50 Shades of Rule Composition

Combining Graph Transformations and Semigroups for Isotopic Labeling Design

The double pushout approach for graph transformation naturally allows an abstraction level of biochemical systems in which individual atoms of molecules can be traced automatically within chemical reaction networks. Aiming at a mathematical rigorous approach for isotope labeling design, we convert chemical reaction networks (represented as directed hypergraphs) into transformation semigroups. Symmetries within molecules correspond to permutations, whereas (not necessarily invertible) chemical reactions define the transformations of the semigroup. An approach for the automatic inference of informative labeling of atoms is presented, which allows to distinguish the activity of different pathway alternatives within reaction networks. To illustrate our approaches, we apply them to the reaction network of glycolysis, which is an important and well-understood process that allows for different alternatives to convert glucose into pyruvate.

Chemical Transformation Motifs-Modelling Pathways as Integer Hyperflows

We present an elaborate framework for formally modelling pathways in chemical reaction networks on a mechanistic level. Networks are modelled mathematically as directed multi-hypergraphs, with vertices corresponding to molecules and hyperedges to reactions. Pathways are modelled as integer hyperflows and we expand the network model by detailed routing constraints. In contrast to the more traditional approaches like Flux Balance Analysis or Elementary Mode analysis we insist on integer-valued flows. While this choice makes it necessary to solve possibly hard integer linear programs, it has the advantage that more detailed mechanistic questions can be formulated. It is thus possible to query networks for general transformation motifs, and to automatically enumerate optimal and near-optimal pathways. Similarities and differences between our work and traditional approaches in metabolic network analysis are discussed in detail. To demonstrate the applicability of the mathematical framework to real-life problems we first explore the design space of possible non-oxidative glycolysis pathways and show that recent manually designed pathways can be further optimized. We then use a model of sugar chemistry to investigate pathways in the autocatalytic formose process. A graph transformation-based approach is used to automatically generate the reaction networks of interest.

An intermediate level of abstraction for computational systems chemistry

Computational techniques are required for narrowing down the vast space of possibilities to plausible prebiotic scenarios, because precise information on the molecular composition, the dominant reaction chemistry and the conditions for that era are scarce. The exploration of large chemical reaction networks is a central aspect in this endeavour. While quantum chemical methods can accurately predict the structures and reactivities of small molecules, they are not efficient enough to cope with large-scale reaction systems. The formalization of chemical reactions as graph grammars provides a generative system, well grounded in category theory, at the right level of abstraction for the analysis of large and complex reaction networks. An extension of the basic formalism into the realm of integer hyperflows allows for the identification of complex reaction patterns, such as autocatalysis, in large reaction networks using optimization techniques.This article is part of the themed issue 'Reconceptualizing the origins of life'.