Synergetics pp 275-303 | Cite as

# Chemical and Biochemical Systems

## Abstract

- 1)
Several chemical reactants are put together at a certain instant, and we are then studying the processes going on. In customary thermodynamics, one usually compares only the reactants and the final products and observes in which direction a process goes. This is not the topic we want to treat in this book. We rather consider the following situation, which may serve as a model for biochemical reactants.

- 2)
Several reactants are continuously fed into a reactor where new chemicals are continuously produced. The products are then removed in such a way that we have steady state conditions. These processes can be maintained only under conditions far from thermal equilibrium. A number of interesting questions arise which will have a bearing on theories of formation of structures in biological systems and on theories of evolution. The questions we want to focus our attention on are especially the following:

- 1)
Under which conditions can we get certain products in large well-controlled concentrations?

- 2)
Can chemical reactions produce spatial or temporal or spatio-temporal patterns?

- a)
deterministic reaction equations without diffusion

- b)
deterministic reaction equations with diffusion

## Keywords

Transition Rate Master Equation Malonic Acid Detailed Balance Soft Mode## Preview

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## References

## Chemical and Biochemical Systems

- Concentration oscillations were reported as early as 1921 by C. H. Bray: J. Am. Chem. Soc.
**43**, 1262 (1921)CrossRefGoogle Scholar - A different reaction showing oscillations was studied by B. P. Belousov: Sb. ref. radats. med. Moscow (1959)Google Scholar
- This work was extended by Zhabotinsky and his coworkers in a series of papers V. A. Vavilin, A. M. Zhabotinsky, L. S. Yaguzhinsky: Oscillatory Processes in Biological and Chemical Systems (Moscow Science Publ. 1967) p. 181Google Scholar
- This work was extended by Zhabotinsky and his coworkers in a series of papers A. N. Zaikin, A. M. Zhabotinsky: Nature
**225**, 535 (1970)ADSCrossRefGoogle Scholar - This work was extended by Zhabotinsky and his coworkers in a series of papers A. M. Zhabotinsky, A. N. Zaikin: J. Theor. Biol.
**40**, 45 (1973)CrossRefGoogle Scholar - A theoretical model accounting for the occurrence of spatial structures was first given by A. M. Turing: Phil. Trans. Roy. Soc. B
**237**, 37 (1952)ADSCrossRefGoogle Scholar - Models of chemical reactions showing spatial and temporal structures were treated in numerous publications by Prigogine and his coworkers. P. Glansdorff, I. Prigogine: Thermodynamik Theory of Structures, Stability and Fluctuations (Wiley, New York 1971)Google Scholar
- with many references, and G. Nicolis, I. Prigogine: Self-organization in Non-equilibrium Systems (Wiley, New York 1977)Google Scholar
- Prigogine has coined the word “dissipative structures”. Glansdorff and Prigogine base their work on entropy production principles and use the excess entropy production as means to search for the onset of an instability. The validity of such criteria has been critically investigated by R. Landauer: Phys. Rev.
**A12**, 636 (1975). The Glansdorff-Prigogine approach does not give an answer to what happens at the instability point and how to determine or classify the new evolving structures. An important line of research by the Brussels school, namely chemical reaction models, comes closer to the spirit of Synergetics.ADSGoogle Scholar - A review of the statistical aspects of chemical reactions can be found in D. McQuarry: Supplementary Review Series in Appl. Probability (Methuen, London 1967)Google Scholar
- A detailed review over the whole field gives the Faraday Symposium 9: Phys. Chemistry of Oscillatory Phenomena, London (1974)Google Scholar
- A detailed review over the whole field gives the Y. Schiffmann: Phys. Rep.
**64**, 88 (1980)MathSciNetADSCrossRefGoogle Scholar - For chemical oscillations see especially G. Nicolis, J. Portnow: Chem. Rev.
**73**, 365 (1973)CrossRefGoogle Scholar

## Deterministic Processes, Without Diffusion, One Variable. 9.3 Reaction and Diffusion Equations

- We essentially follow F. Schlögl: Z. Phys.
**253**, 147 (1972),ADSCrossRefGoogle Scholar - who gave the steady state solution. The transient solution was determined by H. Ohno: Stuttgart (unpublished)Google Scholar

