Enzyme Kinetics: A Modern Approach - Hardcover

Marangoni, Alejandro G.

Hardcover

ISBN 10: 0471159859 ISBN 13: 9780471159858

Publisher: Wiley-Interscience, 2002

Synopsis

Practical Enzyme Kinetics provides a practical how-to guide for beginning students, technicians, and non-specialists for evaluating enzyme kinetics using common software packages to perform easy enzymatic analyses.

About this title

About the Author

A. G. MARANGONI, PhD, is Professor and Canada Research Chair in Food and Soft Materials in the Department of Food Science at the University of Guelph, Ontario, Canada.

From the Back Cover

A modern approach to enzyme kinetics and its applications

As catalysts for the majority of metabolic and biochemical reactions in the body, enzymes are important drug targets as well as useful synthetic catalysts. Enzyme kinetics is the study of the speed of an enzyme-catalyzed reaction and provides useful knowledge that aids in the design of enzyme-based processes. A. G. Marangoni s Enzyme Kinetics: A Modern Approach provides a practical, how-to guide for students, technicians, and nonspecialists to evaluate enzyme kinetics, using common software packages to perform easy enzymatic analyses.

The treatment of enzyme kinetics in this book is radically different from the way the topic is traditionally covered. Marangoni stresses an understanding of how researchers arrive at models, what the models limitations are, and how they can be used in practical ways to analyze enzyme kinetic data. With the advent of computers, linear transformations of models have become unnecessary Enzyme Kinetics does away with all linear transformations of enzyme kinetic models, advancing the use of nonlinear regression techniques. Marangoni develops new ways to carry out analyses of enzyme kinetic data, particularly in the study of pH effects on catalytic activity and multisubstrate enzymes. Other topics addressed include:

• Tools and techniques of kinetic analysis
• Reversible and irreversible enzyme inhibition
• Multisite and cooperative enzymes
• Immobilized and interfacial enzymes
• Characterization of enzyme stability

Enzyme Kinetics is a handy, innovative resource for practicing researchers in the chemical, pharmaceutical, and food science industries.

As catalysts for the majority of metabolic and biochemical reactions in the body, enzymes are important drug targets as well as useful synthetic catalysts. Enzyme kinetics is the study of the speed of an enzyme-catalyzed reaction and provides useful knowledge that aids in the design of enzyme-based processes. A. G. Marangoni’s Enzyme Kinetics: A Modern Approach provides a practical, how-to guide for students, technicians, and nonspecialists to evaluate enzyme kinetics, using common software packages to perform easy enzymatic analyses.

The treatment of enzyme kinetics in this book is radically different from the way the topic is traditionally covered. Marangoni stresses an understanding of how researchers arrive at models, what the models’ limitations are, and how they can be used in practical ways to analyze enzyme kinetic data. With the advent of computers, linear transformations of models have become unnecessary–Enzyme Kinetics does away with all linear transformations of enzyme kinetic models, advancing the use of nonlinear regression techniques. Marangoni develops new ways to carry out analyses of enzyme kinetic data, particularly in the study of pH effects on catalytic activity and multisubstrate enzymes. Other topics addressed include:

• Tools and techniques of kinetic analysis
• Reversible and irreversible enzyme inhibition
• Multisite and cooperative enzymes
• Immobilized and interfacial enzymes
• Characterization of enzyme stability

Enzyme Kinetics is a handy, innovative resource for practicing researchers in the chemical, pharmaceutical, and food science industries.

Excerpt. © Reprinted by permission. All rights reserved.

Enzyme Kinetics

A Modern ApproachBy Alejandro G. Marangoni

John Wiley & Sons

Copyright © 2002 Alejandro G. Marangoni
All right reserved.
ISBN: 978-0-471-15985-8

Chapter One

TOOLS AND TECHNIQUES OF KINETIC ANALYSIS

1.1 GENERALITIES

Chemists are concerned with the laws of chemical interactions. The theories that have been expounded to explain such interactions are based largely on experimental results. Two main approaches have been used to explain chemical reactivity: thermodynamic and kinetic. In thermodynamics, conclusions are reached on the basis of changes in energy and entropy that accompany a particular chemical change in a system. From the magnitude and sign of the free-energy change of a reaction, it is possible to predict the direction in which a chemical change will take place. Thermodynamic quantities do not, however, provide any information on the rate or mechanism of a chemical reaction. Theoretical analysis of the kinetics, or time course, of processes can provide valuable information concerning the underlying mechanisms responsible for these processes. For this purpose it is necessary to construct a mathematical model that embodies the hypothesized mechanisms. Whether or not the solutions of the resulting equations are consistent with the experimental data will either prove or disprove the hypothesis.

