Flexibility can be observed in loop regions, rearrangements between secondary structure elements, and conformational changes between entire domains. However, most protein structure alignment methods treat protein structures as rigid bodies. Thus, these methods fail to identify the equivalences of residue pairs in regions with flexibility. In this study, we considered that the evolutionary relationship between proteins corresponds directly to the residue–residue physical contacts rather than the three-dimensional (3D) coordinates of proteins. Thus, we developed a new protein structure alignment method, contact area-based alignment , which uses the residue–residue contact area to identify regions of similarity. The main purpose of CAB-align is to identify homologous relationships at the residue level between related protein structures. The CAB-align procedure comprises two main steps: First, a rigid-body alignment method based on local and global 3D structure superposition is employed to generate a sufficient number of initial alignments. Then, iterative dynamic programming is executed to find the optimal alignment. We evaluated the performance and advantages of CAB-align based on four main points: (1) agreement with the gold standard alignment, (2) alignment quality based on an evolutionary relationship without 3D coordinate superposition, (3) consistency of the multiple alignments, and (4) classification agreement with the gold standard classification. Comparisons of CAB-align with other state-of-the-art protein structure alignment methods  using our benchmark dataset showed that CAB-align performed robustly in obtaining high-quality alignments and generating consistent multiple alignments with high coverage and accuracy rates, and it performed extremely well when discriminating between homologous and nonhomologous pairs of proteins in both single and multi-domain comparisons. The CAB-align software is freely available to academic users as stand-alone software at  During evolution, protein structures are more highly conserved than nucleotide sequences ,2. ThereHowever, proteins are known to be flexible, and this flexibility has an essential role in their functions, such as catalysis, protein–ligand interactions, and protein–protein interactions –14. RigiRecently, it was shown that residue–residue contacts have strong relationships with correlated mutations in multiple sequence alignments . It was Based on the same concept, Olechnovic et al. introduced a contact area difference (CAD)-based score for evaluating the structural similarity between a protein model and the native structure , where tIn this study, we developed a new protein structure alignment method called contact area-based alignment , which uses the similarity of the residue–residue contact area. The main aim of CAB-align is to identify homologous relationships at the residue level between related protein structures. CAB-align comprises the following three main steps. First, CAB-align employs a rigid-body alignment method based on local structural similarity. Second, structural alignment is performed based on global 3D structure superposition to generate a sufficient number of initial alignments. Finally, a heuristic method (iterative DP) is executed based on the modified CAD-score to obtain the optimal alignment.In various benchmarks for protein structure alignment, the alignment quality (AQ) is defined based on comparisons with manually curated gold standard alignments or using geometrical similarity measures. In general, gold standard alignments are used, such as in SISYPHUS , the ConHowever, as mentioned above, our CAB-align method allows flexible protein structure alignments where proteins are flexible. Therefore, we could not use geometric similarities that employ superposition based on Cartesian coordinates to assess the performance of this alignment method, we employed four main evaluation approaches: (1) agreement with the gold standard alignment, (2) AQ based on an evolutionary relationship without the superposition of 3D coordinates, (3) consistency among multiple alignments, and (4) agreement with the gold standard classification.We compared the performance of CAB-align with the HHalign sequence alignment method and three representative structure alignment methods that each use different algorithms: rigid-body alignment , flexible alignment (FATCAT), and the residue–residue distance matrix-based method , and the HHalign sequence alignment method , using three evaluation approaches: quality of alignment, alignment consistency, and agreement with the SCOPe classification .http://scop.berkeley.edu/). We then excluded low-quality structures  is a subset that represents each FAMILY in the SCOPe database . SCOPe_NR10  is a subset with <10% shared identity from SCOPe. The lists for SCOPe_NR10 and SCOPe_FAMILY were taken from the SCOPe website (CI score  <0.4), NCI score , 14 Mar Nc is the number of correctly aligned positions, Lref is the length of the reference alignment, and Lali is the length of the alignment under evaluation. Positions with gaps were excluded.First, we evaluated the alignments using the gold standard alignments in SISYPHUS_ID10 as the reference. We defined two criteria of agreement and reliability as follows:E-value estimated by HHalign shares high similarity with the observed E-value. To assess the quality of the alignment, we defined AQ as the fraction of correctly aligned positions with estimated confidence values not less than a given threshold, as follows:CRconf≥x and Nconf≥x are the number of correctly aligned positions and the number of positions with confidence values not less than the given value of x in the reference alignment, respectively.Second, we defined correctly aligned positions based on comparisons with the reference alignment, which reflected the evolutionary relationship. To generate the reference alignments, we used the HMM–HMM alignment program , which A–B, B–C, and C–A for proteins A, B, and C. In a consistently aligned position, the following condition:E denotes that position i in protein A is aligned with position j in protein B. To evaluate the consistency, we used the coverage (Cov) and relative rate (Rate) of the consistent position, which are defined as follows:Nconsist is the number of consistent positions among the triplet alignment, NA is the number of residues in protein A, and ALA,B,C denotes the number of positions that are commonly aligned regions among the three alignments. Positions with gaps in any of the three alignments were excluded.According to Sadowski and Taylor , we asses are defined as follows:TPs is the number of true positive pairs that are correctly classified as belonging to the same class  based on the threshold s, FPs is the number of false-positive pairs that are incorrectly classified as belonging to the same class based on the threshold s, TNs is the number of true negative pairs that are correctly assigned as belonging to a different class based on the threshold s, and FNs is the number of false-negative pairs that are incorrectly classified as belonging to a different class based on the threshold s. We used the area under the ROC curve (AUC) and the area under the PRC (AUPRC) as measures of agreement with the SCOPe classification. The AUC corresponds to the probability that the proposed structural similarity score will rank a randomly selected domain pair that belongs to the same class as higher than a randomly selected pair that belongs to a different class. The AUPRC corresponds to the average precision of the proposed structural similarity score.We assessed the ability to discriminate between homologous and nonhomologous pairs of proteins based on the agreement with SCOPe classifications. We used the receiver-operating characteristic (ROC) curve and precision-recall curve (PRC) analysis. The ROC curve plots the recall against the false-positive rate (FPrate). The PRC plots the precision (or reliability) against the recall (or coverage). The precision, recall, and FPrate for a given threshold S (NormS)   with theNormS averages   for the six benchmark datasets. For all datasets (AQ(0). As shown in datasets , CAB-aliTo assess the consistency of the alignments, we used six datasets that comprised triplets of alignments from the aforementioned protein pair datasets. In total, 7,384, 2,173, 50,630, 14,689, 1,403,291, and 790,623 triplets of alignments were obtained from SCOPe_NR10_all, SCOPe_NR10_e10, SCOPe_FAMILY_all, SCOPe_FAMILY_e10, PDB30_e5, and PDB30_e10, respectively.Nconsist), length of a commonly aligned position (AL), coverage (Cov), and relative rate (Rate) for the five alignment methods using the six datasets. Cov against Rate. Based on the Rate, HHalign performed the best with all six datasets. These results correlated with the high reliability of HHalign for the SISYPHUS_ID10 benchmark dataset  does not directly represent the AQ. However, a high degree of consistency, which is derived from structural similarity, is necessary for improving multiple alignments. Cov score with CAB-align was comparable with that of DaliLite.High scores are necessary for To assess the classification performance, we used two benchmark datasets obtained from SCOPe_NR10 and SCOPe_FAMILY, which comprised single domains, where these two benchmark datasets are denoted as NR10 and FAMILY, respectively. Each benchmark dataset contained 249,500 domain pairs (500 versus 500 domains), excluding 500 same-domain pairs. The NR10 dataset contained 120 pairs from the same family, 732 pairs from the same superfamily, and 1,774 pairs with the same fold. The FAMILY dataset contained 410 pairs from the same superfamily and 1,288 pairs with the same fold. According to the SCOPe classification, the domain pairs within the same family or superfamily correspond to an evolutionary relationship. The fold only group proteins with similar structures; therefore, some domain pairs with the same fold only share structural similarity, and no evolutionary relationship has been confirmed. Furthermore, to assess the performance with multidomain proteins, we constructed a benchmark dataset containing 99,792 protein pairs selected from PDB30. In this multidomain benchmark dataset, all of the proteins contained at least two domains assigned by SCOPe. For this multidomain benchmark dataset, we defined a pair of proteins with no common domain class  as negative; otherwise, the pairs were defined as positive. The multidomain benchmark dataset contained 946 pairs from the same family, 3,100 pairs from the same superfamily, and 5,762 pairs with the same fold. We evaluated the agreement between the five alignment methods and the SCOPe classifications at three levels  using ROC and PRC analyses. As shown in To assess the contribution of each step in the CAB-align method, we evaluated two components used in CAB-align denoted as step 1 and step 2. As shown in S and NormS were lower than the values derived from CAB-align. However, step2 performed marginally better than CAB-align in terms of agreement and reliability. The advantage of iterative DP in terms of the AQ was not observed with the SISPHUS_ID10 benchmark dataset. In addition, The results for CAB-align and step 2 are presented in E-value = 1.2 × 10−28) and a reliable alignment for 146 positions between the two calmodulin-like proteins. The alignment obtained by CAB-align had a higher AQ score (AQ(5) = 0.80) than DaliLite (AQ(5) = 0.43). Moreover, Figs Figs S  and three large benchmark datasets, i.e., SCOPe_FAMILY, SCOPe_NR10, and PDB30. Our comparison of CAB-align with other state-of-the-art protein structure alignment methods  showed that CAB-align was robust, and it obtained high-quality alignments for protein pairs with known evolutionary relationships. Moreover, CAB-align generated consistent multiple alignments with high coverage and accuracy rates, which were comparable with those obtained by DaliLite. Finally, CAB-align performed well at discriminating between homologous and nonhomologous pairs