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Field and Noyes (1974), The Oregonator

September 2007, model of the month by Christian Knüpfer
Original model: BIOMD0000000040

Photographs of the progress of a BZ reaction

Photographs showing the progress of a BZ reaction. Picture taken from [6]

If one sees the Belousov-Zhabotinsky (BZ) reaction [1] for the first time in a chemical show, it is both fascinating and unbelievable at the same time: A blue fluid changes its colour to red and afterwards back to blue, again and again! How can this happen?

The BZ reaction is the classical example of a chemical oscillator. It was discovered 1951 by Boris Belousov. The original reaction is the oxidation of malonic acid by bromate ions catalysed by cerium (Ce). A sustained oscillation can be observed between the two states of cerium, Ce3+ and Ce4+. If the reaction is demonstrated today, ferroin is often used instead of cerium. The colour effect is more visible between brick red (Fe2e) and bright blue (Fe3e). [2]

One of the first attempts (1920!) to model such chemical oscillations was done by Lotka [3]. He proposed a hypothetical set of chemical reactions able to exhibit sustained oscillation. But the period and amplitude of the oscillations depended on the initial conditions and they were not stable with respect to perturbations. In other words, the system did not show a stable limit cycle behaviour as observed in the actual BZ reaction. Ilya Prigogine and René Lefever developed a model [4] for chemical oscillation which can exhibit limit cycle behaviour. This model was called "Brusselator" because it was developed in Brussels. In order to overcome the main problem of the Brusselator - the implausible third order of one reaction - a group in Oregon developed BIOMD0000000040, the "Oregonator" [5].

Actually, the BZ reaction is not a biological system and therefore the Oregonator is not, strictly speaking, a "biomodel". But, first lessons in modelling cyclic biological behaviour were learnt through modelling chemical oscillations. So it is worth having a look at the Oregonator model of the fascinating BZ reaction.

The Oregonator is based on a chemical mechanism for the BZ reaction developed by Field, Körös, and Noyes [7]. There are two processes which repeatedly follow each other: process A occurs at high concentrations of bromide, whereas process B occurs below a critical bromide concentration. Because process A consumes bromide and process B produces bromide the system oscillates between the two processes.

From this mechanism, the following abstract reaction scheme of irreversible reactions is derived:

  1. A + Y → X
  2. X + Y → P
  3. B + X → 2X + Z
  4. 2X → Q
  5. Z → fY

This relates to the chemical mechanism by X=HBrO2, Y=Br-, Z=Ce4+. The system is assumed to be open. Therefore the concentrations of the non-intermediates A, B, P, Q can be regarded as constant. The scheme is generalised by means of the variable stoichiometric factor f. This results in a three-dimensional model:

dX/dt = k1AY - k2XY + k3BX - 2k4X2

dY/dt = -k1AY - k2XY + fk5Z

dZ/dt = k3BX - k5Z

with rate constants k1, … k5.

It can be shown numerically that this model indeed exhibits sustained oscillation, which is stable and attracting, i.e. it shows a limit cycle behaviour. For a nice 3D illustration of this behaviour see [8]. In order to prove the existence of a limit cycle analytically and to explore the dependency of this behaviour from parameters, the model can be reduced to two dimensions: It is chemically plausible to assume that bromous acid (HBrO2) is always in steady state, i.e. dX/dt=0. All parameters except f and k5 are given by the chemical mechanism in [7]. For f≈1 this system has a limit cycle for a wide range of the parameter k5.

phase plane plot of log[Ce(IV)] vs log[Br-].

Phase plane plot of log[Ce(IV)] vs log[Br-]. The solid line indicates the limit cycle. Figure taken from [5].

Bibliographic References

  1. http://www.scholarpedia.org/article/Belousov-Zhabotinsky_Reaction
  2. J.D. Murray. Mathematical Biology I - An Introduction. Springer, 3rd edition, 2002.
  3. A.J. Lotka. Undamped oscillations derived from the law of mass action. Journal of the American Chemical Society, 42:1595-1599, 1920.
  4. I. Prigogine and R. Lefever. Symmetry Breaking Instabilities in Dissipative Systems II. Journal of Chemical Physics, 48(4):1695-1700, 1968.
  5. R.J. Field and R.M. Noyes. Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction. Journal of Chemical Physics, 60(5):1877-1884, 1974.
  6. Belousov–Zhabotinsky reaction on Wikipedia
  7. R. J. Field, E. Körös and R. M. Noyes. Oscillations in Chemical Systems II. Thorough analysis of temporal oscillations in the Ce-BrO3- - malonic acid system. Journal of the American Chemical Society, 94: 8649-64, 1972.
  8. http://www.scholarpedia.org/article/Oregonator
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