Toric ideals of phylogenetic invariants.

Statistical models of evolution are algebraic varieties in the space of joint probability distributions on the leaf colorations of a phylogenetic tree. The phylogenetic invariants of a model are the polynomials which vanish on the variety. Several widely used models for biological sequences have transition matrices that can be diagonalized by means of the Fourier transform of an abelian group. Their phylogenetic invariants form a toric ideal in the Fourier coordinates. We determine generators and Gr?bner bases for these toric ideals. For the Jukes-Cantor and Kimura models on a binary tree, our Gr?bner bases consist of certain explicitly constructed polynomials of degree at most four.     

SPIn: model selection for phylogenetic mixtures via linear invariants

In phylogenetic inference, an evolutionary model describes the substitution processes along each edge of a phylogenetic tree. Misspecification of the model has important implications for the analysis of phylogenetic data. Conventionally, however, the selection of a suitable evolutionary model is based on heuristics or relies on the choice of an approximate input tree. We introduce a method for model Selection in Phylogenetics based on linear INvariants (SPIn), which uses recent insights on linear invariants to characterize a model of nucleotide evolution for phylogenetic mixtures on any number of components. Linear invariants are constraints among the joint probabilities of the bases in the operational taxonomic units that hold irrespective of the tree topologies appearing in the mixtures. SPIn therefore requires no input tree and is designed to deal with nonhomogeneous phylogenetic data consisting of multiple sequence alignments showing different patterns of evolution, for example, concatenated genes, exons, and/or introns. Here, we report on the results of the proposed method evaluated on multiple sequence alignments simulated under a variety of single-tree and mixture settings for both continuous- and discrete-time models. In the simulations, SPIn successfully recovers the underlying evolutionary model and is shown to perform better than existing approaches.     

Toric ideals of phylogenetic invariants.

Statistical models of evolution are algebraic varieties in the space of joint probability distributions on the leaf colorations of a phylogenetic tree. The phylogenetic invariants of a model are the polynomials which vanish on the variety. Several widely used models for biological sequences have transition matrices that can be diagonalized by means of the Fourier transform of an Abelian group. Their phylogenetic invariants form a toric ideal in the Fourier coordinates. We determine generators and Gr?bner bases for these toric ideals. For the Jukes-Cantor and Kimura models on a binary tree, our Gr?bner bases consist of certain explicitly constructed polynomials of degree at most four.     

Designer invariants for large phylogenies.

The Cavender-Felsenstein edge-length invariants for binary characters on 4-trees provide the starting point for the development of "customized" invariants for evaluating and comparing phylogenetic hypotheses. The binary character invariants may be generalized to k-valued characters without losing the quadratic nature of the invariants as functions of the theoretical frequencies f(UVXY) of observable character configurations (U at organism 1, V at 2, etc.). The key to the approach is that certain sets of these configurations constitute events which are probabilistically independent from other such sets, under the symmetric Markov change models studied. By introducing more complex sets of configurations, we find the quadratic invariants for 5-trees in the binary model and for individual edges in 6-trees or, indeed, in any size tree. The same technique allows us to formulate invariants for entire trees, but these are cubic functions for 6-trees and are higher-degree polynomials for larger trees. With k-valued characters and, especially, with large trees, the types of configuration sets (events) used in the simpler examples are too rare (i.e., their predicted frequencies are too low) to be useful, and the construction of meaningful pairs of independent events becomes an important and nontrivial task in designing invariants suited to testing specific hypotheses. In a very natural way, this approach fits in with well-known statistical methodology for contingency tables. We explore use of events such as "only transitions occur for character i (i.e., position i in a nucleic acid sequence) in subtree a" in analyzing a set of data on ribosomal RNA in the context of the controversy over the origins of archaebacteria, eubacteria, and eukaryotes.     

The identifiability of tree topology for phylogenetic models, including covarion and mixture models.

