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The ability to accurately predict RNA hairpin structure and stability for different loop sequences and salt conditions is important for understanding, modeling, and designing larger RNA folds. However, traditional RNA secondary structure models cannot treat loop-sequence and ionic effects on RNA hairpin folding. Here, we describe a general, three-dimensional (3D) conformation-based computational method for modeling salt concentration-dependent conformational distributions and the detailed 3D structures for a set of three RNA hairpins that contain a variable, 15-nucleotide loop sequence. For a given RNA sequence, the new, to our knowledge, method integrates a Vfold2D two-dimensional structure folding model with IsRNA coarse-grained molecular dynamics 3D folding simulations and Monte Carlo tightly bound ion estimations of ion-mediated electrostatic interactions. The model predicts free-energy landscapes for the different RNA hairpin-forming sequences with variable salt conditions. The theoretically predicted results agree with the experimental fluorescence measurements, validating the strategy. Furthermore, the theoretical model goes beyond the experimental results by enabling in-depth 3D structural analysis, revealing energetic mechanisms for the sequence- and salt-dependent folding stability. Although the computational framework presented here is developed for RNA hairpin systems, the general method may be applied to investigate other RNA systems, such as multiway junctions or pseudoknots in mixed metal ion solutions.

RNA functions are often kinetically controlled. The folding kinetics of RNAs involves global structural changes and local nucleotide movement, such as base flipping. The most elementary step in RNA folding is the closing and opening of a basepair. By integrating molecular dynamics simulation, master equation, and kinetic Monte Carlo simulation, we investigate the kinetics mechanism of RNA helix-terminal basepairing. The study reveals a six-state folding scheme with three dominant folding pathways of tens, hundreds, and thousands of nanoseconds of folding timescales, respectively. The overall kinetics is rate limited by the detrapping of a misfolded state with the overall folding time of 10(−5) s. Moreover, the analysis examines the different roles of the various driving forces, such as the basepairing and stacking interactions and the ion binding/dissociation effects on structural changes. The results may provide useful insights for developing a basepair opening/closing rate model and further kinetics models of large RNAs.

Metal ions play essential roles in nucleic acids folding and stability. The interaction between metal ions and nucleic acids can be highly complicated because of the interplay between various effects such as ion correlation, fluctuation, and dehydration. These effects may be particularly important for multivalent ions such as Mg 2+ ions. Previous efforts to model ion correlation and fluctuation effects led to the development of the Monte Carlo tightly bound ion model. Here, by incorporating ion hydration/dehydration effects into the Monte Carlo tightly bound ion model, we develop a, to our knowledge, new approach to predict ion binding. The new model enables predictions for not only the number of bound ions but also the three-dimensional spatial distribution of the bound ions. Furthermore, the new model reveals several intriguing features for the bound ions such as the mutual enhancement/inhibition in ion binding between the fully hydrated (diffuse) ions, the outer-shell dehydrated ions, and the inner-shell dehydrated ions and novel features for the monovalent-divalent ion interplay due to the hydration effect.

The interaction between metal ions, especially Mg 2+ ions, and RNA plays a critical role in RNA folding. Upon binding to RNA, a metal ion that is fully hydrated in bulk solvent can become dehydrated. Here we use molecular dynamics simulation to investigate the dehydration of bound hexahydrated Mg 2+ ions. We find that a hydrated Mg 2+ ion in the RNA groove region can involve significant dehydration in the outer hydration shell. The first or innermost hydration shell of the Mg 2+ ion, however, is retained during the simulation because of the strong ion-water electrostatic attraction. As a result, water-mediated hydrogen bonding remains an important form for Mg 2+ -RNA interaction. Analysis for ions at different binding sites shows that the most pronounced water deficiency relative to the fully hydrated state occurs at a radial distance of around 11 A from the center of the ion. Based on the independent 200 ns molecular dynamics simulations for three different RNA structures (Protein Data Bank: 1TRA, 2TPK, and 437D), we find that Mg 2+ ions overwhelmingly dominate over monovalent ions such as Na + and K + in ion-RNA binding. Furthermore, application of the free energy perturbation method leads to a quantitative relationship between the Mg 2+ dehydration free energy and the local structural environment. We find that ΔΔ G hyd , the change of the Mg 2+ hydration free energy upon binding to RNA, varies linearly with the inverse distance between the Mg 2+ ion and the nearby nonbridging oxygen atoms of the phosphate groups, and ΔΔ G hyd can reach −2.0 kcal/mol and −3.0 kcal/mol for an Mg 2+ ion bound to the surface and to the groove interior, respectively. In addition, the computation results in an analytical formula for the hydration ratio as a function of the average inverse Mg 2+ -O distance. The results here might be useful for further quantitative investigations of ion-RNA interactions in RNA folding.

