Probability Steiner trees and maximum parsimony in phylogenetic analysis

The phylogenetic tree (PT) problem has been studied by a number of researchers as an application of the Steiner tree problem, a well-known network optimisation problem. Of all the methods developed for phylogenies the maximum parsimony (MP) method is a simple and commonly used method because it relies on directly observable changes in the input nucleotide or amino acid sequences. In this paper we show that the non-uniqueness of the evolutionary pathways in the MP method leads us to consider a new model of PTs. In this so-called probability representation model, for each site a node in a PT is modelled by a probability distribution of nucleotide or amino acid states, and hence the PT at a given site is a probability Steiner tree, i.e. a Steiner tree in a high-dimensional vector space. In spite of the generality of the probability representation model, in this paper we restrict our study to constructing probability phylogenetic trees (PPT) using the parsimony criterion, as well as discussing and comparing our approach with the classical MP method. We show that for a given input set although the optimal topology as well as the total tree length of the PPT is the same as the PT constructed by the classical MP method, the inferred ancestral states and branch lengths are different and the results given by our method provide a plausible alternative to the classical ones.     

Shrinkage Effect in Ancestral Maximum Likelihood

Ancestral maximum likelihood (AML) is a method that simultaneously reconstructs a phylogenetic tree and ancestral sequences from extant data (sequences at the leaves). The tree and ancestral sequences maximize the probability of observing the given data under a Markov model of sequence evolution, in which branch lengths are also optimized but constrained to take the same value on any edge across all sequence sites. AML differs from the more usual form of maximum likelihood (ML) in phylogenetics because ML averages over all possible ancestral sequences. ML has long been know to be statistically consistent - that is, it converges on the correct tree with probability approaching 1 as the sequence length grows. However, the statistical consistency of AML has not been formally determined, despite informal remarks in a literature that dates back 20 years. In this short note we prove a general result that implies that AML is statistically inconsistent. In particular we show that AML can 'shrink' short edges in a tree, resulting in a tree that has no internal resolution as the sequence length grows. Our results apply to any number of taxa.     

ImOSM: intermittent evolution and robustness of phylogenetic methods

Among the criteria to evaluate the performance of a phylogenetic method, robustness to model violation is of particular practical importance as complete a priori knowledge of evolutionary processes is typically unavailable. For studies of robustness in phylogenetic inference, a utility to add well-defined model violations to the simulated data would be helpful. We therefore introduce ImOSM, a tool to imbed intermittent evolution as model violation into an alignment. Intermittent evolution refers to extra substitutions occurring randomly on branches of a tree, thus changing alignment site patterns. This means that the extra substitutions are placed on the tree after the typical process of sequence evolution is completed. We then study the robustness of widely used phylogenetic methods: maximum likelihood (ML), maximum parsimony (MP), and a distance-based method (BIONJ) to various scenarios of model violation. Violation of rates across sites (RaS) heterogeneity and simultaneous violation of RaS and the transition/transversion ratio on two nonadjacent external branches hinder all the methods recovery of the true topology for a four-taxon tree. For an eight-taxon balanced tree, the violations cause each of the three methods to infer a different topology. Both ML and MP fail, whereas BIONJ, which calculates the distances based on the ML estimated parameters, reconstructs the true tree. Finally, we report that a test of model homogeneity and goodness of fit tests have enough power to detect such model violations. The outcome of the tests can help to actually gain confidence in the inferred trees. Therefore, we recommend using these tests in practical phylogenetic analyses.     

Steiner minimal trees, twist angles, and the protein folding problem

The Steiner Minimal Tree (SMT) problem determines the minimal length network for connecting a given set of vertices in three-dimensional space. SMTs have been shown to be useful in the geometric modeling and characterization of proteins. Even though the SMT problem is an NP-Hard Optimization problem, one can define planes within the amino acids that have a surprising regularity property for the twist angles of the planes. This angular property is quantified for all amino acids through the Steiner tree topology structure. The twist angle properties and other associated geometric properties unique for the remaining amino acids are documented in this paper. We also examine the relationship between the Steiner ratio rho and the torsion energy in amino acids with respect to the side chain torsion angle chi(1). The rho value is shown to be inversely proportional to the torsion energy. Hence, it should be a useful approximation to the potential energy function. Finally, the Steiner ratio is used to evaluate folded and misfolded protein structures. We examine all the native proteins and their decoys at http://dd.stanford.edu. and compare their Steiner ratio values. Because these decoy structures have been delicately misfolded, they look even more favorable than the native proteins from the potential energy viewpoint. However, the rho value of a decoy folded protein is shown to be much closer to the average value of an empirical Steiner ratio for each residue involved than that of the corresponding native one, so that we recognize the native folded structure more easily. The inverse relationship between the Steiner ratio and the energy level in the protein is shown to be a significant measure to distinguish native and decoy structures. These properties should be ultimately useful in the ab initio protein folding prediction.     