For a model of molecular evolution to be useful for phylogenetic inference, the topology of evolutionary trees must be identifiable. That is, from a joint distribution the model predicts, it must be possible to recover the tree parameter. We establish tree identifiability for a number of phylogenetic models, including a covarion model and a variety of mixture models with a limited number of classes. The proof is based on the introduction of a more general model, allowing more states at internal nodes of the tree than at leaves, and the study of the algebraic variety formed by the joint distributions to which it gives rise. Tree identifiability is first established for this general model through the use of certain phylogenetic invariants.     

Dimensional Reduction for the General Markov Model on Phylogenetic Trees

We present a method of dimensional reduction for the general Markov model of sequence evolution on a phylogenetic tree. We show that taking certain linear combinations of the associated random variables (site pattern counts) reduces the dimensionality of the model from exponential in the number of extant taxa, to quadratic in the number of taxa, while retaining the ability to statistically identify phylogenetic divergence events. A key feature is the identification of an invariant subspace which depends only bilinearly on the model parameters, in contrast to the usual multi-linear dependence in the full space. We discuss potential applications including the computation of split (edge) weights on phylogenetic trees from observed sequence data.     

Multidimensional vector space representation for convergent evolution and molecular phylogeny

With growing amounts of genome data and constant improvement of models of molecular evolution, phylogenetic reconstruction became more reliable. However, our knowledge of the real process of molecular evolution is still limited. When enough large-sized data sets are analyzed, any subtle biases in statistical models can support incorrect topologies significantly because of the high signal-to-noise ratio. We propose a procedure to locate sequences in a multidimensional vector space (MVS), in which the geometry of the space is uniquely determined in such a way that the vectors of sequence evolution are orthogonal among different branches. In this paper, the MVS approach is developed to detect and remove biases in models of molecular evolution caused by unrecognized convergent evolution among lineages or unexpected patterns of substitutions. Biases in the estimated pairwise distances are identified as deviations (outliers) of sequence spatial vectors from the expected orthogonality. Modifications to the estimated distances are made by minimizing an index to quantify the deviations. In this way, it becomes possible to reconstruct the phylogenetic tree, taking account of possible biases in the model of molecular evolution. The efficacy of the modification procedure was verified by simulating evolution on various topologies with rate heterogeneity and convergent change. The phylogeny of placental mammals in previous analyses of large data sets has varied according to the genes being analyzed. Systematic deviations caused by convergent evolution were detected by our procedure in all representative data sets and were found to strongly affect the tree structure. However, the bias correction yielded a consistent topology among data sets. The existence of strong biases was validated by examining the sites of convergent evolution between the hedgehog and other species in mitochondrial data set. This convergent evolution explains why it has been difficult to determine the phylogenetic placement of the hedgehog in previous studies.     

Substitution Model of Sequence Evolution for the Human Immunodeficiency Virus Type 1 Subtype B gp120 Gene over the C2-V5 Region

Phylogenetic analyses frequently rely on models of sequence evolution that detail nucleotide substitution rates, nucleotide frequencies, and site-to-site rate heterogeneity. These models can influence hypothesis testing and can affect the accuracy of phylogenetic inferences. Maximum likelihood methods of simultaneously constructing phylogenetic tree topologies and estimating model parameters are computationally intensive, and are not feasible for sample sizes of 25 or greater using personal computers. Techniques that initially construct a tree topology and then use this non-maximized topology to estimate ML substitution rates, however, can quickly arrive at a model of sequence evolution. The accuracy of this two-step estimation technique was tested using simulated data sets with known model parameters. The results showed that for a star-like topology, as is often seen in human immunodeficiency virus type 1 (HIV-1) subtype B sequences, a random starting topology could produce nucleotide substitution rates that were not statistically different than the true rates. Samples were isolated from 100 HIV-1 subtype B infected individuals from the United States and a 620 nt region of the env gene was sequenced for each sample. The sequence data were used to obtain a substitution model of sequence evolution specific for HIV-1 subtype B env by estimating nucleotide substitution rates and the site-to-site heterogeneity in 100 individuals from the United States. The method of estimating the model should provide users of large data sets with a way to quickly compute a model of sequence evolution, while the nucleotide substitution model we identified should prove useful in the phylogenetic analysis of HIV-1 subtype B env sequences. Received: 4 October 2000 / Accepted: 1 March 2001     

