arXiv subjects

Mike Steel

Publications and source records attributed to Mike Steel.

At least 109 records · Page 6Linked to original sources

arXiv2009

Can we avoid 'SIN' in the house of 'No Common Mechanism'?

In 'no common mechanism' (NCM) models of character evolution, each character can evolve on a phylogenetic tree under a partially or totally separate process (e.g. with its own branch lengths). In such cases, the usual conditions that suffice to establish the statistical consistency of tree reconstruction by methods such as maximum likelihood (ML) break down, suggesting that such methods may be prone to statistical inconsistency (SIN). In this paper we ask whether we can avoid SIN for tree topology reconstruction when adopting such models, either by using ML or any other method that could be devised. We prove that it is possible to avoid SIN for certain NCM models, but not for others, and the results depend delicately on the tree reconstruction method employed. We also describe the biological relevance of some recent mathematical results for the more usual 'common mechanism' setting. Our results are not intended to justify NCM, rather to set in place a framework within which such questions can be formally addressed.

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arXiv2009

Trait-dependent extinction leads to greater expected biodiversity loss

We use a classical combinatorial inequality to establish a Markov inequality for multivariate binary Markov processes on trees. We then apply this result, alongside with the FKG inequality, to compare the expected loss of biodiversity under two models of species extinction. One of these models is the generalized version of an earlier model in which extinction is influenced by some trait that can be classified into two states and which evolves on a tree according to a Markov process. Since more than one trait can affect the rates of species extinction, it is reasonable to allow, in the generalized model, k binary states that influence extinction rates. We compare this model to one that has matching marginal extinction probabilities for each species but for which the species extinction events are stochastically independent.

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arXiv2009

Quantifying the Extent of Lateral Gene Transfer Required to Avert a `Genome of Eden'

The complex pattern of presence and absence of many genes across different species provides tantalising clues as to how genes evolved through the processes of gene genesis, gene loss and lateral gene transfer (LGT). The extent of LGT, particularly in prokaryotes, and its implications for creating a `network of life' rather than a `tree of life' is controversial. In this paper, we formally model the problem of quantifying LGT, and provide exact mathematical bounds, and new computational results. In particular, we investigate the computational complexity of quantifying the extent of LGT under the simple models of gene genesis, loss and transfer on which a recent heuristic analysis of biological data relied. Our approach takes advantage of a relationship between LGT optimization and graph-theoretical concepts such as tree width and network flow.

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arXiv2009

Expected length of pendant and interior edges of a Yule tree

The Yule (pure-birth) model is the simplest null model of speciation; each lineage gives rise to a new lineage independently with the same rate $λ$. We investigate the expected length of an edge chosen at random from the resulting evolutionary tree. In particular, we compare the expected length of a randomly selected edge with the expected length of a randomly selected pendant edge. We provide some exact formulae, and show how our results depend slightly on whether the depth of the tree or the number of leaves is conditioned on, and whether $λ$ is known or is estimated using maximum likelihood.

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arXiv2009

Species, Clusters and the 'Tree of Life': A graph-theoretic perspective

A hierarchical structure describing the inter-relationships of species has long been a fundamental concept in systematic biology, from Linnean classification through to the more recent quest for a 'Tree of Life.' In this paper we use an approach based on discrete mathematics to address a basic question: Could one delineate this hierarchical structure in nature purely by reference to the 'genealogy' of present-day individuals, which describes how they are related with one another by ancestry through a continuous line of descent? We describe several mathematically precise ways by which one can naturally define collections of subsets of present day individuals so that these subsets are nested (and so form a tree) based purely on the directed graph that describes the ancestry of these individuals. We also explore the relationship between these and related clustering constructions.

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arXiv2009

Characterizing phylogenetically decisive taxon coverage

Increasingly, biologists are constructing evolutionary trees on large numbers of overlapping sets of taxa, and then combining them into a `supertree' that classifies all the taxa. In this paper, we ask how much coverage of the total set of taxa is required by these subsets in order to ensure we have enough information to reconstruct the supertree uniquely. We describe two results - a combinatorial characterization of the covering subsets to ensure that at most one supertree can be constructed from the smaller trees (whatever trees these may be) and a more liberal analysis that asks only that the supertree is highly likely to be uniquely specified by the tree structure on the covering subsets.

