Site-specific time heterogeneity of the substitution process and its impact on phylogenetic inference
© Roure and Philippe; licensee BioMed Central Ltd. 2011
Received: 15 September 2010
Accepted: 14 January 2011
Published: 14 January 2011
Model violations constitute the major limitation in inferring accurate phylogenies. Characterizing properties of the data that are not being correctly handled by current models is therefore of prime importance. One of the properties of protein evolution is the variation of the relative rate of substitutions across sites and over time, the latter is the phenomenon called heterotachy. Its effect on phylogenetic inference has recently obtained considerable attention, which led to the development of new models of sequence evolution. However, thus far focus has been on the quantitative heterogeneity of the evolutionary process, thereby overlooking more qualitative variations.
We studied the importance of variation of the site-specific amino-acid substitution process over time and its possible impact on phylogenetic inference. We used the CAT model to define an infinite mixture of substitution processes characterized by equilibrium frequencies over the twenty amino acids, a useful proxy for qualitatively estimating the evolutionary process. Using two large datasets, we show that qualitative changes in site-specific substitution properties over time occurred significantly. To test whether this unaccounted qualitative variation can lead to an erroneous phylogenetic tree, we analyzed a concatenation of mitochondrial proteins in which Cnidaria and Porifera were erroneously grouped. The progressive removal of the sites with the most heterogeneous CAT profiles across clades led to the recovery of the monophyly of Eumetazoa (Cnidaria+Bilateria), suggesting that this heterogeneity can negatively influence phylogenetic inference.
The time-heterogeneity of the amino-acid replacement process is therefore an important evolutionary aspect that should be incorporated in future models of sequence change.
With the expansion of genome projects, phylogenomics - the use of numerous genes to infer phylogenetic trees - is becoming a common way to resolve controversial relationships (e.g. [1–5]). Since large datasets increase the amount of phylogenetic signal included in the analysis, phylogenomics is less subject to stochastic errors than single gene phylogenies. Nevertheless, some nodes remain unresolved even at the genome-scale level [6–9]. This can either be due to intrinsic properties of the data, (i.e., short internal branches due to speciation events closely spaced in time) or to inadequate inference methods . In fact, systematic errors may be more pronounced in phylogenomics: in some cases, the gain in phylogenetic signal is masked by an increased level of systematic error, which can attain the same order of magnitude . In the worst case, this leads to erroneous phylogenies with a high statistical support [11, 12]. Molecular sequence evolution exhibits a high complexity that is not fully accounted for in current models of sequence evolution. Since the first substitution model  several simplifying assumptions have been relaxed; the evolutionary process is considered as heterogeneous (i) between character states, (ii) over time and (iii) along the alignment. Models that relaxed these assumptions (e.g. empirical exchangeability matrices , non-stationary nucleotide content , or gamma distribution of rates across sites ) improve fit to the data and phylogenetic accuracy, and are therefore widely used.
More recently, probabilistic models have been developed to take into account heterogeneity of the qualitative aspect of amino-acid replacements along the alignment, by assigning sites to different classes of substitutional processes [17–25]. Of particular interest is the CAT model , a mixture model that infers categories from the data without any a priori biological assumptions and takes advantage of the Dirichlet process prior  to control the number of categories through a set of hyperparameters (i.e. an infinite mixture model). In this model, the substitution process is assumed to be site-independent and is entirely defined by the equilibrium frequencies of amino acids, while their exchangeabilities are assumed equal (i.e. Poisson process). The equilibrium frequencies over the twenty amino acids constitute a good proxy to represent the functional constraints acting on each position during evolution. In the following, we will call such categories substitution profiles, or simply profiles. The number of profiles is often several hundreds, showing that previous models lack flexibility in handling heterogeneity of the substitution process across sites and, for large datasets, the CAT model has a better fit to the data than standard models based on substitution matrices (e.g. JTT, WAG or GTR) [5, 24, 27–30] and renders phylogenetic inference less sensitive to long branch attraction artefact [5, 10, 31–34]. Despite that the inferred number of categories is large, it has been shown, using a posterior predictive approach, that the number of categories estimated by the CAT model is conservative .
