This text covers the different aspects of the main formulae of Enzyme Kinetics. With more emphasis on Michaelis-Menten equation and Lineweaver-Burk plot, this text includes detailed and step-by-step derivations of the major four types of reversible enzyme inhibition (Competitive, Uncompetitive, Non-competitive and Mixed), substrate inhibition, the effect of pH on enzyme activity, thermal denaturation and half-life time, etc. Hope you will get enough information from the text about the major derivations of the formulae involved in enzyme kinetics.

1. 1
SIMPLIFIED AND DETAILED
DERIVATIONS
OF
ENZYME KINETICS (I)
 Michaelis – Menten (MM) Kinetics
 Enzyme Kinetics Plots I – MM Plot & Lineweaver – Burk (LB) Plot
 Enzyme Kinetics Plots II – Eadie-Hofstee (EH) Plot & Hanes-Woolf (HW) Plot
 Competitive Inhibition Kinetics
 Uncompetitive Inhibition Kinetics
 Non-competitive Inhibition Kinetics
 Mixed Inhibition Kinetics
 Applications of MM equation and LB plot
 Enzyme Deactivation Kinetics – Thermal Denaturation and Half-life time

2. 2
Michaelis – Menten (MM) Kinetics
 One of the best-known models of enzyme kinetics.
 Named after German biochemist Leonor Michaelis and Canadian physician Maud Menten in
1913.
 The initial MM model considered only forward reaction and ES dissociation constants:
Equilibrium approximation approach.
 Later, the MM model was modified by British botanist G. E. Briggs and British geneticist J. B. S.
Haldane in 1925, by considering reverse reaction constant additional to the constants
considered in the previous MM model: Quasi-Steady State approximation approach.
 The model is explained based on the MM equation, which describes the rate of enzymatic
reaction.
 MM equation relates the rate of enzymatic reaction to the substrate concentration.
MM Equation:
Keywords (with units in square brackets):
V: Rate of Reaction (Reaction Velocity) [M s-1]
Vmax: Maximum Reaction Velocity [M s-1]
KM: MM Reaction Constant (Briggs-Haldane Model)
( )
[M]
Kd: Dissociation Constant (Michaelis-Menten Model) [M]
kcat: Catalytic Rate Constant [s-1]
[E]: Enzyme Concentration (taking part in the reaction) [M]
[E]0: Total (or initial) Enzyme Concentration [M]
[S]: Substrate Concentration [M]
[ES]: Concentration of Enzyme-Substrate Complex [M]
[P]: Product Concentration [M]
k1: Rate of formation of ES complex (forward reaction) [s-1]
k2: Rate of dissociation of ES complex into P and E (product formation reaction) [s-1]
k-1: Rate of dissociation of ES complex into E and S (reverse reaction) [s-1]
↔ : To be read as
[I]: Inhibitor Concentration [M]
kI (or ki): Rate of formation of EI (or ESI) complex [s-1]
k-I (or k-i): Rate of dissociation of EI (or ESI) complex into E and I (or ES and I) [s-1]
Standard equation for a reaction in equilibrium like ↔ is given by
k1
k-1

3. 3
Derivation: MM Equation: The Quasi-Steady State approximation approach
Let us consider a simple single-enzyme reaction.
↔ →
Formation of ES complex: (1)
Dissociation of ES complex: ( ) (2)
At steady state, (1) = (2)
 ( )

( )
(3)
Total enzyme concentration,
 (4)
Substitute (4) in (3)

( )

 ( )
 (5)
We know that, (6a)
And (6b)
Substitute (6a) and (6b) in (5)

 Michaelis-Menten Equation

4. 4
Michaelis – Menten (MM) Plot
 X-axis: [S] (M)
 Y-axis: V (M s-1)
 The graph is non-linear and reaches saturation at increasing [S]. The saturation indicates that
the enzyme model reached its maximum velocity, Vmax.
 If a perpendicular is drawn from Y-axis from a point where x=0, y= Vmax/2, then the line cuts
the curve at a point.
 If a perpendicular is drawn from the point to the X-axis, it cuts the axis at a point where x=KM,
y=0.
 This indicates that KM (or MM constant) is defined as the substrate concentration required
by a particular enzyme to reach half the maximum reaction velocity in optimum
conditions. Thus, KM is an intrinsic property of an enzyme. Unit: M
 What does KM indicate? The value of KM indicates the association strength or the affinity
between E and S in an ES complex. If more the KM, more will be the affinity between E
and S, which in turn slows down the rate of product formation, which makes the enzyme
less efficient. Thus for an efficient enzyme, the KM should be low.
 That is, when KM = [S],

