Michaelis-Menten kinetic behavior

Fig. 11.2 Reciprocal plots of Michaelis-Menten kinetic behavior (Equation 1E8). The reciprocal rate of product formation for mechanism 11.1 is plotted vs reciprocal substrate concentration (arbitrary units) for the three cases specified in the caption of Fig. 11.1...

A transesterication reaction occurs that results in cleavage of the substrate and ligation of the 3 -portion of the substrate (Tsang and Joyce 1994). Just like in the case of enzyme- or catalytic antibody-catalyzed reactions, the rate depends upon substrate binding affinity and the intrinsic catalytic rate parameters. For example, in ester hydrolysis there is a hyperbolic dependence on the concentration of the ribozyme at low concentration of catalyst the rate of hydrolysis is first order, while at high concentration of catalyst the reaction rate is indepen-dent of ribozyme concentration (Piccirilli et al. 1992). This type of saturation or Michaelis-Menten kinetic behavior is typical of ribozymes and is completely analogous to the enzyme-substrate complex observed for enzymes and catalytic antibodies. 

The kinetic behavior of GMD is quite complex and displays exquisite sensitivity to reaction conditions including the nature of the buffer and even the order of addition of the substrates." In phosphate buffer GMD exhibits Michaelis-Menten kinetic behavior, and the kinetic mechanism is hi uni uni hi ping-pong, with GDP-mannose binding first and GDP-mannuronate dissociating last. There is a single binding site for the pyridine nucleotide cofactor, so after oxidation of GDP-mannose to the aldehyde, NADH dissociates from the enzyme and is replaced by NAD" " so the second oxidative step can take place. 

A Pd-colloid catalyst, stabilized by P-CD, was effective in the catalytic reduction of HCO3 to HCOJ in a system based on deazariboflavin as photosensitizer, methylviologen as electron acceptor, and oxalate as sacrificial donor. The catalyst exhibits active sites for bicarbonate activation and reduction and for Hj evolution as well as Michaelis-Menten kinetic behavior, thus mimicking an enzymatic system [334,345]. These results are reviewed in reference 346. 

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. 

Quite often the asymptotic behavior of the model can aid us in determining sufficiently good initial guesses. For example, let us consider the Michaelis-Menten kinetics for enzyme catalyzed reactions,... 

The kinetic behavior of drugs in the body can generally be accounted for by first-order kinetics that are saturable, i.e., Michaelis-Menten kinetics. A brief review of the principles of Michaelis-Menten kinetics is given next (4). 

An enzyme is said to obey Michaelis-Menten kinetics, if a plot of the initial reaction rate (in which the substrate concentration is in great excess over the total enzyme concentration) versus substrate concentration(s) produces a hyperbolic curve. There should be no cooperativity apparent in the rate-saturation process, and the initial rate behavior should comply with the Michaelis-Menten equation, v = Emax[A]/(7 a + [A]), where v is the initial velocity, [A] is the initial substrate concentration, Umax is the maximum velocity, and is the dissociation constant for the substrate. A, binding to the free enzyme. The original formulation of the Michaelis-Menten treatment assumed a rapid pre-equilibrium of E and S with the central complex EX. However, the steady-state or Briggs-Haldane derivation yields an equation that is iso-... 

In contrast to ABTS, the kinetic behavior of native HRP and Fc HRP toward water-soluble ferrocenes HOOCFc and Me2NCH2Fc is remarkably different. Instead of first-order kinetics observed for native HRP, the reaction rate levels off on increasing the ferrocene concentration for Fc HRP, Fig. 13. The Michaelis Menten kinetics holds for both ferrocene substrates, and the inverse rate vs. [ferrocene]-1 plots are linear. The values of Vm for HOOCFc and... 

It is important to distinguish between the Michaelis-Menten equation and the specific kinetic mechanism on which it was originally based. The equation describes the kinetic behavior of a great many enzymes, and all enzymes that exhibit a hyperbolic dependence of V0 on [S] are said to follow Michaelis-Menten kinetics. The practical rule that... 

Criteria for calling a compound a synthetic enzyme are (i) completion of at least one catalytic cycle (ii) its presence after the catalytic cycle in unchanged form and (iii) a saturation kinetics behavior such as is manifested by Michaelis-Menten kinetics. There is a tetrameric helical peptide that catalyzes the decarboxylation of oxaloacetate with Michaelis-Menten kinetics and accelerates the reaction 103-104-fold faster than n-butylamine as control, a record for a chemically derived artificial enzyme. 

Mean clearance (CL) values for cetuximab are displayed as a function of dose in Fig. 14.3. Mean CL values decreased from 0.079 to 0.018 L/h/m2 after single cetuximab doses of 20 to 500 mg/m2, respectively. In the dose range 20 to 200 mg/m2, CL values decreased with dose. At doses of 200 mg/m2 and greater, CL values leveled off at a value of approximately 0.02 L/h/m2. This biphasic behavior suggests the existence of two elimination pathways. The elimination of cetuximab apparently involves a specific, capacity-limited elimination process that is saturable at therapeutic concentrations, in parallel with a nonspecific first-order elimination process that is non-saturable at therapeutic concentrations. Increasing doses of cetuximab will therefore ultimately lead to the saturation of the elimination process that is capacity-limited and that follows Michaelis-Menten kinetics, whereas the first-order process will become the dominant mechanism of elimination beyond a particular dose range. 