The models that assume different evolutionary processes across sites consider that they are homogeneous over time. However, except for a few constant positions for which the evolutionary constraints are uniformly strong, the functional constraints of most positions are likely to have changed over their evolutionary history [35, 36]. Indeed, some amino acids are replaced at sites without a significant effect on function, but these changes might modify the environment of the protein, the intra- or inter-protein interactions, and so on, leading to changes of the selective pressure at other sites . This results in variation of the site-specific evolutionary rate across time [35, 36], a phenomenon called heterotachy , which is recurrent in biological sequences [38–40]. To handle this heterogeneity, models have been proposed that allow the substitution rate to vary over time [41–46].
In this study, we will extend to the principle of heterotachy to the heterogeneity of the substitution process over time. We call homopecilly (ποικιλλω, pecilly, means to vary in Greek) the hypothesis of an identical substitution process over time at a given site. Since functional constraints acting on proteins change over time, we however expect that not only the rate but also the amino acid substitution process may vary. In particular, the subset of acceptable amino acids or the exchangeability matrix at a given site may change throughout evolutionary time. We will therefore test the hypothesis of homopecilly by evaluating whether the nature of the substitution process varies significantly over time at a given site. Briefly, the substitution process will be characterized by the set of stationary frequencies of amino acids, as estimated by the CAT model . Several large datasets will be divided into monophyletic taxa to test the null hypothesis of a homogeneous substitution process, that is a site should be affiliated to the same CAT category (i.e. a set of stationary frequencies) in all predefined monophyletic groups. We demonstrate not only that this null hypothesis is significantly rejected, but also that heteropecilly might generate phylogenetic artifacts.
Results and Discussion
Evidence for a significant qualitative heterogeneity over time
Altogether, both the FDP that considers all stably affiliated sites in a pairwise comparison and the PIPn that considers every site separately, while comparing all taxonomic groups simultaneously, demonstrate that qualitative time-heterogeneity of the substitution process-which we refer to as heteropecilly-is widespread in real data. Heteropecilly might be due to the comparison of paralogous genes, with different functions. However, the genes encoded in the mitochondrial genome of animals have an extremely high probability of being truly orthologous, since no protein encoding gene duplication is known in these organelles. Similarly, the orthology was carefully checked in the nuclear supermatrix using the protocol described in ; in fact, the proteins considered, mainly ribosomal proteins, are not subject to frequent gene duplications in animals and more importantly the duplicates do not change its function. As a result, it is unlikely that the observed heteropecilly is due to the comparison of paralogous genes.
Relationship between heteropecilly and other biological properties
Distribution of sites according to the substitution number and -ln(PIPn) for the nuc80 dataset
PIPn = 0
Distribution of sites according to the substitution number and -ln(PIPn) for the mt336 dataset
PIPn = 0
Distribution of sites according to the p-value of the heterotachy test and -ln(PIPn) for the nuc80 dataset
PIPn = 0
Distribution of sites according to the p-value of the heterotachy test and -ln(PIPn) for the mt336 dataset
PIPn = 0
Distribution of sites according to the standard deviation of the PHS score and -ln(PIPn) for the nuc80 dataset
PIPn = 0
Distribution of sites according to the standard deviation of the PHS score and -ln(PIPn) for the mt336 dataset
PIPn = 0
Finally, we estimated whether changes between profiles showing similar physico-chemical properties (distributed among these five groups: small, aliphatic, aromatic, charged, other; see Materials and Methods) are more frequent than changes between profiles with different properties. Only sites that are stably affiliated in two different clades are considered. About half of sites (44% and 48%, for mt336 and nuc80, respectively) are affiliated to profiles with different biochemical properties. This suggests that heteropecilly is driven not only by different fine-grained functional constraints, but also by important functional changes. Analyses at the codon level  are nevertheless required to estimate whether heteropecilly is driven by positive selection or is the result of changes in purifying selection properties.
