enzyme kinetics - chapter 4

8/12/2019 Enzyme Kinetics - Chapter 4

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Why study enzyme kinetics?

To quantitate enzyme characteristicsdefine substrate and inhibitor affinities

define maximum catalytic rates

Describe how reaction rates vary with reaction

conditions Provide an understanding of an enzymes role in a

metabolic pathway

Define the conditions under which the rate of thereaction is proportional to the amount of enzymepresentbiochemical and clinical enzyme assays


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Enzyme Kinetics


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First- and Second- order reactions

First-order Second order

PA

][][

Akdt

Adv

][

][ 00

]ln[A

A

t

dtkAd

ekt

oAA ][][

ktAA 0]ln[]ln[

PA2

2][][ AkdtAdv

tA

Adtk

A

Ad

02

][

][

][

][

0

ktAA o

][

1

][

1


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Half-life of first-order reactions

When [A] = 1/2 the initial concentration [A]0

ekt

oAA

][][

ekt

oAA

2/1][

2

][ 0

2/1][

2/][lnthen kt

A

A

o

o

kkt

693.02ln

and 2/1

(For a second-order reaction:

t 1/2= 1/k [A]o)

ktAA 0]ln[]ln[


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e kt

A

A

0][

][

ktAA

A

][1][

][

0

Decay curves for 1st and 2nd orderreactions of same half-life


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First- and Second- order reactions

First-order Second order

Ln[A]

Time Time

1/[A]

ln[A] = ln[A]0-kt 1/A = 1/A0+ ktln[A]0

Slope = -k

1/A0

Slope = k


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[A][B]k

or at equilibrium:

U + PV

G=H- TS

for most biological systems:

U


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QPXBA k

K

'

][']][[][

XkBAkdt

Pd

]][[

][

BA

XK

GKRT ln

]][['][ / BAek

dt

Pd RTG

'k

h

Tkk B

'

RTGB e

h

Tkk /

Thermodynamics of the Transition State

h/TkB


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11/36

Plots of Initial rates and Reactant ConcentrationsDiffer for enzymatic and non-enzymatic reactions

Non-enzymatic enzymatic

[S]

RATE

[A]

RATE


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-ES complex is formed

-[E]


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where Km = (k-1+ k2)/k1

since [S] = [St] early in the reaction and [E] = [Et] - [ES] then

PEESSE k

k

k

2

1

1 0][][]][[][

211

ESESSEdt

ESdkkk

][]][[

21

1 ESSE

kk

k

][)

]][[

121(

ESSE

kkk

K

SEES

m

]][[][

K

SESEES

m

t ]])[[]([][

Michaelis-Menten ModelDerivations


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K

SESEES

m

t ]])[[]([][

K

SES

K

SEES

mm

t ]][[]][[][

][as)][1(

][][][

2 ESv

KS

KSEES k

m

mt

KSEKSES mtm ]][[)][1]([

][Km

][Vmax

][

]][[2

S

S

SKm

SEtv k

Michaelis-Menten ModelDerivations (contd)


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Km and VmaxDerivation and significance

When [S] >> Km then v = Vmax [S]/[S] = Vmax

When v = Vmax/2 x then Vmax = 2 Vmax [S] Km + [S]

or Km + [S] = 2 [S] and Km = [S]

since Km = (K -1+ K2)/K1 thus when K2


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Lineweaver_BurkDouble reciprocal plot

1/Vmax

slope=Km/Vmax

1/ [ S]

1/v

-1/Km

][Km

][Vmax

S

Sv

Vmax

1

][

1

Vmax

Km1

Svbmxy


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Graphical Representations of Changes in

Km and Vmax

v

V m a x

V m a x

[S ]

2

1

K m

v

Vmax

[S]Km Km1 2


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Graphical Representations of Changes in

Km and Vmax (contd)

1/[S]

1/v

Vmax

Km same

Km

Vmax same


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Use of turnover number and catalytic

efficiency as measures of enzyme behavior

For the above reaction at saturatingsubstrate concentrations Vmax = k2[Et]or Vmax = kcat[Et]

v = Vmax [S] Km + [S]

PEESSE k

k

k

2

1

1

kcat= turnover number = Vmax

[Et]

turnover number is the first order rateconstant describing the number ofmolecules of substrate converted toproduct per molecule of enzyme in

units ofsec-1

At low [S], v approaches the value: v= Vmax[S] or kcat[Et] [S] Km Km

v = kcat[Et] [S] = keff [Et] [S]

