Saturation kinetics enzyme reactions

Reversibly fonned micelles have long been of interest as models for enzymes, since tliey provide an amphipatliic environment attractive to many substrates. Substrate binding (non-covalent), saturation kinetics and competitive inliibition are kinetic factors common to botli enzyme reaction mechanism analysis and micellar binding kinetics. 

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]. 

As a simple model for the enzyme penicillinase, Tutt and Schwartz (1970, 1971) investigated the effect of cycloheptaamylose on the hydrolysis of a series of penicillins. As illustrated in Scheme III, the alkaline hydrolysis of penicillins is first-order in both substrate and hydroxide ion and proceeds with cleavage of the /3-lactam ring to produce penicilloic acid. In the presence of an excess of cycloheptaamylose, the rate of disappearance of penicillin follows saturation kinetics as the cycloheptaamylose concentration is varied. By analogy to the hydrolysis of the phenyl acetates, this saturation behavior may be explained by inclusion of the penicillin side chain (the R group) within the cycloheptaamylose cavity prior to nucleophilic attack by a cycloheptaamylose alkoxide ion at the /3-lactam carbonyl. The presence of a covalent intermediate on the reaction pathway, although not isolated, was implicated by the observation that the rate of disappearance of penicillin is always greater than the rate of appearance of free penicilloic acid. 

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 ... 

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. 

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... 

Many enzymes, which transform two different substrates to one or two product(s), could be characterized using equation (8.1), if the concentration of one substrate is high enough to saturate the enzyme. If the two substrate molecules bind to the enzyme independently from each other, the calculated KM values will reflect the affinity of the substrate to the complex of the other substrate molecule and the enzyme. Further, the Vj ax " ill characterize the rate of the reaction at the excess concentrations of both substrates (the enzyme is saturated by both substrates). However, this could be just a coarse approximation, and there are kinetic analytical methods for a more exact characterization of such two-substrate enzymic reactions, which could run on different ways e.g. random Bi-Bi, ping-pong Bi Bi mechanisms (Keleti, 1986 Fersht, 1985 Segel, 1975 Comish-Bowden, 1995). 

Rates for this reaction may easily be measured by disappearance of azide UV absorption. Most importantly, kinetic saturation behavior is noted with sufficient amounts of the reactants cycloaddition velocity becomes independent of substrate concentration. As is familiar from enzyme catalysis, this indicates complete occupancy of all available cucurbituril by reacting species. In actuality, the rate of the catalyzed reaction under conditions of saturation was found to be the same as that for release of the product from cucurbituril. Such a stoichiometric triazole complex was independently prepared and its kinetics of dissociation were examined by the displacement technique previously outlined, giving the identical rate constant of 1.7xl0 s under the standard conditions. (It is not uncommon for product release to be rate-limiting in enzymic reactions). 

At any given instant in an enzyme-catalyzed reaction, the enzyme exists in two forms, the free or uncombined form E and the combined form ES. At low [S], most of the enzyme is in the uncombined form E. Here, the rate is proportional to [S] because the equilibrium of Equation 6-7 is pushed toward formation of more ES as [S] increases. The maximum initial rate of the catalyzed reaction (Prnax) is observed when virtually all the enzyme is present as the ES complex and [E] is vanishingly small. Under these conditions, the enzyme is saturated with its substrate, so that further increases in [S] have no effect on rate. This condition exists when [S] is sufficiently high that essentially all the free enzyme has been converted to the ES form. After the ES complex breaks down to yield the product P, the enzyme is free to catalyze reaction of another molecule of substrate. The saturation effect is a distinguishing characteristic of enzymatic catalysts and is responsible for the plateau observed in Figure 6-11. The pattern seen in Figure 6-11 is sometimes referred to as saturation kinetics. 

In the preceding section, four diagnostic tests of affinity labeling were listed (inactivation inhibited by substrates, pH dependence of inactivation similar to that of catalysis, labeled inhibitor covalently bound in 1 1 stoichiometry, and saturation kinetics obeyed). The same criteria may be used to diagnose suicide inhibition. In addition, tests must be made to detect any diffusion of the activated intermediate I into solution. For example, the addition of —SH reagents that rapidly react with electrophiles and hence scavenge them should not slow down the rate of reaction. The suicide inhibitor should not, in any case, react with the thiol at an appreciable rate in the absence of enzyme. 

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 ... 

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. 

In contrast to [E]free, [E]total is observable. Eq. (2.3) is written with fCM, the Michaelis constant, instead of the equilibrium binding constant Ks unless the enzyme reaction is very fast (Section 2.3.3.) i.e., in almost all cases, fCM = Ks. In Eq. (2.3), the reaction rate is traditionally denoted by v [concentration/time] and fccat is the reaction rate constant [time4]. The equation describes a two-parameter kinetics, with a monotonically rising reaction rate with respect to substrate concentration and saturation at high substrate concentration. The maximum reaction rate at saturation is denoted by vmax, with vmax = fccat[E], The fCM value corresponds to the substrate... 

Most categories of enzymes have been found to be active in organic solvents, not just lipases but also proteases, dehydrogenases, peroxidases, and several others. Enzymatic reactions in organic solvents follow saturation kinetics mechanisms and active sites have been found to be the same as in aqueous solution. Kinetic constants, however, do not correlate with a few simple solvent parameters such as hydrophobicity, dielectric constant, or dipole moment instead, case-by-case correlations are found. 

