Recent years have witnessed an explosive growth in RNA research, as numerous new noncoding RNAs (ncRNAs) have been discovered [1,2], and rich information has been revealed in the various relationships between their structures and cellular functions [3]. It is increasingly evident that RNAs play important roles, far beyond transferring genetic information from DNA to protein. Exploring the structural diversity of the RNA population constitutes a central goal in RNomics [4], which requires new computational methods for the comparison, identification and classification of RNA.

As there remain many difficulties in predicting three-dimensional RNA structure, secondary structures are typically used as a basis for researching RNA conformation. RNA secondary structure can be viewed as a combination of basic structural elements, also known as stems, hairpin loops, bulge loops, interior loops, multiple loops and external loops (the latter five categories are referred to collectively as 'loops'). Mathematical representations of RNA secondary structure are of great importance. Some approaches for deducing these structures have been proposed as planar graphs [5-9]. Among these RNA representations [5-7,9-11] is the homeomorphically irreducible tree (HIT) [10], which contains most of the RNA molecule's original structural information. Each HIT node corresponds to a structural element weighted by its 'size'. The stem elements are weighted by the number of contained base pairs, while the loop elements are weighted by their lengths. The topological nature of a HIT is a vertex-weighted and vertex-colored tree graph, in which the stem and loop vertices are color-coded. Most of the other RNA graphs give unequal prominence to stems and loops in the secondary RNA structures, that is, the stem regions are always represented as adjacent relationships between loop vertices and cannot be reflected directly in the matrix representations and numerical descriptors. The rationality of this abstraction may depend on the opinion that single-stranded regions play important roles in RNA-RNA, RNA-DNA and RNA-protein interactions. However, some studies have revealed that stem regions are of the same importance as loop regions. For example, recent studies show that stem regions in precursors of miRNAs are indispensable for miRNA biogenesis [12-15]. Considering that stems and loops are biochemically different, an ideal RNA graphical representation should distinguish these two element types.

Graphical representation of RNA secondary structure provides the basis for the construction of topological indices. Topological indices are numeric parameters associated with patterns of connectivity among vertices, reflecting the intrinsic nature of a graph. In computational compound design, topological indices have been successfully employed in many applications such as QSAR (quantitative structure-activity relationships) and QSPR (or quantitative structure-property relationships) [16]. For RNA-related research, topological indices based on RNA graphics provide simple solutions for structure comparison, classification and enumeration [5-7,17-20], and are gaining increasing acceptance in the scientific community. In recent innovative works, Schlick et al successfully used topological indices from tree and dual graphs to explore the repertoire of RNA secondary motifs [8,9,21,22], and further uncovered structural diversity in random sequence pools [23].

However, it is difficult to construct topological indices to characterize the colors of the HIT-like fine-grained RNA graphs, because the node colors encoded in the polarity of items in the topological index definition renders the range of topological indices uncontrollable, even unmanageable in extreme cases. On the other hand, ignoring the length of the loop and stem regions can lead to index degeneracy. The RNA topological indices presented herein focused mainly on molecular connectivity descriptions. Although these indices reflect some significant aspect of RNA structure and show good performance in distinguishing between different structural patterns, they may not be appropriate for characterizing structural details. As a consequence, RNAs with different structures may share the same index value. The latent risk of high degeneracy derives mainly from the coarse-grained abstraction in RNA graph construction. Additionally, even for connectivity, no single index is sufficient. A numerical descriptor derived from the spectrum of the Laplacian matrix of the RNA graph, which has been widely used recently [8,9,17-21,24], cannot uniquely determine graph topology when the vertex number is greater than five [25].

In this study, we present a complete and fine scheme to represent RNA molecules graphically. These representations will facilitate the exploration of the numerous detailed facets of each RNA element and their combined patterns in creating RNA secondary structures. Herein we introduce three typical examples of information-rich topological indices that are based on our novel graph representations to characterize the RNA secondary structure. The involvement of the numerical range, distribution and intercorrelations of these indices for their possible rendering of useful RNA topologies are presented, and the applicability of these indices is illustrated by three case studies.

