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`

`13iochemistry
`
`SECOND EDITION
`
`Reginald H. Garrett
`Charles M. Grisham
`University of Virginia
`
`•
`
`Saunders College Publishing
`Harcourt Brace College Publishers
`Fort Worth • Philadelphia • San Diego • New York • Orlando • Austin
`San Antonio • Toronto • Montreal • London • Sydney • Tokyo
`
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`

`Copyright© 1999, 1995 by Saunders College Publishing
`
`All rights reserved. No part of this publication may be reproduced or transmitted in
`any form or by any means, electronic or mechanical, including photocopy, recording,
`or any information storage and retrieval system, without permission in vvriting from
`the publisher.
`
`Requests for permission to make copies of any part of the work should be mailed to:
`Permissions Department, Harcourt Brace & Company, 6277 Sea Harbor Drive,
`Orlando, Florida 32887-6777.
`
`Publisher: Emily Barrosse
`Publisher: John J. Vondeling
`Product Manager: Pauline Mula
`Developmental Editor: Sandra Kiselica
`Project Editors: Linda Boyle Riley, Beth Ahrens, Sarah Fitz-Hugh
`Production Manager: Charlene Catlett Squibb
`Art Director and Text Designer: Carol Clarkson Bleistine
`Cover Designer: Cara Castiglia
`
`On the Cover: The structure of UQ-cytc reductase, also known as the cytochrome bc1
`complex. Image provided by Johann Deisenhofer and Di Xia, reprinted with permis(cid:173)
`sion from Proceedings of the National Academy of Sciences 95: 8028, © 1998 National
`Academy of Sciences, U.S.A
`
`Printed in the United States of America
`
`BIOCHEMISTRY, 2nd edition
`0-03-022318-0
`
`Library of Congress Catalog Card Number: 98-87839
`
`901234567 032 10 98765432
`
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`

