A fast structural multiple alignment method for long RNA sequences

  • Yasuo Tabei1, 2,

    Affiliated with

    • Hisanori Kiryu2,

      Affiliated with

      • Taishin Kin2 and

        Affiliated with

        • Kiyoshi Asai1, 2Email author

          Affiliated with

          BMC Bioinformatics20089:33

          DOI: 10.1186/1471-2105-9-33

          Received: 14 September 2007

          Accepted: 23 January 2008

          Published: 23 January 2008

          Abstract

          Background

          Aligning multiple RNA sequences is essential for analyzing non-coding RNAs. Although many alignment methods for non-coding RNAs, including Sankoff's algorithm for strict structural alignments, have been proposed, they are either inaccurate or computationally too expensive. Faster methods with reasonable accuracies are required for genome-scale analyses.

          Results

          We propose a fast algorithm for multiple structural alignments of RNA sequences that is an extension of our pairwise structural alignment method (implemented in SCARNA). The accuracies of the implemented software, MXSCARNA, are at least as favorable as those of state-of-art algorithms that are computationally much more expensive in time and memory.

          Conclusion

          The proposed method for structural alignment of multiple RNA sequences is fast enough for large-scale analyses with accuracies at least comparable to those of existing algorithms. The source code of MXSCARNA and its web server are available at http://​mxscarna.​ncrna.​org.

          Background

          Non-coding RNAs (ncRNAs) are transcribed RNA molecules that do not encode proteins. Their functions often depend on their 3D-structures rather than their primary sequences. The secondary structures of RNA sequences can be identified by various methods, including minimization of the free energy [13]. However, it is not always possible to obtain the accurate secondary structures. More reliable predictions of the secondary structures are possible if we have a set of RNA sequences with a common secondary structure. For consensus structure prediction, RNAalifold [4], Pfold [5], and McCaskill-MEA [6] are applicable only to sets of aligned RNA sequences. Multiple alignment tools that consider only sequence similarities, e.g. ClustalW [7], Dialign [8], and T-Coffee [9], however, have limited accuracy for RNA sequences with low similarity.

          Simultaneous prediction of the common secondary structure and optimal alignment of RNA sequences is computationally quite expensive, even if pseudo-knotted structures are excluded. For example, the strict algorithm of Sankoff [10] requires O(L 3N ) in time and O(L 2N ) in memory for N sequences of length L. Its faster variants that restrict the distances of the base pairs in the primary sequences are proposed for pairwise alignments [1114].

          Although structural alignment of multiple RNA sequences with reasonable computational cost is difficult, several algorithms have been proposed. Hofacker et al. proposed a method for progressive multiple alignments by direct comparison of the base-pairing probability matrices [12], implemented in PMmulti which was recently reimplemented in FoldalignM [15] and Locarna [16] by Torarinsson et al. and Will et al., respectively. In Stemloc, Holmes et al. incorporated a constraint approach that limits the range of structures and alignments to be considered by pre-processing the sequences [13, 14]. Siebert et al. proposed an approach distantly related to Sankoff's algorithm and implemented it in MARNA [17] that uses the structural information for pairwise alignments before combining them into multiple alignments with T-Coffee [9]. Dalli et al. developed a new scoring approach, StrAl, that takes into account sequence similarities as well as base-pairing probabilities [18]. Xu et al. proposed a new sampling based algorithm that finds the common structure between input sequences by probabilistically sampling aligned stems based on stem conservation calculated from intrasequence base pairing probabilities and intersequence base alignment probabilities, which was implemented in RNASampler [19]. Bauer et al. developed a graph based representation which modeled sequence-structured alignment as an integer linear program (ILP), and implemented it in RNAlara [20]. Kiryu et al. proposed a variant of Sankoff's algorithm with marked reduction of computation, which was implemented in Murlet [21]. All of these methods, however, are still too slow to apply to the RNA sequences longer than 1000 bases. Seibel et al. developed an alignment tool with an editor, which uses the secondary structure information of individual sequences to align multiple RNA sequences with low time complexities (4SALE) [22]. In order to extract the common secondary structure, it is also possible to find the structural motifs without aligning the whole sequences. For structural motif finding, Yao et al. proposed an algorithm based on covariance models (CMfinder) [23], and Hamada et al. proposed a graph mining approach (RNAmine) [24].

          Here we propose a method, implemented in MXSCARNA, for fast multiple alignments of RNA sequences. This method extends our previous work in pairwise alignments (SCARNA) [25] to progressive multiple alignments with improved score functions, and simultaneously construct multiple alignments and the associated common secondary structures. The pairwise alignment in this progressive alignment is an heuristic algorithm that separately aligns 5' parts and 3' parts of the stems with rough consistency considerations.

          In benchmark experiments, our method was at least as accurate as currently available state-of-art multiple alignment methods, but unlike those methods, the computations were fast enough for large-scale analyses, though the accuracies for the alignments of long sequences have not yet been confirmed.

          Results and Discussion

          Algorithm

          Overview of the algorithm

          The proposed method, implemented in MXSCARNA, progressively aligns multiple RNA sequences, in an extension of the pairwise structural alignment algorithm (implemented in SCARNA) of our previous work [25].

          First the guide tree for the progressive alignment is built by Unweighted Pair Group Method with Arithmetic Mean (UPGMA) [26] by using the pairwise similarities of the RNA sequences. Second the base-pairing probability matrices are calculated for all the RNA sequences by McCaskill's algorithm [27]. Those base-pairing probabilities are used for extracting the potential stems and for the matching scores in the Dynamic Programming (DP) of the alignments. Third the RNA sequences are progressively aligned along the guide tree using SCARNA's pairwise alignment algorithm with improved score functions introduced in this paper.

          At the first stage of the progressive alignment, which corresponds to the bottom level of the guide tree, the pairs of RNA sequences are aligned by engineered DP algorithm of SCARNA's pairwise alignment. The pairwise alignment is very fast because the potential stems extracted from the base-pairing probability matrices are decomposed into 5' part and 3' part and those two parts are independently aligned. In each upper-level step of the progressive alignment according to the guide tree, potential stems for groups of RNA sequences are extracted from the averaged base-pairing probability matrices.

          The DP algorithm of the pairwise alignment uses the approximated posterior probabilities as score functions. The approximation uses the product of the pairwise posterior probabilities of Maximum Expected Accuracy (MEA) alignments and the base-pairing probabilities of the sequences. MEA alignment maximizes the expected number of positions where the two nucleotides are correctly aligned. To yield robust alignments, the pairwise posterior probabilities of MEA alignments are modified by the probability consistency transformation.

          Definitions

          Definition 1: Stem candidate

          Given a base-pairing probability matrix for an RNA sequence and a threshold τ (0 <τ < 1), stem candidate is a set of continuous base pairs of which the base-pairing probabilities are greater than τ.

          Definition 2: Stem fragment

          Given a base-pairing probability matrix for an RNA sequence, a threshold τ (0 <τ < 1), and an integer W, stem fragment is a set of continuous base pairs of length W, of which the base-pairing probabilities are greater than τ.

