The evolutionary history of a set of taxa is usually represented by a phylogenetic tree, and this model has greatly facilitated the discussion and testing of hypotheses. However, it is well known that more complex evolutionary scenarios are poorly described by such models. Further, even when evolution proceeds in a tree-like manner, analysis of the data may not be best served by using methods that enforce a tree structure but rather by a richer visualization of the data to evaluate its properties, at least as an essential first step. Thus, phylogenetic networks should be employed when reticulate events such as hybridization, horizontal gene transfer, recombination, or gene duplication and loss are believed to be involved, and, even in the absence of such events, phylogenetic networks have a useful role to play. This article reviews the terminology used for phylogenetic networks and covers both split networks and reticulate networks, how they are defined, and how they can be interpreted. Additionally, the article outlines the beginnings of a comprehensive statistical framework for applying split network methods. We show how split networks can represent confidence sets of trees and introduce a conservative statistical test for whether the conflicting signal in a network is treelike. Finally, this article describes a new program, SplitsTree4, an interactive and comprehensive tool for inferring different types of phylogenetic networks from sequences, distances, and trees.

https://bioinformatics.cs.vt.edu/~easychair/HusonBryant_MBE_2006.pdf

Application of Phylogenetic Networks in Evolutionary Studies

BOLD: The Barcode of Life Data System: Barcoding

KEGG(Kyoto Encyclopedia of Genes and Genomes)〔和文〕 (特集 ゲノム医学の現在と未来--基礎と臨床) -- (データベース)

Gene families are growing rapidly, but standard methods for inferring phylogenies do not scale to alignments with over 10,000 sequences. We present FastTree, a method for constructing large phylogenies and for estimating their reliability. Instead of storing a distance matrix, FastTree stores sequence profiles of internal nodes in the tree. FastTree uses these profiles to implement Neighbor-Joining and uses heuristics to quickly identify candidate joins. FastTree then uses nearest neighbor interchanges to reduce the length of the tree. For an alignment with N sequences, L sites, and a different characters, a distance matrix requires O(N2) space and O(N2L) time, but FastTree requires just O(NLa + N) memory and O(Nlog (N)La) time. To estimate the tree's reliability, FastTree uses local bootstrapping, which gives another 100-fold speedup over a distance matrix. For example, FastTree computed a tree and support values for 158,022 distinct 16S ribosomal RNAs in 17 h and 2.4 GB of memory. Just computing pairwise Jukes–Cantor distances and storing them, without inferring a tree or bootstrapping, would require 17 h and 50 GB of memory. In simulations, FastTree was slightly more accurate than Neighbor-Joining, BIONJ, or FastME; on genuine alignments, FastTree's topologies had higher likelihoods. FastTree is available at http://microbesonline.org/fasttree.

/pdf/fasttree-computing-large-minimum-evolution-trees-with-eo2tre5l8r.pdf

FastTree: Computing Large Minimum Evolution Trees with Profiles instead of a Distance Matrix

Gene families are growing rapidly, but standard methods for inferring phylogenies do not scale to alignments with over 10,000 sequences. We present FastTree, a method for constructing large phylogenies and for estimating their reliability. Instead of storing a distance matrix, FastTree stores sequence profiles of internal nodes in the tree. FastTree uses these profiles to implement neighbor-joining and uses heuristics to quickly identify candidate joins. FastTree then uses nearest-neighbor interchanges to reduce the length of the tree. For an alignment with N sequences, L sites, and a different characters, a distance matrix requires O(N^2) space and O(N^2 L) time, but FastTree requires just O( NLa + N sqrt(N) ) memory and O( N sqrt(N) log(N) L a ) time. To estimate the tree's reliability, FastTree uses local bootstrapping, which gives another 100-fold speedup over a distance matrix. For example, FastTree computed a tree and support values for 158,022 distinct 16S ribosomal RNAs in 17 hours and 2.4 gigabytes of memory. Just computing pairwise Jukes-Cantor distances and storing them, without inferring a tree or bootstrapping, would require 17 hours and 50 gigabytes of memory. In simulations, FastTree was slightly more accurate than neighbor joining, BIONJ, or FastME; on genuine alignments, FastTree's topologies had higher likelihoods. FastTree is available at http://microbesonline.org/fasttree.

/pdf/fast-tree-computing-large-minimum-evolution-trees-with-4grq0gwfpy.pdf

Fast Tree: Computing Large Minimum-Evolution Trees with Profiles instead of a Distance Matrix

The fragile complexity of a comparison-based algorithm is $f(n)$ if each input element participates in $O(f(n))$ comparisons. In this paper, we explore the fragile complexity of algorithms adaptive to various restrictions on the input, i.e., algorithms with a fragile complexity parameterized by a quantity other than the input size n. We show that searching for the predecessor in a sorted array has fragile complexity ${\Theta}(\log k)$, where $k$ is the rank of the query element, both in a randomized and a deterministic setting. For predecessor searches, we also show how to optimally reduce the amortized fragile complexity of the elements in the array. We also prove the following results: Selecting the $k$-th smallest element has expected fragile complexity $O(\log \log k)$ for the element selected. Deterministically finding the minimum element has fragile complexity ${\Theta}(\log(Inv))$ and ${\Theta}(\log(Runs))$, where $Inv$ is the number of inversions in a sequence and $Runs$ is the number of increasing runs in a sequence. Deterministically finding the median has fragile complexity $O(\log(Runs) + \log \log n)$ and ${\Theta}(\log(Inv))$. Deterministic sorting has fragile complexity ${\Theta}(\log(Inv))$ but it has fragile complexity ${\Theta}(\log n)$ regardless of the number of runs.

