6630 Nucleic Acids Research, 2006, Vol. 34, No. 22 Figure 1. Symmetric configurations for two (A) and three (B) DNA helices tethered by one (or two) semiflexible loop(s) can be characterized by two structural parameters: the angle q between the helical axises and the end-end distance x. The DNA helices are produced from the grooved primitive model (see Materials and Methods) (70,72,73). which gives the complete free energy distribution in the con- formational space, enables detailed study of the physical mechanism of the ion-induced helix–helix assembly, such as the stability and the compactness of the collapsed state and the cooperativity of the compaction transition. One of the key issues in the present study is the ion- mediated helix–helix interactions (19,20,28,33). Previous computer simulations (50–54) and analytical calculations (55–60) for similar systems reveal the existence of helix– helix attraction in the presence of multivalent ions (31,51). However, these previous studies are mostly based on simpli- ﬁed nucleic acid models and do not take into account the effects of the tether loops and the added salt in the solution. The mean-ﬁeld PB theory (61–65) predicts no helix–helix attraction even for multivalent ion solutions. The counterion condensation theory (66) can predict the inter-helix attraction arising from the increased ion entropy (67–69) for both monovalent and multivalent ions. Recently, we developed a statistical mechanical model, the tightly bound ion (TBI) model (70–72). The advantage of the model is that it can account for correlations and ﬂuctua- tions, i.e. different ion-binding modes, for the bound ions. The model is based on realistic helical structure of nucleic acids and takes into account the effect of the added salt in the supporting solutions and the realistic ion sizes (70–72). MATERIALS AND METHODS Structural model We focus on two model systems: two helices tethered by a semiﬂexible loop [Figure 1A and (49)] and three helices tethered by two loops (Figure 1B). To simplify the calcula- tion, we treat only symmetric helix–helix conﬁgurations and assume that the axes of the helices are co-planar (2D). As a result, the system can be characterized by two structural parameters: the angle q and the distance x between the heli- ces. The helices are assumed to adopt the B-form structure and are described by the grooved primitive model (70–73), where the charged phosphate groups, the neutral groups and the ion-inaccessible helix core regions are represented by charged spheres, neutral spheres and ion-inaccessible cylin- ders, respectively. The coordinates and sizes of the spheres and the cylinders are determined from the experimental atomic coordinates of a B-form helix; see Supplementary Data. We choose the above described two- (or three-) helix system because (i) the system is simple thus ideal for detailed analysis and (ii) there are available experimental results for a similar system (49). Free energy landscape For a conﬁguration described by (x, q) for the nucleic acid model, the free energy G (x, q) is calculated as 1 2 Gðx‚ qÞ ¼ GEðx‚ qÞ þ Gloopðx‚ LÞ‚ where GE (x, q) is the electrostatic free energy for the helices immersed in ion solutions and Gloop (x, L) (L ¼ loop length) is the entropic free energy of the loop(s). We compute GE (x, q) from TBI model (70–72), which explicitly accounts for the correlation and ﬂuctuation of the bound ions, and we obtain Gloop (x, L) from analytical expres- sions from polymer theories (74): Gloopðx‚ LÞ ¼ kBT ln Ploopðx‚ LÞ‚ where Ploop (x, L), the probability of ﬁnding a loop with end- end distance x is given by an analytical expression derived from the polymer theory (74): Ploopðx‚ LÞ ¼ ðx=LÞ2 ð2lP/L=AÞ Pk1¼1 p2k2ð 1Þkþ1eð lP/LÞp2k2ð1 x/LÞ‚ where A is the normalization factor. We note that calculations based on a different analytical formula for Ploop (x, L) (75) would give very close results for our calculations. A loop of polyethylene or PEG chain used in the exp- eriment (49) is neutral and its persistence length is ion- independent. In contrast, a realistic polynucleotide loop is polyanionic and its persistence length lP (and the loop entropy) is sensitive to the ionic condition (76–79). We account for the ion-dependence of the entropy of a polynu- cleotide loop through an empirical ion-dependent persistence length derived from the experimental measurements (76–79). The distribution of free energy G (x, q) (Equation 1) for different (x, q) states gives the free energy landscape of the system. We explore the landscape for conformations des- cribed by x 2 [22 s, 60 s] and q 2 [0 , 180 ]. Such an ensemble of conformations would include a broad range of different types of conformations, including the compact, the random and the extend conformations. We choose 22 s as the lower limit for x because this is the closest inter-axis distance of approach between two helices. In the actual depic- tion of the landscape, we scan the (x, q) space with incre- ments of 2 s and 20 for x and q, respectively. For each given (x, q) state, we enumerate all the accessible 3D confor- mations through the rotation of the planar rigid body with the ﬁxed x and q. We use such an ensemble of 3D conformations to compute the free energy G (x, q). TBI model For a given helix–helix conﬁguration (x, q), we calculate the electrostatic free energy using the TBI model. The develop- ment of the TBI model is motivated by the need to accurately account for Coulombic and excluded volume correlations between bound ions and between bound ions and helices as well as the ﬂuctuations of the ion-binding modes (70–72); see Supplementary Data and Refs (70–72) for details. In the TBI model, according to the ion–ion correlation strength, we classify the tightly bound (¼ strongly correlated) and the diffusive (¼ weakly correlated) ions/regions (70–72).