Chemical Reaction 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... 

B. D. Agunda and B. L. Clarke, Bistability in chemical reaction networks Theory and application to the peroxidase reaction. J. Chem. Phys. 87(6), 3461 3469 (1987). 

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. 

Sellers (9) developed a theory of chemical reaction networks for the... 

Sellers, P. H., An introduction to a mathematical theory of chemical reaction networks 1. 

Sinanoglu, O., Theory of chemical reaction networks. All possible mechanisms or synthetic pathways with given number of reaction steps or species. J. Am. Chem. Soc. 97, 2309-2320 (1975). 

Sellers, P. (1971). An introduction to a mathematical theory of chemical reaction networks, I. Arch. Rati. Mech. Anal., 44, 23-40. 

Gorban and colleagues (Gorban et al., 2010 Gorban and Radulescu, 2008) developed a general theory of limitation in chemical reaction networks. The main new concept of this theory is an acychc dominant mechanism, which is distinguished within the complex mechanism. The opinion that for a not too complicated mechanism peculiarities of transition regimes can be... 

With so many molecules now being observed in interstellar clouds, chemical reaction models which can explain how these molecules are produced and destroyed are becoming increasingly more valuable. The most modern chemical reaction networks that have been proposed involve following the concentration of several hundred atomic and molecular species as a function of time, and reliable temperature-dependent rate coefficients for several thousand reactions are a vital requirement in such simulations. The role of ion-molecule reactions has been shown to be of particular Importance in these networks as these reactions can have very large rate coefficients at the low temperatures of interstellar clouds [2]. Furthermore, a more limited number of neutral species, particularly radicals and open-shell atoms, can have large rate coefficients at low temperatures [3]. Since only a relatively small number of reactions have been studied in the laboratory at the temperatures relevant to Interstellar chemistry, theory plays an Important role in producing many of the required rate coefficients. 

B.L. Clarke, "pualitative Dynatnics and Stability of Chemical Reactions Networks", in Chemical Applications of Graph Theory and Topology, Ed. R.B. King, (Elsevier, Amsterdam 1983) pp 322-357. 

Rappoport, D., Galvin, C.J., Zubarev, D.Y., Aspuru-Guzik, A. Complex chemical reaction networks fiom heuristics-aided quantum chemistry. J. Chem. Theory Comput. 10, 897-907 (2014)... 

Recently there has been an increasing interest in self-oscillatory phenomena and also in formation of spatio-temporal structure, accompanied by the rapid development of theory concerning dynamics of such systems under nonlinear, nonequilibrium conditions. The discovery of model chemical reactions to produce self-oscillations and spatio-temporal structures has accelerated the studies on nonlinear dynamics in chemistry. The Belousov-Zhabotinskii(B-Z) reaction is the most famous among such types of oscillatory chemical reactions, and has been studied most frequently during the past couple of decades [1,2]. The B-Z reaction has attracted much interest from scientists with various discipline, because in this reaction, the rhythmic change between oxidation and reduction states can be easily observed in a test tube. As the reproducibility of the amplitude, period and some other experimental measures is rather high under a found condition, the mechanism of the B-Z reaction has been almost fully understood until now. The most important step in the induction of oscillations is the existence of auto-catalytic process in the reaction network. 

Several analytical techniques have been used to characterize the polymer/ silane coupling agent interphase. Culler et aL [2] used Fourier transform infrared (FT-IR) spectroscopy to characterize the chemical reactions at the matrix/silane interphase of composite materials. They correlated the extent of reaction of the resin with the coupling agent (as determined by FT-IR) with the extent of interpenetration. Culler et al. [2] have also used observations of improved resistance of the interphase region to solvent attack as indirect evidence to support the interpenetrating network theory. 

Later development in singularity theory, especially the pioneering work of Golubitsky and Schaeffer [19], has provided a powerful tool for analyzing the bifurcation behavior of chemically reactive systems. These techniques have been used extensively, elegantly and successfully by Luss and his co-workers [6-11] to uncover a large number of possible types of bifurcation. They were also able to apply the technique successfully to complex reaction networks as well as to distributed parameter systems. 

A significant simplification of the algorithm is associated with applying chemical kinetic methods taken from the graphs theory. A graph is a geometrical scheme consisting of a set of points connected by lines. It can be a complex electric scheme, a railway network, a plan of constructional works or finally, a complex chemical reaction. 

The scales involved in such a reactor should be defined in a relative manner. For a chemist, the molecule is at the start and catalyst (particle) at the end of the scales. To reveal the reaction mechanism over a catalyst particle, a sequence of network of elementary reactions" will be needed. Accordingly, on the basis of, for example, the molecular collision theory (Turns, 2000), the "global reaction" can be derived in terms of global rate coefficient and reaction order. Here, the resultant reaction mechanism is termed "global" by chemists, because the use of it for a specific problem is normally a "black box" approach, without knowing exactly the underlying networks or structures of chemical routes from reactants to products. On the other hand, for a chemical reaction engineer, the catalyst (particle) is often the start and the reactor is the end. The reaction free of inner-particle and outer-particle diffusions, that is,... 

Balaban AT, ed (1976) Chemical applications of graph theory, Academic Press, London see also Kvasnicka V, Pospichal J (1990) Graph theoretical interpretation fo Ugi s concept of the reaction networks, J Math Chem 5 309, and ref therein... 

Dynamics calculations of reaction rates by semiempirical molecular orbital theory. POLYRATE for chemical reaction rates of polyatomics. POLYMOL for wavefunctions of polymers. HONDO for ab initio calculations. RIAS for configuration interaction wavefunctions of atoms. FCI for full configuration interaction wavefunctions. MOLSIMIL-88 for molecular similarity based on CNDO-like approximation. JETNET for artificial neural network calculations. More than 1350 other programs most written in FORTRAN for physics and physical chemistry. 

Due to the changes in the dynamics, a general relationship for stochastic dynamics is not available like it is for deterministic dynamics. However, for mesoscopic systems, a mesoscopic FR is useful. Therefore, there has been much work on developing stochastic models with different conditions. Andrieux and Gaspard developed a stochastic fluctuation relation for nonequilibrium systems whose dynamics can be described by Schnakenberg s network theory (e.g. mesoscopic electron transport, biophysical models of ion transport and some chemical reactions). Due to early experimental work on protein unfolding and related molecular motors, and their ready treatment by stochastic dynamics, a number of papers have appeared that model these systems and test the or JE for these. FR... 

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