Km

Thus, the catalytic efficiency (keff) ofan enzyme is a second order rate

constant in units of M-1sec-1

For highly efficient enzymes whoserates are nearly diffusion limiting:

keffis near 108to 109M-1sec-1

carbonic anhydrase = 8 x 107M-1sec-1

acetylcholinesterase =1.5 x 108M-1sec-1


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Kinetic constants for some enzymes and

substrates


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Reversibility of the reaction adds complexity

to the rate equation

PEESSE

k

k

k

k

2

2

1

1

P

M

S

M

P

M

r

S

M

f

K

P

K

S

K

PV

K

SV

v][][

1

][][ maxmax

For the simple reversible reaction

The velocity of the reaction is:

WhereT

f EkV ][2max Tr EkV ][1max

1

21

k

kkK

S

M

2

21

k

kkKPM

When [P] =0, i.e. when v = v0then the more familiar

Michelis-Menten form is

evident:

][

][

][1

][max

max

SK

SV

K

S

KSV

vS

M

f

S

M

S

M

f


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3O

F

CH

3CH

3CH

3CH

OH SER-PROTEIN

B:

3 O

F

CH

3CH

3CH

3CH

O

SER-PROTEIN

B:H

-

HO

H

3 OCH

3CH

3CH

3

CH

O

SER-PROTEIN

O H

H

B:

FAST VERY

SLOW

C H - O - P - O - C H C H - O - P - O - C H C H - O - P - O - C H

Enzyme Inhibition:

Irreversible


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Enzyme Inhibition:

Reversible

ENZ ENZ ENZ

S

P

+

FAST

ENZ

I

I

+I

+S

S

Competitive

S

S

P

I

S

P

I

+S

+ I

FAST

Un-competitive


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S

I

S

P

I I

S

P

I

+S

+ I+ I

+S

FAST

Non-competitive

Enzyme Inhibition:

Reversible (mixed)


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Enzyme Inhibition:

Reversible (competitive)

v

V m ax

[S ]

K m K m

-I

+ I

I

m

KI

SK

SVv

][1

][

][max


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1/ [S]

1/v

1 / V m a x

-1/Km

-1

S L O P E = K m V m a x

S L O P E = K m /V m a x

+ I

-I

K m

I

m

K

I

VSV

K

v

][1where

1

][

11

maxmax

Enzyme Inhibition:



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Enzyme Inhibition:



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Enzyme Inhibition:Reversible (uncompetitive)

E E S E + P

E IS E I + P

I

S

[ES][ I ]

[E IS ]K i =

V m a x [S ]

K m + [S ]V=

v

V m a x

V m a x

[S ]K m

-I

+ I

K m /

=(1 + [I]/Ki)


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Enzyme Inhibition:Reversible (uncompetitive)

maxmax ][

11

VSV

K

v

m


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Enzyme Inhibition:Reversible mixed (non-competitive)

E E S E + P

E I E IS E I + P

I I

S

S

[E + ES ] [I]

[E I + E IS ]K i =

V m a x [S ]

K m + [S ]( )V =

v

V m a x

V m a x

[S ]

K m

-I

+ I

=(1 + [I]/Ki)


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K m

[S ]V m a x

1

V m a x+

1

v=

1/ [S]

1/v

-I

+ I

V m a x

1 / V m a x-1/Km


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E E S E + P

E I E IS E I + P

I I

S

S

V m a x [S ]

K m + ' [S]V =

v

V m a x

V m a x

[S ]

K m / '

-I

+ I

'K m

][

]][[

][

]][[

'

ESI

IESK

EI

IEK

I

I

IK

I][1

'

][1'

IK

I



34/36

maxmax

'][

11VSV

Kv

M



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FOR

A= 0.10; t = 1 MIN.; PATH = 1 CM

VOLUME = 1 ML

RATE = (0.16 X 10 M/MIN)(0.001 LITERS)

= 16 NMOLES/MIN

-4

-4

L=

t

I o I

CUVETTE

DETECTOR

ABSORBANCE = A = LOG (I/ I o)

BEER'S LAW A = CL

L

C = CONCENTRATION (MOLAR)L = PATH LENGTH (CM)

= MOLAR EXTINCTION COEFFICIENT

(M CM )-1 -1

= 6.23 X 10 M CM

FOR NADH AT 340 nm

3 -1 -1

A/t C = 0.16 X 10 M/MIN

TIME

ABSORB

ANCE

t

A

LACTATE + NAD+ PYRUVATE +

H+ + NADH

Use of Substrate or Product Absorbance to Measure

Rates of a Reaction

enzyme kinetics - chapter 4

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