Maximum likelihood molecular clock comb: analytic solutions.

Maximum likelihood (ML) is increasingly used as an optimality criterion for selecting evolutionary trees, but finding the global optimum is a hard computational task. Because no general analytic solution is known, numeric techniques such as hill climbing or expectation maximization (EM), are used in order to find optimal parameters for a given tree. So far, analytic solutions were derived only for the simplest model--three taxa, two state characters, under a molecular clock. Four taxa rooted trees have two topologies--the fork (two subtrees with two leaves each) and the comb (one subtree with three leaves, the other with a single leaf). In a previous work, we devised a closed form analytic solution for the ML molecular clock fork. In this work, we extend the state of the art in the area of analytic solutions ML trees to the family of all four taxa trees under the molecular clock assumption. The change from the fork topology to the comb incurs a major increase in the complexity of the underlying algebraic system and requires novel techniques and approaches. We combine the ultrametric properties of molecular clock trees with the Hadamard conjugation to derive a number of topology dependent identities. Employing these identities, we substantially simplify the system of polynomial equations. We finally use tools from algebraic geometry (e.g., Gr?bner bases, ideal saturation, resultants) and employ symbolic algebra software to obtain analytic solutions for the comb. We show that in contrast to the fork, the comb has no closed form solutions (expressed by radicals in the input data). In general, four taxa trees can have multiple ML points. In contrast, we can now prove that under the molecular clock assumption, the comb has a unique (local and global) ML point. (Such uniqueness was previously shown for the fork.).     

Molecular clock fork phylogenies: closed form analytic maximum likelihood solutions

Maximum likelihood (ML) is increasingly used as an optimality criterion for selecting evolutionary trees, but finding the global optimum is a hard computational task. Because no general analytic solution is known, numeric techniques such as hill climbing or expectation maximization (EM) are used in order to find optimal parameters for a given tree. So far, analytic solutions were derived only for the simplest model-three-taxa, two-state characters, under a molecular clock. Quoting Ziheng Yang, who initiated the analytic approach,"this seems to be the simplest case, but has many of the conceptual and statistical complexities involved in phylogenetic estimation."In this work, we give general analytic solutions for a family of trees with four-taxa, two-state characters, under a molecular clock. The change from three to four taxa incurs a major increase in the complexity of the underlying algebraic system, and requires novel techniques and approaches. We start by presenting the general maximum likelihood problem on phylogenetic trees as a constrained optimization problem, and the resulting system of polynomial equations. In full generality, it is infeasible to solve this system, therefore specialized tools for the molecular clock case are developed. Four-taxa rooted trees have two topologies-the fork (two subtrees with two leaves each) and the comb (one subtree with three leaves, the other with a single leaf). We combine the ultrametric properties of molecular clock fork trees with the Hadamard conjugation to derive a number of topology dependent identities. Employing these identities, we substantially simplify the system of polynomial equations for the fork. We finally employ symbolic algebra software to obtain closed formanalytic solutions (expressed parametrically in the input data). In general, four-taxa trees can have multiple ML points. In contrast, we can now prove that each fork topology has a unique(local and global) ML point.     

Maximum likelihood Jukes-Cantor triplets: analytic solutions

Maximum likelihood (ML) is a popular method for inferring a phylogenetic tree of the evolutionary relationship of a set of taxa, from observed homologous aligned genetic sequences of the taxa. Generally, the computation of the ML tree is based on numerical methods, which in a few cases, are known to converge to a local maximum on a tree, which is suboptimal. The extent of this problem is unknown, one approach is to attempt to derive algebraic equations for the likelihood equation and find the maximum points analytically. This approach has so far only been successful in the very simplest cases, of three or four taxa under the Neyman model of evolution of two-state characters. In this paper we extend this approach, for the first time, to four-state characters, the Jukes-Cantor model under a molecular clock, on a tree T on three taxa, a rooted triple. We employ spectral methods (Hadamard conjugation) to express the likelihood function parameterized by the path-length spectrum. Taking partial derivatives, we derive a set of polynomial equations whose simultaneous solution contains all critical points of the likelihood function. Using tools of algebraic geometry (the resultant of two polynomials) in the computer algebra packages (Maple), we are able to find all turning points analytically. We then employ this method on real sequence data and obtain realistic results on the primate-rodents divergence time.     

