The Michaelis-Menten equation

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. ... 

Substituting this relationship into the expression for v gives the Michaelis-Menten equation... 

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]... 

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 ... 

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. 

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),... 

A classical non-linear model of chemical kinetics is defined by the Michaelis-Menten equation for rate-limited reactions, which has already been mentioned in Section 39.1.1 ... 

Usually, one plots the initial rate V against the initial amount X, which produces a hyperbolic curve, such as shown in Fig. 39.17a. The rate and amount at time 0 are larger than those at any later time. Hence, the effect of experimental error and of possible deviation from the proposed model are minimal when the initial values are used. The Michaelis-Menten equation can be linearized by taking reciprocals on both sides of eq. (39.114) (Section 8.2.13), which leads to the so-called Lineweaver-Burk form ... 

Non-linear models, such as described by the Michaelis-Menten equation, can sometimes be linearized by a suitable transformation of the variables. In that case they are called intrinsically linear (Section 11.2.1) and are amenable to ordinary linear regression. This way, the use of non-linear regression can be obviated. As we have pointed out, the price for this convenience may have to be paid in the form of a serious violation of the requirement for homoscedasticity, in which case one must resort to non-parametric methods of regression (Section 12.1.5). 

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