5. 5
Lineweaver – Burk (LB) Plot (Double Reciprocal Plot)
 One major limitation of MM Plot is the difficulty to calculate the precise values of KM and Vmax
because the plot is a curve and not linear.
 To avoid such complications, LB equation was formulated and an effective graphical method
was developed by American physical chemist Hans Lineweaver and American biochemist
Dean Burk in 1934.
 The equation is obtained by simple reciprocation of the MM equation.
That is,
Michaelis-Menten Equation (1)
Taking the reciprocal of (1)

 Lineweaver-Burk Equation (2)
We can express (2) as the standard line equation, where y = 1/V, m = KM/Vmax, x = 1/[S]
and c = 1/Vmax. {y: Y-axis, m: slope, x: X-axis, c: Y-intercept}
 The line cuts Y-axis at x=0, y=1/Vmax, from which the precise value of Vmax can be calculated
 By knowing the value of Vmax, the value of KM can be calculated by using the slope formula,
.
 More easily, KM can be easily calculated by extrapolating the line towards the negative X (X’)
axis. The line cuts the X’-axis at x= (-1/KM), y=0, from which the precise value of KM can be
calculated.

6. 6
Eadie-Hofstee (EH) Plot
 The equation for this plot is obtained by multiplying the L.H.S. and R.H.S. of LB equation with
Vmax.
 The plot helps us in the rapid identification of KM and Vmax.
That is,
Lineweaver-Burk Equation (1)
Multiply with Vmax


 Eadie-Hofstee Equation (2)
We can express (2) as the standard line equation, where y = V, m = -KM, x = V/[S] and c =
Vmax. {y: Y-axis, m: slope, x: X-axis, c: Y-intercept}
 The line cuts Y-axis at x=0, y=Vmax, from which we get the value for Vmax.
 The value of KM can be calculated by using the slope formula, .
 The value of KM can be also calculated if Vmax is known. The line cuts X-axis at x=Vmax/KM, y=0.

7. 7
Hanes-Woolf (HW) Plot
 The equation for this plot is obtained by multiplying the L.H.S. and R.H.S. of LB equation with
[S].
 The plot helps us in determining KM, Vmax and Vmax/KM parameters.
That is,
Lineweaver-Burk Equation (1)
Multiply with [S]


 Hanes-Woolf Equation (2)
We can express (2) as the standard line equation, where y = [S]/V, m = 1/ Vmax, x = [S]
and c = KM/Vmax. {y: Y-axis, m: slope, x: X-axis, c: Y-intercept}
 The line cuts X-axis at x=-KM, y=0 from which we get the value for KM.
 The value of Vmax can be calculated by using the slope formula, .
 The value of Vmax can be also calculated if KM is known. The line cuts Y-axis at x=0, y=KM/Vmax.

8. 8
Enzyme Inhibition Kinetics
 2 main categories of enzyme inhibition:
i. Reversible Inhibition – the action of the inhibitor on the enzyme is temporary and can
be reversed.
ii. Irreversible Inhibition – the action of the inhibitor on the enzyme is permanent and
cause non-functional enzymes [discussed in the second part of this text: DETAILED
DERIVATIONS OF ENZYME KINETICS (II)].
(i) Reversible Inhibition
Mainly of 4 types:
a. Competitive Inhibition – inhibitor binds to the active site of free enzyme and makes it
inactive to catalyse substrate into product. Formation of EI complex.
b. Uncompetitive Inhibition – inhibitor binds to the allosteric site of ES complex and makes it
inactive. Formation of ESI complex.
c. Non-competitive Inhibition – inhibitor binds to both free enzyme as well as ES complex at the
same rate (KI), but has some degree of substrate binding. Most effective inhibition. Formation
of EI and ESI complexes.
d. Mixed Inhibition - inhibitor binds to both free enzyme as well as ES complex, but at different
rates (Ki and KI). Formation of EI and ESI complexes.
Competitive Inhibition Kinetics
Derivation: MM and LB Equations for Competitive Inhibition
Let us consider a single-enzyme – inhibitor model.
↔ →
↔
Formation of ES complex: (1)
Dissociation of ES complex: ( ) (2)
At steady state, (1) = (2)
 ( )

