Oscillating reaction schemes

A well-known oscillating reaction scheme is the Brusselator system, representing a trimolecular model given by... 

Also considered in this chapter are oscillating reactions. These are a class of reactions, still not too numerous, in which the concentrations of the intermediates and the buildup rate of a product fluctuate over time. That is, there are sinusoidal fluctuations in rate and concentration with time. We shall see how these can arise from straightforward, albeit complicated, schemes, often involving the catalysis of one step by a product of another. 

Reaction scheme, defined, 9 Reactions back, 26 branching, 189 chain, 181-182, 187-189 competition, 105. 106 concurrent, 58-64 consecutive, 70, 130 diffusion-controlled, 199-202 elementary, 2, 4, 5, 12, 55 exchange, kinetics of, 55-58, 176 induced, 102 opposing, 49-55 oscillating, 190-192 parallel, 58-64, 129 product-catalyzed, 36-37 reversible, 46-55 termination, 182 trapping, 2, 102, 126 Reactivity, 112 Reactivity pattern, 106 Reactivity-selectivity principle, 238 Relaxation kinetics, 52, 257 -260 Relaxation time, 257 Reorganization energy, 241 Reversible reactions, 46-55 concentration-jump technique for, 52-55... 

The actual schemes of these reactions are very complicated the radicals involved may also react with the metal ions in the system, the hydroperoxide decomposition may also be catalysed by the metal complexes, which adds to the complexity of the autoxidation reactions. Some reactions, such as the cobalt catalysed oxidation of benzaldehyde have been found to be oscillating reactions under certain conditions [48],... 

The schemes considered are only a few of the variety of combinations of consecutive first-order and second-order reactions possible including reversible and irreversible steps. Exact integrated rate expressions for systems of linked equilibria may be solved with computer programs. Examples other than those we have considered are rarely encountered however except in specific areas such as oscillating reactions or enzyme chemistry, and such complexity is to be avoided if at all possible. 

The autocatalytic reaction scheme A + 2B —> 3B, B —> C was introduced in 1983s and has proved itself to be fecund of useful applications in the study of reactor stability and chemical oscillations.6 We shall depart from their notation for we wish to be able to generalize to several species, Au and it is not desirable to use the concentration of A as a reference concentration when it is going to be varied. Similarly, the several species will have different rate constants for the several autocatalytic steps and therefore the first-order rate constant of B — C is most apt for the time scale. 

In this paper I shall discuss, in a general way, some basic dynamical properties of biochemical reaction schemes that are subjected to an external perturbation that oscillates in time. The first part of this paper will deal with the ways in which a weak external oscillating stimulus is able to alter the concentrations of metabolites in a biochemical system. This part will, in particular, consider what happens when the reaction system already oscillates in a limit cycle mode due to non-linearities in its reaction kinetics. It will be shown that such an autonomous biochemical oscillator may exhibit an enhanced sensitivity to a narrow range of externally applied frequencies. 

The Perturbation of Non-linear Reaction Schemes By Oscillating Stimuli... 

For nonlinear reaction schemes, maintained far from chemical equilibrium, a variety of more interesting interactions are possible (2) These include threshold phenomena in which a small transitory external perturbation may induce a permanent change in the steady state concentrations of metabolites. In such a case the magnitude of the change may be independent of that of the stimulus beyond a certain threshold value. Nonlinear reactions may also display a form of resonance when the perturbation oscillates in time. This can be inferred by examining the stability properties of linearized forms of nonlinear reaction schemes (2, 3) A complete description of this form of interaction, however, usually requires numerical computations ( ). I shall now describe the results of some computations in which a nonlinear reaction scheme that is capable of autonomous oscillations was perturbed by an oscillating stimulus applied over a range of frequencies ( ) ... 

An external oscillating perturbation may be introduced into this reaction scheme by allowing one of the rate constants to be a function of time, for example,... 

Some autocatalytic chemical reactions such as the Brusselator and the Belousov-Zhabotinsky reaction schemes can produce temporal oscillations in a stirred homogeneous solution. In the presence of even a small initial concentration inhomogeneity, autocatalytic processes can couple with diffusion to produce organized systems in time and space. 

The net reaction is A I B -> E I F. This reaction scheme has been developed by the Brussels School of Thermodynamics, and consists of a trimolecular collision and an autocatalytic step. This reaction may take place in a well-stirred medium leading to oscillations, or the diffusions of the components A and B may be considered. In the latter case, the system may produce Turing structures. 

The Belousov-Zhabotinsky reaction system is one example leading to such chemical oscillations. One of the interesting phenomena is the effect of the very narrow range of controlling parameter /x on the stability of the Belousov-Zhabotinsky reaction system. The following reactions represent the Belousov-Zhabotinsky reaction scheme ... 

