Michaelis-Menten type equations

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 total transit time is longer because of the third step. If the reciprocals of equation 3.68 are taken, a Michaelis-Menten-type equation is generated. 

Even this scheme represents a complex situation, for ES can be arrived at by alternative routes, making it impossible for an expression of the same form as the Michaelis-Menten equation to be derived using the general steady-state assumption. However, types of non-competitive inhibition consistent with the Michaelis-Menten type equation and a linear Linweaver-Burk plot can occur if the rapid-equilibrium assumption is valid (Appendix S.A3). In the simplest possible model, involving simple linear non-competitive inhibition, the substrate does not affect the inhibitor binding. Under these conditions, the reactions... 

If one substrate was in great excess compared to the other and to the enzyme concentration, the corresponding term could be treated as being constant. Thus, the typical Michaelis-Menten type equation results in both cases. See eqn (4.23), left, for the case when [XO] is in excess and the right... 

As the ammonia uptake rate is an enzymatic step, a Michaelis—Menten-type equation is chosen ... 

Yamada and coworkers carried out characterization of a diaphorase monolayer at a glass surface in the same operationmode SECM [46]. Diaphorase purified from Bacillus stearothermophilus is a membrane protein, which catalyzes the oxidation of NADH in the presence of an electron mediator. Because of the high activity of diaphorase, a Michaelis-Menten type equation was applicable without further simplification for the boundary condition at the substrate surface. In the following studies [47], diaphorase patterns with monolayer-level coverage at flat glass substrates were visualized by feedback mode imaging. 

The respiratory metabolism is governed by enzymatic reactions, and hence, many researchers have used the Michaelis-Menten type equation to describe the relation... 

Under physiological conditions, [S] is seldom saturating, and itself is not particularly informative. That is, the in vivo ratio of [S]/A , usually falls in the range of 0.01 to 1.0, so active sites often are not filled with substrate. Nevertheless, we can derive a meaningful index of the efficiency of Michaelis-Menten-type enzymes under these conditions by employing the following equations. As presented in Equation (14.23), if... 

The Effect of Various Types of Inhibitors on the Michaelis-Menten Rate Equation and on Apparent K, and Apparent F ,ax ... 

Each of the processes shown in Figure 2.8 can be described by a Michaelis-Menten type of biochemical reaction, a standard generalized mathematical equation describing the interaction of a substrate with an enzyme. Michaelis and Men ten realized in 1913 that the kinetics of enzyme reactions differed from the kinetics of conventional... 

Similarly, for the steady-state situation with one Michaelis-Menten type of uptake site, the Best equation (17) still applies, now with a bioconversion capacity given by ... 

The necessity of developing approximate kinetics is unclear. It is sometimes argued that uncertainties in precise enzyme mechanisms and kinetic parameters requires the use of approximate schemes. However, while kinetic parameters are indeed often unknown, the typical functional form of generic rate equations, namely a hyperbolic Michaelis Menten-type function, is widely accepted. Thus, rather than introducing ad hoc functions, approximate Michaelis Menten kinetics can be utilized an approach that is briefly elaborated below. 

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]. 

Determine whether these data can be reasonably fitted by a kinetic equation of the Michaelis-Menten type, or... 

The analytical solutions of these equations are easily obtained for first-order or constant-order reactions, but numerical solutions are required for Michaelis-Menten type reactions. The above equations, in these cases, are usually rewritten in terms of dimensionless variables. For a spherical pellet this equation is ... 

Similarly, for enzyme-catalyzed reactions of the Michaelis-Menten type, we can derive Equation 7.3 from Equation 3.31. 

Consider an idealized simple case of a Michaelis-Menten type bioreaction taking place in a vertical cylindrical packed-bed bioreactor containing immobilized enzyme particles. The effects of mass transfer within and outside the enzyme particles are assumed to be negligible. The reaction rate per dilfcrential packed height (m) and per unit horizontal cross-sectional area of the bed (m ) is given as (cf. Equation 3.28) ... 

