arXiv subjects

Jeremy G. Sumner

Publications and source records attributed to Jeremy G. Sumner.

11 recordsLinked to original sources

arXiv2020

The impracticalities of multiplicatively-closed codon models: a retreat to linear alternatives

A matrix Lie algebra is a linear space of matrices closed under the operation $ [A, B] = AB-BA $. The "Lie closure" of a set of matrices is the smallest matrix Lie algebra which contains the set. In the context of Markov chain theory, if a set of rate matrices form a Lie algebra, their corresponding Markov matrices are closed under matrix multiplication; this has been found to be a useful property in phylogenetics. Inspired by previous research involving Lie closures of DNA models, it was hypothesised that finding the Lie closure of a codon model could help to solve the problem of mis-estimation of the non-synonymous/synonymous rate ratio, $ ω$. We propose two different methods of finding a linear space from a model: the first is the \emph{linear closure} which is the smallest linear space which contains the model, and the second is the \emph{linear version} which changes multiplicative constraints in the model to additive ones. For each of these linear spaces we then find the Lie closures of them. Under both methods, it was found that closed codon models would require thousands of parameters, and that any partial solution to this problem that was of a reasonable size violated stochasticity. Investigation of toy models indicated that finding the Lie closure of matrix linear spaces which deviated only slightly from a simple model resulted in a Lie closure that was close to having the maximum number of parameters possible. Given that Lie closures are not practical, we propose further consideration of the two variants of linearly closed models.

q-bio.PE↗

arXiv2019

The ancient Operational Code is embedded in the amino acid substitution matrix and aaRS phylogenies

The underlying structure of the canonical amino acid substitution matrix (aaSM) is examined by considering stepwise improvements in the differential recognition of amino acids according to their chemical properties during the branching history of the two aminoacyl-tRNA synthetase (aaRS) superfamilies. The evolutionary expansion of the genetic code is described by a simple parameterization of the aaSM, in which (i) the number of distinguishable amino acid types, (ii) the matrix dimension, and (iii) the number of parameters, each increases by one for each bifurcation in an aaRS phylogeny. Parameterized matrices corresponding to trees in which the size of an amino acid sidechain is the only discernible property behind its categorization as a substrate, exclusively for a Class I or II aaRS, provide a significantly better fit to empirically determined aaSM than trees with random bifurcation patterns. A second split between polar and nonpolar amino acids in each Class effects a vastly greater further improvement. The earliest Class-separated epochs in the phylogenies of the aaRS reflect these enzymes' capability to distinguish tRNAs through the recognition of acceptor stem identity elements via the minor (Class I) and major (Class II) helical grooves, which is how the ancient Operational Code functioned. The advent of tRNA recognition using the anticodon loop supports the evolution of the optimal map of amino acid chemistry found in the later Genetic Code, an essentially digital categorization, in which polarity is the major functional property, compensating for the unrefined, haphazard differentiation of amino acids achieved by the Operational Code.

q-bio.PE↗

arXiv2017

Exploring the consequences of lack of closure in codon models

Models of codon evolution are commonly used to identify positive selection. Positive selection is typically a heterogeneous process, i.e., it acts on some branches of the evolutionary tree and not others. Previous work on DNA models showed that when evolution occurs under a heterogeneous process it is important to consider the property of model closure, because non-closed models can give biased estimates of evolutionary processes. The existing codon models that account for the genetic code are not closed; to establish this it is enough to show that they are not linear (meaning that the sum of two codon rate matrices in the model is not a matrix in the model). This raises the concern that a single codon model fit to a heterogeneous process might mis-estimate both the effect of selection and branch lengths. Codon models are typically constructed by choosing an underlying DNA model (e.g., HKY) that acts identically and independently at each codon position, and then applying the genetic code via the parameter $ω$ to modify the rate of transitions between codons that code for different amino acids. Here we use simulation to investigate the accuracy of estimation of both the selection parameter $ω$ and branch lengths in cases where the underlying DNA process is heterogeneous but $ω$ is constant. We find that both $ω$ and branch lengths can be mis-estimated in these scenarios. Errors in $ω$ were usually less than 2% but could be as high as 17%. We also assessed if choosing different underlying DNA models had any affect on accuracy, in particular we assessed if using closed DNA models gave any advantage. However, a DNA model being closed does not imply that the codon model constructed from it is closed, and in general we found that using closed DNA models did not decrease errors in the estimation of $ω$.