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.     

Ancestral maximum likelihood of evolutionary trees is hard

Maximum likelihood (ML) (Neyman, 1971) is an increasingly popular optimality criterion for selecting evolutionary trees. Finding optimal ML trees appears to be a very hard computational task--in particular, algorithms and heuristics for ML take longer to run than algorithms and heuristics for maximum parsimony (MP). However, while MP has been known to be NP-complete for over 20 years, no such hardness result has been obtained so far for ML. In this work we make a first step in this direction by proving that ancestral maximum likelihood (AML) is NP-complete. The input to this problem is a set of aligned sequences of equal length and the goal is to find a tree and an assignment of ancestral sequences for all of that tree's internal vertices such that the likelihood of generating both the ancestral and contemporary sequences is maximized. Our NP-hardness proof follows that for MP given in (Day, Johnson and Sankoff, 1986) in that we use the same reduction from Vertex Cover; however, the proof of correctness for this reduction relative to AML is different and substantially more involved.     

Maximum Parsimony and the Skewness Test: A Simulation Study of the Limits of Applicability

The maximum parsimony (MP) method for inferring phylogenies is widely used, but little is known about its limitations in non-asymptotic situations. This study employs large-scale computations with simulated phylogenetic data to estimate the probability that MP succeeds in finding the true phylogeny for up to twelve taxa and 256 characters. The set of candidate phylogenies are taken to be unrooted binary trees; for each simulated data set, the tree lengths of all (2n − 5)!! candidates are computed to evaluate quantities related to the performance of MP, such as the probability of finding the true phylogeny, the probability that the tree with the shortest length is unique, the probability that the true phylogeny has the shortest tree length, and the expected inverse of the number of trees sharing the shortest length. The tree length distributions are also used to evaluate and extend the skewness test of Hillis for distinguishing between random and phylogenetic data. The results indicate, for example, that the critical point after which MP achieves a success probability of at least 0.9 is roughly around 128 characters. The skewness test is found to perform well on simulated data and the study extends its scope to up to twelve taxa.     

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.     

Analysis of the Distribution of Bootstrap Tree Lengths Using the Maximum Parsimony Method

The bootstrap is an important tool for estimating the confidence interval of monophyletic groups within phylogenies. Although bootstrap analyses are used in most evolutionary studies, there is no clear consensus as how best to interpret bootstrap probability values. To study further the bootstrap method, nine small subunit ribosomal DNA (SSU rDNA) data sets were submitted to bootstrapped maximum parsimony (MP) analyses using unweighted and weighted sequence positions. Analyses of the lengths (i.e., parsimony steps) of the bootstrap trees show that the shape and mean of the bootstrap tree distribution may provide important insights into the evolutionary signal within the sequence data. With complex phylogenies containing nodes defined by short internal branches (multifurcations), the mean of the bootstrap tree distribution may differ by 2 standard deviations from the length of the best tree found from the original data set. Weighting sequence positions significantly increases the bootstrap values at internal nodes. There may, however, be strong bootstrap support for conflicting species groupings among different data sets. This phenomenon appears to result from a correlation between the topology of the tree used to create the weights and the topology of the bootstrap consensus tree inferred from the MP analysis of these weighted data. The analyses also show that characteristics of the bootstrap tree distribution (e.g., skewness) may be used to choose between alternative weighting schemes for phylogenetic analyses.     