The interaction between metal ions, especially Mg(2+) ions, and RNA plays a critical role in RNA folding. Upon binding to RNA, a metal ion that is fully hydrated in bulk solvent can become dehydrated. Here we use molecular dynamics simulation to investigate the dehydration of bound hexahydrated Mg(2+) ions. We find that a hydrated Mg(2+) ion in the RNA groove region can involve significant dehydration in the outer hydration shell. The first or innermost hydration shell of the Mg(2+) ion, however, is retained during the simulation because of the strong ion-water electrostatic attraction. As a result, water-mediated hydrogen bonding remains an important form for Mg(2+)-RNA interaction. Analysis for ions at different binding sites shows that the most pronounced water deficiency relative to the fully hydrated state occurs at a radial distance of around 11 Å from the center of the ion. Based on the independent 200 ns molecular dynamics simulations for three different RNA structures (Protein Data Bank: 1TRA, 2TPK, and 437D), we find that Mg(2+) ions overwhelmingly dominate over monovalent ions such as Na(+) and K(+) in ion-RNA binding. Furthermore, application of the free energy perturbation method leads to a quantitative relationship between the Mg(2+) dehydration free energy and the local structural environment. We find that [Formula: see text] the change of the Mg(2+) hydration free energy upon binding to RNA, varies linearly with the inverse distance between the Mg(2+) ion and the nearby nonbridging oxygen atoms of the phosphate groups, and [Formula: see text] can reach −2.0 kcal/mol and −3.0 kcal/mol for an Mg(2+) ion bound to the surface and to the groove interior, respectively. In addition, the computation results in an analytical formula for the hydration ratio as a function of the average inverse Mg(2+)-O distance. The results here might be useful for further quantitative investigations of ion-RNA interactions in RNA folding.

RNA/RNA interactions are essential for genomic RNA dimerization and regulation of gene expression. Predicting the structure and folding stabilities of RNA/RNA complexes is a biologically significant problem. We recently developed a new web server, VfoldCPX, to predict two-dimensional (2D) structures of RNA/RNA complexes from the sequence. The server is based on a new Vfold model that employs coarse-grained entropy parameters for the different structural motifs. By taking into account both the intra- and inter-molecular interactions, the model can predict RNA/RNA structures with cross-linked tertiary contacts. Furthermore, in addition to the minimum free energy structure, the server can predict multiple energetically stable structures ranked by their stabilities. The predicted structures can offer users with extensive insights into the folding and conformational switches in RNA biological functions. The web server is freely accessible at “http://rna.physics.missouri.edu”.

RNAs are negatively charged molecules residing in macromolecular crowding cellular environments. Macromolecular confinement can influence the ion effects in RNA folding. In this work, using the recently developed tightly bound ion model for ion fluctuation and correlation, we investigate the confinement effect on the ion-mediated RNA structural collapse for a simple model system. We found that, for both Na$^+$ and Mg$^{2+}$, ion efficiencies in mediating structural collapse/folding are significantly enhanced by the structural confinement. Such an enhancement in the ion efficiency is attributed to the decreased electrostatic free energy difference between the compact conformation ensemble and the (restricted) extended conformation ensemble due to the spatial restriction., 22 pages, 5 figures

Recent experiments pointed to the potential importance of ion correlation for multivalent ions such as Mg(2+) ions in RNA folding. In this study, we develop an all-atom model to predict the ion electrostatics in RNA folding. The model can treat ion correlation effects explicitly by considering an ensemble of discrete ion distributions. In contrast to the previous coarse-grained models that can treat ion correlation, this new model is based on all-atom nucleic acid structures. Thus, unlike the previous coarse-grained models, this new model allows us to treat complex tertiary structures such as HIV-1 DIS type RNA kissing complexes. Theory-experiment comparisons for a variety of tertiary structures indicate that the model gives improved predictions over the Poisson-Boltzmann theory, which underestimates the Mg(2+) binding in the competition with Na(+). Further systematic theory-experiment comparisons for a series of tertiary structures lead to a set of analytical formulas for Mg(2+)/Na(+) ion-binding to various RNA and DNA structures over a wide range of Mg(2+) and Na(+) concentrations.

