Mike Steel

• PhD
• Managing Director at University of Canterbury

We consider the following question: how close to the ancestral root of a phylogenetic tree is the most recent common ancestor of $k$ species randomly sampled from the tips of the tree? For trees having shapes predicted by the Yule-Harding model, it is known that the most recent common ancestor is likely to be close to (or equal to) the root of the...

The evolutionary relationships between species are typically represented in the biological literature by rooted phylogenetic trees. However, a tree fails to capture ancestral reticulate processes, such as the formation of hybrid species or lateral gene transfer events between lineages, and so the history of life is more accurately described by a ro...

Motivated by applications in medical bioinformatics, Khayatian et al. (2024) introduced a family of metrics on Cayley trees (the $k$-RF distance, for $k=0, \ldots, n-2$) and explored their distribution on pairs of random Cayley trees via simulations. In this paper, we investigate this distribution mathematically, and derive exact asymptotic descrip...

Phylogenetic networks provide a more general description of evolutionary relationships than rooted phylogenetic trees. One way to produce a phylogenetic network is to randomly place $k$ arcs between the edges of a rooted binary phylogenetic tree with $n$ leaves. The resulting directed graph may fail to be a phylogenetic network, and even when it is...

The evolutionary relationships between species are typically represented in the biological literature by rooted phylogenetic trees. However, a tree fails to capture ancestral reticulate processes, such as the formation of hybrid species or lateral gene transfer events between lineages, and so the history of life is more accurately described by a ro...

Catalytic reaction networks serve as fundamental models for understanding biochemical systems. CatReNet is a novel software designed to facilitate interactive analysis of such networks. It offers fast and exact algorithms for computing various types of self-sustaining autocatalytic subnetworks, including so-called CAFs (constructively autocatalytic...

In a recent paper, the question of determining the fraction of binary trees that contain a fixed pattern known as the snowflake was posed. We show that this fraction goes to 1, providing two very different proofs: a purely combinatorial one that is quantitative and specific to this problem; and a proof using branching process techniques that is les...

A wide variety of stochastic models of cladogenesis (based on speciation and extinction) lead to an identical distribution on phylogenetic tree shapes once the edge lengths are ignored. By contrast, the distribution of the tree's edge lengths is generally quite sensitive to the underlying model. In this paper, we review the impact of different mode...

The concept of an autocatalytic network of reactions that can form and persist, starting from just an available food source, has been formalized by the notion of a reflexively autocatalytic and food-generated (RAF) set. The theory and algorithmic results concerning RAFs have been applied to a range of settings, from metabolic questions arising at t...

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-b...

Gene loss is an important process in gene and genome evolution. If a gene is present at the root of a rooted binary phylogenetic tree and can be lost in one descendant lineage, it can be lost in other descendant lineages as well, and potentially can be lost in all of them, leading to extinction of the gene on the tree. In that case, just before the...

Phylogenetic metrics are essential tools used in the study of ecology, evolution and conservation. Phylogenetic diversity (PD) in particular is one of the most prominent measures of biodiversity, and is based on the idea that biological features accumulate along the edges of phylogenetic trees that are summed. We argue that PD and many other phylog...

Phylogenetics’ is the reconstruction and analysis of phylogenetic (evolutionary) trees and networks based on inherited characteristics. It is a flourishing area of intereaction between mathematics, statistics, computer science and biology. The main role of phylogenetic techniques lies in evolutionary biology, where it is used to infer historical re...

The emergence of an autocatalytic network from an available set of elements is a fundamental step in early evolutionary processes, such as the origin of metabolism. Given the set of elements, the reactions between them (chemical or otherwise), and with various elements catalysing certain reactions, a Reflexively Autocatalytic F-generated (RAF) set...

The genomic era has opened up vast opportunities in molecular systematics, one of which is deciphering the evolutionary history in fine detail. Under this mass of data, analyzing the point mutations of standard markers is often too crude and slow for fine-scale phylogenetics. Nevertheless, genome dynamics (GD) events provide alternative, often rich...

Prokaryotic evolution is often described as the Spaghetti of Life due to massive genome dynamics (GD) events of gene gain and loss, resulting in different evolutionary histories for the set of genes comprising the organism. These different histories, dubbed as gene trees provide confounding signals, hampering the attempt to reconstruct the species...

