Michael D. Hendy

Spectral Analysis of Phylogenetic Data

The spectral analysis of sequence and distance data is a new approach to phylogenetic analysis. For two-state character sequences, the character values at a given site split the set of taxa into two subsets, a bipartition of the taxa set. The vector which counts the relative numbers of each of these bipartitions over all sites is called a sequence spectrum. Applying a transformation called a Hadamard conjugation, the sequence spectrum is transformed to the conjugate spectrum. This conjugation corrects for unobserved changes in the data, independently from the choice of phylogenetic tree. For any given phylogenetic tree with edge weights (probabilities of state change), we define a corresponding tree spectrum. The selection of a weighted phylogenetic tree from the given sequence data is made by matching the conjugate spectrum with a tree spectrum. We develop an optimality selection procedure using a least squares best fit, to find the phylogenetic tree whose tree spectrum most closely matches the conjugate spectrum. An inferred sequence spectrum can be derived from the selected tree spectrum using the inverse Hadamard conjugation to allow a comparison with the original sequence spectrum.A possible adaptation for the analysis of four-state character sequences with unequal frequencies is considered. A corresponding spectral analysis for distance data is also introduced. These analyses are illustrated with biological examples for both distance and sequence data. Spectral analysis using the Fast Hadamard transform allows optimal trees to be found for at least 20 taxa and perhaps for up to 30 taxa.The development presented here is self contained, although some mathematical proofs available elsewhere have been omitted. The analysis of sequence data is based on methods reported earlier, but the terminology and the application to distance data are new.

Mutation and Evolutionary Rates in Adélie Penguins from the Antarctic

Precise estimations of molecular rates are fundamental to our understanding of the processes of evolution. In principle, mutation and evolutionary rates for neutral regions of the same species are expected to be equal. However, a number of recent studies have shown that mutation rates estimated from pedigree material are much faster than evolutionary rates measured over longer time periods. To resolve this apparent contradiction, we have examined the hypervariable region (HVR I) of the mitochondrial genome using families of Adélie penguins (Pygoscelis adeliae) from the Antarctic. We sequenced 344 bps of the HVR I from penguins comprising 508 families with 915 chicks, together with both their parents. All of the 62 germline heteroplasmies that we detected in mothers were also detected in their offspring, consistent with maternal inheritance. These data give an estimated mutation rate (μ) of 0.55 mutations/site/Myrs (HPD 95% confidence interval of 0.29–0.88 mutations/site/Myrs) after accounting for the persistence of these heteroplasmies and the sensitivity of current detection methods. In comparison, the rate of evolution (k) of the same HVR I region, determined using DNA sequences from 162 known age sub-fossil bones spanning a 37,000-year period, was 0.86 substitutions/site/Myrs (HPD 95% confidence interval of 0.53 and 1.17). Importantly, the latter rate is not statistically different from our estimate of the mutation rate. These results are in contrast to the view that molecular rates are time dependent.

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 Γ 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 Γ 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.

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.

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.

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 photosynthesis organelles and single origins for each type of light harvesting complex.

On the distribution of lengths of evolutionary trees

This paper presents the results of the authors’ investigation of a combinatorial problem arising from the study of evolutionary trees. In graph theoretic terms it can be expressed as a problem of colouring vertices of a binary tree. For a given colouring of the pendant vertices of a binary tree there is a simple algorithm for assigning colours to internal vertices minimising the number of edges of the tree whose end vertices have differing colours. This minimal number is called the length of the tree. The question posed is: For given numbers of pendant vertices of assigned colours, how many trees of a particular length can be constructed on those vertices? This question is answered in two special cases. Answers to this problem are needed to establish the distribution of lengths of evolutionary trees, by which the significance of the maximum parsimony principle for selecting evolutionary trees can be judged.

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, Lis method, Maximum Parsimony, Neighbor-joining, Neighborliness, and UPGMA. We also provide a modification to Lis 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 Lis method, perform very poorly.

Reconstructing phylogenies from nucleotide pattern probabilities: a survey and some new results

Abstract The variations between homologous nucleotide sequences representative of various species are, in part, a consequence of the evolutionary history of these species. Determining the evolutionary tree from patterns in the sequences depends on inverting the stochastic processes governing the substitutions from their ancestral sequence. We present a number of recent (and some new) results which allow for a tree to be reconstructed from the expected frequencies of patterns in its leaf colorations generated under various Markov models. We summarize recent work using Hadamard conjugation, which provides an analytic relation between the parameters of Kimuras 3ST model on a phylogenetic tree and the sequence patterns produced. We give two applications of the theory by describing new properties of the popular “maximum parsimony” method for tree reconstruction.

SPECTRAL ANALYSIS AND A CLOSEST TREE METHOD FOR GENETIC SEQUENCES

We describe a new method for estimating the evolutionary tree linking a collection of species from their aligned four-state genetic sequences. This method, which can be adapted to provide a branch-and-bound algorithm, is statistically consistent provided the sequences have evolved according to a standard stochastic model of nucleotide mutation. Our approach exploits a recent group-theoretic description of this model.