          A stem candidate longer than W is represented by a set of overlapping stem fragments of fixed-length W (Figure 1). Smaller values in W or τ increase the sensitivity of the predictions of the stems and decrease the specificity of them. W and τ are set to 2 and 0.01 respectively in all the computational experiments in this paper. For each stem fragment, the 5' stem component and the 3' stem component, which are representatives of the 5' and 3' portions of the stem fragment, respectively, are defined as follows.
          http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_Fig1_HTML.jpg
          Figure 1

          Stem candidates, stem fragments and stem components. A stem candidate (a pair of underlined positions) comprises four overlapping stem fragments. A fragment consists of a 5' (left) component and a 3' (right) component. X i (blue box) and http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_IEq1_HTML.gif (red boxe) are 1-continuous stem components.

          Definition 3: Stem component

          For each stem fragment, a stem component X a , either a 5' stem component or a 3' stem component, is an object that has the following properties:

          • p(X a ): position, the position of the leftmost base of the 5' or 3' part of the stem fragment.

          • s(X a ): sequence, the nucleotide sequence of the 5' or 3' part of the stem fragment.

          • c(X a ): partner component, the complementary (3' or 5') stem component.

          • d(X a ): loop distance, the distance to the complementary (3' or 5') stem component.

          A stem fragment is written as [X a , X a' ] by using the mutually complementary stem components, 5' stem component X a and 3' stem component X a' , which represent the 5' and 3' parts of the stem fragment. X a and X a' satisfyX a = c(X a' ) and X a' = c(X a ).

          The loop distance d(X a ) can be written asd(X a ) = p(c(X a )) - p(X a ) - W.

          Definition 4: stem component sequence (SCS)

          A stem component sequence (SCS) is a sorted sequence of all the stem components of an RNA sequence, in order of their positions and, if the positions are the same, according to their loop distances.

          For i <j, a SCS X = X 1 X 2 ... X m satisfiesp(X i ) <p(X j ) or p(X i ) = p(X j ) &d(X i ) <d(X j ).

          Definition 5: relations of stem fragments without an overlap

          Two stem fragments, [X a , X a' ] and [X b , X b' ] of an RNA sequence are, parallel if and only ifp(X a ) <p(X a' ) <p(X b ) <p(X b' ) or p(X b ) <p(X b' ) <p(X a ) <p(X a' ),

          nested if and only if p(X a ) <p(X b ) <p(X b' ) <p(X a' ) or p(X b ) <p(X a ) <p(X a' ) <p(X b' ),

          pseudo-knotted if and only if p(X a ) <p(X b ) <p(X a' ) <p(X b' ) or p(X b ) <p(X a ) <p(X b' ) <p(X a' ).

          Definition 6: relations of overlapping stem fragments

          Two stem fragments, [X a , X a' ] and [X b , X b' ] of an RNA sequence are, r-continuous if and only if

          r = p(X b ) - p(X a ) = p(X a' ) - p(X b' ),

          ill-continuous if and only if X a overlaps X b and X a' overlaps X b' and

          p(X b ) - p(X a ) ≠ p(X a' ) - p(X b' ),

          contradictory if and only if only one side, either 5' part or 3' part, of the stem fragments overlap.

          The three possible relationships between stem fragments without an overlap: parallel, nested, and pseudo-knotted, may exist in the same secondary structure of an RNA sequence. However, among the three possible relationships between overlapping stem fragments, only r-continuous stem fragments may coexist in the same secondary structure of an RNA sequence. 1-continuous, a special case of r-continuous, means that the two stem fragments are adjacent in the RNA sequence and a part of a stem candidate with a length of W + 1 (Figure 1). As described later, two overlapping stem components in the alignment are controlled to belong to two r-continuous stem fragments in DP.

          Building stem component sequences

          In a base-pairing probability matrix, which is calculated by McCaskill's algorithm [27], a potential stem is located in two symmetry locations as continuous counterdiagonal positions which have high base-pairing probabilities. Therefore, the stem components for each RNA sequence defined in previous section are extracted by scanning counterdiagonal windows of length W in the base-pairing probability matrix and selecting the windows whose elements are greater than τ. Smaller value in W or τ increase the sensitivity of the predictions of the stems and decrease the specificity of them. W and τ are set to 2 and 0.01 respectively in all the computational experiments in this paper.

          The stem components are sorted in order of their positions and loop distances to construct a stem component sequence (SCS).

          For each group alignment in the progressive alignment, the average of the base-pairing probability matrices is calculated directly according to the alignment of the group of RNA sequences. The stem components for the group are extracted from the averaged matrix, and the SCS is constructed by sorting the stem components.

          Alignment of stem component sequences

          Before the pairwise alignments or group alignments, RNA sequences or groups of RNA sequences are represented by their stem component sequences (SCSs). Those two SCSs are aligned by SCARNA's pairwise DP algorithm in each stage of the progressive alignment. The alignment of the two SCSs uses two DP matrices, M(i, j) and G(i, j). For two SCSs, {X i }(i = 1, ... |X|) and {Y j }(j = 1, ... |Y|), M(i, j) is the best score of the alignment of the pair X i and Y j , given that X i matches Y j , and G(i, j) is the best score given that X i mismatches Y j . The recursions for M(i, j) and G(i, j) are written as:
          http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_Equ1_HTML.gif
          (1)
          http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_Equ2_HTML.gif
          (2)

          with the initial conditions; M(0, 0) = 0, M(·, 0) = M(0, ·) = G(0, 0) = G(·, 0) = G(0, ·) = -∞.

          The first term of equation(1) controls the 1-continuous case where the continuous matches of two overlapping stem components form a match of the corresponding stem longer than W. α i /β j are the indices (smaller than i/j) of the components that are 1-continuous with X i /Y j . The positions of http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_IEq2_HTML.gif are adjacent to X i /Y j in the nucleotide sequences (Figures 1 and 2), i.e.
          http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_Fig2_HTML.jpg
          Figure 2

          An example of relations of indices of stem components in SCS alignment. α i /β j and p i /q j in equation (1) are the indices (smaller than i/j) of stem components of X/Y. The red lines separate the stem components into groups which have the same positions in RNA sequences. When X i and Y j match in DP of SCS alignment, the stem components of the adjacent previous match must be either non-overlapping or 1-continuous with X i /Y j . http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_IEq2_HTML.gif are 1-continuous with X i /Y j . http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_IEq3_HTML.gif are the nearest stem components that do not overlap with X i /Y j .

          http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_Equa_HTML.gif

          δ s (i, j) corresponds to the incremental score for the match of the overlapping stem components, which is discussed in the next section.

          The second and third terms of equation(1) keep the stem components in the adjacent DP match from overlapping in the nucleotide sequences. p i /q j are the indices (smaller than i/j) of the nearest components that do not overlap with X i /Y j (Figure 2). s(i, j) is a match score for X i and Y j , which is discussed in the next section.

          Equation(2) refers only adjacent positions in DP matrix because overlaps of X i and Y j with the other stem components are permitted. Because the 5' stem components and the 3' stem components are handled independently, there is no term for bifurcation in secondary structures in equations (1) and (2).

