VfoldMCPX: predicting multistrand RNA complexes

Abstract

Multistrand RNA complexes play a critical role in RNA-related biological processes. The understanding of RNA functions and the rational design of RNA nanostructures require accurate prediction of the structure and folding stability of the complexes, including those containing pseudoknots. Here, we present VfoldMCPX, a new model for predicting two-dimensional (2D) structures and folding stabilities of multistrand RNA complexes. Based on a partition function-based algorithm combined with physical loop free energy parameters, the VfoldMCPX model predicts not only the native structure but also the folding stability of the complex. An important advantage of the model is the ability to treat pseudoknotted structures. Extensive tests on structure predictions show the VfoldMCPX model provides improved accuracy for multistranded RNA complexes, especially for RNA complexes with three or more strands and/or containing pseudoknots. We have developed a freely accessible VfoldMCPX web server at http://rna.physics.missouri.edu/vfoldMCPX2.

INTRODUCTION

Multistrand RNAs can assemble to form complexes through RNA–RNA interactions. RNA–RNA interactions play crucial roles in many biological processes, such as mRNA splicing (Griffith and Swartz 2006), RNA dimerization (Paillart et al. 2004), and prokaryotic/eukaryotic gene regulation (Guil and Esteller 2015; Dutta and Srivastava 2018). Because of the hierarchical characteristics of RNA folding and the versatility/variety of RNA structures, various multistrand RNA complexes have become promising elementary building blocks for RNA nanostructure design (Shu et al. 2011; Afonin et al. 2014a; Li et al. 2015; Sharma et al. 2016). Recent advances in structural biology have led to accurate RNA structure determination for various systems (Chen 2008; Zhang et al. 2013; Sun et al. 2017) and rapid development of RNA nanotechnology (Yingling and Shapiro 2007; Bindewald et al. 2008; Guo 2010; Shukla et al. 2011; Afonin et al. 2014b; Jasinski et al. 2017). For example, Chworos et al. (2004) generated controllable jigsaw puzzle units by predicting and designing artificial RNA building blocks. Dibrov et al. (2011) solved the crystal structure of the square-shaped nanoobject self-assembled from double-stranded RNA. Furthermore, the crystal structures of the pRNA 3WJ (Zhang et al. 2013) and the hexamer structure (Zhang et al. 2013; Saeed et al. 2021) of the phi29 DNA packing motor have been demonstrated to serve as powerful building blocks for RNA nanotechnology.

Motivated by the demand for the accurate prediction of (two-dimensional, 2D) structures of RNA complexes, many computational models have been developed using thermodynamic approaches (Dimitrov and Zuker 2004; Rehmsmeier et al. 2004; Andronescu et al. 2005; Bernhart et al. 2006; Dirks et al. 2007; Busch et al. 2008; Bindewald et al. 2011, 2016; Zadeh et al. 2011; Montaseri et al. 2014; Kato et al. 2017; Legendre et al. 2019). RNAcofold (Hofacker et al. 1994; Bernhart et al. 2006; Lorenz et al. 2011; Bindewald et al. 2016), bifold (Reuter and Mathews 2010), IntaRNA (Busch et al. 2008), and PairFold (Andronescu et al. 2005) enable predictions of pseudoknot-free 2D structures with intra- and interstrand base pairs, while RNAhybrid (Rehmsmeier et al. 2004) and UNAFold (Dimitrov and Zuker 2004) mainly focus on the interstrand base pairs. To treat multistrand RNA complexes, MultiRNAFold (Andronescu et al. 2005, 2007), NUPACK (Dirks et al. 2007; Zadeh et al. 2011), and NanoFolder (Bindewald et al. 2011) programs were developed to predict multistrand structures by linking the input strands in a specific order. RCPred (Legendre et al. 2019) provides a novel method for predicting RNA complexes by modeling the predicted secondary structures and interactions as a graph, and finding the optimal solution to the constrained maximum weight clique problem. However, most of those models, NUPACK for example, can only predict multistrand RNA complexes without crossing base pairs. Because of the huge cost of computational resources and the lack of experimentally determined thermodynamic parameters, very few models can treat multistrand RNA complexes with crossing base pairs or pseudoknots. For the models mentioned above, RCPred can handle the prediction of multiple strands and complex motifs such as pseudoknots. NanoFolder, which does not explicitly account for the energetic contributions from pseudoknotted loops, can also successfully treat pseudoknots.