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arXiv2009

A Hall-type theorem for triplet set systems based on medians in trees

Given a collection $\C$ of subsets of a finite set $X$, let $\bigcup \C = \cup_{S \in \C}S$. Philip Hall's celebrated theorem \cite{hall} concerning `systems of distinct representatives' tells us that for any collection $\C$ of subsets of $X$ there exists an injective (i.e. one-to-one) function $f: \C \to X$ with $f(S) \in S$ for all $S \in \C$ if and and only if $\C$ satisfies the property that for all non-empty subsets $\C'$ of $\C$ we have $|\bigcup \C'| \geq |\C'|$. Here we show that if the condition $|\bigcup \C'| \geq |\C'|$ is replaced by the stronger condition $|\bigcup \C'| \geq |\C'|+2$, then we obtain a characterization of this condition for a collection of 3-element subsets of $X$ in terms of the existence of an injective function from $\C$ to the vertices of a tree whose vertex set includes $X$ and that satisfies a certain median condition. We then describe an extension of this result to collections of arbitrary-cardinality subsets of $X$.

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arXiv2009

An improved bound on the Maximum Agreement Subtree problem

We improve the lower bound on the extremal version of the Maximum Agreement Subtree problem. Namely we prove that two binary trees on the same $n$ leaves have subtrees with the same $\geq c\log\log n$ leaves which are homeomorphic, such that homeomorphism is identity on the leaves.

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arXiv2008

Computing the Distribution of a Tree Metric

The Robinson-Foulds (RF) distance is by far the most widely used measure of dissimilarity between trees. Although the distribution of these distances has been investigated for twenty years, an algorithm that is explicitly polynomial time has yet to be described for computing this distribution (which is also the distribution of trees around a given tree under the popular Robinson-Foulds metric). In this paper we derive a polynomial-time algorithm for this distribution. We show how the distribution can be approximated by a Poisson distribution determined by the proportion of leaves that lie in `cherries' of the given tree. We also describe how our results can be used to derive normalization constants that are required in a recently-proposed maximum likelihood approach to supertree construction.

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arXiv2008

Expected Anomalies in the Fossil Record

The problem of intermediates in the fossil record has been frequently discussed ever since Darwin. The extent of `gaps' (missing transitional stages) has been used to argue against gradual evolution from a common ancestor. Traditionally, gaps have often been explained by the improbability of fossilization and the discontinuous selection of found fossils. Here we take an analytical approach and demonstrate why, under certain sampling conditions, we may not expect intermediates to be found. Using a simple null model, we show mathematically that the question of whether a taxon sampled from some time in the past is likely to be morphologically intermediate to other samples (dated earlier and later) depends on the shape and dimensions of the underlying phylogenetic tree that connects the taxa, and the times from which the fossils are sampled.

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arXiv2008

A basic limitation on inferring phylogenies by pairwise sequence comparisons

Distance-based approaches in phylogenetics such as Neighbor-Joining are a fast and popular approach for building trees. These methods take pairs of sequences from them construct a value that, in expectation, is additive under a stochastic model of site substitution. Most models assume a distribution of rates across sites, often based on a gamma distribution. Provided the (shape) parameter of this distribution is known, the method can correctly reconstruct the tree. However, if the shape parameter is not known then we show that topologically different trees, with different shape parameters and associated positive branch lengths, can lead to exactly matching distributions on pairwise site patterns between all pairs of taxa. Thus, one could not distinguish between the two trees using pairs of sequences without some prior knowledge of the shape parameter. More surprisingly, this can happen for {\em any} choice of distinct shape parameters on the two trees, and thus the result is not peculiar to a particular or contrived selection of the shape parameters. On a positive note, we point out known conditions where identifiability can be restored (namely, when the branch lengths are clocklike, or if methods such as maximum likelihood are used).

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arXiv2008

Phylogenetic information complexity: Is testing a tree easier than finding it?

Phylogenetic trees describe the evolutionary history of a group of present-day species from a common ancestor. These trees are typically reconstructed from aligned DNA sequence data. In this paper we analytically address the following question: is the amount of sequence data required to accurately reconstruct a tree significantly more than the amount required to test whether or not a candidate tree was the `true' tree? By `significantly', we mean that the two quantities behave the same way as a function of the number of species being considered. We prove that, for a certain type of model, the amount of information required is not significantly different; while for another type of model, the information required to test a tree is independent of the number of leaves, while that required to reconstruct it grows with this number. Our results combine probabilistic and combinatorial arguments.