RNAs are highly negatively charged chain molecules. Metal ions play a crucial role in RNA folding stability and conformational changes. In this work, we employ the recently developed tightly bound ion (TBI) model, which accounts for the correlation between ions and the fluctuation of ion distributions, to investigate the ion-dependent free energy landscape for the three-way RNA junction in a 16S rRNA domain. The predicted electrostatic free energy landscape suggests that 1), ion-mediated electrostatic interactions cause an ensemble of unfolded conformations narrowly populated around the maximally extended structure; and 2), Mg(2+) ion-induced correlation effects help bring the helices to the folded state. Nonelectrostatic interactions, such as noncanonical interactions within the junctions and between junctions and helix stems, might further limit the conformational diversity of the unfolded state, resulting in a more ordered unfolded state than the one predicted from the electrostatic effect. Moreover, the folded state is predominantly stabilized by the coaxial stacking force. The TBI-predicted folding stability agrees well with the experimental measurements for the different Na(+) and Mg(2+) ion concentrations. For Mg(2+) solutions, the TBI model, which accounts for the Mg(2+) ion correlation effect, gives more improved predictions than the Poisson-Boltzmann theory, which tends to underestimate the role of Mg(2+) in stabilizing the folded structure. Detailed control tests indicate that the dominant ion correlation effect comes from the charge-charge Coulombic correlation rather than the size (excluded volume) correlation between the ions. Furthermore, the model gives quantitative predictions for the ion size effect in the folding energy landscape and folding cooperativity.

Single-stranded junctions/loops are frequently occurring structural motifs in nucleic acid structures. Due to the polyanionic nature of the nucleic acid backbone, metal ions play a crucial role in the loop stability. Here we use the tightly bound ion theory, which can account for the possible ion correlation and ensemble (fluctuation) effects, to predict the ion-dependence of loop and stem-loop (hairpin) free energies. The predicted loop free energy is a function of the loop length, the loop end-to-end distance, and the ion (Na(+) and Mg(2+) in this study) concentrations. Based on the statistical mechanical calculations, we derive a set of empirical formulas for the loop thermodynamic parameters as functions of Na(+) and Mg(2+) concentrations. For three specific types of loops, namely, hairpin, bulge, and internal loops, the predicted free energies agree with the experimental data. Further applications of these empirical formulas to RNA and DNA hairpin stability lead to good agreements with the available experimental data. Our results indicate that the ion-dependent loop stability makes significant contribution to the overall ion-dependence of the hairpin stability.

The complexity of RNA hairpin folding arises from the interplay between the loop formation, the disruption of the slow-breaking misfolded states, and the formation of the slow-forming native base stacks. We investigate the general physical mechanism for the dependence of the RNA hairpin folding kinetics on the sequence and the length of the hairpin loop and the helix stem. For example, 1), the folding would slow down when a stable GC basepair moves to the middle of the stem; 2), hairpin with GC basepair near the loop would fold/unfold faster than the one with GC near the tail of the stem; 3), within a certain range of the stem length, a longer stem can cause faster folding; and 4), certain misfolded states can assist folding through the formation of scaffold structures to lower the entropic barrier for the folding. All our findings are directly applicable and quantitatively testable in experiments. In addition, our results can be useful for molecular design to achieve desirable fast/slow-folding hairpins, hairpins with/without specific misfolded intermediates, and hairpins that fold along designed pathways.

A DNA triplex is formed through a third strand binding to the major groove of a duplex through hydrogen bonds. Due to the high charge density of a DNA triplex, metal ions are critical for triplex stability. We recently developed the tightly bound ion (TBI) model for ion-nucleic acids interactions. The model accounts for correlation and fluctuations of the ion distribution. We now apply the TBI model to analyze the ion-dependence of thermodynamic stability for DNA triplexes. We investigate the ion-dependent stability for two experimentally studied systems: a 24-bp DNA triplex and a 15-bp triplex with the third strand containing different base sequence. Because a triplex has a higher charge density than a duplex, a triplex attracts more ions and hence causes stronger ion correlation than a duplex. Our results for the number of bound ions indicate that for a DNA triplex in a Mg2+ solution, the TBI model, which accounts for ion correlation, gives much improved predictions than Poisson-Boltzmann Equation (PB), which ignores ion correlation, and the improvement is more significant for a triplex than for a duplex. In addition, we also calculate the stability of a pair of 14-mer triple helices immersed in an ionic solution. The goal is to predict the ion-dependent free energy landscape for different sequences and helix lengths.

