RNA pseudoknots play a critical role in RNA-related biology from the assembly of ribosome to the regulation of viral gene expression. A predictive model for pseudoknot structure and stability is essential for understanding and designing RNA structure and function. A previous statistical mechanical theory allows us to treat canonical H-type RNA pseudoknots that contain no intervening loop between the helices (see S. Cao and S.J. Chen [2006] in Nucleic Acids Research, Vol. 34; pp. 2634–2652). Biologically significant RNA pseudoknots often contain interhelix loops. Predicting the structure and stability for such more-general pseudoknots remains an unsolved problem. In the present study, we develop a predictive model for pseudoknots with interhelix loops. The model gives conformational entropy, stability, and the free-energy landscape from RNA sequences. The main features of this new model are the computation of the conformational entropy and folding free-energy base on the complete conformational ensemble and rigorous treatment for the excluded volume effects. Extensive tests for the structural predictions show overall good accuracy with average sensitivity and specificity equal to 0.91 and 0.91, respectively. The theory developed here may be a solid starting point for first-principles modeling of more complex, larger RNAs.

A three-vector virtual bond model

Because the torsional angles of the C–O bonds (Fig. 1B, ε, β) in the nucleotide backbone tend to adopt the trans isomeric state, Olson (Olson and Flory 1972; Olson 1980) proposed to use a two-vector virtual bond to represent nucleic structures (see Fig. 1A). We recently developed a virtual-bond-based RNA folding model (the Vfold model) for H-type RNA pseudoknots (Cao and Chen 2006b). In Vfold, we model the helix as an A-form RNA helix using the experimentally measured atomic coordinates. For loops, which can be flexible, we use the usual gauche + (g +), trans (t), and gauche −1 (g −1) rotational isomeric states for polymers (Flory 1969) to sample backbone conformations. The fact that the three isomeric states can be exactly configured in a diamond lattice (Cao and Chen 2005, 2006b) suggests that we can effectively configure the loop conformations as random walks of the virtual bonds on a diamond lattice. We note that the rotameric nature of RNA backbone conformations also has been observed for the known RNA structures (Duarte and Pyle 1998; Murthy et al. 1999; Murray et al. 2003; Wadley et al. 2007; Richardson et al. 2008).

The traditional two-vector virtual bond model cannot describe the base orientations. Motivated by the need to explicitly include base orientations in the structural description, we here propose a three-vector virtual bond model by introducing a third virtual bond to describe the base orientation (see Fig. 1B). Specifically, we add the N1 (for pyrimidine) or N9 (for purine) atom to the original P–C4 and C4–P virtual bonds (Fig. 1B). From the PDB database (Michiels et al. 2001; Theimer et al. 2005) for RNA pseudoknots, we find that the distance (D CN) between N1 (N9) and C4 atoms is close to 3.9 Å. In addition, we find that the torsion angle (χ) between plane Pi–C4–Pi +1 and plane Pi–C4–N1 (N9) is close to the g −1 isomeric state; see the distributions for D CN and the torsion angle in Figure 2. The localized distributions for the virtual bond C4–N1 (N9) in Figure 2 suggest that C4–N1 (N9) is quite rigid and can be configured in a diamond lattice. A previous study on RNA molecules also suggested a rigid base orientation (Olson and Flory 1972).

FIGURE 2.

Survey of the (DCN, χ) distributions for two pseudoknot structures: (A) the 47-nt ΔU177 pseudoknot (Theimer et al. 2005) (PDB code: 1YMO) and (B) the 36-nt SRV-1 pseudoknot (Michiels et al. 2001) (PDB code: 1E95). DCN is the length of the C4–N1,9 virtual bond, and χ is the torsion angle between the Pi–C4–Pi +1 plane and the Pi–C4–N1,9 plane.

Figure 1A shows a pseudoknot with an interhelix loop. We use the atomic coordinates of the A-form RNA helix to configure the helices (Arnott and Hukins 1972). The (r, θ, z) coordinates (in a cylindrical coordinate system) for the P, C4, and N1 (or N9) atoms are (8.71 Å, 70.5 + 32.7i, −3.75 + 2.81i), (9.68 Å, 46.9 + 32.7i, −3.10 + 2.81i), and (7.12 Å, 37.2 + 32.7i, −1.39 + 2.81i) (i = 0, 1, 2, …) (Arnott and Hukins 1972), respectively. For the other strand, we need to negate θ and z.

