Nonhomogeneous models of sequence evolution in the Bio++ suite of libraries and programs
 Julien Dutheil^{1}Email author and
 Bastien Boussau^{2}
https://doi.org/10.1186/147121488255
© Dutheil and Boussau; licensee BioMed Central Ltd. 2008
Received: 24 April 2008
Accepted: 22 September 2008
Published: 22 September 2008
Abstract
Background
Accurately modeling the sequence substitution process is required for the correct estimation of evolutionary parameters, be they phylogenetic relationships, substitution rates or ancestral states; it is also crucial to simulate realistic data sets. Such simulation procedures are needed to estimate the nulldistribution of complex statistics, an approach referred to as parametric bootstrapping, and are also used to test the quality of phylogenetic reconstruction programs. It has often been observed that homologous sequences can vary widely in their nucleotide or aminoacid compositions, revealing that sequence evolution has changed importantly among lineages, and may therefore be most appropriately approached through nonhomogeneous models. Several programs implementing such models have been developed, but they are limited in their possibilities: only a few particular models are available for likelihood optimization, and data sets cannot be easily generated using the resulting estimated parameters.
Results
We hereby present a general implementation of nonhomogeneous models of substitutions. It is available as dedicated classes in the Bio++ libraries and can hence be used in any C++ program. Two programs that use these classes are also presented. The first one, Bio++ Maximum Likelihood (BppML), estimates parameters of any nonhomogeneous model and the second one, Bio++ Sequence Generator (BppSeqGen), simulates the evolution of sequences from these models. These programs allow the user to describe nonhomogeneous models through a property file with a simple yet powerful syntax, without any programming required.
Conclusion
We show that the general implementation introduced here can accommodate virtually any type of nonhomogeneous models of sequence evolution, including heterotachous ones, while being computer efficient. We furthermore illustrate the use of such general models for parametric bootstrapping, using tests of nonhomogeneity applied to an already published ribosomal RNA data set.
Keywords
Background
In phylogenetics, simulations have been widely used to study the robustness of inference methods [1] and have been involved in parametric bootstrapping [2]. For instance, simulations have shown that maximum likelihood methods often more accurately reconstructed the evolution of an alignment than distance or parsimony methods [3, 4], but could also fail in conditions where compositional biases (a condition here referred to as nonhomogeneity) or rate heterogeneity along branches (a phenomenon named heterotachy, [5]) were too intense [6–8]. Similarly, simulations have been used to compare topologies with respect to an alignment [9], or to assess the fit of a model to a particular data set [10–13]. In this last case, a model has a good fit to a particular data set if the alignments it generates have properties similar to the properties of the real alignment. Both for investigating reconstruction methods and for parametric bootstrapping, it is highly desirable that simulation methods model as precisely as possible the conditions that shaped biological sequences through evolution. However, widelyused simulation programs cannot be easily tuned to precisely reproduce the peculiar evolution of a particular data set. Noticeably, nonhomogeneity cannot be simulated by SeqGen [14] or PAML [15], even if these phenomena are all known to affect the evolution of many data sets [5, 16–20].
The ability to estimate parameters of sequence evolution with realistic models, and then computationally evolve sequences using these fitted parameters is crucial to better characterize the behavior of reconstruction methods in realistic settings.
Here we introduce extensions to the Bio++ package [21] that permit first to estimate parameters of evolution on a specific data set in a maximum likelihood framework, and second to simulate the evolution of sequences using these estimated parameters. Importantly, nearly any combination of nonhomogeneous (including nonstationary models) and heterotachous models of evolution can be fitted to data, so that simulations may mimic very precisely the evolution of a data set. Such a flexibility should enable one to probe how robust methods of phylogenetic tree or ancestral state reconstruction are to more realistic evolutionary conditions. Moreover, it offers the possibility to compare a large variety of models by assessing through parametric bootstrapping their respective ability to reproduce a given characteristic of interest, measured on a real data set.
