Practical Enzymology E-Book

Hans Bisswanger

• Herausgeber: Wiley-VCH
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 522

Veröffentlichungsjahr: 2013

Beschreibung

The second edition of the perfect companion for practical course work in enzymology. Now with an improved selection of enzymatic assays based on key metabolic reactions, this book provides the detailed background to the enzymatic reaction and the enzyme per se. Particular emphasis is placed on troubleshooting with the described methods, and, in addition, the new four-color layout throughout features detailed protocols highlighted for easy recognition.

www.wiley-vch.de/home/enzymology: A companion website provides animations for all figures together with supplementary material, for deeper understanding of the partially abstract matter.

Leseprobe

Contents

Preface to the Second Edition

Note to the Reader

Abbreviations

1 Introduction

2 General Aspects of Enzyme Analysis

2.1 Basic Requirements for Enzyme Assays

2.2 What Must Be Observed for an Enzyme Assay?

2.3 Instrumental Aspects

2.4 Theory of Coupled Enzyme Reactions

2.5 Substrate Determination

3 Enzyme Assays

3.1 Enzyme Nomenclature

3.2 Practical Considerations for Enzyme Assays

3.3 Special Enzyme Assays

3.4 Assays for Enzyme Characterization

3.5 Enzyme Immunoassays

4 Binding Measurements

4.1 Different Types of Binding

4.2 Binding Measurements by Size Discrimination

4.3 Spectroscopic Methods

4.4 Other Binding Methods

5 Enzymes in Technical Applications

5.1 Modes of Enzyme Immobilization

5.2 Methods for Enzyme Immobilization

5.3 Analysis of Immobilized Enzymes

5.4 Enzyme Reactors

5.5 Biosensors

5.6 Immobilized Enzymes in Therapy

Appendix

Index

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The Author

Prof. Dr. Hans Bisswanger

Interfakultäres Institut für Biochemie

Hoppe-Seyler-Str. 4

72076 Tübingen

Germany

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ISBN: 978-3-527-32076-9

Preface to the Second Edition

The principal concept of the first edition: general aspects of enzymes and presentation of special enzyme assays and related tests such as protein determination and enzyme immobilization, as well as instrumental aspects, remain conserved in the second edition. Additional enzyme assays and tests, for example, for the determination of glycoproteins and inorganic phosphate, have been included, considering the principles of broad interest and diversity of methods. Features of the enzymes, which are of importance for the assay conditions, such as cofactor requirement, molecular mass, state of aggregation, kinetic constants, and pH optimum, are indicated, but it is not intended to present all known data. Actually, an overwhelming quantity of data accumulated within the last years, owing to the violent progress in gene technology, and features of the same enzyme species can differ extremely, depending on the organisms from where it originates. For practical treatment of an enzyme assay, such a variety of data is more disturbing than helpful; therefore, only the features of one or few representative enzyme species, preferentially from human or mammalian origin, are mentioned. Particular attention has been drawn to frequent pitfalls and error detection for the methods described.

Enzymes applied for molecular biology and gene technology, such as restriction enzymes and polymerases, are not taken into consideration. They are extensively described in text books and manuals of the respective field, but the general rules for handling and for the assay conditions are also valid for these enzymes. This holds also for RNA enzymes (ribozymes), antibody enzymes (antizymes), and artificial enzymes derived, for example, from cyclodextrans or crown ethers, and for enzymes modified by site-specific mutations.

The layout has been improved, for example, by introduction of colors; the structure of text is clearer; and the essential points of the sections are summarized in separated boxes. A companion web site (www.wiley-vch.de/home/enzymology) provides animations for all figures together with supplementary material, for deeper understanding of the partially abstract matter.

Especially emphasized are the valuable contributions of Klaus Möschel and Rainer Figura to the chapter of immobilized enzymes.

Tübingen, January 2011

Hans Bisswanger

Note to the Reader

Animations should assist the comprehension of the text. They are principally self-explaining, but knowledge of the corresponding book chapter and the respective figure legend is presupposed. Most animations are subdivided into sequential steps, which are initiated by the cursor button (→) or a mouse click. A green arrow at the bottom of each figure gives the signal for pressing, during the animation run it disappears. ‘X’ indicates the end of the animation before passing to the next figure.

Abbreviations1)

A, B, C

specific binding ligands

{A]

ligand concentration

{A]0

total ligand concentration

A

absorption

ABTS

2,2'-azino-bis-3-ethylbenzothiazoline-6-sulfonic acid

ADH

alcohol dehydrogenase

ANS

anilinonaphthalene sulfonate

BAPNA

N

'-benzoyl-

L

-arginine-

p

-nitroanilide

BCA

bicinchoninic acid

BSA

bovine serum albumine

CDI

carbonyldiimidazol

CoA

coenzyme A

CPG

controlled pore glass

d

density

DMSO

dimethylsulfoxide

DPIP

2,6-dichlorophenolindophenol

DTE

dithioerythritol

DTNB

5,5'-dithio-bis(2-nitrobenzoic acid), Ellman’s reagent

DTT

dithiothreitol, cleland’s reagent

E, [E]

enzyme, enzyme concentration

{E]o

total enzyme concentration

ε

nm

absorption (“extinction”) coefficient at the wavelength indicated

ε

r

dielectric constant

EDTA

ethylenediaminetetraacetic acid

EIA

enzyme immunoassays

ELISA

enzyme-linked immunoadsorbent assays

FAD

flavine adenine dinucleotide

FMN

flavine mononucleotide

GOD

glucose oxidase

h

Planck’s constant

HK

hexokinase

I

light intensity

IU

International enzyme unit (μmol min

–1

)

k

rate constant

kat

Katal (mol s

–1

)

k

cat

catalytical constant

K

d

dissociation constant

K

m

Michaelis constant

LDH

lactate dehydrogenase

MDH

malate dehydrogenase

M

r

relative molecule mass

n

number of subunits

NAD

nicotinamide adenine dinucleotide

2)

