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Andrew Francis

Publications and source records attributed to Andrew Francis.

At least 19 recordsLinked to original sources
arXiv2025

Binary normal networks without near reticulations can be reconstructed from their rooted triples

Normal networks are an important class of phylogenetic networks that have compelling mathematical properties which align with intuition about inference from genetic data. While tools enabling widespread use of phylogenetic networks in the biological literature are still under mathematical, statistical, and computational development, many such results are being assembled, and in particular for normal phylogenetic networks. For instance, it has been shown that binary normal networks can be reconstructed from the sets of three- and four-leaf rooted phylogenetic trees that they display. It is also known that one can reconstruct particular subclasses of normal networks from just the displayed rooted triples. This applies, for instance, to rooted binary phylogenetic trees and to binary level-$1$ normal networks. In this paper we address the question of how much of the class of binary normal networks can be reconstructed from just the rooted triples that they display. We find that all except those with substructures that we call ``near-sibling reticulations'' and ``near-stack reticulations'' can be reconstructed just from their rooted triples. This goes some way to answering the natural question of how much information can be extracted from a set of displayed rooted triples, which are arguably the simplest substructure that one may hope for in a phylogenetic object.

math.CO↗
arXiv2025

Counting spinal phylogenetic networks

Phylogenetic networks are an important way to represent evolutionary histories that involve reticulations such as hybridization or horizontal gene transfer, yet fundamental questions such as how many networks there are that satisfy certain properties are very difficult. A new way to encode a large class of networks, using expanding covers, may provide a way to approach such problems. Expanding covers encode a large class of phylogenetic networks, called labellable networks. This class does not include all networks, but does include many familiar classes, including orchard, normal, tree-child and tree-sibling networks. As expanding covers are a combinatorial structure, it is possible that they can be used as a tool for counting such classes for a fixed number of leaves and reticulations, for which, in many cases, a closed formula has not yet been found. More recently, a new class of networks was introduced, called spinal networks, which are analogous to caterpillar trees for phylogenetic trees and can be fully described using covers. In the present article, we describe a method for counting networks that are both spinal and belong to some more familiar class, with the hope that these form a base case from which to attack the more general classes.

q-bio.PE↗
arXiv2024

Factorizing the Brauer monoid in polynomial time

Finding a minimal factorization for a generic semigroup can be done by using the Froidure-Pin Algorithm, which is not feasible for semigroups of large sizes. On the other hand, if we restrict our attention to just a particular semigroup, we could leverage its structure to obtain a much faster algorithm. In particular, $\mathcal{O}(N^2)$ algorithms are known for factorizing the Symmetric group $S_N$ and the Temperley-Lieb monoid $\mathcal{T}\mathcal{L}_N$, but none for their superset the Brauer monoid $\mathcal{B}_{N}$. In this paper we hence propose a $\mathcal{O}(N^4)$ factorization algorithm for $\mathcal{B}_{N}$. At each iteration, the algorithm rewrites the input $X \in \mathcal{B}_{N}$ as $X = X' \circ p_i$ such that $\ell(X') = \ell(X) - 1$, where $p_i$ is a factor for $X$ and $\ell$ is a length function that returns the minimal number of factors needed to generate $X$.

math.RA↗
arXiv2023

An algebraic model for inversion and deletion in bacterial genome rearrangement

Inversions, also sometimes called reversals, are a major contributor to variation among bacterial genomes, with studies suggesting that those involving small numbers of regions are more likely than larger inversions. Deletions may arise in bacterial genomes through the same biological mechanism as inversions, and hence a model that incorporates both is desirable. However, while inversion distances between genomes have been well studied, there has yet to be a model which accounts for the combination of both deletions and inversions. To account for both of these operations, we introduce an algebraic model that utilises partial permutations. This leads to an algorithm for calculating the minimum distance to the most recent common ancestor of two bacterial genomes evolving by inversions (of adjacent regions) and deletions. The algebraic model makes the existing short inversion models more complete and realistic by including deletions, and also introduces new algebraic tools into evolutionary distance problems.

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arXiv2023

Phylogenetic network classes through the lens of expanding covers

It was recently shown that a large class of phylogenetic networks, the `labellable' networks, is in bijection with the set of `expanding' covers of finite sets. In this paper, we show how several prominent classes of phylogenetic networks can be characterised purely in terms of properties of their associated covers. These classes include the tree-based, tree-child, orchard, tree-sibling, and normal networks.

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arXiv2022

Brauer and partition diagram models for phylogenetic trees and forests

We introduce a correspondence between phylogenetic trees and Brauer diagrams, inspired by links between binary trees and matchings described by Diaconis and Holmes (1998). This correspondence gives rise to a range of semigroup structures on the set of phylogenetic trees, and opens the prospect of many applications. We furthermore extend the Diaconis-Holmes correspondence from binary trees to non-binary trees and to forests, showing for instance that the set of all forests is in bijection with the set of partitions of finite sets.

