Enzyme kinetics
Why study the rate of enzyme catalyzed reactions?
• Study of reaction rates is an important tool to
investigate the chemical mechanism of catalysis
• Kinetic studies provide information on substrate and
product affinity to the enzyme
• Knowledge of the dynamic properties of enzyme
catalysis is a prerequisite for the design of inhibitors
(drugs) directed against a certain enzyme
Chemical reaction kinetics
Reaction order of a chemical process corresponds to the molecularity
of the reaction, which is the number of molecules that must collide
simultaneously to generate a product:
aA + bB + cC
product(s)
The velocity for such a process is given by:
v = k [A]a • [B]b • [C]c
Where v is the velocity of the reaction and k the rate constant of the
reaction; the reaction order is the sum of the exponentials in the rate
equation.
First and Second-order reactions
For a first-order process:
A
P
v =
The reaction velocity, v, is given by:
dA
dP
=
= k [A]
dt
dt
The reaction velocity for such a first-order reaction is proportional to the
concentration of A. Such a reaction is also called a unimolecular reaction.
A second-order reaction can involve the reaction of two identical or different
substrate molecules:
A + A
product(s)
or
A + B
product(s)
The velocity of the reaction is then:
dA = k [A]2
v = dP = dt
dt
or
v =-
dA = - dB = k [A] [B]
dt
dt
The dimension of the rate constant
The dimension of k depends on the reaction
order. This is due to the fact that the velocity of a
reaction is measured in mol product formed per
time unit (e.g. sec).
Therefore the unit of the rate constant of a firstorder reaction is sec-1 and that of a second-order
reaction is M-1 sec-1.
Determination of k for a first-order reaction
For a first-order process:
A
P
As before, the reaction velocity, v, is given by:
v =
dA = k [A]
dt
ln [A]
ln [A]t=0
slope = -k
dA = dln [A] = - k dt
[A]
Integration with [A]t=0 to [A]t yields:
[A]t
t
∫ dln [A] = - k ∫ dt
[A]t=0
t
ln [A]t = ln [A]t=0 - kt
t=0
[A]t = [A]t=0 e-kt
Example: radioactive decay
Relationship of half-life and rate constant
The half-life (t1/2) is a constant (dimension: time). Radioactive decay is usually
expressed in half-life rather than the first-order rate constant:
[A]t=0
The half-life is defined as the time required for [A]t=0 to decrease to
2
[A]t=0
= [A]t=0 e-kt1/2
2
t 1/2 =
ln2
k
Rate constant for a second-order reaction
Consider the following reaction: A + A
v=-
dA = k [A]2
dt
[A]t
∫-
[A]t=0
product(s)
1
[A]t
t
dA
[A]2
= k
∫ dt
t=0
slope = k
1
[A]t=0
1
= 1
+ kt
[A]t
[A]t=0
The half-life for such a reaction is:
t1/2 =
1
•
[A]t=0
1
k
t
This type of plot can be used to
distinguish between a unimolecular
and bimolcular reaction involving A
Note: half-life depends on [A]t=0
The enzyme-substrate complex
Adrian Brown
Brown and Henri investigated the
substrate-dependence of an enzymecatalyzed reaction and found that the
reaction reached a maximum velocity at
high substrate concentrations (Vmax)
Brown and Henri’s conclusion:
„The enzyme works by forming a complex (like a lock and a key)
with the susbtrate and acting on it for a finite period of time.“
binding
E + S
enzyme
+
substrate
reaction
E-S
dissociation
E-P
enzyme-substrate enzyme-product
complex
complex
E + P
enzyme
+
product
