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

Kinetics of Enzyme-Catalyzed Reactions
Enzymatic Reactions in Heterogeneous Systems

Until now, our primary focus has been on The behavior of Enzymes in solution, where they interact with dissolved substrates. However, as previously noted, this is not the only possibility. For instance, Figure 2.27, which schematically depicts a Introduction/4.html">Prokaryotic Cell, illustrated that certain cellular enzymes are bound to cell membranes. Similar structures exist in eukaryotes; in Mitochondria, for example, enzymes responsible for highly complex reaction sequences are associated with an intricate system of internal membranes.

As shown in Fig. 3.31, nature and biochemical engineering offer many other combinations of physical states for enzymes and substrates. In this section, we will examine the Kinetics of Enzymatic reactions between enzyme solutions and insoluble substrates, and in the following chapter, we will study reactions of soluble substrates catalyzed by immobilized enzymes.

If a substrate can exist in several phases simultaneously, sometimes only the portion in solution undergoes the enzymatic reaction. The data presented in Fig. 3.32 serve as an example. Since all substances are soluble in Water to some extent, a small amount of substrate will always be present in solution and, consequently, subject to enzymatic action. At the same time, The rate of this process may be so low that it is of no practical interest. Let us now consider some different Examples.

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FIG. 3.31. Enzymes in various physical states catalyze the transformation of substrates in different forms. The classic case of an enzymatic reaction in solution is only one of many possible enzyme-substrate interaction variants. [Reproduced from: McLaren A. D., Packer L., Some Aspects of Enzyme Reactions in Heterogeneous Systems, Adv. Enzymol. Rel. Sub. Biochem., 33, 245 (1970).]

One such example is the Hydrolysis of methyl butyrate by pancreatic lipase—an enzyme produced in the Human digestive tract capable of breaking down fats. Unlike the previous example, the reaction here does not proceed until the substrate forms an insoluble state in the form of small droplets in the reaction mixture (Fig. 3.33). Clearly, this enzyme can only exhibit activity at the interface between two liquid phases. Given the possibility of enzyme Denaturation due to surface tension, It is interesting to note that—as with the aforementioned foam fractionation method in the presence of detergents—the surface tension on fat droplets can be significantly reduced by the adsorption of Bile acids, which are natural Surfactants secreted into the digestive tract.

FIG. 3.32. The enzymatic reaction occurs only with the soluble form of the substrate. (From: Dixon M., Webb E., Enzymes, Vol. 1—3, — M.: Mir, 1982, Vol. 1, p. 122.)

FIG. 3.33. Reaction at the interface of two liquid phases. [Reproduced with permission from: Sarda L., Desnuelle Р., Action de la Lipase Pancreatique sur les Esters en Emulsion, Biochim. Biophys. Acta, 30, 513 (1958).]

Other Enzymes are active toward both soluble and insoluble forms of the substrate. For example, it has been shown that the proteolytic enzyme Trypsin can cleave both free Lysozyme and lysozyme adsorbed On the surface of kaolinite. Lysozyme itself can interact with both soluble and "insoluble" substrates. As previously noted, lysozyme actively destroys bacterial cell walls; simultaneously, it can catalyze the Cleavage of soluble oligomers derived from Cell wall polymers (Section 3.4.2).

FIG. 3.34. The dependence of the degradation rate of a solid substrate (thiogel) on the Enzyme Concentration in solution. [Data from: Tsuk A. G., Oster G., DETERMINATION OF ENZYME Activity by a Linear Measurement, Nature (London), 190, 721 (1961).]

The Interaction of a dissolved enzyme with an insoluble substrate via adsorption onto the latter's surface can be described using an interesting variation of the kinetic equations discussed earlier. In contrast to previous cases where the reaction rate increased proportionally to the total enzyme concentration, here, as the enzyme concentration increases, the reaction rate initially rises and then approaches a certain limiting value. This behavior of the enzyme-substrate system is clearly illustrated by the data in Fig. 3.34, which reflect The kinetics of the hydrolysis of a solid protein block (in this case, thiogel—cross-linked gelatin) by trypsin.

