FACTORS AFFECTING ENZYME KINETICS

3. FACTORS AFFECTING ENZYME KINETICS

Enzyme kinetics specifically refers to the in-depth study of ‘enzymes in action’. The abnormally high inherent rate of the induced enzyme-catalyzed reactions enormously obviates and facilitates this study. It has been established beyond any reasonable doubt that the physical and chemical nature of the carrier matrices employed, together with the conformational alterations in the enzyme structure after adequate immobilization, ultimately gives rise to the conversion of the ensuing catalyst to heterogenous nature having significant changes in its characteristic features. The various effects caused duly by the altered reactivity essentially includes : activity, pH, stability, optimum temperature, kinetic constants,* and substrate specificity.

* Kinetic constants : Usually come into being from a definite change in the activation energy.

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It has been duly observed that the rate of reaction catalyzed by an enzyme particularly enhances ‘linearity’ with the corresponding increase in the substrate concentration generally upto a certain point. However, it soon approaches the maximum value, usually termed as V max ; and beyond which there is absolutely no further enhancement in the rate of reaction as shown in Fig. 5.11. It is known as saturation.

Just contrary to this, is the rate of a nonenzymatically catalyzed reaction that enhances linearly very much across the entire range of attainable substrate concentrations. Importantly, the prevailing ‘immobilization phenomenon’ does help in the actual conversion of the catalyst from homongeneous (i.e., soluble enzyme) nature to the heterogeneous one, whereby the enzyme is intimately associated either with a particular enveloping matrix or a supporting matrix. Nevertheless, in the course of immobilization phenomenon, the activity of ensuing enzyme is virtually lost by virtue of two vital rea- sons, namely : (a) various reactions involved in the process ; and (b) effective occlusion of active sites in the enzyme support complex.

Examples :

(1) Hem-containing proteins : Haemoglobin and catalase : It has been observed that haemo- globin gets bound to O 2 ; and in doing so several O 2 -molecules may bind and release during one minute, while at any material time only one O 2 molecule becomes intimately associated

with one hem centre. Catalase — an enzyme invariably exerts a cumulative effect. In this particular instance, not more

than one O 2 molecule gets bound per hem ; and when it remains in the ‘bound form’ it has a tendency to react progressively. Consequently, one may distinctly observe a very rapid evolution O 2 (i.e., one million molecules per enzyme molecule per minute).

ENZYMATICALLY CATALYZED A B REACTION

NONENZYMATICALLY CATALYZED

T E REACTION

SUBSTRATE CONCENTRATION SUBSTRATE CONCENTRATION

[S]

[S]

Fig. 5.11. Effect of Initial Substrate Concentration upon Initial Reaction Velocity of Nonenzymatically [A] and Enzymatically [B] Catalyzed Reactions.

S = Substrate Concentration ; V max = Maximum Reaction Rate (Velocity) ; K m = Michaelis – Menten Constant ; E = Enzyme ; P = Products ;

3.1. Enzyme Activity

Enzyme refers to an organic catalyst invariably produced by ‘living cells’ but capable of acting either outside cells or even in vitro. Enzymes are ‘proteins’ that change the rate of chemical reactions without needing an external energy source or being changed themselves ; an enzymes may catalyze a reaction numerous times. Enzymes are highly reaction specific in that they act only on certain sub-

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give rise to a temporary configuration, called an enzyme-substrate complex that essentially involves both physical shape and chemical bonding. The enzyme usually promotes the formation of bonds be-

tween separate substrates, or induces the breaking of bonds in a single substrate to form the product or products of reaction. The human body contains thousands of enzymes, each catalyzing one of the many reactions that eventually occur as part of metabolism.