A recently developed tightly bound ion model can account for the correlation and fluctuation (i.e., different binding modes) of bound ions. However, the model cannot treat mixed ion solutions, which are physiologically relevant and biologically significant, and the model was based on B-DNA helices and thus cannot directly treat RNA helices. In the present study, we investigate the effects of ion correlation and fluctuation on the thermodynamic stability of finite length RNA helices immersed in a mixed solution of monovalent and divalent ions. Experimental comparisons demonstrate that the model gives improved predictions over the Poisson-Boltzmann theory, which has been found to underestimate the roles of multivalent ions such as Mg2+ in stabilizing DNA and RNA helices. The tightly bound ion model makes quantitative predictions on how the Na+-Mg2+ competition determines helix stability and its helix length-dependence. In addition, the model gives empirical formulas for the thermodynamic parameters as functions of Na+/Mg2+ concentrations and helix length. Such formulas can be quite useful for practical applications.

Transitions between the different conforma- tional states play a critical role in many RNA catalytic and regulatory functions. In this study, we use the Kinetic Monte Carlo method to investigate the kinetic mechanism for the conformational switches between bistable RNA hairpins. We find three types of conformational switch pathways for RNA hairpins: refolding after complete unfolding, folding through basepair-exchange pathways and through pseudoknot-assisted pathways, respectively. The result of the competition between the three types of pathways depends mainly on the location of the rate-limiting base stacks (such as the GC base stacks) in the structures. Depending on the structural relationships between the two bistable hairpins, the conformational switch can follow single or multiple dominant pathways. The predicted folding pathways are supported by the activation energy results derived from the Arrhenius plot as well as the NMR spectroscopy data.

One of the key issues in the theoretical prediction of RNA stability and structure is how to predict the loop free energy. Experimental results have shown strong sequence-dependence of the loop free energy. However, most currently available models only account for the loop length-dependence of the loop free energy. We recently developed a three-bead coarse-grained model to generate the three-dimensional conformations for RNA hairpin loops. Based on the pseudo-torsion angles for the coarse-grained structures, we can extract a set of pseudo-torsion angle-based statistical potential parameters from the known structures. The statistical potential parameters enable folding predictions of the low-energy coarse-grained 3D structures from the sequence. Further molecular dynamics computations for an ensemble of decoy conformations about the predicted coarse-grained structures lead to the final all-atom structures. A notable advantage of the approach is the use of the statistical potential for evaluating the loop free energies and guiding the search for the low-energy coarse-grained structures from the sequence.

Transitions between different conformational states, so called conformational switching, are intrinsic to RNA catalytic and regulatory functions. Often, conformational switching occurs on time-scales of several seconds. In combination with the recent real-time NMR experiments (Wenter et al. Angew. Chem. Int. Ed. (2005). 44, 2600; Wenter et al. ChemBioChem. (2006). 7, 417) for the transitions between bistable RNA conformations, we combine the master equation method with the kinetic cluster method to investigate the detailed kinetic mechanism and the factors that govern the folding kinetics. We propose that heat capacity change (ΔCp) upon RNA folding may be important for RNA folding kinetics. In addition, we find that for tetraloop hairpins, noncanonical (tertiary) intraloop interactions are important to determine the folding kinetics. Furthermore, through theory-experiment comparisons, we find that the different rate models for the fundamental steps (i.e., formation/disruption of a base pair or stack) can cause contrasting results in the theoretical predictions.

