Kinetic parameters
E+S
kass
kdiss
E·S
kcat
P
Kinetic parameters describe and quantify properties of an enzyme
in relation with the reactants of a catalyzed chemical reaction.
We make a clear distinction between microscopic and
macroscopic kinetic parameters.
Microscopic parameters are simply the respective rate
constants of distinct steps in a multi-steps catalytic mechanism.
These are intrinsic properties of the enzyme, thus independent
of concentrations, e.g., kass, kdiss and kcat.
Macroscopic parameters are combinations of microscopic
parameters describing effective properties of the
enzyme/reactants/environment system, usually derived from
kinetic studies of a catalyzed reaction, e.g., Km = (kdiss+kcat)/kass
and Vmax = kcat *ET.
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Kinetic parameters (cont)
Using the simple Michaelis-Menten mechanism, while
kass, kdiss and kcat are microscopic parameters, one can
examine the meaning of the macroscopic parameters:
kcat = TON (1/s); catalytic constant, or
turnover number, representing the rate at
which a bound substrate is converted to
product (number of S converted by one E in
one second). Independent of concentrations.
Vmax = kcat ET (M/s), maximal velocity for the
enzyme saturated with substrate.
Dependent on E concentration.
Km = kdiss+kcat/kass (M), Michaelis constant, the
ratio of the rates ‘off ’ to ‘on’ for the complex
enzyme-substrate; this ‘dynamic equilibrium’
constant is inversely proportional to the
effective affinity between E and S.
E+S
kass
kdiss
E·S
kcat
P
Property
Micro
Macro
Rate of
saturated
enzyme
kcat
Vmax
‘Affinity’
kass, kdiss
Km
Overall
k /K
efficiency cat m
Vmax/Km
Independent of concentrations.
kcat/Km (1/M.s), pseudo-second order rate constant; this represents the true catalytic
efficiency of E in the conversion of S. Independent of concentrations.
Vmax/Km (1/s), the corresponding pseudo-first order rate constant for conversion of low
concentrations of S at given E concentration. Dependent on E concentration.
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Quantitative and experimental issues for kinetic analysis
120
• Sampling properly the substrate concentration
(independent variable) range to achieve evenly
distributed velocities (dependent variable).
v - mM/s
100
• The same applies to linearization methods (e.g.,
in Lineweather-Burk; sampling evenly 1/v by
choosing proper values for 1/S).
80
60
40
20
0
0
50
100
150
200
S0 - mM
• Test the validity of the steady-state assumption
(constant v during measurement) under all
conditions (especially at low [S]).
P
v0=dP/dt
vexp=DP/Dt
• In case a steady state is not sustained during
the measurement, don’t panic, there are still
ways to use the data…
t
• There is a point in time where the true velocity
equals the velocity calculated (while assuming
the measurement was made at steady state).
• At this time, the concentrations of S and P are
close to their respective averages interpolated
between the start and the end of the
measurement (practice).
P
vav= dP/dt
Pav
vexp= DP/Dt
tav
t
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Reversible reactions, the Haldane relation
E+S
k1
k-1
E·S
k2
k-2
P+E
At steady state, we write
k  S  k 2  P
ES  1
E
k2  k 1
and
vnet
K mS 
defining
K mP 
Assoc.
Dissoc.
k2  ES  k2  E  P
 v f  vr 
E  ES
k 1  k2 E  S

k1
ES
one may obtain
k2  k1 E  P

k 2
ES
At equilibrium, vnet = 0,
or
vnet
k2
K mS Peq

 K eq
k 1 Seq
K mP
f
r
Vmax
Vmax
S P P
K mS
Km

S
P
1 S  P
Km Km
Where Vmax and Km can be expressed in terms of the microscopic rate constants.
An important conclusion is that, while one may alter kcat’s and Km’s separately by
site directed mutagenesis of active site residues in an enzyme, they will always
remain linked by the chemical equilibrium constant for the reaction being catalyzed.
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Competitive Inhibition - Summary
E+S
+
I
k3
k1
k-1
E·S
k2
Vmax  k 2ET
P
Km 
S
k-3
E
I
EI
ET = E + ES + EI
KI 
v0
ES


Vmax E  ES  EI
v0
Vmax
KI
E I   k 3
EI  k3
E S
S
Km


E S E I
I
E

Km  S  Km
Km
KI
KI
Vmax  S
V S
V S
v0 
 max
 max
app
I
K m (1  )  S K m  S K m  S
KI
I
  1
E S   k 1  k 2
ES 
k1
1
Kmapp    Km  Km
Linear forms
1 K mapp 1
1

 
v 0 Vmax S0 Vmax
v 0  K mapp 
v0
 Vmax
S0
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Graphic behavior
Rectangular hyperbola
Vmax  S
V S
V S
v0 
 max
 max
app
I
K m (1  )  S K m  S K m  S
KI
1.E-06
v0 - M/s
8.E-07
Vmax = 1 E-6 M/s
Km = 2 E-3 M
KI = 4 E-3 M
6.E-07
4.E-07
I0
I4
2.E-07
I12
I24
0.E+00
0.00
[I] = 0, 4, 12, 24 * E-3 M
Lineweather-Burk
1 K
1
1

 
v 0 Vmax S0 Vmax
0.03
S-M
v 0  K mapp 
1.E+07
0.04
0.05
0.06
v0
 Vmax
S0
1.E-06
I0
I4
I12
I24
I0
8.E-07
I4
I12
v - M/s
1/v - s/M
0.02
Eadie-Hofstee
app
m
8.E+06
0.01
6.E-07
6.E+06
I24
4.E-07
4.E+06
2.E-07
2.E+06
0.E+00
0.0E+00
0.E+00
0
200
400
600
800
1.0E-04
2.0E-04
3.0E-04
4.0E-04
5.0E-04
v/S - 1/s
1/S - 1/M
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Replots
1 K mapp 1
1
1



 slope 
 y int
v 0 Vmax S0 Vmax
S0
yint - s/M
K mapp   K m
K
I
slope 

 m (1  )
Vmax
Vmax
Vmax
KI
y int 
1
Vmax
7
6
slope - s,
5
1
Km
Vmax
4
Vmax 3
2
Km 1

Vmax K I
1
0
0.02
0.04
0.06
[I] - M
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