Pre-equilibrium approximation
Product formed only by
second step
Physical Chemistry
Elementary rate
determined by
intermediate’s
concentration

A  B
I
I
Assumption of rapid
equilibrium
Lecture 9
Kinetics of enzymatic reactions
Substitution gives an
expression for the rate in
terms of reactant
concentrations
Effective rate constant
depends on all three rates
K eq
B
fast
fast
slower

kf
kr

[I ]
[ A][ B ]
[ I ]  K eq [ A][ B ]
Effective activation energy
depends on all elementary
activation energies

k
d [ P]
 k2 [ I ]
dt
Gives the intermediate’s
concentration in terms of
reactant concentrations

f

I
kr

A 
kp
 P
d [ P]
 K eq k p [ A][ B ]
dt
Example: formation of NO2
from NO and O2
Biological reactions
Catalytic action
Some reactions proceed with the presence of material that are not
Usually occur in aqueous
solution
Generally multiple-step
reactions
Can be grouped in classes
of reactions that happen by
similar mechanisms






Materials that affect a reaction, but are not among the reactants or
products, are



k
A simple model
A  B  I
k

A  B
Assumes the existence ofI
k
 P
a reactive intermediate I
Requires a difference in
time scale of the
elementary processes
f
r
p


Substrate is a reactant
k1
S  C 
SC
k 1
SC 
S  C
k2
SC  P  C
Pre-equilibrium approximation
Fast establishment of
equilibrium
Slower transformation of
intermediate into product
Requires a difference in
activation energies of the
steps
Catalysts if they promote a reaction
Inhibitors if they slow or stop a reaction
Must interact with the reactants to influence the reaction’s progress
Materials must not be used up in the reaction
Tend to have small concentrations of catalysts compared to the
reactants and products
Simplest catalytic mechanism involves formation of a substrate-catalyst
complex
Enzyme catalysis
Photochemically initiated
reactions
Electron transfer

Reactants
Products
Intermediates in the sense of our definition
fast
fast
slower
Complex formation
Unreactive complex dissociation
Product formation
Simple catalytic kinetics
Complex treated
as an intermediate
Creation of product
determined by last
elementary step
Final rate equation
depends on two
constants
d[P]
 k2[SC]
dt
d[SC]
 k1[S][C]  k1[SC]  k2[SC]  0
dt
[SC] 
k1
[S][C]
k1  k2
d ]P]
k2

[S][C]
dt
KM
k1
S  C 
SC
k 1
SC 
S  C
k2
SC  P  C
Complex formation
Unreactive complex dissociation
Product formation
1
Practical rate expression
In general, one does not
know the catalyst’s and the
complex’s concentrations at
any time
Stoichiometry and
conservation of matter
specify relations among
concentrations
Restrict consideration to
early times


Small product concentration
Quadratic term negligible
Result is an expression for
the initial rate in terms of
the formal concentrations
and rate constants
Michaelis-Menten kinetics
Enzyme
Conservation of matter
[ S ]0
 [ S ]  [ SC ]  [ P ]
[C ]0
 [C ]  [ SC ]


Using the equilibrium relation
K m [ SC ]  [ S ][C ] 
[ S ]0  [ SC ]  [ P][C ]0  [ SC ]
which gives the quadratic equation
[ SC ]2

[ P]  [C ]0  [ S ]0  K m [ SC ]

[ S ]0  [ P][C ]0
 [ S ]0 [C ]0
v0

d [ P]
dt 0
 k2
[ S ]0 [C ]0
K m  [C ]0  [ S ]0
Usual situation in
catalysis
[C]0 << [S]0
d [ P]
dt 0

v0
 k2
regenerate the enzyme and
liberate product
 0
Large excess of the
substrate

substrate complex
 Transformation to
which gives
Limiting behavior

 Formation of an enzyme 0
At early times, [ P]  0 and [ SC ]2 is negligible
[C ]0  [ S ]0  K m [ SC ]
[ S ]0 [C ]0
[ S ]0  K m
E
1
k 2 [C ]0


 S
k1

ES
ES
 E  S
ES
k2

E
k 1


Derivable from
mechanism
Defined in terms of
maximum reaction rate
(vmax) and MichaelisMenten constant (Km)
Lineweaver-Burk plot

Molecule that serves to
catalyze biochemical
reactions
Activity usually interpreted
in terms of the MichaelisMenten mechanism
E
E
 S
k1

ES
ES
k 1

E  S
ES
k2

E
1
v0

[S ]0
[S ]0  Km
K  1
1
  m 
vmax
 vmax  [S ]0
 S
k1

ES
ES
 E  S
ES
k2

E
k 1
 P
Inhibition
Change of the enzyme’s activity
by the presence of another
molecule, inhibitor
Three kinds of inhibition

Competitive

Noncompetitive

Uncompetitive
 Inhibitor blocks the substrate by
 Formation of an enzyme-
substrate complex
 Transformation to
regenerate the enzyme and
liberate product
 vmax
[S ]0
[S ]0  Km
 Linear form
Michaelis-Menten kinetics

v0  k2[E]0
Rate equation
Km
1
k 2 [C ]0 [ S ]0
Enzyme
 P
Michaelis-Menten kinetics
which is usually linearized as
1
v0
Molecule that serves to
catalyze biochemical
reactions
Activity usually interpreted
in terms of the MichaelisMenten mechanism
binding at the active site
 Inhibitor changes the binding site
structure to slow or prevent binding
 Inhibitor binds after the substrate and
changes the activity
 P
2
Competitive inhibition
Inhibition involves binding by
inhibitor, I
Two processes in parallel
Lineweaver-Burk plot has different
slope from uninhibited reaction
Example: methotrexate inhibition
of dihydrofolate reductase
(1)
E  S
( 2)
ES
(3)
E  I
 k2
v0
1
v0
 ES
 P 
E
 EI

Biological systems control chemical
reactivity through catalytic mechanisms

[ S ]0 [ E ]0
 [I ] 

[ S ]0  K m 1 
 Ki 
1
vmax

Summary
 [I] 

K m  1 
K i  1

vmax
[ S ]0

Enzymes
Michaelis-Menten kinetics
Inhibition




A means of manipulation of enzyme activity
Competitive
Noncompetitive
Uncompetitive
Noncompetitive inhibition
Noncompetitive binding
involves two processes in
parallel and in sequence
Lineweaver-Burk plot has


Different slope from
uninhibited reaction
Different intercept from
uninhibited reaction
(1)
E
 S
(2)
 ES
 P 
ES
(3)
E 
I
 EI
(4)
ES 
I
 EIS
v0
 k2
1
v0

E
[ S ]0 [ E ]0
 [I ] 
 [I ] 
[ S ]0 1    K m 1  
 Ki 
 Ki 
1  [I] 
1 
 
K i 
vmax 

[I] 

K m  1 
K i  1

[ S ]0
vmax
Uncompetitive inhibition
Inhibitor binds after the
substrate is bound to
the enzyme
Lineweaver Burk plot


Slope independent of
inhibitor concentration
Intercept depends on
inhibitor concentration
E  S
 ES
 I  EIS
ES
v0
 k2
1
v

[ S ]0 [ E ]0

 [I ]
1 
[ S ]0   K m
K
m


1  [I ] 
1   
vmax  K i 
Km 1
vmax [ S ]0
3