Reversible Michaelis-Menten kinetics

The derivation of the Michaelis-Menten equation in the previous section differs from the standard treatment of the subject found in most textbooks in that the quasi-steady approximation is justified based on the argument that the catalytic cycle kinetics is rapid compared to the overall biochemical reactant kinetics. In 

1 Note that Vmax and Km may be estimated from data on steady state flux and substrate concentration based on a number of different ways of plotting J and [S], Cornish-Bowden illustrates that the Lineweaver-Burk plot (or double-reciprocal plot) is not recommended when one would like to minimize the effect of experimental error on parameter estimates. For a detailed discussion see Section 2.6 of [35]. 

Section 4.2 we explore the quasi-steady approximation with somewhat more mathematical rigor. However, before undertaking that analysis, let us analyze the reversible enzyme mechanism studied in Chapter 3 from the perspective of cycle kinetics. 

In Chapter 3 we determined the conditions under which it is and is not appropriate to treat a reaction as irreversible. Using the notation of cycle kinetics and apparent mass-action constants, the reversible mechanism of Equation (3.25) is represented 

Assuming again that the cycle kinetics are rapid and maintain enzyme and complex in a rapid quasi-steady state, we can obtain the steady state velocity for the reversible Michaelis-Menten enzyme kinetics  

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Figure 3.5 illustrates the comparison between a system governed by the reversible Michaelis-Menten kinetics of Equations (3.31) and (3.32), the irreversible kinetics of Equations (3.35) and (3.36). The parameter values are indicated in the legend. The values used correspond to the same set of values as used in Figure 3.4 with the exception that k-2 is changed from 10 M 1 sec-1 in Figure 3.4 to 1.0 M-1 sec-1 in... 

Fig. 4.15 Global effectiveness factor (mean integral value) of an immobilized enzyme with reversible Michaelis-Menten kinetics in a spherical particle as a function of bulk substrate concentration and Thiele modulus (substrate conversion 0.4 substrate conversion at equihbrium 0.5)...

Abu-Reesh, I. M. 1997. Predicting the Performance of Immobilized Enzyme Reactors Using Reversible Michaelis-Menten Kinetics. Bioprocess Engineering 17 131-137. 

Smith, W. G., 1992. In kinetics and the reversible Michaelis-Menten model. Journal of Chemical Education 12 981 — 984. 

The inactivation is normally a first-order process, provided that the inhibitor is in large excess over the enzyme and is not depleted by spontaneous or enzyme-catalyzed side-reactions. The observed rate-constant for loss of activity in the presence of inhibitor at concentration [I] follows Michaelis-Menten kinetics and is given by kj(obs) = ki(max) [I]/(Ki + [1]), where Kj is the dissociation constant of an initially formed, non-covalent, enzyme-inhibitor complex which is converted into the covalent reaction product with the rate constant kj(max). For rapidly reacting inhibitors, it may not be possible to work at inhibitor concentrations near Kj. In this case, only the second-order rate-constant kj(max)/Kj can be obtained from the experiment. Evidence for a reaction of the inhibitor at the active site can be obtained from protection experiments with substrate [S] or a reversible, competitive inhibitor [I(rev)]. In the presence of these compounds, the inactivation rate Kj(obs) should be diminished by an increase of Kj by the factor (1 + [S]/K, ) or (1 + [I(rev)]/I (rev)). From the dependence of kj(obs) on the inhibitor concentration [I] in the presence of a protecting agent, it may sometimes be possible to determine Kj for inhibitors that react too rapidly in the accessible range of concentration. ... 

Because of the complexity of biological systems, Eq. (1) as the differential form of Michaelis-Menten kinetics is often analyzed using the initial rate method. Due to the restriction of the initial range of conversion, unwanted influences such as reversible product formation, effects due to enzyme inhibition, or side reactions are reduced to a minimum. The major disadvantage of this procedure is that a relatively large number of experiments must be conducted in order to determine the desired rate constants. 

For reversible enzymatic reactions, the Haldane relationship relates the equilibrium constant KeqsNith the kinetic parameters of a reaction. The equilibrium constant Keq for the reversible Michaelis Menten scheme shown above is given as... 

The scaled elasticities of a reversible Michaelis Menten equation with respect to its substrate and product thus consist of two additive contributions The first addend depends only on the kinetic propertiesand is confined to an absolute value smaller than unity. The second addend depends on the displacement from equilibrium only and may take an arbitrary value larger than zero. Consequently, for reactions close to thermodynamic equilibrium F Keq, the scaled elasticities become almost independent of the kinetic propertiesof the enzyme [96], In this case, predictions about network behavior can be entirely based on thermodynamic properties, which are not organism specific and often available, in conjunction with measurements of metabolite concentrations (see Section IV) to determine the displacement from equilibrium. Detailed knowledge of Michaelis Menten constants is not necessary. Along these lines, a more stringent framework to utilize constraints on the scaled elasticities (and variants thereof) as a determinant of network behavior is discussed in Section VIII.E. 

