enzyme and enzyme kinetics

PAMANTASAN NG LUNGSOD NG MAYNILA

College of Engineering and Technology

Chemical Engineering Department

CHE 512 Biochemical Engineering

A WRITTEN REPORT

ENZYME AND ENZYME KINETICS

Submitted by: BESA, Rose Lynn Anne DARANCIANG, Erish

DECAPIA, Handrey LUDOVICE, Bianca Jamila

BSChE 5

Submitted to: Engr. Elaine G. Mission

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Contents ENZYME ......................................................................................................................................................... 3

Brief History of Enzyme ............................................................................................................................. 3

Enzyme Classification and Nomenclature ................................................................................................. 4

Common Enzymes ..................................................................................................................................... 9

pH of some Enzymes ............................................................................................................................... 10

Industrial Applications ............................................................................................................................ 11

Factors affecting Enzyme Activity ........................................................................................................... 13

SIMPLE ENZYME KINETICS........................................................................................................................... 16

Introduction to Enzyme Kinetics ............................................................................................................. 16

Rate of Reaction .................................................................................................................................. 16

Formation of Enzyme-Substrate Complex ........................................................................................... 18

Lock-and-Key Theory ........................................................................................................................... 18

Induced-Fit Model ............................................................................................................................... 18

Michaelis-Menten Kinetic Model ............................................................................................................ 19

Biographies of Michaelis and Menten ................................................................................................ 22

Derivation of Rate Equation (Michaelis-Menten equation) ................................................................ 23

Biographies of Lineweaver and Burk ................................................................................................... 25

Graphical Analysis ............................................................................................................................... 26

ENZYME REACTOR WITH SIMPLE KINETICS ................................................................................................. 31

General Types of Bioreactors .................................................................................................................. 32

Batch Stirred-Tank Reactor .................................................................................................................. 32

Plug-Flow Reactor ............................................................................................................................... 33

Continuous Stirred-Tank Reactor ........................................................................................................ 34

ENZYME INHIBITION ................................................................................................................................... 36

Bibliography ................................................................................................................................................ 37

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ENZYME

Enzymes are biological catalysts. They increase the rate of chemical reactions taking place within living

cells without themselves suffering any overall change. The reactants of enzyme-catalyzed reactions are

called substrates. Each enzyme is quite specific in character, acting on a particular substrate or substrates

to produce a particular product.

All known enzymes are proteins. They therefore consist of one or more polypeptide chains and display

properties that are typical of proteins. Some enzymes require small non-protein molecules, known as

cofactors, in order to function as catalysts.

Enzymes differ from chemical catalysts in several important ways:

1. Enzyme-catalyzed reactions are at least several orders of magnitude faster than chemically-

catalyzed reactions. When compared to the corresponding unanalyzed reactions, enzymes

typically enhance the rates by 106 to 1013 times.

2. Enzymes have far greater reaction specificity than chemically-catalyzed reactions and they rarely

form byproducts.

3. Enzymes catalyze reactions under comparatively mild reaction conditions, such as temperatures

below 100C, atmospheric pressure and pH around neutral. Conversely, high temperatures and

pressures and extremes of pH are often necessary in chemical catalysis.

Brief History of Enzyme

1833 The active agent breaking down the sugar was partially isolated and given the name

distase (now known as amylase). A little later, a substance which digested dietary protein

was extracted from gastric juice and called pepsin. These and other active preparations

were given the general name ferments. Justus von Liebig recognized that these ferments

could be non-living materials obtained from living cells, but Louis Pasteur still maintained

that ferments must contain living material.

1878 The term ferment was gradually replaced by the name enzyme which is proposed by

Wilhelm Kuhne. It comes from a Greek enzume, meaning in yeast.

1897 Brothers Eduard and Hans Buchner showed that sugar fermentation could take place

when a yeast cell extract was added even though no living cells were present.

1926 James Sumner crystallized urease from jack-bean extracts and in the next few years, many

other enzymes were purified and crystallized.

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Enzyme Classification and Nomenclature

Enzymes are classified according to the report of the Nomenclature Committee appointed by the

International Union of Biochemistry (1984). This enzyme commission assigned each enzyme a

recommended name and a four-part distinguishing number.

The first Enzyme Commission, in its report in 1961, devised a system for classification of enzymes that also

serves as a basis for assigning code numbers to them. These code numbers, prefixed by EC, which are now

widely in use, contain four elements separated by points, with the following meaning:

1. The first number shows to which of the six main classes the enzyme belongs,

2. The second figure indicates the type of group involved in the reaction,

3. The third figure denotes the reaction by indicating substrate on which the group acts,

4. The fourth figure is the serial number of the enzyme in its sub-subclass.

Class 1. Oxidoreductases

To this class belong all enzymes catalysing redox reactions. The substrate that is oxidized is regarded as

hydrogen donor. The systematic name is based on donor:acceptor oxidoreductase. The common name will

be dehydrogenase, wherever this is possible; as an alternative, reductase can be used. Oxidase is only used

in cases where O2 is the acceptor.