In a broader sense, enzymes do carry out a plethora of remarkable and much needed rearrangements as well as chemical modifications both in vivo and in vitro. A few typical examples are given below to justify its magnanimous utility in the living biological systems, namely :

1. Casein → Paracasein

Rennin

[Milk Protein]

2. Ethanol → Acetaldehyde

Dehydrogenase

[From Molasses]

3. Glucose → Hydrogen Peroxide + Gluconic Acid

+O 2 +H 2 O

Oxidase

4. Hydrogen Peroxide → Oxygen + Water

Catalase

5. Lactose → Glucose + Fructose

Lactase

[Milk Sugar]

α -Amylase

6. Starch → Glucose + Maltose

[Barley ; Maize]

β -Amylase

7. Starch → Maltose + Dextrin

[Barley ; Maize]

Rennin

Oxidase

8. Sucrose → Glucose → CO 2 +H 2 O

[Starch]

+O 2 +H 2 O

9. Xanthine → Uric Acid + H 2 O

Oxidase

Deaminases

10. Amino Acids → Removal of — NH 2 moieties

[Excess protein hydrolysates]

11. Fatty Acids → Fatty Acids

Lipase

[Long-chain fatty acids]

[Short-chain fatty acids]

The enzyme activity exclusively rests on the ‘concentration of reactant’ ; and, therefore, the rate of enzyme activity gets enhanced progressively with the increase in the concentration. One may

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observe the enhancement in the reaction-rate to maintain a genuine linearity along with increase in the substrate concentration ; however, the linearity subsequently levels off at high concentrations, and, ultimately becomes independent of the prevailing concentration (see Fig. 5.11.B).

Explanation : The above hypothesis may be explained by putting forward the simple argument explicitely that at high concentrations majority of the ensuing enzyme molecules do have their ‘active sites occupied’. Thus, the overall net rate of enzyme activity solely depends on the rate at which the bound reactants are converted into the respective products. Evidently, any extent of further increases in the ‘substrate concentrations’ would exert practically little effect. In fact, the behavioural pattern of enzyme activity in the above context is widely termed as Michaelis-Menten Mechanism.

3.2. Michaelis-Menten Constant [K m ]

Michaelis-Menten hypothesis assumes categorically that an enzyme-catalyzed reaction should normally take its course via several different stages, for instance :

Stage - 1 : Binding of enzyme (E) and substrate (S) would result into the formation of enzyme- substrate complex.

Stage - 2 : Obvious catalysis takes place by virtue of the actual formation of transient enzyme- substrate (ES) complex.

Stage - 3 : In a situation when the substrate concentration stands at a higher ebb, further incor- poration of enzyme would increase the reaction-rate appreciably, till such time the prevailing substrate concentration renders limiting :

Stage - 4 : Being the last stage the final conversion into product (P) predominently takes place very much within the ES-complex (see stage-2) ; and subsequently the desired product gets released, whereas the original free enzyme eventually gets regenerated.

Fig. 5.11(B) clearly shows a distinct direct proportionality existing between the reaction rate (V) and substrate concentration (S) till such time the prevailing enzyme concentration becomes limiting. Thus, at certain material time when the substrate concentration (S) affords almost half the maximal enzyme-reaction rate, the specific value is termed as Michaelis-Menten Constant, and designated as ‘K m ’. It is, however, pertinent to state here that the normal observed Km values for a large experimental

data remain constant, besides found to be absolutely independent with respect to the quantum of en- zymes present. Nevertheless, the usual K

m –7 -values for a host of enzymes reported so far ranges between

10 to 10 molar K m . Significance of K m Values : The various important significance of K m values are as follows : (1) Indicative of substrate concentration (S), (2) Affinity of enzyme with corresponding substrate, (3) Indicative partially of enzyme-substrate concentration prevailing in the cellular compart-

ment i.e., the target where most of the reaction invariably takes palce. (4) K m -values are found to be inversely proportional to the ensuing affinity of the enzyme for its

substrate i.e., higher K m -values give rise to lower stability of the enzyme substrate (ES)- complex apparently.