Implementation
Molecular phylogenetic methods are used by a wide range of biologists, from bioinformaticians willing to characterize and improve models of sequence evolution to molecular biologists trying to grasp the particular evolutionary history of their gene of interest. These different types of users have different needs: the former may benefit from easytoassemble, highlevel objectoriented code to conduct phylogenetic analysis, while the latter likes userfriendly interfaces. However, both demand programs able to run the most recent models of evolution. The newly introduced extensions are available in two flavors that might fit different users' needs: (i) as classes in the Bio++ phylogenetic library, including a special class called SubstitutionModelSet which implements the relationships between models, parameters and branches, and (ii) through the BppML and BppSeqGen programs, which can respectively adjust these models to a data set and simulate data from these models. These programs share a common syntax for model specification and are hence fully interoperational and easytouse.
The SubstitutionModelSet class
Substitution models can be totally independent of each other, or can share any number of parameters. Virtually any nonhomogeneous model can thus be set up, provided the alignment is not a mix of nucleotide, aminoacid or codon sequences. All models available in Bio++ can be used with this class (e.g. K80, T92, HKY85, GTR, JTT92, etc), including heterotachous models (Galtier's model [22] and Tuffley and Steel's model [23]) and any rates across sites model (i.e. Gamma and Gamma + invariant distributions). The developer can also use the SubstitutionModelSet class with his own substitution model through the Bio++ SubstitutionModel interface. The SubstitutionModelSet class can be used in conjunction with other Bio++ classes to reconstruct ancestral states or to map substitutions, and hence allows to perform these analyses in the general nonhomogeneous case.
Estimating parameters
Estimation of numerical parameters is performed using a modified NewtonRaphson optimization algorithm, commonly used in phylogenetics [4, 24, 25], and therefore requires computing derivatives with respect to parameters of the model. Because the use of the cross derivatives leads to numerical instabilities in the optimization (Nicolas Galtier, personal communication), they are set to zero in the Hessian matrix. Derivatives regarding branch lengths are computed analytically, whereas derivatives regarding the rates across sites distribution are computed numerically. Although the substitution model derivatives can be computed analytically in the homogeneous case as well as in Galtier and Gouy's model, they are difficult to compute analytically in the more general case, and are consequently computed numerically in Bio++. To prevent convergence issues due to erroneous derivative values we use, in the last optimization steps, Powell's multidimensions algorithm, which does not rely on parameter derivatives [26].
A general file format to describe nonhomogeneous models
We introduced a new userintuitive property file format to describe nonhomogeneous substitution models. This format is an extension of PAML or NHML property file formats, and uses a syntax of the kind
property_name = property_value
The BppML and BppSeqGen programs
Parameter estimation and simulation procedures are available as dedicated classes in the Bio++ phylogenetic library, and can hence be used in any C++ program. However, for users who would rely on appropriate software rather than program their own tools, the Bio++ program suite was designed. These programs, including BppML (for Bio++ Maximum Likelihood) and BppSeqGen (Bio++ Sequence Generator) are command line driven and fully parametrized using property files, as introduced above. They can thus easily be pipelined with scripting languages as bash, python or perl. In addition to the BppML and BppSeqGen programs, the Bio++ program suite also contains programs for distancebased phylogenetic reconstruction, sequence file format conversion and tree manipulation.
Results and Discussion
Our new general nonhomogeneous model implementation was applied to Boussau and Gouy's data set of concatenated small and large subunit ribosomal RNA sequences and tree [6]. This data set contains 92 sequences and 527 complete sites. We first compare computation time, memory usage and parameter estimation for various models and software. We then show how the general nonhomogeneous model introduced here can be used to study model fit through parametric bootstrapping.
In this section, we use the following model notations:
H Homogeneous model, using a Tamura 1992 substitution model [27].
