salt concentration. At a fixed salt concentration the condensation limit is reached by increasing the length of the line. At moderate salt even very short B-DNA line-model oligomers have condensed fractions not far from the infinite polymer limit. For a long double-helical array with charge coordinates at the phosphates of B-DNA, the limiting condensed fraction appears to be approached at low salt. In contrast to the results for the line of charges, however, the computed condensed fraction varies strongly with salt in the range of experimentally typical concentrations. Salt invariance is restored, in agreement with both the line model and experimental data, when dielectric saturation is considered by means of a distance-dependent dielectric function. For sufficiently long B-DNA line and helical models, as typical salt concentrations, the counterion binding fraction approaches the polymer limit as a linear function of 1/P, where P is the number of phosphate groups of B-DNA.     

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.     

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

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

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.     

Polyelectrolyte theory of charged-ligand binding to nucleic acids

The Poisson-Boltzmann framework provides simple and sound formulae for evaluation of the interactions of polynucleotides with charged ligands. The theory is based on the observation that shape-dependent effects are small. Hence the results for the charged plane may be used as a first appromixation for polyelectrolytes. The counterion distribution and the extent of counterion or ligand binding is derived on the basis of a sumrule for counterion concentrations, combined with the mass-action law. Calculations require no more than a pocket calculator, even in the case of mixed-salt solutions. Significant qualitative results are given, and applied to the problems of site-binding and of repressor-DNA interaction.     

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

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

Magnetic birefringence study of the electrostatic and intrinsic persistence length of DNA

Magnetic birefringence experiments were performed on solutions of DNA of intermediate molecular weight at several concentrations (c_p) over a wide range of ionic strengths (of NaCl and MgCl₂). The specific Cotton-Mouton constant (CM/c_p) is found to be independent of c_p when contributions from c_p to the ionic strength (μ_eff) are taken into account according to the concept of counterion condensation. For μ_eff ? 10^?2M, CM/c_p is also independent of the ionic strength; the plateau value results in an acceptable value of the intrinsic persistence length, when a revised theoretical expression for the magnetic birefringence of wormlike chains is used, combined with new experimental data for the monomeric optical and magnetic anisotropy. For μ_eff < 10^?2M, CM/c_p strongly, or wealky, increases with decreasing μ_eff, depending on the valency of the counterion used (Na⁺ or Mg²⁺, respectively). This increase agrees quantitatively with the variation of the electrostatic persistence length as predicted by Odijk [(1977) J. Polym. Sci. Polym. Phys. Ed. 15 , 477–483], Odijk and Houwaart [(1978) J. Polym. Sci. Polym. Phys. Ed. 16 , 627–639], and by Skolnick and Fixman [(1977) Macromolecules 10 , 944–948]. A comparison with other experimental data seems to reveal the importance of excluded-volume effects, which are particularly pronounced in the low-salt regime.     

Multiscale study of counterion-induced attraction and bundle formation of F-actin using an Ising-like mean-field model

An Ising-like counterion-binding model is developed and solved by a mean-field method. For G-actin, the calculated affinity constants of all the binding sites ranging from loose to tight binding match the experimental data. The model is used to calculate the interaction energy between two F-actin filaments. Within a certain counterion concentration range, a rapidly decaying attractive force between two parallel filaments is produced not only by the correlation of the counterion distributions on the two filaments, but also by the correlation of the configurations of the two filaments with fixed counterion positions, which has been ignored in previous calculations. The bundling energy depends strongly on the configuration of the filaments. Upon bundling, the tightly bound counterion site is not affected, but the medium and loosely bound ones are. The model reproduces the observed minimal divalent counterion concentration for bundling, and naturally predicts the resolubilization of bundles which is seen in recent experiments. At the optimal counterion concentration, we obtain a bundling energy of approximately -0.01 eV per monomer along the filament. The counterion valence strongly affects the optimal counterion concentration, but has only minor effects on the optimal bundling energy. We show that the attractive potential between filaments can be simplified as the sum of interactions between their monomers. This simplification makes it possible to calculate the exact free energy of a two-F-actin-filament system. We are thus able to probe the effects of filament length on F-actin bundling and obtain a critical length for bundling of 59 monomers at 1 microM monomer concentration and pH=7.2.     

Electric dichroism of DNA. Influence of the ionic environment on the electric polarizability

In order to test the diffuse ion atmosphere polarization model recently developed by us, the effects of ionic strength, titrating with Mg2+ and Co(NH3)3+6, and coion charge on the electric polarizability of short fragments of DNA are investigated. The results are consistent with the predictions of the theory and show that the diffuse ion atmosphere polarization contributes significantly to the overall orientation of DNA. At low ionic strengths, we attempt to separate the total dipole moment into two components: one that agrees well with the Debye-Hückel ion atmosphere calculations, while the other, presumably due to condensed counterion polarization, appears to be substantially independent of the ionic strength. At higher salt concentrations, however, a simple separation into dipole components is not possible, perhaps due to a significant coupling of ion flows between the diffuse atmosphere and the condensed counterion layer.     