2nd EC digit: indicates group in the hydrogen donor (substrate oxidized)

The six main classes of enzymes

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3rd EC digit: indicates type of acceptor involved

Example:

Code number: EC 1.1.1.27

Systematic name: (S)-lactate: NAD+ oxidoreductase

Common name: Lactate dehydrogenase

Class 2. Transferases

Transferases are enzymes transferring a group, e.g. a methyl group or a glycosyl group, from one

compound (generally regarded as donor) to another compound (generally regarded as acceptor). The

systematic names are formed according to the scheme donor:acceptor group transferase. The common

names are normally formed according to acceptor group transferase or donor group transferase. In many

cases, the donor is a cofactor (coenzyme) charged with the group to be transferred.

2nd EC digit: indicates group transferred

3rd EC digit: further information on group transferred

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Example:


Systematic name: ATP: D-hexose-6-phosphotransferase

Common name: Hexokinase

Class 3. Hydrolases

These enzymes catalyze the hydrolytic cleavage of C-O, C-N, C-C and some other bonds, including

phosphoric anhydride bonds. Although the systematic name always includes hydrolase, the common name

is, in many cases, formed by the name of the substrate with the suffix -ase. It is understood that the name

of the substrate with this suffix means a hydrolytic enzyme.

2nd EC digit: indicates nature of bond hydrolysed

3rd EC digit: indicates nature of substrate

Example:


Systematic name: Orthophosphoric monoester phosphohydrolase

Common name: Alkaline phosphatase

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Class 4. Lyases

Lyases are enzymes cleaving C-C, C-O, C-N, and other bonds by elimination, leaving double bonds or rings,

or conversely adding groups to double bonds. The systematic name is formed according to the pattern

substrate group-lyase. The hyphen is an important part of the name, and to avoid confusion should not be

omitted, e.g. hydro-lyase not 'hydrolyase'. In the common names, expressions like decarboxylase, aldolase,

dehydratase (in case of elimination of CO2, aldehyde, or water) are used.

2nd EC digit: indicates the bond broken

3rd EC digit: further information on group removed

Example:


Systematic name: L-histidine carboxy-lyase

Common name: Histidine decarboxylase

Note the importance of the hyphen and the extra y in the systematic name, because carboxy-lyase and

carboxylase do not mean the same thing: carboxylase simply refers to the involvement of CO2 in a reaction.

Class 5. Isomerases

These enzymes catalyze geometric or structural changes within one molecule. According to the type of

isomerism, they may be called racemases, epimerases, cis-trans-isomerases, isomerases, tautomerases,

mutases or cycloisomerases.

2nd EC digit: indicates type of isomerism

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3rd EC digit: indicates type of substrate

Example:


Systematic name: Maleate cis-trans isomerase

Common name: Maleate isomerase

Class 6. Ligases

Ligases are enzymes catalyzing the joining of two molecules coupled with the hydrolysis of a diphosphate

bond in ATP or a similar triphosphate.

2nd EC digit: indicates the bond formed

3rd EC digit: (only used in the CN ligases)

Example:


Systematic name: L-glutamate: ammonia ligase

Common name: glutamate-ammonia ligase

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Summary:

Class Type of Reaction Catalyzed Example

(EC number, Common name)

Oxidoreductases Oxidation and reduction (removal or addition of hydrogen atoms)

1.1.1.27 Lactate dehydrogenase

Transferases Transfer of chemical group (phosphate, methyl) from one substance to another

2.7.12 Glucokinase

Hydrolases Hydrolysis (splitting of bonds by water) 3.2.1.22

-D-galactosidase

Lyases Removal of chemical groups from substances

4.2.1.1 Carbonic anhydrase

Isomerases Structural rearrangements of atoms within one molecule

5.3.1.9 Glucose 6-phosphate isomerase

Ligases Joining together of two molecules utilizing energy (ATP)

6.2.1.1 Acetyl CoA synthetase

Common Enzymes

1. Amylase - enzyme that catalyzes the hydrolysis of starch into sugars. Present in the saliva of

humans and some other mammals 2. Protease - enzymes that break the long chainlike molecules of proteins into shorter

fragments (peptides). Present in bacteria and plants but are most abundant in animals. 3. Lipase - enzyme that catalyzes the hydrolysis of fats (lipids). Most lipases act at a specific

position on the glycerol backbone of lipid substrate. 4. Pectinase - enzyme that breaks down pectin, a polysaccharide found in plant cell walls.

Pectinase enzymes are commonly used in processes involving the degradation of plant materials

5. Cellulase - break down the cellulose molecule into monosaccharides ("simple sugars"). These produced chiefly by fungi, bacteria, and protozoans.

Enzyme Function

Amylase Breaks down starch into sugar

Protease Breaks down proteins into amino acids

Lipase Breaks down fats into fatty acids and glycerol

Pectinase Breaks down pectin into simpler molecules

Cellulase Breaks down cellulose into monosaccharides

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pH of some Enzymes

Pepsin is an enzyme released by the chief cells in the stomach and that degrades food proteins

into peptides

Trypsin is a serine protease from the PA clan superfamily, found in the digestive system of many

vertebrates, where it hydrolyses proteins.

