Graph-theoretical method for determining routes of complex chemical reactions

Спивак, С. И.; Ismagilova, A. S.; Khamitova, I. A.

doi:10.1134/s0012501610100040

Cited by 8 publications

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References 2 publications

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“…The connections can often usefully be thought to define a network represented by a graph. [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] The underlying machinery of cell biology and physiology consists of interconnected metabolic, genetic, and signaling networks. [18][19][20][21][22] We can sometimes relate some particular behavior (e.g., multistability or oscillations) to properties of a suitable graphical representation of the network.…”

Section: Introductionmentioning

confidence: 99%

Graph-theoretic analysis of a model for the coupling between photosynthesis and photorespiration

Ruhul

Roussel

2014

Can. J. Chem.

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We develop and analyze a mathematical model based on a previously enunciated hypothesis regarding the origin of rapid, irregular oscillations observed in photosynthetic variables when a leaf is transferred to a low-CO 2 atmosphere. This model takes the form of a set of differential equations with two delays. We review graph-theoretical methods of analysis based on the bipartite graph representation of mass-action models, including models with delays. We illustrate the use of these methods by showing that our model is capable of delay-induced oscillations. We present some numerical examples confirming this possibility, including the possibility of complex transient oscillations. We then use the structure of the identified oscillophore, the part of the reaction network responsible for the oscillations, along with our knowledge of the plausible range of values for one of the delays, to rule out this hypothetical mechanism.Résumé : Nous présentons un modèle mathématique développé à partir d'une hypothèse récente sur l'origine d'oscillations rapides et irrégulières des variables photosynthétiques observées lorsqu'une feuille est introduite à une atmosphère à basse teneur en dioxyde de carbone. Ce modèle consiste d'un système d'équations différentielles avec deux retards. Nous présentons brièvement des méthodes d'analyse de modèles à action de masse représentés par des graphes bipartis, y compris les modèles avec retards. Nous utilisons notre modèle comme exemple, et démontrons à l'aide de ces méthodes que ce modèle est capable d'oscillations induites par le retard. L'existence de telles oscillations est confirmée par simulation numérique. Nous découvrons de plus la possibilité d'oscillations complexes transitoires. Nous utilisons ensuite la structure de l'oscillophore identifié par nos méthodes, donc de la partie du réseau chimique responsable des oscillations, ainsi que notre connaissance de la longueur attendue d'un des retards, pour éliminer ce mécanisme hypothétique.

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Section: Introductionmentioning

confidence: 99%

Graph-theoretic analysis of a model for the coupling between photosynthesis and photorespiration

Ruhul

Roussel

2014

Can. J. Chem.

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“…The major result is one to one cor respondence between the graph cycles and index matrix. Thus, taking into account the theorem relating the route basis and the cycle basis [7] leads to the appropriate algorithm.…”

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“…The algorithm developed in [7] is based on inci dence matrix analysis, while, in the present work, the index matrix is used. The decrease in time is due to the fact that the dimension of the index matrix (the num ber of reactions by the number of intermediates) is sig nificantly lower than the dimension of the incidence matrix (the number of vertices by the number of the graph arcs; the number of vertices is the number of all participants of the mechanism and the number of steps; the arcs are relations between vertices of differ ent types, reaction vertices and substance vertices).…”

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Index method of finding the independent routes of complex chemical reactions