Table 1 Dataset of ncRNA sequences. A dataset of 6,305 ncRNAs taken from different Database are selected as representatives of the RNA world. These 6,305 ncRNA sequences are classified into two classes: one class covers five kinds of ncRNAs with known structures, (more ...)

The distributions of our RNA topological indices based on the dataset are illustrated in Figure 1 [see Additional file 1]. Clearly, the statistical distributions of these indices cannot be well-described by Normal distribution model, since all of the distributions are skewed to some extent. There are four typical candidate distribution models that we considered for modeling the statistical distributions of our indices. These included the Normal and Log-normal distributions, the Gamma distribution, and the Weibull distribution. The parameters of these distribution models were estimated through the maximum likelihood method, and the goodness of model fit was evaluated by Pearson's correlation. The results of the distribution modeling listed in Table 2 revealed that the Weibull distribution (average goodness of fit was 0.94, 0.92 and 0.85 for Wiener indices, Balaban indices and Randić indices, respectively) and Gamma distribution (average goodness of fit were 0.93, 0.93 and 0.84 for Wiener indices, Balaban indices and Randić indices, respectively) fit the statistical distributions of these indices well, while the Log-normal distribution (average goodness of fit were 0.88, 0.89 and 0.77 for Wiener indices, Balaban indices and Randić indices, respectively) and the Normal distribution (average goodness of fitting are 0.84, 0.81 and 0.80 for Wiener indices, Balaban indices and Randić indices, respectively) failed to describe the distributions with sufficient accuracy. These results were verified with the distribution fitting results of the representatives of the three topological index families [see Additional file 2].

Figure 1 Distributions of topological indices. The distributions of the topological indices for the dataset of 6,305 ncRNAs are illustrated. (A) Distributions of Wiener indices. (B) Distributions of Balaban indices. (C) Distributions of Randić indices. (more ...)

Table 2 Correlations between topological indices and their fitting models. Four typical distribution models (normal distribution, Gamma distribution, Weibull distribution and log-normal distribution) are employed here to model the statistical distributions of (more ...)

Since all of the definitions of topological indices (equations (1) ~ (6) in Methods section) contained a summation operation, the topological indices examined herein may include information describing the shape and size of the secondary RNA molecule structure. The Pearson's correlations [see Additional file 3] showed that the Wiener-type and Balaban-type indices did not correlate strongly with the free energies and the lengths of the RNAs, as their values did not increase substantially with RNA size [see Additional file 4, 5, 7, 8]. However, most of the Randić-type indices did correlate strongly with the free energies and the lengths of the RNAs [see Additional file 6 and 9]. Furthermore, the topological indices appeared to be independent of GC contents [see Additional file 10, 11, 12]. Theses results are consistent with the conclusions drawn in computational chemistry [16].

We conducted correlation analysis and principal component analysis (PCA) on the RNA dataset listed in Table 1. These analyses reduce the complexity of the datasets and create new orthogonal variables from combinations of the original variables that describe spatial information. Figure 2 illustrates the Pareto charts of the three topological index families, whereby the primary principal components (PCs) are arranged in descending order, with the first PC, PC1, describing the greatest proportion of the variability being followed by PCs 2, 3, 4 and so on. In addition, the Pearson's correlations among the indices within the index families are presented [see Additional file 3, and 13, 14, 15]. These results indicate that Wiener-type and Randić-type indices are highly correlated within their families, and that the first three PCs of each index family contain more than 99% of the dataset variability, which comprises the information required to construct the indices. The correlation between the Balaban-type indices, however, appears to be weaker, as they require the first five PCs to explain 99% of the information.

