The availability of genome-wide epigenomic datasets enables in-depth studies of epigenetic modifications and their relationships with chromatin structures and gene expression. Various alignment tools have been developed to align nucleotide or protein sequences in order to identify structurally similar regions. However, there are currently no alignment methods specifically designed for comparing multi-track epigenomic signals and detecting common patterns that may explain functional or evolutionary similarities. We propose a new local alignment algorithm, EpiAlign, designed to compare chromatin state sequences learned from multi-track epigenomic signals and to identify locally aligned chromatin regions. EpiAlign is a dynamic programming algorithm that novelly incorporates varying lengths and frequencies of chromatin states. We demonstrate the efficacy of EpiAlign through extensive simulations and studies on the real data from the NIH Roadmap Epigenomics project. EpiAlign is able to extract recurrent chromatin state patterns along a single epigenome, and many of these patterns carry cell-type-specific characteristics. EpiAlign can also detect common chromatin state patterns across multiple epigenomes, and it will serve as a useful tool to group and distinguish epigenomic samples based on genome-wide or local chromatin state patterns. All tissue and cell types, such as embryonic stem cells (ESCs), terminally differentiated tissues, and cultured cell lines, are maintained and controlled by epigenomic regulation and gene expression programs . An epigA series of computational methods, including ChromHMM , Segway Based on existing chromatin state annotations, previous work has studied similarities and differences of human tissue and cell types in terms of epigenomic signals in specific functional genomic elements (e.g. promoters and enhancers), as well as the tissue and cell specificity of these elements, using the Pearson correlation coefficients ,21 or a The comparison of DNA/RNA or protein sequences is based on the sequential information of nucleotides or amino acids. Many sequence alignment methods have been developed over the past decades to measure the similarity between sequences. Earlier work such as the Needleman-Wunsch algorithm  and the Given the fast accumulation of large-scale epigenomic datasets generated in recent years, biological researchers are in great need of a new bioinformatic tool to efficiently retrieve genomic regions similar to an interested query region in terms of epigenomic signals. Motivated by the enormous successes of sequence alignment algorithms in comparing nucleotide and protein sequences , here weabbcc is represented by a compressed state sequence NRG3 has the best alignment with another brain-specific gene GRIA1, among all the protein-coding genes. This result is reasonable as both genes are brain-specific and highly expressed in brain samples. However, as these two genes have vastly different lengths (NRG3 is three times longer than GRIA1) and their chromatin state sequences have long stretches of the quiescent/low state, they are poorly aligned when we apply EpiAlign without the compression step. This result indicates that the compression step, which condenses the epigenetic information encoded in chromatin state sequences, is necessary and effective for finding similar and biologically meaningful chromatin state patterns. Additionally, aligning uncompressed sequences is much more time-consuming (20 times more computation time on average) than aligning their compressed counterparts. Therefore, adding the compression step also increases the computational efficiency of EpiAlign. In the following text, unless specified, all the chromatin state sequences refer to the compressed state sequences.The EpiAlign algorithm aims to find an optimal local alignment between two chromatin state sequences. Our algorithm development is motivated by the classic Smith–Waterman Algorithm . We desitate see , and incS1 and S2, we characterize a possible alignment between S1 and S2 through a set of triplets N denotes the total number of aligned basepairs , fi gives the alignment status between two chromatin states whose positions are ui1 and ui2 in S1 and S2, respectively. We may equivalently write this set of triplets as three equal-length sequences: F = f1f2⋅⋅⋅fN, U1 = u11u12⋅⋅⋅uN1, and U2 = u21u22⋅⋅⋅uN2. Specifically, m for match, n for mismatch, S1, and S2. If ui1-th state of S1 and the ui2-th state of S2; if ui1-th state of S1 and the ui2-th state of S2; if ui1-th state of S1 is aligned to nothing in S2 (ui2 is set to 0); if ui2-th state of S2 is aligned to nothing in S1 (ui1 is set to 0). In an example with S1 and S2 may have different lengths. Also given S1 and S2, it is possible to have more than one alignment results, i.e. sets of Given two chromatin state sequences H( · ) as:h denotes the score of the alignment status fi between the ui1th state in S1 and the ui2th state in S2. Specifically,Now we define the alignment score function h( · ) takes a form that depends on the value of its first argument fi.We will formally define the matching function MF( · ), the mismatching function NF( · ), and the deletion function DF( · ) later in this section. To summarize, the function H:M that stores dynamic alignment results. The matrix element Mk, l is the maximal alignment score of the two subsequences k states of S1 and l states of S2. Let n1 and n2 be the length of S1 and S2, respectively. We update the matrix M using the following rule.Then we consider the alignment problem as an optimization problem where the goal is to find the optimal alignment ng rule.\documentW, which describes the weight of the kth state in a sequence S. The weights can be used to distinguish chromatin states of different importance if we have prior knowledge that some states have more significant biological functions than others at certain positions. We design two sets of weights: (i) equal weights mean that all states are treated equally with the same weight 1 in the sequence S, i.e. W = 1 , k = 1, …, |S|; (ii) frequency-based weights assign larger weights to less common chromatin states (see To define the specific forms of the matching function MF( · ), the mismatching function NF( · ) and the deletion function DF( · ), we first introduce a weight function N and εD are the penalty parameters for a mismatch and a deletion in the alignment, respectively. In EpiAlign, εN and εD can be tuned by users, and the default values are 1.5 and 1, respectively. The choice of εN and εD values depends on how ‘local’ users would like the result to be, i.e. if we set a larger εN or εD value, it means that we penalize more on a mismatch or a gap in the alignment, and the final best alignment result will be shorter or more local. Figure With the weights defined above, we specify the matching function, the mismatching function, and the deletion function as:mentclass2pt{minimWe demonstrate in three aspects that EpiAlign is a useful tool for investigating sequential patterns of chromatin states. First, we demonstrate that EpiAlign can identify common chromatin state patterns within the same epigenome or across different epigenomes. Second, we investigate biological interpretation of the common chromatin state patterns found by EpiAlign. Third, as a technical verification, we show that EpiAlign is able to distinguish real epigenomes from randomized epigenomes. We also demonstrate the superiority of EpiAlign over a naïve method that compares two chromatin sequences only based on chromatin state frequencies. We conduct the above analysis using simulation and real data studies based on the Roadmap epigenomic database . In thisIn this paper, for most analysis, we select ESC, heart and brain samples from the Roadmap dataset as representative examples. The reason is that among all the Roadmap tissue types, these three types are relatively better understood and have well-annotated tissue-specific genes .EpiAlign is a powerful local alignment algorithm to quantify the similarity of two chromatin state sequences in terms of their aligned subsequences. Here we apply EpiAlign to compare chromatin state sequences of the same genomic region in two different epigenomes, a strategy we define as the vertical alignment. The diversity of the same region’s chromatin state sequences represents epigenetic characteristics of various tissues and cell types. As epigenetic characteristics are known to have a strong association with gene expression characteristics , we expeIn the first study, we divide the Roadmap epigenomes into two categories: 51 male samples and 38 female samples. In the second study, we compare the Roadmap epigenomes of two cell types: 10 brain samples and 5 heart samples. In both studies, we compare the chromatin state sequences for each of the 19,935 protein-coding genes between every pair of samples.  are highly relevant to heart/cardiac processes and brain processes . Figure In the second study, we investigate if EpiAlign can help identify cell-type specific genes, which have been previously discovered from gene expression profiles , using oxon test . Here wees Table  are enriSTMN4, which has the lowest P-value from the two-sample one-sided Wilcoxon test described above . Using it as an example, we investigate the chromatin state sequences of STMN4 in all brain and heart samples. From Figure STMN4 as the top differentially aligned gene shows that EpiAlign can successfully identify regions where chromatin state patterns diverge or conserve between cell types.To better illustrate how EpiAlign helps identify common chromatin state patterns, we study a brain-specific gene We also analyze the expression profiles of protein-coding genes. We use DESeq2  and edgehorizontal alignment score. A high score indicates that the region shares a highly similar and non-random chromatin state sequence with another region in the same epigenome, implying that the region’s chromatin state sequence pattern is likely biologically meaningful.Motivated by the fact that similar chromatin state sequences may encode similar biological functions, here we use EpiAlign to analyze frequent chromatin state sequence patterns within an epigenome. We introduce the ‘horizontal alignment’, which takes the chromatin state sequence of a region as the query and searches for its best hit except itself within an epigenome. We first divide a given epigenome into regions of 500 kb length, and then we align the chromatin state sequence of each region (i.e. the ‘query’) to those of other regions to find the best match. It is worth noting that the alignment scores of different query chromatin state sequences are not directly comparable. To normalize the alignment scores, we align every query chromatin state sequence to randomized chromatin state sequences, which serve as a negative control  distance metric. The clustering result matches our expectation: samples from the same tissue are clustered together, confirming that the horizontal alignment scores are indeed consistent across the samples from the same tissue Figure .We further perform a simulation study to technically validate the efficacy of EpiAlign in terms of horizontal alignment. Our goal is to check if EpiAlign is able to distinguish real epigenomes from randomized epigenomes, which serve as a negative control. We calculate horizontal alignment scores using EpiAlign on all the 127 Roadmap samples based on the 15-state ChromHMM annotation. In addition to each real epigenome, we also generate