Chemical kinetics
• Elementary reactionsA → P (Overall stoichiometry)
I1 → I2 (Intermediates)
• Rate equationsaA + bB + … +zZ → PRate = 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 reactionsA → P (First-order reaction)
v =d[P]dt
d[A]dt
= - = k[A]
2A → P (Second-order reaction)
A + B → P (Second-order reaction)
v =d[P]dt
d[A]dt
= - = k[A]2
v =d[P]dt
d[A]dt
= - = k[A][B]d[B]dt
= -
Rate constant for the first-order reaction
kte
ktkt
dtkd
kdtd
kdt
dv
−=
−=−=−
−=
−=
=−=
∫ ∫
0
0
A][
[A]
t
0
]A[]A[
]Aln[A]ln[ln[A]A]ln[
[A][A]
]A[]A[
]A[]A[
0
0
(t)The reactant concentration decreases exponentially with time
kkt
kt
kt
kt
693.02ln21ln
]A[A][ln
]Aln[A]ln[
2/1
2/1
0
0
==
−=
−=
−=−
Half-life is constant for a first-order reaction
For the first-order reaction, half-life is independent of the initial reactant concentration
Second-order reaction with one reactant
2A → P
0
2/1
000
2/1
0
0
A][1
A][1
A][1
A][2
A][1
A][1[A]
[A]]A[
]A[
]A[]A[
A][
[A]
t
02
2
2
kt
kt
kt
dtkd
kdtd
kdt
dv
=
=−=
+=
−=
−=
=−=
∫ ∫
Time (t)
1[A]
1[A]0
Slope= k
Half-life is dependent on the initial reactant concentration
Pseudo first-order reactions
0B][ where [A]=[A],[B]When ]B][A[
kkkv
kv
=′′
>>=
e.g. B is water (55.5M): k'= 55.5k
Multistep reactions have rate-determining steps
Rate-determining step: the slow step (= the higher activation energy)
k1 k2
k1 > k2
k1 < k2
Catalysts reduce the activation energy
Rate enhancement = kcat/kuncat = e ΔEa/RT
e.g. When ΔEa = 5.7 kJ/mol, the reaction rate increases 10 foldWhen ΔEa = 34 kJ/mol, the reaction rate increases 106 fold
ΔEa(the reduction in Ea by the catalyst)
Michaelis-Menten equation
]S[]S[max
0 +=
MKVv
• v0 = the initial rate• Vmax = the maximum rate• KM = the Michaelis constant • [S] = the substrate concentration
Derivatization of Michaelis-Menten equation
]S[]S[
[E] :form [ES] in theentirely is enzyme when theion concentrat substratehigh at occurs )( velocity maximum The
S][[S][E]ES][ velocity initial The
whereS][
[S][E]ES][
S][
[S][E][ES]
[S][E]S])[ES][(
[S][E][S])[ES]()[ES](ES])[S][([E]
[ES][E][E]
ion)approximat state(Steady 0[ES]ES][[E][S] [ES]
ES][][
max0
max T2
max
T220
1
21T
1
21
T
T
1
21
T1121
21T1
T
211
2
+=
=
+==
+=
+=
++
=
=++
=+++=−
+=
=−−=
==
−
−
−
−
−
−
MKVv
kVV
Kkkv
kkkK
K
kkk
kkk
kkkkkkk
kkkdt
d
kdtPdv
M
M
M
E + S ES P + E k1
k-1
k2
Michaelis constant KM
• 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)
KM = [S] at which v0 = Vmax/2
Lineweaver-Burke plot
maxmax
max
max
1][S
11
][S]S[1]S[
][S
0
0
0
VVK
v
VK
v
KVv
M
M
M
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
+=
+=
kcat/KM is a measure of catalytic efficiency
[E][S][S][E][S][S]