On the other hand, the previously developed VfoldCPX model (Xu and Chen 2016a,b) uses the Vfold-derived loop entropic parameters (Cao and Chen 2005, 2006a,b, 2009, 2011, 2012; Cao et al. 2014; Xu et al. 2014; Zhao et al. 2017) to calculate the folding stabilities, and can predict RNA–RNA complex structures with tertiary motifs, such as pseudoknotted and hairpin loop–loop kissed motifs. However, the VfoldCPX model is limited to two-strand RNA complex systems, which hampers its applications in RNA design and predictions of multistrand complexes. In the multistrand system, the intermediate states formed by any two of the strands should be accounted for because the final multistrand native structure may involve dimeric intermediate states for the global optimization of the interactions in the multistrand system (Binzel et al. 2016, 2021). Here, we develop a new model, VfoldMCPX, to treat RNA systems with multiple strands. The VfoldMCPX model is based on a partition function-based algorithm, which includes secondary structural motifs, such as hairpins, bulge/internal loops and multiway junctions (see Fig. 1A), as well as tertiary motifs, such as an H-type pseudoknotted motif (see Figs. 1B and 2). We have developed a VfoldMCPX web server to offer a user-friendly online interface for fully automated predictions of 2D structures and base-pairing probabilities of RNA multistrand complexes. The web server outputs a set of energetically stable structures for RNA complexes with up to ten strands. This new model, VfoldMCPX, as well as the web server, can provide extensive physical insights into the role of RNA–RNA interactions in RNA complexes, and can aid in the design of RNA nanostructures.

FIGURE 1.

Secondary structure (left) and arc (right) diagrams for (A) a four-strand structure without crossing base pairs, (B) a three-strand structure with crossing base pairs (an H-type pseudoknot motif). VfoldMCPX uses the phantom linkers (shown as three red dots) to connect the RNA strands. In the arc diagrams, RNA sequences are shown as black horizontal lines while base pairs are shown as blue arcs. The crossing arcs indicate the crossing base pairs. Specifically labeled motifs are in black boldface.

FIGURE 2.

(A) A schematic diagram for the H-type pseudoknot motif, which is divided into three single-stranded loops (L₁, L₂ and L₃, green and orange) and two helix stems (S₁ and S₂, blue) that may contain an internal/bulge loop. (B–I) Schematic diagrams for the H-type pseudoknot motif with one or more phantom linkers in different positions. Structures that contain the linkers are labeled. The phantom linkers are depicted as red dots.

RESULTS AND DISCUSSION

Test of the VfoldMCPX model for structure prediction

To evaluate the prediction accuracy of VfoldMCPX, we construct a test set by collecting experimentally determined RNA structures that consist of two or more unique strands. We use VfoldMCPX and other available models to predict the structure for each test case and compare the performance of the different models. The structures in the test data set are from the PDB database (Berman et al. 2000), and for each 3D structure, its 2D structure is extracted by RNApdbee (Antczak et al. 2014; Zok et al. 2018). We only consider multistrand RNAs with the total nucleotide length larger than 25 nt. After removing redundant structures (structures with sequence identity larger than $90\text{\%}$), we have 29 pseudoknot-free structures with two unique strands, 4 H-type pseudoknot-containing structures with two unique strands, and five structures with three or more unique strands. We then use these structures to test VfoldMCPX and other four models that are available through web servers or source codes: RNAcofold (Hofacker et al. 1994; Bernhart et al. 2006; Lorenz et al. 2011), bifold (from the RNAStructure suite) (Reuter and Mathews 2010), MultiRNAFold (Andronescu et al. 2005, 2007), and NUPACK (Dirks et al. 2007; Zadeh et al. 2011). The results from RNAcofold and bifold are obtained from the respective web servers http://rna.tbi.univie.ac.at/cgi-bin/RNAWebSuite/RNAcofold.cgi for RNAcofold and https://rna.urmc.rochester.edu/RNAstructureWeb/Servers/bifold/bifold.html for bifold, respectively. The results from MultiRNAFold and NUPACK are obtained by downloading the source code, version 3.2 for NUPACK and version 2.1 for MultiRNAFold, and running the predictions locally. We note that RNAcofold and bifold are limited to pseudoknot-free structures with two RNA strands, and MultiRNAFold and NUPACK can treat RNA structures with three or more strands; however, pseudoknots are not allowed in these predictions.