Maltase is an enzyme that breaks down the disaccharide maltose. Maltase catalyzes the

hydrolysis of maltose to the simple sugar glucose.

Enzyme pH Optimum

Lipase (pancreas) 8

Lipase (stomach) 4 5

Pepsin 1.5 1.6

Trypsin 7.8 8.7

Urease 7

Invertase 4.5

Maltase 6.1 6.8

Amylase (pancreas) 6.7 7

Amylase (malt) 4.6 5.2

Catalase 7

Enzymes involved in Particular Reaction

1. Papaya

Papain - digesting enzyme, single-chained polypeptide with three disulfide bridges

2. Pineapple

Bromelain - protease enzymes, extract derived from the stems of pineapples, contains

chemicals that might interfere with the growth of tumor cells

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Industrial Applications

Detergent Industry

Enzyme used as detergent additives still represents the largest application of industrial enzymes,

both in terms of volume and value. The major component is proteases, but other and very

different hydrolases are introduced to provide various benefits, such as the efficient removal of

specific stains. The most recent introduction of a new enzyme class into a detergent has been the

addition of a mannanase the result of a joint development between Procter and Gamble and

Novozymes. This enzyme helps remove various food stains containing guar gum, a commonly used

stabilizer and thickening agent in food products.

Enzymes for starch conversion

The enzymatic conversion of starch to high fructose corn syrup is a well-established process and

provides a beautiful example of a bioprocess in which the consecutive use of several enzymes is

necessary. The enzymes utilized in the starch industry are also subjected to constant

improvements. The first step in the process is the conversion of starch to oligomaltodextrins by

the action of -amylase. The concomitant injection of steam puts extreme demands on the

thermostability of the enzyme. New -amylases with optimized properties, such as enhanced

thermal stability, acid tolerance, and ability to function without the addition of calcium, have

recently been developed offering obvious benefits to the industry.

Fuel alcohol production

In the alcohol industry, the use of enzymes for the production of fermentable sugars from starch

is also well established. Over the past decade, there has been an increasing interest in fuel alcohol

as a result of increased environmental concern, higher crude oil prices and, more acutely, by the

ban in certain regions of the gasoline additive methyl tert-butyl ether (MTBE), which can be

interchanged directly with ethanol. Intense efforts are currently being undertaken to develop

improved enzymes that can enable the utilization of cheaper and partially utilized substrates such

as lignocellulose, to make bio-ethanol more competitive with fossil fuels. The cost of enzymes

needed to turn lignocellulose into a suitable fermentation feed-stock is a major issue, and current

work focuses both on the development of enzymes with increased activity and stability as well as

on their efficient production.

Enzymes for the food industry

Applications of enzymes in the food industry are many and diverse, ranging from texturizing to

flavoring. Common to more or less all food applications, the enzymes are applied to processed

food products as processing agents upstream from the final product. Several advances have been

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made in the optimization of enzymes for existing applications and in the use of recombinant

protein production to provide efficient mono-component enzymes that do not have potential

detrimental side-effects. Recently, much work has been carried out on the application of

transglutaminase as a texturing agent in the processing of, for example, sausages, noodles and

yoghurt, where cross-linking of proteins provides improved viscoelastic properties of the products.

An obstacle, which may prevent even wider usage, is the currently limited availability of the

enzyme in industrial scale. At present only the transglutaminase from Streptoverticillium sp. Is

commercially available at a reasonable scale, and work is ongoing to increase the availability of

the enzyme by recombinant production in Escherichia coli.

Textile applications

In the textile industry a completely new enzymatic activity has recently been introduced. This

industry is under considerable environmental pressure owing to its large energy and water

consumption and subsequent environmental pollution. One of the most energy- and water-

consuming steps in the processing of cotton is the scouring step, the removal of various remaining

cell-wall components on the cellulose fibers performed at high temperature and under strong

alkaline conditions. An alternative, enzyme-based process performed at much lower temperatures

and using less water has now been developed based on a pectate lyase. The use of these enzymes

has benefited both the textile industry and the environment.

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Factors affecting Enzyme Activity

The activity of an enzyme is affected by its environmental conditions. Changing these alter the rate of reaction caused by the enzyme. In nature, organisms adjust the conditions of their enzymes to produce an Optimum rate of reaction, where necessary, or they may have enzymes which are adapted to function well in extreme conditions where they live.

1. Temperature

Increasing temperature increases the kinetic energy that molecules possess. In a fluid, this means that there are more random collisions between molecules per unit time.

Since enzymes catalyze reactions by randomly colliding with substrate molecules, increasing temperature increases the rate of reaction, forming more product.

However, increasing temperature also increases the vibrational energy that molecules have, specifically in this case enzyme molecules, which puts strain on the bonds that hold them together.

As temperature increases, more bonds, especially the weaker hydrogen and ionic bonds, will break as a result of this strain. Breaking bonds within the enzyme will cause the active site to change shape.

