Biochemical Engineering Fundamentals, Part 1 - Bailey J., Ollis D. 1989

Kinetics of Enzyme-Catalyzed Reactions
Regulation of Enzyme Activity
Analysis of the effect of reversible regulation on the kinetics of enzyme-catalyzed reactions

The primary value of the Michaelis-Menten approach to analyzing Enzymatic Reaction Kinetics lies in its ability to quantitatively describe The Effect of effectors. First, let us assume that the reaction sequence shown below accurately reflects The Mechanism of interaction

between the inhibitor and substrate with the enzyme during (complete) competitive inhibition. Here, as usual, E and S denote the enzyme and substrate, respectively, while the symbols I and EI denote the inhibitor and enzyme-inhibitor complex, respectively.

Equilibrium reaction steps:

Class="center">

Slow step:

In this case, the enzyme can bind either only the substrate or only the inhibitor. Since a portion of the enzyme is bound in the EI complex, not all of the enzyme participates in the catalytic Conversion of the substrate; consequently, the reaction rate decreases in the presence of the inhibitor.

Based on the equilibrium reactions (3.48) and the total enzyme mass balance equation

е + (es) + (ei) = е0      (3.50)

the reaction rate can be expressed in terms of the total Enzyme Concentration e0 and the concentrations of free substrate and inhibitor (s and i, respectively):

(complete) competitive inhibition (3.51)

Comparison with the original form of the Michaelis–Menten Equation shows that the parameter v_max remains unchanged, while the apparent Michaelis constant K_m^app increases in the presence of a competitive inhibitor:

In other words, the reduction in reaction rate caused by a competitive inhibitor can be fully compensated for by a corresponding increase in Substrate Concentration; the competitive inhibitor does not affect the maximum possible reaction rate.

If the sequential reactions (3.48) and (3.49) are supplemented with the equilibrium reactions (3.53a) and (3.53b), a useful model of (complete) noncompetitive inhibition can be obtained:

Here, the inhibitor and substrate can bind to the enzyme simultaneously, forming a ternary EIS (enzyme-inhibitor-substrate) complex. In this simplest model of noncompetitive inhibition, we assume that inhibitor binding does not affect the affinity of the enzyme for the substrate, and substrate binding does not affect the affinity of the enzyme for the inhibitor. It follows that K_s and K_i in equations (3.53) are identical to the corresponding dissociation constants in equation (3.48). We also assume that the EIS complex does not dissociate to form the reaction product P.

Using the previously described method of expressing complex concentrations in terms of e₀, s, and i, we can obtain the following expression for the reaction rate:

(complete) noncompetitive inhibition (3.54)

It is easy to see that in this expression, the Michaelis constant remains unchanged, while the maximum reaction rate decreases to

It follows that in the presence of a noncompetitive inhibitor, adding an arbitrarily large amount of substrate to the reaction mixture will not increase the maximum reaction rate to the value achievable in the absence of the inhibitor.

Competitive and noncompetitive inhibition can be easily distinguished using Lineweaver–Burk plots. In the case of competitive inhibition, adding the inhibitor increases the intercept on the 1/s axis and does not affect the intercept on the 1/v axis. Conversely, in noncompetitive inhibition, only the intercept on the 1/v axis increases. As an example, let us consider the starch Hydrolysis data shown in Fig. 3.21. Comparing the plots of 1/v versus 1/s in the presence of various inhibitors, as well as without any inhibitor, it is easy to see that α-dextrin is a competitive inhibitor, whereas maltose and limit dextrin are noncompetitive inhibitors.

FIG. 3.21. Competitive (α-dextrin) and noncompetitive (maltose, limit dextrin) inhibition of α-amylase. (Reproduced with permission from: Aiba S., Humphrey A. E., Millis N., Biochemical Engineering, 2d ed., p. 101, University of Tokyo Press, Tokyo, 1974; Russian Introduction/27.html">Translation of the 1st edition: Aiba Sh., Humphrey A., Millis N., Biochemical Technology and Apparatus. — Moscow: Pishchevaya Promyshlennost, 1975, p. 81.

As indicated in Table 3.7, other Types of Enzymatic reaction inhibition exist. Below, we will briefly discuss how they can be described by modifying the inhibition models discussed above. In partial competitive inhibition (Type Ib in Table 3.7), the inhibitor and substrate bind to different sites on the enzyme molecule, and the inhibitor affects the affinity of the enzyme for the substrate. To describe this case, we add to The sequence of

reactions (3.48) and (3.49) the following steps, which we will assume to be at equilibrium:

and the slow step:

(Note: it can be shown that K_is = K_sK_si/K_i.)

In another case of noncompetitive inhibition (Type IIb), the model described above is modified to account for the possibility of reaction product formation from the EIS complex; the dissociation of the latter proceeds at a lower rate compared to the ES complex:

One of the models of mixed inhibition, which includes fully competitive and partially competitive types, can be obtained by omitting reaction (3.57) in the above model of partial competitive inhibition. The results of the analysis of all these types of inhibition can be written in the form of the Michaelis–Menten equation:

In this equation, the apparent parameters v_max^app and K_m^app depend on the inhibitor concentration and various kinetic parameters (Table 3.8). Fig. 3.22 schematically shows the effect of Different types of inhibition on various graphical representations of enzyme kinetics data.

One type of enzyme activation by an effector can be described by a Type IIb model, provided that k_i is greater than k. In terms of the reaction mechanism, this means that The rate of product formation from the EIS complex is higher than from the ES complex. Thus, here substance I Functions as an enzyme activator.