Quasi Steady-State Approximation Analysis

One of the fundamental assumptions made in deriving basic Michaelis-Menten kinetics, except in the initial so-called transient phase of the reaction, is the quasi steady state approximation of the [ 5] concentration, i.e., the rate of S5mthesis of the ES complex must equal its rate of consumption imtil 

Let us investigate whether condition (1.38) may arise within the proposed probabilistic enzymatic kinetics and what consequences that has for applicability of the logistic treatment. 

For reaction (1.4) to proceed with a high probability it is necessary that (Putz et al., 2007) 

by combining equation (1.40) with the general in vitro form (1.33), we derive the time dependent equation 

The substrate condition [ S](t) 0 corresponds to the binding case for which Eq. (1.40) is valid under the conditions given in expression (1.30). Applying this substrate condition to Eq. (1.42) during the rate limiting step when 

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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. 

It should be noted that the time dependence of the resin viscosity, t/, appears only through Equation 10. This is because of the quasi-steady-state approximation implicit in the present analysis. Equation 8 reduces to an expression similar to that derived by Aung (4) when k 0... 

The error of the quasi steady-state approximation in spatially distributed systems has recently been studied by Yannacopoulos ef al. [160]. It has been shown qualitatively that QSSA errors, which might decay quickly in homogeneous systems, can readily propagate in reactive flow systems so that the careful selection of QSSA species is very important. A quantitative analysis of QSSA errors has not yet been carried out for spatially distributed systems but would be a useful development. 

The validity of the quasi-steady-state approximation, has already been mentioned in Subsection 4.8.7. A detailed analysis of enzyme kinetics is given in Heineken et al. (1967), Walter (1977) and Segel (1984). The strict mathematical basis of the assumption is based on a theorem by Tikhonov (1952). He investigated the assumptions leading to separation of the fast and slow components of the solutions of the system... 

When a preliminary extended version of the kinetic model is used to identify the B-Zh reaction mechanism, frequently classical approximation methods are used, such as the ratedetermining step and the quasi-steady state approximations (see, for example [22,24-26]). However T. Turanyi and S. Vajda [29], applying the sensitivity analysis method, more precisely the method of the principal components analysis selection, specified numerically a base mechanism that comprise only 9 steps for the B-Zh reaction (see table 8.1), from the conventional model of Edelson-Field-Noes (EFN), which includes 32 steps. In this case, the... 

An overview of the methods used previously in mechanism reduction is presented in Tomlin et al. (1997). The present work uses a combination of existing methods to produce a carbon monoxide-hydrogen oxidation scheme with fewer reactions and species variables, but which accurately reproduces the dynamics of the full scheme. Local concentration sensitivity analysis was used to identify necessary species from the full scheme, and a principle component analysis of the rate sensitivity matrix employed to identify redundant reactions. This was followed by application of the quasi-steady state approximation (QSSA) for the fast intermediate species, based on species lifetimes and quasi-steady state errors, and finally, the use of intrinsic low dimensional manifold (ILDM) methods to calculate the mechanisms underlying dimension and to verify the choice of QSSA species. The origin of the full mechanism and its relevance to existing experimental data is described first, followed by descriptions of the reduction methods used. The errors introduced by the reduction and approximation methods are also discussed. Finally, conclusions are drawn about the results, and suggestions made as to how further reductions in computer run times can be achieved. 

For the casein which [A] is held fixed, we have onlyEq. (13.3-13b) andEq. (13.3-13c) to solve simultaneously. Since we expect an oscillatory solution, we cannot apply the quasi-equilibrium approximation or the quasi-steady-state approximation. The equations must be solved numerically, using standard methods of numerical analysis to obtain the time dependence of [X] and [Y]. The concentrations of X and Y must satisfy the expression... 

Sensitivity analysis seeks to determine and eliminate insignificant reactions and species on the basis of their impact on designated important species. As the number of species of interest increases, sensitivity analysis is less likely to provide substantial model order reduction. For this reason, sensitivity analysis is often used in conjunction with the quasi-steady state approximation. 

Time-scale analysis identifies the different scales over which species react, and the fast reactions and species are assumed to be at steady state. Thus, order reduction is possible as the differential equations for the fast species are replaced by algebraic relations. This is the basis for the quasi-steady state approximation, which has been mathematically formalized by perturbation theory. 

The implication of distinguishing between fast and slow variables is that a short time after the perturbation, the values of the fast variables are determined by the values of the slow ones. Appropriate algebraic expressions to determine the values of the fast variables as functions of the values of the slow ones can therefore be developed. This is the starting point of model reduction methods based on timescale analysis. One such method was introduced in Sect. 2.3 where the quasi-steady-state approximation (QSSA) was demonstrated for the reduction in the number of variables of a simple example. In this case, the system timescales were directly associated with chemical species. We shah see in the later discussion that this need not always be the case. 

Furthermore, mathematical procedures can be applied to the detailed mechanism or the skeletal mechanism which reduces the mechanism even more. These mathematical procedures do not exclude species, but rather the species concentrations are calculated by the use of simpler and less time-consuming algebraic equations or they are tabulated as functions of a few preselected progress variables. The part of the mechanism that is left for detailed calculations is substantially smaller than the original mechanism. These methods often make use of the wide range of time scales and are thus called time scale separation methods. The most common methods are those of (i) Intrinsic Low Dimensional Manifolds (ILDM), (ii) Computational Singular Perturbation CSF), and (iii) level of importance (LOl) analysis, in which one employs the Quasy Steady State Assumption (QSSA) or a partial equilibrium approximation (e.g. rate-controlled constraints equilibria, RCCE) to treat the steady state or equilibrated species. 

Laforge et al. [71] considered the case when solute partitioning and interfacial bimolecular ET simultaneously contribute to the positive feedback current. It was shown that the contributions of these two processes can be quantitatively separated by analysis of the i- d curves. A simple approximate theory for the concentration profile of the partitioning solute in one of two liquid phases was also developed and used to extract the partition coefficient (K) value from the SECM approach curves under quasi-steady-state conditions [71b]. An advantage of this approach is that K can be determined without waiting for the complete equilibration of the solute, which can be very slow if one of the liquid phases (e.g., ionic liquid) is viscous. 

In order to answer this question, a significant source of statistical correlation arising from mutation paths that visit a particularly advantageous mutant more than once must be considered. In the perturbation theory these paths are represented by products of factors involving the mutant replication rates, and it is necessary to remove the strong correlation that arises between these factors where repeated indices are present in order to obtain a tractable statistical analysis of convergence. The Watson renormalization procedure [29], the application of which to the steady-state quasi-species is summarized in Appendix 7, accomplishes just this [30]. The cost is a consecutive modification of the denominator, which may however be simplified to good approximation, as in Eqn. (A7.5). 

Figure 6.118 shows the integrated Eq. (3) for the steady state of quasi-isothermal experiments (A). The measurements needed for this analysis are displayed for polystyrene in Figs. 6.14 and 6.15 and for poly(ethylene terephthalate) in Figs. 4.129 and 4.130. The parameters in Fig. 6.118 were arbitrarily chosen to clarify the three different contributions to the approximation shown in the figure. For a solution fitted to the experiments for poly(ethylene terephthalate), see Fig. 4.131. The parameters A and A represent the amplitude contributions due to the change in x and N with temperature, and P and y are phase shifts. The plotted (N - No)/N is proportional to the heat flow (and thus to ACp). The curve (A), however, is not a sinusoidal response.

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