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.     

Reconstructing genealogies of serial samples under the assumption of a molecular clock using serial-sample UPGMA

Reconstruction of evolutionary relationships from noncontemporaneous molecular samples provides a new challenge for phylogenetic reconstruction methods. With recent biotechnological advances there has been an increase in molecular sequencing throughput, and the potential to obtain serial samples of sequences from populations, including rapidly evolving pathogens, is fast being realized. A new method called the serial-sample unweighted pair grouping method with arithmetic means (sUPGMA) is presented that reconstructs a genealogy or phylogeny of sequences sampled serially in time using a matrix of pairwise distances. The resulting tree depicts the terminal lineages of each sample ending at a different level consistent with the sample's temporal order. Since sUPGMA is a variant of UPGMA, it will perform best when sequences have evolved at a constant rate (i.e., according to a molecular clock). On simulated data, this new method performs better than standard cluster analysis under a variety of longitudinal sampling strategies. Serial-sample UPGMA is particularly useful for analysis of longitudinal samples of viruses and bacteria, as well as ancient DNA samples, with the minimal requirement that samples of sequences be ordered in time.     

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.     

Coalescent-based species tree inference from gene tree topologies under incomplete lineage sorting by maximum likelihood

Incomplete lineage sorting can cause incongruence between the phylogenetic history of genes (the gene tree) and that of the species (the species tree), which can complicate the inference of phylogenies. In this article, I present a new coalescent-based algorithm for species tree inference with maximum likelihood. I first describe an improved method for computing the probability of a gene tree topology given a species tree, which is much faster than an existing algorithm by Degnan and Salter (2005). Based on this method, I develop a practical algorithm that takes a set of gene tree topologies and infers species trees with maximum likelihood. This algorithm searches for the best species tree by starting from initial species trees and performing heuristic search to obtain better trees with higher likelihood. This algorithm, called STELLS (which stands for Species Tree InfErence with Likelihood for Lineage Sorting), has been implemented in a program that is downloadable from the author's web page. The simulation results show that the STELLS algorithm is more accurate than an existing maximum likelihood method for many datasets, especially when there is noise in gene trees. I also show that the STELLS algorithm is efficient and can be applied to real biological datasets.     

Serial evolutionary networks of within-patient HIV-1 sequences reveal patterns of evolution of X4 strains

Background  

The HIV virus is known for its ability to exploit numerous genetic and evolutionary mechanisms to ensure its proliferation, among them, high replication, mutation and recombination rates. Sliding MinPD, a recently introduced computational method [1], was used to investigate the patterns of evolution of serially-sampled HIV-1 sequence data from eight patients with a special focus on the emergence of X4 strains. Unlike other phylogenetic methods, Sliding MinPD combines distance-based inference with a nonparametric bootstrap procedure and automated recombination detection to reconstruct the evolutionary history of longitudinal sequence data. We present serial evolutionary networks as a longitudinal representation of the mutational pathways of a viral population in a within-host environment. The longitudinal representation of the evolutionary networks was complemented with charts of clinical markers to facilitate correlation analysis between pertinent clinical information and the evolutionary relationships.     

Rapid maximum likelihood ancestral state reconstruction of continuous characters: A rerooting‐free algorithm

Ancestral state reconstruction is a method used to study the evolutionary trajectories of quantitative characters on phylogenies. Although efficient methods for univariate ancestral state reconstruction under a Brownian motion model have been described for at least 25 years, to date no generalization has been described to allow more complex evolutionary models, such as multivariate trait evolution, non‐Brownian models, missing data, and within‐species variation. Furthermore, even for simple univariate Brownian motion models, most phylogenetic comparative R packages compute ancestral states via inefficient tree rerooting and full tree traversals at each tree node, making ancestral state reconstruction extremely time‐consuming for large phylogenies. Here, a computationally efficient method for fast maximum likelihood ancestral state reconstruction of continuous characters is described. The algorithm has linear complexity relative to the number of species and outperforms the fastest existing R implementations by several orders of magnitude. The described algorithm is capable of performing ancestral state reconstruction on a 1,000,000‐species phylogeny in fewer than 2 s using a standard laptop, whereas the next fastest R implementation would take several days to complete. The method is generalizable to more complex evolutionary models, such as phylogenetic regression, within‐species variation, non‐Brownian evolutionary models, and multivariate trait evolution. Because this method enables fast repeated computations on phylogenies of virtually any size, implementation of the described algorithm can drastically alleviate the computational burden of many otherwise prohibitively time‐consuming tasks requiring reconstruction of ancestral states, such as phylogenetic imputation of missing data, bootstrapping procedures, Expectation‐Maximization algorithms, and Bayesian estimation. The described ancestral state reconstruction algorithm is implemented in the Rphylopars functions anc.recon and phylopars.     