`

`Chapter 14
`Enzyme Kinetics
`
`"Now! Now! " cried the Queen. "Faster!
`Faster! . .. Now here, you see, it takes all
`the running you can do, to keep in the
`same place. If you want to get somewhere
`else, you must run at least twice as fast as
`that!"
`
`L EWIS WRROI.L, Alict'sildv<mlLLres in Wonderlmul (1865)
`
`O UTUNE
`14.1 • Enzymes-Catalytic Power, Specificity,
`and Regulation
`14.2 • Introduction to Enzyme Kinetics
`14.3 • Kinetics of Enzyme-Catalyzed Reactions
`14.4 • Enzyme Inhibition
`14.5 • Kinetics of Enzyme-Catalyzed Reactions
`Involving Two or More Substrates
`14.6 • RNA and Antibody Molecules as
`Enzymes: Ribozymes and Abzymes
`
`"Alice and the Queen of Hearts," illustrated by John
`Tenniel, The Nursery Alice. (Mary Evans Picture Library,
`London)
`
`Living organisms seethe with metabolic activity. Thousands of chemical reac(cid:173)
`tions are proceeding very rapidly at any given instant within all living cells.
`Virtually all of these transformations are mediated by enzymes, proteins (and
`occasionally RNA) specialized to catalyze metabolic reactions. The substances
`transformed in these reactions are often organic compounds that show little
`tendency for reaction outside the cell. An excellent example is glucose, a sugar
`that can be stored indefinitely on the shelf with no deterioration. Most cells
`quickly oxidize glucose, producing carbon dioxide and water and releasing lots
`of energy:
`C5H 1206 + 6 02 ~ 6 C02 + 6 H 20 + 2870 kJ of energy
`( - 2870 kJ/ mol is the standard free energy change [.1.G" ' ] for the oxidation of
`glucose; see Chapter 3). In chemical terms, 2870 kJ is a large amount of energy,
`and glucose can be viewed as an energy-rich compound even though at ambi-
`
`426
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`14.1 • Enzymes-Catalytic Power; Specificity, and &gulation
`
`427
`
`FlGURE 14.1 • Reac tion profile showing
`large tid for glucose oxidation, free energy
`change of -2,870 kJ/ mol; catalysts lower tlct,
`thereby accelerating rate .
`
`Glucose
`+602
`
`--1 t.ct ,Energyofactivation
`-- -'"'"f'ym"
`
`tiC, Free e nergy
`released
`
`Progress of reaction
`
`enL temperature it is not readily reactive with oxygen outside of cells. Stated
`another way, glucose represents thermodynamic potentiality: its reaction with
`oxygen is strongly exergonic, but it just doesn't occur under normal condi(cid:173)
`tio ns. On the other hand, enzymes can catalyze such thermodynamically favor(cid:173)
`able reactions so that they proceed at extraordinarily rapid rates (Figure 14.1).
`In glucose oxidation and countless other instances, enzymes provide cells with
`the ability to exert kinetic control over thermodynamic potentiality. That is, living sys(cid:173)
`Lems use enzymes to accelerate and control the rates of vitally important bio(cid:173)
`chemical reactions.
`
`Enzymes Are the Agents of Metabolic Function
`Acting in sequence, enzymes form metabolic pathways by which nutrient mole(cid:173)
`cules are degraded, energy is released and converted into metabolically useful
`forms, and precursors are generated and transformed to create the literally
`thousands of distinctive biomolecules found in any living cell (Figure 14.2).
`Situated at key junctions of metabolic pathways are specialized regulatory
`enzymes capable of sensing the momentary metabolic needs of the cell and
`adjusting their catalytic rates accordingly. The responses of these enzymes
`ensure the harmonious integration of the diverse and often divergent meta(cid:173)
`bolic activities of cells so that the living state is promoted and preserved.
`
`14.1 • Enzymes - Catalytic Power, Specificity, and Regulation
`
`Enzymes are characterized by three distinctive features: catalytic power, speci(cid:173)
`ficity, and regulation.
`
`Catalytic Power
`Enzymes display enormous catalytic power, accelerating reaction rates as much
`as 1016 over uncatalyzed levels, which is far greater than any synthetic catalysts
`can achieve, and enzymes accomplish these astounding feats in dilute aqueous
`
`0
`
`Glucose
`
`t H exokinase
`
`Glucose-6-P
`Phosphog,luco(cid:173)
`ISomerase
`
`0
`
`Fructose-6-P
`I Phosph ofructokinase
`v
`t Ald olase
`Fructose-I ,6-bis P
`0
`
`Glyceraldehyde-3-P Dihydroxyacetone-P
`
`A~ Triose-P
`~ ~ 8
`iso memse
`I
`Glyceraldehyde-
`3-P de hydrogenase 't
`1,3-Bisphosphoglycerate
`
`•
`
`Phosphoglycernte
`kinase
`
`Phosphoglycero-
`mutase
`
`3-Phosphoglycerate
`0
`I
`t
`2-Phosphoglycerate
`C!) t Enolase
`6!> t Pyruvate kinase
`
`Phosphoenolpyruvate
`
`Pyruvate
`
`FlGURE 14.2 • The breakdown of glucose by
`gLycolysis provides a prime example of a meta(cid:173)
`bolic pathway. Ten enzymes mediate the reac(cid:173)
`tions of glycolysis. Enzyme 4, fructose I, 6, biphos(cid:173)
`phate aldolase, catalyzes the c-c bond(cid:173)
`breaking reaction in this pathway.
`
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`