Consider the simple reaction A + B [??] C. The law of mass action states that the rate at which the reactant A is converted to product C is proportional to the number of molecules of A available to participate in the chemical reaction. Doubling the concentration of either A or B will double the number of collisions between molecules that lead to product formation.

The stoichiometry of a reaction is the simplest ratio of the number of reactant molecules to the number of product molecules. It should not be mistaken for the mechanism of the reaction. For example, three molecules of hydrogen react with one molecule of nitrogen to form ammonia: [N.sub.2] + 3[H.sub.2] [??] 2N[H.sub.3].

The molecularity of a reaction is the number of reactant molecules participating in a simple reaction consisting of a single elementary step. Reactions can be unimolecular, bimolecular, and trimolecular. Unimolecular reactions can include isomerizations (A -> B) and decompositions (A -> B + C). Bimolecular reactions include association (A + B -> AB; 2A -> [A.sub.2]) and exchange reactions (A + B -> C + D or 2A -> C + D). The less common termolecular reactions can also take place (A + B + C -> P).

The task of a kineticist is to predict the rate of any reaction under a given set of experimental conditions. At best, a mechanism is proposed that is in qualitative and quantitative agreement with the known experimental kinetic measurements. The criteria used to propose a mechanism are (1) consistency with experimental results, (2) energetic feasibility, (3) microscopic reversibility, and (4) consistency with analogous reactions. For example, an exothermic, or least endothermic, step is most likely to be an important step in the reaction. Microscopic reversibility refers to the fact that for an elementary reaction, the reverse reaction must proceed in the opposite direction by exactly the same route. Consequently, it is not possible to include in a reaction mechanism any step that could not take place if the reaction were reversed.

1.2 ELEMENTARY RATE LAWS

1.2.1 Rate Equation

The rate equation is a quantitative expression of the change in concentration of reactant or product molecules in time. For example, consider the reaction A + 3B -> 2C. The rate of this reaction could be expressed as the disappearance of reactant, or the formation of product:

rate = - d]A]/dt = -1/3 d]B]/dt = 1/2 d/dt (1.1)

Experimentally, one also finds that the rate of a reaction is proportional to the amount of reactant present, raised to an exponent n:

rate [varies] [[A].sup.n] (1.2)

where n is the order of the reaction. Thus, the rate equation for this reaction can be expressed as

- d]A]/dt = [k.sub.r][[A].sup.n] (1.3)

where [k.sub.r] is the rate constant of the reaction.

As stated implicitly above, the rate of a reaction can be obtained from the slope of the concentration-time curve for disappearance of reactant(s) or appearance of product(s). Typical reactant concentration-time curves for zero-, first-, second-, and third-order reactions are shown in Fig. 1.1(a). The dependence of the rates of these reactions on reactant concentration is shown in Fig. 1.1(b).

1.2.2 Order of a Reaction

If the rate of a reaction is independent of a particular reactant concentration, the reaction is considered to be zero order with respect to the concentration of that reactant (n = 0). If the rate of a reaction is directly proportional to a particular reactant concentration, the reaction is considered to be first-order with respect to the concentration of that reactant (n = 1). If the rate of a reaction is proportional to the square of a particular reactant concentration, the reaction is considered to be second-order with respect to the concentration of that reactant (n = 2). In general, for any reaction A + B + C + ... -> P, the rate equation can be generalized as

rate = [k.sub.r] [[A].sup.a][[B].sup.b][.sup.c]... (1.4)

where the exponents a, b, c correspond, respectively, to the order of the reaction with respect to reactants A, B, and C.

1.2.3 Rate Constant

The rate constant ([k.sub.r]) of a reaction is a concentration-independent measure of the velocity of a reaction. For a first-order reaction, [k.sub.r] has units of [(time).sup.-1]; for a second-order reaction, [k.sub.r] has units of [(concentration).sup.-1] [(time).sup.-1]. In general, the rate constant of an nth-order reaction has units of [(concentration).sup.-(n-1)][(time).sup.-1].

1.2.4 Integrated Rate Equations

By integration of the rate equations, it is possible to obtain expressions that describe changes in the concentration of reactants or products as a function of time. As described below, integrated rate equations are extremely useful in the experimental determination of rate constants and reaction order.