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arXiv2022

A polynomial invariant for a new class of phylogenetic networks

Invariants for complicated objects such as those arising in phylogenetics, whether they are invariants as matrices, polynomials, or other mathematical structures, are important tools for distinguishing and working with such objects. In this paper, we generalize a complete polynomial invariant on trees to a class of phylogenetic networks called separable networks, which will include orchard networks. Networks are becoming increasingly important for their ability to represent reticulation events, such as hybridization, in evolutionary history. We provide a function from the space of internally multi-labelled phylogenetic networks, a more generic graph structure than phylogenetic networks where the reticulations are also labelled, to a polynomial ring. We prove that the separability condition allows us to characterize, via the polynomial, the phylogenetic networks with the same number of leaves and same number of reticulations by considering their internally labelled versions. While the invariant for trees is a polynomial in Z[x_1,..., x_n,y] where n is the number of leaves, the invariant for internally multi-labelled phylogenetic networks is an element of Z[x_1,..., x_n,lambda_1,...,lambda_r,y], where r is the number of reticulations in the network. When the networks are considered without leaf labels the number of variables reduces to r+2.

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arXiv2021

Encoding and ordering X-cactuses

Phylogenetic networks are a generalization of evolutionary or phylogenetic trees that are commonly used to represent the evolution of species which cross with one another. A special type of phylogenetic network is an {\em $X$-cactus}, which is essentially a cactus graph in which all vertices with degree less than three are labelled by at least one element from a set $X$ of species. In this paper, we present a way to {\em encode} $X$-cactuses in terms of certain collections of partitions of $X$ that naturally arise from $X$-cactuses. Using this encoding, we also introduce a partial order on the set of $X$-cactuses (up to isomorphism), and derive some structural properties of the resulting partially ordered set. This includes an analysis of some properties of its least upper and greatest lower bounds. Our results not only extend some fundamental properties of phylogenetic trees to $X$-cactuses, but also provides a new approach to solving topical problems in phylogenetic network theory such as deriving consensus networks.

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arXiv2021

"Normal" phylogenetic networks may be emerging as the leading class

The rich and varied ways that genetic material can be passed between species has motivated extensive research into the theory of phylogenetic networks. Features that align with biological processes, or with desirable mathematical properties, have been used to define classes and prove results, with the goal of developing the theoretical foundations for network reconstruction methods. We may have now reached the point where a collection of recent results can be drawn together to make one class of network, the \emph{normal} networks, a leading contender, sitting in the sweet spot between biological relevance and mathematical tractability.

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arXiv2021

Phylogenetic Diversity Rankings in the Face of Extinctions: the Robustness of the Fair Proportion Index

Planning for the protection of species often involves difficult choices about which species to prioritize, given constrained resources. One way of prioritizing species is to consider their "evolutionary distinctiveness", i.e. their relative evolutionary isolation on a phylogenetic tree. Several evolutionary isolation metrics or phylogenetic diversity indices have been introduced in the literature, among them the so-called Fair Proportion index (also known as the "evolutionary distinctiveness" score). This index apportions the total diversity of a tree among all leaves, thereby providing a simple prioritization criterion for conservation. Here, we focus on the prioritization order obtained from the Fair Proportion index and analyze the effects of species extinction on this ranking. More precisely, we analyze the extent to which the ranking order may change when some species go extinct and the Fair Proportion index is re-computed for the remaining taxa. We show that for each phylogenetic tree, there are edge lengths such that the extinction of one leaf per cherry completely reverses the ranking. Moreover, we show that even if only the lowest ranked species goes extinct, the ranking order may drastically change. We end by analyzing the effects of these two extinction scenarios (extinction of the lowest ranked species and extinction of one leaf per cherry) for a collection of empirical and simulated trees. In both cases, we can observe significant changes in the prioritization orders, highlighting the empirical relevance of our theoretical findings.

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arXiv2021

Normalising phylogenetic networks

Rooted phylogenetic networks provide a way to describe species' relationships when evolution departs from the simple model of a tree. However, networks inferred from genomic data can be highly tangled, making it difficult to discern the main reticulation signals present. In this paper, we describe a natural way to transform any rooted phylogenetic network into a simpler canonical network, which has desirable mathematical and computational properties, and is based only on the 'visible' nodes in the original network. The method has been implemented and we demonstrate its application to some examples.