NADH

nicotinamide adenine dinucleotide

2)

NADP

nicotinamide adenine dinucleotide phosphate

2)

NADPH

nicotinamide adenine dinucleotide phosphate

2)

ONPG

o

-nitrophenyl

β

-

D

-galactopyranoside

ORD

optical rotatory dispersion

P, Q, R

products

PAGE

polyacrylamide gel electrophoresis

PBS

phosphate buffered saline

PEG

polyethylene glycol

PLP

pyridoxal 5-phosphate

PMSF

phenylmethylsulfonyl fluoride

POD

peroxidase

R

Gas constant (8.3145 J mol

–1

K

–1

)

RIA

radioimmunoassay

RN

recommended name

rpm

rotations per min

RT

room temperature

S

substrate

SA

specific enzyme activity

SDS

sodium dodecyl sulfate

SN

systematic name

ThDP

thiamine diphosphate

TCA

trichloroacetic acid

TRIS

tris(hydroxymethyl)aminomethane

υ

reaction velocity

V

maximum reaction velocity

υ

i

initial reaction velocity

1) Only repeatedly used abbreviations, special abbreviations are defined at the respective section.

2)For simplicity the charge (NAD(P)+), for the reduced form the free proton (NAD(P)H + H+) is omitted.

1

Introduction

Enzymes are the most important catalysts and regulators indispensably involved in each process in living organisms. Any investigation of the cell metabolism requires a thorough understanding of enzyme action. Enzymes are very sensitive markers for correct function and, consequently, also for dysfunction of the metabolism, serving as indicators both for health and manifestation of diseases. Accordingly, they are used as invaluable tools in medical diagnostics. Beyond that, enzymes are applied in many technical operations. They play an essential role in the environmental processes in the microbial world in waters, rivers, lakes, and soil, and are important for filter plants as well as for fermentation procedures in dairies and breweries.

According to current estimates, about 25 000 enzymes are expected to exist in the living world, where more than 3000 are described in detail, and some hundreds are commercially available. Enzymes are extremely efficient catalysts, enhancing the turnover rates of spontaneous reactions by factors between 108 and 1010, sometimes even up to 1012 (Menger, 1993). Orotidine-5’-phosphate decarboxylase is a striking example: the spontaneous reaction proceeds with a half-life of 78 million years and the enzyme increases the velocity by a factor of 1017 (Radzicka and Wolfenden, 1995). Triosephosphate isomerase accelerates the enolization of dihydroxyacetone phosphate by more than 109(Alberty and Knowles, 1976).

Even reactions spontaneously proceeding with a considerable rate, such as the formation of water from hydrogen and oxygen in the respiratory chain, are subject to enzyme catalysis: each reaction step in metabolism is controlled by a special enzyme. Thus the role of enzymes in the metabolism is broader than to act only as biocatalysts. The peculiarity of catalysis is not only restricted to the acceleration of spontaneous reactions, but it also allows controlling reactions. Spontaneous reactions, after initiating, run off to the end and cannot be stopped. Catalyzed reactions, in contrast, proceed only in the presence of the catalyst; its activity and amount determines the reaction rate. Consequently, tuning the activity of an enzyme from the outside by activating or inhibiting mediates an exact control of the velocity. In the living cell, a strictly coordinate network of regulation exists, comprising enzymes whose activity is controlled by the concentration levels of metabolites, hormones, and transmitter substances. The precise interaction of all these components is a prerequisite of life.

The protein nature of enzymes is excellently suited for this dual function as catalyst and regulator; it supplies functional groups of amino acids to form specific binding sites and catalytic centers, and it provides flexibility to promote formation and stabilization of transition states and to induce conformational changes for modulation of the catalytic efficiency. The 20 proteinogenic amino acids with their hydrophilic, hydrophobic, acidic, and basic side chains permit most enzymes to realize both functions such as specific binding of substrates and regulator molecules and catalytic conversion. More difficult catalytic mechanisms cannot be brought forth only by the amino acid side chains; rather, nonproteinogenic compounds are included, which can either be dissociable as coenzymes1) or nondissociable as prosthetic groups. Dissociable coenzymes are NAD(P), thiamine diphosphate, or coenzyme A, while FAD, cytochromes, porphyrins, pyridoxamine, lipoic acid, biotin, and tetrahydrofolic acid function as nondissociable, partly covalently bound prosthetic groups. Often also metal ions are required, both for catalysis and for stability of the enzyme, Mg2+ serves to neutralize the phosphate groups in compounds such as ADP, ATP, and thiamine diphosphate and mediates their binding to the enzyme. Iron (in cytochromes), cobalt (in the corrin ring system), copper (e.g., in cytochrome oxidase and tyrosinase), zinc (in carboanhydase and alcohol dehydrogenase), molybdenum (in nitrogenase), manganese (in arginase and xylose/glucose isomerase), and selenium ion (in glutathion peroxidase) support the enzyme reactions.

The protein nature enables enzymes to adapt their specificity to any desired ligand by mutations. This feature is applied in biotechnology using site-directed mutagenesis to modify the specificity and function of enzymes. By the method of molecular modeling (protein design) distinct modifications are simulated and thereafter the respective mutations are executed. An example is hydroxyisocaproate dehydrogenase, an enzyme catalyzing the reductive conversion of α-oxo acids to chiral hydroxycarbonic acids as hydroxyanalogs of amino acids. Its preferred substrate is α-oxocaproic acid. α-Isocaproic acid, an analogous compound, is accepted only with reduced efficiency. By site-directed mutagenesis the catalytic efficiency (fecat/Km) for this compound has been increased by four orders of magnitude, as compared to the physiological substrate (Feil, Lerch, and Schomburg, 1994).