To develop a reasonably sound model for the kinetics of a heterogeneous reaction, we will begin with an assumption opposite to the one used in the analysis of enzymatic reaction rates in solution: we will now assume that the enzyme adsorbs onto the substrate. Denoting vacant sites on the substrate surface with the symbol А, we can write the following equation:

If the total number of moles of adsorption sites on the substrate surface per unit volume of the reaction mixture is taken as a₀, then

а₀ = а + (еа)      (3.87)

From equation (3.87) and the enzyme adsorption equilibrium equation (3.86), it follows that

Now we only need to assume that the reaction concludes with the irreversible Cleavage of the complex EA. Consequently,

According to the model, е denotes the concentration of free enzyme, which at THE START OF the experiment is related to the total enzyme concentration е₀ by the ratio

е₀ = е + (еа)      (3.90)

If the initial enzyme concentration significantly exceeds the initial Substrate Concentration (е_{0 ≫} а₀), then it can be assumed with a good degree of approximation that

e₀ ≈ e      (3.91)

Therefore

For reactions involving solid substrates, situations where e ≫ a₀ are by no means an exception. For instance, the results presented in Fig. 3.34 were obtained from an experiment where the ratio e₀/a₀ was approximately 4000. In this regard, heterogeneous reactions differ sharply from reactions in solutions, where s0 is typically significantly greater than e₀.

From equation (3.92), it follows that a plot of 1/v versus 1/e₀ (in Lineweaver-Burk double reciprocal coordinates) should yield a straight line. As shown in Fig. 3.35, this is indeed the case. Here, the line in Fig. 3.35, a, represents the same data as presented in Fig. 3.34, but in a different form, while Fig. 3.35, b, shows analogous results obtained for another soluble enzyme — insoluble substrate system.

FIG. 3.35. a — Dependence of the solid substrate (thiogel) degradation rate (1/v) on trypsin concentration (1/e₀) in solution, plotted in Lineweaver-Burk coordinates. b — Dependence of the insoluble substrate (Poly-β-hydroxybutyrate particles) degradation rate by an enzyme solution (depolymerase from R. temoignei). [Reproduced from: McLaren A. D., Packer L., Some Aspects of Enzyme Reactions in Heterogeneous Systems; Adv. Enzymol. Rel. Sub. Biochem, 33, 245 (1970).]

Many nutrient sources for microorganisms exist as solid particles (in wastewater streams, lakes, compost piles, etc.). Evidently, the adsorption and transport of dissolved nutrients across cell membranes must be preceded by the hydrolysis of these particles by extracellular enzymes. Similarly, the hydrolysis of Cellulose by cellulase-type enzymes requires prior breakdown of insoluble particles. Therefore, it is reasonable to assume that the kinetics of heterogeneous enzymatic reactions discussed in this section will be applicable to the design of reactors utilizing such substrates.

In Conclusion, it should be emphasized that the applicability of the kinetic model discussed is not limited to solid substrates; it has also been used to describe insoluble liquid substrates dispersed in an enzyme solution (e.g., the system whose kinetics are depicted in Fig. 3.33). Furthermore, we have not considered here the differences in reactant concentrations in the bulk liquid phase and at the phase interface. We will examine this latter problem in detail in the next chapter.

Exercises

3.1. Determination of k_m and V_max. In Table 3У1.1, the initial rates of an enzyme-catalyzed reaction at various substrate concentrations are presented.

a) Determine v_max and K_m using the Lineweaver-Burk method.

b) Determine the same values graphically using Eadie-Hofstee coordinates.

c) Calculate the standard deviations of the slopes and intercepts for each method.

Table 3У1.1

3.2. Enzymatic Reaction in a Batch Reactor. An enzyme with K_m = 1 ∙ 10^-3 M was incubated with a substrate at an initial concentration of 3∙10^-5 M. After 2 minutes, 5% of the substrate had reacted. What amount of substrate will be transformed within 10, 30, and 60 minutes?

3.3. Kinetics of Enzymatic reactions involving Enzymes with Multiple Active Sites. Assume that the enzyme possesses two active sites, such that the substrate is transformed into the reaction product according to the following sequence of elementary reactions:

Assuming quasi-steady-state conditions for (ES) and (ESS), derive an expression for the rate of P formation.

