Saturation kinetics, Michaelis

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. 

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]. 

A kinetic model describing the HRP-catalyzed oxidation of PCP by H202 should account for the effects of the concentrations of HRP, PCP, and H202 on the reaction rate. To derive such an equation, a reaction mechanism involving saturation kinetics is proposed. Based on the reaction scheme described in Section 17.3.1, which implies that the catalytic cycle is irreversible, the three distinct reactions steps (Equations 17.2 to 17.4) are modified to include the formation of Michaelis-Menten complexes ... 

Mathematically, the Michaelis-Menten equation is the equation of a rectangular hyperbola. Sometimes you ll here reference to hyperbolic kinetics, this means it follows the Michaelis-Menten equation. A number of other names also imply that a particular enzyme obeys the Michaelis-Menten equation Michaelis-Menten behavior, saturation kinetics, and hyperbolic kinetics. 

Equation (4) corresponds to saturation-type (Michaelis-Menten) kinetics and rate constants obtained over a suitable range of [CD], sufficient to reflect the hyperbolic curvature, can be analysed to provide the limiting rate constant, kc, and the dissociation constant, Ks (VanEtten et al., 1967a Bender and Komiyama, 1978 Szejtli, 1982 Sirlin, 1984 Tee and Takasaki, 1985). The rate constant ku is normally determined directly (at zero [CD]), and sometimes Ks can be corroborated by other means (Connors, 1987). 

It has been found experimentally that in most cases v is directly proportional to the concentration of enzyme [.E0] and that v generally follows saturation kinetics with respect to the concentration of substrate [limiting value called Vmax. This is expressed quantitatively in the Michaelis-Menten equation originally proposed by Michaelis and Menten. Km can be seen as an apparent dissociation constant for the enzyme-substrate complex ES. The maximal velocity Vmax = kcat E0. ... 

The rhodium dimer has two Lewis acidic sites and thus the catalyst could coordinate to two substrate molecules under saturation kinetics, which would make the Michaelis-Menten plots complicated. This does not happen and the second site becomes less acidic once the other site is occupied by the substrate. What does happen, though, is that other Lewis bases compete with the substrate, as might be expected. The ligand dissociation reaction may be part of the rate equation of the process. Coordination of one Lewis base reduces already the activity of the catalyst. The solvent of choice is often anhydrous dichloromethane. The polar group may also be part of one of the substrates and in this instance one cannot avoid inhibition. 

Carrier-mediated passage of a molecular entity across a membrane (or other barrier). Facilitated transport follows saturation kinetics ie, the rate of transport at elevated concentrations of the transportable substrate reaches a maximum that reflects the concentration of carriers/transporters. In this respect, the kinetics resemble the Michaelis-Menten behavior of enzyme-catalyzed reactions. Facilitated diffusion systems are often stereo-specific, and they are subject to competitive inhibition. Facilitated transport systems are also distinguished from active transport systems which work against a concentration barrier and require a source of free energy. Simple diffusion often occurs in parallel to facilitated diffusion, and one must correct facilitated transport for the basal rate. This is usually evident when a plot of transport rate versus substrate concentration reaches a limiting nonzero rate at saturating substrate While the term passive transport has been used synonymously with facilitated transport, others have suggested that this term may be confused with or mistaken for simple diffusion. See Membrane Transport Kinetics... 

Pirrang, Liu, and Morehead [22] have elegandy demonstrated the application of saturation kinetics (Michaehs-Menten) to the rhodium(II)-mediated insertion reactions of a-diazo /9-keto esters and a-diazo /9-diketones. Their method used the Eadie-Hofstee plot of reaction velocity (v) versus v/[S] to give and K, the equilibrium constants for the catalytic process. However, they were unable to measure the Michaelis constant (fC ) for the insertion reactions of a-diazo esters because they proved to be too rapid. 

Satisfaction of kinetic order. Carriers follow Michaelis-Menten-type saturation kinetics or first-order kinetics. Ion channels follow the type of respective structure—unimolecular transmembrane channels and bimolecular half-channels follow first- and second-order kinetics, respectively. The kinetic order of supramolecular channels depends on the assembly number. However, this principle can be applied only when the association constants are small. If the association becomes strong, the kinetic order decreases down to zero. Then the validity becomes dubious in view of the absolute criterion of the mechanism. Decreased activation energy compared to the carrier transport mechanism and competitive inhibition by added other cations stand as criteria. 

Non-linear pharmacokinetics are much less common than linear kinetics. They occur when drug concentrations are sufficiently high to saturate the ability of the liver enzymes to metabolise the drug. This occurs with ethanol, therapeutic concentrations of phenytoin and salicylates, or when high doses of barbiturates are used for cerebral protection. The kinetics of conventional doses of thiopentone are linear. With non-linear pharmacokinetics, the amount of drug eliminated per unit time is constant rather than a constant fraction of the amount in the body, as is the case for the linear situation. Non-linear kinetics are also referred to as zero order or saturation kinetics. The rate of drug decline is governed by the Michaelis-Menton equation ... 

Non Michaelis-Menten behavior (i.e., no saturation kinetics in presence of excess HC03) was observed and the second-order rate constants for the catalyzed decarboxylation (/ = k k<>j(k i / 2) for... 