Nucleotide substitution models and estimation of phylogeny

The nucleotide substitution matrix inferred from avian data sets using cytochrome b differs considerably from the models commonly used in phylogenetic analyses. To analyze the possible effects of this particular pattern of change in phylogeny estimation we performed a computer simulation in which we started with a real sequence and used the inferred model of change to produce a tree of 10 species. Maximum parsimony (MP), maximum likelihood (ML), and various distance methods were then used to recover the topology and the branch lengths. We used two kinds of data with varying levels of variation. In addition, we tested with the removal of third positions and different weighting schemes. At low levels of variation, MP was outstanding in recovering the topology (90% correct), while unweighted pair-group method, arithmetic average (UPGMA), regardless of distances used, was poor (40%). At the higher level, most methods had a chance of around 40%-58% of finding the true tree. However, in most cases, the trees found were only slightly wrong, with only one or a few branches misplaced. On the other hand, the use of a "wrong" model had serious effects on the estimation of branch lengths (distances). Although precision was high, accuracy was poor with most methods, giving branch lengths that were biased downward. When seeded with the true distance matrix, Fitch and NJ always found the true tree, while UPGMA frequently failed to do so. The effect of removing third positions was dramatic at low levels of variation, because only one MP program was able to find a true tree at all, albeit rarely, while none of the others ever did so. At higher levels, the situation was better, but still much worse than with the whole data set.     

Approximating subtree distances between phylogenies.

We give a 5-approximation algorithm to the rooted Subtree-Prune-and-Regraft (rSPR) distance between two phylogenies, which was recently shown to be NP-complete. This paper presents the first approximation result for this important tree distance. The algorithm follows a standard format for tree distances. The novel ideas are in the analysis. In the analysis, the cost of the algorithm uses a "cascading" scheme that accounts for possible wrong moves. This accounting is missing from previous analysis of tree distance approximation algorithms. Further, we show how all algorithms of this type can be implemented in linear time and give experimental results.     

Revisiting an Equivalence Between Maximum Parsimony and Maximum Likelihood Methods in Phylogenetics

Tuffley and Steel (Bull. Math. Biol. 59:581–607, 1997) proved that maximum likelihood and maximum parsimony methods in phylogenetics are equivalent for sequences of characters under a simple symmetric model of substitution with no common mechanism. This result has been widely cited ever since. We show that small changes to the model assumptions suffice to make the two methods inequivalent. In particular, we analyze the case of bounded substitution probabilities as well as the molecular clock assumption. We show that in these cases, even under no common mechanism, maximum parsimony and maximum likelihood might make conflicting choices. We also show that if there is an upper bound on the substitution probabilities which is ‘sufficiently small’, every maximum likelihood tree is also a maximum parsimony tree (but not vice versa).     

Maximum tree: a consistent estimator of the species tree

We propose a model based approach to use multiple gene trees to estimate the species tree. The coalescent process requires that gene divergences occur earlier than species divergences when there is any polymorphism in the ancestral species. Under this scenario, speciation times are restricted to be smaller than the corresponding gene split times. The maximum tree (MT) is the tree with the largest possible speciation times in the space of species trees restricted by available gene trees. If all populations have the same population size, the MT is the maximum likelihood estimate of the species tree. It can be shown the MT is a consistent estimator of the species tree even when the MT is built upon the estimates of the true gene trees if the gene tree estimates are statistically consistent. The MT converges in probability to the true species tree at an exponential rate.     

Maximum likelihood estimation of oncogenetic tree models

We present a new approach for modelling the dependences between genetic changes in human tumours. In solid tumours, data on genetic alterations are usually only available at a single point in time, allowing no direct insight into the sequential order of genetic events. In our approach, genetic tumour development and progression is assumed to follow a probabilistic tree model. We show how maximum likelihood estimation can be used to reconstruct a tree model for the dependences between genetic alterations in a given tumour type. We illustrate the use of the proposed method by applying it to cytogenetic data from 173 cases of clear cell renal cell carcinoma, arriving at a model for the karyotypic evolution of this tumour.     