Most problems associated with approximate kinetics are avoided when Michaelis Menten-type rate equations are utilized. Though this choice sacrifices the possibility of analytical treatment, reversible Michaelis Menten-type equations are straightforwardly consistent with fundamental thermodynamic constraints, have intuitively interpretable parameters, are computationally no more demanding than logarithmic functions, and are well known to give an excellent account of biochemical kinetics. Consequently, Michaelis Menten-type kinetics are an obvious choice to translate large-scale metabolic networks into (approximate) dynamic models. It should also be emphasized that simplified Michaelis Menten kinetics are common in biochemical practice almost all rate equations discussed in Section III.C are simplified instances of more complicated rate functions. 

The overall influence of ATP on the rate V (ATP) is measured by a saturation parameter C (—oo, 1]. Note that, when using Eq. (139) as an explicit rate equation, the saturation parameter implicitly specifies a minimal Hill coefficient min > C necessary to allow for the reverse transformation of the parameters. The interval 6 [0,1] corresponds to conventional Michaelis Menten kinetics. For = 0, ATP has no net influence on the reactions, either due to complete saturation of a Michaelis Menten term or, equivalently, due to an exact compensation of the activation by ATP as a substrate by its simultaneous effect as an inhibitor. For < 0, the inhibition by ATP supersedes the activation of the reaction by its substrate ATP. 

Henri and Michaelis-Menten kinetics assumed that the rate of formation of products was much less than that for the back reaction from ES to yield E + S. Van Slyke assumed the reverse. A more rigorous formulation was offered by Briggs and Haldane (1925) using steady-state assumptions previously applied to chemical kinetics by Bodenstein (1913). 

A term (also referred to as prior equilibrium ) denoting any reversible step that precedes an irreversible step or the rate-limiting step in a multistage reaction mechanism. The so-described reaction step must rapidly establish an equilibrium between its reactants and products. The first association/dissociation equilibrium leading to the formation of EX complex from E and S in the Michaelis-Menten treatment is an example of a preequilibrium. See Michaelis-Menten Kinetics... 

Crude and three diethyl ether extracted, acetone treated, fractions were isolated from large-scale cultures of Gambierdiscus toxicus. Crude extracts at. 04 mg/ml inhibited the histamine contraction response in smooth muscle of the guinea pig ileum. Three semi-purified fractions at 5 ng/ml, effectively inhibited the guinea pig ileum preparation. Two of these fractions followed Michaelis-Menten kinetics for a competitive inhibition. The third fraction inhibited in a non-reversible manner. This study has established the presence of three lipid extracted toxins in toxicus, outlined a method for their assay in small quantities, and identified at least two of the effects of these toxic extracts in animals. 

Also characteristic of enzymes that obey Michaelis-Menten kinetics is that suitable inhibitors can compete with the substrate for the enzyme active site, thus impeding the reaction. If the inhibitor binds reversibly to the enzyme active site, then the substrate can compete for the active site leading to competitive inhibition. To test for... 

In binding experiments, the affinity of magnesium ADP to native membranes and to the isolated calcium dependent ATPase was found to be considerably lower than that of magnesium ATP173. On the other hand, from the inhibition of the calcium-dependent ATPase or the activation of calcium release and ATP synthesis apparent affinities for ADP are obtained that are very similar to those of ATP (Fig. 12). The affinity of ADP for the enzyme apparently depends on its functional state. The affinity of ADP for the membranes under conditions of calcium release depends markedly on the pH of the medium. When the medium pH is reduced from 7.0 to 6.0, the affinity drops by a factor of 10. At pH 7.0 the affinity of the membrane for ADP corresponds to the affinity for ATP to the high affinity binding sites in the forward running mode of the pump. In contrast to the complex dependence of the forward reaction on the concentration of ATP, the dependence of the reverse reaction on ADP seems to follow simple Michaelis-Menten kinetics. 

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... 

Decarboxylation of indole-3-acetic add enhanced by thermal polylysine in darkness (without irradiation of white light) follows Michaelis-Menten kinetics, thus indicating a reversible catalyst-substrate interaction 22). 

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 formation of a reversible Michaelis-Menten-type complex of the enzyme and ferrocytochrome c [ES1S2 in Eq. (3) ] can be postulated from initial steady-state kinetics of the cytochrome c peroxidase reaction (17). Since cytochrome c peroxidase and cytochrome c are acidic and basic proteins, respectively, their interaction may be governed principally by electrostatic attraction. This assumption is further supported by the fact that several polycations which reversibly and irreversibly bind cytochrome c peroxidase inhibit its enzymic activity in competition with ferrocytochrome c 17,62). 

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