We generate loop (L1, L2, or L3) conformations through self-avoiding walks in a diamond lattice (Cao and Chen 2005), where a virtual bond is represented by a lattice bond. In the Vfold model, helices are configured off-lattice, while loop conformations are on-lattice. Loops and helices are connected via the six loop–helix interfacial nucleotides (Fig. 1A,C, a1, a2, b1, b2, c1, c2). These six interfacial nucleotides can be configured either as on-lattice loop terminals or as off-lattice helix terminals. We connect the loop and the helix through the minimum RMSD between the on-lattice and off-lattice coordinates for the six terminal nucleotides. Our calculation shows that the minimum RMSD is as small as 0.56 Å for the (Pi, C4, N1 or N9, Pi +1) virtual bond atoms. The small RMSD indicates a smooth connection/transition between the on-lattice loop and the off-lattice helix.

Enumeration of helix orientations

The orientations of helices S1 and S2 are determined by the coordinates of the terminal nucleotides (Fig. 1A, c1, c2) of the loop L3. To enumerate the relative orientations of the helices, we fix the (Pi, C4, N1 or N9, Pi +1) coordinates for c1, then enumerate the viable (Pj, C4, N1 or N9, Pj +1) coordinates for c2. Specifically, we enumerate the loop L3 conformations as self-avoiding random walks of the virtual bonds in a diamond lattice. The number of possible coordinates of the terminal nucleotide c2 (specifically, the coordinates of Pj, C4, N1, or N9, and Pj +1 atoms of nucleotide c2) is much smaller than the number of loop L3 conformations (see Fig. 4A, below). Therefore, the number of helix orientations, as determined by the c1 and c2 positions/coordinates, increases with L3 much more slowly than the number of loop (L3) conformations. For instance, the number of helix orientations grows as 73 → 390 → 1358 → 3208 → 6096 → 10,272 → 15,984 for an increasing interhelix loop length 1 → 2 → 3 → 4 → 5 → 6 → 7 nt. The slow growth of the number of helix orientations makes the exact enumeration of all the possible helix orientations computationally viable.

FIGURE 4.

(A, filled triangle) The number of single-stranded RNA chain conformations grows much faster than (unfilled triangle) the number of the end–end configurations (relative coordinates) of the chain. (B) We study the excluded volume effects on the loop entropy for a given pseudoknot with S1 = 4 bp, S2 = 4 bp, L1 = 4 nt, and L3 = 2 nt. We vary the loop length for L2 from 1 nt to 7 nt and find that we can neglect the loop–loop excluded volume interactions. (Open square) Results without excluded volume interactions; (filled triangle) results without considering the loop–loop excluded volume interactions; (open triangle) results with all the excluded volume interactions fully considered. (C) Comparison between the conformational entropy from the exact computer enumeration and the entropy from our theory. The deviation is small (<1 k BT in free energy).

Enumeration of loop conformations for a given helix orientation

For each relative orientation of the helices S1 and S2, we compute ΩPK in Equation 1 by enumerating the conformations for loops L1 and L2 and loop L3. The key is how to treat the excluded volume effect, i.e., the effect that different atoms cannot bump into each other.

We have two choices to treat the excluded volume effect here. We may fit the off-lattice helix onto the diamond lattice, then both helix and loop are configured in the same lattice, thus the volume exclusion effect can be conveniently treated in the lattice framework. Such an approach is computationally time-consuming because it requires off-lattice → on-lattice fitting for all the helices for each and every helix orientation. Given the large number of helix orientations that we enumerate (Equation 1), the excluded volume treatment based on the above lattice-fitting is highly inefficient.

Alternatively, we can take a different approach that avoids the off-lattice → on-lattice fitting procedure. The strategy of the alternative approach is to keep the off-lattice helix structure. To treat the mixed system with the on-lattice loop conformations and off-lattice helix structure, we introduce a cutoff distance Dc such that atoms separated by a distance below the cutoff are considered to bump into each other, and the corresponding conformation is eliminated. Such a cutoff would allow us to treat the excluded volume effect in a unified framework, irrespective of the on-lattice or off-lattice representation of the conformations. We determine the value of the cutoff distance Dc from the criterion that it gives the same entropy as the one calculated from the off-lattice → on-lattice fitting. We found that the optimal Dc value is 2.8 Å (see Fig. 3C,D). Therefore, in our entropy computation, we use Dc = 2.8 Å as the criterion for volume exclusion.

FIGURE 3.