a randomized epigenome and two hybrid epigenomes for comparison. Here the randomized epigenome is generated in the same way as in the normalization step for calculating horizontal alignment scores (see We use an ESC sample (Roadmap ID E003) as an example and calculate horizontal alignment scores in four epigenomes: the real ESC epigenome, a randomized epigenome, and two hybrid epigenomes. We summarize the distributions of horizontal alignment scores in the real and randomized epigenomes in Figure Q is the number of unique chromatin states in the annotation . The naïve alignment score is a similarity measure defined as MnaïveWe further validate our EpiAlign algorithm with equal weights by comparing it with two alternative approaches. The first is a variant of EpiAlign using frequency-based weights, which are determined by the frequencies of chromatin states , the frequency-based variant of EpiAlign, and the naïve method do not have horizontal alignment scores on the same scale and cannot be compared directly, so we compare the three approaches by evaluating the biological meaning of the regions they find with high scores. Since gene regions are expected to share some common chromatin state patterns , a good alignment method is expected to assign high horizontal alignment scores to gene regions. In other words, genes expressed in a tissue are expected to have high horizontal alignment scores in the tissue’s epigenome. Hence, we design two evaluation criteria: one is the enrichment of known tissue-associated genes, i.e. the non-house-keeping genes highly expressed in a tissue , in regiet al.  and anno th entry Aij represents the normalized alignment score of regions i and j and is defined asAij < 1 for all i ≠ j. We then define a distance matrix i, j)-th entry is Dij = 1 − Aij. We then perform hierarchical clustering with average linkage on the top 200 regions based on As a further investigation, we check if the regions with top horizontal alignment scores share any chromatin state patterns in common. We apply EpiAlign to perform horizontal alignment within the epigenome of the ESC sample E003, and we select the top 200 regions with the highest horizontal alignment scores. To investigate whether common chromatin state patterns exist among these regions, we calculate the pairwise alignment scores between each pair of these top 200 regions. We normalize the pairwise alignment scores and store them in a 200 × 200 symmetric matrix fined as\documentmotifs in the following text. For notation simplicity, we use alphabets ‘a’ to ‘o’ to denote chromatin states 1 to 15.From the heatmap in Figure Using the motif-discovery tool MEME , we findet al.  in genome annotation when transcriptional regulation is of particular importance.N and εD, for penalizing mismatches and gaps in the alignment. Similar parameters are also necessary for classic alignment algorithms designed for DNA and protein sequences such as BLAST. For example, the εD in EpiAlign is analogous to the gap extend penalty in BLAST. The NCBI BLAST, an online tool that implements the BLAST algorithm, sets the Gap Extend Penalty to 1 by default. In EpiAlign, we also set εD to 1 by default. In BLAST, a substitution matrix is used to score matches/mismatches, and multiple substitution matrices have been constructed for users to select based on alignment purposes. In EpiAlign, we set εN to 1.5, which is equivalent to a substitution matrix with diagonal entries as 1 and off-diagonal entries as -1.5. Given that the alignment of epigenetic sequences is new to this field, how to construct more specialized substitution matrices for chromatin states is an important future research question.Fourth, EpiAlign depends on two tuning parameters: εFinally, in some computationally efficient sequence alignment algorithms, hash tables or tree-based data structures are utilized to index the database, and these techniques have greatly increased the efficiency of query retrieval. EpiAlign can also benefit from similar techniques and further improve its computation efficiency.Two other computational methods, EpiCompare  and Chrocis-elements. Specifically, the vertical analysis based on EpiAlign will reveal tissue-specific genes and regulatory regions that share common chromatin state patterns within a tissue type, and such patterns will serve as the basis of defining new regulatory domains. We have also demonstrated that EpiAlign has found meaningful chromatin state motifs. Besides, EpiAlign is able to distinguish tissue-associated genes. These results suggest the potential of EpiAlign as a useful bioinformatic tool to discover tissue-specific gene regulation. Moreover, the alignment scores calculated by EpiAlign can serve as a covariate when constructing functional genomic networks, thus allowing the network to incorporate similarities of chromatin structures as a factor. Further, EpiAlign is applicable to 3D genomic analysis to address the question if there are chromatin state patterns in regions with a specific 3D structure such as a loop.In terms of biological applications, the biggest strength of EpiAlign is its ability to identify common chromatin state patterns and how they are conserved or divergent between cell types. This strength will pave the way for identifying regulatory domains defined by combinatorial effects of strings of gkz287_Supplemental_FileClick here for additional data file.


RNAz-2, (ii) realigned and refined candidate loci with LocARNA-P, (iii) scored candidates again with RNAz-2 in structure alignment mode, and (iv) searched for additional homologous loci in mouse genome that were not covered by genome alignments. The 3’-untranslated regions (3’-UTRs) of protein-coding genes and small noncoding RNAs are enriched for structures, while coding sequences are depleted. Repeat-associated loci make up about 95% of the homologous loci identified and are, as expected, predominantly found in intronic and intergenic regions. Nevertheless, we report the structure elements enriched in specific genome elements, such as 3’-UTRs and long noncoding RNAs (lncRNAs). We provide full access to our results via a custom UCSC genome browser trackhub freely available on our website (http://rna.tbi.univie.ac.at/trackhubs/#RNAz).In this work, we present a computational screen conducted for functional RNA structures, resulting in over 100,000 conserved RNA structure elements found in alignments of mouse (mm10) against 59 other vertebrates. We explicitly included masked repeat regions to explore the potential of transposable elements and low-complexity regions to give rise to regulatory RNA elements. In our analysis pipeline, we implemented a four-step procedure: (i) we screened genome-wide alignments for potential structure elements using  To date, these functional elements remain poorly annotated, since they cannot be easily identified by high-throughput methods.RNAs fulfill a multitude of regulatory functions in the cell which are often mediated by a particular RNA structure. Functional—and therefore evolutionarily conserved—RNA structures are thus abundant throughout the genome, both in the form of structured noncoding RNAs , as wellOne valuable experimental approach to fill this gap is structural probing of the whole transcriptome, as reviewed by Choudhary et al. . An overQRNA [RNAz [EvoFold [CMfinder [A number of computational methods have been proposed that identify functional RNA structures based on their evolutionary conservation, such as QRNA , RNAz [7NA [RNAz , EvoFold[EvoFold , and CMfCMfinder . All of RNAz uses a support vector machine (SVM)-based approach to classify alignment windows as structurally conserved RNA based on structure conservation and covarying base pairs, as well as the thermodynamic stability of the predicted structure. Evofold uses stochastic context-free grammar to model sequence variation of structured RNAs along a phylogenetic tree. CMfinder is based on covariance models (CMs). Covariance models are “probabilistic models of the conserved sequence and secondary structure of an RNA family” [CMfinder goes one step further by creating covariance models directly from a set of (orthologous) sequences with unknown structures, similar to motif discovery in sequences. An iterative process is then used to optimize the CMs and corresponding alignments. family” , which e family” , are buiA few genome-wide screens for conserved RNA structures in vertebrate genomes have been performed using these tools, including ,14,15,16RNAz-2 [In this contribution, we concentrate on the mouse genome as the most important vertebrate model organism, and present a novel screen for functional RNA structures in genome-wide alignments of mouse with other vertebrates. We provide a database of the predicted functional elements for easy use in genome browsers. Furthermore, we introduce a novel and improved pipeline for genome-wide screens using RNAz-2  that empRNAz-2  or the ERNAz-2 , TorarinRNAz-2 , here, wRNAz-2 . There iRNAz-2 . It is tRNAz [RNAz-2 [z-score), as well as a choice of two SVM decision models suitable for sequence alignments and structural alignments, respectively. Our novel pipeline starts with a conventional window-based screen of the input alignments similar to the one described by Washietl and Hofacker [RNAz. In contrast to the original pipeline, however, we mitigate some flaws of this first RNAz screen by introducing additional realignment and re-evaluation steps. We use LocARNA-P [RNAz, including flanking regions. A second run of RNAz then classifies the realigned loci with improved boundaries, again after subsampling. See Starting from an alignment of 59 species against the mouse genome ,19,20, wRNAz ,8, but iHofacker . This meocARNA-P ,23 for sRNAz version 1, analyzing the ENCODE selected regions in a human-based alignment [All genome-wide screening approaches for structured RNA suffer from false positives due to the large number of input alignments analyzed. Therefore, our pipeline aims to successively reduce the number of false positives in each step see b. We estlignment . HoweverLocARNA-P [RNAz evaluation uses these predicted boundaries, avoiding the need for sliding windows. An artefact of the sliding window approach is seen in the length distribution of the initial RNAz screen’s loci  of protein-coding genes, as well as in regions annotated as short noncoding RNA (ncRNA), while coding sequences (CDS) are depleted see . The enrFurthermore, on average, we found more structures at the beginning of the CDS than at its end, and we observe a sharp and dramatic increase in structuredness directly after the stop codon see . This caRepeatMasker [While repetitive regions of the genome are often neglected, they have been implicated as a source for the evolution of novel ncRNAs. We therefore looked at the prevalence of conserved structures in repeat regions identified by atMasker ,27 , as well as UA, UAUAUG, and CAUAUA . These repeats (or their reverse complement) are complementary to themselves on the RNA level and can form very stable structures. In this case, the conserved structure may be a consequence of the sequence pattern, rather than a consequence of selection pressure on the structure level.Among the simple repeats, those which contain complementary nucleotides in the repeated region show the highest enrichment, led by As expected, we found an enrichment in transfer RNAs (tRNAs) and their pseudo-genes. We found 109 loci in regions identified as possible pseudo-genes by tRNAScan-SE  and 80 lOlfr) genes were previously