Figure 2 Pareto charts of topological indices. Pareto charts of three topological index families for the dataset of 6,305 ncRNAs are shown. The primary components in the Pareto chart are arranged in descending order. (A) Pareto charts of Wiener index family. (B) (more ...)

Although almost all pre-miRNAs are characterized by their stem-loop hairpin structures [28,29,35,47], a large number of pre-miRNA-like hairpins in many genomes can be folded. Distinguishing the real pre-miRNAs from other hairpin sequences with similar stem-loops (pseudo pre-miRNAs) is important both for understanding of the nature of miRNAs and for developing prediction methods for identifying miRNAs for which homology is unknown. However, this remains a challenging task. Xue et al. presented an SVM-based method for classifying real and pseudo pre-miRNAs [48]. A recent study distinguished real from pseudo pre-miRNAs using a random forest prediction model with a hybrid feature [49].

As numeric features of RNA structure, topological indices may be used to score candidates based on structure similarity measurements among the folds and structures of the reference miRNAs. We randomly chose 200 real pre-miRNAs from the 1,082 miRNAs in our dataset (Table 1) and generated 1,000 pseudo pre-miRNAs as a reference set using the dinucleotide shuffling method presented in our previous study [50]. To evaluate the potentials of topological indices as features in the miRNA identification procedure, we explored the distribution of the 200 real pre-miRNAs and the corresponding 1,000 pseudo pre-miRNAs in the topological feature space. Figures 3(A), (B) and 3(C) illustrate the 2D mapping results of these real and pseudo pre-miRNAs from the structural space to the topological feature space of the three types of topological indices using the K-means algorithm, respectively. The corresponding ROC curves are plotted in Figure 4.

Figure 3 Mapping results of miRNA identification. The mapping results of miRNA identification using K-means clustering algorithm for the three topological index families are shown. In this application case study, 200 real pre-miRNAs are randomly chosen from the (more ...)

Figure 4 ROC curves for miRNA identification. ROC curves are employed here to evaluate and compare the performance of miRNA identification for three types of topological indices. (A) ROC curve for miRNA identification using Wiener indices. (B) ROC curve for miRNA (more ...)

We ran the K-means algorithm independently for 50 times, and each time randomly chose 200 real pre-miRNAs and generated corresponding 1,000 pseudo pre-miRNAs. The average accuracy of the miRNA identification was 0.968, 0.953 and 0.985 for Wiener indices, Balaban indices and Randić indices, respectively. The sensitivity and specificity exceeded 0.95 for all three types of topological indices. Table 3 shows the details of the evaluation results of the identifications performances, indicating that the performance of Randić indices was much higher than that of the Wiener indices and Balaban indices. This finding may be attributable to the high number of RNA structural details that are encoded into the 12 Randić indices.

Table 3 Evaluation results of miRNA identification. The evaluation results of miRNA identification using K-means clustering algorithm for the three topological index families are shown. In this application case study, the K-means algorithm is run independently (more ...)

We randomly chose 25 sequences from each of the six RNA classes (5S rRNA, riboswitch, miRNA, RNase P, Intron, and tRNA; Table 1). The 2D mapping results of the K-means classification is shown in Figure 5, with the ncRNA centroids demarcated. We ran the K-means algorithm independently 50 times, and randomly chose 25 sequences from each class each time. The average clustering accuracy was about 98.0% for the three types of topological indices.

Figure 5 Mapping results of ncRNA classification. The mapping results of ncRNA classification using K-means clustering algorithm for the three topological index families are shown. In this application case study, 25 sequences of each kind are randomly chosen from (more ...)

In principle, an RNA mutation can be deleterious when it disrupts a functional site involved in catalysis, ligand-binding or protein interactions. Since ncRNA function depends critically on its secondary structure, nucleotide alterations that result in structural changes have great potential to be deleterious. Accordingly, structure analysis should help to identify deleterious mutations. Some structure-based methods and software for RNA deleterious mutation analysis have been reported [17,18,24,61,62].