of proteins in both single- and multidomain comparisons.These results suggest future applications for the CAB-align algorithm. For example, the production of high-quality alignments will facilitate the identification of functionally important positions and the functional annotation of novel proteins. CAB-align will also allow us to discover novel evolutionary relationships at the residue level. The consistent multiple alignments obtained by CAB-align will help to identify structurally conserved regions as well as improving template-based modeling methods based on multiple templates. At present, there are many types of structural classification databases for single proteins . We used Marching Cubes triangulation to define the triangle surface, vertices, and edges. The voxel size was set to 1.0 Å. The extracted surface was used to calculate the residue–residue contact area, where the distance between the surfaces of different residues was less than 2.8 Å. In this study, we defined an inter-residue contact area between the lth and mth residues as the sum of the surface area for the interactions between the mth and lth residues on each residue , and the linear correlation coefficient was determined as 0.995 when taking 49,458 amino acid residues from a random selection of 200 protein structures.Our smooth surface model was used to approximate the solvent excluded surface (SES), which is defined by a spherical probe with a radius of 1.4 Å rolling over all the atom spheres. SES is used widely to represent the surface of a molecule, and it can be calculated by various algorithms ,43. Our URMS) distance between all pairs of heptapeptides, as described in the MAMMOTH algorithm [To generate various initial alignments, we employed the Smith–Waterman DP algorithm  accordinlgorithm , whereasURMSR is the expected minimum URMS between two random heptapeptides, which we set to 0.917. The affine gap penalty is defined as g(k) = α + β(k − 1), where k is the number of gaps, and α and β denote gap open and extension penalties, respectively. We used 11 gap open penalties ranging from 0 to 50 (step size = 5.0) and six extension penalties ranging from 0 to 10 (step size = 2.0). In total, we employed 58 combinations of α and β in the DP phase, where α ≥ β. In our study, the optimal alignments based on local similarity were not sufficient to generate the best alignment with the highest residue–residue contact area similarity. Thus, we also generated four suboptimal alignments [S. During each iteration, S for the previously aligned positions was decreased by 10%, and the forward trace DP matrix was then updated and thus a new alignment was generated, which was changed slightly compared with the previously computed alignment.First, we found 290 initial alignments based on the local structural similarity with various gap open and extension penalties. According to the MAMMOTH algorithm, the score matrix used in the DP phase is defined as follows:ignments  for eachWe obtained more than 290 initial alignments based on the global structural similarity. These 290 alignments were then re-aligned to maximize the TM-score based on a heuristic iteration described in TM-align . In thisA and B, we defined the simple similarity score as follows:i and j denote a pair of aligned residues from A and B as i =  and j = , L is the length of the alignment, and θ is the similarity measure for a residue pair using the residue–residue contact area. For θ, we modified the CAD-score by considering two points: (1) strong overprediction of the contact is better than entirely missing the contact and (2) the similarity measure is a symmetrical function based on proteins A and B. θ is defined as follows:i and j of A and B, respectively, and wi,j is a weight function used to control the overcounting of nearby contact when the sequence separation is less than five.Following the removal of redundancies from the 580 initial alignments, the remaining alignments were subjected to iterative Smith–Waterman DP based on the residue–residue contact area matrix  denotes the similarity between residues kA in protein A and kB in protein B when the other aligned positions i =  are not changed for the given alignment (kA − iA)(kB − iB) ≤ 0.DP was applied to the similarity matrix to find an optimal alignment in a heuristic manner, which was calculated from the given initial alignment and the residue–residue contact area matrix. The preliminary similarity matrix lignment . As showM is too restrictive to improve the alignment by DP and only the local optimal alignments were obtained in our study. Therefore, M is converted into M' by considering neighboring residues using a sliding window. M' is defined as follows:NA and NB are the number of residues in protein A and B, respectively, and Nw defines the size of the window. Thus, 2Nw + 1 corresponds to the window size. M' approximates the similarity between the two fragments, i.e., Nw residues around kA and kB , and M' was then calculated from the given alignment . The training dataset contained 4,144 protein pairs with a significant evolutionary relationship  and a reliable alignment length (≥100). Moreover, to confirm the evolutionary relationship, both proteins in the pair had to belong to the same superfamily. Our parameter optimization indicated that the optimal parameter was γ = 0.5, and the gap open penalty in iterative DP was 90.Starting from the given initial alignment, the heuristic method (iterative DP) was applied to obtain an optimal alignment. Step 3 in Figs lignment . DP was obtained . The ite5, 3, 1} . Finallymeters γ  and the S represents the degree of a common inter-residue contact area between two proteins according to the alignment, but it depends on the size of the proteins. A pair of large proteins usually has a large S value. Due to this size dependency, S cannot be used to compare the similarities of different protein pairs. The main aim of the normalization is to distinguish homologous pairs of proteins from nonhomologous pairs. Thus, we propose a NormS that considers the total inter-residue contact area as follows:S is the similarity score of the alignment  the protein pair with a higher S tends to have a higher NormS, and (2) the higher relative rate of S against TA and TB tends to yield a higher NormS Click here for additional data file.S2 Table(DOCX)Click here for additional data file.


Escherichia coli K-12, as deposited in the KEGG database, and the enzymes belonging to a particular pathway were transformed into enzymatic step sequences by using the breadth-first search algorithm. These sequences represent contiguous enzymes linked to each other, based on their catalytic activities as they are encoded in the Enzyme Commission numbers. In a posterior step, these sequences were compared using a GA in an all-against-all (pairwise comparisons) approach. Individual reactions were chosen based on their measure of fitness to act as parents of offspring, which constitute the new generation. The sequences compared were used to construct a similarity matrix  that was then considered to be clustered by using a k-medoids algorithm. A total of 34 clusters of conserved reactions were obtained, and their sequences were finally aligned with a multiple-sequence alignment GA optimized to align all the reaction sequences included in each group or cluster. From these comparisons, maps associated with the metabolism of similar compounds also contained similar enzymatic step sequences, reinforcing the Patchwork Model for the evolution of metabolism in E. coli K-12, an observation that can be expanded to other organisms, for which there is metabolism information. Finally, our mapping of these reactions is discussed, with illustrations from a particular case.In order to understand how cellular metabolism has taken its modern form, the conservation and variations between metabolic pathways were evaluated by using a genetic algorithm (GA). The GA approach considered information on the complete metabolism of the bacterium  From a global perspective, the comparative analysis of metabolic pathways aims to identify similarities and differences among them, providing insights for the identification of evolutionary events, such as enzyme recruitment and duplication events. In this regard, metabolic pathways exhibit high retention of duplicates within functional modules and a preferential biochemical coupling of reactions. This retention of duplicates may result from the biochemical rules governing substrate-enzyme–product relationships In this context, diverse studies have evaluated the variations among pathways, both intra- and interspecies Escherichia coli K-12, as deposited in the KEGG database, were transformed into linear Enzymatic Step Sequences (ESS), to be compared via a genetic algorithm (GA). The sequences compared were used to construct a similarity matrix to identify groups of conserved reactions based on a k-medoids clustering analysis, and then a multiple-sequence alignment (MSA) GA was optimized to align all the reaction sequences included in a group. Finally, we consider our comparisons in terms of the clues they provide in reinforcing the Patchwork Model in the evolution of metabolism for E. coli K-12 and probably for other organisms beyond this bacterium.In this work, we evaluated whether there are groups of similar reactions in different pathways, which might suggest a transfer of enzymatic activities, and whether these groups can be used to define common and variable regions of an organism's metabolism. This analysis was addressed using EC numbers, coded as a succession of reaction steps. To this end the metabolic maps of the bacterium 22.1E. coli K-12 as of June 2011 were downloaded from the KEGG database (The KGML files (version 0.71) that describe the metabolic maps (pathways) of database . Pathwaydatabase . ESS con2.2The proposed GA starts by creating a random initial population of variable-length chromosomes, which represents the potential for alternative alignments. For each iteration or generation, the population evolves by means of selection, crossover, and mutation. Here, a tournament selection that randomly selected a subset of individuals and chose the best individual of each subset was used.2.3In this work, one-point crossovers were considered; however, it is important to note that for crossover two alternative metabolic pathway sequence alignments, they must be in general, of different sizes (i.e. have a different number of columns). Then, this operator is used to select a position in the first parent and the second parent is cut, keeping in mind the EC numbers conserved for the left side of the first parent. As shown in 2.4A binary codification was used, where 1 represented an enzyme and 0 represented a gap. The gap insertion was highly penalized. The algorithm was designed to find the best alignment with the maximum score. Therefore, if the selected position is a gap, this can be extended or reduced in one unit with an uniform probability; on the contrary, if the selected position corresponds to an EC number, a gap is inserted.2.5Assessment of the quality of an alignment considers the column homogeneity, with penalizations for gap insertions and column increments. Thus, the proposed objective function (O.F.) is composed of three weighted terms, defined in Eq. 2.6The sum of pairs is a popular criterion for column homogeneity evaluation, as it assigns a cost to each pair of aligned codifications in each column of an alignment (substitution cost) and a cost to gap insertions (gap costs). The sum of these costs yields the global cost of the alignment. In this work, rating the grade of diversity in the elements of a given position (column) was evaluated as a measure of column homogeneity.M, where jm is the jth column and j1m, j2m, and j3m are the three levels of the EC number represented by jm:E(mjk) is the normalized entropy for the kth level of the jth column and 1ω, 2ω, and 3ω have values of 15, 10 and 5, respectively. Then, the entropy for each