Read mapping on de Bruijn graphs
 Antoine Limasset^{1}Email author,
 Bastien Cazaux^{2, 3},
 Eric Rivals^{2, 3} and
 Pierre Peterlongo^{1}
https://doi.org/10.1186/s1285901611039
© Limasset et al. 2016
Received: 9 December 2015
Accepted: 26 May 2016
Published: 16 June 2016
Abstract
Background
Next Generation Sequencing (NGS) has dramatically enhanced our ability to sequence genomes, but not to assemble them. In practice, many published genome sequences remain in the state of a large set of contigs. Each contig describes the sequence found along some path of the assembly graph, however, the set of contigs does not record all the sequence information contained in that graph. Although many subsequent analyses can be performed with the set of contigs, one may ask whether mapping reads on the contigs is as informative as mapping them on the paths of the assembly graph. Currently, one lacks practical tools to perform mapping on such graphs.
Results
Here, we propose a formal definition of mapping on a de Bruijn graph, analyse the problem complexity which turns out to be NPcomplete, and provide a practical solution. We propose a pipeline called GGMAP (Greedy Graph MAPping). Its novelty is a procedure to map reads on branching paths of the graph, for which we designed a heuristic algorithm called BGREAT (de Bruijn Graph REAd mapping Tool). For the sake of efficiency, BGREAT rewrites a read sequence as a succession of unitigs sequences. GGMAP can map millions of reads per CPU hour on a de Bruijn graph built from a large set of human genomic reads. Surprisingly, results show that up to 22 % more reads can be mapped on the graph but not on the contig set.
Conclusions
Although mapping reads on a de Bruijn graph is complex task, our proposal offers a practical solution combining efficiency with an improved mapping capacity compared to assemblybased mapping even for complex eukaryotic data.
Keywords
Read mapping De Bruijn graph NGS Sequence graph path Hamiltonian path Genomics Assembly NPcompleteBackground
Next Generation Sequencing technologies (NGS) have drastically accelerated the generation of sequenced genomes. However, these technologies remain unable to provide a single sequence per chromosome. Instead, they produce a large and redundant set of reads, with each read being a piece of the whole genome. Because of this redundancy, it is possible to detect overlaps between reads and to assemble them together in order to reconstruct the target genome sequence.
Even today, assembling reads remains a complex task for which no single piece of software performs consistently well [1]. The assembly problem itself has been shown to be computationally difficult, more precisely NPhard [2]. Practical limitations arise both from the structure of genomes (repeats longer than reads cannot be correctly resolved) and from the sequencing biases (nonuniform coverage and sequencing errors). Applied solutions represent the sequence of the reads in an assembly graph: the labels along a path of the graph encode a sequence. Currently, most assemblers rely on two types of graphs: either the de Bruijn graph (DBG) for the short reads produced by the second generation of sequencing technologies [3], or for long reads the overlap graph (which was introduced in the Celera Assembler [4]) and variants thereof, like the string graph [5]. Then, the assembly algorithm explores the graph using heuristics, selects some paths and outputs their sequences. Due to these heuristics, the set of sequences obtained, called contigs, is biased and fragmented because of complex patterns in the graph that are generated by sequencing errors, and genomic variants and repeats. The set of contigs is rarely satisfactory and is usually postprocessed, for instance, by discarding short contigs.
The most frequent computational task for analyzing a set of reads is mapping them on a reference genome. Numerous tools are available to map reads when the reference genome has the form of a set of sequences (e.g. BWA [6] and Bowtie [7]). The goal of mapping on a finished genome sequence is to say whether a sequence can be aligned to this genome, and in this case, at which location(s). This is mostly done with a heuristic (semiglobal) alignment procedure that authorizes a small edit or Hamming distance between the read and genome sequences. Read mapping process suffers from regions of low mappability [8]. Repeated genomic regions may not be mapped precisely since the reads mapping on these regions have multiple matches. When a genome is represented as a graph, the mappability issue is reduced, as occurrences of each repeated region are factorized, limiting the problem of multiple matches of reads.
