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M D Hendy

Publications and source records attributed to M D Hendy.

At least 19 recordsLinked to original sources
PubMed2004

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.

Algorithms↗
PubMed2003

Outgroup misplacement and phylogenetic inaccuracy under a molecular clock--a simulation study.

We conducted a simulation study of the phylogenetic methods UPGMA, neighbor joining, maximum parsimony, and maximum likelihood for a five-taxon tree under a molecular clock. The parameter space included a small region where maximum parsimony is inconsistent, so we tested inconsistency correction for parsimony and distance correction for neighbor joining. As expected, corrected parsimony was consistent. For these data, maximum likelihood with the clock assumption outperformed each of the other methods tested. The distance-based methods performed marginally better than did maximum parsimony and maximum likelihood without the clock assumption. Data correction was generally detrimental to accuracy, especially for short sequence lengths. We identified another region of the parameter space where, although consistent for a given method, some incorrect trees were each selected with up to twice the frequency of the correct (generating) tree for sequences of bounded length. These incorrect trees are those where the outgroup has been incorrectly placed. In addition to this problem, the placement of the outgroup sequence can have a confounding effect on the ingroup tree, whereby the ingroup is correct when using the ingroup sequences alone, but with the inclusion of the outgroup the ingroup tree becomes incorrect.

Cluster Analysis↗
PubMed2001

An algorithm for constructing local regions in a phylogenetic network.

The groupings of taxa in a phylogenetic tree cannot represent all the conflicting signals that usually occur among site patterns in aligned homologous genetic sequences. Hence a tree-building program must compromise by reporting a subset of the patterns, using some discriminatory criterion. Thus, in the worst case, out of possibly a large number of equally good trees, only an arbitrarily chosen tree might be reported by the tree-building program as "The Tree." This tree might then be used as a basis for phylogenetic conclusions. One strategy to represent conflicting patterns in the data is to construct a network. The Buneman graph is a theoretically very attractive example of such a network. In particular, a characterization for when this network will be a tree is known. Also the Buneman graph contains each of the most parsimonious trees indicated by the data. In this paper we describe a new method for constructing the Buneman graph that can be used for a generalization of Hadamard conjugation to networks. This new method differs from previous methods by allowing us to focus on local regions of the graph without having to first construct the full graph. The construction is illustrated by an example.

Algorithms↗
PubMed2001

Mathematical elegance with biochemical realism: the covarion model of molecular evolution.

There is an apparent paradox in our understanding of molecular evolution. Current biochemically based models predict that evolutionary trees should not be recoverable for divergences beyond a few hundred million years. In practice, however, trees often appear to be recovered from much older times. Mathematical models, such as those assuming that sites evolve at different rates [including a Gamma distribution of rates across sites (RAS)] may in theory allow the recovery of some ancient divergences. However, such models require that each site maintain its characteristic rate over the whole evolutionary period. This assumption, however, contradicts the knowledge that tertiary structures diverge with time, invalidating the rate-constancy assumption of purely mathematical models. We report here that a hidden Markov version of the covarion model can meet both biochemical and statistical requirements for the analysis of sequence data. The model was proposed on biochemical grounds and can be implemented with only two additional parameters. The two hidden parts of this model are the proportion of sites free to vary (covarions) and the rate of interchange between fixed sites and these variable sites. Simulation results are consistent with this approach, providing a better framework for understanding anciently diverged sequences than the standard RAS models. However, a Gamma distribution of rates may approximate a covarion model and may possibly be justified on these grounds. The accurate reconstruction of older divergences from sequence data is still a major problem, and molecular evolution still requires mathematical models that also have a sound biochemical basis.

Evolution, Molecular↗
PubMed2000

Can fast early rates reconcile molecular dates with the Cambrian explosion?

