Rate Michaelis-Menten kinetics

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

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... 

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. 

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... 

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... 

An additional problem arises when the exchange processes are rate-limited. This may be caused by enzymes that become saturated when all their active sites are occupied by the drug, or it may be due to adsorbing proteins that have a limited binding capacity. In such cases, one obtains a type of Michaelis-Menten kinetics of the form ... 

Fig. 39.17. Schematic illustration of Michaelis-Menten kinetics in the absence of an inhibitor (solid line) and in the presence of a competitive inhibitor (dashed line), (a) Plot of initial rate (or velocity) V against amount (or concentration) of substrate X. Note that the two curves tend to the same horizontal asymptote for large values of X. (b) Lineweaver-Burk linearized plot of 1/V against l/X. Note that the two lines intersect at a common intercept on the vertical axis.

The reaction was monitored by UV/Vis spectroscopy by following the product formation at 420 mn. The initial rates were used for analysis of the catalyzed oxidation of 8 into 9 that follows Michaelis-Menten kinetics. Control experiments show a linear increase of the reaction rates with the catalyst concentration at constant substrate concentration. 

Let us consider the determination of two parameters, the maximum reaction rate (rITOIX) and the saturation constant (Km) in an enzyme-catalyzed reaction following Michaelis-Menten kinetics. The Michaelis-Menten kinetic rate equation relates the reaction rate (r) to the substrate concentrations (S) by... 

Pyruvate kinase (PK) is one of the three postulated rate-controlling enzymes of glycolysis. The high-energy phosphate of phosphoenolpyruvate is transferred to ADP by this enzyme, which requires for its activity both monovalent and divalent cations. Enolpyruvate formed in this reaction is converted spontaneously to the keto form of pyruvate with the synthesis of one ATP molecule. PK has four isozymes in mammals M, M2, L, and R. The M2 type, which is considered to be the prototype, is the only form detected in early fetal tissues and is expressed in many adult tissues. This form is progressively replaced by the M( type in the skeletal muscle, heart, and brain by the L type in the liver and by the R type in red blood cells during development or differentiation (M26). The M, and M2 isozymes display Michaelis-Menten kinetics with respect to phosphoenolpyruvate. The Mj isozyme is not affected by fructose-1,6-diphosphate (F-1,6-DP) and the M2 is al-losterically activated by this compound. Type L and R exhibit cooperatively in... 

Lactone3 Dipole moment (P) Rate constant Michaelis-Menten kinetics 1 ... 

Use of a N. globerula R-9 strain was demonstrated for desulfurization of straight run diesel oils. Sulfur reduction from 1807 to 741 mg/dm3 was reported at a desulfurization rate of 5.1 mmol/Kgdcw/h. The desulfurization of model oils containing DBT and 4,6 dimethyl DBT was studied and Michaelis-Menten kinetic parameters were reported. 

In the presence of sucrose alone as the single substrate, initial reaction rates follow Michaelis-Menten kinetics up to 200 mM sucrose concentration, but the enzyme is inhibited by higher concentrations of substrate.30 The inhibitor constant for sucrose is 730 mM. This inhibition can be overcome by the addition of acceptors.31,32 The enzyme activity is significantly enhanced, and stabilized, by the presence of dextran, and by calcium ions. 

The effect of non-participating ligands on the copper catalyzed autoxidation of cysteine was studied in the presence of glycylglycine-phosphate and catecholamines, (2-R-)H2C, (epinephrine, R = CH(OH)-CH2-NHCH3 norepinephrine, R = CH(OH)-CH2-NH2 dopamine, R = CH2-CH2-NH2 dopa, R = CH2-CH(COOH)-NH2) by Hanaki and co-workers (68,69). Typically, these reactions followed Michaelis-Menten kinetics and the autoxidation rate displayed a bell-shaped curve as a function of pH. The catecholamines had no kinetic effects under anaerobic conditions, but catalyzed the autoxidation of cysteine in the following order of efficiency epinephrine = norepinephrine > dopamine > dopa. The concentration and pH dependencies of the reaction rate were interpreted by assuming that the redox active species is the [L Cun(RS-)] ternary complex which is formed in a very fast reaction between CunL and cysteine. Thus, the autoxidation occurs at maximum rate when the conditions are optimal for the formation of this species. At relatively low pH, the ternary complex does not form in sufficient concentration. 

