Michaelis-menten model expression

The above rate equation is in agreement with that reported by Madhav and Ching [3]. Tliis rapid equilibrium treatment is a simple approach that allows the transformations of all complexes in terms of [E, [5], Kls and Kjp, which only deal with equilibrium expressions for the binding of the substrate to the enzyme. In the absence of inhibition, the enzyme kinetics are reduced to the simplest Michaelis-Menten model, as shown in Figure 5.21. The rate equation for the Michaelis-Menten model is given in ordinary textbooks and is as follows 11... 

From the Michaelis-Menten model, there is a relationship between 1/Fo and the initial substrate concentration, expressed as the reciprocal, 1/[Bq]. To develop this relationship we shall repeat Example 9.1 using varying concentrations of B cells. Be sure to subtract the number of Bq cells in each study from the total number of water, D, cells in the setup. 

Often the key entity one is interested in obtaining in modeling enzyme kinetics is the analytical expression for the turnover flux in quasi-steady state. Equations (4.12) and (4.38) are examples. These expressions are sometimes called Michaelis-Menten rate laws. Such expressions can be used in simulation of cellular biochemical systems, as is the subject of Chapters 5, 6, and 7 of this book. However, one must keep in mind that, as we have seen, these rates represent approximations that result from simplifications of the kinetic mechanisms. We typically use the approximate Michaelis-Menten-type flux expressions rather than the full system of equations in simulations for several reasons. First, often the quasi-steady rate constants (such as Ks and K in Equation (4.38)) are available from experimental data while the mass-action rate constants (k+i, k-i, etc.) are not. In fact, it is possible for different enzymes with different detailed mechanisms to yield the same Michaelis-Menten rate expression, as we shall see below. Second, in metabolic reaction networks (for example), reactions operate near steady state in vivo. Kinetic transitions from one in vivo steady state to another may not involve the sort of extreme shifts in enzyme binding that have been illustrated in Figure 4.7. Therefore the quasi-steady approximation (or equivalently the approximation of rapid enzyme turnover) tends to be reasonable for the simulation of in vivo systems. 

Other types of expressions are assumed for the rate law. For instance, enzyme-catalyzed reactions are often described by the Michaelis-Menten model, which leads to a rate law in the following form ... 

Kinetic experiments with synthetic iron oxyhydroxides have shown that the initial microbial reduction rate increases with increasing initial ferric iron concentration up to a given maximum reduction rate (Bonneville et al. 2004). This observation was explained by a saturation of active membrane sites with Fe(III) centers. The respective reaction was best described with a Michaelis-Menten rate expression with the maximum reduction rate per cell positively correlating with the solubility of the iron oxyhydroxides (Bonneville et al. 2004). Kinetic studies involving iron are not only inherently important to describe reaction pathways and to derive rate constants, which can be used in models. Kinetic studies also increasingly focus on iron isotopic fractionation to better understand the iron isotopic composition of ancient sediments, which may assist in the reconstruction of paleo-environments. Importantly, iron isotope fractionation occurs in abiotic and biotic processes the degree of isotopic fractionation depends on individual reaction rates and the environmental conditions, e.g. whether reactions take place within an open or closed system (Johnson et al. 2004). 

Table 3 lists the kinetic rate expressions for each of the hydrolysis and fermentation reaction rates shown in Fig. 5 and in the mass balance equations of Tables 1 and 2. Each of the reaction rates were found to fit the data through trial and error, starting with the simplest model. For the hydrolysis reaction rates (rs,arch and / maltose), the simplest form was the Michaelis-Menten model without inhibition. For all other reaction rates which described fermentation kinetics, the simplest form was the Monod model without inhibition. More descriptive models were found in the literature and tested one by one until the set of kinetic rate equations with the best fit to the experimental data were determined. This was completed with the hydrolysis datasets first, then the complete SSF datasets. 

