Michaelis-Menten enzyme

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 ... 

An exponential function that describes the increase in product during a first-order reaction looks a lot like a hyperbola that is used to describe Michaelis-Menten enzyme kinetics. It s not. Don t get them confused. If you can t keep them separated in your mind, then just forget all that you ve read, jump ship now, and just figure out the Michaelis-Menten description of the velocity of enzyme-catalyzed reaction—it s more important to the beginning biochemistry student anyway. 

R . The extent of substrate saturation of two Michaelis-Menten enzymes with different values will be identical if their R values are the same. For example, if R = 1, then v/ymax = 0.5 for both enzymes. Likewise, R values of 5 and 10 yield respective velocities of... 

Reactions in which the velocity (v) of the process is independent of the reactant concentration, following the rate law v = k. Thus, the rate constant k has units of M sAn example of a zero-order reaction is a Michaelis-Menten enzyme-catalyzed reaction in which the substrate concentration is much larger than the Michaelis constant. Under these conditions, if the substrate concentration is raised even further, no change in the velocity will be observed (since v = Umax)- Thus, the reaction is zero-order with respect to the substrate. However, the reaction is still first-order with respect to total enzyme concentration. When the substrate concentration is not saturating then the reaction ceases to be zero order with respect to substrate. Reactions that are zero-order in each reactant are exceedingly rare. Thus, zero-order reactions address a fundamental difference between order and molecularity. Reaction order is an empirical relationship. Hence, the term pseudo-zero order is actually redundant. All zero-order reactions cease being so when no single reactant is in excess concentration with respect to other reactants in the system. 

Box 12.2 An Enzyme-Catalyzed Reaction (Michaelis-Menten Enzyme Kinetics)... 

In Box 12.2, a simple model for a special kind of catalyzed reaction, the Michaelis-Menten enzyme kinetics, is presented, which leads to the following kinetic expression ... 

KiMM is given the subscript, MM, to remind us that it reflects Michaelis-Menten enzyme kinetics as distinguished from KiM used above to model microbial growth kinetics (see Monod cases above). Note, is the same as KE in Box 12.2 when it s value represents the reciprocal of the equilibrium constant for the binding step. 

Figure 17.16 Relationships of biodegradation rate, v, to substrate concentration, [/], when Michaelis-Menten enzyme kinetics is appropriate (a) when plotted as hyperbolic relationship (Eq. 17-79 in text), or (b) when plotted as inverse equation, Vv =...

The following data were obtained for a Michaelis-Menten enzyme with and without an inhibitor ... 

For a Michaelis-Menten enzyme Rs = 81. For a sigmoidal curve, Rs <81 and the enzyme is said to exhibit positive cooperativity with respect to the substrate. Positive cooperativity implies that the substrate binding or catalytic rate, or both, increases with increasing substrate concentration more than would be expected for a simple Michaelis-Menten enzyme. If Rs >81, the enzyme is said to display negative cooperativity with respect to the substrate substrate binding or catalysis, while increasing, becomes progressively less than would be found with a simple Michaelis-Menten enzyme as substrate concentration is increased. 

Prove that for a Michaelis-Menten enzyme, Rs of Eq. (9.44) is equal to 81. SOLUTION... 

In a Michaelis-Menten enzyme mechanism, what substrate concentrations (relative to Km) are needed for the reaction rate to be (a) 0.1 Vmax (ft) 0.25 Fmax (c) 0.5 Fmax (d) 0.9 Fmax ... 

As an example in this section, we consider the well-known Michaelis-Menten enzyme mechanism ... 

The behavior of the Michaelis-Menten enzyme system is illustrated in Figure 3.4 parameter values and initial conditions are listed in the figure legend. For the parameter values indicated Keq = 10, which corresponds to the final ratio of b/a for a closed system. From the figure it is apparent that for the given set of parameters, the enzyme complex concentration c changes at a rate much smaller in magnitude than da/dt and db/dt. Based on this observation we can introduce the... 

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).

Equation (3.31) is the standard form for the steady state flux though a simple reversible Michaelis-Menten enzyme. This expression obeys the equilibrium ratio arrived at above (b/a)eq = Keq = k+ k+2/(k- k-2), when Jmm(g, b) = 0. In addition, from the positive and negative one-way fluxes in Equation (3.30), we note that the relationship J+/J = Keq(a/b) = e AG/RT is maintained whether or not the system is in equilibrium. Thus, as expected, the general law of Equation (3.12) is obeyed by this reaction mechanism. 

Example simulating Michaelis-Menten enzyme kinetics... 

As we have pointed out in the introduction, our focus in this chapter is on how to build models of biochemical systems, and not on mathematical analysis of models. As an example, consider the system of Equations (3.27), which represents a model for the reactions of Equation (3.25). It is possible to analyze these equations using a number of mathematical techniques. For example Murray [146] presents an elegant asymptotic analysis of a model of an irreversible (with 2 = 0) Michaelis-Menten enzyme. Such analyses invariably yield mathematical insights into the behavior of... 

Figure 4.2 Kinetic mechanism of a Michaelis-Menten enzyme. (A) The reaction mechanism for the irreversible case - Equation (4.1) - is based on a single intermediate-state enzyme complex (ES) and an irreversible conversion from the complex to free enzyme E and product P. (B) The reaction mechanism for the reversible case - Equation (4.7) - includes the formation of ES complex from free enzyme and product P. For both the irreversible and reversible cases, the reaction scheme is illustrated as a catalytic cycle.

Figure 4.3 Lineweaver-Burk, or double-reciprocal, plot for single-substrate irreversible Michaelis-Menten enzyme. A plot of 1/7 versus 1/[S] yields estimates of Vmax and Km, as illustrated in the figure.

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 ... 

For the example of the reversible Michaelis-Menten enzyme catalyzing S P, in the steady state S and P are transported into and out of the system at the constant rate J. The positive and negative fluxes of the catalytic cycle are given by... 

A more cogent mathematical treatment of this problem was given in the 1970s by several mathematical biologists. For details see books by Lin and Segel [130] and Murray [146], Here we provide a brief account of this approach. The approach uses the somewhat advanced mathematical method of singular perturbation analysis, but does provides a deep appreciation of the Michaelis-Menten enzyme kinetics. 

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