Kinetic systems Michaelis-Menten mechanism

The rate form of Eq. 57 and some of its generalizations are used to represent a number of widely different kinds of reactions. For example, in homogeneous systems this form is used for enzyme-catalyzed reactions where it is suggested by mechanistic studies (see the Michaelis-Menten mechanism in Chap. 2 and in Chap. 27). It is also used to represent the kinetics of surface-catalyzed reactions. 

The power-law formalism was used by Savageau [27] to examine the implications of fractal kinetics in a simple pathway of reversible reactions. Starting with elementary chemical kinetics, that author proceeded to characterize the equilibrium behavior of a simple bimolecular reaction, then derived a generalized set of conditions for microscopic reversibility, and finally developed the fractal kinetic rate law for a reversible Michaelis-Menten mechanism. By means of this fractal kinetic framework, the results showed that the equilibrium ratio is a function of the amount of material in a closed system, and that the principle of microscopic reversibility has a more general manifestation that imposes new constraints on the set of fractal kinetic orders. So, Savageau concluded that fractal kinetics provide a novel means to achieve important features of pathway design. 

The simple enzyme kinetic system does not exhibit oscillatory behaviour. The existence, uniqueness and global asymptotic stability of a periodic solution of a Michaelis-Menten mechanism was proved (Dai, 1979) for the case when the reaction occurred in a volume bounded by a membrane. The permeability of the membrane to a given species was specified as the function of another species. The form of the model is... 

As the above discussion indicates, assigning mechanisms to simple anation reactions of transition metal complexes is not simple. The situation becomes even more difficult for a complex enzyme system containing a metal cofactor at an active site. Methods developed to study the kinetics of enzymatic reactions according to the Michaelis-Menten model will be discussed in Section 2.2.4. 

In cases where the depuration of HOCs from BMOs involves enzyme-mediated biotransformations (Eq. 7.4) or active transport mechanisms, and environmental concentrations are high (e.g. near a point source), depuration rates have been shown to follow Michaelis-Menten kinetics (Spade and Hamelink, 1985). Michaelis-Menten kinetics is elicited when an enzyme or active transport system is saturated with a chemical. This type of kinetics is characterized by lower values of keS at sites with high HOC concentrations. If k s are unchanged at high concentration sites, Michaelis-Menten kinetics will result in elevated BAFs. However, if chemical concentrations become toxic, finfish likely avoid the area and sessile organisms such as mussels may close their valves for extended periods (Huckins et al., 2004). 

Some investigators also unnecessarily apply the further restriction that Michaelis-Menten kinetics refers only to enzymes catalyzing the conversion of a single substrate to a single product. Were this taken to its extreme, only isomerases would qualify, because most one-substrate systems utilize water as a second substrate or product. See Michaelis-Menten Equation Uni Uni Mechanism Enzyme Rate Equations (1. The Basics)... 

In relation to enzymic cytochrome P-450 oxidations, catalysis by iron porphyrins has inspired many recent studies.659 663 The use of C6F5IO as oxidant and Fe(TDCPP)Cl as catalyst has resulted in a major improvement in both the yields and the turnover numbers of the epoxidation of alkenes. 59 The Michaelis-Menten kinetic rate, the higher reactivity of alkyl-substituted alkenes compared to that of aryl-substituted alkenes, and the strong inhibition by norbornene in competitive epoxidations suggested that the mechanism shown in Scheme 13 is heterolytic and presumably involves the reversible formation of a four-mernbered Fev-oxametallacyclobutane intermediate.660 Picket-fence porphyrin (TPiVPP)FeCl-imidazole, 02 and [H2+colloidal Pt supported on polyvinylpyrrolidone)] act as an artificial P-450 system in the epoxidation of alkenes.663... 

Substrate-catalyst interaction is also essential for micellar catalysis, the principles of which have long been established and consistently described in detail [63-66]. The main feature of micellar catalysis is the ability of reacting species to concentrate inside micelles, which leads to a considerable acceleration of the reaction. The same principle may apply for polymer systems. An interesting way to concentrate the substrate inside polymer catalysts is the use of cross-linked amphiphilic polymer latexes [67-69]. Liu et al. [67] synthesized a histidine-containing resin which was active in hydrolysis of p-nitrophenyl acetate (NPA). The kinetics curve of NPA decomposition in the presence of the resin was of Michaelis-Menten type, indicating that the catalytic act was accompanied by sorption of the substrate. However, no discussion of the possible sorption mechanisms (i.e., sorption by the interfaces or by the core of the resin beads) was presented. 

