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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 accelerateReactions by loweringG
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.
PEESSE kkkk 4231
PEESSE kkk 231
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)
Total substrate concentration
[ES][E]][Et
PESSS t ][][][
M-M Assumptions Cont:
We can assume – [ES] and [P] are very small compared to [S]
because: [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
very little P exists.
][][ SS t
v0
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]
][2ESv k
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
– Rate of conversion of ES to E + P
– Rate of dissociation of ES to E + S
]][[1
SESE k
][2ESPE k
]][[4
PEPE k
][3ESSE k
Total Concentration Change of ES with Time
(rate of formation of ES)-(rate of breakdown or conversion of ES)
dt
ESd ][
][][]][[][
231ESESSE
dt
ESdkkk
Michaelis-Menten Derivation
Since,
Therefore,
Substituting for [E] in
][][][ ESEE t
][][][ ESEE t
][][]][[][
231ESESSE
dt
ESdkkk
Michaelis-Menten Derivation, Cont:
Therefore,
Briggs and Haldane suggested a steady-state condition where rate of formation = rate of dissociation.
][][]])[[]([][
231ESESSES
dt
ESdkkEk t
0][
dt
ESd
Michaelis-Menten Derivation, Cont:
By substitution from:
Where,
Segregating [Et] and [ES in above equation]
][][]])[[]([][
231ESESSES
dt
ESdkkEk t
)][]([
][][]][[]][[
231
2311
kkkkkkEk
SES
ESESSESSt
0][
dt
ESd
Solving for [ES]
Yields
From
][][
]][[
231
1 ESS
S
kkkEk t
)][]([
][][]][[]][[
231
2311
kkkkkkEk
SES
ESESSESSt
Dividing Numerator and Denominator by k1
We get,
][
][
]][[
1
23
ES
S
S
kkk
E t
Michaelis-Menten Derivation, Cont:
We can define the M-M constant:
Substituting into With
We get,
kkkkm1
23
km
][
][
]][[
1
23
ES
S
S
kkk
E t
km
kE
m
t
S
SES
][
]][[][
Michaelis-Menten Derivation, Cont:
Velocity (v) is defined
as rate of formation
of product Substituting for [ES]
Or rearranging..
][2ESv k
kE
k m
t
S
Sv
][
]][[
2
kEk
m
t
S
Sv
][
]][[2
Michaelis-Menten Derivation, Cont:
At very high saturating substrate concentrations, the enzyme is found essentially all in the [ES] form so that ..
Under these conditions..
][][ E tES
VEk tv
max2][
Michaelis-Menten Derivation, Cont:
By substitution..
Then..
This equation is a hyperbola..
][2max EkV t
kV
mS
Sv
][
][max
][
][
Sb
Sav
Michaelis-Menten Derivation, Cont:
We generally say in M-M derivation that is the [S] where the velocity his half-maximal or…
Dividing both sides by We get..
Where = [S] when velocity is half-maximal.
km
kV
mS
Sv
][
][max
V max
][][][2
][2][][
][
2
1
SSS
SSS
S
k
kkm
mm
km
Michaelis-Menten Derivation, Cont:
Therefore,
is the equilibrium constant for the dissociation of the ES complex
Then, the M-M equation is…
kkk ES
SEs
1
3
][
]][[k s
kVv
sS
S
][
][max
0
Example Problem
Let’s use the form
][
][
max
0
S
S
kVV
m
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
75max
V 600v
1010101010
10101010
54
444
44
4
4
5.260
15
15)60()75(60
75)60(60
1
1
75
60
k
kkk
m
m
m
m
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:Lineweaver-Burk Plot
bmxyfollowing
Sor
S
S
SExpand
S
SByDivide
S
SInvert
S
S
VVk
v
VVk
v
Vk
vV
kvV
kVv
m
m
m
m
m
maxmax0
maxmax0
max0max
0
max
max
0
1
][
11
][
][
][
1
][
][1
][
][:
][
][
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– Blocks active site of enzyme by covalently binding to side
group(s) on amino acids
Mechanism-Based or Suicide Inhibitors– 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 Cyclooxygenase Anti-inflammatory
Allopuranol Xanthine oxidase Gout treatment
5-fluorouracil Thymidylate synthetase
Anti-cancer drug
Paragyline Monoamine oxidase
Anti-hypertensive drug
Penicillin Transpeptidase Anti-bacterial
Sarin Cholinesterase Chemical warfare
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
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)
