Chapter 14. Enzyme Kinetics
Chemical kinetics
•
•
Elementary reactions
A → P
(Overall stoichiometry)
I1 → I2
(Intermediates)
Rate equations
aA + bB + … +zZ → P
Rate = k[A]a[B]b…[Z]z
k: rate constant
•
The order of the reaction (a+b+…+z): Molecularity of the reaction
– Unimolecular (first order) reactions: A → P
– Bimolecular (second order) reactions: 2A → P or A + B → P
– Termolecular (third order) reactions
Rates of reactions
A → P (First-order reaction)
d[P]
v=
dt
=-
d[A]
dt
= k[A]
2A → P (Second-order reaction)
v=
d[P]
dt
=-
d[A]
dt
= k[A]2
A + B → P (Second-order reaction)
v=
d[P]
dt
=-
d[A]
dt
=-
d[B]
dt
= k[A][B]
Rate constant for the first-order reaction
d [A]
= k[A]
dt
d [A]
= −kdt
[A]
[ A] d [A]
t
∫[A]0 [A] = −k ∫0 dt
ln[A] − ln[A]0 = −kt
v=−
ln[A] = ln[A]0 − kt
[A] = [A]0 e − kt
The reactant concentration
decreases exponentially with time
(t)
Half-life is constant for a first-order reaction
ln[A] − ln[A]0 = −kt
[A]
ln
= −kt
[A]0
1
ln = −kt 1 / 2
2
ln 2 0.693
t1 / 2 =
=
k
k
For the first-order reaction, half-life is independent of
the initial reactant concentration
Second-order reaction with one reactant
2A → P
d [A]
= k [ A ]2
dt
d [A]
= −kdt
2
[A]
[ A] d [A]
t
k
=
−
∫[A]0 [A]2
∫0 dt
1
1
=
+ kt
[A] [A]0
v=−
1
Slope= k
[A]
1
[A]0
Time (t)
kt 1 / 2 =
2
1
1
−
=
[A]0 [A]0 [A]0
1
t1 / 2 =
Half-life is dependent on the initial reactant concentration
k[A]0
Pseudo first-order reactions
v = k[A][B]
When [B] >> [A],
v = k ′[A]
where k ′ = k[B]0
e.g. B is water (55.5M): k'= 55.5k
Arrhenius equation
k = Ae-Ea/RT
= Activation energy (Ea)
Multistep reactions have
rate-determining steps
k1 > k2
k1 < k2
k1
k2
Rate-determining step: the slow step (= the higher activation energy)
Catalysts reduce the activation energy
ΔEa
(the reduction
in Ea by the
catalyst)
Rate enhancement = kcat/kuncat = e ΔEa/RT
e.g. When ΔEa = 5.7 kJ/mol, the reaction rate increases 10 fold
When ΔEa = 34 kJ/mol, the reaction rate increases 106 fold
Michaelis-Menten equation
Vmax [S]
v0 =
K M + [S]
•
•
•
•
v0 = the initial rate
Vmax = the maximum rate
KM = the Michaelis constant
[S] = the substrate concentration
Steady state approximation
E+S
k1
k-1
ES
k2
P+E
Derivatization of Michaelis-Menten equation
k1
E+S
ES
k2
P+E
k-1
v=
d [ P]
= k 2[ES]
dt
d [ES]
= k 1[E][S] − k − 1[ES] − k 2[ES] = 0 (Steady state approximation)
dt
[E]T = [E] + [ES]
k 1([E]T − [ES])[S] = (k − 1 + k 2)[ES]
(k − 1 + k 2 + k 1[S])[ES] = k 1[E]T[S]
k − 1 + k2
+ [S])[ES] = [E]T[S]
k1
[E]T[S]
[ES] =
k − 1 + k2
+ [S]
k1
[E]T[S]
k − 1 + k2
[ES] =
where KM =
KM + [S]
k1
k 2[E]T[S]
The initial velocity v 0 = k 2[ES] =
KM + [S]
The maximum velocity (Vmax) occurs at high substrate concentration
(
when the enzyme is entirely in the [ES] form : Vmax = k 2[E]T
v0 =
Vmax [S]
K M + [S]
Michaelis constant KM
