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(1)

ZOOLOGY: SEM- III, PAPER- C7T: FUNDAMENTALS OF BIOCHEMISTRY, UNIT 5: ENZYMES

E E n n z z y y m m e e s s K K i i n n e e t t i i c c s s

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D D R R . . P P O O U U L L A A M M I I A A D D H H I I K K A A R R Y Y M M U U K K H H E E R R J J E E E E A A S S S S I I S S T T A A N N T T P P R R O O F F E E S S S S O O R R

D D E E P P A A R R T T M M E E N N T T O O F F Z Z O O O O L L O O G G Y Y

N N A A R R A A J J O O L L E E R R A A J J C C O O L L L L E E G G E E

(2)

ZOOLOGY: SEM- III, PAPER- C7T: FUNDAMENTALS OF BIOCHEMISTRY, UNIT 5: ENZYMES

Enzyme kinetics is the study of the chemical reactions that

are catalysed by enzymes. In enzyme kinetics, the reaction

rate is measured and the effects of varying the conditions of

the reaction are investigated. Studying an enzyme's kinetics in

this way can reveal the catalytic mechanism of this enzyme, its

role in metabolism, how its activity is controlled, and how

a drug or an agonist might inhibit the enzyme.

(3)

Enzymes are usually protein molecules that manipulate other molecules—the enzymes' substrates. These target molecules bind to an enzyme's active site and are transformed into products through a series of steps known as the enzymatic mechanism:

E + S ⇄ ES ⇄ ES* ⇄ EP ⇄ E + P

These mechanisms can be divided into single-substrate and

multiple-substrate mechanisms. Kinetic studies on enzymes

(4)

that only bind one substrate, such as triosphosphate

isomerase, aim to measure the affinity with which the enzyme

binds this substrate and the turnover rate. Some other

examples of enzymes are phosphofructokinase and

hexokinase, both of which are important for cellular

respiration (glycolysis).

(5)

Michaelis–Menten kinetics is one of the best-known models

of enzyme kinetics. It is named after German

biochemist Leonor Michaelis and Canadian physician Maud

Menten. The model takes the form of an equation describing

the rate of enzymatic reactions, by relating reaction rate ὑ

(rate of formation of product[P]) to the concentration of

a substrate S [S]. Its formula is given by:

(6)

This equation is called the Michaelis–Menten equation.

Here, represents the maximum rate achieved by the system,

happening at saturating substrate concentration. The value of

the Michaelis constant is numerically equal to the substrate

concentration at which the reaction rate is half of V

max

(7)

Michaelis–Menten saturation curve for an enzyme reaction showing

the relation between the substrate concentration and reaction rate.

(8)

ZOOLOGY: SEM- III, PAPER- C7T: FUNDAMENTALS OF BIOCHEMISTRY, UNIT 5: ENZYMES

D D e e r r i i v v a a t t i i o o n n o o f f t t h h e e M M i i c c h h a a e e l l i i s s - - M M e e n n t t e e n n E E q q u u a a t t i i o o n n : :

A simple model of enzyme action:

We would like to know how to recognize an enzyme that behaves

according to this model. One way is to look at the enzyme's kinetic

behaviour at how substrate concentration affects its rate. Therefore,

we want to know what rate law such an enzyme would obey. If a

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ZOOLOGY: SEM- III, PAPER- C7T: FUNDAMENTALS OF BIOCHEMISTRY, UNIT 5: ENZYMES

newly discovered enzyme obeys the rate law derived from this model, then it's reasonable to assume that the enzyme reacts according to this model. It's not proof that the model is correct, but at least it tells us that kinetics does not rule it out.

For derivation of a rate law from this model.

For this model, let V

0

be the initial velocity of the reaction. Then V

0

= k

cat

[ES]...(2)

The maximum velocity V

max

occurs when the enzyme is saturated --

that is, when

(10)

all enzyme molecules are tied up with S, or [ES] = [E]

total

So V

max

= k

cat

[E]

total

...(3)

We want to express V

0

in terms of measurable quantities, like [S]

and [E]

total

, so we can see how to test the mechanism by

experiments in kinetics. So we must replace [ES] in (2) with

measurables.

(11)

During the initial phase of the reaction, as long as the reaction velocity remains constant, the reaction is in a steady state, with ES being formed and consumed at the same rate. During this phase, the rate of formation of [ES] equals its rate of consumption. According to model (1),

Rate of formation of ES = k

1

[E][S].

Rate of consumption of ES = k

-1

[ES] + k

cat

[ES].

