Bacterial syntenies: an exact approach with gene quorum
 YvesPol Deniélou^{1}Email author,
 MarieFrance Sagot^{1, 2},
 Frédéric Boyer^{3} and
 Alain Viari^{1}Email author
https://doi.org/10.1186/1471210512193
© Deniélou et al; licensee BioMed Central Ltd. 2011
Received: 24 January 2011
Accepted: 24 May 2011
Published: 24 May 2011
Abstract
Background
The automatic identification of syntenies across multiple species is a key step in comparative genomics that helps biologists shed light both on evolutionary and functional problems.
Results
In this paper, we present a versatile tool to extract all syntenies from multiple bacterial species based on a clearcut and very flexible definition of the synteny blocks that allows for gene quorum, partial gene correspondence, gaps, and a partial or total conservation of the gene order.
Conclusions
We apply this tool to two different kinds of studies. The first one is a search for functional gene associations. In this context, we compare our tool to a widely used heuristic  IADHO RE  and show that at least up to ten genomes, the problem remains tractable with our exact definition and algorithm. The second application is linked to evolutionary studies: we verify in a multiple alignment setting that pairs of orthologs in synteny are more conserved than pairs outside, thus extending a previous pairwise study. We then show that this observation is in fact a function of the size of the synteny: the larger the block of synteny is, the more conserved the genes are.
Background
The increasing number of fully sequenced microbial genomes (more than 1200 genomes are now completed and available) is fuelling our knowledge on the architecture and dynamics of these bacterial genomes [1]. In this context, the identification of conserved genomic regions across several species is of prime importance. These conserved blocks of genes are referred to in the literature as (micro)syntenies, (conserved) synteny blocks, (conserved) gene clusters, or gene teams. From an evolutionary standpoint, they are important for understanding how, as species diverge, their gene order is progressively shuffled [2, 3]. From a more practical genome annotation perspective, it is also important to identify which regions are resisting this continuous shuffling because these regions are likely to be subjected to stronger functional constraints. In prokaryotic genomes these constraints are of course related to the operon structure. Operons, although subjected as well to gene shuffling [4] appear to be more robust, especially in the case of physically interacting gene products [5].
Whole genomes alignement methods in the literature
edition operations  

Name  nb genomes  correspondence  INV/TRANS  DUPL  DEL  algorithm  reference 
GRIMM  pairwise  onetoone  yes  yes  yes  minimise evolutionary distance  [6] 
CINTENY  pairwise  manytomany  yes  yes  yes  minimise evolutionary distance  [7] 
UNO AND TAGIURA  pairwise  onetoone  yes  no  no  find common intervals  [8] 
HEBER AND STOYE  multiple  onetoone  yes  no  no  find common intervals  [9] 
DIDIER  pairwise  manytomany  yes  yes  no  find common intervals  [10] 
GENE TEAMS  multiple  equivalence  yes  no  yes  divide and conquer  [11] 
HOMOLOGY TEAMS  pairwise  equivalence  yes  yes  yes  divide and conquer  [12] 
DOMAIN TEAMS  multiple  equivalence  yes  yes  yes  divide and conquer  [13] 
MCGS  multiple  equivalence  yes  yes  yes  divide and conquer  [14] 
MCPA GE  pairwise  equivalence  yes  yes  yes  divide and conquer  [15] 
MCMU SE C  multiple  equivalence  yes  yes  yes  divide and conquer  [16] 
C3PART  multiple  manytomany  yes  yes  yes  partition the NAM  [17] 
FISH  pairwise  manytomany  local  no  yes  dynamic programming  [18] 
DAGCHAINER  pairwise  manytomany  no  no  yes  dynamic programming  [19] 
COLINEAR SCAN  pairwise  manytomany  no  no  yes  dynamic programming  [20] 
SYNTENATOR  multiple  manytomany  no  yes  yes  dynamic programming on POG  [21] 
CYNTENATOR  multiple  manytomany  no  yes  yes  same + phylogeny  [22] 
ADHO RE  pairwise  manytomany  no  tandem  yes  clustering  [30] 
IADHO RE  multiple  manytomany  no  tandem  yes  greedy heuristic  [26] 
The first category contains methods that aim at reconstructing evolutionary scenarios using sorting by reversal to recover the minimal number of permutations needed to transform one genome (modelled as a signed or unsigned permutation of integers) into the other. In such approaches, of which the two main ones are GRIMM [6] and CINTENY[7], the identification of blocks of syntenies is, in some way, a byproduct of the algorithm, not its main goal. Some authors also formalise the problem of genome alignment as an exact search for common intervals between permutations. The first article using this formalism was Uno et al. [8]. Two important extensions were proposed later on, an extension to multiple alignment by Hebert and Stoye [9] and another extension to a manytomany relation by Didier et al. [10].