Combined nuclear and mitochondrial DNA sequences resolve generic relationships within the Cracidae (Galliformes,Aves)

The Cracidae is one of the most endangered and distinctive bird families in the Neotropics, yet the higher relationships among taxa remain uncertain. The molecular phylogeny of its 11 genera was inferred using 10,678 analyzable sites (5,412 from seven different mitochondrial segments and 5,266 sites from four nuclear genes). We performed combinability tests to check conflicts in phylogenetic signals of separate genes and genomes. Phylogenetic analysis showed that the unrooted tree of ((curassows, horned guan) (guans, chachalacas)) was favored by most data partitions and that different data partitions provided support for different parts of the tree. In particular, the concatenated mitochondrial DNA (mtDNA) genes resolved shallower nodes, whereas the combined nuclear sequences resolved the basal connections among the major clades of curassows, horned guan, chachalacas, and guans. Therefore, we decided that for the Cracidae all data should be combined for phylogenetic analysis. Maximum parsimony (MP), maximum likelihood (ML), and Bayesian analyses of this large data set produced similar trees. The MP tree indicated that guans are the sister group to (horned guan, (curassows, chachalacas)), whereas the ML and Bayesian analysis recovered a tree where the horned guan is a sister clade to curassows, and these two clades had the chachalacas as a sister group. Parametric bootstrapping showed that alternative trees previously proposed for the cracid genera are significantly less likely than our estimate of their relationships. A likelihood ratio test of the hypothesis of a molecular clock for cracid mtDNA sequences using the optimal ML topology did not reject rate constancy of substitutions through time. We estimated cracids to have originated between 64 and 90 million years ago (MYA), with a mean estimate of 76 MYA. Diversification of the genera occurred approximately 41-3 MYA, corresponding with periods of global climate change and other Earth history events that likely promoted divergences of higher level taxa.     

Improving the efficiency of SPR moves in phylogenetic tree search methods based on maximum likelihood

MOTIVATION: Maximum likelihood (ML) methods have become very popular for constructing phylogenetic trees from sequence data. However, despite noticeable recent progress, with large and difficult datasets (e.g. multiple genes with conflicting signals) current ML programs still require huge computing time and can become trapped in bad local optima of the likelihood function. When this occurs, the resulting trees may still show some of the defects (e.g. long branch attraction) of starting trees obtained using fast distance or parsimony programs. METHODS: Subtree pruning and regrafting (SPR) topological rearrangements are usually sufficient to intensively search the tree space. Here, we propose two new methods to make SPR moves more efficient. The first method uses a fast distance-based approach to detect the least promising candidate SPR moves, which are then simply discarded. The second method locally estimates the change in likelihood for any remaining potential SPRs, as opposed to globally evaluating the entire tree for each possible move. These two methods are implemented in a new algorithm with a sophisticated filtering strategy, which efficiently selects potential SPRs and concentrates most of the likelihood computation on the promising moves. RESULTS: Experiments with real datasets comprising 35-250 taxa show that, while indeed greatly reducing the amount of computation, our approach provides likelihood values at least as good as those of the best-known ML methods so far and is very robust to poor starting trees. Furthermore, combining our new SPR algorithm with local moves such as PHYML's nearest neighbor interchanges, the time needed to find good solutions can sometimes be reduced even more.     

Multiple maxima of likelihood in phylogenetic trees: an analytic approach

Maximum likelihood (ML) is a widely used criterion for selecting optimal evolutionary trees. However, the nature of the likelihood surface for trees is still not sufficiently understood, especially with regard to the frequency of multiple optima. Here, we initiate an analytic study for identifying sequences that generate multiple optima. We concentrate on the problem of optimizing edge weights for a given tree or trees (as opposed to searching through the space of all trees). We report a new approach to computing ML directly, which we have used to find large families of sequences that have multiple optima, including sequences with a continuum of optimal points. Such data sets are best supported by different (two or more) phylogenies that vary significantly in their timings of evolutionary events. Some standard biological processes can lead to data with multiple optima, and consequently the field needs further investigation. Our results imply that hill-climbing techniques as currently implemented in various software packages cannot guarantee that one will find the global ML point, even if it is unique.     