So in the steady state, k

-1

[ES] + k

cat

[ES] = k

1

[E][S]...(4)

(12)

Remember that we are trying to solve for [ES] in terms of measurables, so that we can replace it in (2). First, collect the kinetic constants in (4):

(k

-1

+ k

cat

) [ES] = k

1

[E][S],

and (k

-1

+ k

cat

)/k

1

= [E][S]/[ES]... (5)

To simplify (5), first group the kinetic constants by defining them as

K

m

:

(13)

K

m

= (k

-1

+ k

cat

)/k

1

...(6) and then express [E] in terms of [ES] and [E]total:

[E] = [E]

total

- [ES] ...(7) Substitute (6) and (7) into (5):

Km = ([E]

total

- [ES]) [S]/[ES] ...(8) Solve (8) for [ES]: First multiply both sides by [ES]:

[ES] K

m

= [E]

total

[S] - [ES][S]

(14)

Then collect terms containing [ES] on the left:

[ES] K

m

+ [ES][S] = [E]

total

[S]

Factor [ES] from the left-hand terms:

[ES](K

m

+ [S]) = [E]

total

[S]

and finally, divide both sides by (K

m

+ [S]):

[ES] = [E]

total

[S]/(K

m

+ [S]) ...(9)

Substitute (9) into (2): V

0

= k

cat

[E]

total

[S]/(K

m

+ [S]) ...(10)

(15)

Recalling (3), substitute V

max

into (10) for k

cat

[E]

total

:

V

0

= V

max

[S]/(K

m

+ [S]) ...(11)

This equation expresses the initial rate of reaction in terms of a measurable quantity, the initial substrate concentration. The two kinetic parameters, V

max

and K

m

, will be different for every enzyme- substrate pair.

Equation (11), the Michaelis-Menten equation, describes the kinetic

behavior of an enzyme that acts according to the simple model (1).

(16)

Equation (11) is of the form y = ax/(b + x)

This is the equation of a rectangular hyperbola, just like the saturation equation for the binding of dioxygen to myoglobin.

Equation (11) means that, for an enzyme acting according to the

simple model (1), a plot of V

0

versus [S] will be a rectangular

hyperbola. When enzymes exhibit this kinetic behavior, unless we

find other evidence to the contrary, we assume that they act

according to model (1), and call them Michaelis-Menten enzymes.

(17)

Another form;

For the enzyme catalyzed reaction:

E + S --k1--> ES complex --k3--> E + P ... <--k2--

V= k3*[ES]

Rate of formation of ES = k1 * [E]*[S]

Rate of breakdown of ES = (k2 + k3) * [ES]

(18)

At steady state, the formation and the breakdown are equal. This steady state would only be temporary.

k1 * [E]*[S] = (k2 + k3) * [ES]

rearranging:

[ES] = [E]*[S] / ( (k2 + k3)/(k1))

We can lump these constants to make a new constant called KM = (k2+k3)/k1

[ES] = [E][S]/ K

M

(19)

[ET] = [E] + [ES] (The total amount of enzyme equals the free and that bound to substrate)

Substituting in [ET] - [ES] for [E]

[ES] = ([ET] - [ES]) [S]/ KM

Solving for [ES] leads to [ES] = ([ET] (([S]/ KM)/(1 + [S]/ KM )) Which simplifies to

[ES] = ([ET] *([S]/([S] + KM )

(20)

Multiplying both sides by the kinetic constant k3 gives the velocity of the reaction

v = k3 * [ES] = k3*[ET] *(([S]/([S] + KM )

and substituting Vmax for k3*[ET] leads to the familiar form of the Michaelis Menten Equation

v = Vmax *[S]/([S] + KM )

(21)

L L i i n n e e w w e e a a v v e e r r – – B B u u r r k k p p l l o o t t : :

The Lineweaver–Burk plot (or double reciprocal plot) is a graphical representation of the Lineweaver–Burk equation of enzyme kinetics, described by Hans Lineweaver and Dean Burk in 1934.

D D e e r r i i v v a a t t i i o o n n : :

The plot provides a useful graphical method for analysis of

the Michaelis–Menten equation, as it is difficult to determine

precisely the V

max

of an enzyme-catalysed reaction:

(22)

Taking the reciprocal gives:

(23)

where V is the reaction velocity (the reaction rate), K

m

is

the Michaelis–Menten constant, V

max

is the maximum reaction

velocity, and [S] is the substrate concentration.

(24)

An example of a Lineweaver-Burk plot.

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Use:

The Lineweaver–Burk plot was widely used to determine important

terms in enzyme kinetics, such as K

m

and V

max

, before the wide

availability of powerful computers and non-linear

regression software. The y-intercept of such a graph is equivalent to

the inverse of V

max

; the x-intercept of the graph represents −1/K

m

. It

also gives a quick, visual impression of the different forms

of enzyme inhibition.

(26)

The double reciprocal plot distorts the error structure of the data, and it is therefore unreliable for the determination of enzyme kinetic parameters. Although it is still used for representation of kinetic data, non-linear regression or alternative linear forms of the Michaelis–Menten equation such as the Hanes-Woolf plot or Eadie–Hofstee plot are generally used for the calculation of parameters.

When used for determining the type of enzyme inhibition, the

Lineweaver–Burk plot can distinguish competitive, non-

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ZOOLOGY: SEM- III, PAPER- C7T: FUNDAMENTALS OF BIOCHEMISTRY, UNIT 5: ENZYMES

competitive and uncompetitive inhibitors. Competitive inhibitors

have the same y-intercept as uninhibited enzyme (since V

max

is

unaffected by competitive inhibitors the inverse of V

max

also doesn't

change) but there are different slopes and x-intercepts between the

two data sets. Non-competitive inhibition produces plots with the

same x-intercept as uninhibited enzyme (K

m

is unaffected) but

different slopes and y-intercepts. Uncompetitive inhibition causes

different intercepts on both the y- and x-axes.

(28)

Enzyme Inhibition displayed using Lineweaver-Burk

(29)

T T H H A A N N K K Y Y O O U U