A second, different, formalism used by several authors is well illustrated by the concept of GENE TEAMS[11], proposed originally by Bergeron et al. This defines syntenies as sets of genes such that on each genome no pair of genes from the team are separated by more than delta genes, the main limitation being that, in the original formulation, the homology relation is supposed to be onetoone.
This idea was developed in several papers, He and Goldwasser [12] proposed the notion of HOMOLOGY TEAMS, which allows duplications on a genome; Pasek et al. [13] chose to cut the genes into protein domains to force an equivalence relation. Some other extensions of the model were suggested by Kim et al. (relaxing the proximity constraint on several genomes [14]), and Ling et al. (allowing for overlapping units, for example sequence anchors: [15, 16]). More generally these approaches consider the question of finding syntenies as a problem of combinatorial optimisation and often rely on graph theory both to formalise and to solve it. The main difficulty is to limit the combinatorial explosion when the genetogene relationship is not onetoone. The common components approach proposed by Boyer et al. [17] in a more general biological context belongs to the same family. The approach developed in this paper is an extension of this latter method and will therefore be detailed later on.
In a third category of methods, the problem of genome alignment is treated in a similar way as what is done more classically for pairwise or multiple sequence alignments and relies on a dynamic programming approach to solve it. Three typical examples of such an approach are FISH [18], DAGCHAINER[19] and COLINEAR SCAN[20]. The two strong advantages brought by these approaches are i) they do not require an explicit genetogene relationship and can cope with an arbitrary similarity (i.e. substitution) score between genes; ii) as scorebased techniques they are more amenable to a statistical evaluation of the significance of the predicted regions. On the other hand, the most important difficulty is to handle inversions in the dynamic programming framework.
Of course, this classification is not perfect and several hybrid methods have already been proposed that belong to several classes. For instance, SYNTENATOR[21] (and its most recent version CYNTENATOR[22]) uses both dynamic programming and a partial order graph representation to detect conserved gene orders in multiple genomes.
In this paper, we introduce an extension of the general graph alignment algorithm presented in [17] and [23] successively. In [17] we introduced the idea of a merged representation of a graph alignment as a multigraph (termed Network Alignment Multigraph (NAM)). This corresponds to an idea that was previously more informally stated in [24] and was also developed in [25]. The definition of blocks of synteny (allowing for gene gaps and permutations) then follows from simple properties of this multigraph. This first approach however suffered from two limitations: i) it required the explicit construction of the NAM, therefore facing the problem of combinatorial explosion in case of multiple genomes or of a very degenerated genetogene relationship and ii) it assumed that genes are in correspondence when they form a clique (i.e. are all pairwise related). In [23] we proposed, in the context of proteinproteininteraction (PPI) networks, to address the first issue by a more careful exploration of the search space, using a depthfirst search and "onthefly" construction of the NAM. We also introduced alternative ways of grouping genes such as stars or, simply, as connected components instead of as complete cliques. In this paper, we increase the flexibility even more by introducing a quorum i.e., when dealing with multiple (n > 2) genomes, we do not require genes to be present on all but only on at least q (≤ n) of them. Notice that the algorithm presented in [23] is not restricted to genomes but applies to any kind of graphs. Our extension will apply as well in the general case. In this paper, we shall therefore keep all definitions as general as possible but restrict the illustrations and interpretations to the case of linear graphs representing genomes.
This paper is organised as follows. First, we describe the approach, starting with an informal presentation before going through the definitions, stating precisely what objects we are going to look for, and then what algorithm we are going to use. In the next section, we describe our results, a comparison to an existing method called IADHO RE[26], and an illustration of the interest of the approach for studies on bacterial evolution.
Description of the approach
Informal presentation of the approach
In this section, we first give a brief summary of the approach without quorum, then explain informally how to introduce it.
Given n chromosomes represented as interval graphs (i.e. vertices are genes and two genes are connected when they are adjacent on the chromosome, or, more generally, when there are less than δ_{ gap } intervening genes between them), the first step is to define a pairwise correspondence relation (noted S) between genes from different chromosomes. Ideally S could be homology (i.e. having a common ancestor) or isofunctionality (i.e. having the same function). In practice, both are traditionally approximated by sequence similarity, for instance by thresholding a BLASTP score or using more sophisticated orthologs detection techniques [27, 28]. Note that, by contrast to homology and isofunctionality that are both transitive, thresholded sequence similarity and orthology are not necessarily transitive relations.
With this pairwise genetogene correspondence at hand, the next step is to extend it to a multiple (nway) correspondence. This is done by specifying a topology constraint on S. The strongest constraint is that genes connected by S form a clique [17] and the loosest constraint is that they merely form a connected component. Intermediate constraints, such as quasicliques are also possible [23]. Whatever the choice of this constraint, we end up with ntuples of genes representing the nway gene correspondence between the n genomes.
