Enzyme kinetics
d[ P]
v
 k [ES]
2
dt
[ E]  [ E0 ]  [ES]
d[ ES]
 k1[E][S ]  k1[ES]  k2[ES]
dt
How to use independent variable [S] to represent v?
Enzyme kinetics
At this point, an assumption is required to
achieve an analytical solution.
- The rapid equilibrium assumption
Michaelis - Menten Approach.
- The quasi-steady-state assumption.
Briggs and Haldane Approach.
Michaelis - Menten Approach
The rapid equilibrium assumption:
- Assumes a rapid equilibrium between the
enzyme and substrate to form an [ES]
complex.
K1
E+S
K-1
k2
ES  P  E
k1[ E ][S ]  k1[ ES ]
Michaelis - Menten Approach
' can be expressed by
The equilibrium constant K m
the following equation in a dilute system.
K1
E+S
k2
ES  P  E
K-1
k 1 [ E ][ S ]
'
Km 

k1
[ ES ]
Michaelis - Menten Approach
Then rearrange the above equation,
[ E ][ S ]
[ ES ] 
'
Km
Substituting [E] in the above equation with enzyme
mass conservation equation
[ E ]  [ E0 ]  [ ES ]
yields,
([ E0 ]  [ ES ])[ S ]
[ ES ] 
'
Km
Michaelis - Menten Approach
[ES] can be expressed in terms of [S],
[ E0 ][ S ]
[ ES ] 
'  [S ]
Km
Then the rate of production formation v can be
expressed in terms of [S],
k [ E0 ][S ]
Vm [S ]
d [ P]
2
v
 k [ ES ] 

2
'
'
dt
K m  [S ]
K m  [S ]
Where Vm  k 2 [E0 ]
represents the maximum forward rate of reaction (e.g.moles/L-min).
Michaelis - Menten Approach
Vm [ S ]
d [ P]
v

'  [S ]
dt
Km
Michaelis-Menten equation
• Vm changes with initial enzyme concentration.
• It is determined by the rate constant K2 of the
product formation
• But it is not affected by the substrate
concentration.
Michaelis - Menten Approach
'
Km
is often called the Michaelis-Menten constant.
- the prime reminds us that it was derived by assuming rapid
equilibrium in the step of enzyme-substrate complex
formation.
-Low value indicates high affinity of enzyme to the substrate.
k 1
'
Km

k1
- It corresponds to the substrate concentration, giving the halfmaximal reaction velocity.
v
Vm [ S ]
1
Vm 
'  [S ]
2
Km
Re-arrange the above equation,
'  [S ]
Km
1
v

Vm
When
2
Briggs-Haldane Approach
The quasi-steady-state assumption:
- A system (batch reactor) is used in which the
initial substrate concentration [S0] greatly
exceeds the initial enzyme concentration [E0].
since [E0] was small,
d[ES]/dt ≈ 0
- It is shown that in a closed system the quasisteady-state hypothesis is valid after a brief
transient if [S0]>> [E0].
The quasi-steady-state hypothesis is valid after a
brief transient in a batch system if [S0]>> [E0].
Briggs-Haldane Approach
With such assumption, the equation representing the
accumulation of [ES] becomes
d[ ES]
 k1[ E][S ]  k1[ ES]  k2[ES]  0
dt
Solving this algebraic equation yields
k1[ E ][ S ]
[ ES ] 
k1  k2
Briggs-Haldane Approach
Substituting the enzyme mass conservation equation
[ E ]  [ E0 ]  [ ES ]
in the above equation yields
k1 ([ E0 ]  [ ES ])[ S ]
[ ES ] 
k 1  k 2
Using [S] to represent [ES] yields
[ E0 ][ S ]
[ ES ] 
k 1  k 2
 [S ]
k1
Briggs-Haldane Approach
Then the product formation rate becomes
k 2 [ E0 ][ S ]
d [ P]
v
 k 2 [ ES ] 
k 1  k 2
dt
 [S ]
k1
Then,
Vm [S ]
v
K m  [S ]
Where Vm  k [E0 ] same as that for rapid equilibrium assumption.
2
k 1  k 2 when K2 << k-1, K m  K m '  k 1
Km 
k1
k
1
Comparison of the Two Approaches
Michaelis-Menten
Assumption: k1[ E ][S ]  k1[ ES ]
Equation:
Maximum forward
reaction rate:
Constant:
Briggs-Haldane
d[ES]/dt ≈ 0
Vm [ S ]
v
'  [S ]
Km
Vm [S ]
v
K m  [S ]
Vm  k 2 [E0 ]
Vm  k 2 [E0 ]
k 1
'
Km 
k1
k 1  k 2
Km 
k1
when k2 << k-1,
k 1  k 2
Km  Km '
k1
Experimentally Determining Rate
Parameters for
Michaelis-Menten Type Kinetics.
Vm [S ]
v
K m  [S ]
To determine the rate parameters:
- predict a specific enzyme catalysis system.
- Design bioreactor
The determination of Vm and Km are typically
obtained from initial-rate experiments.
-A batch reactor is charged with known initial concentration
of substrate [So] and enzyme [Eo] at specific conditions
such as T, pH, and Ionic Strength.
- The product or substrate concentration is plotted against
time.
- The initial slope of this curve is estimated.
v=(d[P]/dt) t=0 , or = - (d[S]/dt) t=0 .
This value v depends on the values of [E0] and [S0].
- Many such experiments an be used to generate many
pairs of V and [S] data, these data can be plotted as v[S].
There are several methods to obtain the rate parameters.
- Lineweaver-Burk plot (Double-reciprocal plot).
Vm [S ]
v
K m  [S ]
Linearizing it in double-reciprocal form:
Km 1
1
1


