Mike Steel

The complexity of reconstructing trees from qualitative characters and subtrees

In taxonomy and other branches of classification it is useful to know when tree-like classifications on overlapping sets of labels can be consistently combined into a parent tree. This paper considers the computation complexity of this problem. Recognizing when a consistent parent tree exists is shown to be intractable (NP-complete) for sets of unrooted trees, even when each tree in the set classifies just four labels. Consequently determining the compatibility of qualitative characters and partial binary characters is, in general, also NP-complete. However for sets of rooted trees an algorithm is described which constructs the “strict consensus tree” of all consistent parent trees (when they exist) in polynomial time. The related question of recognizing when a set of subtrees uniquely defines a parent tree is also considered, and a simple necessary and sufficient condition is described for rooted trees.

Recovering a tree from the leaf colourations it generates under a Markov model

Abstract We describe a simple transformation that allows for the fast recovery of a tree from the probabilities such a tree induces on the colourations of its leaves under a simple Markov process (with unknown parameters). This generalizes earlier results by not requiring the transition matrices associated with the edges of the tree to be of a particular form, or to be related by some fixed rate matrix, and by not insisting on a particular distribution of colours at the root of the tree. Applications to taxonomy are outlined briefly in three corollaries.

A few logs suffice to build (almost) all trees (l): part I

Inferring evolutionary trees is an interesting and important problem in biology that is very difficult from a computational point of view as most associated optimization problems are NP-hard. Although it is known that many methods are provably statistically consistent (i.e. the probability of recovering the correct tree converges on 1 as the sequence length increases), the actual rate of convergence for different methods has not been well understood. In a recent paper we introduced a new method for reconstructing evolutionary trees called the Dyadic Closure Method (DCM), and we showed that DCM has a very fast convergence rate. DCM runs in O(n^5 log n) time, where n is the number of sequences, so although it is polynomial it has computational requirements that are potentially too large to be of use in practice. In this paper we present another tree reconstruction method, the Witness-Antiwitness Method, or WAM. WAM is significantly faster than DCM, especially on random trees, and converges at the same rate as DCM. We also compare WAM to other methods used to reconstruct trees, including Neighbor Joining (possibly the most popular method among molecular biologists), and new methods introduced in the computer science literature.

A supertree method for rooted trees

The amalgamation of leaf-labelled (phylogenetic) trees on overlapping leaf sets into one (super)tree is a central problem in several areas of classification, particularly evolutionary biology. In this paper, we describe a new technique for amalgamating rooted phylogenetic trees. This appears to be the first such method to provably exhibit particular desirable properties which we list and establish.

Acquisition of 1,000 eubacterial genes physiologically transformed a methanogen at the origin of Haloarchaea

Archaebacterial halophiles (Haloarchaea) are oxygen-respiring heterotrophs that derive from methanogens—strictly anaerobic, hydrogen-dependent autotrophs. Haloarchaeal genomes are known to have acquired, via lateral gene transfer (LGT), several genes from eubacteria, but it is yet unknown how many genes the Haloarchaea acquired in total and, more importantly, whether independent haloarchaeal lineages acquired their genes in parallel, or as a single acquisition at the origin of the group. Here we have studied 10 haloarchaeal and 1,143 reference genomes and have identified 1,089 haloarchaeal gene families that were acquired by a methanogenic recipient from eubacteria. The data suggest that these genes were acquired in the haloarchaeal common ancestor, not in parallel in independent haloarchaeal lineages, nor in the common ancestor of haloarchaeans and methanosarcinales. The 1,089 acquisitions include genes for catabolic carbon metabolism, membrane transporters, menaquinone biosynthesis, and complexes I–IV of the eubacterial respiratory chain that functions in the haloarchaeal membrane consisting of diphytanyl isoprene ether lipids. LGT on a massive scale transformed a strictly anaerobic, chemolithoautotrophic methanogen into the heterotrophic, oxygen-respiring, and bacteriorhodopsin-photosynthetic haloarchaeal common ancestor.