Most catalytic cycles are characterized by the fact that, prior to the rate-determining step [18], intermediates are coupled by equilibria in the catalytic cycle. For that reason Michaelis-Menten kinetics, which originally were published in the field of enzyme catalysis at the start of the last century, are of fundamental importance for homogeneous catalysis. As shown in the reaction sequence of Scheme 10.1, the active catalyst first reacts with the substrate in a pre-equilibrium to give the catalyst-substrate complex [20]. In the rate-determining step, this complex finally reacts to form the product, releasing the catalyst... 

Because of the complexity of biological systems, Eq. (1) as the differential form of Michaelis-Menten kinetics is often analyzed using the initial rate method. Due to the restriction of the initial range of conversion, unwanted influences such as reversible product formation, effects due to enzyme inhibition, or side reactions are reduced to a minimum. The major disadvantage of this procedure is that a relatively large number of experiments must be conducted in order to determine the desired rate constants. 

There are two limiting cases of Michaelis-Menten kinetics. Beginning from Eq. (1) at high substrate excesses (or very small Michaelis constants) Eq. (4 a) results. This corresponds to a zero-order reaction with respect to the substrate, the rate of product formation being independent of the substrate concentration. In contrast, very low substrate concentrations [26] (or large Michaelis constants) give the limiting case of first-order reactions with respect to the substrate, Eq. (4b) ... 

The rate law for two diastereomeric catalyst-substrate complexes -symmetric ligands) resulting from Michaelis-Menten kinetics (Eq. (11)) has already been utilized by Halpern et al. for the kinetic analysis of hydrogenations according to Scheme 10.2, and corresponds to Eq. (3) of this study. 

A more detailed analysis, however, shows that such comparisons of activity can be completely misleading, because Michaelis-Menten kinetics are principally described by two constants. The Michaelis constant contains information regarding the pre-equilibria, the rate constants quantify the product formation from the intermediates. 

Figure 3. Schematic view of the substrate uptake rate versus concentration relationship as described by the whole-cell Michaelis-Menten kinetics. Q is the substrate uptake rate, <2max the biologically determined maximum uptake rate per biomass, c the substrate concentration, and Kj the whole-cell Michaelis constant, i.e. the concentration resulting in 2max/2 (mass of substrate per volume). At c

Bosma et al. [1] have proposed including the details of extracellular substrate transport in the calculation of whole-cell Michaelis-Menten kinetics. For the situation of a quasi-steady-state (i.e. when the transport flux and the rate of degradation of the substrate are equal) the consumption of substrate by a microorganism is represented as a function of the distant, and effectively unavailable, substrate concentration ca ... 

Characteristically, within certain concentration limits, if a chemical is absorbed by passive diffusion, then the concentration of toxicant in the gut and the rate of absorption are linearly related. However, if absorption is mediated by active transport, the relationship between concentration and rate of absorption conforms to Michaelis-Menten kinetics and a Lineweaver-Burk plot (i.e., reciprocal of rate of absorption plotted against reciprocal of concentration), which graphs as a straight line. 

Figure 9. Various types of inhibition that occur for Michaelis Menten kinetics. Shown is competitive (A), uncompetitive (B), and noncompetitive inhibition (C). The corresponding rate laws are listed in Table II, (see text for details).

Aiming at a computer-based description of cellular metabolism, we briefly summarize some characteristic rate equations associated with competitive and allosteric regulation. Starting with irreversible Michaelis Menten kinetics, the most common types of feedback inhibition are depicted in Fig. 9. Allowing all possible associations between the enzyme and the inhibitor shown in Fig. 9, the total enzyme concentration Er can be expressed as... 

Taking into account the typical functional form of Michaelis Menten kinetics (see Section III.C), the rate of consumption will usually increase with increasing... 

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