When the reference is not a finished genome sequence, but a redundant set of contigs, the situation differs. The mapping may correctly determine whether the read is found in the genome, but multiple locations may for instance not be sufficient to conclude whether several true locations exist. Conversely, an unfruitful mapping of a read may be due to an incomplete assembly or to the removal of some contigs during postprocessing. In such cases, we argue it may be interesting to consider the assembly graph as a (less biased and/or more complete) reference instead of the set of contigs. Then mapping on the paths of this graph is needed to complement mapping on set of contigs. This motivates the design and implementation of BGREAT.
In this context, we explore the problem of mapping reads on a graph. Aligning or mapping sequences on sequence graphs (a generic term meaning a graph representing sequences along its paths) has already been explored in the literature in different application contexts: assembly, read correction, or metagenomics.
In the context of assembly, once a DBG has been built, mapping the reads back to the graph can help in eliminating unsupported paths or in computing the coverage of edges. To our knowledge, no practical solution has been designed for this task. Cerulean assembler [9] mentions this possibility, but only uses regular alignment on assembled sequences. AllpathsLG [10] also performs a similar task to resolve repeats using long noisy reads from third generation sequencing techniques. Its procedure is not generic enough to suit the mapping of any read set on a DBG. From the theoretical view point, the question is related to the NPhard readthreading problem (also termed Eulerian superpath problem [2, 11]), which consists in finding a read coherent path in the DBG (a path that can be represented as a sequence of reads as defined in [5]). The assembler called SPADES [12] threads the reads against the DBG by keeping track of the paths used during construction, which requires a substantial amount of memory. Here, we propose a more general problem, termed De Bruijn Graph Read Mapping Problem (DBGRMP), as we aim at mapping to a graph any source of NGS reads, either those reads used for building the graph or other reads.
Recently, the hybrid error correction of long reads using short reads has become a critical step to leverage the third generation of sequencing technologies. The error corrector LoRDEC [13] builds the DBG of the short reads, and then aligns each long read against the paths of the DBG by computing their edit distance using a dynamic programming algorithm (which is slow for our purposes). For shorts reads correction, several tools that evaluate the kmer spectrum of reads to correct the sequencing errors use a probabilistic or an exact representation of a DBG as a reference [14, 15].
In the context of metagenomics, Wang et al. [16] have estimated the taxonomic composition of a metagenomics sample by mapping reads on a DBG representing several genomes of closelyrelated bacterial species. In fact, the graph collapses similar regions of these genomes and avoids redundant mapping. Their tool maps the read using BWA on the sequence resulting from the random concatenation of unitigs of the DBG. Hence, a read cannot align over several successive nodes of the graph (ER: il y a un pb ce n’est pas vrai). Similarly, several authors have proposed to store related genomes into a single, less repetitive, DBG [17–19]. However, most of these tools are efficient only when applied to very closely related sequences that result in flat graphs. The BlastGraph tool [19], is specifically dedicated to the mapping of reads on graphs, but is unusable on real world graphs (see Results section).
Here, we formalize the mapping of reads on a De Bruijn graph and show that it is NPcomplete. Then we present the pipeline GGMAP and dwell on BGREAT, a new tool which enables to map reads on branching paths of the DBG (Section GGMAP: a method to map reads on de Bruijn Graph). For the sake of efficiency, BGREAT adopts a heuristic algorithm that scales up to huge sequencing data sets. In Section Results, we evaluate GGMAP in terms of mapping capacity and of efficiency, and compare it to mapping on assembled contigs. Finally, we discuss the limitations and advantages the of GGMAP and give some directions of future work (Section Discussion).
Methods
We formally define the problem of mapping reads on a DBG and investigate its complexity (Section Complexity of mapping reads on the paths of a DBG). Besides, we propose a pipeline called GGMAP to map short reads on a representation of a DBG (Section GGMAP: a method to map reads on de Bruijn Graph). This pipeline includes BGREAT, a new algorithm mapping sequences on branching paths of the graph (Section BGREAT: mapping reads on branching paths of the CDBG).
Complexity of mapping reads on the paths of a DBG
In this section, we present the formal problem we aim to solve and prove its intractability. First, we introduce preliminary definitions, then formalize the problem of mapping reads on paths of a DBG, called the De Bruijn Graph Read Mapping Problem (DBGRMP), and finally prove it is NPcomplete. Our starting point is the wellknown Hamiltonian Path Problem (HPP); we apply several reductions to prove the hardness of DBGRMP.
Definition 1 (de Bruijn graph).
Definition 2 (Walk and Path of a directed graph).

