12 - substrates modified Michaelis-Menten equation

The kinetic data from phosphate [28] and proton [58] release experiments were analyzed quantitatively by means of a modified Michaelis-Menten equation assuming activation when only one substrate molecule, S, is bound and inhibition when two substrate molecules are bound to P-gp as described by the following scheme ... 

Kinetic Analysis. The kinetic parameters were obtained by iterative non-linear curve fitting of raw data (current generated versus the substrate concentration).The data fitted a modified Michaelis-Menten equation ... 

A modified Michaelis-Menten equation that relates the velocity of the reaction in the presence of inhibitor to the concentrations of substrate and inhibitor can be derived ... 

The mechanisms of enzyme inhibition fall into three main types, and they yield particular forms of modified Michaelis-Menten equations. These can be derived for single-substrate/single-product enzymic reactions using the steady-state analysis of Sec. 5.10, as follows. 

Table 18.2 lists 30 of the molecules used in this study that are known to be substrates for active transport or active efflux. The mechanistic ACAT model was modified to accommodate saturable uptake and saturable efflux using standard Michaelis-Menten equations. It was assumed that enzymes responsible for active uptake of drug molecules from the lumen and active efflux from the enterocytes to the lumen were homogeneously dispersed within each luminal compartment and each corresponding enterocyte compartment, respectively. Equation (5) is the overall mass balance for drug in the enterocyte compartment lining the intestinal wall. 

Equation 10.2-9 is known as the Michaelis-Menten equation. It represents the kinetics of many simple enzyme-catalyzed reactions which involve a single substrate (or if other substrates are in large excess). The interpretation of Km as an equilibrium constant is not universally valid, since the assumption that step (1) is a fast equilibrium process often does not hold. An extension of the treatment involving a modified interpretation of Km is given next. 

Under many circumstances, the behavior of a simple unireactant enzyme system cannot be described by the Michaelis-Menten equation, although a v versus [S] plot is still hyperbolic and can be described by a modified version of the equation. For example, as will be discussed later, when enzyme activity is measured in the presence of a competitive inhibitor, hyperbolic curve fitting with the Michaelis-Menten equation yields a perfectly acceptable hyperbola, but with a value for Km which is apparently different from that in the control curve O Figure 4-7). Of course, neither the affinity of the substrate for the active site nor the turnover number for that substrate is actually altered by the presence of a competitive... 

Table 2.4 shows the SAS NLIN specifications and the computer output. You can choose one of the four iterative methods modified Gauss-Newton, Marquardt, gradient or steepest-descent, and multivariate secant or false position method (SAS, 1985). The Gauss-Newton iterative methods regress the residuals onto the partial derivatives of the model with respect to the parameters until the iterations converge. You also have to specify the model and starting values of the parameters to be estimated. It is optional to provide the partial derivatives of the model with respect to each parameter, b. Figure 2.9 shows the reaction rate versus substrate concentration curves predicted from the Michaelis-Menten equation with parameter values obtained by four different... 

This equation indicates that when v is equal to one-half of V, the equilibrium constant Km is numerically equal to S. A plot of the reaction rates at different substrate concentrations can be used to determine Km. Because it is not always possible to attain the maximum reaction rate at varying substrate concentrations, the Michaelis-Menten equation has been modified by using reciprocals and in this... 

This is the Michaelis-Menten equation, but with K , modified by a term including the inhibitor concentration and inhibitor constant. is unaltered. Therefore, curves of v against [5] in the presence and absence of inhibitor reach the same limiting value at high substrate concentrations, but when the inhibitor is present, JC, is apparently greater. Plots of 1/v against 1/[S] with and without inhibitor cut the ordinate at the same point but have different slopes and intercepts on the abscissa (Figure 8-9). 

A model for enzyme kinetics that has found wide applicability was proposed by Michaelis and Menten in 1913 and later modified by Briggs and Haldane. The Michaelis-Menten equation relates the initial rate of an enzyme-catalyzed reaction to the substrate concentration and to a ratio of rate constants. This equation is a rate equation,... 

In textbooks dealing with enzyme kinetics, it is customary to distinguish four types of reversible inhibitions (i) competitive (ii) noncompetitive (iii) uncompetitive and, (iv) mixed inhibition. Competitive inhibition, e.g., given by the product which retains an affinity for the active site, is very common. Non-competitive inhibition, however, is very rarely encountered, if at all. Uncompetitive inhibition, i.e. where the inhibitor binds to the enzyme-substrate complex but not to the free enzyme, occurs also quite often, as does the mixed inhibition, which is a combination of competitive and uncompetitive inhibitions. The simple Michaelis-Menten equation can still be used, but with a modified Ema, or i.e. ... 

This is essentially identical with the form of the Michaelis-Menten equation (8.8), although the meaning of the corresponding biocatalytic parameters is slightly modified. The kinetic scheme upon which this derivation is based is clearly limited to a single-substrate biocatalyst that operates with two independent catalytic sites. Such a biocatalyst could be monomeric with two catalytic sites, or else homodimeric with one catalytic site per subunit. With reference to the biocatalyst examples described in Section 8.1, the chemical biology reader should be able to see that Equation (8.25) and the Uni Uni kinetic scheme for two catalytic sites seems... 

If we take Eq. (19.13) into account, Eq. (19.10) yields a modified version of the Michaelis-Menten equation that is valid for arbitrary substrate concentration ... 

In our previous work [63], we studied the hydrolysis kinetics of lipase from Mucor javanicus in a modified Lewis cell (Fig. 4). Initial hydrolysis reaction rates (uri) were measured in the presence of lipase in the aqueous phase (borate buffer). Initial substrate (trilinolein) concentration (TLj) in the organic phase (octane) was between 0.05 and 8 mM. The presence of the interface with octane enhances hydrolysis [37]. Lineweaver-Burk plots of the kinetics curve (1/Uj.] = f( /TL)) gave straight lines, demonstrating that the hydrolysis reaction shows the expected kinetic behavior (Michaelis-Menten). Excess substrate results in reaction inhibition. Apparent parameters of the Michaelis equation were determined from the curve l/urj = f /TL) and substrate inhibition was determined from the curve 1/Uj.] =f(TL) ... 

Microbial Biotransformation. Microbial population growth and substrate utilization can be described via Monod s (35) analogy with Michaelis-Menten enzyme kinetics (36). The growth of a microbial population in an unlimiting environment is described by dN/dt = u N, where u is called the "specific growth rate and N is microbial biomass or population size. The Monod equation modifies this by recognizing that consumption of resources in a finite environment must at some point curtail the rate of increase (dN/dt) of the population ... 

Unfortunately, most enzymes do not obey simple Michaelis-Menten kinetics. Substrate and product inhibition, presence of more than one substrate and product, or coupled enzyme reactions in multi-enzyme systems require much more complicated rate equations. Gaseous or solid substrates or enzymes bound in immobilized cells need additional transport barriers to be taken into consideration. Instead of porous spherical particles, other geometries of catalyst particles can be apphed in stirred tanks, plug-flow reactors and others which need some modified treatment of diffusional restrictions and reaction technology. 

Equation 8.4a gives the expression for the rate of formation of product by a modified version of the Michaelis-Menten mechanism in which the second step is also reversible. Derive the expression and find its limiting behavior for large and small concentrations of substrate. 

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