3 enzyme kinetics
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ENZYME KINETICS
Uni-substrate kinetics
A simple model to illustrate a kinetic plotof a chemical reaction and an enzymaticreaction
Chemical reaction Enzymatic reaction
R P S P
v= k[R] v= ?
v [R]
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Michaelis-Menten model to characteriseenzyme kinetics
k1 k3E + S ES E + P
k2
k1, k2, k3= rate constants
Assumptions
[P] 0 (P never accumulateSubstantially)
Reverse reaction does not occur
v= v0 (initial rate)
[E]
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Michaelis Constant
Rapid equilibrium - Michaelis-Menten
k1[E][S] = k2[ES], when k3
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To derive Michaelis-Mentens Equation
[E][S] = Km[ES]
[E] = Km[ES][S]
- According to conservation theory ofenzyme,
e = [E] + [ES]
= Km[ES] + [ES][S]
= [ES] ( 1 + Km )[S]
[ES] = e( 1 + Km )
[S]
- The rate of reaction ( v) is defined as :
v = [P] = k3[ES] t
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- Substituting the value of [ES] :
v = k3[ES]
v = k3e( 1 + Km )
[S]
- Maximum rate constant ( Vm ) is attainedwhen all the active sites are saturatedwith substrate, ie
Vm = k3e
- So that, when substituting the value ofVm
v = Vm ( 1 + Km )
[S]
Rearrange the equation :
v= Vm[S] Michaelis-Menten[S] + Km Equation
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Validation of the Equation
v = Vm[S]
[S] + Km
i. When [S] Km, Km is negligible
v = Vm[S] = Vm [S]
iii. when [S] = Km
v = Vm[S] = Vm[S] + [S] 2
V
V
Vm
m
m
K[ S ]
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The significance ofKm value
Km value is the equivalent concentration ofsubstrate at the half of maximum rate ofreaction ( Vm)
- If the Km value is known, then theFraction of active site (fES) being filled upcould be calculated according to the
equation:
fES = v = [S] Vm [S] + Km
Km is related to the rate constants of theenzymatic reactions
k1 k3E + S E-S E + P
k2
- Forsteady-state Briggs-Haldane,
k1[E][S] = k2[ES] + k3[ES]
= (k2 + k3)[ES]
[E][S] = (k2 + k3) = Km
[ES] k1
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- ForRapid Equilibrium Michaelis-Menten
k1[E][S] = k2[ES], k3 negligible
because k2>> k3
[E][S] = k2 = Km[ES] k1
- In this condition :
Km = value ofdissociation constant(Kdis) ofES complex
- The denominator of the two equations isk1 which is the value ofassociation orformation of ES complex. Thus Km value
reflects the affinity for ES complexformation oraffinity/specificityof anenzyme towards substrate.
The lower the value ofKm , the morespecific/affinity an enzyme towards the
substrate.
The Significance of Turnover Number (k3)
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Vm eVm = k3e
k3 = Vme
= Turnover NumberorCatalytic Number
- It shows enzyme efficiency in catalysisof a substrate.
For example :
A 10-6 M Carbonic anhydrase catalyses theproduction of 0.6M H2CO3 in a second atsubstrate saturation.Thus the turnover number is 6 X 105 s-1.
k3 = Vm = 0.6 = 6 X 105 s-1
e 10-6
The Importance ofKm
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Km and Vm could be determined from:
a. Plot Michaelis-Mentenb. Plot Lineweaver-Burkc. Plot Eadie-Hofsteed. Plot Hanes-Woolf
Plot Michaelis-Menten
v = Vm[S][S] + Km
V
V
Vm
m
m
K[ S ]
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Plot Lineweaver-Burk
v = Vm[S][S] + Km
- Invert the MM equation MM
1 = [S] + Km = [S] + Kmv Vm[S] Vm[S] Vm[S]
1 = Km . 1 + 1v Vm [S] Vm
- compare with y = mx + c
1V
V
K
- 1
1[ S ]
1m
m
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Plot Eadie-Hofstee
v = Vm[S][S] + Km
1 = Km . 1 + 1v Vm [S] Vm
- Multiply by factor vVm
1 (v)Vm = Km 1 v(Vm) + 1v(Vm)(v) Vm [S] Vm
v = - Km . v + Vm
[S]
V
m
m- K
mV
K
mV
[S]
V
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Plot Hanes-Woolf
1 = Km . 1 + 1v Vm [S] Vm
- Multiply by factor [S]
1 [S] = Km 1 [S] + 1 [S]
v Vm [S] Vm
[S] = Km + [S] v Vm Vm
[S] = 1 . [S] + Km
v Vm Vm
mV
mK
mV
1
m
- K
[ S ]
[ S ]
V
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EISENTHAL and CORNISH-BOWDEN PLOT
Cornish-Bowden (1974) proposed a different approach to get the kinetics constantsbased on Michaelis-Menten equation. At a constant/fixed [E] the equation could be
arranged:
1 = Km + [So]
vo Vm[So]
Vm = Km + [So] = Km + 1
vo [So] [So]
At a constant vo and [So] , the plot ofVmagainst Km is linear. It would rather beconfusing as to why a constant could be plotted against a constant.
