Chapter 14. Enzyme Kinetics

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

Arrhenius equation

= Activation energy (Ea)

k = Ae-Ea/RT

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

Steady state approximation

E + S ES P + E k1

k-1

k2

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

Competitive inhibitors affect KM

αα

of regardless ;[S] ]S[

]S[

max0

max0

VvKVv

M

→∞→+

=

Determination of KIof the competitive inhibitor

maxmax

M 1[S]11

0 VVK

v+⎟

⎠⎞

⎜⎝⎛=

α

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

Determination of KI’of the uncompetitive inhibitor

maxmax

M '[S]11

0 VVK

vα

+⎟⎠⎞

⎜⎝⎛=

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

Lineweaver-Burk plot of a mixed inhibition

maxmax

M '[S]11

0 VVK

vαα

+⎟⎠⎞

⎜⎝⎛=

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 inhibitors on Vmax and KMof the Michaelis-Menten equation

]S[]S[max

0 += app

M

app

KVv

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