Electrostatic interaction between helical macromolecules in dense aggregates: an impetus for DNA poly- and meso-morphism

Abstract

DNA exhibits a surprising multiplicity of structures when it is packed into dense aggregates. It undergoes various polymorphous transitions (e.g., from the B to A form) and mesomorphous transformations (from hexagonal to orthorhombic or monoclinic packing, changes in the mutual alignment of nearest neighbors, etc). In this report we show that such phenomena may have their origin in the specific helical symmetry of the charge distribution on DNA surface. Electrostatic interaction between neighboring DNA molecules exhibits strong dependence on the patterns of molecular surface groups and adsorbed counter-ions. As a result, it is affected by such structural parameters as the helical pitch, groove width, the number of base pairs per helical turn, etc. We derive expressions which relate the energy of electrostatic interaction with these parameters and with the packing variables characterizing the axial and azimuthal alignment between neighboring macromolecules. We show, in particular, that the structural changes upon the B-to-A transition reduce the electrostatic energy by approximately kcal/mol per base pair, at a random adsorption of counter ions. Ion binding into the narrow groove weakens or inverts this effect, stabilizing B-DNA, as it is presumably the case in Li+-DNA assemblies. The packing symmetry and molecular alignment in DNA aggregates are shown to be affected by the patterns of ion binding.

Figure 1

A sketch of the horizontal projection of a hexagonal aggregate of helical molecules. Shaded circles represent molecular backbones. All molecules are similar. Positive and negative charges, for simplicity shown only around the central molecule, are intrinsic surface charges, counter-ions, or fractional water charges. We assume that all essential charges are located within cylindrical shells shown by the dashed lines and neglect possible effects of charges, which may be located in interstices between the shells (a sample interstice is painted black). The thin solid circle with the radius a shows the radial center of mass of all charges associated with the central helix. The two-dimensional radius vectors R₁ and R₂ point from the coordinate origin to the central axes of the molecules 1 and 2, respectively, R = R₂ − R₁.

Figure 2

An azimuthal (A) and axial (B) alignment of two neighboring B-DNA molecules. The sketch A is constructed as follows. We select the plane z = 0 at an arbitrary height. We then project the positions of two phosphates on molecule 1, each belonging to a different strand and nearest to the z = 0 plane. These projections are shown by the small, filled circles numbered as #1 and #2. We repeat the same projection procedure for molecule 2. We define the phosphate #1 on each molecule as the phosphate located at the major-to-minor groove crossover upon the counter clockwise rotation around the corresponding molecular axis. The angles φ′₁ and φ′₂ between the phosphates #1 and the direction of the vector R, defined as shown, give the azimuthal orientation of each molecule. The continuous helical lines in B are drawn through the centers of phosphate groups along the two strands of each molecule. H is the helical pitch, Δz is the axial shift needed to superimpose these helical lines after the lateral translation merging the axes of the molecules. The value of Δz characterizes the axial alignment.

Figure 3

Pair-interaction potential landscape for the B-to-A transition in DNA fibers. E_int⁽¹⁾ in k_BT units per base pair (bp) is plotted at a fixed interaxial separation R = 23 Å as a function of the axial alignment (Δz) and of the effective “reaction coordinate” (x) of the transition (x = 0 in the B-form and x = 1 in the A-form). The energy has two minima at Δz/H ≈ ±0.15 in the B form and one minimum at Δz = 0 in the A form. The energy scale is color coded as shown at the bottom. The landscape is symmetric with respect to the Δz = 0 plane. Note that the “reaction trajectory” along the two valleys, most favorable with respect to the electrostatic pair-interaction potential, may differ from the real optimal trajectory because the plotted energy landscape does not include the many-body effects and the internal mechanical energy of the molecules.

Figure 4

Effect of counter-ion adsorption pattern on the B-to-A transition. The change in E_int⁽¹⁾ upon the transition is plotted vs. the fraction of ions (f) preferentially adsorbed in the middle of the narrow groove, at R = 23 Å and at the optimal Δz-alignment (f = 0 corresponds to completely random distribution of counter-ions along molecular surfaces). The optimal Δz in each DNA form is plotted in the insert as the function of f. The optimal Δz is a continuous function of f in B-DNA. It exhibits a discontinuous behavior in A-DNA, Δz = 0 at f < f_* and Δz/H > 0.1 at f > f_*, where f_* ≈ 0.24.

Figure 5

Dependence of electrostatic interaction energy on azimuthal orientation of neighboring DNA molecules. The dependence of E_int⁽²⁾ + E_int⁽³⁾ on φ′₁ and φ′₂ is shown at R = 23 Å and at optimal Δz for A-DNA helices at f = 0 (A), B-DNA helices at f = 0 (B), and B-DNA helices at f = 0.5 (C). The contours represent lines of constant energy. The values of the energy in the units of thermal energy (k_BT) per base pair are shown by the color code. Different energy landscapes in the pair interaction may lead to different lateral packing of helices in these three cases.