Phylogenetic structure of heteropecilly
With the mt336 dataset, most of the known clades are recovered with the GTR+Γ4 model (Figure 3), many of them with high posterior probabilities (PP), e.g. Bilateria, Pancrustacea, Vertebrata, Amphibia, Sauria, Archosauria, Lepidosauria, Theria and Eutheria. The only problematic case is the relative position of Amphibia, Mammalia and Sauria. The mono/paraphyly of Deuterostomia is known to be a difficult question, even using a hundred nuclear genes . Although we don't know which statistical model has to be used for analyzing the artificial recoded sequences, recovering at least some clades is reassuring. To test the possibility of a correct grouping of clades by chance, we applied the recoding protocol to the ten datasets simulated under a homopecillous model (sites should therefore be affiliated to the same profile whatever the clade considered, except because of stochastic error which should not be phylogenetically structured): except in one case, the expected monophyletic groups are not recovered or only recovered with very small posterior probabilities (Additional file 2: Table S7). Similar results were obtained for the nuc80 dataset (data not shown), but are less clear because only five clades are available.
The congruence of the phylogeny inferred from recoded data with current taxonomy demonstrates the presence of a strong phylogenetic signal in changes of the evolutionary process, especially since only few parsimony informative positions are available (353 and 120 for the recoded mitochondrial and nuclear datasets, respectively). Changes in the evolutionary process are therefore relatively rare since sites remain affiliated to the same profile over hundred millions of years. The same observation has been made for evolutionary rates, leading to local molecular clock  or auto-correlated relaxed molecular clock models . Nevertheless, the incongruence observed in a few cases indicates that homoplasy is present in recoded data and is not correctly handled by the CAT+Γ model; not surprisingly, when a profile affiliation changes, it can either revert to the ancestral state or converge toward a state independently acquired in a distantly related clade. The misleading effect of homoplasy is enhanced by the very heterogeneous rate of change of profile affiliations: for the mitochondrial dataset, the rate is high on the branch leading to Bilateria (Figure 3) and for the nuc80 dataset, nematodes and platyhelminthes evolved several times faster than the other bilaterians (data not shown).
Since the amino acid composition is influenced by the genomic nucleotide composition , heteropecilly could be the result of the heterogeneity of the mutation process over time, i.e. the frequency of some profiles will increase in a clade because their most frequent amino acids are becoming more frequent due to changes in the nucleotide bias. This hypothesis predicts that profiles will be heterogeneously distributed across clades, especially for the fast evolving positions, which are most likely to reflect mutational bias. The profiles are indeed heterogeneously distributed across clades for both mitochondrial and nuclear datasets (Figure 4 and Additional file 1: Figure S10, Additional file 2: Table S8) but, when the fastest evolving positions are considered, this remains highly significant for the mitochondrial alignment only. For this alignment, when one compares an equal number of the most and the least heteropecillous positions, the p-value is slightly lower for the former than for the latter, even if the profiles are significantly heterogeneously distributed across clades. This suggests that changes in the mutational process over time, albeit particularly marked in the mitochondrial genome, are likely a minor cause of heteropecilly.
In summary, the time heterogeneity of the amino acid evolutionary process (heteropecilly) is a general phenomenon in animal evolution, present in both nuclear and mitochondrial coding genomes. Although the rate of change of the substitution process is sufficiently low to allow the recovery of a phylogenetic signal in the recoded sequences, homoplasy is present as suggested by the different rate of evolution across the tree (Figure 3), and by the heterogeneity of profile frequencies across clades (Figure 4 and S10). We do not recommend using the recoding protocol to avoid model violation in cases where heteropecilly is suspected; even if some phylogenetic signal can be captured in the recoded sequences, the signal is probably too weak to obtain an accurate phylogeny.
Heteropecilly may generate phylogenetic reconstruction artefacts
The observations presented so far demonstrate that the assumption of homopecilly, i.e. no change of the substitution process over time, is violated, potentially leading to tree reconstruction artifacts. As a case study, we chose the relationships between Porifera, Cnidaria and Bilateria (Protostomia+Deuterostomia), which are difficult to resolve with mitochondrial genomes . Since slow-evolving Porifera and Cnidaria are grouped together , the monophyly of Eumetazoa (Cnidaria+Bilateria), long proposed by morphologists and recovered with nuclear genes , is not observed for the mitochondrial data. We make the assumption that this is due to a long-branch attraction (LBA) artifact between the fast-evolving Bilateria and the distant outgroup (Choanoflagellata). This could be aggravated when using the CAT model by the very long branch observed at the base of Bilateria in recoded sequences (Figure 3), which indicates a large amount of profile affiliation changes, and hence a serious violation of one hypothesis of the CAT model (i.e. the time-homogeneity of the evolutionary process).