In this NAM, we then simply define blocks of syntenies called syntons as the maximal subgraphs that are connected on each of the n colours.
This definition of syntenies matches the intuition that they are made of corresponding neighbourhoods: the selection of spines ensures the genetogene correspondence, and the connectivity condition on each colour ensures that on every genome a synton involves connected genes. Also notice that this definition allows for any permutation in the gene order along the chromosomes.
An important property that follows from this definition is that syntons form a partition of the vertices of the NAM. This means that efficient partitioning algorithms can be put into play, provided that the NAM has been already built.
But therein lies the rub: in the general case, the NAM itself may grow exponentially with the number of genomes, both in terms of vertices and of edges.
In Deniélou et al. [23] we proposed to avoid the explicit construction of the NAM by building "on the fly" only the parts of the multigraph we need. The idea is to add the genomes progressively in a depthfirst search (and construction) of the multigraph. To do this, we basically alternate two procedures: Split and Expand. Split partitions the current multigraph on the previously treated genomes and Expands adds a new genome.
In this paper, we show that the same kind of strategy also works when allowing for missing genes on several genomes.
The basic idea is that instead of ntuples we now allow for ktuples, with 2 ≤ q ≤ k ≤ n, where q is a quorum fixed by the user. With these ktuples, we can define a Partial Network Alignment Multigraph (PNAM) in a similar way as we defined the NAM before, and syntons are maximal subsets of ktuples that are connected on the k colours.
Intuitively, this means that a synton concerns exactly k genomes (with q ≤ k ≤ n). On these genomes, a synton is made of corresponding neighbourhoods as with the previous definition and the n  k remaining genomes are simply ignored.
The next sections explain the formalisation and algorithm in more detail.
Layered data graph
The layered data graph (also called layered alignment graph in [25]) provides the simplest representation of the primary data at hand.
Definition . (adapted from [25]) Given a set of n primary graphs G_{ i } = (V_{ i }, E_{ i } ), i ∈ [1, n] and a correspondence relation S between the elements of distinct sets V_{ i }, the layered data graph is the graph D = (V, E) with
Observe that there are two kinds of edges in E: edges in E_{ P } correspond to the original sets E_{ i } (hereafter called intralayer edges) and the other ones (E_{ S } ) connect vertices from different layers (hereafter called interlayer edges) (see Figure 1).
In the specific case of genomes, V_{ i } represents the set of genes in the i^{ th } genome and E_{ i } represents gene contiguity: (u_{ i }, v_{ i } ) ∈ E_{ i } ⇔ rank(u_{ i } )  rank(v_{ i } ) ≤ δ_{ gap } where rank is the rank of the gene on the chromosome and δ_{ gap } is a gap parameter (the formula can easily be adapted to circular genomes as well).
nway partial correspondence and don't care element
As mentioned earlier, dealing with n ≥ 2 genomes requires to define formerly how genes from different genomes are aggregated.
The definition of an ncorrespondence without a quorum (i.e. for q = n) is the following:
Definition (without quorum). An nway correspondence between elements of V_{1}, V_{2},..., V_{ n }is defined as a restriction of the cartesian product, denoted by .
Several practical cases of such an aggregation are discussed in [23].
For instance, a clique aggregation requires that all elements of an ntuple are pairwise related: , v_{i}S v_{ j }.
Conceptually the introduction of a quorum q consists of working with ktuples of genes (with q ≤ k ≤ n) instead of ntuples. However for the sake of simplicity, we shall continue to work with ntuples by introducing a don't care element, *.
Given , we introduce a function called cover which for a given ntuple v returns the set of nodes in v that are not don't care elements. Formally, , cover(v) = {v_{ i } ≠ _{*}, i ∈ [1, n]}.
One can now define an nway partial correspondence.
Definition (with quorum). An nway partial correspondence between elements of V_{1}, V_{2},..., V_{ n }for a quorum q is defined as an nway correspondence such that , cover(v) ≥ q.
We shall in the following use the notation .
As before, this restriction is computed by using the S relation. For instance a clique aggregation for a quorum q would be expressed as follows and ∀i, j, (v_{ i }S v_{ j }) or (v_{ i }= _{*}) or (v_{ j }= _{*}).
Partial Network Alignment MultiGraph
The PNAM (Partial Network Alignment Multigraph) is an extension of the NAM (Network Alignment Multigraph) presented in [23]. It summarises both the nway partial correspondence and the connectivity in the genomes (i.e. gene neighbourhood).
Definition (with quorum). A partial network alignment multigraph (PNAM) for n primary graphs G_{ i }= (V_{ i }, E_{ i }) is a graph such that:
In other words, the vertices of the multigraph are ntuples of genes and don't care elements and there is an edge e ∈ ℰ _{ i } (that is, an edge of colour i) between two vertices if they have genes at the i^{ th } position that are neighbours in the genome G_{ i } . In the following, we refer to such an edge as an edge of colour i.
Defining syntons in the PNAM
With the previous definition of a PNAM at hand, several definitions of synteny are possible, depending on the properties one is looking for (see Table 1).
For instance, without quorum (i.e. q = n) and if strict gene order has to be conserved, syntons are simply the connected components of the PNAM in which all edges that do not have all of the n colours are removed. Note that it is also possible to specify a partial gene order conservation, stating that, in a given synton, two spines consecutive for a colour should not be separated by more than δ_{ shuffle } spines for the other colours. This idea will not be developed in the present paper.
Since we want to allow all gene permutations, we shall use the most general definition, which is the following.
Definition (without quorum). A synton is a maximal subgraph of the PNAM such that ∀i ∈ [1, n], is connected for .
To introduce a quorum, one has to modify the previous definition in order to cope with the presence of don't care elements in the PNAM vertices.
Definition (with quorum). A synton is a maximal subgraph of the PNAM such that ∃I ⊆ [1, n], I ≥ q such that ∀i ∈ I, is connected for and , cover(v) = I.
Informally, the first part of the definition ensures that the result will be a synton when restricted to the colours I, and the second part makes sure that for the colours i ∉ I, all the vertices of correspond to a don't care element. In the previous definition, the term "maximal" naturally means that no other vertex of the PNAM can be added without breaking the connectivity conditions. With this definition, it is easy to show that syntons form a partition of the PNAM vertices.
The PNAM given in Figure 2 is thus partitioned into 4 syntons: {(a_{1}, a_{2}, a_{3})} (I = {1, 2, 3}), {(a_{1}, _{*}, a_{3})} (I = {1, 3}), {(_{*}, a_{2}, a_{3})} (I = {2, 3}) and {(a_{1}, a_{2}, _{*}), (b_{1}, b_{2}, _{*})} (I = {1, 2}).
However, one can notice that although each class of this partition is maximal in the previous sense, it may not be maximal in terms of the genes involved. When projecting a synton onto the layered data graph, it may occur that the set of genes obtained is included into another projection (this occurs for instance when a synton on k genomes has the same boundaries as a synton on k + 1 genomes). It is therefore advisable to add another constraint to syntons in order to remove some redundancy in the results in terms of genes.
Definition of⊑. Let us denote by ⊑ the relation defined as follows: for u, u ⊑ v ⇔ cover(u) ⊆ cover(v).
This means that all vertices in the spine u that are not "don't care" elements are also elements of the spine v.
This definition can be extended to subgraphs of the PNAM.
Definition of⊑. For two subgraphs of the PNAM C_{1} = (U_{1}, F_{1},... F_{ n }) and , C_{1} ⊑ C_{2} ⇔ ∀u_{1} ∈ U_{1}, ∃u_{2} ∈ U_{2}, such that u_{1} ⊑ u_{2}.
We call then a maximal synton a synton that is maximal for the relation ⊑.
On the example of Figure 2, we have for instance {(a_{1}, _{*}, a_{3})} ⊑ {(a_{1}, a_{2}, a_{3})} which means {(a_{1}, _{*}, a_{3})} is not maximal. The only two maximal syntons in that example are {(a_{1}, a_{2}, a_{3})} and {(a_{1}, a_{2}, _{*} ), (b_{1}, b_{2}, _{*})}.
Note that this constraint is not embedded into the algorithm but is added as a final filter on the results.
Algorithm
The most natural approach to compute syntons with quorum would be to build the PNAM, and then to use one of the graph partitioning algorithms at our disposal [17, 29].
However, we already showed in [23] that when the correspondence relation is not onetoone, the size of the NAM may grow exponentially with the number of graphs. The situation for PNAM is even worse since for each vertex of the NAM we now have additional vertices in the PNAM. This means that avoiding the explicit construction of the PNAM is an even bigger priority.
In this paper, we choose to extend the graph partitioning algorithm described in [23]. The idea is to conduct a depth first search (DFS) of the classes starting with the connected components of the first colour (genome), and to add colours incrementally. Therefore the multigraph is not computed explicitly but instead smaller parts of it (classes) are computed on the fly in each branch of the DFS.
The basic algorithm is fully described in [23] and a summary pseudocode is given below. The two main operations are:

SPLIT_{1...i}that splits a class on colours 1 to i;

EXPAND_{i+1}that adds the (i + 1)^{ th }colour to the current network alignment multigraph.
Algorithm 1: OTF.
Global: Layered Data Graph D = (V, E)
for the primary graphs G_{ i }= (V_{ i }, E_{ i }), i ∈ [1, n]
Input: Multigraph C
/* current class: initialised with */
Integer i/* current layer: initialised with 1 */
Variables: Partition P
/* current partition on the i first layers */
(1) begin /* partition on the i first layers */
(2) P ← SPLIT_{1...i}(C),
(3) if (P ≠ 1) then /* C is split */
(4) for NewC ∈ P do
(5) OTF(NewC, i)
(6) end for
(7) else if (i < n) then /* C is stable */
(8) NC ← EXPAND_{i+1}(C),
(9) OTF(NC, i + 1),
(10) else /* C is stable for all colours */
(11) PRINT(C)
(12) end if
(13) end
Initialisation
As in [23], the algorithm is initialised with all connected components on the first genome. In order to cope with missing genes on the first genome, one has only to add an initial singleton class containing a don't care vertex.
Expand
The procedure called EXPAND_V ERTICES_{i+1}expands the current set C of PNAM vertices to two kinds of new (i + 1)tuples.
The first case is similar to the noquorum condition and expands each vertex v = {v_{1}, v_{2},... v_{ i }} ∈ C by genes v_{i+1}∈ V_{i+1}. These genes are called terminals of v. Ideally, a terminal v_{i+1}is such that there exists an ntuple such that ∀j ∈ [1, i + 1], u_{ j }= v_{ j }(i.e. v is a prefix of u). In practice, the efficient computation of these terminals greatly depends upon the chosen aggregation function. For instance, for the clique, this is a simple task: (v_{1},..., v_{i+1}) is required to be a clique too (don't care elements are ignored) since this is a necessary condition for the final ntuple to be a clique. Using the ordering G 1, G 2, G 3 in the example given in Figure 2, this allows us to build the vertex (a 1, a 2, a 3) from the vertex (a 1, a 2).
The second case is to extend v = (v_{1},... v_{ i }) by a don't care element if this is allowed by the quorum condition, i.e. the total number of don't care elements is less than n q. These added are intended to introduce the missing genes on genome i + 1. For instance, in Figure 2, with the same ordering G_{1}, G_{2}, G_{3}, the vertex (c_{1}, c_{2}, _{*}) is recovered thanks to the introduction of don't care elements as terminals (it is an expansion of the vertex (c_{1}, c_{2})).
Split
As in [23], the SPLIT_{1i}operation computes the connected components on each colour in turn. If, for a colour j, the class is split, then it returns the split parts.
The main difference is that the connected components for a colour j are now computed on P_{ j } (V), the restriction of the set of vertices to those that do not have a don't care element at position j.
 1.
removing temporarily all tuples having a don't care element at position j,
 2.
computing the connected components of the resulting set of vertices,
 3.
adding back the previously removed set in each connected component.
Finally, just as in [23], when a class C is such that SPLIT_{1i}(C) = C then it is stable for colours 1 to i and needs expansion to the (i + 1) ^{ th } colour.
Results and Discussion
In order to illustrate our approach in practice, we performed two different experiments. The first one is a comparison to a popular heuristics with a similar aim (IADHO RE). The second one is a study of the evolution rate of genes in synteny that generalises previous observations on this subject. In the remaining of this paper, we shall refer to our algorithm as OTFQ (OTF was the name of the algorithm in [23] and Q stands for quorum).
Comparison to IADHO RE
In this section, we compare the syntenies recovered by our approach to those found by IADHO RE[26], a popular program used to recover syntenies with missing genes.
ADHO RE and IADHO RE
ADHO RE[30] (Automatic Detection of Homologous Regions) looks for genomic regions between two genomes where the gene order is strictly conserved. The genetogene (homology) relation S is, as in our approach, boolean and provided as input to the program together with the genes location. The algorithm basically proceeds by clustering of the homologous genes pairs based on the linear distance of the corresponding genes on both chromosomes (two pairs are close if they lie close together on the same diagonal of the alignment matrix) and a measure of the cluster linearity (how the set of pairs fits to a diagonal). The procedure allows for gaps (maximum number of intervening nonhomologous genes between two pairs) but not explicitly for genes permutations although a limited amount of permutations is possible by considering them as gaps.
In 2004, the authors extended this method to the case of multiple genomes: the new version is called IADHO RE[26]. IADHO RE starts by collecting the results of ADHO RE obtained on each pair of genomes. The results are called "multiplicons of level 2". This series of multiplicons initialises a set ℳ that will eventually constitute the solution set.
The procedure starts with the largest (in terms of gene pairs) multiplicon and represents it as a "profile" i.e. a series of aligned positions without permutation. Next, this profile is aligned to the whole set of chromosomes using a variant of ADHO RE. One important point is that there is a match between a gene and a given position in the profile when this position contains at least one gene that matches. In other terms, in IADHO RE, spines are connected components of the S relation.
The results of this alignment are multiplicons of level 3 (they can be extensions of part or the totality of the original multiplicon). These new multiplicons are then put back in the ℳ set and the algorithm iterates. There is no need in IADHO RE for the quorum parameter we previously defined, instead the results are presented in the form of an arborescence of multiplicons (from level 2 to, possibly, level n) from which the spines involving at least q genomes can be easily retrieved.
Finally, this algorithm was improved in 2007: in this latest version  IADHO RE 2.0 [31]  a new alignment profile is recomputed for all segments of the current multiplicon after extension. This implies that some erroneous decisions taken at the beginning of the execution could be corrected later on.
Because IADHO RE is a heuristic (basically a greedy algorithm), its main advantage is that it is extremely quick. The main drawback is that the definition of a multiplicon is procedural rather than formal. This makes the comparison to other approaches more difficult since we do not know exactly what is to be found and what is missed. Because of this, in the following experiment we decided to compare the results in terms of genes involved in syntons (OTFQ) versus multiplicons (IADHO RE).
Data and parameters
The four groups of bacterial species used in this study
bacteria  Bacteria  

Acnum  Species  Mb  NbGenes 
AE000512_GR  Thermotoga maritima (strain JCM 10099/DSM 3109)  1.9  1853 
AE009951_GR  Fusobacterium nucleatum nucleatum (strain JCM 8532)  2.2  2069 
AL009126_GR  Bacillus subtilis (strain 168)  4.2  4237 
BA000022_GR  Synechocystis sp. (strain PCC 6803)  3.6  3166 
U00096_GR  Escherichia coli (strain K12)  4.6  4320 
AE000520_GR  Treponema pallidum (strain Nichols)  1.1  1028 
AE001273_GR  Chlamydia trachomatis (strain D/UW3/Cx)  1.0  895 
AM398681_GR  Flavobacterium psychrophilum (strain JIP02/86)  2.8  2432 
BX248353_GR  Corynebacterium diphtheriae (strain NCTC 13129)  2.5  2317 
CP000359_GR  Deinococcus geothermalis (strain DSM 11300)  2.5  2330 
proteo  Bacteria; Proteobacteria  
acnum  Species  Mb  NbGenes 
AE005673_GR  Caulobacter crescentus (strain CB15/ATCC 19089)  4.0  3738 
AE016825_GR  Chromobacterium violaceum (strain IFO 12614)  4.7  4407 
AE017282_GR  Methylococcus capsulatus (strain Bath/NCIMB 11132)  3.3  2960 
CP000661_GR  Rhodobacter sphaeroides (strain ATCC 17025)  3.2  3111 
U00096_GR  Escherichia coli (strain K12)  4.6  4320 
CP000112_GR  Desulfovibrio desulfuricans (strain G20)  3.7  3775 
AE001439_GR  Helicobacter pylori (strain J99)  1.6  1491 
CP000251_GR  Anaeromyxobacter dehalogenans (strain 2CPC)  5.0  4346 
CP000744_GR  Pseudomonas aeruginosa (strain PA7)  6.6  6286 
CP000814_GR  Campylobacter jejuni (subsp. jejuni, serovar O:6)  1.6  1626 
gamma  Bacteria; Proteobacteria; Gammaproteobacteria  
acnum  Species  Mb  NbGenes 
AE017282_GR  Methylococcus capsulatus (strain Bath/NCIMB 11132)  3.3  2960 
CP000127_GR  Nitrosococcus oceani (strain ATCC 19707/NCIMB 11848)  3.5  2976 
CP000462_GR  Aeromonas hydrophila (subsp. hydrophila, ATCC 7966)  4.7  4122 
CP000744_GR  Pseudomonas aeruginosa (strain PA7)  6.6  6286 
U00096_GR  Escherichia coli (strain K12)  4.6  4320 
CP000675_GR  Legionella pneumophila (strain Corby)  3.6  3204 
CP000681_GR  Shewanella putrefaciens (strain CN32/ATCC BAA453)  4.7  3972 
CP001091_GR  Actinobacillus pleuropneumoniae (serovar 7, AP6/AP76)  2.3  2131 
CP001132_GR  Acidithiobacillus ferrooxidans (strain ATCC 53993)  2.9  2826 
AM920689_GR  Xanthomonas campestris (pathovar campestris)  5.1  4510 
enterob  Bacteria; Proteobacteria; Gammaproteobacteria; Enterobacteriales; Enterobacteriaceae  
acnum  Species  Mb  NbGenes 
AE006468_GR  Salmonella typhimurium (strain ATCC 700720)  4.9  4455 
AE009952_GR  Yersinia pestis (biovar Mediaevalis, strain KIM5)  4.6  4104 
CP000822_GR  Citrobacter koseri (strain ATCC BAA895)  4.7  5003 
CP000964_GR  Klebsiella pneumoniae (strain 342)  5.6  5425 
U00096_GR  Escherichia coli (strain K12)  4.6  4320 
AE005674_GR  Shigella flexneri (serovar 2a, strain 301)  4.6  4395 
AP008232_GR  Sodalis glossinidius (strain morsitans)  4.2  2432 
BX470251_GR  Photorhabdus luminescens laumondii (strain TT01)  5.7  4897 
BX950851_GR  Erwinia carotovora (subsp. atroseptica, ATCC BAA672)  5.1  4491 
CP000653_GR  Enterobacter sp. (strain 638)  4.5  4115 
The genetogene relation S is provided by BLASTP by selecting pairs of gene products with p  value ≤ 1e  10, %identity ≥ 40 and with the alignment covering at least 80% of the smallest protein.
OTFQ was run with the following parameters: gaps of at most 3 genes (deltagap = 3), all gene permutations allowed (deltashuf = ∞), minimum synton size of 3 (mineltsize = 3) and a quorum q of at least 2 genomes. Spines are defined simply as connected components (CC) of S. Note that OTFQ allows other definitions of spines, as cliques or γquasicliques of S (for a spine made of n nodes, it means that for each node v we must have degree(v) ≥ γ_{*} (n  1)), those definitions are not used in this test.
For IADHO RE, we used the default parameters, except for "gap_size", "tandem_gap_size" et "cluster_gap" which were all set to 3 in order to be as consistent as possible with the deltagap parameter used in OTF. The "anchor_points" parameter, corresponding to the minimal number of genes in a cluster, was set to 3 in order to fit with the "mineltsize" parameter of OTF. Additionally, we set up the q_{ value } parameter very close to 0 and the prob_{ cutoff } parameter to 1, in order to disable any filtering and keep as many multiplicons as possible.
Notice that the chosen parameters are not optimal (both for IADHO RE and OTF) but were selected in order to make the comparison as consistent as possible. In particular, a quorum of 2 out of 10 genomes is usually too low for OTF since it potentially leads to an exponential growth of the solution size. However, we keep this value since IADHO RE naturally reports these multiplicons too.
Execution times
Comparison of execution times and results of IADHO RE and OTFQ
IADHO RE  OTFQ  

Set  execution time (s)  
bacteria  5  36  20  
10  122  55  
proteo  5  110  28  
10  428  197  
gamma  5  232  42  
10  825  373  
entero  5  632  273  
10  2881  NA  
number of genes found  size of ∩  
bacteria  5  460 (3%)^{1}  572  457 (99%)^{2} 
10  831 (3%)  919  756 (91%)  
proteo  5  2012 (11%)  2449  1991 (99%) 
10  4117 (11%)  4935  4088 (99%)  
gamma  5  4266 (21%)  4777  4246 (100%) 
10  8005 (22%)  9039  7977 (100%)  
entero  5  15376 (66%)  15792  15355 (100%) 
10  29306 (66%)  NA  NA 
We notice that for the more heterogeneous groups (bacteria, proteo, gamma), the OTFQ execution times are short, and indeed, shorter than for IADHO RE. Both are anyway shorter than the time needed to perform the initial allagainstall BLASTP comparisons needed to compute the S relation. However, this situation is inverted when working with close species and a larger number of genomes. For 10 enterobacteria, OTFQ runs out of memory (the limit was 2 Gb). Indeed a closer examination of running times shows that IADHO RE behaves roughly quadratically with the number of genomes (as expected if the computation time is dominated by the initial pairwise comparison and not by the following greedy search) whereas the OTFQ runtime grows exponentially with this number. However, Table 3 shows that for up to 10 genomes, the problem remains amenable to an exact algorithm and there is therefore no need to resort to heuristics. We should observe however that IADHO RE was specifically designed to deal with large eukaryotic genomes, whereas our definitions and algorithm are better suited to the alignment of bacterial genomes (one of the differences, for instance, is that we do not look for intragenomic syntenies).
Comparison of results
As mentioned before, in order to compare more easily the results of IADHO RE (multiplicons) and OTFQ (syntons), we choose to project them back onto the layered data graph (i.e. genomes) and, therefore, to compare sets of genes involved in multiplicons versus syntons. In this comparison, "gap" genes (i.e. genes not involved in any spine for OTFQ and non "anchor" genes for IADHO RE) are ignored. This is shown in Table 3. We first observe that the number of genes found in multiplicons and syntons are very close, they range from 3% of the total number of genes for distant species (bacteria group), and to up to 66% for very close species (entero group). Moreover, OTFQ constantly finds slightly more genes than IADHO RE (from 10 to 20% irrespective of the phylogenetic group) and, except for the 10bacteria group where the overlap is 91%, the set of genes found in multiplicons are almost totally included in the sets found in syntons (from 99 to 100%). A closer examination of these results shows that missed genes come from three main reasons, all of them being related to the linearity and collinearity constraint in IADHO RE. The first, and more important, reason comes from the definition of "gap" genes. In OTFQ "gaps" are simply genes that do not belong to a spine within the considered synton (they may have no homologous genes or they may be involved in a spine from another synton). In IADHO RE the definition is similar but multiplicons have an additional linearity constraint that may be broken when the number of "gaps" on one genome is not counterbalanced by an equivalent number of "gaps" on the other one. With 3 "gaps" for instance, a synteny can be missed when two genes are adjacent on one chromosome and their homologous genes are separated by more than 2 gaps on the other chromosome. This linearity condition makes the interpretation of the "gaps" parameter more delicate. Unfortunately, we did not find any IADHO RE parameter to change this behaviour but we believe this could be fixed easily. The second reason is due to the existence of gene order permutations and is therefore more fundamentally related to the IADHO RE algorithm, that enforces strict collinearity of genes. Finally, a third case marginally arises from an inversion of the orientation of one gene. This comes from the fact that IADHO RE considers gene orientation in its definition whereas OTFQ ignores it.
Synteny and gene evolutionary rates
As mentioned in the introduction, bacterial syntenies can provide useful information about evolutionary processes. For instance, Lemoine et al. have shown [36] that genes in synteny groups are subjected to stronger evolutionary pressure than genes outside of synteny groups. They started by identifying pairs of orthologous genes between two bacterial genomes. A given pair is then classified as a POG (Positional Ortholog Group) if it is strictly adjacent to (at least) another pair of orthologous genes. Finally, the PAM distances (as computed with the DARWIN package [37]) between the elements of a pair are computed for the POG and nonPOG groups and their distribution are compared.
Using this approach on several pairs of bacterial species, from closely related enterobacteria (E. coli and S. enterica) to more distant species (E. coli and B. subtilis), they showed that the mean d_{ PAM } in POG s is significantly lower than the mean d_{ PAM } in nonPOG, thus indicating that genes within synteny groups are generally more conserved than genes outside of synteny groups.
In this section, we attempt to generalise this result in two ways: first by working with more than two genomes, and second by investigating if the same evolutionary trends is observed as a function of the size of the synteny group.
Conclusions
In this paper, we presented an extension of the graph alignment algorithm proposed in [23] based on a clearcut definition of bacterial syntenies, as well as an exact algorithm to find them. The main purpose of this new extension is to allow for missing genes. More precisely, we now require genes to be present only on a quorum q ≤ n of the genomes under study. Together with a very flexible definition of block of synteny (nway genes aggregation in spines, presence of gap genes, partial or total conservation of the gene order), this resulted in a versatile tool to study syntenies in bacteria, that may be adapted to various kinds of studies. We presented two typical applications. The first one is related to the search of functional gene associations (for instance to the purpose of genome annotation). In this context, one should choose a quite loose genetogene relationship based on sequence similarity and relaxed synteny parameters. We compared our approach to the widely used heuristics IADHO RE. Execution times and results are very similar, showing that, at least up to ten genomes, the problem is still tractable with an exact definition and algorithm. The second application is related to evolutionary studies. In this context, it is better to work with a tighter definition of syntenies (orthology relation, clique association of genes, absence of gaps). We showed that the syntons retrieved by the algorithm presented an already documented feature: isolated pairs of orthologs are less conserved than the ones involved in larger blocks of synteny. However, we could extend this observation in two ways: first by working with multiple comparisons (i.e. by focusing on syntenies occurring on at least 3 genomes) and, second, by showing that the larger the block is, the more conserved the genes are.
Finally, we would like to stress that, although this paper and its illustrations focus on the question of syntenies, the definitions and the algorithm presented herein are applicable as well to the more general context of graphs alignments. For biological data, other possible applications could therefore concern the alignment of metabolic graphs, of PPI graphs or even of mixed data (e.g. metabolic versus genomic data [17]).
Availability
The algorithm has been implemented in Java, is platform independent and is distributed as opensource (GPL). Source code, user's documentation and samples files are available for download at: http://www.inrialpes.fr/helix/people/viari/lxgraph
Declarations
Acknowledgements and Funding
We would like to thank Anne Morgat from the Swiss Institute of Bioinformatics for helpful discussions about the glycogen and biotin operons. YPD, AV and MFS are supported by INRIA. FB is supported by INRA. This work was funded by the French project ANR MIRI BLAN081335497 and the ERC Advanced Grant SISYPHE.
Authors’ Affiliations
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