SUMMARY: 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 food-generated networks), RAFs (reflexively autocatalytic food-generated networks), and pseudo-RAFs. It provides dynamic visualizations to aid exploration and understanding. AVAILABILITY AND IMPLEMENTATION: This open-source Java application runs on Linux, MacOS, and Windows. It is available at https://github.com/husonlab/catrenet under a GPL3 license.

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, and this process has since been mathematically analyzed. In this paper, we provide an exact expression for the expected number of self-sustaining autocatalytic networks that will form in a general chemical reaction system, and the expected number of these networks that will also be uninhibited (by some molecule produced by the system). Using these equations, we are able to describe the patterns of catalysis and inhibition that maximize or minimize the expected number of such networks. We apply our results to derive a general theorem concerning the trade-off between catalysis and inhibition, and to provide some insight into the extent to which the expected number of self-sustaining autocatalytic networks coincides with the probability that at least one such system is present.

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 reactions under widely applied model assumptions. Conclusions from earlier studies rely on either approximations or asymptotic counting, and we show that the exact counts lead to similar, though not always identical, asymptotic results. In the second part of the paper, we investigate a novel model assumption whereby polymers are invariant under spatial rotation. We outline the biochemical relevance of this condition and again give exact enumerative and asymptotic formulae for the number of reactions.

A central theme in phylogenetics is the reconstruction and analysis of evolutionary trees from a given set of data. To determine the optimal search methods for reconstructing trees, it is crucial to understand the size and structure of the neighborhoods of trees under tree rearrangement operations. The diameter and size of the immediate neighborhood of a tree have been well-studied; however, little is known about the number of trees at distance two, three, or (more generally) $k$ from a given tree. In this paper we provide a number of exact and asymptotic results concerning these quantities and identify some key aspects of tree shape that play a role in determining these quantities. We obtain several new results for two of the main tree rearrangement operations---nearest neighbor interchange and subtree prune and regraft---as well as for the Robinson--Foulds metric on trees.

We prove lower bounds on the expected size of the maximum agreement subtree of two random binary phylogenetic trees under both the uniform distribution and the Yule--Harding distribution and prove upper bounds under the Yule--Harding distribution. This positively answers a question posed in earlier work. Determining tight upper and lower bounds remains an open problem.

We use a classical combinatorial inequality to establish a Markov inequality for multivariate binary Markov processes on trees. We then apply this result, alongside the Fortuin–Kasteleyn–Ginibre (FKG) inequality, to compare the expected loss of biodiversity under two models of species extinction. One of these models is the generalized version of an earlier model in which extinction is influenced by some trait that can be classified into two states and which evolves on a tree according to a Markov process. Since more than one trait can affect the rates of species extinction, it is reasonable to allow, in the generalized model, k binary states that influence extinction rates. We compare this model to one that has matching marginal extinction probabilities for each species but for which the species extinction events are stochastically independent.

The Robinson-Foulds (RF) distance is by far the most widely used measure of dissimilarity between trees. Although the distribution of these distances has been investigated for 20 years, an algorithm that is explicitly polynomial time has yet to be described for computing the distribution for trees around a given tree. In this paper, we derive a polynomial-time algorithm for this distribution. We show how the distribution can be approximated by a Poisson distribution determined by the proportion of leaves that lie in "cherries" of the given tree. We also describe how our results can be used to derive normalization constants that are required in a recently proposed maximum likelihood approach to supertree construction.

The seven papers in this special section focus on phylogenetics and these four themes: new data types and algorithms in phylogenetics; reticulate evolution; constructing large trees; and mathematical modeling of evolution.

Ancestral maximum likelihood (AML) is a method that simultaneously reconstructs a phylogenetic tree and ancestral sequences from extant data (sequences at the leaves). The tree and ancestral sequences maximize the probability of observing the given data under a Markov model of sequence evolution, in which branch lengths are also optimized but constrained to take the same value on any edge across all sequence sites. AML differs from the more usual form of maximum likelihood (ML) in phylogenetics because ML averages over all possible ancestral sequences. ML has long been know to be statistically consistent--that is, it converges on the correct tree with probability approaching 1 as the sequence length grows. However, the statistical consistency of AML has not been formally determined, despite informal remarks in a literature that dates back 20 years. In this short note we prove a general result that implies that AML is statistically inconsistent. In particular we show that AML can 'shrink' short edges in a tree, resulting in a tree that has no internal resolution as the sequence length grows. Our results apply to any number of taxa.

Given a set X of taxa, a phylogenetic X-tree T that is only partially resolved, and a collection of characters on X, we consider the problem of finding a resolution (refinement) of T that minimizes the parsimony score of the given characters. Previous work has shown that this problem has a polynomial time solution provided certain strong constraints are imposed on the input. In this paper we provide a new algorithm for this problem, and show that it is fixed parameter tractable under more general conditions.