q-bio.PE↗

arXiv2017

Distinguishing between convergent evolution and violation of the molecular clock

We give a non-technical introduction to convergence-divergence models, a new modeling approach for phylogenetic data that allows for the usual divergence of species post speciation but also allows for species to converge, i.e. become more similar over time. By examining the $3$-taxon case in some detail we illustrate that phylogeneticists have been "spoiled" in the sense of not having to think about the structural parameters in their models by virtue of the strong assumption that evolution is treelike. We show that there are not always good statistical reasons to prefer the usual class of treelike models over more general convergence-divergence models. Specifically we show many $3$-taxon datasets can be equally well explained by supposing violation of the molecular clock due to change in the rate of evolution along different edges, or by keeping the assumption of a constant rate of evolution but instead assuming that evolution is not a purely divergent process. Given the abundance of evidence that evolution is not strictly treelike, our discussion is an illustration that as phylogeneticists we often need to think clearly about the structural form of the models we use.

q-bio.PE↗

arXiv2017

Lie-Markov models derived from finite semigroups

We present and explore a general method for deriving a Lie-Markov model from a finite semigroup. If the degree of the semigroup is $k$, the resulting model is a continuous-time Markov chain on $k$ states and, as a consequence of the product rule in the semigroup, satisfies the property of multiplicative closure. This means that the product of any two probability substitution matrices taken from the model produces another substitution matrix also in the model. We show that our construction is a natural generalization of the concept of group-based models.

math.GR↗

arXiv2014

A new hierarchy of phylogenetic models consistent with heterogeneous substitution rates

When the process underlying DNA substitutions varies across evolutionary history, the standard Markov models underlying standard phylogenetic methods are mathematically inconsistent. The most prominent example is the general time reversible model (GTR) together with some, but not all, of its submodels. To rectify this deficiency, Lie Markov models have been developed as the class of models that are consistent in the face of a changing process of DNA substitutions. Some well-known models in popular use are within this class, but are either overly simplistic (e.g. the Kimura two-parameter model) or overly complex (the general Markov model). On a diverse set of biological data sets, we test a hierarchy of Lie Markov models spanning the full range of parameter richness. Compared against the benchmark of the ever-popular GTR model, we find that as a whole the Lie Markov models perform remarkably well, with the best performing models having eight parameters and the ability to recognise the distinction between purines and pyrimidines.

q-bio.PE↗

arXiv2013

Lie Markov models with purine/pyrimidine symmetry

Continuous-time Markov chains are a standard tool in phylogenetic inference. If homogeneity is assumed, the chain is formulated by specifying time-independent rates of substitutions between states in the chain. In applications, there are usually extra constraints on the rates, depending on the situation. If a model is formulated in this way, it is possible to generalise it and allow for an inhomogeneous process, with time-dependent rates satisfying the same constraints. It is then useful to require that there exists a homogeneous average of this inhomogeneous process within the same model. This leads to the definition of "Lie Markov models", which are precisely the class of models where such an average exists. These models form Lie algebras and hence concepts from Lie group theory are central to their derivation. In this paper, we concentrate on applications to phylogenetics and nucleotide evolution, and derive the complete hierarchy of Lie Markov models that respect the grouping of nucleotides into purines and pyrimidines -- that is, models with purine/pyrimidine symmetry. We also discuss how to handle the subtleties of applying Lie group methods, most naturally defined over the complex field, to the stochastic case of a Markov process, where parameter values are restricted to be real and positive. In particular, we explore the geometric embedding of the cone of stochastic rate matrices within the ambient space of the associated complex Lie algebra. The whole list of Lie Markov models with purine/pyrimidine symmetry is available at http://www.pagines.ma1.upc.edu/~jfernandez/LMNR.pdf.

