Benchmarking Multi-Rate Codon Models

The single rate codon model of non-synonymous substitution is ubiquitous in phylogenetic modeling. Indeed, the use of a non-synonymous to synonymous substitution rate ratio parameter has facilitated the interpretation of selection pressure on genomes. Although the single rate model has achieved wide acceptance, we argue that the assumption of a single rate of non-synonymous substitution is biologically unreasonable, given observed differences in substitution rates evident from empirical amino acid models. Some have attempted to incorporate amino acid substitution biases into models of codon evolution and have shown improved model performance versus the single rate model. Here, we show that the single rate model of non-synonymous substitution is easily outperformed by a model with multiple non-synonymous rate classes, yet in which amino acid substitution pairs are assigned randomly to these classes. We argue that, since the single rate model is so easy to improve upon, new codon models should not be validated entirely on the basis of improved model fit over this model. Rather, we should strive to both improve on the single rate model and to approximate the general time-reversible model of codon substitution, with as few parameters as possible, so as to reduce model over-fitting. We hint at how this can be achieved with a Genetic Algorithm approach in which rate classes are assigned on the basis of sequence information content.     

Mutation-selection models of codon substitution and their use to estimate selective strengths on codon usage

Current models of codon substitution are formulated at the levels of nucleotide substitution and do not explicitly consider the separate effects of mutation and selection. They are thus incapable of inferring whether mutation or selection is responsible for evolution at silent sites. Here we implement a few population genetics models of codon substitution that explicitly consider mutation bias and natural selection at the DNA level. Selection on codon usage is modeled by introducing codon-fitness parameters, which together with mutation-bias parameters, predict optimal codon frequencies for the gene. The selective pressure may be for translational efficiency and accuracy or for fine-tuning translational kinetics to produce correct protein folding. We apply the models to compare mitochondrial and nuclear genes from several mammalian species. Model assumptions concerning codon usage are found to affect the estimation of sequence distances (such as the synonymous rate d(S), the nonsynonymous rate d(N), and the rate at the 4-fold degenerate sites d(4)), as found in previous studies, but the new models produced very similar estimates to some old ones. We also develop a likelihood ratio test to examine the null hypothesis that codon usage is due to mutation bias alone, not influenced by natural selection. Application of the test to the mammalian data led to rejection of the null hypothesis in most genes, suggesting that natural selection may be a driving force in the evolution of synonymous codon usage in mammals. Estimates of selection coefficients nevertheless suggest that selection on codon usage is weak and most mutations are nearly neutral. The sensitivity of the analysis on the assumed mutation model is discussed.     

Choosing appropriate substitution models for the phylogenetic analysis of protein-coding sequences

Although phylogenetic inference of protein-coding sequences continues to dominate the literature, few analyses incorporate evolutionary models that consider the genetic code. This problem is exacerbated by the exclusion of codon-based models from commonly employed model selection techniques, presumably due to the computational cost associated with codon models. We investigated an efficient alternative to standard nucleotide substitution models, in which codon position (CP) is incorporated into the model. We determined the most appropriate model for alignments of 177 RNA virus genes and 106 yeast genes, using 11 substitution models including one codon model and four CP models. The majority of analyzed gene alignments are best described by CP substitution models, rather than by standard nucleotide models, and without the computational cost of full codon models. These results have significant implications for phylogenetic inference of coding sequences as they make it clear that substitution models incorporating CPs not only are a computationally realistic alternative to standard models but may also frequently be statistically superior.     

