Zero-order enzyme kinetics

HCN is detoxified to thiocyanate (SCN ) by the mitochondrial enzyme rhodanese rhodanese catalyzes the transfer of sulfur from thiosulfate to cyanide to yield thiocyanate, which is relatively nontoxic (Smith 1996). The rate of detoxification of HCN in humans is about 1 pg/kg/min (Schulz 1984) or 4.2 mg/h, which, the author states, is considerably slower than in small rodents. This information resulted from reports of the therapeutic use of sodium nitroprusside to control hypertension. Rhodanese is present in the liver and skeletal muscle of mammalian species as well as in the nasal epithelium. The mitochondria of the nasal and olfactory mucosa of the rat contain nearly seven times as much rhodanese as the liver (Dahl 1989). The enzyme rhodanese is present to a large excess in the human body relative to its substrates (Schulz 1984). This enzyme demonstrates zero-order kinetics, and the limiting factor in the detoxification of HCN is thiosulphate. However, other sulfur-containing substrates, such as cystine and cysteine, can also serve as sulfur donors. Other enzymes, such as 3-mercapto-pyruvate sulfur transferase, can convert... 

Interestingly, at very low concentrations of micellised Qi(DS)2, the rate of the reaction of 5.1a with 5.2 was observed to be zero-order in 5.1 a and only depending on the concentration of Cu(DS)2 and 5.2. This is akin to the turn-over and saturation kinetics exhibited by enzymes. The acceleration relative to the reaction in organic media in the absence of catalyst, also approaches enzyme-like magnitudes compared to the process in acetonitrile (Chapter 2), Cu(DS)2 micelles accelerate the Diels-Alder reaction between 5.1a and 5.2 by a factor of 1.8710 . This extremely high catalytic efficiency shows how a combination of a beneficial aqueous solvent effect, Lewis-acid catalysis and micellar catalysis can lead to tremendous accelerations. 

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]. 

FIGURE 14.7 Substrate saturation curve for au euzyme-catalyzed reaction. The amount of enzyme is constant, and the velocity of the reaction is determined at various substrate concentrations. The reaction rate, v, as a function of [S] is described by a rectangular hyperbola. At very high [S], v= Fnax- That is, the velocity is limited only by conditions (temperature, pH, ionic strength) and by the amount of enzyme present becomes independent of [S]. Such a condition is termed zero-order kinetics. Under zero-order conditions, velocity is directly dependent on [enzyme]. The H9O molecule provides a rough guide to scale. The substrate is bound at the active site of the enzyme. 

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.)... 

Case 1 When [S] is large compared to K , the enzyme is saturated with S and the kinetics are zero-order in S. 

Alcohol dehydrogenase is a cytoplasmic enzyme mainly found in the liver, but also in the stomach. The enzyme accomplishes the first step of ethanol metabolism, oxidation to acetaldehyde, which is further metabolized by aldehyde dehydrogenase. Quantitatively, the oxidation of ethanol is more or less independent of the blood concentration and constant with time, i.e. it follows zero-order kinetics (pharmacokinetics). On average, a 70-kg person oxidizes about 10 ml of ethanol per hour. 

Zero-order kinetics describe the time course of disappearance of drugs from the plasma, which do not follow an exponential pattern, but are initially linear (i.e. the drug is removed at a constant rate that is independent of its concentration in the plasma). This rare time course of elimination is most often caused by saturation of the elimination processes (e.g. a metabolizing enzyme), which occurs even at low drug concentrations. Ethanol or phenytoin are examples of drugs, which are eliminated in a time-dependent manner which follows a zero-order kinetic. 

The kinetics should be zero-order during the initial portion of the reaction, over a practical working range that includes the usually encountered enzyme activities. 

In this case, most of the catalyst is in the form of M A and the reaction is zero order with respect to A. Thus, the kinetics move from first order at low cA toward zero order as cA increases. This feature of the rate saturating or reaching a plateau is common to many catalytic reactions, including surface catalysis (Section 8.4) and enzyme catalysis (Chapter 10). 

Silverman has pointed out that several criteria must be met to demonstrate that a compound is a true suicide substrate 1101 (1) Loss of enzyme activity must be time-dependent, and it must be first-order in [inactivator] at low concentrations and zero-order at higher concentrations (saturation kinetics), (2) substrate must protect the enzyme from inactivation (by blocking the active site), (3) the enzyme must be irreversibly inactivated and be shown to have a 11 stoichiometry of suicide substrate active site (dialysis of enzyme previously treated with radiolabeled suicide substrate must not release radiolabel into the buffer), (4) the enzyme must unmask the suicide substrate s potent electrophile via a catalytic step,1121 and (5) the enzyme must not be covalently labeled with the activated form of the suicide substrate following its escape from the active site (the presence of bulky scavenging thiol nucleophiles in the buffer must not decrease the observed rate of inactivation). 

Figure 9.9 Kinetic scheme for MBI.

The expression for the effectiveness factor q in the case of zero-order kinetics, described by the Michaelis-Menten equation (Eq. 8) at high substrate concentration, can also be analytically solved. Two solutions were combined by Kobayashi et al. to give an approximate empirical expression for the effectiveness factor q [9]. A more detailed discussion on the effects of internal and external mass transfer resistance on the enzyme kinetics of a Michaelis-Menten type can be found elsewhere [10,11]. 

In zone a of Figure 2.5, the kinetics are first order with respect to [S], that is to say that the rate is limited by the availability (concentration) of substrate so if [S] doubles the rate of reaction doubles. In zone c however, we see zero order kinetics with respect to [S], that is the increasing substrate concentration no longer has an effect as the enzyme is saturated zone b is a transition zone. In practice it is difficult to demonstrate the plateau in zone c unless very high concentrations of substrate are used in the experiment. Figure 2.5 is the basis of the Michaelis-Menten graph (Figure 2.6) from which two important kinetic parameters can be approximated ... 