The prediction accuracy is evaluated by the positive predictive value (PPV), sensitivity (STY) values, and the F₁ score:

$$STY=TP/\left(TP+FN\right);PV=TP/\left(TP+FP\right)$$

$$F_1\mathrm{score}=2\times \left(STY\times PPV\right)/\left(STY+PPV\right),$$

where TP, FN, and FP denote the true positive, false negative, and false positive base pairs, respectively. PPV and STY measure the precision (positive predictive value) and sensitivity (recall) of the model, respectively, and F₁ score is a balanced measurement between PPV and STY, defined as the harmonic mean of them.

We first compare the performance of VfoldMCPX and other four models on the 29 pseudoknot-free structures with two unique strands. As shown in Figure 3, five models are ranked from top to bottom according to the mean STY, PPV, and F₁ score, respectively. The mean values are labeled and depicted as dots in the figure. The error bars are depicted using confidence intervals at 95% confidence level. The upper and lower bounds of the error bar reflect the statistical difference; hence a narrow error bar indicates less random errors and better accuracy. As shown in Figure 3, all the models show nearly the same performance with VfoldMCPX giving slightly higher mean STY, PPV, and F₁ score, as well as narrower error bars. The result suggests that the VfoldMCPX model is on par with the other models and provides slightly improved predictions for structures containing two strands.

FIGURE 3.

Comparison of the positive predictive value (PPV), sensitivity (STY), and F₁ score between the predicted structures from VfoldMCPX, RNAcofold (Hofacker et al. 1994; Bernhart et al. 2006; Lorenz et al. 2011), bifold (Reuter and Mathews 2010), MultiRNAFold (Andronescu et al. 2005, 2007), and NUPACK (Dirks et al. 2007; Zadeh et al. 2011). The data set consists of 29 RNA-only, pseudoknot-free structures with two unique strands, obtained from the PDB database (Berman et al. 2000). Mean values are labeled and depicted as dots. Error bars are depicted using confidence intervals at 95% confidence level. Models are ranked from top to bottom according to the mean values. All the models show nearly the same performance with VfoldMCPX slightly outperforming other models in terms of mean PPV, STY, and F₁ score.

To evaluate the performance of VfoldMCPX on more complicated RNA complex systems, we test VfoldMCPX on five structures with three or more unique strands, and four H-type pseudoknot-containing structures (PDB ID shown in Table 1). Two models that can treat three or more RNA strands are also tested (MultiRNAFold and NUPACK). The result is reported in Table 1, boldface numbers indicate the models that give better or the same results. The test result suggests that VfoldMCPX outperforms NUPACK and performs on par with MultiRNAFold on structures with three or more unique strands. We note that with STY, PPV, and F₁ score all equal to 1.00, VfoldMCPX makes the exact prediction for PDB structure 4KZ2. For the prediction of structures with H-type pseudoknots, as shown in Table 1, VfoldMCPX gives high STY, PPV, and F₁ score in all four cases. We note that among the tested models, VfoldMCPX is the only one that can treat pseudoknots, and all the test scores for VfoldMCPX are higher than 0.80 except for the STY in PDB 6YML. The predicted structures from VfoldMCPX and the native structures extracted from the 3D crystal structures are drawn in Supplemental Figure S3 for structures with three or more strands, and in Supplemental Figure S4 for structures containing H-type pseudoknots. We note that VfoldMCPX gives low scores for structure PDB-4G6R. The native structure of PDB-4G6R contains a 7 × 9 internal loop and a 4 × 4 internal loop; the incorrect prediction for the two large internal loops causes the low scores for VfoldMCPX. The crystal structure of 4G6R indicates strong intra- and interloop interactions for the two internal loops. Currently, VfoldMCPX cannot explicitly treat the loop sequence-dependence and cannot consider tertiary structural motifs such as intraloop kissing motifs, which may cause inaccuracy in the predictions.

TABLE 1.

Comparisons between the structure prediction results from VfoldMCPX, MultiRNAFold (Andronescu et al. 2005, 2007), and NUPACK (Dirks et al. 2007; Zadeh et al. 2011) on five structures with three or more unique strands and structure prediction results from VfoldMCPX on four H-type pseudoknot-containing structures

We then compare VfoldMCPX with NanoFolder. In the comparison, we use the data set in the published work of NanoFolder (Bindewald et al. 2011). The data set contains 10 pseudoknot-free structures with two strands (PDB ID shown in Supplemental Table S1). The results are shown as the Box-Whisker plot of the Matthews correlation coefficients (Supplemental Fig. S1), where the results of NanoFolder are from the published work (Bindewald et al. 2011). The Matthews correlation coefficient (MCC) is defined from the true positive (TP), false negative (FN), false positive (FP), and true negative (TN) base pairs:

$$MCC={\displaystyle \frac{TP\times TN-FP\times FN}{\sqrt{\left(TP+FP\right)\left(TP+FN\right)\left(TN+FP\right)\left(TN+FN\right)}}.}$$

MCC measures the quality of binaryclassifications.The result suggests that VfoldMCPX can make predictions on this data set with MCC for all the tested structures higher than 0.80. We note that VfoldMCPX gives higher median MCC (0.94) and a better minimum score (0.80).

