Michaelis-Menten model

2 MODELS OF ENZYME KINETICS 10.2.1 Michaelis-Menten Model 

The kinetics of the general enzyme-catalyzed reaction (equation 10.1-1) may be simple or complex, depending upon the enzyme and substrate concentrations, the presence/absence of inhibitors and/or cofactors, and upon temperature, shear, ionic strength, and pH. The simplest form of the rate law for enzyme reactions was proposed by Henri (1902), and a mechanism was proposed by Michaelis and Menten (1913), which was later extended by Briggs and Haldane (1925). The mechanism is usually referred to as the Michaelis-Menten mechanism or model. It is a two-step mechanism, the first step being a rapid, reversible formation of an enzyme-substrate complex, ES, followed by a slow, rate-determining decomposition step to form the product and reproduce the enzyme  

Henri and Michaelis and Menten assumed that the first at equilibrium, such that the concentration of the complex, step is a fast reaction virtually ES, may be represented by  

The ratio k 1/k1 is effectively the dissociation equilibrium constant3 for ES in step (1), and is usually designated by Km (Michaelis constant, with units of concentration). The rate of formation of the product, P, is determined from step (2) 4 

3In the biochemical literature, equilibrium constants are expressed as dissociation constants (for complexes), rather than as association constants. 

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Smith, W. G., 1992. In kinetics and the reversible Michaelis-Menten model. Journal of Chemical Education 12 981 — 984. 

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... 

The functioning of enzymes produces phenomena driving the processes which impart life to an organic system. The principal source of information about an enzyme-catalyzed reaction has been from analyses of the changes produced in concentrations of substrates and products. These observations have led to the construction of models invoking intermediate complexes of ingredients with the enzyme. One example is the Michaelis-Menten model, postulating an... 

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. 

The simplest kinetic model applied to describe lipase catalyzed reactions is based on the classic Michaelis-Menten mechanism [10] (Table 3). To test this model Belafi-Bakd et al. [58] studied kinetics of lipase-catalyzed hydrolysis of tri-, di-, and mono-olein separately. All these reactions were found to obey the Michaelis-Menten model. The apparent parameters (K and V ) were determined for global hydrolysis. 

In kinetic studies of enzymatic reactions, rate data are usually tested to determine if the reaction follows the Michaelis-Menten model of enzyme-substrate interaction. Weetall and Havewala [Biotechnol. and Bioeng. Symposium 3 (241), 1972] have studied the production of dextrose from cornstarch using conventional... 

A strain P. delafeildii R-8 was reported to desulfurize DBT giving 2-HBP as the end product [92], This strain was isolated from sewage pool of Shanghai oil field. The report described the effect of cell density, oil/water phase ratio, which was very similar to that of IGTS8. These parameters will be discussed in Section 2.2.10. A Michaelis-Menten model was used to describe the kinetics and the Vmax and Km were reported for DBT to be 13mmol/kg dcw/h and 1.3 mM, respectively. 

The three models used are described by Eq. (6-8) below. The Eqn. (6) is the first-order model based on Michaelis-Menten model, Eqn. (7) is the second-order model, and the Eqn. (8) is the competitive-substrate model. Rso represents the initial specific reaction rate for the substrate S. 

The model most often invoked to rationalize cooperative behavior is the MWC (Monod-Wyman-Changeaux), or concerted, model. This model is 1.5 times more complicated than the Michaelis-Menten model and took three people to develop instead of two. Most texts describe it in detail. In the absence of substrate, the enzyme has a low affinity for substrate. The MWC folks say that the enzyme is in a T (for tense or taut) state in the absence of substrate. Coexisting with this low-affinity T state is another conformation of the enzyme, the R (for relaxed) state, that has a higher affinity for substrate. The T and R states coexist in the absence of substrate, but there s much more of the T state than the R. This has always seemed backward, since one would expect the enzyme to be more tense in the presence of substrates when some work is actually required. In keeping with the tradition of biochemistry, the MWC folks obviously wanted this to be backward too (Fig. 8-8). 

As the above discussion indicates, assigning mechanisms to simple anation reactions of transition metal complexes is not simple. The situation becomes even more difficult for a complex enzyme system containing a metal cofactor at an active site. Methods developed to study the kinetics of enzymatic reactions according to the Michaelis-Menten model will be discussed in Section 2.2.4. 

