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Enzyme Kinetics
Michaelis-Menten Theory
Dehaloperoxidase: Multi-functional Enzyme
NC State University
Michaelis-Menton kinetics
The rate of an enzyme catalyzed reaction in which
substrate S is converted into products P depends on
the concentration of the enzyme E even though the
enzyme does not undergo any net change.
k aon
k bcat
E + S ↔ ES → P + E
k
′
aoff
Michaelis-Menton rate equations
ka
kb
E + S ↔ ES → P + E
k
′
a
Steps in the Michaelis-Menton
mechanism
Step 1. Bimolecular formation of the enzyme E and
and substrate S:
E+S
ES
rate of formation of ES = kon[E][S]
Step 2. Unimolecular decomposition of the complex:
ES
E + S rate of decomposition of ES = -koff[ES]
Step 3. Formation of products and release from the
enzyme:
ES
P+E
rate of formation of P = kcat[ES]
The rate law of interest is the formation of the
product in terms of E and S.
The enzyme substrate complex
can be eliminated
The enzyme substrate complex is formed
transiently and can be approximated using the
steady state approximation.
The result of this approximation is
Pseudo-first order
Michaelis-Menton kinetics
In an experiment we know the total enzyme
concentration [E]0 and not the unbound enzyme [E].
The total concentration of enzyme [E]0 = [E] + [ES].
which rearranges to
Pseudo-first order
Michaelis-Menton kinetics
At this point it is convenient to define the Michaelis
constant
and to rearrange the equations as
Michaelis-Menton parameters
The rate of formation of product can be written
where Km is the Michaelis constant and kcat is the
maximum turnover number. We often make the
definitions
which permit us to write the equation as
Limiting conditions of
enzyme reactivity
• Maximal rate: If there is excess substrate present
the rate is limited by the rate at which the ES complex
falls apart. The rate of formation of products is a
maximum and vmax = kcat[E]0is called the maximum velocity.
• Second order regime: If [S] << KM then the rate of formation
of products is d[P]/dt = kcat/Km [E]0[S]. The rate depends
on [S] as well as [E]0.
• A plot of 1/k yields kcat and Km but not the rate constants
kon and koff. The latter rate constants can be obtained
from stopped-flow experiments.
General expression
for reaction velocity
Based on the previous analysis the velocity at an
arbitrary substrate concentration is:
Lineweaver-Burke Plots
• The Michaelis-Menton expression is non-linear.
• The Lineweaver-Burke plot is linearized plot of data.
1 = K M + [S ] = 1 + K M 1
v
[S ]vmax
vmax
vmax [S ]
• This expression has the form of an equation for a line:
y = intercept + slope x
• Such plots are not necessary today with common non-linear
fitting programs.
Transition State Stabilization
The original idea of the enzyme having maximum
complementarity to the TS was put forward by
Linus Pauling in 1946. It wasn't until the early 70's
that the idea was put on a more solid grounding.
As put forward by Lienhard and Wolfenden the idea
is as follows:
Kn‡
kn
E+S
E + S‡
E+P
Ks
ES
Kc‡
Kt
ES‡
kc
E+P
Transition State Stabilization
Defining the equilibrium constants as association constants:
Kn‡ = [S‡]/[S] , Kt = [ES‡]/[E][S‡]
from TS theory: ∆G‡ = -RT ln K‡ and kobs = (kBT/h)e-∆G‡/RT
Thus, kn = (kBT/h)Kn‡ and kc = (kBT/h)Kc‡
where c means catalyzed and n means uncatalyzed.
From the scheme you can see that Ks Kc‡ = Kn‡ Kt
hence Kt/Ks = Kc‡/Kn‡ however, kc/kn = Kc‡/Kn‡
Therefore the observed rate enhancement
kc/kn = Kt/Ks >> 1
Therefore the transition state geometry S‡ must bind
more tightly than the substrate S in its equilibrium geometry!
Transition State Analogs
The transition state stabilization hypothesis was tested by
designing so-called transition state analogs, molecules
which mimick the real TS as closely as possible. One of the
first enzymes examined was proline racemase:
-
N
H
COO
H
N
H
H
COO-
N
COO-
H
The compound on the right is a planar TS state analog.
