Michaelis-Menten-kinetics

The Michaelis-Menten model shown below is the simplest mechanism for describing the kinetics of enzyme catalysed reactions  

According to this mechanism, the rate of the reaction depends on the rate constants k12, k2i, and kcal. In the simple mechanism shown above and with the assumption that ES is in a steady state KM is defined as KM = (kcal + k21)/ku. The dimensions of are concentration and (time)-1 respectively. The rate of the reaction v (dimension concentration/time) is given by the expression 9.14 and vmax is equal to kcal - cEtotai. The dependence of the reaction rate on substrate concentration is given by Eqn. 9.14, from which it can be seen that the kM value is the concentration of substrate than gives half of the maximum rate vmax= kcai cEtolai. (cf. eq. 8.22) 

To evaluate KM and kcal, the rate of reaction is measured as a function of substrate concentration and the two kinetic parameters are determined using Eqn. 9.14. The classical method of doing this is by fitting the data to a linearised form of Eqn. 9.14 such as the Lineweaver-Burk plot shown in Eqn. 9.15 below (cf. eq. 8.24) 

In the attached exercises we discuss three methods for analysing Michaelis-Menten kinetics  

The fundamental cornerstone of the kinetic characterization of enzymatic reactions has been and remains the Michaelis-Menten equation (Eq. 4.1). 

FIGURE 4.1 Representative hyperbolic plot indicative of a reaction following Michaelis-Menten kinetics. = substrate concentration necessary to achieve one-half maximum velocity. Vmax = maximum velocity. 

Fmax is an estimation of the maximum velocity of the reaction (Fig. 4.1) and is the product of kcat and where kcat is the capacity of the enzyme-substrate complex to form product and cq is the enzyme concentration. The cat parameter is also known as the catalytic constant or the turnover number and refers to the number of catalytic cycles or the number of molecules of substrate that one molecule of enzyme can convert to product per unit time. As stated above, Fmax is only an estimation of the maximum velocity of the reaction, since the true maximum velocity is never reached at a finite substrate concentration. 

Apart from simply obtaining kinetic parameters and understanding reaction rates, the estimation of and Umax has several important applications in the drug discovery and development processes. Typically, it is assumed that in vivo, enzymatic reactions take place when substrate concentrations are much lower 

This intrinsic clearance (CLjnt) is analogous to the in vivo intrinsic clearance in that it is the ability of the enzyme to clear (metabolize) drug in the absence of blood flow or protein binding restrictions. Equation 4.2 is analogous to Equation 4.3 and thus equality is frequently assumed such that 

Using the linear reaction flows given in Eq. (8.234), and minimizing the entropy production by setting d t /i). I, = 0, for the independent affinities of. I, and A2, we find 

These equations support the kinetic equations at stationary states 

All chemical reactions in a biological cell take place with the direct participation of enzymes as catalysts. Enzymes are proteins, which are macromolecules composed of a combination of the 20 amino acids. Enzymes, as catalysts, are highly efficient and selective in binding small molecular species called ligands. A ligand that is acted upon by an enzyme to cause a chemical reaction is called a substrate. Only a certain, small portion of the amino acids that comprise an enzyme is involved in the catalytic reaction. This region is called the active site, and is directly involved in the formation of product(s). For example, the amino acid residues of proteins are greatly influenced by their local pH values, and the activity of proton acceptors and donors occurs in the active site. 

In 1913, Michaelis and Menten assumed that the enzyme and substrate react reversibly to form a substrate-enzyme complex. Later, the complex dissociates to form the free enzyme and product(s). The reactions are as follows 

Since P = -S coincides with the reaction rate. /r, we obtain 

Suppose the reaction S P occurs using an enzyme as a catalyst. The following reaction mechanism is postulated  

The two equilibrium relations and the rate expression allow the unknown surface concentrations [S E], [SE], and [E] to be eliminated. The result is 

This equation gives 0l b) = 0, a maximum at i = JKmIK2, and 0k(oo) = 0. The assumed mechanism involves a first-order surface reaction with inhibition of the reaction if a second substrate molecule is adsorbed. A similar functional form for can be ob- 

SOLUTION This case is different than any previously considered site models since product can be formed by two distinct reactions. 

