4. ENZYME KINETICS
Enzyme kinetics
Investigation of enzymatic reaction rate, identification of pa- rameters.
E + S ↔ E + P
For stoichiometric calculations all components should be gi- ven in moles or grams. But: enzymes are not pure proteins!
→ amount of enzymes is measured through their catalytic effect → ACTIVITY
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Enzyme kinetics
One UNIT is the amount of the enzyme which consumes 1 µmol substrate or forms 1 µmol product during 1 minute at given reac- tion circumstances.
SI: 1 Katal: 1 mol substrate (product) during 1 s.
(too huge!!) → nKat = 10-9Kat (nanoKatal)
1 Kat = 6*107U, 1U =1.666*10-8Kat, 1U= 1/60 µKat
Specific activity: U/mass or U/volume → U/mg, U/ml
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Michaelis-Menten kinetics
Conditions:
k-2= 0 (the second step is irreversible)
the first step reaches the equilibrium quickly = RAPID EQUILIBRIUM:
Dissociation constant of (ES):
stable ES complex, EP complex negligible
k1SE = k-1(ES)
(ES) S.E k
K k
1 1
s = − =
E + S ES E + P
k1 k-1
k2 k-2
one active centre, one substrate
concentration can be applied (instead of activity) (S) >> (E0) i.e. E0 / S << 1
Reaction rate:
Mass balance for E:
Divide these equations!
V dP
dt k (ES)2
= =
E
+( ES )
=E
oMichaelis-Menten kinetics
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E + S ES E + P
k1 k-1
k2 k-2
Michaelis-Menten kinetics
Divide the two equations:
substitute:
Rearrange:
because
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V E
k (ES) E (ES)
o
= 2
+
V E
k S K E E S
K E
o 2
s
s
= +
1 s
1
k S .E
K k ( E S )
= − =
S K
S K
1 S K
S E
k V
s s s o
2 = +
= +
V
max= k E
2 o V dPdt k (ES)2
= =
The rate equation:
or
V V
S K 1 S
K
max
s
s
= +
V V S
K S
max s
= +
Michaelis-Menten kinetics
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Leonor Michaelis 1875-1949 Maud Menten
1879-1960
Michaelis, L., Menten, M. (1913) Die kinetik der invertinwirkung, Biochemische Zeitung 49, 333-369
M és M
Briggs-Haldane kinetics
The same differential equtions but the condition:
(quasi) steady state:
(S) >> (E0) i.e. E0/S << 1 k1ES > k-1(ES) ill. k1ES > k2(ES)
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E + S ES E + P
k1 k-1
k2 k-2
( )
( ) ( ) ( )
( )
1 1
1 1 2
2
dS k ES k ES
dt
d ES k ES k ES k ES
dt
dP k ES dt
−
−
= − +
= − −
=
d(ES)/dt =0
Briggs, G. E., and Haldane, J. B. (1925) A Note on the Kinetics of Enzyme Action, Bio-chem J 19, 338-339.
Briggs-Haldane kinetics
After a short transition peri- od (pre-steady state) the rate is almost constant (quasi-steady state).
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Quasi st-st
time Pre- st-st
( )
1 1( )
2( )
d ES k E S k ES k ES 0
dt
= ⋅ ⋅ − − − =Km= (k-1+ k2) / k1
E+(ES)= Eo
( )( )
( ) ( )
1 1 2
1
1 2
k E S k k ES
k E S
ES k k
−
−
⋅ ⋅ = +
= ⋅ ⋅ +
Michaelis constant
S K
V S S K
S E V k
m max m
o 2
= +
= +
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Briggs-Haldane kinetics
Michaelis-Menten Briggs-Haldane
V V S
K S
max s
= +
V V S
K S
max m
= +
1 2
m
1
k k
K k
− +
=
if (k1) >> (k2) the two constants are equal!
Discussion
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m s
1
K K k
= + k
1 s
1
K k k
= −
1thorder range
S E k S K E
V k 0 0
S
2 = ′
≈
catalytic effectivity = specifity constant
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S >>Ks
S <<Ks
Discussion
Vmax=k2E0
rectangular hiperbole:
V=(Vmax/Ks)S
V V S
K S
max s
= +
zero order range
Hiperbole
V
S Vmax
-Km 0
Km
m m m s m
m s m s
m V
K S
V K K
S
V K V K S V V
x y a
+ +
− + =
−
= +
=
Experimental area
In M-M and B-H equations V means initial reaction rate (V0 → extrapolated to t=0).
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How to measure reaction rate?
Parameter estimation
Linearised diagrams are used:
Calculation of nonlinear regression was complicated without computers
It provides additional info about enzyme inhibition 1. Lineweaver-Burk plot
1/v – 1/S
1 1
m1
max max
K
V = V + V
⋅S
Linearised forms
2. Hanes-Langmuir plot S/v – S
3. Eady-Hofstee plot v/S – v
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V
V = V max − K m S 1
S K m S
V V V
max max
= + ⋅
tg α= k2
Effect of enzyme concentration
If vmax = k2.E0, then:
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Vmax: its not a climax, but limit → border of rate Its not an enzyme feature, it depends on E0:
Vmax= k2 . E0 → = ACTIVITY
k2 is the real enzyme feature = turnover number [s-1] → transformation frequency
Extending to every enzymes and every kinetics:
kcat[ s-1 ]: Turnover frequency of one enzyme molecule (at S-saturation): how many substra- te molecules are transformed in one second by one enzyme molecule.
Vmax= kcat. E0
Interpretation of kinetic parameters
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Kinetic parameters: K
s, K
mAffinity of az enzyme to substrate
Usually the S concentration in a living cell – easy adap- tion to changes
KShas changed → Inhibitor? Activator?
Enzyme analytics:
- activity measurement:
S>>KS v=vmax
- substrate measurement:
S<<KS linear range
unsensitive
very sensitive
k1 107-1010 dm3mol-1min-1[max. value (~1011) limited by diffusivity of small molecules] k-1 102-106min-1
k2 50-107min-1
Km 10-6- 10-2 mol/dm3
Interpretation of kinetic parameters
Molecular switches Metabolic enzymes
Restriction enzymes
Many enzyme catalysed reactions - mainly biopolymer hyd- rolysis - are highly shifted to the right hand side, practically k-2may really be neglected.
But conversions like
glucose fructose
(glucose isomerase)
~50 : 50 %
are of reversible character.
Reversible reactions
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K k k
k
K k k
k
ms
2 1
1
mp
2 1
2
= +
= +
−
−
−
V k E
V k E
maxs 2 o
maxp 1 o
=
=
−2 1
2 1 2 1 um) eq(uilibri
2 2 2 1
1 1
k k
k K k
K K
k K k k
K k
−
−
−
−
=
=
=
=
1/KS KP
Reversible reactions
While k-2= 0 in both kinetic models reactions seems to be irre- versible. Models for reversible (equilibrium) reactions are built up from models of two countercurrent irreversible reaction.
E + S ES E + P k1
k-1
k2 k-2
E
WHAT WILL HAPPEN?
S → P or P → S
?
S P
V
netto= V
foreward- V
backward= k
2(ES) - k
-2(EP)
V
V S P
K
K 1 P
K S
V S
K V P
K
1 S
K
P K
netto
maxs
eq
ms
mp
maxs ms
maxP mP
ms mP
=
−
+
+
=
−
+ +
What does it depend on?
Reversible M-M equation
Reversible reactions
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