[S][E] [E]ly consequent and formed is ES littlevery
[S] When
[E]
:enzymean ofnumber r or turnoveconstant Catalytic
MM
T
M
max
M
max0
T
, M
T
max
⎟⎠⎞
⎜⎝⎛≈=≈
+=
≈>>
=
Kk
Kk
KV
KVv
K
Vk
catcat
cat
Catalytic perfection
when maximal is ratio The
1
M
,12
21
21
M
2
M
kKk
kkkk
kkKk
Kk
cat
cat
=
>>+
==
−
−
(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
N
NHN
N
N
O
H2NH
NH
O
CO2-
CO2-
HN
NN
N
N
NH2
H2NCH3
NH
O
CO2-
CO2-
Dihydrofolate(Dihydrofolate reductase substrate)
Methotrexate(Dihydrofolate reductase inhibitor,
anticancer drug )
Modes of the reversible inhibition• Competitive inhibitors
– Binds to the substrate binding site
• Uncompetitive inhibitors– Binds to enzyme-
substrate complex• Non-competitive inhibitors
– Binds to a site different from the substrate binding site
• Mixed inhibitors– Binds to the substrate-
binding site and the enzyme-substrate
Competitive inhibition
I
T2
I
T220
I
T
II
T
T
II
1
21
211
I
[I]1 e wher]S[
]S[,[E] Since
S][[I]1
[S][E]ES][
S][[I]1
[S][E][ES]
1[I]1[S]
[ES][ES][S]
ES][I][[S]
ES][[E]
[ES][EI][E][E][S]
ES][I][[E][I][EI]
[S]ES][
[S]ES][[E]
ion)approximat state(Steady 0[ES]ES][[E][S] [ES][EI]
[E][I]
max0
max
KKVv
kVK
K
kkv
KK
KK
KKK
KK
K
Kk
kk
kkkdt
d
K
M
M
M
MMM
M
M
+=+
=
=
+⎟⎠⎞
⎜⎝⎛ +
==
+⎟⎠⎞
⎜⎝⎛ +
=
⎭⎬⎫
⎩⎨⎧
+⎟⎠⎞
⎜⎝⎛ +=++=
++=
==
=⎟⎠⎞
⎜⎝⎛ +
=
=−−=
=
−
−
αα
Uncompetitive inhibition
'[I]1' where
]S['
]S['
]S[']S[
,[E] Since
S]['
[I]1
[S][E]ES][
S]['
[I]1
[S][E][ES]
'[I]1
[S][ES]
'[ES][I][ES]
[S]ES][[E]
[ESI][ES][E][E]'
[ES][I][ESI]
[S]ES][[E]
[ESI][ES][I]'
I
T2
I
T220
I
T
II
T
T
I
I
max
max0
max
KK
V
KVv
kVK
K
kkv
KK
KK
KK
K
K
K
MM
M
M
MM
M
+=+
=+
=
=
⎟⎠⎞
⎜⎝⎛ ++
==
⎟⎠⎞
⎜⎝⎛ ++
=
⎟⎟⎠
⎞⎜⎜⎝
⎛++=++=
++=
=
=
=
α
α
αα
Uncompetitive inhibitors decrease both Vmax and KM
'[I]1'
IK+=α
Vmax
KMKMI
VmaxI
v0
[S]
no inhibitor (α'=1)Increasing [I]
]S[ [S];
'
;[S]
]S['
]S['
max0
max0
max
0
MM
M
KVvK
Vv
K
V
v
→>>
→∞→
+=
α
α
α
(Effects of an uncompetitive inhibitor become negligible)
Vmax/2
VmaxI/2
Mixed inhibition
'[I]1' and [I]1 where
]S[']S[
,[E] Since S]['
[S][E]ES][
S]['[S][E][ES]
']S[
]ES[']ES[[S][ES][E]
']ES[]E[[E]'
[I]1]ES[[I]1]E[[E]
[ESI][ES]]EI[[E][E]'
[ES][I][ESI]
[E][I][EI]
[ESI][ES][I]'
[EI][E][I]
II
T2
T220
T
T
T
II
T
T
I
I
II
max0
max
KKKVv
kVKkkv
K
KK
KK
K
K
KK
M
M
M
MM
+=+=+
=
=+
==
+=
⎟⎟⎠
⎞⎜⎜⎝
⎛+=+=
+=
⎟⎠⎞
⎜⎝⎛ ++⎟
⎠⎞
⎜⎝⎛ +=
+++=
=
=
==
αααα
αα
αα
αααα
αα
Noncompetitive inhibition
A special mixed inhibition when KI = KI’
]S[
]S[
])S[(]S[
' ,'When '
[I]1' and [I]1 where]S['
]S[
max
max0
max0
II
II
+=
+=
==
+=+=+
=
MM
M
K
V
KVv
KKKKK
Vv
αα
αα
αααα
Noncompetitive inhibitors affect not KM but Vmax
α
α
max0
max
0
;[S]
]I[1α where]S[
]S[
I
Vv
KK
V
vM
→∞→
+=+
=
I
[I]1K