Owing to their protein nature, enzymes are very sensitive to environmental influences such as pH, ionic strength, and temperature and, consequently, to attain optimum activity, stringent conditions must be established. In the physiological milieu of the living cell, these conditions are maintained as far as possible, although with respect to temperature, this cannot be permanently guaranteed (with the exception of warm-blooded vertebrates). However, enzymes are remarkablyable to adapt to extreme conditions. Although proteins are regarded as being very temperature sensitive, distinct microorganisms such as Thermus, Thermotoga, and Thermoplasma, including their complete enzymatic inventory, persist in temperatures up to 100 °C. It must be assumed that during evolution the ancient organisms have had to bear much higher temperatures. The ancient precursors of the present enzymes must have all been thermophilic, but obviously they lost this feature with the decrease in environmental temperature. This can also explain the fact that proteins, instead of the more stable nucleic acids, are preferred by nature as biocatalysts, although some catalytic activities are retained in RNA.

As an introduction to the practical work with enzymes, at least some fundamental theoretical rules must be discussed. They will be addressed in the first part, followed by a description of the general features of enzymes, which must be considered when dealing with them. This is followed by a presentation of the most important techniques. This general part should enable the reader to work with enzymes; for instance, to develop an assay for a newly isolated enzyme without further need to consult the literature. The following special part presents detailed descriptions of enzyme assays and related methods such as protein determination. A multitude of assays corresponding to the immense number of different enzymes exists, which cannot all be considered within the scope of a laboratory manual; rather, only a selection can be presented. Criteria for the selection are not only the frequency of application, but also the broad variety of enzyme types and methods. Certainly, such a selection cannot satisfy all expectations and the choice will sometimes appear rather arbitrary. For further information the reader is referred to standard books and databases of enzymology (see References section below). Procedures for immobilization of enzymes and special aspects of analysis of immobilized enzymes, principles of enzyme reactors, and enzyme electrodes are presented in separate sections of the book.

References

Alberty, W.J. and Knowles, J.R. (1976) Biochemistry,15, 5631–5640.

Feil, I.K., Lerch, H.P., and Schomburg, D. (1994) Eur.J. Biochem.,223, 857–863.

Menger, F.M. (1993) Acc. Chem. Res.,26, 206–212.

Radzicka, A. and Wolfenden, R. (1995) Science,267, 90–93.

Standard Books, Series, and Databases

Advances in Enzymology and Related Areas of Molecular Biology, John Wiley & Sons, Inc.,New York.

Bergmeyer, H.U. (ed.) (1983) Methods of Enzymatic Analysis, 3rd edn, Verlag Chemie, Weinheim.

Methods in Enzymology, Academic Press, San Diego.

Schomburg, D. (ed.) (2001f) Springer Handbook of Enzymes, Springer, Berlin.

www.expasy.org/enzymes.

www.brenda-enzymes.org.

1) The terms coenzyme and cosubstrate are not always clearly differentiated. Coenzymes, in contradistinction to cosubstrates, are supposed to support the catalytic mechanism and should not be converted. For example, pyridoxal phosphate in transamination reactions accepts an amino residue becoming pyridoxamine phosphate, but in the second step of the reaction the amino group is transferred to an α-oxoacid and the coenzyme regains its original form at the end of the reaction. NAD(P), on the other hand, is reduced in a dehydrogenase reaction and must be reoxidized by a separate enzyme reaction, therefore it is more a cosubstrate than a coenzyme.

2

General Aspects of Enzyme Analysis

2.1 Basic Requirements for Enzyme Assays

The task of enzymes as biocatalysts is to render feasible reactions, which cannot proceed in their absence. Therefore, the first requirement when dealing with a special enzyme is to study its reaction. In a simple generalization it can be stated that one compound (or more than one), designated as the substrate, gets converted into another compound, the product, with the aid of the enzyme. Thus, to identify an enzyme its reaction must be demonstrable or measurable, that is, a method must be developed for the quantitative detection of the reacting components. The prerequisite for this is a detectable signal for the reacting components. But a signal alone is not sufficient; rather, a clear distinction between the substrate and the product is necessary. Absorption ultraviolet and visible (UV/Vis) spectroscopy may be an illustrative example. It is an easy and convenient detection method and in fact, each biological substance shows absorption at least in the UV region. Therefore, this method is principally suited for the quantification of every substance and, thus, may be the method of choice for any enzyme assay. However, in most cases, the substrate and the product of the same reaction show similar absorption features. So, even if the compounds possess pronounced absorption spectra, they are not useful to detect the reaction. This is the case with sugars, such as glucose and fructose, which cannot be distinguished by absorption spectroscopy and so this method is not applicable here.

Hence, the first step is to find a clear signal for detection of the substrate and/or the product, and the second step is to uncover differences between both compounds. The compound showing the clearer difference signal will be used. Principally, this is irrespective of whether the substrate or the product will be detected, as it can be assumed, that the amount of substrate converted corresponds exactly to the amount of product formed. Observing the decay of the substrate or the formation of the product must give the same result, only changing the sign. However, if possible, product formation is preferred because of practical reasons. At the start of the reaction the product concentration and, consequently, its signal, is zero and any increase is a direct indication of the progressing reaction. Conversely, the concentration of the substrate and thus its signal is highest at the beginning of the reaction. This can influence the detection method. Each method shows some scatter and usually higher signals cause stronger scatter, and small changes, for example in the case of slow reaction rates are difficult to detect. Some substrates are unstable and decay spontaneously appearing to be an enzyme-catalyzed reaction; an effect, which is also more pronounced at high concentrations.