This change in shape means that the active site is less complementary to the shape of the substrate, so that it is less likely to catalyze the reaction. Eventually, the enzyme will become denatured and will no longer function.

As temperature increases, more enzymes' molecules' active sites' shapes will be less complementary to the shape of their substrate, and more enzymes will be denatured. This will decrease the rate of reaction.

In summary, as temperature increases, initially the rate of reaction will increase, because of increased Kinetic Energy. However, the effect of bond breaking will become greater and greater, and the rate of reaction will begin to decrease.

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The temperature at which the maximum rate of reaction occurs is called the enzyme's Optimum Temperature. This is different for different enzymes. Most enzymes in the human body have an Optimum Temperature of around 37.0 C.

2. pH

H+ and OH- Ions are charged and therefore interfere with hydrogen and ionic bonds that hold together an enzyme, since they will be attracted or repelled by the charges created by the bonds. This interference causes a change in shape of the enzyme, and importantly, its active site.

Different enzymes have different optimum pH values. This is the pH value at which the bonds within them are influenced by H+ and OH- Ions in such a way that the shape of their active site is the most complementary to the shape of their substrate.

Any change in pH above or below the optimum will quickly cause a decrease in the rate of reaction, since more of the enzyme molecules will have active sites whose shapes are not (or at least are less) complementary to the shape of their substrate.

Small changes in pH above or below the optimum do not cause a permanent change to the enzyme, since the bonds can be reformed. However, extreme changes in pH can cause enzymes to denature and permanently lose their function.

Enzymes in different locations have different optimum pH values since their environmental conditions may be different. For example, the enzyme Pepsin functions best at around pH2 and is found in the stomach, which contains Hydrochloric Acid.

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3. Concentration

Changing the enzyme and substrate concentrations affect the rate of reaction of an enzyme-catalyzed reaction. Controlling these factors in a cell is one way that an organism regulates its enzyme activity and so its metabolism.

Changing the concentration of a substance only affects the rate of reaction if it is the limiting factor: that is, it the factor that is stopping a reaction from preceding at a higher rate.

If it is the limiting factor, increasing concentration will increase the rate of reaction up to a point, after which any increase will not affect the rate of reaction. This is because it will no longer be the limiting factor and another factor will be limiting the maximum rate of reaction.

As a reaction proceeds, the rate of reaction will decrease, since the substrate will get used up. The highest rate of reaction, known as the initial reaction rate is the maximum reaction rate for an enzyme in an experimental situation.

Substrate Concentration

Increasing substrate concentration increases the rate of reaction. This is because more substrate molecules will be colliding with enzyme molecules, so more product will be formed.

However, after a certain concentration, any increase will have no effect on the rate of reaction, since substrate concentration will no longer be the limiting factor. The enzymes will effectively become saturated, and will be working at their maximum possible rate.

Enzyme Concentration

Increasing Enzyme Concentration will increase the rate of reaction, as more enzymes will be colliding with substrate molecules.

However, this too will only have an effect up to a certain concentration, where the Enzyme Concentration is no longer the limiting factor.

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SIMPLE ENZYME KINETICS Introduction to Enzyme Kinetics Enzyme kinetics deals with the rate of enzyme reaction and how it is affected by various chemical and physical conditions. Kinetic studies of enzymatic reactions provide information about the basic mechanism of the enzyme reaction and other parameters that characterize the properties of the enzyme. The rate equations developed from the kinetic studies can be applied in calculating reaction time, yields, and optimum economic condition, which are important in the design of an effective bioreactor.

Important Vocabulary

Chemistry Biochemistry

Reactant Substrate

Catalyst Enzyme

Rate of Reaction Assume that a substrate (S) is converted to a product (P) with the help of an enzyme (E) in a reactor as

SE P

If you measure the concentrations of substrate and product with respect to time, the product concentration will increase and reach a maximum value, whereas the substrate concentration will decrease as shown in the graph.

The rate of reaction can be expressed in terms of either the change of the substrate Cs or the product concentrations Cp as follows:

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In order to understand the effectiveness and characteristics of an enzyme reaction, it is important to know how the reaction rate is influenced by reaction conditions such as substrate, product, and enzyme concentrations. If we measure the initial reaction rate at different levels of substrate and enzyme concentrations, we obtain a series of curves like the one shown in the graph.

From these curves we can conclude the following:

1. The reaction rate is proportional to the substrate concentration (that is, first-order reaction) when the substrate concentration is in the low range.

2. The reaction rate does not depend on the substrate concentration when the substrate

concentration is high, since the reaction rate changes gradually from first order to zero order as the substrate concentration is increased.

3. The maximum reaction rate rmax is proportional to the enzyme concentration within the

range of the enzyme tested. French chemist Victor Henri (18721940) observed this behavior in 1902 and proposed the rate equation

where rmax and KM are kinetic parameters which need to be experimentally determined. This equation expresses the three preceding observations fairly well. The rate is proportional to Cs (first order) for low values of Cs, but with higher values of Cs, the rate becomes constant (zero order) and equal to rmax. Since this equation describes the experimental results well, we need to find the kinetic mechanisms which support this equation.