A genetic algorithm for maximum-likelihood phylogeny inference using nucleotide sequence data

Phylogeny reconstruction is a difficult computational problem, because the number of possible solutions increases with the number of included taxa. For example, for only 14 taxa, there are more than seven trillion possible unrooted phylogenetic trees. For this reason, phylogenetic inference methods commonly use clustering algorithms (e.g., the neighbor-joining method) or heuristic search strategies to minimize the amount of time spent evaluating nonoptimal trees. Even heuristic searches can be painfully slow, especially when computationally intensive optimality criteria such as maximum likelihood are used. I describe here a different approach to heuristic searching (using a genetic algorithm) that can tremendously reduce the time required for maximum-likelihood phylogenetic inference, especially for data sets involving large numbers of taxa. Genetic algorithms are simulations of natural selection in which individuals are encoded solutions to the problem of interest. Here, labeled phylogenetic trees are the individuals, and differential reproduction is effected by allowing the number of offspring produced by each individual to be proportional to that individual's rank likelihood score. Natural selection increases the average likelihood in the evolving population of phylogenetic trees, and the genetic algorithm is allowed to proceed until the likelihood of the best individual ceases to improve over time. An example is presented involving rbcL sequence data for 55 taxa of green plants. The genetic algorithm described here required only 6% of the computational effort required by a conventional heuristic search using tree bisection/reconnection (TBR) branch swapping to obtain the same maximum-likelihood topology.      

A heuristic approach of maximum likelihood method for inferring phylogenetic tree and an application to the mammalian SOX-3 origin of the testis-determining gene SRY

Applying the tree bisection and reconnection (TBR) algorithm, we have developed a heuristic method (maximum likelihood (ML)-TBR) for inferring the ML tree based on tree topology search. For initial trees from which iterative processes start in ML-TBR, two cases were considered: one is 100 neighbor-joining (NJ) trees based on the bootstrap resampling and the other is 100 randomly generated trees. The same ML tree was obtained in both cases. All different iterative processes started from 100 independent initial trees ultimately converged on one optimum tree with the largest log-likelihood value, suggesting that a limited number of initial trees will be quite enough in ML-TBR. This also suggests that the optimum tree corresponds to the global optimum in tree topology space and thus probably coincides with the ML tree inferred by intact ML analysis. This method has been applied to the inference of phylogenetic tree of the SOX family members. The mammalian testis-determining gene SRY is believed to have evolved from SOX-3, a member of the SOX family, based on several lines of evidence, including their sequence similarity, the location of SOX-3 on the X chromosome and some aspects of their expression. This model should be supported directly from the phylogenetic tree of the SOX family, but no evidence has been provided to date. A recently published NJ tree shows implausibly remote origin of SRY, suggesting that a more sophisticated method is required for understanding this problem. The ML tree inferred by the present method showed that the SRYs of marsupial and placental mammals form a monophyletic cluster which had diverged from the mammalian SOX-3 in the early evolution of mammals.     

Accelerated likelihood surface exploration: the likelihood ratchet

The existence of multiple likelihood maxima necessitates algorithms that explore a large part of the tree space. However, because of computational constraints, stepwise addition-based tree-searching methods do not allow for this exploration in reasonable time. Here, I present an algorithm that increases the speed at which the likelihood landscape can be explored. The iterative algorithm combines the computational speed of distance-based tree construction methods to arrive at approximations of the global optimum with the accuracy of optimality criterion based branch-swapping methods to improve on the result of the starting tree. The algorithm moves between local optima by iteratively perturbing the tree landscape through a process of reweighting randomly drawn samples of the underlying sequence data set. Tests on simulated and real data sets demonstrated that the optimal solution obtained using stepwise addition-based heuristic searches was found faster using the algorithm presented here. Tests on a previously published data set that established the presence of tree islands under maximum likelihood demonstrated that the algorithm identifies the same tree islands in a shorter amount of time than that needed using stepwise addition. The algorithm can be readily applied using standard software for phylogenetic inference.     

Stochastic search strategy for estimation of maximum likelihood phylogenetic trees

The maximum likelihood (ML) method of phylogenetic tree construction is not as widely used as other tree construction methods (e.g., parsimony, neighbor-joining) because of the prohibitive amount of time required to find the ML tree when the number of sequences under consideration is large. To overcome this difficulty, we propose a stochastic search strategy for estimation of the ML tree that is based on a simulated annealing algorithm. The algorithm works by moving through tree space by way of a "local rearrangement" strategy so that topologies that improve the likelihood are always accepted, whereas those that decrease the likelihood are accepted with a probability that is related to the proportionate decrease in likelihood. Besides greatly reducing the time required to estimate the ML tree, the stochastic search strategy is less likely to become trapped in local optima than are existing algorithms for ML tree estimation. We demonstrate the success of the modified simulated annealing algorithm by comparing it with two existing algorithms (Swofford's PAUP* and Felsenstein's DNAMLK) for several theoretical and real data examples.     