These properties are likely to have an important influence on the behavior of intact biochemical systems, e.g., within the living cell, enzymes do not function in dilute homogeneous conditions isolated from one another. The postulates of the Michaelis-Menten formalism are violated in these processes and other formalisms must be considered for the analysis of kinetics in situ. The intracellular environment is very heterogeneous indeed. Many enzymes are now known to be localized within 2-dimensional membranes or quasi 1-dimensional channels, and studies of enzyme organization in situ [26] have shown that essentially all enzymes are found in highly organized states. The mechanisms are more complex, but they are still composed of elementary steps governed by fractal kinetics. 

Due to the formation of an intermediate complex, this type of reaction mechanism was described as being analogous to Michaelis-Menten kinetics [39]. A common error made when examining the behaviour of systems of this type is to use the Koutecky-Levich equation to analyse the rotation speed-dependence of the current. This is incorrect because the Koutecky-Levich analysis is only applicable to surface reactions obeying strictly first-order kinetics. Applying the Koutecky-Levich analysis to situations where the surface kinetics are non-linear, as in this case, leads to erroneous values for the rate constants. Below, we present the correct treatment for this problem based on an extension of a model originally developed by Albery et al. [42]... 

Figure 3.4 Simulation of Michaelis-Menten enzyme mechanism kinetics in closed system. Solid lines correspond to solution of Equations (3.27) with parameter values k+ = 1000 M-1 sec-1, k = 1.0 sec-1, k+2 = 0.1 sec-1, k-2 = 10M-1sec-1, and = 0.1mM. The initial conditions are a(0) = 1 mM, b 0) = 0, and c(0) = 0. Dashed lines correspond to the solution obtained by Equations (3.32).

Obtaining quasi-steady approximations for fluxes through reaction mechanisms, including mechanisms more complex than the simple Michaelis-Menten system studied in this section, is a major component of the study of enzyme kinetics. This topic will be treated in some detail in Chapter 4. 

While the majority of these concepts are introduced and illustrated based on single-substrate single-product Michaelis-Menten-like reaction mechanisms, the final section details examples of mechanisms for multi-substrate multi-product reactions. Such mechanisms are the backbone for the simulation and analysis of biochemical systems, from small-scale systems of Chapter 5 to the large-scale simulations considered in Chapter 6. Hence we are about to embark on an entire chapter devoted to the theory of enzyme kinetics. Yet before delving into the subject, it is worthwhile to point out that the entire theory of enzymes is based on the simplification that proteins acting as enzymes may be effectively represented as existing in a finite number of discrete states (substrate-bound states and/or distinct conformational states). These states are assumed to inter-convert based on the law of mass action. The set of states for an enzyme and associated biochemical reaction is known as an enzyme mechanism. In this chapter we will explore how the kinetics of a given enzyme mechanism depend on the concentrations of reactants and enzyme states and the values of the mass action rate constants associated with the mechanism. 

The quasi-steady approximation, which was introduced in Section 3.1.3.2 and justified on the basis of rapid cycle kinetics in Section 4.1.1, forms the basis of the study of enzyme mechanisms, a field with deep historical roots in the subject of biochemistry. In later chapters of this book, our studies make use of this approximation in building models of biochemical systems. Yet there remains something unsatisfying about the approximation. We have seen in Section 3.1.3.2 that the approximation is not perfect. Particularly during short-time transients, the quasisteady approximation deviates significantly from the full kinetics of the Michaelis-Menten system described by Equations (3.25)-(3.27). Here we mathematically analyze the short timescale kinetics of the Michaelis-Menten system and reveal that a different quasi-steady approximation can be used to simplify the short-time kinetics. 

Often the key entity one is interested in obtaining in modeling enzyme kinetics is the analytical expression for the turnover flux in quasi-steady state. Equations (4.12) and (4.38) are examples. These expressions are sometimes called Michaelis-Menten rate laws. Such expressions can be used in simulation of cellular biochemical systems, as is the subject of Chapters 5, 6, and 7 of this book. However, one must keep in mind that, as we have seen, these rates represent approximations that result from simplifications of the kinetic mechanisms. We typically use the approximate Michaelis-Menten-type flux expressions rather than the full system of equations in simulations for several reasons. First, often the quasi-steady rate constants (such as Ks and K in Equation (4.38)) are available from experimental data while the mass-action rate constants (k+i, k-i, etc.) are not. In fact, it is possible for different enzymes with different detailed mechanisms to yield the same Michaelis-Menten rate expression, as we shall see below. Second, in metabolic reaction networks (for example), reactions operate near steady state in vivo. Kinetic transitions from one in vivo steady state to another may not involve the sort of extreme shifts in enzyme binding that have been illustrated in Figure 4.7. Therefore the quasi-steady approximation (or equivalently the approximation of rapid enzyme turnover) tends to be reasonable for the simulation of in vivo systems. 