( )
(3)
Total enzyme concentration,
 (4)
We need all the parameters in terms of the variables of the original MM equation. Therefore, we need
to express [EI] in terms of [E], [I] and kI.
We can express (5)

9. 9
Substitute (5) in (4)


 ( )

( )
(6)
Substitute (6) in (3)

(
( )
)
( )
( )
 ( )
 (( ) )

( )
 Michaelis-Menten Equation for Competitive Inhibition
 Lineweaver-Burk Equation for Competitive Inhibition
* Salient Features
 Vmax remains unaltered, whereas KM increases.
 KM is increased to K’M. ( )

10. 10
Uncompetitive Inhibition Kinetics
Derivation: MM and LB Equations for Uncompetitive Inhibition
Let us consider a single-enzyme – inhibitor model.
↔ →
↔
Formation of ES complex: (1)
Dissociation of ES complex: ( ) (2)
At steady state, (1) = (2)
 ( )

( )
(3)
Total enzyme concentration,
 (4)
We need all the parameters in terms of the variables of the original MM equation. Therefore, we need
to express [EI] in terms of [E], [I] and kI.
We can express (5)
Substitute (5) in (4)

 ( ) (6)
Substitute (6) in (3)

( ( ) )
 ( )
 ( ( ) )

( )

( )

11. 11

( )
(7)
To reach an equation close to the standard line equation (for LB Plot), the X-axis should be changed to
[S]. For this, divide the numerator and denominator of (5) by( ).

( )
( ( ) )
( )
( )
( )
 Michaelis-Menten Equation for Uncompetitive Inhibition
 Lineweaver-Burk Equation for Uncompetitive Inhibition
* Salient Features
 Both KM and Vmax decreases proportionally.
 Why the LB plot is parallel for uncompetitive inhibition?
This is because of similar slope. That is:
( )
( )
 KM is decreased to K’M.
( )
 Vmax is decreased to V’max.
( )

12. 12
Non-competitive Inhibition Kinetics
Derivation: MM and LB Equations for Non-competitive Inhibition
Let us consider a single-enzyme – inhibitor model.
↔ →
↔
↔
Formation of ES complex: (1)
Dissociation of ES complex: ( ) (2)
At steady state, (1) = (2)
 ( )

( )
(3)
Total enzyme concentration,
 (4)
We need all the parameters in terms of the variables of the original MM equation. Therefore, we need
to express [EI] in terms of [E], [I] and kI.
We can express (5a)
And (5b)
Substitute (5a) and (5b) in (4)


 ( ) ( )

( )
( )
(6)
Substitute (6) in (3)

(
( )
( )
)
( ( ) )
( )

13. 13
 ( ) ( )
 (( ) ( ) )
 ( ) ( )

( ) ( )
( )

( )
 Michaelis-Menten Equation for Non-competitive Inhibition
 Lineweaver-Burk Equation for Non-competitive Inhibition
* Salient Features
 KM remains unaltered, whereas Vmax decreases.
 Vmax is decreased to V’max.
( )

14. 14
Mixed Inhibition Kinetics
Derivation: MM and LB Equations for Mixed Inhibition
Let us consider a single-enzyme – inhibitor model.
↔ →
↔
↔
Formation of ES complex: (1)
Dissociation of ES complex: ( ) (2)
At steady state, (1) = (2)
 ( )

( )
(3)
Total enzyme concentration,
 (4)
We need all the parameters in terms of the variables of the original MM equation. Therefore, we need
to express [EI] in terms of [E], [I] and kI.
We can express (5a)
And (5b)
Substitute (5a) and (5b) in (4)


 ( ) ( )

( )
( )
(6)
Substitute (6) in (3)

(
( )
( )
)
( ( ) )
( )
 ( ) ( )

15. 15
 (( ) ( ) )

( ) ( )

( ) ( )
(7)
To reach an equation close to the standard line equation (for LB Plot), the X-axis should be changed to
[S]. For this, divide the numerator and denominator of (5) by( ).

