Foundations of chemical reaction network theory
type: Item
instance of: scholarly article
title: Foundations of chemical reaction network theory
zbMATH Open document ID: 1420.92001
DOI: 10.1007/978-3-030-03858-8
author: Martin Feinberg
published in: Applied Mathematical Sciences
publication date: 2018-11-21T00:00:00Z
review text: 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 s…
reviewed by: Dieter Erle
Mathematics Subject Classification ID: 92-02
Mathematics Subject Classification ID: 92E20
Mathematics Subject Classification ID: 92C40
Mathematics Subject Classification ID: 92C42
Mathematics Subject Classification ID: 92C45
Mathematics Subject Classification ID: 76V05
Mathematics Subject Classification ID: 93D05
Mathematics Subject Classification ID: 05C90
zbMATH DE Number: 6982458
zbMATH Keywords: periodic solution
zbMATH Keywords: rank
zbMATH Keywords: Lyapunov function
zbMATH Keywords: reaction
zbMATH Keywords: equilibrium
zbMATH Keywords: bistability
zbMATH Keywords: instability
zbMATH Keywords: bipartite graph
zbMATH Keywords: complex
zbMATH Keywords: species
zbMATH Keywords: deficiency
zbMATH Keywords: reversible reaction
zbMATH Keywords: Helmholtz free energy
zbMATH Keywords: kinetic system
zbMATH Keywords: deficiency zero theorem
zbMATH Keywords: deficiency one theorem
zbMATH Keywords: chemical reaction network
zbMATH Keywords: rate constant
zbMATH Keywords: stoichiometric coefficient
zbMATH Keywords: stoichiometric subspace
zbMATH Keywords: mass action kinetics
zbMATH Keywords: stoichiometric compatibility class
zbMATH Keywords: linkage class
zbMATH Keywords: asymptotically stable
zbMATH Keywords: absolute concentration robustness
zbMATH Keywords: concordant network
zbMATH Keywords: nondegenerate network
zbMATH Keywords: isothermal continuous-flow tank-reactor
zbMATH Keywords: species reaction graph
zbMATH Keywords: quasi-thermodynamic kinetic system
zbMATH Keywords: complex balancing
zbMATH Keywords: weakly monotonic kinetics
zbMATH Keywords: even cycle
zbMATH Keywords: conservative network
zbMATH Keywords: knot graph theorem
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-030-03858-8
OpenAlex ID: W2911585371
description: scientific article; zbMATH DE number 6982458