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

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

Electrostatic effects on the stability of condensed DNA in the presence of divalent cations.

Cylindrical cell model Poisson-Boltzmann (P-B) calculations are used to evaluate the electrostatic contributions to the relative stability of various DNA conformations (A, B, C, Z, and single-stranded (ss) with charge spacings of 3.38 and 4.2 A) as a function of interhelix distance in a concentrated solution of divalent cations. The divalent ion concentration was set at 100 mM, to compare with our earlier reports of spectroscopic and calorimetric experiments, which demonstrate substantial disruption of B-DNA geometry. Monovalent cations neutralize the DNA phosphates in two ways, corresponding to different experimental situations: 1) There is no significant contribution to the ionic strength from the neutralizing cations, corresponding to DNA condensation from dilute solution and to osmotic stress experiments in which DNA segments are brought into close proximity to each other in the presence of a large excess of buffer. 2) The solution is uniformly concentrated in DNA, so that the neutralizing cations add significantly to those in the buffer at close DNA packing. In case 1), conformations with lower charge density (Z and ssDNA) have markedly lower electrostatic free energies than B-DNA as the DNA molecules approach closely, due largely to ionic entropy. If the divalent cations bind preferentially to single-stranded DNA or a distorted form of B-DNA, as is the case with transition metals, the base pairing and stacking free energies that stabilize the double helix against electrostatic denaturation may be overcome. Strong binding to the bases is favored by the high concentration of divalent cations at the DNA surface arising from the large negative surface potential; the surface concentration increases sharply as the interhelical distance decreases. In case 2), the concentration of neutralizing monovalent cations becomes very large and the electrostatic free energy difference between secondary structures becomes small as the interhelical spacing decreases. Such high ionic concentrations will be expected to modify the stability of DNA by changing water activity as well as by screening electrostatic interactions. This may be the root of the decreased thermal stability of DNA in the presence of high concentrations of magnesium ions.     

An electrostatic model of B-DNA for its stability against unwinding

In single molecule experiments Smith et al. (Science 271 (1996) 795) have unwound the B-DNA helix by stretching it in an aqueous salt solution. They found that the stretching force required for the transition decreases significantly with lowering of the salt concentration. We show that the observed salt effect is consistent with a uniformly charged cylinder model of DNA surrounded by a Poisson-Boltzmann ionic atmosphere. We also derive a simple connection between the sharpness of the center part of the transition and its cooperativity in terms of an average block size of base pairs that unwinds or rewinds.     

Comparison of hard-cylinder and screened coulomb interactions in the modeling of supercoiled DNAs

A 1000 base pair (bp) model supercoiled DNA is simulated using spherical screened Coulomb interactions between subunits on one hand and equivalent hard-cylinder interactions on the other. The amplitudes, or effective charges, of the spherical screened Coulomb electrostatic potentials are chosen so that the electrostatic potential surrounding the middle of a linear array of 2001 subunits (31.8 Å diameter) closely matches the solution of the nonlinear Poisson-Boltzmann equation for a cylinder with 12 Å radius and the full linear charge density of DNA at all distances beyond the 24 Å hard-core diameter. This superposition of spherical screened Coulomb potentials is practically identical to the particular solution of the cylindrical linearized Poisson-Boltzmann equation that matches the solution of the nonlinear Poisson-Boltzmann equation at large distances. The interaction energy between subunits is reckoned from the effective charges according to the standard DLVO expression. The equivalent hard-cylinder diameter is chosen following Stigter's protocol for matching second virial coefficients, but for the full linear charge density of DNA. The electrostatic persistence length of the model with screened Coulomb interactions is extremely sensitive to the (arbitrarily) chosen subunit length at the higher salt concentrations. The persistence length of the hard-cylinder model is adjusted to match that of the screened Coulomb model for each ionic condition. Simulations for a superhelix density σ = -0.05 using a spherical screened Coulomb interaction plus a 24 Å hard-cylinder core (SCPHC) potential indicate that the radius of gyration of this 1000 bp DNA actually undergoes a slight increase as the NaCl concentration is raised from 0.01 to 1.0M. Thus, merely softening the potential from hard-cylinder to screened Coulomb form does not produce a large decrease in radius of gyration with increasing NaCl concentration for DNAs of this size. Radii of gyration, static structure factors, and diffusion coefficients obtained using the equivalent hard-cylinder (EHC) potential agree well with those obtained using the SCPHC potential in 1.0M NaCl, but in 0.1M NaCl the agreement is not as good, and in 0.01M NaCl the agreement is definitely unsatisfactory. These conclusions differ in significant respects from those obtained in previous studies. © 1997 John Wiley & Sons, Inc. Biopoly 42: 455–470, 1997     

