Abstract
There are a series of graphs which are able to represent chemical reactions so as to lead to consequences on their (static or dynamic) behavior. We start with the Feinberg–Horn–Jackson graph, the one which may be the most well known for the chemists. Strictly connected to this concept is reversibility and weak reversibility and also the concept of central importance: deficiency. The Volpert graph is almost as well known, especially for those involved in metabolism research. Indexing of the Volpert graph has relevant consequences on the time evolution of the concentrations. Relationships between the two graphs are studied. Finally, the species–complex–linkage class graph and the complex graph are finally introduced. Further graphs (such as the influence diagram, the graph of atomic fluxes, the state space of the stochastic model) even more closely related to time-dependent behavior can and will only be introduced in later chapters.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Banaji M, Craciun G (2009) Graph-theoretic approaches to injectivity and multiple equilibria in systems of interacting elements. Commun Math Sci 7(4):867–900
Banaji M, Craciun G (2010) Graph-theoretic criteria for injectivity and unique equilibria in general chemical reaction systems. Adv Appl Math 44(2):168–184
Brochot C, Tóth J, Bois FY (2005) Lumping in pharmacokinetics. J Pharmacokinet Pharmacodyn 32(5–6):719–736
Busacker RG, Saaty TL (1965) Finite graphs and networks. McGraw-Hill, New York
Crăciun G (2002) Systems of nonlinear equations deriving from complex chemical reaction networks. PhD thesis, Department of Mathematics, The Ohio State University, Columbus, OH
Crăciun G, Feinberg M (2006) Multiple equilibria in complex chemical reaction networks: Ii. The species reaction graph. SIAM J Appl Math 66:1321–1338
De Leenheer P, Angeli D, Sontag ED (2006) Monotone chemical reaction networks. J Math Chem 41(3):295–314
Donnell P, Banaji M (2013) Local and global stability of equilibria for a class of chemical reaction networks. SIAM J Appl Dyn Syst 12(2):899–920
Edelstein BB (1970) Biochemical model with multiple steady states and hysteresis. J Theor Biol 29(1):57–62
Epstein I, Pojman J (1998) An introduction to nonlinear chemical dynamics: oscillations, waves, patterns, and chaos. Topics in physical chemistry series. Oxford University Press, New York. http://books.google.com/books?id=ci4MNrwSlo4C
Feinberg M (1972) Lectures on chemical reaction networks. http://www.che.eng.ohio-state.edu/~feinberg/LecturesOnReactionNetworks/
Feinberg M, Horn FJM (1977) Chemical mechanism structure and the coincidence of the stoichiometric and kinetic subspaces. Arch Ratl Mech Anal 66(1):83–97
Goss PJE, Peccoud J (1998) Quantitative modeling of stochastic systems in molecular biology using stochastic Petri nets. Proc Natl Acad Sci USA 95:6750–6755
Györgyi L, Field RS (1991) Simple models of deterministic chaos in the Belousov–Zhabotinskii reaction. J Phys Chem 95(17):6594–6602
Harary F (1969) Graph theory. Addison–Wesley, Reading
Horn F (1973a) On a connexion between stability and graphs in chemical kinetics. II. Stability and the complex graph. Proc R Soc Lond A 334:313–330
Horn F (1973b) Stability and complex balancing in mass-action systems with three short complexes. Proc R Soc Lond A 334:331–342
Ivanciuc O, Balaban AT (1998) Graph theory in chemistry. In: Schleyer PvR, Allinger NL, Clark T, Gasteiger J, Kollman PA, Schaefer HFI, Schreiner PR (eds) Encyclopedia of computational chemistry. Wiley, Chichester, pp 1169–1190
Jacquez J (1999) Modeling with compartments. BioMedware, Ann Arbor, MI
Kiss IZ, Miller JC, Simon PL (2017) Mathematics of epidemics on networks. From exact to approximate models. Springer, Cham
Lovász L (2007) Combinatorial problems and exercises. AMS Chelsea Publishing, Providence, RI
Nagy I, Tóth J (2012) Microscopic reversibility or detailed balance in ion channel models. J Math Chem 50(5):1179–1199
Ohtani B (2011) Photocatalysis by inorganic solid materials: revisiting its definition, concepts, and experimental procedures. Adv Inorg Chem 63:395–430
Øre O (1962) Theory of graphs. AMS Colloquium Publications, vol 38. AMS, Providence
Othmer HG (1981) A graph-theoretic analysis of chemical reaction networks. I. Invariants, network equivalence and nonexistence of various tpes of steady states. Course notes, 1979
Pólya G (1937) Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen. Acta Math 68(1):145–254
Rácz I, Gyarmati I, Tóth J (1977) Effect of hydrophilic and lipophilic surfactant materials on the salicylic-acid transport in a three compartment model. Acta Pharm Hung 47:201–208 (in Hungarian)
Schlosser PM, Feinberg M (1994) A theory of multiple steady states in isothermal homogeneous CFSTRs with many reactions. Chem Eng Sci 49(11):1749–1767
Siegel D, Chen YF (1995) The S-C-L graph in chemical kinetics. Rocky Mt J Math 25(1):479–489
Tóth J, Nagy AL, Zsély I (2015) Structural analysis of combustion mechanisms. J Math Chem 53(1):86–110
Volpert AI (1972) Differential equations on graphs. Mat Sb 88(130):578–588
Volpert AI, Hudyaev S (1985) Analyses in classes of discontinuous functions and equations of mathematical physics. Martinus Nijhoff Publishers, Dordrecht. Russian original: 1975
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Science+Business Media, LLC, part of Springer Nature
About this chapter
Cite this chapter
Tóth, J., Nagy, A.L., Papp, D. (2018). Graphs of Reactions. In: Reaction Kinetics: Exercises, Programs and Theorems. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-8643-9_3
Download citation
DOI: https://doi.org/10.1007/978-1-4939-8643-9_3
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-8641-5
Online ISBN: 978-1-4939-8643-9
eBook Packages: Chemistry and Materials ScienceChemistry and Material Science (R0)