See Table 1. Table 1 illustrates that SPAdes compares well to other assemblers on multicell and, particularly, single-cell datasets. SPAdes assembled contigs totaling 5,, bp vs. Since the complete genome of Deltaproteobacterium SAR is unknown, we used long ORFs to estimate the number of genes longer than bp, as a proxy for assembly quality see Chitsaz et al. The running time of Stage 1 is roughly proportional to the number of iterations in the construction of multisized assembly graph. The time per iteration varied between 30 to 40 minutes, slightly exceeding the running time for Velvet.
Stage 2 k -bimer adjustment took 42 minutes. Stage 3 paired assembly graph took 9 minutes. Stage 4 outputting contigs took under a minute. The total time was approximately 3 hours. Peak memory was 1. To account for double-strandedness, we assemble all reads and their reverse complements together.
Every edge in the graph has a reverse complementary edge, although a small number of edges may be their own reverse complement. These dual pairs of edges are kept in sync through all graph transformations. Errors in the start or end of a read may lead to a chain of stray edges protruding from the assembly graph but not connecting to other reads; this is called a tip Fig. To determine if an h-path P from hub u to hub v is a tip, we consider its topology, length, and coverage:.
If u is a source of outdegree 1 and v has indegree at least 2, one of the other h-paths ending at v usually there is just one other is chosen as the alternative h-path Q.
Similarly, if v is a sink of indegree 1 and u has outdegree at least 2, one of the other h-paths starting at u is chosen as the alternative h-path Q. For Illumina reads of length , we use a maximum tip length of This filters out stray h-paths from bad read ends while retaining long contigs that terminate in a source or sink. SPAdes iterates through all h-paths of the graph in order of increasing length terminating at the maximum tip length threshold.
When it's deleted, the hub to which it was joined may become a vertex with indegree 1 and outdegree 1, requiring us to recompute the h-paths in that portion of the graph. As with other graph simplification procedures, we update the list of h-paths on the spot.
This allows us to remove all tips from the graph in one pass, while removing as few nucleotides as possible. Chimeric h-paths arise from chimeric reads and from chance short overlaps between reads. We use a variety of tests to determine if an h-path P from hub u to hub v may be chimeric. Basic gradual h-path removal considers three criteria:. Since coverage varies widely within SCS datasets, some chimeric junctions may be amplified in the reads. We thus developed additional heuristics to delete some chimeric junctions with high coverage, based only on topology and length rather than coverage.
However, there may still be chimeric junctions not detected by these heuristics. Bacterial genomes typically have a small number of long repeats several kb long. If all three h-paths are correct, then P 2 must be a long repeat of multiplicity at least 2. However, P 0 satisfies the topology and length conditions for a chimeric h-path, and it is more likely that P 0 is chimeric but the chimeric junction was amplified, so we delete P 0.
Note that Q may have additional hubs along the path, so Q is a path but not necessarily an h-path. By default, the length of P and Q is small if it is at most and the lengths P , Q are similar if either or. The numeric values are parameters that can be changed. To correct a bulge, SPAdes removes path P from the graph, and substitutes each edge a of P by an edge projection a in Q.
If a is at offset i in P , then projection a is in Q at offset. After all other graph simplification procedures, we remove isolated h-paths with length below Although the graph simplification algorithm is universal, bookkeeping operations are performed that allow us to map reads back to their positions in the assembly graph after graph simplification.
This is used in Stage 2 to map bireads to the simplified graph, and may be output in Stage 4 for downstream applications. The bookkeeping details presented here are specific for assembling a de Bruijn graph from reads and simplifying the graph. The details may change for other A -Bruijn graph applications.
For a de Bruijn graph with k -mer edges, the graph editing operations can be described as either projecting one k -mer x onto another, y , or deleting x. Let S be the set of all k -mers edge labels in the de Bruijn graph. We define , where we repeatedly apply the M ap function until the value stabilizes.
While the presentation in this article used edges representing k -mers, many steps in the assembler are implemented in terms of condensed edges representing sequences of varying length. Each h-path in the assembly graph consisting of many vertices and edges is represented as one condensed edge. As the graph is simplified, some condensed edges need to be combined into longer condensed edges. Note that no k -mer is ever contained in more than one condensed edge. Bookkeeping with the condensed edge representation of the assembly graph is implemented as follows.
To recombine a path comprised of segments into a single new condensed edge e :. This vertex was previously a hub but is no longer a hub. The positions of the k -mers are sufficient to compute the h-biedge histograms described in Stage 2.
For downstream applications, a more detailed alignment of reads to contigs may be required. Before graph simplification, the list of k -mers in a read maps to a sequence of edges forming a path in the de Bruijn graph.
After graph simplification, there may be some disruptions in continuity of the path, due to deletion of edges and due to bulge corremovals involving paths of slightly different lengths arising from indels. However, the approximate positions in the graph are sufficient to realign the read to the contigs E dge S tring of each condensed edge in the graph.
We illustrate the construction of the paired assembly graph in Figure 4. We treat the case of all reads at a fixed genomic distance. This small example illustrates the definitions, rather than covering all the complexities that may arise. In the general case, distances would vary in each biread, and the reads in a biread would not overlap, but such an example is too large to show.
Each biread from G enome contributes information about genomic distances that is collected on h-biedges. For the reader's convenience, these are listed and numbered in the order traversed by the cycle C , although it is not known in advance. Applying gluing rule H2 to the 26 bivertices arising from these 13 rectangles h-biedges forms a single cycle that reconstructs G enome.
