Transcript Introduction to enzymes

Enzyme Kinetics
9/1/2004
General Properties of Enzymes
•Increased reaction rates sometimes 106 to 1012 increase
Enzymes do not change DG just the reaction rates.
•Milder reaction conditions
•Great reaction specificity
•Can be regulated
Substrate specificity
The non-covalent bonds and forces are maximized to bind substrates with
considerable specificity
•Van der Waals forces
•electrostatic bonds (ionic interactions)
•Hydrogen bonding
•Hydrophobic interaction
•Stereospecificity
•Geometrically specific
enz
A+B
Substrates
P+ Q
Products
Enzymatic catalysis and mechanisms
•A. Acid - Base catalysis
•B. Covalent catalysis
•C. Metal ion aided catalysis
•D. Electrostatic interactions
•E. Orientation and Proximity effects
•F. Transition state binding
General Acid Base
Rate increase by partial proton abstraction by a
Bronsted base or
Rate increase by partial proton donation by a
Bronsted Acid
Enzyme Kinetics
Rates of Enzyme Reactions
How fast do reactions take place
•Reaction rates
Thermodynamics says I know the difference between
state 1 and state 2 and DG = (Gf - Gi)
But
Changes in reaction rates in response to differing
conditions is related to path followed by the reaction
and
is indicative of the reaction mechanism!!
Chemical kinetics and Elementary
Reactions
A simple reaction like A  B may proceed through several
elementary reactions like A  I1  I2  B Where I1 and I2 are
intermediates.
The characterization of elementary reactions comprising an
overall reaction process constitutes its mechanistic
description.
Rate Equations
Consider aA + bB + • • • + zZ. The rate of a reaction is
proportional to the frequency with which the reacting
molecules simultaneously bump into each other
Rate  kA B    Z
a
b
z
The order of a reaction = the sum of exponents
Generally, the order means how many molecules have to bump into
each other at one time for a reaction to occur.
A first order reaction one molecule changes to
another
AB
A second order reaction two molecules react
A+BP+Q
or
2A  P
3rd order rates A + B + C  P + Q + R rarely occur
and higher orders are unknown.
Let us look at a first order rate
AB
dA  dP 
v

dt
dt
dA
v
 k A
dt
n = velocity of the reaction
in Molar per min.
or
moles per min per volume
k = the rate constant of the
reaction
Instantaneous rate: the rate of reaction at any specified
time point that is the definition of the derivative.
We can predict the shape of the curve if we know the
order of the reaction.
A second order reaction: 2A  P
d A 
2
v
 k A 
dt
Or for A + B  P + Q
d A 
d B
v

 k A B
dt
dt
Percent change in A (ratio ) versus time in first and
second order reactions
It is difficult to determine if the reaction is either first or
second order by directly plotting changes in
concentration.
dA 

 k A 
dt
A
t
dA

k
dt
A A
0
o
d A 
  kdt
A 
lnA  lnAo  kt
A  Ao e
-kt
However, the natural log of the concentration is
directly proportional to the time.
- for a first order reactionThe rate constant for the
first order reaction has
units of s-1 or min-1 since
velocity = molar/sec
and v = k[A] : k = v/[A]
Gather your data and plot
ln[A] vs time.
The half-life of a first order reaction

A o
A  
2
Plugging in
to rate equation
 Ao

ln 2
 A




  kt1

2


ln 2 0.693
t1 

k
k
2
The half-life of a first order reaction is the time for half of
the reactant which is initially present to decompose or react.
a common radioactive isotope, emits an energetic b
particle and has a half-life of 14 days. 14C has a half life of
5715 years.
32P,
A second order reaction such like 2A  P
dA o

k
dt
2


A o A 
0
A 
t
1
1

 kt
A Ao
When the reciprocal of the concentration is plotted verses time a
second order reaction is characteristic of a straight line.
1
t1 
The half-life of a second order reaction is


k
A
o
2
and shows a dependents on the initial concentration
The Transition State
A bimolecular reaction A + B C
A B + C at some point
in the reaction coordinate an intermediate ternary complex will exist
A
B
C
This forms in the process of bond formation and bond breakage
and is called a transition state.
Ha + Hb
Hc
Ha
Hb +
Hc
This is a molecule of H2 gas reforming by a collision
An energy contour of
the hydrogen reaction
as the three molecules
approach the transition
state at location c.
This is called a saddle
point and has a higher
energy than the
starting or ending
point.
Energy diagrams for the
transition state using the
hydrogen molecule
Transition state diagram
for a spontaneous reaction.
X‡ is the symbol for the
species in the transition
state
For the reaction
‡
dP
 kA B  k' X 
dt

