Michaelis-Menten rate law

Repeat the derivation of the Michaelis-Menten rate law, assuming that there is a pre-equilibrium between the bound and the unbound states of the substrate. 

Show that the results conform to the Michaelis-Menten rate law, and determine the values of the kinetics parameters Km, and kr. 

The full Michaelis-Menten rate law that one can derive on this basis is,... 

What are the assumptions made when describing a catalyzed reaction by a Michae-lis-Menten type rate law Write down the Michaelis-Menten rate law and discuss the various terms by using a graphical representation. 

This ratio is of fundamental importance in the relationship between enzyme kinetics and catalysis. In the analysis of the Michaelis-Menten rate law (equation 5.8), the ratio cat/Km is an apparent second-order rate constant and, at low substrate concentrations, only a small fraction of the total enzyme is bound to the substrate and the rate of reaction is proportional to the free enzyme 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. 

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 ... 

Often the key entity one is interested in obtaining in modeling enzyme kinetics is the analytical expression for the turnover flux in quasi-steady state. Equations (4.12) and (4.38) are examples. These expressions are sometimes called Michaelis-Menten rate laws. Such expressions can be used in simulation of cellular biochemical systems, as is the subject of Chapters 5, 6, and 7 of this book. However, one must keep in mind that, as we have seen, these rates represent approximations that result from simplifications of the kinetic mechanisms. We typically use the approximate Michaelis-Menten-type flux expressions rather than the full system of equations in simulations for several reasons. First, often the quasi-steady rate constants (such as Ks and K in Equation (4.38)) are available from experimental data while the mass-action rate constants (k+i, k-i, etc.) are not. In fact, it is possible for different enzymes with different detailed mechanisms to yield the same Michaelis-Menten rate expression, as we shall see below. Second, in metabolic reaction networks (for example), reactions operate near steady state in vivo. Kinetic transitions from one in vivo steady state to another may not involve the sort of extreme shifts in enzyme binding that have been illustrated in Figure 4.7. Therefore the quasi-steady approximation (or equivalently the approximation of rapid enzyme turnover) tends to be reasonable for the simulation of in vivo systems. 

A number of methods have been developed to estimate the parameters in the Michaelis-Menten rate law. In each instance, the data consist of initial rate determinations for different values of the reactant concentrations. In one method, the rate is plotted as a function of substrate concentration in rectangular coordinates. In this case the parameter is the asymptotic value of the rate for high substrate concentrations. The parameter is given by the concentration of substrate that produces a reaction rate equal to V /2. This method is illustrated in Figure 3. An accurate estimation of the high-substrate asymptote for this type of curve often is difficult to obtain as a result, other methods of estimating the parameters have been developed. 

Figure 3. Michaelis-Menten rate law expressed in Cartesian coordinates. The maximum velocity of the reaction V is the asymptotic value of the rate v at high concentrations of the substrate S. The parameter /C is given by the value of S that yields half the maximum velocity, or V /2.

Figure 4. Michaelis-Menten rate law expressed in double-reciprocal coordinates. In this plot, which is attributed to Lineweaver-Burk, 1 /V is given by the intercept on the //vaxis. The parameter K can be obtained from the slope of the straight line or the intercept on the negative i/S axis.

Finally, it should be noted that while the Michaelis-Menten Formalism may be appropriate for many isolated enzymes in vitro, this does not imply that the resulting rate law for the reaction will be the classical Michaelis-Menten rate law [Eqn. (22)]. Hill et al. (1977) have made a careful assessment of this issue and, on the basis of their results, have come to question whether the simple Michaelis-Menten rate law fits any enzyme that is examined with sufficient care. The tendency to ignore inconsistencies, and continue to treat rate laws as if they were the classical case, indicates that the grip of the conventional Michaelis-Menten paradigm is very strong. We shall examine this point from another perspective in the following section. 

The Michaelis-Menten Formalism did not anticipate the type of enzyme-enzyme organization described above. One of its fundamental assumptions has been that complexes do not occur between different forms of an enzyme or between different enzymes (Segal, 1959 Webb, 1963 Cleland, 1970 Segel, 1975 Wong, 1975). From the derivation of the classical Michaelis-Menten rate law, it can be seen that such complexes must be excluded or they will destroy the linear structure of the kinetic equations. 

Behari et al. (1982b) find that ceric oxidation of cyclohexanone and methylcyclohex-anone in sulfuric acid solutions do conform to the Michaelis-Menten rate law. By comparing the rate of oxidation with that for enolization (determined by reaction rate for the substrate with iodine), the authors establish that the reaction must involve the ketonic form of the substrate. Earlier results reported by Benson (1976) agree with this assessment and suggest further that a C-H bond is broken in the rate-determining step. The authors are unable to resolve the rate and equilibrium parameters for electron transfer and precursor complex stability. The cyclic ketones are oxidized nearly an order of magnitude faster and under milder conditions than the aliphatic ketones. 

Although perhaps more properly considered among the organic substrates, the following are reports of the oxidation of a few organosulfur compounds. Dimethyl sulfoxide (DMSO) is a common polar nonaqueous solvent which solvates the lanthanides quite admirably. The rate of oxidation of DMSO by Ce(lV) in perchloric acid solutions is the subject of a report from Pratihari et al. (1976). The reaction conforms to the Michaelis-Menten rate law and the rate is enhanced by increasing acidity. The acid concentration dependence is attributed to hydrolysis of... 

An equation equivalent to the Michaelis-Menten rate law has been derived earlier by Henri (1902,1903) indeed, Michaelis and Menten in their original publication in 1913 honored the contribution of Henri. Therefore, in appreciation of the work of Henri, it is often also referred to as the Henri-Michaelis-Menten equation. 

Because of the rapid interconversion, we may assume that a steady state is established for [W ] so that d[Wk]/dt 0 (Such an assumption is used, for example, in obtaining the Michaelis-Menten rate law for enzyme kinetics.) Thi implies... 

It can be seen that if the concentration of one substrate is much larger than the other and remains essentially constant, then equation 9.19 will behave as a Michaelis-Menten rate law. The partieipation of a cofactor in a singlesubstrate enzymatic reaction (or a dual-substrate enzymatic reaction with [5]. STjg) can be modeled via the sequence given in steps 9.4-9.9. If the substrate concentration is considered to be essentially constant, then equation 9.19 exhibits a Michaelis-Menten dependence on cofactor concentration. 

Analysis This example demonstrated how to evaluate the parameters V m and Ky in the Michaelis-Menten rate law from enzymatic reaction data. Two techniques were used a Lineweaver-Burk plot and non-linear regression. It ras also shown how (he analysis could be carried out using Hanes-Woolf and Eadie-Hofstee plots. 

Analysis This example shows a straightforward Chapter 3 type calculation of the hatch reactor time to achieve a certain conversion X for an enzymatic-reaction with a Michaelis-Menten rate law. This batch reaction time is v y shml consequently, a continuous flow reactor would be better suited for this reaction. 

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