`428
`
`Chapter 14 • Enzyme Kinetics
`
`100
`
`75
`
`50
`
`35
`
`25
`
`...,
`-o
`·;:._
`..,
`<::
`<,) ... ..,
`
`"-<
`
`0
`
`0
`
`2
`
`3
`
`7
`5 6
`4
`Reaction step
`
`8
`
`9 10
`
`FIGURE 14.3 • A 90% yield over 10 steps,
`for example, in a metab olic pathway, gives an
`overall yield of 35%. Therefore, yields in bio(cid:173)
`logical reactions must be substantially greater; oth(cid:173)
`erwise, unwanted by-products would accumu(cid:173)
`late to unacceptable levels.
`
`solutions under mild conditions of temperature and pH. For example, the
`enzyme jack bean urease catalyzes th e hydrolysis of urea:
`
`0
`II
`H2N- C-NH2 + 2 H20 + H + ----7 2 NH4 + + HC03-
`At 20°C, the rate con stant for the enzyme-catalyzed reaction is 3 X 104 /sec; the
`rate constant for the uncatalyzed hydrolysis of urea is 3 X 10- Jo /sec. Thus, 1014
`is the ratio of the catalyzed rate to the uncatalyzed rate of reaction. Such a
`ratio is defined as the relative catalytic power of an enzyme, so the catalytic
`power of urease is 1014
`.
`
`Specificity
`A given enzyme is very selective, both in the substances with which it interacts
`and in the reaction that it catalyzes. The substances upon which an en zyme
`acts are traditionally called substrates. In an enzyme-catalyzed reaction , none
`of the substrate is diverted into nonproductive side-reactions, so no wasteful
`by-products are produced. It follows then that the products formed by a given
`enzyme are also very specific. This situation can be contrasted with your own
`experiences in the organic ch emistry labora tory, where yields of 50% or even
`30% are viewed as substantial accomplishments (Figure 14.3) . The selective
`qualities of an enzym e are collectively recognized as its specificity. Intimate
`interaction between an enzyme and its substrates occurs through molecular
`recognition based on structural complementarity; su ch mutu al recognition is
`the basis of specificity. The specific site on the enzyme where substrate binds
`and catalysis occurs is called the active site.
`
`Regulation
`Regulation of enzyme activity is achieved in a variety of ways, ranging from con(cid:173)
`trols over the amount of en zyme protein produced by the cell to more rapid,
`reversible interactions of the enzyme with metabolic inhibitors and activators.
`Chapter 15 is devoted to discussions of enzyme regulation. Because most
`enzymes are proteins, we can anticipate that the functional attributes of
`enzymes are due to the remarkable versatility found in protein structures.
`
`Enzynte Nomenclature
`Traditionally, enzymes often were named by adding the suffix -ase to th e name
`of the substrate upon which they acted, as in urease for th e urea-hydrolyzing
`enzyme or phosphatase for enzymes hydrolyzin g phosphoryl groups from organ(cid:173)
`ic phosphate compounds. Other enzymes acquired names bearing little re(cid:173)
`semblance to their activity, such as the peroxide-decomposing enzyme catalase
`or the proteolytic enzymes (proteases) of the digestive tract, trypsin and pepsin.
`Because of the confusion that arose from these trivial designations, an
`International Commission on Enzymes was established in 1956 to create a sys(cid:173)
`tematic basis for enzyme nomenclature. Although common names for many
`enzymes remain in u se, all enzymes now are classified and formally named
`according to the reaction they catalyze. Six classes of reactions are recognized
`(Table 14.1). Within each class are subclasses, and under each subclass are sub(cid:173)
`subclasses within which individual enzymes are listed. Classes, subclasses, sub(cid:173)
`subclasses, and individual entries are each numbered, so that a series of fo ur
`numbers serves to specify a particular enzyme. A systematic name, d escriptive
`
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`

`Table 14.1
`
`Systematic Classification of Enzymes According to the Enzyme Commission
`
`E.C. Number
`
`Systematic Name and Subclasses
`
`14.1 • Enzymes- Catalytic Power; Specificity, and Regulation
`
`429
`
`1.1
`1.1 .1
`1.1.3
`1.2
`1.2.3
`1.3
`1.3.1
`2
`2.1
`2.1.1
`2.1.2
`2.1.3
`2.2
`2.3
`2.4
`2.6
`2.6.1
`2.7
`2.71
`3
`3.1
`3.1.1
`3.1.3
`3.1.4
`4
`4.1
`4.1.1
`4.1.2
`4.2
`4.2.1
`4.3
`4.3.1
`5
`5.1
`5.1.3
`5.2
`6
`6.1
`6.1.1
`6.2
`6.3
`6.4
`6.4.1
`
`Oxidoreductases (oxidation- reduction reactions)
`Acting on CH- OH group of donors
`With NAD or NADP as acceptor
`Wi th 0 2 as acceptor
`f
`.
`"
`Actmg on the C=O group o donors
`With 0 2 as (cceptor
`Acting on the CH- CH group of donors
`With NAD or NADP as acceptor
`Transjerases (transfer of functional grou ps)
`Transferring C-1 groups
`Methyltransferases
`Hydroxymethyltransferases and formyltransferases
`Carboxyltransferases and carbamoyltransferases
`Transferring aldehydic or ketonic residues
`Acyltransferases
`Glycosyltransferases
`Transferring N-containing grou ps
`Aminotransferases
`Transferring P-containing groups
`With an alcohol group as acceptor
`Hydrolases (hydrolysis reactions)
`Cleaving ester linkage
`Carboxylic ester hydrolases
`Phosphoric monoester hydrolases
`Phosphoric diester hydrolases
`Lyases (addition to double bonds)
`C= C lyases
`Carboxy lyases
`Aldehyde lyases
`C=O lyases
`Hydro lases
`C=N lyases
`Ammonia-lyases
`Isomerases (isomerization reactions)
`Racemases and epimerases
`Acting on carbohydrates
`Cis-trans isomerases
`Ligases (formation of bonds with ATP cleavage)
`Forming C-0 bonds
`Amino acid-RNA ligases
`Forming C- S bonds
`Forming C- N b onds
`Forming C-C bonds
`Carboxy lases
`
`of the reaction, is also assigned to each entry. To illustrate, consider the enzyme
`that catalyzes this reaction:
`ATP + n-glucose -----7 ADP + n-glu cose-6-phosphate
`A p h osphate group is transferred from ATP to the C-6-0H group of glucose,
`so the enzyme is a transferase (Class 2·; Table 14.1). Subclass 7 of transferases is
`
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`