X
‡
K
AB
‡
Where [X] is the
concentration of the
transition state species
k' = rate constant for the decomposition of the activated complex
‡
- RTlnK  DG
‡
DG‡ is the Gibbs free energy of the activated
complex.
‡
The greater the DG‡, the more unstable the transition
state and the slower the reaction proceeds.
d P
 k' e
dt
DG
RT
AB
This hump is the activation barrier or kinetic barrier for a reaction.
The activated complex is held together by a weak bond that would
fly apart during the first vibration of the bond and can be
expressed by k' = kn where n is the vibrational frequency of the
bond that breaks the activated complex and k is the probability
that it goes towards the formation of products.
Now we have to define n. E = hn and n = E/h where h
is Planks constant relating frequency to Energy. Also
through a statistical treatment of a classical
oscillator E= KbT where Kb is Boltzmann constant.
By putting the two together
And
K bT
k
e
h
K bT
k 
h
DG

RT
‡
The rate of reaction decreases as its free energy of
activation, DG‡ increases
or
the reaction speeds up when thermal energy is added
Multi-step reactions have rate determining steps
Consider
A  I  P
k1
k2
If one reaction step is much slower than all the rest this step
acts as a “bottleneck” and is said to be the rate-limiting step
Catalysis lowers the activation energy
Preferential transition state binding
Binding to the transition state with greater affinity to
either the product or reactants.
RACK MECHANISM
Strain promotes faster rates
The strained reaction more closely resembles the
transition state and interactions that preferentially
bind to the transition state will have faster rates
kN
S 
 P
kE
ES 
 EP
kN for uncatalyzed reaction
and
kE for catalyzed reaction
K N‡
‡
E  S  S  E  P  E
 KR
‡
 KT
KE
ES 
 ES

ES
KR 
ES

ES 
KE 
ES
‡
‡
‡


EP
‡

ES 
KT 
‡
ES 
‡
‡
K T SES  K E
 ‡

‡
K R S ES K N
‡
‡
DG N  DGE  
kE

RT 
 exp
kN
The enzyme binding
of a transition state
(ES‡ ) by two
hydrogen bonds that
cannot form in the
Michaelis Complex
(ES) should result in
a rate enhancement
of 106 based on this
effect alone
106 rate enhancement
requires a 106 higher
affinity which is 34.2
kJ/mol
Catalysts act to lower the activation barrier of the reaction being
catalyzed by the enzyme.
Where DDG‡cat = DG‡uncat- DG‡cat
The rate of a reaction is increased by
DDG cat
e
RT
DDG‡cat = 5.71 kJ/mol is a ten fold increase in rate.
This is half of a hydrogen bond!!
DDG‡cat = 34.25 kJ/mol produces a million fold
increase in rate!!
Rate enhancement is a sensitive function of DDG‡cat
Kinetics of Enzymes
Enzymes follow zero order kinetics when substrate
concentrations are high. Zero order means there is no increase
in the rate of the reaction when more substrate is added.
Given the following breakdown of sucrose to glucose and
fructose
Sucrose + H20
Glucose + Fructose
H
H
H
H OH
O
HO
H
H
H
O
HO
OH
H
HO
H
H
OH
OH
OH
OH
H
HO
H
k1
E  S  ES  E  P
k2
k -1
E = Enzyme S = Substrate P = Product
ES = Enzyme-Substrate complex
k1 rate constant for the forward reaction
k-1 = rate constant for the breakdown of the ES to
substrate
k2 = rate constant for the formation of the
products
When the substrate concentration becomes large
enough to force the equilibrium to form completely
all ES the second step in the reaction becomes rate
limiting because no more ES can be made and the
enzyme-substrate complex is at its maximum value.
d P 
v
 k 2 ES 
dt
[ES] is the difference between the
rates of ES formation minus the
rates of its disappearance. 1
d ES 
 k1 E S  k 1 ES   k 2 ES 
dt
Assumption of equilibrium
k-1>>k2 the formation of product is so
much slower than the formation of the ES
complex. That we can assume:
k1 ES
Ks 

