Michaelis-Menten kinetics assumptions

Hehre and coworkers showed that beta amylase from sweet potatoes, an inverting, a-specific exo-(l 4)-glucanase, catalyzes the hydrolysis of jS-maltosyl fluoride with complex kinetics which indicated the participation of two substrate molecules in the release of fluoride ion. Furthermore, the reaction was strongly accelerated by the addition of methyl ) -maltoside. Hydrolysis of a-maltosyl fluoride, on the other hand, obeyed Michaelis-Menten kinetics. The main product with both a- and yj-maltosyl fluoride was )S-maltose. The results with )3-maltosyl fluoride were interpreted by the assumption of a glycosylation reaction preceding hydrolysis by which a malto-tetraoside is formed by the replacement of fluoride ion by a second substrate molecule or added methyl -maltoside (see Scheme 5). 

Sato et al. (1991) expanded their earlier PBPK model to account for differences in body weight, body fat content, and sex and applied it to predicting the effect of these factors on trichloroethylene metabolism and excretion. Their model consisted of seven compartments (lung, vessel rich tissue, vessel poor tissue, muscle, fat tissue, gastrointestinal system, and hepatic system) and made various assumptions about the metabolic pathways considered. First-order Michaelis-Menten kinetics were assumed for simplicity, and the first metabolic product was assumed to be chloral hydrate, which was then converted to TCA and trichloroethanol. Further assumptions were that metabolism was limited to the hepatic compartment and that tissue and organ volumes were related to body weight. The metabolic parameters, (the scaling constant for the maximum rate of metabolism) and (the Michaelis constant), were those determined for trichloroethylene in a study by Koizumi (1989) and are presented in Table 2-3. 

The dependence (1 TP of v, on ATP is modeled as in the previous section, using an interval C [—00,1] that reflects the dual role of the cofactor ATP as substrate and as inhibitor of the reaction. All other reactions are assumed to follow Michaelis Menten kinetics with ()rs E [0, 1], No further assumption about the detailed functional form of the rate equations is necessary. Given the stoichiometry, the metabolic state and the matrix of saturation parameter, the structural kinetic model is fully defined. An explicit implementation of the model is provided in Ref. [84],... 

Henri and Michaelis-Menten kinetics assumed that the rate of formation of products was much less than that for the back reaction from ES to yield E + S. Van Slyke assumed the reverse. A more rigorous formulation was offered by Briggs and Haldane (1925) using steady-state assumptions previously applied to chemical kinetics by Bodenstein (1913). 

Furthermore, it can be shown that, in the limiting cases of first-order kinetics [Equation (11.35) also holds for this case] and zero-order kinetics, the equal and optimal sizes are exactly the same. As shown, the optimal holding times can be calculated very simply by means of Equation (11.40) and the sum of these can thus be used as a good approximation for the total holding time of equal-sized CSTRs. This makes Equation (11.31) an even more valuable tool for design equations. The restrictions are imposed by the assumption that the biocatalytic activity is constant in the reactors. Especially in the case of soluble enzymes, for which ordinary Michaelis-Menten kinetics in particular apply, special measures have to be taken. Continuous supply of relatively stable enzyme to the first tank in the series is a possibility, though in general expensive. A more attractive alternative is the application of a series of membrane reactors. 

There is an interesting parallel between substrate binding and adsorption. Since each enzyme molecule has one active site, and since these active sites all have the same structure, we can think of enzyme molecules in solution as a surface with many equivalent adive sites. In this case, k2 in the Michaelis-Menten kinetics (Eq. 5.1 see Chapter 2 for a detailed discussion) represents the rate of adsorption, k x the rate of desorption, and k2 the rate of the surface readion followed by fast product desorption. Moreover, this system fits the assumptions of the Langmuir isotherm (all sites identical, one molecule per site, no lateral interadions) even better than the adive sites on some real solid catalysts ... 