Graphs obtained from a binary leaf labeled ("phylogenetic") tree by adding an edge so as to introduce a cycle provide a useful representation of hybrid evolution in molecular evolutionary biology. This class of graphs (which we call "unicyclic networks") also has some attractive combinatorial properties, which we present. We characterize when a set of binary phylogenetic trees is displayed by a unicyclic network in terms of tree rearrangement operations. This leads to a triple-wise compatibility theorem and a simple, fast algorithm to determine 1-cycle compatibility. We also use generating function techniques to provide closed-form expressions that enumerate unicyclic networks with specified or unspecified cycle length, and we provide an extension to enumerate a class of multicyclic networks.

It was recently shown that just five characters (functions on a finite set X) suffice to convexly define a trivalent tree with leaf set X. Here we show that four characters suffice which, since three characters are not enough in general, is the best possible.

UNLABELLED: Comparing gene content between species can be a useful approach for reconstructing phylogenetic trees. In this paper, we derive a maximum-likelihood estimation of evolutionary distance between species under a simple model of gene genesis and gene loss. Using simulated data on a biological tree with 107 taxa (and on a number of randomly generated trees), we compare the accuracy of tree reconstruction using this ML distance measure to an earlier ad hoc distance. We then compare these distance-based approaches to a character-based tree reconstruction method (Dollo parsimony) which seems well suited to the analysis of gene content data. To simplify simulations, we give a formal proof of the well-known 'fact' that the Dollo parsimony score is independent of the choice of root. Our results show a consistent trend, with the character-based method and ML distance measure outperforming the earlier ad hoc distance method. AVAILABILITY: http://www.ab.informatik.uni-tuebingen.de/software/genecontent/welcome_en.html

MOTIVATION: Supertree methods have been often identified as a possible approach to the reconstruction of the 'Tree of Life'. However, a limitation of such methods is that, typically, they use just leaf-labelled phylogenetic trees to infer the resulting supertree. RESULTS: In this paper, we describe several new supertree algorithms that extend the allowable information that can be used for phylogenetic inference. These algorithms have been recently implemented and we describe here two illustrative applications. AVAILABILITY: These new algorithms are freely available for application at http://darwin.zoology.gla.ac.uk/cgi-bin/build.pl.

In practice, one is often faced with incomplete phylogenetic data, such as a collection of partial trees or partial splits. This paper poses the problem of inferring a phylogenetic super-network from such data and provides an efficient algorithm for doing so, called the Z-closure method. Additionally, the questions of assigning lengths to the edges of the network and how to restrict the "dimensionality" of the network are addressed. Applications to a set of five published partial gene trees relating different fungal species and to six published partial gene trees relating different grasses illustrate the usefulness of the method and an experimental study confirms its potential. The method is implemented as a plug-in for the program SplitsTree4.

In this paper we study inverting random functions under the maximum likelihood estimation (MLE) criterion in the discrete setting. In particular, we consider how many independent evaluations of the random function at a particular element of the domain are needed for reliable reconstruction of that element. We provide explicit upper and lower bounds for MLE, both in the nonparametric and parametric setting, and give applications to coin-tossing and phylogenetic tree reconstruction.

The reconstruction of evolutionary trees (phylogenies) from DNA sequence data is a central problem in biology. We describe simple sufficient conditions for two tree reconstruction methods (maximum parsimony and maximum compatibility) to correctly reconstruct a tree when applied to sufficiently many sequence sites generated under a simple stochastic model.

Evolutionary tree reconstruction is a challenging problem with important applications in Biology and Liiguistics.In Biology, one of the most promising approaches to tree reconstruction is to 6nd the "maximum parsimony" tree, while in Liignistics, the use of the "m&mum compatibility" method has been very useful.However, these problems are NP-hard, and current approaches to solving these problems amount to heuristic searches through the space of possible tree topologies (a search which can, on large trees, take months to complete).In this paper, we present a new technique, Uptimmnl l+ee Refinement, for reconstructing very large trees.Our technique is motivated by recent experimental studies which have shown that certain polynomial time methods reliably return contractions of the true tree.We study the use of this technique in solving maximum parsimony and maximum compatibility and present both hardness results and polynomial time algorithms.

An assignment of colors to objects induces a natural integer weight on each tree that has these objects as leaves. This weight is called "parsimony length" in biostatistics and is the basis of the "maximum parsimony" technique for reconstructing evolutionary trees. Equations for the average value (over all binary trees) of the parsimony length of both fixed and random colorations are derived using generating function techniques. This leads to asymptotic results that extend earlier results confined to just two colors. A potential application to DNA sequence analysis is outlined briefly.

The classical result that characterizes metrics induced by paths in a labeled tree having positive real edge weights is generalized to allow the edge weights to take values in any cancellative abelian monoid satisfying the additional requirement that $x + x = y + y$ implies $x = y$. This includes the case of arbitrary real-valued edge weights, which applies to distance-hereditary graphs, thus yielding (unique) weighted tree representations for the latter.

The symmetric difference metric has been useful in comparing phylogenetic trees derived from DNA sequence data. The main result shown here is that the frequency of pairs of binary trees a given distance apart is described by a limiting Poisson distribution, with $e^{ -1 / 8} \approx 88$ percent of all pairs maximally distant. Asymptotic bounds on the distribution are derived, and the asymptotic mean and variance of the normalized metric on the class of all phylogenetic trees is also calculated. The results rely on simple combinatorial constructions and analytic properties of appropriate generating functions.