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arXiv2020

A path-deformation framework for determining weighted genome rearrangement distance

Measuring the distance between two bacterial genomes under the inversion process is usually done by assuming all inversions to occur with equal probability. Recently, an approach to calculating inversion distance using group theory was introduced, and is effective for the model in which only very short inversions occur. In this paper, we show how to use the group-theoretic framework to establish minimal distance for any weighting on the set of inversions, generalizing previous approaches. To do this we use the theory of rewriting systems for groups, and exploit the Knuth--Bendix algorithm, the first time this theory has been introduced into genome rearrangement problems. The central idea of the approach is to use existing group theoretic methods to find an initial path between two genomes in genome space (for instance using only short inversions), and then to deform this path to optimality using a confluent system of rewriting rules generated by the Knuth--Bendix algorithm.

math.CO↗
arXiv2020

How tree-based is my network? Proximity measures for unrooted phylogenetic networks

Tree-based networks are a class of phylogenetic networks that attempt to formally capture what is meant by "tree-like" evolution. A given non-tree-based phylogenetic network, however, might appear to be very close to being tree-based, or very far. In this paper, we formalise the notion of proximity to tree-based for unrooted phylogenetic networks, with a range of proximity measures. These measures also provide characterisations of tree-based networks. One measure in particular, related to the nearest neighbour interchange operation, allows us to define the notion of "tree-based rank". This provides a subclassification within the tree-based networks themselves, identifying those networks that are "very" tree-based. Finally, we prove results relating tree-based networks in the settings of rooted and unrooted phylogenetic networks, showing effectively that an unrooted network is tree-based if and only if it can be made a rooted tree-based network by rooting it and orienting the edges appropriately. This leads to a clarification of the contrasting decision problems for tree-based networks, which are polynomial in the rooted case but NP complete in the unrooted.

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arXiv2020

A mean first passage time genome rearrangement distance

This paper introduces a new way to define a genome rearrangement distance, using the concept of mean first passage time from probability theory. Crucially, this distance estimate provides a genuine metric on genome space. We develop the theory and introduce a link to a graph-based zeta function. The approach is very general and can be applied to a wide variety of group-theoretic models of genome evolution.

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arXiv2019

A partial order and cluster-similarity metric on rooted phylogenetic trees

Metrics on rooted phylogenetic trees are integral to a number of areas of phylogenetic analysis. Cluster-similarity metrics have recently been introduced in order to limit skew in the distribution of distances, and to ensure that trees in the neighbourhood of each other have similar hierarchies. In the present paper we introduce a new cluster-similarity metric on rooted phylogenetic tree space that has an associated local operation, allowing for easy calculation of neighbourhoods, a trait that is desirable for MCMC calculations. The metric is defined by the distance on the Hasse diagram induced by a partial order on the set of rooted phylogenetic trees, itself based on the notion of a hierarchy-preserving map between trees. The partial order we introduce is a refinement of the well-known refinement order on hierarchies. Both the partial order and the hierarchy-preserving maps may also be of independent interest.

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arXiv2019

The space of tree-based phylogenetic networks

Phylogenetic networks are generalizations of phylogenetic trees that allow the representation of reticulation events such as horizontal gene transfer or hybridization, and can also represent uncertainty in inference. A subclass of these, tree-based phylogenetic networks, have been introduced to capture the extent to which reticulate evolution nevertheless broadly follows tree-like patterns. Several important operations that change a general phylogenetic network have been developed in recent years, and are important for allowing algorithms to move around spaces of networks; a vital ingredient in finding an optimal network given some biological data. A key such operation is the Nearest Neighbor Interchange, or NNI. While it is already known that the space of unrooted phylogenetic networks is connected under NNI, it has been unclear whether this also holds for the subspace of tree-based networks. In this paper we show that the space of unrooted tree-based phylogenetic networks is indeed connected under the NNI operation. We do so by explicitly showing how to get from one such network to another one without losing tree-basedness along the way. Moreover, we introduce some new concepts, for instance ``shoat networks'', and derive some interesting aspects concerning tree-basedness. Last, we use our results to derive an upper bound on the size of the space of tree-based networks.

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arXiv2019

Tree-metrizable HGT networks

Phylogenetic trees are often constructed by using a metric on the set of taxa that label the leaves of the tree. While there are a number of methods for constructing a tree using a given metric, such trees will only display the metric if it satisfies the so-called "four point condition", established by Buneman in 1971. While this condition guarantees that a unique tree will display the metric, meaning that the distance between any two leaves can be found by adding the distances on arcs in the path between the leaves, it doesn't exclude the possibility that a phylogenetic network might also display the metric. This possibility was recently pointed out and "tree-metrized" networks --- that display a tree metric --- with a single reticulation were characterized. In this paper, we show that in the case of HGT (horizontal gene transfer) networks, in fact there are tree-metrized networks containing many reticulations.

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