We treat diffusive ions using non-linear PB and treat the tightly bound ions separately by exhaustively enumerating all the possible distributions of the tightly bound ions (¼ ion-binding modes). Each mode M is described by a set of numbers with each number mi equal to the number of the tightly bound ions surrounding the ith phosphate. For each mode, we compute the partition function ZM by evaluating the energy and entropy of the diffusive ions, the tightly bound ions, and the interactions between the diffusive and the tightly bound ions (70–72). In the calculation, the many- particle Coulombic and excluded volume correlations between the tightly bound ions and between the tightly bound ions and the helices are accounted for. Sum over all the possible ion- binding modes gives the total partition function Zðx‚ qÞ ¼ PM ZM for a given nucleic acid structure, from which the electrostatic free energy is obtained. GEðx‚ qÞ ¼ kBT ln Zðx‚ qÞ ¼ kBT ln XZM: M We assume Na+ and Mg2+ ions are hydrated and have radii 3.5 s and 4.5 s (70–72), respectively. When solving the non-linear PB, we add a thin charge-free layer of one cation radius to the molecular surface to account for the volume exclu- sion of the cations (61,70–72). The dielectric constant e is assumed to be 12 [experimental value (80)] for the helix interior and 78 (of bulk water at 25 C) elsewhere in the solution. SAXS profile SAXS can effectively reveal the shape and size of macro- molecules (42–49,81,82). To compute the SAXS proﬁle I (Q) with the grooved primitive helix model, we use the following equation for I (Q) (83): IðQÞ ¼ M0 M0 X X i¼1 j¼1 sinQrij f i f j Qrij ‚ where M0 is the total number of the charged and neutral spheres in the model, and rij is the distance between two spheres. The scattering vector Q is equal to 4psin(q)/l, where q is the Bragg angle and l is the wavelength of the radiation. f is the scattering factor (84) f ðQÞ ¼ 1 þ cosðpQ=QmaxÞ, where Q is in the range between 0 and Qmax, the largest scattering vector in reciprocal space (84). For simplicity, we use the same scattering factor f (Q) for all spheres in the model (84). Qmax is taken as 3 s 1 in the calculations. Our tests with different Qmax (&2 s 1) give quite robust results. RESULTS AND DISCUSSIONS We consider a system of loop-tethered DNA helices immersed in an ionic solution. Each helix is assumed to have a ﬁnite length of 10 bp. To simplify the analysis, we assume that the helices are symmetrically oriented, so the system can be characterized by two structural parameters: angle q and distance x between the helices (see Figure 1). The stability of a state described by (x, q) is determined by its free energy G (x, q) (see Equation 1). G (x, q) for all the possible (x, q) states with x 2 [20 s, 60 s] and q 2 [0 , 180 ] gives the free energy landscape. 3 4 Nucleic Acids Research, 2006, Vol. 34, No. 22 6631 From the free energy landscape G (x, q), we can identify all the possible stable states, which are represented by the min- ima on the landscape, and the transitions between the differ- ent stable states. We characterize the compactness of the helix assembly by a Boltzmann-weighted mean value Rg for the distance rg between the centers of the helix pair: 5 Rg ¼ Pðx‚qÞ rge Gðx‚qÞ/kBT Pðx‚qÞ e Gðx‚qÞ/kBT : As described in Methods, in our calculation, we choose x 2 [22 s, 60 s] and q 2 [0 , 180 ] with increments of 2 s and 20 for x and q, respectively. According to the compactness of the structure, we distin- guish three types of states: (i) the extended state E for conﬁgurations with large x and large q, (ii) the random ﬂuctuation state R for conﬁgurations with random x and q and (iii) the collapsed state C for conﬁgurations with x 2 [22, 26]A˚ and q 2 [0 , 40 ] (69). Our calculation shows that the collapsed state C has a mean compactness of Rg ’ 30 A, which agrees with the upper limit for the helix– helix separation for a collapsed (aggregated) DNA array (18–28). It is important to note that such an electrostatically collapsed state is less compact than the ﬁnal RNA native structure (44), where further compaction can be induced by speciﬁc tertiary interactions that are ignored in the present study. The key to understanding ion-induced structural transition is the free energy landscape G (x, q). As shown in Equation 1 in ‘Methods’, G (x, q) of the system consists of two parts: (i) the electrostatic free energy GE (x, q) deﬁned as the loop-free free energy for the helices and the ionic solution and (ii) the loop-free energy Gloop (x) for the loop conforma- tional entropy. Before including the loop effect, we ﬁrst focus on the effect of the electrostatic free energy landscape GE (x, q). Without the tether loop Na+ ions. In Figure 2A–D we show density plots for the elec- trostatic free energy landscapes GE (x, q) for two helices immersed in a Na+ solution. At very low [Na+] ( 1 mM), the favorable low-free energy state is the extended state E with the largest x and q so that the repulsive helices can maxi- mally avoid each other. As [Na+] is increased, the favorable state would ‘smear out’ to reach the region of smaller x and q as shown in Figure 2B and C for 0.01 M and 0.1 M NaCl, respectively. When [Na+] becomes very high, such as 1 M, the favorable conformations can spread over nearly the whole region of x and q. In Figure 3A, the solid lines show the compactness Rg as a function of [Na+]. We ﬁnd that the increase of [Na+] can induce the compaction process. The rapid decrease in Rg occurs for [Na+] 0.3 M. For [Na+] >0.3 M, the helix–helix system approaches the random ﬂuc- tuation state R. Physically, as [Na+] increases, with the decrease in the entropic penalty for ion-binding, more Na+ ions can bind to the helices, resulting in a weakened repulsion between the helices. The predictions show that Na+ can cause a random ﬂuctuation state at high [Na+] (42–49), and the random ﬂuctuation state is more compact than the electro- statically repulsive unfolded state at low [Na+]. Such high