A walk of G is an alternating sequence of nodes and connecting edges of G.

A path of G is a walk of G without repeated node.

A Hamiltonian path is a path that that visits each node of G exactly once.
Definition 3 (Sequence generated by a walk in a d B G _{ k }).
Let G be a de Bruijn graph of order k. A walk of G composed of l nodes (v _{1},…,v _{ l }) generates a sequence of length k+l−1 obtained by the concatenation of v _{1} with the last character of v _{2}, of v _{3},…, of v _{ l }.
We define the de Bruijn Graph Read Mapping Problem (DBGRMP) as follows:
Definition 4 (De Bruijn Graph Read Mapping Problem).

S, a set of strings over Σ,

k, an integer such that k≥2,

q:=q _{1}…q _{q} a word of Σ ^{∗} such that q≥k,

a cost function \(F : \Sigma \times \Sigma \rightarrow \mathbb {N}\), and

a threshold \(t\in \mathbb {N}\),
decide whether there exists a path of the d B G _{ k }(S) composed of q−k+1 nodes (generating a word m:=m _{1}…m _{q}∈Σ ^{q}) such that the cost \(C(m,q) := \sum _{i=1}^{{\vert q \vert }} F(m_{i},q_{i}) \leq t\).
We recall the definition of the Hamiltonian Path Problem (HPP), which is NPcomplete [20].
Definition 5 (Hamiltonian Path Problem (HPP)).
Given a directed graph G, the HPP consists in deciding whether there exists a Hamiltonian path of G.
To prove the NPcompleteness of DBGRMP we introduce two intermediate problems. The first problem is a variant of the Asymmetrical Travelling Salesman Problem.
Definition 6 (Fixed Length Asymmetric Travelling Salesman Problem (FLATSP)).

l be an integer,

G:=(V,A,c) be a directed graph whose edges are labeled with a nonnegative integer cost (given by the function \(c : A \rightarrow \mathbb {N}\)),

\(t \in \mathbb {N}\) be a threshold.
FLATSP consists in deciding whether there exists a path p:=(v _{1},…,v _{ l }) of G composed of l nodes whose cost \(c(p) := \sum _{j = 1}^{l1} c((v_{j},v_{j+1}))\) satisfies c(p)≤t.
We consider the restriction of FLATSP to instances having a unit cost function (i.e., where c(a)=1 for any a∈A) and where l equals both the threshold and the number of nodes in V. This restriction makes FLATSP very similar to HPP, and the hardness result quite natural.
Proposition 1.
FLATSP is NPcomplete even when restricted to instances with a unit cost function and satisfying l=V=t.
Proof.
We reduce HPP to an instance of FLATSP where the cost function c simply counts the edges in the path, and where the path length l equals the threshold t and the number of nodes in V.
Let G=(V,A) be a directed graph, which is an instance of HPP. Let H=(V,A,c:A→{1}), and l:=V and t:=l. Thus (H,l,t) is an instance of FLATSP.
Let us now show that there is an equivalence between the existence of a Hamiltonian path in G and the existence of a path p=(v _{1},…,v _{ l }) of H such that c(p)≤t. Assume that G has a Hamiltonian path p. In this case, p is also a path in H of length V, and then the cost of p equals its length, i.e. \(c(p) = \sum _{i=1}^{\vert V \vert } 1 = \vert V \vert \). Hence, there exists a path p of H such that c(p)≤t=V.
Assume that there exists a path p=(v _{1},…,v _{V}) of H such that c(p)≤t. As p is a path it has no repeated nodes, and as by assumption l=V, one gets that p is a Hamiltonian path of H, and thus also a Hamiltonian path of G, since G and H share the same set of nodes and edges.
The second intermediate problem is called the Read Graph Mapping Problem (GRMP) and is defined below. It formalizes the mapping on a general sequence graph. Hence, DBGRMP is a specialization of GRMP, since it considers the case of the de Bruijn graph.
Definition 7 (Graph Read Mapping Problem).