Various methods are available to search for an appropriate difference signal for an enzyme assay. Any analytic method for identifying substances such as the substrate and the product can be considered and usually the method yielding the clearest difference signal will be chosen; however, other criteria must also be considered. A simple, but practical aspect, is the availability of an appropriate instrument. For enzyme analysis, frequent assays series must be performed and the appropriate instrument should be permanently accessible. Accordingly, it must be affordable and handling should be easy. Such demands limit the kind of methods that can be employed. One important aspect concerns the mode of registration. As discussed in detail later, a progressive reaction should be pursued continuously (continuous assay) as far as possible, while various methods allow only detection of single points of the reaction after defined time periods (stopped assay,Box 2.1). In fact, often a method enabling continuous registration is superior to a method allowing only stopped assays even if a weaker signal must be accepted.

Box 2.1: Fundamental Demands for Enzyme Assays

Product formation (increasing reaction) or substrate consumption (decreasing reaction) can be detected, alternativelyEnzyme activity is defined as the amount of product formed – substrate consumed – within a distinct time unit (second, minute, hour). During this time unit the reaction must proceed strictly linearly (zero order)Continuous monitoring of the reaction (e.g., spectral change, pH change)Stopped assay (if continuous monitoring is not possible): stop of the reaction after a defined time and subsequent analysis of the amount of product formed – substrate consumed – within the time unit

The most frequently used methods for enzyme assays are summarized in Box 2.2. All spectroscopic methods (absorption, fluorescence, circular dichroism (CD), optical rotatory dispersion (ORD), turbidimetry) and electrochemical techniques such as pH stat, can principally be performed as a continuous test (but only if the substrate or the product can be identified directly), while trapping and separation methods allow only stopped assays. Therefore, such methods will be used only if the other methods do not work. For routine assays, simple devices and instruments belonging to the standard equipment of an analytical or a biochemical laboratory are preferred. These criteria are fulfilled in the best manner by absorption (UV/Vis) spectroscopy, which is an easy method with various applications, for example, determinations of proteins, nucleic acids, and phosphate (cf. Box 2.15). Suitable apparatus are available at moderate prices and computer-controlled instruments with monitoring and calculation modes make the evaluation of reactions easy so that photometric enzyme assays are the first choice. Other spectroscopic methods, such as fluorescence, CD, and ORD are for special applications and thus usually not present as standard equipment in laboratories. Manipulation is more difficult, intense knowledge for appropriate operation is required, and high-quality instruments are rather expensive; all these are aspects not supporting their application. Nevertheless, these spectroscopic methods possess significant advantages. They are more selective and, especially for fluorescence, much more sensitive compared with absorption spectroscopy, a feature important especially for enzyme studies. If such an instrument is not available, the work must be done in specialized laboratories, which usually provide not only the appropriate apparatus, but also a thorough knowledge of the technique, indispensable to avoid inappropriate procedures and misinterpretations. These considerations hold also for other instruments such as the pH stat, a very useful device, if demanded by the type of assay, such as the digestion of lipids, but superfluous if not really required. The main instruments applied for enzyme tests are described in detail in Section 2.3.

Box 2.2: How to Determine the Enzyme Activity.?

Substrate A and product P must differ in at least one detectable feature.

2.2 What Must Be Observed for an Enzyme Assay?

As already mentioned, for an enzyme assay the progression of the reaction (progress curve) is decisive and should be carefully observed. For normal enzyme reactions this curve should obey a common pattern, that is, it should be a straight line reflecting linear progression of the reaction proportional to time. In reality, however, nonlinear behavior is often observed, which is frequently a smooth curvature, but sometimes even irregular deviations. Evaluation of such behavior requires some knowledge of the theoretical background; the essential rules are discussed in the following chapter.

2.2.1 Order of Reactions

The progression of a reaction is determined by its order. The simplest chemical reaction is the conversion of a substance A (in chemical terms: educt) into the product P, as the spontaneous decomposition of instable substances, for example, the radioactive decay:

The velocity v of this reaction depends on the initial concentration of A and is expressed as:

(2.1)

t is the time, k the rate constant with the dimension of s−1. It is obvious that the higher the amount of A, the faster the reaction. Because A decays during the reaction, the velocity declines permanently, and the reaction follows a curve, which is steepest at the start and decreases steadily (Figure 2.1a). A similar curve, only in a positive sense, is obtained, when the formation of P is observed. Mathematically, this curve is described by an exponential relationship ([A]0 is the initial substrate concentration):

(2.2)

This is the equation for a first-order reaction, because only one substrate is involved. Hence, an exponential curve is indicative of a first-order reaction. However, an exponential progression of a reaction is not easy to recognize unequivocally, because other reaction types (higher orders) show similar nonlinear curves. Although they follow no simple exponential relationship, in practice they are often difficult to discern from real exponential curves. In such cases of ambiguity it is a good principle to transform the nonlinear relationship into a linear form, where only dependencies obeying the original relationship will yield straight lines, while others show characteristic deviations. By transformation of the first-order (equation 2.2) into a half logarithmic form

(2.3)

the curves in Figure 2.1a become linear if a logarithmic ordinate scale is chosen (Figure 2.1b).

Figure 2.1 Progress curves of various reaction orders. (a) Direct plotting and (b) semilogarithmic plotting.

In nature, spontaneous decays are rather seldom.1) More frequent are reactions initiated by collisions of two or more reactive substances. The number of substrates involved determines the reaction order,2) for example, a reaction:

is of second order.3) The velocity of a second-order reaction depends on two variables. As shown in Figure 2.1a, nonlinear behavior is observed and no straight line results in the half logarithmic plot (Figure 2.1b). This feature allows the distinction of first and second orders (and similarly for higher orders, which are not dealt with here). When performing experiments, the dependency of the second-order reaction on two independent variables is impracticable. Under normal conditions both substrates may be present in comparable amounts, but this is not a necessary condition. If we assume that one component (e.g., B) is present in surplus in comparison with the other one (A), conversion of the lesser amount of A will not essentially change the higher amount of B, so that its concentration can be considered as constant. Under this condition, the reaction depends only on one, the minor component (A), and becomes similar to a first-order reaction, following an exponential time course, which now becomes linear in the half logarithmic plot (Figure 2.1b). As this reaction is only formally first order, but second order in reality, it is designated as pseudo-first order.