column is estimated as follows,The EC numbers  are represented by three levels (subcolumns), evaluating the normalized entropy of each level based on Shannon's entropy. These results are weighted, giving a higher value to the first level. Gaps are considered one more symbol, and a penalty is applied to compensate for the possibility of “false homogeneity” indicated by a high number of gaps in a column. The following equations give details for the evaluation of the homogeneity of each column. Given alignment a in column jk is the ratio of the symbol a counter in column jk and the sum of all symbol counters a′ (number of sequences). Gaps are also considered as a symbol The probability of symbol 2.7GP corresponds to the total number of individual gaps in the alignment. It is important to mention that initial and final gaps are not taken into account for the gaps penalization. If there are no gaps in the alignment, the GC value is zero.The gaps concentration criterion is used to penalize the number of gap blocks in the whole alignment. This criterion is defined as:C0 is the length of the longest sequence in the unaligned matrix and C1 corresponds to the number of columns of the aligned matrix. Thus, the defined objective function (O.F.) has to be minimized to attain better alignments.This criterion serves to reward alignments where gap codifications are more concentrated, that is, where there are few larger blocks of gaps rather than blocks of smaller lengths, in contrast to Equation 2.8k-medoids algorithm is used to cluster the sequences previously compared and included in the similarity matrix. The number of clusters is settled by using the elbow criterion and plotting the cluster quality from k = 2 to k = 100. A total of 20 replicates for each k value are evaluated, and the best result is selected. To define the cutoff for the clustering, the second derivative associated with this function is used, and when k is maximized, this value represents the greater slope, i.e., the greater difference between two k′ values.The 33.1E. coli K-12, a collection of ESS was generated and compared using a GA. In this work, an ESS was defined as a linear collection of consecutive enzymatic reactions from a given substrate to a given product, similar to the method used for a previously proposed definition of metabolic pathways E. coli K-12 were obtained In order to evaluate the conserved and variable catalytic steps in metabolic pathways of the bacterium Therefore, it seems that the numbers and lengths of sequences of enzymatic steps depend on the size of the metabolic map and reflect the number of alternative pathways that can be traced, beginning from the start nodes, i.e., large metabolic maps generate more ESS than small metabolic maps as a consequence of the complexity of the map. In this regard, the pyruvate metabolism is represented by a complex map, as it is the endproduct of glycolysis and the starting point for gluconeogenesis, and it can be generated by transamination of alanine. It can be converted by the pyruvate dehydrogenase complex to acetyl-CoA 3.2To evaluate the significance of the alignments, 10 random sets of ESS were constructed by shuffling the EC numbers from the real sequences, where the length and the EC composition were conserved . Because we were interested in evaluating the most significant scores, a threshold of fitness of 0.4 that considers “Alternatively, we assessed the consistency of the GA approach in a different scenario, one which considered a Dynamic Programing (DP) Needleman–Wunsch algorithm for the pairwise alignment of ESS. This algorithm uses the same objective function as the GA, minimizing the score of the alignment. The DP algorithm exhibits a similar (Gumbel-like) distribution as the GA fitness values, suggesting that our approach is consistent with the previous results described .3.3E. coli K-12, 452 ESS were used to carry out all-against-all pairwise alignments. Therefore, the GA previously described was applied based on the objective function (O.F.), with which we evaluated the entropy per column using a population of 100 chromosomes, a 1% mutation rate and 90% crossover rate. Mutation is applied per individual and it means that an individual will be mutated each generation. Therefore, an individual in the population is a particular possible solution to the ESS alignment. The algorithm was concluded when 20 generations were reached with no changes in the objective function. Ten replicates of the GA were performed, and the best result was chosen. The GA evaluates the alignment by using a normalized entropy-based function defined above in Eq. k-medoids algorithm and identified groups with similar ESS that may share similar catalytic properties. To determine the number of clusters (k) that could be generated, the elbow criterion was used. In addition, diverse k values were used to construct different clusters: k = 21, 27, and 34, as these exhibited the higher peaks in the analysis plot. As a criterion to discriminate between the clustering at k = 21, 27, and 34, the clusters were depurated, eliminating any sequence whose mean fitness with the rest of the sequences of its cluster was above 0.4. After depuration, 93, 71, and 37 sequences were eliminated for k = 21, 27, and 34, respectively. In this way, we selected a k value of 34 to minimize the number of sequences eliminated during depuration.In order to evaluate the similarity of the metabolic maps for the bacterium From this analysis, similar ESS associated with diverse pathways were clustered together. For instance, the ESS constructed from the pyruvate map were differentially clustered in nine groups with sequences belonging to different pathways, including those for the citrate cycle (TCA cycle), glycolysis/gluconeogenesis, and lysine, glycine, serine, and threonine metabolism . This fiAlternatively, cluster 4 contains sequences from different maps related to lipid metabolism, including lipopolysaccharide, glycerolipid, glycerophospholipid, and steroid metabolism, pyrimidine metabolism, selenium amino acids, the pentose phosphate pathway, and galactose metabolism. The similar catalytic activities in all these pathways suggest the phenomenon of recruitment events. Finally, cluster 23 contains exclusively ESS from the glycerolipid and glycerophospholipid metabolic pathways, suggesting a common origin shared by these two metabolic pathways. These results correlate with the origin of fatty acid metabolism by gene duplication Patchwork Model of evolution.Similar results were observed in the ESS included in clusters 18, 24, 22, 25, and 30, which included reactions from diverse amino acid metabolism pathways, whereas clusters 2, 11, 17, 21, 23, and 29 included ESS from carbohydrate metabolism. In particular, clusters 17 and 21 included ESS belonging to galactose, starch, and sucrose metabolic pathways; cluster 12 included ESS from glycosphingolipid biosynthesis; and cluster 21 included ESS from the galactose, pentose, sucrose, mannose, and glycolysis/gluconeogenesis pathways. All these findings suggest that the assimilation of these carbon sources involved identical enzymatic processes that may have arisen from recruitment events, according to the 3.4core of ESS includes an isomerization (EC 5.3.1) reaction followed by a phosphorylation (EC 2.7.1) reaction, suggesting that these reactions are ancestral to all these pathways or that they were recruited in small catalytic modules In order to maximize the identification of functional similarities among sequences included in a common cluster, MSAs were performed using a progressive GA alignment and an objective function based on entropy. This function considered three main factors, determined empirically, to evaluate the enzymatic number level . The progressive GA works as follows: the ESS are sorted according to their similarity values, and this order is used to guide the sequence alignment, considering the more similar pair and adding a third one, and so on. This algorithm increases column homogeneity among the sequences aligned and penalizes the insertion of gaps; hence, the fitness function has to be minimized to obtain better alignments. From this analysis, in Nevertheless, it is expected that similar enzymatic steps found in different metabolic maps actually represent different proteins. Therefore, to examine this phenomenon, five ESS were selected from cluster 21 that represented each of the five metabolic maps, and the information for protein sequences associated with each EC number was obtained. Based on this approach, the ESS from the glycolysis/gluconeogenesis, fructose and mannose metabolism, galactose metabolism, and pentose and glucoronate metabolism pathways, representing different enzymes, as shown in Patchwork EvolutionBased on these data, it was suggested that the similar enzymatic steps identified by this strategy may share a common origin, i.e., they have been recruited from different pathways. To evaluate the probable common origin of these enzymes, their amino acid sequences were analyzed using the domain identification system based on Superfamily database assignations Based on this analysis we identified that three proteins  share a common structural domain, identified by superfamily, the aldolase domain, and two else (FucA and AraD) the AraD/HMO-PK domain-like, which catalyze dissimilar reactions. These proteins might represent two cases of divergent evolution, in which two enzymes from a common ancestor diverged at different functions. Therefore, according to our data, it is clear that convergent evolution events are also frequently associated with enzymes devoted to metabolism; for instance, four nonhomologous enzymes  catalyze the isomerization reaction 5.3.1.x, conserved in all the sequences of the cluster. This finding correlates with previous reports that showed that catalytic convergent evolution is a common phenomenon Finally, it is interesting that in the case of the sequences representing the fructose and mannose metabolism pathways and the pentose and glucoronate metabolism pathways, the FucK–FucA and XylB–AraD enzymatic steps are catalyzed by enzymes sharing a common structural domain, suggesting that the approach described here is able to identify probable patchwork events as well as duplication and convergent enzymes associated with the evolution of a microorganisms's metabolism.4In general, two main approaches to compare metabolic pathways have been proposed. In one approach entails methods based on the alignment of complete or fragmented networks In this regard, sequence alignment is a problem in bioinformatics where it is important to identify conserved regions in a set of sequences, and as a consequence, to find the differences between them based on a criterion to assess the quality of an alignment. Thus, a GA was proposed in order to optimize the quality criterion of potential alignments considering maximization of homogeneity and, at the same time, minimization of gaps. The proposed GA used a variable binary representation of individual enzymes  where 1 represents EC numbers and 0 indicates the gaps.In the present work, the strategy to transform enzymatic reaction sequences from a metabolic pathway allowed us to implement a sequence alignment method in an efficient and easy fashion, where regions sharing a similar succession of EC steps were identified, suggesting common catalysis that is not easy to identify when using traditional computational tools. The progressive GA proposed method has shown efficient results, providing good alignments after less than 100 iterations. These results allow us to make comparative studies of metabolic pathways to elucidate the functions of newly discovered pathways, increase our understanding of evolutionary traits, and identify potential missing pathway elements. It is interesting that some of the enzymatic steps common to various sequences may, indeed, represent the same reactions, such as the first 10 sequences in Patchwork Model for studying the evolution of metabolism in E. coli K-12, where similar consecutive enzymatic steps may have different origins, in agreement with biochemical restrictions in enzymatic recruitment In summary, we consider that the strategy described in this work allowed us to identify similar ESS from different metabolic maps, as shown in