Shaw and Bell (1991) examined this effect in the case of competition between radiocaesium and the K+ and NH4+ ions during root uptake by wheat (Triticum aestivum). These authors formalised the observed relationships in terms of classical Michaelis-Menten kinetics which necessitates the assumption that each of these ions is taken up by identical sites associated with the root plasmalemma. Lembrechts et al. (1990) found a similar negative and non-linear relationship between the concentration of Ca either in soil or in solution culture and the degree of radiostrontium uptake by lettuce Lactuca saliva). The principle of competitive exclusion of a radionuclide by an ion analogue may be exploited, with varying degrees of success, as a post-con-... 

Enzyme kinetics and the mode of inhibition are well described by transformation of the Michaelis-Menten equation. The binding affinity of the inhibitor to the enzyme is defined as the inhibition constant Ki, whereas the affinity, with which the substrate binds, is referred to the Michaelis-Menten coefficient Km. Michaelis-Menten kinetics base on three assumptions ... 

Can a phosphorylation-dephosphorylation switch be more sensitive to the level of kinase concentration than n = 1 as given in Equation 5.12 We note that the kinetic scheme in Equation (4.7) is obtained under the assumption of no Michaelis-Menten saturation. Since this assumption may not be realistic, let us move on to study the enzyme kinetics in Figure (5.2) in terms of saturable Michaelis-Menten kinetics. The mechanism by which saturating kinetics of the kinase and phosphatase leads to sensitive switch-like behavior is illustrated in Figure 5.4. The reaction fluxes as a function of / (the ratio [S ]/Sc) for two cases are plotted. The first case (switch off)... 

The assumption of uniformity is in fact justified for some realistic kinetic schemes, such as Langmuir isotherm catalyzed reactions, Michaelis-Menten kinetics, and others (Aris, 1989 Cicarelli et al, 1992). The assumption bears a more than superficial analogy with those systems termed pseudo-monomolecular by Wei and Prater (1962). Mathematically, it is a very powerful assumption By crossing out the dependence of F[ ] on x, its value has been reduced from an infinite-dimensional vector (a function of x) to a scalar. This simplification makes Eq. (102) a quasilinear one, and it can be integrated explicitly by introducing a warped time scale t(0. (0) = 0. The solution, as can be verified by inspection, is... 

The results of this section have shown that neither the assumption of Michaelis-Menten functional response nor the assumption of only two competitors is essential for the main results of Chapter 1 to hold. In much of what follows we retain the Michaelis-Menten kinetics, since the parameters are readily measurable in the laboratory. 

A rigorous kinetic description of interfacial catalysis has been hampered by the ill-defined physical chemistry of the lipid—water interface (Martinek et ai, 1989). Traditional kinetic assumptions are undermined by the anisotropy and inhomogeneity of the substrate aggregate. For example, the differential partitioning of reactants (enzyme, calcium ion, substrate) and products (lysolecithins, fatty acids) between the two bulk phases prevents direct measurement of enzyme and substrate concentrations. This complicates dissection of the multiple equilibria that contribute to the observed rate constants. Only recently has it become possible to describe clearly the activity of SPLA2S in terms of traditional Michaelis— Menten kinetics. Such a description required the development of methods to reduce experimentally the number of equilibrium states available to the enzyme (Berg etai, 1991). 

The Michaelis-Menten kinetics, represented by Eq. (18), may be extended to more complicated reactions by looking at the structure of the adsorption term. This procedure shown below is valid as long as the rapid equilibrium assumption is made. This is not valid in all cases. 

In the following sections the extension of Eq. (18) to more complex reaction schemes is described. Again the rapid equilibrium assumption is used to show how more complex rate equations are derived from simple Michaelis-Menten kinetics. Attention is focused on some typical rate equations that are useful to describe enzyme kinetics with respect to a desired process optimization. The whole complexity of enzyme kinetics is of importance for a basic understanding of the enzyme mechanism, but it is not necessary for the fitting of kinetic data and the calculation of reactor performance. 

The assumption that microbial communities will always rapidly adapt to the available environment and substrate supply is a fundamental assumption in using first order kinetics, as in Kq. (1). Most biogeochemical processes arc catalyzed, however, and cat-alv/.ed reactions invariably show. Michaelis- Menten kinetics. 