1.2.4.1 Zero-Order Integrated Rate Equation

The reactant concentration-time curve for a typical zero-order reaction, A [right arrow] products, is shown in Fig. 1.1(a). The rate equation for a zero-order reaction can be expressed as

d]A]/dt = -[k.sub.r] [A.sup.0] (1.5)

Since [[A].sup.0] = 1, integration of Eq. (1.5) for the boundary conditions A = [A.sub.0] at t = 0 and A = [A.sub.t] at time t,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)

yields the integrated rate equation for a zero-order reaction:

[[A.sub.t]] = [[A.sub.0]] - [k.sub.r]t (1.7)

where [[A.sub.t]] is the concentration of reactant A at time t and [[A.sub.0]] is the initial concentration of reactant A at t = 0. For a zero-order reaction, a plot of [[A.sub.t]] versus time yields a straight line with negative slope -[k.sub.r] (Fig. 1.2).

1.2.4.2 First-Order Integrated Rate Equation

The reactant concentration-time curve for a typical first-order reaction, A [right arrow] products, is shown in Fig. 1.1(a). The rate equation for a first-order reaction can be expressed as

d]A]/dt = - [k.sub.r]]A] (1.8)

Integration of Eq. (1.8) for the boundary conditions A = [A.sub.0] at t = 0 and A = [A.sub.t] at time t,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.9)

yields the integrated rate equation for a first-order reaction:

ln [[A.sub.t]]/[[A.sub.0]] = - [k.sub.r]t (1.10)

or

[[A.sub.t]] = [[A.sub.0]] [e.sup.-[k.sub.r]t] (1.11)

For a first-order reaction, a plot of ln([[A.sub.t]]/[[A.sub.0]]) versus time yields a straight line with negative slope -[k.sub.r] (Fig. 1.3).

A special application of the first-order integrated rate equation is in the determination of decimal reduction times, or D values, the time required for a one-[log.sub.10] reduction in the concentration of reacting species (i.e., a 90% reduction in the concentration of reactant). Decimal reduction times are determined from the slope of [log.sub.10]([[A.sub.t]]/[[A.sub.0]]) versus time plots (Fig. 1.4). The modified integrated first-order integrated rate equation can be expressed as

[log.sub.10] [[A.sub.t]]/[[A.sub.0]] = - t/D (1.12)

or

[[A.sub.t]] = [[A.sub.0]] x [10.sup.-(t/D)] (1.13)

The decimal reduction time (D) is related to the first-order rate constant ([k.sub.r]) in a straightforward fashion:

D = 2.303/[k.sub.r] (1.14)

1.2.4.3 Second-Order Integrated Rate Equation

The concentration-time curve for a typical second-order reaction, 2A [right arrow] products, is shown in Fig. 1.1(a). The rate equation for a second-order reaction can be expressed as

d]A]/dt = -[k.sub.r]]A.sup.2] (1.15)

Integration of Eq. (1.15) for the boundary conditions A = [A.sub.0] at t = 0 and A = [A.sub.t] at time t,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.16)

yields the integrated rate equation for a second-order reaction:

1/[[A.sub.t]] = 1/[[A.sub.0]] + [k.sub.r]t (1.17)

or

[[A.sub.t]] = [[A.sub.0]]/1 + [[A.sub.0]][k.sub.r]t (1.18)

For a second-order reaction, a plot of 1/[A.sub.t] against time yields a straight line with positive slope [k.sub.r] (Fig. 1.5).

For a second-order reaction of the type A + B [right arrow] products, it is possible to express the rate of the reaction in terms of the amount of reactant that is converted to product (P) in time:

d]P]/dt = [k.sub.r]][A.sub.0] - P][[B.sub.0] - P] (1.19)

Integration of Eq. (1.19) using the method of partial fractions for the boundary conditions A = [A.sub.0] and B = [B.sub.0] at t = 0, and A = [A.sub.t] and B = [B.sub.t] at time t,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.20)

yields the integrated rate equation for a second-order reaction in which two different reactants participate:

1/[[A.sub.0] - [B.sub.0]] ln [[B.sub.0]][[A.sub.t]]/[[A.sub.0]][[B.sub.t]] = [k.sub.r]t (1.21)

where [[A.sub.t]] = [[A.sub.0] - [P.sub.t] and [[B.sub.t]] = [[B.sub.0] - [P.sub.t]]. For this type of second-order reaction, a plot of (1/[[A.sub.0] - [B.sub.0]]) ln([[B.sub.0]][[A.sub.t]]/[[A.sub.0]][[B.sub.t]]) versus time yields a straight line with positive slope [k.sub.r].