The concept of an autocatalytic network of reactions that can form and persist, starting from just an available food source, has been formalised by the notion of a Reflexively-Autocatalytic and Food generated (RAF) set. The theory and algorithmic results concerning RAFs have been applied to a range of settings, from metabolic questions arising at t...

The current rapid extinction of species leads not only to their loss but also the disappearance of the unique features they harbour, which have evolved along the branches of the underlying evolutionary tree. One proxy for estimating the feature diversity (FD) of a set S of species at the tips of a tree is 'phylogenetic diversity' (PD): the sum of t...

Biodiversity is a concept most naturally quantified and measured across sets of species. However, for some applications, such as prioritising species for conservation efforts, a species-by-species approach is desirable. Phylogenetic diversity indices are functions that apportion the total biodiversity value of a set of species across its constituen...

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-b...

Given a set of elements, the reactions between them (chemical or otherwise), and certain elements catalysing certain reactions, a Reflexively Autocatalytic F-generated (RAF) set is a subset $R'$ of reactions that is self-generating from a given food set, and with each reaction in $R'$ being catalysed from within $R'$. RAF theory has been applied to...

Phylogenetic networks are mathematical representations of evolutionary history that are able to capture both tree-like evolutionary processes (speciations) and non-tree-like 'reticulate' processes such as hybridization or horizontal gene transfer. The additional complexity that comes with this capacity, however, makes networks harder to infer from...

The conservation of evolutionary history has been linked to increased benefits for humanity and can be captured by phylogenetic diversity (PD). The Evolutionarily Distinct and Globally Endangered (EDGE) metric has, since 2007, been used to prioritise threatened species for practical conservation that embody large amounts of evolutionary history. Wh...

A bstract Phylogenetic networks are mathematical representations of evolutionary history that are able to capture both tree-like evolutionary processes (speciations), and non-tree-like “reticulate” processes such as hybridization or horizontal gene transfer. The additional complexity that comes with this capacity, however, makes networks harder to...

Biodiversity is a concept most naturally quantified and measured across sets of species. Yet for some applications, such as prioritising species for conservation efforts, a species-by-species approach is desirable. Phylogenetic diversity indices are functions that apportion the total biodiversity value of a set of species across its constituent mem...

The idea that chemical evolution led to the origin of life is not new, but still leaves open the question of how exactly it could have led to a coherent and self-reproducing collective of molecules. One possible answer to this question was proposed in the form of the emergence of an autocatalytic set: a collection of molecules that mutually catalyz...

In the simplest phylogenetic diversification model (the pure-birth Yule process), lineages split independently at a constant rate λ for time t. The length of a randomly chosen edge (either interior or pendant) in the resulting tree has an expected value that rapidly converges to 12λ as t grows, and thus is essentially independent of t. However, the...

We show how uncertainty and insight can be modeled using Reflexively Autocatalytic Foodset-generated (RAF) networks. RAF networks have been used to model the self-organization of adaptive networks associated with the origin and early evolution of both biological life, and the kind of cognitive structure necessary for cultural evolution. The RAF app...

A bstract Phylogenetic metrics are essential tools in ecology, evolution and conservation, and Phylogenetic Diversity (PD) is one of the most prominent measures of biodiversity. PD is based on the idea that biological features accumulate along the branches of phylogenetic trees, and that these features are of biological importance. We argue that PD...

In conservation biology, phylogenetic diversity (PD) provides a way to quantify the impact of the current rapid extinction of species on the evolutionary ‘Tree of Life’. This approach recognises that extinction not only removes species but also the branches of the tree on which unique features shared by the extinct species arose. In this paper, we...

The global biodiversity crisis threatens the natural world and its capacity to provide benefits to humans into the future. The conservation of evolutionary history, captured by the measure phylogenetic diversity (PD), is linked to the maintenance of these benefits and future options. The Evolutionarily Distinct and Globally Endangered (EDGE) metric...

Monophyly is a feature of a set of genetic lineages in which every lineage in the set is more closely related to all other members of the set than it is to any lineage outside the set. Multiple sets of lineages that are separately monophyletic are said to be reciprocally monophyletic, or jointly monophyletic. The prevalence of reciprocal monophyly,...

