Recently a maximum likelihood method was employed to study gene insertions and deletions assuming constant rates across genes in a given genome [13]. However, the assumption of constant insertion/deletion rates among genes is unrealistic. For example, it has been shown that informational genes, such as those involved in transcription and translation, are less likely to be laterally transferred than are operational genes responsible for metabolic processes [14,15]. This observation forms the basis of the "complexity hypothesis". Unfortunately, causes of rate variation for insertions/deletions beyond the difference between informational genes and operational genes still remain unclear. A study of rate variation for gene insertions/deletions making use of the maximum likelihood method, therefore, becomes useful to address questions on rate variation for gene insertions/deletions.

Here, a Γ-distribution has been incorporated into a maximum likelihood estimation of gene insertion/deletion rates (Figure (Figure1).1). After incorporating rate variation for gene insertions/deletions among genes, the likelihood was improved significantly over a constant rate model using the same set of data from the Bacillus group as in [13]. The method was applied to 173 complete bacterial genomes in 25 phylogenetic groups; 20 groups showed significantly better fits to the data with rate variation for gene insertions/deletions. The remaining five groups did not show significantly better fits to the data with rate variation. Furthermore, the removal of informational genes from the likelihood estimation contributes little change in terms of the rate variation parameter α for gene insertions/deletions. This is the case despite informational genes having significantly lower rates of insertions/deletions than non-informational genes.

The likelihood of the models was improved significantly by incorporating a Γ-distribution in rate variation models compared with relevant constant rate models (χ² = 2ΔLnL 3.84 with d.f. = 1, see Table 1). Of the three models, the likelihood of the single-constant rate model was improved the most by incorporating rate variation. In the single rate model with a Γ-distribution, the MLE (maximum likelihood estimation) rate of insertions/deletions is 5.29, which is much greater than the rate 0.51 of the constant rate model. There is a high degree of rate variation for gene insertions/deletions, since the rate variation parameter α is 0.37; indicating that there is a small subset of genes with rapid gene turn-over.

The two-rate and three-rate models both assumed different rates on certain parts of the phylogeny. After incorporating rate variation, they both showed significantly better fits to the data than did the single rate model (Table 1), but the MLE rates are similar to those estimated from the constant rate models. In the two-rate model, the rates with rate variation in a Γ-distribution are 4.73, 0.37 versus 4.42, 0.35 with constant rates. Similarly, in the three-rate model, the rates with rate variation are 4.49, 0.32, 1.67 versus 3.92, 0.28, 1.23 with constant rates. Both models consistently support that recently transferred genes tend to have high rates of gene insertions/deletions as noted in [13]. Both cases showed lower levels of rate variation (greater α values) compared with the single-rate model (5.83 and 3.04 versus 0.37 of α values), even though, incorporating rate variation also improved the likelihood significantly.

To gain a clearer picture of rate variation of lateral gene transfer in the domain of bacteria, the study was expanded to 173 complete bacterial genomes in 25 phylogenetic groups (Tables 2 and 3). For each phylogenetic group, two phylogenies were constructed. One (the select-genes tree) is based on a group of selected genes, the other (the common-genes tree) is based on genes present among the relevant taxa as described in the Methods section. Of the 25 phylogenetic groups, 20 groups showed a fairly high level of rate variation for gene insertions/deletions among genes, and the confidence interval falls into a small range for each α value (Table 4). It is also striking that estimates using phylogenies on different set of genes or using different phylogenies are similar (Tables 2, 3, and 4). Five groups did not show a significant level of rate variation for gene insertions/deletions (α is ∞). The five groups are Candidatus, Ehrlichia, Lactobacillus, Mycoplasma, and Synechococcus. The lower boundary of each infinite α value was also estimated. The Ehrlichia group shows a very broad interval range for the "maximum" likelihood value and the lower boundary of α value is 0.39 (Table 4). The undistinguished difference between the rate variation model and the constant rate model in Ehrlichia might be due to the limited size of data (gene families) and/or accelerated evolution at the sequence level in this intracellular group. The relationship between the rate variation parameter α in a Γ distribution and the average branch length of each group was examined. Figure Figure33 shows that there is a positive correlation between the rate variation parameter α and the average branch length of each group. Closely related groups tend to have a higher degree of rate variation for gene insertions/deletions among genes, while distantly related groups tend to have a lower level of rate variation for gene insertions/deletions. This suggests that the observation of rate variation is a strong local phenomenon and becomes blurry over evolutionary time.