The oxidation step is approximated with the reaction above. Concentration of the organic compounds (B) is assumed constant. Effective stoichiometry (/ ) is a variable, and the oscillations occur when/varies in the range 0.5-2.4. Representative kinetic equations of the Belousov-Zhabotinsky reaction scheme based on Eqs. (13.21)— (13.25) are... 

Lotka-Volterra reaction scheme A + X— 2X X y — 2Y Y— Z structurally unstable oscillation... 

Example 12 The simplest system with time fading auto oscillations the scheme with an autocatalytic reaction and one "buffer" intermediate... 

Just as a negative differential resistance can have different chemical origins, there are different reaction schemes possible that possess the characteristic features of HNDR oscillators. Another hypothetical mechanism discussed in the literature is the potential-dependent adsorption of the reactant, that is, the following reaction steps are considered ... 

At this point, it should be emphasized that in contrast to the lrf+/SCN system in which the physicochemical mechanism is very likely described by the reaction scheme underlying Eq. (16), the individual reaction steps that cause the oscillations during Cu dissolution are far from being... 

In the literature there is a small number of reactions exhibiting oscillations, observed experimentally, which motivated a vast number of studies either devising a model for the reaction scheme or analyzing the small variations thereof. Although oscillatory behavior has been recognized in the past by a handful of chemists, it is recently that oscillatory behavior of chemical systems attracted considerable attention. As a result, studies carried out by various groups of researchers have been reviewed and summarized in review articles. Some of these reviews are more comprehensive than others and cover multiple examples of oscillatory reactions. A partial list of these articles is given in Table II with some annotations. 

In a series of articles, Rossler proposed abstract models which exhibit increasingly complicated oscillations. Most of these reaction schemes are in three dimensions and are accompanied by rate equations in discussing the behavior of the system. 

Termonia and Ross (1981-1,2) developed a reaction scheme coupling phosphofructo-kinase and pyruvate kinase reactions by referring to experimental observations of known activations and inhibitions of enzymes by metabolites. By numerical analysis of the rate equations they confirmed the oscillations in the concentrations of fructose-6-phosphate, pyruvate, phosphoenolpyruvate, fructose 1,6-biphosphate, and ADP. 

This section will illustrate how one can use the implicit method to simulate several interesting kinetic schemes, such as an autocatalytic reaction, and heterogeneous catalysis. Then we will see the ramifications of an often used (and sometimes misused) simplification, the steady-state assumption. Finally, we will simulate a prototypical oscillating reaction. 

The qualitative and, in a a large measure, quantitative agreement of this model with experimental observations should not hide the fact that it is based on a certain number of simplifying assumptions or as yet unverified conjectures as to the precise mechanism of some steps of reaction scheme (5.5). Thus, one or more G-proteins play a role in the activation or inhibition of adenylate cyclase after binding of extracellular cAMP to the active and desensitized receptor states (Van Haastert, 1984 Janssens Van Haastert, 1987 Snaar-Jagalska Van Haastert, 1990) this issue is discussed further below in seetion 5.9. Ca could play a role in the control of adenylate cyclase and could contribute to the termination of a cAMP pulse by inhibiting the enzyme this as yet unverified conjecture is at the basis of an alternative model proposed for cAMP relay and oscillations in D. discoideum (Othmer, Monk Rapp, 1985 Rapp, Monk Othmer, 1985 Monk Othmer, 1989). The latter model, however, does not take into account the phenomenon of desensitization of the cAMP receptor, which plays an essential role here. 

Building on similar ideas but starting from a more detailed reaction scheme, Tyson (1991) proposed a model for the mitotic oscillator based on the formation of a complex between cycUn and cdc2 kinase, followed by the activation of this complex. Essential to the oscillatory mechanism is the assumption that the active complex, i.e. MPF, promotes its own activation in a nonlinear memner. The kinetic equations, of a polynomial form, reduce under some simplifying assumptions to the equations of the two-variable Brusselator model. Inactivation of MPF is not... 

Traditional physical chemistry deals largely with linear systems, ot at least linearises systems that are not already linear—we effectively linearise kinetic rate equations by imposing a steady state assumption. However, many real systems are nonlinear, and cannot be linearised. An obvious example is an oscillating reaction. With numerical integration techniques, linearisation becomes a thing of the past, and kinetic schemes can be explored fully and fruitfully. 

Before we discuss the mechanism of the B-Z reaction, it would be worthwhile to review the earlier hypothetical reaction schemes, which generate oscillations by taking recourse to computer solutions of differential equations. These were pioneering efforts since earlier it was difficult to conceive how a reaction system could generate oscillations. In fact, the principles and paradigms emanating from such analyses paved the way for formulating the reaction mechanism, as we shall see in the next section. 

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