6632 Nucleic Acids Research, 2006, Vol. 34, No. 22 Figure 2. The normalized probability exp[ GE (x, q)/kBT] for loop-free helix–helix system. Here GE (x, q) is the electrostatic free energy landscape for different (x, q) configurations in a solution of NaCl (A–D), MgCl2 (E–H), or divalent ion (M2+) with (small) radius ¼ 3.5 s (I–L). The red and blue colors represent the low and high free energies, respectively. concentration Na+-induced compaction has been observed in experiments for RNAs (33–43). For larger RNA molecules, experiments also show that Na+ can induce the partial forma- tion of some native tertiary contacts (33–43). In each panel of Figure 3, we show three curves, each corresponding to a different constraint for the maximum helix–helix distance xmax. Physically, xmax is determined by the length of the linker (loop) that tethers the helices. The results for different xmax’s indicate how the spatial constraint from the ﬁnite loop length affects the two-helix electrostatic free energies and the resultant compactness of the system. In general, for low ion concentration, the two-helix conﬁgura- tion tends to become extended. So the compactness is more sensitive to xmax (¼ the upper limit of the extension). Larger xmax causes a more extended state. For high ion con- centration, the system becomes more compact and thus the compaction is less sensitive to xmax. As a result, larger xmax causes a more abrupt (ion-induced) change in the compact- ness, i.e. a larger xmax causes a sharper compaction transition. Mg2+ ions. For [Mg2+] below 1 mM, as [Mg2+] is increased, due to the stronger charge screening and consequently the weakened repulsion, helices become more compact as shown by the shift of the free energy minima from Figure 2E to F. For [Mg2+] above 1 mM, more ions are bound and the correlation effect becomes strong. The bound ions can self-organize to reach correlated low-free energy conﬁgura- tions. Such correlated energies decrease rapidly as helices approach each other, leading to an inter-helix attractive force (45,72). Higher [Mg2+] causes more bound ions and stronger inter-helix attraction (72), and hence a more compact state (as shown by the shift of the free energy minima from Figure 2F to H and the change of the compactness in Figure 3B). Because the inter-helix attraction is strongest for two par- allel helices (53), the free energy minima occur at q ¼ 0 in Figure 2F–H. As shown in Figure 3B, the Mg2+-induced com- pact state is more compact than the random ﬂuctuation state R but less compact than the collapsed state C. Such effect is more notable for larger xmax; see Figure 3 for the curves for three different xmax. Moreover, for very close approach between the helices, the parallel helices would repel each other due to the Coulombic repulsions between helices and crowded bound ions (72). In such close-packing case, the helices could be stabilized through rotation in order to mini- mize the repulsion; see the (rotated) local minimum at (q, x) ’ (40 , 22 A˚ ) in Figure 2H. As [Mg2+] exceeds 0.1 M, the helices may be over-neutralized by the Mg2+ ions and the compaction becomes saturated and close to the random