Fig. 4 Mechanism of action (MOA) and inhibition studies of ML119 (compound 1) with HePTP and HePTP mutants, (a) Progress curves of HePTP (6.25 nM) activity in the presence of different doses of compound 1 (0, 0.078,0.156,0.313,0.625,1.25 /jM) and 0.3 mM OMFP in 20 mM Bis-Tris, pH 6.0,150 mM NaCI, 1 mM DH, and 0.005 % Tween-20 in 20 /jL totai assay voiume in biack 384-weii microtiter plates. No time-dependent inhibition was observed as demonstrated by the linear progress curves of the HePTP phosphatase reaction, (b) Eadie-Hofstee plot of the Michaelis-Menten kinetic study with compound I.The HePTP-catalyzed hydrolysis of OMFP was assayed at room temperature in a 60 /jL 96-well format reaction system in 50 mM Bis-Tris, pH 6.0 assay buffer containing 1.7 mM DTT, 0.005 % Tween-20, and 5 % DMSO. Recombinant HePTP (5 nM) was preincubated with various fixed concentrations of inhibitor (0,0.1,0.2,0.4,0.8,1.6 /jM) for 10 min. The reaction was initiated by addition of various concentrations of substrate (0,12.5,25,50,100,200,400 pM) to the...

The use of kinetics to characterize the behavior of integrated biochemical systems is a more recent and less developed practice. One of the more important issues in this integrative context is the selection of an appropriate formal representation. The most common approach is simply to adopt the Michaelis-Menten Formalism that has served so well for the elucidation of isolated reaction mechanisms. However, as the discussion above showed, there are difficulties in estimating the parameters of this formalism in general, and there is a combinatorial explosion in the amount of data required to characterize the rate law by kinetic means. Thus, even if the Michaelis-Menten Formalism were appropriate in principle, there... 

It has been reported that GABA was absorbed in the gastrointestinal tract as a specialized transport mechanism which obeys Michaelis-Menten and first-order kinetics [52]. Since the blood-brain barrier is impermeable to GABA [53], the hypotensive effect of peripherally administered GABA is due to the actions of the peripheral nervous system and those of several... 

Enzyme systems do not always follow Michaelis-Menten kinetics. Sometimes they show a sigmoidal variation of rate with substrate concentration, as shown schematically in Figure 10.12. There has recently been considerable interest in this kind of behavior, and it turns out that a number of different types of mechanisms can give rise to it. 

Fig. 7.1 Chemical reaction mechanism representing a biochemical NAND gate. At steady state, the concentration of species 85 is low if and only if the concentrations of both species Ii and I2 are high. All species with asterisks are held constant by buffering. Thus, the system is formally open although there are two conservation constraints. The first constraint conserves the total concentration of S3 -F 84 -F 85, and the second conserves -F 87. All enzyme-catalyzed reactions in this model are governed by simple Michaelis-Menten kinetics. Lines ending in over an enzymatic reaction step indicate that the corresponding enzyme is inhibited (noncom-petitively) by the relevant chemical species. We have set the dissociation constants, Kp j, of each of the enzymes Ei-Eg, from their respective substrates equal to 5 concentration units. The inhibition constants, K i and K 2, for the noncompetitive inhibition of E1 and 7 by 11 and I2, respectively, are both equal to 1 unit. The Vmax for both Ej and E2 is set to 5 units, and that for E3 and E4 is 1 unit/s. The Vmax s for E5 and Eg are 10 and 1 units/s, respectively. (From [1].)...

The analytical data show that gold catalysis and enzymatic catalysis allow fast and selective aerobic oxidation of glucose according to the same stoichiometry characterized by the formation of hydrogen peroxide as the by-product (Eq. (21.1)) [8]. However, it is not surprising that completely different catalytic systems adopt different reaction mechanisms as shown by the kinetic studies on commercial enzymatic preparations containing /wcose oxidase and catalase [13]. The results of the research support a Michaelis-Menten type mechanism where the kinetic... 

A to the first line, Rossler (1976) was the first to provide a chemical model of chaos. It was not a mass-action-type model, but a three-variable system with Michaelis-Menten-type kinetics. Next Schulmeister (1978) presented a three-variable Lotka-type mechanism with depot. This is a mass-action-type model. In the same year Rossler (1978) presented a combination of a Lotka-Volterra oscillator and a switch he calls the Cause switch showing chaos. This model was constructed upon the principles outlines by Rossler (1976a) and is a three-variable nonconservative model. Next Gilpin (1979) gave a complicated Lotka-Volterra-type example. Arneodo and his coworkers (1980, 1982) were able to construct simple Lotka-Volterra models in three as well as in four variables having a strange attractor. 

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