Michaelis-Menten equation limitations

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 ... 

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 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 ... 

A major limitation of the linearized forms of the Michaelis-Menten equation is that none provides accurate estimates of both Km and Vmax. Furthermore, it is impossible to obtain meaningful error estimates for the parameters, since linear regression is not strictly appropriate. With the advent of more sophisticated computer tools, there is an increasing trend toward using the integrated rate equation and nonlinear regression analysis to estimate Km and While this type of analysis is more complex than the linear approaches, it has several benefits. First, accurate nonbiased estimates of Km and Vmax can be obtained. Second, nonlinear regression may allow the errors (or confidence intervals) of the parameter estimates to be determined. 

In the steady-state approach (equations (35) and (36)), no attempt is made to isolate the adsorption step from the internalisation of solutes. In this case, a Langmuir adsorption via membrane carriers is coupled to an irreversible and rate-limiting internalisation of the solute carrier complex [186], The process can be described by the Michaelis-Menten equation ... 

It has been found experimentally that in most cases v is directly proportional to the concentration of enzyme [.E0] and that v generally follows saturation kinetics with respect to the concentration of substrate [limiting value called Vmax. This is expressed quantitatively in the Michaelis-Menten equation originally proposed by Michaelis and Menten. Km can be seen as an apparent dissociation constant for the enzyme-substrate complex ES. The maximal velocity Vmax = kcat E0. ... 

The only rate-limiting factor in a coupled assay should be the concentration of the initial and linking products and all other reagents should be in excess. The role of the auxiliary and indicator enzymes is essentially that of a substrate assay system and under optimum assay conditions the rate of the indicator reaction should be equal to the rate of formation of the initial product. The indicator reaction must be capable of matching the different test reaction rates and its velocity can be defined by the Michaelis-Menten equation in the usual way ... 

THE MICHAELIS-MENTEN EQUATION AS A LIMITING CASE OF THE STEADY STATE EQUATION. To achieve a rapid equilibrium between E and EX, ki[S] and k2 must each be much greater than ks. [Note the rate constant ki is a bimolecular rate constant with units of molarity seconds, and we must use ki[S]... 

This rule has found a number of uses in enzymology. A simple example applies to the standard Michaelis-Menten equation, v = Lmax[S]/(ii m + [S]). Here, /(x) = Lmax[S] and g(x) = + [S]. Hence, limit... 

A key point should be to identify the rate-limiting step of the polymerization. Several studies indicate that the formation of the activated open monomer is the rate-limiting step. The kinetics of polymerization obey the usual Michaelis-Menten equation. Nevertheless, all experimental data cannot be accounted for by this theory. Other studies suggest that the nature of the rate-limiting step depends upon the structure of the lactone. Indeed, the reaction of nucleophilic hydroxyl-functionalized compounds with activated opened monomers can become the rate-limiting step, especially if stericaUy hindered nucleophilic species are involved. 

In chapter 8 the most generally nsed kinetic eqnations for describing the consnmption of snbstrate as a resnlt of biocatalysis have been given and/or derived. In biocatalysis, in the absence of limitation of the rate of consnmption by diffusion of substrate, the Michaelis-Menten equation usually is a good description ... 

The rate of cell growth is influenced by temperature, pH, composition of medium, rate of air supply, and other factors. In the case that all other conditions are kept constant, the specific growth rate may be affected by the concentration of a certain specific substrate (the limiting substrate). The simplest empirical expression for the effect ofthe substrate concentration on the specific growth rate is the following Monod equation, which is similar in form to the Michaelis-Menten equation for enzyme reactions ... 

This is the Michaelis-Menten equation, the rate equation for a one-substrate enzyme-catalyzed reaction. It is a statement of the quantitative relationship between the initial velocity V0, the maximum velocity Vnmx, and the initial substrate concentration [S], all related through the Michaelis constant Km. Note that Km has units of concentration. Does the equation fit experimental observations Yes we can confirm this by considering the limiting situations where [S] is very high or very low, as shown in Figure 6-12. 

In this case, most of the enzyme is in the EP form at saturation, and Fmax = /c3[Et]. It is useful to define a more general rate constant, kcat, to describe the limiting rate of any enzyme-catalyzed reaction at saturation. If the reaction has several steps and one is clearly rate-limiting, fccat is equivalent to the rate constant for that limiting step. For the simple reaction of Equation 6-10, kCat = k2. For the reaction of Equation 6-25, kcat = k3. When several steps are partially rate-limiting, kcat can become a complex function of several of the rate constants that define each individual reaction step. In the Michaelis-Menten equation, kcat = Fmax/[Et], and Equation 6-9 becomes... 

Km and Umax have different meanings for different enzymes. The limiting rate of an enzyme-catalyzed reaction at saturation is described by the constant kcat, the turnover number. The ratio kcat/Km provides a good measure of catalytic efficiency. The Michaelis-Menten equation is also applicable to bisubstrate reactions, which occur by ternary-complex or Ping-Pong (double-displacement) pathways. 