2.2.2 Significance of the Reaction Order for Enzyme Reactions

It has already been mentioned that enzyme reactions should, ideally, proceed in a linear manner, while we now see that the simplest chemical reaction, the first-order reaction, is already exponential. Are enzyme reactions simpler than simple? Linear progression can only be expected if the reaction rate is completely independent of the substrate concentration, so that the amount of product formed per time unit remains constant, irrespective of whether low or high substrate concentrations are present:

(2.4)

(2.5)

To explain this apparent contradiction let us turn to enzyme reactions. The simplest enzyme reaction is the conversion of one substrate catalyzed by the enzyme

(2.6)

Box 2.3: How Much Enzyme Is Required for an Assay.?

The velocity of enzyme-catalyzed reactions is strictly proportional to the enzyme amount:

For substrate saturation:

Deviation from linear relationship is an indication for nonideal conditions.

The enzyme concentration [E]05)in the assay must adhere to the following rules:

concentration should be as low as possible, according to the steady-state theory [enzyme] [substrate]however, it must be sufficient to detect the initial velocity

(2.7)

(2.8)

(2.9)

(2.10)

The overall reaction velocity v is defined as the rate of product formation (Eq. (2.10)). In addition to these four equations the mass conservation relationships

(2.11)

(2.12)

(2.13)

The derivation of the Michaelis–Menten equation on the basis of constancy of the EA complex accentuates its strict limitation to the linear zero-order range. Nonlinear deviations are indications for nonvalidity of this relationship and it can now be understood that linear progress curves are a prerequisite for analyzing enzymes.

The Michaelis–Menten equation describes the dependency of the substrate concentration on the reaction velocity v. However, it has been stated above, that the zero-order range should be independent of the substrate concentration, depending only on the enzyme amount. How can this contradiction be understood? To explain this, the term saturation is introduced. If a very small amount of enzyme is added to a surplus of substrate to establish the condition [E] [A], it may intuitively be assumed, that the enzyme must be saturated, that is, the substrate occupies all available enzyme molecules. If this will be the case, the reaction indeed becomes completely independent of the substrate concentration. However, in reality, the enzyme will not be completely saturated; only a part of the enzyme molecules binds the substrate, while the other part remains unoccupied and does not contribute to the reaction. Only the part actually binding substrate, not the total enzyme amount, determines the reaction rate. To understand this, it must be considered that the degree of binding is determined by the binding affinity, expressed by the dissociation constant Kd. Its value expresses just the concentration of substrate required for half saturation of the enzyme; low substrate amounts cause a lower degree of saturation and more substrate amounts a higher degree. For example, consider a Kd value of 10−5 M. Substrate is added in this concentration to a 10−9 M enzyme solution. As the substrate concentration is the same as the Kd value, the enzyme is only half saturated – in spite of a 10 000-fold surplus of substrate. Variation of the enzyme concentration will not influence the degree of saturation, but will change the reaction velocity by the same factor, because the amount of enzyme molecules occupied with substrate changes to the same extent. On the other hand, change of the substrate concentration at constant enzyme amount alters the degree of saturation corresponding to the Michaelis–Menten law and the velocity changes accordingly. This demonstrates the mutual dependence of the velocity on both the enzyme and the substrate concentration, the first one being strictly linear and the second one being dependent on the Michaelis–Menten law. For our example, increase of the enzyme concentration by a factor of 10 increases the velocity 10-fold, while a 10-fold increase of substrate raises the degree of saturation, and consequently the reaction velocity, from 50 to 90.9%, which is less than twofold!

2.2.3 Determination of the Velocity of Enzyme Reactions

In the following section, the basic principles of the Michaelis–Menten equation are discussed. The dependency of the velocity of an enzyme reaction on the substrate concentration is described, but before going into details one must understand how velocity is to be determined. For this, let us recall the statement, that an enzyme-catalyzed reaction should proceed in a linear manner, with the progress curve forming a straight line. The slope of this line, expressed by the amount of product formed and substrate converted within a defined time unit, respectively, comprises the actual value of the velocity v (Figure 2.3).

2.2.3.1 Enzyme Units

Box 2.4: Expressions for the Enzyme Activity

For exact determination of any enzyme unit the enzyme assay must be carried out under standard conditions. They are discussed in detail in the following sections. On the other hand, for investigation of enzyme features, such as determination of the Michaelis constant, temperature stability or inhibition mechanisms, distinct parameters, for example, the substrate concentration, must be varied and in this respect standard conditions can no longer be maintained. In such cases the values obtained from velocity determinations correspond no longer to defined enzyme units, but the actual enzyme velocity should still be indicated in the respective dimensions of moles/second or micromoles/minute.

Figure 2.2 Schematic representation of the three phases of a progress curve.

2.2.3.2 Progress Curves

The progression of an enzyme reaction is schematically shown in Figure 2.2. Three phases can be discerned: initially a steep, nonlinear pre-steady-state phase, which is too short (~μs) to be detected in normal enzyme assays. Immediately thereafter the linear steady-state phase follows until the reaction ceases after the nonlinear phase of substrate depletion.

