Michaelis-Menten equation

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Transcript Michaelis-Menten equation

KAPITOLA 3
Enzymová katalýza I
• katylytická aktivita enzymů
• interakce enzym - substrát
• koenzymy
• vazebné místo pro substrát
• fyzikálně chemické vlastnosti
• nomenklatura enzymů
An imaginary enzyme (stickase) designed to catalyze the breaking of a metal stick.
(a) To break, the stick must first be bent (the transition state). In the stickase, magnetic interactions take
the place of weak-bonding interactions between enzyme and substrate.
(b) An enzyme with a magnet-lined pocket complementary in structure to the stick (the substrate) will
stabilize this substrate . Bending will be impeded by the magnetic attraction between stick and
stickase.
(c) An enzyme complementary to the reaction transition state will help to destabilize the stick,
resulting in catalysis of the reaction. The magnetic interactions provide energy that compensates
for the increase in free energy required to bend the stick. Reaction coordinate diagrams show the
energetic consequences of complementarity to substrate versus complementarity to transition
state. The term GM represents the energy contribued by the magnetic interactions between the
stick and stickase. When the enzyme is complementary to the substrate, as in (b), the ES
complex is more stable and has less free energy in the ground state than substrate alone. The
result is an increase in the activation energy. For simplicity, the EP complexes are not shown.
Free energy, G
Reaction coordinate
The role of binding energy in catalysis.
To lower the activation energy for a reaction, the system must acquire an amount of energy
equivalent to the amount by which DG# is lowered. This energy comes largery from binding
energy ( GB) contribued by formation of weak noncovalent interactions between substrate
and enzyme in the transition state. The role of  GB is analogous to that of  GM.
Many organic reactions are promoted by proton donors (general acids) or proton acceptors
(general bases). The active sites of some enzymes contain amino acid functional groups, such as
those shown here, that can participe in the catalytic process as proton donors or proton acceptors.
Enzymes
What?
Proteins which catalyse biological reactions
Properties?
Highly specific
Unchanged after reaction
Can accelerate reactions >106 fold
Turnover
Cannot alter reaction equilibria
Conversion of substrate to product:
S
P
2 factors determine the appearance of product:
i. Thermodynamic
ii. Kinetic
Enzyme Classification
Reaction profile for a simple reaction
Rawn Biochemistry, International edition
Reaction profile for an enzyme-catalysed reaction
Uncatalysed
Enzyme catalysed
Rawn Biochemistry, International edition
Enzyme kinetics
Change in [S] and [P] with time
1200
Concentration (micromolar)
1000
800
S
600
400
P
200
0
0
50
100
150
200
Time (secs)
250
300
350
Impact of changing enzyme concentration
Rate versus [enzyme]
3
Rate (v) micromolar per sec
2.5
2
1.5
1
0.5
0
0
1
2
3
4
[Enzyme] micromolar
5
6
Impact of changing substrate concentration
Rate ve rsus [substrate]
1
Rate (v) micomolar per sec
0.8
0.6
0.4
0.2
0
0
200
400
600
800
[Substrate] micromolar
1000
1200
Derivation of the Michaelis-Menten equation
k1
k2
E+P
ES
E +S
k -1
Rate of reaction:
v  k 2 [ES ]
Equation (1)
During the steady-state, [ES] is constant
i.e. the rate of formation = rate of breakdown
k 1[E ][S]  k
Therefore:
 1[ES
]  k 2[ES ]
k 1[E ][S]
[ES ] 
k  1 k2
k 1[E ][S]
[ES ] 
k  1 k2
If we introduce a constant KM defined as
This equation becomes
KM 
[E ][S]
[ES ] 
KM
k
1k 2
k1
Equation (2)
The total concentration of enzyme introduced at the start of the reaction is
still present, it is either bound to substrate or it is not, i.e.
[E0] = [E] + [ES]
Or rearranged:
[E] = [E0] - [ES]
Substituting this into Equation (2) gives us:
([E 0 ]  [ES ])[S ]
[ES ] 
KM
[ES] =
([E0] - [ES])[S]
KM
Opening the bracket on the right-hand side gives:
[ES] =
[E0][S] - [ES][S]
KM
Therefore:
KM [ES] = [E0][S] - [ES][S]
Collect [ES]:
KM [ES] + [S][ES] = [E0][S]
(KM + [S])[ES] = [E0][S]
Therefore:
[ES] =
[E0][S]
[S] + K M
[ES] =
[E0][S]
[S] + K M
v  k 2 [ES ]
Substituting this back into equation (1):
Gives us:
v = k2
[E0][S]
Equation (3)
[S] + KM
The maximum rate for the enzyme (Vmax) will be achieved when all of the
Enzyme is saturated with substrate, i.e.
[ES] = [E0]
Putting this again into equation (1) we get:
Vmax = k2 [E0]
Notice that the term k2[E0] features in equation (3), which can now be written
v=
Vmax [S]
KM + [S]
Michaelis-Menten
equation
What does the KM actually tell us?
Consider the case when the reaction rate (v) is half of the maximum
rate (Vmax). Under these conditions the MM equation becomes:
Vmax
2
=
Vmax [S]
KM + [S]
This leaves us with:
2[S] = KM + [S]
Which simplifies to:
[S] = KM
KM is the substrate concentration at which the reaction
proceeds at half of the maximum rate (Vmax)
Plot of rate (V) against substrate concentration ([S])
V max
Initial
rate, V
V max/
2
KM
Substrate concentration, [S]
v=
Vmax [S]
KM + [S]
Taking the reciprocal of both sides can give
1
v
=
KM
.
Vmax
This is in the general form
1
[S]
1
+
Vmax
y=mx+c
i.e. a plot of 1/v against 1/[S] will give straight line
This is a Lineweaver-Burk plot
Lineweaver-Burk plot
1/V
-1/K
M
slope =
KM/ V
max
1/V
max
1/[S]
KAPITOLA 4
Enzymová katalýza II
• kinetika reakcí katalyzovaných enzymy
• analýza reakční rychlosti
• inhibice a regulace katalytické aktivity
• metody studia
Vm a x
2