a directed graph G=(V,A,x), whose edges are labeled by symbols of the alphabet (x:A→Σ),

q:=q _{1}…q _{q} a word of Σ ^{∗},

a cost function \(F : \Sigma \times \Sigma \rightarrow \mathbb {N}\),

a threshold \(t \in \mathbb {N}\),
GRMP consists in deciding whether there exists a path p:=(v _{1},…,v _{q+1}) of G composed of q+1 nodes, which generates a word m:=m _{1}…m _{q}∈Σ ^{q} such that m _{ i }:=x((v _{ i },v _{ i+1})), and which satisfies \(\sum _{i=1}^{\vert q \vert } F(m_{i},q_{i}) \leq t\). Here, m is called the word generated by p.
Proposition 2.
GRMP is NPcomplete.
Proof.
We reduce FLATSP to GRMP.
Let \((G = (V,A,c:A\rightarrow \mathbb {N}),l \in \mathbb {N},t \in \mathbb {N})\) be an instance of FLATSP. Let Σ={y _{1},…,y _{Σ}} an alphabet larger than the largest value of c(A), and let s be the application such that s:{0,…,Σ}→Σ and such that for each i in {0,…,Σ}, s(i)=y _{ i }. Let H=(V,A,x:=s∘c) and let α be a letter that does not belong to Σ, let q=α ^{ l−1} and F such that for each i in {0,…,Σ}, F(α,y _{ i })=i. Thus, we obtain q=l−1.
Now, let us show that there is an equivalence between the existence of a path p=(v _{1},…,v _{ l }) of G such that c(p)≤t and the existence of a path p ^{′}=(u _{1},…,u _{q+1}) of H composed of q+1 nodes, which generates a word m=m _{1}…m _{q} of Σ ^{q}, where each m _{ j }=x((u _{ j },u _{ j+1})), and such that \(\sum _{j=1}^{\vert q \vert } F\left (m_{j},q_{j}\right) \leq t\). Assume that there exists a path p=(v _{1},…,v _{ l }) of G such that c(p)≤t. By definition, p is a path in H. Let m be the word generated by p. Thus we have \(\sum _{j=1}^{\vert q \vert } F\left (m_{j},q_{j}\right) = \sum _{j=1}^{l1} F(m_{j},\alpha) = \sum _{j=1}^{l1} c((v_{j}, v_{j+1})) \leq t\).
Now, suppose that there exists a path p ^{′}=(u _{1},…,u _{q+1}) of H composed of q+1 nodes, which generates a word m=m _{1}…m _{q} of Σ ^{q}, where each m _{ j }=x((u _{ j },u _{ j+1})), and such that \(\sum _{j=1}^{\vert q \vert } F\left (m_{j},q_{j}\right) \leq t\). By the construction of H, p ^{′} is a path in G of length q+1=l. Hence, we obtain \(\sum _{j=1}^{l1} c((u_{j}, u_{j+1})) = \sum _{j=1}^{\vert q \vert } F(m_{j},\alpha) = \sum _{j=1}^{l1} F\left (m_{j},q_{j}\right) \leq t\).
Theorem 1.
DBGRMP is NPcomplete.
Proof.
Let us now reduce GRMP to DBGRMP.
Any letter of a word in V ^{′} is a symbol of Σ ^{′} numbered by a node of V. Moreover, if that symbol is taken from V then it labels an edge of A that goes out a node, say i, of V, and the number associated to that symbol is i. In fact, V ^{′} is the union of three sets (see Eq. 1):
set 1 considers the cases of an edge of A labeled α followed by an edge labeled β, sets 2 and 3 contain the cases of an edge of A labeled α that is not preceded by another edge of A; for each such edge one creates two words: Δ _{ i } $ _{ i } in set 2 and $ _{ i } α _{ i } in set 3.
Let H be the 2dBG of V ^{′}; note that Σ ^{′} is the alphabet of the words of V ^{′}. Now let z be the application from V ^{′} to Σ that for any α _{ i } of V ^{′} satisfies z(α _{ i })=α. (Note that in this equation, the right term is a shortcut meaning the symbol of α _{ i } without its numbering i; this shortcut is used only for the sake of legibility, but can be properly written with a heavier notation). Let \(F' : \Sigma ' \times \Sigma \rightarrow \mathbb {N}\) be the application such that ∀(α _{ i },β)∈Σ ^{′}×Σ, F ^{′}(α _{ i },β)=F(z(α _{ i }),β)=F(α,β).
Let us show that this reduction is a bijection that transforms a positive instance of GRMP into a positive instance of DBGRMP. Assume there exists a path p:=(v _{1},…,v _{q+1}) of G which generates a word m=m _{1}…m _{q}∈Σ ^{q} satisfying m _{ i }=x((v _{ i },v _{ i+1})) and such that \(\sum _{i=1}^{\vert q \vert } F(m_{i},q_{i}) \leq t\). We show that there exists a path p ^{′} of G ^{′} which generates a word m ^{′}=m1′…mq′∈Σ ^{′} ^{q} such that \(\sum _{i=1}^{\vert q \vert } F'(m'_{i},q_{i}) \leq t\).