Upper bounds on maximum likelihood for phylogenetic trees

We introduce a mechanism for analytically deriving upper bounds on the maximum likelihood for genetic sequence data on sets of phylogenies. A simple 'partition' bound is introduced for general models. Tighter bounds are developed for the simplest model of evolution, the two state symmetric model of nucleotide substitution under the molecular clock. This follows earlier theoretical work which has been restricted to this model by analytic complexity. A weakness of current numerical computation is that reported 'maximum likelihood' results cannot be guaranteed, both for a specified tree (because of the possibility of multiple maxima) or over the full tree space (as the computation is intractable for large sets of trees). The bounds we develop here can be used to conclusively eliminate large proportions of tree space in the search for the maximum likelihood tree. This is vital in the development of a branch and bound search strategy for identifying the maximum likelihood tree. We report the results from a simulation study of approximately 10(6) data sets generated on clock-like trees of five leaves. In each trial a likelihood value of one specific instance of a parameterised tree is compared to the bound determined for each of the 105 possible rooted binary trees. The proportion of trees that are eliminated from the search for the maximum likelihood tree ranged from 92% to almost 98%, indicating a computational speed-up factor of between 12 and 44.     

A tree island approach to inferring phylogeny in the ant subfamily Formicinae,with especial reference to the evolution of weaving

The ant subfamily Formicinae is a large assemblage (2458 species (J. Nat. Hist. 29 (1995) 1037), including species that weave leaf nests together with larval silk and in which the metapleural gland-the ancestrally defining ant character-has been secondarily lost. We used sequences from two mitochondrial genes (cytochrome b and cytochrome oxidase 2) from 18 formicine and 4 outgroup taxa to derive a robust phylogeny, employing a search for tree islands using 10000 randomly constructed trees as starting points and deriving a maximum likelihood consensus tree from the ML tree and those not significantly different from it. Non-parametric bootstrapping showed that the ML consensus tree fit the data significantly better than three scenarios based on morphology, with that of Bolton (Identification Guide to the Ant Genera of the World, Harvard University Press, Cambridge, MA) being the best among these alternative trees. Trait mapping showed that weaving had arisen at least four times and possibly been lost once. A maximum likelihood analysis showed that loss of the metapleural gland is significantly associated with the weaver life-pattern. The graph of the frequencies with which trees were discovered versus their likelihood indicates that trees with high likelihoods have much larger basins of attraction than those with lower likelihoods. While this result indicates that single searches are more likely to find high- than low-likelihood tree islands, it also indicates that searching only for the single best tree may lose important information.     

RAxML-VI-HPC: maximum likelihood-based phylogenetic analyses with thousands of taxa and mixed models

RAxML-VI-HPC (randomized axelerated maximum likelihood for high performance computing) is a sequential and parallel program for inference of large phylogenies with maximum likelihood (ML). Low-level technical optimizations, a modification of the search algorithm, and the use of the GTR+CAT approximation as replacement for GTR+Gamma yield a program that is between 2.7 and 52 times faster than the previous version of RAxML. A large-scale performance comparison with GARLI, PHYML, IQPNNI and MrBayes on real data containing 1000 up to 6722 taxa shows that RAxML requires at least 5.6 times less main memory and yields better trees in similar times than the best competing program (GARLI) on datasets up to 2500 taxa. On datasets > or =4000 taxa it also runs 2-3 times faster than GARLI. RAxML has been parallelized with MPI to conduct parallel multiple bootstraps and inferences on distinct starting trees. The program has been used to compute ML trees on two of the largest alignments to date containing 25,057 (1463 bp) and 2182 (51,089 bp) taxa, respectively. AVAILABILITY: icwww.epfl.ch/~stamatak     

Phylogeny and divergence of the pinnipeds (Carnivora: Mammalia) assessed using a multigene dataset

