- Methodology article
- Open Access
A novel representation of RNA secondary structure based on element-contact graphs
- Wenjie Shu^{1, 2},
- Xiaochen Bo^{1}Email author,
- Zhiqiang Zheng^{2} and
- Shengqi Wang^{1}Email author
https://doi.org/10.1186/1471-2105-9-188
© Shu et al; licensee BioMed Central Ltd. 2008
Received: 04 September 2007
Accepted: 11 April 2008
Published: 11 April 2008
Abstract
Background
Depending on their specific structures, noncoding RNAs (ncRNAs) play important roles in many biological processes. Interest in developing new topological indices based on RNA graphs has been revived in recent years, as such indices can be used to compare, identify and classify RNAs. Although the topological indices presented before characterize the main topological features of RNA secondary structures, information on RNA structural details is ignored to some degree. Therefore, it is necessity to identify topological features with low degeneracy based on complete and fine-grained RNA graphical representations.
Results
In this study, we present a complete and fine scheme for RNA graph representation as a new basis for constructing RNA topological indices. We propose a combination of three vertex-weighted element-contact graphs (ECGs) to describe the RNA element details and their adjacent patterns in RNA secondary structure. Both the stem and loop topologies are encoded completely in the ECGs. The relationship among the three typical topological index families defined by their ECGs and RNA secondary structures was investigated from a dataset of 6,305 ncRNAs. The applicability of topological indices is illustrated by three application case studies. Based on the applied small dataset, we find that the topological indices can distinguish true pre-miRNAs from pseudo pre-miRNAs with about 96% accuracy, and can cluster known types of ncRNAs with about 98% accuracy, respectively.
Conclusion
The results indicate that the topological indices can characterize the details of RNA structures and may have a potential role in identifying and classifying ncRNAs. Moreover, these indices may lead to a new approach for discovering novel ncRNAs. However, further research is needed to fully resolve the challenging problem of predicting and classifying noncoding RNAs.
Keywords
Background
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.
Results
Statistical properties of topological indices
Numerical range and distribution of topological indices
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, and the other class is made up of six kinds of ncRNAs with predicted structures by Vienna RNA package.
Category | Number | Length (nt) | dG (Kcal/mol) | GC% | ||
---|---|---|---|---|---|---|
Mean ± SD | Min | Max | ||||
The sequences with known structures | ||||||
5S ^{(1)} | 147 | 121 ± 3 | 113 | 135 | -49.97 ± 10.08 | 0.58 ± 0.05 |
16S ^{(1)} | 647 | 1532 ± 284 | 612 | 2741 | -556.15 ± 156.85 | 0.49 ± 0.08 |
Intron ^{(1)} | 144 | 615 ± 418 | 210 | 2630 | -191.53 ± 99.46 | 0.46 ± 0.11 |
RNase P ^{(2)} | 466 | 332 ± 49 | 189 | 486 | -139.83 ± 35.4 | 0.57 ± 0.09 |
tRNA ^{(3)} | 1272 | 76 ± 5 | 56 | 94 | -29.28 ± 5.31 | 0.58 ± 0.06 |
The sequences with predicted structures | ||||||
tmRNA ^{(4)} | 140 | 359 ± 30 | 251 | 423 | -117.44 ± 25.45 | 0.47 ± 0.09 |
5.8S ^{(4)} | 1168 | 146 ± 27 | 29 | 180 | -41.33 ± 11.99 | 0.49 ± 0.06 |
SRP ^{(4)} | 262 | 225 ± 96 | 78 | 339 | -95.71 ± 45.65 | 0.57 ± 0.08 |
miRNA ^{(4)} | 1082 | 89 ± 16 | 55 | 153 | -37.55 ± 8.89 | 0.46 ± 0.08 |
Guide ^{(4)} | 977 | 141 ± 47 | 66 | 459 | -47.69 ± 23.52 | 0.51 ± 0.11 |
Total | 6305 | 296 ± 445 | 29 | 2741 | -106.93 ± 165.81 | 0.52 ± 0.09 |
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 the three topological index families. The parameters of these distribution models are estimated through maximum likelihood method, and the goodness of model fitting is evaluated by Pearson's correlation coefficient. For each topological index, the highest Pearson's correlation coefficient is in bold.
