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

Kinetics of Enzyme-Catalyzed Reactions
Kinetics of Simple Single- and Two-Substrate Enzymatic Reactions

In this section, we will attempt to derive mathematical expressions that characterize the rates of enzyme-catalyzed reactions. Naturally, the most critical criterion for the validity of such an expression is the agreement between calculated and experimentally determined rates. To avoid potential misunderstandings, we will first outline some experimental Methods used to determine reaction rates.

First, let us define what we mean by reaction rate. Consider the reaction

The rate of this reaction, under the quasi-equilibrium state approximation (Section 3.2.1), is defined as

Class="center"> (3.1)

where the symbols s and p denote the molar concentrations of the substrate S and reaction product P, respectively. It follows that v is expressed in moles per unit volume per unit time. Reaction rate is an intensive property that has a specific value at every point in the reaction mixture. Therefore, if concentrations or other intensive variables vary from point to point, the reaction rates at those points will also differ. In experimental kinetics studies, well-mixed reactors are often used to ensure a uniform reaction rate throughout the entire volume of the reaction mixture.

As in the modeling of any other chemical engineering processes, the word "point" in the definition of reaction rate is not used in a strictly geometric sense. Rather, a point here refers to a certain volume containing many molecules, which is nevertheless very small compared to the volume of the entire reaction mixture. This seemingly minor clarification is very important to keep in mind when transitioning from hypothetical models to concrete biological systems. For example, when modeling molecular processes in a single isolated Cell, The concepts of the concentration of a given substance A1 and its rate of conversion at a specific point may be completely inapplicable if The Cell contains only a small number of A1 molecules. We will return to this issue later when studying cell growth kinetics (Ch. 7).

A typical experiment to study the Kinetics of Enzymatic reactions (i.e., Reactions Catalyzed by Enzymes) is usually conducted as follows. At zero time, solutions of the substrate and the corresponding purified enzyme are mixed under vigorous stirring in a closed vessel maintained at a constant Temperature and containing a buffer solution required to maintain a specific pH value. Subsequently, at specified time intervals, the concentrations of the substrate and/or reaction product are determined. Various methods are used for this purpose, including spectrophotometric, manometric, electrode, and polarimetric methods, as well as sampling techniques. Usually, only initial reaction rate data are used. Since reaction conditions, including enzyme and substrate concentrations, are most accurately known at zero reaction time, the initial slope of the curve representing The change in substrate or product concentration as a function of time is determined from the following data:

Figure 3.5 shows data obtained from an experiment of the described type. Note that at t = 0, some reaction product is already present in the mixture; this implies that zero time is not actually the true start time of the reaction. This is one of the difficulties inherent in the METHOD OF DETERMINING initial reaction rates; other drawbacks of the method are described in specialized literature on chemical kinetics. Nevertheless, the initial rate method allows for the determination of enzymatic activity and substance concentrations at the initial moment of the reaction with sufficient reproducibility, and therefore can be successfully applied in practical work.

FIG. 3.5. Experimental Determination of the rate of Hydrolysis of a 30% starch solution by glucoamylase in a batch Reactor (60 °C, e₀ = 11,600 units, reactor volume 1 L).

Structure/149.html">The problem of reproducibility in determining enzymatic activity is important from many Perspectives, including the design of reactors in which processes involving isolated enzymes are carried out. In previous sections, we have already mentioned that Proteins easily denature under conditions that differ from their natural biological environment; in this regard, it is not surprising that an isolated enzyme in an "unusual" aqueous environment may gradually lose its catalytic activity (Fig. 3.6). It is known that such inactivation is characteristic of many enzymes; at the same time, in many textbooks on enzyme kinetics, this phenomenon is at best only mentioned. The problem of gradual loss of catalytic activity by enzymes is not as significant in vivo (in an intact living Organism), where the decrease in enzyme concentrations due to inactivation is compensated for by their Biosynthesis in the required amounts. Conversely, Enzyme inactivation in vitro (outside the living cell) must be taken into account both when studying reaction kinetics and when designing enzymatic reactors. We will return to this topic in Section 3.7.

In the caption of Fig. 3.5 (in parentheses), we encounter for the first time another feature typical of enzyme kinetics. Note that The amount of enzyme used in this experiment is expressed in certain "units." What are these mysterious units, and why can the amount of enzyme not be expressed in a more understandable and definite way, such as in moles or mass units?

FIG. 3.6. Enzyme inactivation in solution. Within 1 h at 37 °C, the aspartase solution loses more than 5% of its initial activity. [Reproduced from: Tosa T. et al., Continuous Production of L-Aspartic Acid by Immobilized Aspartase, Biotech. Bioeng., 15, 68 (1973)].

First, let us answer the second part of the question. The amount of enzyme is usually not expressed in mass units, for example, because enzyme preparations are typically mixtures of various proteins, in which the enzyme of interest is only one of several components. The content of the target enzyme in this mixture is often unknown; more so, it can vary from batch to batch. To characterize the preparation, the enzyme content is usually expressed in units of catalytic activity per unit mass of the enzyme preparation.

A unit, or activity unit, is defined as the amount of enzyme that exhibits a specific catalytic activity under standard conditions specified for that enzyme. For example, in the starch hydrolysis experiment shown in Fig. 3.5, one unit of glucoamylase activity was defined as the amount of enzyme that produces 1 µmol of glucose per minute in a 4% Lintner starch solution at pH 4.5 and 60 °C. It follows that the activity unit for each specific enzyme will have its own unique definition, and, furthermore, a single enzyme may have multiple Definitions of activity depending on the specific conditions of the reactions it catalyzes. Therefore, to avoid errors in interpreting or using the results of enzyme kinetics studies, one should always carefully verify the Definition of the enzyme activity unit applied in each case. Of course, the likelihood of error is significantly reduced if The activity of a highly purified enzyme is known.