v Vm Vm S
- Lineweaver-Burk plot (Double-reciprocal plot).
- a slope equals to Km/Vm
- y-intercept is 1/Vm.
- More often used as it shows the independent variable [S]
and dependent variable v.
-1/v approaches infinity as [S] decreases
- gives undue weight to inaccurate measurement
made at low concentration
- give insufficient weight to more accurate
measurements at high concentration.
Eadie-Hofstee plot
v
v  Vm  K m
[S ]
- the slope is –Km
- y-axis intercept is Vm.
-Can be subject to large error since both coordinates contain
dependent variable v,
but there is less bias on points at low [s].
Hanes-Woolf (Langmuir) plot
[S ] K m
1


[S ]
v
Vm Vm
- the slope is –1/Vm
- y-axis intercept is Km/Vm
- better fit: even weighting of the data
Nonlinear regression
Vm [S ]
v
K m  [S ]
-Better fit
- Initial trial values
Vm
Vm  k 2 [E0 ]
-The unit of Vm is the same as that of a reaction rate
(moles/l-min, g/l-s)
-The dimension of K2 must reflect the units of [E0]
-if the enzyme is highly purified, it may be possible to
express [E0] in mol/l, g/l, then K2 in 1/time.
- if the enzyme is crude, its concentration is in units.
A “unit” is the amount of enzyme that gives a predetermined
amount of catalytic activity under specific conditions.
(Textbook, Bioprocessing Engineering, M. Shuler, p.66-67)
If Vm is mmol/ml-min, [E0] is units/ml, then K2 should be in
mmol/unit-min.
Summary of Simple Saturation
Kinetics
• Michaelis-Menten Approach
• Briggs-Haldane Approach
• Use these two approaches to derive
enzyme catalytic reaction.
• Use experimental data to obtain
parameters of Michaelis-Menten kinetics.
Inhibited Enzyme Kinetics
• Inhibitors may bind to enzyme and reduce
their activity.
• Enzyme inhibition may be reversible or
irreversible.
• For reversible enzyme inhibition, there are
- competitive
- noncompetitive
- uncompetitive
Inhibited Enzyme Kinetics
• Competitive inhibitors (I)
- substrate analogs
- compete with substrate for the active
site of the enzyme
K1
E+S
+
I
KI
EI
K-1
k2
ES  P  E
Inhibited Enzyme Kinetics
• Competitive inhibitors (I)
Assume rapid equilibrium and with the
definitions of
d[ P]
v
 k [ES]
2
dt
k 1 [ E ][ S ]
'
Km 

k1
[ ES ]
[ E ][ I ]
KI 
[ EI ]
[ E0 ]  [ E ]  [ ES ]  [ EI ]
Inhibited Enzyme Kinetics
• Competitive inhibitors (I),
we can obtain,
Vm [ S ]
v
' (1  [ I ] / K )  [ S ]
Km
I
v
Vm [S ]
'
K m, app  [ S ]
'
'
'
'
K m, app  K m (1  [ I ] / K I ) When [I] =0, K m, app  K m
Vm
remains same as that in Michaelis-Menten equation.
Inhibited Enzyme Kinetics
• Noncompetitive inhibitors (I)
- not substrate analogs
- bind on sites other than the active site and
reduce enzyme affinity to the substrate
Km’
E+S
k2
ES  P  E
+
+
I
I
KI
EI+S
Km’
KI
ESI
Inhibited Enzyme Kinetics
• Noncompetitive inhibitors (I)
Assume:
- rapid equilibrium
- same equilibrium constants of inhibitor
binding to E and ES
KI
- same equilibrium constants of substrate
Km’
binding to E and EI
Inhibited Enzyme Kinetics
• Noncompetitive inhibitors (I)
d[ P]
v
 k [ES]
2
dt
k 1 [ E ][ S ] [ EI ][ S ]
'
Km 