The genomic era has opened up vast opportunities in molecular systematics, one of which is deciphering the evolutionary history in fine detail. Under this mass of data, analysing the point mutations of standard markers is too crude and slow for fine-scale phylogenetics. Nevertheless, genome dynamics events provide far richer information. The synten...

Monophyly is a feature of a set of genetic lineages in which every lineage in the set is more closely related to all other members of the set than it is to any lineage outside the set. Multiple sets of lineages that are separately monophyletic are said to be reciprocally monophyletic, or jointly monophyletic. The prevalence of reciprocal monophyly,...

In conservation biology, phylogenetic diversity (PD) provides a way to quantify the impact of the current rapid extinction of species on the evolutionary `Tree of Life'. This approach recognises that extinction not only removes species but also the branches of the tree on which unique features shared by the extinct species arose. In this paper, we...

In reflexively autocatalytic foodset (RAF)-generated networks, nodes are not only passive transmitters of activation, but they also actively galvanize, or “catalyze” the synthesis of novel (“foodset-derived”) nodes from existing ones (the “foodset”). Thus, RAFs are uniquely suited to modeling how new structure grows out of currently available struc...

Two recent high profile studies have attempted to use edge (branch) length ratios from large sets of phylogenetic trees to determine the relative ages of genes of different origins in the evolution of eukaryotic cells. This approach can be straightforwardly justified if substitution rates are constant over the tree for a given protein. However, suc...

Tree‐child networks are a class of directed acyclic graphs that have recently risen to prominence in phylogenetics. Although these networks have numerous, attractive mathematical properties, many combinatorial questions concerning them remain intractable. We show that endowing tree‐child networks with a biologically relevant ranking structure yield...

A bstract In the simplest phylodynamic model (the pure-birth Yule process), lineages split independently at a constant rate λ for time t . The length of a randomly chosen edge (either interior or pendant) in the resulting tree has an expected value that rapidly converges to as t grows, and thus is essentially independent of t . However, the behavio...

Phylogenetic trees from real-world data often include short edges with very few substitutions per site, which can lead to partially resolved trees and poor accuracy. Theory indicates that the number of sites needed to accurately reconstruct a fully resolved tree grows at a rate proportional to the inverse square of the length of the shortest edge....

Natural selection successfully explains how organisms accumulate adaptive change despite that traits acquired over a lifetime are eliminated at the end of each generation. However, in some domains that exhibit cumulative, adaptive change—e.g. cultural evolution, and earliest life—acquired traits are retained; these domains do not face the problem t...

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...

A bstract Phylogenetic trees from real-world data often include short edges with very few substitutions per site, which can lead to partially resolved trees and poor accuracy. Theory indicates that the number of sites needed to accurately reconstruct a fully resolved tree grows at a rate proportional to the inverse square of the length of the short...

Attention Schema Theory (AST) is a recent proposal to provide a scientific explanation for the basis of subjective awareness. In AST, the brain constructs a representation of attention taking place in its own (and others') mind ('the attention schema'). Moreover, this representation is incomplete for efficiency reasons. This inherent incompleteness...

Polymer models are a widely used tool to study the prebiotic formation of metabolism at the origins of life. Counts of the number of reactions in these models are often crucial in probabilistic arguments concerning the emergence of autocatalytic networks. In the first part of this paper, we provide the first exact description of the number of react...

The emergence of self-sustaining autocatalytic networks in chemical reaction systems has been studied as a possible mechanism for modeling how living systems first arose. It has been known for several decades that such networks will form within systems of polymers (under cleavage and ligation reactions) under a simple process of random catalysis, a...

Two recent high profile studies have attempted to use edge (branch) length ratios from large sets of phylogenetic trees to determine the relative ages of genes of different origins in the evolution of eukaryotic cells. This approach can be straightforwardly justified if substitution rates are constant over the tree for a given protein. However, suc...

Rooted phylogenetic networks provide a more complete representation of the ancestral relationship between species than phylogenetic trees when reticulate evolutionary processes are at play. One way to reconstruct a phylogenetic network is to consider its ‘ancestral profile’ (the number of paths from each ancestral vertex to each leaf). In general,...

Discontinuities permeate culture, and present a formidable challenge to mathematical models of cultural evolution. Cultural discontinuities have their origin in cognitive processes that include metaphor, analogy, cross-domain transfer, and self-organized criticality. This paper shows how cultural discontinuities can be accommodated by a theory of c...

