80 A60 h t g n e lce40 n e t s i s r ep20 0 0.001 0.01 0.1 ionic strength M 1 Figure 5. The persistence length lP 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. transition. Furthermore, as we discuss below, a polynucleotide loop (instead of an electrically neutral loop), which has an ion-dependent ﬂexibility and entropy, would also lead to a sharper transition. Nevertheless, despite the different types of loops considered, we ﬁnd the compact state is much less compact than the collapsed state. Long neutral loop. To further investigate the loop effect, we consider longer polyethylene (-CH2-) tether loops (49) with lP 6 A˚ (86,87). As shown in Figure 3 for different loop length (maximum helix–helix end-to-end distance), the loop promotes compaction in both Na+ and Mg2+ solutions in order to gain larger loop entropy, which occurs for smaller x. For example, at 10 mM [Mg2+] in Figure 2F, our free energy landscape shows that the inter-helix distance x for the most stable state decreases from 34 A˚ (without loop) to 32 A˚ with a loop of length L ¼ 60 A˚ . As the loop length is decreased, the loop entropy effect is weakened because of the reduced conformational entropy of the loop (74,75). As a caveat, we note that if the loop is too short, the helices would be restricted to the small x region to form a rotated state, which is less compact (larger Rg) than the parallel state, as discussed above. With a polynucleotide loop As described in Materials and Methods, a realistic polynu- cleotide loop is polyanionic. Unlike neutral loops, such as polyethylene or PEG chains, polynucleotide persistence length lP and the loop entropy are sensitive to the ionic con- dition (76–79). Based on the experimental data for the ion-dependence of lP (76–79), we obtain an empirical expres- sion for lP as a function of the ionic strength I (in M) in the solutions (see Figure 5). lP ’ 5 þ 1:5/pﬃIﬃ ðAÞ: The above expression is mainly derived from the experi- mental data for Na+ due to the limited experimental data for lP in Mg2+ solution. As an approximation, we use Equation 7 for both Na+ and Mg2+ solutions. 7 Nucleic Acids Research, 2006, Vol. 34, No. 22 6635 As shown in Figure 3A and B for the ion-dependence of the compactness, for low ion concentration the loop is stiff (with large lP, so the system is much more extended than the neutral loop case, especially for large loops. For high ion concentra- tion, the loop is ﬂexible (with small lP) (76), so the helices collapse to a state as compact as the neutral polyethylene loop case. Therefore, the polynucleotide loop greatly enhances the sharpness of the compaction transition. Such effect is much more pronounced for Mg2+ due to the stronger ion concentration-dependence of lP. The Mg2+-induced com- paction transition occurs in the [Mg2+] range of 1–10 mM. In addition, since Equation 7, which is ﬁtted from the experi- mental data mainly for Na+, may underestimate the [Mg2+]- dependence of lP (79), the realistic [Mg2+]-induced transition could be even sharper than that predicted here. With more helices and loops Since realistic RNA-folding often involves multiple (>2) helices, it would be useful to investigate how the multiplicity of helices affects the compaction process. We consider a simpliﬁed system consisting of three helices tethered by two loops as shown in Figure 1B. A rigorous treatment for the system requires a full account of all the possible bound ion conﬁgurations for the three-helix system for all the possible (x, q) conﬁgurations, which would be extremely demanding in computational time. So we parse the problem into two simpliﬁed sub-problems. We ﬁrst treat the full (x, q) conﬁgurational ensemble by assuming pairwise addit- ive electrostatic free energy for the three-helix system: GEADDðx‚ qÞ ¼ Pi3¼1Pj>i GðEijÞ, where GðEijÞ is the electrostatic free energy for a system of two helices i and j. We then exam- ine the effect of the three-helix correlation by considering the system of three parallel helices (q ¼ 0), where the enumera- tion for all the possible triple-helix (rather than pairwise) cor- related bound ion modes can be performed. Pairwise approximation. With the pairwise (additivity) approximation, for each given symmetric three-helix conﬁg- uration (x, q), the ion-dependence of the electrostatic free GEADDðx‚ qÞ ’ 3GðEijÞ as well the loop-free energy energy Gloop(x, q) ’ 2Gsingle loop(x) are ‘ampliﬁed’. Therefore, the three-helix system shows a qualitatively similar but quantita- tively much stronger ion-dependence than the two-helix system; see Figure 3A–D. For example, at low [Mg2+], the extended state becomes more extended due to the enhanced helix–helix repulsion and the effect from the extended rigid loops, and at high [Mg2+], the compact state becomes more compact due to the stronger electrostatic attraction and the enhanced loop entropy effect. Triple-helix correlation. By considering the interactions between the three helices (and their associated bound ions), we compute the triple-helix correlated electrostatic free energy landscape DGE(x, 0 ) ¼ GE(x, 0 ) GE(60 A˚ , 0 ). As shown in Figure 6, the pairwise approximation slightly underestimates and overestimates the stability of the free energy minima for [Mg2+] below 1 mM and above 10 mM, respectively. For high [Mg2+], the helices are highly neutralized. For example, at 0.1 M Mg2+, tightly bound Mg2+ can neutralize about 92% of the phosphate charges for separated DNA