Electrostatic free energy landscapes for nucleic acid helix assembly

Abstract

Metal ions are crucial for nucleic acid folding. From the free energy landscapes, we investigate the detailed mechanism for ion-induced collapse for a paradigm system: loop-tethered short DNA helices. We find that Na+ and Mg2+ play distinctive roles in helix-helix assembly. High [Na+] (>0.3 M) causes a reduced helix-helix electrostatic repulsion and a subsequent disordered packing of helices. In contrast, Mg2+ of concentration >1 mM is predicted to induce helix-helix attraction and results in a more compact and ordered helix-helix packing. Mg2+ is much more efficient in causing nucleic acid compaction. In addition, the free energy landscape shows that the tethering loops between the helices also play a significant role. A flexible loop, such as a neutral loop or a polynucleotide loop in high salt concentration, enhances the close approach of the helices in order to gain the loop entropy. On the other hand, a rigid loop, such as a polynucleotide loop in low salt concentration, tends to de-compact the helices. Therefore, a polynucleotide loop significantly enhances the sharpness of the ion-induced compaction transition. Moreover, we find that a larger number of helices in the system or a smaller radius of the divalent ions can cause a more abrupt compaction transition and a more compact state at high ion concentration, and the ion size effect becomes more pronounced as the number of helices is increased.

Figure 1

Symmetric configurations for two (A) and three (B) DNA helices tethered by one (or two) semiflexible loop(s) can be characterized by two structural parameters: the angle θ between the helical axises and the end-end distance x. The DNA helices are produced from the grooved primitive model (see Materials and Methods) (70,72,73).

Figure 2

The normalized probability exp[−G_E (x, θ)/k_BT] for loop-free helix–helix system. Here G_E (x, θ) is the electrostatic free energy landscape for different (x, θ) configurations in a solution of NaCl (A–D), MgCl₂ (E–H), or divalent ion (M²⁺) with (small) radius = 3.5 Å (I–L). The red and blue colors represent the low and high free energies, respectively.

Figure 3

(A and B) Ion concentration-dependence of the compactness ${\overline{R}}_g$ (= the mean distance between helix centers) for two helices without loop (solid lines), with neutral polyethylene loop (dotted lines) and polynucleotide loop (dashed lines) in NaCl (A) and MgCl₂ (B) solutions. (C and D) The compactness ${\overline{R}}_g$ for three helices. Results for different end-to-end distance maximum (= loop length L): x_max = 60 Å, 44 Å and 28 Å (from the upper to lower) are plotted, respectively. In A–D, the gray solid lines denote the compactness ${\overline{R}}_g$ of the random fluctuation states R for different x_max's and gray dashed lines denote the compactness of the collapsed state C. The compactness is computed from the random averaging over x ∈ [22,26]Å and θ ∈ [0°, 40°] (69) for state C and x ∈ [22 Å, x_max] and θ ∈ [0, θ_max] (θ_max = 180° for a two-helix system and 120° for a three-helix system) for state R. We note that ${\overline{R}}_g\simeq 30\AA$ for state C, and the result is in agreement with the helix–helix separation for an (least compact) aggregated DNA array (–28).

Figure 4

The SAXS profiles for the two-helix model with neutral PEG loop for different ionic conditions: diamond 20 mM NaCl; plus 0.6 M MgCl₂ and square 1.2 M NaCl. (A) The predicted SAXS profiles weighted by the free energy G(x, θ) (see Materials and Methods); (B) The experimental SAXS profiles (49). The dashed lines, solid lines and dotted lines represent the standard SAXS profiles for the collapsed C, random R, and extended E states of the tethered helices, respectively.

Figure 5

The persistence length l_P of a single-stranded polynucleotide chain as a function of ionic strength I. The solid line is the fitted function Equation 7. The symbols are the experimental data: filled square (76), filled diamond (77) and filled triangle (78) are for Na⁺ solutions and open square (79) is for mixed Na⁺/Mg²⁺ solution.

Figure 6

The electrostatic free energy ΔG_E(x, 0°) = G_E(x, 0°)−G_E(60 Å, 0°) (in k_BT) as functions of the helix–helix separation x for three parallel DNA helices (θ = 0) at different Mg²⁺ concentrations: 0.1, 1, 10 and 100 mM (from the upper to lower). Solid lines: calculated from the pairwise additivity approximation. Dotted lines: calculated with three-helix correlation in the TBI model.

Figure 7

Compactness ${\overline{R}}_g$ of two loop-free helices in Mg²⁺ (radius ≃ 4.5 Å) and smaller divalent ion (M²⁺, radius = 3.5 Å) solutions, respectively. Results for different end-to-end distance maximum: x_max = 60 Å, 44 Å and 28 Å (from the upper to lower) are plotted separately. The gray lines denote the compactness ${\overline{R}}_g$ of the random fluctuation states R for different x_max's.