Quasi steady state approximation

The quasi steady state approximation is a powerful method of transforming systems of very stiff differential equations into non-stiff problems. It is the most important, although somewhat contradictive technique in chemical kinetics. Before a general discussion we present an example where the approximation certainly applies. 

Example 5.4A Detailed model of the fumarase reaction The basic mechanism of enzyme reactions is 

With the initial step size H = .1 s M72 gives the following results  

The enzyme - substrate complex concentration reaches its maximum value in a very short time, and decays very slowly afterwards. To explain this special behavior of the concentration [ES], write its kinetic equation in the form 

Replacing (5.52) by the algebraic equation (5.54) we can solve (5.54) for [ES]ss, the quasi steady state concentration of the enzyme - substrate. The solution depends on the actual value [P], therefore [ES]ss is not at all constant, and hence the usual equation 

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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. 

Develop a suitable rate expression using the Michaelis-Menten rate equation and the quasi-steady-state approximations for the intermediate complexes formed. 

Frequently function R can be written as a single term having the simple form of equation 1. For Instance, with the aid of the long chain approximation (LCA) and the quasi-steady state approximation ((JSSA), the rate of monomer conversion, I.e., the rate of polymerization, for many chain-addition polymerizations can be written as... 

Hie quasi steady state approximation can be conveniently applied to equations 19 to 21, without any significant loss of accuracy, due to tlie high reactivity of tlie reacting species in aqueous solution. Hms, the system of ordinary differential equations is readily reduced to a system of algebraic non linear equations. 

The assumption made is called the quasi-steady-state approximation (QSSA). It is valid here mainly because of the great difference in densities between the reacting species (gaseous A and solid B). For liquid-isolid systems, this simplification cannot be made. 

Using the quasi-steady-state approximation and the conservation of total enzyme Et = E) + [ZsS], the concentration of the complex is given as... 

A. Ciliberto, F. Capuani, and J. J. Tyson2, Modeling networks of coupled enzymatic reactions using the total quasi steady state approximation. PLoS Comput. Biol. 3(3), e45 (2007). 

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. 

The quasi-steady-state approximation then allows the concentration time derivatives to be set equal to zero,... 

Simulations show negligible differences in the transient temperature and concentration profiles as a result of this quasi-steady-state approximation. The major advantage of this assumption should be apparent in control system design, where a reduction in the size of the state vector is computationally beneficial or in the time-consuming simulations of the full nonlinear model. 

This ratio provides the criterion for the applicability of the quasi-steady-state approximation for the concentration as shown in Table V. 

The first two sections of Chapter 5 give a practical introduction to dynamic models and their numerical solution. In addition to some classical methods, an efficient procedure is presented for solving systems of stiff differential equations frequently encountered in chemistry and biology. Sensitivity analysis of dynamic models and their reduction based on quasy-steady-state approximation are discussed. The second central problem of this chapter is estimating parameters in ordinary differential equations. An efficient short-cut method designed specifically for PC s is presented and applied to parameter estimation, numerical deconvolution and input determination. Application examples concern enzyme kinetics and pharmacokinetic compartmental modelling. 

Existence and uniqueness of the particular solution of (5.1) for an initial value y° can be shown under very mild assumptions. For example, it is sufficient to assume that the function f is differentiable and its derivatives are bounded. Except for a few simple equations, however, the general solution cannot be obtained by analytical methods and we must seek numerical alternatives. Starting with the known point (tD,y°), all numerical methods generate a sequence (tj y1), (t2,y2),. .., (t. y1), approximating the points of the particular solution through (tQ,y°). The choice of the method is large and we shall be content to outline a few popular types. One of them will deal with stiff differential equations that are very difficult to solve by classical methods. Related topics we discuss are sensitivity analysis and quasi steady state approximation. 

The following treatment applies to the case where the solids are stationary in a shallow packed bed, so that they can be considered to be in well-mixed conditions, and that the solute initially saturates the solid, as in the case of vegetable oil in crushed seeds. For the quasi--steady-state approximation, Brunner [51] derived a practical equation ... 

A. Quasi-Stationary and Quasi-Steady-State Approximations. 105... 

The governing equation is therefore identical with that for the irrotational flow of an ideal fluid through a circular aperture in a plane wall. The stream lines and equipotential surfaces in this rotationally symmetric flow turn out to be given by oblate spheroidal coordinates. Since, from Eq. (157), the rate of deposition of filter cake depends upon the pressure gradient at the surface, the governing equation and boundary conditions are of precisely the same form as in the quasi-steady-state approximation... 

A quasi-steady-state approximation may be used to describe the variation of the polymer thickness with time and hence to calculate the time for onset of disintegration ... 

At bg -> oo, reactions involving the participation of the greatest number of gas molecules (k = raax) become predominant. When choosing a new time scale t = we can go to the quasi-steady-state approximation at s - 0... 

Consequently, starting from some sufficiently low bs, the quasi-steady-state approximation can be applied after a certain period of time ("boundary layer )... 

At e = SI V -> 0, the Tikhonov theorem is applicable, hence starting from sufficiently small , we can use the quasi-steady-state approximation. 

Of great importance is the fact that the quasi-steady-state approximation is the solution asymptote of the initial system at e -> 0, but it is applied at finite e. To establish a starting value from which this approximation can be used with the prescribed accuracy is a rather difficult problem in each particular case. 

Fig. 15.6(c)]. At the center of each element there is a node. The nodes of adjacent elements are interconnected hy links. Thus, the total flow field is represented by a network of nodes and links. The fluid flows out of each node through the links and into the adjacent nodes of the network. The local gap separation determines the resistance to flow between nodes. Making the quasi-steady state approximation, a mass (or volume) flow rate balance can be made about each node (as done earlier for one-dimensional flow), to give the following set of algebraic equations... 

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