3.4. Formation of Multiple Enzyme-Substrate Complexes. During enzyme-catalyzed reactions, several intermediate complexes can sometimes form. Derive the corresponding rate expressions by applying a) the Michaelis equilibrium approach and b) the quasi-steady-state approximation for the complexes, given the following sequence of reactions:

3.5. Relaxation Kinetics with Sinusoidal Perturbations. As shown in Fig. 3У5.1, if the parameters of a reacting system at equilibrium (or in a steady state) are subjected to a small sinusoidal perturbation, the concentrations of the reacting species will also change according to the same law.

FIG. 3У5.1.

Derive the equations relating the observed concentration fluctuations to the kinetic parameters for the reaction

assuming the average Temperature varies according to the equation

T = T₀(1 + asin ωt), where a ≪ 1

Show how k₁ and k₂ can be determined given the responses a and b to periodic perturbations.

3.6. Reversible Reactions. Suppose we are studying a reversible reaction:

a) Show that the equilibrium will be significantly shifted to the right only if k₁k₂ ≫ k_-1k_-2.

b) Show that the parameters v_s, v_p, K_s, and K_p are not independent.

c) Under what conditions will a graphical solution to the equation in part (b) using Lineweaver-Burk coordinates yield useful results?

d) By integrating dp/dt for the given reaction sequence, determine the dependence of p on t and the value of p at equilibrium

e) Show that B = 0 if K_s = K_p.

3.7. Enzyme inactivation. Suppose an enzyme is irreversibly inactivated according to equation (3.77).

a) Show that in such a process, only v_max changes, while K_m remains constant.

b) Show that for enzymes acting on insoluble substrates [equation (3.92)], one might erroneously arrive at a conclusion opposite to that in part (a) if The change in the concentration of the active enzyme form is not taken into account.

3.8. pH Dependence. Protons are generated during an enzyme-catalyzed irreversible reaction according to the following equation:

H₂O + E + S+ → E + SOH + H⁺

Assume that e^- is the concentration of the active enzyme form, and that e^-/e = 1.0 at pH = 6.0 = pK₁, and e^2-/e^- = 1.0 at pH = 10.0 = pK₂.

a) Show that the reaction rate at pH 7.0 can be approximately expressed by the equation

b) By integrating the previous equation, show that the change in s over time follows the law

where a and ß are parameters depending on the initial pH and substrate concentration.

3.9. Inhibition Kinetics. A pesticide inhibits The activity of enzyme A; therefore, the enzyme can be used to determine the pesticide concentration in samples of unknown composition.

a) Initial reaction rates were measured in laboratory experiments; the measurement results are presented in Table 3U9.1. Is this pesticide a competitive or non-competitive inhibitor? Determine The values of K_i, v_max, and K_m.

Table 3U9.1

b) A 50 mL enzyme solution, as described in problem (a), was mixed with 50 mL of an 8∙10^-4 M substrate solution and a 25 mL sample; the initial reaction rate was 18 µmol/(min∙L). What is the inhibitor concentration in the sample, assuming it contains no other inhibitors (besides the pesticide) or substrates?

3.10. Kinetics of enzymatic reactions involving an ionizable cofactor. For catalytic activity, the enzyme requires a bound cofactor. The cofactor-enzyme binding is very strong. The cofactor group essential for activity has a pK equal to pK_c; the cofactor activates the enzyme only in its deprotonated form. The primary pK values of the enzyme's Active Site on either side of the pH of maximum enzyme activity are pK₁ and pK₂. Derive (or simply state) the corresponding expressions for the rate of individual substrate conversion to product under the following conditions:

a) |рК_с| ≪ |pK₁| ≪ |рК₂|

b) |pK| ≪ |pK_c| ≪ |рК₂|

c) рК_с = рК₁

3.11. Substrate Activation. Derive the equations describing the rate of an enzymatic reaction with substrate activation:

As usual, assume that a) intermediate compounds are in a quasi-steady state and b) the substrate concentration significantly exceeds the enzyme concentration.