Gene finding with a hidden Markov model of genome structure and evolution

Motivation: A growing number of genomes are sequenced. The differences in evolutionary pattern between functional regions can thus be observed genome-wide in a whole set of organisms. The diverse evolutionary pattern of different functional regions can be exploited in the process of genomic annotation. The modelling of evolution by the existing comparative gene finders leaves room for improvement. Results: A probabilistic model of both genome structure and evolution is designed. This type of model is called an Evolutionary Hidden Markov Model (EHMM), being composed of an HMM and a set of region-specific evolutionary models based on a phylogenetic tree. All parameters can be estimated by maximum likelihood, including the phylogenetic tree. It can handle any number of aligned genomes, using their phylogenetic tree to model the evolutionary correlations. The time complexity of all algorithms used for handling the model are linear in alignment length and genome number. The model is applied to the problem of gene finding. The benefit of modelling sequence evolution is demonstrated both in a range of simulations and on a set of orthologous human/mouse gene pairs. AVAILABILITY: Free availability over the Internet on www server: http://www.birc.dk/Software/evogene.     

Modeling sequence evolution in acute HIV-1 infection

We describe a mathematical model and Monte Carlo (MC) simulation of viral evolution during acute infection. We consider both synchronous and asynchronous processes of viral infection of new target cells. The model enables an assessment of the expected sequence diversity in new HIV-1 infections originating from a single transmitted viral strain, estimation of the most recent common ancestor (MRCA) of the transmitted viral lineage, and estimation of the time to coalesce back to the MRCA. We also calculate the probability of the MRCA being the transmitted virus or an evolved variant. Excluding insertions and deletions, we assume HIV-1 evolves by base substitution without selection pressure during the earliest phase of HIV-1 infection prior to the immune response. Unlike phylogenetic methods that follow a lineage backwards to coalescence, we compare the observed data to a model of the diversification of a viral population forward in time. To illustrate the application of these methods, we provide detailed comparisons of the model and simulations results to 306 envelope sequences obtained from eight newly infected subjects at a single time point. The data from patients were in good agreement with model predictions, and hence compatible with a single-strain infection evolving under no selection pressure. The diversity of the samples from the other two patients was too great to be explained by the model, suggesting multiple HIV-1-strains were transmitted. The model can also be applied to longitudinal patient data to estimate within-host viral evolutionary parameters.     

DPRml: distributed phylogeny reconstruction by maximum likelihood

MOTIVATION: In recent years there has been increased interest in producing large and accurate phylogenetic trees using statistical approaches. However for a large number of taxa, it is not feasible to construct large and accurate trees using only a single processor. A number of specialized parallel programs have been produced in an attempt to address the huge computational requirements of maximum likelihood. We express a number of concerns about the current set of parallel phylogenetic programs which are currently severely limiting the widespread availability and use of parallel computing in maximum likelihood-based phylogenetic analysis. RESULTS: We have identified the suitability of phylogenetic analysis to large-scale heterogeneous distributed computing. We have completed a distributed and fully cross-platform phylogenetic tree building program called distributed phylogeny reconstruction by maximum likelihood. It uses an already proven maximum likelihood-based tree building algorithm and a popular phylogenetic analysis library for all its likelihood calculations. It offers one of the most extensive sets of DNA substitution models currently available. We are the first, to our knowledge, to report the completion of a distributed phylogenetic tree building program that can achieve near-linear speedup while only using the idle clock cycles of machines. For those in an academic or corporate environment with hundreds of idle desktop machines, we have shown how distributed computing can deliver a 'free' ML supercomputer.     

