On the statistical interpretation of site-specific variables in phylogeny-based substitution models

Evaluation of Methods to Detect Shifts in Directional Selection at the Genome Scale

Identifying the footprints of selection in coding sequences can inform about the importance and function of individual sites. Analyses of the ratio of nonsynonymous to synonymous substitutions (dN/dS) have been widely used to pinpoint changes in the intensity of selection, but cannot distinguish them from changes in the direction of selection, that is, changes in the fitness of specific amino acids at a given position. A few methods that rely on amino-acid profiles to detect changes in directional selection have been designed, but their performances have not been well characterized. In this paper, we investigate the performance of six of these methods. We evaluate them on simulations along empirical phylogenies in which transition events have been annotated and compare their ability to detect sites that have undergone changes in the direction or intensity of selection to that of a widely used dN/dS approach, codeml's branch-site model A. We show that all methods have reduced performance in the presence of biased gene conversion but not CpG hypermutability. The best profile method, Pelican, a new implementation of Tamuri AU, Hay AJ, Goldstein RA. (2009. Identifying changes in selective constraints: host shifts in influenza. PLoS Comput Biol. 5(11):e1000564), performs as well as codeml in a range of conditions except for detecting relaxations of selection, and performs better when tree length increases, or in the presence of persistent positive selection. It is fast, enabling genome-scale searches for site-wise changes in the direction of selection associated with phenotypic changes.

A Bayesian Mutation-Selection Framework for Detecting Site-Specific Adaptive Evolution in Protein-Coding Genes

In recent years, codon substitution models based on the mutation–selection principle have been extended for the purpose of detecting signatures of adaptive evolution in protein-coding genes. However, the approaches used to date have either focused on detecting global signals of adaptive regimes—across the entire gene—or on contexts where experimentally derived, site-specific amino acid fitness profiles are available. Here, we present a Bayesian site-heterogeneous mutation–selection framework for site-specific detection of adaptive substitution regimes given a protein-coding DNA alignment. We offer implementations, briefly present simulation results, and apply the approach on a few real data sets. Our analyses suggest that the new approach shows greater sensitivity than traditional methods. However, more study is required to assess the impact of potential model violations on the method, and gain a greater empirical sense its behavior on a broader range of real data sets. We propose an outline of such a research program.

Efficient inference, potential, and limitations of site-specific substitution models

Natural selection imposes a complex filter on which variants persist in a population resulting in evolutionary patterns that vary greatly along the genome. Some sites evolve close to neutrally, while others are highly conserved, allow only specific states, or only change in concert with other sites. On one hand, such constraints on sequence evolution can be to infer biological function, one the other hand they need to be accounted for in phylogenetic reconstruction. Phylogenetic models often account for this complexity by partitioning sites into a small number of discrete classes with different rates and/or state preferences. Appropriate model complexity is typically determined by model selection procedures. Here, we present an efficient algorithm to estimate more complex models that allow for different preferences at every site and explore the accuracy at which such models can be estimated from simulated data. Our iterative approximate maximum likelihood scheme uses information in the data efficiently and accurately estimates site-specific preferences from large data sets with moderately diverged sequences and known topology. However, the joint estimation of site-specific rates, and site-specific preferences, and phylogenetic branch length can suffer from identifiability problems, while ignoring variation in preferences across sites results in branch length underestimates. Site-specific preferences estimated from large HIV pol alignments show qualitative concordance with intra-host estimates of fitness costs. Analysis of these substitution models suggests near saturation of divergence after a few hundred years. Such saturation can explain the inability to infer deep divergence times of HIV and SIVs using molecular clock approaches and time-dependent rate estimates.