Here k2 and also /c4 and k5, are second-order rate constants. The release of product, as determined by /c4 and k5, may be rate-limiting. At zero time the reverse reactions may be ignored, and steady-state analysis shows that the Michaelis-Menten equation (Eq. 9-16b) will be replaced by Eq. 9-39. Here, D is a constant and A is also constant if X is present at a fixed concentration. 

At very high substrate concentrations deviations from the classical Michaelis-Menten rate law are observed. In this situation, the initial rate of a reaction increases with increasing substrate concentration until a limit is reached, after which the rate declines with increasing concentration. Substrate inhibition can cause such deviations when two molecules of substrate bind immediately, giving a catalytically inactive form. For example, with succinate dehydrogenase at very high concentrations of the succinate substrate, it is possible for two molecules of substrate to bind to the active site and this results in non-functional complexes. Equation S.19 gives one form of modification of the Michaelis-Menten equation. 

However, the further increases in the substrate concentration to 15mM decreased the initial reaction rate. This behavior may be due to substrate or product inhibition. Since the Michaelis-Menten equation does not incorporate the inhibition effects, we can drop the last two data points and limit the model developed for the low substrate concentration range only (Cs < lOmM). Figure 2.8 shows the three plots prepared from the given data. The two data points which were not included for the linear regression were noted as closed circles. 

When the rate of diffusion is very slow relative to the rate of reaction, all substrate will be consumed in the thin layer near the exterior surface of the spherical particle. Derive the equation for the effectiveness of an immobilized enzyme for this diffusion limited case by employing the same assumptions as for the distributed model. The rate of substrate consumption can be expressed by the Michaelis-Menten equation. 

If tj - 1 the reaction is not, or not significantly, influenced by pore diffusion. If tj pore diffusion is the sole dominating rate-limiting step. For the determination of Tj, the combined diffusion and reaction equation has to be solved. With a sequential model of the two rate phenomena, diffusion and reaction, and with the assumption of spherical geometry and validity of the Michaelis-Menten equation for the en2yme kinetics, r = kcat[E] [S]/(JCM + [S]), Eq. (5.58) results. 

The equations describing increase in cell density [Eqs. (8.3)-(8.8)] so far do not contain any information about the nature and concentration of any substrate such as the C-source. As the specific growth rate /i tends to depend on quality and amount of substrate, however, we require a growth model which provides the function /i = jU([S]). The most widely used growth model is the Monod model (Monod, 1950) which assumes that only one substrate limits cell growth and proliferation. The corresponding equation [Eq. (8.9), in which /imax is the maximum specific growth rate [h-1]] reads very similarly to the Michaelis-Menten equation. 

Describing the reaction in kinetic terms, let us apply to the fact that the intermediate perFTPhPFe3+00H/Al203 formation stage (7.7) is fast, the epoxide formation stage (7.8) is slow and, consequently, limiting. For kinetic simulation of propylene oxidation to epoxide, this gives an opportunity to apply the Michaelis-Menten equation in Linuver-Berk coordinates ... 

Several drugs, including salicylate (in overdose), alcohol, and possibly some hydrazines and other drugs which are metabolised by acetylation, have saturable elimination kinetics, but the only significant clinical example is phenytoin. With this drug, capacity-limited elimination is complicated further by its low therapeutic index. A 50% increase in the dose of phenytoin can result in a 600% increase in the steady-state blood concentration, and thus expose the patient to potential toxicity. Capacity-limited pathways of elimination lead to plasma concentrations of drugs which can be described by a form of the Michaelis-Menten equation. In such cases, the plasma concentration at steady state is given by... 

The Henri-Michaelis-Menten equation describes the curve obtained when initial velocity is plotted versus substrate concentration. The curve shown in Figure 4-7 is a right rectangular hyperbola with limits of and - K . The curvature is fixed regardless of the values of and V mxx- Consequently, the ratio of substrate concentrations for any two fractions of Vj m is constant for all enzymes that obey Henri-Michaelis-Menten kinetics. For example, the ratio of substrate required for 90% of Vmat to the substrate required for... 

This is the Michaelis-Menten equation, but with K , modified by a term including the inhibitor concentration and inhibitor constant. is unaltered. Therefore, curves of v against [5] in the presence and absence of inhibitor reach the same limiting value at high substrate concentrations, but when the inhibitor is present, JC, is apparently greater. Plots of 1/v against 1/[S] with and without inhibitor cut the ordinate at the same point but have different slopes and intercepts on the abscissa (Figure 8-9). 

If we wish to determine the amount of enzyme, the region of high substrate concentration is employed—where Equation 29-21 applies—and the rate is independent of substrate concentration. The limiting rate of the reaction at large values of [S] is the maximum rate that can be achieved at a given enzyme concentration, Vmax us indicated in the figure. It can be shown that the value of the substrate concentration at exactly VraioJ is equal to the Michaelis constant /fa,. Example 29-4 illustrates the use of the Michaelis-Menten equation. 

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