ES
k1
Ks is the dissociation constant for the ES complex.
Assumption of steady state
Transient phase where in the course of a reaction the
concentration of ES does not change
2
d ES 
0
dt
ET  E  ES
3
Combining 1 + 2 + 3
k1 ET - ESS  k -1  k 2 ES
rearranging
ESk -1  k 2  k1S  k1ET S
Divide by k1 and solve for [ES]

E T S
ES 
K M  S
Where
k -1  k 2
KM 
k1
k2 ET S
 d P 
vo  
  k2 ES 
K M  S
 dt t 0
vo is the initial velocity when the reaction is just starting out.
And
Vmax  k 2 ET
Vmax S
vo 
K M  S
is the maximum velocity
The Michaelis - Menten
equation
The Km is the substrate concentration where vo equals
one-half Vmax
The KM widely varies among different enzymes
The KM
can be expressed as:
k 1 k 2
k2
KM 
  Ks 
k1 k1
k1
As Ks decreases, the affinity for the substrate
increases. The KM can be a measure for substrate
affinity if k2<k-1
There are a wide range of KM, Vmax , and efficiency
seen in enzymes
But how do we analyze kinetic data?
The double reciprocal plot
1  KM
 
vo  Vmax
 1
1
 
 S Vmax
Lineweaver-Burk plot: slope = KM/Vmax,
1/vo intercept is equal to 1/Vmax
the extrapolated x intercept is equal to -1/KM
For small errors in at low [S] leads to large errors in 1/vo
kcat
Vmax

ET
kcat is how many reactions an
enzyme can catalyze per second
The turnover number
For Michaelis -Menton kinetics k2= kcat
When [S] << KM very little ES is formed and [E] = [E]T
and
k cat
k2
ET S  ES
vo 
KM
KM
Kcat/KM is a measure of catalytic efficiency
What is catalytic perfection?
When k2>>k-1 or the ratio
Then
kcat
 k1
KM
k1k 2
k 1  k 2
is maximum
Or when every substrate that hits
the enzyme causes a reaction to
take place. This is catalytic
perfection.
Diffusion-controlled limit- diffusion rate of a substrate
is in the range of 108 to 109 M-1s-1. An enzyme lowers
the transition state so there is no activation energy
and the catalyzed rate is as fast as molecules collide.
Kinetic data cannot unambiguously
establish a reaction mechanism.
Although a phenomenological description can be
obtained the nature of the reaction intermediates
remain indeterminate and other independent
measurements are needed.
Reaction Mechanisms
A: Sequential Reactions
• All substrates must combine with enzyme
before reaction can occur
Bisubstrate reactions
Random Bisubstrate Reactions
Ping-Pong Reactions
• Group transfer reactions
• One or more products released before all
substrates added
Inhibition kinetics
There are three types of inhibition kinetics competitive,
mixed and uncompetitive.
•Competitive- Where the inhibitor competes with the
substrate.
Competitive Inhibition

E I
KI 
EI
Vmax S
vo 
K M  S
 I 
  1  
 KI 
HIV protease inhibitors
Competitive Inhibition: Lineweaver-Burke Plot
Uncompetitive Inhibition
Uncompetitive Inhibition: Lineweaver-Burke Plot
Mixed inhibition
Mixed inhibition is when the inhibitor binds to the
enzyme at a location distinct from the substrate
binding site. The binding of the inhibitor will either
alter the KM or Vmax or both.

ESI
KI 
ESI 

E I
KI 
EI
Vmax S
vo 
K M   S


I 
   1  
 K I 
The effect of pH on kinetic parameters