Background  

Phylogenetic comparative methods are often improved by complete phylogenies with meaningful branch lengths (e.g., divergence dates). This study presents a dated molecular supertree for all 34 world pinniped species derived from a weighted matrix representation with parsimony (MRP) supertree analysis of 50 gene trees, each determined under a maximum likelihood (ML) framework. Divergence times were determined by mapping the same sequence data (plus two additional genes) on to the supertree topology and calibrating the ML branch lengths against a range of fossil calibrations. We assessed the sensitivity of our supertree topology in two ways: 1) a second supertree with all mtDNA genes combined into a single source tree, and 2) likelihood-based supermatrix analyses. Divergence dates were also calculated using a Bayesian relaxed molecular clock with rate autocorrelation to test the sensitivity of our supertree results further.     

Phylogenetic networks based on the molecular clock hypothesis

A classical result in phylogenetic trees is that a binary phylogenetic tree adhering to the molecular clock hypothesis exists if and only if the matrix of distances between taxa is ultrametric. The ultrametric condition is very restrictive. In this paper we study phylogenetic networks that can be constructed assuming the molecular clock hypothesis. We characterize distance matrices that admit such networks for 3 and 4 taxa. We also design two algorithms for constructing networks optimizing the least-squares fit.     

Reconstructing the evolutionary history of the Lorisidae using morphological, molecular, and geological data

Major aspects of lorisid phylogeny and systematics remain unresolved, despite several studies (involving morphology, histology, karyology, immunology, and DNA sequencing) aimed at elucidating them. Our study is the first to investigate the evolution of this enigmatic group using molecular and morphological data for all four well-established genera: Arctocebus, Loris, Nycticebus, and Perodicticus. Data sets consisting of 386 bp of 12S rRNA, 535 bp of 16S rRNA, and 36 craniodental characters were analyzed separately and in combination, using maximum parsimony and maximum likelihood. Outgroups, consisting of two galagid taxa (Otolemur and Galagoides) and a lemuroid (Microcebus), were also varied. The morphological data set yielded a paraphyletic lorisid clade with the robust Nycticebus and Perodicticus grouped as sister taxa, and the galagids allied with Arctocebus. All molecular analyses maximum parsimony (MP) or maximum likelihood (ML) which included Microcebus as an outgroup rendered a paraphyletic lorisid clade, with one exception: the 12S + 16S data set analyzed with ML. The position of the galagids in these paraphyletic topologies was inconsistent, however, and bootstrap values were low. Exclusion of Microcebus generated a monophyletic Lorisidae with Asian and African subclades; bootstrap values for all three clades in the total evidence tree were over 90%. We estimated mean genetic distances for lemuroids vs. lorisoids, lorisids vs. galagids, and Asian vs. African lorisids as a guide to relative divergence times. We present information regarding a temporary land bridge that linked the two now widely separated regions inhabited by lorisids that may explain their distribution. Finally, we make taxonomic recommendations based on our results.     

SuperFine: fast and accurate supertree estimation

Many research groups are estimating trees containing anywhere from a few thousands to hundreds of thousands of species, toward the eventual goal of the estimation of a Tree of Life, containing perhaps as many as several million leaves. These phylogenetic estimations present enormous computational challenges, and current computational methods are likely to fail to run even on data sets in the low end of this range. One approach to estimate a large species tree is to use phylogenetic estimation methods (such as maximum likelihood) on a supermatrix produced by concatenating multiple sequence alignments for a collection of markers; however, the most accurate of these phylogenetic estimation methods are extremely computationally intensive for data sets with more than a few thousand sequences. Supertree methods, which assemble phylogenetic trees from a collection of trees on subsets of the taxa, are important tools for phylogeny estimation where phylogenetic analyses based upon maximum likelihood (ML) are infeasible. In this paper, we introduce SuperFine, a meta-method that utilizes a novel two-step procedure in order to improve the accuracy and scalability of supertree methods. Our study, using both simulated and empirical data, shows that SuperFine-boosted supertree methods produce more accurate trees than standard supertree methods, and run quickly on very large data sets with thousands of sequences. Furthermore, SuperFine-boosted matrix representation with parsimony (MRP, the most well-known supertree method) approaches the accuracy of ML methods on supermatrix data sets under realistic conditions.     