identified to have dramatically fewer predicted secondary structures and a higher average AU-content [Many of our structured elements are associated with protein-coding genes. This allows us to relate the prevalence of conserved structures with the annotated function of the protein product, and to ask which classes of protein-coding genes are more likely to be regulated at the RNA level. We therefore performed an enrichment analysis for the RNAz hit coverage across protein-coding transcripts associated with various gene ontology (GO) terms. While the 5’-UTRs of the protein coding genes did not show a significant enrichment for RNAz hits, 3’-UTRs were significantly enriched. Moreover, this enrichment was correlated with specific GO terms. A similar but less pronounced enrichment pattern was also reflected in the coding regions. For example, 3’-UTRs of protein-coding genes annotated with the Molecular Function GO term see  “binding-content .Similarly, we observed enrichments and depletions of GO terms representing Biological Processes see  or CelluE-value Of particular interest are regulatory structures that occur in several transcripts. In order to identify such RNA families, we built covariance models (CMs) for a subset of 14,119 high-confidence loci with at least three species and a maximum length of 200 nt (see Methods). We then scanned the entire (unmasked) mouse genome, yielding over 11 million significant hits , 74% of our high-confidence hits are only conserved in mouse and one other species, mostly mouse and rat. As expected, a large number of predicted elements are specific to rodents. Despite the use of a stricter RNAz screening step (data not shown). Within snoRNAs and small Cajal body-specific RNAs (scaRNAs), we have a higher sensitivity for HACA-box RNAs, which is expected. We did not find any U6 snRNA, which is not surprising, considering that the predicted consensus structure of this spliceosomal compound shows only a few base pairs, and that U6 can give rise to pseudo-genes [To estimate the sensitivity of our approach to different types of structured RNAs, we investigated the number of annotated small non-coding RNAs in our results. do-genes .RNAz cannot find any base pairs and thus does not correctly classify the RNA.Additionally, it can clearly be seen that we miss more than half of the annotated RNAs because they are split between several alignment blocks. Obviously, if a structural element, such as a hairpin, is divided into two alignment blocks, CMfinder. Of the 30,973 high-confidence loci that are not specific to Glires, 2220 overlap at least 50% of a locus  from Seemann et al. The lack of a larger overlap is not surprising, since RNAz and CMfinder have the best sensitivity for different alignment characteristics [RNAz. We therefore argue that the low overlap is due to limited sensitivity, rather than high false positive rates.We provide a list of high-confidence predictions of structured RNA loci centered on mouse. Since all previous screens of vertebrate genomes used human as the reference genome, the overlap between their loci and our predicted loci is necessarily small. Of particular interest is the very recent screen of Seemann et al. , which ueristics . The medLocARNA took around 2 months wall-clock time for the realignment of both strands. It was the only step where more than 2 GB RAM per core were occasionally used, and thus again not all cores could be fully utilized. Boundary prediction and the second RNAz classification step took less than a day each. Calibration of the CMs and subsequent genome screening took about 2 weeks. Most of the computation time was used by very few CMs originating from loci in highly repetitive regions.Once the whole pipeline was set up and tested, the computation of the screen took almost 3 months wall-clock time on our in-house cluster with a total of 1304 cores on 33 nodes, and on 96 private cores on the Vienna Scientific Cluster. For the first step of the pipeline, File-IO was limiting, and we used only one physical node (up to 48 cores) per chromosome as a practical way to limit file access over the network. With this setup, the first step took 11 days. The realignment step with Neat1.We built a publicly available track hub to visualize our results via the UCSC genome browser and made the underlying data easily accessible to the research community. The trackhub ‘RNAz screen by ViennaRNA’ contains three tracks , each holding a collection of subtracks which can be activated individually. In total, we provided 207 subtracks, allowing users to browse and search through our data in a variety of ways. cmscan performed with models built from high-confidence loci. The subtracks here show cmscan hits based on hit quality and number of hits found with each CM model. This way, users can, for instance, easily select frequently occurring high-confidence hits. The trackhub file (http://rna.tbi.univie.ac.at/trackhubs/RNAz/RNAzhub.txt) can easy be loaded into the UCSC genome browser (http://genome-euro.ucsc.edu/cgi-bin/hgHubConnect) and displayed along side with standard or custom tracks to put them in context with other genome annotations and elements. For all RNAz hits, we provided detailed structure information  of consensus structure, structure conservation index (SCI), GC content, number of species in alignment) and two images of the structure plot and alignment, both color-coding sequence-structure conservation. All information can be downloaded track by track via the table browser.The track ‘RNAzGenome’ lists all RNAz hits identified in mm10/60 vertebrate alignments and is split into subsets with 2–5 and 6–10 species in the input alignment. These sets are further subdivided into genome partitions. We therefore projected gene elements onto the genome in a hierarchical fashion, thereby creating intervals of unique identity—of particular interest in cases of overlapping transcript isoforms or antisense transcripts. RNAz hits were assigned to genome partitions (min. overlap 60%) and presented in individual subtracks. The ‘RNAzRepeats’ track holds all those hits found in repeat regions, as annotated in the UCSC genome browser ‘repeats’ track. For instance, for the ‘RNAzGenome’ track, a locus must show at least 60% overlap with a repeat element. Tracks are subdivided by repeat class and number of species in the alignment. The ‘CMstat’ track contains all hits of the search with RNAz score cutoff for loci with only two species, we were able to capture evolutionarily young structures which we would have missed otherwise.We performed a genome-wide prediction of evolutionarily conserved RNA structures in mouse, which we believe will be a valuable resource for researchers interested in regulatory structures, particularly those who use mouse as a model organism. The need for a mouse-centered screen is clearly illustrated by the fact that the majority of predicted loci are not conserved beyond rodents. By including regions only alignable between mouse and rat and carefully adjusting the CMFinder [Our improved pipeline, including a realignment step and boundary prediction, tries to ameliorate the effects of windowing and alignment errors. This was achieved to a certain extent, as can be seen from the length distribution of our loci and improved FDR, especially for regions with low sequence identity. Nevertheless, the quality of genome-wide input alignments still has a strong impact on our results. For reasons of computational efficiency, we used structural alignments only in the second stage of the pipeline, where the number of alignments was much smaller. Moreover, realignment could not help in cases where a structured region in the reference genome has been misaligned to a region without the conserved structure in another genome; see  for furtCMFinder .We chose to use the published 60-way multiz alignment from UCSC as starting point for our analysis instead of creating our own alignment. In addition to the obvious benefit of saving computation time and research time for refining the parameters, there is a more fundamental reason for this choice: this 60-way alignment is the only multiple species alignment for the newest mouse version (mm10) provided by UCSC  and the community standard. By using this alignment, we ensured our results remain comparable to other research projects based on the same alignment.RNAz screen and before realignment, but, for future screens, we recommend merging of alignment blocks before the first filtering. Unfortunately, merging of alignment blocks is not trivial, as there are many parameters that can be optimized. On the one hand, a balance between length of the merged block and the number of species has to be found, since merging usually requires the removal of species from the alignment. On the other hand, the maximum number of nucleotides to be fetched for individual sequences between alignment blocks must be determined. Whenever nucleotides are fetched from the genomic sequence of a species, these nucleotides are not aligned and, thus, an additional realignment step would be necessary, which comes at additional computational cost. Finally, a greedy algorithm might be suboptimal for merging, since there will be cases where an alignment block can be either merged with its upstream neighbor or its downstream neighbor, but not with both. Finally, merging cannot be restricted to only pairs of blocks, since we frequently find the case where longer blocks are separated by multiple blocks of only one or two nucleotides. While we find it highly desirable to test such a merging before the first step, it still requires additional research and parameter tuning. Thus, we have resorted to just merging blocks after an initial classification step with RNAz. This makes a huge difference, because now we can work under the assumption that the blocks indeed belong together. Furthermore, now potential merging only effects less than 1% of the initial loci, thus, one could argue that suboptimal merging parameters do not make much difference in the big picture.The use of this published multiz alignment, however, comes with certain disadvantages. The way these multiz alignments are constructed allows for the same sequence of other species to be aligned against multiple regions of the mouse genome. This might be especially relevant for repeat regions without clear orthologs. We found that, for mouse and rat, roughly 90% of the aligned nucleotides are present only once in the alignment. Thus, we do not believe that sequences aligned multiple times caused a noticeable bias in our results. Another effect of the alignment characteristics had much stronger influence on our results: the fact that the alignment contains so many species and tries to include as many species as possible as often as possible leads to fragmentation of the alignment into many, short alignment blocks. Twenty-five percent of the alignment is in blocks with less than 50 columns, which we had to filter out in the beginning of our screen. We did perform merging of alignment blocks after the initial For alignment blocks with more than 10 species, we resorted to subsampling the alignment (see Methods). This limited our ability to assess precisely in which clades of the phylogenetic tree a structure is conserved. Moreover, alignments with many species carry a greater risk of including sequences that do not share a common structure (even though they may exhibit some sequence similarity). In these cases, tools that can split an alignment into subparts