Molecular dates consistently place the divergence of major metazoan lineages in the Precambrian, leading to the suggestion that the 'Cambrian explosion' is an artefact of preservation which left earlier forms unrecorded in the fossil record. While criticisms of molecular analyses for failing to deal with variation in the rate of molecular evolution adequately have been countered by analyses which allow both site-to-site and lineage-specific rate variation, no analysis to date has allowed the rates to vary temporally. If the rates of molecular evolution were much higher early in the metazoan radiation, molecular dates could consistently overestimate the divergence times of lineages. Here, we use a new method which uses multiple calibration dates and an empirically determined range of possible substitution rates to place bounds on the basal date of divergence of lineages in order to ask whether faster rates of molecular evolution early in the metazoan radiation could possibly account for the discrepancy between molecular and palaeontological date estimates. We find that allowing basal (interphylum) lineages the fastest observed substitution rate brings the minimum possible divergence date (586 million years ago) to the Vendian period, just before the first multicellular animal fossils, but excludes divergence of the major metazoan lineages in a Cambrian explosion.

Animals↗
PubMed2000

The power of relative rates tests depends on the data.

One of the most useful features of molecular phylogenetic analyses is the potential for estimating dates of divergence of evolutionary lineages from the DNA of extant species. But lineage-specific variation in rate of molecular evolution complicates molecular dating, because a calibration rate estimated from one lineage may not be an accurate representation of the rate in other lineages. Many molecular dating studies use a "clock test" to identify and exclude sequences that vary in rate between lineages. However, these clock tests should not be relied upon without a critical examination of their effectiveness at removing rate variable sequences from any given data set, particularly with regard to the sequence length and number of variable sites. As an illustration of this problem we present a power test of a frequently employed triplet relative rates test. We conclude that (1) relative rates tests are unlikely to detect moderate levels of lineage-specific rate variation (where one lineage has a rate of molecular evolution 1.5 to 4.0 times the other) for most commonly used sequences in molecular dating analyses, and (2) this lack of power is likely to result in substantial error in the estimation of dates of divergence. As an example, we show that the well-studied rate difference between murid rodents and great apes will not be detected for many of the sequences used to date the divergence between these two lineages and that this failure to detect rate variation is likely to result in consistent overestimation the date of the rodent-primate split.

Evolution, Molecular↗
PubMed2000

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.

Evolution, Molecular↗
PubMed1999

Mammalian evolution: timing and implications from using the LogDeterminant transform for proteins of differing amino acid composition.

We explore the tree of mammalian mtDNA sequences, using particularly the LogDet transform on amino acid sequences, the distance Hadamard transform, and the Closest Tree selection criterion. The amino acid composition of different species show significant differences, even within mammals. After compensating for these differences, nearest-neighbor bootstrap results suggest that the tree is locally stable, though a few groups show slightly greater rearrangements when a large proportion of the constant sites are removed. Many parts of the trees we obtain agree with those on published protein ML trees. Interesting results include a preference for rodent monophyly. The detection of a few alternative signals to those on the optimal tree were obtained using the distance Hadamard transform (with results expressed as a Lento plot). One rearrangement suggested was the interchange of the position of primates and rodents on the optimal tree. The basic stability of the tree, combined with two calibration points (whale/cow and horse/rhinoceros), together with a distant secondary calibration from the mammal/bird divergence, allows inferences of the times of divergence of putative clades. Allowing for sampling variances due to finite sequence length, most major divergences amongst lineages leading to modern orders, appear to occur well before the Cretaceous/Tertiary (K/T) boundary. Implications arising from these early divergences are discussed, particularly the possibility of competition between the small dinosaurs and the new mammal clades.

Amino Acids↗
PubMed1996

Complete families of linear invariants for some stochastic models of sequence evolution, with and without the molecular clock assumption.