`430
`
`Chapter 14 • Enzyme Kinetics
`
`enzymes transferring phosphorus-containing groups, and sub-subclass 1 covers those
`phosphotramferases with an alcohol group as an acceptor. Entry 2 in this sub-subclass
`isATP: o-glucose-6-phosphotransferase, and its classification number is 2.7.1.2.
`In use, this number is written preceded by the letters E. C., denoting the Enzyme
`Commission. For example, entry 1 in the same sub-subclass is E.C.2.7.1.1, ATP:
`o-hexose-6-phosphotransferase, an ATP-dependent enzyme that transfers a
`phosphate to the 6-0H of hexoses (that is, it is nonspecific regarding its hex(cid:173)
`ose acceptor). These designations can be cumbersome, so in everyday usage ,
`trivial names are employed frequently. The glucose-specific enzyme, E.C.2.7.1.2 ,
`is called glucokinase and the nonspecific E.C.2.7.1.1 is known as hexokinase.
`Kinase is a trivial term for enzymes that are ATP-dependent phosphotrans(cid:173)
`ferases.
`
`Coenzymes
`Many enzymes carry out their catalytic function relying solely on their protein
`structure. Many others require nonprotein components, called cofactors (Table
`14.2). Cofactors may be metal ions or organic molecules referred to as coenzymes.
`Cofactors, because they are structurally less complex than proteins, tend to be
`stable to heat (incubation in a boiling water bath). Typically, proteins are dena(cid:173)
`tured under such conditions. Many coenzymes are vitamins or contain vitamins
`as part of their structure. Usually coenzymes are actively involved in the catalytic
`reaction of the enzyme, often serving as intermediate carriers of functional groups
`in the conversion of substrates to products. In most cases, a coenzyme is firmly
`associated with its enzyme, perhaps even by covalent bonds, and it is difficult to
`
`Table 14.2
`Enzyme Cofactors: Some Metal Ions and Coenzymes and the Enzymes with Which They Are Associated
`
`Coenzymes Serving as Transient Carriers
`of Specific Atoms or Functional Groups
`
`Representative Enzymes
`Using Coenzymes
`
`Metal Ions and Some
`Enzymes That Require
`Them
`
`Metal
`Ion
`
`Enzyme
`
`Coenzyme
`
`Cytochrome oxidase
`Catalase
`Peroxidase
`Cytochrome oxidase
`DNA polymerase
`Carbonic anhydrase
`
`Thiamine pyrophosphate (TPP)
`Flavin adenine dinucleotide (FAD)
`Nicotinamide adenine dinucleotide
`(NAD)
`Coenzyme A ( CoA)
`Pyridoxal phosphate (PLP)
`
`Entity Transferred
`
`Aldehydes
`Hydrogen atoms
`Hydride ion (H - )
`
`Acyl groups
`Amino groups
`
`Pyruvate dehydrogenase
`Succinate dehydrogenase
`Alcohol dehydrogenase
`
`Acetyl-CoA carboxylase
`Aspartate
`aminotransferase
`Methylmalonyl-CoA
`mutase
`
`Alcohol dehydrogenase
`
`5'-Deoxyadenosylcobalamin (vitamin B12 )
`
`H atoms and alkyl groups
`
`Hexokinase
`Glucose-6-phosphatase
`
`Biotin (biocytin)
`
`Tet.rahydrofolate (THF)
`
`Arginase
`Pyruvate kinase
`(also requires Mg2+ )
`Urease
`Nitrate reductase
`Glutathione peroxidase
`
`Ni2+
`Mo
`Se
`
`C02
`
`Propionyl-CoA
`carboxylase
`Other one-carbon groups Thymidylate synthase
`
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`