          The traceback pointer keeps the triplets, indices of X, Y, and the selection of M or G, in the recursion (1) and (2). The first term of the triplet, the index of X, can be either α i , p i , i, or i - 1, and the second term of the triplet, the index of Y, can be either β j , q j , j, or j - 1. In the traceback of the DP, M(i, j) and G(i, j) are used jointly to obtain the optimal path and to select M or G, which gives the maximum score of the alignment. The alignments of SCSs are constructed by selecting the stem components that appear in the path with the selected M. All of the mismatched stem components are excluded from the alignment. The algorithm makes the adjacent DP matches of stem components either not overlapping in the nucleotide sequences or consistently overlapping (1-continuous) as a match of the stems longer than W. Pairwise alignment of the SCSs requires only O(|X||Y|) in time and in memory. That computational complexities are evaluated as (L 2) for two RNA sequences of length L because the number of the stem components is regarded as a linear function of the length of the nucleotide sequence [25].

          The pairwise alignment of SCSs allows some inconsistent matches by ignoring strict treatments of the complementary components. For two stem fragments, [X a , X a' ] and [Y b , Y b' ], if X a matches Y b in the SCS alignment, X a' should match Y b' . Let us define such a match as left-right consistent. Because 5' stem components and 3' stem components are aligned independently, left-right consistency is not guaranteed in general. Any match which is not left-right consistent is removed as a post process. If any two of the stem components of a same SCS appear in the SCS alignment and their complementary components overlap (i.e. contradictory in Definition 6), those complementary components do not appear together in the alignment because the alignment of complementary components are controlled to be either nonoverlapping or r-continuous. Therefore, the post process also guarantees that no pair of contradictory stem fragments appears in the alignment [25].

          The score function using the MEA alignment

          In our previous work [25], a function of the RIBOSUM [28] score, loop distance, base-pairing probabilities, and the stacking energy were used as the score s(i, j) in recursion (1). In MXSCARNA, the score function is replaced by an approximated posterior probability according to the principle of Maximum Expected Accuracy (MEA). Recent studies have shown that the accuracy of the resulting sequence alignment and secondary structure predictions is better than that of predictions made by the conventional maximum likelihood estimation (MLE) algorithms [21, 2932].

          In the following, for nucleotide sequences x and y, x i ~ y j means that x i x and y i y are aligned on the same column in the alignment, and x i x j means that x i , x j x form a base pair. For two RNA sequences, x, y and k, l ∈ {1, ···, |x|}, m, n ∈ {1, ···, |y|}, let P(x k ~ y m , x l ~ y n , x k x l , y m y n |x, y) be the posterior probability, i.e. the sum of the probabilities that two positions of the sequences, x k and y m , x l and y n , are aligned in the alignment, and that two pairs of the nucleotides, x k and x l , y m and y n , form base pairs in the secondary structures as well; this is computed by the inside-outside algorithm of the pair Stochastic Context Free Grammar (pair SCFG) [5] for structural pairwise alignments of RNA sequences. We wanted to use posterior probability as the score function s(i, j), but the computational costs, O(L 6) in time and O(L 4) in memory for sequences of length L, are impractical. We instead used the following approximated posterior probability introduced by Kiryu et al. [21].
          http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_Equb_HTML.gif

          P(x k x l |x) and P(y m y n |y) are the base-pairing probabilities that the particular positions x k and x l , y m and y n , respectively, form base pairs; these probabilities are computed by McCaskill's algorithm [27].

          http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_IEq4_HTML.gif (x k ~ y m |x, y) and http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_IEq4_HTML.gif (x l ~ y n |x, y) are the posterior probabilities modified by probability consistency transformation [32], which is computed as follows.
          http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_Equ3_HTML.gif
          (3)

          where S is the set of RNA sequences to be aligned. In this transformation, the probability of specific nucleotides of two sequences being aligned are replaced by the average over the products of probabilities that the two nucleotides are aligned to the same nucleotides in arbitrary third sequences. This calculation requires O(N 3 L 3) in time and O(N 2 L 2) in memory. The probability consistency transformations are applied twice in current implementation.

          P(x k ~ z r |x, z) is the posterior probability, i.e. the sum of the probabilities that particular positions of the two sequences, x k and z r , are aligned in some alignment; this is computed by the forward-backward algorithm of the pair Hidden Markov Model (pair HMM) [31] for pairwise alignment of the sequences. Our new matching scores in (1) are defined as follows.
          http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_Equ4_HTML.gif
          (4)
          http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_Equ5_HTML.gif
          (5)

          where http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_IEq5_HTML.gif are the complementary stem components of X i /Y j .

          The sum of the probabilities, not the logarithms of the probabilities, is used for the matching score, in an effort to maximize the number of correctly aligned bases including the implicit prediction of the base pairs (MEA principle).

          Alignment of loop region

          The remaining loop regions (except the selected common stems) are aligned by using the consistency-transferred posterior probabilities, http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_IEq4_HTML.gif (x k ~ y m |x, y), as the matching scores. The probabilities, not the logarithms of the probabilities, again are used, according to the MEA principle. The recursion is shown following.
          http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_Equc_HTML.gif
          Emission and transition probabilities for the pair HMM in MXSCARNA (Figure 3) were trained via Expectation-Maximization (EM) on a set of unaligned sequences that is extracted from the Rfam database and that do not overlap the sequences of the dataset for subsequent experiments.
          http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_Fig3_HTML.jpg
          Figure 3

          A pair-HMM for pairwise sequence alignment. A pair-HMM is used for alignment of loop regions and calculation of the posterior probabilities in score function. The state M has emission probability distribution http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_IEq6_HTML.gif for emitting an aligned pair x i and y j . The state I x has distributions http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_IEq7_HTML.gif for emitting symbol x i against a gap. The state I y has distributions http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_IEq8_HTML.gif for emitting symbol y j against a gap. The parameters δ and ε are the state transition probabilities.

          Computational Experiments

          Datasets

          To test the empirical performance of MXSCARNA, we used three datasets for the benchmark multiple alignments: an original multiple alignment dataset, the BRAlibaseII multiple alignment dataset [33], and Kiryu et al.'s multiple alignment dataset [21].

          Our original dataset comprised 1669 multiple alignments of 5 sequences, the secondary structures of which have been published, obtained from the Rfam 7.0 database [34]. There are 27 families of RNA sequences in the dataset and the sequence identities varied from 35% to 100%. Sequences that included bases other than A, C, G, and U were removed because some of the alignment programs were unable to align them. The BRAlibaseII benchmark dataset included 481 multiple alignments of 5 sequences. The sequences of each multiple alignment were extracted from tRNA, Intron_gpII, 5S_rRNA, and U5 families in the Rfam 5.0 database and the signal recognition particle RNA family (SRP) in the SRPDB database [35]. Because the dataset did not include consensus secondary structure annotations to the alignments, we used the secondary structure annotations recovered by Kiryu et al. [21].

          Kiryu et al.'s multiple alignment benchmark dataset was generated from selected seed alignments in the Rfam 7.0 database that have published consensus structures [21]. For each sequence family, as many as 1000 random combinations of 10 sequences were generated. The alignments whose mean pairwise sequence identity exceeded 95% and whose gap characters accounted for more than 30% of the total number of characters aligned were removed. As such, this dataset consisted of 85 multiple alignments of 10 sequences, generated from 17 sequence families, with five alignments for each. The dataset was reasonably divergent, and its mean length varied from 54 to 291 bases, and mean pairwise sequence identities varied from 40% to 94%.