When Km = 0 , Vm = vo and when Vm = 0 , Km = - [So]. Therefore for each pair ofvoand [So] could be generated a line to note voat Vmaxis and -[So] at Km axis; the twopoints is joined to extrapolate a straight line. The lines for all of each pair at constant [Eo]be plotted and the intercept of the lines are the true values ofKm and Vm.. However dueto experimental errors the intercepts occur at a set of values (see figure). It would belogical to get the average of values or the middle of all values.
Vm
Plot Eisenthal-Corning-Bowden
Vm*the best estimated value Vm*
Km*the best estimated value (vo)n
Vm = Km vo + vo(vo)1 [ So ]
[So]n [So]1 Km* Km
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Reversible reaction
k1 k3E + S ES/EP E + P
k2 k4
At equilibrium the rate of association toform the complex ES is equal to the ratedissociation of the complex ES
k2[ES] = k1[E] [S] , that is
[ES] = k1 [S][E] k2
k3[ES] = k4[E] [P] , that is
[ES] = k4 [P][E] k3
From both equations,
k1 [S] = k4 [P]k2 k3
[P] = k1k3 = Keq[S] k2k4
= Equilibrium Constant
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k1 k3E + S ES E + P
k2
Vms = k3e Kms = k2 + k3k1
k3E + S EP E + P
k2 k4
Vmp = k2e Kmp = k2 + k3 k4
At equilibrium
Vms = k3 Kmp = k1Vmp k2 Kms k4
Keq = k1 k3 = VmsKmp k2 k4 VmpKms
Keq = VmsKmp Haldane RelationshipVmpKms
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enzyme conservation :
e = [E] + [ES] + [EP]
thus,
vnet = k3[ES] - k2[EP] e [E] + [ES] + [EP]
Kmsand Kmp values are :
Kms = [S][E] , thus [ES] = [S][E][ES] Kms
Kmp = [P][E] , thus[EP] = [P][E][EP] Kmp
substituting the value of [ES] and [EP]
vnet = k3[S][E] - k2[P][E]e Kms Kmp
[E] + [S][E] + [P][E] Kms Kmp
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Vms and Vmp values :
Vms = k3e Vmp = k2e
thus,
vnet = Vms[S] - Vmp[P] Kms Kmp
1 + [S] + [P] Kms Kmp
vnet = VmsKmp[S] - VmpKms [P] KmsKmp + Kmp[S] + Kms[P]
when [P] 0
v = Vm[S]
Km + [S]
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Determination of the initial rate of reaction ( vo )
EISENTHAL and CORNISH-BOWDEN PLOT
K e c e r u n a n a
M a s aTime ( t )
Initial gradient
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Cornish-Bowden (1974) proposed a different approach to get the kinetics constantsbased on Michaelis-Menten equation. At a constant/fixed [E] the equation could bearranged:
1 = Km + [So]
vo Vm[So]
Vm = Km + [So] = Km + 1
vo [So] [So]
At a constant vo and [So] , the plot ofVmagainst Km is linear. It would rather beconfusing as to why a constant could be plotted against a constant.
When Km = 0 , Vm = vo and when Vm = 0 , Km = - [So]. Therefore for each pair ofvoand [So] could be generated a line to note voat Vmaxis and -[So] at Km axis; the twopoints is joined to extrapolate a straight line. The lines for all of each pair at constant [Eo]be plotted and the intercept of the lines are the true values ofKm and Vm.. However dueto experimental errors the intercepts occur at a set of values (see figure). It would belogical to get the average of values or the middle of all values.
Vm
Plot Eisenthal-Corning-Bowden
Vm*the best estimated value Vm*
Km*the best estimated value (vo)n
Vm = Km vo + vo(vo)1 [ So ]
[So]n [So]1 Km* Km
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