Feinberg network theory

Besides the two most well-known cases, the local bifurcations of the saddle-node and Hopf type, biochemical systems may show a variety of transitions between qualitatively different dynamic behavior [13, 17, 293, 294, 297 301]. Transitions between different regimes, induced by variation of kinetic parameters, are usually depicted in a bifurcation diagram. Within the chemical literature, a substantial number of articles seek to identify the possible bifurcation of a chemical system. Two prominent frameworks are Chemical Reaction Network Theory (CRNT), developed mainly by M. Feinberg [79, 80], and Stoichiometric Network Analysis (SNA), developed by B. L. Clarke [81 83]. An analysis of the (local) bifurcations of metabolic networks, as determinants of the dynamic behavior of metabolic states, constitutes the main topic of Section VIII. In addition to the scenarios discussed above, more complicated quasiperiodic or chaotic dynamics is sometimes reported for models of metabolic pathways [302 304]. However, apart from few special cases, the possible relevance of such complicated dynamics is, at best, unclear. Quite on the contrary, at least for central metabolism, we observe a striking absence of complicated dynamic phenomena. To what extent this might be an inherent feature of (bio)chemical systems, or brought about by evolutionary adaption, will be briefly discussed in Section IX. 

Closely related to the approach considered here are the formal frameworks of Feinberg and Clarke, briefly mentioned in Section II. A. Though mainly devised for conventional chemical kinetics, both, Chemical Reaction Network Theory (CRNT), developed by M. Feinberg and co-workers [79,80], as well as Stoichiometric Network Analysis (SNA), developed by B. L. Clarke [81 83], seek to relate aspects of reaction network topology to the possibility of various... 

Feinberg, M., Some recent results in chemical reaction network theory. In Patterns and Dynamics in Reactive Media, IMA Volumes on Mathematics and its Applications, Springer-Verlag, Berlin, 1991a. 

Feinberg, M., Applications of chemical reaction network theory in heterogeneous catalysis. In Chemical Reactions in Complex Systems (A. V. Sapre and F. J. Krambeck, eds.). Van Nostrand Reinhold, New York, 1991b, p. 179. 

In continuous stirred-tank reactors (CSTRs), complex kinetics may give rise to multiple steady states even in isothermal operation, especially in heterogeneous catalysis. However, to unravel the causes may be difficult. Here, Feinberg s network theory can help [3]. It operates with a deficiency index that is a readily calculated zero or positive integer. The most useful result of the theory is ... 

Feinberg s network theory [3-7]. The deficiency of a network is calculated from its number of complexes, n, linkage classes, Z, and rank, s. Here, a "complex" is defined as the reactant(s) or product(s) of a step. For example, the complexes of a step A + B K are A+B and K. A network consists of one or more "linkage classes," defined as portions consisting of complexes connected with one another by arrows, but not with complexes of another class For example, a network... 

Feinberg s network theory is based on a number of premises ... 

Instability typically arises from the interaction of two phenomena with different dependences on a reaction parameter In a nonisothermal reaction, the dependence on temperature is exponential for heat generation by the reaction, but linear for heat loss to the cooling coil or environment in a reaction with chain branching, the dependence on radical population is exponential for acceleration by branching, but quadratic for chain termination. A reaction is unstable if acceleration outruns retardation. This can cause an explosion or, in a CSTR, lead to multiple steady states. Feinberg s network theory can help to assess whether an isothermal reaction admits multiple steady states in a CSTR. 

An alternate approach to the analysis of the network geometry from the viewpoint of multiplicity and stability is due to Beretta and his coworkers (1979, 1981) the latter approach is based on knot theory. The Schlosser-Feinberg theory reproduces some of the Beretta-type theory results concerning stability from a somewhat different viewpoint. 

Finally, Feinberg considers the rank of the stoichiometric matrix, R. He then defines a deficiency of the reaction network, 5,as6 = N — N — R. In his series of papers, Feinberg obtains a number of strong results for systems of deficiency 0 and deficiency 1 these are not discussed here, and only very concisely in the next section. For the whole strength of the theory the reader is referred to the original Feinberg papers. 

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