Transcript powerpoint

Enzyme Kinetics and Enzyme
Regulation
Robert F. Waters, PhD.
Level one with some calculus.
Michaelis-Menten Equation

Describes enzymatic activity of enzymes that
are NOT allosterically controlled
–
Allosterically controlled enzymes have sigmoidal
curve
Enzymes Cannot Alter Equilibria

Exergonic versus Endergonic
Activation Energy and Delta G

Enzymes lower
activation energy
Enzymes accelerate
Reactions by lowering
G
–

G (Gibbs Free Energy
of activation)
-G=exergonic
+ G=endergonic
Transition Configurations of [ES]

May be multiple transition states in a reaction
Maximal Velocity Vm

At a constant [E], the reaction rate increases with
increasing [S] concentration until Vmax is reached.
–
When [s] concentration is sufficiently high, then the highest
probability of [ES] formed and reaches Vmax



Saturation of enzyme active sites
This is indirect proof of ES complexes
Note:
–
–
–
1st order kinetics
Mixed order kinetics
Zero order kinetics
Analysis of Enzymatic Reactions



NMR (Nuclear Magnetic Resonance)
ESR (Electron Spin Resonance)
Fluorescent Spectroscopy
Example: Fluorescent
Spectroscopy (Bacteria)

Spectroscopic changes during different ES
configurations
–
–
Bacterial tryptophan synthetase
with pyridoxal phosphate prosthetic
group forms L-tryptophan from
L-Serine and indole
Note fluorescence differences
Enzyme Active Sites





Active sites are relatively small areas of
enzyme structure
Active sites are 3-dimensional
Substrates are bound to active sites by multiple
weak interactions
Active sites are crevices or protrusions
Specificity depends upon atomic arrangement
at the active site
Derivation of Michaelis-Menten

The dissociation constant k4 is dropped
because of the small amount of E + P forming
ES.
E  S   ES   E  P
k1
k3
k2
k4
E  S   ES  E  P
k1
k3
k2
M-M Variables





E=uncombined enzyme
ES=enzyme combined with substrate
S=substrate
P=product
k1,k2,k3,k4
–
Association and dissociation constants
Assumptions


[ ] = molar concentration
Total enzyme concentration (All Forms)
[Et ]  [E]  [ES]

Total substrate concentration
[S t ]  [S ]  [ ES ]  P
M-M Assumptions Cont:

We can assume
–
[ES] and [P] are very small compared to [S]
because:

–
[S t ]  [S ]
[S]>>[ES] since [S] concentrations are always much
greater than [E] or [S]>>>[E]
Velocity and rate measurements are usually
conducted as soon as possible after enzyme and
substrate are mixed. THEREFORE, at
0
very little P exists.
v
Velocity and M-M Equation


Velocity is related to the rate of formation of the
product. Product (P) is formed from ES.
The velocity of the reaction is proportional to
[ES]
v  k 2 [ ES ]
M-M Equation Continued

Express ES in terms of rate of formation and
breakdown.
–
Rate of formation of ES from E + S
–
Rate of formation of ES from E + P

–
–
Small amount so neglected
E  S  k1[ E ][ S ]
E  P  k 4 [ E ][ P]
E  P  k 2 [ ES ]
Rate of dissociation of ES to E + S
E  S  k 3 [ ES ]
Rate of conversion of ES to E + P
Total Concentration Change of ES
with Time
d [ ES ]

dt
(rate of formation of ES)-(rate of breakdown or
conversion of ES)
d [ ES ]

 k 1[ E ][ S ]  k 3 [ ES ]  k 2 [ ES ]
dt
Michaelis-Menten Derivation
[ E t ]  [ E ]  [ ES ]

Since,

Therefore,

Substituting for [E] in
[ E ]  [ E t ]  [ ES ]
d [ ES ]

 k 1[ E ][ S ]  k 3 [ ES ]  k 2 [ ES ]
dt
Michaelis-Menten Derivation, Cont:

Therefore,
d [ ES ]

 k 1 ([ E t ]  [ ES ])[ S ]  k 3 [ ES ]  k 2 [ ES ]
dt

Briggs and Haldane suggested a steady-state
condition where rate of formation = rate of
dissociation.
d [ ES ]
0
dt
Michaelis-Menten Derivation, Cont:



By substitution from:
d [ ES ]
 k 1 ([ E t ]  [ ES ])[ S ]  k 3 [ ES ]  k 2 [ ES ]
dt
Where,
d [ ES ]
0
dt
k 1[ E t ][ S ]  k 1[ ES ][ S ]  k 3 [ ES ]  k 2 [ ES ] 
[ ES ]( k 1[ S ]  k 3  k 2)
Segregating [Et] and [ES in above equation]
Solving for [ES]

Yields
k [ E ][ S ]
[
S
]


k
k k
1

 [ ES ]
t
1
3
2
From
k [ E ][ S ]  k [ ES ][ S ]  k [ ES ]  k
[ ES ]( k [ S ]  k  k )
1
t
1
1
3
3
2
2
[ ES ] 
Dividing Numerator and
Denominator by k1

We get,
[ E t ][ S ]
 [ ES ]

k
k
3
2
[S ] 
k
1
Michaelis-Menten Derivation, Cont:

We can define the M-M constant:
k



m

k3  k2
k
1
Substituting into
With
We get,
k
m
[ ES ] 
[ Et ][ S ]
k
m
[ E t ][ S ]
 [ ES ]
 k2
k
3
[S ] 
[S ]  k m
k
1
Michaelis-Menten Derivation, Cont:


k

v  k 2 [ ES ]
Velocity (v) is defined
as rate of formation
of product
Substituting for [ES] v
Or rearranging..

