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.     

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.     

Integrated gene and species phylogenies from unaligned whole genome protein sequences

MOTIVATION: Most molecular phylogenies are based on sequence alignments. Consequently, they fail to account for modes of sequence evolution that involve frequent insertions or deletions. Here we present a method for generating accurate gene and species phylogenies from whole genome sequence that makes use of short character string matches not placed within explicit alignments. In this work, the singular value decomposition of a sparse tetrapeptide frequency matrix is used to represent the proteins of organisms uniquely and precisely as vectors in a high-dimensional space. Vectors of this kind can be used to calculate pairwise distance values based on the angle separating the vectors, and the resulting distance values can be used to generate phylogenetic trees. Protein trees so derived can be examined directly for homologous sequences. Alternatively, vectors defining each of the proteins within an organism can be summed to provide a vector representation of the organism, which is then used to generate species trees. RESULTS: Using a large mitochondrial genome dataset, we have produced species trees that are largely in agreement with previously published trees based on the analysis of identical datasets using different methods. These trees also agree well with currently accepted phylogenetic theory. In principle, our method could be used to compare much larger bacterial or nuclear genomes in full molecular detail, ultimately allowing accurate gene and species relationships to be derived from a comprehensive comparison of complete genomes. In contrast to phylogenetic methods based on alignments, sequences that evolve by relative insertion or deletion would tend to remain recognizably similar.     

Markov invariants, plethysms, and phylogenetics

We explore model-based techniques of phylogenetic tree inference exercising Markov invariants. Markov invariants are group invariant polynomials and are distinct from what is known in the literature as phylogenetic invariants, although we establish a commonality in some special cases. We show that the simplest Markov invariant forms the foundation of the Log–Det distance measure. We take as our primary tool group representation theory, and show that it provides a general framework for analyzing Markov processes on trees. From this algebraic perspective, the inherent symmetries of these processes become apparent, and focusing on plethysms, we are able to define Markov invariants and give existence proofs. We give an explicit technique for constructing the invariants, valid for any number of character states and taxa. For phylogenetic trees with three and four leaves, we demonstrate that the corresponding Markov invariants can be fruitfully exploited in applied phylogenetic studies.     

Performance of a new invariants method on homogeneous and nonhomogeneous quartet trees

An attempt to use phylogenetic invariants for tree reconstruction was made at the end of the 80s and the beginning of the 90s by several researchers (the initial idea due to Lake [1987] and Cavender and Felsenstein [1987]). However, the efficiency of methods based on invariants is still in doubt (Huelsenbeck 1995; Jin and Nei 1990). Probably because these methods only used few generators of the set of phylogenetic invariants. The method studied in this paper was first introduced in Casanellas et al. (2005) and it is the first method based on invariants that uses the "whole" set of generators for DNA data. The simulation studies performed in this paper prove that it is a very competitive and highly efficient phylogenetic reconstruction method, especially for nonhomogeneous models on phylogenetic trees.     

A comprehensive vertebrate phylogeny using vector representations of protein sequences from whole genomes

We recently developed a method for producing comprehensive gene and species phylogenies from unaligned whole genome data using singular value decomposition (SVD) to analyze character string frequencies. This work provides an integrated gene and species phylogeny for 64 vertebrate mitochondrial genomes composed of 832 total proteins. In addition, to provide a theoretical basis for the method, we present a graphical interpretation of both the original frequency matrix and the SVD-derived matrix. These large matrices describe high-dimensional Euclidean spaces within which biomolecular sequences can be uniquely represented as vectors. In particular, the SVD-derived vector space describes each protein relative to a restricted set of newly defined, independent axes, each of which represents a novel form of conserved motif, termed a correlated peptide motif. A quantitative comparison of the relative orientations of protein vectors in this space provides accurate and straightforward estimates of sequence similarity, which can in turn be used to produce comprehensive gene trees. Alternatively, the vector representations of genes from individual species can be summed, allowing species trees to be produced.     

