Foundations of chemical reaction network theory
A book like this one has been urgently needed and can only be classified as overdue. It affords a description of the theoretical aspects of reaction network theory in chemistry from the beginnings in the 1970's by the foundational work of Horn, Jackson, and Feinberg all the way through the present state of the art. A prominent position is occupied by the famous theorems on deficiency zero and deficiency one. Before entering into an overall appreciation of the work, it appears worthwhile to get across an idea of these special results by going through a fictitious example of a reaction network: Let \( \; X + Y \rightleftarrows 2Y\), \(2X + Y \rightleftarrows 4X\) be a reaction. Here, \(X\) and \(Y\) are the species (e.g., reacting molecules) and \( \; X + Y, \; 2Y, \; 2X + Y, \; 4X \; \) the so-called complexes. The stoichiometric coefficients in front of the species appearing in the complexes of the reaction, here \(\, 1\)'s, \(2\)'s, and \(4 \;\), as well as various \( \, 0\)'s, form vectors spanning the stoichiometric subspace \(S\) and appear as exponents in monomials of the polynomial ODE system governing the reaction if the latter follows mass action kinetics. (The intersections of parallel translations of the stoichiometric subspace with the positive cone generally enjoy invariance, i.e., that they cannot be left by solutions and are called stoichiometric compatibility classes.) An equivalence relation among the complexes leads to the linkage classes of the network, whose number \( \ell\) is \(2\) in our example, due to the reversibility of the \(2\) reactions (double arrows). With \(n\) the number of complexes (\(4\) in the example) and \( \,s = 2 \, \) the rank of the stoichiometric subspace \(S\), the famous deficiency of the reaction network is \( \, \delta = n - \ell - s \), i.e., \(4 - 2 - 2 = 0 \, \) in the example, and the deficiency zero theorem applies. The main part of it certifies that, in a weakly reversible mass action reaction network, in every stoichiometric compatibility class, there is exactly one equilibrium, and this is asymptotically stable. Remarkably, this is independent of the rate constants of the individual mass action reactions and thus guarantees a desirable behavior through a wide class of systems. For the deficiency one theorem, the notion of deficiency has to be refined to apply to the individual linkage classes. Then, under appropriate conditions, the existence of an equilibrium with nonsingular Jacobian is guaranteed. The actual propositions are a lot more general. Naturally, the book has far more to offer than just the deficiency theorems. We can here only summarily touch on the most important points, leaving out many interesting details of the contents. Part I establishes the framework of reaction network theory, introducing the basic notions and describing the problems to be studied, thereby presenting examples and explaining the historical development of the field. What a chemist most appreciates is predictibility and reliability of the reaction systems considered. As reaction networks in practice will tend to be very complicated, it is desirable to find out certain aspects of their behavior without beforehand knowing too much about them, especially about the kinetics. It is therefore important to look in the first place for cases of what is here called dull behavior, e.g., unique existence of equilibria and their stability. Most of what follows is devoted to this aim. The central Part II first deals with the aforementioned deficiency theorems and discusses the influence of the rate constants on the existence of equilibria in cases not covered by these theorems. Then, a special form of reliable behavior is studied: For mass action systems, conditions are found that guarantee that a certain species concentration at equilibria is independent of the stoichiometric stability class (absolute concentration robustness in the context of deficiency one). Two following chapters present fairly recent and very powerful results in the framework of the above-mentioned aim. The first one uses the concept of concordance in reaction networks; numerous related notions, like nondegenerate networks, are discussed, involving also the kinetics of the networks. As it turns out, concordance is a common property, and this is taken as an explanation of the general experience that isothermal continuous-flow tank-reactors usually show surprisingly stable behavior. Though a rather technical notion, concordance is a property that can be checked via extant software. More important, however, is that concordance is the consequence of properties of the species-reaction graph of the next chapter, properties which are more easily accessible. The species-reaction graph of a network is a bipartite graph with vertices for species and reactions, with edges connecting reaction-vertices with the vertices of the occurring species where the edges are equipped with labelings by the pertinent complexes. Thus, the species-reaction graph mirrors all the information of the reaction network. Impressively, concordance can often be determined by purely graph theoretic properties of the species-reaction graph, a fact leading to statements about equilibria, their uniqueness and their stability. The final Part III takes a wider view at the foundations and digs more deeply into the theory. A chapter on thermodynamics detects Lyapunov functions leading to the above-mentioned stability statements, inspired by Helmholtz free energy. This holds in particular for quasi-thermodynamic kinetic systems. Then, complex balancing, a notion referring to the kinetics installed on a reaction network, is introduced; it serves as key to the deficiency zero theorem in that it is a consequence of deficiency zero and implies quasi-thermodynamicity. A chapter deals with the proof of the deficiency one theorem, and another one with the implication of absolute concentration robustness by deficiency one. The final chapter outlines proofs in the context of the species-reaction graph. This book comprehensively describes network theory applied to chemical reactions, delivering powerful conclusions surprisingly following from hypotheses that are essentially formulated in terms of linear algebra and graph theory. These results aim at providing means to deduce uniqueness and stability of an equilibrium state; nevertheless, cases of instability and periodicity of solutions are displayed. The presentation is rich in material including motivating examples, applications to practical problems, guide to the literature, and mathematical proofs. It deserves a salient place in the section on mathematical chemistry of any library.
scientific article; zbMATH DE number 6982458