`14.2 • Introduction lo Enzyme Kinetics
`
`431
`
`separate the two. Such tightly bound coenzymes are referred to as prosthetic
`groups of the enzyme. The catalytically active complex of protein and prosthetic
`group is called the holoenzyme. The protein without the prosthetic group is called
`the apoenzyme; it is catalytically inactive.
`
`14.2 • Introduction to Enzyme Kinetics
`
`Kinetics is the branch of science concerned with the rates of chemical reac(cid:173)
`tions. The study of enzyme kinetics addresses the biological roles of enzymatic
`catalysts and how they accomplish their remarkable feats. In enzyme kinetics,
`we seek to determine the maximum reaction velocity that the enzyme can attain
`and its binding affinities for substrates and inhibitors. Coupled with studies on
`the structure and chemistry of the enzyme, analysis of the enzymatic rate under
`different reaction conditions yields insights regarding the enzyme's mechanism
`of catalytic action. Such information is essential to an overall understanding of
`metabolism.
`Significantly, this information can be exploited to control and manipulate
`the course of metabolic events. The science of pharmacology relies on such a
`strategy. Pharmaceuticals, or drugs, are often special inhibitors specifically tar(cid:173)
`geted at a particular enzyme in order to overcome infection or to alleviate ill(cid:173)
`ness. A detailed knowledge of the enzyme's kinetics is indispensable to ratio(cid:173)
`nal drug design and successful pharmacological intervention.
`
`Review of Chemical Kinetics
`Before beginning a quantitative treatment of enzyme kinetics, it will be fruit(cid:173)
`ful to review briefly some basic principles of chemical kinetics. Chemical kinet(cid:173)
`ics is the study of the rates of chemical reactions. Consider a reaction of over(cid:173)
`all stoichiometry
`
`A----l-P
`Although we treat this reaction as a simple, one-step conversion of A to P, it
`more likely occurs. through a sequence of elementary reactions, each of which
`is a simple molecular process, as in
`
`A~I~J-~P
`where I and J represent intermediates in the reaction. Precise description of
`all of the elementary reactions in a process is necessary to define the overall
`reaction mechanism for A~ P.
`Let us assume that A~ P is an elementary reaction and that it is sponta(cid:173)
`neous and essentially irreversible. Irreversibility is easily assumed if the rate of
`P conversion to A is very slow or the concentration of P (expressed as [P]) is
`negligible under the conditions chosen. The velocity, v, or rate, of the reac(cid:173)
`tion A~ P is the amount of P formed or the amount of A consumed per unit
`time, t. That is,
`
`d[P]
`v = dt or
`
`-d[A]
`v=---
`dt
`The mathematical relationship between reaction rate and concentration of
`reactant(s) is the rate law. For this simple case, the rate law is
`v= - d[A] = k[A]
`dt
`
`(14.1)
`
`(14.2)
`
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`

`432
`
`Chapter 14 • Enzyme Kinetics
`
`From this expression, it is obvious that the rate is proportional to the concen(cid:173)
`tration of A, and k is the proportionality constant, or rate constant. k has the
`. vis a function of [A] to the first power, or, in
`units of (time) -J , usually sec - 1
`the terminology of kinetics, v is first-order with respect to A. For an elemen(cid:173)
`tary reaction, the order for any reactant is given by its exponent in the rate
`equation. The number of molecules that must simultaneously interact is
`defined as the molecularity of the reaction. Thus, the simple elementary reac(cid:173)
`tion of A~ P is a first-order reaction. Figure 14.4 portrays the course of a first(cid:173)
`order reaction as a function of time . The rate of decay of a radioactive isotope ,
`like 14C or 3 2P, is a first-order reaction, as is an intramolecular rearrangement,
`such as A~P. Both are unimolecular reactions (the molecularity equals 1) .
`
`Bimolecular Reactions
`Consider the more complex reaction, where two molecules must react to yield
`products:
`
`A+B---+P + Q
`Assuming this reaction is an elementary reaction , its molecularity is 2; that is,
`it is a bimolecular reaction. The velocity of this reaction can be determined
`from the rate of disappearance of either A or B, or the rate of appearance of
`P or Q:
`
`d[Q]
`d[P]
`- d[B]
`- d[A]
`v= ------ = ------ = ---- = ----
`dt
`dt
`dt
`dt
`
`(14.3)
`
`The rate law is
`
`(14.4)
`v = k[A][B]
`The rate is proportional to the concentrations of both A and B. Because it is
`proportional to the product of two concentration terms, the reaction is sec(cid:173)
`ond-order overall, first-order with respect to A and first-order vvith respect to
`B. (Were the elementary reaction 2A ~ P + Q, the rate law would be v = k[A] 2
`second-order overall and second-order with respect to A.) Second-order rate
`constants have the units of (concentration) - 1 (time) - \ as in l f1sec- 1
`Molecularities greater than two are rarely found (and greater than three ,
`never). When the overall stoichiometry of a reaction is greater than two (for
`example, as in A + B + C ~ or 2A + B ~), the reaction almost always proceeds
`via uni- or bimolecular elementary steps, and the overall rate obeys a simple
`first- or second-order rate law.
`
`,
`
`FIGURE 14.4 • Plot of the course of a first(cid:173)
`order reaction. The half-time, t11 2, is the time
`for one-half of the starting amount of A to dis(cid:173)
`appear.
`
`100
`
`bO c
`'2
`'(;j s 50
`..,
`...
`<:
`ll'(
`
`Slope of tangent to the
`line at any point = - tl[A]! tlt
`
`/
`
`o._ ........................ - . . . . . . . . . . . .
`
`2t ,/2
`Time -------------1~
`
`4t ll,
`
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`

