6636 Nucleic Acids Research, 2006, Vol. 34, No. 22 helices. Compared to the two-helix case, the freedom of the bound ions for the helix triad is much more restricted due to the ion–helix and ion–ion Coulombic and excluded volume interactions. Such triple-helix restriction tends to pre- vent the (frustrated) bound ions from self-organizing to reach the lowest energy states that would be realized for a two-helix system. As a result, the resultant most compact state is less stable and slightly less compact than that predicted from the pairwise additive model. On the other hand, for lower [Mg2+], the number of bound ions is smaller and the charge neutralization is weaker, so the bound Mg2+ ions can self- organize relatively freely to reach the correlated low-energy states (88). In such case, due to the higher charge density for the three-helix case, the system can reach an energy lower than that predicted from the additive two-helix model (88). Our above predictions agree with the previous analysis based on simpliﬁed models (68,88), which showed that the non-additivity effect is not strong for three-body system, and the non-additivity can result in slightly stronger attraction Figure 6. The electrostatic free energy DGE(x, 0 ) ¼ GE(x, 0 ) GE(60 A˚, 0 ) (in kBT) as functions of the helix–helix separation x for three parallel DNA helices (q ¼ 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. than the pairwise approximation for low ion concentration (68,88). Ion size effect Ion size has been shown experimentally to play important roles in RNA compaction (35). To investigate the size effect, we compare the free energy landscapes and the compactness for two divalent ions of different sizes: Mg2+ with radius 4.5 s (70,72) and divalent ion (denoted by M2+) with a smaller radius 3.5 s. Due to the lack of the information about the ion size-dependence of the loop persistence length, we would focus on the loop-free system. Two-helix system. By comparing the free energy landscapes in Figure 2E–L and the compactness in Figure 7A for Mg2+ and the small divalent ions, we ﬁnd that the smaller ions can induce stronger helix–helix attraction and the resultant stable states are more compact (72). Therefore, small ions are more efﬁcient to induce the compaction. Physically, smaller ions can bind to the helices at a smaller ion-helix distance. Stron- ger ion–helix interaction leads to a higher ion-binding afﬁnity and hence a larger number of bound ions, which, as a result, causes a stronger helix–helix attraction (72) and thus a more compact state. Three-helix system. As shown in Figure 7B for the ion- dependence of the compactness, the higher efﬁciency of the smaller ions is much more pronounced for the three-helix sys- tem. Notably, the three-helix system with high concentration of smaller ions can reach a very compact state—the collapsed state C. Such a collapsed state can never be reached by a two- helix system or a three-helix system in the solution of the (more bulky) Mg2+ ions. Physically, this is caused by the stronger helix–helix attraction for smaller ions and the effect is further ‘ampliﬁed’ by the larger number of helices. CONCLUSIONS AND DISCUSSIONS Using the TBI model (70–72) to account for the correlation and ﬂuctuation effects of bound ions, we explore the electro- static free energy landscape for nucleic acid helix–helix assembly for several paradigm systems: two and three helices with and without tether loop(s). The computed free energy Figure 7. Compactness Rg of two loop-free helices in Mg2+ (radius ’ 4.5 A˚) and smaller divalent ion (M2+, radius ¼ 3.5 A˚ ) solutions, respectively. Results for different end-to-end distance maximum: xmax ¼ 60 A˚, 44 s and 28 s (from the upper to lower) are plotted separately. The gray lines denote the compactness Rg of the random fluctuation states R for different xmax’s.