logo in the name of god kinetics of enzyme & immobilized enzymes by sara madani

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IN THE NAME OF GOD

KINETICS OF ENZYME & IMMOBILIZED ENZYMES

BY SARA MADANI

Contents

Enzyme

Immobilized Enzymes

Kinetics of immobilized enzyme

Enzyme kinetics

Method of Immobilization

Introduction

Enzymes are usually proteins of high molecular weight (15,000 < MW < several million daltons ) that act as catalysts.

Enzymes are specific versatile, biological catalysts, resulting in much higher reaction rates as compared to chemically catalyzed reaction under ambient condition.

Enzymes are substrate specific and are classified

according to the reaction the catalyze.

Substrate & Enzyme

The substrate is a relatively small molecule and fit into a certain region on the enzyme molecule, which is a much larger molecule.

The simplest model describing this interaction is the lock-and-key model.

Immobilized Enzymes

The restriction of enzyme mobility in a fixed space is known as enzyme immobilization .

Why Immobilization?

Immobilization Immobilization

Advantages

Lower capital cost

enzyme reutilization

elimination of recovery & purification

provide a better

environment

METHODS OF IMMOBILIZATION

D

B

C

ABinding to Carriers

Immobilization by binding

Cross-linking

Matrix Entrapment

Membrane Enclosure

Immobilization by

Physical retention

Classification of enzyme immobilization method

Immobilized Enzyme

Kinetics of Enzyme in Solution

Enzyme Kinetics

Kinetic of Immobilized Enzyme

Enzyme Kinetic

IntroductionA mathematical model of the kinetics of single-

substrate-enzyme-catalyzed reaction was first developed by V. C. R. Henri and by L. Michaelli and M. L. Menten.Kinetics of simple enzyme-catalysed reaction are often referred to as Michaelis-Menten kinetics.

Mechanistic Models for Simple Enzyme Kinetics

Two major approaches used in developing a rate expression for the enzyme catalyzed reactions are:

rapid-equilibrium approach quasi-study-state approach.

Mechanistic Models for Simple Enzyme Kinetics

The rate of product formation:

The rate of variation of ES complex:

The eqn on the enzyme yields:

Both of them are the same in initial steps in deriving a rate expression.

The rapid equilibrium assumption

Assuming a rapid equilibriume between the enzyme & Substrate to form an [ES] complex.

The equilibriume constant is:

For [ES]:

Finaly:

where

The quasi- steady-state assumption

By applying this assumption to eqn 3 we find:

Subs enzyme eqn in eqn 9 yields:

Subs above eqn in to eqn 2 yields

Where

Kinetic of Immobilized Enyme

Many factors can cause the kinetic parameters of immobilized enzymes to differ from those of soluble enzymes.

2

Electrostatic and partitioing effects

3

Diffusional ,or mass-transfer effect

1

Conformational effects

This factors can be classified as follows

Effects Of the Electrostatic potential

The equilibrium condition requires that the electrochemical potential of the hydrogen ions in the particle equals with bulk phase

The distribution of the charged substrate between the particle and the bulk phase is then:

HHi 0

~~ AvoHH

i ZN

0

RT

NZaa AvoHi

H

lnln 0

TkZi

BeSS /0

TkZmapp

BeKK /

im

i

SK

Svv

max

TkZm

TkZ

B

B

eSK

eSvv /

0

/0max

Assume that the reaction catalyzed by the immobilized enzyme obey Michaelis-Menten kinetics

The rate of reaction expressed in terms of the local substrate concentration by:

the apparent Michaelis-Menten of the immobilized enzyme is:

Effects Of the Electrostatic potential

Effect of External Mass Transfer

Uncharged SupportSuppose the enzyme is immobilized to the surface of an

nonporous particleThe average flux of substrate to the fluid-solid interface can

be written :

At steady state, the enzymatic reaction rate must be exactly balanced by the rate of substrate transport to the catalyst surface ;therefore ,

This eqn can be cast into dimensionless form by introducing the following dimensionless variables

)( *0 SSkN ss

)( *0*

*max SSkSK

Svv s

m

0

max

00

** ;;

Sk

vDa

S

Kv

S

Sx

s

m

Thus we can write

Where Da,an important dimensionless group known as the Damköhler numbers To determine the significant effect of external diffusion resistance on the rate of enzyme catalytic reaction rate we use Da.The physical interpretation:

Da

x

vx

x *

*

* 1

vx

xvv

*

*max

osSk

vDa

'

diffusion of rate maximum

reaction of rate maximum max


When Da >> 1, the external diffusion rate is limiting; Da << 1, the reaction rate is limiting; Da ≈ 1, the external diffusion and reaction resistances are comparable.