Спивак

Ismagilova

2015

Dokl Phys Chem

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The route is a vector such that multiplication of its components by the corresponding step of the complex reaction mechanism with subsequent summation of the steps leads to the overall reaction equation that does not contain intermediates [1].The concept of "reaction route" has been intro duced in classical works by J. Horiuti [1] and M.I. Temkin [2] while developing a steady state reac tion theory. The route theory played an important role in construction of complex reaction kinetics theory [2]. The concept of "route" and its application have again attracted much attention in analysis of complex reactions of metal complex catalysis [3].When formulating the steady state reaction theory, Temkin [2] has suggested graphical interpretation of the complex reaction mechanism. This approach has been used for exclusion of intermediates and con struction of general kinetic equations for the cases when the stationarity equation set is linear with respect to the intermediates. This is the case when all elemen tary reactions are first order in intermediates. This "linearity" has considerably limited the application of the graph theory.A general description of chemical reaction schemes based on the graph theory has been intro duced by A.I. Volpert [4]. A specific feature here is the fact that all results can be used for arbitrary nonlinear reaction mechanisms.Real complex reaction mechanisms often involve a large number of substances and reactions between them [5]. Intermediates are formed and consumed in the course of reaction. At the initial instant of time, their concentrations are zero. The vertex index is the minimum order of the derivative with respect to con centration nonzero at the initial instant of time [4]. In other words, information on indices characterizes intermediates on the basis of the rate of their involve ment in the course of a complex reaction.Any of the sets found according to the algorithm described in [2] can be selected as the route basis. A question arises as to how to select a route basis each component of which is characterized by a part of a complex reaction mechanism that has physicochemi cal meaning. In this context, the rate of involvement of a substance in the reaction provides important specific information for selection of the basic route. This can be of importance for finding a unique solution of inverse problems of identification of complex reaction mechanisms [6].This brings up a question of constructing algo rithms for route selection based on information on vertex indices. The present study is aimed at creating an algorithm for independent route basis selection, using the information on vertex indices of the chemi cal reaction graph. The major result is one to one cor respondence between the graph cycles and index matrix. Thus, taking into account the theorem relating the route basis and the cycle basis [7] leads to the appropriate algorithm.Let us describe the algorithm of searching for inde pendent routes and deriving overall equations of com plex chemical reactions.(1) Construction...

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“…In the construction of an algorithm for the deter mination of routes from the Vol'pert graph [16], it was necessary to introduce the concept of edge weight. The edge weight is equal to the stoichiometric coefficient.…”

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Computer analysis of the graphs of complex chemical reactions

2015

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The development of scientifically substantiated methods for studying the mechanisms of complex reactions with the use of computer programs remains a problem of considerable current interest to the inter national scientific community. Methods for construct ing the mathematical models of the mechanisms of complex chemical reactions were described [1]. The mechanisms of catalytic reactions were studied with the use of graph theory methods [2,3]. A theoretical graph approach was also proposed [4] for determining balance relationships between the kinetic equations of biochemical systems. The DRGEP method for con structing a kinetic model and removing redundant species from the mechanism was automated and described [5]. The graph theory apparatus was also used to describe chemical equilibrium in complex chemical reactions [6,7]. A theoretical graph approach was used for abridging the system of chemi cal reaction networks [8]. To a larger degree, this is due to the fact that currently available computer technologies make it possible to obtain reliable technological solutions and to develop a conve nient interface for specialists in the design of chemical processes.The inverse problems of chemical kinetics-the determination of the rate constants of chemical reac tions based on kinetic measurements-occupy a spe cial position in the mathematical simulation of the kinetics of complex chemical reactions [9]. The absence of experimental information on intermediate substances leads to the ambiguous solutions of the inverse problems. The information content of kinetic measurements was analyzed previously [10]. An algo rithm was constructed for the analysis of information content based on the theory of implicit functions and the functional Jacobi matrices. The problem consists in the complexity of the test systems as a result of their large dimensionality and the nonlinearity of mathe matical descriptions.Spivak and coauthors [11,12] posed the problem of analyzing the information content of kinetic measure ments based on the decomposition (paralleling) of the original problem into subproblems of smaller dimen sionality, which have a physicochemical interpreta tion. The system of independent routes became the basis for the decomposition. Spivak and Ismagilova [11] described a methodology for the analysis of the information content of kinetic measurements. As a result of the proposed algebraic interpretation, algo rithms were developed for the determination of the number and form of independent nonlinear paramet ric functions of kinetic constants, which can be unam biguously evaluated based on the accessible kinetic information.The aim of this work was to develop methods for the computer analysis of graphs that correspond to the mechanisms of complex chemical reactions. This mathematical support and software will become the basis of the wide use of these algorithms.Determination of the basis of routes in the mecha nism of a complex chemical reaction. The development of computer analysis methods can be based on the the ory of stead...

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Graph-theoretical method for determining routes of complex chemical reactions

Cited by 8 publications

References 2 publications

Graph-theoretic analysis of a model for the coupling between photosynthesis and photorespiration

Graph-theoretic analysis of a model for the coupling between photosynthesis and photorespiration

Index method of finding the independent routes of complex chemical reactions

Computer analysis of the graphs of complex chemical reactions

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