A comprehensive whole genome bacterial phylogeny using correlated peptide motifs defined in a high dimensional vector space

As whole genome sequences continue to expand in number and complexity, effective methods for comparing and categorizing both genes and species represented within extremely large datasets are required. Methods introduced to date have generally utilized incomplete and likely insufficient subsets of the available data. We have developed an accurate and efficient method for producing robust gene and species phylogenies using very large whole genome protein datasets. This method relies on multidimensional protein vector definitions supplied by the singular value decomposition (SVD) of a large sparse data matrix in which each protein is uniquely represented as a vector of overlapping tetrapeptide frequencies. Quantitative pairwise estimates of species similarity were obtained by summing the protein vectors to form species vectors, then determining the cosines of the angles between species vectors. Evolutionary trees produced using this method confirmed many accepted prokaryotic relationships. However, several unconventional relationships were also noted. In addition, we demonstrate that many of the SVD-derived right basis vectors represent particular conserved protein families, while many of the corresponding left basis vectors describe conserved motifs within these families as sets of correlated peptides (copeps). This analysis represents the most detailed simultaneous comparison of prokaryotic genes and species available to date.     

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.     

基于DNA序列K-tuple分布的一种非序列比对分析

文章在基因组K-tuple分布的基础上, 给出了一种推测生物序列差异大小的非序列比对方法。该方法可用于衡量真实DNA序列和随机重排序列在K-tuple分布上的差异。将此方法用于构建含有26种胎盘哺乳动物线粒体全基因组的系统树时, 随着K的增大, 系统树的分类效果与生物学一致公认的结果愈加匹配。结果表明, 用此方法构建的系统进化树比用其他非序列比对分析方法构建的更加合理。     

Comparison of Tree-Child Phylogenetic Networks

Phylogenetic networks are a generalization of phylogenetic trees that allow for the representation of nontreelike evolutionary events, like recombination, hybridization, or lateral gene transfer. While much progress has been made to find practical algorithms for reconstructing a phylogenetic network from a set of sequences, all attempts to endorse a class of phylogenetic networks (strictly extending the class of phylogenetic trees) with a well-founded distance measure have, to the best of our knowledge and with the only exception of the bipartition distance on regular networks, failed so far. In this paper, we present and study a new meaningful class of phylogenetic networks, called tree-child phylogenetic networks, and we provide an injective representation of these networks as multisets of vectors of natural numbers, their path multiplicity vectors. We then use this representation to define a distance on this class that extends the well-known Robinson-Foulds distance for phylogenetic trees and to give an alignment method for pairs of networks in this class. Simple polynomial algorithms for reconstructing a tree-child phylogenetic network from its path multiplicity vectors, for computing the distance between two tree-child phylogenetic networks and for aligning a pair of tree-child phylogenetic networks, are provided. They have been implemented as a Perl package and a Java applet, which can be found at http://bioinfo.uib.es/~recerca/phylonetworks/mudistance/.     

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.     

Markov invariants and the isotropy subgroup of a quartet tree

The purpose of this article is to show how the isotropy subgroup of leaf permutations on binary trees can be used to systematically identify tree-informative invariants relevant to models of phylogenetic evolution. In the quartet case, we give an explicit construction of the full set of representations and describe their properties. We apply these results directly to Markov invariants, thereby extending previous theoretical results by systematically identifying linear combinations that vanish for a given quartet. We also note that the theory is fully generalizable to arbitrary trees and is equally applicable to the related case of phylogenetic invariants. All results follow from elementary consideration of the representation theory of finite groups.     

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.     

A support vector machine based test for incongruence between sets of trees in tree space

ABSTRACT: BACKGROUND: The increased use of multi-locus data sets for phylogenetic reconstruction has increased the need to determine whether a set of gene trees significantly deviate from the phylogenetic patterns of other genes. Such unusual gene trees may have been influenced by other evolutionary processes such as selection, gene duplication, or horizontal gene transfer. RESULTS: Motivated by this problem we propose a nonparametric goodness-of-fit test for two empirical distributions of gene trees, and we developed the software GeneOut to estimate a p-value for the test. Our approach maps trees into a multi-dimensional vector space and then applies support vector machines (SVMs) to measure the separation between two sets of pre-defined trees. We use a permutation test to assess the significance of the SVM separation. To demonstrate the performance of GeneOut, we applied it to the comparison of gene trees simulated within different species trees across a range of species tree depths. Applied directly to sets of simulated gene trees with large sample sizes, GeneOut was able to detect very small differences between two set of gene trees generated under different species trees. Our statistical test can also include tree reconstruction into its test framework through a variety of phylogenetic optimality criteria. When applied to DNA sequence data simulated from different sets of gene trees, results in the form of receiver operating characteristic (ROC) curves indicated that GeneOut performed well in the detection of differences between sets of trees with different distributions in a multi-dimensional space. Furthermore, it controlled false positive and false negative rates very well, indicating a high degree of accuracy. CONCLUSIONS: The non-parametric nature of our statistical test provides fast and efficient analyses, and makes it an applicable test for any scenario where evolutionary or other factors can lead to trees with different multi-dimensional distributions. The software GeneOut is freely available under the GNU public license.     