Let m ^{′} denote the word generated by p ^{′}. Clearly, one sees that \(\phantom {\dot {i}\!}m' = (m_{1})_{v_{1}} \ldots (m_{\vert q \vert })_{v_{\vert q \vert }}\), and since \(\phantom {\dot {i}\!}m_{i} = z((m_{i}')_{v_{i}})\), one gets that z(m ^{′})=m and \(\sum _{i=1}^{\vert q \vert } F'\left (m'_{i},q_{i}\right) = \sum _{i=1}^{\vert q \vert } F\left (m_{i},q_{i}\right) \leq t\).
In the other direction, the proof is similar since our construction is a bijection.
GGMAP: a method to map reads on de Bruijn Graph
We propose a practical solution for solving DBGRMP. We consider the case of short (hundred of base pairs) reads with a low error rate (1 % of substitution), which is a good approximation of widely used NGS reads. Since errors are mostly substitutions, mapping is computed using the Hamming distance.
In a CDBG, the nodes are not necessarily kmers, words of length k, but unitigs, with some unitigs being longer than reads. Thus, while mapping on a CDBG, one distinguishes between two mapping situations: i/ the reads mapping completely on a unitig of the graph, and ii/ the reads whose mapping spans two or more unitigs. For the latter, we say that the read maps on a branching path of the graph.
GGMAP takes as inputs a query set of reads and a reference DBG. To avoid including sequencing errors in the DBG, we construct the reference DBG after filtering out all kmers whose coverage lies below a userdefined threshold c. This error removal step is a classical preprocessing step that is performed in kmer based assemblers. The unitigs of the CDBG are computed using BCALM2 (the parallel version of BCALM [21]), using the kmers having a coverage ≥c. GGMAP uses such a set of unitigs as DBG.
We now propose a detailed description of BGREAT.
BGREAT: mapping reads on branching paths of the CDBG
As previously mentioned, BGREAT is designed for mapping reads on branching paths of a CDBG, using reasonable resources both in terms of time and memory. Our approach follows the usual “seed and extend” paradigm. More generally, the proposed implementation applies heuristic schemes, both regarding the indexing and the alignment phases.
Indexing heuristic
We remind that our algorithm maps reads that span at least two distinct unitigs. Such mapped reads inevitably traverse one or more DBG edge(s). In a CDBG, edges are represented by the prefix and suffix of size k−1 of each unitig. We call such sequences the overlaps. In order to limit the index size and the computation time, our algorithm indexes only overlaps that are later used as seeds. Those overlaps are good anchors for several reasons: they are long enough (k−1) to be selective, they cannot be shared by more than eight unitigs (four starting and four ending with the overlap), and a CDBG usually has a reasonable number of unitigs and then of overlaps. For instance, the CDBG in our experiment with human data has 70 million unitigs and 87 million overlaps for 3 billion kmers). In our implementation, the index is a minimal perfect hash table indicating for each overlap the unitig(s) starting or ending with this (k−1)mer. Using a minimal perfect hash function limits the memory footprint, while keeping efficient query times (see Table 3).
Read alignment
This mapping procedure is performed only if the two extremities of the read are mapped by two unitigs. The extreme overlaps of the read enables BGREAT to quickly filter out unmappable reads. For doing this, the first (resp. last) overlap of the read is used to align the read to the first (resp. last) unitig. Note that, as polymorphism exists between the read and the graph, some of the overlaps present on the read may be spurious. In this case the alignment fails, and the algorithm continues with the next (resp. previous) overlap. At most n alignment failures are authorized in each direction. If a read cannot be anchored neither on the left, nor on the right, it is considered as not aligned to the graph.

