Menten kinetics

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. 

If the initial concentration of the enzyme is [E]0, then at any timet, [E]0= [E] + [ES]. Thus the reaction rate depends on the enzyme concentration, even though the enzyme remains unchanged at the end of the cycle. The product formation rate and the change in concentration of the catalytic intermediate ES are then given by Eq. (2.36). 

Assuming the steady-state approximation for [ES] (which is a reasonable assumption because ES is a catalytic intermediate), we obtain Eq. (2.37). 

We see that the rate of the enzyme-catalyzed reaction depends linearly on the enzyme concentration, and in a more complicated way on the substrate concentration. Thus, when [S] Km, (Eq. (2.41)) reduces to v = k2[E]0, and the reaction is zero order in [S], This means that there is so much substrate that all of the enzyme s active sites are occupied. It also means that [S] remains effectively unchanged, even though products are formed. This situation is known as saturation kinetics. The value k2[E]0 is also called the maximum velocity of the enzymatic reaction, and written as vmax. 

Conversely, if [S] C Km, (Eq. (2.42)) reduces to v = (k2/Km)[E]o[S]. This means that the active sites on the enzyme are effectively unoccupied. The ratio k2/Km is also known as the enzyme s specificity constant, a measure of the enzyme s affinity for different substrates. Thus, if the same enzyme can catalyze the reaction of two substrates, S and S, the relative rates of these two reactions are compared using (k2/Km)s (k2/Km)s-. Because the specificity constant reflects both affinity and catalytic ability, it is also used for comparing different enzymes. 

A simple example is the so-called Michaelis-Menten kinetics for enzymatic reactions A + E +C- B + E, which, when the pseudo-steady-state hypothesis is invoked, gives for the concentration of A, for instance, a, 

Moreover, because t — as a — 0, the last term dominates after a time and so gives an explicit approximation 

This shows the obvious limits of first-order reaction with rate constant kIK if K a0 and zero-order as K - 0. 

Interestingly, a fully appropriate model was developed at the same time as the Langmuir model using a similar basic approach. This is the Michaelis-Menten equation which has proved to be so useful in the interpretation of enzyme kinetics and, thereby, understanding the mechanisms of enzyme reactions. Another advantage in using this model is the fact that a graphical presentation of the data is commonly used to obtain the reaction kinetic parameters. Some basic concepts and applications will be presented here but a more complete discussion can be found in a number of texts.  

As with the Langmuir treatment discussed above, the development of the Michaelis-Menten equation also begins with the rapid and reversible formation of a reactive complex, E-S, between the enzyme, E, or active site and the substrate, S. As shown in Eqn. 7.14, this complex then reacts to give the product, P, regenerating the enzyme for further reaction. 

The first order rate constant for this reaction is kj-gj while the rate of the reaction, v, is given by Eqn. 7.15. 

The stability of the E-S complex is determined by the magnitude of the dissociation constant, Kj, as defined by Eqn. 7.16. 

It is at this point that the Langmuir and Michaelis-Menten approaches diverge. The development of the Langmuir isotherm (Eqn. 7.7) was based on the adsorption coefficient of the catalyst-substrate complex (Eqn. 7.4) while the Michaelis-Menten equation is based on the dissociation constant of this complex (Eqn. 7.16). 

Experimentally, the initial rate, v, of enzyme catalysed reactions is found to show saturation kinetics with respect to the concentration of the substrate, S. At low concentrations of substrate the initial rate increases with increasing concentration of S but becomes independent of [S] at high or saturating concentrations of S (Fig. 2). This observation was interpreted by Michaelis and Menten in terms of the rapid and reversible formation of a non-covalent complex (ES), from the substrate (S) and enzyme (E), which then decomposes into products (P) (Eqn. 1). 

This scheme led to the familiar Michaelis-Menten equation. 

At high concentrations of S, where [S] v is given by Eqn. 4 and becomes independent of S. 

Although the Michaelis-Menten equation (Eqn. 2) is valid for several enzyme-catalysed reactions the mechanism (Eqn. 1) is not always followed. The measured and k values are not always equal to the dissociation constant, K, for the 

When the activation energies for the catalytic steps have been sufficiently lowered, the binding of the substrate or the desorption of the product may become at least partially rate-limiting. 

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Fig. 1. Reaction velocity as a function of substrate concentration for a reaction obeying MichaeHs-Menten kinetics.

Since the rates for MichaeHs-Menten kinetics at the steady state are described by... 

Fig. 1. Free-energy profile for a kinetic resolution depicted by equation 1 that follows Michaelis-Menten kinetics.

Figure 11-1a. Simple Michaelis-Menten kinetics. At low substrate concentration...

The Michaelis constant is equal to substrate concentration at which the rate of reaction is equal to one-half the maximum rate. The parameters and characterize the enzymatic reactions that are described by Michaelis-Menten kinetics. is dependent on total... 

Michaelis-Menten kinetics Kineties of eonversion of substrates in enzyme-eatalyzed reaetions. 

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]. 

On the other hand, for an enzyme that obeys Michaelis-Menten kinetics, the reaction is viewed as being first-order in S at low S and zero-order in S at high S. (See Chapter 14, where this distinction is discussed.)... 