For various models of sequence evolution, the set of linear functions of the frequencies of the nucleotide patterns forms a vector space, the invariant space. Here we distinguish between the model of nucleotide substitution, and the phylogenetic tree T describing the paths on which these changes occur. We describe a procedure to construct a basis of the invariant space for those models that are extensions of models incorporating Kimura's three substitution model of nucleotide change, including both the Jukes-Cantor and Cavender-Farris models. The dimension of the invariant space is determined, for those models where it is independent of the tree topology, as a function of the number of sequences. These are calculated where the nucleotide distribution at the root is unspecified, and both with, and without, the assumption of the molecular clock hypothesis. The invariants have a number of potential applications, including tree identification, and testing the fit of models (which could include the molecular clock) to sequence data.

Biological Evolution↗
PubMed1995

Improved analyses of human mtDNA sequences support a recent African origin for Homo sapiens.

New quantitative methods are applied to the 135 human mitochondrial sequences from the Vigilant et al. data set. General problems in analyzing large numbers of short sequences are discussed, and an improved strategy is suggested. A key feature is to focus not on individual trees but on the general "landscape" of trees. Over 1,000 searches were made from random starting trees with only one tree (a local optimum) being retained each time, thereby ensuring optima were found independently. A new tree comparison metric was developed that is unaffected by rearrangements of trees around many very short internal edges. Use of this metric showed that downweighting hypervariable sites revealed more evolutionary structure than studies that weighted all sites equally. Our results are consistent with convergence toward a global optimum. Crucial features are that the best optima show very strong regional differentiation, a common group of 49 African sequences is found in all the best optima, and the best optima contain the 16 !Kung sequences in a separate group of San people. The other 86 sequences form a heterogeneous mixture of Africans, Europeans, Australopapuans, and Asians. Thus all major human lineages occur in Africa, but only a subset occurs in the rest of the world. The existence of these African-only groups strongly contradicts multiregional theories for the origin of Homo sapiens that require widespread migration and interbreeding over the entire range of H. erectus. Only when the multiregional model is rejected is it appropriate to consider the root, based on a single locus, to be the center of origin of a population (otherwise different loci could give alternative geographic positions for the root). For this data, several methods locate the root within the group of 49 African sequences and are thus consistent with the recent African origin of H. sapiens. We demonstrate that the time of the last common ancestor cannot be the time of major expansion in human numbers, and our results are thus also consistent with recent models that differentiate between the last common ancestor, expansion out of Africa, and the major expansion in human populations. Such a two-phase model is consistent with a wide range of molecular and archeological evidence.

Africa↗
PubMed1994

A discrete Fourier analysis for evolutionary trees.

Discrete Fourier transformations have recently been developed to model the evolution of two-state characters (the Cavender/Farris model). We report here the extension of these transformations to provide invertible relationships between a phylogenetic tree T (with three probability parameters of nucleotide substitution on each edge corresponding to Kimura's 3ST model) and the expected frequencies of the nucleotide patterns in the sequences. We refer to these relationships as spectral analysis. In either model with independent and identically distributed site substitutions, spectral analysis allows a global correction for all multiple substitutions (second- and higher-order interactions), independent of any particular tree. From these corrected data we use a least-squares selection procedure, the closest tree algorithm, to infer an evolutionary tree. Other selection criteria such as parsimony or compatibility analysis could also be used; each of these criteria will be statistically consistent for these models. The closest tree algorithm selects a unique best-fit phylogenetic tree together with independent edge length parameters for each edge. The method is illustrated with an analysis of some primate hemoglobin sequences.

Animals↗
PubMed1994

The effects of sequence length, tree topology, and number of taxa on the performance of phylogenetic methods.