All overlaps on which the read should map contain errors, in this case the read is not anchored or only badly anchored and thus not mapped.

The n first or n last overlaps of the read are spurious, in this case the begin or end is not found and the read is not mapped. By default and in all experiments n=2.

The greedy choices made during the path selection are wrong.
We implemented BGREAT as a dependencefree tool in C++ available at github.com/Malfoy/BGREAT.
Results
Beforehand we give details about the data sets (Subsection Data sets and CDBG construction), then we perform several evaluations of GGMAP and of BGREAT. First, we compare graph mapping to mapping on the contigs resulting from an assembly (Subsection Graph mapping vs assembly mapping). Second, we assess how many reads are mapped on branching paths vs on unitigs (Subsection Mapping on branching paths usefulness). Third, we evaluate the efficiency of BGREAT in both terms of throughput and scalability (Subsection GGMAP performances), then assess the quality of the mapping itself (Subsection GGMAP accuracy). All BGREAT alignments were performed authorizing up to two mismatches.
CDBG used in this study
CDBG Id  Reads Id  k  c  Number of unitigs  Mean length of unitigs 

E.coli  SRR959239  31  3  42,843  134 
C.elegans_norm  SRR065390  31  3  1,627,335  93 
C.elegans_cpx  SRR065390  21  2  8,273,338  34 
Human  SRR345593  31  10  69,932,343  70 
SRR345594 
Data sets and CDBG construction
For our experiments we used publicly available Illumina read data sets from species of increasing complexity: from the bacterium E.coli, the worm C.elegans, and from Human. Detailed information about the data sets are given in Additional file 1: Table S1 (identifiers, read length, read numbers, and coverages – from 70x to 112x–).
For each of these three data sets, we generated a CDBG using BCALM. From the C.elegans read set, we additionally generated an artificially complex graph, by using small k and c values (respectively 21 and 2). This particular graph, called C.elegans_cpx, contains lot of small unitigs. We used it to assess situations of highly complex and/or low quality sequencing data. The characteristics of the CDBG obtained on each of these data sets are given in Table 1.
Graph mapping vs assembly mapping
We compared GGMAP to the popular approach consisting in mapping the reads to the reference contigs computed by an assembler. For testing this approach, for each of the three sets used, we first assembled them and then we mapped back the reads on the obtained set of contigs. We used two different assemblers, the widely used Velvet [22], and Minia [23], a memory efficient assembler based on Bloom filters. Finally, we used Bowtie2 for mapping the reads on the obtained contigs.
Percentage of mapped reads, either mapping on contigs (here obtained thank to the Minia assembler) or mapping on CDBG with GGMAP
Set  % mapped on contigs  % mapped on CDBG 