k1
[ ES ]
[ ESI ]
[ E ][ I ] [ ES ][ I ]
KI 

[ EI ]
[ ESI ]
[ E0 ]  [ E]  [ ES]  [ EI ]  [ ESI]
Inhibited Enzyme Kinetics
• Noncompetitive inhibitors (I)
we can obtain,
Vm [ S ]
v
'  [S ])
(1  [ I ] / K I )( K m
Vm, app[S ]
v
'  [S ]
Km
Vm, app  Vm /(1  [ I ] / K I ) When [I] =0, Vm, app  Vm
' remains same as that in Michaelis-Menten equation.
Km
Inhibited Enzyme Kinetics
• Uncompetitive inhibitors (I)
- have no affinity for the enzyme itself
- bind only to the ES complex
Assume rapid equilibrium,
Km’
E+S
k2
ES  P  E
+
I
KI
ESI
Inhibited Enzyme Kinetics
• Uncompetitive inhibitors (I)
d[ P]
v
 k [ES]
2
dt
k 1 [ E ][ S ]
'
Km 

k1
[ ES ]
[ ES ][ I ]
KI 
[ ESI ]
[ E0 ]  [ E ]  [ ES ]  [ ESI ]
Inhibited Enzyme Kinetics
• Uncompetitive inhibitors (I)
we can obtain,
v
(Vm /(1  [ I ] / K I ))[ S ]
' /(1  [ I ] / K ))  [S ]
(K m
I
Vm, app[ S ]
v
'
Km
, app  [ S ]
Vm, app  Vm /(1  [ I ] / K I )
'
' /(1  [ I ] / K )
Km

K
, app
m
I
'
'
K m, app / Vm, app  K m / Vm -slope in Lineweaver-Burk Plot
Inhibited Enzyme Kinetics
• Uncompetitive substrate inhibitors
- can cause inhibition at high substrate
concentration
- bind only to the ES complex
Assume rapid equilibrium,
Km’
k2
E+S
ES  P  E
+
S
K SI
ES2
Substrate inhibition
Inhibited Enzyme Kinetics
• Uncompetitive substrate inhibitors (I)
d[ P]
v
 k [ES]
2
dt
k 1 [ E ][ S ]
'
Km 

k1
[ ES ]
[ ES ][ S ]
K SI 
[ ES 2]
[ E0 ]  [ E ]  [ ES ]  [ ES 2]
Inhibited Enzyme Kinetics
• Uncompetitive substrate inhibitors (I)
we can obtain,
v
Vm [S ]
'  [ S ]  [ S ]2 / K
Km
SI
At low substrate concentration [S ]2 / K <<1
SI
Vm [ S ]
v
'  [S ]
Km
Michaelis-Menten Equation
'
At high substrate concentration K m /[S ]  1
v
Vm
1  [ S ] / K SI
Inhibited Enzyme Kinetics
• Uncompetitive substrate inhibitors (I)
The substrate concentration resulting in the
maximum reaction rate can be determined by
setting dv/d[S]=0, [S]max is given by
dv / d [ S ]  d (
Vm [ S ]
'  [ S ]  [ S ]2 / K
Km
SI
' K )1/ 2
[S ]max  ( K m
SI
) / d[S ]  0
dv / d [ S ]  d (
Vm [ S ]
'  [ S ]  [ S ]2 / K
Km
SI
) / d[S ]  0
Vm [ S ](1  2[ S ] / K SI )
(
)0
'
2
'
2
2
K m  [ S ]  [ S ] / K SI
( K m  [ S ]  [ S ] / K SI )
Vm
'  [ S ]  [ S ]2 / K )  V [ S ](1  2[ S ] / K )
Vm ( K m
m
SI
SI
(
)0
'
2
2
( K m  [ S ]  [ S ] / K SI )
'  V [ S ]2 / K
Vm K m
m
SI
'  [ S ]  [ S ]2 / K ) 2
(K m
SI
)0
'  V [ S ]2 / K  0
Vm K m
m
SI
' K )1 / 2
[ S ]max  ( K m
SI
Inhibition Estimation
• Product formation rate v ~ [S]: v has a peak?
If yes, then it’s substrate inhibition.
- get [S]max from the plot of v~[s].
- at low substrate concentration, obtain Vm and Km’
graphicallyor through direct calculation.
- calculate KI through
' K )1/ 2
[S ]max  ( K m
SI
If no, then examine the data with and without inhibitors in
1/v ~ 1/[S] plot (Lineweaver-Burk Plot).
Estimation of inhibited enzyme
kinetics
Estimation of inhibited enzyme
kinetics
• Determine the type of inhibition.
• Determine the parameters for MichaelisMenten equation without inhibition.
• Determine the parameter of KI for inhibited
kinetics.
Factors Affecting Enzyme Kinetics
• pH effects
- on enzymes
- enzymes have ionic groups on their active sites.
- Variation of pH changes the ionic form of the active
sites.
- pH changes the three-Dimensional structure of
enzyme.
- on substrate
- some substrates contain ionic groups
- pH affects the ionic form of substrate
affects the affinity of the substrate to the
enzyme
Factors Affecting Enzyme Kinetics
• Temperature
- on the rate enzyme catalyzed reaction
d[ P]
v
 k [ ES]
2
dt
k2=A*exp(-Ea/R*T)
T
k2
v
- enzyme denaturation
T
Summary of Inhibited Kinetics
• For reversible enzyme inhibition, there are
- competitive
- noncompetitive
- uncompetitive
- substrate inhibition
• Determine parameters for all these types
of inhibition kinetics.