q-bio.PE↗

arXiv2012

Lie geometry of 2x2 Markov matrices

In recent work discussing model choice for continuous-time Markov chains, we have argued that it is important that the Markov matrices that define the model are closed under matrix multiplication (Sumner 2012a, 2012b). The primary requirement is then that the associated set of rate matrices form a Lie algebra. For the generic case, this connection to Lie theory seems to have first been made by Johnson (1985), with applications for specific models given in Bashford (2004) and House (2012). Here we take a different perspective: given a model that forms a Lie algebra, we apply existing Lie theory to gain additional insight into the geometry of the associated Markov matrices. In this short note, we present the simplest case possible of 2x2 Markov matrices. The main result is a novel decomposition of 2x2 Markov matrices that parameterises the general Markov model as a perturbation away from the binary-symmetric model. This alternative parameterisation provides a useful tool for visualising the binary-symmetric model as a submodel of the general Markov model.

math.ST↗

arXiv2012

A tensorial approach to the inversion of group-based phylogenetic models

Using a tensorial approach, we show how to construct a one-one correspondence between pattern probabilities and edge parameters for any group-based model. This is a generalisation of the "Hadamard conjugation" and is equivalent to standard results that use Fourier analysis. In our derivation we focus on the connections to group representation theory and emphasize that the inversion is possible because, under their usual definition, group-based models are defined for abelian groups only. We also argue that our approach is elementary in the sense that it can be understood as simple matrix multiplication where matrices are rectangular and indexed by ordered-partitions of varying sizes.

q-bio.PE↗

arXiv2012

Tensor Rank, Invariants, Inequalities, and Applications

Though algebraic geometry over $\mathbb C$ is often used to describe the closure of the tensors of a given size and complex rank, this variety includes tensors of both smaller and larger rank. Here we focus on the $n\times n\times n$ tensors of rank $n$ over $\mathbb C$, which has as a dense subset the orbit of a single tensor under a natural group action. We construct polynomial invariants under this group action whose non-vanishing distinguishes this orbit from points only in its closure. Together with an explicit subset of the defining polynomials of the variety, this gives a semialgebraic description of the tensors of rank $n$ and multilinear rank $(n,n,n)$. The polynomials we construct coincide with Cayley's hyperdeterminant in the case $n=2$, and thus generalize it. Though our construction is direct and explicit, we also recast our functions in the language of representation theory for additional insights. We give three applications in different directions: First, we develop basic topological understanding of how the real tensors of complex rank $n$ and multilinear rank $(n,n,n)$ form a collection of path-connected subsets, one of which contains tensors of real rank $n$. Second, we use the invariants to develop a semialgebraic description of the set of probability distributions that can arise from a simple stochastic model with a hidden variable, a model that is important in phylogenetics and other fields. Third, we construct simple examples of tensors of rank $2n-1$ which lie in the closure of those of rank $n$.

math.AG↗

arXiv2012

Low-parameter phylogenetic estimation under the general Markov model

In their 2008 and 2009 papers, Sumner and colleagues introduced the "squangles" - a small set of Markov invariants for phylogenetic quartets. The squangles are consistent with the general Markov model (GM) and can be used to infer quartets without the need to explicitly estimate all parameters. As GM is inhomogeneous and hence non-stationary, the squangles are expected to perform well compared to standard approaches when there are changes in base-composition amongst species. However, GM includes the IID assumption, so the squangles should be confounded by data generated with invariant sites or with rate-variation across sites. Here we implement the squangles in a least-squares setting that returns quartets weighted by either confidence or internal edge lengths; and use these as input into a variety of quartet-based supertree methods. For the first time, we quantitatively investigate the robustness of the squangles to the breaking of IID assumptions on both simulated and real data sets; and we suggest a modification that improves the performance of the squangles in the presence of invariant sites. Our conclusion is that the squangles provide a novel tool for phylogenetic estimation that is complementary to methods that explicitly account for rate-variation across sites, but rely on homogeneous - and hence stationary - models.

q-bio.QM↗