To test how our topological indices can help with deleterious RNA mutation analysis, we analyzed the precursor of human miRNA miR-30a (pre-miR-30a), a stem-loop of 71 nt (Figure 6A). Figure 6B shows its mountain representation [63]. The dissimilarity of the secondary structures between the wild-type RNAs and those with possible single point mutations are measured by computing the differences between the weighted first order Randić indices. The mean structural differences among the wild-type and the possible mutants at each position were extracted into a structural deleteriousness profile [62] and plotted as waveforms (Figure 6C); the sites that were crucial for structure determination are represented by peaks with high structural deleteriousness within the profile.

Figure 6 Deleterious mutation analysis of miRNA. The results of deleterious mutation analysis for miRNA miR-30a precursor are shown. (A) The secondary structure of wild-type miR-30a. (B) The mountain representation plot of the structure of wild-type miR-30a. (C) (more ...)

It appeared that the mutations opening the base stem of the precursor led to marked differences in RNA structure, while the mutations in the terminal loop and bulges seemed to be less deleterious. This finding indicates that the base-pairing at the base of the precursor stem is of critical importance to RNA structure determination compared to the internal loops, terminal loops and bulges. These results are in good accord with the same conclusions drawn in previous experimental studies [12,13,15].

Figure 7 Element-contact graph representations for three typical RNA secondary structures. Three typical RNA secondary structures and their element-contact graph representations are illustrated. (A) Secondary structures of three typical RNAs (miRNA lin-4, SAM (more ...)

Formally, these three types of ECGs can be represented as ordered triples of disjoint sets G_SL = (V_SL, E_SL, W_SL), GS = (V_S, E_S, W_S) and G_L = (V_L, E_L, W_L), respectively, where V_SL, V_S, V_L are a set of vertices, E_SL, E_S, E_L are a set of edges, and W_SL, W_S, W_L are a set of weights. The group of these three ECGs G_E = {G_SL, G_S, G_L} forms a complete and superlative description of RNA secondary structure, which facilitates the definition of topological indices. Although there are some redundancies, all of these ECGs contribute importantly to the final analysis.

As the first non-trivial topological index, the Wiener index has become one of the most widely utilized and investigated topological indices, as it is simple to compute and offers good structure-property correlations in QSAR and QSPR studies. The Wiener index of a graph G is the half-sum of all entries in the distance matrix D = [d_ij], i.e.

Wiener-type indices can be defined for all molecular graph matrices with the Wiener operator. Suggested by Merris [64,65] and tested by Barash's group [17,18], the Wiener index has been introduced into a fine-grained RNA graph, in which each nucleotide becomes a node of the graph [66,67]. Thus, the classic Wiener indices increase rapidly with the magnitude of a graph, especially for the weighted Wiener indices. This may be the main reason why Avihoo and Barash limit their Wiener index to fine-grained RNA graphs that characterize only small RNAs (≤ 50 nt) [67]. In this study, similar to the work on the connectivity index [68], we generalized the Wiener indices by assigning α = -0.5 to the exponent of each item in the equation (1) to reduce their range.

The Balaban index of a graph G also is a distance-based graph connectivity index, defined as

where D_i and D_j denote the distance sums of the vertices v_i and v_j, and can be easily computed by summarizing corresponding rows or columns in the distance matrix, q is the number of edges in the molecular graph, μ is the cyclomatic number and the summation goes over all edges in the graph.

The Randić indices of ECGs encode aspects of element connectivity for RNA secondary structure. The m th order Randić index of a graph G is given as

where δ_v is the degree of vertex v and the summation is over the total number of sub-graphs of order m. The first two order Randić indices, ⁰χ and ¹χ, are employed in this study.

As vertex-weighted RNA graphs, ECGs offer convenience for constructing weighted numerical descriptors aimed at detailed structure characterization. The method presented by Zmazek and Zrovnik [69] is employed for extending the indices mentioned above, and the properties of vertices in equation (1) ~ (3) are multiplied by their weights. The weighted Wiener index, and the Balaban index and Randić indices of a graph are given as:

where w_i and w_j are the weights of vertex v_i and v_j, respectively.