Site-Specific Amino Acid Distributions Follow a Universal Shape

In many applications of evolutionary inference, a model of protein evolution needs to be fitted to the amino acid variation at individual sites in a multiple sequence alignment. Most existing models fall into one of two extremes: Either they provide a coarse-grained description that lacks biophysical realism (e.g., dN/dS models), or they require a large number of parameters to be fitted (e.g., mutation-selection models). Here, we ask whether a middle ground is possible: Can we obtain a realistic description of site-specific amino acid frequencies while severely restricting the number of free parameters in the model? We show that a distribution with a single free parameter can accurately capture the variation in amino acid frequency at most sites in an alignment, as long as we are willing to restrict our analysis to predicting amino acid frequencies by rank rather than by amino acid identity. This result holds equally well both in alignments of empirical protein sequences and of sequences evolved under a biophysically realistic all-atom force field. Our analysis reveals a near universal shape of the frequency distributions of amino acids. This insight has the potential to lead to new models of evolution that have both increased realism and a limited number of free parameters.

Hybrid of Restricted and Penalized Maximum Likelihood Method for Efficient Genome-Wide Association Study

In genome-wide association studies, linear mixed models (LMMs) have been widely used to explore the molecular mechanism of complex traits. However, typical association approaches suffer from several important drawbacks: estimation of variance components in LMMs with large scale individuals is computationally slow; single-locus model is unsatisfactory to handle complex confounding and causes loss of statistical power. To address these issues, we propose an efficient two-stage method based on hybrid of restricted and penalized maximum likelihood, named HRePML. Firstly, we performed restricted maximum likelihood (REML) on single-locus LMM to remove unrelated markers, where spectral decomposition on covariance matrix was used to fast estimate variance components. Secondly, we carried out penalized maximum likelihood (PML) on multi-locus LMM for markers with reasonably large effects. To validate the effectiveness of HRePML, we conducted a series of simulation studies and real data analyses. As a result, our method always had the highest average statistical power compared with multi-locus mixed-model (MLMM), fixed and random model circulating probability unification (FarmCPU), and genome-wide efficient mixed model association (GEMMA). More importantly, HRePML can provide higher accuracy estimation of marker effects. HRePML also identifies 41 previous reported genes associated with development traits in Arabidopsis, which is more than was detected by the other methods.

Conditional Approximate Bayesian Computation: A New Approach for Across-Site Dependency in High-Dimensional Mutation-Selection Models

A key question in molecular evolutionary biology concerns the relative roles of mutation and selection in shaping genomic data. Moreover, features of mutation and selection are heterogeneous along the genome and over time. Mechanistic codon substitution models based on the mutation-selection framework are promising approaches to separating these effects. In practice, however, several complications arise, since accounting for such heterogeneities often implies handling models of high dimensionality (e.g., amino acid preferences), or leads to across-site dependence (e.g., CpG hypermutability), making the likelihood function intractable. Approximate Bayesian Computation (ABC) could address this latter issue. Here, we propose a new approach, named Conditional ABC (CABC), which combines the sampling efficiency of MCMC and the flexibility of ABC. To illustrate the potential of the CABC approach, we apply it to the study of mammalian CpG hypermutability based on a new mutation-level parameter implying dependence across adjacent sites, combined with site-specific purifying selection on amino-acids captured by a Dirichlet process. Our proof-of-concept of the CABC methodology opens new modeling perspectives. Our application of the method reveals a high level of heterogeneity of CpG hypermutability across loci and mild heterogeneity across taxonomic groups; and finally, we show that CpG hypermutability is an important evolutionary factor in rendering relative synonymous codon usage. All source code is available as a GitHub repository (https://github.com/Simonll/LikelihoodFreePhylogenetics.git).

Detecting amino acid preference shifts with codon-level mutation-selection mixture models

BACKGROUND

In recent years, increasing attention has been placed on the development of phylogeny-based statistical methodologies for uncovering site-specific changes in amino acid fitness profiles over time. The few available random-effects approaches, modelling across-site variation in amino acid profiles as random variables drawn from a statistical law, either lack a mechanistic codon-level formulation, or pose significant computational challenges.

RESULTS

Here, we bring together a few existing ideas to explore a simple and fast method based on a predefined finite mixture of amino acid profiles within a codon-level substitution model following the mutation-selection formulation. Our study is focused on the detection of site-specific shifts in amino acid profiles over a known sub-clade of a tree, using simulations with and without shifts over the sub-clade to study the properties of the method. Through modifications of the values of the amino acid profiles, our simulations show different levels of reliability under different forms of finite mixture models. Sites identified by our method in a real data set show obvious overlap with those identified using previous methods, with some notable differences.