Two controls were performed. First, since a correlation exists between evolutionary rate and heteropecilly (Tables 1 and 2), improvements in phylogenetic inference could be due to the removal of fast evolving sites [12, 53]. When fast evolving sites are progressively removed according to the SF method , support for the incorrect grouping of Cnidaria and Porifera slightly increases (Additional file 1: Figure S11). This is in sharp contrast with the removal of heteropecillous positions (Figure 5). Second, since the negative effect of heteropecilly may constitute a model violation more important for the CAT+Γ4 model than for site-homogeneous models (the site-specific variation of stationary frequencies over time should have less effect -i.e. averaged- when the same stationary frequencies are used for all the sites), the GTR+Γ4 or mtREV+Γ4 models are also used. Results are completely different: the removal of heteropecillous positions does not affect phylogenetic inferences conducted with GTR+Γ4 or mtREV+Γ4 models, for which high support for Cnidaria + Porifera is always observed (Additional file 2: Table S9).
This result is not unexpected because, while heteropecilly constitutes a model violation for the CAT model, its effect on the site-homogeneous GTR model is less clear. To understand the different behavior of these two models, one has to distinguish two model violations, heterogeneity of the substitution process across sites and over time. The first violation is known to seriously exacerbate LBA artifacts, because the amount of homoplasy is underestimated . It is not expected to decrease with the removal of heteropecillous positions and probably dominates when using the site-homogeneous GTR and mtREV models. The effect of the second model violation can only be observed with the CAT model, which handles heterogeneity across sites. Removing heteropecillous positions will reduce the time-homogeneous violation and increase the accuracy of the CAT model. This hypothesis is corroborated by an evaluation of model fit by cross-validation (Additional file 2: Table S10). The GTR+Γ4 model fits the complete mitochondrial mt68 dataset better than the CAT+Γ4 (GTR vs. CAT: 70.7 ± 57.7). This is probably due to the large number of parameters of the latter model and the limited amount of positions, since, with the larger nuclear dataset, the CAT+Γ4 model has a better fit than the two models based on exchange matrices (GTR vs. CAT: -1610.3 ± 139.2, WAG vs. CAT: -2615.7 ± 94.4). Importantly, after removal of most heteropecillous sites, the CAT+Γ4 model appears to best fit the data (GTR vs. CAT: -45.2 ± 47.0). This indicates that heteropecilly constitutes a serious violation of the CAT model, because the best fit is obtained despite the limited number of sites (1240).
Numerous heterogeneities of the evolutionary process have been discovered. Most have a clear negative impact on phylogenetic inference when not adequately handled: heterogeneity of rate across lineages , of substitution type , of rate across sites , of composition across taxa  and of substitution process across sites [23, 24]. Some of them, such as heterotachy, seem to have a more limited effect on phylogenetic accuracy . Further studies are needed to know in which category heteropecilly has to be classified. An important prerequisite is the development of models that handle the fact that substitution properties change over time. This could be achieved via a Markov modulated CAT model, similar to the covarion model , or via the use of breakpoints . To choose between these two approaches, it would be important to estimate whether sites generally change their properties in a collective manner or not, since only the second approach can model this aspect. The costs of these improved models (number of parameters, computational time) could be major and would be useful for phylogenetic inference only if heteropecilly turned out to seriously impair accuracy. In any case such improved models would be helpful to advance our knowledge of protein evolution, since in most cases one can select a set of species for whose relationships are confidently known, which drastically simplifies the problem and allows the use of complex but computationally demanding models.