with/without a conserved structure would have to be employed .Conserved RNA structures are not distributed uniformly over the genome. Instead, we observed a notable enrichment of conserved structures in 3’-UTRs, while coding sequences contain fewer structures than the genomic background. Some classes of repetitive regions also seem to harbor conserved structures, with satellites showing the strongest enrichment. The mRNAs of different classes of proteins also seem to behave differently: mRNAs and 3’-UTRs of proteins involved in RNA and DNA binding, as well as transcription regulation, seem to be particularly prone to harboring conserved structures, while the mRNAs of G-protein coupled receptors seem to avoid them.rnazWindosws.pl, which comes with RNAz [–max-gap=0.7 option. Since RNAz does not work well with alignments containing more than 10 species, we then generated up to five different samples of 10 species using the –num-samples=5 option of rnazWindosws.pl whenever too many species were present in a window. Finally, windows or samples were filtered to remove those with less than a 50-column length or a very atypical mean pairwise identity (MPI) (below 40% or exactly 100%). We did not filter soft-masked genomic regions. These filtering criteria are less strict than the default parameters, since further steps of our pipeline would work as additional filtering steps. Blocks with a length below 50 nucleotides made up around 25% of the aligned region, while the other filters rarely kicked in.We obtained multiple sequence alignments of 60 vertebrate species to the mouse genome  from the UCSC genome browser ,19,20. Tith RNAz , we slicRNAz on both strands using a dinucleotide shuffled background model and a cutoff for RNA class probability of 0.5. Overlapping or “book-ended” RNAz hits were grouped into loci, irrespective of the strand. Whenever loci extended over multiple alignment blocks, we tried to merge these alignment blocks by either fetching up to 20 missing nts directly from the genome or by removing some species from the alignment. We merged when: at least two species would remain in the alignment, at most five species were in the initial blocks, and the resulting merged block would be below 400 alignment columns. If merging was not possible, the locus was split into two or more loci at the boundaries of the alignment blocks.These windows were scored with LocARNA-P. Since the boundary prediction will always shrink the alignment, we added 20 nts of flanking genomic regions at both sides of the alignment for each species. Structural realignment was performed with LocARNA-P using the suggested parameters from the manual [LocARNA-P’s boundary prediction was performed, allowing for multiple reliably aligned regions per locus. All such regions which at least partly overlapped the original locus without flanking regions were merged by including the nucleotides in between. These regions of improved boundaries were again passed through rnazWindows.pl for sampling in cases where there were more than 10 species per alignment, and for removing loci with fewer than 50 alignment columns (but without cutting the region into sliding windows), and reclassified on their respective strands using RNAz Next, we performed structural realignments and boundary prediction using e manual . Each loRNAz score above 0.5 in the second classification step.The set of raw loci contains all regions with at least one sample (if sampling of species was performed) having a We split the original input alignment into chunks of 100 alignment blocks and shuffled every fifth chunk using multiperm . This waThe final loci in the shuffled alignment have virtually no overlap with the loci in the unshuffled alignment, which shows that shuffling efficiently destroys the original structureRNAz class probability is above 0.99.Loci with only two species were only included if the For loci with 3–10 species, we used a score cutoff of 0.9.For loci with more species, where samples were drawn, we required that at most one sample has a score below 0.5, and at least one sample has a score above 0.90.Since a lot of data are available for genome-wide screens, we can estimate the false discovery rate for subsets of the alignment while still retaining enough data points to avoid overfitting. Unsurprisingly, the FDR of the raw loci was strongly dependent on the number of species in the input alignment see , which ibedtools. miRNAs were restricted to loci listed in miRBase [tRNA-Scan-SE [bedtools.To test the sensitivity of our screen, small ncRNA types ‘snoRNA’, ‘scaRNA’, ‘snRNA’, and ‘miRNA’ were extracted from the GENCODE mouse M17 annotation (assembly GRCm38) and intersected with RNAz loci using  miRBase ,40, agai miRBase  and Rfam miRBase  were int-Scan-SE . Small Rmatplotlib [ggplot [RepeatMasker was obtained from RepBase [Data analysis was performed using bedtools , and custplotlib  and ggpl [ggplot . Informa RepBase ,47. The The false discovery rate (FDR) was calculated as the fraction of nucleotides predicted in the random control multiplied by 5 (because only 20% of the genome was shuffled), divided by the number of nucleotides in the true predicted loci. This estimation was only performed for the plus strand and assumed to be the same on the minus strand. For the subgroups of the genomic alignment based on MPI or the number of species, a factor of 5 was confirmed to be almost exactly the fraction of nucleotides in the initial windows of the unshuffled genome over the number of nucleotides in the initial windows of the random control.To perform the genomic partitions, we used the transcript biotype annotations available from the GENCODE release M17  and descThe annotated transcript biotypes considered to represent the superclass of mRNA, lncRNA, sncRNA (short non-coding RNA), and other can be found in the p > 0.5) against the respective genomic parts (minimum of 1 nucleotide overlap), and compared the fraction of total nucleotides of RNAz hits in a given part against the fraction of nucleotides of the genome in this part that are in alignments and used as input to RNAz. If the RNAz hits were evenly distributed in all genomic partitions, then the fraction of total nucleotides of RNAz hits in a given part should be the same as the fraction of nucleotides of the aligned genome (used as input to RNAz) in this part.In order to see if the coverage of RNAz hits is enriched in a specific genomic part, we intersected all the raw RNAz hits  by comparing: (1) the proportions of coverage of RNAz hits (minimum of 1 nucleotide overlap) in genes annotated as a given GO term divided by the total nucleotides covered by RNAz hits in all GO terms, against the total length of genes in this GO term divided by the total length of genes in all GO terms; (2) the proportions of the total number of genes with at least one RNAz hit (minimum overlap of 50% nucleotides of RNAz hit) for a given GO term divided by the total No. of genes with at least one RNAz hits in all GO terms, against the total number of genes in this ontology term divided by the total number of genes in all ontology terms.p-values were computed using Fisher’s exact test and were adjusted for multiple testing using the Benjamini–Hochberg procedure, as implemented in the p.adjust function in R, with a false discovery rate of 5%.The enrichment analysis was performed separately for the three extensive ontologies—molecular function, biological process, and cellular component. cmbuild and cmcalibrate programs from the INFERNAL toolsuite [LocARNA-P predicted boundaries, alongside its consensus RNA secondary structures as calculated by RNAalifold-r from the Vienna RNA package [In order to find genomic loci that show secondary structures strongly similar to our RNAz hits, covariance models (CMs) were built, using the oolsuite . For all package .Since construction and subsequent calibration of CMs from a multiple sequence alignment and its consensus secondary structure is a computationally highly resource-consuming task, we preselected a subset of loci derived from the total set of high-confidence loci for CM building and subsequent screening against the full mouse genome. In order to be considered for a CM, a high-confidence locus was required to have (1) at least three sequences, (2) a total number of columns between 50 and 200. Enforcing constraint 1 ensured that all multiple sequence alignments showed at least a minimum amount of sequence variation and excluded non-informative alignments with only two species. Imposing restrictions on alignment length was necessary, since very small loci usually carry structural features that can be found multiple times throughout the genome and therefore render the CM unspecific, whereas the building of CMs from long loci takes a disproportionate amount of computational time and was therefore omitted.cmpress and cmscan utilities, the CMs were used to screen both strands of the entire mouse genome (mm10) for structurally similar loci using default parameters. For further downstream analysis, only hits below an E-value significance cutoff of Subsequent to successful calibration, using INFERNAL’s ary in R .cmsearch hits with E-value bedtools. RNAz hits were intersected with genome partitions (bedtools intersect-f 0.6). For the track ‘RNAzRepeats’, repeat classes [bedtools intersect -f 0.6). The cmsearch hits (‘CMstat’ track) were subdivided based on cmsearchE-values and the number of hits within each of these sets; bedplus files with RNAz hits were converted to bigBed format [http://rna.tbi.univie.ac.at/trackhubs/#RNAz.We built a UCSC genome browser track hub  with thr classes  were merd format . The tra


Escherichia coli MG-1655 and Halophilic archaeon DL31. To our knowledge, this is the first genome-wide metabolic network alignment at both the metabolite level and the enzyme level. These results demonstrate that it can supply quite a few valuable insights into metabolic networks. First, this method can access the driving force of organic reactions through the chemical evolution of metabolic network. Second, this method can incorporate the chemical information of enzymes and structural changes of compounds to offer new way defining reaction class and module, such as those in KEGG. Third, as a vertex-focused treatment, this method can supply novel structural and functional annotation for ill-defined molecules. The related source code is available on request.Biological network alignment aims to discover important similarities and differences and thus find a mapping between topological and/or functional components of different biological molecular networks. Then, the mapped components can be considered to correspond to both their places in the network topology and their biological attributes. Development and evolution of biological network alignment methods has been accelerated by the rapidly increasing availability of such biological networks, yielding a repertoire of tens of methods based upon graph theory. However, most biological processes, especially the metabolic reactions, are more sophisticated than simple pairwise interactions and contain three or more participating components. Such multi-lateral relations are not captured by graphs, and computational methods to overcome this limitation are currently lacking. This paper introduces hypergraphs and association hypergraphs to describe metabolic networks and their potential alignments, respectively. Within this framework, metabolic networks are aligned by identifying the maximal Z-eigenvalue of a