First-order chemical reaction networks I: theoretical considerations

• Published in: Journal of mathematical chemistry Vol. 54; no. 9; pp. 1863 - 1878
• Main Authors: Tóbiás, Roland, Stacho, László L., Tasi, Gyula
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.10.2016; Springer Nature B.V
• Subjects: Algebra; Chemical reactions; Chemistry; Chemistry and Materials Science; Eigenvalues; Exact solutions; Incompatibility; Math. Applications in Chemistry; Networks; Original Paper; Physical Chemistry; Theoretical and Computational Chemistry; Network decomposition; Multiplicity of the zero eigenvalue; First-order reaction network; Algebraic model; Mass incompatibility; Marker network
• DOI: 10.1007/s10910-016-0655-2
• ISSN: 0259-9791

Our former study Tóbiás and Tasi (J Math Chem 54:85, 2016 ) is continued, where a simple algebraic solution was given to the kinetic problem of triangle, quadrangle and pentangle reactions. In the present work, after defining chemical reaction networks and their connectedness, first-order chemical reaction networks ( FCRN s) are studied on the basis of the results achieved by Chellaboina et al. (Control Syst 29:60, 2009 ). First, it is proved that an FCRN is disconnected iff its coefficient matrix is block diagonalizable. Furthermore, mass incompatibility is used to interpret the reducibility of subconservative networks. For conservative FCRN s, the so-called marker network is introduced, which is linearly conjugate to the original one, to describe the zero eigenvalue associated to the coefficient matrix of an FCRN . Instead of using graph-theoretical concepts, simple algebraic tools are applied to present and solve these problems. As an illustration, an industrially important ten-component (formal) FCRN is presented which has algebraically exact solution.