NH1 Onethetaperbranch nonhomogeneous model [24]. This model uses Tamura's 1992 substitution model, with one θ (equilibrium G+C content) per branch in the tree, whereas κ (transitions/transversions ratio) is shared by all branches.
NH2 Onethetaperkingdom nonhomogeneous model. In this general model, we allowed each kingdom (Bacteria, Eukaryotes or Archaea) to have its own equilibrium G+C content, while sharing the same transitions/transversions ratio.
NH3 Same as NH2, but in addition the (hyper)thermophilic Bacteria on one hand, and the eukaryote G+Crich genus Giardia on the other hand were allowed to have their own equilibrium G+C content.
NH4 Onekappaperbranch nonhomogeneous model. This model has one κ per branch in the tree, whereas θ is shared by all branches.
Performance
Comparison of the NHML, (NH)PhyML and BppML programs. Likelihood:  log (likelihood) of the optimized parameters, with a fixed tree topology.
Likelihood  

Rate  Constant  Γ(4)  Γ(4) + I  Covarion  
Model  H  NH1  NH3  H  NH1  NH3  H  NH1  NH3  H  NH1  NH3 
NHML  15307  15034    14145  13828          13750  13397   
PhyML  15187  15011    14141  13824    14128           
BppML  15187  14920  15109  14141  13821  14029  14128  13810  14018  13747  13399  13615 
Time  
Rate  Constant  Γ(4)  Γ(4) + I  Covarion  
Model  H  NH1  NH3  H  NH1  NH3  H  NH1  NH3  H  NH1  NH3 
NHML  0:01:40  0:02:28    0:03:07  00:02:13          0:19:24  0:19:09   
PhyML  0:00:07  0:01:43    0:00:34  00:02:29    0:00:35           
BppML  0:00:27  0:11:57  0:01:12  0:00:47  00:35:46  0:00:48  0:01:01  0:29:40  0:01:38  0:02:52  1:14:32  0:14:27 
Memory  
Rate  Constant  Γ(4)  Γ(4) + I  Covarion  
Model  H  NH1  NH3  H  NH1  NH3  H  NH1  NH3  H  NH1  NH3 
NHML  16.38  20.48    55.30  65.54          55.30  65.54   
PhyML  10.24  28.67    30.73  77.82    30.72           
BppML  08.19  08.19  08.19  14.34  14.34  14.34  14.34  16.38  16.38  12.29  14.34  12.29 
Example of application: parametric bootstrap and Bowker's test for nonhomogeneity
As most phylogenetic reconstruction models are homogeneous, they do not properly model the evolution of homologous sequences that vary widely in their compositions. Analyzing compositionally heterogeneous data sets with homogeneous models of sequence evolution may therefore lead to incorrect inferences, provided the heterogeneity is large enough. Several tests have been developed to assess the amount of heterogeneity present in a data set (see [29] for a review).
Estimating the amount of compositional heterogeneity in a data set
Most commonly, a matrix is assembled that contains compositions in all characters for all sequences, and this matrix is analyzed through χ^{2} statistics [29]. However, this approach usually does not distinguish between constant and variable sites, and therefore may underestimate the true amount of heterogeneity in a data set [29].
Recently, Ababneh et al. [30] reintroduced Bowker's pairwise test [31] for symmetry. Given two aligned sequences S_{1} and S_{2} on a given alphabet of size n and characters x_{{1,2...n}}, it compares the numbers of substitutions between x_{ i }in S_{1} and x_{ j }in S_{2}, {i,j} ∈ [1 : n], with the numbers of substitutions between x_{ j }in S_{1} and x_{ i }in S_{2}. If these pairs of numbers are equal for all {i, j} ∈ [1 : n], the two sequences may have evolved according to two identical processes. Otherwise, the two processes were necessarily different.
Bowker's test therefore permits to assess whether compositional differences have accumulated between two sequences through nonhomogeneous evolution. To apply it to more than two sequences, RodriguezEzpeleta et al. [32] computed all pairwise Bowker's tests in their alignment and computed the median value; one could also have counted the number of Bowker's tests that are significant at a 5% threshold according to a χ^{2} table.
