Michaelis-Menten rates

The Effect of Various Types of Inhibitors on the Michaelis-Menten Rate Equation and on Apparent K, and Apparent F ,ax ... 

For Sa << Km, the rate model is reduced to a first-order rate equation The Michaelis-Menten rate equation is ... 

Develop a suitable rate expression using the Michaelis-Menten rate equation and the quasi-steady-state approximations for the intermediate complexes formed. 

The Michaelis-Menten rate equation for enzyme reactions is typically written as the rate of formation of product (Eq. 19a). This equation implies that 1/Rate (where rate is the rate of formation of product) depends linearly on the inverse of the substrate concentration [S]. This relation allows KM to be determined. Derive this equation and sketch 1/Rate against 1/[S]. Label the axes, the y-intercept, and the slope with their corresponding functions. 

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. 

Although equation 7.3.28 and, in particular, equation 7.3.29 are known as Michaelis-Menten rate expressions, these individuals used a somewhat different approach to arrive at this mathematical form for an enzymatic rate expression (35). 

Assuming that the reactions are reversible and that a one-substrate enzyme-catalyzed reaction is being studied, one can derive the Michaelis-Menten rate ... 

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,... 

P4.04.32. MICHAELIS-MENTEN RATE EQUATION For a reaction with rate equation,... 

To obtain an expression for the Michaelis Menten rate equation, the dissociation of the product from the complex needs to be evaluated. Using a... 

Figure 7. The Michaelis Menten rate equation as a function of substrate concentration S (in arbitrary units). Parameters are Km 1 and Vm 1. A A linear plot. B A semilogarithmic plot. At a concentration S Km, the rate attains half its maximal value Vm.

For power-law functions the (scaled) elasticities do not depend on the substrate concentration, that is, unlike Michaelis Menten rate equations, power-law functions will not saturate for increasing substrate concentration. 

Figure 27. Interpretation of the saturation parameter. Shown is a Michaelis Menten rate equation (solid line) and the corresponding saturation parameter d (dashed line). For small substrate concentration S Km the reaction acts in the linear regime. For increasing concentrations the saturation parameter d ... 

The generalized parameters are invariant with respect to different functional forms of the rate equation. All results hold for a large class of biochemical rate functions [84], For example, the Michaelis Menten rate function used in Eq. (133) is not the only possible choice. A number of alternative rate equations are summarized in Table VI. Although in each case the specific kinetic parameters may differ, each rate equation is able to generate a specified partial derivative and is thereby consistent with results obtained from an analysis of the corresponding Jacobian. Note that, obviously, not each rate equation is capable to generate each possible Jacobian. However, vice versa, for each possible Jacobian there exists a class of rate equations that is consistent with the Jacobian. 

Given the Michaelis-Menten rate form to represent enzyme-substrate reactions (or catalyst-reactant reaction)... 

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. 

Originally published in 1913 as a rate law for enzymatic sugar inversion [19], the Michaelis-Menten rate equation is also used frequently for describing homogeneously catalyzed reactions. It describes a two-step cycle (Eqs. (2.34) and (2.35)) the catalyst (the enzyme, E) first reacts reversibly with the substrate S, forming an enzyme-substrate complex ES (a catalytic intermediate). Subsequently, ES decomposes, giving the enzyme E and the product P. This second step is irreversible. 

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. 

The epoxidation of alkenes by sodium hypochlorite in the presence of manganese porphyrins under phase-transfer conditions has been thoroughly studied. Kinetic studies of this reaction revealed a Michaelis-Menten rate equation. As in Scheme 12, the active oxidant is thought to be a high-valent manganese( V)-oxo-porphyrin complex which reversibly interacts with the alkene to form a metal oxo-alkene intermediate which decomposes in the rate determining step to the epoxide and the reduced Mn porphyrin. Shape selective epoxidation is achieved when the sterically hindered complex Mn(TMP)Cl is used as the catalyst in the hypochlorite oxidation. ... 

The time-dependent transformation of N compounds by microbial oxidation and reduction reactions can also be described by Michaelis-Menten rate expression... 

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.

---