          Evaluation measures

          The qualities of the alignments were evaluated by the Sum-of-Pairs Score (SPS) for the accuracy of the alignments and by the Matthews Correlation Coefficient (MCC) [36] for the accuracy of the secondary structure predictions. The SPS and MCC of the alignment to be evaluated (named as a test alignment) for the reference alignment were defined as follows. The SPS was defined as the proportion of correctly aligned nucleotide pairs:
          http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_Equd_HTML.gif
          where I is the number of columns in the test alignment, J is the number of columns in the reference alignment, on column i in the test alignment http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_IEq9_HTML.gif is the total number of "correct" nucleotide pairs which also appear in the reference alignment, on column j in the reference alignment http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_IEq10_HTML.gif is the total number of nucleotide pairs. The MCC was defined as
          http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_Eque_HTML.gif

          where TP indicates the number of correctly predicted base pairs, TN the number of base pairs that were correctly predicted as unpaired, FP the number of incorrectly predicted base pairs, and FN the number of true base pairs that were not predicted. The term ξ accounts for predicted base pairs that were not present in the reference structure but were compatible with it. Compatible base pairs are not true positives but have to be neither inconsistent (one or both nucleotides being a part of a different base pair in the reference structure) nor pseudo-knotted with respect to the reference structure [37]. In order to calculate MCC for each test alignment, the reference alignment and the "correct" consensus secondary structure are taken from the database. In order to compare the accuracies of the alignments in terms of the implicitly predicted common secondary structures, the common secondary structures for each test alignment by the alignment programs were predicted by the Pfold program [5].

          Comparison of accuracies with those of other aligners

          To compare the accuracies of the alignment methods we used a Linux machine with an AMD Opteron processor (2 GHz and 4 GB RAM).

          We compared the performance of MXSCARNA with that of Murlet [21], ProbCons [32], MAFFT [38], ClustalW [7], StrAl [18], MARNA [17], RNASampler [19], RNAlara [20], FoldalignM [15], Locarna [16], PMmulti [12], and Stemloc [13] on the three datasets described earlier. Whereas ProbCons, MAFFT, and ClustalW align RNA sequences on the basis of sequence similarities only, StrAl, MARNA, RNASampler, RNAlara, FoldalignM, Locarna, PMmulti, Stemloc, and Murlet weigh both sequence similarities and secondary structures. The command line options for the programs in the experiments are shown in Table 1. The results for the original dataset are shown in Table 2. Because MARNA, Locarna, FoldalignM, PMmulti, and Stemloc impose high time and memory demands, those programs were executed only on families of which the average sequence lengths were less than or equal to 100 bases. The SPS of MXSCARNA was comparable to those of Murlet and ProbCons, which currently are the best performing aligners [21]. In addition, the MCC of MXSCARNA was one of the highest among aligners. In particular, the MCC of MXSCARNA is similar to that of Stemloc, which aligns only short sequences that have simple secondary structures.
          Table 1

          Command line options for the programs in the experiments. This table summarizes multiple alignment programs and their command line options used in the paper.

          Program

          Command

          MXSCARNA

          ./mxscarna <input_filename>

          Murlet

          ./murlet -max_time = 100 <input_filename>

          ProbCons

          ./probcons <input_filename>

          MAFFT

          ./mafft <input_filename>

          ClustalW

          ./clustalw <input_filename>

          StrAl

          ./stral <input_filename>

          RNASampler

          perl RNASampler_driver.pl -p <input_dir> -q <input_filename> -i 15 -S 100

          RNAlara

          ./lara -i <input_filename>

          Locarna

          ./mlocarna -struct-local = false -sequ-local = false <input_filename>

          FoldalignM-Foldalign

          perl FoldalignM_Foldalign.pl -f <input_filename>

          FoldalignM-McCaskill

          java FoldalignM_McCaskill <input_filename>

          MARNA

          perl marna.pl -g 2 <input_filename>

          PMmulti

          perl pmmulti.pl <input_filename>

          stemloc

          ./stemloc -g -m -slow <input_filename>

          Table 2

          Accuracies for the original multiple alignment dataset. SPS and MCC values (%) for the original multiple alignment dataset are presented. Each family has 5 RNA sequences. Family: Rfam family name. %id: average sequence identity. length: average sequence length (bases) in each family. Average(all): the average SPS or MCC for all families. Average(sub): the average SPS or MCC for the subset of families with an average sequence length of less than or equal to 100 bases. Because PMmulti and Stemloc were unable to align all data, the proportion of data that was aligned is given in parentheses as no. of sequences aligned/total no. of sequences. FoldalignM consists of two modes: FoldalgnM_FoldalignM and FoldalignM_McCaskill, which are separately evaluated and indicated as FoldalignM(1) and FoldalignM(2) respectively.

          SPS:Family

          %id

          length

          MXSCARNA

          Murlet

          ProbCons

          MAFFT

          ClustalW

          StrAl

          RNASampler

          IRE

          62

          29

          98

          96

          93

          89

          77

          83

          92

          s2m

          78

          43

          96

          96

          96

          96

          96

          96

          95

          UnaL2

          78

          54

          95

          98

          98

          96

          92

          94

          82

          Hammerhead_3

          71

          55

          96

          94

          93

          89

          83

          89

          91

          SECIS

          42

          64

          65

          66

          65

          52

          45

          56

          46

          sno_14q_I_II

          71

          74

          90

          95

          96

          93

          93

          87

          71

          tRNA

          48

          76

          87

          87

          87

          84

          76

          82

          87

          ctRNA_pGA1

          74

          80

          88

          86

          85

          88

          84

          86

          89

          Tymo_tRNA

          70

          83

          84

          83

          82

          75

          75

          79

          77

          Y

          64

          95

          71

          72

          72

          73

          65

          64

          65

          SRP_bact

          52

          95

          71

          71

          70

          66

          70

          62

          66

          Purine

          55

          100

          74

          77

          77

          78

          75

          81

          69

          5S_rRNA

          60

          117

          88

          89

          89

          86

          86

          86

          83

          S_box

          66

          130

          79

          80

          78

          78

          68

          72

          67

          U4

          67

          141

          79

          81

          80

          79

          79

          78

          69

          RFN

          65

          150

          86

          87

          87

          88

          81

          80

          81

          5_8S_rRNA

          67

          154

          91

          93

          93

          90

          88

          89

          78

          U1

          60

          158

          81

          81

          79

          79

          79

          76

          74

          Telomerase_cil

          56

          171

          48

          50

          49

          43

          38

          41

          37

          Lysine

          50

          180

          78

          81

          79

          72

          70

          76

          72

          U2

          66

          185

          76

          76

          75

          71

          71

          73

          71

          U17

          75

          214

          91

          93

          93

          89

          89

          87

          81

          U3

          51

          246

          43

          44

          44

          47

          41

          40

          34

          SRP_euk_arch

          46

          294

          50

          56

          47

          42

          42

          48

          44

          tmRNA

          46

          373

          48

          50

          50

          47

          46

          39

          42

          RnaseP_bact_b

          64

          387

          82

          80

          79

          78

          74

          74

          66

          Telomerase_vert

          66

          463

          69

          70

          69

          69

          66

          64

          65

          Average(all)

            

          78

          79

          78

          75

          72

          73

          70

          Average(sub)

            

          85

          85

          85

          82

          78

          80

          78

          SPS:Family

          RNAlara

          Locarna

          FoldalignM(1)

          FoldalignM(2)

          MARNA

          PMmulti

          Stemloc

            

          IRE

          85

          88 (100/100)

          87 (99/101)

          85 (97/100)