What are the main reasons for these shifts in profile affiliation? The correlation between the PIPn criterion and the substitution number might suggest a neutral explanation (e.g. change in mutation pressure) of the variation in substitutional profile over time. Since the PIPn criterion is also correlated with a change in hydropathy, this neutral explanation could be insufficient. These changes in the site-specific substitution process could be related to functional shifts, such as adaptation of organisms to new environments (e.g. higher growth temperature) or of the protein to a new cellular environment (e.g. new interactome). To answer these questions, it would be particularly relevant to study paralogous genes in which functional shifts are well characterized.
Three large datasets have been used: (i) 13 proteins encoded in mitochondria from 336 metazoan species divided into 15 clades (Additional file 2: Table S1), named mt336 dataset, (ii) 111 nucleus encoded proteins from 80 metazoan species divided into 5 clades (Additional file 2: Table S2), named nuc80 dataset, and (iii) 13 mitochondrion encoded proteins from 68 species (66 Metazoa and 2 Choanoflagellata) grouped into 5 clades (Additional file 2: Table S3), named mt68 dataset. All sequences have been downloaded from the GenBank database. For each dataset, proteins have been aligned by ClustalW , manually refined using ED , and then concatenated into a super-matrix using SCaFoS . Orthology of nuclear proteins was verified using the congruence approach described in . Ambiguously aligned positions have been removed using Gblocks , with some manual refinements (necessary because of an inadequate stringency of Gblocks in the presence of missing data). Since we are not interested in constant or quasi-constant positions, which obviously have the same evolutionary properties in all clades, only the parsimony informative positions have been retained, resulting in 1,851, 12,608 and 1,927 positions (from originally 2,547, 22,082 and 2,382 unambiguously aligned positions) for mt336, nuc80 and mt68 datasets, respectively, which allows us to reduce computational costs. The species were selected in order to obtain the most homogeneous taxonomic diversity, that is, monophyletic groups of a similar size (i.e. similar tree length) represented by about twenty species. Two groups have more species (Actinopterygii and Primates) with respect to the tree length criterion.
Topologies were inferred by maximum likelihood separately for each monophyletic group under a WAG  or a mtREV  model with four gamma discrete categories using Treefinder , for the nuc80 and the mitochondrial datasets respectively. As these topologies are biologically reasonable (see Additional file 2: Table S4), all subsequent analyses were performed under these fixed topologies in order to reduce the CPU burden. We verified, in the case of the mt336 dataset, that the same results were obtained under free topology (data not shown).
A scheme of the protocol, described only for two clades for clarity, is shown in the Additional file 1: figure S4. For each clade, the CAT model  implemented in the program Phylobayes inferred substitution profiles. However, comparing the profile affiliation across groups is not straightforward since (1) the CAT model infers profiles independently from each clade resulting in different sets of profiles, (2) the number and nature of profiles varies during the MCMC, and (3) different profiles can be affiliated to a given site during the Monte Carlo Markov Chain (MCMC). To achieve our comparison between clades, we need identical profiles in the different runs, and therefore need to define a set of common profiles to which sites can be affiliated whatever the clade. In a first step, the CAT model freely inferred the profiles separately for each clade under a fixed topology; the phylobayes program performed a total of 10,000 cycles, the 1,000 first cycles being discarded as "burn-in". Profiles and their affiliation to positions are repeatedly updated during the MCMC, so different profiles, which are themselves potentially unstable, can be assigned to a same position; we focus on profiles that are the most stable. Stable profiles were identified according to the protocol described in  with a threshold value of 0.035 for quadratic distance and a threshold value of 4 for the minimum of profile affiliation to site number. Only profiles that are present >50% of draws from the posterior were retained. Among the stable profiles identified in various clades, some are generally highly similar and need to be further clustered to avoid redundancy. For each pair of profiles, the quadratic distance over the twenty amino acid frequencies was calculated to compute a distance matrix as an input for clustering using UPGMA as implemented in NEIGHBOR . A threshold on the quadratic distance was chosen in order to obtain about 25 clusters (Additional file 1: Figure S1-3), a number of profiles known to provide the greatest step in model fit improvement . Within each cluster, a common profile was defined as the average over the twenty equilibrium frequencies weighted by the affiliation frequency of each initial profile included in the cluster. Twenty-six, twenty-four and twenty-four common profiles were obtained for the large, the small mitochondria encoded proteins and the nuc80 datasets, respectively.
