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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
Effect of External Mass Transfer
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
Effect of External Mass Transfer
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
Effect of External Mass Transfer
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
Effect of External Mass Transfer
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
Effect of External Mass Transfer
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
Effect of External Mass Transfer
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
Effect of External Mass Transfer
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
Effect of External Mass Transfer
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
Effects of Intraparticle Diffusion
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
Effects of Intraparticle Diffusion
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
Effects of Intraparticle Diffusion
And definition of effectiveness factor is
gradientsionconcentratrapelletofabsencetheinrate
ratereactionobservedl int
)]1/(1[3
)|/(
)]//(1/[
|/(32
1
max2
1
r
mo
reffml
rddx
KvR
rddxDK
Effects of Intraparticle Diffusion
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
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