Fragile Complexity of Adaptive Algorithms

We present a deterministic local routing algorithm that is guaranteed to find a path between any pair of vertices in a half-$\theta_6$-graph (the half-$\theta_6$-graph is equivalent to the Delaunay triangulation where the empty region is an equilateral triangle). The length of the path is at most $5/\sqrt{3} \approx 2.887$ times the Euclidean distance between the pair of vertices. Moreover, we show that no local routing algorithm can achieve a better routing ratio, thereby proving that our routing algorithm is optimal. This is somewhat surprising because the spanning ratio of the half-$\theta_6$-graph is 2, meaning that even though there always exists a path whose lengths is at most twice the Euclidean distance, we cannot always find such a path when routing locally. 
Since every triangulation can be embedded in the plane as a half-$\theta_6$-graph using $O(\log n)$ bits per vertex coordinate via Schnyder's embedding scheme (SODA 1990), our result provides a competitive local routing algorithm for every such embedded triangulation. Finally, we show how our routing algorithm can be adapted to provide a routing ratio of $15/\sqrt{3} \approx 8.660$ on two bounded degree subgraphs of the half-$\theta_6$-graph.

Optimal local routing on Delaunay triangulations defined by empty equilateral triangles

Understanding the underlying chemistry of a catalytic process is essential for advancing of its medical and industrial applications. A well defined and compact representation of a catalytic process is becoming exceedingly valuable in face of the growth in computer assisted development strategies. A step-wise partition of a catalytic process is traditional in organic chemistry. We argue, however, that such a description remains incomplete, especially in face of automation and formal methods. In this contribution—based on enzymatic reaction mechanisms—we fully formalize the step-wise notion of a catalytic process within the mathematical framework of graph transformation, and propose a concise representation as an overlay graph (OG). We formally define this concept in a dual form as an OG itself and an overlay rule. The former representing a static summary of the process, and the latter being an executable in the graph transformation formalism. We demonstrate that OGs readily expose the underlying chemistry and facilitate the interpretation of catalytic processes. The emergence of electron flow patterns, for instance, provides an excellent stage for classification efforts. We further show how the executable overlay rules can be employed to automatically assign a fully specified mechanism to a reaction with only an overall description, i.e. educt and product molecules. Introduction In chemistry, a given overall reaction is often known to proceed as a sequence of smaller reaction steps, collectively termed the reaction mechanism. Each step in a reaction mechanism typically consists of the relocation of a small set of electrons. In particular, this is the case for catalytic reactions. Catalytic reactions are reactions where an additional reagent—the catalyst—opens up a route between the starting materials and the products with a lower activation barrier than the corresponding uncatalyzed process having the same overall reaction. Alternatively, the catalyst might also be able to accelerate an existing (non-catalyzed) pathway if the transition state exhibits a stronger binding than the substrate. In such a scenario the activation energy would also be lowered. The catalyst is not itself consumed in the process. It participates in intermediate reaction steps, but is in the end regenerated and comes out of the process structurally unchanged. For catalytic reactions in biochemistry, the number of steps in a reaction mechanism may be quite large: the Mechanism and Catalytic Site Atlas (M-CSA) database lists close to thousand enzymes and for many of these it gives suggestions for reaction mechanisms, some of which contain over ten steps. More importantly, most of the molecular changes induced in intermediate steps are often not reflected in the overall reaction at all. Thus, for catalytic reactions in general, and for enzymatic reactions in particular, the over1 ar X iv :2 20 1. 04 51 5v 1 [ cs .D M ] 1 2 Ja n 20 22 all reaction is often too simple to capture the essential chemical components involved in the reaction. For instance, the overall reaction is unable to distinguish between different enzymatic mechanisms for a given reaction. Our aim in this paper is to develop a richer, yet still succinct, notion of overall reaction which expresses major aspects of the underlying reaction mechanism. Our claim is that this may be a useful tool when describing, analyzing, searching for, and designing catalytic reactions, in particular when combined with computational methods. In the second part of the paper, we give some first use cases justifying this claim. We call our notion Overlay Graphs (OG). Our starting point is the so-called Imaginary Transition Structures (ITS) of Fujita. ITS are structural formulas with three bond types: bonds only seen in the educt molecules, bonds only seen in the product molecules, and bonds appearing in both. We propose to augment the concept of ITS with two additional bonds types capturing transient bond changes in the underlying reaction mechanism. The first new type is for bonds that appear in both educts and products, but are transiently absent in the mechanism. The second new type is for bonds that do not appear in neither educts nor products, but are transiently present in the mechanism. The concept of OG is quite straight-forward on the intuitive level—see Figure 1 for an example. However, we take the effort to also define the concept of OG more rigorously by expressing it in the context of graph transformation. As we outline now, this effort comes with a number of benefits. It is standard to describe molecules by chemical graphs, i.e., undirected graphs with vertices labeled by atom types and edges labeled by bond types. As chemical reaction constitute a change of educt molecules into product molecules, formal methods for specifying transformations of graphs are well suited for modeling chemical reactions. In this paper, we use the so-called double push-out (DPO) formalism for graph transformation when developing our notion of OG. Besides ensuring a high level of precision in our definitions, it also provides an interface to the computational framework MØD. This framework is able to test for matches between graphs and graph transformation rules, and in case of a match to calculate the effect of applying the transformation rule to the graph. In a chemical context, this amounts to being able to execute (in silico) a set of specified reactions on a set of given substrates. The DPO formalism is very flexible, since graph transformation rules used in matches can specify the full educt molecule, just the core reaction center, or anything in between, which allows for modeling classes of reactions with a single rule. One further benefit is that individual atoms can be tracked across reactions. An enzyme, although topologically unchanged by the reaction that it catalyzes, may well swap atoms with the substrate, so tracking individual atoms is relevant for instance when investigating biochemical pathways via stable isotope labeling of molecules. Clearly, being able to computationally model, manipulate, search, and execute chemical reactions on the level of DPO opens up many avenues of investigation. The second part of this paper exemplifies this in the context of OG. In short, our contributions contain the extension of the notion of ITS graphs to OG by means of a strict generalization. That is, if a mechanisms consists of a single step, the obtained OG is identical to the ITS graph. We additionally capture the notion of OG in full mathematical formalism, allowing us to utilize the concept within MØD, and consequently perform computational analyses. To counteract the lack of knowledge of the exact activate site for many enzymatic reactions, we propose a reduced version of an OG, called substrate OG, which only specifies the substrates. This version allows us to investigate enzymatic reactions without explicit knowledge of the catalytic pocket. Finally, we construct OGs for a number of carboxylic ester hydrolases from the M-CSA database. We use the obtained OGs to demonstrate not only their use in interpreting the underlying chemistry, but also in searching for reactions which could be explained by the same mechanism. The latter application further motivates the possibility of using OGs as