CONCLUSION

Overall, our results show that when a site-specific shift in amino acid profile is strongly pronounced, involving two clearly different sets of profiles, the method performs very well; but shifts between profiles that share many features are difficult to correctly identify, highlighting the challenging nature of the problem.

Evolution of Viral Genomes: Interplay Between Selection, Recombination, and Other Forces

Natural selection is a fundamental force shaping organismal evolution, as it both maintains function and enables adaptation and innovation. Viruses, with their typically short and largely coding genomes, experience strong and diverse selective forces, sometimes acting on timescales that can be directly measured. These selection pressures emerge from an antagonistic interplay between rapidly changing fitness requirements (immune and antiviral responses from hosts, transmission between hosts, or colonization of new host species) and functional imperatives (the ability to infect hosts or host cells and replicate within hosts). Indeed, computational methods to quantify these evolutionary forces using molecular sequence data were initially, dating back to the 1980s, applied to the study of viral pathogens. This preference largely emerged because the strong selective forces are easiest to detect in viruses, and, of course, viruses have clear biomedical relevance. Recent commoditization of affordable high-throughput sequencing has made it possible to generate truly massive genomic data sets, on which powerful and accurate methods can yield a very detailed depiction of when, where, and (sometimes) how viral pathogens respond to various selective forces.Here, we present recent statistical developments and state-of-the-art methods to identify and characterize these selection pressures from protein-coding sequence alignments and phylogenies. Methods described here can reveal critical information about various evolutionary regimes, including whole-gene selection, lineage-specific selection, and site-specific selection acting upon viral genomes, while accounting for confounding biological processes, such as recombination and variation in mutation rates.

A Not-So-Long Introduction to Computational Molecular Evolution

In this chapter, we give a not-so-long and self-contained introduction to computational molecular evolution. In particular, we present the emergence of the use of likelihood-based methods, review the standard DNA substitution models, and introduce how model choice operates. We also present recent developments in inferring absolute divergence times and rates on a phylogeny, before showing how state-of-the-art models take inspiration from diffusion theory to link population genetics, which traditionally focuses at a taxonomic level below that of the species, and molecular evolution. Although this is not a cookbook chapter, we try and point to popular programs and implementations along the way.

Modeling site-specific amino-acid preferences deepens phylogenetic estimates of viral sequence divergence

Molecular phylogenetics is often used to estimate the time since the divergence of modern gene sequences. For highly diverged sequences, such phylogenetic techniques sometimes estimate surprisingly recent divergence times. In the case of viruses, independent evidence indicates that the estimates of deep divergence times from molecular phylogenetics are sometimes too recent. This discrepancy is caused in part by inadequate models of purifying selection leading to branch-length underestimation. Here we examine the effect on branch-length estimation of using models that incorporate experimental measurements of purifying selection. We find that models informed by experimentally measured site-specific amino-acid preferences estimate longer deep branches on phylogenies of influenza virus hemagglutinin. This lengthening of branches is due to more realistic stationary states of the models, and is mostly independent of the branch-length extension from modeling site-to-site variation in amino-acid substitution rate. The branch-length extension from experimentally informed site-specific models is similar to that achieved by other approaches that allow the stationary state to vary across sites. However, the improvements from all of these site-specific but time homogeneous and site independent models are limited by the fact that a protein’s amino-acid preferences gradually shift as it evolves. Overall, our work underscores the importance of modeling site-specific amino-acid preferences when estimating deep divergence times—but also shows the inherent limitations of approaches that fail to account for how these preferences shift over time.