We further test VfoldMCPX against RCPred. In the test, we use the structure data set in the published work of RCPred (Legendre et al. 2019). The data set consists of 23 PDB structures (Berman et al. 2000) (PDB ID shown in Supplemental Table S2) each containing two or more unique strands with a chain length longer than 25 nt. The structures predicted by RCPred are obtained by the RCPred home page at https://evryrna.ibisc.univ-evry.fr/evryrna/RCPred/RCPred_home. The native 2D structures are extracted from the 3D crystal structures using RNApdbee (Antczak et al. 2014; Zok et al. 2018). The prediction accuracy is evaluated by the PPV, STY, and F₁ score. As shown in Supplemental Figure S2, the mean values are depicted as dots while the error bars are shown with confidence intervals at a 95% confidence level. The result suggests that VfoldMCPX can give reliable predictions with all the mean values larger than 0.88. The two models show similar performances with VfoldMCPX giving slightly higher mean PPV, STY, and F₁ score. We note that RCPred relays on available thermodynamic models for generating the input secondary structures and RNA–RNA interactions, while VfoldMCPX can directly predict folding thermodynamics and thermodynamic properties such as base-pairing probabilities.

Predicting melting thermodynamics for two-strand systems

For two-strand systems, we are able to predict heat capacity melting curves and melting temperatures using VfoldMCPX. We compute the heat capacity melting curve C(T) from the partition function Q(T):$C\left(T\right)={\displaystyle \mathrm{\partial }/\mathrm{\partial }T\left[k_B{T}^2{\displaystyle \mathrm{\partial }/\mathrm{\partial }T\ln Q\left(T\right)}\right],}$where the partition function Q(T) is calculated from the sum over all the possible 2D structures Q_total. We include the concentration-dependent free energy term ΔG_{concentration} = −k_B Tln(C_T/a) in the free eenrgy calculation. Here C_T denotes the total concentration of RNA strands, k_B is the Boltzmann constant, a is equal to 1 for self-complementary strands and 4 for non-self-complementary strands (Xia et al. 1998).

As an example, we investigate the folding thermodynamics for the 3WJ-forming two-strand system (Liu et al. 2011). The thermodynamic measurements are available for the system by optical melting and fluorescence competition experiments. When predicting the melting curves, we use the sequences, the order of strands, and the solution condition (23.5 µM of total RNA concentration) obtained from the experiment (Liu et al. 2011). The predicted melting curve is reported in Figure 4A, which shows two peaks at T = 59.5°C and T = 80.5°C, respectively. We note that the heat capacity melting temperature T = 59.5°C predicted by VfoldMCPX is close to the experiment-determined melting temperature T_m = 57.6°C. The second peak at T = 80.5°C is out of the experiment range. As shown in Figure 4B, structures are inferred from the base-pairing probabilities at four different temperatures, T = 37°C, T = 55°C, T = 70°C, and T = 85°C. The structure predicted at T = 37°C agrees with the experimental determined structure, which shows a 3WJ structure with Stem 1, Stem 2, and Stem 3 formed. Our calculation further indicates that the unfolding of Stem 2 occurs between T = 55°C and T = 70°C. The unfolding of Stem 1 occurs between T = 70°C and T = 85°C.

FIGURE 4.

Predicted melting curve (A) and secondary structures (B) for the 3WJ-forming two-strand system (Liu et al. 2011). (A) The heat capacity is computed from the total partition function of all the possible 2D structures and the total concentration of two strands (23.5 µM). The first peak at T = 59.5°C matches the experiment-determined melting temperature T = 57.6°C (Liu et al. 2011). (B) The base-pairing probability-inferred structures at T = 37°C, T = 55°C, T = 70°C, and T = 85°C, respectively. Two strands are shown in blue and red.

Multistrand RNA structure prediction

VfoldMCPX provides a new approach for predicting multistrand RNA complexes that may contain pseudoknots. In this section, we show the applications of the model by predicting 2D structures for two experimentally studied systems: the core domain of pRNA (Sharma et al. 2016) and the body region of snRNA (Nguyen et al. 2016). To perform structural predictions, we use the sequences and the order of strands from the nanostructure assembly experiments (Sharma et al. 2016) for the pRNA system and from the 3D crystal structure for the snRNA system.

