Michaelis-Menten concepts

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... 

Substrate-limited growth in terms of reduced availability of both the electron donor and the electron acceptor is common in wastewater of sewer systems. Based on the concept of Michaelis-Menten s kinetics for enzymatic processes, Monod (1949) formulated, in operational terms, the relationship between the actual and the maximal specific growth rate constants and the concentration of a limiting substrate [cf. Equation (2.14)] ... 

It would be distinctly arrogant to say that we understand how enzymes work. At best we catch glimpses of their action. One model involves the key-and-lock concept—an attempt to rationalize their specificity. A much simplified presentation is shown in Fig. 7.115. The idea is that certain shapes in the enzyme structure are precise fits for a part of the reactant molecule. A famous formulation of this is the Michaelis-Menten kinetics. If E is the enzyme and R is some part of a reactant (a complex biomolecule),... 

There are methods used Lo study enzymes other than those of chemical instrumental analysis, such as chromatography, that have already been mentioned. Many enzymes can be crystallized, and their structure investigated by x-ray or electron diffraction methods. Studies of the kinetics of enzyme-catalyzed reactions often yield useful data, much of this work being based on the Michaelis-Menten treatment. Basic to this approach is the concept (hat the action of enzymes depends upon the formation by the enzyme and substrate molecules of a complex, which has a definite, though transient, existence, and then decomposes into the products, of the reaction. Note that this point of view was the basis of the discussion of the specilicity of the active sites discussed abuve. 

Steady-state approximation is based on the concept that the formation of [ES] complex by binding of substrate to free enzyme and breakdown of [ES] to form product plus free enzyme occur at equal rates. A graphical representation of the relative concentrations of free enzyme, substrate, enzyme-substrate complex, and product is shown in figure 7.8 in the text. Derivation of the Michaelis-Menten expression is based on the steady-state assumption. Steady-state approximation may be assumed until the substrate concentration is depleted, with a concomitant decrease in the concentration of [ES]. 

The Michaelis-Menten equation, especially if derived with the steady-state concept as above, is a rigorous rate law which not only fits almost all one-substrate enzyme kinetics, except in the case of inhibition (see Section 5.3), but also allows identification of the kinetic constants with the elementary steps in Eq. (5.1). 

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

This concept should be applied to a Michaelis-Menten kinetic [30] ... 

There is almost no biochemical reaction in a cell that is not catalyzed by an enzyme. (An enzyme is a specialized protein that increases the flux of a biochemical reaction by facilitating a mechanism [or mechanisms] for the reaction to proceed more rapidly than it would without the enzyme.) While the concept of an enzyme-mediated kinetic mechanism for a biochemical reaction was introduced in the previous chapter, this chapter explores the action of enzymes in greater detail than we have seen so far. Specifically, catalytic cycles associated with enzyme mechanisms are examined non-equilibrium steady state and transient kinetics of enzyme-mediated reactions are studied an asymptotic analysis of the fast and slow timescales of the Michaelis-Menten mechanism is presented and the concepts of cooperativity and hysteresis in enzyme kinetics are introduced. 

While the majority of these concepts are introduced and illustrated based on single-substrate single-product Michaelis-Menten-like reaction mechanisms, the final section details examples of mechanisms for multi-substrate multi-product reactions. Such mechanisms are the backbone for the simulation and analysis of biochemical systems, from small-scale systems of Chapter 5 to the large-scale simulations considered in Chapter 6. Hence we are about to embark on an entire chapter devoted to the theory of enzyme kinetics. Yet before delving into the subject, it is worthwhile to point out that the entire theory of enzymes is based on the simplification that proteins acting as enzymes may be effectively represented as existing in a finite number of discrete states (substrate-bound states and/or distinct conformational states). These states are assumed to inter-convert based on the law of mass action. The set of states for an enzyme and associated biochemical reaction is known as an enzyme mechanism. In this chapter we will explore how the kinetics of a given enzyme mechanism depend on the concentrations of reactants and enzyme states and the values of the mass action rate constants associated with the mechanism. 

The regulation of enzymes by metabolites leads to the concept of allostenc regulation. Allosteric means other structure. Allosteric modulators can bind at a site other than the active site in question and cause activation or inhibition. These modulators can include the substrate itself, which binds at another active site in a multi-subunit enzyme. In fact, allosterically modulated enzymes almost always have a complex quaternary structure (multiple subunits) and exhibit non-Michaelis-Menten kinetics. 

Interestingly, a fully appropriate model was developed at the same time as the Langmuir model using a similar basic approach. This is the Michaelis-Menten equation which has proved to be so useful in the interpretation of enzyme kinetics and, thereby, understanding the mechanisms of enzyme reactions. Another advantage in using this model is the fact that a graphical presentation of the data is commonly used to obtain the reaction kinetic parameters. Some basic concepts and applications will be presented here but a more complete discussion can be found in a number of texts. ... 

From this concept, the Michaelis-Menten equation was derived... 