To compare the profile affiliation in different monophyletic groups, phylobayes was re-run with the set of common profiles for each clade separately (CAT model, fixed topology, 1,100 cycles, removing of 100 first cycles). Under these conditions, only the profile affiliations, branch lengths and site-specific rates were free parameters. This allowed to compute pik(c), the posterior probability of affiliation of the profile k to site i for clade c. An affiliation was considered stable if k exists such that pik(c)>0.75.
where ndif is the number of positions with two different profiles in the two clades, and nid is the number of positions with two identical profiles. For a threshold of 75% of stability across the MCMC, only 28% and 11% of sites are on average stably affiliated, for the nuc80 dataset and the mt336 dataset respectively. This statistic cannot be extended for comparing all clades simultaneously, since only 1% of the sites are always stably affiliated for the 15 clades of the later dataset. Moreover, the FDP criterion does not give information at the site level.
where K is the number of profiles and n the number of clades. This criterion will take a high value when the site shares the same profiles in different clades. In contrast, a small PIPn corresponds to a site that has different evolutionary profiles in the taxonomic groups under consideration. Indeed, even a site with unstable affiliations (i.e. affected to a different set of profiles within a clade) can be compared and shows a PIPn value close to zero. For instance, the site can belong to various categories containing hydrophobic profiles in one clade and to various categories containing charged amino acid profiles in other clades. For computational reasons, we have limited the phylobayes runs to 1,100 cycles. This choice increases the number of sites with a PIPn value equal to 0: a low frequency of affiliation for a given profile (e.g. 10-5) would artificially be estimated at 0 in the posterior distribution. The more cycles performed, the less sites show a zero PIPn (Additional file 2: Table S5). However, precision of PIPn estimated with 1,000 points is sufficient for the aim of this work (see R2 = 0.999 for comparison between 1000 and 5000 points on Additional file 1: Figure S8). As PIPn values are small, in the subsequent analysis, we will use -ln(PIPn), except for sites with a PIPn value of zero. Because of this latter constraint, results are presented by binning sites into four classes of equal size plus one class for PIPn = 0 (Tables 1, 2, 3, 4, 5 and 6, and Additional file 1: Figure S7).
Evaluation of the protocol
The statistical significance was evaluated by comparing results obtained from real and simulated data. With phylobayes, we performed simulations for each dataset according to the posterior predictive principle, i.e. we took 10 points from the posterior distribution obtained with the real complete dataset (i.e. all species simultaneously) under the CAT+Γ4 model (burn-in discarded) and simulated 10 new sequence alignments according to the parameter values of each point, in particular the profiles (for more details see [24, 31]). Using the same sets of predefined profiles (obtained from real data), the previously described protocol was used to calculate the FDP and the PIPn for each replicate.
Second, we tested that our results were robust vis-à-vis various aspects of our protocol. (i) The use of different stability thresholds yielded virtually identical results in the FDP analysis (Additional file 1: Figure S6B). (ii) To test that low PIPn values were due to unstable profile affiliations related to an insufficient number of taxa, we randomly removed half and three quarter of the species in each clade of the large mitochondrial dataset and recomputed PIPn. As expected, the less species considered, the less profile affiliations were stable (data not shown), because the phylogenetic signal became insufficient. However, this instability led to a sharp decrease of PIPn values equal to 0 (Additional file 1: Figure S5A), indicating that instability was not responsible for low PIPn values. (iii) Three additional sets of profiles were used in the case of mt336 alignment: 45 profiles defined using a different threshold on the UPGMA tree, the 25 stable profiles obtained from the analysis of the complete alignment, and the 20 profiles obtained by Le et al. . Similar results were obtained for both PIPn (Additional file 1: Figure S5B) and FDP (Additional file 1: Figure S6A) criteria.