Schematic representation of an enzyme-catalyzed reaction
Kinetics of enzyme reactions
Recall: Brown observed that the rate at which sucrose is degraded by
invertase shows saturation behavior, that is at high sucrose concentrations the
rate becomes independent of the sucrose concentration („zero-order“ reaction
with respect to sucrose). It was concluded that an enzyme-catalyzed reaction
proceeds in two steps:
[E] + [S]
k1
[ES]
k2
[P] + [E]
k-1
When [S] >> [E] then all of the enzyme is in the complex [ES] and the
formation of product is given by:
V =
d[P] = k [ES]
2
dt
The concentration of [ES] is a complex function depending on the individual
rate constants k1, k-1 and k2:
d[ES]
dt
= k1 [E] [S] - k-1 [ES] - k2 [ES]
Two approaches
1. Michaelis-Menten kinetic (1913)
(„rapid equilibrium“ assumption)
2. Briggs-Haldane kinetic (1925)
(„steady-state“ assumption)
Title page of
Michaelis &
Menten’s original
paper in
“Biochemische
Zeitschrift” in
1913
Leonor Michaelis
(1875-1940)
Maud L. Menten
(1879-1960)
The Michaelis-Menten approach
k1
[E] + [S]
[ES]
k2
[P] + [E]
k-1
Assumption: k-1 >> k2 i.e. the equilibrium of [E], [S] and [ES] is not
affected by k2:
KS = dissociation constant
k-1
[E] [S]
KS =
k1
=
[ES]
[ES] = „Michaelis-Menten“ complex
Since we assume equilibrium it follows:
[E] [S] k1 = [ES] k-1 solving for [E] =
In addition we know that:
k-1
k1
[ES]
[S]
(1)
[E]total = [E] + [ES] (2)
This relationship is called the „enzyme conservation equation“
The Michaelis-Menten approach
[E] =
k-1
k1
[ES]
[S]
(1)
[E]total = [E] + [ES] (2)
Solving equation (2) for [E] and substituting [E] in equation (1):
[E]total = [ES] (1 +
k-1
)
(3)
k1 [S]
We also know that the velocity of the reaction equals:
v = k2 [ES]
(4)
Solving equation (3) and (4) for [ES] and then substituting [ES] in
equation (3) with [ES] = v / k2 then yields:
v=
k2
(1 +
[E]total
k-1
=
k2 [E]total [S]
k-1
[S] +
k1
)
k1 [S]
We define k-1/ k1 as KM, the Michaelis-Menten constant and the
maximal velocity as vmax = k2 [E]total
This simplifies the above equation to:
v =
vmax
[S]
[S] + KM
if [S] >> KM then v = vmax
vmax
if [S] = KM then v =
2
Therefore KM can be viewed as the substrate concentration with halfmaximal velocity (dimension M, typically mM to nM)
Michaelis-Menten plot
v
Linear plot of substrate concentration versus velocity
yields a hyperbolic relationship:
vmax
vmax
1st order
zero order
2
KM
[S]
The Briggs-Haldane approach
[E] + [S]
k1
k-1
Assumption: k-1 ~ k2 i.e.
during substrate turnover the
concentration of [ES] is
constant („steady-state“
assumption). The assumption is
less restrictive than the „rapidequilibrium“ assumption by
Michaelis-Menten.
d[ES]
=0
dt
[ES]
k2
[P] + [E]
The Briggs-Haldane approach
d[ES]
= 0 = k1 [E] [S] - k-1 [ES] - k2 [ES]
dt
k1 [E] [S] = k-1 [ES] + k2 [ES]
We also know that:
[E]total = [E] + [ES], solving this equation
For [E] and substituting in the „steady-state“ equation yields:
k1 ([E]total - [ES]) [S] = k-1 [ES] + k2 [ES]
k1 [E]total [S] - k1 [ES] [S] = k-1 [ES] + k2 [ES]
k1 [E]total [S] = (k-1 + k2) [ES] + k1 [ES] [S]
: k1
[E]total [S] =
(k-1 + k2)
[ES] + [ES] [S]
k1
Solving this equation for [ES] yields:
[ES] =
[E]total [S]
(k-1 + k2)
k1
[ES] =
[E]total [S]
KM =
(k-1 + k2)
+ [S]
and with v = k2 [ES]
KM + [S]
k2 [E]total [S]
v =
KM + [S]
Same equation!