Edges resp. In this section, we present an abstraction for assembly graphs and other A -Bruijn graphs for which edge labels e. For the sake of simplicity, we address the case of a unichromosomal circular genome corresponding to a cycle in the graph. Definition 1. The distance d G u , v between vertices is the length of a shortest directed path starting at u and ending at v. Definition 2. Definition 3. Definition 4. Definition 5. Definition 6. Below we describe some properties of an unknown cycle C representing a solution to the UGAP problem that naturally guides its reconstruction; the proofs are omitted.
Property 1. For every edge , there exist vertex instances such that. We define L eft a and R ight a as arbitrary vertex instances satisfying Property 1. Example of parallel paths and bulges. Edges are labeled as vectors and vertices are labeled as scalars. The second property ensures that the cycle C cannot be shortened by rerouting some of its subpaths:.
Property 2. For example, in Figure 6 , if C passes through edges , as well as through , then it would violate Property 2 unless or. The third property ensures that cycle C obeys the prescribed distances for biedges in BE:.
Property 3. For any biedge ,. Several breakthroughs in single-cell genomics in have opened the possibilities of performing genome-wide haplotyping Fan et al. While this article is limited to bacterial sequencing, the goal is to extend SPAdes for assembling structural variations in human SCS projects.
For multicell datasets, Quake and Hammer produce similar results. However, we use the Eulerian assembly framework Idury and Waterman, ; Pevzner et al. An alternative formulation is to consider Chinese Postman cycles. Compare with Pevzner and Tang Moreover, the maximal error in distance estimate for each individual h-biedge can be bounded and these bounds may vary across various h-biedges. SPAdes, in contrast, does not consider such contigs as the final truth and uses all reads at each iteration of the multisized assembly graph construction.
This is important since contigs for smaller k have an elevated number of local misassemblies usually manifested as small indels as compared to contigs for larger k. For example, reducing vertex size from 55 to 31 default parameter in Velvet significantly increases the number of erroneous indels.
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Abstract The lion's share of bacteria in various environments cannot be cloned in the laboratory and thus cannot be sequenced using existing technologies. Key words: assembly, de Bruijn graph, single cell, sequencing, bacteria. Introduction M ost bacteria in environments ranging from the human body to the ocean cannot be cloned in the laboratory and thus cannot be sequenced using existing Next Generation Sequencing NGS technologies. Terminology Since various assembly articles use widely different terminology, below we specify a terminology that is well suited for PDBGs.
Open in a separate window. Standard de Bruijn graphs An n-mer is a string of length n. Multisized de Bruijn graphs The choice of k affects the construction of the de Bruijn graph.
B-transformation Consider a pair of reads r 1 and r 2 at approximate genomic distance d 0 inferred from the nominal insert length and their mapping described in Sec.
Biedge graphs To explain the logic of constructing the paired assembly graph from the set of adjusted h-biedges A H B B ireads , we first describe the simpler biedge graph construction on the set of adjusted biedges E A H B B ireads. Bulge corremoval versus bulge removal Existing assembers often use two complementary approaches to deal with errors in reads: error correction in reads Pevzner et al.
Gradual h-path removal Velvet and some other assemblers use a fixed coverage cutoff threshold for h-paths in the de Bruijn graph to prune out low-coverage and likely erroneous h-paths.
Results 7. Assembly datasets We used three datasets from Chitsaz et al. How accurate are the distance estimates for h-biedges? Table 1. SOAPdenovo 1. Statistics in this table differ slightly from statistics presented in Chitsaz et al. The percentage of the E. Additional Details on Assembly Graph Construction 8. Double strandedness To account for double-strandedness, we assemble all reads and their reverse complements together.
Tip removal Errors in the start or end of a read may lead to a chain of stray edges protruding from the assembly graph but not connecting to other reads; this is called a tip Fig.
Gradual chimeric h-path removal Chimeric h-paths arise from chimeric reads and from chance short overlaps between reads. Gradual bulge corremoval Paths P and Q connecting the same hubs form a simple bulge if i P is an h-path and ii the lengths of P and Q are small and similar.
Isolated h-path removal After all other graph simplification procedures, we remove isolated h-paths with length below Backtracking edges relocated during graph simplification Although the graph simplification algorithm is universal, bookkeeping operations are performed that allow us to map reads back to their positions in the assembly graph after graph simplification. Example of Constructing Paired Assembly Graph We illustrate the construction of the paired assembly graph in Figure 4.
Read-pairs sampled from a circular 24 bp genome. Universal Genome Assembly In this section, we present an abstraction for assembly graphs and other A -Bruijn graphs for which edge labels e.
Footnotes 1 Chimeric reads are formed by concatenation of distant substrings of the genome, and chimeric read-pairs are formed by reads at a distance significantly different from the insert length, as well as by read pairs with an incorrect orientation.
Acknowledgements This work was supported by the Government of the Russian Federation grant Disclosure Statement No competing financial interests exist. References Bandeira N. Clauser K. Pevzner P. Shotgun protein sequencing: assembly of peptide tandem mass spectra from mixtures of modified proteins.
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Azimi N. Skiena S. Crystallizing short-read assemblies around seeds. BMC Bioinform. Reinert K. Myers E. The greedy path-merging algorithm for contig scaffolding. Waterman M. A new algorithm for DNA sequence assembly. Fazayeli F. Ilie S. Hitec: accurate error correction in high-throughput sequencing data.
Woyke T. Stepanauskas R. Genomic sequencing of single microbial cells from environmental samples. Kjallquist U. Moliner A. Characterization of the single-cell transcriptional landscape by highly multiplex RNA-seq. Schatz M. Salzberg S. Quake: quality-aware detection and correction of sequencing errors. Genome Biol. Vederas J. Drug discovery and natural products: end of an era or an endless frontier?
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