Vm a x [ S ]
Km  [ S ]
The Michaelis-Menten equation
Dependence of initial velocity on substrate concentration, showing the kinetic parameters that define
the limits of the curve at hight and low [S]. At low [S], Km>>[S], and the [S] term in the denominator of
the Michaelis-Menten equation becomes insignificant; the equation simplifies to V0 = Vmax[S]/Km,
and V0 exhibits a linear dependence on [S], as observed. At high [S], where [S]>>Km, the Km term in
the denominator of the Michaelis-Menten equation becomes insignificant, and the equation simplifies
to V0=Vmax; this is consistent with the plateau observed at high [S]. The Michaelis-Menten equation is
therefore consistent with the observed dependence of V0 on [S], with the shape of the curve defined by
the terms Vmax/Km at low [S] and Vmax at high [S].
1  Km 1  1
V0 Vm ax [ S ] Vm ax
The Lineweaver-Burk equation
A double-reciprocal, or Lineweaver-Burk, plot.
Common mechanisms for enzyme-catalyzed bisubstrate reactions.
(a) the enzyme and both substrates come together to form a ternary complex. In ordered binding,
substrate 1 must be bound before substrate 2 can bind productively.
(b) an enzyme-substrate complex forms, a product leaves the complex, the altered enzyme forms a
second complex with another substrate molecule, and the second product leaves, regenerating the
enzyme. Substrate 1 may transfer a functional group to the enzyme (forming E´), which is
subsequently transferred to substrate 2. This is a ping-pong or double-displacement mechanism.
A
B
A
shows a set of double-reciprocal plots obtained in the absence of the inhibitor and with two
different concentrations of a competitive inhibitor. Increasing inhibitor concentration [I] results in the
production of a family of lines with a common intercept on the 1/V0 axis but with different slopes.
Because the intercept on the 1/V0 axis is equal to 1/Vmax, we can see that Vmax is unchanged by a
presence of a competitive inhibitor. That is, regardless of the concentration of a competitive inhibitor,
there is always some high substrate concentration that will displace the inhibitor from the enzyme´s
active site.
In noncompetitive inhibition, similar plots of the rate data give the family of lines shown in B, having
a common intercept on the 1/[S] axis. This indicates that Km for the substrate is not altered by a
noncompetitive inhibitor, but Vmax decreases.
Three types of reversible inhibition.
Competitive inhibitors bind to the enzyme´s
active site.
Noncompetitive inhibitors generally bind at a
separate site.
Uncompetitive inhibitors also bind at a
separate site, but they bind only to the ES
complex. KI is the equilibrum constant for
inhibitor binding.
Some well-studied examples of enzyme modification reactions.
Allosteric Enzymes
Obrázek Lehninger
3 obr grafy z PDF
Lehninger str 228