However, none of these tests permit to estimate if the amount of heterogeneity that they detect in a given data set is sufficient to bias inferences made using homogeneous models, although this is likely the question an average user would like to answer.
Assessment of the fit of evolutionary models with respect to compositional heterogeneity
Here, we describe a method to reveal the ability of evolutionary models to account for the compositional heterogeneity in a sequence alignment, which we measure using the median of all Bowker's pairwise statistics, or the number of significant Bowker's pairwise tests (in the following, we note the measure of compositional heterogeneity h). This method is treebased, and uses parametric bootstrapping [10–12]. In this respect, it is similar to the method recently introduced in [13] in the Bayesian setting. Our approach requires 5 steps to estimate the fit of a model M to a data set D.
1. Compute the compositional heterogeneity measure h for the data set D.
2. Estimate the parameters of model M based on the data set D according to the Maximum Likelihood criterion.
3. Simulate a large number of data sets D' using the model M previously estimated.
4. Compute the compositional heterogeneity measure h' for each alignment D'.
4. Compare the measure h obtained on data set D to measures h' obtained on data sets D'. If h is outside 95% of the distribution of h', the model does not properly reproduce the heterogeneity of data set D.
Using such an approach, any model can be compared with others with respect to their ability to handle the compositional heterogeneity of a given data set: the closest the distribution of h' is from h, the highest is the fit. Ideally, the distribution of measures h' obtained on the parametric bootstrap replicates of a good model should be centered around the value obtained for the real alignment h, with a very low variance. If one neglects potential problems linked with overparametrization, the inferences of the best model should be preferentially trusted compared to a model that fails to account for an important feature of a data set. Overall, our approach can be used for model selection, although contrary to criteria such as AIC or BIC [28] this approach does not take into account the number of parameters; more importantly, it can also be used for estimating model adequacy.
Application to an rRNA data set
Our approach to assess the compositionwise fit of evolutionary models to a data set was applied to an alignment containing ribosomal RNA sequences from Archaea, Bacteria and Eukaryotes [6]. First, several homogeneous and nonhomogeneous models were fitted to the data set, using a Tamura 1992 model of substitution with a four classes Gamma + invariant distribution of rates across sites. Then, 10,000 artificial data sets were simulated in each case using these estimated parameters. Eventually, the real data set and the simulated data sets were compared with respect to their compositional heterogeneity: models able to simulate data sets with similar amounts of heterogeneity as the real data set appropriately account for this specific aspect of the data.
Model comparisons.
Model  lnL  k  LRT  AIC  BIC  Bowker  

H  NH2  NH3  # tests  median  
H  14110.628293  185  28591.26  29380.69  0.0008  0.0028  
NH1  13810.371502  368  600.51  556.74  416.97  28356.74  29927.07  0.7010  0.8110 
NH2  14088.739682  189  43.78  28555.48  29361.98  0.0040  0.0141  
NH3  14018.854234  191  183.55  139.77  28419.71  29234.74  0.1448  0.2672  
NH4  13970.841467  368  279.57  28677.68  30248.01  0.0015  0.0047 
Conclusion
Bio++ is a growing set of libraries designed for sequence, phylogenetic and molecular evolution analyzes. In this article extensions allowing to implement a wide variety of nonhomogeneous models of sequence evolution were introduced. Combined with support for rates across sites and heterotachous models of evolution, and with routines for optimizing parameters and tree topology in the maximum likelihood framework, they provide a comprehensive platform for phylogenetic studies, either for bioinformaticians willing to develop their own software, or for biologists characterizing the evolution of a particular set of sequences using the BppML and BppSegGen programs. Whilst being a generalist program implementing a large variety of models, BppML was shown to be of a similar quality as programs dedicated to particular homogeneous or nonhomogeneous models of evolution, achieving higher likelihood scores with smaller memory requirements while conserving reasonable runningtimes. Its joint use with BppSeqGen permits to precisely study the evolution of a particular data set through parametric bootstrapping, and may be used to generate realistic artificial data sets to study the robustness of phylogenetic reconstruction methods in the presence of heterogeneity and heterotachy. Further developments may involve methods to optimize the number of models necessary to account for the heterogeneity in a data set, or methods to explore the space of tree topologies with a broad range of nonhomogeneous models of sequence evolution.