Michaelis-Menten kinetics, in 1913 L. Michaelis and M. Men ten realized that the rate of an enzymatic reaction... 

Most enzymes catalyse reactions and follow Michaelis-Menten kinetics. The rate can be described on the basis of the concentration of the substrate and the enzymes. For a single enzyme and single substrate, the rate equation is ... 

The values of the Michaelis-Menten kinetic parameters, Vj3 and C,PP characterise the kinetic expression for the micro-environment within the porous structure. Kinetic analyses of the immobilised lipase in the membrane reactor were performed because the kinetic parameters cannot be assumed to be the same values as for die native enzymes. 

The initial reaction rate (v0) obtained from each substrate concentration was fitted to Michaelis-Menten kinetics using enzyme kinetics. Pro (EKP) Software (ChemSW product,... 

Continnons Infnsion, Zero Order, and Michaelis-Menten Kinetics... 

Membrane, 141, 178 Mercury electrodes, 62, 108 Mercury film electrode, 76, 110 Metals, 75, 81 Metal complexes, 64 Methyl viologen, 43 Michaelis-Menten kinetics, 175 Microbalance, 52, 53 Microcells, 102 Microchip, 194, 195... 

One reaction scheme that leads to Michaelis-Menten kinetics is known as the Briggs-Haldane scheme. It consists of these reactions ... 

Almost every reaction scheme that gives rise to Michaelis-Menten kinetics will proceed at a rate directly proportional to [E]j. It is customary to express Emax as... 

Michaelis-Menten kinetics. Consider the hydrolysis of phenyl acetate catalyzed by acetyl cholinesterase,... 

Runge-Kutta. Consider the disappearance of substrate in an enzyme-catalyzed reaction that follows Michaelis-Menten kinetics ... 

FIGURE 12.1 Effects of substrate (reactant) concentration on the rate of enzymatic reactions (a) simple Michaelis-Menten kinetics (b) substrate inhibition (c) substrate activation. 

Example 12.3 Suppose S P according to first-order, Michaelis-Menten kinetics. Find Sout for a CSTR. 

Solution Most enzyme reactors use such high concentrations of water that the fluid density is constant. Applying Michaelis-Menten kinetics to the component balance for a steady-state CSTR gives... 

Most biochemical reactors operate with dilute reactants so that they are nearly isothermal. This means that the packed-bed model of Section 9.1 is equivalent to piston flow. The axial dispersion model of Section 9.3 can be applied, but the correction to piston flow is usually small and requires a numerical solution if Michaehs-Menten kinetics are assumed. 

If the enzyme charged to a batch reactor is pristine, some time will be required before equihbrium is reached. This time is usually short compared with the batch reaction time and can be ignored. Furthermore, 5o Eq is usually true so that the depletion of substrate to establish the equilibrium is negligible. This means that Michaelis-Menten kinetics can be applied throughout the reaction cycle, and that the kinetic behavior of a batch reactor will be similar to that of a packed-bed PFR, as illustrated in Example 12.4. Simply replace t with thatch to obtain the approximate result for a batch reactor. 

The initial condition for [SE] assumes that the enzyme was charged to the reactor in pristine condition. It makes no difference whether the enzyme is free or immobilized provided the reaction follows Michaelis-Menten kinetics. 

Hehre and coworkers showed that beta amylase from sweet potatoes, an inverting, a-specific exo-(l 4)-glucanase, catalyzes the hydrolysis of jS-maltosyl fluoride with complex kinetics which indicated the participation of two substrate molecules in the release of fluoride ion. Furthermore, the reaction was strongly accelerated by the addition of methyl ) -maltoside. Hydrolysis of a-maltosyl fluoride, on the other hand, obeyed Michaelis-Menten kinetics. The main product with both a- and yj-maltosyl fluoride was )S-maltose. The results with )3-maltosyl fluoride were interpreted by the assumption of a glycosylation reaction preceding hydrolysis by which a malto-tetraoside is formed by the replacement of fluoride ion by a second substrate molecule or added methyl -maltoside (see Scheme 5). 

The inactivation is normally a first-order process, provided that the inhibitor is in large excess over the enzyme and is not depleted by spontaneous or enzyme-catalyzed side-reactions. The observed rate-constant for loss of activity in the presence of inhibitor at concentration [I] follows Michaelis-Menten kinetics and is given by kj(obs) = ki(max) [I]/(Ki + [1]), where Kj is the dissociation constant of an initially formed, non-covalent, enzyme-inhibitor complex which is converted into the covalent reaction product with the rate constant kj(max). For rapidly reacting inhibitors, it may not be possible to work at inhibitor concentrations near Kj. In this case, only the second-order rate-constant kj(max)/Kj can be obtained from the experiment. Evidence for a reaction of the inhibitor at the active site can be obtained from protection experiments with substrate [S] or a reversible, competitive inhibitor [I(rev)]. In the presence of these compounds, the inactivation rate Kj(obs) should be diminished by an increase of Kj by the factor (1 + [S]/K, ) or (1 + [I(rev)]/I (rev)). From the dependence of kj(obs) on the inhibitor concentration [I] in the presence of a protecting agent, it may sometimes be possible to determine Kj for inhibitors that react too rapidly in the accessible range of concentration. ... 

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