3.2.1. Kinetics of ES-Complex Formation

Importantly, the kinetics of enzyme-substrate (ES) complex formation has been profusely dealt with in literatures based on the laws of physical chemistry. In this context only a few fundamental

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From Fig. 5.11(B) one may have the following expression : E+S WXXXX XXXXXV

Fast k + 1 X Slow k

ES → + E P

...(a) where,

E = Enzyme ; S = Substrate ; P = Product ; k +1 and k +2 = Velocity Constant of ‘fast’ and ‘slow’ reactions.

In Eqn. (a), the very first step of this reaction is reversible, while the second step is irreversible. One may assume that only a single substrate is present, and also only a single product is formed ; and thus, the enzymatic phenomenon goes ahead predominantly to accomplish completion, and the concen- tration of the substrate renders the prevailing enzyme to saturation.

In case, V f , designates the velocity for the formation of the ES-complex, we may have :

V f =k +1 [S][E] – [ES] ...(b) where, [E], [S] and [ES] represent the molar concentration of enzyme, substrate, and the ES-complex

respectively. Importantly, the velocity at any material time solely depends upon the concentration of enzyme and substrate. In realtiy the concentration of the former (i.e., enzyme) is almost equivalent to the original concentration minus the concentration of the complex i.e., {[E] – [ES]} ; whereas the latter (i.e., substrate) is observed to be the same as that of the original concentration [S], because it is so appreciably large that its corresponding fraction duly present in ES-complex is practically insignificant and hence negligible.

Nevertheless, the velocity of disappearance designated by Vd of the resulting ES-complex predominantly depends upon two vital factors, namely :

(i) its subsequent dissociation into E and S respectively (i.e., reverse reactive process), and (ii) its inherent decomposition into enzyme E, and S as product P. Therefore, we may have :

...(c) In the perfect steady state we have

V d =k –1 [ES] + k +2 [ES]

V f =V d ...(d) Substituting the values of V f from Eqn. (b) and V d from Eqn. (c) in Eqn. (d) we have :

...(e) Rearrangement of Eqn. (e) we may have :

k +1 [S] {[E] – [ES]} = k –1 [ES] + k +2 [ES] = [ES] (k –1 +k +2 )

[S]{[E] [ES]} −

This ‘K m ’ is termed as Michaelis constant in honour of the name of the scientist who first and foremost postulated the well accepted ‘theory of complex formation’. Evidently, the Michalis con- stant serves as one of the most vital, glaring, and pivotal parameters ; and it invariably represents the characteristic feature of an individual enzyme.

Thus, Eqn. (f) may be rewritten as :

[S] {[E} – [ES]} = K m × [ES]

or

[S] [E] – [S] [ES] = K m × [ES]

or K m × [ES] + [S] [ES] = [S] [E]

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or [ES] {K m + [S]} = [S] [E]

It has been duly proved and established that the experimentally determined velocity, V, is rightly displayed by the actual decomposition of the prevailing ES-complex, and hence we have :

...(h) Putting the value of [ES] from Eqn. (g) we have :

V=k +2 [ES]

V max i.e., the maximum velocity is accomplished when all the available enzyme gets bound to the corresponding substrate (see Fig. 5.11.B), and under this parameters we have :

[E] = [ES]

Therefore,

...(j) Substituting the values from Eqn. (j) in Eqn. (i) we have :

From Eqn. (k) it is abundantly obvious that both components viz., V max and K m are constants ; and, therefore, the plot between V against [S] must be a rectangular hyperbola as illustrated Fig. 5.11.B obtained experimentally.

