Nucleic Acids Research, 2006, Vol. 34, No. 22 6633 Figure 3. (A and B) Ion concentration-dependence of the compactness Rg (¼ 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 Rg for three helices. Results for different end-to-end distance maximum (¼ loop length L): xmax ¼ 60 A˚, 44 s and 28 s (from the upper to lower) are plotted, respectively. In A–D, the gray solid lines denote the compactness Rg of the random fluctuation states R for different xmax’s and gray dashed lines denote the compactness of the collapsed state C. The compactness is computed from the random averaging over x 2 [22,26] A˚ and q 2 [0 , 40 ] (69) for state C and x 2 [22 A˚, xmax] and q 2 [0, qmax] (qmax ¼ 180 for a two-helix system and 120 for a three-helix system) for state R. We note that Rg ’ 30 A for state C, and the result is in agreement with the helix–helix separation for an (least compact) aggregated DNA array (18–28). ﬂuctuation state (R) (especially for smaller xmax), due to the Coulombic and excluded volume repulsions between the bound Mg2+ ions (27,28,72). Both Na+ and Mg2+ can facilitate helix–helix compaction, but through different mechanisms. High concentration of Na+ causes compaction through strong ionic screening for helix charges, while Mg2+ ions can cause the compaction through helix–helix attraction (72). Moreover, Na+-induced compact state is heterogeneous as shown in Figure 2D while Mg2+-induced compact states are ordered as shown in Figure 2G and H. The overwhelming efﬁciency of Mg2+ over Na+ is much more pronounced for tertiary than secondary structure fold- ing. For example, we ﬁnd that for a 9 bp single helix, 10 mM Mg2+ and 1 M Na+ can provide the same stability (71). For helix–helix assembly as a tertiary fold, however, our results show that a 10-fold dilute [Mg2+], i.e. 1 mM [Mg2+], would be sufﬁcient to provide the same stability as 1 M Na+. This suggests that Mg2+ tends to promote the formation of the tertiary structure so the free energy can be more efﬁciently lowered by the Mg2+ ions. The result is in accordance with the experiments on Na+ and Mg2+-mediated RNA compaction (33–43). It is important to note that the ‘anomalous’ efﬁciency of the Mg2+ cannot be merely attribu- ted to the more efﬁcient charge screening, as shown by the PB-predicted electrostatic free energy landscape for Mg2+ in Supplementary Figure 9, which gives a disordered and much less compact state even at high [Mg2+]. It is important to note that, however, the Mg2+-mediated helix–helix interaction is not strong enough and x0, the inter-helix distance at the free energy minimum, is not small enough to cause a collapsed state; see Supplementary Figure 10. In Supplementary Figure 10, we show the magni- tude of ion-mediated helix–helix interaction in NaCl and MgCl2 solutions. For example, DGE ¼ GE GE (extended state) for 0.1 M Mg2+ is the lowest at (x0, q) ﬃ (32 s, 0 ) with DGE 3.3kBT. As the helices further approach each other (with x < x0), DGE for the parallel helices increases, and the minimum DGE is shifted from parallel helices (q ¼ 0) to a rotated helix–helix conﬁguration with q 6¼ 0 in order to avoid electrostatic repulsion between the helices. For example, for x ¼ 24 s, the minimum DGE ( 2.4kBT) occurs for a V-shaped conformation with q ¼ 40 . The rota- tion angle is larger for smaller x. For example, the minimum DGE (most stable conﬁguration) occurs at q ’ 60 for x ¼ 20 A˚ . When Mg2+ exceeds 0.1 M, the Mg2+-mediated attraction becomes weaker due to the increased likelihood for over-neutralization (70,72). For example, for [Mg2+] ¼ 0.6 M, the free energy minimum occurs at (x0, q0) ’ (32 A˚ , 0 ) with DGE 2.7kBT, which is weaker than 3.3kBT, the minimum free energy for 0.1 M Mg2+. Similar multivalent ion-induced destabilization has been observed experimentally (22,27,28,30,53). For example, packed DNA assembly has been found to be dissolved by high multivalent ion concentration (22,27,28). The predicted DGE is slightly stronger than the experimentally derived value (49). The