An empirical codon model for protein sequence evolution

In the past, 2 kinds of Markov models have been considered to describe protein sequence evolution. Codon-level models have been mechanistic with a small number of parameters designed to take into account features, such as transition-transversion bias, codon frequency bias, and synonymous-nonsynonymous amino acid substitution bias. Amino acid models have been empirical, attempting to summarize the replacement patterns observed in large quantities of data and not explicitly considering the distinct factors that shape protein evolution. We have estimated the first empirical codon model (ECM). Previous codon models assume that protein evolution proceeds only by successive single nucleotide substitutions, but our results indicate that model accuracy is significantly improved by incorporating instantaneous doublet and triplet changes. We also find that the affiliations between codons, the amino acid each encodes and the physicochemical properties of the amino acids are main factors driving the process of codon evolution. Neither multiple nucleotide changes nor the strong influence of the genetic code nor amino acids' physicochemical properties form a part of standard mechanistic models and their views of how codon evolution proceeds. We have implemented the ECM for likelihood-based phylogenetic analysis, and an assessment of its ability to describe protein evolution shows that it consistently outperforms comparable mechanistic codon models. We point out the biological interpretation of our ECM and possible consequences for studies of selection.     

Models of amino acid substitution and applications to mitochondrial protein evolution

Models of amino acid substitution were developed and compared using maximum likelihood. Two kinds of models are considered. "Empirical" models do not explicitly consider factors that shape protein evolution, but attempt to summarize the substitution pattern from large quantities of real data. "Mechanistic" models are formulated at the codon level and separate mutational biases at the nucleotide level from selective constraints at the amino acid level. They account for features of sequence evolution, such as transition-transversion bias and base or codon frequency biases, and make use of physicochemical distances between amino acids to specify nonsynonymous substitution rates. A general approach is presented that transforms a Markov model of codon substitution into a model of amino acid replacement. Protein sequences from the entire mitochondrial genomes of 20 mammalian species were analyzed using different models. The mechanistic models were found to fit the data better than empirical models derived from large databases. Both the mutational distance between amino acids (determined by the genetic code and mutational biases such as the transition-transversion bias) and the physicochemical distance are found to have strong effects on amino acid substitution rates. A significant proportion of amino acid substitutions appeared to have involved more than one codon position, indicating that nucleotide substitutions at neighboring sites may be correlated. Rates of amino acid substitution were found to be highly variable among sites.      

Advantages of a mechanistic codon substitution model for evolutionary analysis of protein-coding sequences

BACKGROUND: A mechanistic codon substitution model, in which each codon substitution rate is proportional to the product of a codon mutation rate and the average fixation probability depending on the type of amino acid replacement, has advantages over nucleotide, amino acid, and empirical codon substitution models in evolutionary analysis of protein-coding sequences. It can approximate a wide range of codon substitution processes. If no selection pressure on amino acids is taken into account, it will become equivalent to a nucleotide substitution model. If mutation rates are assumed not to depend on the codon type, then it will become essentially equivalent to an amino acid substitution model. Mutation at the nucleotide level and selection at the amino acid level can be separately evaluated. RESULTS: The present scheme for single nucleotide mutations is equivalent to the general time-reversible model, but multiple nucleotide changes in infinitesimal time are allowed. Selective constraints on the respective types of amino acid replacements are tailored to each gene in a linear function of a given estimate of selective constraints. Their good estimates are those calculated by maximizing the respective likelihoods of empirical amino acid or codon substitution frequency matrices. Akaike and Bayesian information criteria indicate that the present model performs far better than the other substitution models for all five phylogenetic trees of highly-divergent to highly-homologous sequences of chloroplast, mitochondrial, and nuclear genes. It is also shown that multiple nucleotide changes in infinitesimal time are significant in long branches, although they may be caused by compensatory substitutions or other mechanisms. The variation of selective constraint over sites fits the datasets significantly better than variable mutation rates, except for 10 slow-evolving nuclear genes of 10 mammals. An critical finding for phylogenetic analysis is that assuming variable mutation rates over sites lead to the overestimation of branch lengths.     