Single column discrepancy and dynamic max-mini optimizations for quickly finding the most parsimonious evolutionary trees

MOTIVATION: In the maximum parsimony (MP) method, the tree requiring the minimum number of changes (discrepancy) to explain the given set of DNA or amino acid sequences is chosen to represent their evolutionary relationships. To find the MP tree, the branch-and-bound algorithm is normally used. For a partial phylogenetic-tree (one that has a subset of the organisms) the traditional algorithm assigns a cost equal to the discrepancy of the partial phylogenetic-tree. We propose a single column discrepancy heuristic which increases this cost by predicting a minimum additional discrepancy needed to attach the sequences yet to be added to the partial phylogenetic-tree. A dynamic Max-mini order of sequence addition is also proposed to quickly terminate branch-and-bound search paths that are guaranteed to lead to suboptimal solutions. RESULTS: We studied the running time of 47 problems generated from 17 data sets. The use of single column discrepancy heuristic speeded up the computation to 2.4-fold for static and 18.2-fold for dynamic search order. The improvement appeared to increase exponentially with the number of sequences. The proposed strategies are also likely to be useful in speeding up the MP tree search using heuristic searches that are based on branch-and-bound-like algorithms.CONTACT: s.kumar@asu.edu     

Distances that perfectly mislead

Given a collection of discrete characters (e.g., aligned DNA sites, gene adjacencies), a common measure of distance between taxa is the proportion of characters for which taxa have different character states. Tree reconstruction based on these (uncorrected) distances can be statistically inconsistent and can lead to trees different from those obtained using character-based methods such as maximum likelihood or maximum parsimony. However, in these cases the distance data often reveal their unreliability by some deviation from additivity, as indicated by conflicting support for more than one tree. We describe two results that show how uncorrected (and miscorrected) distance data can be simultaneously perfectly additive and misleading. First, multistate character data can be perfectly compatible and define one tree, and yet the uncorrected distances derived from these characters are perfectly treelike (and obey a molecular clock), only for a completely different tree. Second, under a Markov model of character evolution a similar phenomenon can occur; not only is there statistical inconsistency using uncorrected distances, but there is no evidence of this inconsistency because the distances look perfectly treelike (this does not occur in the classic two-parameter Felsenstein zone). We characterize precisely when uncorrected distances are additive on the true (and on a false) tree for four taxa. We also extend this result to a more general setting that applies to distances corrected according to an incorrect model.     

The average common substring approach to phylogenomic reconstruction.

We describe a novel method for efficient reconstruction of phylogenetic trees, based on sequences of whole genomes or proteomes, whose lengths may greatly vary. The core of our method is a new measure of pairwise distances between sequences. This measure is based on computing the average lengths of maximum common substrings, which is intrinsically related to information theoretic tools (Kullback-Leibler relative entropy). We present an algorithm for efficiently computing these distances. In principle, the distance of two l long sequences can be calculated in O(l) time. We implemented the algorithm using suffix arrays our implementation is fast enough to enable the construction of the proteome phylogenomic tree for hundreds of species and the genome phylogenomic forest for almost two thousand viruses. An initial analysis of the results exhibits a remarkable agreement with "acceptable phylogenetic and taxonomic truth." To assess our approach, our results were compared to the traditional (single-gene or protein-based) maximum likelihood method. The obtained trees were compared to implementations of a number of alternative approaches, including two that were previously published in the literature, and to the published results of a third approach. Comparing their outcome and running time to ours, using a "traditional" trees and a standard tree comparison method, our algorithm improved upon the "competition" by a substantial margin. The simplicity and speed of our method allows for a whole genome analysis with the greatest scope attempted so far. We describe here five different applications of the method, which not only show the validity of the method, but also suggest a number of novel phylogenetic insights.     

Do model‐based phylogenetic analyses perform better than parsimony? A test with empirical data

The use of model‐based methods to infer a phylogenetic tree from a given data set is frequently motivated by the truism that under certain circumstances the parsimony approach (MP) may produce incorrect topologies, while explicit model‐based approaches are believed to avoid this problem. In the realm of empirical data from actual taxa, it is not known (or knowable) how commonly MP, maximum‐likelihood or Bayesian inference are inaccurate. To test the perceived need for “sophisticated” model‐based approaches, we assessed the degree of congruence between empirical phylogenetic hypotheses generated by alternative methods applied to DNA sequence data in a sample of 1000 recently published articles. Of 504 articles that employed multiple methods, only two exhibited strongly supported incongruence among alternative methods. This result suggests that the MP approach does not produce deviant hypotheses of relationship due to convergent evolution in long branches. Our finding therefore indicates that the use of multiple analytical methods is largely superfluous. We encourage the use of analytical approaches unencumbered by ad hoc assumptions that sap the explanatory power of the evidence. © The Willi Hennig Society 2010.