One way to achieve a more accurate phylogeny is to make use of a large number of genes in comprehensive phylogenetic studies, such as supermatrics (concatenated genes) and supertrees [26-30]. The more data included in a phylogenomic analysis, the more likely to overcome possible stochastic errors [27,31]. In this study, the common-genes tree was not always favored by the supertree over the select-genes trees. Indeed, there are two groups that the select-genes tree is supported by the supertree (Additional file 4). Slow evolving genes are sometimes more informative for phylogeny construction, since fast evolving genes might cause problems such as long branch attraction [28,32]. The tree length of each gene was computed from each phylogeny and plotted in (Additional file 5). It is clear that the selected genes for phylogeny construction have relatively slow evolutionary rates compared with all common genes. There are 4 groups whose supertree does not support the select-genes tree or the common-genes tree. The lack of congruence in these groups is likely due to insufficient taxon sampling. [27]. More accurate trees might be obtained as more complete genome sequences become available. Importantly, the maximum likelihood estimates based on the common-genes trees are remarkably similar with the estimates based on the select-genes trees (Tables 2, 3, 4, 5 and Figures Figures3,3, ,4).4). For the 4 groups that do not have an identical topology between the select-genes tree and the common-genes tree, the results reported are based on the supertree topology (branch lengths were estimated from the selected genes). The results are again similar to those based on the select gene tree and those based on the common-genes tree (data not shown).

To further explore how much, if at all, different phylogenies might alter the results, maximum likelihood estimation based on possible alternative topologies was investigated. One hundred bootstraps from the alignment of the common genes were generated for each group. For the select-genes, possible alternative topologies were obtained from the MRBAYES output. If there are more than 10 distinct topologies, the top 10 ones according to their likelihood were chosen for further maximum likelihood estimation. The maximum likelihood estimates are shown in Additional files 6 and 7. It is clear that the α values are similar among different phylogenies. The removal of informational genes results in little change on the rate variation parameter α, and this holds true for each phylogeny. Furthermore, the likelihood estimations in the Bacillus group based on a phylogeny constructed from different genes (Tables 2 and 3) are similar with those based on the phylogeny of the previous study (Tables 1). The slight difference is due to the removal of short sequences in this study and differences in the phylogenies constructed from different sequences. The results do not, therefore, seem to be an artifact of the genes included or the phylogeny reconstruction. Informational genes are known to be less likely to undergo lateral gene transfer [14], which is also the core of the "complexity hypothesis" [15,33]. In this study, informational genes were found to have lower rates of gene insertions/deletions compared with non-informational genes (Additional file 1). However, no group has an insertion/deletion rate equal to 0, suggesting that informational genes are not completely free of gene movement. In fact, several ribosomal protein coding genes are deleted from Streptococcus mutans (Additional file 8). The rate variation parameter, α, change after excluding informational genes is similar to the α change after randomly removing genes rather than the α change after excluding the most conserved genes (Additional file 9). Furthermore, different cutoffs used in identifying informational genes only resulted in variation of the number of informational genes but did not affect the degree of rate variation for insertions/deletions in non-informational genes (Additional file 10). There is a great deal of rate variation for gene insertions/deletions in non-informational genes and also there is a significant level of rate variation in informational genes. In other words, the different rates between informational genes and non-informational genes as shown in the "complexity hypothesis" can only explain a small part of rate variation for gene insertions/deletions. Similarly, our simulation study showed that the high level of rate variation can not be explained solely by the fast turn-over rates of recently transferred genes (Additional file 11).