`

`14.2 • Introduction to Enzyme Kinetics
`
`433
`
`At this point, it may be useful to remind ourselves of an important caveat
`ohat is the first principle of kinetics: Kinetics cannot prove a hypothetical mechanism.
`Kinetic experiments can only rule out various alternative hypotheses because
`theY don't fit the data. However, through thoughtful kinetic studies, a process
`of ~ limination of alternative hypotheses leads ever closer to the reality.
`
`Free Energy of Activation and the Action of Catalysts
`In a first-order chemical reaction, the conversion of A to P occurs because, at
`any given instant, a fraction of the A molecules has the energy necessary to
`ach ieve a reactive condition known as the transition state. In this state, the
`probability is very high that the particular rearrangement accompanying the
`A~ P transition will occur. This transition state sits at the apex of the energy
`profile in the energy diagram describing the energetic relationship between A
`and P (Figure 14.5). The average free energy of A molecules defines the ini(cid:173)
`ti al state and the average free energy of P molecules is the final state along the
`reaction coordinate. The rate of any chemical reaction is proportional to the
`co ncentration of reactant molecules (A in this case) having this transition-state
`energy. Obviously, the higher this energy is above the average energy, the
`smaller the frac tion of molecules that will have this energy, and the slower the
`reaction will proceed. The height of this energy barrier is called the free energy
`of activation, ~G.t. Specifically, ~G! is the energy required to raise the average
`energy of one mole of reactant (at a given temperature) to the transition-state
`energy. The relationship between activation energy and the rate constant of
`the reaction, k, is given b y the Arrhenius equation:
`
`(14.5)
`
`(a)
`
`(b)
`
`Average free
`energy of A
`
`- - - - - j - - - · Transition state
`I
`(uncatalyzed)
`.:>ct
`uncatalyzed
`
`-: j_-1\ ~ T:~-;:;:, ·~~
`
`Ll ct catalyzed
`
`Average free energy
`of P at T1
`
`Average free
`energy ofP
`
`Progress of reaction
`
`Progress of reaction
`
`FIGURE 14.5 • Energy diagram for a chemical reaction (A~ P) and the effects of (a)
`raising the temperature from T1 to T2 or (b) adding a catalyst. Raising the temperature
`raises the average energy of A molecules, which increases the population of A molecules
`having energies equal to the activation energy for the reaction, thereby increasing the
`reaction rate. In contrast, the average free energy of A molecules remains the same in
`uncatalyzed versus catalyzed reactions (conducted at the same temperature) . The effect of
`the catalyst is to lower the free energy of activation for the reaction.
`
`Par Pharmaceutical, Inc. Ex. 1032
`Par v. Horizon, IPR of Patent No. 9,561,197
`Page 11 of 37
`
`

