Foundations of Chemical Reaction Network Theory pp 153-204 | Cite as

# Concordant Reaction Networks: Architectures That Promote Dull, Reliable Behavior Across Broad Kinetic Classes

Chapter

First Online:

## Abstract

In this chapter we’ll resume a narrative that ran through Chapters 7 and 8. There our interest was largely in the relationship between the structure of a reaction network and its capacity (or lack of it) to give rise to instabilities or multiple equilibria within a positive stoichiometric compatibility class. The principal theorems centered mostly on a reaction network’s deficiency or, in some cases, on the deficiencies of the network’s linkage classes. In this chapter, however, deficiencies will play almost no role at all.

## References

- 9.Aris, R.: Introduction to the Analysis of Chemical Reactors. Prentice-Hall, Inc., Englewood Cliffs, NJ (1965)Google Scholar
- 12.Arnold, V.I.: Ordinary Differential Equations. Springer, New York (1992)Google Scholar
- 14.Banaji, M., Craciun, G.: Graph-theoretic approaches to injectivity and multiple equilibria in systems of interacting elements. Communications in Mathematical Sciences
**7**(4), 867–900 (2009)MathSciNetCrossRefGoogle Scholar - 16.Banaji, M., Donnell, P., Baigent, S.: P-matrix properties, injectivity, and stability in chemical reaction systems. SIAM Journal on Applied Mathematics
**67**(6), 1523 (2007)MathSciNetCrossRefGoogle Scholar - 44.Craciun, G.: Systems of nonlinear differential equations deriving from complex chemical reaction networks. Ph.D. thesis, The Ohio State University (2002)Google Scholar
- 46.Craciun, G., Feinberg, M.: Multiple equilibria in complex chemical reaction networks. I. The injectivity property. SIAM Journal on Applied Mathematics
**65**(5), 1526–1546 (2005)MathSciNetCrossRefGoogle Scholar - 47.Craciun, G., Feinberg, M.: Multiple equilibria in complex chemical reaction networks: extensions to entrapped species models. IEE Proc. Syst. Biol
**153**(4), 179–186 (2006)CrossRefGoogle Scholar - 48.Craciun, G., Feinberg, M.: Multiple equilibria in complex chemical reaction networks. II. The species-reaction graph. SIAM Journal on Applied Mathematics
**66**(4), 1321–1338 (2006)CrossRefGoogle Scholar - 49.Craciun, G., Feinberg, M.: Multiple equilibria in complex chemical reaction networks: semi-open mass action systems. SIAM Journal on Applied Mathematics
**70**(6), 1859–1877 (2010)MathSciNetCrossRefGoogle Scholar - 52.Craciun, G., Tang, Y., Feinberg, M.: Understanding bistability in complex enzyme-driven reaction networks. Proceedings of the National Academy of Sciences
**103**(23), 8697–8702 (2006)CrossRefGoogle Scholar - 53.De Kepper, P., Boissonade, J.: Theoretical and experimental analysis of phase diagrams and related dynamical properties in the Belousov–Zhabotinskii system. The Journal of Chemical Physics
**75**(1), 189–195 (1981)CrossRefGoogle Scholar - 62.Ellison, P., Ji, H., Knight, D., Feinberg, M.: The Chemical Reaction Network Toolbox, Version 2.3 (2014). Available at crnt.osu.edu
- 71.Feinberg, M.: Lectures on Chemical Reaction Networks (1979). Written version of lectures given at the Mathematical Research Center, University of Wisconsin, Madison, WI Available at http://crnt.osu.edu/LecturesOnReactionNetworks
- 84.Feliu, E., Wiuf, C.: Preclusion of switch behavior in networks with mass-action kinetics. Applied Mathematics and Computation
**219**(4), 1449–1467 (2012)MathSciNetCrossRefGoogle Scholar - 89.Geiseler, W., Bar-Eli, K.: Bistability of the oxidation of cerous ions by bromate in a stirred flow reactor. The Journal of Physical Chemistry
**85**(7), 908–914 (1981)CrossRefGoogle Scholar - 94.Golubitsky, M., Langford, W.F.: Classification and unfoldings of degenerate Hopf bifurcations. Journal of Differential Equations
**41**(3), 375–415 (1981)MathSciNetCrossRefGoogle Scholar - 95.Golubitsky, M., Schaeffer, D.: Singularities and Groups in Bifurcation Theory: Volume I. Springer, New York (1984)zbMATHGoogle Scholar
- 97.Graziani, K.R., Hudson, J.L., Schmitz, R.A.: The Belousov-Zhabotinskii reaction in a continuous flow reactor. The Chemical Engineering Journal