Simulations were used to study the performance of several character-based and distance-based phylogenetic methods in obtaining the correct tree from pseudo-randomly generated input data. The study included all the topologies of unrooted binary trees with from 4 to 10 pendant vertices (taxa) inclusive. The length of the character sequences used ranged from 10 to 10(5) characters exponentially. The methods studied include Closest Tree, Compatibility, Li's method, Maximum Parsimony, Neighbor-joining, Neighborliness, and UPGMA. We also provide a modification to Li's method (SimpLi) which is consistent with additive data. We give estimations of the sequence lengths required for given confidence in the output of these methods under the assumptions of molecular evolution used in this study. A notation for characterizing all tree topologies is described. We show that when the number of taxa, the maximum path length, and the minimum edge length are held constant, there it little but significant dependence of the performance of the methods on the tree topology. We show that those methods that are consistent with the model used perform similarly, whereas the inconsistent methods, UPGMA and Li's method, perform very poorly.

Base Sequence↗
PubMed1994

Reconstructing trees when sequence sites evolve at variable rates.

For a sequence of colors independently evolving on a tree under a simple Markov model, we consider conditions under which the tree can be uniquely recovered from the "sequence spectrum"-the expected frequencies of the various leaf colorations. This is relevant for phylogenetic analysis (where colors represent nucleotides or amino acids; leaves represent extant taxa) as the sequence spectrum is estimated directly from a collection of aligned sequences. Allowing the rate of the evolutionary process to vary across sites is an important extension over most previous studies-we show that, given suitable restrictions on the rate distribution, the true tree (up to the placement of its root) is uniquely identified by its sequence spectrum. However, if the rate distribution is unknown and arbitrary, then, for simple models, it is possible for every tree to produce the same sequence spectrum. Hence there is a logical barrier to accurate, consistent phylogenetic inference for these models when assumptions about the rate distribution are not made. This result exploits a novel theorem on the action of polynomials with non-negative coefficients on sequences.

Markov Chains↗
PubMed1993

Neighbor-joining uses the optimal weight for net divergence.

A class of phylogenetic clustering methods which calculate net divergences from distance data, but assign differing weights to the net divergences, is defined. The class includes the Neighbor-Joining Method and the Unweighted Pair-Group Method with Arithmetic Mean. The accuracy of some of these methods is studied by computer simulation for the case of four taxa under the additive tree hypothesis. Of these methods and under this hypothesis, it is proved that Neighbor-Joining uses the only weighting for net divergence which is consistent, so that it is the only method in the class which is expected to converge to the correct tree as more data are added. Neighbor-Joining is then compared with Closest Tree on Distances for five taxa by simulation. It is proved that Closest Tree on Distances is equivalent to Neighbor-Joining for four taxa, though it is not when more than four taxa are considered.

Cluster Analysis↗
PubMed1992

Controversy on chloroplast origins.

Controversy exists over the origins of photosynthetic organelles in that contradictory trees arise from different sequence, biochemical and ultrastructural data sets. We propose a testable hypothesis which explains this inconsistency as a result of the differing GC contents of sequences. We report that current methods of tree reconstruction tend to group sequences with similar GC contents irrespective of whether the similar GC content is due to common ancestry or is independently acquired. Nuclear encoded sequences (high GC) give different trees from chloroplast encoded sequences (low GC). We find that current data is consistent with the hypothesis of multiple origins for photosynthetic organelles and single origins for each type of light harvesting complex.

Base Composition↗
PubMed1989

Influenza viruses, comets and the science of evolutionary trees.

The study of phylogeny is becoming increasing scientific in that hypotheses can be tested quantitatively. We report a method of estimating the probabilities of obtaining a tree of a given length from nucleic acid sequence data. The method is applied to the hypothesis of Hoyle & Wickramasinghe that the earth is being continually bombarded by influenza (and other) viruses which originate from comets. A quantitative analysis of sequences from the H1 strain of human influenza viruses contradicts three versions of the Hoyle-Wickramasinghe model. One non-evolutionary version of their model has less than one chance in 10(66) of being correct. A version that allowed extraterrestrial evolution has less than one change in 10(6) of being correct. The sequence data is in agreement with the biological (evolutionary) model. The results are discussed from the aspect of the falsifiability of evolutionary theory.

Base Sequence↗