It has been suggested that different cutoff thresholds for identifying homologues might affect the identification of some gene gains [34], but different thresholds result in little change on the number of gene families [35] and the rates of gene insertions/deletions [36]. In this study, results using different thresholds were similar (data not shown). It is important to note that gene duplication was not taken into consideration in this study, since our focus was on insertions/deletions of gene families rather than intraspecific gene family duplication. This avoids the difficulty of distinguishing some gene transfer from gene duplication [4,11]. Recently, some studies have suggested that duplicated genes or genes that have a high duplicability propensity might be more likely to be involved in lateral gene transfer [37,38]. Methods incorporating gene duplication information are desirable for future studies.

For the 20 groups that showed a significant improvement in likelihood by adding rate variation, there is a positive association between the rate variation parameter α and the average branch length. Higher degrees of rate variation for gene insertions/deletion are expected to be observed in closely related groups. The seven closely related Bacillus genomes in the Bc group were analyzed separately and, as expected, a high degree of rate variation was observed (data not shown). Similarly, the five groups that have an infinite α value show fairly high levels of divergence within the group in terms of the average branch length. An acceleration of sequence evolution in the endosymbiont genomes has been acknowledged [39]. The accelerated rates of evolution might affect the branch lengths of the phylogeny used in the analyses and might also affect the identification of homologues within each phylogenetic group. Four endosymbiont groups showed strong accelerated rates of evolution. They are Candidatus, Ehrichia, Mycoplasma, and Richettsia. Analyses after the removal of these four groups also showed similar results (data not shown).

There are two possible explanations for the correlation between the average branch length and the α value. First, it is possible that the observed correlation is due to the lack of power of maximum likelihood estimation in distantly related groups. Previously, it has been shown that comparison among distantly related species tends to infer lower rates of insertions/deletions [13]. On the other hand, if maximum likelihood estimation in the study has enough power in distantly related groups, the results might suggest that rate variation for insertions/deletions has a strong local effect and becomes weaker as evolution in progress. This strong local effect might be, at least partially, due to a high variability in recently transferred genes. It is known that many of recently transferred genes are under faster rates of evolution and might be eliminated from the genome rapidly [8,13], and while some transferred genes that play roles in long term adaptation might become fixed [40-43] and integrated into the functional network [44,45]. Maximum likelihood estimations from simulated data showed support for these explanations. When the number of insertions/deletions increased, a larger proportion of insertions/deletions became undetectable, at the same time, sister taxa shared less common genes and have more unique genes (Additional file 11). If this correlation holds true, after a long enough time period, one should expect rate variation becomes undetectable. Hence, over a long time period, genes would have roughly the same chance to be transferred. This has been shown in some recent studies. By examining the Cyanobacteria group, Zhaxybayeva et al. reported that genes from all functional categories are subject to gene transfer [46]. In addition, it was suggested that among all sequenced gene families, at least two-thirds and probably all, have been affected by LGT at some time in their evolutionary past [35].

Since the genes absent in all of the taxa are unobservable, the results must be corrected for missing data. Hao and Golding (2006) used the correction for missing data as was used for missing restriction sites in [58], and the results are then made conditional on observing the gene present in at least one species. This is

Here L_- is the likelihood of a gene being absent in all taxa while L₊ is the likelihood of the gene present in at least one genome from the observed data. After incorporating a discrete Γ model, the likelihood of observing the pattern of gene family i will be

At the root of the tree we can compute the overall likelihood as

Here N is the total number of gene families. To estimate the maximum likelihood, the ins/del rates together with the rate variation parameter α in a Γ distribution were optimized to find those rates/values that maximized the likelihood of observing the gene patterns.