The multistrand pRNA structure motif can be utilized as the core scaffold for constructing RNA dendrimers (Shu et al. 2011; Sharma et al. 2016). The pRNA-3WJ core is a three-way junction (3WJ) system that consists of three unique strands: 3WJ-a, 3WJ-b, and 3WJ-c (see Fig. 5A). Experimental observations have suggested that the pRNA-3WJ can be reversed by switching the 5′- and 3′-ends. The reversed system is denoted as pRNA-3WJ_reverse, which contains three reversed strands: 3WJ-a_reverse, 3WJ-b_reverse, and 3WJ-c_reverse (see Fig. 5B). We use VfoldMCPX to compute the base-pairing probabilities and determine the most probable structures for the pRNA-3WJ and the pRNA-3WJ_reverse. The results are reported in Figure 5A,B. We find that the predicted 2D structures are consistent with the 3WJ structures observed in the experiments. Furthermore, the base-pairing probabilities for the most probable structure (Fig. 5A) are around 99%. The high probability is consistent with the experimental finding, which suggests that the pRNA-3WJ motif is ultra-stable.

FIGURE 5.

Predicted 2D structures for (A) pRNA-3WJ, (B) pRNA-3WJ_reverse, and (C) “Module-2” agree with the experimentally determined structures (Sharma et al. 2016). (A) The predicted structure for pRNA-3WJ with three RNA strands: 3WJ-a (blue), 3WJ-b (red), and 3WJ-c (green). (B) The predicted structure for pRNA-3WJ_reverse with three RNA strands: 3WJ-a_reverse (blue), 3WJ-b_reverse (red), and 3WJ-c_reverse (green). (C) The predicted structure for “Module-2” with nine RNA strands shown in blue and red.

To further illustrate the application of the VfoldMCPX model, we investigate the “Module-2” (Sharma et al. 2016), a functional module that is assembled from three different units with the pRNA-3WJ core harboring 3WJ-a_reverse, 3WJ-b_reverse, and 3WJ-c_reverse, respectively. The large number of RNA strands in “Module-2” makes the structure prediction and experimental structure assembly quite challenging. Experimentally, a step-wise assembly is used to ensure the formation of “Module-2” and to reduce the probability of misfolding. Here, we treat the “Module-2” as a RNA complex with nine strands, and predict the most probable structure using VfoldMCPX. The result is reported in Figure 5C, with nine strands shown in blue and red. The predicted “Module-2” structure is consistent with the structure designed and observed in the experiment. Despite the large number of strands, VfoldMCPX well predicts the designed structure, which suggests that this model is promising for predicting large RNA complexes.

Multistrand RNA complexes with tertiary contacts frequently occur in the spliceosome (Will and Lührmann 2011). Here, we investigate the body region of the yeast U4/U6.U5 tri-snRNP, which is an essential part of the spliceosome. The complex consists of three small nuclear RNAs (snRNAs): U4, U6, and U5. Figure 6A shows the 3D and 2D structures extracted from the cryo-electron microscopy structure (PDB 5GAP) (Nguyen et al. 2016). The 2D structure and the tertiary interactions of the three snRNAs are critical for revealing the activation mechanism of the spliceosome (Will and Lührmann 2011). Here, we treat the system as an RNA complex with three strands and predict the most probable pseudoknot-containing structure using VfoldMCPX. Three snRNAs are connected using two phantom linkers in the strand order U5–U6–U4. As shown in Figure 6B, the predicted structure is in overall agreement with the experimentally determined native structure: Both the theoretically predicted and the experimentally determined structures contain the H-type pseudoknot with crossing base pairs between U5 and U6 RNAs and two stems with crossing base pairs between U4 and U6 RNAs. We note that in the crystal structure, the body region of the yeast U4/U6.U5 tri-snRNP contains multiple proteins interacting with the three snRNAs. For example, in the cryo-electron microscopy structure, the conserved hairpin loop in U5 RNA points toward the conserved surface of the Prp8 protein due to RNA–protein interactions. Such RNA–protein interactions may affect the inaccuracy in our predictions.

FIGURE 6.

Experimentally determined (A) and theoretically predicted (B) structures of the body region of the yeast U4/U6.U5 tri-snRNP, which contains three RNA strands: U4 snRNA (green), U5 snRNA (blue), and U6 snRNA (red). Structures in A are extracted from the cryo-electron microscopy structure (PDB 5GAP) (Nguyen et al. 2016). Experimentally determined 2D structure in A is similar to the predicted 2D structure in B.