KM = [S] at which v0 = Vmax/2
•
•
If an enzyme has a small value of KM, it achieves maximal catalytic efficiency
at low substrate concentrations
Measure of the enzyme’s binding affinity for the substrate
(The lower KM, the higher affinity)
Lineweaver-Burke plot
Vmax [S]
v0 =
K M + [S]
1 K M + [S]
=
v 0 Vmax [S]
1 ⎛ KM ⎞ 1
1
⎟⎟
= ⎜⎜
+
v 0 ⎝ Vmax ⎠ [S] Vmax
kcat/KM is a measure of catalytic efficiency
Catalytic constant or turnover number of an enzyme :
Vmax
kcat =
[E]T
When KM >> [S],
very little ES is formed and consequently [E] ≈ [E]T
Vmax[S] Vmax[S] kcat[E]T[S] ⎛ kcat ⎞
v0 =
≈
=
≈ ⎜ ⎟[E][S]
KM + [S]
KM
KM
⎝ KM ⎠
Catalytic perfection
kcat k 2
k 1k 2
=
=
KM K M k − 1 + k 2
The ratio is maximal when k 2 >> k − 1,
kcat
= k1
KM
(Diffusion-controlled limit: 108 to 109 M-1s-1)
Inhibitors
• Substances that reduce an enzyme’s activity
– Study of enzymatic mechanism
– Therapeutic agents
• Reversible or irreversible inhibitors
CO2-
O
O
N
HN
H2N
N
N
N
H
CO2-
O
CO2-
N
Dihydrofolate
(Dihydrofolate reductase substrate)
N
N
H
H
N
H
NH2
H2N
N
N
CO2-
N
CH3
Methotrexate
(Dihydrofolate reductase inhibitor,
anticancer drug )
Modes of the reversible inhibition
• Competitive inhibitors
– Binds to the substrate
binding site
• Uncompetitive inhibitors
– Binds to enzymesubstrate complex
• Non-competitive inhibitors
– Binds to a site different
from the substrate
binding site
• Mixed inhibitors
– Binds to the substratebinding site and the
enzyme-substrate
Competitive inhibition
KI =
[E][I]
[EI]
d [ES]
= k 1[E][S] − k − 1[ES] − k 2[ES] = 0 (Steady state approximation)
dt
⎛ k − 1 + k 2 ⎞ [ES] KM [ES]
=
[E] = ⎜
⎟
[S]
⎝ k 1 ⎠ [S]
[E][I] KM [ES][I]
=
[EI] =
KI
[S]KI
[E]T = [E] + [EI] + [ES]
[E]T =
⎧ KM ⎛ [I] ⎞ ⎫
KM [ES] KM [ES][I]
+
+ [ES] = [ES]⎨ ⎜1 + ⎟ + 1⎬
[S]
[S]KI
⎩ [S] ⎝ KI ⎠ ⎭
[E]T[S]
⎛ [I] ⎞
KM ⎜1 + ⎟ + [S]
⎝ KI ⎠
k 2[E]T[S]
v 0 = k 2[ES] =
⎛ [I] ⎞
KM ⎜1 + ⎟ + [S]
⎝ KI ⎠
Since Vmax = k 2[E]T,
[ES] =
v0 =
Vmax [S]
αK M + [S]
where α = 1 +
[I]
KI
Competitive inhibitors affect KM
v0 =
Vmax [S]
αK M + [S]
[S] → ∞; v0 → Vmax regardless of α
Determination of KI
of the competitive inhibitor
1 ⎛ α KM ⎞ 1
1
+
=⎜
⎟
v0 ⎝ V max ⎠ [S] V max
Uncompetitive inhibition
[ES][I]
[ESI]
KM [ES]
[E] =
[S]
[ES][I]
[ESI] =
KI '
[E]T = [E] + [ES] + [ESI]
KI ' =
[E]T =
⎛ KM
[I] ⎞
KM [ES]
[ES][I]
+ [ES] +
= [ES]⎜⎜
+ 1 + ⎟⎟
KI '
KI ' ⎠
[S]
⎝ [S]
[E]T[S]
⎛ [I] ⎞
KM + ⎜1 + ⎟[S]
⎝ KI ' ⎠
k 2[E]T[S]
v 0 = k 2[ES] =
⎛ [I] ⎞
KM + ⎜1 + ⎟[S]
⎝ KI ' ⎠
Since Vmax = k 2[E]T,
[ES] =
Vmax
[S]
Vmax [S]
α
'
=
v0 =
K M + α '[S] K M + [S]
α'
where α ' = 1 +
[I]
KI '
Uncompetitive inhibitors
decrease both Vmax and KM
Vmax
[S]
'
v0 = α
KM
+ [S]
α'
Vmax
α'
V
K M >> [S]; v0 → max [S]
KM
[S] → ∞; v0 →
Vmax
(Effects of an uncompetitive inhibitor
become negligible)
no inhibitor (α'=1)
Increasing [I]