## React ion-Diffusion Model with Two or Three Variables; the Brusselator and the Oregonator

- We give here our own nonlinear treatment (A. Wunderlin, H. Haken, unpublished) of the reaction-diffusion equations of the “Brusselator” reaction, originally introduced by Prigogine and coworkers, l.c. For related treatments see J. F. G. Auchmuchty, G. Nicolis: Bull. Math. Biol.
**37**, 1 (1974)Google Scholar - We give here our own nonlinear treatment (A. Wunderlin, H. Haken, unpublished) of the reaction-diffusion equations of the “Brusselator” reaction, originally introduced by Prigogine and coworkers, l.c. For related treatments see Y. Kuramoto, T. Tsusuki: Progr. Theor. Phys.
**52**, 1399 (1974)ADSCrossRefGoogle Scholar - We give here our own nonlinear treatment (A. Wunderlin, H. Haken, unpublished) of the reaction-diffusion equations of the “Brusselator” reaction, originally introduced by Prigogine and coworkers, l.c. For related treatments see M. Herschkowitz-Kaufmann: Bull. Math. Biol.
**37**, 589 (1975)CrossRefGoogle Scholar - The “Oregonator” model reaction was formulated and treated by R. J. Field, E. Korös, R. M. Noyes: J. Am. Chem. Soc.
**49**, 8649 (1972)CrossRefGoogle Scholar - The “Oregonator” model reaction was formulated and treated by R. J. Field, R. M. Noyes: Nature
**237**, 390 (1972)ADSCrossRefGoogle Scholar - The “Oregonator” model reaction was formulated and treated by R. J. Field, R. M. Noyes: J. Chem Phys.
**60**, 1877 (1974)ADSCrossRefGoogle Scholar - The “Oregonator” model reaction was formulated and treated by R. J. Field, R. M. Noyes: J. Am. Chem. Soc.
**96**, 2001 (1974)CrossRefGoogle Scholar

## Stochastic Model for a Chemical Reaction Without Diffusion. Birth and Death Processes. One Variable

- A first treatment of this model is due to V. J. McNeil, D. F. Walls: J. Stat. Phys.
**10**, 439 (1974)ADSCrossRefGoogle Scholar

## Stochastic Model for a Chemical Reaction with Diffusion. One Variable

- The master equation with diffusion is derived by H. Haken: Z. Phys.
**B20**, 413 (1975)ADSGoogle Scholar - We essentially follow C. H. Gardiner, K. J. McNeil, D. F. Walls, I. S. Matheson: J. Stat. Phys.
**14**, 4, 307 (1976)ADSCrossRefGoogle Scholar - Related to this chapter are the papers by G. Nicolis, P. Aden, A. van Nypelseer: Progr. Theor. Phys.
**52**, 1481 (1974)ADSCrossRefGoogle Scholar - Related to this chapter are the papers by M. Malek-Mansour, G. Nicolis: preprint Febr. 1975Google Scholar

## Stochastic Treatment of the Brusselator Close to Its Soft Mode Instability

- We essentially follow H. Haken: Z. Phys.
**B20**, 413 (1975)ADSGoogle Scholar

## Chemical Networks

- Related to this chapter are G. F. Oster, A. S. Perelson: Chem. Reaction Dynamics. Arch. Rat. Mech. Anal.
**55**, 230 (1974)MathSciNetCrossRefGoogle Scholar - Related to this chapter are A. S. Perelson, G. F. Oster. Chem. Reaction Dynamics, Part II; Reaction Networks. Arch Rat. Mech. Anal.
**57**, 31 (1974/75)MathSciNetCrossRefGoogle Scholar - with further references. G. F. Oster, A. S. Perelson, A. Katchalsky: Quart. Rev. Biophys.
**6**, 1 (1973)CrossRefGoogle Scholar - O. E. Rössler: In Lecture Notes in Biomathematics, Vol. 4 (Springer, Berlin-Heidelberg-New York 1974) p. 419Google Scholar
- O. E. Rössler: Z. Naturforsch.
**31a**, 255 (1976)Google Scholar