Methods
Data and phylogeny reconstruction
RNA sequences from the small and the large subunit of the ribosome were aligned and concatenated. Sequences coming from 22 Archaea, 34 Bacteria and 36 Eukaryotes were selected to yield a data set containing 92 sequences and 527 complete sites, with G+C contents ranging from 43% to 71%. A phylogenetic tree was built with nhPhyML [6]. For additional information, please refer to [6].
Comparing likelihood optimizations
The NHML, (NH)PhyML and BppML programs were used to compare optimization performances. The programs were run on the data set from [6], after all columns in the alignment containing at least either a gap or an unknown character had been removed. The phylogenetic tree from [6] was used as a fixed topology, and the branch lengths used as initial values for the optimization. To allow the comparison between the three programs, the Kimura two parameters model of substitution [35] was used for homogeneous models and models derived from Tamura's 1992 model [27] for nonhomogeneous models. Initial values were set to 1 and 0.5 for the κ and θ parameters respectively. A Gamma (4 classes) + invariant rates across sites distribution was also tested, with initial value set to 0.5 for the Gamma shape parameter, and 0.2 for the proportion of invariants. Galtier's 2001 [22] heterotachous model was also tested, with 4 rate classes, initial values of the shape parameter set to 0.5, and initial value of the rate change parameter set to 0.5. The precision in the optimization algorithm was set to 0.000001 for the three programs. The total length of execution was corrected according to the average CPU usage, and the memory usage corresponds to the maximum reached during program execution, as reported by the Unix "top" command. All calculations were performed on a 64 bits Intel(R) Core(TM)2 Duo, CPU 2.66 GHz.
Assessing the convergence of the optimization procedure
Different initial values were used as initial guesses for the optimization algorithm. The GC frequencies and the proportion of invariant sites were chosen randomly from a uniform distribution between 0 and 1. The transitions/transversions ratio and the alpha parameter of the rate distribution were picked from a [0, 5] and [0.2, 2] uniform distributions, respectively. Branch lengths were taken from a uniform distribution between 0 and 0.1.
Computing pvalues for Bowker tests
Alignmentwise tests for nonhomogeneity were performed using two types of statistics:

The number of 5% significant pairwise tests,

The median of pairwise statistics.
where N_{1} is the number of simulations performed under the null model, and N_{2} is the number of values of the statistic in the simulations that were greater or equal to the observed one, measured from the real data set. In this study, N_{1} was set to 10,000.
Program source code for performing Bowker's test is provided as Additional file 2. The data and scripts to run the analyses are in Additional file 3.
Availability and requirements
Project name: The Bio++ libraries (version 1.6) and programs suite (version 1.0).
Project home page: http://kimura.univmontp2.fr/BioPP and http://home.gna.org/bppsuite
Operating systems: Any platform with a C++ compiler and supporting the Standard Template Library
Programming language: C++
Other requirements: The C++ Standard Template Library
License: The CeCILL free software license (GNU compatible)
Declarations
Acknowledgements
The authors would like to thank Manolo Gouy, Nicolas Galtier, Mathieu Emily and Matthew Spencer for helpful comments on this manuscript.