From the distribution of the low-lying states on the energy landscape, we recently developed a new computational method for the prediction of RNA folding kinetics [1]. The method can treat long sequences because it is based on a small ensemble of seed states. Additionally, the method is based on an analytic formalism and thus enables predictions for long-time folding kinetics. Applications of the new kinetic model to mechanical folding of RNA hairpins reveal distinct kinetic behaviors for a wide range of different force regimes, from zero force to forces much stronger than the critical force for the folding-unfolding transition. In the low force limit, folding can be initiated (nucleated) at any position by the formation of the first base stack and folding can proceed through many parallel pathways. In contrast, for a higher force, the folding/unfolding would predominantly proceed along a single zipping/unzipping pathway. Studies for different hairpin-forming sequences indicate that depending on the nucleotide sequence, a kinetic intermediate can emerge in the low force regime but disappear in high force regime, and a new kinetic intermediate, which is absent in the low and high force regimes, can emerge in the medium force range. Variations of the force lead to changes in folding cooperativity and rate-limiting steps. For TAR RNA sequence, we predicted two parallel dominant pathways. The rate-limiting folding steps (at f = 8 pN) are the formation of specific base pairs that are 2-4 base pairs away from the loop. At a higher force (f = 11 pN), the folding rate is controlled by the formation of the bulge loop. The predicted rates and transition states are in good agreement with the experimental data for a broad force regime.[1] Cao and Chen, 2009, Biophys. J., 96, 4024-4034.

Depending on the nucleotide sequence, the temperature, and other conditions, RNA hairpin-folding kinetics can be very complex. The complexity with a wide range of cooperative and noncooperative kinetic behaviors arises from the interplay between the formation of the loops, the disruption of the misfolded states, and the formation of the rate-limiting base stacks. With a rate constant model and a kinetic-cluster theory, we explore the broad landscape for RNA hairpin-folding kinetics. The model is validated through direct tests against several experimental measurements. The general kinetic folding mechanisms and the predicted great variety of folding kinetics are directly applicable and quantitatively testable in experiments. The results from this study suggest that 1), previous experimental findings based on the individual hairpins revealed only a small fraction of much broader and more complex RNA hairpin-folding landscapes; 2), even for structures as simple as hairpins, universal folding timescales and pathways do not exist; and 3), to treat the loop size as the sole factor to determine the hairpin-folding rate is an oversimplification.

Salt ions are essential for the folding of nucleic acids. We use the tightly bound ion (TBI) model, which can account for the correlations and fluctuations for the ions bound to the nucleic acids, to investigate the electrostatic free-energy landscape for two parallel nucleic acid helices in the solution of added salt. The theory is based on realistic atomic structures of the helices. In monovalent salt, the helices are predicted to repel each other. For divalent salt, while the mean-field Poisson-Boltzmann theory predicts only the repulsion, the TBI theory predicts an effective attraction between the helices. The helices are predicted to be stabilized at an interhelix distance approximately 26-36 A, and the strength of the attractive force can reach -0.37 k(B)T/bp for helix length in the range of 9-12 bp. Both the stable helix-helix distance and the strength of the attraction are strongly dependent on the salt concentration and ion size. With the increase of the salt concentration, the helix-helix attraction becomes stronger and the most stable helix-helix separation distance becomes smaller. For divalent ions, at very high ion concentration, further addition of ions leads to the weakening of the attraction. Smaller ion size causes stronger helix-helix attraction and stabilizes the helices at a shorter distance. In addition, the TBI model shows that a decrease in the solvent dielectric constant would enhance the ion-mediated attraction. The theoretical findings from the TBI theory agree with the experimental measurements on the osmotic pressure of DNA array as well as the results from the computer simulations.

Based on an ensemble of kinetically accessible conformations, we propose a new analytical model for RNA folding kinetics. The model gives populational kinetics, kinetic rates, transition states, and pathways from the rate matrix. Applications of the new kinetic model to mechanical folding of RNA hairpins such as trans-activation-responsive RNA reveal distinct kinetic behaviors in different force regimes, from zero force to forces much stronger than the critical force for the folding-unfolding transition. In the absence of force or a low force, folding can be initiated (nucleated) at any position by forming the first base stack and there exist many pathways for the folding process. In contrast, for a higher force, the folding/unfolding would predominantly proceed along a single zipping/unzipping pathway. Studies for different hairpin-forming sequences indicate that depending on the nucleotide sequence, a kinetic intermediate can emerge in the low force regime but disappear in high force regime, and a new kinetic intermediate, which is absent in the low and high force regimes, can emerge in the medium force range. Variations of the force lead to changes in folding cooperativity and rate-limiting steps. The predicted network of pathways for trans-activation-responsive RNA suggests two parallel dominant pathways. The rate-limiting folding steps (at f = 8 pN) are the formation of specific basepairs that are 2–4 basepairs away from the loop. At a higher force (f = 11 pN), the folding rate is controlled by the formation of the bulge loop. The predicted rates and transition states are in good agreement with the experimental data for a broad force regime.

