Reaction kinetics: 1st order reactions
A
k1
•
B (+ C)
•
Decay reactions, like radio-activity;
SN1 reactions
Rate: -
d[A]
dt
Rewriting: -
[A]
Integration gives:
•
••
[A]
= k 1 [A]
d[A]
•
= k 1 dt
t
1
 [A]
0
So: ln[A]t – ln[A]0 = -kt
t
t
d[A]    ktdt
0
or:
ln
[A ]t
[ A ]0
=
-kt
••
•
•
•
The time for half of the
reactant initially present to
decompose, its half-time or
half-life, t1/2 , is a constant
and hence independent of the
initial
concentration
of
reactant.
By substituting the relationship [A] = [A0] / 2
when t = t1/2 into ln [A]=ln [A]0 - kt
and rearranging:
t 1/2 = ln2/k = 0.693/k
Second-order reaction
2A
The half-time for a second order reaction is expressed
P
t 1/2 = 1/k [A]0
and therefore, in contrast to a
first order reaction depends on the initial reactant concentration.
A+B
P
Here, the reaction is said to be first order in A and first order in B.
Unimolecular and bimolecular reactions are common.
Termolecular reactions are unusual because the simultaneous
collision of three molecules is a rare event. Fourth and higher
order reactions are unknown.
Enzyme Kinetics
ß-fructofuranosidase:
Sucrose + H2O
glucose + fructose
When [S] » [E] : the rate is zero order with respect to sucrose
The Michaelis-Menten Equation
This equation cannot be explicitly integrated, however, without simplifying assumptions,
two possibilities are
1. Assumption of equilibrium. Leonor Michaelis and Maud Menten, building on the
work of Victor Henri, assumed that k-1 » k2, so that the first step of the reaction reaches
equilibrium.
Ks is the dissociation constant of the first step in the enzymatic reaction
The Michaelis-Menten Equation
1.
Assumption of steady-state. Figure illustrates the progress curves of the various
participants in reaction
under the physiologically common conditions that substrate is in great excess over
Enzyme ([S] » [E]).
ES maintains a steady state and [ES] can be
treated as having a constant value:
The so called steady state assumption, a more
general condition than that of equilibrium, was
first proposed in 1925 by G. E. Briggs and B. S.
Haldane
The Michaelis-Menten Equation
Letting [E] = [E]T - [ES] and rearranging yields
The Michaelis constant, KM ,
is defined as
Solving for [ES],
The Michaelis-Menten Equation
The expression of the initial velocity (v0) of the reaction, the velocity
at t=0, thereby becomes
The maximal velocity of a reaction, Vmax occurs at high substrate concentrations when
the enzyme is saturated, that is, when it is entirely in the ES form
Therefore, combining the last two equations, we obtain:
This expression, the Michaelis-Menten equation, is the basic equation of enzyme kinetic.
Significance of the Michaelis Constant
The Michaelis constant, KM, has a simple operational definition. At the substrate
concentration at which [S] = KM, this equation
yields v0 = Vmax/2 so that
KM is the substrate concentration at which the reaction velocity is half maximal
Lineweaver-Burk or double-reciprocal plot
Analysis of Kinetic Data
S >> Km
vi=Vmax
Vmax= k2Et
Vmax= 10 M/seg Km=10 x10-5 M
Si en el ensayo se usaron 5mg/L de preparación enzimática, entonces:
v= Vmax = k2 ET
k2= 10/5 = 2 moles/mg seg
¿Qué predicciones podemos hacer a partir de esta información?
Significance of the Michaelis Constant
The magnitude of KM varies widely with the identity of the enzyme and the nature of the substrate.
It is also a function of temperature and pH. The Michaelis constant can be expressed as
Since Ks is the dissociation constant of the Michaelis complex, as Ks decreases, the enzyme’s affinity
for substrate increases. KM in therefore also a measure of the affinity of the enzyme for its substrate,
provided k2/k1 is small compared to Ks, that is k2 ‹ k-1 so that the ES
P reaction proceeds more
slowly than ES reverts to E + S
kcat/KM Is a Measure of Catalytic Efficiency
We can define the catalytic constant, kcat, of an enzyme as
This quantity is also known as the turnover number of an enzyme because it is the number of
reaction processes (turnovers) that each active site catalyzes per unit time.