DNA-dependent RNA polymerase subunit B as a tool for phylogenetic reconstructions: Branching topology of the archaeal domain

The branching topology of the archaeal (archaebacterial) domain was inferred from sequence comparisons of the largest subunit (B) of DNA-dependent RNA polymerases (RNAP). Both the nucleic acid sequences of the genes coding for RNAP subunit B and the amino acid sequences of the derived gene products were used for phylogenetic reconstructions. Individual analysis of the three nucleotide positions of codons revealed significant inequalities with respect to guanosine and cytosine (GC) content and evolutionary rates. Only the nucleotides at the second codon positions were found to be unbiased by varied GC contents and sufficiently conserved for reliable phylogenetic reconstructions. A decision matrix was used for the combination of the results of distance matrix, maximum parsimony, and maximum likelihood methods. For this purpose the original results (sums of squares, steps, and logarithms of likelihoods) were transformed into comparable effective values and analyzed with methods known from the theory of statistical decisions. Phylogenetic invariants and statistical analysis with resampling techniques (bootstrap and jackknife) confirmed the preferred branching topology, which is significantly different from the topology known from phylogenetic trees based on 16S rRNA sequences. The preferred topology reconstructed by this analysis shows a common stem for the Methanococcales and Methanobacteriales and a separation of the thermophilic sulfur archaea from the methanogens and halophiles. The latter coincides with a unique phylogenetic location of a characteristic splitting event replacing the largest RNAP subunit of thermophilic sulfur archaea by two fragments in methanogens and halophiles. This topology is in good agreement with physiological and structural differences between the various archaea and demonstrates RNAP to be a suitable phylogenetic marker molecule. Correspondence to: H.-P. Klenk     

Phylogenetic Constraints Do Not Explain the Rarity of Nitrogen-Fixing Trees in Late-Successional Temperate Forests

Background

Symbiotic nitrogen (N)-fixing trees are rare in late-successional temperate forests, even though these forests are often N limited. Two hypotheses could explain this paradox. The ‘phylogenetic constraints hypothesis’ states that no late-successional tree taxa in temperate forests belong to clades that are predisposed to N fixation. Conversely, the ‘selective constraints hypothesis’ states that such taxa are present, but N-fixing symbioses would lower their fitness. Here we test the phylogenetic constraints hypothesis.

Methodology/Principal Findings

Using U.S. forest inventory data, we derived successional indices related to shade tolerance and stand age for N-fixing trees, non-fixing trees in the ‘potentially N-fixing clade’ (smallest angiosperm clade that includes all N fixers), and non-fixing trees outside this clade. We then used phylogenetically independent contrasts (PICs) to test for associations between these successional indices and N fixation. Four results stand out from our analysis of U.S. trees. First, N fixers are less shade-tolerant than non-fixers both inside and outside of the potentially N-fixing clade. Second, N fixers tend to occur in younger stands in a given geographical region than non-fixers both inside and outside of the potentially N-fixing clade. Third, the potentially N-fixing clade contains numerous late-successional non-fixers. Fourth, although the N fixation trait is evolutionarily conserved, the successional traits are relatively labile.

Conclusions/Significance

These results suggest that selective constraints, not phylogenetic constraints, explain the rarity of late-successional N-fixing trees in temperate forests. Because N-fixing trees could overcome N limitation to net primary production if they were abundant, this study helps to understand the maintenance of N limitation in temperate forests, and therefore the capacity of this biome to sequester carbon.     

Nephele: genotyping via complete composition vectors and MapReduce

Background

Current sequencing technology makes it practical to sequence many samples of a given organism, raising new challenges for the processing and interpretation of large genomics data sets with associated metadata. Traditional computational phylogenetic methods are ideal for studying the evolution of gene/protein families and using those to infer the evolution of an organism, but are less than ideal for the study of the whole organism mainly due to the presence of insertions/deletions/rearrangements. These methods provide the researcher with the ability to group a set of samples into distinct genotypic groups based on sequence similarity, which can then be associated with metadata, such as host information, pathogenicity, and time or location of occurrence. Genotyping is critical to understanding, at a genomic level, the origin and spread of infectious diseases. Increasingly, genotyping is coming into use for disease surveillance activities, as well as for microbial forensics. The classic genotyping approach has been based on phylogenetic analysis, starting with a multiple sequence alignment. Genotypes are then established by expert examination of phylogenetic trees. However, these traditional single-processor methods are suboptimal for rapidly growing sequence datasets being generated by next-generation DNA sequencing machines, because they increase in computational complexity quickly with the number of sequences.