Using the Mutation-Selection Framework to Characterize Selection on Protein Sequences

When mutational pressure is weak, the generative process of protein evolution involves explicit probabilities of mutations of different types coupled to their conditional probabilities of fixation dependent on selection. Establishing this mechanistic modeling framework for the detection of selection has been a goal in the field of molecular evolution. Building on a mathematical framework proposed more than a decade ago, numerous methods have been introduced in an attempt to detect and measure selection on protein sequences. In this review, we discuss the structure of the original model, subsequent advances, and the series of assumptions that these models operate under.

Modeling Site Heterogeneity with Posterior Mean Site Frequency Profiles Accelerates Accurate Phylogenomic Estimation

Proteins have distinct structural and functional constraints at different sites that lead to site-specific preferences for particular amino acid residues as the sequences evolve. Heterogeneity in the amino acid substitution process between sites is not modeled by commonly used empirical amino acid exchange matrices. Such model misspecification can lead to artefacts in phylogenetic estimation such as long-branch attraction. Although sophisticated site-heterogeneous mixture models have been developed to address this problem in both Bayesian and maximum likelihood (ML) frameworks, their formidable computational time and memory usage severely limits their use in large phylogenomic analyses. Here we propose a posterior mean site frequency (PMSF) method as a rapid and efficient approximation to full empirical profile mixture models for ML analysis. The PMSF approach assigns a conditional mean amino acid frequency profile to each site calculated based on a mixture model fitted to the data using a preliminary guide tree. These PMSF profiles can then be used for in-depth tree-searching in place of the full mixture model. Compared with widely used empirical mixture models with $k$ classes, our implementation of PMSF in IQ-TREE (http://www.iqtree.org) speeds up the computation by approximately $k$/1.5-fold and requires a small fraction of the RAM. Furthermore, this speedup allows, for the first time, full nonparametric bootstrap analyses to be conducted under complex site-heterogeneous models on large concatenated data matrices. Our simulations and empirical data analyses demonstrate that PMSF can effectively ameliorate long-branch attraction artefacts. In some empirical and simulation settings PMSF provided more accurate estimates of phylogenies than the mixture models from which they derive.

Parallel Evolution of Metazoan Mitochondrial Proteins

Amino acid propensities at amino acid sites change with time due to epistatic interactions or changing environment, affecting the probabilities of fixation of different amino acids. Such changes should lead to an increased rate of homoplasies (reversals, parallelisms, and convergences) at closely related species. Here, we reconstruct the phylogeny of twelve mitochondrial proteins from several thousand metazoan species, and measure the phylogenetic distances between branches at which either the same allele originated repeatedly due to homoplasies, or different alleles originated due to divergent substitutions. The mean phylogenetic distance between parallel substitutions is ∼20% lower than the mean phylogenetic distance between divergent substitutions, indicating that a variant fixed in a species is more likely to be deleterious in a more phylogenetically remote species, compared with a more closely related species. These findings are robust to artefacts of phylogenetic reconstruction or of pooling of sites from different conservation classes or functional groups, and imply that single-position fitness landscapes change at rates similar to rates of amino acid changes.

Sequence entropy of folding and the absolute rate of amino acid substitutions

Adequate representations of protein evolution should consider how the acceptance of mutations depends on the sequence context in which they arise. However, epistatic interactions among sites in a protein result in time and spatial substitution rate heterogeneity beyond the capabilities of current models. Here, we exploit parallels between amino acid substitutions and chemical reaction kinetics to develop an improved theory of protein evolution. We constructed a mechanistic framework for modelling amino acid substitution rates that employs the formalisms of statistical mechanics, with population genetics principles underlying the analysis. Theoretical analyses and computer simulations of proteins under purifying selection for thermodynamic stability show that substitution rates and the stabilisation of resident amino acids (the ‘evolutionary Stokes shift’) can be predicted from biophysics and the effect of sequence entropy alone. Furthermore, we demonstrate that substitutions predominantly occur when epistatic interactions result in near neutrality; substitution rates are determined by how often epistasis results in such nearly neutral conditions. This theory provides a general framework for modelling protein sequence change under purifying selection, potentially explains patterns of convergence and mutation rates in real proteins that are incompatible with previous models, and provides a better null model for the detection of adaptive changes.