The site balance is the same as in Example 12.1. Eliminating the unknown surface concentrations gives 

Since the second step here is rate-determining, the rate of this reaction will be as given in Eq. 9.49. Using the steady state approximation for [E S] gives Eq. 9.50, where we let [E] equal to [E]o - [E S] and [E]o is the total concentration of enzyme. Solving for [E S] and substituting the solution in Eq. 9.49 gives Eq. 9.51. Now, Km is defined as (fccat + and is called the 

Michaelis constant, which leads to Eq. 9.52, which is called the Michaelis-Menten equation. This equation predicts a kinetic scenario that will show saturation behavior when [S] Km-Under this condition, the rate of the reaction is equal to /Ccai[E]o/ which is called the maximum velocity (Vmax)- If is the fastest that the catalytic reaction can occur, because all the catalyst has been converted to the catalyst-substrate complex (E S). The catalyst/enzyme is considered to be saturated with the substrate. 

There are several ways to measure fccat and Km, which we leave to a textbook devoted to biochemistry. Here, we want to understand the meaning of these constants, so that the student can easily read about enzyme kinetics, and also apply the notions to other forms of catalysis. 

The meaning of fccat is the easiest to understand. This is the rate constant for the conversion of the substrate to the product within the active site of the catalyst, and is often called the turnover number. Note that it is a unimolecular rate constant, with units of s All the factors that we have examined in this chapter that can impart transition, state stabilization will influence fccat—namely, proximity, acid-base catalysis, electrostatic considerations, covalent catalysis, and the relief of strain. This rate constant is for the chemical step of catalysis, and it is thus the focus of efforts to interpret transition state binding relative to the substrate binding. Note that the scheme in Eq. 9.48 is for a simple, single-step conversion. As in other kinetic analyses, if multiple chemical steps are involved in converting the substrate to the 

The meaning of the Michaelis constant (K ) is more complex. There are two extremes. When /c, then /fci, which is the dissociation constant (Kj) for the enzyme- 

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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... 

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. ... 

Sato et al. (1991) expanded their earlier PBPK model to account for differences in body weight, body fat content, and sex and applied it to predicting the effect of these factors on trichloroethylene metabolism and excretion. Their model consisted of seven compartments (lung, vessel rich tissue, vessel poor tissue, muscle, fat tissue, gastrointestinal system, and hepatic system) and made various assumptions about the metabolic pathways considered. First-order Michaelis-Menten kinetics were assumed for simplicity, and the first metabolic product was assumed to be chloral hydrate, which was then converted to TCA and trichloroethanol. Further assumptions were that metabolism was limited to the hepatic compartment and that tissue and organ volumes were related to body weight. The metabolic parameters, (the scaling constant for the maximum rate of metabolism) and (the Michaelis constant), were those determined for trichloroethylene in a study by Koizumi (1989) and are presented in Table 2-3. 

Kemp and Waters also found the oxidations of cyclohexanone and of mandelic, malonic and a-hydroxyisobutyric acids by Cr(VI) to be Mn(II)-catalysed. In these cases, as with oxalic acid, the [Cr(VI)] versus time plots are almost linear and the reaction becomes first order in substrate (or involves Michaelis-Menten kinetics), and, except at lowest catalyst concentrations, approximately first order in [Mn(II)]. Detailed examination of the initial rate of oxidation of a-hydroxyrobutyric acid as a function of oxidant concentration revealed, however, that the dependence is... 

All the oxidants convert primary and secondary alcohols to aldehydes and ketones respectively, albeit with a great range of velocities. Co(III) attacks even tertiary alcohols readily but the other oxidants generally require the presence of a hydrogen atom on the hydroxylated carbon atom. Spectroscopic evidence indicates the formation of complexes between oxidant and substrate in some instances and this is supported by the frequence occurrence of Michaelis-Menten kinetics. Carbon-carbon bond fission occurs in certain cases. 

On the other hand, the macrolides showed unusual enzymatic reactivity. Lipase PF-catalyzed polymerization of the macrolides proceeded much faster than that of 8-CL. The lipase-catalyzed polymerizability of lactones was quantitatively evaluated by Michaelis-Menten kinetics. For all monomers, linearity was observed in the Hanes-Woolf plot, indicating that the polymerization followed Michaehs-Menten kinetics. The V, (iaotone) and K,ax(iaotone)/ m(iaotone) values increased with the ring size of lactone, whereas the A (iactone) values scarcely changed. These data imply that the enzymatic polymerizability increased as a function of the ring size, and the large enzymatic polymerizability is governed mainly by the reachon rate hut not to the binding abilities, i.e., the reaction process of... 

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