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arXiv2008

Sequence length bounds for resolving a deep phylogenetic divergence

In evolutionary biology, genetic sequences carry with them a trace of the underlying tree that describes their evolution from a common ancestral sequence. The question of how many sequence sites are required to recover this evolutionary relationship accurately depends on the model of sequence evolution, the substitution rate, divergence times and the method used to infer phylogenetic history. A particularly challenging problem for phylogenetic methods arises when a rapid divergence event occurred in the distant past. We analyse an idealised form of this problem in which the terminal edges of a symmetric four--taxon tree are some factor ($p$) times the length of the interior edge. We determine an order $p^2$ lower bound on the growth rate for the sequence length required to resolve the tree (independent of any particular branch length). We also show that this rate of sequence length growth can be achieved by existing methods (including the simple `maximum parsimony' method), and compare these order $p^2$ bounds with an order $p$ growth rate for a model that describes low-homoplasy evolution. In the final section, we provide a generic bound on the sequence length requirement for a more general class of Markov processes.

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arXiv2008

Markovian log-supermodularity, and its applications in phylogenetics

We establish a log-supermodularity property for probability distributions on binary patterns observed at the tips of a tree that are generated under any 2--state Markov process. We illustrate the applicability of this result in phylogenetics by deriving an inequality relevant to estimating expected future phylogenetic diversity under a model of species extinction. In a further application of the log-supermodularity property, we derive a purely combinatorial inequality for the parsimony score of a binary character. The proofs of our results exploit two classical theorems in the combinatorics of finite sets.

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arXiv2008

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 known 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.

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arXiv2007

Mixed-up trees: the structure of phylogenetic mixtures

In this paper we apply new geometric and combinatorial methods to the study of phylogenetic mixtures. The focus of the geometric approach is to describe the geometry of phylogenetic mixture distributions for the two state random cluster model, which is a generalization of the two state symmetric (CFN) model. In particular, we show that the set of mixture distributions forms a convex polytope and we calculate its dimension; corollaries include a simple criterion for when a mixture of branch lengths on the star tree can mimic the site pattern frequency vector of a resolved quartet tree. Furthermore, by computing volumes of polytopes we can clarify how ``common'' non-identifiable mixtures are under the CFN model. We also present a new combinatorial result which extends any identifiability result for a specific pair of trees of size six to arbitrary pairs of trees. Next we present a positive result showing identifiability of rates-across-sites models. Finally, we answer a question raised in a previous paper concerning ``mixed branch repulsion'' on trees larger than quartet trees under the CFN model.

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arXiv2007

Maximum Likelihood Supertrees

We analyse a maximum-likelihood approach for combining phylogenetic trees into a larger `supertree'. This is based on a simple exponential model of phylogenetic error, which ensures that ML supertrees have a simple combinatorial description (as a median tree, minimising a weighted sum of distances to the input trees). We show that this approach to ML supertree reconstruction is statistically consistent (it converges on the true species supertree as more input trees are combined), in contrast to the widely-used MRP method, which we show can be statistically inconsistent under the exponential error model. We also show that this statistical consistency extends to an ML approach for constructing species supertrees from gene trees. In this setting, incomplete lineage sorting (due to coalescence rates of homologous genes being lower than speciation rates) has been shown to lead to gene trees that are frequently different from species trees, and this can confound efforts to reconstruct the species phylogeny correctly.

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arXiv2007

Distribution of phylogenetic diversity under random extinction

Phylogenetic diversity is a measure for describing how much of an evolutionary tree is spanned by a subset of species. If one applies this to the (unknown) subset of current species that will still be present at some future time, then this `future phylogenetic diversity' provides a measure of the impact of various extinction scenarios in biodiversity conservation. In this paper we study the distribution of future phylogenetic diversity under a simple model of extinction (a generalized `field of bullets' model). We show that the distribution of future phylogenetic diversity converges to a normal distribution as the number of species grows (under mild conditions, which are necessary). We also describe an algorithm to compute the distribution efficiently, provided the edge lengths are integral, and briefly outline the significance of our findings for biodiversity conservation.

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