COMPARISONS OF OBSERVED PHYLOGENETIC TOPOLOGIES WITH NULL EXPECTATIONS AMONG THREE MONOPHYLETIC LINEAGES

Three null models have been proposed to predict the relative frequencies of topologies of phylogenetic trees. One null model assumes each distinguishable n-member tree is equally likely (proportional-to-distinguishable-arrangements model). A second model assumes that each topological type is equally likely (equiprobable model). A third model assumes that the probability of each topological type is determined by random speciation (Markov model). We sampled published phylogenetic trees from three major groups of organisms: division Angiospermae, class Insecta, and superclass Tetrapoda. Our sampling was more restricted than previous studies and was designed to test whether observed topological frequencies were distinguishable from those predicted by the three null models. The pattern of evolution reflected in five-member phylogenetic trees is different from predictions of the equiprobable and Markov model but is indistinguishable from the proportional-to-distinguishable-arrangements model. This indicates that 1) speciation (and/or extinction) is not equally likely among all taxa, even for small phylogenies; or 2) systematists' attempts at reconstructing small phylogenies are, on average, indistinguishable from those expected if they had merely selected a tree at random from the pool of all possible trees. The topology frequencies were not different among the three groups of organisms, suggesting that factors shaping patterns of speciation and extinction are consistent among major taxonomic groups.     

Phylogenetic mixture models can reduce node-density artifacts

We investigate the performance of phylogenetic mixture models in reducing a well-known and pervasive artifact of phylogenetic inference known as the node-density effect, comparing them to partitioned analyses of the same data. The node-density effect refers to the tendency for the amount of evolutionary change in longer branches of phylogenies to be underestimated compared to that in regions of the tree where there are more nodes and thus branches are typically shorter. Mixture models allow more than one model of sequence evolution to describe the sites in an alignment without prior knowledge of the evolutionary processes that characterize the data or how they correspond to different sites. If multiple evolutionary patterns are common in sequence evolution, mixture models may be capable of reducing node-density effects by characterizing the evolutionary processes more accurately. In gene-sequence alignments simulated to have heterogeneous patterns of evolution, we find that mixture models can reduce node-density effects to negligible levels or remove them altogether, performing as well as partitioned analyses based on the known simulated patterns. The mixture models achieve this without knowledge of the patterns that generated the data and even in some cases without specifying the full or true model of sequence evolution known to underlie the data. The latter result is especially important in real applications, as the true model of evolution is seldom known. We find the same patterns of results for two real data sets with evidence of complex patterns of sequence evolution: mixture models substantially reduced node-density effects and returned better likelihoods compared to partitioning models specifically fitted to these data. We suggest that the presence of more than one pattern of evolution in the data is a common source of error in phylogenetic inference and that mixture models can often detect these patterns even without prior knowledge of their presence in the data. Routine use of mixture models alongside other approaches to phylogenetic inference may often reveal hidden or unexpected patterns of sequence evolution and can improve phylogenetic inference.     

Efficient selection of branch-specific models of sequence evolution

The analysis of extant sequences shows that molecular evolution has been heterogeneous through time and among lineages. However, for a given sequence alignment, it is often difficult to uncover what factors caused this heterogeneity. In fact, identifying and characterizing heterogeneous patterns of molecular evolution along a phylogenetic tree is very challenging, for lack of appropriate methods. Users either have to a priori define groups of branches along which they believe molecular evolution has been similar or have to allow each branch to have its own pattern of molecular evolution. The first approach assumes prior knowledge that is seldom available, and the second requires estimating an unreasonably large number of parameters. Here we propose a convenient and reliable approach where branches get clustered by their pattern of molecular evolution alone, with no need for prior knowledge about the data set under study. Model selection is achieved in a statistical framework and therefore avoids overparameterization. We rely on substitution mapping for efficiency and present two clustering approaches, depending on whether or not we expect neighbouring branches to share more similar patterns of sequence evolution than distant branches. We validate our method on simulations and test it on four previously published data sets. We find that our method correctly groups branches sharing similar equilibrium GC contents in a data set of ribosomal RNAs and recovers expected footprints of selection through dN/dS. Importantly, it also uncovers a new pattern of relaxed selection in a phylogeny of Mantellid frogs, which we are able to correlate to life-history traits. This shows that our programs should be very useful to study patterns of molecular evolution and reveal new correlations between sequence and species evolution. Our programs can run on DNA, RNA, codon, or amino acid sequences with a large set of possible models of substitutions and are available at http://biopp.univ-montp2.fr/forge/testnh.     

