Michaelis-Menten equations

A number of multimeric enzymes do not obey the classic Michaelis-Menten kinetics, since the value of their kinetic properties, and kcat, depend on the specific binding of small molecules called effectors. Such regulatory enzymes have, in addition to the catalytic sites, regulatory sites which bind effectors and alter so the properties of the catalytic site. If these effectors are the same as S, they are called homotropic or allosteric (Monod et al., 1963). 

Homotropic effectors enhance the binding of subsequent S molecules, i.e. they exhibit a cooperative S binding, resulting in a sigmoid relationship between v and [S]. 

Methods to determine the parameters of the Michaelis-Menten equation 

Thus Km and are determined from the slope and the intercept of the straight line by measuring the concentration of the S used (or P produced) several times during the reaction. The usefulness of this approach has been shown repeatedly (e.g. Duggleby and Morrison, 1978). Systematic errors may change the value of 

Another method to obtain estimates for Km and is the rearrangement of the Michaelis-Menten equation to a linear form. The estimation for the initial velocities, Vo, from progress curves is not a particularly reliable method. A better way to estimate Vn is by the integrated Michaelis-Menten equation (Cornish-Bowden, 1975). Nevertheless, the graphical methods are popular among enzymolo-gists. The three most common linear transformations of the Michaelis-Menten equation are the Lineweaver-Burk plot of 1/Vo vs. 1/[S] (sometimes called the double-reciprocal plot), the Eadie-Hofstee plot, i.e. v vs. vo/[S], and the Hanes plot, i.e., [SJ/vo vs. [S] (Fig. 9.3). 

Because the reaction of urea and urea.se is carried out in aqueous solution, water is, of course, in excess, and the concentration of water is therefore considered constant. Let 

Dividing the numerator and denominator of Equation (7-24) by jtj, obtain a form of the Michaelis-Menten equation.  

The parameter (, .31 is also referred to as the turnover ntonber. It is the number of sub.strate molecules converted to product in a given time on a single-enzyme molecule when the enzyme is saturated with substrate (i.e,. ail the active sites on the enzyme are occupied. S Kvi). For example, turnover number for the decomposition H Oi by the enzyme catalase is 40 x 10 s . That is, 40 million molecules of are decomposed every second on a single-enzyme molecule saturated wdth HiO . The constant (mol/dm ) is called the Michael is constant and for simple. systems is a measure of the 

Micfiaaiis attraction of the enzyme for its substrate, so it s also called the affinin- c conuant stant. The Michaelis constant. Ky/[. for the decomposition of H2O2 discus earlier is 1.1 M while that for chymotrypsin is 0.1 M.  

in addition, we let represent the maximum rate of reaction fc given total enzyme concentration, 

Let us consider a single-substrate reaction. Enzyme E reacts with substrate A to form an in- 

In order to determine the catalytic activity of the enzyme, the decrease in substrate concentration or the increase in product concentration as a function of time can be measured. The activity curve obtained (Fig. 2.21) has the following regions  

The maximum activity which occurs for a few msec until an equilibrium is reached between the rate of enzyme-substrate formation and rate of breakdown of this complex. 

Measurements in this pre-steady state region which provide an insight into the reaction steps and mechanism of catalysis are difficult and time consuming. Hence, further analysis fo the pre-steady state will be ignored. 

The usual procedure is to measure the enzyme activity when a steady state has been reached. In the steady state the intermediary complex concentration remains constant while the concentration of the substrate and end product are changing. For this state, the following is valid  

A plot of this type is sometimes called a Michaelis-Menten plot. At low substrate concentration, (S), 

What does represent Consider the case when the substrate concentration is such that the reaction rate is equal to one-half the maximum rate, 

In 1902, Adrian Brown proposed an explanation for the rate of hydrolysis of sucrose to glucose and fructose catalysed by the yeast enzyme j5-fructofuranosidase, based on the formation of a complex between the enzyme and its substrate [2]. Today, the mechanism involving such an intermediate is expressed as 

In general, the concentrations of free enzyme and of the enzyme-substrate complex are not known, but the total concentration of enzyme is equal to its initial concentration, [E]q, and 

The value of [E] can be eliminated from the above two equations to give 

This can be presented in a more convenient form dividing the numerator and the denominator of the fraction by 

When the concenhation of the subshate is sufficiently small, [S] and the reaction rate is first order with respect to the substrate 

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Since the El complex does not yield product P, and I competes with S for E, there is a state of competitive inhibition. By analogy to the Michaelis-Menten equation ... 

Figure 11.1 A plot of the reaction rate as a function of the substrate concentration for an enzyme catalyzed reaction. Vmax is the maximal velocity. The Michaelis constant. Km, is the substrate concentration at half Vmax- The rate v is related to the substrate concentration, [S], by the Michaelis-Menten equation ...

Equation 1-108 can be considered as the Michaelis-Menten equation, where is the Michaelis constant and represented as... 

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. 

The Michaelis-Menten Equation 11-15 is not well suited for estimation of the kinetic parameters and Reananging Equation 11-15 gives various options for plotting and estimating the parameters. 

Saturation kinetics are also called zero-order kinetics or Michaelis-Menten kinetics. The Michaelis-Menten equation is mainly used to characterize the interactions of enzymes and substrates, but it is also widely applied to characterize the elimination of chemical compounds from the body. The substrate concentration that produces half-maximal velocity of an enzymatic reaction, termed value or Michaelis constant, can be determined experimentally by graphing r/, as a function of substrate concentration, [S]. 