Codon substitution models based on residue similarity and their applications

Codon models are now widely used to draw evolutionary inferences from alignments of homologous sequence data. Incorporating physicochemical properties of amino acids into codon models, two novel codon substitution models describing the evolution of protein-coding DNA sequences are presented based on the similarity scores of amino acids. To describe substitutions between codons a continue-time Markov process is used. Transition/transversion rate bias and nonsynonymous codon usage bias are allowed in the models. In our implementation, the parameters are estimated by maximum-likelihood (ML) method as in previous studies. Furthermore, instantaneous mutations involving more than one nucleotide position of a codon are considered in the second model. Then the two suggested models are applied to five real data sets. The analytic results indicate that the new codon models considering physicochemical properties of amino acids can provide a better fit to the data comparing with existing codon models, and then produce more reliable estimates of certain biologically important measures than existing methods.     

Investigating protein-coding sequence evolution with probabilistic codon substitution models

This review is motivated by the true explosion in the number of recent studies both developing and ameliorating probabilistic models of codon evolution. Traditionally parametric, the first codon models focused on estimating the effects of selective pressure on the protein via an explicit parameter in the maximum likelihood framework. Likelihood ratio tests of nested codon models armed the biologists with powerful tools, which provided unambiguous evidence for positive selection in real data. This, in turn, triggered a new wave of methodological developments. The new generation of models views the codon evolution process in a more sophisticated way, relaxing several mathematical assumptions. These models make a greater use of physicochemical amino acid properties, genetic code machinery, and the large amounts of data from the public domain. The overview of the most recent advances on modeling codon evolution is presented here, and a wide range of their applications to real data is discussed. On the downside, availability of a large variety of models, each accounting for various biological factors, increases the margin for misinterpretation; the biological meaning of certain parameters may vary among models, and model selection procedures also deserve greater attention. Solid understanding of the modeling assumptions and their applicability is essential for successful statistical data analysis.     

Bayesian comparisons of codon substitution models

In 1994, Muse and Gaut (MG) and Goldman and Yang (GY) proposed evolutionary models that recognize the coding structure of the nucleotide sequences under study, by defining a Markovian substitution process with a state space consisting of the 61 sense codons (assuming the universal genetic code). Several variations and extensions to their models have since been proposed, but no general and flexible framework for contrasting the relative performance of alternative approaches has yet been applied. Here, we compute Bayes factors to evaluate the relative merit of several MG and GY styles of codon substitution models, including recent extensions acknowledging heterogeneous nonsynonymous rates across sites, as well as selective effects inducing uneven amino acid or codon preferences. Our results on three real data sets support a logical model construction following the MG formulation, allowing for a flexible account of global amino acid or codon preferences, while maintaining distinct parameters governing overall nucleotide propensities. Through posterior predictive checks, we highlight the importance of such a parameterization. Altogether, the framework presented here suggests a broad modeling project in the MG style, stressing the importance of combining and contrasting available model formulations and grounding developments in a sound probabilistic paradigm.     

Selection of evolutionary models for phylogenetic hypothesis testing using parametric methods

Recent molecular studies have incorporated the parametric bootstrap method to test a priori hypotheses when the results of molecular based phylogenies are in conflict with these hypotheses. The parametric bootstrap requires the specification of a particular substitutional model, the parameters of which will be used to generate simulated, replicate DNA sequence data sets. It has been both suggested that, (a) the method appears robust to changes in the model of evolution, and alternatively that, (b) as realistic model of DNA substitution as possible should be used to avoid false rejection of a null hypothesis. Here we empirically evaluate the effect of suboptimal substitution models when testing hypotheses of monophyly with the parametric bootstrap using data sets of mtDNA cytochrome oxidase I and II (COI and COII) sequences for Macaronesian Calathus beetles, and mitochondrial 16S rDNA and nuclear ITS2 sequences for European Timarcha beetles. Whether a particular hypothesis of monophyly is rejected or accepted appears to be highly dependent on whether the nucleotide substitution model being used is optimal. It appears that a parameter rich model is either equally or less likely to reject a hypothesis of monophyly where the optimal model is unknown. A comparison of the performance of the Kishino–Hasegawa (KH) test shows it is not as severely affected by the use of suboptimal models, and overall it appears to be a less conservative method with a higher rate of failure to reject null hypotheses.     