In the equations describing enzyme kinetics in this chapter, the notation varies a bit from other chapters. Thus v is accepted in the biochemical literature as the symbol for reaction rate while Vmax is used for the maximum rate. Furthermore, for simplification frequently Vmax is truncated to V in complex formulas (see Equations 11.28 and 11.29). Although at first glance inconsistent, these symbols are familiar to students of biochemistry and related areas. The square brackets indicate concentrations. Vmax expresses the upper limit of the rate of the enzyme reaction. It is the product of the rate constant k3, also called the turnover number, and the total enzyme concentration, [E]o. The case u, = Vmax corresponds to complete saturation of all active sites. The other kinetic limit, = (Vmax/KM)[S], corresponds to Km >> [S], in other words Vmax/KM is the first order rate constant found when the substrate concentration approaches zero ... 

Figure 11.1 illustrates the behavior of Equation 11.6. By the assumption of rapid equilibrium the rate determining step is the unimolecular decomposition. At high substrate composition [S] KM and the rate becomes zero-order in substrate, v = Vmax = k3 [E0], the rate depends only on the initial enzyme concentration, and is at its maximum. We are dealing with saturation kinetics. The most convenient way to test mechanism is to invert Equation 11.6... 

The kinetics of this relationship are straightforward the effector system (enzyme activity) is a zero order process (i.e. the substrate S saturates the enzyme Ej whereas the inactivation reaction E2 is a first order process. The consequences of the kinetics of this system is that the magnitude of the change in Ei results in precisely the same quantitative change in the concentration of X. For example a fivefold increase in Ei produces a fivefold increase in the concentration of X. This relationship is shown in Box 12.2. Three examples are given. 

Ei is a zero order process whose activity can be changed. X is the messenger molecule. Reaction (E2) is catalysed by an enzyme that obeys Michaelis Menten kinetics, the ATn of which for X is assumed to be 0.1 arbitrary units. 

The best-known exception to exponential kinetics is the elimination of alcohol (ethanol), which obeys a linear time course (zero-order kinetics), at least at blood concentrations > 0.02 %. It does so because the rate-limiting enzyme, alcohol dehydrogenase, achieves half-saturation at very low substrate concentrations, i.e at about 80 mg/L (0.008 %). Thus, reaction velocity reaches a plateau at blood ethanol concentrations of about 0.02 %, and the amount of drug eliminated per unit of time remains constant at concentrations above this level. 

Any enzymic reaction that supplies substrate to a metabolic pathway. For all subsequent steps to maintain their steady-state concentrations, flux-generating reactions must exhibit zero-order kinetics. 

Ethanol is metabolized primarily in the liver by at least two enzyme systems. The best-studied and most important enzyme is zinc dependent alcohol dehydrogenase. Salient features of the reaction can be seen in Fig. 35.1. The rate of metabolism catalyzed by alcohol dehydrogenase is generally linear with time except at low ethanol concentrations and is relatively independent of the ethanol concentration (i.e., zero-order kinetics). The rate of metabolism after ingestion of different amounts of ethanol is illustrated in Fig. 35.2. 

Furthermore, it can be shown that, in the limiting cases of first-order kinetics [Equation (11.35) also holds for this case] and zero-order kinetics, the equal and optimal sizes are exactly the same. As shown, the optimal holding times can be calculated very simply by means of Equation (11.40) and the sum of these can thus be used as a good approximation for the total holding time of equal-sized CSTRs. This makes Equation (11.31) an even more valuable tool for design equations. The restrictions are imposed by the assumption that the biocatalytic activity is constant in the reactors. Especially in the case of soluble enzymes, for which ordinary Michaelis-Menten kinetics in particular apply, special measures have to be taken. Continuous supply of relatively stable enzyme to the first tank in the series is a possibility, though in general expensive. A more attractive alternative is the application of a series of membrane reactors. 

When only a fixed amount of drug is eliminated in a given interval of time, because enzymes for biotransformation and elimination are saturated, the kinetics of drug elimination are zero order (16). Alcohol is the classic example, where blood levels rise exponentially with increased amounts ingested, because elimination mechanisms are saturated and only a certain fraction of the total dose taken can be eliminated before the next dose is ingested. 

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 ... 

A coating bearing one enzyme (papain) is produced on the surface of a glass pH electrode by the method previously introduced (co-crosslinking). The papain reaction decreases the pH, and the pH-activity variation gives an autocatalytic effect for pH values greater than the optimum under zero-order kinetics for the substrate (benzoyl arginine ethyl ester) the pH inside the membrane is studied as a function of the pH in the bulk solution in which the electrode is immersed. A hysteresis effect is observed and the enzyme reaction rate depends not only on the metabolite concentrations, but also on the history of the system. 

Michaelis—Menten kinetics kinetics describing processes such as the majority of Enzyme-mediated reactions in which the initial reaction rate at low substrate concentrations is first order but at higher substrate concentrations becomes saturated and zero order. Can also apply to excretion for some compounds. 

In Chapter 8, we addressed proton transfer reactions, which we have assumed to occur at much higher rates as compared to all other processes. So in this case we always considered equilibrium to be established instantaneously. For the reactions discussed in the following chapters, however, this assumption does not generally hold, since we are dealing with reactions that occur at much slower rates. Hence, our major focus will not be on thermodynamic, but rather on kinetic aspects of transformation reactions of organic chemicals. In Section 12.3 we will therefore discuss the mathematical framework that we need to describe zero-, first- and second-order reactions. We will also show how to solve somewhat more complicated problems such as enzyme kinetics. 

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