The same set of data from the Bacillus group in [13] were used to test the performance of the likelihood model with rate variation. As was done in the previous study, three models were examined (Figure (Figure2)2) in an attempt to capture some of the major differences among the genomes. In brief, the rate on the branches of the Bc group (including Ba, Bc, and Bt) is μ₁, the rate on the branch leading to the Bc group is distinguished as μ₂, the rate on the remaining branches is μ₃. Three models were examined; one with all branches evolving at the same rate (μ₁ = μ₂ = μ₃), one with just the Bc group evolving at a separate rate (μ₁, μ₂ = μ₃) and one with each of the three groups of branches evolving at separate rates (μ₁, μ₂, μ₃). The likelihood of each model incorporating rate variation was compared with that of the relevant constant rate model. In each case, the likelihood of the constant rate model was improved significantly by incorporating a Γ-distribution for insertion/deletion rates (Table 1).

To apply the improved maximum likelihood estimation to a broad spectrum in the bacterial domain, 173 complete bacterial genomes in 25 phylogenetic groups (including Bacillus) were examined. Genomes were selected to be within the same group based on the same genus name in the NCBI taxonomy database and whenever at least four genomes from the same genus were completely sequenced. Following previous studies, Oceanobacillus iheyensis and Geobacillus kaustophilus were included in the Bacillus group [13] and Ureaplasma urealyticum was included in the Mycoplasma group [8]. Since some highly diverged Synechococcus species are closely related to Prochlorococcus species [59], the group of Synechococcus in this study only includes Synechococcus sp. strains. Genome sequences were obtained from the NCBI database [60]. Sixteen non-ribosomal protein coding genes from commonly present genes [21,22] were chosen for phylogeny construction, and they are argS, gcp, gltX, hisS, infB, ksgA, lysS, metG, nusA, nusG, pheS, proS, rpoA, secY, serS, and ychF. In each group, any duplicated genes of these 16 genes were excluded from phylogeny construction for that group. The phylogeny of each group was constructed from the concatenated DNA sequences of these genes using MRBAYES [61] (200,000 generations sampled every 100 generations with a Γ distribution model and invariant class). For convenience, this tree is called the select-genes tree. The species information of each group together with outgroup information, genes used for phylogeny construction of each group, and the best supported phylogenetic tree are given as supplementary information at [25]. If more than one possible phylogeny was generated for a group, all possible phylogenies were used for further analysis, weighed by their posterior probabilities.

The robustness of these phylogenies was further assessed by concatenating all common genes from each group (labelled the common-genes tree to distinguish it from the select-genes tree). As for selected genes, common genes that have paralogs were excluded from the analysis, to avoid the confounding effects of duplication. The number of genes (and characters) from each group are given in Additional file 4. Sequence alignment was performed individually for each gene using MUSCLE [62]. Aligned sequences were concatenated for phylogenetic analysis. Since MRBAYES [61] has a limitation for the maximum number of characters, DNAML in the PHYLIP package was used instead and the rate variation parameter alpha was estimated using the PUZZLE program [63].

A supertree method was then employed for the groups in which the select-genes tree and common-genes tree are not topologically identical. Genes present in at least 4 taxa were used for phylogeny construction. A supertree was computed by assuming equal weight on all phylogenies using the CLANN program [64]. When the supertree does not support either the select-genes tree or the common-genes tree, the supertree topology is additionally as supplementary information at [25]. Please note that reconstructed supertrees themselves do not have branch length information. When needed, branch length information was estimated from the selected genes by forcing a supertree topology.

Average branch length was used as an indicator for the degree of the divergence in that group. The method to identify members of a gene family has been described in [8]. This study focuses on the presence/absence pattern of each gene family rather than individual gene; thus, varied number of genes (e.g. duplicated genes) in a gene family within the group of organisms would not be taken into consideration in the analysis. Non-annotated genes were recovered from the whole genome DNA sequences using a TBLASTN search [65] with annotated genes as query sequences, and predicted ORFs that are present in only one genome but do not have homologues detected in any other complete genomes by BLAST were removed from further analysis. In addition, genes encoding proteins that are less than 100 amino acids in length were removed from further analysis in this study, since a similarity search using BLAST has less power to detect homologues in short sequences [65].