+=α
Vmax
KM
VmaxI
v0
[S]
no inhibitor (α=1)Increasing [I]
Vmax/2
VmaxI/2
Determination of KIof the noncompetitive inhibitor
maxmax
M
[S]11
0 VVK
vαα
+⎟⎠⎞
⎜⎝⎛=
I
]I[1αK
+=
0 1/[S]
Slope=αKM/Vmax
α= 1(no inhibitor)
MK1
−
α = 2
α= 1.5Increasing[I]
1/v0
α/Vmax
Effects of pH• Binding of substrate to enzyme• Catalytic activity of enzyme• Ionization of substrate• Variation of protein structure
(only at extreme pHs)
EH + S ESH P + EH k1 k2
k-1
ES-
ESH2+
E-
EH2+
H+ H+
H+H+
KE2
KE1
KES2
KES1
]H[1]H[
]H[1]H[
where)/(and/
]S[]S[
ES2
ES1
2
E2
E1
1
21 2'
max'
max
'
'max
0
+
+
+
+
++=
++=
==
+=
KK
f
KK
f
ffKKfVV
KVv
MM
M
Approximate identity of catalytic amino acid residues
pKa ~4 → Catalytic Asp or Glu residuepKa ~6 → Catalytic His residue
pKa ~10 → Catalytic Lys residueCaution 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
OrderedRandom
• 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
Ordered sequential
Randomsequential
Competitive inhibition → Substrate and inhibitor competitively bind to the same site of the enzyme
Ping Pong
A vs Q: CompetitiveB vs P: CompetitiveA vs P: NoncompetitiveB vs Q: Noncompetitive
A vs Q: CompetitiveB vs P: NoncompetitiveA vs P: NoncompetitiveB vs Q: Noncompetitive
Under assumption of dead-end complex formation (A is similar with Q and B is similar with P)A vs Q: CompetitiveB vs P: CompetitiveA vs P: NoncompetitiveB vs Q: Noncompetitive
Dead-end complexes
Dead-end complex (no chemistry)
B
Q P
A
Q
E
B
A P no chemistry
ATP + Creatine ADP + Creatine-phosphate
competitive
competitive
Differentiating Bi Bi mechanisms by isotope exchange
Ping Pong MechanismA*→ P isotope exchange is possible without B
B* → Q isotope exchange is possible without A
or
Sequential MechanismTwo substrates are required for the isotope exchange
Glucose-fructose + phosphate Glucose-1-phosphate + fructose
Sucrose phosphorylase
E
Glucose-fructose + fructose* Glucose-fructose* + fructose
Glucose-1-phosphate + phosphate* Glucose-1-phosphate* + phosphate
Glucose-fructose + E Fructose + Glucose-E
Glucose-E + phosphate Glucose-1-phosphate + E
E
E
E
E
Ping Pong Mechanism (Double displacement)
Isotope exchanges in a ping pong mechanism
Isotope exchange experiments
(Sucrose)
Isotope exchanges in a sequential mechanism
Maltose phosphorylase
Glucose-glucose + phosphate
Glucose-1-phosphate* + glucoseE
Glucose-glucose + glucose*
Glucose-glucose + phosphate*
ENo isotope exchange
E
Sequential Mechanism (Single displacement)
Glucose-1-phosphate + phosphate*
Glucose-1-phosphate + glucose
No isotope exchange
E(Maltose)
Isotope exchange experiments
EGlucose-1-phosphate + glucose* Glucose-glucose + phosphate