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

Kinetics of enzyme-catalyzed reactions
Enzyme inactivation
Modeling and kinetics of inactivation processes

In the simplest inactivation model, active enzyme molecules (Ea) undergo irreversible structural or chemical changes, leading to an inactive form (E_і):

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The rate of this reaction, r_a, is proportional to the concentration of the active form of the enzyme:

rd = k_de_a      (3.76)

Therefore, in a closed, well-mixed system (assuming that the reaction mixture contains no substrate, product, inhibitor, or other effector), The change in the concentration of the active form of the enzyme over time is described by the equation

so that

This expression is consistent, in particular, with the experimental data shown in Fig. 3.27 for the determination of adenosine triphosphatase activity under various conditions. The Temperature dependence of the inactivation constant k_d can be described by Transition State Theory [for example, Eq. (3.72) with a = 1] in a form slightly different from the Arrhenius equation [Eq. (3.66)] for the narrow temperature range of interest for biological systems. Table 3.10 lists the activation energy E and Entropy ∆S* values for the Denaturation of several common Enzymes. As can be seen from the data in the table, Protein Denaturation processes are characterized by very large activation energies.

In the absence of enzymatic activity, Cell viability is impossible. In some cases, a cell dies after the destruction of only a very small fraction of its intracellular enzymes. In this regard, it becomes clear why heat Treatment can be used for sterilization (i.e., the destruction of all forms of microorganisms) of gases, liquids, or solids. In Chapter 7, we will learn in more detail why only a few enzymes (and, accordingly, few microorganisms) are able to withstand prolonged heating without harm.

FIG. 3.27. Time dependence of adenosine triphosphatase inactivation at various pH values and temperatures. [Reproduced with permission from: Pelletier G. E., Quellet L., Influence of Temperature and pH on Myosin Inactivation; Can. J. Chem., 39, 265 (1961).]

Data on Enzyme Inactivation are usually obtained by incubating the enzyme for some time under denaturing conditions in the absence of substrate, after which the denaturing factors are removed, standard conditions are established, substrate is added, and the activity is determined from the initial rate of the enzymatic reaction. In the presence of substrate, the rate of enzyme inactivation can differ significantly from the rate determined as described above, if, for example, the free enzyme and the enzyme-substrate complex (or the enzyme-product complex) are inactivated at different rates, or if the substrate and/or reaction product themselves initiate enzyme inactivation. We will demonstrate one approach to analyzing such systems with a simple example. Let us assume that substrate binding stabilizes the enzyme (such an effect has indeed been observed to some extent in several systems). Thus, in our example, only the free enzyme will undergo inactivation.

Table 3.10. Activation energy and entropy of enzyme denaturation^a

⁸ Reproduced from: Laidler K. J., Bunting P. S., The Chemical Kinetics of Enzyme Action, 2nd ed., p. 430, Oxford University Press, London, 1973.

Combining such a model inactivation system with a simple Michaelis-Menten elementary reaction sequence [Eq. (3.4)], we obtain

If we further reasonably assume that the inactivation process occurs much slower than reactions (3.79a), then, applying the quasi-steady-state approximation for the complex (E_aS), we obtain the following expression for the reaction rate:

where e_tot,a is the total concentration of the active enzyme, both in the free form and as a complex. The rate of change of e_tot,a is expressed by the equation

Returning to the quasi-steady-state kinetics calculations used in deriving Eq. (3.80), we can express e_a in terms of e_tot,a and the parameters of the catalytic reaction; then Eq. (3.81) takes the following form:

It follows from Eqs. (3.80) and (3.82) that the rates of substrate conversion and enzyme inactivation are interrelated. In particular, the rate of enzyme inactivation depends on the Substrate Concentration. On the other hand, if E_a and (E_aS) are inactivated at the same rates, then under enzymatic reaction conditions, the enzyme activity will decrease in exactly the same way as in the absence of substrate.