Planar,cylindrical and spherical electrical double layers in biological systems. The effect of counterion size

The effect of counterion size on the electrical properties of an electrolyte solution in contact with charged planar, cylindrical and spherical surfaces is considered. Electrostatic interaction is considered by means of the mean electrostatic field, while the finite size of particles constituting the electrolyte solution is considered via the excluded volume effect within the lattice statistics. Different sizes of counterion are described by different values of the lattice constant. It is shown that the excluded volume effect considerably decreases the calculated number density of counterions near the charged surface. This effect is more pronounced in cylindrical geometry than in spherical geometry, and less pronounced than in planar geometry.     

Counter-Ion Condensation and System Dimensionality

Abstract

We review some of the characteristic properties of the structure of polyelectrolyte solutions: the condensed layer of counterions that forms abruptly at a critical threshold charge density on the polymer chain; the more diffuse Debye-Hückel cloud, which is spatially distinct from the condensed layer; and the entropie release of counterions from the condensed layer as a driving force for the binding of oppositely charged ligands. We present a reminder of the basis of our current understanding in a variety of experiments, simulations, and theories; and we attempt as well to clarify some misunderstandings. We present a new analysis of a lattice model that suggests why the limiting laws for polyelectrolyte thermodynamics have proved to be accurate despite the neglect of polymer-polymer interactions in their original derivation. We sketch recent progress in constructing a potential between counterion and polyion. A counterion located in the interface between condensed layer and Debye cloud is repelled from the polyion, creating a sharp boundary between the two counterion populations.     

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

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

Ligand/Receptor Interactions —The Influence of the Microenvironment on Macroscopic Properties. Electrostatic Interactions with the Membrane Phase

Abstract

The heterogeneous environment in which ligand/receptor interactions occur often leads to complex binding behaviour. We consider here the ligand/membrane interaction, emphasizing the possibilities of electrostatic modulation of the overall binding characteristics. The binding of Substance P to neutral or negatively charged planar lipid bilayers was monitored using the capacitance minimization technique. The electrostatic attraction to the charged bilayer potentiates the interaction by more than two orders of magnitude and leads to a nonlinearity in the Scatchard plot of bound vs. bulk concentrations. The Boltzmann accumulation factor, along with the direct measurement of the surface potential, provides an easy explanation of the effect. The general importance of electrostatic accumulation (or repulsion) at surfaces is discussed and the concept applied to examples from the literature.     

Specific interactions versus counterion condensation. 1. Nongelling ions/polyuronate systems

The characteristics of the interaction between nongelling divalent cations (typically Mg(2+)) and polyuronates have been explored by means of isothermal calorimetry. In particular, three polyuronates mimicking separately guluronan (polyguluronate, polyG), mannuronan (polymannuronate, polyM), and polyalternating (polyMG), the three block-components of natural alginate samples, have been treated with divalent ions, and the enthalpy of mixing was determined for different values of the [M(2+)]/[Polym](rep.unit) ratio. Despite the absence of a site-specific chemical bonding between the two, as confirmed by circular dichroism spectroscopy, a substantial deviation of the experimental enthalpy of mixing from the theoretical behavior, as predicted by the classical counterion condensation (CC) theory, was observed. Such deviation has been interpreted in terms of a "generic" nonbonding affinity of the condensed divalent counterion for the polyelectrolytes. The mathematical formalism of the CC theory was extended to include a contribution to the (reduced) free energy and enthalpy arising from the counterion affinity, g(aff,0) and h(aff,0), and allowed the parametrical calculation of the fraction of divalent counterions condensed as function of the reduced thermodynamic quantity g(aff,0). A best fit procedure of the experimental enthalpy of mixing allowed the g(aff,0) and h(aff,0) pair to be estimated for each of the different polyuronates considered, revealing differences in the three samples. In qualitative terms, the results obtained seem to suggest a notable contribution of the desolvation process (i.e., release of structured water as a consequence of the interaction between the divalent counterion and the uronate group) to the enthalpy of affinity for polyM which is counterbalanced and overcome by an ion pairing term (i.e., partial formation of ion-ion and/or ion-dipole bonds) for polyG and polyMG, respectively.     

Electrostatic effects on polynucleotide transitions. I. Behavior at neutral pH

An approximate analytical expression for the electrostatic free energy of a polynucleo-tide in any of its possible ordered or random conformations is derived by integration of the screened-Coulomb potential energy function over all charge pairs in the structure. The electrostatic free energy of any form is found to be a linear function of the logarithm of the monovalent counterion concentration, in the range of low salt concentrations. Hence the electrostatic free energy difference between ordered and disordered forms in a polynucleotide structural transition is a linear function of the logarithm of the monovalent counterion concentration. A free energy balance applied to a two-state model for the transition then yields a linear dependence of the transition temperature T_m upon the logarithm of the counterion concentration. Calculation of the quantity dT_m/d log M, where M is the monovalent counterion concentration, shows it to be a characteristic constant for a given transition, with a magnitude and sign proportional to the charge density difference between the ordered and disordered forms. Use of any one of several alternate, simple assumptions yields predicted dT_m/d log M values in good agreement with experimental data for various polynucleotide transitions.     

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.