Test of the excluded volume effects with different distance cutoffs (Dc). (A) Using a pseudoknot motif as the test system, we find that the cutoff distance of 2.8 Å can best reproduce the conformational count from exact enumeration of on-lattice conformations. (B) The triangles are the calculated entropies with cutoff distance varying from 2.2 Å to 3.4 Å. The line denotes the entropies from the on-lattice exact computer enumeration. (C) We also test the cutoff distances for different stem lengths while keeping the loop lengths fixed and (D) for different loop lengths while keeping the stem length fixed. (Dashed lines) The results with off-lattice helix and the cutoff distance; (solid lines) the results with on-lattice exact conformational enumerations. (B–D) The y-axes are the numbers of loop (L2) conformations (in log scale).

For a fixed helix–helix orientation, we enumerate the loop conformations through self-avoiding random walks in the diamond lattice. The excluded volume between helix and helix, helix and loop, and loop and loop are explicitly considered. The treatment here for the excluded volume effect is more rigorous than previous Gaussian chain-based models (Gultyaev et al. 1999; Isambert and Siggia 2000; Bon et al. 2008), which ignore the excluded volume effect.

Using the three-vector virtual bond conformational model developed here, we can test the strengths of the different excluded volume effects (helix–helix, helix–loop, and loop–loop) (see Fig. 4B). We find that the loop–helix excluded volume interaction is strong. In contrast, the loop–loop excluded volume effect is rather weak (Fig. 4B), suggesting that we can neglect loop–loop correlations when enumerating the conformations. This leads to the “independent loop approximation”: we can ignore the presence of the other loops when we count the conformations for a loop.

Tests for the above method against exact computer enumeration suggest that our method can give reliable conformational entropies (see Fig. 4C). We have computed and tabulated the loop entropies ΔS(S1, S2, L1, L2, L3) for stems S1 and S2 as ≤10 bp; loops L1, L2 as ≤7 nt; and loop L3 as between 2 nt and 6 nt. For L3 ≤ 1 nt, we use the entropy table in Cao and Chen (2006b). The results for ΔS(S1, S2, L1, L2, L3) are compiled as an entropy table (see Supplemental Table S1). On a Dell EM64T cluster system with 48 Intel(R) Xeon(R) 5150 (2.66 GHz) processors, the computation of the full entropy table took less than a week.

For large loops L1 > 7 nt or L2 > 7 nt, we use the following fitted formula (Serra and Turner 1995) for the entropy ΔS:

where l is the loop length and a and b are fitted parameters; see Supplemental Tables S2 and S3 for the a and b parameters for different loop sizes.

Comparison with other models

We measure the accuracy of structure predictions by two parameters: (1) the sensitivity parameter SE, defined as the ratio between the number of correctly predicted base pairs and number of the base pairs in the experimentally determined structure; and (2) the specificity parameter SP, defined as the ratio between the number of correctly predicted base pairs and the total number of predicted base pairs. Our tests for structural predictions indicate that the model developed here gives better results than other models that we have tested (see Tables 1, ​,2).2). Specifically, our model gives the highest SE value for 15 sequences among the total 18 sequences, and the highest SP value for 13 sequences. In addition, our model gives higher overall average SE (0.91) and SP (0.91) than other models.

TABLE 1.

The sensitivity (SE) values for the structures predicted from seven different models

TABLE 2.

The specificity (SP) values for the predicted structures from seven different models

H-type pseudoknot

LP-PK1 and SRV-1 are two H-type pseudoknots. LP-PK1 is a PK1 domain of Legionella pneumophila tmRNA (Zwieb et al. 1999), and Simian retrovirus type-1 (SRV-1) (ten Dam et al. 1995) forms a pseudoknot that promotes the ribosomal frameshifting. For the two H-type pseudoknots, our Vfold model gives the highest SE value (see Fig. 6A,B; Table 1). We note that the ILM model gives a false prediction for the SRV-1 pseudoknot. The failure of the ILM model may be due to the fact that the model does not account for the loop entropy for pseudoknots.

FIGURE 6.

The predicted structures for three pseudoknots. (A) For LP-PK1, Hotknots (Ren et al. 2005), ILM (Ruan et al. 2004), pknotsRE (Rivas and Eddy 1999), STAR (Gultyaev et al. 1995), and pknots-RG (Reeder and Giegerich 2004) all give poor predictions for the structure (SE = 0.5). NUPACK (Dirks and Pierce 2003) gives a relatively high SE value (SE = 0.8). Our Vfold model gives the highest accuracy with SE = 0.9. (B) For the SRV-1 pseudoknot, the ILM model fails to predict the native structure of the SRV-1 pseudoknot. (C) For the 70.8 anti-HIV aptamer, we predict a pseudoknot with a 3-nt interhelix loop. In the calculations, the temperature is 37°C for A and B and 20°C for C according to the experimental condition (Held et al. 2006a,b). Also shown in the figures are the density plot for the base-pairing probability Pij (Equation 4). In the density plots, the horizontal and vertical axes denote the indices of the nucleotides i and j.