1.2.4.4 Third-Order Integrated Rate Equation

The reactant concentration-time curve for a typical second-order reaction, 3A [right arrow] products, is shown in Fig. 1.1(a). The rate equation for a third-order reaction can be expressed as

d]A]/dt = -[k.sub.r]][A].sup.3] (1.22)

Integration of Eq. (1.22) for the boundary conditions A = [A.sub.0] at t = 0 and A = [A.sub.t] at time t,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.23)

yields the integrated rate equation for a third-order reaction:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.24)

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.25)

For a third-order reaction, a plot of 1/(2[[[A.sub.t]].sup.2]) versus time yields a straight line with positive slope [k.sub.r] (Fig. 1.6).

1.2.4.5 Higher-Order Reactions

For any reaction of the type nA [right arrow] products, where n > 1, the integrated rate equation has the general form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.26)

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.27)

For an nth-order reaction, a plot of 1/[(n - 1)[[[A.sub.t]].sup.n-1]] versus time yields a straight line with positive slope [k.sub.r].

1.2.4.6 Opposing Reactions

For the simplest case of an opposing reaction A [??] B,

d]A]/dt = -[k.sub.1][A] + [k.sub.-1][B] (1.28)

where [k.sub.1] and [k.sub.-1] represent, respectively, the rate constants for the forward (A [right arrow] B) and reverse (B [right arrow] A) reactions. It is possible to express the rate of the reaction in terms of the amount of reactant that is converted to product (B) in time (Fig. 1.7a):

d]B]/dt = [k.sub.1][[A.sub.0] - B] - [k.sub.-1][B] (1.29)

At equilibrium, d]B]/dt = 0 and [B] = [[B.sub.e]], and it is therefore possible to obtain expressions for [k.sub.-1] and [k.sub.1][[A.sub.0]]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.30)

Substituting the [k.sub.1][[A.sub.0] - [B.sub.e]]/[[B.sub.e] for [k.sub.-1] into the rate equation, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.31)

Summing together the terms on the right-hand side of the equation, substituting ([k.sub.-1] + [k.sub.1])[[B.sub.e]] for [k.sub.1][[A.sub.0]], and integrating for the boundary conditions B = 0 at t = 0 and B = [B.sub.t] at time t,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.32)

yields the integrated rate equation for the opposing reaction A [??] B:

In [[B.sub.e]]/[[B.sub.e] - [B.sub.t]] = ([k.sub.1] + [k.sub.-1])t (1.33)

or

[[B.sub.t]] = [[B.sub.e]] - [[B.sub.e]] [e.sup.-([k.sub.1]+[k.sub.- 1])t] (1.34)

A plot of ln([[B.sub.e]]/[[B.sub.e] - B]) versus time results in a straight line with positive slope ([k.sub.1] + [k.sub.-1]) (Fig. 1.7b).

The rate equation for a more complex case of an opposing reaction, A + B [??] P, assuming that [[A.sub.0]] = [[B.sub.0]], and [P] = 0 at t = 0, is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.35)

The rate equation for an even more complex case of an opposing reaction, A + B [??] P + Q, assuming that [[A.sub.0]] = [[B.sub.0]], [P] = [Q], and [P] = 0 at t = 0, is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.36)

1.2.4.7 Reaction Half-Life

The half-life is another useful measure of the rate of a reaction. A reaction half-life is the time required for the initial reactant(s) concentration to decrease by 1/2. Useful relationships between the rate constant and the half-life can be derived using the integrated rate equations by substituting 1/2 [A.sub.0] for [A.sub.t].

The resulting expressions for the half-life of reactions of different orders (n) are as follows:

n = 0 ... [t.sub.1/2] = 0.5[[A.sub.0]]/[k.sub.r] (1.37)

n = 1 ... [t.sub.1/2] = ln 2/[k.sub.r] (1.38)

n = 2 ... [t.sub.1/2] = 1/[k.sub.r]][A.sub.0]] (1.39)

n = 3 ... [t.sub.1/2] = 3/2[k.sub.r]][[A.sub.0]].sup.2] (1.40)

The half-life of an nth-order reaction, where n > 1, can be calculated from the expression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.41)

(Continues...)

Excerpted from Enzyme Kineticsby Alejandro G. Marangoni Copyright © 2002 by Alejandro G. Marangoni. Excerpted by permission.
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Edition details

• Publisher: Wiley-Interscience
• Publication date: 2002
• Language: English
• ISBN 10: 0471159859
• ISBN 13: 9780471159858
• Binding: Hardcover
• Edition number: 1
• Number of pages: 248