Excess counterion binding and ionic stability of kinked and branched DNA.

We compute the excess number of counterions associated with kinked and branched DNA, and the ionic stabilities of these structures as a function of chain length and both sodium and magnesium salt concentration, using numerical counterion condensation theory. The DNA structures are modeled as two or more finite lines of phosphate charges radiating from the kink or junction center. The number of excess counterions around the (40-90 degrees) kinked duplex is very small (at most four). The geometries of large three- and four-way DNA junctions (with > 50 base pairs per arm) in solutions containing low to moderate NaCl concentrations, by contrast, accumulate a substantial number of excess sodium ions (> 20) but no more than 15 magnesium counterions. The excess number of counterions surrounding the kinked linear chain and the branched DNA structures either remains invariant or increases with chain length, tending to reach a plateau value. Open configurations, such as the planar Y-shaped three-way junction (with three 120 degrees inter-arm angles) and the 90 degrees cross-shaped four-way junction, are ionically more stable than compact geometries, such as pyramidal three-way junctions and X-shaped four-way junctions, over the entire range of salt concentration considered (10(-5)-10(-1) M NaCl or MgCl2). The ionic stabilities of the compact forms increase with increasing salt concentration and become comparable to those of the extended geometries at high salt (especially when magnesium is the supporting salt).

Electrostatic effects in short superhelical DNA.

We present Monte Carlo simulations of the equilibrium configurations of short closed circular DNA that obeys a combined elastic, hard-sphere, and electrostatic energy potential. We employ a B-spline representation to model chain configuration and simulate the effects of salt on chain folding by varying the Debye screening parameter. We obtain global equilibrium configurations of closed circular DNA, with several imposed linking number differences, at two salt concentrations (specifically at the extremes of no added salt and the high salt regime), and for different chain lengths. Minimization of the composite elastic/long-range potential energy under the constraints of ring closure and fixed chain length is found to produce structures that are consistent with the configurations of short supercoiled DNA observed experimentally. The structures generated under the constraints of an electrostatic potential are less compact than those subjected only to an elastic term and a hard-sphere constraint. For a fixed linking number difference greater than a critical value, the interwound structures obtained under the condition of high salt are more compact than those obtained under the condition of no added salt. In the case of no added salt, the electrostatic energy plays a dominant role over the elastic energy in dictating the shape of the closed circular DNA. The DNA supercoil opens up with increasing chain length at low salt concentration. A branched three-leaf rose structure with a fixed linking number difference is higher in energy than the interwound form at both salt concentrations employed here.

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.

A numerical counterion condensation analysis of the B-Z transition of DNA.

We examine the salt dependence of the B-Z transition in DNA by means of the counterion condensation theory adapted to structurally realistic coordinates of the phosphate groups. The ionic contribution to the free energy difference delta G is computed for both the ZI and ZII conformations over broad ranges of NaCl and MgCl2 concentrations and polymer lengths. For the solvent we employ both a constant-dielectric model (dielectric constant set to 78.3) and a dielectric saturation model (distance-dependent dielectric constant). Where comparison can be made, the results for the constant-dielectric model are similar to those obtained by other workers for the same model but with different computational methods. The existence of a low-salt transition, and its location when it does occur, depends strongly on the DNA length and on the dielectric model. The behavior of ZI and ZII are qualitatively similar throughout the entire salt range for the constant-dielectric model, but qualitatively different if dielectric saturation is simulated, as we think is necessary for a realistic description. The ionic delta G, in the presence of dielectric saturation, bears comparison with the high-salt trend of the measured total delta G if "Z-DNA" is predominantly ZI, but not if it is predominantly ZII.