Protein sequence similarities to any types of non-globular segments  are pertinent sources for mistaken homologies. Regretfully, these considerations regularly escape attention in large-scale annotation studies since, often, there is no substitute to manual handling of these cases. Quantitative criteria are required to suppress events of function annotation transfer as a result of false homology assignments.The sequence homology concept is based on the similarity comparison between the structural elements, the basic building blocks for conferring the overall fold of a protein. We propose to dissect the total similarity score into fold-critical and other, remaining contributions and suggest that, for a valid homology statement, the fold-relevant score contribution should at least be significant on its own. As part of the article, we provide the DissectHMMER software program for dissecting HMMER2/3 scores into segment-specific contributions. We show that DissectHMMER reproduces HMMER2/3 scores with sufficient accuracy and that it is useful in automated decisions about homology for instructive sequence examples. To generalize the dissection concept for cases without 3D structural information, we find that a dissection based on alignment quality is an appropriate surrogate. The approach was applied to a large-scale study of SMART and PFAM domains in the space of seed sequences and in the space of UniProt/SwissProt.Sequence similarity core dissection with regard to fold-critical and other contributions systematically suppresses false hits and, additionally, recovers previously obscured homology relationships such as the one between aquaporins and formate/nitrite transporters that, so far, was only supported by structure comparison. The modus operandi of the modern day sequence homology concept ,2 is foup → q). Whereas this first implication appears quite acceptable , the second one is by far not obvious. In the proof of the second implication where structure/function similarity is concluded from high sequence similarity , the conserved key amino acids in the uncharacterized sequence for concluding similarity to the structure/function of the well-studied protein need to be those that correspond to the hydrophobic patterns responsible for the 3D structure formation and the residues critical for binding/catalysis/etc. To note, in both cases of inductive proofs, the proteins under scrutiny were soluble, globular proteins of limited size without non-globular segments.In both proofs, there are some crucial, yet problematic assumptions . In the Thus, homology has the precise meaning of “having a common evolutionary origin” but it also carries the loose meaning of “possessing sequence similarity or being matched”. In addition, homology between sequences is always a hypothesis while similarity, being a measurable fact, can be attributed to either chance, convergent evolution or common ancestry -13. In op → q into q → p since the issue of distinguishing between convergent evolution and common ancestry among hits of high similarity is non-trivial. As a guide, similarities to any types of non-globular segments  are pertinent sources for mistaken homologies [Fortunately, sequence similarity by chance can be eliminated via stringent statistical criteria like E-value cutoffs in Blast  or HMMERmologies ,17-19. Amologies , regretfmologies ,19-21.q → p is a non-equivalent converse statement of the original proof p → q. Indeed, this is the bane of current sequence search approaches that, frequently, lead to wrongful protein function predictions or annotations, especially when one attempts to extrapolate very deep into sequence space [Standard alignment tools  ace space -29.To alleviate the abovementioned issue, we reiterate that the working principle of the sequence homology concept is based on the similarity comparison between the structural elements, the basic building blocks for conferring the overall fold of a protein which in turn characterizes its biological function . To noteTo further emphasize, the concept of a globular domain has a deeply-rooted notion where it implies a sequence segment  having an independent tertiary structure , it folds and melts autonomously. Its sequence evolves as a unit in phylogeny . The unsIn this work, we achieved four main objectives. First, we created an algorithm and the software tool DissectHMMER (provided as supplement to this MS ) that caThird, to generalize the dissection framework to domains without PDB/DSSP representation, the quality score based on alignment quality was introduced. Out of 635 SMART and 5876 Pfam domains with structures, 537 SMART and 4771 Pfam domains were found to be enriched with structural residues in their high-quality segments. This was more than 80% of the statistically testable cases. Thus, the quality score is justifiable surrogate for estimating fold-related and remnant segments in domain models. Importantly, this and similar criteria can be applied to segmenting HMM models in domain libraries without having the domain alignments to be re-edited or the HMMER searches to be rerun.Finally, the application of the dissection framework  on the seed alignments of SMART and Pfam domains gave an average positive concordance rates of almost 100% and a negative one of less than 1%. The latter implies that almost all of the seed sequences were recognized correctly as true hits. Meanwhile, the dissection of alignment results from searches against the UniProt/SwissProt for these SMART and Pfam domains returned average false-positive rates of less than 1% but average false-negative (FN) rates of 7.63% (SMART) and 4.86% (Pfam). The latter presents an opportunity to recover previously obscured homologous relationship between the FN hits and its associated domain model. Filtering for domain models that have exceptionally high error rates also allows finding those cases where reconsidering the seed alignment might be useful.In the current implementation of the HMMER packages  scores and transition  scores where M/D/I are match, delete and insert states. In the case of the invariant scores, they account for the fixed transition entry scores  and exit scores  for each domain-to-sequence alignment. These are added to the positional scores to give the final log-odd score of the alignment. As a rule, these positional and position-invariant components are retrievable from the respective HMM model files provided with domain libraries. Then, the reconstruction of the HMMER scores follows the straightforward arithmetic computations as described in equation (1) (see Methods).In fact, the score reconstruction has already been applied on HMMER2 glocal  and global  outputs in one of our earlier works . TherefoHowever, two issues ensue to complicate the straightforward procedure. First, the current implementation of HMMER3  lacks su1, B- > D1; see Figure K- > E, DK- > E; see Figure 2..K-1 shown by blue lines in Figure This problem stems from the exclusion of certain invariant score parameters during the conversion of HMMER3 model files to HMMER2 format. Regretfully, the conversion is necessary to export the HMMER3 null model parameters (as part of the log-odd score parameters) since they are embedded in the HMMER3 program code, the second major issue. In contrast, the HMMER2 null model parameters are already captured in their model files. To note, the HMMER3 software suite only allows for model conversion (via hmmconvert -2) from the HMMER3 local model to the HMMER2 glocal/global model. In the process, only the first HMMER state  and the HMM model file as inputs independent on the HMMER suite version used (2 or 3). The algorithmic detail is described in the Methods section. The code is provided as Additional file To summarize, the score calculation in the various HMMER versions is a complicated routine with some parts not explicitly documented in the literature. Besides algorithmic assumptions, numerical issues such as rounding errors also play a role. Thus, it cannot be expected that the reconstructed scores exactly match the scores reported by HMMER but it is close enough for the purpose of reconstructing the segmental contributions to the total score.To test the score reconstruction workflow, the seed alignments from SMART version 6 and Pfam release 27 were used. In comparison to SMART, the current Pfam library is about 12 times larger and, hence, the rigor of the scores reproduction was truly being tested in this case. In total, 735 SMART domains (excluding 73 domains with less than 5 seed sequences) and 12121 Pfam domains (excluding 2711 domains with less than 5 seed sequences) were examined.For each domain alignment, the HMMER model is first built (using hmmbuild with null2 option off) and, then, it is searched against (using hmmsearch -F) the same set of seed sequences. For each seed sequence, the alignments reported are considered true hits. By this constraint, both HMMER2 and HMMER3 share the same search space and, hence, the alignments generated by both are expected to be similar . Next, the HMMER log-odd scores for the total alignment were reconstructed as described in Methods (see equations (1 and 2)).r2) as goodness of fit, is plotted for both SMART (version 6) and Pfam (release 27) domains in Figure Once this computation was completed for all seed sequences of a given domain, linear regression analysis was performed against the original scores (see equations (3 and 4) in Methods). The regression analysis output, in terms of slope (ination r as goodnination r as goodnination r as goodnr2), for the original versus reconstructed HMMER2 and HMMER3 scores are depicted in Figure r2 of 1.000 (SMART/Pfam) and small standard deviations of 0.003 (SMART) and 0.007 (Pfam). HMMER3 reconstruction closely followed at an average r2 of 0.997 (SMART) and 0.998 (Pfam) over a slightly larger variation of 0.007 (SMART/Pfam). Taken together, the general trend where all values of r2 converges to one with little variation, implies that the reconstruction workflow for HMMER2/3 scores are highly reliable and reproducible. The reconstruction works well for the relatively small SMART library as well as for the huge Pfam library.Next, the relative error estimates per SMART/Pfam domain were examined  but not equivalent. To note, this work was not planned as a comparative study between the two tools and we strived as much as possible to focus on conclusions supported by either program.This large scale study of seed sequence scores also allows comparing some aspects of HMMER2 and HMMER3 program behaviors. Figure The idea of dissecting a HMM score into several segments of a larger alignment stems from the observation that the influence of well conserved, truly homologous alignment segments on the score can be overwhelmed by score contribution from spurious alignment extensions. In our previous work ,19, we hAt the same time, it is well accepted that structural elements are the basic building blocks for conferring the overall fold of a protein which in turn characterizes its biological function. Therefore, for the purpose of inferring homology, one should evaluate the score of the structural, fold-relevant segments independently from the score associated with remnant segments. Figure In retrospect, all hit examples  were retrieved from the corresponding DSSP annotations  with theBased on a collective view of the standard HMMER output scores/E-values in Table However, once the fold-critical and remnant scores  where the fold-related scores were larger than the remnant scores. For hit 13 (TIP12_MAIZE), the difference between its fold-related and remnant scores was marginal. The corresponding fold-related E-values of 1.2e-4 to 2.2e-6 (HMMER2) and 3.1e-3 to 1.5e-9 (HMMER3) were more significant than the remnant segments’ E-values of 9.4e-5 to 1.4e + 2 (HMMER2) and 3.1e + 2 to 5.3e + 5 (HMMER3). Thus, the latter three hits are rather true homologies in the segment representing the protein fold.Furthermore, to investigate the difference in magnitudes between the fold-critical and remnant E-values, their ratios  describes the N-terminus of the homeobox-containing transcription factor HNF-1 (Hepatocyte nuclear factor 1). The latter contains a dimerization sequence and an acidic region which is implicated in transcription activation . In contMeanwhile, the model T2SL  describes protein L, an inner membrane protein of the bacterial type II secretion system that serves as a critical link between the cytoplasmic and the periplasmic part of the Eps-system . In contFinally, the model ELL  is a RNA polymerase II elongation factor that regulates the polymerase II . Yet, thFor the alleged true hits, the justification of sequence similarity between the hit and domain model is best shown by fold similarity, especially for cases of distant homologs  where more sequence divergence is expected. Therefore, structure alignment was performed on each pair of representative PDB structures from the hit and the domain model using the jCE algorithm   is a family of ATPases associated with various cellular activities. The ATP-dependent metal binding core of the domain’s representative PDB structure (1LV7|A) consists of the characteristic Walker A or P-loop motif, Walker B motif and sensor motif, each extending beyond a ß-strand . The hit+ and its ß-ketoacyl substrates [+/NADPH binding structure.Next, the model adh_short  is a family of NADP-dependent oxidoreductases. Its representative PDB structure (3MJC|A) is an A-type ketoreductases consisting of two subdomains, a N-terminal sub-structural domain and a C-terminal catalytic subdomain that binds NADPbstrates . On the bstrates . The staFinally, the model Form_Nir_trans  describes the multi-membrane formate/nitrite transporter (PDB: 3KCU|E) of bacteria that facilitates the formate/nitrite transport essential for anaerobic respiration . On the Taken together, we have illustrated that the dissection framework provides the segment-based scores (e.g. the fold-related and other segments’ scores) for a more concise assessment of sequence similarity as evidence for homology. To emphasize, filtering of compositionally-bias sequence segment might be unnecessary since false hits will be occluded under this framework when their non-fold-related segments appeared significant statistically. Most importantly, the framework provides an opportunity to elucidate the obscured true hits hidden among the false ones in the twilight E-value range of 0.1 to 10.In an ideal situation, the combined PDB/DSSP data provides the best information for dissecting a domain model into its fold-related and remaining segments for score reconstruction. But currently, only a small portion of domain models have an associated PDB structure. As such, one needs a surrogate for estimating the potentially more conserved elements and remaining segments for the dissection framework to be applicable on a larger set of domain models.For this purpose, the alignment quality measure  that assesses sequence conservation in CLUSTALX  was inveα = 0.05. Next, each alignment position is classified as high or low-quality based on the appropriate thresholds  in Methods). The quality score thresholds are at least 0.06  and 0.14  for SMART and Pfam domains respectively. Finally, the high-quality and low-quality segments per domain alignment were derived using equation (18).First, the quality score per position for each domain alignment in SMART and Pfam were computed  in the Methods). Alignments with less than 5 sequences were not considered for the analysis due to insufficient statistical power at a significance level of Separately, matching structures were searched for. The global-mode HMM models were built using the HMMER2 software suite to maximize for full coverage. The HMM models were searched against the PDB FASTA sequences to obtain the most significant hit  with the associated secondary structure residues resolved for each alignment position using the DSSP annotations , and theα = 0.05, 537 (out of 635) SMART domains and 4771 (out of 5876) Pfam domains were enriched with structural residues in their respective high-quality segments  SMART and 5876 (out of 14831) Pfam domains were able to retrieve a significant PDB hit that covers the model’s full length. Each of these domain models was then subjected to the Fisher’s exact test (see equation (9) in the Methods section “Fisher’s exact (one-tailed) test …” and also the table therein) to determine if there is an enrichment of structural residues in the high-quality segments against the low-quality segments. Interestingly, at a significance level of It is also noteworthy to mention that the dissection framework using the quality score were applied to the 13 validated examples from the preceding section. The conclusions were similar to that of Table In this section, the behavior of the score dissection framework when applied to the hits’ alignments returned by domain models from HMMER searches was examined. For this purpose, 285 (out of 537) SMART and 2381 (out of 4771) Pfam domain models were taken from the preceding section after filtering domain models with low-quality segments of less than 10 alignment positions. Furthermore, to avoid potential bias in outcome due to differences in search sensitivity by either HMMER2 or HMMER3, both were used for the generation of the initial result sets where only common hits by both HMMER2 and HMMER3 were dissected.First, each selected SMART/Pfam domain was searched (via HMMER2/3 hmmsearch) against (i) the seed database consisting of seed sequences and (ii) UniProt/SwissProt database to generate altogether 4 sets of scores: HMMER2/seed, HMMER3/seed, HMMER2/SwissProt and HMMER3/SwissProt score sets. Next, for each score set, the hit alignments were dissected into high-quality  and low-quality segments. The corresponding sub-scores as well as the total score were statistically evaluated in terms of E-values independently for both high- and low-quality parts  in the Methods section). This implies that the HMMER3 E-values were adjusted since their original E-values were computed based on the size of the returned set. On the other hand, the peculiarity in HMMER2 E-value calculation previously reported in . SubsequFinally, the hits from HMMER2/seed and HMMER3/seed score sets were paired as long as the hits shared a common sequence segment to create a unified set between HMMER2 and HMMER3 results. The same was repeated for HMMER2/SwissProt and HMMER3/SwissProt score sets. Among the paired hits, they can be sub-classified into the concordance and discordance class. Accordingly, the concordance class contains hit results agreed upon by both HMMER2 and HMMER3 where the positive concordance class suggests that the hits are true while the negative concordance class suggests that the hits are false. On the other hand, the discordance class contains the results where HMMER2 and HMMER3 disagreed upon. Fundamentally, this class arises due to the differences in model parameterization and search/alignment algorithm attributed by the two flavors of HMMER. It is beyond the scope of this work to resolve which version of HMMER is better suited for the purpose. In addition, unmatched or orphaned hits are also excluded since this touches on the issue of search sensitivity and it is again not the focus of this work on score dissection  and PF13894.1 (zf-C2H2_4), it was found that the number of orphaned hits found by HMMER2 only  and false-positive (FP) hits from the results of the unified SwissProt score set generated earlier  and this is not a matter to worry. It is more important in this context that the FP rate is extreme low (<1%) for most domain models. The few exceptional models with high FP rates deserve re-visiting and some modifications in their seed alignment. However, it is important to bear in mind that the error rates estimated here are suggestive of baseline rates since the searches have been performed over UniProt/SwissProt, which is a relatively small database. The expected error rates might be higher when a larger database such as NCBI’s non-redundant protein database is considered.The sequence homology concept is backed by an inductive proof. It originates from the observation that homologous proteins share a high degree of sequence similarity, protein fold and biological function. The key to sharing a similar fold, implying a similar function, between the homologs is dependent on the similarity between the more conserved parts, most importantly the structural elements. As such, the evidence for homology should stem from the similarity between the aligned structural elements and key functional motifs with less emphasis from the other sequence segments. As we delve deeper into the search space, higher sequence divergence is to be expected and it will dilute overall sequence similarity and consequently, the homology signal. Therefore, the emphasis on similarity between the structural elements in alignments is the key to finding the homologs (both the close and the distant ones) while keeping the false ones at bay.Despite its simplicity and elegance, the sequence homology concept is not readily computable since homology has no direct measure. It can at best be formulated into a hypothesis to be tested from the sequence similarity which is a necessary but insufficient condition for concluding homology. Although similarity by chance can be removed by some statistical criterion like E-value, often, the main issue is dealing with the statistically significant similarities of any aligned pieces (as the program outputs) that are concluded as homologous instead of convergence as alternative. Since current sequence search packages can only operate strictly in similarity space, this has a tendency to promote, to some extent, the fallacy that ‘high sequence similarity implies homology’.Even in current times, this fallacy is still being extensively discussed by several authors, e.g. by Varshavsky and coworkers who coined the term “sequelog” in an attempt to differentiate homology from high sequence similarity  and by TThus, the sequence homology concept has yet to be fully implemented in current sequence homology search packages because mindful distinction between contributions from evolutionary important pieces versus spurious similarity pieces was never explicitly dealt with; hence, this necessitates for the dissection of an alignment for explicit segments to be reevaluated. As we emphasized in the Introduction, a (globular) domain is a special protein sequence unit with structural (autonomous hydrophobic core), thermodynamic (independent folding and melting) and evolutionary (domain shuffling) implications . ProteinIn our proposed dissection framework, an alignment is dissected into its high-quality segments (representing fold-relevant residues) and low-quality segments (representing other residues) with the subsequent purpose of statistically evaluating the two segment-based score sums. Together with the original scores/E-values, these segment-based sums provide a new level of granularity to the dissection framework for determining if a hit is true  or false . In a nutshell, the dissection framework has created a new paradigm in which homology can be evaluated more concisely and, at the same time, more faithful to the sequence homology concept. And for the purist of the homology concept, sequence searches now have a better chance to escape the fallacy of ‘high sequence similarity implies homology’.For the true-positives of the domain model, the dissection framework can reassert their validity as legit hits with respect to the domain. Indeed, when the framework was applied to the seed sequences of 285 SMART and 2381 Pfam domain models , they