Quasi-steady state, hypothesis

The free-radical concentrations will be small—and the quasi-steady state hypothesis will be justified— whenever the initiation reaction is slow compared with the termination reaction, kj /f[CH3CHO]. 

The rates of the elementary steps can be formulated in a conventional manner, and the quasi-steady state hypothesis is applied to the adsorbed substrate (A ). The... 

Example 4. Let us return to the catalytic isomerization reaction described in example 1 and give it a complete consideration without using the suggestion about the low amount of the catalyst and the quasi-steady state hypothesis (in contrast to example 3). Substances for this reaction are isomers Ai and A2 surface compounds A3 = Z (active size) A4 = A,Z A5 = A2Z. There exist two laws of conservation under conservation are the overall number of isomers (both in the gas and on the surface) and the overall number of active sites... 

So far the quasi-steady-state hypothesis introduced in 1913 has remained the most favourable approach to operating with chemical kinetic equations. In short (and not quite strictly), its most applicable version can be formulated as follows. During the reaction, the concentrations of some (usually intermediate) compounds are the concentration functions of the other (usually observed) substances and "adapt to their values as if they were steady-state values. 

In the intriguingly entitled publication "The steady-state approximation, fact or fiction by Farrow and Edelson [41] presents calculated data on the unsteady-state behaviour of a complex chemical reaction including 81 steps. The reaction mixture consists of 50 substances. Numerical calculation shows a great variety of unsteady-state characteristics of a complex reaction. This variety cannot be interpreted in the narrow framework of the quasi-steady-state hypothesis. Nevertheless, the authors discriminate between the ranges of parameters and time intervals within which this hypothesis is confirmed by numerical experiments. 

Figure 2.3 True solution versus approximation using the quasi-steady-state hypothesis.

When the concentration of the enzyme is much smaller than those of the substrate and product, the enzyme forms evolve much more rapidly than the metabolites. A quasi-steady-state hypothesis can then be made for the enzyme (Heineken, Tsuchiya Aris, 1967 Reich Sel kov, 1974 Segel, 1988). It is useful, at this point, to normalize the concentration of the substrate by dividing it by the dissociation constant for the enzyme in the R state similarly, the concentration of the product is divided by its dissociation constant for the regulatory site of the enzyme in the R state. Thus normalized, the concentrations of substrate and product become dimensionless ... 

The study of the nine-variable system (5.6) is of course rather cumbersome. It would be useful to simplify it by reducing the number of variables. Such a reduction is often possible because certain variables evolve more rapidly than others. A quasi-steady-state hypothesis adopted for the fast variables then allows us to transform the corresponding kinetic equations into algebraic relations. Such an approach was followed in chapter 2 for the reduction of an 11-variable system, obtained in the case of a dimeric enzyme model for glycolytic oscillations, to the form of system (2.7), which contains only two variables. 

Thus the question arises as to whether a further reduction of system (5.12) is possible, which would allow the description of the dynamics of cAMP signalling in terms of two variables only. For sufficiently large values of parameters q, and k, the variation of jS in the course of time is much faster than that of pj and y in the three-variable system (5.12). In such conditions, a quasi-steady-state hypothesis could be justified for )3, whose kinetic equation would then reduce to the algebraic relation ... 

We now require that, after an initial transient phase, the differential equations for the fastest variables Y, c, e, and reduce to algebraic equations corresponding to the quasi-steady-state hypothesis for these receptor and enzymic forms. This condition leads to ... 

The time evolution of the concentrations of the substrate ATP (a), of intracellular (j8) and extracellular (y) cAMP, and of the different complexes formed by adenylate cyclase and by the cAMP receptor is then governed by a system of nine differential equations, as in the slightly different model studied in chapter 5. When a quasi-steady-state hypothesis is adopted for the enzyme-substrate complexes formed by adenylate cyclase in its free (C) and activated (E) states, the dynamics is described by the system of seven differential equations (6.2). In these equations, variables and parameters are defined as in eqns (5.6) (see table 5.3), but for dimensional reasons, /3 and y represent the concentrations of intracellular and extracellular cAMP divided by moreover, c - ( Cr/A d) and - (1 + a) (Martiel Goldbeter, 1984 Goldbeter, Decroly Martiel, 1984). 

In order to reduce the number of variables down to three, it is thus desirable to eliminate some variable other than ATP, if we wish to retain the possibility of complex oscillations. As indicated in chapter 5, a quasi-steady-state hypothesis for variable /3, justified by the large value of parameter q, allows the transformation of the kinetic equation for p into an algebraic relation. It is precisely such a reduction that led... 

Multiple regulation, see Model based on two instability mechanisms Multiple time scales effect on birhythmicity, 108 effect on bursting, 146 effect on oscillations, 66-8,73 see also Quasi-steady-state hypothesis Muscle, glycolytic oscillations, 37,344 Mutants of circadian rhythm, see frq gene per gene tim... 

Quantal Ca release, 391 Quasi-steady-state hypothesis, 47,184,214... 

For rapidly reacting intermediates, the quasi-steady-state hypothesis can be applied to eliminate the concentrations of the intermediates from the rate equations. For rapid reaction steps, the quasi-equilibrium hypothesis is used to eliminate the concentrations of the intermediates. 

FIGURE 2.1 Quasi-steady-state hypothesis applied to the reaction system A where R is a rapidly reacting intermediate in a batch reactor. 

If the reaction mechanism is nonlinear with respect to the intermediates, the solution of Equation 2.18 becomes more complicated and an iterative procedure is applied in most cases. It should be noticed that an assumption of each rapid intermediate reduces the number of adjustable rate parameters by one. For example, the application of the quasi-steady-state hypothesis in the system A R S implies that... 

From the general solution obtained with the quasi-steady-state hypothesis, the solutions corresponding to the quasi-equilibrium hypothesis can be obtained as special cases. If step I is much more rapid than step II, fc i fc+2CB in Equation 2.27, the reaction rate becomes... 

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