Authors’ Affiliations
References
 Williams PD, Pollock DD, Blackburne BP, Goldstein RA: Assessing the accuracy of ancestral protein reconstruction methods. PLoS Comput Biol. 2006, 2: e69e69. 10.1371/journal.pcbi.0020069.PubMed CentralView ArticlePubMedGoogle Scholar
 Goldman N: Statistical tests of models of DNA substitution. J Mol Evol. 1993, 36: 182198. 10.1007/BF00166252.View ArticlePubMedGoogle Scholar
 Kuhner MK, Felsenstein J: A simulation comparison of phylogeny algorithms under equal and unequal evolutionary rates. Mol Biol Evol. 1994, 11: 459468.PubMedGoogle Scholar
 Guindon S, Gascuel O: A simple, fast, and accurate algorithm to estimate large phylogenies by maximum likelihood. Syst Biol. 2003, 52: 696704. 10.1080/10635150390235520.View ArticlePubMedGoogle Scholar
 Lopez P, Casane D, Philippe H: Heterotachy, an important process of protein evolution. Mol Biol Evol. 2002, 19: 17.View ArticlePubMedGoogle Scholar
 Boussau B, Gouy M: Efficient likelihood computations with nonreversible models of evolution. Syst Biol. 2006, 55: 756768. 10.1080/10635150600975218.View ArticlePubMedGoogle Scholar
 Kolaczkowski B, Thornton JW: Performance of maximum parsimony and likelihood phylogenetics when evolution is heterogeneous. Nature. 2004, 431: 980984. 10.1038/nature02917.View ArticlePubMedGoogle Scholar
 Philippe H, Zhou Y, Brinkmann H, Rodrigue N, Delsuc F: Heterotachy and longbranch attraction in phylogenetics. BMC Evol Biol. 2005, 5: 5050. 10.1186/14712148550.PubMed CentralView ArticlePubMedGoogle Scholar
 Goldman N, Anderson JP, Rodrigo AG: Likelihoodbased tests of topologies in phylogenetics. Syst Biol. 2000, 49: 652670. 10.1080/106351500750049752.View ArticlePubMedGoogle Scholar
 Bollback JP: Bayesian model adequacy and choice in phylogenetics. Mol Biol Evol. 2002, 19: 11711180.View ArticlePubMedGoogle Scholar
 Foster PG: Modeling compositional heterogeneity. Syst Biol. 2004, 53: 485495. 10.1080/10635150490445779.View ArticlePubMedGoogle Scholar
 Lartillot N, Brinkmann H, Philippe H: Suppression of longbranch attraction artefacts in the animal phylogeny using a siteheterogeneous model. BMC Evol Biol. 2007, 7 (Suppl 1): S4S4. 10.1186/147121487S1S4.PubMed CentralView ArticlePubMedGoogle Scholar
 Blanquart S, Lartillot N: A Site and TimeHeterogeneous Model of AminoAcid Replacement. Mol Biol Evol. 2008Google Scholar
 Rambaut A, Grassly NC: SeqGen: an application for the Monte Carlo simulation of DNA sequence evolution along phylogenetic trees. Cabios. 1997, 13: 235238.PubMedGoogle Scholar
 Yang Z: PAML 4: phylogenetic analysis by maximum likelihood. Mol Biol Evol. 2007, 24: 15861591. 10.1093/molbev/msm088.View ArticlePubMedGoogle Scholar
 Sueoka N: On the genetic basis of variation and heterogeneity of DNA base composition. Proc Natl Acad Sci USA. 1962, 48: 582592. 10.1073/pnas.48.4.582.PubMed CentralView ArticlePubMedGoogle Scholar
 Galtier N, Lobry JR: Relationships between genomic G+C content, RNA secondary structures, and optimal growth temperature in prokaryotes. J Mol Evol. 1997, 44: 632636. 10.1007/PL00006186.View ArticlePubMedGoogle Scholar
 Foster PG, Jermiin LS, Hickey DA: Nucleotide composition bias affects amino acid content in proteins coded by animal mitochondria. J Mol Evol. 1997, 44: 282288. 10.1007/PL00006145.View ArticlePubMedGoogle Scholar