symmetric tensor. A shifted higher-order power method was utilized to identify a solution. A rotational strategy has been introduced to accelerate the tensor-vector product by 250-fold on average and reduce the storage cost by up to 1,000-fold. The algorithm was implemented on a spark-based distributed computation cluster to significantly increase the convergence rate further by 50- to 80-fold. The parameters have been explored to understand their impact on alignment accuracy and speed. In particular, the influence of initial value selection on the stationary point has been simulated to ensure an accurate approximation of the global optimum. This framework was demonstrated by alignments among the genome-wide metabolic networks of  Together with continuous improvements in determining the interactions among biological molecules, this sequencing effort has produced a huge number of biological networks at different scales for various species. Biological networks such as metabolic networks, protein-protein interaction networks and gene regulation networks can be used to describe the composition, status, and operation of biological systems3. Analysis of biological network, especially the genome-scale network, could systematically provide the collective patterns and common features of massive amounts of genome information can be studied as well as new biometric features and emergent phenomena4.In recent years, whole-genome sequencing has been gradually completed for thousands of organisms, enabling a deeper and broader understanding of the functions represented by gene sequences7. Biological network alignment aims to find a mapping between topological and/or functional components of different biological molecular networks. It can successfully address many essential biological questions, including the following8: which biological molecular interactions or groups of interactions are likely to have equivalent or conserved functions across species? In light of these similarities, can we predict novel functional information about components and interactions that are poorly characterized? Do these relationships inform us about the dynamics and evolution of molecules, networks and entire species?The flood of increasingly rich biological networks has accelerated the development and evolution of biological network alignment methods15. For instance, Kelley and Sharan et al. matched two networks by searching high-ranking seeds in a dynamic programming method and extending around the seeds using a greedy strategy9. Pache et al. proposed a pairwise alignment approach with connected components as seeds13. Flannick et al. developed a multiple network aligner, Graemlin, which uses an incremental alignment approach by implementing successively pairwise alignments on the closest graph pairs16. Singh et al. used the idea of PageRank as the definition of similarities between vertices from different networks. And, they used a spectral graph method to rapidly identify the highest-ranking match from all possible matches in terms of the total score of all the aligned vertices17. Pržulj et al. exploited graphlet counts as topological node similarity scores and a greedy seed-and-extend method as the alignment strategy11. Heymans et al. performed metabolic network alignment by identifying a maximum weight matching of the enzyme similarity bipartite graphs. Pinter et al. converted the metabolic graph matching problem into a simple tree homeomorphism problem for alignment. Ay et al. proposed the SubMAP method, in which pathways are represented as compound-enzyme bipartite graphs and the alignment is converted into a conventional optimization problem18. Ay and his team again proposed a method that included a compression and decompression process of the pathways followed by the SubMAP. These efforts lead to a family of alignment methods that have swiftly evolved from a few early approaches into a repertoire of tens of methods20. The resulting methods have driven exploration and enhanced our understanding of the functional and organizational principles of different cellular processes.Accordingly, numerous algorithms and tools have been developed for biological network alignment over the past decade21. Such multi-lateral relations cannot be captured by a graph.These methods all rely on a graph representation. However, graph representation does not fully conform to reality of metabolic network, since metabolic reactions are obviously more complicated than can be described in simple graphs. A fundamental attribute of a graph is that each edge links two vertices. In contrast, metabolic reactions involve more than two participating components and are therefore not always bilateral21. Klamt et al. and Mithani et al. proposed using a hypergraph to represent biological networks22. Michoel et al. have used hypergraph-based spectral clustering to perform protein-protein interaction networks classification23. Mohammadi et al. introduce a Triangular Alignment (TAME), which attempts to maximize the number of aligned triangles for 3-order protein-protein interaction network alignment based upon a tensor approach24.Hypergraphs offer a framework to overcome such difficulties; biological networks can be intuitively described using the hypergraph modelEscherichia coli MG1655 and Halophilic archaeon DL31.This contribution introduces a hypergraph framework for metabolic network representation and develops a fast and easy alignment method through mathematical and computer improvements. Within this framework, metabolic networks are matched by identifying the maximal Z-eigenvalue of a symmetric tensor. A shifted symmetric higher-order power method was used to identify a solution that accurately approximates the global optimum. A rotational calculation strategy was designed to traverse all possible hyper-edges in Mohammadi’s implicit kernel of tensor-vector products for speed acceleration. The corresponding algorithm was realized with a Spark-based distributed memory computation on a cluster of 35 workers. These efforts attain a hundreds-fold increase in the convergence rate. Impact of certain parameters have been tested for this method. And, the influence of initial value on the convergent point has been investigated and a uniform vector has been found to be an appropriate choice. To demonstrate the framework, we apply it to aligning genome-scale metabolic networks of This framework offers a completely intuitive, accurate, and comprehensive basis for the processing, management and analysis of metabolic networks. Because it is compatible with numerous tensor-based algorithms, this method will be benefit to a large family of downstream tools that could provide more in-depth insight into metabolic systems. The related source code is available on request.First, the difference of representing a metabolic network by a graph or by a hypergraph has been illustrated in Fig. Instead, it can only be represented by a reformatted simple graph using enzymes as the vertices and metabolites as the edges of the network, as shown in Fig. Fortunately, a hypergraph, in which edges can join more than two vertices (please see Method section), realize a more precise and comprehensive representation of metabolic network. This can be shown in Fig. Gab  to account for empty alignment or an absent compound, such as V1N. Therefore, the association hypergraph  of the super-symmetric score tensor of Fig. As a graph represented by a matrix, the association hypergraph can be described by a tensor (please see Method section). A mathematical tensor is a multidimensional array of numerical values organized by their subscripts. For instance, a 2-dimentional tensor is a matrix. The element of such tensor is corresponding to the hypervertices and hyperedges of the association hypergraph. The value of each element is the similarity score of related association components. As such, association hypervertex is denoted by diagonal element and association hyperedge by non-diagonal element. This tensor is symmetric since non-diagonal elements’ value will not change when the subscripts are permutated. Figure 26. Then, hypergraph alignment determines a vector x that maximizes the tensor multiplication according to certain constraints. The task for such a problem is seeking this vector and then discretizes it via a greedy algorithm27. This seeking task remains a NP-hard problem. Currently, we can only achieve an optimal approximate solution in the actual calculation28.As described in the Methods section, the score of hypergraph alignment can be represented as a modal tensor-vector product29. However, this method is computationally intensive and can only be used for networks with few vertices. Alternatively, tensor power iterations can be used to solve the problem31. This approach seeks the maximum Z-eigenvector and the eigenvalue of the tensor. Because the tensor is super-symmetric, this approach is equivalent to identifying the best symmetric rank-1 approximation of a symmetric tensor, and we adopt the shifted symmetric higher-order power method (SS-HOPM), which was introduced by Kolda et al.30. The corresponding procedure is provided in algorithm 1 (see Method section). The super-symmetric tensor H corresponds to the association hypergraph. The elements of x represent the alignment of vertices between two hypergraphs.Several methods are available for this task. For example, maximizing the n-mode tensor product can be converted into a semi-definite programming problem via semi-definite relaxation, and a solution can be obtained via the primal-dual interior point method32. To ensure an accurate tensor-vector product, the hyperedge of the association hypergraph should be proliferated by permutating its subscript and filled into a symmetry tensor. A full representation of this symmetric tensor will introduce a K! fold increase in the number of variables. Such combinatorial explosion leads to massive storage need and, in particular, explosive computation requirement, which prevents the alignment on large networks.Generally, genome-scale metabolic networks encompass hundreds to thousands of reactions, resulting in association hypergraphs of up to millions of hyperedgesTo this end, computational efforts have been realized to ensure the practicability of network alignment on a genome scale. The combinatorial explosion stemmed from the high-dimensional structure of tensor. After a careful study of the process of tensor-vector n-mode product, we found that this difficulty could be addressed by taking advantage of the symmetry of the tensor (Algorithm 2). We call this a rotational tensor-vector product for a super-symmetric tensor. In the process of production, the tensor could be fully represented by the elements with their size just as one upper hypertriangular region. Thus, it is only necessary to reconstruct such elements and store them in memory. An element in one position will contribute to elements on multiple positions of the resulting vector according to its subscript. To account for this feature, the elements are rotated according to its subscript to generate its K equivalents, each of which contribute to one position . In addition, different elements contribute to the same position of the resulting vector as long as the last K-1 subscripts of these elements belong to the permutations of the same set of numbers. To this end, the new elements generated by rotation were multiplied by the permutation number of its last K-1 subscripts . The flowchart of the entire process is