Published online 1 December 2006 Nucleic Acids Research, 2006, Vol. 34, No. 22 6629–6639 doi:10.1093/nar/gkl810 Electrostatic free energy landscapes for nucleic acid helix assembly Zhi-Jie Tan and Shi-Jie Chen* Department of Physics and Astronomy and Department of Biochemistry, University of Missouri, Columbia, MO 65211, USA Received July 31, 2006; Revised October 1, 2006; Accepted October 3, 2006 ABSTRACT Metal ions are crucial for nucleic acid folding. From the free energy landscapes, we investigate the detailed mechanism for ion-induced collapse for a paradigm system: loop-tethered short DNA helices. We find that Na+ and Mg2+ play distinctive roles in helix–helix assembly. High [Na+] (>0.3 M) causes a reduced helix–helix electrostatic repulsion and a subsequent disordered packing of helices. In con- trast, Mg2+ of concentration >1 mM is predicted to induce helix–helix attraction and results in a more compact and ordered helix–helix packing. Mg2+ is much more efficient in causing nucleic acid com- paction. In addition, the free energy landscape shows that the tethering loops between the helices also play a significant role. A flexible loop, such as a neutral loop or a polynucleotide loop in high salt concentration, enhances the close approach of the helices in order to gain the loop entropy. On the other hand, a rigid loop, such as a polynu- cleotide loop in low salt concentration, tends to de-compact the helices. Therefore, a polynucleotide loop significantly enhances the sharpness of the ion-induced compaction transition. Moreover, we find that a larger number of helices in the system or a smaller radius of the divalent ions can cause a more abrupt compaction transition and a more com- pact state at high ion concentration, and the ion size effect becomes more pronounced as the number of helices is increased. INTRODUCTION Nucleic acid (DNA and RNA) structures and structural changes are critical for biological functions (1,2). Due to the polyanionic nature of the nucleic acid molecules, the ther- mal stability and folding kinetics of nucleic acids are strongly dependent on the ionic condition of the solution (1–14). A rudimentary question in nucleic acid folding is how the negatively charged helices pack against each other to form a compact 3D structure, and how different ions and ion con- centrations affect the structure and stability of the helix assembly. Experiments have suggested that multivalent ion- mediated force plays an important role in DNA condensation (15–28) and compaction of other polyelectrolyte molecules (29–32). Moreover, extensive experiments have demonstrated the particularly efﬁcient role of multivalent ions in RNA compaction (33–39). For example, it was found that [Mg2+] in millimolar (mM) or [Na+] in molar (M) can stabilize a compact intermediate state (33–44), and the compact state induced by Mg2+ is more compact and ordered (39,41). Recent experimental studies have signiﬁcantly advanced our understanding of RNA-ion interactions. For example, several experiments demonstrated that cations with higher valency and smaller size are more efﬁcient in RNA-folding and such ions could also slow down the folding by stabilizing possible misfolded states (8,33–36). Moreover, a series of experiments indicated that charge neutralization of the nucleotide backbone may be responsible for the initial com- paction of RNA (39–43). A recent small angle X-ray scatter- ing (SAXS) measurement for a simple two-helix system suggested the existence of an attractive inter-helix force at high [Mg2+] and a repulsive force at very low [Mg2+] (45). Furthermore, a different SAXS experiment for loop-tethered helix system showed that the inter-helix force at high [Mg2+] is not sufﬁcient to cause a collapsed state for the sys- tem (46). Based on the (mean-ﬁeld) non-linear Poisson- Boltzmann (PB) theory, a thermodynamic framework for Mg2+–RNA interaction suggested that the predominant elec- trostatic contribution to RNA-folding stability comes from the diffusively bound ions (9,46–48). Inspired by the experiments on the ion effect in nucleic acid folding (15–43,45,49), we investigate the ion-dependence of the folding free energy landscape for a paradigm system: two or three loop-tethered helices immersed in an ionic solu- tion. The motivation of the present study is to quantitatively understand (i) the physical mechanism of the ion-induced compaction of the helix assembly and (ii) how the length and the electrostatic properties of loops, the multiplicity of the helices, and the ion size determine the stability and structural changes of the system. The free energy landscape, *To whom correspondence should be addressed. Tel: +1 573 882 6626; Fax: +1 573 882 4195; Email: chenshi@missouri.edu 2006 The Author(s). This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/ by-nc/2.0/uk/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.