This is known as Michaelis-Menten or saturation kinetics. The processes that involve specific interactions between chemicals and proteins such as plasma protein binding, active excretion from the kidney or liver via transporters, and metabolism catalyzed by enzymes can be saturated. This is because there are a specific number of binding sites that can be fully occupied at higher doses. In some cases, cofactors are required, and their concentration may be limiting (see chap. 7 for salicylate, paracetamol toxicity). These all lead to an increase in the free concentration of the chemical. Some drugs, such as phenytoin, exhibit saturation of metabolism and therefore nonlinear kinetics at therapeutic doses. Alcohol metabolism is also saturated at even normal levels of intake. Under these circumstances, the rate of... 

However, active uptake mechanisms have now been found in some bacteria for various xenobiotic organic anions. These include 4-chlorobenzoate (Groenewegen et al., 1990), 4-toluene sulfonate (Locher et al., 1993), 2,4-D (Leveau et al., 1998), mecoprop and dichlorprop (Zipper et al., 1998), and even aminopolycarboxylates (Egli, 2001). Such active uptake appears to be driven by the proton motive force (i.e., accumulation of protons in bacterial cytoplasm). These transport mechanisms exhibit saturation kinetics (e.g., Zipper et al., 1998), and so their quantitative treatment is the same as other enzyme-limited metabolic processes (discussed below as Michaelis-Menten cases). 

Fiqure 3.1 Reaction rate v plotted against substrate concentration [S] for a reaction obeying Michaelis-Menten (or saturation) kinetics. 

It is possible to devise kinetic curiosities that give Michaelis-Menten kinetics without the enzyme being saturated with the substrate. For example, in the following scheme—where the active form of the enzyme reacts with the substrate in a second-order reaction to give the products and an inactive form of the enzyme, E, which slowly reverts to the active form—apparent saturation kinetics are followed with cat = k2 and Ku = k2/k1. Equation 3.24 applies to this example if E is treated as a bound form of the enzyme ... 

This scheme is analogous to that of the Michaelis-Menten mechanism, and the reaction should thus show saturation kinetics with increasing inhibitor concentration. The kinetics were solved in Chapter 4, equation 4.46. For the simple case of pre-equilibrium binding followed by a slow chemical step, the solution reduces to... 

In an enzyme reaction, initially free enzyme E and free substrate S in their respective ground states initially combine reversibly to an enzyme-substrate (ES) complex. The ES complex passes through a transition state, AGj, on its way to the enzyme-product (EP) complex and then on to the ground state of free enzyme E and free product P. From the formulation of the reaction sequence, a rate law, properly containing only observables in terms of concentrations, can be derived. In enzyme catalysis, the first rate law was written in 1913 by Michaelis and Menten therefore, the corresponding kinetics is named the Michaelis-Menten mechanism. The rate law according to Michaelis-Menten features saturation kinetics with respect to substrate (zero order at high, first order at low substrate concentration) and is first order with respect to enzyme. 

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. 

More complex enzymatic reactions usually display Michaelis-Menten kinetics and can be described by Eq. (2). However, the forms of constants Km and Vm can be very complicated, consisting of many individual rate constants. King and Altman (7) have provided a method to readily derive the steady-state equations for enzymatic reactions, including the forms that describe Km and Vm. The advent of symbolic mathematics programs makes the implementation of these methods routine, even for very complex reaction schemes. The P450 catalytic cycle (Fig. 2) is an example of a very complicated reaction scheme. However, most P450-mediated reactions display standard hyperbolic saturation kinetics. Therefore, although the rate constants that determine Km and Vm are... 

Substrate A has a hyperbolic saturation curve Enzymes that bind to only one substrate molecule will show hyperbolic saturation kinetics. However, the observation of hyperbolic saturation kinetics does not necessarily mean that only one substrate molecule is interacting with the enzyme (see discussion of non-Michaelis-Menten kinetics in sec. IV). 

Several reports have shown that the kinetics of P-gp transport activity can be sufficiently described by one-site Michaelis-Menten saturable kinetics (199-206). Where JP.g ) is the flux mediated by P-gp transport activity,, /max is the maximal flux mediated by P-gp transport activity, Km is the Michaelis-Menten constant, and Ct is the concentration of substrate present at the target (binding) site of P-gp. When donor concentration is used in place of Ct, apparent Km and Jmax values are obtained. Binding affinity of the substrate to P-gp and the catalytic (ATPase) activity of P-gp combine to determine Km, and, /max is determined by the catalytic (ATPase) activity of P-gp and the expression of P-gp in the system (concentration of P-gp protein). It has recently been noted that since substrates must first partition or cross the membrane to access the binding site, accurate assessing of P-gp kinetics can be difficult (207). Furthermore, the requirement of first partitioning into the membrane has been shown to produce asymmetric apparent kinetics in polarized cells where AP and BL membrane compositions may be sufficiently different (206). 

In the kinetic considerations discussed above, a plot of 1 /V0 vs 1/[S0] yields a straight line, and the enzyme exhibits Michaelis-Menten (hyperbolic or saturation) kinetics. It is implicit in this result that all the enzyme-binding sites have the same affinity for the substrate and operate independently of each other. However, many enzymes exist as oligomers containing subunits or domains that function in the regulation of the catalytic site. Such enzymes do not exhibit classic Michaelis-Menten saturation kinetics. 

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