This molecule was found to be a good inhibitor, with Ki some
two orders of magnitude smaller than Km.
The Role of Entropy
In a seminal paper Page and Jencks showed that the loss
in entropy in going from a bimolecular to a unimolecular
reaction, i.e. E + S <=> ES, could account for as much as
108 of the observed rate enhancement. In other words,
this much free energy would come from the intrinsic
binding energy. The entropy loss arises from the loss of
translational and rotational degrees of freedom when the
substrate is bound. The configurational entropy is:
S = kB lnW
where W is the number of degrees of freedom available to
a molecule.
Inhibition
An inhibitor is any compound that causes a decrease in the
catalytic rate. We will consider non-covalent ligands that
can bind to the enzyme. The general scheme is shown below:
S
E
I
EI
S
kc
ES
I
EIS
ki
E+P
EI + P
I = inhibitor
Inhibition occurs if
ki[EIS] < kc[ES]
Competitive Inhibition
Competitive inhibition results from the direct competition
between the I and S for the substrate binding site. There
is an additional equilibrium constant:
EI
E +I
[E ][I ]
KI =
[EI ]
The velocity under these conditions turns out to be:
[S ]vmax
v=
αK M + [S ]
α=1+
[I ]
KI
Uncompetitive Inhibition
Uncompetitive inhibition arises when I can bind at site
that is not the same as the substrate binding site. There
is an additional equilibrium constant:
EI
E +I
[E ][I ]
KI =
[EI ]
Here the complex IE indicates that the inhibitor does not
bind in the same site as the substrate.
The velocity under these conditions is:
vmax[S ]
v=
K M + α[S ]
α=1+
[I ]
KI
DHP has a natural peroxidase function
Engineered globin peroxidases
Mauk group
Watanabe group
DHP
O
X
-
O
X
+ H2O2
X
Trihalogenated Phenol
(X = I, Br, Cl, F)
DHP
X
X
+ H2 O + X
O
Horseradish Peroxidase
Dihalogenated Quinone
-
Structural model of inhibitor and substrate binding
based on X-ray, NMR and resonance Raman.
Inhibitor
Bound
Resting
State
Substrate
Bound
TXP
4-XP
(Int)
H55
(equilibrium)
5cHS/6cHS
1494 5cHS
ν3
c
ν3
1480 6cHS
b
ν3
1483 6cHS
a
H55
(closed)
6cHS Heme
1494 5cHS
H55
(open)
5cHS Heme
4BP
DHP
TCP
1460
1500
1480
Wavenumber (cm-1)
1460
1500
1480
Wavenumber (cm-1)
1460
1500
1480
Wavenumber (cm-1)
Kinetic analysis showing competitive
inhibition under native conditions
Absorbance
Product/4-BP
Product
2,4,6-TBP
2,4,6-TBP
Wavelength (nm)
Wavelength (nm)
OH
Br
Br
+ H2O2
Br
OH
OH
Br
Br
+
Br
+ H2O2
Br
Kinetic analysis showing
competitive inhibition mechanism
a 1.5
b 1.5
1.0
2,4,6-TCP
0.5
0.0
2,6-DCQ
250
300
350
0.2
400
2,4,6-TCP
0.5
250
300
350
400
Wavelength (nm)
Vmax
30
Vo
20
1/Vo
10
Fe(IV)=O
inhibited
360
Product/4-BP
1.0
d
DHP
DHP Cmp ES
Cmp ES + 20µM 4BP
Cmp ES + 40µM 4BP
Cmp ES + 100µM 4BP
Cmp ES + 200µM 4BP
Cmp ES + 400µM 4BP
0.4
DHP + Substrate + Inhibitor
0.0
400
Wavelength (nm)
c 0.6
Absorbance
DHP + Substrate
Absorbance
Absorbance
TCP
Substrate
Substrate + Inhibitor 1/[S]
0
440
Wavelength (nm)
480
0
5
10
15
Concentration TCP (mM)
20