The nonlinearity with respect to Uj in Eq.(11) is to be noted particularly, which may eliminate the exclusive bang-bang optimality result. Thus simultaneous utilization of substrates could also be predicted by such models. If ample resources are present, then the above Michaelis-Menten kinetic expression in Eq. (11) prevents over-allocation for any particular enzyme. 

In kinetic smdies of enzymatic reactions, rate data are usually tested to determine if the reaction follows the Michaelis-Menten model of enzyme-substrate interactions. H. H. WeetaU and N. B. Havewala [Biotechnol. Bioeng. Symp., 3, 241 (1972) studied the production of dextrose from cornstarch using conventional glucoamylase and an immobihzed version thereof. Their goal was to obtain the data necessary to design a commercial facility for dextrose production. Their studies were carried out in a batch reactor at 60°C. Compare the data below with those predicted from a Michaelis-Menten model with a rate expression of the form... 

Known after Monod (1950), this expression is best understood in terms of the representation shown in Table 20.7 along with the Michaelis-Menten model for comparison (see Levenspiel, 1993). It can be readily seen that... 

The basic principle of this method is to react one of the enantiomers preferentially with an enzyme (free or immobilized, largely the latter). Expressions can be developed for the enantiomeric excess of the desired enantiomer (the one left unreacted) as a function of the extent of conversion of the racemate as a whole and the parameters of the Michaelis-Menten models for the two enantiomers (see Chen et al., 1982 Rakels et al. 1994 Wu, 1997). 

The results from this chapter on zeolite catalysis provide a good reference point for the discussion presented later Chapter 8 where we compare heterogeneous catalysis and biocatalysis. The similarity between the Michaelis-Menten kinetic expression for enzyme catalysis and the Langmuir-Hinshelwood kinetic models for heterogeneous catalysis are noted. This ultimately derives from the conservation in the number of active reaction centers for both systems. However, the more refined synergy of the activation of molecular bonds by the enzyme will become apparent as a major difference between the two. 

The Michaelis-Menten rate expression for a monomolecular enzyme-catalyzed reaction is very similar to the Langmuir kinetic expression that we discussed previously for heterogeneous catalyzed systems. The Michaelis-Menten relationship is readily deduced for a simple model where the enzyme molecule (E) equilibrates with substrate molecule (S) to form the enzyme substrate complex (ES). The enzyme-substrate complex then reacts to the product molecule (P) and regenerates the active site of the enzyme (E) in what is considered to be the rate-limiting step of the proposed scheme ... 

To make an appropriate assessment of the pattern of inhibition, one need only compare the pattern of reaction velocity versus [S] observed relative to the pattern predicted from an application of the hyperbolic kinetics model. This requires making an estimate of V ax and from the data available. Transforming the original data to a Lineweaver-Burke plot (despite the aforementioned limitations) indicates that only four data points (at low [S]) can be used to estimate Vmax and Km (as 3.58 units and 0.48 mM, respectively. Fig. 14.10). The predicted (uninhibited) behavior of the enzyme activity can now be calculated by applying the rectangular hyperbola [Eq. (14.5)] (yielding the upper curve in Fig. 14.11), and it becomes clear that inhibition was obvious at [S] <1 mM. The degree of inhibition is expressed appropriately as the difference between observed and predicted activity at any [S] value, if one makes interpretations within the context of the Michaelis-Menten model. 

The operational model allows simulation of cellular response from receptor activation. In some cases, there may be cooperative effects in the stimulus-response cascades translating activation of receptor to tissue response. This can cause the resulting concentration-response curve to have a Hill coefficient different from unity. In general, there is a standard method for doing this namely, reexpressing the receptor occupancy and/or activation expression (defined by the particular molecular model of receptor function) in terms of the operational model with Hill coefficient not equal to unity. The operational model utilizes the concentration of response-producing receptor as the substrate for a Michaelis-Menten type of reaction, given as... 

In Box 12.2, a simple model for a special kind of catalyzed reaction, the Michaelis-Menten enzyme kinetics, is presented, which leads to the following kinetic expression ... 