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Formation of Enzyme-Substrate Complex

British professor Adrian John Brown (1852 1919) proposed that an enzyme forms a complex with its substrate. The complex then breaks down to the products and regenerates the free enzyme. The mechanism of one substrate-enzyme reaction can be expressed as

Brown's kinetic inference of the existence of the enzyme-substrate complex was made long before the chemical nature of enzymes was known, 40 years before the spectrophotometric detection of such complexes. One of the original theories to account for the formation of the enzyme-substrate complex is the lock-and-key theory.

Lock-and-Key Theory In 1894, long before the first enzyme structure had been determined and several decades before it had been shown that enzymes are proteins, German chemist Emil Fischer (18521919) proposed that a substrate molecule fits into the enzyme active site like a key in a lock. An enzyme usually contains one or more active sites, where reactions with substrates take place. An active site may comprise only a few amino acid residues; the rest of the protein is required for maintaining the three-dimensional integrity of the network. The specificity of enzymes for substrates varies from molecule to molecule. Many enzymes exhibit stereochemical specificity in that they catalyze the reactions of one conformation but not the other. Also, enzymes promote catalysis by positioning key acidic or basic groups and metal ions in the right position for catalysis. This theory has been widely embraced, in part because it appears to explain the exquisite substrate specificity of enzymes. Just as only the right key will fit into a lock, only the right substrate fits into the enzyme.

Induced-Fit Model One limitation of the Lock and Key Model is that it does not explain why the reaction actually occurs, and another is that enzymes are flexible and not rigid as this theory implies. The Induced-Fit Model overcomes some of the limitations of the Lock and Key Model. In this model, the substrate still needs to fit into the enzyme like a key, but instead of simply fitting into the keyhole, some type of modification is induced in the substrate, enzyme, or both. The modification begins the process of the reaction.

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When the enzyme binds its substrate to form an enzymesubstrate complex, the structure of the substrate is distorted and pulled into the transition state conformation. This reduces the energy required for the conversion of a given reactant into a product and increases the rate of a reaction by lowering the energy requirement and therefore increasing the number of effective collisions that can result in the formation of the product.

Products can only be formed when colliding reactants have sufficient activation energy. The greater the activation energy for a given reaction is, the lower the number of effective collisions. In reality, the free energy diagram for an enzyme-catalyzed reaction is considerably more complicated than the example. Typically an enzyme-catalyzed reaction will involve multiple steps, each with an activation energy that is markedly lower than that for the uncatalyzed reaction. Michaelis-Menten Kinetic Model

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The reaction rate equation (Michaelis-Menten equation) can be derived from the preceding mechanism based on the following assumptions:

1. The total enzyme concentration stays constant during the reaction, that is

where CE0 is the total enzyme concentration CES is the enzyme-substrate complex concentration CE is the free-enzyme concentration

2. The amount of an enzyme is very small compared to the amount of substrate. Therefore, the formation of the enzyme-substrate complex does not significantly deplete the substrate.

3. The product concentration is so low that product inhibition may be considered negligible. In addition to the preceding assumptions, there are three different approaches to derive the rate equation:

1. Michaelis-Menten approach. It is assumed that the product-releasing step is much slower than the reversible reaction and the slow step determines the rate, while the other is at equilibrium. This assumption is known as the Rapid Equilibrium Method. This is an assumption which is often employed in heterogeneous catalytic reactions in chemical kinetics.

2. Briggs-Haldane approach. The change of the intermediate concentration with respect to time is assumed to be negligible, that is

This is also known as the pseudo-steady-state (or quasi-steady-state) assumption or simply Steady-state Approximation in chemical kinetics and is often used in developing rate expressions in homogeneous catalytic reactions.

3. Numerical solution. Solution of the simultaneous differential equations developed from product-releasing step and reversible reaction without simplification. From the mechanism described by these step and reaction, three rate equations can be written for C, CES and Cs as

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These equations plus the initial assumption that

can be solved simultaneously without simplification. Since the analytical solution of the preceding simultaneous differential equations are not possible, we need to solve them numerically by using a computer. Among many software packages that solve simultaneous differential equations include

Advanced Continuous Simulation Language (ACSL, 1975)

Mathematica (Wolfram Research, Inc., Champaign, IL) or

MathCad (MathSoft, Inc., Cambridge, MA).

Comparison of the Michaelis-Menten and Briggs-Haldane Approaches

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Biographies of Michaelis and Menten Leonor Michaelis (January 16, 1875 October 8, 1949) was born in Berlin, Germany and graduated from the humanistic Koellnisches Gymnasium in 1893 after passing the Abiturienten Examen. It was here that Michaelis interest in physics and chemistry was first sparked as he was encouraged by his teachers to utilize the relatively unused laboratories at his school. With concerns about the financial stability of a pure scientist, he commenced his study of medicine at Berlin University in 1893. Among his instructors were Emil du Bois-Reymond for physiology, Emil Fischer for chemistry, and Oskar Hertwig for histology and embryology.