Phylogenetic analysis of RAD‐seq data: examining the influence of gene genealogy conflict on analysis of concatenated data

One of the major issues in phylogenetic analysis is that gene genealogies from different gene regions may not reflect the true species tree or history of speciation. This has led to considerable debate about whether concatenation of loci is the best approach for phylogenetic analysis. The application of Next‐generation sequencing techniques such as RAD‐seq generates thousands of relatively short sequence reads from across the genomes of the sampled taxa. These data sets are typically concatenated for phylogenetic analysis leading to data sets that contain millions of base pairs per taxon. The influence of gene region conflict among so many loci in determining the phylogenetic relationships among taxa is unclear. We simulated RAD‐seq data by sampling 100 and 500 base pairs from alignments of over 6000 coding regions that each produce one of three highly supported alternative phylogenies of seven species of Drosophila. We conducted phylogenetic analyses on different sets of these regions to vary the sampling of loci with alternative gene trees to examine the effect on detecting the species tree. Irrespective of sequence length sampled per region and which subset of regions was used, phylogenetic analyses of the concatenated data always recovered the species tree. The results suggest that concatenated alignments of Next‐generation data that consist of many short sequences are robust to gene tree/species tree conflict when the goal is to determine the phylogenetic relationships among taxa.     

Constructing and counting phylogenetic invariants.

The method of invariants is an approach to the problem of reconstructing the phylogenetic tree of a collection of m taxa using nucleotide sequence data. Models for the respective probabilities of the 4m possible vectors of bases at a given site will have unknown parameters that describe the random mechanism by which substitution occurs along the branches of a putative phylogenetic tree. An invariant is a polynomial in these probabilities that, for a given phylogeny, is zero for all choices of the substitution mechanism parameters. If the invariant is typically non-zero for another phylogenetic tree, then estimates of the invariant can be used as evidence to support one phylogeny over another. Previous work of Evans and Speed showed that, for certain commonly used substitution models, the problem of finding a minimal generating set for the ideal of invariants can be reduced to the linear algebra problem of finding a basis for a certain lattice (that is, a free Z-module). They also conjectured that the cardinality of such a generating set can be computed using a simple "degrees of freedom" formula. We verify this conjecture. Along the way, we explain in detail how the observations of Evans and Speed lead to a simple, computationally feasible algorithm for constructing a minimal generating set.     

Identifying evolutionary trees and substitution parameters for the general Markov model with invariable sites

The general Markov plus invariable sites (GM+I) model of biological sequence evolution is a two-class model in which an unknown proportion of sites are not allowed to change, while the remainder undergo substitutions according to a Markov process on a tree. For statistical use it is important to know if the model is identifiable; can both the tree topology and the numerical parameters be determined from a joint distribution describing sequences only at the leaves of the tree? We establish that for generic parameters both the tree and all numerical parameter values can be recovered, up to clearly understood issues of 'label swapping'. The method of analysis is algebraic, using phylogenetic invariants to study the variety defined by the model. Simple rational formulas, expressed in terms of determinantal ratios, are found for recovering numerical parameters describing the invariable sites.     

A structural EM algorithm for phylogenetic inference.