Interpreting Statistical Evidence with Empirical Likelihood Functions

There has been growing interest in the likelihood paradigm of statistics, where statistical evidence is represented by the likelihood function and its strength is measured by likelihood ratios. The available literature in this area has so far focused on parametric likelihood functions, though in some cases a parametric likelihood can be robustified. This focused discussion on parametric models, while insightful and productive, may have left the impression that the likelihood paradigm is best suited to parametric situations. This article discusses the use of empirical likelihood functions, a well‐developed methodology in the frequentist paradigm, to interpret statistical evidence in nonparametric and semiparametric situations. A comparative review of literature shows that, while an empirical likelihood is not a true probability density, it has the essential properties, namely consistency and local asymptotic normality that unify and justify the various parametric likelihood methods for evidential analysis. Real examples are presented to illustrate and compare the empirical likelihood method and the parametric likelihood methods. These methods are also compared in terms of asymptotic efficiency by combining relevant results from different areas. It is seen that a parametric likelihood based on a correctly specified model is generally more efficient than an empirical likelihood for the same parameter. However, when the working model fails, a parametric likelihood either breaks down or, if a robust version exists, becomes less efficient than the corresponding empirical likelihood.     

Context-Dependent Evolutionary Models for Non-Coding Sequences: An Overview of Several Decades of Research and an Analysis of Laurasiatheria and Primate Evolution

The past decade has seen a growing interest in evolutionary models that relax the assumption of site-independent evolution for non-coding sequences. While phylogenetic inference using such so-called context-dependent models is currently computationally prohibitive, these models have been shown to yield significant increases in model fit compared to site-independent evolutionary models, which remain the most widely used evolutionary models to study substitution patterns and perform phylogenetic inference. Context-dependent models have been shown to be suited to study the spontaneous deamination of cytosine in mammalian sequences. In this paper, I discuss various approaches presented in recent years to model context-dependent evolution. I start with discussing the empirical research and results that have led to the development of these models. To accurately estimate the context-dependent substitution patterns that arise from these models, accurate sampling of substitution histories under such models is required. Further, appropriate model selection techniques to assess model performance has become more important than ever, given the drastic increase in parameters of context-dependent models and the tendency of older model selection techniques to prefer parameter-rich models. I also present new results on two mammalian datasets (Primate and Laurasiatheria data) to shed a light on so-called lineage-dependent context-dependent evolution. I conclude this paper with a discussion on current challenges in the development of context-dependent modeling approaches.     

CodonTest: Modeling Amino Acid Substitution Preferences in Coding Sequences

Codon models of evolution have facilitated the interpretation of selective forces operating on genomes. These models, however, assume a single rate of non-synonymous substitution irrespective of the nature of amino acids being exchanged. Recent developments have shown that models which allow for amino acid pairs to have independent rates of substitution offer improved fit over single rate models. However, these approaches have been limited by the necessity for large alignments in their estimation. An alternative approach is to assume that substitution rates between amino acid pairs can be subdivided into rate classes, dependent on the information content of the alignment. However, given the combinatorially large number of such models, an efficient model search strategy is needed. Here we develop a Genetic Algorithm (GA) method for the estimation of such models. A GA is used to assign amino acid substitution pairs to a series of rate classes, where is estimated from the alignment. Other parameters of the phylogenetic Markov model, including substitution rates, character frequencies and branch lengths are estimated using standard maximum likelihood optimization procedures. We apply the GA to empirical alignments and show improved model fit over existing models of codon evolution. Our results suggest that current models are poor approximations of protein evolution and thus gene and organism specific multi-rate models that incorporate amino acid substitution biases are preferred. We further anticipate that the clustering of amino acid substitution rates into classes will be biologically informative, such that genes with similar functions exhibit similar clustering, and hence this clustering will be useful for the evolutionary fingerprinting of genes.     