In general form the observed reaction rate is:

But in the case of no diffusional limitations;that is whenAnd hence

vx

xvv

*

*max

0* SS

1* x

v

v

SK

Svv

mSS i

1

max

0

0max*Da << 1


Therefore, the observed rate can be related to the rate that whould be obtained in the absence of external diffusional limitations by

Where is known as the external effectivenss factor defined as

In terms of dimensionless quantities

v

vv E

1max

Soionconcentratsubstractebulkatevaluatedrate

ratereactionobservedE

vx

xvE

*

*)1(

E


The external effectiveness factor is a numerical measure of the influence of external mass transfer resistance on the observed reaction rate.

when the external mass transfer resistance is limiting.

When the external mass transfer rate is not limiting

1E

1E


For diffusion-limited regime

Therefore,the observed rate of reaction

At the other extreme, we have:

and

Da

vE

1

01Skv sDa

11

DaE v

vvDa

1max

1


c Charged SupportThe steady-state molar flux of charged substrate to planar

charged support can be written as :

Fickian diffusion Migration due to the gradient of the

electrostatic potential

dz

d

RT

FZDS

dz

dSDN sss


Integration and subsequent manipulation yields :

and the enzymatic reaction rate yields the steady-state relation

where

)( *0

eSS

MDN s

RT

ZF 0

00

1 )(exp

1dz

zM

*

*max*

0 )(SK

SveSSMkN

mss


We defind a modified Damköhler numbers as:

And apparent Michaelis-Menten constant,that accounts for both electrostatic and external mass transfer effect:

0

max

SMk

vDa

sM

*

*max*

0 )(SK

SveSSMkN

mss

)()1(

*

**

vx

xDaex M

)(

10

max,

eKSMk

veKK

msmappm


Michaelis-Menten Kinetics Enzymes are often immobilized to porous materials

with larg internal surface areas, All of the effective factor are typically incorporated into

a single diffusion coefficient, the effective diffusivity,

And

Effects of Intraparticle Diffusion

HDD pseff

)948.089.21044.21()1( 532 H

The general differential eqn for mass transfer is

If we assume that diffusion occurs in the radial direction only,the s-s material balance becomes

And dimensionless form is

)( isi SvNt

S

effim

i

eff

iii

DSK

Sv

D

Sv

drr

dS

dr

Sd

)(

)(2 max2

2

x

xDKvR

SD

SvR

rd

dx

rrd

xd effm

eff

i

1

)/()(2 max2

0

2

2

2


We combine main factor that exist in the eqn in a dimensionless parameter,Thiele modulus, defined for Michaelis-Menten Kinetics by:

2/1

max

3

effmDK

vR

x

x

SD

SvR

rd

dx

rrd

xd

eff

i

1

9)(2

0

2

2

2

0;10

1

rr rd

dxx

1

0 1max2

)(

)(3 rd

rx

rxvrvobs


And definition of effectiveness factor is

gradientsionconcentratrapelletofabsencetheinrate

ratereactionobservedl int

)]1/(1[3

)|/(

)]//(1/[

|/(32

1

max2

1

r

mo

reffml

rddx

KvR

rddxDK


Simultaneous External &Internal Mass-Transfer Resistances& Partitioning Effects

We begin by reconsidering the steady –state intraparticle mass balance for substrate in a spherical immobilized enzyme pellet:

Boundary conditions for eqn are

x

xDKvR

SD

SvR

rd

dx

rrd

xd effm

eff

i

1

)/()(2 max2

0

2

2

2

0| 0rrd

dx)1(| *

1 xBird

dxr

The new parameter appearing here is the Biot number,Bi defined as

In the presence of partitioning effects,the equilibrium concentration of substrate with in the pellet wil differ from that in outside liquid.

If Bi>100 the effects of external resistance are not significant

ratediffusioneerapasticlsticcharacteri

ratetransportfilmsticcharacteri

D

RkBi

eff

s

int

** SKS pi

oP

i

r SK

SBi

rd

dx *

11|

Simultaneous External &Internal Mass-Transfer Resistances& Partitioning Effects

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Reference

[1] Shuler,ML. and Kargi, F.”Bioprocess Engineering:Basic Concepts”2nd Edition, 2005.prentice-Hall Inc.

[2] Harvey W.Blanch. And Douglas S. Clark.”Biochemical Engineering”1996,MARCEL DEKKER,INC., New York,USA.

[3] http://www.wsu.edu/~jmlee/eBioCheSample.pdf

[4]http://www.cheric.org/ippage/e/ipdata/2004/05/file/e200405-1101.pdf

www.themegallery.com

http://www.wsu.edu/~jmlee/eBioCheSample.pdf

http://www.cheric.org/ippage/e/ipdata/2004/05/file/e200405-1101.pdf

logo in the name of god kinetics of enzyme & immobilized enzymes by sara madani

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