With deeper understanding of the rate laws applicable to these hydrolases, now we need to deduce the parameters that combine to give corresponding khl0 values for Michaelis-Menten cases (Eq. 17-80). We may now see that the mathematical form we used earlier to describe the biodegradation of benzo[f]quinoline (Eq. 17-82) could apply in certain cases. Further we can rationalize the expressions used by others to model the hydrolysis of other pollutants when rates are normalized to cell numbers (e.g., Paris et al., 1981, for the butoxyethylester of 2,4-dichlorophenoxy acetic acid) or they are found to fall between zero and first order in substrate concentration (Wanner et al., 1989, for disulfoton and thiometon). 

Even this scheme represents a complex situation, for ES can be arrived at by alternative routes, making it impossible for an expression of the same form as the Michaelis-Menten equation to be derived using the general steady-state assumption. However, types of non-competitive inhibition consistent with the Michaelis-Menten type equation and a linear Linweaver-Burk plot can occur if the rapid-equilibrium assumption is valid (Appendix S.A3). In the simplest possible model, involving simple linear non-competitive inhibition, the substrate does not affect the inhibitor binding. Under these conditions, the reactions... 

The most straightforward way is to plot r against Cs as shown in Figure 2.2. The asymptote for r will be rmax and KM is equal to Cs when r = 0.5 rmax. However, this is an unsatisfactory plot in estimating rmax and KM because it is difficult to estimate asymptotes accurately and also difficult to test the validity of the kinetic model. Therefore, the Michaelis-Menten equation is usually rearranged so that the results can be plotted as a straight line. Some of the better known methods are presented here. The Michaelis-Menten equation, Eq. (2.11), can be rearranged to be expressed in linear form. This can be achieved in three ways ... 

When the rate of diffusion is very slow relative to the rate of reaction, all substrate will be consumed in the thin layer near the exterior surface of the spherical particle. Derive the equation for the effectiveness of an immobilized enzyme for this diffusion limited case by employing the same assumptions as for the distributed model. The rate of substrate consumption can be expressed by the Michaelis-Menten equation. 

Relative recovery can be mathematically expressed by Fick s law of diffusion as modified by Jacobson (Jacobson et al., 1985). In this relationship, recovery is the ratio of the concentration in the perfusate to the concentration extracellular. For the mass recovery, an expression similar to a Michaelis-Menten formula for enzymatic reactions was derived (Ekblom et al., 1992). A number of other mathematical models for quantitative microdialysis have been proposed and are reviewed elsewhere (Justice 1993 Kehr, 1993b). 

Last, population pharmacokinetics of sibrotuzumab, a humanized monoclonal antibody directed against fibroblast activation protein (FAP), which is expressed in the stromal fibroblasts in >90% of malignant epithelial tumors, were analzyed in patients with advanced or metastatic carcinoma after multiple IV infusions of doses ranging from 5 mg/m to a maximum of 100 mg (78). The PK model consisted of two distribution compartments with parallel first-order and Michaelis-Menten elimination pathways from the central compartment. Body weight was significantly correlated with both central and peripheral distribution volumes, the first-order elimination clearance, and ymax of the Michaelis-Menten pathway. Of interest was the observation that body surface area was inferior to body weight as a covariate in explaining interpatient variability. 

A plot of V versus [S] using this expression, the Michaelis-Menten equation, gives a rectangular hyperbola—a curve of the form depicted in Fig. 2-3. This lends support to the contention that our kinetic model is descriptive of the reaction mechanism. 

A final test of the intracellular fluxes determined by metabolite balancing was provided through comparison with the predictions of a first-order kinetic model describing the oxidation of pulsed [ C]-indene to all detectable indene derivatives in steady state cells. Assuming Michaelis-Menten kinetics for a typical reaction depicted in Fig. 4, the rate of labeled metabolite conversion by that reaction can be expressed as... 

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