Turn Over Numbers of Enzymes
kcat (s-1)
Enzymes
Substrate
Catalase
H2O2
Carbonic anhydrase
HCO3-
400,000
Acetylcholinesterase
Acetylcholine
140,000
b-Lactamase
Benzylpenicillin
Fumarase
Fumarate
RecA protein (ATPase)
ATP
40,000,000
2,000
800
0.4
The number of product transformed from substrate
by one enzyme molecule in one second
Adapted from Nelson & Cox (2000) Lehninger Principles of Biochemistry (3e) p.263
kcat/KM Is a Measure of Catalytic Efficiency
When [S] « KM, very little ES is formed. Consequently, [E] ≈ [E]T, so
reduces to a second-order rate equation:
The quantity kcat/KM is a measure of an enzyme’s catalytic efficiency.
There is an upper limit to the value of kcat/KM : It can be not greater than k1; that is, the decomposition of ES
to E + P can occur no more frequently than E and S come together to form ES. The most efficient enzymes
have kcat/KM values near to the diffusion-controlled limit of 108 to 109 M-1.s-1
Chymotrypsin Has Distinct kcat /Km to Different Substrates
=
–
– –
=
O
R O
H3C–C–N–C–C–O–CH3
H H
kcat / Km
R=
Glycine
–H
1.3 ╳ 10-1
Norvaline
–CH2–CH2–CH3
3.6 ╳ 102
Norleucine
–CH2–CH2–CH2–CH3
3.0 ╳ 103
Phenylalanine –CH2–
1.0 ╳ 105
(M-1 s-1)
Adapted from Mathews et al (2000) Biochemistry (3e) p.379
Al iniciar:
t = 0, S = So
A cualquier tiempo:
T=t S=S
X = (So-S)/So
- dS/dt = vi = So dX/dt
Temperature Dependence of Enzymes
•
•
•
•
As is the case with most reactions, an increase in
temperature will result in an increase in kcat for an
enzymatic reaction.
From general principles, it can be determined that the
rate of any reaction will typically double for every
10°C increase in temperature.
Many enzymes display maximum temperatures
around 40°C, which is relatively close to body
temperature.
There are enzymes that are isolated from
thermophilic organisms that display maxima around
100°C, and some that are isolated from psychrophilic
organisms that display maxima around 10°C.
Enzyme Inhibition (Mechanism)
Equation and Description
Cartoon Guide
I
Competitive
I
Non-competitive
Substrate
E
S
S
E
I
Compete for
Inhibitor active site
S
I
I
S
Uncompetitive
E
I
I
Different site
E + S←
→ ES → E + P
+
I
↓↑
EI
E + S←
→ ES → E + P
+
+
I
I
↓↑
↓↑
EI + S →EIS
[I] binds to free [E] only,
and competes with [S];
increasing [S] overcomes
Inhibition by [I].
[I] binds to free [E] or [ES]
complex; Increasing [S] can
not overcome [I] inhibition.
S
I
E + S←
→ ES → E + P
+
I
↓↑
EIS
[I] binds to [ES] complex
only, increasing [S] favors
the inhibition by [I].
Juang RH (2004) BCbasics
Competitive Inhibition
Product
C-OO-
Substrate
Competitive Inhibitor
Succinate
Glutarate
Malonate
C-OO-
C-OO-
C-OO-
C-H
H-C-H
H-C-H
C-H
H-C-H
H-C-H
C-OO-
C-OO-
H-C-H
Oxalate
C-OOC-OO-
C-OO-
H-C-H
C-OO-
Succinate Dehydrogenase
Adapted from Kleinsmith & Kish (1995) Principles of Cell and Molecular Biology (2e) p.49
Sulfa Drug Is Competitive Inhibitor
Domagk (1939)
Para-aminobenzoic acid (PABA)
H2N-
-COOH
Bacteria needs PABA for
the biosynthesis of folic acid
Folic
acid
Precursor
H2N-
-SONH2
Tetrahydrofolic acid
Sulfa drugs has similar
structure with PABA, and
inhibit bacteria growth.