Third, we performed a cross-validation test to evaluate the fit of different models on the various datasets: CAT, GTR and WAG/mtREV models were compared using cross-validation as described in Lartillot et al. . An alignment was randomly split in two slices: one tenth for use as a test dataset, and nine-tenths for use as a "training", or "learning" dataset. The parameters were estimated on the learning sets for each model (fixed topology; 21,000 and 11,000 cycles, the first 11,000 and 1,000 cycles discarded, for CAT and others models, respectively) and used to calculate the cross-validation log-likelihood scores of the test sets. Scores were averaged over 10 replicates.
Influence of profile change on phylogenetic inference
Profiles are representative of the functional constraints acting on a given site in a given clade; if a change of profile occurred in the common ancestor of two clades, the same substitutional profile should be shared by the two sister clades. Hence this can be viewed as a synapomorphy. In other words, the variation of substitution profiles across clades may contain a phylogenetic signal (or noise if the same profile has been independently acquired). A simple recoding approach might capture this putative phylogenetic signal. For reasons of compatibility with available inference tools, each profile is encoded as a one-letter amino acid, therefore only the twenty most frequent profiles have been conserved for this analysis. More precisely, a new sequence is created for each clade according to the following rule: the site is encoded as an amino acid when profile affiliation is stable, and by a question mark otherwise. Under these conditions, the percent of un-encoded sites in the alignments is 59% and 54% for the mt336 and the nuc80 dataset, respectively. It is difficult to know which model of sequence evolution should be used on this artificial alignment. Since we do not know a priori which exchangeability rate between profiles should be applied, the resulting file is analyzed with a GTR+Γ4 model to infer a phylogenetic tree using phylobayes. To test the effect of the model, we also made inferences with the CAT+Γ4 model. To verify the significance of the results, we performed the same analysis with the 10 simulated alignments obtained by using a posterior predictive approach for the mtp336 dataset, as described above.
Progressive removal of heteropecillous sites
The mt68 dataset was used to evaluate the potential misleading effect of the detected model violations on phylogenetic inference. More precisely, we made the hypothesis that the observed grouping of Cnidaria and Porifera to the exclusion of Bilateria was due to a long branch attraction artifact. We followed the same approach as for heterotachous positions [71, 72], by removing the most heterogeneous sites, as estimated by PIPn. At each step, we removed ~10% of the positions and stopped when 1,039 positions remained in order to keep a sufficient amount of phylogenetic signal. The five steps corresponded to the exclusion of sites with PIPn = 0, and -ln(PIPn) higher than 12, 8, 6, and 4.5, respectively. Phylogenetic trees were inferred from these reduced dataset with the CAT+Γ4 model using phylobayes. The reduced datasets were also analyzed with GTR+Γ4 and mtREV+Γ4 models using RAxML , the robustness was evaluated with 100 bootstrap replicates.
To compare the qualitative heterogeneity studied here (heteropecilly) with the quantitative rate heterogeneity (heterotachy) over time, we looked for heterotachous positions. The number of substitutions per position and per clade was calculated by phylobayes under the CAT+Γ4 model. Subsequently, heterotachous positions were identified by the test of Lopez et al.  (the improved test of Baele et al.  is not implemented for amino acid sequences). Eventually, the coefficient of correlation between heterotachy p-values and PIPn values over sites was computed.
Biochemical constraint estimation
We want to know whether the time-variation in profiles corresponds to change in physico-chemical properties of the amino acids involved in the profiles. Profiles were classified into five groups with similar physico-chemical properties (small, aliphatic, aromatic, charged, other) according to the properties of the two amino acids with the highest equilibrium frequencies in the profile. Only sites stably affiliated (threshold = 75%) to two different profiles in clade pairwise comparison were considered. The numbers of sites for which the two profiles were in the same physico-chemical group were counted over all pairwise comparisons.
To estimate the existence of a hydropathy change over time, the standard deviation of PHS(c,i) across all clades was calculated.
We wish to thank Henner Brinkmann, Claudia Kleinman, Nicolas Lartillot, Nicolas Rodrigue, Guy Baele and two anonymous referees for their helpful comments and suggestions. We gratefully acknowledge the financial support provided by NSERC, the Canadian Research Chair Program and the Université de Montréal, and the Réseau Québecois de Calcul de Haute Performance for computational resources. B.R. has been supported by 'Bourses d'Excellence biT' a strategic program of the Canadian CIHR.
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