VmaxI
v0
Vmax/2
α ' = 1+
VmaxI/2
KMI KM
[S]
[I]
KI '
Determination of KI’
of the uncompetitive inhibitor
1 ⎛ KM ⎞ 1
α'
=⎜
+
⎟
v0 ⎝ V max ⎠ [S] V max
Mixed inhibition
KI =
[E][I]
[EI]
KI ' =
[ES][I]
[ESI]
[E][I]
KI
[ES][I]
[ESI] =
KI '
[E]T = [E] + [EI] + [ES] + [ESI]
[EI] =
⎛ [I] ⎞
⎛ [I] ⎞
[E]T = [E]⎜1 + ⎟ + [ES]⎜1 + ⎟
⎝ KI ' ⎠
⎝ KI ⎠
[E]T = [E]α + [ES]α '
[E]T =
⎞
⎛ αKM
KM [ES]
α + [ES]α ' = [ES]⎜⎜
+ α ' ⎟⎟
[S]
⎠
⎝ [S]
[E]T[S]
αKM + α '[S]
k 2[E]T[S]
v 0 = k 2[ES] =
αKM + α '[S]
Since Vmax = k 2[E]T,
[ES] =
v0 =
Vmax [S]
αK M + α '[S]
where α = 1 +
[I]
[I]
and α ' = 1 +
KI
KI '
Lineweaver-Burk plot of a mixed inhibition
1 ⎛ αKM ⎞ 1
α'
+
=⎜
⎟
v0 ⎝ V max ⎠ [S] V max
Noncompetitive inhibition
A special mixed inhibition when KI = KI’
v0 =
Vmax [S]
αK M + α '[S]
where α = 1 +
[I]
[I]
and α ' = 1 +
KI
KI '
When KI = KI' , α = α '
Vmax
[S]
Vmax [S]
v0 =
= α
α ( K M + [S]) K M + [S]
Noncompetitive inhibitors affect not KM but Vmax
Vmax
v0 = α
[S]
K M + [S]
where α = 1 +
[S] → ∞; v0 →
[ I]
KI
Vmax
α
Vmax
no inhibitor (α=1)
Vmax
v0
Increasing [I]
I
Vmax/2
VmaxI/2
α = 1+
KM
[S]
[I]
KI
Determination of KI
of the noncompetitive inhibitor
α
1 ⎛ α KM ⎞ 1
+
=⎜
⎟
v0 ⎝ V max ⎠ [S] V max
1/v0
α=2
α= 1.5
Increasing
[I]
α= 1
(no inhibitor)
Slope=αKM/Vmax
α/Vmax
−
1
KM
0
α = 1+
[ I]
KI
1/[S]
Effects of inhibitors on Vmax and KM
of the Michaelis-Menten equation
app
Vmax
[S]
v0 = app
K M + [S]
Effects of pH
• Binding of substrate to enzyme
• Catalytic activity of enzyme
'
Vmax
= Vmax / f 2 and K M' = K M ( f 1 / f 2)
• Ionization of substrate
where
KE2
[H + ]
f1=
+1+ +
KE1
[H ]
• Variation of protein structure
(only at extreme pHs)
EH+
EH + S
KE1
KES2
[H + ]
f2=
+1+ +
KES1
[H ]
ES-
KE2
H+
EH2+
KES2
k1
k-1
H+
ESH
KES1
H+
ESH2+
'
Vmax
[S]
v0 = '
K M + [S]
k2
P + EH
Approximate identity of catalytic amino acid residues
pKa ~4 → Catalytic Asp or Glu residue
pKa ~6 → Catalytic His residue
pKa ~10 → Catalytic Lys residue
Caution should be taken because pKa of amino acid residues are environmentally sensitive
Bisubstrate reactions
• 60% of biochemical reactions involve two substrates and two products
• Transfer reactions and oxidation-reduction reactions
Cleland’s nomenclature system
for the enzymatic reactions
•
Substrates: A, B, C, D… in the order that they add to the enzyme
•
Products: P, Q, R, S… in the order that they leave the enzyme
•
Inhibitors: I, J, K, L…
•
Stable enzyme complexes: E, F, G, H… with E being the free
enzyme
Numbers of reactants and products: Uni (one), Bi (two), Ter (three),
and Quad (four)
e.g. Bi Bi reaction: a reaction that requires two substrates and yields
two products
•
Types of Bi Bi Reactions
•
Sequential reactions (single displacement reactions):
all substrates bind before chemical event
Ordered mechanism