VMV pseudoknot

From a recent biochemical study, Pennell and Brierley and colleagues found that the stimulatory RNA for VMV frameshifting forms a pseudoknot structure (Pennell et al. 2008) instead of a stem–loop structure. Moreover, the pseudoknot is quite unique because it contains a long interhelix loop. We perform the structural prediction for this 67-nt RNA. Figure 7A shows that the predicted structure agrees exactly with the experimental structure, with SE and SP values both equal to 1.

FIGURE 7.

The density plots and the predicted structures for the 67-nt 5′ VMV pseudoknot at different temperatures. Stem S1 is the most stable stem, which is the last one to be unzipped. At T = 37°C, our predicted structure shows the highest SE = 1.0 and SP = 1.0 as tested against the experimental structure (Pennell et al. 2008).

R2 retrotransposon pseudoknot

The 5′ header of the R2 retrotransposon controls the R2 protein binding and cleavage of the DNA target (Christensen et al. 2006). Based on the NMR spectra and computational models (Hart et al. 2008), Hart and Turner and colleagues found a knotted structure in the 74-nt header of the R2 retrotransposon. In this study, we use the Vfold model developed here to predict the secondary structure for the 74-nt header. Figure 8A shows our predicted structure. The predicted structure is a pseudoknot with four stems. All four stems have been found in the experiments (Hart et al. 2008). The predicted structure shows a high accuracy with SE = 1.0 and SP = 1.0. In the calculation, we have added the base-stacking energy for the Watson-Crick base pairs between nucleotides 48CG49 and 62CG63 (see the dashed lines in Fig. 8A). This tertiary interaction has been confirmed in previous NMR measurement (Ferré-D'Amaré et al. 1998).

Anti-HIV RNA aptamer

Recently experiments suggested that the interhelix loop may be essential for efficient ribosomal frameshifting (Brierley et al. 2008; Giedroc and Cornish 2008). Moreover, previous experimental studies on the anti-HIV RNA aptamer (Held et al. 2006a,b) suggested that the interhelix loop, which causes flexible helix orientations in the pseudoknot aptamers, may play an important functional role in accommodating aptamer binding to the HIV reverse transcriptase. For example, for an aptamer (labeled as 70.8 according to the notations used in the literature) (Held et al. 2006a,b), the proposed native structure contains a 3-nt interhelix loop. The predicted structure from our model (Fig. 6C), indeed, shows a 3-nt interhelix loop. The structure has a high accuracy of SE = 0.9 and SP = 1.0 if we treat the experimentally proposed structure (Held et al. 2006a,b) as the “experimental” structure.

The 70.8 aptamer

As the temperature increases, stems S1 and S2 of the pseudoknot (see Fig. 6A) is disrupted at nearly the same temperature. At T = 80°C, both stems are unfolded. Thus, stems S1 and S2 have the comparable thermal stability.

VMV pseudoknot

A recent combined biochemical and NMR experiment (Pennell et al. 2008) showed that the VMV pseudoknot contains a 6-nt interhelix loop. Our predicted unfolding pathway suggests that at T = 80°C, stem S2 is the first stem to be unzipped, and stem S1 is the last one to be unzipped (see Fig. 7B,C). Our prediction agrees with the experimental finding (Pennell et al. 2008), which suggested that S1 is the most stable and S2 is disrupted at a temperature around 76.8°C.

R2 retrotransposon pseudoknot

The native structure of the R2 retrotransposon pseudoknot contains four stems (Fig. 8A). The structure shows high stability against temperature increase. At T = 80°C, stem S1 is the first stem to be unfolded, resulting in an intermediate state that contains stem S3 and a partially unfolded stem S2. As the temperature is further increased to 90°C, stem S2 becomes completely unzipped since the hairpin with S2 is destabilized by the large loop. Stem S3 is the most robust stem and is the last stem to be unzipped. The melting temperature for S3 is ∼100°C.

We thank Professor Donald H. Burke for useful discussions. The research was supported by NIH through grant GM063732 (to S.-J.C.). Most of the computations involved in this research were performed on the HPC resources at the University of Missouri Bioinformatics Consortium (UMBC).

Article published online ahead of print. Article and publication date are at http://www.rnajournal.org/cgi/doi/10.1261/rna.1429009.

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