displayed in Fig. |Eab| edges, |Vab| vertices, K orders and its corresponding tensor. For a normal strategy, these features will result in an |Eab| * K! + |Vab| computation requirement and storage need. In comparison, for the rotational strategy the algorithm requires a computation cost of about |Eab| * K + |Vab| and the storage required is approximately |Eab| * K. Here, to trade space for time, we didn’t express the tensor for associate hypergraph as the Kronecker product of the tensors for original graph, which will achieve maximal storage cost saving as Mohammadi proposed.We theoretically compared the computation and storage cost of the normal strategy and the rotational strategy. Given an association hypergraph with The algorithm incorporating rotational multiplication, which we call Rotational SS-HOPM (R-SS-HOPM), was first implemented in a stand-alone version. Further, distributed memory computing has been realized to ensure the feasibility of alignment for larger network, which we called Distributed SS-HOPM. The algorithm was implemented with Java APIs of the Spark framework on a cluster of up to 35 nodes (32G memory and 8 cores of 6700 K CPU). We compared their calculation speed with those of normal strategy on a stand-alone setting using selected network pairs. The speed-up ratio of the R-SS-HOPM over the normal one is reported in Fig. The speed-up ratio of the DR-SS-HOPM over R-SS-HOPM is reported in Fig. Escherichia coli (eco01100) and Saccharomyces cerevisiae (hah01100) which cannot be achieved by other methods in a short time.Since some metabolic reactions will encompass many metabolites, most genome-scale metabolic networks will have a large order. In aiding of the rotational multiplication strategy, the Distributed Rotational SS-HOPM becomes the most efficient and fast method to successfully treat such alignment tasks. We successfully aligned the complete metabolic networks of 12. Alignment score is the matching score of objective function in Eq. A number of parameters can impact the performance of the algorithm. We tested their effects on the alignment speed and accuracy. Each network pair used for this purpose include one big network and one small network  and recorded the convergence time and alignment accuracy with λ = 0.9 and the same random x0. The convergence time and accuracy for each α were normalized by the corresponding values at α = 0.1. Figure First, we tested the effects of the shift parameter α (please see Method section). An output alignment vector will be generated by iterative power multiplication on an input vector during each round of iteration. The algorithm will use both the output vector and the input vector to generate a vector as the new input vector. α represents the fraction of the old input vector retained in the new input vector. α is just like a sort of step size, less α means larger step size. The addition of the positive shift parameter α leads to a definite convexity of the objective function and ensures that the power method can convergeWe then tested the impact of the balance factor λ on the convergence speed and alignment accuracies. The similarity score itself is dimensionless, so it is hard to weight the impact of vertices and edges in directing the optimization process (please see Method section). The factor λ has been introduced to adjust the weight of the two components. In final formula, hyperedge similarity was multiplied by (1 − λ) while hypervertex by λ. Higher λ lead to increasing weight of hypervertex in alignment score. So, this parameter will directly change the objective score and thus the direction of optimization process.x0 and α = 0.01, we recorded changes in the convergence speed and alignment accuracy for λ values in a list from 0 to 0.9 with a step of 0.1. The alignment score was normalized by the corresponding values at λ = 0.9 Fig. For each of the selected pairs, given For all the alignments, the curves of EC and VC versus λ looks two-valued and there is a threshold of λ for the curve. When λ is less than the threshold, EC and VC are relatively low. When λ becomes greater than the threshold, the alignment reaches relatively high EC and VC values. Meanwhile, increase of λ cause the drop of alignment speed of all networks. The reason for such phenomenon is because one pair of enzymes will give birth to a great deal of non-diagonal elements with equivalent values through the combination of the two sets of metabolites. This results in a large fraction of equivalent non-diagonal elements in the tensor, which greatly reduces the ability to identify correct alignments. Thus, it is difficult to converge to a high-quality alignment when λ is less than the threshold and non-diagonal elements exert a dominant influence on the alignment result. Conversely, this could explain why increased λ results in good alignment. This feature suggests that metabolite matching outperforms enzyme matching in determining the ultimate alignment result in current setting. In the following part, λ was set to 0.9.34. Therefore, the value of λ is only a relative value and must be adjusted for different purposes.In addition, many different score function can be used to align enzymes and metabolitesx0 as the initial point. The non-negativity of hypergraph tensor ensures the non-negativity of both the initial point and the optimum. In addition, a preliminary test shows that a uniform vector will always achieve an optimum with very high alignment accuracy. So, we generate the initial x0 by adding a non-negative random vector to a uniform vector xu. The fraction of the uniform component is controlled by coefficient β. In the following calculation, β ranges from 0 to 0.9 in a step of 0.1. For each β, 30 initial points were randomly generated as x0. For each network pair, the discretized score was normalized by the corresponding value based upon xu.To a certain degree, the ultimate stationary points depend on initial values. To accurately approximate the global optimum, the impact of initial values on stationary points has been carefully explored with multiple rounds of parallel optimization using a random x0 and the distribution of alignment accuracy. Specifically, the x-axis represents values of the L1-norm of (x0–xu) ranging from 0 to 1, which actually represents the distance from x0 to xu, whereas the y-axis represents the alignment accuracy. Each dot is a stationary point of a single optimization process. The shape of the dots presents a downward trend. The alignment accuracy  always peaks when the L1-norm approaches zero. And, it decreases sharply as x0 moves away from xu. This finding suggests that optimization processes starting from xu converge to the approximated maximum xm in the space near xu given that this stationary point is a satisfactory approximation of the global optimum in all feasible space. So, a uniform vector is the preferred initial point candidate.Figure xu is necessary to achieve a satisfactory approximation of the global optimum. For other experiments, xu is chosen as the preferred initial point.We also studied the relationship between the objective function score and its discretized counterpart. This relationship is represented by the scatter plots in Fig. 35. The true alignment is known because the networks are constructed using the same set of nodes.We performed self-alignment with noise to assess the robustness of hypergraph-based method. The similarity score of each component in the association hypergraph was modified by simulated noise. The noise intensity was set at 5%, 10%, 15%, 20% and 25% of the mean intensity of the original alignmentThe manner in which the alignments score varies as noise is shown in Fig. Escherichia coli MG-1655 (eco01100) and Halophilic archaeon DL31 (hah01100), both of which were obtained from KEGG PATHWAY module36.To assess the biological relevance produced by our method, we aligned the genome-scale metabolic networks of The basic statistics of the alignment are listed in Table Escherichia coli MG-1655, 498 enzymes and 329 metabolites were not aligned or correctly aligned to any of those in hah01100 of Halophilic archaeon DL31. A considerable proportion of the unaligned reactions and metabolites could be grouped together by their linkages, forming several separate pathways with clear biological function. There is no correct hit in hah0110 for enzymes including ATP:D-xylulose 5-phosphotransferase, D-xylose aldose-ketose-isomerase, sedoheptulose-7-phosphate:D-glyceraldehyde-3-phosphate glycolaldehyde transferase, D-xylose xylohydrolase, L-ribulose-5-phosphate 4-epimerase, L-ribulose-5-phosphate 3-epimerase, 3-dehydro-L-gulonate-6-phosphate carboxylyase, ATP:L-ribulose 5-phosphotransferase, beta-D-Fructose 6-phosphate:D-glyceraldehyde-3-phosphate glycolaldehyde transferase, sedoheptulose-7-phosphate:D-glyceraldehyde-3-phosphate glyceronetransferase, D-Ribulose-5-phosphate 3-epimerase and for metabolites including L-ribose, L-arabinose, D-xylose, D-xylulose, D-xylulose-5-phosphate, L-ribulose-5-phosphate, L-xylulose-5-phosphate. As shown in Fig. For eco0110 of As shown in Fig. In addition, a number of cofactor pathways such as Vitamin B6, Vitamin B12 and biotin, as well as lipopolysaccharide biosynthesis pathway, have been found to be not fully aligned in eco01100 and thus to be partly missing in hah01100. These alignments show that hah01100 use modified versions of the pathways or it is inefficient in corresponding cofactors de novo synthesis and cell wall blocks producing, if hah01100 represents a complete annotation for the genome.Halophilic archaeon DL31, 112 enzymes and 94 metabolites were not aligned or correctly aligned to any of those in Escherichia coli MG-1655. There is no correct hit for enzymes of 5,10-Methenyl tetrahydro methanopterin 10-hydrolase, 5,10-Methylene tetrahydro methanopterin:coenzyme-F420 oxidoreductase and metabolites of 5,10-Methylene tetrahydro methanopterin, Reduced coenzyme F420, 5-Methyl-5,6,7,8-tetrahydro methanopterin, Coenzyme F420, 5,10-Methenyltetrahydromethanopterin, 5-Formyl-5,6,7,8-tetrahydro methanopterin in methane metabolism . It is not surprising in that as one extremophile, 37, which in turn demonstrate the value of our method.In a word, our alignment reveals the difference in nutrition requirement and metabolic capacity between eco01100 and hah01100 in details. Regardless of incomplete genome annotation, this will be a short reflection of their natural habitat and evolutionary history. Meanwhile, most of the results are also evidenced by rigorous experimentsA hypergraph is a powerful tool with more profundity and applicability than a graph. Most complex relationships in the real world can be represented by hypergraphs. Therefore, it has increasing practical significance for extending the alignment of the metabolic network from a graph-based method to a hypergraph.We adopted a power iteration method as the optimization strategy given its briefness and efficiency. Numerous modified versions have been generated during its evolution, such as the shifted power method or adaptive power method. In the power method without a shift, the iteration will always be rapid but will occasionally fail to converge. In contrast, sometimes the convergence rate is relatively slow for the adaptive method and the normal shifted power method with a large shift parameter. Thus, a compromise that satisfies both sides involves utilizing a small and fixed shifted parameter.The greatest challenge of hypergraph alignment is the implementability of the entire