A central task in the study of molecular evolution is the reconstruction of a phylogenetic tree from sequences of current-day taxa. The most established approach to tree reconstruction is maximum likelihood (ML) analysis. Unfortunately, searching for the maximum likelihood phylogenetic tree is computationally prohibitive for large data sets. In this paper, we describe a new algorithm that uses Structural Expectation Maximization (EM) for learning maximum likelihood phylogenetic trees. This algorithm is similar to the standard EM method for edge-length estimation, except that during iterations of the Structural EM algorithm the topology is improved as well as the edge length. Our algorithm performs iterations of two steps. In the E-step, we use the current tree topology and edge lengths to compute expected sufficient statistics, which summarize the data. In the M-Step, we search for a topology that maximizes the likelihood with respect to these expected sufficient statistics. We show that searching for better topologies inside the M-step can be done efficiently, as opposed to standard methods for topology search. We prove that each iteration of this procedure increases the likelihood of the topology, and thus the procedure must converge. This convergence point, however, can be a suboptimal one. To escape from such "local optima," we further enhance our basic EM procedure by incorporating moves in the flavor of simulated annealing. We evaluate these new algorithms on both synthetic and real sequence data and show that for protein sequences even our basic algorithm finds more plausible trees than existing methods for searching maximum likelihood phylogenies. Furthermore, our algorithms are dramatically faster than such methods, enabling, for the first time, phylogenetic analysis of large protein data sets in the maximum likelihood framework.     

Slicing hyperdimensional oranges: the geometry of phylogenetic estimation

A new view of phylogenetic estimation is presented where data sets, tree evolution models, and estimation methods are placed in a common geometric framework. Each of these objects is placed in a vector space where the character patterns are the basis vectors. This viewpoint allows intuitive understanding of various complex properties of the phylogeneticestimation problem structure. This is illustrated with examples discussing data set combinations, mixture models, consistency, and phylogenetic invariants.     

Heterogeneity of nucleotide frequencies among evolutionary lineages and phylogenetic inference

A major assumption of many molecular phylogenetic methods is the homogeneity of nucleotide frequencies among taxa, which refers to the equality of the nucleotide frequency bias among species. Changes in nucleotide frequency among different lineages in a data set are thought to lead to erroneous phylogenetic inference because unrelated clades may appear similar because of evolutionarily unrelated similarities in nucleotide frequencies. We tested the effects of the heterogeneity of nucleotide frequency bias on phylogenetic inference, along with the interaction between this heterogeneity and stratified taxon sampling, by means of computer simulations using evolutionary parameters derived from genomic databases. We found that the phylogenetic trees inferred from data sets simulated under realistic, observed levels of heterogeneity for mammalian genes were reconstructed with accuracy comparable to those simulated with homogeneous nucleotide frequencies; the results hold for Neighbor-Joining, minimum evolution, maximum parsimony, and maximum-likelihood methods. The LogDet distance method, specifically designed to deal with heterogeneous nucleotide frequencies, does not perform better than distance methods that assume substitution pattern homogeneity among sequences. In these specific simulation conditions, we did not find a significant interaction between phylogenetic accuracy and substitution pattern heterogeneity among lineages, even when the taxon sampling is increased.     

Bayesian phylogenetic inference via Markov chain Monte Carlo methods

We derive a Markov chain to sample from the posterior distribution for a phylogenetic tree given sequence information from the corresponding set of organisms, a stochastic model for these data, and a prior distribution on the space of trees. A transformation of the tree into a canonical cophenetic matrix form suggests a simple and effective proposal distribution for selecting candidate trees close to the current tree in the chain. We illustrate the algorithm with restriction site data on 9 plant species, then extend to DNA sequences from 32 species of fish. The algorithm mixes well in both examples from random starting trees, generating reproducible estimates and credible sets for the path of evolution.     

Bayesian phylogenetics using an RNA substitution model applied to early mammalian evolution

We study the phylogeny of the placental mammals using molecular data from all mitochondrial tRNAs and rRNAs of 54 species. We use probabilistic substitution models specific to evolution in base paired regions of RNA. A number of these models have been implemented in a new phylogenetic inference software package for carrying out maximum likelihood and Bayesian phylogenetic inferences. We describe our Bayesian phylogenetic method which uses a Markov chain Monte Carlo algorithm to provide samples from the posterior distribution of tree topologies. Our results show support for four primary mammalian clades, in agreement with recent studies of much larger data sets mainly comprising nuclear DNA. We discuss some issues arising when using Bayesian techniques on RNA sequence data.