L. Michaelis (left) and M. Menten (right)

Maud Menten (March 20, 1879 July 26, 1960) was born in Port Lambton, Ontario and studied medicine at the University of Toronto (B.A. 1904, M.B. 1907, M.D. 1911). She was among the first women in Canada to earn a medical doctorate. She completed her thesis work at University of Chicago. At that time women were not allowed to do research in Canada, so she decided to do research in other countries such as the United States and Germany. In 1912 she moved to Berlin where she worked with Leonor Michaelis and co-authored their paper in Biochemische Zeitschrift (1913;49:333369) which showed that the rate of an enzyme-catalyzed reaction is proportional to the amount of the enzyme-substrate complex. This relationship between reaction rate and enzyme-substrate concentration is known as the Michaelis-Menten equation.

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Derivation of Rate Equation (Michaelis-Menten equation)

Rapid Equilibrium Method If the slower reaction

determines the overall rate of reaction, the rate of product formation and substrate consumption is proportional to the concentration of the enzyme-substrate complex as:

From the assumption that the first reversible reaction

is in equilibrium, then the forward reaction is equal to the reverse reaction so that

. Assume that the total enzyme contents are conserved such that

. Perform substitution and rearrangement.

This results in the final rate equation

where KM is the Michaelis constant, which characterizes the interaction of an enzyme with a given substrate; and rmax is the maximum reaction rate, which is proportional to the initial enzyme concentration. (Because of the difficulty of expressing the enzyme concentration in molar unit, k3 and CEO are lumped into one parameter rmax. Whatever unit is adopted for CEO, the unit for k3CEO should be the same as r.)

This equation is known as the Michaelis-Menten equation and is identical to the empirical expression proposed by Henri.

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Steady-state Approximation

From the mechanism

the rates of product formation and of substrate consumption are

Assume that the change of CES with time is negligible compared to that of C or CS.

Substitution confirms that the rate of product formation and that of the substrate consumption are the same, that is,

Assume that the total enzyme contents are conserved such that

Perform substitution and rearrangement.

This results in the final rate equation

which is the same as the Michaelis-Menten equation, except that in the rapid equilibrium method, KM is equal to the dissociation constant k2/k1 while in the steady-state approximation, it is equal to (k2 + k3)/k1. This can be resolved when k2>>k3, which means means that the product-releasing step is much slower than the enzyme-substrate complex dissociation step. Since the formation of the complex involves only weak interactions, it is likely that the rate of dissociation of the complex will be rapid. The breakdown of the complex to yield products will involve the making and breaking of chemical bonds, which is much slower than the enzyme-substrate complex dissociation step.

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Biographies of Lineweaver and Burk Hans Lineweaver (December 25, 1907 June 10, 2009) was an American physical chemist, who developed the LineweaverBurk plot in 1934 while still a graduate student, working as a laboratory assistant under Burk at the US Department of Agriculture in Washington, D.C. The paper containing the equation, also known as the Double Reciprocal Plot, was co-authored by Dr. Dean Burk, and was titled "The Determination of Enzyme Dissociation Constants (1934)". It remains the most frequently cited paper ever to appear in the Journal of the American Chemical Society. Dean Burk (March 21, 1904 October 6, 1988) was an American biochemist: a co-discoverer of biotin, medical researcher, and a cancer researcher at the Kaiser Wilhelm Institute and the National Cancer Institute. Burk joined the Department of Agriculture in 1929 working in the Fixed Nitrogen Research Laboratory. In 1934, he developed the LineweaverBurk plot together with Hans Lineweaver.

Microapparatus used for studying bacterial nitrogen fixation. Image shows the Warburg-Barcroft microrespiration apparatus, the Micro-

Kjeldahl distilling apparatus; multiple gas-mixing flow meter; nephelometer; and bacterial culture bottle. On the left is Dr. Burk and to the right is Mr. Lineweaver.

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Graphical Analysis In order to estimate the values of the kinetic parameters, we need to make a series of batch runs with different levels of substrate concentration. Then the initial reaction rate can be calculated as a function of initial substrate concentrations. The results can be plotted graphically so that the validity of the kinetic model can be tested and the values of the kinetic parameters can be estimated. The most straightforward way is to plot r against CS. The asymptote for r will be rmax and is equal to CS when r = 0.5 rmax. However, this is an unsatisfactory plot in estimating rmax and because it is difficult to estimate asymptotes accurately and also difficult to test the validity of the kinetic model. Therefore, the Michaelis-Menten equation is usually rearranged so that the results can be plotted as a straight line.

Lineweaver-Burk plot Before the invention of computers, the determination of KM and Vmax was a tedious process. Today curve-fitting programs allow rapid analysis of the data to determine these values. However, a relatively simple method allows a relatively accurate determination of these two constants. This method is to construct a Lineweaver-Burk plot, also known as a double-reciprocal plot. The basis of a Lineweaver-Burk plot comes from the manipulation of the Michaelis-Menten equation to the form:

This equation has the form y = mx + b, and describes a straight line. A plot of the reciprocal of the rate, 1 / V, versus the reciprocal of the substrate concentration, 1 / [S], gives a line with a y-intercept equal to 1 / Vmax and an x-intercept of 1 / KM.