Both the weighted and the unweighted topological indices are examined in this study to evaluate their utility and potential in structure determination applications. The exponents of each item in equations (1) ~ (6) are assigned to α = -0.5, β = -0.5, and γ = -0.5 to reduce their ranges, respectively. The symbols representing these indices are listed in Table 4.

Table 4 The symbols of the three topological index families. The symbols of the three topological index families based on element-contact graphs are shown.

We used the clustering accuracy as a measure of a clustering result. Given the final number of clusters, K, clustering accuracy r is defined as

where n is the number of instances in the data set and r_i is the number of instances partitioned into the correct cluster i. For miRNA identification, we use receiver operating characteristic (ROC) curves to evaluate and compare the classification performance. The ROC curve provides a convenient graphical display of the trade-off between true- and false-positive rates. Additional terms associated with ROC curves are sensitivity and specificity [75].

This paper presents a complete and fine-grained topological description for representing RNA graphs, and establishes a new basis for constructing RNA topological indices. Distinct from other methods, RNA secondary structure is represented by a combination of three vertex-weighted element-contact graphs. Based on the opinion that the stem and loop regions in RNA molecules have similar importance in biochemical processes, the stem and loop topologies are described in stem-contact and loop-contact graphs, respectively, while the overall pattern of the structure is abstracted into a stem-loop-contact graph. In addition, these graphs can be selected according to the needs of a particular application. Three typical topological index families defined with ECGs are described.

To investigate the relationship between the topological indices and RNA secondary structures, we constructed a detailed analysis on a dataset of 6,305 ncRNA sequences downloaded from different databases, and explored the numerical features of these indices. We then employed the topological indices to quantify the structural aspects of the selected RNAs, and utilized them to identify miRNAs, classify ncRNAs and conduct deleterious mutation analyses. Based on the applied small dataset, we find that the topological indices can distinguish true from pseudo pre-miRNAs with about 96% accuracy, and cluster known types of ncRNAs with about 98% accuracy. The results indicate that the topological indices can characterize RNA structure details, and show high potential for identifying and classifying ncRNAs. Importantly, while difficult, the successful identification and classification of ncRNAs may provide a new approach for discovering new ncRNAs. The difficulty of correctly identifying and classifying these molecules is underscored by the fact that the predictions of both Evofold [76] and RNAz [77] differ to some extent from that of the ENCODE [78] experimental data. Further research is needed to fully resolve the challenging problem of predicting and classifying ncRNAs.

The utility test and the application examples of typical topological indices defined on the ECGs illustrate their latent utility for RNA structure analysis. With the aid of topological indices, it is now possible for biologists to explore 'RNA spaces' visually, as exemplified by the three case studies presented herein. Characterizing RNA molecules using topological indices may open a door to studying the structure-function relationships of RNA molecules by combining many application algorithms for pattern recognition and classification, most of which are based on feature space. Further applications of these topological indices are represented by our studies on robustness analysis of RNA secondary structure [50,79], whereby the topological indices are employed as distance measures for secondary structures to evaluate the robustness of RNAs.

The author(s) declare that they have no competing interests.

WS wrote the programs, analyzed the results and drafted the manuscript. XB and ZZ helped in analysis and discussion, and gave useful comments. SW and XB guided the project. All authors read and approved the final manuscript.

The authors would like to thank the editors and the reviewers of the paper for their constructive comments and suggestions which contributed to an improved presentation. The authors would also like to thank the Super Biomed Computation Center at Beijing Institute of Health Administration and Medicine Information for providing computing resources. This work is supported by a grant from the National High Technology Research and Development Program of China (No. 2007AA02Z311) and a grant from the National Nature Science Foundation of China (No. 30700139).