The Quasi-Steady State Hypothesis

Many reactions involve short-lived intermediates that are so reactive that they never accumulate in large quantities and are difficult to detect. Their presence is important in the reaction mechanism and may dictate the functional form of the rate equation. Consider the following reaction  

This system contains only first-order steps. An exact but somewhat cumbersome analytical solution is available. 

Suppose that B is highly reactive. When formed, it rapidly reverts back to A or transforms into C. This implies kr kf and kB kf. The quasi-steady hypothesis assumes that B is consumed as fast as it is formed so that its time rate of change is zero. More specifically, we assume that the concentration of B rises quickly and achieves a dynamic equilibrium with A, which is consumed at a much slower rate. To apply the quasi-steady hypothesis to component B, we set dbjdt = 0. The ODE for B then gives 

The quasi-steady hypothesis is used when short-lived intermediates are formed as part of a relatively slow overall reaction. The short-lived molecules are hypothesized to achieve an approximate steady state in which they are created at nearly the same rate that they are consumed. Their concentration in this quasi-steady state is necessarily small. A typical use of the quasi-steady 

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

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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. 

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 ... 

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. 

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... 

This procedure constitutes an application of the steady-state approximation [also called the quasi-steady-state approximation, the Bodenstein approximation, or the stationary-state hypothesis]. It is a powerful method for the simplification of complicated rate equations, but because it is an approximation, it is not always valid. Sometimes the inapplicability of the steady-state approximation is easily detected for example, Eq. (3-143) predicts simple first-order behavior, and significant deviation from this behavior is evidence that the approximation cannot be applied. In more complex systems the validity of the steady-state approximation may be difficult to assess. Because it is an approximation in wide use, much critical attention has been directed to the steady-state hypothesis. 

As for the quasi (pseudo)-steady-state case, the basic assumption in deriving kinetic equations is the well-known Bodenshtein hypothesis according to which the rates of formation and consumption of intermediates are equal. In fact. Chapman was first who proposed this hypothesis (see in more detail in the book by Yablonskii et al., 1991). The approach based on this idea, the Quasi-Steady-State Approximation (QSSA), is a common method for eliminating intermediates from the kinetic models of complex catalytic reactions and corresponding transformation of these models. As well known, in the literature on chemical problems, another name of this approach, the Pseudo-Steady-State Approximation (PSSA) is used. However, the term "Quasi-Steady-State Approximation" is more popular. According to the Internet, the number of references on the QSSA is more than 70,000 in comparison with about 22,000, number of references on PSSA. 

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 ... 

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... 

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

Such a scheme for the catalytic isomerization of n-butenes over A1203 has been studied in detail previously [11]. Each reaction has a rate that is a function of both the gas composition and the surface state. In this case the assumption that the concentration of surface intermediates on the catalyst is a function of the gas composition is often used. It is a hypothesis about a quasi-steady state that is considered in detail in what follows. According to this hypothesis, for the reaction under study there exist three functions of the gas composition, w1w2, and w3, so that the kinetic equations can be written as... 

In an attempt to better understand the role of receptor internalization, a minimal model has been developed using hypothesis testing [10]. The model is based on experimental data on autophosphorylation of the IR. Upon addition of insulin to intact adipocytes, the IR rapidly autophosphorylates with an overshoot peak before t = 0.9 min, and then slowly declines to a quasi-steady state at around 15 min. 

By application of the steady-state hypothesis, a site balance and the assumption that the surface dissociation is rate determining while the other steps are in quasi-equilibrium, the following rate expression is derived ... 

This two-variable system (Goldbeter et al, 1978) presents the additional advantage of being formally identical with the system of eqns (2.7) studied in chapter 2 for glycolytic oscillations. This similarity stems from the basic structure common to the two models a substrate, injected at a constant rate, is transformed in a reaction catalysed by an allosteric enzyme activated by the reaction product. In the cAMP-synthesizing system in D. discoideum, activation is indirect as extracellular cAMP enhances the synthesis of intracellular cAMP, which is then transported into the extracellular medium. However, the hypothesis of a quasi-steady state for intracellular cAMP is tantamount to considering that the variation of )8 is so fast that the enzyme is, de facto, activated directly by its apparent product, extracellular cAMP. 

As indicated in table 5.2, if the value of parameter q is sufficiently high to justify the hypothesis of a quasi-steady state for variable /3, the situation is somewhat different for the experimental values of k, and ki-The latter are too low for the hypothesis to hold rigorously. It is for this reason that the preceding results on oscillations and relay were obtained by means of the three-variable system (5.12). It is nevertheless interesting to study the behaviour of the two-variable system (5.15), in order to provide an estimate for the error made upon reducing the number of variables to only two. This error could well turn out to be minute, and thus acceptable, in comparison with the advantage provided by the possibility of resorting to the phase plane analysis of a two-variable system. 

The growing radicals are very reactive intermediate species that conform to what in chemical kinetics is called a quasi-steady state (QSS) or stationary state hypothesis [11, 12]. This means that the rate of formation and consumption of that species become nearly equal in a very short timescale as a consequence, the absolute value of the derivative becomes very small and negligible compared with the derivatives of other species in the system (e.g., d[M]/df) and for practical purposes can be approximated as zero. Note, however, that this does not imply constancy of the value of [P], as sometimes interpreted by some authors, but this will be more clear later. By making the QSS approximation in Equation 4.5... 

Tip 9 Radical stationary state hypothesis. A few practical steps to check for the validity of the (quasi-) stationary (steady-) state hypothesis (QSSH or simply SSH) for radicals are as follows (i) determine the rate of change (with time) of the total radical concentration (ii) find the maximum rate in (i) (c) divide (ii) by the rate of initiation and (iv) if the ratio is much less than unity, the QSSH (or QSSA, A here stands for assumption) is valid. 

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