          92

          90 (101/101)

          87 (101/101)

            

          s2m

          96

          99 (43/50)

          95 (50/50)

          95 (47/50)

          94

          98 (44/50)

          93 (50/50)

            

          UnaL2

          93

          93 (78/89)

          89 (75/89)

          86 (87/89)

          88

          87 (75/89)

          75 (89/89)

            

          Hammerhead_3

          88

          80 (100/100)

          79 (99/100)

          77 (97/100)

          89

          87 (77/100)

          94 (100/100)

            

          SECIS

          48

          47 (62/63)

          46 (63/63)

          44 (63/63)

          49

          39 (63/63)

          60 (60/63)

            

          sno_14q_I_II

          78

          76 (98/98)

          73 (98/98)

          61 (97/98)

          67

          73 (97/98)

          89 (97/98)

            

          tRNA

          91

          82 (103/103)

          90 (100/103)

          78 (97/103)

          59

          86 (103/103)

          88 (103/103)

            

          ctRNA_pGA1

          86

          74 (20/28)

          75 (28/28)

          74 (27/28)

          79

          72 (20/28)

          84 (28/28)

            

          Tymo_tRNA

          79

          78 (49/59)

          68 (59/59)

          68 (56/59)

          66

          71 (49/59)

          57 (57/59)

            

          Y

          56

          49 (21/24)

          50 (24/24)

          47 (24/24)

          60

          43 (11/24)

          68 (23/24)

            

          SRP_bact

          63

          62 (70/70)

          64 (70/70)

          56 (67/70)

          56

          61 (63/70)

          60 (67/70)

            

          Purine

          64

          65 (45/45)

          64 (45/45)

          62 (45/45)

          64

          65 (45/45)

          37 (45/45)

            

          5S_rRNA

          85

                  

          S_box

          57

                  

          U4

          71

                  

          RFN

          78

                  

          5_8S_rRNA

          81

                  

          U1

          74

                  

          Telomerase_cil

          32

                  

          Lysine

          59

                  

          U2

          71

                  

          U17

          79

                  

          U3

          32

                  

          SRP_euk_arch

          42

                  

          tmRNA

          34

                  

          RnaseP_bact_b

          66

                  

          Telomerase_vert

          51

                  

          Average(all)

          68

                  

          Average(sub)

          77

          74

          73

          69

          72

          73

          74

            

          MCC:Family

          %id

          length

          MXSCARNA

          Murlet

          ProbCons

          MAFFT

          ClustalW

          StrAl

          RNASampler

          IRE

          62

          29

          90

          91

          75

          72

          65

          43

          89

          S2m

          78

          43

          84

          83

          84

          84

          84

          84

          81

          UnaL2

          78

          54

          52

          70

          51

          53

          51

          50

          51

          Hammerhead_3

          71

          55

          99

          95

          93

          86

          72

          79

          96

          SECIS

          42

          64

          76

          60

          55

          35

          23

          37

          73

          sno_14q_I_II

          71

          74

          93

          98

          93

          93

          91

          91

          95

          tRNA

          48

          76

          91

          89

          86

          84

          76

          83

          95

          ctRNA_pGA1

          74

          80

          96

          93

          88

          89

          81

          92

          94

          Tymo_tRNA

          70

          83

          87

          85

          75

          73

          72

          80

          90

          Y

          64

          95

          95

          85

          86

          83

          67

          83

          94

          SRP_bact

          52

          95

          81

          72

          54

          50

          58

          57

          83

          Purine

          55

          100

          90

          94

          90

          86

          80

          84

          91

          5S_rRNA

          60

          117

          75

          79

          69

          70

          69

          70

          70

          S_box

          66

          130

          90

          87

          86

          81

          76

          81

          86

          U4

          67

          141

          75

          71

          62

          62

          54

          65

          67

          RFN

          65

          150

          84

          83

          84

          84

          82

          83

          82

          5_8S_rRNA

          67

          154

          58

          51

          47

          45

          41

          46

          52

          U1

          60

          158

          70

          68

          61

          56

          60

          57

          71

          Telomerase

          56

          171

          65

          41

          28

          21

          24

          31

          60

          Lysine

          50

          180

          87

          90

          76

          66

          63

          71

          89

          U2

          66

          185

          73

          76

          58

          51

          62

          59

          77

          U17

          75

          214

          79

          80

          78

          76

          75

          72

          72

          U3

          51

          246

          46

          26

          22

          19

          46

          21

          39

          SRP_euk_arch

          46

          294

          72

          75

          46

          37

          35

          49

          72

          tmRNA

          46

          373

          51

          54

          50

          49

          43

          42

          49

          RNaseP_bact_b

          64

          387

          73

          58

          63

          58

          53

          60

          37

          Telomerase

          66

          463

          64

          51

          47

          44

          40

          36

          53

          Average(all)

            

          78

          74

          67

          63

          61

          63

          74

          Average(sub)

            

          86

          85

          77

          74

          68

          72

          86

          MCC:Family

          RNAlara

          Locarna

          FoldalignM(1)

          FoldalignM(2)

          MARNA

          PMmulti

          Stemloc

            

          IRE

          81

          89

          85

          89

          82

          89

          81

            

          s2m

          84

          83

          85

          86

          79

          85

          84

            

          UnaL2

          53

          53

          53

          51

          51

          50

          45

            

          Hammerhead_3

          95

          94

          87

          91

          95

          91

          98

            

          SECIS

          63

          74

          74

          75

          57

          67

          78

            

          sno_14q_I_II

          92

          87

          92

          84

          81

          93

          85

            

          tRNA

          95

          87

          95

          87

          67

          90

          95

            

          ctRNA_pGA1

          97

          95

          96

          96

          95

          94

          95

            

          Tymo_tRNA

          85

          75

          87

          82

          62

          82

          83

            

          Y

          86

          87

          94

          89

          87

          83

          93

            

          SRP_bact

          65

          87

          86

          83

          63

          80

          69

            

          Purine

          77

          89

          88

          88

          82

          86

          89

            

          5S_rRNA

          72

                  

          S_box

          72

                  

          U4

          60

                  

          RFN

          79

                  

          5_8S_rRNA

          46

                  

          U1

          60

                  

          Telomerase

          39

                  

          Lysine

          58

                  

          U2

          63

                  

          U17

          63

                  

          U3

          21

                  

          SRP_euk_arch

          42

                  

          tmRNA

          40

                  

          RNaseP_bact_b

          65

                  

          Telomerase

          28

                  

          Average(all)

          65

                  

          Average(sub)

          81

          83

          85

          83

          75

          83

          83

            
          The results from the BRAlibaseII benchmark multiple alignment dataset are shown in Table 3. Because of their prohibitive requirements for memory and time, Stemloc, FoldalignM, PMmulti, and MARNA were not applied to the SRP family data. Again, MXSCARNA was comparable to Murlet and ProbCons in terms of SPS and one of the best performers among multiple aligners according to the MCC. These trends continue in Table 4, which contains the results from Kiryu et al.'s benchmark dataset comprising 10 sequences for each alignment.
          Table 3

          Accuracies for the BRAlibaseII multiple alignment dataset. SPS and MCC values (%) for the BRAlibaseII multiple alignment dataset are presented. Each family has 5 RNA sequences. Family: Rfam family name. %id: average sequence identity. length: average sequence length (bases) in each family. Average(all): the results of the average value of the SPS or MCC for all families. Average(sub): the average SPS or MCC for the subset of families with an average sequence length of less than or equal to 100 bases. Because PMmulti and Stemloc were unable to align all data, the proportion of data that was aligned is given in parentheses as no. of sequences aligned/total no. of sequences. FoldalignM consists of two modes: FoldalgnM_FoldalignM and FoldalignM_McCaskill, which are separately evaluated and indicated as FoldalignM(1) and FoldalignM(2) respectively.

