Chemical Oscillators are some of the most complicated and least explored chemical systems in existence. Their rate equations are immensely complex and it is impossible to solve such systems analytically. The dynamical equations governing such systems are essentially nonlinear in character and the only way is to resort to numerical techniques.
In this article, after delving into some history, we shall carry out a complete nonlinear analysis of one such system and also take a look at Glycolysis. We have tried to keep it as less mathematical as possible. Nonetheless, the interested can refer to the materials provided at the end.
History:
The 2nd Law of Thermodynamics is the holiest of the laws of physics. Mathematically it takes the form,
dQ/T ≤ dS
For an isolated system, the LHS is zero and we have 0 ≤ dS, with equality holding only for a reversible process. This means that for an irreversible process, the entropy of the system must always increase towards a maximum value Smax at which equilibrium is attained. Isolated chemical oscillating systems violate the 2nd Law. Self-sustained oscillations of an isolated chemical system involve deviations from the eventual equilibrium conditions, which is against what the 2nd law mandates. On account of this even in the first half of the 20th century, the majority of the scientific community believed that it was impossible for chemical oscillators to exist, even in non-isolated systems, as it was thought that equilibrium is always attained monotonically. Nonetheless, such reactions had been demonstrated by a few researchers, although their findings never garnered the attention of the scientific community. One such scientist was Boris Belousov.
Boris Belousov was a Soviet biochemist. In the early 1950s, while trying to artificiate the Krebs Cycle in his lab, he mixed citric acid and bromate ions in a solution of sulfuric acid, catalyzed by cerium and observed that the solution first became yellow, then colourless, then yellow once again with a period of one minute and this carried on for nearly half an hour before the mixture settled to equilibrium. Belousov’s paper was rejected by one journal after another. However, his reaction became quite famous, despite being mocked at by the scientific community. In 1961, a graduate student named Zhabotinsky took up the problem, and in 1968 showed that Belousov was indeed right. The reaction came to be known as the BZ reaction and in 1980, Belousov and Zhabotinsky jointly received the Lenin Prize, the Soviet Union’s highest medal. Belousov had passed away ten years earlier. He was a legend.
The physics behind oscillating chemical reactions is quite complicated. Such reactions cannot be explained by Classical Equilibrium Thermodynamics. Although, isolated chemical oscillators can never exist, in tandem with the 2nd Law, open and closed systems do not violate the physical laws, provided we use Non-Equilibrium Thermodynamics to explain them. The foundations of Non-Equilibrium Thermodynamics were laid down by Ilya Prigogine and is yet a developing discipline. The earlier belief that equilibrium is attained monotonically is not true. Also, chemical oscillations can be seen in such systems which are prevented from attaining equilibrium through a continuous supply of reactants and energy. The best example of such systems are organisms, including us. In such biological systems, equilibrium is never attained as a continuous supply of reactants is always maintained through the nutrient intake. If equilibrium is reached, the organism shall perish.
The BZ reaction is extremely complex and involves more than twenty elementary steps. However, most of these equilibrate and the analysis can be reduced to three steps only. See Tyson, 1985.
The remainder of the article is dedicated to carrying out a mathematical analysis of an oscillating system and drawing a parallel with Glycolysis.
Chlorine Dioxide-Iodine-Malonic Acid Reaction:
Lengyel and his colleagues in 1990 proposed this model of an elegant oscillating chemical system. See Lengyel et al, 1990, for a derivation of the rate equations. The following three equations capture the behaviour of the system,
The three equations above cannot be solved by analytical means. A nonlinear analysis can be carried out, but being a three-dimensional system it’s quite involved. A further simplification can be carried out, by observing that the concentrations of MA, I2 and ClO2, vary quite slowly during the reaction and can be treated as constants wrt I– and ClO2--. The resulting equations after non-dimensionalization are,
Here x is the concentration of I– and y denotes the concentration of ClO2-. The analysis of this 2-D system is quite easy. A phase plane analysis means that we can easily do away with any possibility of chaos and it is also easier to analyze the stability of any fixed point. Further, we shall show that there exists a stable limit cycle, using the Poincare-Bendixson Theorem. For those unfamiliar with the term, limit cycles are isolated, closed orbits, and are the characteristics of all oscillating systems.