Results

Nephele is a suite of tools that uses the complete composition vector algorithm to represent each sequence in the dataset as a vector derived from its constituent k-mers by passing the need for multiple sequence alignment, and affinity propagation clustering to group the sequences into genotypes based on a distance measure over the vectors. Our methods produce results that correlate well with expert-defined clades or genotypes, at a fraction of the computational cost of traditional phylogenetic methods run on traditional hardware. Nephele can use the open-source Hadoop implementation of MapReduce to parallelize execution using multiple compute nodes. We were able to generate a neighbour-joined tree of over 10,000 16S samples in less than 2 hours.

Conclusions

We conclude that using Nephele can substantially decrease the processing time required for generating genotype trees of tens to hundreds of organisms at genome scale sequence coverage.     

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.     

Multigene Phylogenetics Reveals Temporal Diversification of Major African Malaria Vectors

The major vectors of malaria in sub-Saharan Africa belong to subgenus Cellia. Yet, phylogenetic relationships and temporal diversification among African mosquito species have not been unambiguously determined. Knowledge about vector evolutionary history is crucial for correct interpretation of genetic changes identified through comparative genomics analyses. In this study, we estimated a molecular phylogeny using 49 gene sequences for the African malaria vectors An. gambiae, An. funestus, An. nili, the Asian malaria mosquito An. stephensi, and the outgroup species Culex quinquefasciatus and Aedes aegypti. To infer the phylogeny, we identified orthologous sequences uniformly distributed approximately every 5 Mb in the five chromosomal arms. The sequences were aligned and the phylogenetic trees were inferred using maximum likelihood and neighbor-joining methods. Bayesian molecular dating using a relaxed log normal model was used to infer divergence times. Trees from individual genes agreed with each other, placing An. nili as a basal clade that diversified from the studied malaria mosquito species 47.6 million years ago (mya). Other African malaria vectors originated more recently, and independently acquired traits related to vectorial capacity. The lineage leading to An. gambiae diverged 30.4 mya, while the African vector An. funestus and the Asian vector An. stephensi were the most closely related sister taxa that split 20.8 mya. These results were supported by consistently high bootstrap values in concatenated phylogenetic trees generated individually for each chromosomal arm. Genome-wide multigene phylogenetic analysis is a useful approach for discerning historic relationships among malaria vectors, providing a framework for the correct interpretation of genomic changes across species, and comprehending the evolutionary origins of this ubiquitous and deadly insect-borne disease.     

Phylogenetic status and matrilineal structure of the biting midge, Culicoides imicola, in Portugal, Rhodes and Israel

The biting midge Culicoides imicola Kieffer (Diptera: Ceratopogonidae) is the most important Old World vector of African horse sickness (AHS) and bluetongue (BT). Recent increases of BT incidence in the Mediterranean basin are attributed to its increased abundance and distribution. The phylogenetic status and genetic structure of C. imicola in this region are unknown, despite the importance of these aspects for BT epidemiology in the North American BT vector. In this study, analyses of partial mitochondrial cytochrome oxidase subunit I gene (COI) sequences were used to infer phylogenetic relationships among 50 C. imicola from Portugal, Rhodes, Israel, and South Africa and four other species of the Imicola Complex from southern Africa, and to estimate levels of matrilineal subdivision in C. imicola between Portugal and Israel. Eleven haplotypes were detected in C. imicola, and these formed one well-supported clade in maximum likelihood and Bayesian trees implying that the C. imicola samples comprise one phylogenetic species. Molecular variance was distributed mainly between Portugal and Israel, with no haplotypes shared between these countries, suggesting that female-mediated gene flow at this scale has been either limited or non-existent. Our results provide phylogenetic evidence that C. imicola in the study areas are potentially competent AHS and BT vectors. The geographical structure of the C. imicola COI haplotypes was concordant with that of BT virus serotypes in recent BT outbreaks in the Mediterranean basin, suggesting that population subdivision in its vector can impose spatial constraints on BT virus transmission.