Rooted phylogenetic networks provide a more complete representation of the ancestral relationship between species than phylogenetic trees when reticulate evolutionary processes are at play. One way to reconstruct a phylogenetic network is to consider its `ancestral profile' (the number of paths from each ancestral vertex to each leaf). In general,...

Attention Schema Theory (AST) is a recent proposal to provide a scientific explanation for the basis of subjective awareness. Essentially, AST posits that conscious experience in the minds of humans and some other animals is the result of evolution having provided the mind with its own internal model of attention taking place within their own (and...

This paper proposes a model of the cognitive mechanisms underlying the transition to behavioural and cognitive modernity in the Upper Palaeolithic using autocatalytic networks. These networks have been used to model life’s origins. More recently, they have been applied to the emergence of cognitive structure capable of undergoing cultural evolution...

Metabolism across all known living systems combines two key features. First, all of the molecules that are required are either available in the environment or can be built up from available resources via other reactions within the system. Second, the reactions proceed in a fast and synchronized fashion via catalysts that are also produced within th...

This paper proposes a model of the cognitive mechanisms underlying the transition to behavioural and cognitive modernity in the Upper Palaeolithic using autocatalytic networks. These networks have been used to model life's origins. More recently, they have been applied to the emergence of cognitive structure capable of undergoing cultural evolution...

Autocatalytic networks have been used to model the emergence of self-organizing structure capable of sustaining life and undergoing biological evolution. Here, we model the emergence of cognitive structure capable of undergoing cultural evolution. Mental representations (MRs) of knowledge and experiences play the role of catalytic molecules, and in...

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...

The extent to which phylogenetic diversity (PD) captures feature diversity (FD) is a topical and controversial question in biodiversity conservation. In this short paper, we formalise this question and establish a precise mathematical condition for FD (based on discrete characters) to coincide with PD. In this way, we make explicit the two main rea...

The emergence of self-sustaining autocatalytic networks in chemical reaction systems has been studied as a possible mechanism for modelling how living systems first arose. It has been known for several decades that such networks will form within systems of polymers (under cleavage and ligation reactions) under a simple process of random catalysis,...

Tree-child networks are a recently-described class of directed acyclic graphs that have risen to prominence in phylogenetics (the study of evolutionary trees and networks). Although these networks have a number of attractive mathematical properties, many combinatorial questions concerning them remain intractable. In this paper, we show that endowin...

A bstract This paper proposes a model of the cognitive mechanisms underlying the transition to behavioral and cognitive modernity in the Upper Paleolithic using autocatalytic networks. These networks have been used to model life’s origins. More recently, they have been applied to the emergence of cognitive structure capable of undergoing cultural e...

Autocatalytic networks have been used to model the emergence of self-organizing structure capable of sustaining life and undergoing biological evolution. Here, we model the emergence of cognitive structure capable of undergoing cultural evolution. Mental representations of knowledge and experiences play the role of catalytic molecules, and interact...

Metabolism across all known living systems combines two key features. First, all of the molecules that are required are either available in the environment or can be built up from available resources via other reactions within the system. Second, the reactions proceed in a fast and synchronised fashion via catalysts that are also produced within th...

A bstract Terraces in phylogenetic tree space are sets of trees with identical optimality scores for a given data set, arising from missing data. These were first described for multilocus phylogenetic data sets in the context of maximum parsimony inference and maximum likelihood inference under certain model assumptions. Here we show how the mathem...

A bstract The extent to which phylogenetic diversity (PD) captures feature diversity (FD) is a topical and controversial question in biodiversity conservation. In this short paper, we formalise this question and establish a precise mathematical condition for FD (based on discrete characters) to coincide with PD. In this way, we make explicit the tw...

A key step in the origin of life is the emergence of a primitive metabolism. This requires the formation of a subset of chemical reactions that is both self-sustaining and collectively autocatalytic. A generic approach to study such processes (‘RAF theory’) has provided a precise and computationally effective way to address these questions, both on...

Modern cells embody metabolic networks containing thousands of elements and form autocatalytic sets of molecules that produce copies of themselves. How the first self-sustaining metabolic networks arose at life's origin is a major open question. Autocatalytic sets smaller than metabolic networks were proposed as transitory intermediates at the orig...