Equation (3-150) is the Michaelis-Menten equation, Vm is the maximum velocity (for the enzyme concentration ,), and is the Michaelis constant. 

The Michaelis-Menten equation is, like Eq. (3-146), a rectangular hyperbola, and it can be cast into three linear plotting forms. The double-reciprocal form, Eq. (3-152), is called the Lineweaver-Burk plot in enzyme kinetics. ... 

The Michaelis-Menten equation (14.23) describes a curve known from analytical geometry as a rectangular hyperbola. In such curves, as [S] is increased,... 

Linear Plots Can Be Derived from the Michaelis-Menten Equation... 

Because of the hyperbolic shape of versus [S] plots, Vmax only be determined from an extrapolation of the asymptotic approach of v to some limiting value as [S] increases indefinitely (Figure 14.7) and is derived from that value of [S] giving v= V(nax/2. However, several rearrangements of the Michaelis-Menten equation transform it into a straight-line equation. The best known of these is the Lineweaver-Burk double-reciprocal plot ... 

Taking the reciprocal of both sides of the Michaelis-Menten equation. Equation (14.23), yields the equality... 

The Hanes-Woolf plot is another rearrangement of the Michaelis-Menten equation that yields a straight line ... 

FIGURE 14.10 A Hanes-Wolff plot of [S]/l/versus [S], another straight-line rearrangement of the Michaelis-Menten equation. 

If the kinetics of the reaction disobey the Michaelis-Menten equation, the violation is revealed by a departure from linearity in these straight-line graphs. We shall see in the next chapter that such deviations from linearity are characteristic of the kinetics of regulatory enzymes known as allosteric enzymes. Such regulatory enzymes are very important in the overall control of metabolic pathways. 

It is revealing to compare the equation for the uninhibited case. Equation (14.23) (the Michaelis-Menten equation) with Equation (14.43) for the rate of the enzymatic reaction in the presence of a fixed concentration of the competitive inhibitor, [I]... 

If the three-parameter Michaelis-Menten equation is divided by C i, it becomes the same as the three-parameter Langmuir-I linshelwood equation where 1/Cm = Ka. Both these rate equations can become quite complex when more than one species is competing with the reactant(s) for the enzyme or active sites on the solid catalyst. 

In evaluation of kinetic parameters, the double reciprocal method is used for linearisation of the Michaelis-Menten equation (5.7.3). 

The above equation can be transformed into the Michaelis-Menten equation by multiplying the numerator and denominator by Km ... 

Membrane module, 369-373 Methylophilus methylotrophus, 338 Michaelis-Menten equation, 109, 137 Microfiltration, 357... 

With the Michaelis-Menten equation, there is no integrated solution for the concentration, but only for the time... 

Kinetic data fitting the rate equation for catalytic reactions that follow the Michaelis-Menten equation, v = k A]/(x + [A]), with[A]0 = 1.00 X 10 J M, k = 1.00 x 10 6 s 1, and k = 2.00 X 10-J molL1. The left panel displays the concentration-time profile on the right is the time lag approach. 

The rates of many catalyzed reactions depend upon substrate concentrations, as shown in Fig. 4-7. The rate at high substrate concentrations is zeroth-order with respect to [S], falling until it shows a first-order dependence in the limit of low [S], This pattern is that of a rectangular hyperbola, defined by an empirical relation known as the Michaelis-Menten equation. 

A noncompetitive inhibitor is one that binds to both E and E S. If both dissociation constants are the same, the Michaelis-Menten equation is... 

Enzyme kinetics. Data for reactions that follow the Michaelis-Menten equation are sometimes analyzed by a plot of v,/tA]o versus l/[A]o. This treatment is known as an Eadie-Hofstee plot. Following the style of Fig. 4-7b, sketch this function and label its features. 

Michaelis-Menten equation, 35, 90-94 Microscopic reversibility, principle of, 172-175... 

A sink flux that has a weaker than proportional dependence on the content M of the emitting reservoir is often described by the Michaelis-Menten equation ... 

Coe and Bessell and coworkers studied the metabolic fates of 2-deoxy-2-fluoro-D-glucose (2DFG) and related compounds by using yeast hexokinase (as a model for mammalian hexokinase), and determined the kinetic constants K and V ) of the Michaelis-Menten equation D-glucose 0.17 (K in mAf)> 1 00 (relative value, D-glucose taken as 1) 2DG 0.59 0.11, 0.85 2DFG 0.19 0.03, 0.50 2-deoxy-2-fluoro-D-mannose (2DFM) 0.41 0.05, 0.85 2-deoxy-2,2-difluoro-D-nraZ>//Jo-hexose... 

The Michaelis-Menten equation (29) illustrates in mathematical terms the relationship between initial reaction velocity V and substrate concentration [S], shown graphically in Figure 8-3. 

The Michaelis constant is the substrate concentration at which is half the maximal velocity (V 3 /2) attainable at a particular concentration of enzyme. thus has the dimensions of substrate concentration. The dependence of initial reaction velocity on [S] and may be illustrated by evaluating the Michaelis-Menten equation under three conditions. 

A Linear Form of the Michaelis-Menten Equation Is Used to Determine... 

The direct measurement of the numeric value of and therefore the calculation of often requires im-practically high concentrations of substrate to achieve saturating conditions. A linear form of the Michaelis-Menten equation circumvents this difficulty and permits and to be extrapolated from initial velocity data obtained at less than saturating concentrations of substrate. Starting with equation (29),... 

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