This brings us to the final mechanism we need to consider for a 2-substrate reaction, namely a random-order mechanism. We have assumed that we would be alerted to the possibility of a steady-state random-order mechanism by non-linear primary or secondary plots, but it is possible to get linear kinetics with a random-order mechanism. If we make the assumption that the further reaction of the ternary complex EAB is much slower than the network of reactions connecting E to EAB via EA and EB, then there are only 4 kinetically significant complexes and their concentrations are related to one another by substrate concentrations and dissociation constants. This is the rapid-equilibrium random-order mechanism, and the assumption made is analogous to the Michaelis-Menten equilibrium assumption for a 1-substrate mechanism. 

An important difference between enzymes and nanoreactors is product release from the cavity. The Michaelis-Menten kinetics assumes that product release occurs rapidly (this assumption is also done in equation 4). However, this assumption does not always counts for nanoreactors because they can display product inhibition (vide infra). 

There has been a substantial effort to model the kinetics of enzymatic hydrolysis of cellulose and (pretreated) lignocellulosic substrates. Bansal et al. (2009) provide a comprehensive review of many of the models that have been developed. Most of these models are strictly empirical or based on highly simplified Michaelis-Menten concepts. Unfortunately, the assumptions commonly used with Michaelis-Menten kinetics models, namely, that reaction takes place in solution and there is a single... 

Langmuir chemo kinetics should be equivelent to Michaelis-Menten kinetics since the total number of sites (the number of enzyme molecules) or the number of reaction sites on a surface is a constant. In addition, the often physically unrealistic assumption is made that ki, k2andk are independent of concentration. 

For an application of (2.6) to enzyme experiments, one usually assumes P 0, which means an elimination of P from the reaction by a very fast transport process, or k2 0, which means that the binding site of the enzyme has a high attractivity only to the substrate S but not to the product P. With this assumption, the plot of T versus S has the form given in Fig. 1 (Michaelis-Menten kinetics)... 

These kinetic studies on mandelate racemase were interpreted by assuming a simple Michaelis-Menten kinetic scheme for this one-substrate enzymic reaction. This assumption has recently received some experimental verification. ... 

Calcium entry into a cell frequently has a number of secondary effects such as initiation of contraction, release of neurotransmitters, and modulation of membrane ion channels. This is usually accompHshed by the binding of calcium ions to calcium receptors inside the cell. Michaelis-Menten kinetic schemes with steady-state assumptions are used to model this binding and therefore expressions of the form f = [Cfl++]7/([Cfl++]7 + K) are frequently employed. Here/is the fraction of calcium that is bound to the receptor and K is the dissociation constant for the reaction, and n is the number of calcium ions that bind to each receptor molecule. 

To illustrate the complexity, we will take a simple case in which the enzymes typically follow the Michaelis-Menten kinetic mechanism. We will also make the following assumptions (a) only one ionic form of the enzyme is active (mono-protic), (b) no change occurs in RDS due to ionization, and (c) the enzyme maintains active conformation within the experimental pH range. For this type of enzyme, an observable quantity, e.g., rate constant (k), depends on the pH of the system as follows (i) the plot of k against pH resembles a titration curve, yielding at the inflection point the pK of the acid involved, (ii) the follows the ionization of the enzyme—substrate complex, (iii) the fC follows the ionizations of both the free enzyme and the enzyme—substrate complex, and (iv) the follows the ionization of the free enzyme only, even when there are multiple intermediates on the reaction pathway. 

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... 

Equation 11-15 is known as the Michaelis-Menten equation. It represents the kinetics of many simple enzyme-catalyzed reactions, which involve a single substrate. The interpretation of as an equilibrium constant is not universally valid, since the assumption that the reversible reaction as a fast equilibrium process often does not apply. 

Another term used to indicate a rapid equilibrium assumption in a kinetic process. The most prominent example in biochemistry is the assumption that enzyme and substrate rapidly form a preequilibrium enzyme-substrate complex in the Michaelis-Menten treatment. 

UNI UNI ENZYME KINETIC MECHANISM MICHAELIS-MENTEN EQUATION ISO UNI UNI MECHANISM RAPID EQUILIBRIUM ASSUMPTION Unpaired electron,... 

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