Application of polyelectrolyte theory to the study of the B-Z transition in DNA (1)

We have used the polyelectrolyte theory to study the ionic strength dependence of the B-Z equilibrium in DNA. A DNA molecule is molded as an infinitely long continuously charged cylinder of radius a with reduced linear charge density q. The parameters a and q for the B and Z forms were taken from X-ray data: aB = 1nm, qB = 4.2, aZ = 0.9 nm and qZ = 3.9. A simple theory shows that at low ionic strengths (when Debye screening length rD much greater than a) the electrostatic free energy difference FelBZ = FelZ - FelB increases with increasing ionic strength since qB greater than qZ. At high ionic strengths (when rD much less than a) the FelBZ would go on growing with increasing ionic strength if the inequality qB/aB greater than qZ/aZ were valid. In the converse case when qZ/qB greater than aZ/aB the FelBZ value decreases with increasing salt concentration at high ionic strength. Since X-ray data correspond to the latter case, theory predicts that the FelBZ value reaches a maximum at an intermediate ionic strength of about 0.1 M (where rD approximately a). We also performed rigorous calculations based on the Poisson-Boltzmann equation. These calculations have confirmed the above criterion of nonmonotonous behaviour of the FelBZ value as a function of ionic strength. Different theoretical predictions for the B-Z transition in linear and superhelical molecules are discussed. Theory predicts specifically that at a very low ionic strength the Z form may prove to be more stable than the B form.(ABSTRACT TRUNCATED AT 250 WORDS)     

Poisson-Boltzmann analysis of the lambda repressor-operator interaction.

A theoretical study of the ion atmosphere contribution to the binding free energy of the lambda repressor-operator complex is presented. The finite-difference form of the Poisson-Boltzmann equation was solved to calculate the electrostatic interaction energy of the amino-terminal domain of the lambda repressor with a 9 or 45 base pair oligonucleotide. Calculations were performed at various distances between repressor and operator as well as at different salt concentrations to determine ion atmosphere contributions to the total electrostatic interaction. Details in the distribution of charges on DNA and protein atoms had a strong influence on the calculated total interaction energies. In contrast, the calculated salt contributions are relatively insensitive to changes in the details of the charge distribution. The results indicate that the ion atmosphere contribution favors association at all protein-DNA distances studied. The theoretical number of ions released upon repressor-operator binding appears to be in reasonable agreement with experimental data.     

A unified theory of the B-Z transition of DNA in high and low concentrations of multivalent ions

We showed recently that the high-salt transition of poly[d(G-C)]. poly[d(G-C)] between B-DNA and Z-DNA (at [NaCl] = 2.25 M or [MgCl(2)] = 0.7 M) can be ascribed to the lesser electrostatic free energy of the B form, due to better immersion of the phosphates in the solution. This property was incorporated in cylindrical DNA models that were analyzed by Poisson-Boltzmann theory. The results are insensitive to details of the models, and in fair agreement with experiment. In contrast, the Z form of the poly[d(G-m5C)] duplex is stabilized by very small concentrations of magnesium. We now show that this striking difference is accommodated quantitatively by the same electrostatic theory, without any adjustable parameter. The different responses to magnesium of the methylated and nonmethylated polymers do not come from stereospecific cation-DNA interactions: they stem from an experimentally derived, modest difference in the nonelectrostatic component of the free energy difference (or NFED) between the Z and B forms. The NFED is derived from circular DNA measurements. The differences between alkaline earth and transition metal ions are explained by weak coordination of the latter. The theory also explains the induction of the transition by micromolar concentrations of cobalt hexammine, again without specific binding or adjustable parameters. Hence, in the case of the B-Z transition as in others (e.g., the folding of tRNA and of ribozymes), the effect of multivalent cations on nucleic acid structure is mediated primarily by nonspecific ion-polyelectrolyte interactions. We propose this as a general rule for which convincing counter-examples are lacking.     

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.     

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.     

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.     