Phylogenetic diversity indices provide a formal way to apportion ‘evolutionary heritage’ across species. Two natural diversity indices are Fair Proportion (FP) and Equal Splits (ES). FP is also called ‘evolutionary distinctiveness’ and, for rooted trees, is identical to the Shapley Value (SV), which arises from cooperative game theory. In this pape...

The dramatic decrease in time and cost for generating genetic sequence data has opened up vast opportunities in molecular systematics, one of which is the ability to decipher the evolutionary history of strains of a species. Under this fine systematic resolution, the standard markers are too crude to provide a phylogenetic signal. Nevertheless, amo...

A key step in the origin of life is the emergence of a primitive metabolism. This requires the formation of a subset of chemical reactions that is both self-sustaining and collectively autocatalytic. A generic theory to study such processes (called 'RAF theory') has provided a precise and computationally effective way to address these questions, bo...

Reconstructing ancestral characters and traits along a phylogenetic tree is central to evolutionary biology. It is the key to understanding morphology changes among species, inferring ancestral biochemical properties of life, or recovering migration routes in phylogeography. The goal is twofold: to reconstruct the character state at the tree root (...

A bstract In phylogenetics, a set of gene trees is often summarized by a consensus tree, such as the majority consensus, which is based on the set of all splits that are present in more than 50% of the input trees. A “consensus network” is obtained by lowering the threshold and considering all splits that are contained in 10% of the trees, say, and...

Modern cells embody metabolic networks containing thousands of elements and form autocatalytic molecule sets that produce copies of themselves. How the first self-sustaining metabolic networks arose at life's origin is a major open question. Autocatalytic molecule sets smaller than metabolic networks were proposed as transitory intermediates at the...

Stochastic birth-death models provide the foundation for studying and simulating evolutionary trees in phylodynamics. A curious feature of such models is that they exhibit fundamental symmetries when the birth and death rates are interchanged. In this paper, we first provide intuitive reasons for these known transformational symmetries. We then sho...

In phylogenetic studies, biologists often wish to estimate the ancestral discrete character state at an interior vertex v of an evolutionary tree T from the states that are observed at the leaves of the tree. A simple and fast estimation method—maximum parsimony—takes the ancestral state at v to be any state that minimises the number of state chang...

Rooted phylogenetic networks provide an explicit representation of the evolutionary history of a set X of sampled species. In contrast to phylogenetic trees which show only speciation events, networks can also accommodate reticulate processes (for example, hybrid evolution, endosymbiosis, and lateral gene transfer). A major goal in systematic biolo...

A feature of human creativity is the ability to take a subset of existing items (e.g. objects, ideas, or techniques) and combine them in various ways to give rise to new items, which, in turn, fuel further growth. Occasionally, some of these items may also disappear (extinction). We model this process by a simple stochastic birth--death model, with...

Phylogenetic networks generalise phylogenetic trees and allow for the accurate representation of the evolutionary history of a set of present-day species whose past includes reticulate events such as hybridisation and lateral gene transfer. One way to obtain such a network is by starting with a (rooted) phylogenetic tree $T$, called a base tree, an...

Systems chemistry deals with the design and study of complex chemical systems. However, such systems are often difficult to investigate experimentally. We provide an example of how theoretical and simulation-based studies can provide useful insights into the properties and dynamics of complex chemical systems, in particular of autocatalytic sets. W...

Phylogenetic diversity indices provide a formal way to apportion 'evolutionary heritage' across species. Two natural diversity indices are Fair Proportion (FP) and Equal Splits (ES). FP is also called 'evolutionary distinctiveness' and, for rooted trees, is identical to the Shapley Value (SV), which arises from cooperative game theory. In this pape...

Self-sustaining autocatalytic networks play a central role in living systems, from metabolism at the origin of life, simple RNA networks and the modern cell, to ecology and cognition. A collectively autocatalytic network that can be sustained from an ambient food set is also referred to more formally as a 'reflexively autocatalytic food-generated'...

Rooted phylogenetic networks provide an explicit representation of the evolutionary history of a set $X$ of sampled species. In contrast to phylogenetic trees which show only speciation events, networks can also accommodate reticulate processes (for example, hybrid evolution, endosymbiosis, and lateral gene transfer). A major goal in systematic bio...