Maximum likelihood of evolutionary trees: hardness and approximation

MOTIVATION: Maximum likelihood (ML) is an increasingly popular optimality criterion for selecting evolutionary trees. Yet the computational complexity of ML was open for over 20 years, and only recently resolved by the authors for the Jukes-Cantor model of substitution and its generalizations. It was proved that reconstructing the ML tree is computationally intractable (NP-hard). In this work we explore three directions, which extend that result. RESULTS: (1) We show that ML under the assumption of molecular clock is still computationally intractable (NP-hard). (2) We show that not only is it computationally intractable to find the exact ML tree, even approximating the logarithm of the ML for any multiplicative factor smaller than 1.00175 is computationally intractable. (3) We develop an algorithm for approximating log-likelihood under the condition that the input sequences are sparse. It employs any approximation algorithm for parsimony, and asymptotically achieves the same approximation ratio. We note that ML reconstruction for sparse inputs is still hard under this condition, and furthermore many real datasets satisfy it.     

Nuclear ribosomal ITS sequences and phylogeny in East AsianAconitum subgenusAconitum (Ranunculaceae), with special reference to extensive polymorphism in individual plants

Internal transcribed spacer (ITS) sequences of nuclear ribosomal DNA (nrDNA) were used to examine the phylogeny of East Asian aconites. Individual aconites were discovered to contain as many as eight different ITS sequences after cloning and PCR-SSCP (single-stranded conformational polymorphisms) analysis. We identified eight putative ITS pseudogenes from four taxa with low predicted secondary structure stability and high substitution rates. Maximum likelihood (ML) and neighbor-joining (NJ) methods were used for phylogenetic reconstruction. The ITS trees agree with the previous chloroplast DNA (cpDNA) tree for the vast majority of the taxa. We found two East Asian clades in the ITS trees: 1) a clade with the Chinese diploid,Aconitum volubile and East Asian tetraploids, and 2) a clade of East Asian diploids and Siberian tetraploids. In the former clade, most tetraploid taxa appear to be polyphyletic; sequences from individual plants did not correspond to recognized taxonomic units. This indicates a recent divergence of the East Asian tetraploids.     

Phylogenetic relationships of agaric fungi based on nuclear large subunit ribosomal DNA sequences

Phylogenetic relationships of mushrooms and their relatives within the order Agaricales were addressed by using nuclear large subunit ribosomal DNA sequences. Approximately 900 bases of the 5' end of the nucleus-encoded large subunit RNA gene were sequenced for 154 selected taxa representing most families within the Agaricales. Several phylogenetic methods were used, including weighted and equally weighted parsimony (MP), maximum likelihood (ML), and distance methods (NJ). The starting tree for branch swapping in the ML analyses was the tree with the highest ML score among previously produced MP and NJ trees. A high degree of consensus was observed between phylogenetic estimates obtained through MP and ML. NJ trees differed according to the distance model that was used; however, all NJ trees still supported most of the same terminal groupings as the MP and ML trees did. NJ trees were always significantly suboptimal when evaluated against the best MP and ML trees, by both parsimony and likelihood tests. Our analyses suggest that weighted MP and ML provide the best estimates of Agaricales phylogeny. Similar support was observed between bootstrapping and jackknifing methods for evaluation of tree robustness. Phylogenetic analyses revealed many groups of agaricoid fungi that are supported by moderate to high bootstrap or jackknife values or are consistent with morphology-based classification schemes. Analyses also support separate placement of the boletes and russules, which are basal to the main core group of gilled mushrooms (the Agaricineae of Singer). Examples of monophyletic groups include the families Amanitaceae, Coprinaceae (excluding Coprinus comatus and subfamily Panaeolideae), Agaricaceae (excluding the Cystodermateae), and Strophariaceae pro parte (Stropharia, Pholiota, and Hypholoma); the mycorrhizal species of Tricholoma (including Leucopaxillus, also mycorrhizal); Mycena and Resinomycena; Termitomyces, Podabrella, and Lyophyllum; and Pleurotus with Hohenbuehelia. Several groups revealed by these data to be nonmonophyletic include the families Tricholomataceae, Cortinariaceae, and Hygrophoraceae and the genera Clitocybe, Omphalina, and Marasmius. This study provides a framework for future systematics studies in the Agaricales and suggestions for analyzing large molecular data sets.     

