Rewriting Modulo Traced Comonoid Structure

In this paper we adapt previous work on rewriting string diagrams using hypergraphs to the case where the underlying category has a traced comonoid structure, in which wires can be forked and the outputs of a morphism can be connected to its input. Such a structure is particularly interesting because any traced Cartesian (dataflow) category has an underlying traced comonoid structure. We show that certain subclasses of hypergraphs are fully complete for traced comonoid categories: that is to say, every term in such a category has a unique corresponding hypergraph up to isomorphism, and from every hypergraph with the desired properties, a unique term in the category can be retrieved up to the axioms of traced comonoid categories. We also show how the framework of double pushout rewriting (DPO) can be adapted for traced comonoid categories by characterising the valid pushout complements for rewriting in our setting. We conclude by presenting a case study in the form of recent work on an equational theory for sequential circuits: circuits built from primitive logic gates with delay and feedback. The graph rewriting framework allows for the definition of an operational semantics for sequential circuits

Analysis of Generative Chemistries

For the modelling of chemistry we use undirected, labelled graphs as explicit models of molecules and graph transformation rules for modelling generalised chemical reactions. This is used to define artificial chemistries on the level of individual bonds and atoms, where formal graph grammars implicitly represent large spaces of chemical compounds. We use a graph rewriting formalism, rooted in category theory, called the Double Pushout approach, which directly expresses the transition state of chemical reactions. Using concurrency theory for transformation rules, we define algorithms for the composition of rewrite rules in a chemically intuitive manner that enable automatic abstraction of the level of detail in chemical pathways. Based on this rule composition we define an algorithmic framework for generation of vast reaction networks for specific spaces of a given chemistry, while still maintaining the level of detail of the model down to the atomic level. The framework also allows for computation with graphs and graph grammars, which is utilised to model non-trivial chemical systems. The graph generation relies on graph isomorphism testing, and we review the general individualisation-refinement paradigm used in the state-of-the-art algorithms for graph canonicalisation, isomorphism testing, and automorphism discovery. We present a model for chemical pathways based on a generalisation of network flows from ordinary directed graphs to directed hypergraphs. The model allows for reasoning about the flow of individual molecules in general pathways, and the introduction of chemically motivated routing constraints. It further provides the foundation for defining specialised pathway motifs, which is illustrated by defining necessary topological constraints for both catalytic and autocatalytic pathways. We also prove that central types of pathway questions are NP-complete, even for restricted classes of reaction networks. The complete pathway model, including constraints for catalytic and autocatalytic pathways, is implemented using integer linear programming. This implementation is used in a tree search method to enumerate both optimal and near-optimal pathway solutions. The formal methods are applied to multiple chemical systems: the enzyme catalysed beta-lactamase reaction, variations of the glycolysis pathway, and the formose process. In each of these systems we use rule composition to abstract pathways and calculate traces for isotope labelled carbon atoms. The pathway model is used to automatically enumerate alternative non-oxidative glycolysis pathways, and enumerate thousands of candidates for autocatalytic pathways in the formose process

Functorial String Diagrams for Reverse-Mode Automatic Differentiation

We formulate a reverse-mode automatic differentiation (RAD) algorithm for (applied) simply typed lambda calculus in the style of Pearlmutter and Siskind [Barak A. Pearlmutter and Jeffrey Mark Siskind, 2008], using the graphical formalism of string diagrams. Thanks to string diagram rewriting, we are able to formally prove for the first time the soundness of such an algorithm. Our approach requires developing a calculus of string diagrams with hierarchical features in the spirit of functorial boxes, in order to model closed monoidal (and cartesian closed) structure. To give an efficient yet principled implementation of the RAD algorithm, we use foliations of our hierarchical string diagrams

GraphProt: modeling binding preferences of RNA-binding proteins

We present GraphProt, a computational framework for learning sequence- and structure-binding preferences of RNA-binding proteins (RBPs) from high-throughput experimental data. We benchmark GraphProt, demonstrating that the modeled binding preferences conform to the literature, and showcase the biological relevance and two applications of GraphProt models. First, estimated binding affinities correlate with experimental measurements. Second, predicted Ago2 targets display higher levels of expression upon Ago2 knockdown, whereas control targets do not. Computational binding models, such as those provided by GraphProt, are essential for predicting RBP binding sites and affinities in all tissues. GraphProt is freely available at http://www.bioinf.uni-freiburg.de/Software/GraphProt