exhibited the average positive and negative concordance rates of 99% and almost 0% respectively. These results imply that the seed sequences were recognized correctly by the framework as true hits of the domains.On the other hand, cases of false hits  will be occluded by the framework due to their significant low-quality scores/E-values. This scenario was played out by the case study of the 10 false hits  where their original HMMER2 E-values were insignificant yet significant for HMMER3. Despite a contradictory conclusion from the HMMER variants, their remnant segment-based E-values were indisputably significant for both HMMER versions. Thus, HMMER3 hits were tagged as false-positives while the same hits by HMMER2 were labeled as true-negatives. In both cases, they were considered as false hits by the framework. Interestingly, pre-filtering of compositionally-bias sequence segment may become less critical under the dissection framework since these hits will anyhow exit as false hits due to their significant remnant segments’ E-values. This also meant that the ‘null2 model correction’ and the ‘nobias’ option in HMMER2/3 can be turned off to maximize for search sensitivity to allow more hits.Given the results in this work, a quantitative criterion for assessing segmented HMM scores in annotation pipelines might include the expectation (i) for the fold-relevant contribution resulting in a low E-value  independently of the E-value for the total alignment and/or (ii) for the ratio between the E-value of the fold-critical part versus that of the remnant contribution clearly below 1.To emphasize, score dissection with regard to fold-critical and other segments is a generic concept that can be applied to any sequence or multiple aligment comparison technique. This idea can be easily extended, for example, to the BLAST-based approach with minor adaptations: first, the extraction of the EVD parameters from the blast statistics and second, the parameters used for score reconstruction need to be extracted from BLOSUM/PAM for BlastP algorithm and PSSM for PSI-Blast algorithm.Most importantly, the dissection itself should aim squarely at approximating the location of globular domains by applying either tertiary structure finding algorithms or any tools for detecting non-globular segments. We can only warn against applying non-physical, non-evolutionary dissection principles such as cutting sequences arithmetically first in two parts, then in four and then, maybe, in eight as many might be tempted to. This approach is likely to distribute fold-critical residues to many of the segments, hence diluting evolutionary information instead of enriching it in one class.Perhaps, the most interesting additional capability of the dissection framework, aside from being able to isolate false hits, is its proposal of unexplored homologous relationships between the hits and domain models. This means the recovery of hits presently being falsely labeled as negatives. When the dissection framework was applied to the search results against UniProt/SwissProt for these 285 SMART and 2381 Pfam domains, it revealed an overall average false-positive rate of less than 1% but the average false-negative rates of 7.63% for SMART and 4.86% for Pfam. Although the low false-positive rate implies that the current domain models have generally very low false hits inclusion, the moderate false-negative rates suggest that there are many potential true hits that are obscured by bad E-values. This situation was exemplified by our case study where the previously insignificant true hits  were obscured as a result of heavy score penalties on the low-quality alignment segments. However, they were subsequently rescued by their significant fold related segments’ E-values.In particular, the discovery of the homologous relationship between the plant aquaporin (TIP12_MAIZE) and formate/nitrate transporter (PF01226.12), which indicates that the latter is actually a channel, was essentially exclusive to the structure-alignment based approaches. Even though certain sequence search methods might detect some level of sequence similarity between aquaporin and formate transporter but their E-values remain statistically insignificant . For the cases of these 2 hits, this signifies that quality score is an overestimate of fold-critical segments and as a result, it tends to underestimate the false-negative hits by adding part of the negative remnant sum to the fold-critical sum. Indeed, a scrutiny on the high-quality segments of the associated domain models for these 2 hits revealed that some of these segments were covered by loop residues when compared against the PDB/DSSP annotations.In hindsight though, one should err on the side of conservativeness; i,e., one needs to be more stringent with claiming a true hit. Therefore, the quality-score is still a reasonable estimate for partitioning the fold-relevant and remnant segments. Nevertheless, one can easily add more estimates like low-complexity/disorder predictors  that will otherwise be missed by the quality-score. Hence some combination of the two predictors makes sense.In any case, the creation of a catalogue of segmentations for existing protein domain libraries such as Pfam or SMART will be necessary in the absence of complete PDB/DSSP information for a foreseeable future and it will be considered in our future work.As sequence homology can only be concluded inductively and overall sequence similarity is a measurable, necessary but insufficient criterion to justify homology, additional considerations are required to decide about homology relationships between biomolecular sequences. To distinguish the true cases from the false background might be possible in a manual study for individual cases; yet, a computerized pipeline for large-scale annotation requires quantitative conditions.The complex hydrophobic/hydrophilic sequence pattern necessary for fold formation and conserved during evolution can be used for this purpose by dissecting the similarity score into fold-critical contributions and other parts originating from non-globular segments, long loops, etc. This work serves as a proof of concept for this idea. The dissection framework and the software tools provided with this article are useful for systematically suppressing otherwise generated false-positive hits in sequence similarity searches.The dissection approach allows also extracting more value out of existing protein domain model databases without the need to re-edit them simply by defining segmental contribution and, thus enhancing or deemphasizing certain parts of the seed alignments.Surprisingly, this approach was also successful in recovering hitherto hidden homology relationships by stripping away the noise created by score contributions from non-fold-critical, non-globular protein regions.v between the HMM hidden sequence X and an observed hit sequence Y of length L can be re-computed by summing up a set of emission, transition and a fixed score f. The general equation for the total score of an alignment, where eHMM, tHMM and enull, tnull are the emission and transition parameters of the hidden and null model respectively, is given as:Generally speaking, the log-odd score of an alignment The respective transition and emission (match or insert state) score for each position can be retrieved from the respective HMM model file (created by hmmbuild). In the case of HMMER3 model files, we added an additional step to convert them to HMMER2 format (via hmmconvert −2) prior to the reconstruction step. Note that the fixed score is independent of the alignment and it is essentially constant for the same domain model. The fixed score is made up of the additional special transition scores  and annotated in ‘XT’ line of the model file.For the computation of E-value, the maximum Gumbel extreme value distribution is used and is given as:N is the size of the database that was searched against,  are the summary statistics of the HMM domain model file .where For the creation of the domain models, the following command and options were used:(HMMER2) hmmbuild -F --amino --fast --gapmax 1hmmcalibrate --seed 0 --num 5000(HMMER3) hmmbuild --amino --fast --symfrac 0.0hmmconvert −2For searching domain models against sequence databases, the following command and options were used:(HMMER2) hmmsearch --null2 -E 10(HMMER3) hmmsearch --nonull2 --nobias -E 10As an initial consideration, the ‘null2 correction model’ and the ‘nobias’ options were turned off since (i) it was unclear how these penalties were calculated and on which part of the alignment, particularly for HMMER3, and (ii) it improves search sensitivity according to the manuals ,39.W = v is tested to affirm the reproducibility of the HMMER scores. For each domain, a linear regression (without intercept) is performed between a set of original scores v and reconstructed scores W for each domain (with P hits) and the associated slope r2 is computed.Here, the linear relationship It is important to note that the regression will be performed on a set of seed sequences’ scores per domain. Therefore, it is inevitable that these scores would cluster closely. As such, an extra point at the origin  is added to each set of scores to alleviate the bias towards the high scores. For a set of scores that is well spread, the additional point has little impact.The slope r2 is given as:The coefficient of determination W can be subjected to (i) rounding errors, (ii) parameter conversion estimation and (iii) unavailability of local model parameters ((ii) and (iii) applies to HMMER3 hmmconvert, see also Figure ϵ can be derive to quantify the approximation error where W is related to the original score v by:With respect to a given domain model, an alignment between the HMM emitted sequence and the hit sequence can be recomputed by summing the appropriate emission, transition and fixed scores taken from the HMMER2/3 model parameters. This reconstructed score ϵ are given as:It follows that the mean and variance of the component-wise error model P pairs of original and reconstructed scores.for As a measure against the representative domain score, the error estimate can be written as a relative measure given as:where the representative domain score is estimated by RS = {H, B, E, G, I, T, S} where H = alpha helix, B = residue in isolated beta-bridge, E = extended strand that participates in beta ladder, G = 3-helix (3/10 helix), I = 5 helix (pi helix), T = hydrogen bonded turn and S = bend. On the other hand, let the unstructured set be denoted by RU = {'',−} where ‘’ and – represent loop residue and alignment gap respectively. Furthermore, let the total high-quality and low-quality residue counts be R1 and R2 respectively while the total structure and non-structural residue counts be C1 and C2 respectively  of between residue r and residue t. At the same time, the score vector of residue aij for sequence i at position j over R residues is given as:where i and position j, the distance measure between the consensus column j and the residue aij over R residues is then given as:For each sequence Q for column j over M sequences is given as:Finally, the quality score, Q as a distance measure is expected to be near zero for high consensus while large for low consensus, it would be more intuitive to invert and limit the range of Q as follows:Since quality score j is normalized by multiplying the ratio of amino acids (less gaps) over the total sequences given as:Finally, inverted quality score k is the count of valid amino acid residues.where M be number of sequences (excluding gaps in the particular column) and let k be the sum of Bernoulli random variable I (an indicator variable). The indicator variable emits either a value of one for a positive prediction or zero for a negative prediction. Collectively, this can be written as the Binomial random variable.Given an alignment, for each position, let Ho : p ≤ 0.5, HA : p > 0.5 to be tested at a significance level of α = 0.05. Under this setup, the