E.coli  95.57  97.16 
C.elegans_norm  80,60  93,24 
C.elegans_cpx  56,33  89,15 
Human  63,16  85,70 
We notice that the more complex the graph, the higher the advantage of mapping on the CDBG. This is due to the inherent difficulty of assembling with huge and highly branching graphs. This is particularly prominent in the results obtained on the artificially complex C.elegans_cpx CDBG.
We also highlight that our approach is resource efficient compared to most assembly processes. For instance, Velvet used more than 80 gigabytes of memory to compute the contigs for the C. elegans data set with k=31. On this data set, our workflow used at most 4 GB memory (during kmer counting). In terms of throughput, using BGREAT and then Bowtie2 on long unitigs is comparable to using Bowtie2 on contigs alone. See Section GGMAP performances for more details about GGMAP performances.
Mapping on branching paths usefulness
Non reflexive mapping on a CDBG
The GGMAP approach is also suitable for mapping a distinct read set from the one used for constructing the DBG. We mapped another read set from C.elegans (SRR1522085) on the C.elegans_norm CDBG. Results in this situation are similar to those observed when performing reflexive mapping (i.e., when mapping the reads used to construct this graph): among 89 % of mapped reads, 15 % were mapped on branching paths of the graph (See Fig. 5).
GGMAP performances
Time and memory footprints of BGREAT and BOWTIE2
BGREAT  BOWTIE2  

CDBG Id  Mapped set  Wall clock time  CPU time  Memory  Wall clock time  CPU time  Memory 
(nb reads)  
E.coli  SRR959239  28 s  1m40  19 MB  1m17  3m53  29 MB 
(5,128,790)  
C.elegans_cpx  SRR065390  19m21  72m31  975 MB  8m12  33m  1.66 GB 
(67,155,743)  
C.elegans_norm  ′′  13m03  51m28  336 MB  17m49  72m31  493 MB 
C.elegans_norm  SRR1522085  1m54  7m13  336 MB  3m29  14m12  493 MB 
(22,509,110)  
Human  SRR345593  4h30  87 h  9.7 GB  4h38  90h15  21 GB 
SRR345594  
(2,967,536,821) 
GGMAP accuracy
To measure the impact of the read alignment heuristics, we forced the tool to explore exhaustively all potential alignment paths once a read is anchored on the graph. Results on the E.coli dataset show that the greedy approach is much faster than the exhaustive one (38 × faster), while the mapping capacity is little impacted: the overall number of mapped reads increases by only 0.03 % with the exhaustive approach. We thus claim that the choice of the greedy strategy is a satisfying tradeoff.
GGMAP mapping results on simulated reads from the reference of the human chromosome 1 with default parameters
% Errors in simulated reads  Distance to optimum of BGREAT mapped reads (percentage)  