`

`434
`
`Chapter 14 • Enzyme Kinetics
`
`Reactant concen tration, [A]
`
`FIGURE 14.6 • A plot of v versus [A] for the
`unimolecular chemical reaction, A--+ P, yie lds a
`su·aight line having a slope equal to h.
`
`where A is a constant for a particular reaction (not to be confused with the
`reactant species, A, th at we 're discussing). Another way of writing this is
`1/ k = ( 1/ A) et>G!! RT That is, k is inversely proportional to e~>ct; wr. Therefore,
`if the energy of activation decreases, the reaction rate increases.
`
`Decreasing ..:1 G t Increases Reaction Rate
`We are familiar with two general ways that rates of chemical reactions may be
`accelerated. First, the temperature can be raised. Th is will in crease the aver(cid:173)
`age energy of reactant molecules, which in effect lowers the energy needed to
`reach the transition state (Figure 14.5a). The rates of many chemical reactions
`are doubled by a lOoC rise in temperature . Second, th e rates of chemical reac(cid:173)
`tions can also be accelerated by catalysts. Catalysts work by lowering the energy
`of activation rather than by raising the average energy of the reactants (Figure
`14.5b). Catalysts accomplish this remarkable feat by combining transiently with
`the reac tants in a way that promotes their entry into the reactive, transition(cid:173)
`sta te condition. Two aspects of catalysts are worth noting: (a) they are regen(cid:173)
`e rated after each reaction cycle (A---+ P), and so can be used over and over
`again; and (b) catalysts have no effect on the overall free energy chan ge in th e
`reaction, the free energy difference between A and P (Figure 14.5b).
`
`14.3 • Kinetics of Enzyme-Catalyzed Reactions
`
`Examination of the change in reaction velocity as the reactant concentration
`is varied is one of the primary measurements in kinetic analysis. Returning to
`A---+ P, a plot of the reaction ra te as a function of the concentration of A yields
`a straight line whose slope is k (Figure 14.6) . The more A that is avail able, the
`greater th e rate of the reaction, v. Similar analyses of enzyme-catalyzed reac(cid:173)
`tions involving only a single substrate yield remarkably different results (Figure
`14.7) . At low concentrations of the subs trate S, vis proportional to [S], as
`expected for a first-order reaction . However, v does not increase proportion(cid:173)
`ally as [S] increases, but instead begins to level off. At high [S], v becomes vir(cid:173)
`tually independent of [S] and approaches a maximal limit. The value of vat
`this limit is written Vmax· Because rate is no longer dependent on [S] at these
`high concentrations, the enzyme-catalyzed reaction is now obeying zero-order
`
`v =Vm~ ·-- ---------- --- -------- -- ------- ---- ---------- --
`
`FIGURE 14.7 • Substrate saturation curve fo r
`an enzyme-catalyzed reaction. The amoun t of
`enzyme is constant, and the velocity of the
`reaction is determined at various substrate con(cid:173)
`cen trations. The reaction rate , v, as a function
`of [S] is described by a rectangular hyperbola.
`At very high [S], v = Vmax· That is, the velocity
`is limited only by conditions (temperature, pH,
`ionic strength) and by the amount of enzyme
`present; v becomes independent of [S] . Such a
`condition is termed zero-order kinetics. Under
`zero-order conditions, velocity is d irectly depen(cid:173)
`dent on [enzyme]. The H 20 molecule provides
`a rough guide to scale. The substrate is bound
`at the active site of the enzyme.
`
`v
`
`_........ACLive site
`
`Enzyme
`molecu le
`
`Substrate concentration, [S]
`
`Par Pharmaceutical, Inc. Ex. 1032
`Par v. Horizon, IPR of Patent No. 9,561,197
`Page 12 of 37
`
`