Identification of positive selection in genes is greatly improved by using experimentally informed site-specific models

Background

Sites of positive selection are identified by comparing observed evolutionary patterns to those expected under a null model for evolution in the absence of such selection. For protein-coding genes, the most common null model is that nonsynonymous and synonymous mutations fix at equal rates; this unrealistic model has limited power to detect many interesting forms of selection.

Results

I describe a new approach that uses a null model based on experimental measurements of a gene’s site-specific amino-acid preferences generated by deep mutational scanning in the lab. This null model makes it possible to identify both diversifying selection for repeated amino-acid change and differential selection for mutations to amino acids that are unexpected given the measurements made in the lab. I show that this approach identifies sites of adaptive substitutions in four genes (lactamase, Gal4, influenza nucleoprotein, and influenza hemagglutinin) far better than a comparable method that simply compares the rates of nonsynonymous and synonymous substitutions.

Conclusions

As rapid increases in biological data enable increasingly nuanced descriptions of the constraints on individual protein sites, approaches like the one here can improve our ability to identify many interesting forms of selection in natural sequences.

Reviewers

This article was reviewed by Sebastian Maurer-Stroh, Olivier Tenaillon, and Tal Pupko. All three reviewers are members of the Biology Direct editorial board.

Electronic supplementary material

The online version of this article (doi:10.1186/s13062-016-0172-z) contains supplementary material, which is available to authorized users.

Finding Direction in the Search for Selection

Tests for positive selection have mostly been developed to look for diversifying selection where change away from the current amino acid is often favorable. However, in many cases we are interested in directional selection where there is a shift toward specific amino acids, resulting in increased fitness in the species. Recently, a few methods have been developed to detect and characterize directional selection on a molecular level. Using the results of evolutionary simulations as well as HIV drug resistance data as models of directional selection, we compare two such methods with each other, as well as against a standard method for detecting diversifying selection. We find that the method to detect diversifying selection also detects directional selection under certain conditions. One method developed for detecting directional selection is powerful and accurate for a wide range of conditions, while the other can generate an excessive number of false positives.

Changing preferences: deformation of single position amino acid fitness landscapes and evolution of proteins

The fitness landscape-the function that relates genotypes to fitness-and its role in directing evolution are a central object of evolutionary biology. However, its huge dimensionality precludes understanding of even the basic aspects of its shape. One way to approach it is to ask a simpler question: what are the properties of a function that assigns fitness to each possible variant at just one particular site-a single position fitness landscape-and how does it change in the course of evolution? Analyses of genomic data from multiple species and multiple individuals within a species have proved beyond reasonable doubt that fitness functions of positions throughout the genome do themselves change with time, thus shaping protein evolution. Here, I will briefly review the literature that addresses these dynamics, focusing on recent genome-scale analyses of fitness functions of amino acid sites, i.e. vectors of fitnesses of 20 individual amino acid variants at a given position of a protein. The set of amino acids that confer high fitness at a particular position changes with time, and the rate of this change is comparable with the rate at which a position evolves, implying that this process plays a major role in evolutionary dynamics. However, the causes of these changes remain largely unclear.

Detecting Adaptation in Protein-Coding Genes Using a Bayesian Site-Heterogeneous Mutation-Selection Codon Substitution Model

Codon substitution models have traditionally attempted to uncover signatures of adaptation within protein-coding genes by contrasting the rates of synonymous and non-synonymous substitutions. Another modeling approach, known as the mutation–selection framework, attempts to explicitly account for selective patterns at the amino acid level, with some approaches allowing for heterogeneity in these patterns across codon sites. Under such a model, substitutions at a given position occur at the neutral or nearly neutral rate when they are synonymous, or when they correspond to replacements between amino acids of similar fitness; substitutions from high to low (low to high) fitness amino acids have comparatively low (high) rates. Here, we study the use of such a mutation–selection framework as a null model for the detection of adaptation. Following previous works in this direction, we include a deviation parameter that has the effect of capturing the surplus, or deficit, in non-synonymous rates, relative to what would be expected under a mutation–selection modeling framework that includes a Dirichlet process approach to account for across-codon-site variation in amino acid fitness profiles. We use simulations, along with a few real data sets, to study the behavior of the approach, and find it to have good power with a low false-positive rate. Altogether, we emphasize the potential of recent mutation–selection models in the detection of adaptation, calling for further model refinements as well as large-scale applications.