Random mechanism
•
Ping pong reactions (double displacement reactions):
chemistry occurs prior to binding of all substrates
Differentiating bisubstrate mechanisms
• Measure rates
• Change concentration of substrates and
products
• Lineweaver-Burk plot
– Intercept (1/Vmax): the velocity at saturated substrate
concentration → It changes when the substrate A
binds to a different enzyme form with the substrate B
– Slope (KM/Vmax): the rate at low substrate
concentration → It changes when both A and B
reversibly bind to an enzyme form
Ping Pong Bi Bi Mechanism
• Intercept changes because A and B bind to the different enzyme forms E and
F, respectively
• Slope remains same because the binding of A and B is irreversible due to the
release of the product (P)
Sequential Bi Bi Mechanism
or
Ordered
Random
• Intercept changes because A and B binds to the different enzyme forms
(E or EB) and (EA or E), respectively
• Slope changes because the binding of A and B is reversible
Differentiating Bi Bi mechanisms
by product inhibition
Competitive inhibition → Substrate and inhibitor competitively bind to the same
site of the enzyme
Ping Pong
Ordered
sequential
Random
sequential
A vs Q: Competitive
B vs P: Competitive
A vs P: Noncompetitive
B vs Q: Noncompetitive
A vs Q: Competitive
B vs P: Noncompetitive
A vs P: Noncompetitive
B vs Q: Noncompetitive
Under assumption of dead-end complex formation
(A is similar with Q and B is similar with P)
A vs Q: Competitive
B vs P: Competitive
A vs P: Noncompetitive
B vs Q: Noncompetitive
Dead-end complexes
A
B
E
Q
P
Q
B
A
P
Dead-end complex (no chemistry)
no chemistry
competitive
ATP + Creatine
ADP + Creatine-phosphate
competitive
Differentiating Bi Bi mechanisms
by isotope exchange
Ping Pong Mechanism
A*→ P isotope exchange is possible without B
B* → Q isotope exchange is possible without A
or
Sequential Mechanism
Two substrates are required for the isotope exchange
Isotope exchanges in a ping pong mechanism
Sucrose phosphorylase
(Sucrose)
Glucose-fructose + phosphate
E
Glucose-1-phosphate + fructose
Isotope exchange experiments
Glucose-fructose + fructose*
E
Glucose-1-phosphate + phosphate*
Glucose-fructose + E
Glucose-E + phosphate
Glucose-fructose* + fructose
E
Glucose-1-phosphate* + phosphate
E
E
Fructose + Glucose-E
Glucose-1-phosphate + E
Ping Pong Mechanism (Double displacement)
Isotope exchanges in a sequential mechanism
Maltose phosphorylase
(Maltose)
Glucose-glucose + phosphate
E
Glucose-1-phosphate + glucose
Isotope exchange experiments
Glucose-glucose + glucose*
E
Glucose-1-phosphate + phosphate*
Glucose-glucose + phosphate*
Glucose-1-phosphate + glucose*
E
E
No isotope exchange
E
No isotope exchange
Glucose-1-phosphate* + glucose
Glucose-glucose + phosphate
Sequential Mechanism (Single displacement)