Empirical codon substitution matrix

Background  

Codon substitution probabilities are used in many types of molecular evolution studies such as determining Ka/Ks ratios, creating ancestral DNA sequences or aligning coding DNA. Until the recent dramatic increase in genomic data enabled construction of empirical matrices, researchers relied on parameterized models of codon evolution. Here we present the first empirical codon substitution matrix entirely built from alignments of coding sequences from vertebrate DNA and thus provide an alternative to parameterized models of codon evolution.     

Exploring among-site rate variation models in a maximum likelihood framework using empirical data: effects of model assumptions on estimates of topology,branch lengths,and bootstrap support

We have investigated the effects of different among-site rate variation models on the estimation of substitution model parameters, branch lengths, topology, and bootstrap proportions under minimum evolution (ME) and maximum likelihood (ML). Specifically, we examined equal rates, invariable sites, gamma-distributed rates, and site-specific rates (SSR) models, using mitochondrial DNA sequence data from three protein-coding genes and one tRNA gene from species of the New Zealand cicada genus Maoricicada. Estimates of topology were relatively insensitive to the substitution model used; however, estimates of bootstrap support, branch lengths, and R-matrices (underlying relative substitution rate matrix) were strongly influenced by the assumptions of the substitution model. We identified one situation where ME and ML tree building became inaccurate when implemented with an inappropriate among-site rate variation model. Despite the fact the SSR models often have a better fit to the data than do invariable sites and gamma rates models, SSR models have some serious weaknesses. First, SSR rate parameters are not comparable across data sets, unlike the proportion of invariable sites or the alpha shape parameter of the gamma distribution. Second, the extreme among-site rate variation within codon positions is problematic for SSR models, which explicitly assume rate homogeneity within each rate class. Third, the SSR models appear to give severe underestimates of R-matrices and branch lengths relative to invariable sites and gamma rates models in this example. We recommend performing phylogenetic analyses under a range of substitution models to test the effects of model assumptions not only on estimates of topology but also on estimates of branch length and nodal support.     

Empirical models for substitution in ribosomal RNA

Empirical models of substitution are often used in protein sequence analysis because the large alphabet of amino acids requires that many parameters be estimated in all but the simplest parametric models. When information about structure is used in the analysis of substitutions in structured RNA, a similar situation occurs. The number of parameters necessary to adequately describe the substitution process increases in order to model the substitution of paired bases. We have developed a method to obtain substitution rate matrices empirically from RNA alignments that include structural information in the form of base pairs. Our data consisted of alignments from the European Ribosomal RNA Database of Bacterial and Eukaryotic Small Subunit and Large Subunit Ribosomal RNA ( Wuyts et al. 2001. Nucleic Acids Res. 29:175-177; Wuyts et al. 2002. Nucleic Acids Res. 30:183-185). Using secondary structural information, we converted each sequence in the alignments into a sequence over a 20-symbol code: one symbol for each of the four individual bases, and one symbol for each of the 16 ordered pairs. Substitutions in the coded sequences are defined in the natural way, as observed changes between two sequences at any particular site. For given ranges (windows) of sequence divergence, we obtained substitution frequency matrices for the coded sequences. Using a technique originally developed for modeling amino acid substitutions ( Veerassamy, Smith, and Tillier. 2003. J. Comput. Biol. 10:997-1010), we were able to estimate the actual evolutionary distance for each window. The actual evolutionary distances were used to derive instantaneous rate matrices, and from these we selected a universal rate matrix. The universal rate matrices were incorporated into the Phylip Software package ( Felsenstein 2002. http://evolution.genetics.washington.edu/phylip.html), and we analyzed the ribosomal RNA alignments using both distance and maximum likelihood methods. The empirical substitution models performed well on simulated data, and produced reasonable evolutionary trees for 16S ribosomal RNA sequences from sequenced Bacterial genomes. Empirical models have the advantage of being easily implemented, and the fact that the code consists of 20 symbols makes the models easily incorporated into existing programs for protein sequence analysis. In addition, the models are useful for simulating the evolution of RNA sequence and structure simultaneously.     