( )
( )
( )
 Michaelis-Menten Equation for Mixed Inhibition
 Lineweaver-Burk Equation for Mixed Inhibition
* Salient Features
 Both KM and Vmax changes non-proportionately. The degree of alteration of KM is decided
by kI and ki.
 Vmax is decreased to V’max.
( )
 KM is increased or decreased to K’’M.
( )
( )

16. 16
* Other notations
 Some texts represent the common term ( ) as .
 Therefore, in competitive inhibition, 
 In uncompetitive inhibition,

and

 In non-competitive inhibition,

 In mixed inhibition, we can express ( ) as  and ( ) as ’. Therefore,

and


.
* Summary

17. 17
Applications of MM equation and LB plot
 As we have discussed, MM equation and LB plot are the soul and heart of enzyme kinetics. Any
enzymatic reaction is incomplete without expressing the kinetic parameters like KM, Vmax, etc.
 The derivations can be used to answer many complicated processes, like the effect of pH in
enzyme activity, thermal deactivation, etc.
 Here we will be seeing the effect of pH on enzyme activity by deriving an equation using the
MM derivation process.
Derivation: MM and LB Equations for the substrate inhibition
At very high substrate concentrations, the substrate can inhibit the enzyme in an uncompetitive
manner that is by binding to the allosteric site of the ES complex.
For deriving the expression, let’s consider a simple uncompetitive enzyme inhibition model.
( )
( )
Michaelis-Menten Equation for Uncompetitive Inhibition
When substrate concentration is high, it plays as the inhibitor. Therefore, [I]=[S], kI=kS and due to high
[S], KM becomes negligible  KM0

( )
( )

( )
MM Equation for Substrate Inhibition
 LB Equation for Substrate Inhibition
Derivation: MM and LB Equations for the effect of pH on enzyme activity
Let us consider a single-enzyme – inhibitor model. For making the reaction simpler, let us consider the
enzyme to be electrically stable and we can now express the enzyme as EH.
↔ →

18. 18
In acidic conditions, enzymes tend to gain extra H+ ion(s). That is,
↔
Similarly, in basic conditions, enzymes tend to loose H+ ion(s). That is,
↔
Formation of EHS complex: (1)
Dissociation of EHS complex: ( ) (2)
At steady state, (1) = (2)
 ( )

( )
(3)
Total enzyme concentration,
 (4)
We need all the parameters in terms of the variables of the original MM equation. Therefore, we need
to express [EH2+] and [E-] in terms of [EH], [H+], ka and kb.
We can express (5a)
And (5b)
Substitute (5a) and (5b) in (4)

 ( )

(
[ ]
[ ]
)
(6)
Substitute (6) in (3)

(
(
[ ]
[ ]
)
)
( )
(
[ ]
[ ]
)
 ( )
 (( ) )

(
[ ]
[ ]
)

19. 19

(
[ ]
[ ]
)
(7)
 Michaelis-Menten Equation for the effect of pH on enzyme activity
 Lineweaver-Burk Equation for the effect of pH on enzyme activity
( ) , this increases KM of the enzyme.
Considering the equation for pH, that is ( ), we can express it as
( ). With this expression, we can express KM’’’ as
(
( )
( ))

20. 20
Enzyme Deactivation Kinetics
Derivation: Thermal denaturation (or deactivation) of enzyme and calculation for enzyme half-
life time
 Enzymes are proteins. Like every other protein, the 3-D structure of the enzyme determines
and facilitates its function.
 The main bond involved in structural conformation of enzymes is H-bond.
 Hydrogen bond is a strong interaction but is thermally labile.
 As the temperature increases, the bond stability gradually destabilises and the 3-D structure of
the protein is disrupted.
 Thermal deactivation is a permanent deactivation of the enzyme. The temperature at which
complete denaturation of the enzyme (or any other proteins) occurs is termed as the
denaturation temperature of that enzyme.
Consider a system involving an “ideal” enzyme E which is subjected to its denaturation temperature.
The enzyme will get denatured to form the denatured enzyme (ED) at a denaturation rate of kD.
→
The rate of thermal denaturation of the enzyme can be expressed as, .
By removing proportionality,
By rearranging and integrating, ∫ ∫

 (1a)
 (1)
By taking the exponent of L.H.S. and R.H.S. of (1b) that is,
 Thermal Denaturation Kinetics of Enzymes
When the enzyme reaches its half-life, [E]=[E]0/2 and t=t1/2. (2)
Reconsidering (1a) and substituting (2)



 Half-life time of an enzyme
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