 Zeldovich KB, Berezovsky IN, Shakhnovich EI: Protein and DNA sequence determinants of thermophilic adaptation. PLoS Comput Biol. 2007, 3: e5e5. 10.1371/journal.pcbi.0030005.PubMed CentralView ArticlePubMedGoogle Scholar
 Wang HC, Spencer M, Susko E, Roger AJ: Testing for covarionlike evolution in protein sequences. Mol Biol Evol. 2007, 24: 294305. 10.1093/molbev/msl155.View ArticlePubMedGoogle Scholar
 Dutheil J, Gaillard S, Bazin E, Glémin S, Ranwez V, Galtier N, Belkhir K: Bio++: a set of C++ libraries for sequence analysis, phylogenetics, molecular evolution and population genetics. BMC Bioinformatics. 2006, 7: 188188. 10.1186/147121057188.PubMed CentralView ArticlePubMedGoogle Scholar
 Galtier N: Maximumlikelihood phylogenetic analysis under a covarionlike model. Mol Biol Evol. 2001, 18: 866873.View ArticlePubMedGoogle Scholar
 Tuffley C, Steel M: Modeling the covarion hypothesis of nucleotide substitution. Math Biosci. 1998, 147: 6391. 10.1016/S00255564(97)000813.View ArticlePubMedGoogle Scholar
 Galtier N, Gouy M: Inferring pattern and process: maximumlikelihood implementation of a nonhomogeneous model of DNA sequence evolution for phylogenetic analysis. Mol Biol Evol. 1998, 15: 871879.View ArticlePubMedGoogle Scholar
 Felsenstein J: PHYLIP (Phylogeny Inference Package) version 3.6. Distributed by the author. 2005Google Scholar
 Press WH, Teukolsky SA, Vetterling WT, Flannery BP: Numerical Recipes in C. The Art of Scientific Computing. 1992, Cambridge University Press, secondGoogle Scholar
 Tamura K: The rate and pattern of nucleotide substitution in Drosophila mitochondrial DNA. Mol Biol Evol. 1992, 9: 814825.PubMedGoogle Scholar
 Felsenstein J: Inferring Phylogenies. 2004, Sinauer Associates, IncGoogle Scholar
 Jermiin L, Ho SY, Ababneh F, Robinson J, Larkum AW: The biasing effect of compositional heterogeneity on phylogenetic estimates may be underestimated. Syst Biol. 2004, 53: 638643. 10.1080/10635150490468648.View ArticlePubMedGoogle Scholar
 Ababneh F, Jermiin LS, Ma C, Robinson J: Matchedpairs tests of homogeneity with applications to homologous nucleotide sequences. Bioinformatics. 2006, 22: 12251231. 10.1093/bioinformatics/btl064.View ArticlePubMedGoogle Scholar
 Bowker A: A test for symmetry in contingency tables. J Am Stat Assoc. 1948, 43: 572574. 10.2307/2280710.View ArticlePubMedGoogle Scholar
 RodríguezEzpeleta N, Brinkmann H, Roure B, Lartillot N, Lang BF, Philippe H: Detecting and overcoming systematic errors in genomescale phylogenies. Syst Biol. 2007, 56: 389399. 10.1080/10635150701397643.View ArticlePubMedGoogle Scholar
 Yang Z, Roberts D: On the Use of Nucleic Acid Sequences to Infer Branchings in the Tree of Life. Mol Biol Evol. 1995, 12: 451458.PubMedGoogle Scholar
 Blanquart S, Lartillot N: A Bayesian Compound Stochastic Process for Modeling Nonstationary and Nonhomogeneous Sequence Evolution. Mol Biol Evol. 2006, 23: 20582071. 10.1093/molbev/msl091.View ArticlePubMedGoogle Scholar
 Kimura M: A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences. J Mol Evol. 1980, 16: 111120. 10.1007/BF01731581.View ArticlePubMedGoogle Scholar
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