Structural conserved moiety splitting of a stoichiometric matrix

Characterising biochemical reaction network structure in mathematical terms enables the inference of functional biochemical consequences from network structure with existing mathematical techniques and spurs the development of new mathematics that exploits the peculiarities of biochemical network structure. The structure of a biochemical network may be specified by reaction stoichiometry, that is, the relative quantities of each molecule produced and consumed in each reaction of the network. A biochemical network may also be specified at a higher level of resolution in terms of the internal structure of each molecule and how molecular structures are transformed by each reaction in a network. The stoichiometry for a set of reactions can be compiled into a stoichiometric matrix N is an element of Z(mxn), where each row corresponds to a molecule and each column corresponds to a reaction. We demonstrate that a stoichiometric matrix may be split into the sum of m - rank(N) moiety transition matrices, each of which corresponds to a subnetwork accessible to a structurally identifiable conserved moiety. The existence of this moiety matrix splitting is a property that distinguishes a stoichiometric matrix from an arbitrary rectangular matrix. (C) 2020 Elsevier Ltd. All rights reserved.Analytical BioScience

String Diagrams for λ\lambda-calculi and Functional Computation

This tutorial gives an advanced introduction to string diagrams and graph languages for higher-order computation. The subject matter develops in a principled way, starting from the two dimensional syntax of key categorical concepts such as functors, adjunctions, and strictification, and leading up to Cartesian Closed Categories, the core mathematical model of the lambda calculus and of functional programming languages. This methodology inverts the usual approach of proceeding from syntax to a categorical interpretation, by rationally reconstructing a syntax from the categorical model. The result is a graph syntax -- more precisely, a hierarchical hypergraph syntax -- which in many ways is shown to be an improvement over the conventional linear term syntax. The rest of the tutorial focuses on applications of interest to programming languages: operational semantics, general frameworks for type inference, and complex whole-program transformations such as closure conversion and automatic differentiation

Visualization of Metabolic Networks

The metabolism constitutes the universe of biochemical reactions taking place in a cell of an organism. These processes include the synthesis, transformation, and degradation of molecules for an organism to grow, to reproduce and to interact with its environment. A good way to capture the complexity of these processes is the representation as metabolic network, in which sets of molecules are transformed into products by a chemical reaction, and the products are being processed further. The underlying graph model allows a structural analysis of this network using established graphtheoretical algorithms on the one hand, and a visual representation by applying layout algorithms combined with information visualization techniques on the other. In this thesis we will take a look at three different aspects of graph visualization within the context of biochemical systems: the representation and interactive exploration of static networks, the visual analysis of dynamic networks, and the comparison of two network graphs. We will demonstrate, how established infovis techniques can be combined with new algorithms and applied to specific problems in the area of metabolic network visualization. We reconstruct the metabolic network covering the complete set of chemical reactions present in a generalized eucaryotic cell from real world data available from a popular metabolic pathway data base and present a suitable data structure. As the constructed network is very large, it is not feasible for the display as a whole. Instead, we introduce a technique to analyse this static network in a top-down approach starting with an overview and displaying detailed reaction networks on demand. This exploration method is also applied to compare metabolic networks in different species and from different resources. As for the analysis of dynamic networks, we present a framework to capture changes in the connectivity as well as changes in the attributes associated with the network’s elements

Graph-based modeling and evolutionary analysis of microbial metabolism

Microbial organisms are responsible for most of the metabolic innovations on Earth. Understanding microbial metabolism helps shed the light on questions that are central to biology, biomedicine, energy and the environment. Graph-based modeling is a powerful tool that has been used extensively for elucidating the organising principles of microbial metabolism and the underlying evolutionary forces that act upon it. Nevertheless, various graph-theoretic representations and techniques have been applied to metabolic networks, rendering the modeling aspect ad hoc and highlighting the conflicting conclusions based on the different representations. The contribution of this dissertation is two-fold. In the first half, I revisit the modeling aspect of metabolic networks, and present novel techniques for their representation and analysis. In particular, I explore the limitations of standard graphs representations, and the utility of the more appropriate model---hypergraphs---for capturing metabolic network properties. Further, I address the task of metabolic pathway inference and the necessity to account for chemical symmetries and alternative tracings in this crucial task. In the second part of the dissertation, I focus on two evolutionary questions. First, I investigate the evolutionary underpinnings of the formation of communities in metabolic networks---a phenomenon that has been reported in the literature and implicated in an organism's adaptation to its environment. I find that the metabolome size better explains the observed community structures. Second, I correlate evolution at the genome level with emergent properties at the metabolic network level. In particular, I quantify the various evolutionary events (e.g., gene duplication, loss, transfer, fusion, and fission) in a group of proteobacteria, and analyze their role in shaping the metabolic networks and determining the organismal fitness. As metabolism gains an increasingly prominent role in biomedical, energy, and environmental research, understanding how to model this process and how it came about during evolution become more crucial. My dissertation provides important insights in both directions