and with vmax = k2 [E]total
v =
vmax [S]
KM + [S]
k1
Michaelis-Menten vs. Briggs-Haldane
Although both approaches yield the
same basic equation for the velocity of
an enzyme catalysed reaction, the
meaning of the Michaelis-Menten
parameter, KM, differs:
In the „rapid-equilibrium“approach by
Michaelis-Menten KM is equivalent to
the true dissociation constant Ks
In the „steady-state“ approach by
Briggs-Haldane the rate of the
„chemical step“, k2, is part of the KM
and hence it is not equivalent to the
dissociation constant.
v =
KM =
KM =
vmax [S]
KM + [S]
k-1
k1
(k-1 + k2)
k1
Analysis of kinetic data - the Lineweaver-Burk plot
We have seen that at high substrate concentration, the initial velocity of the
reaction approaches the maximal velocity vmax asymptotically. In practice this
asymptotic value is difficult to determine in a direct (hyperbolic) plot of velocity
vs. substrate concentration. Therefore Hans Lineweaver and Dean Burk have
linearized the Michaelis-Menten equation to:
1
=
v
KM
vmax
( ) [S]1 + v1
max
In this double-reciprocal
plot 1/v is plotted vs 1/
[S]. The y-axis intercept
yields 1/vmax whereas the
x-axis intercept yields 1/KM. The slope of the
straight line is equivalent
to KM/vmax.
Catalytic efficiency of enzymes
For an enzyme that obeys the Michaelis-Menten kinetics: vmax = k2 [E]total
k2 is also called kcat or turnover number because it reflects directly the
commitment to catalysis and therefore we can also write:
vmax = kcat [E]total or
kcat =
vmax
[E]total
When [S] << KM then [E] ~ [E]total
v =
k2 [E]total [S]
KM + [S]
kcat
v =
KM
[E] [S]
kcat/KM is a 2nd order rate constant (M-1 sec-1)
and as such reflects the efficiency of „ E to
react with S“
These consideration also allow us to determine how fast an enzyme catalyzed
reaction can proceed:
Maximal velocity of an enzyme-catalyzed reaction
kcat
KM
k1 k2
k2
=
=
k-1 + k2
KM
In the case of efficient catalysis k2 >> k-1
This means that the Michaelis-Menten complex decays rapidly to the
product(s) and the back-reaction to free enzyme and substrate are much
slower („enzyme is committed to catalysis“). In such a case the equation
above can be rewritten as:
kcat
= k1
KM
Since k1 is the rate at which the Michaelis-Menten
complex forms, the limiting value for this rate constant is
the rate of encounter of enzyme and substrate, i.e. the rate
limited by diffusion. This rate is of the order 108-109 M-1
sec-1. Hence enzymes that operate in this range have
achieved maximal velocity (catalase: 4 x 108 M-1 sec-1)
Determination of rates
• From steady-state measurements two enzyme parameters are obtained:
1)
The Michaelis-Menten Parameter (KM) which may be
equivalent to the enzyme-substrate dissociation constant
2)
kcat (turnover number) which may be a microscopic rate
constant or a combination of several
• To observe individual rates the approach to steady-state needs to be
observed („pre-steady-state kinetics“); the rates are typically on the order
of 1- 10-7 sec!
The Continuous Flow Method
• Hartridge & Roughton (1923)
• Reactants are compressed at a constant rate generating a constant
flow
• At a constant flow rate the age of the solution is linearly proportional
to the distance down the flow tube
from „Structure and mechanism in protein science“, Alan Fersht
The Stopped Flow Method
• Roughton (1934); improved by Chance (1940)
• Amenable for reactions that undergo spectral changes (UV-Vis absorbance,
fluorescence, CD)
from „Structure and mechanism in protein science“, Alan Fersht
The Rapid Quench Flow Technique
• Require quenching of the
reaction and concomittant
analysis of the sample
collected by an appropriate
analytical method (HPLC,
GC-MS, etc.)
from „Structure and mechanism in protein science“, Alan Fersht