Complexity of the simplest phylogenetic estimation problem

The maximum-likelihood (ML) solution to a simple phylogenetic estimation problem is obtained analytically The problem is estimation of the rooted tree for three species using binary characters with a symmetrical rate of substitution under the molecular clock. ML estimates of branch lengths and log-likelihood scores are obtained analytically for each of the three rooted binary trees. Estimation of the tree topology is equivalent to partitioning the sample space (space of possible data outcomes) into subspaces, within each of which one of the three binary trees is the ML tree. Distance-based least squares and parsimony-like methods produce essentially the same estimate of the tree topology, although differences exist among methods even under this simple model. This seems to be the simplest case, but has many of the conceptual and statistical complexities involved in phylogeny estimation. The solution to this real phylogeny estimation problem will be useful for studying the problem of significance evaluation.     

Property and efficiency of the maximum likelihood method for molecular phylogeny

Summary The maximum likelihood (ML) method for constructing phylogenetic trees (both rooted and unrooted trees) from DNA sequence data was studied. Although there is some theoretical problem in the comparison of ML values conditional for each topology, it is possible to make a heuristic argument to justify the method. Based on this argument, a new algorithm for estimating the ML tree is presented. It is shown that under the assumption of a constant rate of evolution, the ML method and UPGMA always give the same rooted tree for the case of three operational taxonomic units (OTUs). This also seems to hold approximately for the case with four OTUs. When we consider unrooted trees with the assumption of a varying rate of nucleotide substitution, the efficiency of the ML method in obtaining the correct tree is similar to those of the maximum parsimony method and distance methods. The ML method was applied to Brown et al.'s data, and the tree topology obtained was the same as that found by the maximum parsimony method, but it was different from those obtained by distance methods.     

Rapid evolutionary divergences in reef fishes of the family Acanthuridae (Perciformes: Teleostei)

A phylogenetic analysis of the sugeonfish family Acanthuridae was conducted to investigate: (a) the pattern of divergences among outgroup and basal ingroup taxa, (b) the pattern of species divergences within acanthurid genera, (c) monophyly in the genus Acanthurus, and (d) the evolution of thick-walled stomach morphology in the genera Acanthurus and Ctenochaetus. Fragments of the 12S, 16S, t-Pro, and control region mitochondrial genes were sequenced for 21 acanthurid taxa (representing all extant genera) and four outgroup taxa. Unweighted parsimony analysis produced two optimal trees. Both of these were highly incongruent with a previous morphological phylogeny, especially with regard to the placement of the monotypic outgroups Zanclus and Luvarus. The maximum likelihood tree and the morphological phylogeny were not significantly different and the conflicting branches were very short. Split decomposition analysis identified conflict in the placement of long basal branches separated by short internodes, providing further evidence that long branch attraction is an important cause of disagreement between molecular and morphological trees. Parametric bootstrapping rejected hypotheses of monophyly of: (a) the genus Acanthurus and (b) a group containing representatives of Acanthurus/Ctenochaetus with thick-walled stomachs. The branching pattern of the likelihood and split decomposition trees indicates that evolution in the acanthurid clade has involved at least three periods of intense speciation.     

Reconstructing phylogeny by quadratically approximated maximum likelihood

Maximum likelihood (ML) for phylogenetic inference from sequence data remains a method of choice, but has computational limitations. In particular, it cannot be applied for a global search through all potential trees when the number of taxa is large, and hence a heuristic restriction in the search space is required. In this paper, we derive a quadratic approximation, QAML, to the likelihood function whose maximum is easily determined for a given tree. The derivation depends on Hadamard conjugation, and hence is limited to the simple symmetric models of Kimura and of Jukes and Cantor. Preliminary testing has demonstrated the accuracy of QAML is close to that of ML.