A fixed point is a point in the phase space of the system, such that if the system is initially at a fixed point, it continues to stay there indefinitely, provided it is not perturbed. They are obtained by setting the time derivatives to zero. For our system, the fixed points are x*=a/5 and y* = 1+(x*)^2 = 1+(a/5)^2 . A fixed point is stable if a slight perturbation brings the system back to the fixed point. To analyze the stability we linearise the system about (x*,y*) via a Taylor Expansion. The resulting Jacobian Matrix has a determinant, Δ, and a trace, τ, given by,
As Δ>0, the fixed point can never be a saddle(one stable direction, and another unstable direction). For the fixed point to be a repeller(unstable), must be greater than zero which means b<bc = (3a/5 – 25/a). For b>bc, the fixed point is stable and the system eventually spirals into it, reaching equilibrium.
We now construct a trapping region as shown in the figure below(a trapping region is a region in the phase space such that, anywhere on its boundary the flow is directed into the region).
The existence of a stable limit cycle for b < bc is guaranteed by the Poincare-Bendixson Theorem, which is applied to the entirety of the trapping region minus the fixed point. Its proof is highly intricate and this theorem is valid only for two dimensional systems.
To verify our theoretical predictions we have numerically integrated the system using a 5th order Runge Kutta module in python for 100 seconds, with a=10 yielding bc =3.5. The relevant plots for two b values are given below.
Plots for b=2.0. Notice the limit cycle and the periodic oscillations of x and y.
Plots for b = 4.0 . Notice the equilibrium concentrations and the stability of the fixed point.
We, therefore, conclude that our numerical simulations are in tandem with our theoretical predictions. It is quite fascinating that we have been able to deduce the behaviour of the system, solely through simple calculus and graphical methods.
The final section draws a parallel with the biological process of Glycolysis.
Glycolysis:
A simple model for oscillations of the chemical components involved in glycolysis was proposed by Sel’kov in 1968. The governing equations in dimensionless forms are,
Here is the x concentration of ADP and y is that of F6P (fructose -6 – phosphate), with and being kinetic parameters. The analysis is similar to the one carried out for the previous system. The above equations were integrated using an RK-5 module for 100 seconds with a=0.08 and b=0.6. The phase plane diagram is plotted below.
From the phase plane plot, we see that the system eventually approaches a stable limit cycle whereby the concentrations of F6P and ADP oscillate periodically. The dynamics is indeed parameter dependent.
Conclusion:
Chemical oscillators are therefore at the intersection of chemistry, physics, and mathematics. These systems can be biological as well as artificial. As we saw, these systems are quite difficult to analyze and their physical properties are not fully understood. Although this article does not include any discussion regarding the thermodynamic potentials of the system, it is imperative that we mention that the Gibbs Free Energy of the system must be in the form of a double-well potential. It is truly fascinating that it is these systems that drive most if not all life on earth.
References:
- Nonlinear Dynamics and Chaos-Strogatz.
- Oscillating Chemical Reactions-Epstein; Kustin; De Kepper and Orban.
- Chemical Oscillator-Wikipedia.org,
By Soumyajit Chatterjee, Department of Physics, IISER Bhopal.
About the author: Soumyajit is presently a 4th-year physics major at IISER Bhopal. His areas of interest include Non-Linear Dynamics, Quantum Information Theory, Statistical Mechanics, Machine Learning and Relativistic Dynamics. Besides being a lover of physics, he is passionately zealous about the armed forces of India and wishes to be a part of it someday. He likes to sketch, dance and code and plays football, badminton and carrom. Soumyajit is a great fan of Batman, John Constantine and Phir Hera Pheri.
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