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. 

We see that the rate of the enzyme-catalyzed reaction depends linearly on the enzyme concentration, and in a more complicated way on the substrate concentration. Thus, when [S] Km, (Eq. (2.41)) reduces to v = k2[E]0, and the reaction is zero order in [S], This means that there is so much substrate that all of the enzyme s active sites are occupied. It also means that [S] remains effectively unchanged, even though products are formed. This situation is known as saturation kinetics. The value k2[E]0 is also called the maximum velocity of the enzymatic reaction, and written as vmax. 

The P450 enzymes are found primarily in the outer membrane of the endoplasmic reticulum. Enzyme activity requires that the enzyme be integrated into a membrane that contains P450 reductase and, for some reactions, cytochrome b5. Characterization of the saturation kinetics for the P450 enzymes can be determined using a variety of enzyme preparations, including tissue slices, whole cells, microsomes, and reconstituted, purified enzymes. The more intact the in vitro preparation, the more it is likely that the environment of the enzyme will represent the in vivo environment. However, intact cell preparations do not... 

The developed H+ concentration gradient plus an electric potential across the membrane supply the driving force for ATP synthesis from ADP and Pi, a thermodynamically unfavorable reaction catalyzed by ATP synthase (Karrasch and Walker, 1999). The latter is a mitochondrial enzyme located on, and spanning, the inner mitochondrial membrane. At least when in submitochondrial particles, ATP synthase saturation kinetics involve ADP positive site-site interactions in catalysis. One group has proposed that ADP saturation in vivo also shows site-site interactions ( , the interaction or Hill coefficient increasing from 1, meaning no interaction, to 2) however, others have not found this, so this issue at this time must be considered to remain unresolved. 

LiP catalyzes the oxidation of 3,4-dimethoxybenzyl alcohol (veratryl alcohol) to veratryl aldehyde. Since this reaction can be easily followed at 310 nm, it is the basis for the standard assay for this enzyme (26,27). The enzyme exhibits normal saturation kinetics for both veratryl alcohol and H202 (28,43). Steady-state kinetic results Indicate a ping-pong mechanism in which H202 first oxidizes the enzyme and the oxidized intermediate reacts with veratryl alcohol (43). The enzyme has an extremely low pH optimum ( 2.5) for a peroxidase (43,44) however, the rate of formation of compound I (kx, Fig. 2) exhibits no pH dependence from 3.0-7.0 (45,46). Addition of excess veratryl alcohol at pH 3.0 results in the rapid conversion of... 

Vmax is the velocity of an enzyme-catalyzed reaction when the enzyme is saturated with all of its substrates and is equal to the product of the rate constant for the rate-limiting step of the reaction at substrate saturation (kCiU) times the total enzyme concentration, Ex, expressed as molar concentration of enzyme active sites. For the very simple enzyme reaction involving only one substrate described by Equation II-4, kCM = . Elowever, more realistic enzyme reactions involving two or more substrates, such as described by Equations II-11 and 11-12, require several elementary rate constants to describe their mechanisms. It is not usually possible to determine by steady-state kinetic analysis which elementary rate constant corresponds to kcat. Nonetheless, it is common to calculate kcat values for enzymes by dividing the experimentally determined Fmax, expressed in units of moles per liter of product formed per minute (or second), by the molar concentration of the enzyme active sites at which the maximal velocity was determined. The units of cat are reciprocal time (min -1 or sec - x) and the reciprocal of cat is the time required for one enzyme-catalyzed reaction to occur. kcat is also sometimes called the turnover number of the enzyme. 

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)... 

The rate of hydrolysis of 3H-phenyl-cocaine in the presence and absence of each monoclonal antibody as a function of substrate concentration was determined. Production of radiolabeled benzoic acid at time points corresponding to < 5% reaction extent provided initial rates. A saturation kinetics and a linear Lineweaver-Burk plot for each artificial enzyme were plotted. The first-order rate constants (kcat) and Michaelis constants (Km) of selected antibodies are provided in Table 2. 

As shown in Figure 8.4, the synthesis of NAD from tryptophan involves the nonenzymic cyclization of aminocarhoxymuconic semialdehyde to quinolinic acid. The alternative metahoUc fate of aminocarhoxymuconic semialdehyde is decarboxylation, catalyzed hy picolinate carboxylase, leading into the oxidative branch of the pathway, and catabolism via acetyl coenzyme A. There is thus competition between an enzyme-catalyzed reaction that has hyperbolic, saturable kinetics, and a nonenzymic reaction thathas linear, first-order kinetics. 

Reactions orders between zero and one in accordance with one-plus rate equations are very common in enzyme catalysis, even if the cycle is more complex and involves additional reactants or products. The plots just described thus are more broadly applicable. On the other hand, straight lines in such plots are only evidence of saturation kinetics, not an indication that the catalyst cycle has only one intermediate. 

For historical interest and to illustrate a general facet of systems with arbitrary distribution of catalyst material over free catalyst and reaction intermediates, the classical models of enzyme catalysis are briefly reviewed. They show "saturation kinetics " An increase in reactant concentration causes a shift of catalyst material from free catalyst to an intermediate, so that the rate has an asymptotic limit that can at most be approached even at the highest reactant concentrations. 

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