There are still other causes of nonlinearities than (apparent or real) higher-order transformation kinetics. In Section 12.3 we discussed catalyzed reactions, especially the enzyme kinetics of the Michaelis-Menten type (see Box 12.2). We may also be interested in the modeling of chemicals which are produced by a nonlinear autocatalytic reaction, that is, by a production rate function, p(Q, which depends on the product concentration, C,. Such a production rate can be combined with an elimination rate function, r(C,), which may be linear or nonlinear and include different processes such as flushing and chemical transformations. Then the model equation has the general form ... 

In concomitance with the displacement observed by i.r., an evolution of the catalytic activity has been observed while studying the liquid-phase epoxidation of cyclohexene in the presence of (EGDA)- Mo(VI), freshly prepared or after four months of conditioning at room temperature under inert atmosphere. As usual, the appearance of epoxide was followed by gas chromatographic analyses or by direct titration of oxirane oxygen and the disappearance of hydroperoxide was monitored by iodometric titration. In figure we report concentration-time for typical runs in ethylbenzene at 80°C obtained with the experimental procedure already described (ref. 9). It may be seen that with a freshly prepared catalyst an induction period is observed which lowers the initial catalytic activity. Our modified Michaelis-Menten type model equation (ref. 9) cannot adequately fit the kinetic curves obtained due to the absence of kinetic parameters which account for the apparent initial induction period (see Figure). 

In the oxidation of alkanethiols to disulfides with chloramine-T (CAT), in alkaline solution, the proposed reactive species are hypochlorous acid and TsNCl- anion. A correlation of reaction rate with Taft s dual substituent parameter equation yielded p = -5.28 and 5 = -2.0, indicating the rate-enhancing effect of electron-donating substituents.133 Michaelis-Menten-type kinetics have been observed in the oxidation of atenolol with CAT in alkaline solutions. TsNHCl is assumed to be reactive species. A mechanism has been suggested and the activation parameters for the rate-determining step were calculated.134 The Ru(III)-catalysed oxidation of diphenyl... 

Phytoplankton production RpA in environment A is a function of solar radiation Ea, concentration of nutrients nA, temperature TA, phytoplankton biomass pA, and concentration of pollutants A. There are many models that describe the photosynthesis process (Legendre and Legendre, 1998 Legendre and Krapivin, 1992). For the description of this function in the present study, an equation of Michaelis-Menten type is used (block MFB) ... 

Oxidation of thio acids by tetrabutylammonium tribromide showed Michaelis-Menten-type kinetics with respect to the thio acid. The effect of solvent composition was analysed using the Grunwald-Winstein equation. A mechanism involving the formation of an intermediate complex in the pre-equilibrium and its subsequent decomposition in a slow step is proposed.129... 

Equation (2.16) clearly shows that the inhibited kinetics are still of the Michaelis-Menten type. Thus, extending the argument of Albery et al. [42] we can use our original equations (2.5)-(2.12) to analyse the data simply by substituting in the new definitions for k cax and K. From equations (2.17) and (2.18). it is clear that when Kf is large, corresponding to little product inhibition, k cat = kcal and Km = Km and the inhibited solution reverts to the solution found previously for the situation where there was no product inhibition as required. 

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. 

Both active and passive fluxes across the cellular membranes can occur simultaneously, but these movements depend on concentrations in different ways (Fig. 3-17). For passive diffusion, the unidirectional component 7jn is proportional to c°, as is indicated by Equation 1.8 for neutral solutes [Jj = Pj(cJ — cj)] and by Equation 3.16 for ions. This proportionality strictly applies only over the range of external concentrations for which the permeability coefficient is essentially independent of concentration, and the membrane potential must not change in the case of charged solutes. Nevertheless, ordinary passive influxes do tend to be proportional to the external concentration, whereas an active influx or the special passive influx known as facilitated diffusion—either of which can be described by a Michaelis-Menten type of formalism—shows saturation effects at higher concentrations. Moreover, facilitated diffusion and active transport exhibit selectivity and competition, whereas ordinary diffusion does not (Fig. 3-17). 

Michaelis-Menten-type biochemical reactions that also resemble the adsorption isotherm. Therefore, a model such as that described by an equation rooted in biology would seem to be pharmacologically relevant. The fit to such a model is shown in Figure 12.7A. However, a better mathematical fit can be obtained by a complex mathematical function of the form... 

For the irreversible first-order reaction and the Michaelis-Menten type reaction, the following Equations 7.6 and 7.7 hold, respectively ... 

---