3.2.2. Determination of K m

The precise and accurate determination of the Michaelis-Menten constant, Km, may be carried out by the help of following two methods, namely :

(a) Lineweaver and Burk Method : The researchers suggested a method to obtain an expres- sion by rearranging Eqn. (k) above :

The above is a ‘linear equation’ of the pattern y = ax + b. As it is known that both K m and V max

are constants ; therefore, the plot between

(along ordinate) and

(along abscissa) will give forth

V [S]

a straight line as shown in Fig. 5.12. Importantly, its intercept on the Y-axis is , and that on the

V max

X-axis (i.e., after extrapolation) is

, having the slope

ENZYME IMMOBILIZATION

Fig. 5.12. Determination of K m * by Liveweaver-Burk Method

(b) V max Method : We may rewrite Eqn. (k) as below :

V max [S] K m + [S] =

Now, when V max = 2V, we may have :

Therefore, one may explain the above expression by saying that the substrate concentration corresponding to half the maximum velocity is equivalent to K m numerically, as could be seen in Fig. 5.11(B).

3.2.3. Kinetic Characteristic Features

It is a well known fact, based upon the ‘transition state theory’, that whenever a chemical interaction takes place specifically between two reactant molecules, their free energy level usually should

be distinctly elevated much above the threshold level so as to boost them up to an extremely unstable, and high-energy state termed as the transition state. Furthermore, the ‘free energy’ essentially required to raise a molecule from its initial ‘ground state’ to the attained unstable ‘transition state’ invariably

+ known as the activation energy, designated as G + as depicted in Fig. 5.13.

From Fig. 5.13 it is quite evident that the prevailing rate of a chemical reaction solely depends upon the number of available reactant entities (molecules) that do possess enough energy to attain criti- cally the ‘transition state’ pertaining particularly to the ‘slowest step’ in the entire reaction profile, and

is mostly known as the ‘rate determining step’. Importantly, one must take cognizance of the fact that only one-half of all the available molecules ultimately approaching the transition state would react to

form the desired products ; whereas, the remaining half shall be returning safely to the low-energy ground-state reactant molecules. Based on a generalized rule, only a rather scanty molecules would gainfully possess enough energy to reach the so called transition state.

+ It is, however, pertinent to mention here that the inherent magnitude of G + denotes the major

+ factor limiting the rate of a chemical reaction i.e., higher the value of G + the lower would be the

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prevailing rate of reaction. Interestingly, the — ‘enzymes predominantly minimize the value of ∆ G + for

a specific reaction, thereby overwhelmingly enhancing the ultimate rate of reactions’ , (see Fig. 5.13).

NON-ENZYME CATALYZED ENZYME CATALYZED

P REACTION COORDINATE

REACTION COORDINATE

Fig. 5.13. Reaction Coordinate Diagrams of Non-Enzyme Catalyzed Reaction [A] and a Enzyme Catalyzed Reaction [B].

[R = Reactants ; P = Products ; GS = Ground-state, TS = Transition State ; ∆ G = Free energy of the reaction

+ ; ∆ G + = Activation energy of the reaction ; ES = Enzyme-Substrate Complex] Important Points : There are three important points that must be taken into consideration,

namely : (1) Energy levels and the stabilities of the products and reactants in a chemical reaction, when

compared with those in the same reaction but catalyzed by an enzyme are more or less identical.

(2) Rate of reverse reaction is increased by an almost same order of magnitude as that of the

forward reaction.

(4) Enzymes when compared to other innumerable catalysts, fails to alter the equilibrium ratio of the reactants as well as the products.

3.2.4. Parameters Governing Enzymatic Reactions

There are several cardinal parameters which categorically govern the enzymatic reactions, such as :

(i) Maximum reaction velocity, (V max ) (ii) pH Activity, (iii) Optimum temperature, (T opt ) (iv) Stability,

(v) Dissociation constants for substrate (K s ) and product (K p ), (vi) Turnover number or Rate constant (K cat ), (vii) Michaelis constant, (K m ) (viii) Specifically constant, (K cat /K m )

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(ix) Rate enhancement, (K cat /K non ), and (x) Catalytic proficiency [(K cat /K m )/K non ]. The above parameters shall now be treated individually in the sections that follows :

3.2.4.1. Maximum Reaction Velocity (V max )

The maximum reaction velocity of an immobilized enzyme is found to be practically the same as that obtained for the free enzyme. However, the precise effectiveness depends solely upon the ensuing enzyme activity together with diffusion limitation in the course of immobilization.