          SPS:Family

          %id

          length

          MXSCARNA

          Murlet

          ProbCons

          MAFFT

          ClustalW

          StrAl

          RNASampler

           

          tRNA

          69

          76

          91

          91

          91

          89

          85

          89

          92

           

          Intron_gpII

          64

          80

          79

          80

          80

          77

          75

          79

          74

           

          5S_rRNA

          70

          117

          89

          90

          90

          89

          88

          89

          90

           

          U5

          72

          118

          74

          75

          76

          72

          72

          73

          78

           

          SRP

          67

          300

          88

          88

          88

          87

          87

          86

          82

           

          Average(all)

            

          84

          85

          85

          83

          81

          83

          83

           

          Average(sub)

            

          83

          84

          84

          82

          80

          82

          83

           

          SPS:Family

          RNAlara

          Locarna

          FoldalignM(1)

          FoldalignM(2)

          MARNA

          PMmulti

          Stemloc

             

          tRNA

          95

          93 (98/98)

          94 (97/98)

          90 (91/98)

          79 (98/98)

          90 (89/98)

          88 (98/98)

             

          Intron_gpII

          75

          71 (89/92)

          70 (92/92)

          67 (89/92)

          76 (92/92)

          77 (61/92)

          77 (92/92)

             

          5S_rRNA

          93

          92 (89/89)

          92 (88/89)

          89 (89/89)

          58 (78/89)

          85 (89/89)

          72 (89/89)

             

          U5

          80

          77 (109/109)

          72 (108/109)

          69 (107/109)

          85 (74/109)

          56 (105/109)

          64 (109/109)

             

          SRP

          82

          83 (84/93)

                  

          Average(all)

          85

          83

                  

          Average(sub)

          86

          83

          82

          79

          74

          77

          75

             

          MCC:Family

          %id

          length

          MXSCARNA

          Murlet

          ProbCons

          MAFFT

          ClustalW

          StrAl

          RNASampler

          RNAlara

          tRNA

          69

          76

          94

          92

          91

          90

          83

          88

          94

          93

          Intron_gpII

          64

          80

          82

          80

          77

          76

          74

          74

          80

          79

          5S_rRNA

          70

          117

          71

          69

          67

          68

          67

          69

          69

          70

          U5

          72

          118

          80

          75

          70

          66

          66

          69

          77

          72

          SRP

          67

          300

          75

          72

          68

          67

          68

          65

          71

          63

          Average(all)

            

          80

          78

          75

          73

          72

          73

          78

          76

          Average(sub)

            

          82

          79

          76

          75

          72

          75

          80

          79

          MCC:Family

          Locarna

          FoldalignM(1)

          FoldalignM(2)

          MARNA

          PMmulti

          Stemloc

              

          tRNA

          92

          96

          92

          80

          93

          93

              

          Intron_gpII

          80

          76

          80

          78

          76

          76

              

          5S_rRNA

          71

          72

          71

          59

          70

          68

              

          U5

          74

          70

          69

          60

          61

          78

              

          SRP

          73

                   

          Average(all)

          78

                   

          Average(sub)

          79

          79

          78

          69

          75

          79

              
          Table 4

          Accuracies for Kiryu et al.'s dataset. SPS and MCC values (%) for Kiryu et al.'s dataset are presented. Each family has 10 RNA sequences. Family: Rfam family name. %id: average sequence identity. length: average sequence length (bases) in each family. Average(all): the results of the average value of the SPS or MCC for all families. Average(sub): the average SPS or MCC for the subset of families with an average sequence length of less than or equal to 100 bases. Because PMmulti and Stemloc were unable to align all data, the proportion of data that was aligned is given in parentheses as no. of sequences aligned/total no. of sequences. FoldalignM consists of two modes: FoldalgnM_FoldalignM and FoldalignM McCaskill, which are separately evaluated and indicated as FoldalignM(1) and FoldalignM(2) respectively.

          SPS:Family

          %id

          length(nt)

          MXSCARNA

          Murlet

          ProbCons

          MAFFT

          ClustalW

          StrAl

          RNAlara

          UnaL2

          73

          54

          92

          95

          95

          91

          84

          86

          87

          SECIS

          41

          64

          70

          73

          68

          44

          35

          59

          53

          tRNA

          45

          73

          87

          90

          87

          76

          62

          75

          91

          sno_14q_I_II

          64

          75

          82

          92

          92

          91

          80

          75

          72

          SRP_bact

          47

          93

          58

          61

          61

          60

          61

          48

          56

          THI

          55

          105

          77

          83

          82

          78

          58

          65

          65

          S_box

          66

          107

          86

          88

          88

          82

          82

          77

          77

          5S_rRNA

          57

          116

          84

          85

          85

          81

          82

          79

          83

          Retroviral_psi

          92

          117

          97

          97

          97

          97

          96

          97

          97

          RFN

          66

          140

          89

          91

          90

          91

          83

          80

          86

          5_8S_rRNA

          61

          154

          85

          88

          87

          84

          78

          81

          75

          U1

          59

          157

          74

          77

          75

          73

          71

          66

          66

          Lysine

          49

          181

          75

          77

          75

          66

          60

          68

          59

          U2

          62

          182

          71

          74

          73

          68

          65

          67

          69

          T-box

          45

          244

          44

          50

          50

          43

          34

          32

          15

          IRES_HCV

          94

          261

          96

          96

          96

          96

          96

          83

          96

          SRP_euk_arch

          40

          291

          42

          42

          40

          36

          34

          40

          39

          Average(all)

            

          77

          80

          79

          74

          68

          69

          70

          Average(sub)

            

          82

          86

          85

          80

          73

          75

          77

          SPS:Family

          RNASampler

          Locarna

          FoldalignM(1)

          FoldalignM(2)

          MARNA

          PMmulti

          Stemloc

            

          UnaL2

          72

          88

          68

          60

          83 (5/5)

          69 (5/5)

          82 (5/5)

            

          SECIS

          54

          49

          42

          41

          47 (5/5)

          35 (5/5)

          82 (5/5)

            

          tRNA

          82

          79

          84

          66

          54 (5/5)

          69 (5/5)

          91 (5/5)

            

          sno_14q_I_II

          64

          57

          45

          34

          49 (5/5)

          39 (3/5)

          77 (5/5)

            

          SRP_bact

          54

          52

          55

          51

          43 (5/5)

          36 (4/5)

          47 (3/5)

            

          THI

          68

          65

          65

          62

          62 (4/5)

          58 (5/5)

          71 (5/5)

            

          S_box

          76

          63

          57

          57

          78 (5/5)

          44 (5/5)

          84 (5/5)

            

          5S_rRNA

          77

          79

          74

          70

          71 (5/5)

          57 (5/5)

          77 (3/5)

            

          Retroviral_psi

          96

          95

          91

          91

          96 (5/5)