process. Although the power method itself is relatively simple, it has not been successfully applied for large-scale and high-order biological networks such as metabolic networks in the real world. This limitation originates from the existence of too many elements in association with the hypergraph tensor. In constructing the association hypergraph, an alignment between any two hyperedges will produce association hyperedges with a number close to the factorial of the order. An alignment of two hyperedges includes both the alignment between two hyperedges and the alignment between the hypervertices belonging to each of the hyperedges. The number of possible alignments is close to a factorial of the order between the two sets of hypervertices.Furthermore, since current eigen-pair enumeration methods are only accessible for super-symmetric tensors, we must permutate the subscripts of the association hyperedge to fill in the corresponding tensor. This process also leads to the explosive growth of the tensor elements. Consequently, the resultant tensor exhibits amazing memory consumption, and tensor-vector multiplication is extremely time-consuming for large-scale networks.To this end, various efforts have been devoted to overcome these limitations. One effort involves implementing our algorithm within the architecture of distributed memory computing. This method possesses good scalability, and its ability can be enhanced with an increase in computing resources. It is suitable for the hypergraph with massive components, but not very high-order massive components. Compared with the stand-alone algorithm, the maximal speed-up ratio has reached the value of 80 by utilizing the full resources in this paper. However, for higher-order problems, the method will be difficult.Another effort is to improve the algorithm performance using subscript rotational tensor-vector multiplication. This method is appropriate for speed acceleration and storage reduction of a high-order network. It could increase algorithm speeds by 250-fold while saving memory by tens of thousands-fold.A joint method combining both of these efforts can achieve a speed-up ratio of 1000 and solve the problem that could not be addressed exclusively by either the stand-alone rotational method or distributed computing.Several parameters have been examined to understand their influences on algorithm performance. α is one of the most dominant parameters. This parameter impacts the convexity of the objective function and then the convergence speed of the algorithm. α also affects the stationary point of the algorithm in our test. In practice, an α between 0.0001 and 0.01 is the preferred choice.Another important parameter is λ. Strictly, λ is the parameter of the objective function instead of our algorithm. Its role is to balance the relative weight of the metabolites (hypervertex) over enzymes (hyperedge). In this paper’s test, λ exerts a great influence on the speed as well as alignment result. A larger λ indicates more weight of the metabolites, a faster convergence rate and a higher alignment score. The smaller the λ, the larger the weight of enzymes. Since enzymes  have a very flat shape due to the equivalence of their values, the algorithm is not easy to reach a good stationary point when enzymes dominate the alignment. It is relatively difficult for an optimization with a small λ to converge. Although the weight of metabolites and enzymes is determined mainly by the specific structure and topology of the analyzed networks, the metabolites would have a stronger impact on alignment, from the formula of optimization algorithm itself. It is because that only if all the metabolites belonging to one enzyme  correctly aligned, the algorithm wills generate an alignment of this enzyme. Additionally, the selection of similarity score function will also impact the alignment and interfere with λ. For instance, sequence homologous score of enzyme usually have a large value and will often let enzyme dominate the alignment. So, the influence of λ is complicated.In essence, current power methods cannot guarantee the identification of global optima—they can only achieve a sufficient approximation of global optima. So, the challenge is how to ensure that the algorithm converges to such approximations. The choice of the initial vector itself has little influence on the alignment score, they give similar alignments with good approximation. However, our hypergraph alignment problem is subject to 0–1 constraints and must discretize the corresponding solution vector to obtain the ultimate alignment result. The discretized results are very sensitive to the stationary points before discretization. A little change in the alignment before discretization will lead to very different alignment result. In practice, the uniform vector is capable of converging to a very good approximation. Therefore, the uniform vector should be a preferred candidate for initial value selection.x0. The alignment with the best score could be considered as an approximation of the global optimum. The number of samplings for achieving a global optimum may be associated with the size of the network itself. However, it is difficult to describe the relationship between the sampling number and the distance from the obtained optimal point to global optimum or the relationship between the sampling number and the probability to obtain the global optimum. The issue of how to determine an appropriate sampling number remains an open question.Another possible method for achieving a satisfactory approximation of the global optimum might involve performing multiple rounds of parallel optimization using random As an example, an alignment among the metabolic networks of hah01100 and eco01100 was performed. The alignment difference was clarified in terms of enzymes and metabolites. Numerous significant biologically relevant differences were observed in non-oxidative pentose phospahte exchange and pentose and glucuronate interconversions pathway, MEX/DOXP pathway and some cofactor pathways. This numerical alignment has an excellent fit to the experimental results, which underscores the accuracy and power of this method.This study describes the background and a methodological framework for using hypergraphs to represent metabolic networks and tensors to align the networks. A hypergraph is suitable for intuitively, accurately and comprehensively describing metabolic networks. Association hypergraphs can be used to represent all possible alignments of the networks, whereas the corresponding tensor can store the similarity scores of the association hypergraph. This method provides an intuitive, accurate and comprehensive mathematical framework for the alignment of metabolic networks. A shifted symmetric higher-order power method was implemented on a spark-based computation cluster to solve the problem, significantly increasing the convergence rate up to 80-fold. A rotational tensor-vector product algorithm was introduced to accelerate the optimization by an average of 250-fold. The parameters have been simulated to determine their influence on alignment performance. In particular, the impact of the initial value on the convergence point has been tested to identify an accurate approximation of the global maximum. For the first time, this method achieved a genome-wide metabolic network alignment at both the metabolite level and the enzyme level. Similar to previous methods based upon a simple graph, a wide range of broad biological significance can be obtained using this hypergraph-based approach. In addition, it provides numerous valuable insights into bio-molecular networks. First, this method can access the driving force of chemical logic of organic reactions through the chemical evolution of metabolic network. Second, this method can incorporate the chemical information of enzymes and structural changes of compounds to offer new way defining reaction class and module, such as those in KEGG. Third, as a vertex-focused treatment, our method can supply novel structural and functional annotation for ill-defined molecules. Because this framework is compatible with numerous mathematical methods applicable to the tensor eigenvector problem, we believe it will form a set of tools with extensive applicability to metabolic network alignment and provide more in-depth insight into biological systems.G . Generally, such a hypergraph G is a pair G where V is a set of elements called vertices and E is a set of non-empty subsets of V called hyperedges. In this hypergraph, the set E = {ej, j = 1, …, m} represents enzyme reactions in the metabolic network, whereas V = {vi, = 1, …, n} represents metabolites in the metabolic network. The metabolic network can be represented by stoichiometric matrix S, in which a row corresponds to a metabolite and a column corresponds to a reaction. The element of S is the stoichiometric coefficient of a metabolite in a reaction with positive value standing for product and minus value standing for reactant. The stoichiometric matrix S can also be transformed into hypergraph matrix G after binarization of the stoichiometric coefficient. The order of hypergraph K is defined as the number of compounds of the reaction connected to the most compounds.A metabolic network containing m reactions and n metabolites can be accurately represented by hypergraph Ga =  and Gb =  can be enumerated by constructing an association hypergraph Gab of the two hypergraphs . This phenomenon necessitates the presentation of alignments between different numbers of vertices, such as the alignment between H to represent the association hypergraph39. For simplicity, we define the tensor H that corresponds to the association hypergraph as super-symmetric because the similarity of a hyperedge does not depend on the order of the vertices of the hyperedge.We then introduce the tensor H is K, and the dimension of H is consistent with the vertices of the association hypergraph. The elements of H, VNN the orders of the two matching hyperedges are smaller than the order of the hypergraph. The function S acts as a degree of similarity measurement between the two hyperedges h are equal, i.e., elements on the tensor trace, represent the similarity measurements between the vertices of the two hypergraphs.The order of 41. Specifically, we selected the similarity score calculated by ChemMine tools, which have an R interface to recognize the CID number of the compounds42. When a similarity score is missing for a compound pair, the value is set as an average of all compound pairs  and Enzyme (1.1.1.2), the lowest class is (1.1.1.