Linear progress curves as an indication for the prevalence of steady-state conditions and the validity of the Michaelis–Menten equation can best be expected at substrate saturation. Total saturation, however, exists only at infinite substrate concentration, while under real assay conditions saturation is not complete and decreases during the reaction course due to the substrate conversion. Therefore, strict linearity can only be expected at the start of the reaction, when the substrate concentration is nearly saturating. During the progression of the reaction, the velocity slows down and deviates from linearity (Figures 2.2 and 2.3). Therefore, the value of the velocity should not be estimated directly from the progress curve, rather a tangent to the initial linear range must be aligned as shown in Figure 2.3. Its slope, expressed as the concentration of substrate or product converted or formed per time unit, respectively gives the actual velocity (tangent method). If the substrate concentration is reduced within an experimental series, or if higher substrate concentrations cannot be realized (e.g., due to low solubility), the situation becomes more difficult, the linear range becomes shorter, and the deviation more pronounced (Figure 2.3, curves to the right). Even in such cases the velocity is obtained from tangents aligned to the initial range. If, however, the linear range becomes so short that it is hardly detectable, the prevalence of steady-state conditions and the validity of the Michaelis–Menten equation are no longer verified. As long as high accuracy is not demanded it may be assumed that steady-state conditions exist at least at the start of the reaction. This simplification, however, does not hold for exact determinations, for which some restrictions must be considered.

Figure 2.3 Determination of velocities of enzyme reactions from progress curves. The progress curves (red) from left to right are measured with decreasing substrate concentrations. Tangents (green) are aligned to the initial range; 1 min is taken as the time unit for the evaluation of the velocity v according to the definition of IU.

2.2.3.3 Difficulties in Determination of Initial Velocities

The most serious restriction in the determination of enzyme reaction velocities is the dead time. From the start of an enzyme reaction to the beginning of detection, that is, onset of registration, a certain time interval, at least some seconds, passes. This is no problem as long as the progress curve is linear (Figure 2.4a), but with nonlinear progress curves registration starts when the initial velocity has already passed. Aligning a tangent to the now apparent initial range causes a severe underestimation of the velocity (Figure 2.4b) and an attempt should be made to make the linear range visible. The reason for nonlinear progress curves is frequently a too high enzyme concentration. If an enzyme assay does not work immediately there is a tendency to add more and more enzyme. But too much enzyme converts the substrate instantly during the dead time, so that after start of recording no reaction will be observed at all. Such a situation is easily misinterpreted as lack of activity, not recognizing that the problem is in fact too much activity. As a rule, the enzyme amount should be as low as possible. The lower limit is given by the sensitivity of the method, when a very slow turnover cannot be detected within the scatter (Box 2.5). The higher the enzyme concentration the shorter the steady-state range. In Figure 2.5, a nonlinear progress curve due to a high enzyme concentration is shown. Dilution of the enzyme (e.g., 10-fold) reduces the velocity by just this factor extending the steady-state range by the same factor. If, after enzyme dilution, the velocity becomes too slow, the assay time can be prolonged. The same amount of product is produced within 10 min as in 1 min with a 10-fold enzyme concentration. Principally there exists no general rule for the assay time. Usually short times (e.g., 1 min) are preferred, but with low enzyme activities, assay times of hours or even days can be chosen. Only the stability of the assay components and, in particular, of the enzyme must be established for such long time periods.

Figure 2.4 Disturbance of velocity determination by the dead time. Progress curve (red); tangent to real initial velocity (green); and tangent to progress curve after dead time (blue). (a) Linear progress curve and (b) nonlinear progress curve.

Figure 2.5 Linearity of progress curve depends on the enzyme amount: less enzyme (1 × [E]) yields slower velocities, but longer linearity than more enzyme (10 × [E]).

Box 2.5: Errors in Determination of Initial Velocities

Particular care on linear progression must be observed with stopped assays. As long as the complete progress curve is continuously recorded, any deviation will be recognized. In stopped assays, however, there is only one measuring point and the velocity is calculated from the slope of a connecting line between the start of the reaction (usually the blank) and the measure point (Figure 2.6). Any deviation occurring during this time interval cannot be detected. Repeated measurements can only compensate for the common scatter, but not for extraordinary or systematic deviations. To reduce this problem, the time intervals should be chosen as short as possible and several measuring points after particular time intervals should be taken to confirm the expected linearity. In cases where this cannot be realized (because of scarcity of time or substances) at least a standard series with sufficient measuring point must be performed (Figure 2.7). Even if such a procedure demonstrates that the chosen time interval lies fairly within the linear range, it must be considered, that this is valid only for actual test conditions. Any change, such as temperature, pH, substrate or enzyme concentration, can shorten the linear range so that the chosen time interval is no longer appropriate. Possible error sources in determination of initial velocities are summarized in Box 2.5.

Figure 2.6 Stopped assay performed under nonideal conditions. The measurements are carried out beyond the linear range so that the determined velocity is lower than the real initial velocity.

Figure 2.7 Stopped assay performed under ideal conditions. A series of measurements allows the control of the linear range. The first measurement of the stopped test lies within this range, the second one is still outside.

An alternative procedure for evaluation of progress curves is the integrated Michaelis–Menten equation, which is obtained by integrating Eq. (2.14) with respect to time

(2.14)

(2.15)

The integrated Michaelis–Menten equation (Eq. (2.15)) describes the complete progress curve, including the nonlinear range, so that there is obviously no necessity to obtain the linear initial range. According to this equation, the Michaelis constant and the maximum velocity can be derived from one single progress curve. By rearranging

(2.16)

the curve is linearized and V and Km obtained from the ordinate intercept and the slope, respectively. The particular advantage of this procedure is the fact that in computer-controlled instruments the data can be directly transformed according to this equation and the constants can be displayed immediately after the experiment. This method appears tempting, but it is only reliable if the respective progress curve obeys the Michaelis–Menten equation completely. Unfortunately, this is usually not the case, because the product formed influenced the reaction by inhibiting the enzyme and inducing the backward reaction. Therefore, for accurate measurements the determination of initial velocities from the linear part of progress curves is emphatically recommended.

