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.

To establish a comprehensive basis for new RNA structure descriptors, and to avoid the use of colored graphics, we used three distinct non-colored ECGs compensated by one another to characterize the secondary structure of an RNA molecule. Similar to the classical HIT, the topology of all structural elements in RNA secondary structure are represented in a stem-loop-contact graph (SLCG), in which the stems and loops are all assigned as vertices (□) without differences, and the edges (-) represent connection relationships. Two other ECGs derived from SLCG are stem-contact graph (SCG) and loop-contact graph (LCG), describing stem and loop topology, respectively. The relationship between the usual form of typical RNA secondary structures and their element-contact graph representations are illustrated in Figure 7. In a LCG, as with some classical RNA graphs 569, stem elements are abstracted into the edges (-) between loop elements, while loop elements are represented as vertices (○). In a SCG, however, the stem topology cannot be obtained by simply abstracting the loops into vertices (●), and stems into edges (-) conversely, since the branches of the RNA graph always end with loop elements. Only the loops between two or more stems can be described as edges; hairpins and external loops cannot be described in the SCG. Stems connected with multiple loops are considered to be adjacent to each other and therefore joined with edges. The stem elements in the SLCG and SCG, distinct from the HIT, are all weighted by the number of nucleotides included.

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.

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'.

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.

W ( G ) = ∑ i < j ( d i j ) α MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4vaCLaeiikaGIaem4raCKaeiykaKIaeyypa0ZaaabuaeaacqGGOaakcqWGKbazdaWgaaWcbaGaemyAaKMaemOAaOgabeaakiabcMcaPmaaCaaaleqabaGaeqySdegaaaqaaiabdMgaPjabgYda8iabdQgaQbqab0GaeyyeIuoaaaa@3EBA@

Wiener-type indices can be defined for all molecular graph matrices with the Wiener operator. Suggested by Merris 6465 and tested by Barash's group 1718, the Wiener index has been introduced into a fine-grained RNA graph, in which each nucleotide becomes a node of the graph 6667. 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

J ( G ) = q μ + 1 ∑ v i v j ( D i D j ) β MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOsaOKaeiikaGIaem4raCKaeiykaKIaeyypa0tcfa4aaSaaaeaacqWGXbqCaeaacqaH8oqBcqGHRaWkcqaIXaqmaaGcdaaeqbqaaiabcIcaOiabdseaenaaBaaaleaacqWGPbqAaeqaaOGaemiraq0aaSbaaSqaaiabdQgaQbqabaGccqGGPaqkdaahaaWcbeqaaiabek7aIbaaaeaacqWG2bGDdaWgaaadbaGaemyAaKgabeaaliabdAha2naaBaaameaacqWGQbGAaeqaaaWcbeqdcqGHris5aaaa@479A@

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

χ m ( G ) = ∑ v i 1 v i 2 ⋯ v i m + 1 ( δ i 1 δ i 2 ⋯ δ i m + 1 ) γ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@5AB1@

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:

W ( G ) = ∑ i < j ( w i w j d i j ) α MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4vaCLaeiikaGIaem4raCKaeiykaKIaeyypa0ZaaabuaeaacqGGOaakcqWG3bWDdaWgaaWcbaGaemyAaKgabeaakiabdEha3naaBaaaleaacqWGQbGAaeqaaOGaemizaq2aaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGGPaqkdaahaaWcbeqaaiabeg7aHbaaaeaacqWGPbqAcqGH8aapcqWGQbGAaeqaniabggHiLdaaaa@44CC@

J ( G ) = q μ + 1 ∑ v i v j ( w i D i ⋅ w j D j ) β MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOsaOKaeiikaGIaem4raCKaeiykaKIaeyypa0tcfa4aaSaaaeaacqWGXbqCaeaacqaH8oqBcqGHRaWkcqaIXaqmaaGcdaaeqbqaaiabcIcaOiabdEha3naaBaaaleaacqWGPbqAaeqaaOGaemiraq0aaSbaaSqaaiabdMgaPbqabaGccqGHflY1cqWG3bWDdaWgaaWcbaGaemOAaOgabeaakiabdseaenaaBaaaleaacqWGQbGAaeqaaOGaeiykaKYaaWbaaSqabeaacqaHYoGyaaaabaGaemODay3aaSbaaWqaaiabdMgaPbqabaWccqWG2bGDdaWgaaadbaGaemOAaOgabeaaaSqab0GaeyyeIuoaaaa@4FF6@

χ m ( G ) = ∑ v i 1 v i 2 ⋯ v i m + 1 ( w i 1 δ i 1 ⋅ w i 2 δ i 2 ⋅ ⋯ ⋅ w i m + 1 δ i m + 1 ) γ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaWraaSqabeaacqWGTbqBaaGccqaHhpWycqGGOaakcqWGhbWrcqGGPaqkcqGH9aqpdaaeqbqaaiabcIcaOiabdEha3naaBaaaleaacqWGPbqAdaWgaaadbaGaeGymaedabeaaaSqabaGccqaH0oazdaWgaaWcbaGaemyAaK2aaSbaaWqaaiabigdaXaqabaaaleqaaOGaeyyXICTaem4DaC3aaSbaaSqaaiabdMgaPnaaBaaameaacqaIYaGmaeqaaaWcbeaakiabes7aKnaaBaaaleaacqWGPbqAdaWgaaadbaGaeGOmaidabeaaaSqabaGccqGHflY1cqWIVlctcqGHflY1cqWG3bWDdaWgaaWcbaGaemyAaK2aaSbaaWqaaiabd2gaTjabgUcaRiabigdaXaqabaaaleqaaOGaeqiTdq2aaSbaaSqaaiabdMgaPnaaBaaameaacqWGTbqBcqGHRaWkcqaIXaqmaeqaaaWcbeaakiabcMcaPmaaCaaaleqabaGaeq4SdCgaaaqaaiabdAha2naaBaaameaacqWGPbqAdaWgaaqaaiabigdaXaqabaaabeaaliabdAha2naaBaaameaacqWGPbqAdaWgaaqaaiabikdaYaqabaaabeaaliabl+UimjabdAha2naaBaaameaacqWGPbqAdaWgaaqaaiabd2gaTjabgUcaRiabigdaXaqabaaabeaaaSqab0GaeyyeIuoaaaa@7066@

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.

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.

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.

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

r = ∑ i = 1 K r i n MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOCaiNaeyypa0tcfa4aaSaaaeaadaaeWbqaaiabdkhaYnaaBaaabaGaemyAaKgabeaaaeaacqWGPbqAcqGH9aqpcqaIXaqmaeaacqWGlbWsaiabggHiLdaabaGaemOBa4gaaaaa@3A1C@

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.