Extensively Parameterized Mutation-Selection Models Reliably Capture Site-Specific Selective Constraint

The mutation-selection model of coding sequence evolution has received renewed attention for its use in estimating site-specific amino acid propensities and selection coefficient distributions. Two computationally tractable mutation-selection inference frameworks have been introduced: One framework employs a fixed-effects, highly parameterized maximum likelihood approach, whereas the other employs a random-effects Bayesian Dirichlet Process approach. While both implementations follow the same model, they appear to make distinct predictions about the distribution of selection coefficients. The fixed-effects framework estimates a large proportion of highly deleterious substitutions, whereas the random-effects framework estimates that all substitutions are either nearly neutral or weakly deleterious. It remains unknown, however, how accurately each method infers evolutionary constraints at individual sites. Indeed, selection coefficient distributions pool all site-specific inferences, thereby obscuring a precise assessment of site-specific estimates. Therefore, in this study, we use a simulation-based strategy to determine how accurately each approach recapitulates the selective constraint at individual sites. We find that the fixed-effects approach, despite its extensive parameterization, consistently and accurately estimates site-specific evolutionary constraint. By contrast, the random-effects Bayesian approach systematically underestimates the strength of natural selection, particularly for slowly evolving sites. We also find that, despite the strong differences between their inferred selection coefficient distributions, the fixed- and random-effects approaches yield surprisingly similar inferences of site-specific selective constraint. We conclude that the fixed-effects mutation-selection framework provides the more reliable software platform for model application and future development.

Detection and sequence/structure mapping of biophysical constraints to protein variation in saturated mutational libraries and protein sequence alignments with a dedicated server

Background

Protein variability can now be studied by measuring high-resolution tolerance-to-substitution maps and fitness landscapes in saturated mutational libraries. But these rich and expensive datasets are typically interpreted coarsely, restricting detailed analyses to positions of extremely high or low variability or dubbed important beforehand based on existing knowledge about active sites, interaction surfaces, (de)stabilizing mutations, etc.

Results

Our new webserver PsychoProt (freely available without registration at http://psychoprot.epfl.ch or at http://lucianoabriata.altervista.org/psychoprot/index.html) helps to detect, quantify, and sequence/structure map the biophysical and biochemical traits that shape amino acid preferences throughout a protein as determined by deep-sequencing of saturated mutational libraries or from large alignments of naturally occurring variants.

Discussion

We exemplify how PsychoProt helps to (i) unveil protein structure-function relationships from experiments and from alignments that are consistent with structures according to coevolution analysis, (ii) recall global information about structural and functional features and identify hitherto unknown constraints to variation in alignments, and (iii) point at different sources of variation among related experimental datasets or between experimental and alignment-based data. Remarkably, metabolic costs of the amino acids pose strong constraints to variability at protein surfaces in nature but not in the laboratory. This and other differences call for caution when extrapolating results from in vitro experiments to natural scenarios in, for example, studies of protein evolution.

Conclusion

We show through examples how PsychoProt can be a useful tool for the broad communities of structural biology and molecular evolution, particularly for studies about protein modeling, evolution and design.

Electronic supplementary material

The online version of this article (doi:10.1186/s12859-016-1124-4) contains supplementary material, which is available to authorized users.

Causes of evolutionary rate variation among protein sites

It has long been recognized that certain sites within a protein, such as sites in the protein core or catalytic residues in enzymes, are evolutionarily more conserved than other sites. However, our understanding of rate variation among sites remains surprisingly limited. Recent progress to address this includes the development of a wide array of reliable methods to estimate site-specific substitution rates from sequence alignments. In addition, several molecular traits have been identified that correlate with site-specific mutation rates, and novel mechanistic biophysical models have been proposed to explain the observed correlations. Nonetheless, current models explain, at best, approximately 60% of the observed variance, highlighting the limitations of current methods and models and the need for new research directions.