Examples :

(1) Glucoamylase : It has been amply proved that the V max value for an immobilized glucoamylase in the presence of maltose as substrate is 10 fold higher in comparison to the soluble- enzyme (i.e., native enzyme).

(2) βββββ -D-Fructofuranocidase : The gel-entrapped βββββ -D-fructofuranocidase exhibits almost

10 times lower value of V max in comparison to the free-enzyme.

3.2.4.2. pH Activity

One may explicitely observe the variation in the pH profile of the immobilized enzymes which prominently depends upon the prevailing charge either of the enzyme molecule or of the support. In the specific instance related to the support binding method, when the matrix is adequately charged, the kinetic behaviour of the immobilized enzyme would differ appreciably than that of the free enzyme. However, the ensuing difference is largely attributed to the prevailing partition effects which may give rise to concentration variants in the charged species viz., products, substrates, H + ion, OH – ion etc. Furthermore, the above characteristic features may be observed in the microenvironment of the prevail- ing immobilized enzymes, besides in the bulk solution as well by virtue of the electrostatic interactions bearing rather fixed charges on the support. The partition effect has an overall tendency with respect to

a corresponding shift in the pH optima, essentially exhibiting a definite displacement activity profile of the immobilized enzyme towards alkaline or acidic pH values for the negatively or positively charged matrices respectively.

Salient Features : The various salient features of pH activity profile are as follows : (a) Degree of shift of the pH profile shall entirely depend on the partition coefficient for the

respective H + ions of the ‘polymer matrix’ e.g., for polymer anion the value is > 1 ; and for polymer cation it is < 1.

(b) Broader pH profile may be observed in the event when the enzymes are coupled covalently to the respective support or entrapped in a gel securedly. However, enzyme entrapment spe- cifically in neutral polymers fails to exhibit any shift at optimum pH.

(c) H + ions exert a marked and pronounced effect upon the enzyme activity ; and this depends

upon exclusively whether the initial pH is above or lies below the pH optimum. (d) Microenvironment pH would fall in a situation when the accumulation of H + ions stands

above the pH optimum ; and this in turn shall enhance the ensuing rate of enzyme reaction. (e) Substrate diffusion limitation may also induce a recognizable effect in the specific response

of an enzyme to change in pH.

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3.2.4.3. Optimum Temperature (T opt )

It has been duly established that enzymes do possess a temperature optimum, designated as T opt ; and are found to be fairly active only in a narrow range of temperatures beyond that they are invariably get denatured readily and inactivated ultimately. A large segment of the enzymes are viable between 30 and 80°C.

Arrhenius Equation : Evidently, the enzyme activity as a function of temperature closely obeys the Arrhenius Equation :

k = Ae

RT

where, k = Rate of reaction,

H = Energy of activation ; T = Absolute temperature ;

R = Gas constant ;

A = Arhenius constant ; It has been proved beyond any reasonable doubt that the usual denaturation of enzyme may be

adequately protected by virtue of two physical phenomena, namely : (a) diffusional effects in case of entrapment ; and (b) binding on porous supports. Obviously, the inherent optimum temperature of the prevailing enzymes turn out to be higher than that of the native enzymes (soluble enzymes). Diffusional effects, that predominantly afford protection to the enzyme against the heat denaturation finally give rise to deactivation energies amalgamated with lower rate of activation. In reality, these critical mass

transfer effects enhance with the increase in ensuing temperature. In actual practice, the immobilized enzymes accomplished via different means and ways prominently exhibits lower activation energies, such as : (i) entrapment in gel ; (ii) entrapment in fibre-structure ; and (iii) support binding upon porous matrix.

