Michaelis-Menten mechanisms, of enzyme

The kinetics of enzyme reactions were first studied by the German chemists Leonor Michaelis and Maud Menten in the early part of the twentieth century. They found that, when the concentration of substrate is low, the rate of an enzyme-catalyzed reaction increases with the concentration of the substrate, as shown in the plot in Fig. 13.41. However, when the concentration of substrate is high, the reaction rate depends only on the concentration of the enzyme. In the Michaelis-Menten mechanism of enzyme reaction, the enzyme, E, and substrate, S, reach a rapid preequilibrium with the bound enzyme-substrate complex, ES ... 

Fig. 7.1 15. The basis of the Michaelis-Menten mechanism of enzyme action. Only a fragment of the large enzyme molecule E is shown. (Reprinted with permission from P. W. Atkins, Physical Chemistry, 5th ed., W. H. Freeman, 1994, p. 890, Fig. 25.12.)...

According to the Michaelis-Menten mechanism of enzyme-catalysed reactions in solution, the enzyme E binds reversibly to the substrate S in a pre-equilibrium to yield an intermediate solvated host/guest complex (E S)g, which then converts the substrate to the solvated products P, according to the following equation ... 

The Michaelis-Menten mechanism of enzyme activity models the enzyme with one active site that, weakly and reversibly, binds a substrate in homogeneous solution. It is a three-step mechanism. The first and second steps are the reversible formation of the enzyme-substrate complex (ES). The third step is the decay of the complex into the product. The steady-state approximation is applied to the concentration of the intermediate (ES) and its use simplifies the derivation of the final rate expression. However, the justification for the use of the approximation with this mechanism is suspect, in that both rate constants for the reversible steps may not be as large, in comparison to the rate constant for the decay to products, as they need to be for the approximation to be valid. The simplest form of the mechanism applies only when A h 2> k. Neverthele.ss, the form of the rate equation obtained does seem to match the principal experimental features of enzyme-catalyzed reactions it explains why there is a maximum in the reaction rate and provides a mechanistic understanding of the turnover number. The model may be expanded to include multisubstrate reaction rate and provides a mechanistic understanding of the turnover number. The model may be expanded to include multisubstrate reactions and inhibition. 

This appears to be a very formidable expression, but it can be simplified by noting the following. The formation of a precursor and successor complex is very similar to the well-known Michaelis-Menten mechanism of enzyme kinetics. ... 

Discuss the features and limitations of the Michaelis-Menten mechanism of enzyme action. 

Figure 4.4 Plot of enzyme complex concentration as a function of time for the Michaelis-Menten mechanism of Equations (4.22). The concentration of ES predicted from a kinetic simulation of Equations (4.22) is plotted as a solid line. The parameter values used are k+ = 1000M-1 sec-1,k i = 1.0sec-1,k+2 = 0.1 sec-1, and E0 = 0.1 mM. The left plot illustrates the fast-time kinetics. The fast-time variable n(r) predicted by Equation (4.29) is plotted as a dashed line.

In this so-called Michaelis-Menten mechanism, the enzyme E reacts reversibly with the substrate S to form an enzyme-substrate complex ES. This complex then decomposes irreversibly to form the product(s) and the regenerated enzyme. The rate law for this mechanism assumes one of two forms, depending on the relative rates of the two steps. In the most general case, the rates of the two steps are fairly comparable in magnitude. Here, ES decomposes as rapidly as it is... 

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. 

According to this expression, a plot of 1/v, versus l/[SJo will yield a straight line if the data follow the Michaelis-Menten mechanism. This line has a slope given by Km/Vmax, a y intercept of 1/Vmax, and an x intercept of -1 fKm. This is also illustrated in Fig. 4-7. Again, this treatment is valid when Eq. (4-107) applies whether or not the catalyst is an enzyme. The Lineweaver-Burk plot, Fig. 4-lb, is convenient for visualization but statistically unreliable for data fitting the form in Eq. (4-107) should be used for numerical analysis. 

Michaelis—Menten mechanism A model of enzyme catalysis in which the enzyme and its substrate reach a rapid pre-equilibrium with the bound substrate-enzyme complex. 

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 ... 

Note that written in this form Eq. (106) retains the linear dependency of the rate on the total enzyme concentration Ej, typical for most Michaelis Menten mechanisms. The dependence on the substrate concentrations is approximated by a sum of nonlinear logarithmic terms [85, 86, 318, 320],... 

The rate form of Eq. 57 and some of its generalizations are used to represent a number of widely different kinds of reactions. For example, in homogeneous systems this form is used for enzyme-catalyzed reactions where it is suggested by mechanistic studies (see the Michaelis-Menten mechanism in Chap. 2 and in Chap. 27). It is also used to represent the kinetics of surface-catalyzed reactions. 

For an enzyme-substrate system obeying the simple Michaelis Menten mechanism, the rate of product formation when the substrate concentration is very large, has the limiting value 0.02 mol dmJ. At a substrate concentration of250 mg dnu, the rate is half this value, K/K assuming that K2 K j, calculated ... 

Analytic solution of the Michaelis-Menten kinetic equation The simplest mechanism of enzyme reactions is of the form... 