-). Then, we calculate the similarity score as the inverse of the numbers of all enzymes belonging to this class.The values of the elements in the association hypergraph are the similarity measurements between the corresponding elements. For similarity among the vertices, we used a compound (metabolites) similarity score for which numerous metrics are available34. The hyperedge similarity was multiplied by a factor of (1 − λ), whereas that of the hypervertex was λ. This parameter will directly affect the direction.We introduce the parameter λ ranging from 0 to 1 to balance the weights between the vertices and hyperedgesX {xi = 0, 1 | xi ∈ X}44. The dimension of X is na * nb, which represents the vertices in the two hypergraphs. After the transformation of this matrix, we obtain a binary vector X. The elements of X are arranged in accordance with the sequence of the vertices of the two networks ith vertex of Ga and the jth vertex of Gb. ith vertex of Ga and the jth vertex of Gb. Then, the score of any alignment is a tensor product that can be represented by the following formula:The alignment of two metabolic hypergraphs can be represented using the matrix xm that maximizes the tensor product under the above constraints, which is formalized as follows:Thus, the hypergraph matching problem can be considered as a vector 45. Despite relaxation, this problem remains a NP-hard problem, and we can only seek an optimal approximate solution in the actual calculation45. These approaches seek the maximum Z-eigenvector and the eigenvalue of this tensor. Given that the tensor is supersymmetric, this approach is equivalent to identifying the best symmetric rank-1 approximation of a symmetric tensor47, and the problem can then be solved using tensor power iterations based on spectral matching48. Here, we used the shifted symmetric higher-order power method (SS-HOPM), which was introduced by Kolda et al.30. Specifically, after the constraints of Eq.  of Saccharomyces cerevisiae, hah01110 (168 reactions) and hah01100 (537 reactions) of Halophilic archaeon, and randomly knocked out certain reactions in the pathways. To construct randomly knockout pathways, we label the enzymes in the pathway with positive integer and randomly choose the integer, then delete the corresponding enzyme from the pathways. This gives birth to a series of different networks including eco00010-01 (21 reactions), eco00010-02 (15 reactions), eco00010-03 (9 reactions), eco00010-04 (6 reactions), sce00010-01 (21 reactions), sce00010-02 (15 reactions), sce00010-03 (12 reactions), hah01110-01 (100 reactions). Please see Supplementary Data 36.The metabolic networks used in this study were obtained from the Kyoto Encyclopedia of Genes and Genomes (KEGG) pathway database using a KEGG API. Specifically, we obtained the pathways eco00010 (31 reactions), eco01110 (242 reactions) and eco01100 (923 reactions) of Supplementary data 1Supplementary data 2 and 4Supplementary data 3


We show that the σ values produced by MODELLER are on average weakly correlated to the true level of structural divergence between target-template pairs and that increasing this correlation greatly improves the program’s predictions, especially in multiple-template modeling. Secondly, we have inquired into how the incorporation of statistical potential terms  in the MODELLER’s objective function impacts positively 3D modeling quality by providing a small but consistent improvement in metrics such as GDT-HA and lDDT and a large increase in stereochemical quality. Python modules to harness this second strategy are freely available at https://github.com/pymodproject/altmod. In summary, we show that there is a large room for improving MODELLER in terms of 3D modeling quality and we propose strategies that could be pursued in order to further increase its performance.The most frequently used approach for protein structure prediction is currently homology modeling. The 3D model building phase of this methodology is critical for obtaining an accurate and biologically useful prediction. The most widely employed tool to perform this task is MODELLER. This program implements the “modeling by satisfaction of spatial restraints” strategy and its core algorithm has not been altered significantly since the early 1990s. In this work, we have explored the idea of modifying MODELLER with two effective, yet computationally light strategies to improve its 3D modeling performance. Firstly, we have investigated how the level of accuracy in the estimation of structural variability between a target protein and its templates in the form of  Proteins are fundamental biological molecules that carry out countless activities in living beings. Since the function of proteins is dictated by their three-dimensional atomic structures, acquiring structural details of proteins provides deep insights into their function. Currently, the most frequently used computational approach for protein structure prediction is template-based modeling. In this approach, a target protein is modeled using the experimentally-derived structural information of a template protein assumed to have a similar structure to the target. MODELLER is the most frequently used program for template-based 3D model building. Despite its success, its predictions are not always accurate enough to be useful in Biomedical Research. Here, we show that it is possible to greatly increase the performance of MODELLER by modifying two aspects of its algorithm. First, we demonstrate that providing the program with accurate estimations of local target-template structural divergence greatly increases the quality of its predictions. Additionally, we show that modifying MODELLER’s scoring function with statistical potential energetic terms also helps to improve modeling quality. This work will be useful in future research, since it reports practical strategies to improve the performance of this core tool in Structural Bioinformatics. In silico protein structure prediction constitutes an invaluable tool in Biomedical Research, since it allows to obtain structural information on a large number of proteins currently lacking an experimentally-determined 3D structure Reviewer's Responses to QuestionsComments to the Authors:Please note here if the review is uploaded as an attachment.Reviewer #1: The manuscript describes an analysis of the potential to improve homology modelling using “satisfaction of spatial restraints” in the widely used MODELLER package. If the structural divergence between target and template is optimally modelled there is a large room for improvement in particular for multiple templates. Improvements (2% for single and 11% for multiple templates) are expected if the predicted structural divergence correlate >0.6 compared to the true divergence. In addition, the authors also investigate the possibility to include a statistical potential in the objective function of the MODELLER and show that using the build in DOPE statistical potential yields a small but consistent improvement in model quality. Code to for the latter is provided in a git repo.Overall this manuscript was a nice read and present results that can be used to consistently improve modelling results. It also establish an upper limit on model quality that can be gained if the templates are used optimally. However, if I should provide some criticism, a large part of the analysis is done using information from the native structure when setting the local weights. If you know exactly which residues to move and to which degree, you should expect large improvements.1) The authors provide some perturbation analysis by randomly changing some fraction of weights from their optimal value, thereby reducing the correlation from 1.0 to 0.0 and show that improvements are expected when the correlation is >0.6 . Looking at Fig S2, 0.6 correlation corresponds to changing only 30% (0.3) of the residues. Thus, 70% of the residues have their structural divergence at their optimal and 30% are random. This scenario is highly unrealistic; 0.6 correlation doesn’t seem too high, but since correlation is really effected by small number of random points buried among perfect predictions, the actual required prediction performance might be much higher. I suggest that the perturbation analysis is performed in a more rigorous way that sample distributions more likely to originate from a model quality assessment prediction method, where each estimate would have some uncertainty, Or better use a proper model quality assessment program, like ProQ3D or QMEAN to get realistic estimates.2) Best performance gain can potentially be obtained using multiple templates. However, again here, the fact the authors are using the knowledge on which template is optimal obfuscates the true value of this results. We already know that if you are able to always pick the best model you would outperform any group in CASP, the problem is to pick the best model/template. Thus, it is crucial that effect of errors in the estimates are investigated more throughly.3) For multiple templates how are the delta(d_ij) for different templates distributed through the target sequence? Is it different templates that dominates in different regions, or are they intertwined? i.e. is it effectively using more than one template for any given region or is it more picking the best template for each region?4) It is a bit unclear on how the local weights are implemented, do you provide a custom made restraints file to Modeller? or did you find any other API to interface with the Modeller functions? Anyway I think it would be useful if you in addition to the code you already provide, also provide code for running Modeller with the optimal weight (if native is available), or user-specified given a list of local predicted CA-CA distances.Reviewer #2: This manuscript describes a study of MODELLER, a widely-used software tool for protein homology modeling.Given that MODELLER is widely used, further study of this program and even a small improvement are always desirable.In this manuscript, the authors have studied the relationship between modeling quality and estimation of the difference of an inter-atom distance between target and template. The authors claim that 1) a more accurate estimation of the difference may improve modeling accuracy and 2) modeling quality may be increased by incorporating some statistical potentials into MODELLER. These findings are not very new, but the manuscript provides sufficient data and analysis to back up them, which to the best of my knowledge is not widely available in the literature. The authors have also released source code for the incorporation of DOPE and DFIRE into MODELLER, although this is not very new (Similar code is available at the MODELLER website).Some minor concerns:1) lines 46-47, it is fine to say that template-based modeling is the most popular, but I am not sure if it is fine to claim that the most successful approach is template-based modeling since this method fails on the modeling of many membrane proteins. In particular, in the past 2-3 years template-free modeling has made a very good progress and now its accuracy is comparable or even better as long as the target protein does not have very good templates. Further, currently the best template-free modeling also works well on membrane proteins.2) lines 69-71, some revision is needed here. In addition to algorithm advance, the enlargement of both sequence and structure databases is also a very important factor for the improvement of HM modeling.**********Have all data underlying the figures and results presented in the manuscript been provided?PLOS Computational Biologydata availability policy, and numerical data that underlies graphs or summary statistics should be provided in spreadsheet form as supporting information.Large-scale datasets should be made available via a public repository as described in the Reviewer #1: YesReviewer #2: Yes**********what does this mean?). If published, this will include your full peer review and any attached files.PLOS authors have the option to publish the peer review history of their article .One of the goals of PLOS is to make science accessible to educators and the public. PLOS staff issue occasional press releases and make early versions of PLOS Computational Biology articles available to science writers and journalists. 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