3.2.4.4. Stability

The actual ‘retention of enzyme activity’ for a relatively longer duration under the influence of proper storage conditions is of paramount importance. Stability, is governed by several vital factors, namley : (a) nature of the surface of ‘carrier’ to which it gets bound intimately ; (b) prevention of conformational inactivation ; and (c) meticulous protection of active moieties on enzymes from the correpsonding reactive groups in solutions increases the stability of immobilized enzymes.

Loss of Enzyme Activity : Enzyme activity may be impaired (lost) for several identified rea- sons, for instance :

(1) Under usual operational parameters, the enzyme activity would suffer an appreciable loss on account of two distinct counts viz., (i) enzyme denaturation ; and (ii) physical loss from carrier.

(2) Loss incurred via matrix could result either due to carrier erosion or from gel-entrapment

matrix or from cleavage of existing bonds between enzyme and carrier.

(3) It is an universal truth that ‘covalent bonding’ certainly leads to fairly stable enzyme prepa- rations ; however, in several identified situations e.g., physical absorption, feeble bonding, and ionically bound enzymes — the activity is lost significantly into solutions depending upon temperature, ionic strength, pH, and above all the prevailing physical operating parameters.

(4) Type of ‘Bioreactor’ being employed also contributes a major determining factor for stability

ENZYME IMMOBILIZATION

Examples : Abrasion* or attrition** may tantamount to noticeable loss of activity for the

immobilized enzymes strategically positioned on the respective organic support. (5) Microbial contamination would give rise to ‘loss in activity’ under usual standard experi-

mental parameters. In a specific instance when the immobilized enzymes are critically diffu- sion limited, they will display a rather slower apparent loss of activity in comparison to a

situation when almost little mass-transfer resistance were present. (6) pH exerts its influence of enzyme stability. The ‘partition effects’ may appreciably increase

the pH stability of an ensuing immobilized enzyme. Examples : Stability gets increased : (a) Towards ‘alkaline range’ — with immobilized enzymes on polyanionic supports, and

(b) Towards ‘acidic range’ — with immobilized enzymes on polycationic supports. (7) Heat Stability : The plausible increment in ‘heat stability’ may be accomplished either by

entrapment techniques or covalent coupling ; whereas, ionic-linkage technique or physical adsoprtion methods afford a distinct decline in the heat stability of enzymes.

(8) Storage stability : The storage stability of enzymes may be enhanced via covalent coupling invariably at relatively low temperature i.e., 40°C.

3.2.4.5. Dissociation Constants of Substrate (K s ) and Product (K p )

It is a measure of dissociation. The dissociation constant of substrat is designated as K s , whereas the dissociation constant of the product as K p . The low values of K s and K p invariably give rise to inhibition of enzyme activity.

3.2.4.6. Turnover Number or Rate Constant [K cat ]

The turnover number refers to the specific activity of the enzyme and is usually quantified by K cat . It may be obtained when V max is divided by the enzyme concentration [E]. This designates the number of substrate molecules processed per enzyme molecule per second, and its value is generally 1000 but could be even much greater.

3.2.4.7. Michaelis Constant [K m ]

Michaclis constant or Michaelis Menten constant represented as K m has already been discussed under section 5.3.2.

3.2.4.8. Specificity Constant [K cat /K m ]

The specificity constant or catalytic efficiency of an enzyme is determined by the maximum specificity constant (K cat /K m ). It precisely indicates the accurate rate of assoication of enzyme and substrate, besides the specificity for competing substrates.

3.2.4.9. Rate Enhancement [K cat /K non ]

It has been observed that one may compare the rate constant of an uncatalysed reaction (K non ) with that of the corresponding catalysed reaction (K cat ), and subsequently calculate the increase in reaction rate which an enzyme eventually yields. It is termed as the rate enhancement and designated as [K cat /K non ].

* Scrapping away.

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3.2.4.10. Catalytic Proficiency [(K cat /K m )/K non )]

The proficiency of the catalytic ability of an enzyme is known as the catalytic proficiency and may be represented as [(K cat /K m )/K non ].