Comparison of different approaches for calculation of polyelectrolyte free energy

We consider the problem of the mean field (Poisson-Boltzmann) calculation of the electrostatic free energy for a strongly charged polyelectrolyte such as DNA in a salt solution. We compare two approaches to calculate the free energy: (i) direct one starting from the statistical-mechanical expression for the electrostatic free energy and (ii) the polyion charge variation method. In the infinite dilution limit (in respect to polyion) and in excess salt (IDLES) the two approaches are fully equivalent. This is shown by straight forward algebra. We have performed specific calculations of the free energy difference for the case of B-Z transition in DNA as a function of ionic strength. As expected, the two approaches led to identical results. The ionic strength dependence of the B-to-Z free energy proves to be concaved up and as a result Z-DNA is stabilized at low ionic concentration as well as at high salt, in full agreement with our previous results (M.D.Frank-Kamenetskii et al., J. Biomol. Struct. Dyn. 3, 35-42 (1985]. Our data quantitatively agree with the results of Soumpasis (D.M.Soumpasis, J. Biomol. Struct. Dyn. 6, 563-574 (1988]. However, his claim about the absence of the effect of stabilization of Z-DNA at low salt proves to be groundless, and the criticism of our earlier approach seems to be irrelevant.     

Generalized Poisson-Boltzmann calculation of the distribution of electrolyte ions around the B- and Z-conformers of DNA

A solution to the three-dimensional Poisson-Boltzmann equation, generalized to include the finite size of the ions, is presented for the environment of DNA in the B- and Z-conformations. The results clearly indicate that despite the lower linear charge density of the left-handed Z-conformer, there is a higher concentration of Na⁺ at the immediate surface of the Z-from than at the surface of the B-form. The average concentration of counterions within a 12-Å radius of the DNA is, nonetheless, higher for the B- than for the Z-form. Calculations of the electrostatic interactions of these conformers with an environment of 0.01M monovalent salt show that the salt exerts a greater stabilizing effect on the left-handed conformer than on the right-handed form.     

Comparison of Different Approaches for Calculation of Polyelectrolyte Free Energy

Abstract

We consider the problem of the mean field (Poisson-Boltzmann) calculation of the electrostatic free energy for a strongly charged polyelectrolyte such as DNA in a salt solution. We compare two approaches to calculate the free energy: (i) direct one, starting from the statistical-mechanical expression for the electrostatic free energy and (ii) the polyion charge variation method. In the infinite dilution limit (in respect to polyion) and in excess salt (IDLES) the two approaches are fully equivalent. This is shown by straight forward algebra. We have performed specific calculations of the free energy difference for the case of B-Z transition in DNA as a function of ionic strength. As expected, the two approaches led to identical results. The ionic strength dependence of the B-to-Z free energy proves to be concaved up and as a result Z-DNA is stabilized at low ionic concentration as well as at high salt in full agreement with our previous results (M.D. Frank- Kamenetskii et al, J. Biomol. Struct. Dyn. 3, 35–42 (1985)). Our data quantitatively agree with the results of Soumpasis (D. M. Soumpasis, J. Biomol. Struct. Dyn. 6, 563–574 (1988)). However, his claim about the absence of the effect of stabilization of Z-DNA at low salt proves to be groundless, and the criticism of our earlier approach seems to be irrelevant.     

DAPI binding to the DNA minor groove: a continuum solvent analysis

A continuum solvent model based on the generalized Born (GB) or finite-difference Poisson-Boltzmann (FDPB) approaches has been employed to compare the binding of 4'-6-diamidine-2-phenyl indole (DAPI) to the minor groove of various DNA sequences. Qualitative agreement between the results of GB and FDPB approaches as well as between calculated and experimentally observed trends regarding the sequence specificity of DAPI binding to B-DNA was obtained. Calculated binding energies were decomposed into various contributions to solvation and DNA-ligand interaction. DNA conformational adaptation was found to make a favorable contribution to the calculated total interaction energy but did not change the DAPI binding affinity ranking of different DNA sequences. The calculations indicate that closed complex formation is mainly driven by nonpolar contributions and was found to be disfavored electrostatically due to a desolvation penalty that outbalances the attractive Coulomb interaction. The calculated penalty was larger for DAPI binding to GC-rich sequences compared with AT-rich target sequences and generally larger for the FDPB vs the GB continuum model. A radial interaction profile for DAPI at different distances from the DNA minor groove revealed an electrostatic energy minimum a few Angstroms farther away from the closed binding geometry. The calculated electrostatic interaction up to this distance is attractive and it may stabilize a nonspecific binding arrangement.     