The simple Michaelis-Menten model does not deal with all aspects of enzyme-catalyzed reactions. The model must be modified to treat the phenomena of inhibition and... 

Determine the kinetics parameters Km and Vmax, assuming that the standard Michaelis-Menten model applies to this system, (a) by nonlinear regression, and (b) by linear regression of the Lineweaver-Burk form. 

Biological. In activated sludge, <0.1% mineralized to carbon dioxide after 5 d (Freitag et al., 1985). The half-life of pentachlorobenzene in an anaerobic enrichment culture was 24 h (Beurskens et al., 1993). In an enrichment culture derived from a contaminated site in Bayou d lnde, LA, pentachlorobenzene underwent reductive dechlorination to 1,2,4,5- and 1,2,3,5-tetrachlorobenzene at relative molar yields of 9 and 91%, respectively. The maximum dechlorination rate, based on the recommended Michaelis-Menten model, was 131 nM/d (Pavlostathis and Prytula, 2000). 

In an enrichment culture derived from a contaminated site in Bayou d lnde, LA, 1,2,4,5-tetrachlorobenzene underwent reductive dechlorination yielding 1,2,4-trichlorobenzene. The maximum dechlorination rate, based on the recommended Michaelis-Menten model, was 208 nM/d (Pavlostathis and Pry tula, 2000). 

STEADY STATE TREATMENT. While the Michaelis-Menten model requires the rapid equilibrium formation of ES complex prior to catalysis, there are many enzymes which do not exhibit such rate behavior. Accordingly, Briggs and Haldane considered the case where the enzyme and substrate obey the steady state assumption, which states that during the course of a reaction there will be a period over which the concentrations of various enzyme species will appear to be time-invariant ie., d[EX]/dr s 0). Such an assumption then provides that... 

What would be the corresponding behavior of an enzyme obeying the Michaelis-Menten model For the answer, let us restate the Michaelis-Menten model in the form of a saturation function ... 

The Michaelis-Menten model is commonly employed in describing nonallosteric enzyme reactions. The overall model can be pictured as follows (Equation 16.25) where E represents the enzyme, M the reacting molecule(s), E + M EM is associated with ki and the reverse reaction associated with k i and EM E + P is associated with 2-... 

The initial rate of product formation, Rx, for the Michaelis-Menten model depends only on the rate of complex breakdown, i.e. 

The basic kinetic model for enzyme catalysed conversions in water and in w/o-microemulsions is based on the theory of MichaeHs and Menten [83]. Although the Michaelis-Menten-model is often sufficient to describe the kinetics, the bi-bi-models (e. g. random bi-bi, orderedbi-bi or ping-pongbi-bi), which describe the sequences of substrate bindings to the enzyme are the more accurate kinetic models [84]. 

Figure 2.3 Comparison of the Michaelis-Menten model for a minimal kinetic scheme (bottom equation) with the pseudo second-order format (top equation). Relationship between the kinetic barriers for the formation of the Michaelis complex and the chemical transformation S -> P, and the Gibbs free energy of the (virtual) barrier for the pseudo second-order reaction S + —> P + E.

Unfortunately, many transformations purported to linearize the model also interchange the role of dependent and independent variables. Important examplE are the various linearization transformations of the simple steady-state Michaelis-Menten model... 

Such a mechanism is also used for the simplest model of enzyme kinetics—the Michaelis-Menten model. 

The Michaelis-Menten equation represents a mechanistic model because it is based upon an assumed chemical reaction mechanism of how the system behaves. If the system does indeed behave in the assumed manner, then the mechanistic model is adequate for describing the system. If, however, the system does not behave in the assumed manner, then the mechanistic model is inadequate. The only way to determine the adequacy of a model is to carry out experiments to see if the system does behave as the model predicts it will. (The design of such experiments will be discussed in later chapters.) In the present example, if substrate inhibition occurs, the Michaelis-Menten model would probably be found to be inadequate a different mechanistic model would better describe the behavior of the system. 

Receptor theories strive to explain how the various types of ligands give rise to a response. The variety of receptors and the signaling pathways to which they are linked pose a significant challenge to receptor theory. Receptor response pathways are far more complex than the activity of enzymes, and therefore the modeling of receptors is more complex than the Michaelis-Menten model seen in Chapter 4. 

The Michaelis-Menten model uses the following concept of enzyme catalysis ... 

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