Accurate quantification of the ionic contribution to RNA folding stability could greatly enhance our ability to understand and predict RNA functions. Recently, motivated by the potential importance of ion correlation and fluctuation in RNA folding, we developed the tightly bound ion (TBI) model. Extensive experimental tests showed that the TBI model can lead to better treatment of multivalent ions than the Poisson-Boltzmann equation. In this study, we use the model to quantify the contribution of salt (Na(+) and Mg(2+)) to the RNA tertiary structure folding free energy. Folding of the RNA tertiary structure often involves intermediates. We focus on the folding transition from an intermediate state to the native state, and compute the electrostatic folding free energy of the RNA. Based on systematic calculations for a variety of RNA molecules, we derive a set of formulas for the electrostatic free energy for tertiary structural folding as a function of the sequence length and compactness of the RNA and the Na(+) and Mg(2+) concentrations. Extensive comparisons with experimental data suggest that our model and the extracted empirical formulas are quite reliable.

Nucleic acids are highly charged polyanionic molecules; thus, the ionic conditions are crucial for nucleic acid structural changes such as bending. We use the tightly bound ion theory, which explicitly accounts for the correlation and ensemble effects for counterions, to calculate the electrostatic free energy landscapes for DNA helix bending. The electrostatic free energy landscapes show that DNA bending energy is strongly dependent on ion concentration, valency, and size. In a Na+ solution, DNA bending is electrostatically unfavorable because of the strong charge repulsion on backbone. With the increase of the Na+ concentration, the electrostatic bending repulsion is reduced and thus the bending becomes less unfavorable. In contrast, in an Mg2+ solution, ion correlation induces a possible attractive force between the different parts of the helical strands, resulting in bending. The electrostatically most favorable and unfavorable bending directions are toward the major and minor grooves, respectively. Decreasing the size of the divalent ions enhances the electrostatic bending attraction, causing an increased bending angle, and shifts the most favorable bending to the direction toward the minor groove. The microscopic analysis on ion-binding distribution reveals that the divalent ion-induced helix bending attraction may come from the correlated distribution of the ions across the grooves in the bending direction.

A previous statistical mechanical theory predicted the structure and stability for canonical H-type RNA pseudoknot that contains no intervening junction between the helices (Cao and Chen 2006, Nucleic Acids Research, 34: 2634-2652). In the present study, we develop a model to treat H-type RNA pseudoknot involving a single-stranded junction between the helices. Specifically, we predict (a) the conformational entropy by explicitly consider the helix-helix, helix-loop, and loop-loop excluded volume interactions, and (b) the full free energy landscape from the RNA sequence. We find that loop-helix volume exclusion severely restricts the number of accessible loop conformations. Our extensive tests for the native structures from RNA sequences show good accuracy with overall sensitivity and specificity equal to 0.91 and 0.91, respectively.

We develop a statistical mechanical model to predict the sequence-dependent folding stability from the sequence for simple RNA hairpin loops. We use a recently developed 3-vector virtual bond-based RNA folding model which enables rigorous computation of RNA chain entropy. Enumeration of all the possible arrangements of base pairs and the corresponding conformational entropies, through exhaustive self-avoiding random walks of the virtual bonds in a diamond lattice, gives the partition function and free energy for a given RNA sequence. The new model developed here has two unique advantages. First, the model is based on the complete conformational ensemble, including loop conformations that contain all the possible intra-loop base pairs and/or terminal mismatches. Therefore, intra-loop base pairs and mismatches are the results predicted from the model instead of known input information for the model. Second, the model gives accurate estimation for the dramatic entropic changes caused by the formation of the intra-loop base pairs and mismatches. Our first principles calculations for the chain entropy for each given set of base pairs, in combination with the empirical energy function provided from Turner rules, provide ab initio predictions for the sequence-dependent loop stability from RNA sequence. Tests against experimental data indicate that the theory can give improved predictions for the sequence-dependent RNA hairpin loop stability than the nearest-neighbor model.