Sodium and chlorine ions as part of the DNA solvation shell.

The distribution of sodium and chlorine ions around DNA is presented from two molecular dynamics simulations of the DNA fragment d(C(5)T(5)). (A(5)G(5)) in explicit solvent with 0.8 M additional NaCl salt. One simulation was carried out for 10 ns with the CHARMM force field that keeps the DNA structure close to A-DNA, the other for 12 ns with the AMBER force field that preferentially stabilizes B-DNA conformations (, Biophys. J. 75:134-149). From radial distributions of sodium and chlorine ions a primary ion shell is defined. The ion counts and residence times of ions within this shell are compared between conformations and with experiment. Ordered sodium ion sites were found in minor and major grooves around both A and B-DNA conformations. Changes in the surrounding hydration structure are analyzed and implications for the stabilization of A-DNA and B-DNA conformations are discussed.     

Calculation of electrostatic effects at the amino terminus of an alpha helix.

It is generally believed that the electrostatic field arising from the dipolar charge distribution in alpha helices is important for protein structure and function. We report a calculation of the electrostatic potential and field at the amino terminus of an alpha helix in water, obtained from a finite difference solution to the Poisson-Boltzmann equation. This method takes into account the detailed helix shape and charge distribution, as well as solvent, and generalized ionic strength effects. The calculated potential and field are found to be in good agreement with the experimentally observed helix-induced Stark effect and pKa shifts of a probe at the N-terminus of a stable, monomeric alpha-helical peptide (Lockhart and Kim, 1992, 1993). Ionic screening effects are reproduced at low salt concentrations. Deviations at higher salt concentrations may result from specific ion effects (specific ion-solute and/or ion-solvent interactions). The FDPB method was used to analyze the contributions from each residue, charged side chains, and solvent to the helix potential and field. Backbone contributions come primarily from the first one to two helical turns. Charged side chains contribute to helix-induced pKa shifts for certain probe-peptide combinations, even at relatively large distances from the probe (> 14 A).     

A Computational Framework for Mechanical Response of Macromolecules: Application to the Salt Concentration Dependence of DNA Bendability

A computational framework is presented for studying the mechanical response of macromolecules. The method combines a continuum mechanics (CM) model for the mechanical properties of the macromolecule with a continuum electrostatic (CE) treatment of solvation. The molecules are represented by their shape and key physicochemical characteristics such as the distribution of materials properties and charge. As a test case, we apply the model to the effect of added salt on the bending of DNA. With a simple representation of DNA, the CM/CE framework using a Debye-Hückel model leads to results that are in good agreement with both analytical theories and recent experiments, including a modified Odijk-Skolnick-Fixman theory that takes the finite length of DNA into consideration. Calculations using a more sophisticated CE model (Poisson-Boltzmann), however, suffer from convergence problems, highlighting the importance of balancing numerical accuracy in the CM and CE models when dealing with very large systems, particularly those with a high degree of symmetry.     

Applying the stochastic difference equation to DNA conformational transitions: a study of B-Z and B-A DNA transitions

Despite the existence of numerous models to account for the B-Z DNA transition, experimenters have not yet arrived at a conclusive answer to the structural and dynamical features of the B-Z transition. By applying the stochastic difference equation to simulate the B-Z DNA transition, we have shown that the stretched intermediate model of the B-Z transition is more probable than other B-Z transition models such as the Harvey model. This is accomplished by comparing potential energy profiles of various B-Z DNA transition models and calculating relative probabilities based on the stochastic difference equation with respect to length (SDEL) formalism. The results garnered in this article allow for new approaches in determining the structural transition of B-DNA to Z-DNA experimentally. We have also simulated the B-A DNA transition using the stochastic difference equation. Unlike the B-Z DNA transition, the mechanism for the B-A DNA transition is well established. The variation in the pseudorotation angle during the transition is in good agreement with experimental results. Qualitative features of the simulated B-A transition also agree well with experimental data. The SDEL approach is thus a suitable numerical technique to compute long-time molecular dynamics trajectory for DNA molecules.     

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.     

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.