In VfoldMCPX, the input RNA strands are connected as an effective single-strand sequence connected by the phantom linkers. However, the permutation of the input strands directly affects the effective single-strand RNA sequence and hence impacts the result of structure prediction. Therefore, we develop an algorithm to test all the possible strand orders and find the one that gives the most stable structure: (1) Enumerate all the possible permutations of the N input strands; (2) for each permutation, predict a set of energetically stable structures, including the structures with and without H-type pseudoknots; (3) combine all the predicted structures and rank the structures according to the predicted free energies. The structure (with the corresponding strand order) of the lowest free energy is predicted to be the most stable structure.

We have tested the above algorithm on six RNA structures with three or more strands obtained from the PDB database (Berman et al. 2000). We use RNApdbee (Antczak et al. 2014; Zok et al. 2018) to extract the 2D structures, identify the strand IDs and the native strand orders from the 3D crystal structures. The test results and the structures are reported in Table 2. Out of the six tested structures, the VfoldMCPX predicted the correct strand order for the most stable structure for five cases. Furthermore, VfoldMCPX predicts that the most stable structures do not contain H-type pseudoknots, which also agrees with the experimentally determined structures. However, the large number of strand permutations may lead to exceedingly long computational time for the structure prediction for all the possible strand orders (N! for N strands).

TABLE 2.

Comparisons of the different strand orders between the predicted most stable structures and experimentally determined structures

Web server

We have developed a freely accessible web server to provide a user-friendly interface for using VfoldMCPX to predict multistrand 2D structures and base-pairing probabilities. The input of the web server is a set of RNA strands. The user can specify the number of strands N, the folding temperature, and the base stacking energy parameters either from Turner's parameters (Turner and Mathews 2010) or from the MFOLD parameter set (Zuker 2003). Furthermore, the user has the option to specify the type of the structure included in the prediction: (1) No pseudoknots are allowed to form; (2) H-type pseudoknots are allowed to form; (3) The (virtual) connection of the RNA strands follows a user-defined sequential order (from RNA 1 to RNA N); (4) All the different sequential orders of the strands are enumerated.

After submitting the job, the user receives a page displaying the job information. Each submitted job has a unique job ID, containing the job name provided by the user and a four-letter unique job code randomly generated by the server. The user can bookmark the page to check the job status. The page is automatically updated and displays the results once the computation is finished. Figure 7 shows a screenshot of the result page for the prediction of the phi29 pRNA 3WJ core (PDB 4KZ2) (Zhang et al. 2013). The VfoldMCPX web server outputs two types of files: (1) a set of energetically stable structures, ranked by the free energies and visualized by the VARNA applet (Darty et al. 2009); (2) base-pairing probabilities. Note that the results for structures with H-type pseudoknots are listed separately. Because multistrand RNA complexes are prone to the formation alternative folds with similar stabilities, it is important for the user to identify possible alternative structures from the predicted base-pairing probabilities. The URL link for the web server is http://rna.physics.missouri.edu/vfoldMCPX2.

FIGURE 7.

A screenshot of the 2D structure prediction result for phi29 pRNA 3WJ core (PDB 4KZ2) (Zhang et al. 2013), generated by VfoldMCPX web server. The web server displays the job information, the predicted structures, and the links to the downloadable files for the predicted structures and the corresponding base-pairing probabilities. The links and the corresponding files are indicated with arrows and boxes. The results for structures with and without pseudoknots are listed separately.

Computational efficiency and restrictions

The computational time for the prediction without pseudoknots ranges from seconds to minutes for a multistrand RNA system with the total nucleotide length less than 300 nt. Predictions for structures with H-type pseudoknots are computationally time-consuming, and could take several hours for RNA complexes with the total nucleotide length around 150 nt.

To avoid extremely long computational time, we restrict the total number of strands to nine and the total length of the strands to L_tot ≤ 300 nt for predictions without pseudoknots and L_tot ≤ 150 nt for predictions with H-type pseudoknots.

Conclusions

We have developed VfoldMCPX, a new, physics-based model for predicting 2D structures and folding thermodynamics of multistrand RNA complexes, with and without pseudoknots. Theory-experiment comparisons indicate that VfoldMCPX can provide accurate predictions for pseudoknot-forming RNA complexes with multiple strands. Comparisons between different models show that VfoldMCPX gives improved structure predictions for multistrand RNA complexes. VfoldMCPX and the web server may aid in the design of RNA nanostructures and provide novel physical insights into RNA–RNA interactions. Currently, VfoldMCPX can treat RNA complex systems with H-type pseudoknots; future development of the model should consider more complicated tertiary structural motifs such as intramolecular kissing motifs.