2
[ E t ][ S ]
[S ]  k m
[ E ][ S ]
k
v
[S ]  k
2
t
m
Michaelis-Menten Derivation, Cont:

At very high saturating substrate
concentrations, the enzyme is found essentially
all in the [ES] form so that .. [ ES ]  [
]
E

Under these conditions..
v  k 2 [ Et ]  V max
t
Michaelis-Menten Derivation, Cont:

By substitution..
V
V
v
max
max
 k 2 [ Et ]
[S ]

Then..

This equation is a hyperbola..
[S ]  k m
a[ S ]
v
b  [S ]
Michaelis-Menten Derivation, Cont:

k
We generally say in M-M derivation that
is the [S] where the velocity his half-maximal or…
v V


V
Dividing both sides by
1
[S ]
We get..


[S ]  k m
2

k
k
m
k
m
[S ]
[S ]  k m
max
 [ S ]  2[ S ]
 2[ S ]  [ S ]  [ S ]
Where
m
= [S] when velocity
is half-maximal.
max
m
Michaelis-Menten Derivation, Cont:


Therefore,
k
s
[ E ][ S ] k 3
k s  [ ES ] 
k1
is the equilibrium constant for the dissociation
of the ES complex
Then, the M-M equation is…
[S ]
v
0
V

max
[S ]  k s
Example Problem

Let’s use the form
V
V
0
max
[S ]

k m  [S ]
Determine Vmax and Km
Initial [S] in M(Moles) V0 (moles/L)
1 x 10-2
75.0
1 x 10-3
74.9
1 x 10-4
60.0
7.5 x 10-5
56.25
6.25 x 10-6
15.0
Assumptions and Computations
V
60

75
max
 75

60
v0
4
110
k
m
4
 110

4
4
60 k m  (60 10 )  75 10
4
4
4
60 k m  (75 10 )  (60 10 )  15 10
4
15 10
5
k m  60  2.5 10
Pick Another Initial Velocity and
Compute Km

What will the maximum velocity be?
Comparison of Enzymes



Same substrate with two separate enzymes.
Higher the Km the lower the affinity.
Differences in first order and mixed order
kinetics.
Linear Representation:LineweaverBurk Plot
v
V
0
max
[S ]

Invert : V max 
k m  [S ]
v0
k
m
[S ]
 [S ]
k
Divide By V  
v V [S ]
1
[S ]
k
Expand  

v V [S ] V [S ]
1
1
1
k
or  


v V [S ] V
1
m
max
0
max
m
0
max
max
m
0
max
following  y  mx  b
max
 [S ]
Lineweaver-Burk Representation
Competitive Inhibition




Inhibitor binds to same site as substrate.
Reversible
Vmax is the same.
Km increases with inhibitor.
Example of Competitive Inhibitor

Malonate with Succinate
–
Malonate 3 carbon dicarboxylate
Non-Competitive Inhibitor


Km is unchanged.
Vmax decreases with inhibitor.
Uncompetitive Inhibitor



Km increases
Vmax decreases.
Inhibitors bind to the ES complex not to the
dissociated enzyme alone.
Example: Inhibitor in Medicine
IRREVERSIBLE Competitive
Inhibitors

Affinity Labels
–

Mechanism-Based or Suicide Inhibitors
–

Blocks active site of enzyme by covalently binding to side
group(s) on amino acids
Product of ES complex will inhibit the active site of the enzyme
itself.
Transition-state analogs
–
Are NOT covalently bound, however, resemble substrates so
closely they bind very tightly to enzyme active site and
enzymatic activity is lost
Examples of Irreversible Inhibitors
Inhibitor
Target Enzyme
Effect
Aspirin
Allopuranol
Cyclooxygenase Anti-inflammatory
Xanthine oxidase Gout treatment
5-fluorouracil
Thymidylate
synthetase
Monoamine
oxidase
Anti-cancer drug
Penicillin
Transpeptidase
Anti-bacterial
Sarin
Cholinesterase
Chemical warfare
Paragyline
Anti-hypertensive
drug
Regulation of Enzymes





pH (Optimum pH based on pseudo-bell curve)
Temperature (Optimum temperature)
Product Inhibition (Affects enzyme itself)
– Feedback control (Modulator)
Covalent Modification
– Phosphorylation
– Proteolytic cleavage (Zymogen System)
 Blood clotting
Allosteric Control (Allosteric means “other site”) (Genetic)
– Effectors (Modifiers, Modulators)
 Activators
 Inhibitors
– Feedback inhibition (e.g., hemin and ALA--aminolevulinate
synthase (-delta)
Some Diagnostic Enzymes



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
Acid Phosphatase (Prostate Cancer)
Alanine Aminotransferase (Viral Hepatitis, Liver
Damage)
Alkaline Phosphatase (Liver disease, Bone Disorders)
Amylase (Acute Pancreatitis)
Creatine Kinase (Muscle Disorders, Heart Attack)
Lactate Dehydrogenase (Heart Attack)