Tree-based maximal likelihood substitution matrices and hidden Markov models

There has been considerable interest in the problem of making maximum likelihood (ML) evolutionary trees which allow insertions and deletions. This problem is partly one of formulation: how does one define a probabilistic model for such trees which treats insertion and deletion in a biologically plausible manner? A possible answer to this question is proposed here by extending the concept of a hidden Markov model (HMM) to evolutionary trees. The model, called a tree-HMM, allows what may be loosely regarded as learnable affine-type gap penalties for alignments. These penalties are expressed in HMMs as probabilities of transitions between states. In the tree-HMM, this idea is given an evolutionary embodiment by defining trees of transitions. Just as the probability of a tree composed of ungapped sequences is computed, by Felsenstein's method, using matrices representing the probabilities of substitutions of residues along the edges of the tree, so the probabilities in a tree-HMM are computed by substitution matrices for both residues and transitions. How to define these matrices by a ML procedure using an algorithm that learns from a database of protein sequences is shown here. Given these matrices, one can define a tree-HMM likelihood for a set of sequences, assuming a particular tree topology and an alignment of the sequences to the model. If one could efficiently find the alignment which maximizes (or comes close to maximizing) this likelihood, then one could search for the optimal tree topology for the sequences. An alignment algorithm is defined here which, given a particular tree topology, is guaranteed to increase the likelihood of the model. Unfortunately, it fails to find global optima for realistic sequence sets. Thus further research is needed to turn the tree-HMM into a practical phylogenetic tool.     

Topological Estimation Biases with Covarion Evolution

Covarion processes allow changes in evolutionary rates at sites along the branches of a phylogenetic tree. Covarion-like evolution is increasingly recognized as an important mode of protein evolution. Several recent reports suggest that maximum likelihood estimation employing covarion models may support different optimal topologies than estimation using standard rates-across-sites (RAS) models. However, it remains to be demonstrated that ignoring covarion evolution will generally result in topological misestimation. In this study we performed analytical and theoretical studies of limiting distances under the covarion model and four-taxon tree simulations to investigate the extent to which the covarion process impacts on phylogenetic estimation. In particular, we assessed the limits of an RAS model-based maximum likelihood method to recover the phylogenies when the sequence data were simulated under the covarion processes. We find that, when ignored, covarion processes can induce systematic errors in phylogeny reconstruction. Surprisingly, when sequences are evolved under a covarion process but an RAS model is used for estimation, we find that a long branch repel bias occurs.     

Estimating phylogenetic trees from pairwise likelihoods and posterior probabilities of substitution counts

The field of phylogenetic tree estimation has been dominated by three broad classes of methods: distance-based approaches, parsimony and likelihood-based methods (including maximum likelihood (ML) and Bayesian approaches). Here we introduce two new approaches to tree inference: pairwise likelihood estimation and a distance-based method that estimates the number of substitutions along the paths through the tree. Our results include the derivation of the formulae for the probability that two leaves will be identical at a site given a number of substitutions along the path connecting them. We also derive the posterior probability of the number of substitutions along a path between two sequences. The calculations for the posterior probabilities are exact for group-based, symmetric models of character evolution, but are only approximate for more general models.     

Homology assessment in parsimony and model‐based analyses: two sides of the same coin

The present paper is mainly concerned with homology assessment through phylogenetic analyses. It raises a fundamental question: What are the epistemological differences between modern parsimony and model‐based analyses in relation to homology assessment and phylogenetic inference? Although these methods usually achieve concordant topological results, they may generate discordant inferences of character evolution from the same datasets. This indicates that method selection has serious implications for evolutionary scenarios and classificatory arrangements. Notwithstanding that parsimony and model‐based approaches use the Hennigian concepts of monophyly and synapomorphy, they employ different epistemological ways of dealing with the monophyly/synapomorphy relationship. Independently of their differences, these analyses should take into account all relevant evidence in support of the phylogenetic inferences. A focus on morphological homologues means that they must be included in data matrices, evaluated as part of the phylogenetic analysis, and cannot be ignored in calculation of the tree(s) length (parsimony), maximum‐likelihood (maximum‐likelihood), and posterior probabilities (Bayes).     

Modelling the evolution and spread of HIV immune escape mutants

During infection with human immunodeficiency virus (HIV), immune pressure from cytotoxic T-lymphocytes (CTLs) selects for viral mutants that confer escape from CTL recognition. These escape variants can be transmitted between individuals where, depending upon their cost to viral fitness and the CTL responses made by the recipient, they may revert. The rates of within-host evolution and their concordant impact upon the rate of spread of escape mutants at the population level are uncertain. Here we present a mathematical model of within-host evolution of escape mutants, transmission of these variants between hosts and subsequent reversion in new hosts. The model is an extension of the well-known SI model of disease transmission and includes three further parameters that describe host immunogenetic heterogeneity and rates of within host viral evolution. We use the model to explain why some escape mutants appear to have stable prevalence whilst others are spreading through the population. Further, we use it to compare diverse datasets on CTL escape, highlighting where different sources agree or disagree on within-host evolutionary rates. The several dozen CTL epitopes we survey from HIV-1 gag, RT and nef reveal a relatively sedate rate of evolution with average rates of escape measured in years and reversion in decades. For many epitopes in HIV, occasional rapid within-host evolution is not reflected in fast evolution at the population level.