MATERIALS AND METHODS

VfoldMCPX transforms the input multistrand RNA to an effective single-strand RNA sequence using the phantom linkers. The phantom linker is a 3-nt linker marked as “XXX” in the connected sequence and depicted as red dots in Figure 1. From the input N RNA strands, the model determines the permutation of the strands through the strand number (from strand 1 to strand N) and inserts N − 1 phantom linkers to connect the strands. The nucleotides “X” in the phantom linkers are not allowed to participate in any base-pairing. The strand concentration-dependent free energy for the initiation of strand association is assumed to be independent of the RNA sequence, therefore, all the predicted multistrand RNA complexes have the same constant initiation energy. As a result, the different structures for the complexation of a given number of strands would involve the same initiation, thus we do not include the constant initiation energy term in the free energy calculation. Using the Vfold model-derived (Cao and Chen 2006a, 2011, 2012; Cao et al. 2010, 2014; Xu et al. 2014) loop entropy parameters to calculate the free energies for each sampled structure, VfoldMCPX can predict 2D structures for multistrand RNA complexes without crossing base pairs and with H-type pseudoknots. The main features and the algorithms of the model are described below.

Structures without crossing base pairs (secondary structures)

After linking the input RNA strands to a single-strand sequence, VfoldMCPX enumerates all the possible base pair arrangements (2D structures). Non-pseudoknotted structural motifs were enumerated by the model, including helices, hairpin loops, bulge/internal loops, and multiway junctions (see Fig. 1A). VfoldMCPX computes the enthalpy and entropy for the helices based on Turner's experimental data (Mathews et al. 2004) for the canonical and mismatched base stacks, and uses the Vfold-derived entropies for the loops. Because the nucleotides in the phantom linkers are not allowed to form base pairs, they can only appear in loops. The phantom linkers are not physical linkers; they break the chain connectivity, hence we treat linker-containing loops as loops with zero entropic contributions. This treatment allows us to perform structural predictions following the procedure below:

1. Enumerate all the possible base pair arrangements for the connected single-strand sequence.

2. Calculate the free energy for each helix by adding the free energies of all the constituent base stacks: $\mathrm{\Delta }{G}_{\mathrm{helix}}=\sum _{\mathrm{stack}}\mathrm{\Delta }{G}_{\mathrm{stack}}$, where ΔG_stack are given by Turner's experimental data (Mathews et al. 2004).

3. Enumerate all the possible intraloop (mismatched) base pairs for the loops and calculate the loop free energy by

   $$\mathrm{\Delta }{G}_{\mathrm{loop}}=-k_{B}T\ln \sum _{\mathrm{mismatch}}{e}^{-\left(\mathrm{\Delta }{G}_{\mathrm{mismatch}}-T\mathrm{\Delta }{S}_{\mathrm{loop}}\right)/\left(k_{B}T\right)},$$

   where the loop partition function is computed from the Boltzmann sum over all the possible intraloop mismatched base stacks. ΔG_mismatch denotes the free energy for mismatched base stacks, ΔS_loop denotes the loop entropy and is given by the pretabulated Vfold-derived parameters if the loop does not contain any phantom linkers, and ΔS_loop = 0 otherwise.
4. Compute the total partition function from all the possible 2D structures by

   $$Q_{\mathrm{total}}=\sum _{\mathrm{all}\mathrm{structures}}{e}^{-\left(\mathrm{\Delta }{G}_{\mathrm{helix}}+\mathrm{\Delta }{G}_{\mathrm{loop}}\right)/\left(k_{B}T\right)}.$$

5. Compute the base-pairing probability P_ij for nucleotides i and j from the conditional partition function Q_ij and the total partition function Q_total :P_ij = Q_ij/Q_total, where Q_ij is computed for all the possible structures that contain base pair (i, j) between nucleotides i and j. We predict the most probable and the alternative structures from P_ij for all the possible (i, j) pairs.