Distributions of cherries for two models of trees

Null models for generating binary phylogenetic trees are useful for testing evolutionary hypotheses and reconstructing phylogenies. We consider two such null models - the Yule and uniform models - and in particular the induced distribution they generate on the number C(n) of cherries in the tree, where a cherry is a pair of leaves each of which is adjacent to a common ancestor. By realizing the process of cherry formation in these two models by extended Polya urn models we show that C(n) is asymptotically normal. We also give exact formulas for the mean and standard deviation of the C(n) in these two models. This allows simple statistical tests for the Yule and uniform null hypotheses.

Progress with methods for constructing evolutionary trees

Evolutionists dream of a tree-reconstruction method that is efficient (fast), powerful, consistent, robust and falsifiable. These criteria are at present conflicting in that the fastest methods are weak (in their use of information in the sequences) and inconsistent (even with very long sequences they may lead to an incorrect tree). But there has been exciting progress in new approaches to tree inference, in understanding general properties of methods, and in developing ideas for estimating the reliability of trees. New phylogenetic invariant methods allow selected parameters of the underlying model to be estimated directly from sequences. There is still a need for more theoretical understanding and assistance in applying what is already known.

Kaikoura tree theorems: computing the maximum agreement subtree

Abstract The Maximum Agreement Subtree Problem was posed by Finden and Gordon in 1985, and is as follows: given a set S = {S1, S2,…, Sn} and two trees P and Q leaf-labelled by the elements of S, find a maximum cardinality subset S0 of S such that P|S0 = Q|S0. This problem arises in evaluationary tree construction, where different methods or data yield (possibly) different trees for the same species on which the trees agree. A superpolynomial time algorithm for finding a maximum agreement subtree of two binary trees was found by Kubicka et al. In this paper, we will present an O(n4.5 log n + V) algorithm to determine the largest agreement subtree of two trees on n leaves, where V is the maximum number of nodes in the trees. For the case of trees of maximum degree k, there are two algorithms presented: one has running time O(k!n2 + V) while the other has running time O(k2.5n2 log n + V). These algorithms also apply when the trees are constrained to be rooted; in this case a maximum agreement subtree is also constrained to be rooted. Each of the algorithms we present can be modified to produce a maximum agreement subtree, rather than computing only the size. Thus, we can solve the searchproblem in the same running time as above.

A few logs suffice to build (almost) all trees: part II

Abstract Inferring evolutionary trees is an interesting and important problem in biology, but one that is computationally difficult as most associated optimization problems are NP-hard. Although many methods are provably statistically consistent (i.e. the probability of recovering the correct tree converges to 1 as the sequence length increases), the actual rate of convergence for different methods has not been well understood. In a recent paper we introduced a new method for reconstructing evolutionary trees called the dyadic closure method (DCM), and we showed that DCM has a very fast convergence rate. DCM runs in O( n 5 log n ) time, where n is the number of sequences, and so, although polynomial, the computational requirements are potentially too large to be of use in practice. In this paper we present another tree reconstruction method, the witness-antiwitness method (WAM). WAM is faster than DCM, especially on random trees, and converges to the true tree topology at the same rate as DCM. We also compare WAM to other methods used to reconstruct trees, including Neighbor Joining (possibly the most popular method among molecular biologists), and new methods introduced in the computer science literature.

Properties of phylogenetic trees generated by Yule-type speciation models.

We investigate some discrete structural properties of evolutionary trees generated under simple null models of speciation, such as the Yule model. These models have been used as priors in Bayesian approaches to phylogenetic analysis, and also to test hypotheses concerning the speciation process. In this paper we describe new results for three properties of trees generated under such models. Firstly, for a rooted tree generated by the Yule model we describe the probability distribution on the depth (number of edges from the root) of the most recent common ancestor of a random subset of k species. Next we show that, for trees generated under the Yule model, the approximate position of the root can be estimated from the associated unrooted tree, even for trees with a large number of leaves. Finally, we analyse a biologically motivated extension of the Yule model and describe its distribution on tree shapes when speciation occurs in rapid bursts.