**12**(1), 9–21 (1976)CrossRefGoogle Scholar - 99.Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (2002)zbMATHGoogle Scholar
- 104.Hirsch, M.W., Smale, S., Devaney, R.L.: Differential Equations, Dynamical Systems, and an Introduction to Chaos, Third Edition. Academic Press (2012)zbMATHGoogle Scholar
- 107.Horn, F.: Necessary and sufficient conditions for complex balancing in chemical kinetics. Archive for Rational Mechanics and Analysis
**49**(3), 172–186 (1972)MathSciNetCrossRefGoogle Scholar - 109.Horn, F., Jackson, R.: General mass action kinetics. Archive for Rational Mechanics and Analysis
**47**(2), 81–116 (1972)MathSciNetCrossRefGoogle Scholar - 111.Hudson, J.L., Mankin, J.C.: Chaos in the Belousov - Zhabotinskii reaction. The Journal of Chemical Physics
**74**(11), 6171–6177 (1981)CrossRefGoogle Scholar - 115.Ji, H.: Uniqueness of equilibria for complex chemical reaction networks. Ph.D. thesis, The Ohio State University (2011)Google Scholar
- 117.Knight, D.: Reactor behavior and its relation to chemical reaction network structure. Ph.D. thesis, The Ohio State University (2015)Google Scholar
- 118.Knight, D., Shinar, G., Feinberg, M.: Sharper graph-theoretical conditions for the stabilization of complex reaction networks. Mathematical Biosciences
**262**(1), 10–27 (2015)MathSciNetCrossRefGoogle Scholar - 122.Lee, E., Salic, A., Krüger, R., Heinrich, R., Kirschner, M.W.: The roles of APC and axin derived from experimental and theoretical analysis of the Wnt pathway. PLoS Biology
**1**(1), e10 (2003)CrossRefGoogle Scholar - 123.Leib, T., Rumschitzki, D., Feinberg, M.: Multiple steady states in complex isothermal CFSTRs. I: General considerations. Chemical Engineering Science
**43**(2), 321–328 (1988)Google Scholar - 138.Rawlings, J., Ekerdt, J.: Chemical Reactor Analysis and Design Fundamentals, 2nd edn. Nob Hill Publishing, Madison, Wisconsin (2013)Google Scholar
- 141.Reidl, J., Borowski, P., Sensse, A., Starke, J., Zapotocky, M., Eiswirth, M.: Model of calcium oscillations due to negative feedback in olfactory cilia. Biophysical Journal
**90**(4), 1147–1155 (2006)CrossRefGoogle Scholar - 144.Rumschitzki, D.: On the theory of multiple steady states in isothermal CSTR’s. Ph.D. thesis, University of California, Berkeley [Work performed at the University of Rochester] (1983)Google Scholar
- 145.Rumschitzki, D., Feinberg, M.: Multiple steady states in complex isothermal CFSTRs II. Homogeneous reactors. Chemical Engineering Science
**43**(2), 329–337 (1988)CrossRefGoogle Scholar - 147.Schlosser, P.M.: A graphical determination of the possibility of multiple steady states in complex isothermal CFSTRs. Ph.D. thesis, University of Rochester (1988)Google Scholar
- 148.Schlosser, P.M., Feinberg, M.: A theory of multiple steady states in isothermal homogeneous CFSTRs with many reactions. Chemical Engineering Science
**49**(11), 1749–1767 (1994)CrossRefGoogle Scholar - 149.Schmitz, R.A., Graziani, K.R., Hudson, J.L.: Experimental evidence of chaotic states in the Belousov-Zhabotinskii reaction. The Journal of Chemical Physics
**67**(7), 3040–3044 (1977)CrossRefGoogle Scholar - 151.Seydel, R.: Practical Bifurcation and Stability Analysis, 3rd edn. Springer, New York (2009)zbMATHGoogle Scholar
- 157.Shinar, G., Feinberg, M.: Concordant chemical reaction networks. Mathematical Biosciences
**240**(2), 92–113 (2012)MathSciNetCrossRefGoogle Scholar - 158.Shinar, G., Feinberg, M.: Concordant chemical reaction networks and the species-reaction graph. Mathematical Biosciences
**241**(1), 1–23 (2013)MathSciNetCrossRefGoogle Scholar - 164.Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Second Edition. Westview Press (2014)zbMATHGoogle Scholar
- 166.Tyson, J., Kauffman, S.: Control of mitosis by a continuous biochemical oscillation: synchronization; spatially inhomogeneous oscillations. Journal of Mathematical Biology
**1**(4), 289–310 (1975)CrossRefGoogle Scholar - 177.Wiuf, C., Feliu, E.: Power-law kinetics and determinant criteria for the preclusion of multistationarity in networks of interacting species. SIAM Journal on Applied Dynamical Systems
**12**(4), 1685–1721 (2013)MathSciNetCrossRefGoogle Scholar

## Copyright information

© Springer Nature Switzerland AG 2019