          87 (5/5)

          75 (5/5)

            

          RFN

          82

          72

          73

          63

          77 (5/5)

          58 (4/5)

          80 (5/5)

            

          5_8S_rRNA

          69

          75

          56

          31

          64 (5/5)

          58 (5/5)

          73 (1/5)

            

          U1

          63

          68

            

          50 (5/5)

              

          Lysine

          71

          55

            

          58 (5/5)

              

          U2

          65

          64

            

          65 (1/5)

              

          T-box

          32

          15

            

          22 (5/5)

              

          IRES_HCV

          93

          75

            

          92 (3/5)

              

          SRP_euk_arch

          33

          40

            

          37 (5/5)

              

          Average(all)

          68

          64

            

          62

              

          Average(sub)

          72

          70

          64

          57

          52

          60

          76

            

          MCC:Family

          %id

          length(nt)

          MXSCARNA

          Murlet

          ProbCons

          MAFFT

          ClustalW

          StrAl

          RNAlara

          UnaL2

          73

          54

          42

          41

          46

          36

          24

          32

          44

          SECIS

          41

          64

          78

          78

          59

          20

          23

          45

          70

          tRNA

          45

          73

          93

          97

          91

          85

          65

          85

          97

          sno_14q_I_II

          64

          75

          87

          91

          91

          91

          66

          75

          87

          SRP_bact

          47

          93

          66

          56

          46

          49

          54

          52

          69

          THI

          55

          105

          71

          70

          70

          62

          38

          48

          58

          S_box

          66

          107

          90

          89

          87

          79

          77

          75

          76

          5S_rRNA

          57

          116

          75

          67

          62

          64

          53

          66

          69

          Retroviral_psi

          92

          117

          86

          86

          86

          84

          86

          86

          86

          RFN

          66

          140

          67

          71

          72

          73

          71

          60

          70

          5_8S_rRNA

          61

          154

          38

          43

          35

          16

          14

          26

          33

          U1

          59

          157

          69

          61

          57

          56

          61

          52

          56

          Lysine

          49

          181

          83

          81

          71

          33

          52

          61

          64

          U2

          62

          182

          74

          71

          56

          38

          39

          58

          68

          T-box

          45

          244

          72

          78

          80

          51

          41

          26

          0

          IRES_HCV

          94

          261

          63

          62

          62

          63

          26

          34

          63

          SRP_euk_arch

          40

          291

          70

          63

          40

          21

          23

          38

          33

          Average(all)

            

          72

          71

          65

          54

          48

          54

          61

          Average(sub)

            

          72

          72

          68

          60

          52

          59

          69

          MCC:Family

          RNASampler

          Locarna

          FoldalignM(1)

          FoldalignM(2)

          MARNA

          PMmulti

          Stemloc

            

          UnaL2

          51

          42

          57

          50

          36

          18

          39

            

          SECIS

          78

          80

          75

          80

          40

          64

          77

            

          tRNA

          94

          96

          95

          89

          51

          91

          98

            

          sno_14q_I_II

          84

          95

          95

          86

          79

          77

          87

            

          SRP_bact

          73

          74

          80

          75

          47

          44

          64

            

          THI

          72

          60

          50

          51

          59

          41

          77

            

          S_box

          86

          82

          83

          87

          83

          55

          88

            

          5S_rRNA

          62

          75

          72

          73

          59

          58

          66

            

          Retroviral_psi

          86

          88

          76

          89

          84

          73

          87

            

          RFN

          69

          69

          67

          70

          65

          58

          72

            

          5_8S_rRNA

          38

          40

          43

          41

          21

          14

          34

            

          U1

          64

          73

            

          45

              

          Lysine

          86

          80

            

          66

              

          U2

          79

          65

            

          66

              

          T-box

          44

          1

            

          2

              

          IRES_HCV

          63

          47

            

          61

              

          SRP_euk_arch

          62

          64

            

          52

              

          Average(all)

          70

          66

            

          54

              

          Average(sub)

          72

          73

          72

          72

          60

          54

          72

            
          All results are summarized in Table 5.
          Table 5

          Summary of accuracies for all three datasets. The summary of SPS and MCC values (%) for all three multiple alignment datasets are presented. Average(all): the results of the average value of the SPS or MCC for all families. Average(sub): the average SPS or MCC for the subset of families. FoldalignM consists of two modes: FoldalgnM_FoldalignM and FoldalignM_McCaskill, which are separately evaluated and indicated as FoldalignM(1) and FoldalignM(2) respectively.

          Dataset

           

          MXSCARNA

          Murlet

          ProbCons

          MAFFT

          ClustalW

          StrAl

          RNASampler

          original dataset

          Average(all)

          78/78

          79/74

          78/67

          75/63

          72/61

          73/63

          70/74

           

          Average(sub)

          85/86

          85/85

          85/77

          82/74

          78/68

          80/72

          78/86

          BRAlibaseII

          Average(all)

          84/80

          85/78

          85/75

          83/73

          81/72

          83/73

          83/78

           

          Average(sub)

          83/82

          84/79

          84/76

          82/75

          80/72

          82/75

          83/80

          Kiryu et al.'s dataset

          Average(all)

          77/72

          80/71

          79/65

          74/54

          68/48

          69/54

          68/70

           

          Average(sub)

          82/72

          86/72

          85/68

          80/60

          73/52

          75/59

          72/72

            

          RNAlara

          Locarna

          FoldalignM(1)

          FoldalignM(2)

          MARNA

          PMmulti

          Stemloc

          original dataset

          Average(all)

          68/65

                
           

          Average(sub)

          77/81

          74/83

          73/85

          69/83

          72/75

          73/83

          74/83

          BRAlibaseII

          Average(all)

          85/76

          83/78

               
           

          Average(sub)

          86/79

          83/79

          82/79

          79/78

          74/69

          77/75

          75/79

          Kiryu et al.'s dataset

          Average(all)

          70/61

          64/66

            

          62/54

            
           

          Average(sub)

          77/69

          70/73

          64/72

          57/72

          60/60

          52/54

          76/72

          Evaluation of new score function

          In order to evaluate the performance of our new score function (4), we compared it in pairwise alignment with the previous score function of SCARNA, which is a linear combination of RIBOSUM score, stacking energy, loop-distance penalty, base-pairing probability. Dowell's dataset [39], which consists of R100 dataset and percid dataset, are used for the evaluation. R100 is a dataset which consists of 100 pairwise alignments chosen randomly from tRNA and 5SrRNA families in Rfam 7.0 database [34] and percid is a balanced dataset of 100 pairwise alignments from the same families.