2.2.3.4 A Short Discussion on Error

Errors severely affect scientific experiments and their avoidance would be a desirable goal. However, errors will always occur and the only realistic aim is to minimize them as far as possible. Here, potential sources of errors, their consequences, and possible methods to limit their perturbing influences on enzyme assays are discussed. At first, a diagnosis of the error should be undertaken to detect its origin. Generally, spontaneous and systematic errors can be differentiated. Spontaneous errors arise from inexact manipulation (e.g., pipetting). Any handling bears intrinsically a certain inaccuracy, which may be reduced by careful performance, but cannot be eliminated completely. Spontaneous errors cause randomly distributed deviations from the true value in the positive and negative directions, both to a comparable extent (Figure 2.8a). Such errors will be observed in any experiment, but as long as the deviations are not too large, they can be mastered by common error calculations and regression analysis, which are described in the relevant literature of statistics. In contrast, systematic errors are not randomly distributed, but rather cause deviations in a particular direction and, therefore, distort the results and can lead to wrong interpretations. Causes of such errors may be imperfect operation of the instrument (e.g., incorrectly adjusted apparatus, such as a defective pipette used for the same step always) or faulty manipulation by the operator (making the same mistake always in routine assays).

Before discussing both error types in detail, one important argument must be mentioned. Errors are regarded as artificial, and, thus, unintended deviations from the real value. The aim is to approach the true value, for example, by calculating a medium value out of a series of repeated measurements, or to adapt a curve, such as a straight line or an exponential function, where it is generally understood, that nature will make no jumps, and physiological processes will always follow clear, smooth functions. Albeit, it must be kept in mind, that any such function, even based on statistical treatments, is an interpretation of experimental results. It is the task of the scientist to demonstrate convincingly the appropriateness of the fit to the data, while the experimental data are the only stringent result. So, outliers, which are values obviously not fitting into the system, are commonly assumed to originate from unusual strong artificial deviations. Because outliers distort the regression analysis dramatically so that the calculated function will no longer fit the majority of data, one feels authorized to suppress such strongly deviating values. This may, in many cases, be justified, but, principally, elimination of particular values out of a data series is an arbitrary selection revealing the preference of the scientist for a particular solution, neglecting any other interpretation. Therefore, it is a strict rule in science to present all original data together with the interpretation (i.e., the fitted function), indicating, if particular data points remained unconsidered for regression analysis, so that an independent observer can judge whether the applied function is an adequate description for the presented data.

Figure 2.8 Error distributions. (a) Direct plotting of experimental data; (b) residual plots of the same data; error bars and deviations from the adapted curve are enlarged.

Even for spontaneous errors different possibilities of expression exist. It may be supposed, that during an experimental series the extent of error, that is, the error limits to both directions, may be constant throughout (Figure 2.8a, left diagram). For such a series often one parameter is varied, such as the substrate concentration for enzyme kinetic determinations, and the question is, whether the error remains independent or whether the varied parameter influences the error.6)Figure 2.8a shows different error developments. They can be easier visualized by residual diagrams, where the deviation of the data from the assumed function is enlarged and plotted around the x axis (Figure 2.8b). For equally distributed errors, the error bars should be the same throughout (Figure 2.8a,b, left diagrams), but the error can increase or decrease during the test series (Figure 2.8a,b, middle and right diagrams, respectively).

It must be considered that the dependent variable (velocity) increases with the substrate concentration and if the error limits remain constant throughout (Figure 2.8a left), they actually increase relative to the velocity in the lower concentration range. The behavior of the error limits depends on the error source. For example, scatter of the instrument will cause a constant error, while pipetting of different volumes influences the extent of the error, with very small volumes producing higher errors than larger ones. In this case the error decreases with increasing substrate concentration.

Pipetting is one of the severest error sources; careful pipetting improves considerably the accuracy of experiments. Various pipette systems are available, producing different error types. Automatic pipettes with variable volumes are mostly used. If they are of good quality they are very precise, but some general rules must be regarded (common manipulation is described by the producer’s instructions and is not mentioned here). It is common usage to pour out the pipette by dipping the tip into the assay solution and to mix the solution thereafter with the tip. This is commodious and time saving, but not a very accurate (Figure 2.9, I). When extracting an aliquot from the stock solution some liquid will adhere to the outer surface of the tip (Figure 2.9, II) and get into the assay solution (Figure 2.9, II). Wiping the tip is not recommended, as tissue fibers can extract some liquid from inside the tip (Figure 2.10). When dipping into the assay solution for ejecting and mixing, the assay solution can penetrate into the tip and wash out the residual liquid, which remains inevitably in the tip after emptying (this remaining portion is already considered in the normal pipetting process, Figures 2.9, III; 2.10). Such an error is more severe when the pipetting volume is smaller.Very small volumes (<5μl) cannot be pipetted with high accuracy, even if such volumes are within the range of the pipette. On the other hand small volumes have the advantage of avoiding significant dilution of the assay solution.7) Volumes of 10–20 μl are recommended for pipetting, they are easy to manipulate and do not essentially change the common assay volumes of 1 or a few milliliters. The aliquot should not be directly added into the assay solution, but placed on the flat end of a (commercially available) plastic spatula (Figure 2.9, IV), which can also be used to mix the assay solution (Figure 2.9, V).

Figure 2.9 Pipetting and mixing in assay solutions. I–III, pipetting of a sample from the stock to the assay solution and direct mixing. A drop of the stock solution remains attached at the outside of the tip and gets into the assay solution. IV and V, placing the sample on a spatula, inserting of the sample, and mixing of the assay solution.

Figure 2.10 Plunging the pipette directly into the assay solution: some solution penetrates into the tip and more of the stock solution gets into the assay solution than intended. A microliter syringe is shown, the inner canal is enlarged.