A recent paper (Manceau and Lambert, 2016) developed a novel approach for describing two well-defined notions of 'species' based on a phylogenetic tree and a phenotypic partition. In this paper, we explore some further combinatorial properties of this approach and describe an extension that allows an arbitrary number of phenotypic partitions to be...

Reconstructing ancestral characters and traits along a phylogenetic tree is central to evolutionary biology. It is the key to understanding morphology changes among species, inferring ancestral biochemical properties of life, and recovering migration routes in phylogeography. The goal is twofold: to reconstruct the character state at the tree root...

Stochastic birth–death models provide the foundation for studying and simulating evolutionary trees in phylodynamics. A curious feature of such models is that they exhibit fundamental symmetries when the birth and death rates are interchanged. In this paper, we explain and formally derive these transformational symmetries. We also show that these t...

Life is more than the sum of its constituent molecules. Living systems depend on a particular chemical organization, i.e., the ways in which their constituent molecules interact and cooperate with each other through catalyzed chemical reactions. Several abstract models of minimal life, based on this idea of chemical organization and also in the con...

Phylogenetic inference aims to reconstruct the evolutionary relationships of different species based on genetic (or other) data. Discrete characters are a particular type of data, which contain information on how the species should be grouped together. However, it has long been known that some characters contain more information than others. For in...

In phylogenetic studies, biologists often wish to estimate the ancestral discrete character state at an interior vertex $v$ of an evolutionary tree $T$ from the states that are observed at the leaves of the tree. A simple and fast estimation method --- maximum parsimony --- takes the ancestral state at $v$ to be any state that minimises the number...

Two related but somewhat different approaches have been proposed to formalize the notion of a self-sustaining chemical reaction network. One is the notion of collectively autocatalytic sets, formalized as RAF theory, and the other is chemical organization theory. Both formalisms have been argued to be relevant to the origin of life. RAF sets and ch...

Self-sustaining autocatalytic networks play a central role in living systems, from metabolism at the origin of life, simple RNA networks, and the modern cell, to ecology and cognition. A collectively autocatalytic network that can be sustained from an ambient food set is also referred to more formally as a `Reflexively Autocatalytic F-generated' (R...

Phylogenetic networks are a type of directed acyclic graph that represent how a set X of present-day species are descended from a common ancestor by processes of speciation and reticulate evolution. In the absence of reticulate evolution, such networks are simply phylogenetic (evolutionary) trees. Moreover, phylogenetic networks that are not trees...

Trees with labelled leaves and with all other vertices of degree three play an important role in systematic biology and other areas of classification. A classical combinatorial result ensures that such trees can be uniquely reconstructed from the distances between the leaves (when the edges are given any strictly positive lengths). Moreover, a line...

A recent paper (Manceau and Lambert, 2016) developed a novel approach for describing two well-defined notions of 'species' based on a phylogenetic tree and a phenotypic partition. In this paper, we explore some further combinatorial properties of this approach and describe an extension that allows an arbitrary number of phenotypic partitions to be...

The reconstruction of large phylogenetic trees from data that violates clocklike evolution (or as a supertree constructed from any m input trees) raises a difficult question for biologists-how can one assign relative dates to the vertices of the tree? In this paper we investigate this problem, assuming a uniform distribution on the order of the inn...

The extinction of species at the present leads to the loss of 'phylogenetic diversity' (PD) from the evolutionary tree in which these species lie. Prior to extinction, the total PD present can be divided up among the species in various ways using measures of evolutionary isolation (such as 'fair proportion' and 'equal splits'). However, the loss of...

A variety of evolutionary processes in biology can be viewed as settings where organisms 'catalyse' the formation of new types of organisms. One example, relevant to the origin of life, is where transient biological colonies (e.g. prokaryotes or protocells) give rise to new colonies via lateral gene transfer. In this short note, we describe and ana...

Given a collection $\tau$ of subsets of a finite set $X$, we say that $\tau$ is {\em phylogenetically flexible} if, for any collection $R$ of rooted phylogenetic trees whose leaf sets comprise the collection $\tau$, $R$ is compatible (i.e. there is a rooted phylogenetic $X$--tree that displays each tree in $R$). We show that $\tau$ is phylogenetica...