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Lineweaver-Burk Plot Derivation The derivation starts with the Michaelis-Menten equation:

=[]

+ []

Getting the reciprocal of both sides of the equation, we have:

1

=

+ []

[]

We can write this equation as the sum of two fractions over a least common denominator

1

=

[]

+[]

[]

which can be simplified as

=

[]+

The derived equation follows the format for the linear regression equation

= +

where is equal to 1

is equal to

is equal to 1

[]

is equal to 1

Setting the value of as 0

= + = 0 the x-intercept can be calculated as follows

=

=

= 1

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The Lineweaver-Burk plot is the most widely used graphical technique for the determination of

KM and Vmax. However, there are other methods.

Woolf (or Hanes-Woolf or Langmuir) plot The Woolf plot uses the equation:

Plotting [S] / V versus [S] gives a straight line.

Eadie-Hofstee plot

An Eadie-Hofstee plot uses the equation

Plotting V versus V / [S] gives a straight-line.

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Woolf Plot Derivation The derivation starts with the Michaelis-Menten equation:

=[]

+ []

Getting the reciprocal of both sides of the equation, we have:

1

=

+ []

[]

The substrate concentration value present in the denominator of the right-hand side of the equation can be transposed into the left-hand side of the equation to give us the value of the ratio of the substrate concentration to the reaction rate.

[]

=

+ []

This can be rewritten as

[]

=

[] +

which follows the format for the linear regression equation

= +

where is equal to []

is equal to 1

is equal to []

is equal to



=

=

=

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Eadie-Hofstee Plot Derivation The derivation starts with the Michaelis-Menten equation:

=[]

+ []

The sum of the Michaelis constant and the substrate concentration value present in the denominator of the right-hand side of the equation can be transposed into the left-hand side of the equation and this results to:

+ [] = [] The resulting equation can be further manipulated as follows.

[] = + []

= []

+[]

[]

This can be rewritten as

=

[]+

which follows the format for the linear regression equation

= +

where is equal to

is equal to

is equal to

[]

is equal to



=

=

=

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ENZYME REACTOR WITH SIMPLE KINETICS A bioreactor is a device within which biochemical transformations are caused by the action of enzymes or living cells. The bioreactor is frequently called a fermenter whether the transformation is carried out by living cells or in vivo cellular components. The chemical process in the bioreactor can either be aerobic or anaerobic. These bioreactors are commonly cylindrical, ranging in size from liters to cubic meters, and are often made of stainless steel.

Industrial and Laboratory Bioreactor

Organisms growing in bioreactors may be submerged in liquid medium or may be attached to the surface of a solid medium. Submerged cultures may be suspended or immobilized. Suspension bioreactors can use a wider variety of organisms, since special attachment surfaces are not needed, and can operate at much larger scale than immobilized cultures. However, in a continuously operated process the organisms will be removed from the reactor with the effluent. Immobilization is a general term describing a wide variety of cell or particle attachment or entrapment. It can be applied to basically all types of biocatalysis including enzymes, cellular organelles, and animal and plant cells. Immobilization is useful for continuously operated processes, since the organisms will not be removed with the reactor effluent, but is limited in scale because the microbes are only present on the surfaces of the vessel.

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General Types of Bioreactors Batch Stirred-Tank Reactor The simplest reactor configuration for any enzyme reaction is the batch mode. A batch enzyme reactor is normally equipped with an agitator to mix the reactant, and the pH of the reactant is maintained by employing either a buffer solution or a pH controller. An ideal batch reactor is assumed to be well mixed so that the contents are uniform in composition at all times.

Batch Stirred-Tank Reactor

Assume that an enzyme reaction is initiated at t = 0 by adding enzyme and the reaction mechanism can be represented by the Michaelis-Menten equation:

=

+ (1)

An equation expressing the change of the substrate concentration with respect to time can be obtained by integrating equation (1), as follows:

(+

) =

0

0

(2)

and

0

+ (0 ) = (3)

This equation shows how CS is changing with respect to time. With known values of rmax and KM, the change of CS with time in a batch reactor can be predicted from this equation.

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Plug-Flow Reactor In a plug-flow enzyme reactor or tubular-flow enzyme reactor, the substrate enters one end of a cylindrical tube which is packed with immobilized enzyme and the product stream leaves at the other end. The long tube and lack of stirring device prevents complete mixing of the fluid in the tube. Therefore, the properties of the flowing stream will vary in both longitudinal and radial directions. Since the variation in the radial direction is small compared to that in the longitudinal direction, it is called a plug-flow reactor. If a plug-flow reactor is operated at steady state, the properties will be constant with respect to time. The ideal plug-flow enzyme reactor can approximate the long tube, packed-bed, and hollow fiber, or multistage reactor.