          The SPS and MCC are shown in Table 6. It is observed that the new score function of MXSCARNA outperformed the previous score function of SCARNA.
          Table 6

          Accuracy of new score function. The comparison of new score function of MXSCARNA and the old one which was used in SCARNA in terms of pairwise alignment. The SPS and MCC values (%) are used as accuracy measure for alignments. R100 is a dataset which consists of 100 pairwise alignments chosen randomly from tRNA and 5SrRNA families in Rfam 7.0 database [4] and percid is a sequence identitly balanced dataset which also consists of 100 pairwise alignments from these families.

          dataset

          score function

          SPS

          MCC

          R100

          MXSCARNA

          90

          77

           

          SCARNA

          84

          74

          percid

          MXSCARNA

          79

          71

           

          SCARNA

          78

          69

          Time and memory

          The computational complexities of the proposed method for N sequences of length L were evaluated as follows. The construction of the guide tree using the alignments of all pairs of the sequences required O(N 2 L 2) in time and O(L 2 + N 2) in memory. The calculation of base-pairing probability matrices for N sequences by McCaskill's algorithm [27] required O(NL 3) in time and O(NL 2) in memory. The probability consistency transformation (see (3) in Method) required O(N 3 L 3) in time and O(N 2 L 2) in memory. Pairwise alignment of stem component sequences required O(N 2 L 2) in time and memory as is explained in Method. Therefore, the total computational complexities were O(N 3 L 3) in time and O(N 2 L 2) in memory. For the base-pairing probabilities, the computational time for each sequence can be reduced to O(LW 2) by restricting the maximum distance of the base pairs to a fixed constant W [40]. The computation of probability consistency transformation for a pair of sequences can also be calculated in O(L 2) time by restricting the effective width of transformation to a fixed value. Those reductions reduce total time complexity to O(N 3 L 2). We will address those improvements in future work.

          Comparisons of alignment tools in regard to execution time for nucleotide sequences of various lengths are presented in Figures 4 and 5. Randomly generated sequences were allocated into groups of the same lengths and were used for alignment. Stemloc aligned sequences of not more than 100 bases; FoldalignM and Locarna were faster than Stemloc and aligned sequences of 500 bases or less. Because the lengths of the sequences were the same in each alignment task, the banded Dynamic Programming (DP) technique of these methods was effective. Although the Murlet program returned results for sequences as long as 4000 bases in the best case, it was much slower than MXSCARNA. MXSCARNA required only 17 seconds to align 5 sequences of 500 bases and returns alignments for sequences as long as 5000 bases, though the accuracies for sequences longer than 500 bases have not yet been evaluated. Similar comparisons for various numbers of the sequences are presented in Figure 6. The execution time of MXSCARNA is acceptable even for 50 sequences.
          http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_Fig4_HTML.jpg
          Figure 4

          Comparison of multiple alignment tools in execution time for various lengths of the sequences (maximal sequence length, 500 bases). The relationships between the length of the sequences (maximum, 500 bases) and the execution time for several multiple alignment tools are plotted. A set of randomly generated sequences of the same length is used for each alignment.

          http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_Fig5_HTML.jpg
          Figure 5

          Comparison of multiple alignment tools in execution time for various lengths of the sequences (maximal sequence length, 5000 bases). The relationships between the length of the sequences (maximum, 5000 bases) and the execution time for MXSCARNA, Murlet and Stemloc are plotted. A set of randomly generated sequences of a same length is used for each alignment. The number of the sequences used for the alignment is indicated after the names of the tools. The accuracies for the sequences longer than 500 bases have not been evaluated.

          http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_Fig6_HTML.jpg
          Figure 6

          Comparison of various multiple alignment tools in execution time and the number of sequences. The relationships between the number of the sequences and the execution time for MXSCARNA, Murlet and Stemloc are plotted. A set of randomly generated sequences of a same length is used for each alignment. The lengths of the sequences used for the alignment is indicated after the names of the tools.

          Sequence identities and alignment accuracies

          Alignment methods based only on sequence similarities often fail to capture common secondary structures among their alignments, especially when the similarities between sequences are low. In contrast, current alignment methods that rely on information about secondary structures tend to produce inaccurate alignments for sequences of moderate to high similarity by putting too much weight on common secondary structures. The relationships between accuracy and sequence identity for three alignment tools MXSCARNA, ProbCons, and Stemloc are shown in Figures 7 and 8. ProbCons, one of the best of the aligners that ignore information regarding secondary structure, maintains a high SPS throughout low to high sequence similarities, but MCC markedly drops for low sequence identities. Stemloc, one of the best structural aligners (as seen in the previous section), achieved robust accuracies in MCC but failed to compete among the other aligners in regard to SPS for moderate sequence identities. MXSCARNA, which incorporates information on Maximum Expected Accuracy (MEA) alignment in its structural alignments, yielded robust accuracies in terms of both SPS and MCC throughout the tested range of sequence similarities.
          http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_Fig7_HTML.jpg
          Figure 7

          Relationship between sequence similarities and SPS. The relationship between sequence similarities and accuracies according to sum-of-pairs score (SPS) is shown. Lines are smoothed by local weighted regression.

          http://static-content.springer.com/image/art%3A10.1186%2F1471-2105-9-33/MediaObjects/12859_2007_Article_2018_Fig8_HTML.jpg
          Figure 8

          Relationship between sequence similarities and MCC. The relationship between sequence similarities and accuracies according to the Matthews correlation coefficient (MCC) is shown. Lines are smoothed by local weighted regression.

          Availability and requirements

          Project name: ncRNA.org project;

          Project home page: http://​www.​ncrna.​org/​;

          Operating systems: Linux with gcc 3.0 and Cygwin with gcc 3.4;

          Programming language: C++;

          License: free software, except for inclusion to comertical software;

          The source code of MXSCARNA and its web server, the dataset and its references are available at http://​mxscarna.​ncrna.​org. On the web server W and τ correspond to "SCSLENGTH" and "BASEPROBTHRESHHOLD" respectively, and "BASEPAIRSCORECONST" is a parameter of McCaskill-MEA [6] used for the secondary structure prediction, which controls the sensitivity and the specificity of the prediction (α in equation4 in [6]).

          Conclusion

          We have developed MXSCARNA, a new structural multiple aligner of RNA sequences, which progressively applies the pairwise alignment algorithm used in SCARNA. The accuracies of MXSCARNA in terms of SPS and MCC were evaluated for three datasets: an original dataset, the BRAlibaseII benchmark multiple alignment dataset, and Kiryu et al.'s multiple alignment dataset. MXSCARNA's accuracies were at least comparable to those of current state-of-art aligners. In addition, the accuracies of MXSCARNA were robust over a broad range of sequence similarities, whereas the other aligners tested showed reductions in SPS or MCC. The computational complexities of MXSCARNA were evaluated as O(N 3 L 3) in time and O(N 2 L 2) in memory for N sequences of length L. In the comparison of execution time for benchmark datasets, MXSCARNA was by far the fastest among the structural aligners and was fast enough for large-scale analyses. MXSCARNA aligns even 5000-base RNA sequences with acceptable computational costs though the accuracies of alignments for long sequences are not yet confirmed. The source code of MXSCARNA and its web server are available at the web site [41].

          Declarations

          Acknowledgements

          This work was supported in part by a Grant-in-Aid for Scientific Research on Priority Areas "Comparative Genomics" from the Ministry of Education, Culture, Sports, Science and Technology of Japan and by the "Functional RNA Project" funded by the New Energy and Industrial Technology Development Organization (NEDO) of Japan. The authors thank the Japan Biological Informatics Consortium (JBIC) for its support through the "Functional RNA Project" and Michiaki Hamada, Kengo Sato, and colleagues in the Computational Biology Research Center (CBRC) for useful discussions.

          Authors’ Affiliations

          (1)
          Graduate School of Frontier Science, University of Tokyo
          (2)
          Computational Biology Research Center (CBRC), National Institute of Advanced Industrial Science and Technology (AIST)

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          Copyright

          © Tabei et al. 2008

          This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.