Excerpt

Introduction

Chemical systems, that is, molecules and their reactions, are just time-dependent multiparticle quantum systems. As such, they are completely described by fundamental principles of physics, expressed in terms of quantum field theory (QFT) (Weinberg 2005). This level of description, however, is of little practical use to a chemist because the computational efforts to obtain answers to essentially all questions that a chemist might ask exceed, by far, the limits of present-day technology. It is possible, however, to arrive at more useful levels of description by means of a hierarchy of approximations and simplifications, making use of specific properties that distinguish chemical reactions for arbitrary quantum systems (see, e.g., fig. 3.1; Andersen et al. 2017a). These include the immutability of atomic nuclei and the idea that chemical reactions comprise only a redistribution of electrons. Furthermore, the Born– Oppenheimer approximation (Born and Oppenheimer 1927) postulates a complete separation of the wave function of nuclei and electrons due to the large difference in their masses, which leads to the concept of potential energy surfaces that determine the geometry of molecules and make it possible to view chemical reactions as classical paths on this surface (Mezey 1987; Heidrich, Kliesch, and Quapp 1991). The computation of potential energy surfaces by solving the Schrödinger equation is one of the key problems in quantum chemistry, for which a wide variety of approximation methods have been developed with different trade-offs between accuracy and computational effort. Semiempirical methods capitalize on the empirical fact that chemical bonds are usually formed by pairs of electrons to further simplify the electronic wave function. Molecular modeling and molecular dynamics (McCammon, Gelin, and Karplus 1977; Burkert and Allinger 1982) abandon quantum mechanics and model the potential energy surface as a sum of empirical contributions for pair bonds and electrostatic effects. This simplifies the computations sufficiently to treat macromolecules and supramolecular complexes that are intractable with quantumchemical methods.

Even coarser approximations have been developed for particular classes of molecules. Aromatic ring systems, for instance, are well described in terms of purely graph-theoretical models known as Hückel theory (Hückel 1931; Hoffmann 1963). Nucleic acids can be coarse grained even further by aggregating their molecular building blocks (nucleotides) into single vertices. Watson–Crick base pairs then become edges in the graph representation known as secondary structure (Zuker and Stiegler 1981).

In this contribution, we adopt labeled graphs as the level of description of choice, that is, the level of chemical formulas and reaction schemes most familiar to chemists. We use this formalism to develop an algebraic description of chemical reaction networks that is consistent with traditional, flux-based methods, such as flux balance analysis or elementary mode analysis, and, at the same time, is sufficiently detailed to capture the constructive aspects of reorganizing atoms through arbitrary reaction systems.

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