0  1  2  3  ≥4  
0  100  0  0  0  0 
0.1  99.31  0.52  0.09  0.04  0.04 
0.2  98.79  0.91  0.21  0.07  0.02 
0.5  97.2  2.17  0.41  0.17  0.05 
1  94.88  3.72  0.92  0.41  0.07 
2  90.85  6.43  1.79  0.83  0.1 
Discussion
We proposed a formal definition of the de Bruijn graph Read Mapping Problem (DBGRMP) and proved its NPcompleteness. We proposed a heuristic algorithm offering a practical solution. We developed a tool called BGREAT implementing this algorithm using a compacted de Bruijn graph (CDBG) as a reference.
From the theoretical viewpoint, the problem DBGRMP considers paths rather than walks in the graph. The current proof of its hardness does not seem to be adaptable to the cases of walks. A perspective is to extend the hardness result to that more general case.
We emphasize that our proposal does not enable genome annotation. It has been designed for applications aiming at a precise quantification of sequenced data, or a set of potential variations between the reads and the reference genome. In this context, it is essential to map as much reads as possible. Experiments show that a significant proportion of the reads (between ≈13and ≈66 % depending on the experiment) can be only mapped on branching paths of the graph. Hence, mapping only on the nodes of the graph or on assembled contigs is thus insufficient. This statement holds true when mapping the reads used for building the graph, but also with reads from a different experiment. Moreover, our results show that a potentially large number of reads (up to ≈32%) that are mapped on a CDBG cannot be mapped on a classical assembly.
With GGMAP, the mapping quality is very high: using Human chromosome 1 as a reference and reads with a realistic error rate (similar to that of Illumina technology), over 99 % of the reads are correctly mapped. The same experiment also pointed out the limits of mapping reads on a divergent graph reference (≥2 % substitutions): approximately 10 % of the reads are mapped at a suboptimal position.
A weak point of BGREAT lies in its anchoring technique. Reads mapped with BGREAT must contain at least one exact k−1mer that is an arc of the CDBG, i.e., an overlap between two connected nodes. This may be a serious limitation when the original read set diverges greatly from the reads to be mapped. Improving the mapping technique may be done by using not only unitig overlaps as anchors at the cost of higher computational resources. Another solution may consist in using a smarter anchoring approach, like spaced seeds, which can accommodate errors in the anchor [24].
A natural extension consists in adapting BGREAT for mapping, on the CDBG obtained from short reads, the long (a few kilobases in average) and noisy reads produced by the third generation of sequencers, whose error rate reaches up to 15 % (with mostly insertion and deletion errors for e.g. Pacific Biosciences technology). Such adaptation is not straightforward because of our seeding strategy, which requires long exact matches. The anchoring process must be very sensitive and very specific, while the mapping itself must implement a Blastlike heuristic or an alignmentfree method. However, mapping such long reads on a DBG could be of interest for correcting these reads as in [13], or for solving repeats, if long reads are mapped on the walks (which main include cycles) of the DBG. Our NPcompleteness proof only considers mapping on (acyclic) paths. Proving the hardness of the problem of mapping reads on walks of a DBG remains open.
Incidentally, using the same read set for constructing the CDBG and for mapping opens the way to major applications. Indeed, the graph and the exact location of each read on it may be used for i/ read correction as in [15], by detecting differences between reads and the mapped area of the graph in which low support kmers likely due to sequencing errors are absent, or for ii/ read compression by recording additionally the mapping errors, or for iii/ both correction and compression by conserving only for each read its mapping location on the graph.
Having for each read (used for constructing the graph or not) its location on the CDBG also provides the opportunity to design algorithms for enriching the graph, for instance enabling a quantification that is sensitive to local variations. This would be valuable for applications such as variant calling, analysis of RNAseq variants [25], or of metagenomic reads [26].
Additionally, BGREAT results provide pieces of information for distant kmers in the CDBG, about their cooccurrences in the mapped read data sets. This offers a way for the resolution, in the de Bruijn graph, of repeats larger than k. It could also allow to phase the polymorphisms and to reconstruct haplotypes.
Conclusion
A take home message is that read mapping can be significantly improved by mapping on the structure of an assembly graph rather than on a set of assembled contigs (respectively ≈22 % and ≈32 % of additional reads mapped for the Human and a complex C.elegans data sets). This is mainly due to the fact that assembly graphs retains more genomic information than assembled contigs, which also suffer from errors induced by the complexity of assembly. Moreover, mapping on a compacted De Bruijn Graph can be fast. The availability of BGREAT opens the door to its application to fundamental tasks such as read error correction, read compression, variant quantification, or haplotype reconstruction.
Abbreviations
CDBG, Compacted De Bruijn graph; DBG, De Bruijn graph; DBGRMP, De Bruijn graph read mapping problem; FLATSP, fixed length assymetric travelling salesman problem; GRMP, graph read mapping problem; HPP, Hamiltonian path problem
Declarations
Acknowledgements
We would like to thank Yannick Zakowski, Claire Lemaitre and Camille Marchet for proofreading the manuscript and discussions.
Funding
This work was funded by French ANR12BS020008 Colib’read project, by ANR11BINF0002, and by a MASTODONS project.
Availability of data and materials
Our implementations are available at github.com/Malfoy/BGREAT. In addition to the following pieces of information, Additional file 1: Table S1 presents the main characteristics of these datasets.
SRR959239 http://www.ncbi.nlm.nih.gov/sra/?term=SRR959239
SRR065390 http://www.ncbi.nlm.nih.gov/sra/?term=SRR065390
SRR1522085 http://www.ncbi.nlm.nih.gov/sra/?term=SRR1522085
SRR345593 and SRR345594 http://www.ncbi.nlm.nih.gov/sra/?term=SRR345593.
Authors’ contributions
PP initiated the work and designed the study. AL, BC and ER designed the formalism and the proofs of NPhardness. AL designed the algorithmic framework, implemented the BGREAT and performed the tests. All authors wrote and accepted the final version of the manuscript.
Competing interests
The authors declare that they have no competing interests.
Ethics approval and consent to participate
Not applicable.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
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