Summarizing these arguments, it is easy to conclude that because of the differences in inactivation rates of different enzyme forms (free enzyme, various complexes, ionized states, etc.), the overall reaction rate depends on any parameter (substrate and inhibitor concentrations, pH, etc.) that affects the relative amounts of the different enzyme forms. In particular, the following model was developed to describe the complex pH dependence of the inactivation rate:

As an example of such a complex process, Fig. 3.28 shows the pH dependence of the ricin inactivation rate (ricin is not an enzyme; the content of the active protein form was determined by measuring the soluble protein fraction). For several other Proteins, a minimum inactivation rate was also observed at a specific pH value.

FIG. 3.28. Dependence of the first-order rate constant of ricin inactivation on temperature and pH. [Reproduced with permission from: Levy M., Benaglia A. E., METABOLISM/18.html">The Influence of Temperature and pH upon the Rate of Denaturation of Ricin, J. Biol. Chem., 186, 829 (1950).]

The decrease in enzymatic activity over time does not always follow a first-order reaction, i.e., Eq. (3.78). For A number of proteins, non-linear relationships between the logarithm of activity and time have been found, sometimes with two distinct linear regions on the corresponding curves (Fig. 3.29, a). Several models have been proposed to explain and analyze such systems. One of these, involving parallel reversible and irreversible inactivations, has been used to explain and analyze the loss of activity in certain enzymes:

In particular, this model, which assumes first-order kinetics for each of these elementary reactions, was successfully used to interpret the results of the luciferase inactivation study shown in Fig. 3.29, a. For example, the inactivation curve at 45 °C was calculated using the parameters k_d1 = k_d2 = 1.02 and k_r = 0.02 h^-1.

FIG. 3.29. a — inactivation of a luciferase preparation at pH 6.8 and the indicated temperature. [Reproduced with permission from: Chase A. M., Studies on Cell Enzyme Systems. IV. The kinetics of Heat Inactivation of Cypridina Luciferase, J. Gen. Physiol., 33, 535 (1950).] b — inactivation of a-Chymotrypsin in solution at pH 7.8, temperature 40 °C, Ca²⁺ concentration of 10^-3 M, and initial Enzyme Concentration of 7.3∙10^-7 M (A), 3.65∙10^-6 M (B), 1.46∙10^-5 M (C), and 2.92∙10^-5 M (D). [Reproduced with permission from: Kawamura Y., Nakanishi K., Matsuno R., Kamikubo T., Stability of Immobilized a-Chymotrypsin, Biotech. Bioeng., 23, 1219 (1981).]

The loss of activity in protease solutions is a more complex process because proteases catalyze their own Hydrolysis. This feature (autolysis) must be taken into account in the corresponding inactivation model. The protease inactivation model for a-chymotrypsin shown below is a variation of model (3.83), supplemented by an autolysis step:

inactive hydrolysis products (Peptides) (3.84)

Some features of this reaction sequence have been confirmed by chemical Methods. Note, in particular, that only the reversibly inactivated form E_i of this very well-studied enzyme is susceptible to attack and hydrolysis by the active protease form E_a. A more detailed analysis of this model and its application to calculating the curves shown in Fig. 3.29, b is proposed in Exercise 3.14.

To conclude the review of enzyme inactivation kinetics, we should mention approaches to analyzing irreversible enzyme inactivation by poisons. In the simplest case, we have

E_a + poison → E_d r_d = k_d∙e_a∙(poison)  (3.85)

Since, however, poisons often act on the Active Site of the enzyme, access to which can be blocked by the enzyme-bound substrate, the analysis must be modified in the presence of the latter. Obviously, we need to consider the processes of catalysis, such as reaction (3.79a), and enzyme poisoning (3.85) together, where in the latter equation E_a will represent the free, uncomplexed enzyme. Similar modified models, which we will not discuss here, can also be developed for the inactivation processes described by Eqs. (3.83) and (3.84).