A combined empirical and mechanistic codon model

The evolutionary selection forces acting on a protein are commonly inferred using evolutionary codon models by contrasting the rate of synonymous to nonsynonymous substitutions. Most widely used models are based on theoretical assumptions and ignore the empirical observation that distinct amino acids differ in their replacement rates. In this paper, we develop a general method that allows assimilation of empirical amino acid replacement probabilities into a codon-substitution matrix. In this way, the resulting codon model takes into account not only the transition-transversion bias and the nonsynonymous/synonymous ratio, but also the different amino acid replacement probabilities as specified in empirical amino acid matrices. Different empirical amino acid replacement matrices, such as secondary structure-specific matrices or organelle-specific matrices (e.g., mitochondria and chloroplasts), can be incorporated into the model, making it context dependent. Using a diverse set of coding DNA sequences, we show that the novel model better fits biological data as compared with either mechanistic or empirical codon models. Using the suggested model, we further analyze human immunodeficiency virus type 1 protease sequences obtained from drug-treated patients and reveal positive selection in sites that are known to confer drug resistance to the virus.     

Synonymous substitutions substantially improve evolutionary inference from highly diverged proteins

Codon-and amino acid-substitution models are widely used for the evolutionary analysis of protein-coding DNA sequences. Using codon models, the amounts of both nonsynonymous and synonymous DNA substitutions can be estimated. The ratio of these amounts represents the strength of selective pressure. Using amino acid models, the amount of nonsynonymous substitutions is estimated, but that of synonymous substitutions is ignored. Although amino acid models lose any information regarding synonymous substitutions, they explicitly incorporate the information for amino acid replacement, which is empirically derived from databases. It is often presumed that when the protein-coding sequences are highly divergent, synonymous substitutions might be saturated and the evolutionary analysis may be hampered by synonymous noise. However, there exists no quantitative procedure to verify whether synonymous substitutions can be ignored; therefore, amino acid models have been arbitrarily selected. In this study, we investigate the issue of a statistical comparison between codon-and amino acid-substitution models. For this purpose, we propose a new procedure to transform a 20-dimensional amino acid model to a 61-dimensional codon model. This transformation reveals that amino acid models belong to a subset of the codon models and enables us to test whether synonymous substitutions can be ignored by using the likelihood ratio. Our theoretical results and analyses of real data indicate that synonymous substitutions are very informative and substantially improve evolutionary inference, even when the sequences are highly divergent. Therefore, we note that amino acid models should be adopted only after carefully investigating and discarding the possibility that synonymous substitutions can reveal important evolutionary information.     

Comparing PAR transmission models for forest understorey vegetation

Question: Do Beer's Law models, multi‐layer scattering models, and a semi‐empirical model for predicting PAR transmission through understorey vegetation give comparable results? Do different driving variables (LAI, PLAI and percentage cover) give different results? How do the models vary when fit with species‐specific, species‐average and the ‘default’ parameters recommended in the literature? Location: Upland boreal forests of western North America. Methods: In calibration and validation plots, PAR transmission was measured, total cover visually estimated, and leaf dispersion, PLAI and cover estimated for each species using a point‐frame. Leaf inclination was measured by clinometer. PAR transmission was modelled using empirically‐fit Beer's Law models, a semi‐empirical model based on hemispherical gap fraction and first‐order scattering, and a multi‐layer model allowing multiple scattering. All models were modified to use leaf area index (LAI), vertically projected leaf area index (PLAI), or percentage cover data. Results: The empirical Beer's Law models had the least bias and best precision in predicting PAR transmission. The semi‐empirical model also had little bias and good precision, since the scattering coefficient compensated for problems in the estimation of gap fraction. The multi‐layer model consistently underestimated transmission. There was little benefit in accounting for species separately. LAI and PLAI‐based models were the most precise, but percentage cover models also provided reasonable predictions of PAR transmission. Conclusions: PAR transmission through forest understories can be simply modelled with Beer's Law using one empirical coefficient representing the average understorey species. More complex scattering models are less effective, likely because they fail to account for the complexity of the dispersion of this vegetation layer and its effect on radiation scattering.