2.2.3.5 Preparation of Dilution Series

If lower concentrations of the added sample are required, instead of reducing the volume, the stock solution should be diluted to avoid pipetting of very small volumes. When the concentration of substrate or cofactor must be varied during a test series, this is after realized by modifying the added volume of the stock solution correspondingly with an automatic pipette (Figure 2.11a). This, however, is the less exact procedure not only because of inaccuracy of repeated pipette adjusting and varying dilution of the assay solution, but, because of the dependency of the extent of the error on the pipetted volume. Therefore, the preparation of a dilution series with varying concentrations of the respective components is strongly recommended. Such series must be prepared very carefully. From such a series, an equal volume (e.g., 10 μl) is always given to the assay solution to yield the desired final assay concentration. For this the pipette volume must not be changed and the aliquot added to the assay has always the same volume and can easily be considered for calculation of the final assay volume. There are two modes to prepare a dilution series. The first is to prepare the dilutions in a stepwise manner (Figure 2.11b). A defined volume of the stock solution (e.g., 0.2 ml) is taken in a test tube and filled up to the final sample volume (e.g., 1 ml) with a buffer solution (resulting in a fivefold dilution of the stock solution). After mixing a similar aliquot (0.2 ml) is removed from this sample and taken in a second tube and also filled up to the final sample volume with buffer (resulting in a fivefold dilution of the previous solution). This procedure is repeated several times until the desired dilution range is covered. This is an easy procedure. However, any erroneous deviation in one step will be carried forward to all following steps and all errors during the procedure accumulate to the end. For the second mode of preparing a dilution series, different volumes of the stock solution are added to the test tubes and they are brought up to an equal final volume (e.g., 1ml) with a buffer solution (Figure 2.11c). This procedure is more reliable; an error concerns only the respective sample and not all the following ones. A disadvantage is that for a broad dilution range also very small volumes are required. To avoid this intermediated stock dilutions, for example, for each order of magnitude, should carefully be prepared.

More accurate than automatic pipettes, especially, for very small volumes are microliter glass syringes (Figure 2.10). They are available for different volume ranges, but the criteria discussed above must be regarded for these pipettes also. They consist of a glass cylinder with a central hole, ending in a steel tubing. A stainless steel piston fitted to the hole serves to draw up the solution. Care must be taken to ensure that the pipette is completely filled with liquid without any air bubbles included.

Mixing is a further source of error. Besides the problems discussed above, inappropriate handling can cause two particular errors. Insufficient mixing, especially with thin tips, produces inhomogeneities in the assay solution with regions of lower and higher concentrations. The result of such an effect in a photometric assay is shown in Figure 2.12, where areas of higher and lower velocity float through the light path consecutively, simulating an oscillating behavior. During the progression of the reaction, the oscillation smoothes slowly approaching the actual reaction velocity – the real initial rate, however, has already been passed. Too vigorous mixing, on the other hand, causes the enclosure of air bubbles, disturbing especially optical methods, and of oxygen, favoring oxidative processes. Various mixing devices are commercially available, such as small magnetic stirrers, which can be installed in optical instruments for direct mixing in the cuvette. Special apparatus, for example, the stopped-flow device, enable rapid mixing within parts of seconds.

Figure 2.11 Different modes for preparation of a dilution series. (a) Direct pipetting into the assay solution has the lowest accuracy. (b) Stepwise dilution from one tube to the next, an error will be propagated into the following samples. (c) Dilution from the same stock solution, best method, no error propagation.

Figure 2.12 Artificial deviations from the progress curve, due to insufficient mixing. The initial velocity taken from the recorded curve deviates considerably from the real one.

2.2.3.6 Considerations about Statistical Treatments

It is particularly requested to perform repeating experiments, from which statistically established mean values can be calculated to avoid misinterpretations of the final result by one or few wrong measurements. So one should not be acquiesced by one single determination of an enzyme assay, rather several independent measurements should be performed. How much repeats are required? It may be asserted, the more determinations, the better the security, but the principle of parsimony dictates a limit both with respect to material (especially enzyme) and time, and accuracy does not essentially increase with a higher number of repetitions. For routine tests, three independent determinations can be regarded as sufficient, and five should be enough for higher accuracy. It is often advantageous to perform repeated assays not under completely identical conditions, but to modify one parameter, which causes strictly proportional changes. For enzyme assays the amount of enzyme (not of substrate!), and for protein tests that of protein may be changed. The expected linear dependency of the values is a further control for the method. Deviations from linearity, especially when steadily inclining in the same direction, are indications for errors. Any assay is reliable only within a particular range. Too low values disappear in the scatter, too high values fall outside the linearity of the method (due to depletion of assay components or the limited range of the Lambert–Beer law in optical assays). Linear dependency establishes that the assay has been performed completely within the reliable range of the method. A mean value can be calculated, regarding the different amounts of the parameter that is varied.

For a test series, such as the determination of the Michaelis constant, numerous (~10) measurements with increasing substrate concentrations will be required. For a series to be confirmed by three independent measurements, 30 assays must be carried out. If each single test needs 5 min, the whole series will last 2.5 h. This is not only the question of time, but also of the stability of the assays, especially of the enzyme. Most enzymes are not stable for longer durations in diluted solutions, as required for enzyme assays. Therefore, the test series must be performed within a short duration to establish equal conditions for all measurements. Interruptions for hours or over night must be avoided. If for example, an enzyme loses 10% of its activity per hour, the value of the last measurement of the test series will be 25% lower than that of the first one under otherwise identical conditions. Obviously, the improvement in statistical reliability by repeated measurements is obtained at the price of a systematic error. Constants derived from such an experiment will be seriously underestimated, although statistical treatment demonstrates more confidence. If the same series is performed with only single determinations, less than 1 h is needed with an activity difference of only 9% between the start and end of the experiment. In this case, it will be better to repeat the whole series three times and to calculate the constants independently for each series and derive the mean value for each constant. Such a procedure has the further advantage that any systematic deviation, caused either artificially or as a special feature of the system, can be easily identified by its independent repeated occurrence. Indeed systematic deviations are often overlooked and taken for usual scatter.