`

`14.3 • Kinetics of Enzyme-Catalyzed Reactions
`
`435
`
`kinetics; that is, the ra te is independent of the reactant (substrate) concentra(cid:173)
`tion. This behavior is a saturation effect: when v shows no increase even though
`~S ) is increased, the system is saturated with substrate. Such plots are called
`substrate saturation curves. The physical interpretation is that every enzyme
`mol ecule in the reaction mixture has its substrate-binding site occupied by S.
`Indeed, such curves were the initial clue that an enzyme interacts directly with
`als substrate by binding it.
`
`'!fhe Michaelis- Menten Equation
`
`[:;enore Michaelis and Maud L. Men ten proposed a general theory of enzyme
`acti o n in 1913 consistent with observed enzyme kinetics. Their theory was based
`on the assumption that the enzyme , E, and its substrate, S, associate reversibly
`c0 form an enzyme-substrate complex, ES:
`
`kl
`E+S ~ES
`k_l
`
`(14.6)
`
`This association/ dissociation is assumed to be a rapid equilibrium, and Ks is
`the enzyme: substrate dissociation constant. At equilibrium,
`
`(14.7)
`
`[E] [S]
`k_ 1
`K = -- = -
`s
`k1
`[ES]
`P, is formed in a second step when ES breaks down to yield E + P.
`
`(14.8)
`
`kl
`E+S~ES~E+P
`L1
`
`j,_
`
`(14.9)
`
`E is then free to interact with another molecule of S.
`
`Steady-State Assumption
`
`The interpretations of Michaelis and Menten were refined and extended in
`1925 by Briggs and Haldane, by assuming the concentration of the enzyme(cid:173)
`su bstrate complex ES quickly reaches a constant value in such a dynamic sys(cid:173)
`tem. That is, ES is formed as rapidly from E + S as it disappears by its two pos(cid:173)
`sible fates: dissociation to regenera te E + S, and reaction to form E + P. This
`assumption is termed the steady-state assumption and is expressed as
`
`d[ES ] = O
`dt
`
`(14.10)
`
`T hat is, the change in concentration of ES with time, t, is 0. Figure 14.8 illus(cid:173)
`trates the time course for formation of the ES complex and establishment of
`the steady-state condition.
`
`Initial Velocity Assumption
`
`One other simplification will be advantageous. Because enzymes accelerate the
`rate of the reverse reaction as well as the forward reaction, it would be help(cid:173)
`ful to ignore any back reaction by which E + P might formES. The velocity of
`this back reaction would be given by v = k_ 2 [E] [P]. However, if we observe
`only the initial velocity for the reaction immediately after E and S are mixed in
`the absence of P, the rate of any back reaction is negligible because its rate will
`
`k
`0
`
`"::l g
`k ..,
`u c
`0 u
`
`§
`'§
`c
`"' u c
`0 u
`
`Time
`
`Time
`
`F1GURE 14.8 • Time course for the con(cid:173)
`sumption of substrate, the formation of prod(cid:173)
`uct, and the establishment of a stea<;iy-state
`level of the enzyme-substrate [ES] complex for
`a typical enzyme obeying the Michdelis(cid:173)
`Menten, Briggs-Haldane models for enzyme
`kinetics. The early stage of the time course is
`shown in greater magnification in the bottom
`graph.
`
`Par Pharmaceutical, Inc. Ex. 1032
`Par v. Horizon, IPR of Patent No. 9,561,197
`Page 13 of 37
`
`

`

`436
`
`Chapter 14 • Enzyme Kinetics
`
`be proportional to [P], and [P] is essentially 0. Given such simplification, we
`now analyze the system described by Equation (14.9) in order to describe the
`initial velocity vas a function of [S] and amount of enzyme.
`The total amount of enzyme is fixed and is given by the formula
`Total enzyme, [E1~ = [E] + [ES]
`(14.11)
`where [E) = free enzyme and [ES] = the amount of enzyme in the enzyme(cid:173)
`substrate complex. From Equation (14.9), the rate of [ES] formation is
`[ES]) [S]
`Vj= hi ( [Ey]
`
`where
`
`[Er] -
`
`(ES] = [E]
`
`From Equation (14.9), the rate of [ES] disappearance is
`vd= k_ 1[ES] + ~[ES] = (k- 1 + l12)[ES]
`At steady state, d[ES]/ dt = 0, and therefore, Vf''"' vd.
`So,
`
`h1 ( [Er] -
`
`[ES]) [S] = (k- 1 + I~) [ES]
`
`Rearranging gives
`
`-'-'( ["-E-'-"r ]:.._-_..:.:_[E_• S=])-=-[ S--=-] =
`[ES]
`
`(14.12)
`
`( 14.13)
`
`(14.14)
`
`(14.15)
`
`The Michaelis Constant, Km
`The ratio of constants (k- 1 + 1!2)/k1 is itself a constant and is defined as the
`Michaelis constant, Km
`
`K, = (k_l + ~)
`''I
`Note from (14.15) that K,n is given by the ratio of two concentrations
`[ES]) and [S]) to one ( [ES]), so K.n has the units of molarity. From
`( ( [Er] -
`Equation (14.15), we can write
`
`(14.16)
`
`[ES]) [S] = K,
`( [Ey] -
`[ES]
`
`(14.17)
`
`which rearranges to'
`
`[ES] = [Er] [S]
`K,. + [S]
`Now, the most important parameter in the kinetics of any reaction is the rate
`of product formation. This rate is given by
`d[P]
`v= - -
`dt
`
`(14.19)
`
`(14.18)
`
`and for this reacti