Index family | Indices | Normal | Gamma | Weilbull | Log-normal |
---|---|---|---|---|---|
Wiener | ${W}_{SL}^{w}$ | 0.96 | 0.95 | 0.97 | 0.88 |
W _{ SL } | 0.83 | 0.93 | 0.93 | 0.83 | |
${W}_{S}^{w}$ | 0.83 | 1.00 | 0.99 | 0.96 | |
W _{ S } | 0.77 | 0.93 | 0.93 | 0.92 | |
${W}_{L}^{w}$ | 0.93 | 0.94 | 0.95 | 0.85 | |
W _{ L } | 0.74 | 0.85 | 0.85 | 0.81 | |
Balaban | ${J}_{SL}^{w}$ | 0.91 | 0.99 | 0.99 | 0.95 |
J _{ SL } | 0.76 | 0.91 | 0.91 | 0.89 | |
${J}_{S}^{w}$ | 0.82 | 0.98 | 0.96 | 0.93 | |
J _{ S } | 0.81 | 0.91 | 0.89 | 0.89 | |
${J}_{L}^{w}$ | 0.92 | 0.93 | 0.94 | 0.86 | |
J _{ L } | 0.64 | 0.84 | 0.84 | 0.82 | |
Randić | $\begin{array}{c}0\end{array}{\chi}_{SL}^{w}$ | 0.93 | 0.80 | 0.86 | 0.69 |
$\begin{array}{c}1\end{array}{\chi}_{SL}^{w}$ | 0.84 | 0.93 | 0.93 | 0.88 | |
^{0} χ _{ SL } | 0.72 | 0.74 | 0.75 | 0.69 | |
^{1} χ _{ SL } | 0.82 | 0.79 | 0.81 | 0.71 | |
$\begin{array}{c}0\end{array}{\chi}_{S}^{w}$ | 0.95 | 0.96 | 0.96 | 0.89 | |
$\begin{array}{c}1\end{array}{\chi}_{S}^{w}$ | 0.82 | 0.99 | 0.98 | 0.94 | |
^{0} χ _{ S } | 0.57 | 0.57 | 0.58 | 0.51 | |
^{1} χ _{ S } | 0.87 | 0.91 | 0.91 | 0.86 | |
$\begin{array}{c}0\end{array}{\chi}_{L}^{w}$ | 0.92 | 0.96 | 0.97 | 0.87 | |
$\begin{array}{c}1\end{array}{\chi}_{L}^{w}$ | 0.80 | 0.99 | 0.99 | 0.95 | |
^{0} χ _{ L } | 0.52 | 0.53 | 0.54 | 0.51 | |
^{1} χ _{ L } | 0.81 | 0.85 | 0.86 | 0.80 |
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].
Intercorrelations of topological indices
Clearly, no single topological index is sufficient to characterize the broad range of structure-function relationship studies on RNA molecule formation. Considering that various structural features of RNAs are usually correlated, the intercorrelations among topological indices should be examined when multiple topological indices are used. Moreover, it is useful to reduce the redundancy and create an orthogonal structural space.
Application case studies
After defining the topological indices based on ECGs and analyzing their statistical properties, the questions naturally arose to regarding the potential utility of the knowledge of these indices. The answers came from the following three application case studies of our topological indices, in which they have been employed to quantify the structural aspects of RNA molecules.
Identification of miRNAs
Novel ncRNAs are difficult to detect experimentally, due to their short lengths, low expression levels, tissue specificity and lack of polyadenylation. Therefore, the most effective method for discovering ncRNAs may be computational identification of ncRNA candidates followed by biochemical verification [26]. Because of the strong inter-dependence between structure and function, incorporating structural features into ncRNA scanning programs could improve the accuracy of candidate identification. Based on secondary structure conservation, RNA structural information has been used in several ways in recently published works to identify microRNA (miRNA) candidates in select genomes [27–35]. The miRNAs molecules are abundant endogenous ~22-nucleotide (nt) noncoding RNAs that can play important roles in gene regulation at the post-transcriptional level. Roles include cleavage or translational repression through the binding of a minimal-recognition 'seed' sequence [36–39]. The miRNAs are transcribed as long primary molecules, which are processed into ~70 nt miRNA precursors (pre-miRNAs) that fold into a stem-loop hairpin structures via nuclear RNase III Drosha [12]. Mature miRNAs (~22 nt) are cleaved from pre-miRNAs through the action of Dicer endonuclease [40–42]. Throughout the miRNA biogenesis procedure, the hairpin structure of the pre-miRNA plays a crucial role, acting as the structure motif for expotin-5 in nuclear-cytoplasm transportation, and as a substrate for Dicer enzyme [13, 41, 43–46].
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].
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 for 50 times. For each test, 200 real pre-miRNAs are randomly chosen from the 1,082 miRNAs in dataset of Table 1 and the corresponding 1,000 pseudo pre-miRNAs are generated as reference set. The clustering accuracy, sensitivity and specificity are employed here to evaluate the performance of the identification results for Wiener indices, Balaban indices and Randić indices, respectively.