Modeling evolution using the probability of fixation: history and implications

Many models of evolution calculate the rate of evolution by multiplying the rate at which new mutations originate within a population by a probability of fixation. Here we review the historical origins, contemporary applications, and evolutionary implications of these "origin-fixation" models, which are widely used in evolutionary genetics, molecular evolution, and phylogenetics. Origin-fixation models were first introduced in 1969, in association with an emerging view of "molecular" evolution. Early origin-fixation models were used to calculate an instantaneous rate of evolution across a large number of independently evolving loci; in the 1980s and 1990s, a second wave of origin-fixation models emerged to address a sequence of fixation events at a single locus. Although origin fixation models have been applied to a broad array of problems in contemporary evolutionary research, their rise in popularity has not been accompanied by an increased appreciation of their restrictive assumptions or their distinctive implications. We argue that origin-fixation models constitute a coherent theory of mutation-limited evolution that contrasts sharply with theories of evolution that rely on the presence of standing genetic variation. A major unsolved question in evolutionary biology is the degree to which these models provide an accurate approximation of evolution in natural populations.

An experimentally informed evolutionary model improves phylogenetic fit to divergent lactamase homologs

Phylogenetic analyses of molecular data require a quantitative model for how sequences evolve. Traditionally, the details of the site-specific selection that governs sequence evolution are not known a priori, making it challenging to create evolutionary models that adequately capture the heterogeneity of selection at different sites. However, recent advances in high-throughput experiments have made it possible to quantify the effects of all single mutations on gene function. I have previously shown that such high-throughput experiments can be combined with knowledge of underlying mutation rates to create a parameter-free evolutionary model that describes the phylogeny of influenza nucleoprotein far better than commonly used existing models. Here, I extend this work by showing that published experimental data on TEM-1 beta-lactamase (Firnberg E, Labonte JW, Gray JJ, Ostermeier M. 2014. A comprehensive, high-resolution map of a gene’s fitness landscape. Mol Biol Evol. 31:1581–1592) can be combined with a few mutation rate parameters to create an evolutionary model that describes beta-lactamase phylogenies much better than most common existing models. This experimentally informed evolutionary model is superior even for homologs that are substantially diverged (about 35% divergence at the protein level) from the TEM-1 parent that was the subject of the experimental study. These results suggest that experimental measurements can inform phylogenetic evolutionary models that are applicable to homologs that span a substantial range of sequence divergence.

An experimentally determined evolutionary model dramatically improves phylogenetic fit

All modern approaches to molecular phylogenetics require a quantitative model for how genes evolve. Unfortunately, existing evolutionary models do not realistically represent the site-heterogeneous selection that governs actual sequence change. Attempts to remedy this problem have involved augmenting these models with a burgeoning number of free parameters. Here, I demonstrate an alternative: Experimental determination of a parameter-free evolutionary model via mutagenesis, functional selection, and deep sequencing. Using this strategy, I create an evolutionary model for influenza nucleoprotein that describes the gene phylogeny far better than existing models with dozens or even hundreds of free parameters. Emerging high-throughput experimental strategies such as the one employed here provide fundamentally new information that has the potential to transform the sensitivity of phylogenetic and genetic analyses.

On the validity of evolutionary models with site-specific parameters

Evolutionary models that make use of site-specific parameters have recently been criticized on the grounds that parameter estimates obtained under such models can be unreliable and lack theoretical guarantees of convergence. We present a simulation study providing empirical evidence that a simple version of the models in question does exhibit sensible convergence behavior and that additional taxa, despite not being independent of each other, lead to improved parameter estimates. Although it would be desirable to have theoretical guarantees of this, we argue that such guarantees would not be sufficient to justify the use of these models in practice. Instead, we emphasize the importance of taking the variance of parameter estimates into account rather than blindly trusting point estimates – this is standardly done by using the models to construct statistical hypothesis tests, which are then validated empirically via simulation studies.