Many enzymes catalyze reactions with two interacting substrates, and although the kinetics of these reactions are more complex than those of one-substrate reactions, they still obey Michaelis-Menten kinetics. Reactions of the type A + B <= P + Q usually fall within either of two classes with respect to kinetic behavior and mechanism of action. 

Pharmacokinetic studies are in general less variable than pharmacodynamic studies. This is so since simpler dynamics are associated with pharmacokinetic processes. According to van Rossum and de Bie [234], the phase space of a pharmacokinetic system is dominated by a point attractor since the drug leaves the body, i.e., the plasma drug concentration tends to zero. Even when the system is as simple as that, tools from dynamic systems theory are still useful. When a system has only one variable a plot referred to as a phase plane can be used to study its behavior. The phase plane is constructed by plotting the variable against its derivative. The most classical, quoted even in textbooks, phase plane is the c (f) vs. c (t) plot of the ubiquitous Michaelis-Menten kinetics. In the pharmaceutical literature the phase plane plot has been used by Dokoumetzidis and Macheras [235] for the discernment of absorption kinetics, Figure 6.21. The same type of plot has been used for the estimation of the elimination rate constant [236]. 

Michaelis-Menten kinetics — is the dependence of an initial -> reaction rate upon the concentration of a substrate S that is present in large excess over the concentration of an enzyme or another catalyst (or reagent) E with the appearance of saturation behavior following the Michaelis-Menten equation,... 

Another type of kinetic behavior that is very common for enzyme-catalyzed reactions (Michaelis-Menten kinetics) has also been observed in a number of homogeneous catalytic systems. The rate expression in such cases is given by... 

Can a phosphorylation-dephosphorylation switch be more sensitive to the level of kinase concentration than n = 1 as given in Equation 5.12 We note that the kinetic scheme in Equation (4.7) is obtained under the assumption of no Michaelis-Menten saturation. Since this assumption may not be realistic, let us move on to study the enzyme kinetics in Figure (5.2) in terms of saturable Michaelis-Menten kinetics. The mechanism by which saturating kinetics of the kinase and phosphatase leads to sensitive switch-like behavior is illustrated in Figure 5.4. The reaction fluxes as a function of / (the ratio [S ]/Sc) for two cases are plotted. The first case (switch off)... 

Such behavior is generally referred to as Michaelis-Menten kinetics or saturation kinetics. As will be seen, it is a rather common feature of trace-level catalysis, not restricted to cycles as simple as 8.14. 

The one-plus rate equation 8.22 is of the same algebraic form as the Michaelis-Menten equation 8.18, only the physical significance of the coefficients is different [instead of the constant K, the expression kAX /(k + kXP) now appears]. Accordingly, the behavior is the same as for Michaelis-Menten kinetics, and that name is often used for Briggs-Haldane kinetics as well. 

When faced with characterizing the kinetic behavior of an enzyme or a complex of enzymes, one usually pulls out a textbook on Michaelis-Menten kinetics and applies it to the system at hand. For beta-glucosidase, which hydrolyzes the soluble substrate cellobiose to glucose, this approach is fine. Unfortunately, for cellulase enzymes producing cellobiose from cellulose, this exercise is inadequate. 

The Michaelis-Menten kinetic model explains several aspects of the behavior of many enzymes. Each enzyme has a Km value characteristic of that enzyme under specified conditions. 

With respect to benzaldehyde, (R)-oxynitrilase exhibits saturation kinetics (Michaelis Menten kinetics, see Sect. 7.4.2.1) and a maximum reaction rate is reached above a concentration of about 5 mmol L 1. The chemical reaction presents a linear increase of the reaction rate with increasing benzaldehyde concentration, representing first order kinetics, when the concentration of HCN is kept constant (see Fig. 7-13). As a consequence the enzymatic reaction becomes more dominating at lower concentrations of the substrate benzaldehyde (for HCN as substrate the same kinetic behavior occurs, data not shown). Accordingly an enzyme reactor would be suitable that works under minimum average substrate concentrations. These requirements are satisfied by the continuous stirred tank reactor (CSTR). In Sect. 7.5.2.1 this aspect of enzyme reaction engineering will be discussed further. 

Hydrolysis by the enzyme for dsDNA (1) and 3 -free ssDNA (5) was also investigated. When ATP concentrations injected in step 2 were changed in the range of 2.5-200 jiM, the initial rates of step 2 showed the saturation behavior of Michaelis-Menten kinetics. The catalytic hydrolysis rate constant (fccat)> fCm for ATP, and apparent second-order rate (fccat/ffm) are also summarized in Table 2. The kcat/ffm value for (1) was fourfold larger than that for (4) due to the large kcat and the constant Km values for (1). Thus, the binding ability of ATP to the enzyme-DNA complex was independent of the DNA structure, and the DNase can efficiently hydrolyze the dsDNA compared with the ssDNA. 

Some negative regulators act in the opposite way. The free enzyme shows Michaelis-Menten kinetics and the regulator shifts it to positive cooperativity so that the [S]0.5 is too high for much activity. An example of this behavior is given by amidophosphoribosyltransferase. This enzyme is the initial step in purine synthesis in which control by negative feedback occurs. The... 

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