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Biochemical Kinetics

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2 2

2

2

2

2

2

2 2

2 2 2

2 D

D

M + O M-O

[ ][ ]Dissociation Constant,

[ ]

[ ]Fraction of binding site occupied,

[ ] [ ]

[ ][ ][ ]

[ ][ ] [ ][ ]

When pO = K , =0.5:

K is the oxygen tenstion at

D

D

D D

D

M OK

MO

MO

O M

M OK O pO

M O O K pO K M

K

θ

θ

θ

=

=+

= = ≈+ ++

2

2 50

which myoglobin

is half saturated, pO

pO p

θ =

+

Kinetics Binding of Oxygen to Myoglobin

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Kinetics Binding of Oxygen to Myoglobin

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Transfer of Respiratory Gases

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Myoglobin Can Not Be Used For O2 Transport

Tissue Lung

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Hemoglobin Is Used For O2 Transport

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O2 Binding – Hb vs. Mb

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O2 Transport Capability, Hb vs Mb

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K = 1

n =1

n =2n =4

1

n

n

n

n

x

K y x

K

=

+

Behavior of Sigmoidal Binding Equation

Sharp and Faster kinetics

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1

n

n

n

n

x

K y x

K

=

+

K = 1

K = 3

n = 3

Behavior of Sigmoidal Binding Equation

Decrease in affinity

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2

0.5

2

0.5

2

0.5

1

1

n

n

n

n

n

n

pO

p

pO

p

pO p

θ

θ

θ

=

+

=−

This is called Hill Equation. Hill Coefficient = nH Hill Constant = KD

nH = 1: No cooperation; nH > 1: Positive cooperation

•For Hemoglobin n = 3 not 4

•n is represented as nH

• nH represent cooperativity

in binding , not the exact

stoichiometry

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Cooperativity Model (Allosteric Model) For Hemoglobin

Monod, Wyman, and Changeaux (MWC)

• Only T and R conformations exist • T has lower affinity for O 2 .• The two states are in equilibrium

• T

R transition involves shift in equilibrium constant

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Hemoglobin Has T and R States

T (Low Affinity) R (High Affinity)

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Buffer in Physiological System

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pKa = 6.86

pKa of Histidine side chain = 6


Phosphate is the most common buffer

Proteins also act as buffer

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pKa = 6.3


Carbonic acid also acts as buffer

Status of carbonic acid depends upon 2 other equilibriums

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Transport of Carbon Dioxide

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Bohr Effect

Low pH His146 of β chain is protonated Stabilizes T state

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Bohr Effect

• CO2 reacts with N terminal of all 4 chains of Hb.

• Release of H+, induces Bore Effect

• Carbamate also stabilizes T state

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Bohr Effect

Haldane effect: Deoxygenation of the blood increases its ability to carry carbon

dioxide;

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BPG Happens in RBC

BPG is formed only in RBC.

Production of BPG increases in Hypoxia.

BPG binds to T state of Hb and stabilizes that

Another Allosteric Control of Hemoglobin

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BPG helps to adopt to high altitude and in hypoxia

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Effect of Temperature on Hemoglobin

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Transport of Carbon Dioxide

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Fetal hemoglobin has 2 and 2 chains

• The γ chain is 72% identical to the β chain.

• The binding affinity of fetal hemoglobin for BPG is significantly lower thanthat of adult hemoglobin

• Thus, the O2 saturation capacity of fetal hemoglobin is greater than that

of adult hemoglobin

Fetal Hemoglobin

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Enzyme Kinetics

E

S P

E + Sk 1

<<<<−−−−−−−−>>>>

k −−−−1

ESk 2

<<<<−−−−−−−−>>>> P + E

An enzyme-catalyzed reaction of substrate S to product P, can bewritten as

In terms of mechanism the same reaction can be written as

k -2

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Assumptions

1.1. EquilibriumEquilibrium: the association and dissociation of the substrate and: the association and dissociation of the substrate and

enzyme is assumed to be a rapid equilibriumenzyme is assumed to be a rapid equilibrium

2.2. Steady stateSteady state: the enzyme substrate complex ES is at a constant: the enzyme substrate complex ES is at a constantvalue.value.

For this to happen [S] >> [E]For this to happen [S] >> [E]TT..

This is true if we only study theThis is true if we only study the initial velocityinitial velocity..

3. P ES not considered

4. Rate limiting step: ES P, (as the E + S ES is a rapid equilibrium)

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E + S

k 1

<<<<−−−−−−−−>>>>

k −−−−1

ES

k 2

−−−−−−−−>>>> P + E

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E + S

k 1

<<<<−−−−−−−−>>>>k −−−−1

ES

k 2

−−−−−−−−>>>> P + E

1 1 2[ ] .[ ].[ ] .[ ] .[ ]d ES k E S k ES k ESdt

−= − −

[ ]0 :

d ES

dt =

As we considered steady state for ES:

1 1 2

1 1 2 1 2

.[ ].[ ] .[ ] .[ ] 0;