minimum number of sequences per alignment position is determined to be at least 5 since there is insufficient power to reject the null hypothesis for sequences below 4. This is because the smallest p-values for M = 4 is P(X ≥ 4) = 0.0625, M = 3 is P(X ≥ 3) = 0.125, M = 2 is P(X ≥ 2) = 0.25 and M = 1 is P(X ≥ 1) = 0.5. All these p-values are larger than the significance level of α = 0.05.Under equal chance condition, the null and alternate hypotheses are stated as Here, the appropriate cutoff to declare if a quality score is high or low is determined. With respect to a domain alignment, (i) the quality score per position and (ii) the number of valid amino acids per position ignoring gaps are first determined. Then, each quality score per position is classified into the following two classes: (i) if the alignment column has less than 5 valid amino acids and (ii) if alignment column has at least 5 or more amino acids.The distributions of the two classes of quality score for SMART version 6) is shown in Figure  is shownSimilarly, the same procedure was performed on Pfam release 27). In a similar fashion, Figure . In a siN. Hence each segment can be written in set notation such that:Consequently, we are interested to find segments in a domain alignment of length where In the proposed comparative analysis, the hits are first generated from both HMMER2 and HMMER3 using the same domain alignment and searched against a common database (e.g. UniProt). In addition, only hits with E-value of 0.1 and below  are considered.Using this E-value criterion, one can then define each hit (whether HMMER2 or 3) as true positive (TP), false negative (FN), true negative (TN) and false positive (FP) based on the E-values of its total score, high-quality segment score and low-quality segment score. Essentially, the TP and FN hits belong to a positive set while the FP and TN hits belongs to a negative set.The type of hits and associated conditions are listed in Table Consequently, the intersection of HMMER2 and HMMER3 hits will result in mainly two large groups: a paired group and an orphaned group. To elaborate, a paired hit is a hit covering the same sequence segment by both HMMER2 and HMMER3. An orphaned hit is (i) a hit scored on the same sequence but non-overlapping segments by HMMER2 and HMMER3; or (ii) a hit covered by either HMMER2 or HMMER3 only.In the paired group, one can further sub-divide the HMMER2/3 hits into four classes of (i) positive concordance hits where both HMMER2/3 mark the hits as positive, (ii) negative concordance hits where both HMMEr2/3 mark the hits as negative (iii) discordance type 1 where HMMER2 marks the hits as positive but HMMER3 marks them as negative and (iv) discordance type 2 hits where HMMER2 marks the hits as negative but HMMER3 marks them as positive. The orphaned groups contain mutually exclusive hits that are found by either HMMER2 or HMMER3. See Table Meanwhile, classes that contain the FN and FP hits are of high interest in this work. A FN hit is a positive hit that has been obscured due to a need to score an alignment for the low-quality segment while a FP hit is a negative hit that has been carried over to significance due to the high-scoring low-quality segments. To quantify the false-negative and false-positive rates in a given domain model, the formulas are given as:For each seed sequence in a domain alignment, the gaps were first removed and then predicted using the SEG low-complexity sequence predictor  with thekexp is calculated as:If a residue is flagged as low-complexity by SEG, then its corresponding position in the domain alignment is marked as 0 to indicate a negative prediction, otherwise, it takes a value of 1 to indicate a positive prediction. Essentially, each column in the alignment will be marked by 1’s or 0’s and can be viewed as a sum of Bernoulli random variables. Then to test for the significance of positive predictions in each alignment column, a p-value (see equation (15)) is calculated and tested at a significance level of 0.05. If the null hypothesis is rejected, the expected positive prediction count kexp is set to zero. Finally, the per-column probability indicating that the consensus column (with M sequences) is representative of a high-complexity residue  is given as:Otherwise, pexp ≥ cutoff implies high-complexity while pexp < cutoff implies low-complexity).Consequently, the SEG-derived segments of the domain alignment can be obtained via equation (18) at a cutoff of 0.8  and validated true hits from 3 Pfam domains . The segmentation of domain models is based on the alignment quality score. The data presented is complementary to Table This table contains the examples of validated false hits from 5 Pfam domains (Click here for file


RNA pseudoknots play important roles in many biological processes. Previous methods for comparative pseudoknot analysis mainly focus on simultaneous folding and alignment of RNA sequences. Little work has been done to align two known RNA secondary structures with pseudoknots taking into account both sequence and structure information of the two RNAs.In this article we present a novel method for aligning two known RNA secondary structures with pseudoknots. We adopt the partition function methodology to calculate the posterior log-odds scores of the alignments between bases or base pairs of the two RNAs with a dynamic programming algorithm. The posterior log-odds scores are then used to calculate the expected accuracy of an alignment between the RNAs. The goal is to find an optimal alignment with the maximum expected accuracy. We present a heuristic to achieve this goal. The performance of our method is investigated and compared with existing tools for RNA structure alignment. An extension of the method to multiple alignment of pseudoknot structures is also discussed.The method described here has been implemented in a tool named RKalign, which is freely accessible on the Internet. As more and more pseudoknots are revealed, collected and stored in public databases, we anticipate a tool like RKalign will play a significant role in data comparison, annotation, analysis, and retrieval in these databases.The online version of this article (doi:10.1186/s12859-015-0464-9) contains supplementary material, which is available to authorized users. RNA pseudoknots are formed by pairing bases on single-stranded loops, such as hairpin and internal loops, with bases outside the loops ,2. They Analysis and detection of RNA pseudoknots has been an active area of research. Many published articles in this area were focused on pseudoknot alignment -14. In tRKalign adopts the partition function methodology to calculate the posterior probabilities or log-odds scores of structural alignments. The idea of using posterior probabilities to align biomolecules originated from ,20 whereSeveral tools are available for RNA sequence-structure alignment -25. ThesThe above methods were concerned with aligning a pseudoknot structure with a sequence or genome. Through the alignment, the sequence is folded and its structure is predicted. Xu et al.  presenteThere are two groups of algorithms which are also capable of aligning two known RNA structures. The first group is concerned with aligning two RNA three-dimensional (3D) structures, possibly containing pseudoknots. Ferre et al.  presenteThe second group of algorithms is concerned with aligning two RNA secondary structures without pseudoknots. These algorithms employed general edit-distance alignment  or tree The method that is most closely related to ours is an option offered by the CARNA tool . Like RKA and B, we adopt the partition function methodology to calculate the posterior probabilities or log-odds scores of the alignments between bases or base pairs in A and B respectively. After calculating the posterior log-odds scores, we then compute the expected accuracy of an alignment between structure A and structure B. The goal is to find an optimal alignment between A and B where the alignment has the maximum expected accuracy. We will present a heuristic to achieve this goal.In this section, we present algorithmic details of RKalign. To align two RNA pseudoknot structures i, j) is a base pair of pseudoknot structure A and  is a base pair of pseudoknot structure B. We use score, ) to represent the score of aligning  with  where the score is obtained from the log-odds RIBOSUM matrix  . Let A and B. We focus on the case in which both  and  are base pairs. The case for aligning single bases is simpler, and thus omitted.Suppose  A[c] is aligned with B[d]; (ii) B[d] is aligned to a gap; and (iii) A[c] is aligned to a gap. Let A and B where A[c] is aligned with B[d]. Let A and B where B[d] is aligned to a gap. Let A and B where A[c] is aligned to a gap. Then Zc,d can be calculated by Equation  is a single base and B[d]  is the right base of some base pair, A[c] be the right base of some base pair  and let B[d] be the right base of some base pair . Following  is aligned to a gap, we know that A[c] must be aligned with B[d-1]. Therefore,When calculating score) is the gap penalty value obtained by aligning base pair  to gaps. Thus,where A[c] is aligned to a gap, B[d] must be aligned with A[c-1]. Therefore,score,−) is the gap penalty value obtained by aligning base pair  to gaps. Thus,When calculating j < c ≤ m and q < d ≤ n. There are three cases to be considered: (i) A[c] is aligned with B[d]; (ii) B[d] is aligned to a gap; and (iii) A[c] is aligned to a gap. Let A and B where A[c] is aligned with B[d]. Let A and B where B[d] is aligned to a gap. Let A and B where A[c] is aligned to a gap. Then Next, we show how to calculate Equation .8\documeA[c] or B[d] is the right base of some base pair. If A[c]  is a single base and B[d]  is the left base of some base pair, A[c] be the left base of some base pair  and let B[d] be the left base of some base pair . Following  is aligned to a gap, A[c] must be aligned with B[d + 1]. Therefore,When calculating A[c] is aligned to a gap, B[d] must be aligned with A[c + 1]. Therefore,When calculating i < c < j and p < d < q. There are three cases to be considered: (i) A[c] is aligned with B[d]; (ii) B[d] is aligned to a gap; and (iii) A[c] is aligned to a gap. Let A and B where A[c] is aligned with B[d]. Let A and B where B[d] is aligned to a gap. Let A and B where A[c] is aligned to a gap. Then Finally, we show how to calculate mentclasspt{minimaA[c] or B[d] is the left base of some base pair. If A[c]  is a single base and B[d]  is the right base of some base pair, A[c] be the right base of some base pair  and let B[d] be the right base of some base pair . If x < i + 1 or y < p + 1, we ignore and skip the computation of . We consider only the case where x ≥ i + 1 and y ≥ p + 1. Following  is aligned to a gap, A[c] must be aligned with B[d-1]. Therefore,When calculating A[c] is aligned to a gap, B[d] must be aligned with A[c-1]. Therefore,When calculating i, j) of structure A with base pair  of structure B, denoted by Prob ~ ).There are four cases to be considered when calculating the posterior probability or log-odds score of aligning base pair  doesn’t cross another base pair and  doesn’t cross another base pair. That is, for any base pair , i < v < j if and only if i < u < j. Furthermore, for any base pair , p < y < q if and only if p < x < q. Consequently, the alignment between structure A and structure B can be divided into the following three parts: (i) the alignment between A and B; (ii) the alignment between A and B; and (iii) the alignment between A and B. Following [Base pair  crosses another base pair whereas  doesn’t cross another base pair. That is, there exists a base pair  in A such that (i) i < v < j and u < i, or (ii) i < u < j and v > j. Furthermore, for any base pair , p < y < q if and only if p < x < q. In this case,  crosses , which forms a pseudoknot in structure A, while  doesn’t form a pseudoknot in structure B.Base pair  is ignored when calculating Zi − 1,p − 1 in Equation  is ignored when calculating u, v) will