Characterizing Catalytic Mechanisms with Overlay Graphs

The fragile complexity of a comparison-based algorithm is f(n) if each input element participates in O(f(n)) comparisons. In this paper, we explore the fragile complexity of algorithms adaptive to various restrictions on the input, i.e., algorithms with a fragile complexity parameterized by a quantity other than the input size n. We show that searching for the predecessor in a sorted array has fragile complexity $\varTheta (\log k)$, where k is the rank of the query element, both in a randomized and a deterministic setting. For predecessor searches, we also show how to optimally reduce the amortized fragile complexity of the elements in the array. We also prove the following results: Selecting the kth smallest element has expected fragile complexity $O(\log \log k)$ for the element selected. Deterministically finding the minimum element has fragile complexity $\varTheta (\log (\mathrm {Inv}))$ and $\varTheta (\log (\mathrm {Runs}))$, where $\mathrm {Inv}$ is the number of inversions in a sequence and $\mathrm {Runs}$ is the number of increasing runs in a sequence. Deterministically finding the median has fragile complexity $O(\log (\mathrm {Runs}) + \log \log n)$ and $\varTheta (\log (\mathrm {Inv}))$. Deterministic sorting has fragile complexity $\varTheta (\log (\mathrm {Inv}))$ but it has fragile complexity $\varTheta (\log n)$ regardless of the number of runs.

Motivation: The design of enzymes is as challenging as it is consequential for making chemical synthesis in medical and industrial applications more efficient, cost-effective and environmentally friendly While several aspects of this complex problem are computationally assisted, the drafting of catalytic mechanisms, ie the specification of the chemical steps-and hence intermediate states-that the enzyme is meant to implement, is largely left to human expertise The ability to capture specific chemistries of multi-step catalysis in a fashion that enables its computational construction and design is therefore highly desirable and would equally impact the elucidation of existing enzymatic reactions whose mechanisms are unknown Results: We use the mathematical framework of graph transformation to express the distinction between rules and reactions in chemistry We derive about 1000 rules for amino acid side chain chemistry from the M-CSA database, a curated repository of enzymatic mechanisms Using graph transformation we are able to propose hundreds of hypothetical catalytic mechanisms for a large number of unrelated reactions in the Rhea database We analyze these mechanisms to find that they combine in chemically sound fashion individual steps from a variety of known multi-step mechanisms, showing that plausible novel mechanisms for catalysis can be constructed computationally

Rolf Fagerberg

Papers

Fragile Complexity of Adaptive Algorithms

Optimal local routing on Delaunay triangulations defined by empty equilateral triangles

Characterizing Catalytic Mechanisms with Overlay Graphs

Fragile Complexity of Adaptive Algorithms

Graph Transformation for Enzymatic Mechanisms