landscapes lead to several predictions for the ion-dependent compaction of the helices. (i) Adding Na+ to the solution would induce the nucleic acid helices to undergo a transition from an extended state to a more compact random fluctuation state. However, Na+-mediated electrostatic interactions cannot cause the helices to form the collapsed state even at very high concentrations. This prediction is in accordance with the experiments on Na+-mediated RNA compaction (42–49). (ii) Mg2+ can mediate helix–helix attraction, which causes the helix–helix assembly to form a state more compact than the random fluctuation state but less compact than the collapsed state. Such effect would be reduced for shorter helix–helix linker. Mg2+ ions are much more efficient than Na+ to stabilize nucleic acid helix–helix assembly. For example, 1 mM Mg2+ can induce a state equally compact as that induced by 1 M Na+. The maximum compact state occurs at [Mg2+ around 0.1 M. (iii) The tether loop(s) can facilitate the compaction transi- tion. While an electrically neutral loop enhances compaction for both low and high ion concentrations, a polynucleotide loop can significantly enhance the sharpness of the ion-induced compaction transition due to the ion-dependence of loop flexibility. Such effect is more pronounced for Mg2+ (than Na+) and for larger loops. (iv) Multiplicity of the helices and the loops can significantly enhance the sharpness of the compaction transition and the compactness of the compact state. (v) Smaller ions are more efficient to induce the electro- static compaction and such effect is much more pronounced for a multiple helix assembly. Our treatment here is limited by several simpliﬁcations and approximations. First, we have neglected the possible par- tially or completely (dehydrated) ion site-binding (9,89) and possible preferential interactions between ions and speciﬁc groups of the nucleic acids (1). Ion site-binding to speciﬁc pockets with very strong negative electric potential can help stabilize speciﬁc tertiary structures (46,47) and can be essential for RNA catalytic function (90). The preferential ion-binding to speciﬁc groups, e.g. Mg2+ binds to phosphate groups or Mn2+ binds to both phosphates and bases (1), would affect the spatial distribution of ions on nucleic acid surface and contribute to the ion-mediated helix–helix interactions. However, the overall contribution of ion site-binding to the global stability may be small, because the large dehydration penalty may reduce the likelihood of speciﬁc binding of the dehydrated ions (46,47). Second, in order to simplify the computation, we assume each helix pair to be symmetric and planar. Such a system covers a full spectrum of typical conﬁgurations, including compact, random and extended states. Thus, our predicted free energy landscape reveals the main feature of the ion- dependent helix assembly. We have also ignored the effect of different rotational phase of the helices about their axes. Previous analysis showed that the free energy landscape is not sensitive to such rotations if the helix–helix separation is large (e.g. x > 28 A˚ ) (91). However, for two closely approached helices (x < 24 A˚ ), such rotation would play Nucleic Acids Research, 2006, Vol. 34, No. 22 6637 an important role because of the detailed geometrical com- patibility between helices and the related charge distributions (69,92). For very closely approached helices, parallel conﬁg- uration is unfavorable because of the strong electrostatic repulsion. In the present model, the helices would be stabi- lized through a rotation to form a V-shaped structure. In realistic full 3D conformational space, the two helices can be stabilized through off-plane rotations to form a typical X-shaped structure (69,92,93) with two possible packing modes: groove-into-groove mode where the major groove of one helix ﬁts in the major groove of the other helix (69), and backbone-into-groove mode where the nucleotide strand of a helix ﬁts in the major groove of the other helix (92). Helix rotation about its axis is important for the formation of such X-shaped structures (69,92,93). Third, we have ignored the sequence effect and speciﬁc tertiary interactions, which are essential for RNAs to form speciﬁc functional structures. In this study, we only investi- gated the ion electrostatic effects on the electrostatic collapse process. The quantitative modeling of further folding steps requires the inclusion of the sequence-dependent tertiary interactions. Finally, we have neglected the detailed loop structures and possible deformation and unfolding of helices at different ionic conditions. Nevertheless, the predicted free energy landscapes, the detailed compaction mechanisms for Na+ and Mg2+ ions, and the effects of different loops, multiplicity of helices and ion size may be useful for understanding the ion-dependent nucleic acid helix–helix assembly. SUPPLEMENTARY DATA Supplementary Data are available at NAR online. ACKNOWLEDGEMENTS This research was supported by NIH through grant GM063732 (to S.-J.C.). Funding to pay the Open Access publication charges for this article was provided by NIH. Conflict of interest statement. None declared. REFERENCES 1. Bloomfield,V.A., Crothers,D.M. and Tinoco,I.,Jr (2000) Nucleic Acids: Structure, Properties and Functions. University Science Books, Sausalito, CA. 2. Tinoco,I. and Bustamante,C. (1999) How RNA folds. J. Mol. Biol., 293, 271–281. 3. 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