.[ ].[ ] .[ ] .[ ] ( ).[ ];

k E S k ES k ES

k E S k ES k ES k k ES

−

− −

− − =

= + = +

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( [ ])[ ] [ ] .[ ]

[ ] .[ ][ ]

( [ ])

m T

T

m

K S ES E S

E S ES

K S

+ =

=+

1 1 2 1 2

1 2m

1

.[ ].[ ] .[ ] .[ ] ( ).[ ];

( )Define: K =

k E S k ES k ES k k ES

k k

k

− −

−

= + = +

+

1 1 2.[ ].[ ] ( ).[ ];

.[ ] [ ].[ ]

=([ ] [ ]).[ ]

m

T

k E S k k ES

K ES E S

E ES S

− = +

=

−

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As ES P is rate limiting step:

2Initial velocity of product formation, k .[ES]ov =

2 2 [ ] .[ ].[ ] .( [ ])

T o

m

E Sv k ES k K S

∴ = =+

max

max 2

Maximum initial velocity, v is achived when allenzymes are in use:

.[ ]T v k E =

max2

.[ ][ ] .[ ]. =( [ ]) ( [ ])

T o

m m

v S E Sv k

K S K S∴ =

+ +

Michaelis–Menten equation

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max

1

if [ ], 2m oK S v v= =

max.[ ]

( [ ])om

v Sv

K S= +

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LineweaverLineweaver--BurkBurk (double reciprocal plot)(double reciprocal plot)

max

max max

.[ ]

( [ ])

1 1 1.[ ]

om

m

o

v Sv

K S

K

v v S v

= +

= +

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E + S

k 1

<<<<−−−−−−−−>>>>k −−−−1

ES

k 2

−−−−−−−−>>>> P + E1 2

m

1

( )K =

k k

k −

+

2 1

1 2 1m

1 1

If :

( )K = D

k k

k k k K

k k

−

− −

<<+

= =

In this case, Km is the measure of affinity of the Substrate for Enzyme

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Measuring efficiency of an enzyme:

Turnover number, kcat = vmax /[E]T

(Rate normalized over amount used in assay;

Unit is 1/t; )

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Specificity Constant = kcat /Km

High affinity of substrate for enzyme Low KD Low Km

Require low amount of S to reach ½. Vmax

High turnover of reaction High kcat

+

High Specificity

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max

m

.[ ]

( [ ])

.[ ] .[ ]

( [ ])

for [S]<<K : [ ] .[ ]

o

m

cat T

m

cat

o T m

v Sv

K S

k E S

K S

k

v E SK

=+

=+

=

Kcat /Km is a second order rate constant. The upper limit of Kcat /Km is fixed by

the rate of diffusion of moleculesUpper limit is ~ 108 to 109 1/M.Sec

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•• KKmm relates to affinity ;relates to affinity ;

•• VVmaxmax relates to efficiencyrelates to efficiency::

•• KKmm tell how much substrate to use in an assaytell how much substrate to use in an assay

•• If more than one enzyme share the same substrate, KIf more than one enzyme share the same substrate, Kmm

determines how to decide which pathway the substrate will takedetermines how to decide which pathway the substrate will take

VVmaxmax helps to identify rhelps to identify rate limiting enzyme/step in pathwayate limiting enzyme/step in pathway

KKmm andand VVmaxmax can be used to determine effectiveness of inhibitorscan be used to determine effectiveness of inhibitors

and activators for enzymeand activators for enzyme

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Competitive Inhibition

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max

max max

.[ ]

( [ ])

1 1 1

.[ ]

Where,

[ ] [ ][ ]1 ;

[ ]

Apperent

o

m

m

o

I

I

m m

v Sv

K S

K

v v S v

I E I K

K EI

K K

α

α

α

α

=+

= +

= + =

=

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Uncompetitive Inhibition

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max max

maxmax

1 1 '.[ ]

Where,

[ ] [ ][ ]' 1 ; '

' [ ]

apperent'

m

o

I

I

K v v S v

I ES I K

K ESI

vv

α

α

α

= +

= + =

=

Uncompetitive Inhibition

Mixed Inhibition

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Mixed Inhibition

Mixed Inhibition

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Mixed Inhibition

max max

1 1 '.[ ]

Where,

[ ] [ ][ ]1 ;

[ ]

[ ] [ ][ ]' 1 ; '' [ ]

m

o

I

I

I

I

K

v v S v

I E I K

K EI

I ES I K K ESI

α α

α

α

= +

= + =

= + =

Irreversible Inhibition

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Irreversible Inhibition

Bacterial Cell wall is made up of

peptidoglycans

Tanspeptidase catalyze cross

linking of peptidoglycans

β-lactum antibiotics irreversibly

bind (react) to transpeptidase

and stops the reaction.

Allosteric Regulation

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Example: Aspartate

carbamoyltransferase (ATCase)

Catalyze one step in pyrimidine

biosynthesis

Made up of two polypeptides,catalytic (C) and regulatory (R):

ATCase C3R2


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CTP inhibit

ATP Stimulate

ATCase does not follow

MichaelisMichaelis--MentenMenten

kineticskinetics rather have

kinetics similar to Hill

kinetics

kinetics 4

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