One of the most useful models in the systematic investigation of enzyme rates was proposed by Leonor Michaelis and Maud Menten in 1913. The concept of the enzyme-substrate complex, first enunciated by Victor Henri in 1903, is central to Michaelis-Menten kinetics. When the substrate S binds in the active site of an enzyme E, an intermediate complex (ES) is formed. During the transition state, the substrate is converted into product. After a brief time, the product dissociates from the enzyme. This process can be summarized as follows ... 

Tissues continuously adjust the rate at which different proteins are synthesized to vary the amount of different enzymes present. The expression for in the Michaelis-Menten equation incorporates the concept that the rate of a reaction is proportional to the amount of enzyme present. Thus, the maximal capacity of a tissue can change with increased protein synthesis, or with increased protein degradation. 

The term should be used for enzymes that display Michaelis-Menten kinetics. Thus, it is not used with allosteric enzymes. Technically, competitive and noncompetitive inhibition are also terms that are restricted to Michaelis-Menten enzymes, although the concepts are applicable to any enzyme. An inhibitor that binds to an allosteric enzyme at the same site as the substrate is similar to a classical competitive inhibitor. One that binds at a different site is similar to a noncompetitive inhibitor, but the equations and the graphs characteristic of competitive and noncompetitive inhibition don t work the same way with an allosteric enzyme. 

Other concepts follow from the Michaelis-Menten equation. When the velocity of an enzymatic reaction is one-half the maximal velocity ... 

The magnitude of EDR can be conveniently expressed by means of the effectiveness factor. The effectiveness factor is a general concept that represents the ratio of rates of a phenomenon under the influence of a factor and freed from that influence. For the present case, it is defined as the ratio of the reaction rate under EDR and that attainable in its absence, this is, the ratio of effective to intrinsic reaction rate. For simple Michaelis-Menten kinetics ... 

Supramolecular chemistry has been a very popular research topic for three decades now. Most applications are foreseen in sensors and opto-electronical devices. Supramolecular catalysis often refers to the combination of a catalyst with a synthetic receptor molecule that preorganizes the substrate-catalyst complex and has also been proposed as an important possible application. The concept, which has proven to be powerful in enzymes, has mainly been demonstrated by chemists that investigated hydrolysis reactions. Zinc and copper in combination with cyclodextrins as the receptor dramatically enhance the rate ofhydrolysis. So far, the ample research devoted to transition metal catalysis has not been extended to supramolecular transition metal catalysis. A rare example of such a supramolecular transition metal catalyst was the results of the joined efforts of the groups of Nolte and Van Leeuwen [SO], They reported a basket-shaped molecule functionalized with a catalytically active rhodium complex that catalyzed hydrogenation reactions according to the principles of enzymes. The system showed substrate selectivity, Michaelis Menten kinetics and rate enhancement by cooperative binding of substrate molecules. The hydroformylation of allyl catachol substrates resulted in a complex mixture of products. 

The kinetics of enzymatic reactions in microemulsions obey, as a rule, the classic Michaelis-Menten equation [6,26,35], but difhculties arise in interpreting the results because of the distribution of reactants, products, and enzyme molecules among the microphases of the microemulsion [8,36-38], In addition, there are some enzymes in reverse micelles that exhibit enhanced activity as compared to that expressed in water this has given rise to the concept of superactivity [6,26,39], The superactivity has been explained in terms of the state of water in reverse micelles, the increased rigidity of the enzymes caused by the surfactant layer, and the enhanced substrate concentration at the enzyme microenvironment [36,40],... 

The concept of the active site in a floating, mobile, water-soluble enzyme can be extended to biological receptors that are fixed in cell membranes and a similar analysis can be applied to competitors to natural substrates. This means that it is important to have an analysis procedure for competitive inhibition. Consider the Michaelis-Menten equations with an inhibitor 1 ... 

While we are still self-constrained to limit our treatment to what we believe is essential to physical chemistry, we have added further examples to the Chapter 7 treatment of reaction kinetics, which include some aspects of multistep mechanisms and introduced the steady-state approximation. The steady-state concept was then extended to the Eyring transition-state concept and used again for the critical step in the Michaelis-Menten treatment of enzyme kinetics. This has been a fast tour of some complicated algebra but in our experience students who learn the derivations have a deeper appreciation for the concepts. Casual interviews of students from past classes have revealed that the Michaehs-Menten derivations have been the most useful aspect of this chapter. 

Fig. 6.2 Illustration of the steady-state concept in enzymatic catalysis. (From http //chemwiki.ucdavis. edu/ api/deki/files/54230/512px-Michaelis Menten S P E ES.svg.png revision—l).

The concept of an enzyme-substrate complex is fundamental to the appreciation of enzyme reactions and was initially developed in 1913 by Michaelis and Menten, who derived an equation that is crucial to enzyme studies. Subsequent to Michaelis and Menten several other workers approached the problem from different viewpoints and although their work is particularly useful in advanced kinetic and mechanistic studies, they confirmed the basic concepts of Michaelis and Menten. 

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