be considered when calculating Prob  ~ ). Thus, our algorithm doesn’t miss the calculation of the posterior log-odds score of aligning any two base pairs from structure A and structure B respectively.When (i) is true, since Equation . In addiEquation . Base pav > j, we have j + 1 ≤ v ≤ m. Furthermore, since u < j < j + 1,  is ignored when calculating u, v) will be considered when calculating Prob  ~ ).When (ii) is true, since Equation . Base pap, q) crosses another base pair whereas  doesn’t cross another base pair. This case is similar to Case 2 above.Base pair  crosses another base pair and  also crosses another base pair. That is, there exists a base pair  in A such that (i) i < v < j and u < i, or (ii) i < u < j and v > j. Furthermore, there exists a base pair  in B such that (iii) p < y < q and x < p, or (iv) p < x < q and y > q. In this case,  crosses , which forms a pseudoknot in structure A. Furthermore  crosses , which also forms a pseudoknot in structure B.Base pair  is ignored when calculating Zi − 1,p − 1 and x, y) is also ignored when calculating Zi − 1,p − 1 and u, v) is ignored when calculating Zi − 1,p − 1 and x, y) is also ignored when calculating u, v) is ignored when calculating Z'j + 1,q + 1 (Case 2 (ii));  is also ignored when calculating Zi − 1,p − 1 and u, v) is ignored when calculating x, y) is also ignored when calculating When (i) and (iii) are true,  and  are single bases, i.e. i = j and p = q, the value of Prob ~ ).When both  ~  ∈ aA,B) means  ∈ A is aligned with  ∈ B in aA,B, Prob ~ ) is the posterior log-odds score of aligning  ∈ A with  ∈ B as defined in Equation (h (k respectively) is the number of single bases plus the number of base pairs in A (B respectively).Let  follows :17\documEquation , and h optimal alignment. From the previous subsection, we are able to construct the posterior log-odds score matrix for aligning structure A with structure B where the matrix contains Prob ~ ) for all  ∈ A and  ∈ B. Our heuristic is an iterative procedure. In the first step, we select two bases or base pairs with the largest score from this matrix to build the first alignment line between A and B where the alignment line connects the selected bases or base pairs. Then, we select the second largest score from the matrix to construct the next alignment line provided that the newly constructed alignment line satisfies the following two constraints: A base  can be aligned with at most one base .i, j) is aligned with  and  is aligned with . The alignment lines between  and  do not cross the alignment lines between  and  if and only if the following conditions hold: (i) i ' < i iff p' < p, (ii) i < i ' < j iff p < p ' < q, (iii) i ' > j iff p ' > q, (iv) j ' < i iff q ' < p, (v) i < j ' < j iff p < q ' < q and (vi) j ' > j iff q ' > q. The newly constructed alignment lines do not cross the alignment lines built in the previous steps. Specifically, suppose  ~ ), we need to compute Zi − 1,p − 1, Z'j + 1,q + 1; cf. Equation  time. Since we need to calculate Prob ~ ) for all  ∈ A and  ∈ B, the time complexity of the pairwise alignment algorithm is O(m2n2). At any moment, we maintain a two-dimensional matrix for storing Zi − 1,p − 1, O(mn) space. Since the total number of bases and base pairs in structure A (B respectively) is at most m (n respectively), we use a two-dimensional matrix to store Prob ~ ), which also requires O(mn) space. Thus, the space complexity of the algorithm is O(mn). Notice that the time complexity derived here is a very pessimistic upper bound since in calculating the partition functions, some base pairs are ignored as described in the previous subsections. During our experiments, we tested over 200 alignments and the running times of our algorithm ranged from 16 ms to roughly 7 minutes, where the lengths of the aligned structures ranged from 22 nt to 1,553 nt.In calculating Equation . ComputiRKalign is implemented in Java. The program accepts as input two pseudoknotted RNAs where each RNA has both sequence data (i.e. nucleotides or bases) and structure data (i.e. base pairs), and produces as output an alignment between the two pseudoknotted RNAs. Popular benchmark datasets such as BRAliBase , RNase PThe first dataset, denoted Dataset1, contains 38 RNA pseudoknot structures chosen from the PDB  and RNA aA,B between structure A and structure B is defined as the number of base mismatches in aA,B divided by the total number of alignment lines in aA,B, multiplied by 100%. Statistically significant performance differences between alignment methods are calculated using Wilcoxon signed rank tests [A good structural alignment tends to align a base pair with another base pair rather than with two single bases ,36. We tnk tests , which ank tests -51. As ink tests -51 we coWe conducted a series of experiments to evaluate the performance of RKalign and compare it with related methods, where the performance measure used was the base_mismatch ratio. In the first experiment, we selected 106 pairs of RNA pseudoknot structures from Dataset1 and applied our method to aligning the two molecules in each pair. The two molecules in a pair belonged to the same pseudoknot type, as it is biologically meaningless to align RNA molecules that lack consensus ,52. The In addition, we also ran CARNA , RNA SamFigure For example, consider Figure In the second experiment, we compared RKalign, CARNA, RNA Sampler and DAFS using the RNA structures in Dataset2. As in the first experiment, we selected 124 pairs of molecules from Dataset2 where the two molecules in a pair belonged to the same pseudoknot type. The average base_mismatch ratio calculated by RKalign for the selected 124 pairs was 35.89%, compared to the average base_mismatch ratio, 81.56%, for all pairs of molecules in Dataset2. We applied each of the four tools to the molecules to produce 124 pairwise alignments. The 124 alignments produced by RKalign can be found in Additional file Figure A and B. A stem sA ∈ A is said to match a stem sB ∈ B if (i) sA,sB are aligned together and they cannot be aligned with other stems, and (ii) for every base pair x ∈ sA and base pair y ∈ sB, a base of x is aligned with a base of y if and only if the other base of x is aligned with the other base of y; otherwise, there is a stem mismatch between sA and sB. The stem_mismatch ratio of an alignment aA,B between structure A and structure B is defined as (1 – M) where M is the number of matched stems in aA,B divided by the total number of stems in A and B, multiplied by 100%.Based on the above experimental results, there is no statistically significant difference between RKalign and CARNA in terms of base_mismatch ratios. As described in ,54, a goFigure Now, observe that the base C at position 14 in 1L2X is aligned with the base U at position 13 in 1RNK, but the base G at position 7 in 1L2X is not aligned with the base G at position 28 in 1RNK; instead the base G at position 7 in 1L2X is aligned with the single base G at position 7 in 1RNK. Thus, there is a stem mismatch between the blue stem of 1L2X and the red stem of 1RNK, a situation that is not favored when performing pseudoknot alignment . This siComparing Figure In this paper, we present a novel method  for comparing two known RNA pseudoknot structures. The method adopts the partition function methodology to calculate the posterior log-odds scores of the alignments between bases or base pairs of the RNAs with a dynamic programming algorithm. The posterior log-odds scores are then used to calculate the expected accuracy of an alignment between the RNAs. The goal is to find an optimal alignment with the maximum expected accuracy. We present a heuristic to achieve this goal. Experimental results demonstrate the good performance of the proposed RKalign method.New pseudoknotted structures are found periodically, as exemplified by the recently determined ribosomal CCR5 frameshift pseudoknot  and the Our pairwise alignment method can be extended to align multiple RNA pseudoknot structures by utilizing a guide tree. Specifically, we treat each structure as a cluster and use the expected accuracy defined in Equation  as the mAn alignment of two clusters is actually an alignment of two profiles, where each cluster is treated as a profile. Initially, each profile contains a single RNA pseudoknot structure. As the guide tree grows, a profile may contain multiple RNA pseudoknot structures; more precisely, the profile is a multiple alignment of these RNA structures. A single base of a profile is a column of the profile where the column contains single bases or gaps; a base pair of a profile includes two columns of the profile where the left column contains left bases or gaps and the right column contains corresponding right bases or gaps, and left bases and corresponding right bases form base pairs.A' and profile B', which amounts to aligning two multiple alignments. Let R (S respectively) be an RNA pseudoknot structure in profile A' (B' respectively) and let   respectively) be a base pair of R (S respectively). Let  represent a base pair of profile A' and let  represent a base pair of profile B'. We use  ∈   ∈  respectively) to represent that   respectively) occurs in the column(s) of base pair   respectively) of profile A' (B' respectively). Equation  ~ ), which represents the transformed probability of aligning base pair  of profile A' with base pair  of profile B'.Suppose we want to align profile Equation  below shProb ~ ) is defined in Equation optimal alignment, found by the heuristic described in the ‘A and structure B. We use the expected accuracy or similarity values to construct the guide tree for the set SS, to determine the order in which two structures or profiles are aligned. To align two profiles A' and B', we use the same heuristic as described in the ‘Prob '  ~ ) defined in Equation  ~ ) of structures A and B defined in Equation (O(k2n4) where k is the number of structures in the alignment and n is the maximum of the lengths of the structures; the space complexity of the algorithm is O(k2n2).The multiple alignment algorithm can now be summarized as follows. The input of the algorithm is a set Equation  where aEquation  replacinEquation . The timMA is defined as the sum of base_mismatch ratios of all pairs of structures in MA divided by the total number of structure pairs in MA, multiplied by 100%. The average base_mismatch ratio of RKalign  was 26.01% . RKalign and CARNA were not statistically different ; the two methods were significantly better than RNA Sampler and DAFS .We tested our algorithm by selecting 30 groups each having 3, 4, or 5 pseudoknot structures of the same type from the datasets used in this study, and by performing multiple alignment in each group. The multiple alignments produced by our algorithm can be found in Additional file Both our pairwise alignment and multiple alignment programs are available in the RKalign tool. This tool is capable of accepting as input pseudoknotted RNAs with both sequence (nucleotides or bases) and structure data (base pairs), and producing as output an alignment between the pseudoknotted RNAs. As more and more pseudoknots are revealed, collected and stored in public databases, we anticipate a tool like RKalign will play a significant role in data comparison, annotation, analysis, and retrieval in these databases.RKalign is designed to align known RNA pseudoknot structures. A different approach is to simultaneously fold and align RNA sequences without known structures, as adopted by several existing tools -13. WhenPseudoknots are part of RNA tertiary motifs . There aWhile the work reported here focuses on pseudoknot alignment, it can also be applied to RNA secondary structures without pseudoknots. We applied RKalign to 102 pairs of pseudoknot-free structures taken from RNA STARND where the pseudoknot-free structures belonged to Rfam  (see AddThe work that is most closely related to RKalign is CARNA . Both mehttp://bioinformatics.njit.edu/RKalign.The latest version of RKalign can be downloaded at: