Several models describing the divergence of gene order in evolution have been proposed 18 32 ; in Table 1 we summarize them and we add three new models fortesting. The first one, (Hill in Table 1), is a sigmoid function with easily interpretable parameters: k, the activation coefficient, corresponds to the x value at which the function takes value 1/2 (half-maximum) and n determines the steepness of the shift from high to low levels (the degree of cooperativeness). We also derived two generalizations of the model used by Rocha (Eq.1 and 32 ): we relaxed the assumption about the partition of a genome into two equally populated groups of fast and slow rearranging gene pairs in the following way:

BS ̂ = f s p s x + ( 1 - f s ) p f x ,

In the table, a, b, k, n, f _f , f _i , f _{f f} , p _f and p _s are fitted parameters and x corresponds to the phylogenetic distance. Values are ordered by increasing AIC value. Models are fitted to the results obtained with the compressed networks.

with p _s ,p _f ,f _s ∈[0, 1]; f _s (1-f _s ) is the fraction of slowly (fast) rearranging gene pairs. The fitting performed with this formula returned a parameter p _s fixed at 1 by the algorithm; this allowed us to reduce the model (Eq. R o c h a+2p in Table 1):

BS ̂ = f i + ( 1 - f i ) p x ,

Following this model, a certain fraction f _i of edges is invariant, whereas the remaining are maintained with probability p. A further extension of the model considered the presence of a third category of very labile gene pairs, such that the probability of conservation is negligible (p _{f f} =0) at the time resolution of the model (Eq. R o c h a+3p in Table 1):

BS ̂ = f i + ( 1 - f i - f ff ) p x ,

Results of the non linear fitting are reported in Table 1. The R o c h a+3p model has the minimum AIC and best explains the data. The Akaike weights indicate that its probability of being the correct model is much higher than for the others. The first observation on estimated parameters is that p _f takes the same value in the two R o c h a models, around 0.24, and so does the f _f fraction, indicating that about 95% of the gene pairs change quite frequently.Following the R o c h a+3p model, a small fraction of edges (f _{f f} =0.02) changes very fast. These might involve transposases and other mobile elements. It should be noticed that the original R o c h a model gives p _f =0.17 and p _s =0.37, not too far from what we obtained here.

In Figure 4 we plot the backbone stability for all intra-species comparisons; genomes above the model predictions are more stable than expected: Sulcia muelleri, Buchnera aphidicola and Prochlorococcus marinus are the most evident cases. In the case of Sulcia, and despite the large phylogenetic distance dividing these genomes, the backbone is almost completely conserved. The age of the symbiosis between Sulcia and its host (260 Mya 37 ) might explain this high stability. Moreover, the fact that these strains have completely conserved gene orders suggests that they maintained an intact chromosome structure since the time of their separation. In agreement with available information, we also detected increased genome stability for the other endosymbiont of our dataset, B. aphidicola. However, the conservation of gene organization is not marked as in Sulcia and moreover we noticed that B. aphidicola Cc diverges with respect to the other strains (as indicated by its average stability value, B S _{C c} in Figure 4). This strain has peculiar features with respect to other Buchneras: its host (the aphid Cinara cedri) harbors two additional symbionts that are as abundant as Buchnera, suggesting the possibility of metabolic replacement 38 . It seems therefore plausible that the presence of other symbionts has relaxed the selection on some of the activities provided by Buchnera, promoting their loss 38 , and on some of the neighborhoods as suggested by our analysis.

Prochlorococcus marinus was identified as unstable in Rocha 32 on the basis of the comparisons of three P. marinus and five other Cyanobacteria. This discrepancy derives from the different phylogenetic ranges of the comparisons here and in 32 . By looking more in detail at the behavior of two additional Cyanobacteria present in our genus dataset (see Methods and Additional file 1), we found that the P. marinus genomes have indeed higher stability (Additional file 2). This reinforces the idea that by considering multiple genomes of the same species we can define the behavior of each group of organisms in a better way. Our analysis moreover shows that P. marinus is relatively unstable on the long-term, in agreement with 32 .

The intra-species analysis shows that all species have more or less stable gene organization. To identify aberrant genomes within a species, we obtained Z-scores for the residuals of each genome with respect to its conspecific and the model predictions (Figure 4C); genomes with the largest deviation from the mean are discussed below in the light of previous reports about genome organization. Despite the only marginal sequence divergence within the analyzed isolates, strain Angola appeared to be highly rearranged with respect to the remaining Y. pestis genomes. A further analysis indicated that this strain shares 92% of the gene pairs with Y. pseudotubercolosis IP_32953, which is almost the same as with the other Y. pestis genomes. The similarity in gene order among the other Y. pestis genomes is instead much higher (they share on average 99% of the gene pairs) as it is higher the similarity of these strains with Y. pseudotubercolosis IP_32953 (about 96% of the gene pairs are in common). This suggests that strain Angola experienced a period of intense reorganization after the separation from Y. pseudotubercolosis and independently from other Y. pestis strains, in agreement with the high degree of intrachromosomal rearrangements detected in a dedicated comparative analysis 39 .

The Bacillus anthracis CI genome was previously analyzed by a comparative genomics approach leading to the conclusion that it has evolved from a B. cereus strain and established a B. anthracis lifestyle 40 . This is also reflected in a markedly different gene organization with respect to the other genomes of the anthracis clade, as it appears from Figure 4C.

Sela et al. 41 found an abundance of mobile genetic elements in the genome of Bifidobacterium longum ATCC15697 relative to other sequenced bifidobacteria, a feature positively affecting rearrangement frequencies 42 . Acinetobacter baumannii is the source of numerous nosocomial infections in humans and is often multidrug resistant. Comparative genomics revealed that strain SDF is highly divergent from strains ADP1 and AYE and that it harbors over 400 insertion sequences, much more than other strains 43 . This, along with our stability analysis, suggests that this strain is undergoing an intense rearrangement of gene order, perhaps as a consequence of the adaptation to the human host or the new challenges imposed by drug treatment.

For the inter-species analysis, we selected one stable genome per species on the basis of the previous analysis and we compared them all against all. Genomes more stable than expected at such large phylogenetic distances were those of B. aphidicola, S. muelleri and Coxiella burnetii (not shown). The latter is a widespread bacterium causing Q fever in humans whereas it does not normally cause overt disease in its reservoirs (cattle, sheep, goat). Its genome stability seems to be in contrast with the phenotype in humans but it agrees with its obligate intracellular life-style. It was indeed shown to have rearranged genomes with large syntenic blocks 44 .

The presence of the two endosymbionts suggests that rearrangements played a minor role in their genome evolution, indicating that these endosymbionts diverged from their ancestors by eliminating superfluous genes and joining the remaining without much rearrangements.

Some of the genomes that were stable following the intra-species analysis showed here instability. We observed that 8/9 of the genomes with highest instability in these comparisons belong to animal pathogens, whereas this category represents about 62% of the genomes in our dataset; however, there is a significant association of instability with the taxonomic affiliation of the genomes, since 6 of these genomes come from the Firmicutes. We checked if Firmicutes tend to be less stable than other genomes and this is indeed the case (Figure 5, p=0.0022 in a Wilcoxon rank sum test), therefore the observed difference between pathogens and non-pathogens may be related to the Firmicutes in our dataset being mostly pathogens. To clarify this point, we focused on Proteobacteria by testing for equality of the median residuals of pathogens and non-pathogens; we obtained a weak but significant difference (Figure 5, p=0.042, when Buchnera is not included in the non-pathogens). Our analysis therefore indicates that pathogens tend to be less stable than non-pathogens and that they experienced past periods of rearrangements, plausibly during adaptation to their new life style. Since these genomes are not particularly unstable on the short evolutionary time, instability was transitory, highlighting the importance of considering multiple time-scales for these comparisons. It should be noticed however that these signals may be related to other taxonomic effects, as also noticed before 32 , and that can be avoided only with much larger and balanced datasets.

When chromosomes are compared following our strategy, the diameter can be another useful proxy for GOS, especially when the distances are short. In Figure 6, experimental data points are mostly located to the right of the simulations, suggesting that the majority of gene translocations involve distant loci. All experimental points located very far from the bulk of the data correspond to comparisons involving Buchnera. The genomes of this species show a very anomalous relationship of the diameter with stability, in-between a pattern of only deletions and local gene movements. Since the B. aphidicola genomes evolved mainly by deleting genes from an E. coli-like ancestral genome 45 , this analysis, together with the previous ones, strongly suggest that the process is still ongoing: different Buchnera strains are independently deleting genes as a consequence of the selective pressure experienced in specific hosts. It is an open question if they will stabilize on similar gene contents, or if they will show signatures of the different metabolic pressures experienced in different hosts, as suggested by the peculiarities of B. aphidicola Cc. The genomes of the other endosymbiont, Sulcia, have instead large and almost constant diameters, in agreement with the extreme stability of their genome backbone.

The relationship between backbone stability (Eq. 3, BS) and genome organization stability (Eq. 4, GOS) provides insights about the relative importance of gene order rearrangements and genome content dynamics for genome organization divergence: BS is affected only by rearrangements, while GOS is a combination of the two. The two stability measures are identical only when there are no accessory genes. At short phylogenetic distances, most of the neighborhoods are broken by genome content dynamics: (BS is very close to 1 while GOS falls from 1 to 0.4-0.5, data not shown), hence genome order rearrangements have a minor effect on genome organization divergence at these phylogenetic distances. Since the two variables are linearly related in intra-species comparisons (Additional file 3), we use the slope of this relationship as a measure of the importance of the two processes for genome organization evolution: small coefficients correspond to a larger contribution of gene content dynamics, whereas larger ones imply more rearrangements (see the inset in Figure 7). We built linear regression models for each species separately using the intra-species comparisons (Additional file 3). We show the sorted coefficients and their standard error in Figure 7. Most non-pathogens are grouped at the bottom of the plot, corresponding to larger coefficients. The probability of sampling 6 non-pathogens species in our dataset can be calculated, giving p=2.8E-04 (even by excluding Buchnera). The only pathogen within these species is Clostridium botulinum for which however there are two groups of points biasing the regression estimate (Additional file 3); even by including this species the result is still highly significant (p=0.001). This suggests that at short phylogenetic distances, non-pathogens have slower gene content dynamics than pathogens, with rearrangements playing a major role in genome organization evolution.

Since GOS is affected by rearrangements of core genes and by the number of accessory gene insertions/deletions whereas genomic fluidity (Eq. 6 and 33 ) is affected by the number of accessory genes, the relationship between the two indicates how strong is the effect of adding accessory genes on GOS. In other words, this relationship is informative on the size of accessory components: when N accessory genes integrate in the genome one by one, their effect on GOS is maximal because N integration events interrupt N core-core neighborhoods; the genomic fluidity is affected similarly. On the other extreme, i.e. when N accessory genes are inserted as a large gene cluster, the effect on GOS is small because only one core-core neighborhood is broken; the genomic fluidity is instead unchanged with respect to the previous example. The relationship between GOS (Eq. 4) and σ (Eq. 6) is therefore informative about the typical length of accessory gene clusters. The relationship between the log transformed variables is linear (Additional file 4), allowing an easy comparison of the behavior of each species and the whole dataset by focusing on the regression coefficients (β) of the model G O S=α+β σ. In particular, given e.g. β=x for the whole intra-species dataset, the species with β<x are characterized by larger accessory gene clusters whereas those with β>x by smaller accessory clusters. Our results (Figure 8) suggest that there are wide differences in the size of accessory gene clusters, with Chlamydophila pneumoniae, Mycobacterium tubercolosis and Yersinia pestis having the smaller accessory gene clusters, whereas Ralstonia solanacearum, Burkholderia pseudomallei, Shewanella baltica and Bifidobacterium longum have the largest ones. Several species in the plot (in gray), have only a few accessory genes, making impossible to get the right parameter estimates, comprising the endosymbionts of our dataset and Chlamydia trachomatis, a pathogen with an obligate intracellular life-style. The 10 species with the largest coefficient are significantly enriched in pathogen species (p=0.024) suggesting that pathogens tend to modify their genome content by gaining and removing blocks of genes that are on average smaller than for non-pathogens.

To explore this issue more in detail, we investigated the distribution of the size of accessory gene clusters present in only one of the genomes within a species (singleton gene clusters), which are enriched in horizontally transferred genes 46 47 48 . We found a linear distribution in double logarithmic plots (Additional file 5): most of the clusters are therefore small but the probability of large clusters is greater than for a normal distribution with the same mean. By analysing the average size of the singleton clusters we find that the non-pathogens tend to have larger singleton components (there are 7 non-pathogens out of 10 species, p=0.003). A Wilcoxon rank sum test supported a separation of pathogens and non-pathogens based on the average size of singleton gene clusters (p=0.046) and the significance increased when further considering free-living non-pathogens only (p=0.003, Figure 9). When considering singleton gene clusters of at least 2 genes, the significance of this difference vanishes; pathogens have therefore more frequently isolated singletons, suggesting that differences may exist in the preferred way to acquire foreign genes.

The analysis on stability reveals a few stable gene associations even at large phylogenetic distances. To go further into the problem, one may wonder which are the genes involved in such associations, their function and phylogenetic distribution. Here we summarize an additional output of our framework and we focus on the gene clusters present in Escherichia coli and in at least 50% of the genomes of the inter-species dataset. We set this threshold to highlight gene clusters whose maintenance responds to widespread selective pressures. We obtained 69 gene clusters of 2 to 22 genes; only 8 gene clusters have more than 4 genes. We use the fraction of edges of the gene cluster that are present in the genome as a conservation score (CS), which also provides an indication about partial occurrences (e.g. C S=0.5 means that half of the gene pairs of the gene cluster are also present in the genome). We plot the results in Figure 10: clusters with 4 or more genes are often only partially conserved in other species, with a trend of increasing score towards E. coli and its closest relatives. Most of the genes of the conserved clusters form larger operons in E. coli, leading to hypothesize they might represent the building blocks of larger and eventually lineage specific gene clusters and operons.

There are 7 and 4 gene clusters that are also present in the Archaea Sulfolobus islandicus and Methanococcus maripaludis, respectively; among those only one is common (gene cluster 23, comprising two genes involved in tryptophan biosynthesis). These gene clusters code for interacting proteins and have metabolic roles (e.g. enzymes for tryptophan, arginine, leucine and pyrimidine nucleotides biosynthesis, glycine cleavage for serine biosynthesis), in addition to the Phe-tRNA synthetase subunits α and β (gene cluster 25). The only gene cluster present in the Archaea and comprising more than two genes codes for the PhoU regulator and three subunits of a phosphate transporter (cluster 61, partially conserved in S. islandicus), underlining the importance of this limiting nutrient across the prokaryotic kingdoms. The 3 gene clusters with the wider phylogenetic distribution code for three subunits of cytochrome oxydase (cluster 50), the elongation factor G plus two ribosomal proteins (cluster 53) and two ribosomal proteins (cluster 38). The functions encoded are in this case more housekeeping but similarly to the previous case, all of them encode interacting proteins. These results lead to the testable hypothesis about protein interaction being the major driving force for gene clustering, facilitating the initial assembly of gene clusters that are then combined to form larger gene aggregates during evolution.

We classify orthologs with the BBH (“Bidirectional-Best-Hit”) criterion 54 by comparing all the proteins coming from a group of genomes at once. After an all-against-all blast, we build a similarity network where two proteins are connected if they are reciprocal best hits. We define a cluster in such a network as an orthologous group (OG). Since we require a one-to-one orthologous mapping for assigning unique gene neighborhoods to all genes in a genome, we systematically excluded all proteins belonging to an OG that contains multiple proteins from a same organism. When reconstructing the gene neighborhood graph, we skip the corresponding genes, i.e. the gene cluster A→B→C→D, in case B and C have been assigned to multiple orthologs group, becomes A→D. Removed genes correspond mainly to transposases present in multiple (almost) identical copies in most of the genomes under analysis (Additional file 6) and for which a one-to-one mapping is almost impossible without adding some information.

The gene neighborhood network of each genome is built using the information about protein coding gene coordinates. Each gene is connected to the following one in the genome table, with no threshold on their distance and taking into account the circularity of the chromosome. Genes corresponding to removed proteins (see Orthologous mapping) are deleted at this stage by joining together predecessor and successor nodes. Taking advantage of the orthologous mapping and the gene ordering information encoded in the genome table files, all gene neighborhoods are stored in compatible adjacency matrices (the genome specific neighborhood networks, GSN), i.e. proteins belonging to the same OG occupy the same place in matrices corresponding to different chromosomes. The GSN is encoded as un undirected graph. Once the GSN for all genomes have been built, they can be compared on a pairwise basis (see Figure 1 and Figure 2). We call GGN the network for a given comparison, which is obtained by taking the sum of the adjacency matrices of the two GSNs under analysis. By summing all the GSNs from a group it is possible to extract connected components that are present in a given fraction of the genomes i.e. evolutionary conserved gene clusters. To this purpose we use the Dulmage-Mendelsohn decomposition performed by the matlab function dmperm.

We ask for at least 4 genomes when selecting species for the short term analysis for two main reasons. First in such a way, we have some statistical power for intra-species comparisons allowing to identify deviant genomes with respect to the average species behavior. This is important if we want to identify genomes that changed their stability recently. Second, this within-species analysis allows to select the genomes for the comparisons between species. These genomes were chosen so that they have a stability which is in line with the other genomes belonging to the same species. It should be noticed that if genomes for the long term analysis are picked randomly within a species, we could end up using a biased genome as the prototypical species genome, affecting the subsequent analysis and interpretation of the results. Thus choosing genomes whose stability pattern corresponds to some sort of average of the species allows to obtain more accurate stability values for the species in the long-term analysis.