A penalized-likelihood method to estimate the distribution of selection coefficients from phylogenetic data

We develop a maximum penalized-likelihood (MPL) method to estimate the fitnesses of amino acids and the distribution of selection coefficients (S = 2Ns) in protein-coding genes from phylogenetic data. This improves on a previous maximum-likelihood method. Various penalty functions are used to penalize extreme estimates of the fitnesses, thus correcting overfitting by the previous method. Using a combination of computer simulation and real data analysis, we evaluate the effect of the various penalties on the estimation of the fitnesses and the distribution of S. We show the new method regularizes the estimates of the fitnesses for small, relatively uninformative data sets, but it can still recover the large proportion of deleterious mutations when present in simulated data. Computer simulations indicate that as the number of taxa in the phylogeny or the level of sequence divergence increases, the distribution of S can be more accurately estimated. Furthermore, the strength of the penalty can be varied to study how informative a particular data set is about the distribution of S. We analyze three protein-coding genes (the chloroplast rubisco protein, mammal mitochondrial proteins, and an influenza virus polymerase) and show the new method recovers a large proportion of deleterious mutations in these data, even under strong penalties, confirming the distribution of S is bimodal in these real data. We recommend the use of the new MPL approach for the estimation of the distribution of S in species phylogenies of protein-coding genes.

An amino acid substitution-selection model adjusts residue fitness to improve phylogenetic estimation

Standard protein phylogenetic models use fixed rate matrices of amino acid interchange derived from analyses of large databases. Differences between the stationary amino acid frequencies of these rate matrices from those of a data set of interest are typically adjusted for by matrix multiplication that converts the empirical rate matrix to an exchangeability matrix which is then postmultiplied by the amino acid frequencies in the alignment. The result is a time-reversible rate matrix with stationary amino acid frequencies equal to the data set frequencies. On the basis of population genetics principles, we develop an amino acid substitution-selection model that parameterizes the fitness of an amino acid as the logarithm of the ratio of the frequency of the amino acid to the frequency of the same amino acid under no selection. The model gives rise to a different sequence of matrix multiplications to convert an empirical rate matrix to one that has stationary amino acid frequencies equal to the data set frequencies. We incorporated the substitution-selection model with an improved amino acid class frequency mixture (cF) model to partially take into account site-specific amino acid frequencies in the phylogenetic models. We show that 1) the selection models fit data significantly better than corresponding models without selection for most of the 21 test data sets; 2) both cF and cF selection models favored the phylogenetic trees that were inferred under current sophisticated models and methods for three difficult phylogenetic problems (the positions of microsporidia and breviates in eukaryote phylogeny and the position of the root of the angiosperm tree); and 3) for data simulated under site-specific residue frequencies, the cF selection models estimated trees closer to the generating trees than a standard Г model or cF without selection. We also explored several ways of estimating amino acid frequencies under neutral evolution that are required for these selection models. By better modeling the amino acid substitution process, the cF selection models will be valuable for phylogenetic inference and evolutionary studies.

Site-heterogeneous mutation-selection models within the PhyloBayes-MPI package

Motivation: In recent years, there has been an increasing interest in the potential of codon substitution models for a variety of applications. However, the computational demands of these models have sometimes lead to the adoption of oversimplified assumptions, questionable statistical methods or a limited focus on small data sets.

Results: Here, we offer a scalable, message-passing-interface-based Bayesian implementation of site-heterogeneous codon models in the mutation-selection framework. Our software jointly infers the global mutational parameters at the nucleotide level, the branch lengths of the tree and a Dirichlet process governing across-site variation at the amino acid level. We focus on an example estimation of the distribution of selection coefficients from an alignment of several hundred sequences of the influenza PB2 gene, and highlight the site-specific characterization enabled by such a modeling approach. Finally, we discuss future potential applications of the software for conducting evolutionary inferences.

Availability and implementation: The models are implemented within the PhyloBayes-MPI package, (available at phylobayes.org) along with usage details in the accompanying manual.

Contact:nicolas.rodrigue@ucalgary.ca