Some results about nonlinear chemical systems represented by trees and cycles

Using linear stability analysis, the qualitative stability properties of open nonlinear chemical systems, in which reactions of any order may occur, will be studied. Systems will be classified in three fundamental classes: trees, cycles and loops, according to their knot graphs. The study of the Jacobian matrix for the kinetic equations of the system shows that the symmetrizability by a particular procedure (calledD-symmetrizability) is a sufficient condition for stability. It has been proved that tree-graphs always satisfy the above condition. For the cycle-graphs, theD-symmetrizability condition leads to a cyclic relation between forward and reverse steady state flows. The stability may be assured, even if the cyclic relation is not satisfied, providing that the “symmetry breaking” be lower than an upper bound; further alternative criteria for stability of cycles have been derived. All these results are independent of the number of diffusive exchanges with the environment.

[1]  The tree graphs theory for enzymatic reactions: a theorem for the reactions among the non-enzymatic species. , 1977, Journal of theoretical biology.

[2]  L. Bittner.  S. H. Lehnigk, Stability Theorems For Linear Motions. (International Series in Applied Mathematics.) XI + 251 S. m. Fig. Englewood Cliffs. N. J. 1966. Prentice‐Hall, Inc. Preis geb. 96.– s. net , 1971 .

[3]  J. Hearon,et al.  The kinetics of linear systems with special reference to periodic reactions , 1953 .

[4]  Valeurs propres des systèmes de transformation représentables par des graphes en arbres , 1973 .

[5]  B. L. Clarke.  Stability of the bromate–cerium–malonic acid network. II. Steady state formic acid case , 1976 .

[6]  J. Hearon,et al.  THEOREMS ON LINEAR SYSTEMS * , 1963, Annals of the New York Academy of Sciences.

[7]  L. Glass,et al.  The logical analysis of continuous, non-linear biochemical control networks. , 1973, Journal of theoretical biology.

[8]  E. Beretta,et al.  The tree graphs theory for enzymatic reactions. , 1976, Journal of theoretical biology.

[9]  M. Feinberg,et al.  Dynamics of open chemical systems and the algebraic structure of the underlying reaction network , 1974 .

[10]  R. Jackson,et al.  General mass action kinetics , 1972 .

[11]  F. Horn,et al.  On a connexion between stability and graphs in chemical kinetics II. Stability and the complex graph , 1973, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[12]  O. Sǐnanoğlu.  Theory of chemical reaction networks. All possible mechanisms or synthetic pathways with given number of reaction steps or species , 1975 .

[13]  B. L. Clarke.  Stability of the bromate–cerium–malonic acid network. I. Theoretical formulation , 1976 .

[14]  L. Glass.  Combinatorial and topological methods in nonlinear chemical kinetics , 1975 .

[15]  F. HoRNt,et al.  On a connexion between stability and graphs in chemical kinetics. I. Stability and the reaction diagram , 1973 .

[16]  Frank Harary,et al.  Graph Theory , 2016 .

[17]  L. Glass.  Classification of biological networks by their qualitative dynamics. , 1975, Journal of theoretical biology.