3 enzyme kinetics

8/3/2019 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

3 enzyme kinetics

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