Structures containing H-type pseudoknots

A multistrand RNA can fold into structures with pseudoknotted base pairs. VfoldMCPX separates out pseudoknot-containing structures from pseudoknot-free structures when computing the partition functions. VfoldMCPX can predict 2D structures with H-type pseudoknots using the available thermodynamic parameters. A schematic figure for the H-type pseudoknot motif is given in Figure 2A. The H-type pseudoknot motif consists of three single-stranded loops (L₁, L₂, and L₃) and two helix stems (S₁ and S₂). The helix stem may contain a short internal/bulge loop with loop lengths of (l₁₁, l₁₂) and (l₂₁, l₂₂). We consider loop lengths ≤3 nt and assume that with such short loops, stems S₁ and S₂ remain quasi-linear in the form of an A-form helix. Therefore, considering the 3D helical shape of the strands, we use effective lengths $S_1^{\mathrm{eff}}$ and $S_2^{\mathrm{eff}}$ for the two stems (Cao and Chen 2009):

$$S_1^{\mathrm{eff}}=l_{12}+h_{12}+h_{11};S_2^{\mathrm{eff}}=l_{21}+h_{21}+h_{22}.$$

We use l₁₂ and l₂₁ for calculating the effective lengths because they closely positioned upstream and downstream from the center loop L₂, which is the core loop of the motif that directly determines the relative orientation of the two stems. Therefore, the pseudoknot motif shown in Figure 2A can be described using five parameters: $\left(L_1,L_2,L_3,S_1^{\mathrm{eff}},S_2^{\mathrm{eff}}\right)$. The $\mathrm{\Delta }S\left(S_1^{\mathrm{eff}},S_2^{\mathrm{eff}},L_1,L_2,L_3\right)$ parameters for various pseudoknot structures as characterized by $\left(S_1^{\mathrm{eff}},S_2^{\mathrm{eff}},L_1,L_2,L_3\right)$ have been computed and tabulated by the Vfold model-based computations (Cao and Chen 2009, 2011, 2012; Cao et al. 2010). However, as shown in Figure 2B–I, depending on the positions in the structure, the phantom linkers can alter/break a pseudoknot, therefore we treat the different scenarios separately. Specifically, VfoldMCPX predicts 2D structures with H-type pseudoknots following the procedure below:

1. Generate a helix pool for all the possible helices. Enumerate all the possible pseudoknot structures assembled from the helices.

2. Assign entropy ΔS_pseudoknot to the H-type pseudoknot motif according to the numbers and positions of the phantom linkers:

   • If no phantom linkers (Fig. 2A,B) are present in the loops, we treat the structure as an intact H-type pseudoknot motif: ΔS_pseudoknot = ΔS(S₁, S₂, L₁, L₂, L₃), where ΔS(S₁, S₂, L₁, L₂, L₃) is the Vfold-derived entropy for a single-strand H-type pseudoknot.
   • If a phantom linker is located in loop L₁ or L₃ (Fig. 2C,D), we treat the structure as a hairpin with an effective reduced loop size by counting only the unpaired nucleotides for the loop. Due to lack of entropy parameters, the current version of the model does not consider structures where the phantom linkers are located in loop L₂ (Fig. 2E).
   • If there exist two or more phantom linkers (Fig. 2F–I) in the pseudoknot loops, the structure is effectively decomposed to form a secondary motif without crossing base pairs.
3. Compute the pseudoknot free energy:

   $$\mathrm{\Delta }{G}_{\mathrm{pseudoknot}}=\mathrm{\Delta }{G}_{\mathrm{helix}\mathrm{stem}}\left(S_1\right)+\mathrm{\Delta }{G}_{\mathrm{helix}\mathrm{stem}}\left(S_2\right)-T\mathrm{\Delta }{S}_{\mathrm{pseudoknot}},$$

   where ΔG_{helix stem} (S₁) (ΔG_{helix stem} (S₂)) is the free energy of the helix stem S₁ (S₂).

4. Calculate the total partition function Q_total from all the possible structures that contain pseudoknots. Compute the base-pairing probability P_ij from the conditional partition function Q_ij. Predict the most probable structures and alternative structures from P_ij.

Due to the availability of thermodynamic parameters, currently we use the tabulated parameters for S₁ and S₂ ≤10 base pairs, L₁ and L₃ ≤7 nt, and L₂ ≤6 nt. For large L₁ and L₃ loops (>7 nt), we use the formula ΔS = alog(l) + b to compute the loop entropy, where l is the loop length, and a and b are fitted parameters (Serra and Turner 1995). Furthermore, we allow nested helices in three loops (L₁, L₂, and L₃) and calculate the effective loop length as the sum of the number of unpaired nucleotides and the number of nested helices.

SUPPLEMENTAL MATERIAL

Supplemental material is available for this article.