3.12. Heat Generation in Enzymatic Reactions. The maximum temperature rise in a plug flow reactor can be determined using the heat balance equation for substrate conversion during a single pass through the reactor.

a) Given that The amount of heat generated is greater than or equal to the amount of heat transferred to the liquid medium, express the maximum temperature rise in a single-enzyme catalyzed reaction in terms of the heat of reaction per mole of reactant ∆H_r, the fractional Conversion of the reactant δ, the inlet Reactant Concentration s₀, the liquid heat capacity C_p, and ∆T = T_outlet—T_inlet.

b) Show that for 80% hydrolysis of a 20% lactose solution [∆H_r = -7100 cal/(g∙mol)], the maximum temperature rise is only a few degrees Celsius.

c) If a thin-walled enzyme reactor is equipped with a cooling jacket, the temperature at the centerline of the outlet can be approximately determined by the equation

where z is the Column length, t is the residence time of the reaction mixture in the reactor, and a = k_t/pC_p, where k_t is the overall thermal conductivity, p is the density of the packed bed, and erf is the error function (tables of this function for various arguments can be found in any handbook). Show that for the conditions described in part (b) of this exercise, T* rapidly approaches unity (calculate T* for z = 1, 2, 3, 4 inches). (Pitcher W. Н., Immobilized Enzymes for Industrial Reactors, p. 151, Academic Press, New York, 1975.)

3.13. Hill Equation for Cooperative Binding. The following simple model is often used to describe the cooperative (n > 1) binding of substrates (S) to Oligomeric Proteins (E_n):

Propose a simple graphical METHOD FOR DETERMINING n, given a series of experimentally determined E_nS_n values.

3.14. Inactivation of α-Chymotrypsin. Consider the processes of thermal denaturation and autolysis of α-chymotrypsin, occurring simultaneously according to equation (3.84), where K is the Equilibrium Constant for the reaction between E_a and E_i1, and K_m is the dissociation constant of the E_aE_i1 complex. Assuming that (e_ae_i1) ≪ e₀ + e_i1, determine the time dependence of the concentration of potentially active enzyme e (e = e_a + e_i1) in a batch reactor.

References

All biochemistry textbooks listed at the end of Chapter 1 provide an introduction to enzyme kinetics. This topic is covered in particular detail and at a modern level in Mahler and Cordes' book (reference [4] in Chapter 2). For more detailed information, the reader may consult the following books and articles.

1. Bernhard A., Structure and function of Enzymes, W. A. Benjamin, New York, 1968. A concise and highly accessible introduction to enzymology.

2. Dixon M., Webb E., Enzymes, Vol. 1—3. — M.: Mir, 1982. The most comprehensive single-volume guide to enzyme chemistry and biology, an essential reference for any enzymologist. It includes an extensive chapter (200 pages) dedicated to enzyme kinetics. Additionally, the monograph covers many other problems in enzymology, provides a large table of enzymes and the reactions they catalyze, and offers an extensive bibliography.

3. Laidler К. J., Bunting Р. S., The Chemical Kinetics of Enzyme Action, 2d ed., Oxford University Press, London, 1973. This monograph provides a more detailed examination of the kinetics of Enzymatic Catalysis and presents extensive experimental data supporting various mathematical expressions proposed for determining reaction rates.

4. Schmid R. D., Stabilized Soluble Enzymes, in Advances in Biochemical Engineering, Vol. 12: Immobilized Enzymes II, p. 41, Ghose T. K., Fiechter A., Blakebrough N. (eds.), Springer-Verlag, Berlin, 1979. An excellent review article providing an in-depth Analysis of the causes and factors leading to Protein Denaturation, as well as a comprehensive Overview of Methods for Enhancing enzyme stability.

5. Johnson F. Н., Eyring Н., Polissar М. J., The Kinetic Basis of Molecular Biology, John Wiley and Sons, Inc., New York, 1954. A superb monograph containing extensive data on the effects of temperature, pH, pressure, and inhibitors on enzymatic and Other forms of biological activity, along with the interpretation of these data. Particular emphasis is placed on the application of Thermodynamics and absolute rate theory.

6. Joly M., Physical Chemistry of Protein Denaturation. — Moscow: Mir, 1968. Although this monograph naturally does not include many of the most recent findings, it covers all the fundamental phenomena related to the causes, manifestations, detection, and study of protein denaturation. All these phenomena are examined from a chemist's perspective.

7. McLaren A. D., Packer L., Some Aspects of Enzyme Reactions in Heterogeneous Systems, Advan. Enzymol., 33, 245 (1970). An excellent review article examining numerous heterogeneous enzymatic reactions occurring in biological systems under natural conditions. This work is dedicated to the important but relatively understudied behavior of insoluble substrates and associated enzymes in Cells and soil.