Schematic Diagram of a Plug-Flow Reactor

Equation 3 can also be applied to an ideal steady-state plug-flow reactor, even though the plug-flow reactor is operated in continuous mode. However, the time t in Equation 3 should be replaced with the residence time in the plug-flow reactor. Rearranging Equation 3 results in the following useful linear equation which can be plotted:

0

ln (0/ )= +

ln (0/ ) (4)

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Continuous Stirred-Tank Reactor A continuous stirred-tank reactor (CSTR) is an ideal reactor which is based on the assumption that the reactor contents are well mixed. Therefore, the concentrations of the various components of the outlet stream are assumed to be the same as the concentrations of these components in the reactor. Continuous operation of the enzyme reactor can increase the productivity of the reactor significantly by eliminating the downtime. It is also easy to automate in order to reduce labor costs. Recently, CSTRs are used to optimize feasible and reliable bioprocess system in order to treat hydrocarbon-rich industrial wastewaters.

Schematic Diagram of a Continuous Stirred-Tank Reactor

The substrate balance of a CSTR can be set up, as follows:

Input - Output + Generation = Accumulation

+ =

(5)

where F is the flow rate and V is the volume of the reactor contents. It should be noted that rS is the rate of substrate consumption for the enzymatic reaction, while dCS /dt is the change of the substrate concentration in the reactor. As can be seen in Equation 5, rS is equal to dCS /dt when F is zero, which is the case in batch operation. For the steady-state CSTR, the substrate concentration of the reactor should be constant. Therefore, dCS /dt is equal to zero. If the Michaelis-Menten equation can be used for the rate of substrate consumption (rS), Equation can be rearranged as:

= =

1

=

(0 )(+) (6)

where D is known as dilution rate, and is equal to the reciprocal of the residence time (). Equation (6) can be rearranged to give the linear relationship:

= + 0

Michaelis-Menten kinetic parameters can also be estimated by running a series of steady-state CSTR runs with various flow rates and plotting CS versus (CS)/(CS0-CS). Summary of the Types of Bioreactors

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Category Batch Stirred-Tank

Reactor Plug-Flow Reactor

Continuously Stirred-Tank Reactor

Operation

assumed to be well mixed; contents are uniform in composition at all times equipped with an agitator to mix the reactant; buffer solution or pH controller for pH

the substrate enters one end of a cylindrical tube which is packed with immobilized enzyme and the product stream leaves at the other end The long tube and lack of stirring device prevents complete mixing of the fluid in the tube

based on the assumption that the reactor contents are well mixed the concentrations of the various components of the outlet stream are assumed to be the same as the concentrations of these components in the reactor

Application Composting Waste Treatment Industrial Wastewater

Treatment

Equations

= +

(+

)

0

=

0

0

+ (0 )

=

0 ln (0/ )

=

+

ln (0/ )

+

=

= =

1

=

(0 )( + )

= + 0

ENZYME INHIBITION Inhibitor

- is a modulator (substance which can combine with enzymes to alter their catalytic activities) which decreases enzyme activity. It can decrease the rate of reaction either competitively, noncompetitively / partially competitively, or uncompetitively.

TYPE OF INHIBITION

Characteristics Reaction Scheme Modification to

Michaelis-Menten Eq. Consequences

Competitive

- competitive inhibitor has a strong structural resemblance to the substrate - both the inhibitor and substrate

compete for the active site of an enzyme

- inhibitor only binds to the free enzyme

KI = [E][I] / [EI]

- Vmax is unchanged

- KM is increased

Non-Competitive

- inhibitor binds to both E and ES - Inhibitor binds close to the active site, or

by binding elsewhere on E has an influence on the active site

- Vmax is decreased - KM is increased

Uncompetitive

- inhibitor binds directly to the ES complex

- inhibitor does not have to bind at the active site

- inhibitor does not have to resemble the substrate

- Vmax is decreased - KM is unchanged

Plots based on Lineweaver-Burk:

Figure 1. Competitive Inhibition Figure 2. Non-competitive Inhibition Figure 3. Uncompetitive Inhibition

Bibliography

Bailey, J., & Ollis, D. (1986). Biochemical Engineering Fundamentals. New York: McGraw-Hill Book Co.

Chang, R. (2005). Physical Chemistry for the Biosciences. California: University Science Books.

Dutta, R. (2008). Fundamentals of Biochemical Engineering. New Delhi, India: Ane Books India.

Marangoni, A. G. (2003). Enzyme Kinetics: A Modern Approach. New Jersey: John Wiley & Sons, Inc.

Moore, J. T., & Langley, R. (New Jersey). Biochemistry For Dummies. 2008: Wiley Publishing, Inc.

Pratt, C., & Cornely, K. (2014). Essential Biochemistry (3rd ed.). New Jersey: John Wiley & Sons, Inc.

Rogers, A., & Gibon, Y. (2009). Enzyme Kinetics: Theory and Practice. In J. Schwender (Ed.), Plant

Metabolic Networks (pp. 71-103). Springer Science+Business Media, LLC.

enzyme and enzyme kinetics

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