Index | Clustering accuracy | Sensitivity | Specificity | ||||||
---|---|---|---|---|---|---|---|---|---|
Mean ± SD | Min | Max | Mean ± SD | Min | Max | Mean ± SD | Min | Max | |
Balaban | 0.9534 ± 0.005 | 0.9475 | 0.9658 | 0.9597 ± 0.0072 | 0.955 | 0.980 | 0.9621 ± 0.0068 | 0.953 | 0.968 |
Wiener | 0.968 ± 0.0014 | 0.9658 | 0.9708 | 0.9623 ± 0.0026 | 0.957 | 0.985 | 0.9891 ± 0.002 | 0.986 | 0.993 |
Randić | 0.9849 ± 0.0026 | 0.9808 | 0.9908 | 0.9842 ± 0.0047 | 0.975 | 0.990 | 0.9851 ± 0.0033 | 0.979 | 0.992 |
Classification of ncRNAs
With the rapidly increasing knowledge of the cellular roles of RNA molecules [51, 52], the expanding repertoire of known functional RNAs has spurred renewed efforts to catalogue and classify RNA structures. An understanding of structural diversity in RNA populations is crucial for identifying novel RNA structures and pursuing RNA genomics initiatives. Since RNA secondary topologies are remarkably well conserved across functional classes, their topological characteristics provide a basis for organizing RNA secondary structures on a broad scale [53]. In this report, we used topological indices to catalogue and to classify RNA structures based on the correlations between conserved RNA secondary structures and topological indices. This method is similar to that of RNA-As-Graphs (RAG) [8, 21], which classifies RNA structures based on the topological properties of their secondary motifs using graph theory results.
Deleterious mutation analysis of RNA
Mutations in RNA genes may lead to striking alterations in the 2D RNA structures that impair cellular functions, resulting in certain diseases [54]. For example, mutations of tRNAs in mitochondria were reported to harbor more than half of the known mitochondrial pathogenic mutations [55]. Recent research has further shown that mutations in miRNA genes and their flanking sequences may contribute to cancer [56–58]. On the other hand, deleterious RNA mutations in pathogenic species can be exploited. Yassin et al demonstrated that deleterious mutations in bacterial rRNAs can serve as hallmarks of antibiotic sites [59]. Additionally, in their study on influenza viruses, Herlocher et al. found a nonsense mutation on a PB2 segment that caused monumental differences in the RNA secondary structure; a finding that can be used to make a live vaccine [60].
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].
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].
Methods
Element-contact graph representations for RNA secondary structure
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.
Classical topological indices based on ECGs
Most topological indices used in computational chemistry can be extended easily into ECGs to characterize RNA secondary structure. In our study, three of the most widely used topological indices were redefined in ECGs for application testing, comprised of the Wiener, Randić and Balaban indices. These indices are essentially the mathematical properties of a graph characterizing its 'compactness'.
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.
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.
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, ^{0}χ and ^{1}χ, are employed in this study.
where w_{ i }and w_{ j }are the weights of vertex v_{ i }and v_{ j }, respectively.
The symbols of the three topological index families. The symbols of the three topological index families based on element-contact graphs are shown.
Index family | Indices based on SLCG | Indices based on SCG | Indices based on LCG | |||
---|---|---|---|---|---|---|
weighted | unweighted | weighted | unweighted | weighted | unweighted | |
Wiener | ${W}_{SL}^{w}$ | W _{ SL } | ${W}_{S}^{w}$ | W _{ S } | ${W}_{L}^{w}$ | W _{ L } |
Balaban | ${J}_{SL}^{w}$ | J _{ SL } | ${J}_{S}^{w}$ | J _{ S } | ${J}_{L}^{w}$ | J _{ L } |
Randić | $\begin{array}{c}0\end{array}{\chi}_{SL}^{w}$, $\begin{array}{c}1\end{array}{\chi}_{SL}^{w}$ | ^{0}χ_{ SL }, ^{1}χ_{ SL } | $\begin{array}{c}0\end{array}{\chi}_{S}^{w}$, $\begin{array}{c}1\end{array}{\chi}_{S}^{w}$ | ^{0}χ_{ S }, ^{1}χ_{ S } | $\begin{array}{c}0\end{array}{\chi}_{L}^{w}$, $\begin{array}{c}1\end{array}{\chi}_{L}^{w}$ | ^{0}χ_{ L }, ^{1}χ_{ L } |
Dataset of ncRNAs
To explore the relationship among the topological indices and RNA secondary structures, we have selected a dataset of 6,305 ncRNAs as representatives of the human RNA population and have evaluated their topological indices. We divided these 6,305 ncRNAs into two classes. One class covers five ncRNA types with known structures, obtained from the Comparative RNA Web Site [70], RNase P database [71] and the Genomic tRNA Database [72]. The second class is composed of six ncRNA types with secondary structures predicted by the Vienna RNA package [73]. All of these ncRNAs were obtained from Rfam [53]. Table 1 provides a detailed description of the dataset.
Clustering algorithm, and its performance evaluation and visualization
The K-means algorithm [74] is one of the most important and most widespread approaches to prototype-based clustering. The K-means methodology is based on the idea that a center point can represent a cluster. Thus, K-means defines a prototype in terms of a centroid, which is usually the mean or median point of a group of points. Herein, we used the PCA mapping method to visualize the 'RNA spaces' of the clustering results, which is very useful in the analysis and visualization of the correlated high-dimensional data.
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].
Discussion and Conclusion
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.
Declarations
Acknowledgements
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).
Authors’ Affiliations
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