Sulfanilamide
Sulfa drug (anti-inflammation)
Adapted from Bohinski (1987) Modern Concepts in Biochemistry (5e) p.197
Enzyme Inhibition
Many substances alter the activity of an enzyme by reversibly combining with it in a way
what influence the binding of substrate and/or its turnover number. Substances that reduce
an enzyme’s activity in this way are known as inhibitors
Competitive Inhibition
A substance that competes directly with a normal substrate for an enzyme’s substratebinding site is known as a competitive inhibitor.
Here it is assumed that I, the inhibitor, bind
reversibly to the enzyme and is in a rapid
equilibrium with it so that
And EI, the enzyme-inhibitor complex, is
catalytically inactive. A competitive inhibitor
therefore reduces the concentration of free
enzyme available for substrate binding.
Enzyme Inhibition
Competitive Inhibition
This
is
the
Michaelis-Menten
equation that has been modified by a
factor, , which is defined as
 Is a function of the inhibitor’s concentration and
its affinity for the enzyme. It cannot be less than 1.
Enzyme Inhibition
Competitive Inhibition
Recasting
in the double-reciprocal form yields
A plot of this equation is linear and has a slope of KM/Vmax, a 1/[S] intercept
of -1/ KM, and a 1/v0 intercept of 1/ Vmax
Enzyme Inhibition
Uncompetitive Inhibition
In uncompetitive inhibition, the inhibitor binds directly to the enzyme-substrate complex
but not to the free enzyme
In this case, the inhibitor binding step has the dissociation constant
The uncompetitive inhibitor, which need not resemble the substrate, presumably distorts
the active site, thereby rendering the enzyme catalytically inactive.
Enzyme Inhibition
Uncompetitive Inhibition
The double-reciprocal plot consists of a family of parallel lines with slope KM/Vmax, 1/v0
intercepts of ’/Vmax and 1/[S] intercept of -’/KM
Enzyme Inhibition
Mixed Inhibition (noncompetitive inhibition)
A mixed inhibitor binds to enzyme sites that participate in both substrate binding and
catalysis. The two dissociation constants for inhibitor binding
Double-reciprocal
plots
consist of lines that have
the slope  KM/Vmax, with a
1/v0 intercept of ’/Vmax
and 1/[S] intercept of -’/
 KM
Enzyme Inhibition (Plots)
I
Competitive
I
Non-competitive
Direct Plots
Vmax
vo
vo
I
Double Reciprocal
Km Km’
I
[S], mM
Km = Km’
I
Uncompetitive
Vmax
Vmax
Vmax’
Vmax’
[S], mM
I
Km’ Km
[S], mM
Vmax unchanged
Km increased
Vmax decreased
Km unchanged
Both Vmax & Km decreased
1/vo
1/vo
1/vo
Intersect
at Y axis
1/Km
I
I
I
Two parallel
lines
1/ Vmax
1/[S]
Intersect
at X axis
1/Km
1/ Vmax
1/[S]
1/ Vmax
1/Km
1/[S]
Juang RH (2004) BCbasics
Bisubstrate Reactions
Almost all of these so called bisubstrate reactions are either transferase reactions in
which enzyme catalyzed the transfer of a specific functional group, X, from one of
the substrates to the other:
or oxidation-reduction reactions in which reducing equivalents are transferred
between two substrates.
Sequential Reactions
Reactions in which all substrates
must combine with the enzyme
before a reaction can occur and
products be released are known as
Sequential reactions
Bisubstrate Reactions
Sequential Reactions
Ordered bisubstrate reaction
A and B : substrates in order that they add to the
enzyme
P and Q : products in order that they leave the
enzyme
Random bisubstrate reaction
Ping Pong Reactions
Group-transfer reactions in which one or
more products are released before all
substrates have been added are known as
Ping Pong reactions