The quantity Vmax also varies greatly from one enzyme to the next. If an enzyme reacts by the two-step Michaelis-Menten mechanism, Vmax = k2[Et], where k2 is rate-limiting. However, the number of reaction steps and the identity of the rate-limiting step(s) can vary from enzyme to enzyme. For example, consider the quite common situation where product release, EP — E + P, is rate-limiting. Early in the reaction (when [P] is low), the overall reaction can be described by the scheme... 

In the simple Michaelis-Menten mechanism in which there is only, one enzyme-substrate complex and all binding steps are fast, cat is simply the first-order rate constant for the chemical conversion of the ES complex to the EP... 

The most-studied enzyme in this context is chymotrypsin. Besides being well characterized in both its structure and its catalytic mechanism, it has the advantage of a very broad specificity. Substrates may be chosen to obey the simple Michaelis-Menten mechanism, to accumulate intermediates, to show nonproductive binding, and to exhibit Briggs-Haldane kinetics with a change of rate-determining step with pH. 

Although it is only for the simple Michaelis-Menten mechanism or in similar cases that Ku = Ks, the true dissociation constant of the enzyme-substrate complex, Km may be treated for some purposes as an apparent dissociation constant. For example, the concentration of free enzyme in solution may be calculated from the relationship... 

If an inhibitor I binds reversibly to the active site of the enzyme and prevents S binding and vice versa, I and S compete for the active site and I is said to be a competitive inhibitor. In the case of the simple Michaelis-Menten mechanism (equation 3.4, where Ku = Ks), an additional equilibrium must be considered, i.e.,... 

The theory predicts that unless there is a change of rate-determining step with pH, the pH dependence of kcJKM for all non-ionizing substrates should give the same pKa that for the free enzyme. With one exception, this is found (Table 5.2). At 25°C and ionic strength 0.1 M, the pKa of the active site is 6.80 0.03. The most accurate data available fit very precisely the theoretical ionization curves between pH 5 and 8, after allowance has been made for the fraction of the enzyme in the inactive conformation. The relationship holds for amides with which no intermediate accumulates and the Michaelis-Menten mechanism holds, and also for esters with which the acylenzyme accumulates. 

The simplest mechanism of enzyme action is formation of a temporary compound ES of enzyme E and substrate S followed by decomposition of ES into E and a product P. When ES is assumed to attain equilibrium, the corresponding rate equation is named after Michaelis and Menten. It is derived in problem P8.04.02. 

Enzymes are a special kind of catalyst, proteins of MW 6,000—400,000 which are found in living matter. They have two remarkable properties (1) they are extremely selective to the given substrate and (2) they are extraordinarily effective in increasing the rates of reactions. Thus, they combine the recognition and amplification steps. A general, enzymatically catalyzed reaction can be described by the Michaelis-Menten mechanism, in which E is the enzyme, S is the substrate, and P is the product, formed from the intermediate complex ES. 

In an enzyme reaction, initially free enzyme E and free substrate S in their respective ground states initially combine reversibly to an enzyme-substrate (ES) complex. The ES complex passes through a transition state, AGj, on its way to the enzyme-product (EP) complex and then on to the ground state of free enzyme E and free product P. From the formulation of the reaction sequence, a rate law, properly containing only observables in terms of concentrations, can be derived. In enzyme catalysis, the first rate law was written in 1913 by Michaelis and Menten therefore, the corresponding kinetics is named the Michaelis-Menten mechanism. The rate law according to Michaelis-Menten features saturation kinetics with respect to substrate (zero order at high, first order at low substrate concentration) and is first order with respect to enzyme. 

What reactions have linear mechanisms Primarily these are enzyme reactions [43, 44]. A typical scheme for enzyme catalysis is the Michaelis Menten mechanism (1) E + A -> ES (2) ES - P + S, where S and P are the initial substrate and product, respectively, and E and ES are various forms of enzymes. 

In the same vein and under dimensionally restricted conditions, the description of the Michaelis-Menten mechanism can be governed by power-law kinetics with kinetic orders with respect to substrate and enzyme given by noninteger powers. Under quasi-steady-state conditions, Savageau [25] defined a fractal Michaelis constant and introduced the fractal rate law. The behavior of this fractal rate law is decidedly different from the traditional Michaelis-Menten rate law ... 

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. 

For this example of the simple Michaelis-Menten mechanism, we hypothesize that an intermediate complex ES is formed between one molecule of sucrose and one molecule of invertase. This complex then reacts with a molecule of water to yield the products P] and 2 and to regenerate a molecule of the free enzyme. Thus the rate of product formation is given by... 

A number of questions might be addressed in the discussion of the results. How reproducible are the initial rate measurements (Note that runs D5 and E3 are duplicates also, runs El and the standard assay for enzyme activity have identical initial concentrations.) Are the enzyme-catalyzed data compatible with the Michaelis-Menten mechanism Do the data from both runs C and D follow apparent zero-order kinetics, and how does this agree with expectations based on comparing (S) with KJ Which of the two types of analysis, Lineweaver-Burk or Eadie-Hofstee, seems to give the better results and why How does 2 agree with the estimate of the turnover number based on the specific activity Are the acid-catalyzed data consistent with the rate law given in Eq. (10) ... 

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