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Kinetic model for oligosaccharide hydrolysis using suspended and immobilized enzymes
Biochemical
Engineering
Journal
1 ( 1998)
Biochemical Engineering Journal
201-209
Kinetic model for oligosaccharide hydrolysis using suspended and immobilized enzymes Naomi Shibasaki-Kitakawa a, Benjamas Cheirsilpa a, Ken-Ichiro Iwamura a, Masanori Kushibiki b, Akio Kitakawa ‘, Toshikuni Yonemoto a7* ‘Department
’ Departmeni of Chemical Engineering, Tohoku University, Sendai, Japan hAomori Advanced industrial Technology Center, Aomori, Japan of Materials Science and Engineering, Miyagi National College of Technology, Received
6 October
1997; accepted
13 January
Natori,
Japan
1998
Abstract A novel kinetic model that describes the hydrolysis of oligosaccharide usingthe suspended enzyme has been constructed by introducing the selectivity of the enzyme for cleaving each (Y-1,6 glycosidic bondof the substrate. The Michaelis-Mententype kineticconstants,K,,, and V maxI and the selectivity coefficient, cr, are estimated by fitting the model with the experimental data obtained under various conditions. The
newconstant,a, is estimatedat 0.411,andalmostthe sameasthe experimentalvalues.Themodelhasbeenextendedfor the immobilized enzyme system by taking into account the intraparticle
mass transfer resistance. The model constants are estimated similarly to the case of
the suspended enzymesystem.aimm( LYfor the immobilizedenzymesystem)is a little higherthan that in the suspended enzymesystem. Kmmm and V,,‘-‘, aremuchgreaterthanthosefor the suspended enzymesystem.The estimatedvaluesof the effective diffusivitiesin the supportparticlearea few orderof magnitudesmallerthanthosein aqueoussolution.The modelwell simulates both experimentalresults obtainedin the suspended andimmobilizedenzymesystems. 0 1998ElsevierScienceS.A. All rightsreserved. Keywords:
Hydrolysis
of oligosaccharide;
Suspended
enzyme;
Immobilized
Recently, it hasbeenreported that oligosaccharideshaving physiological activity can be obtained by the enzymatic hydrolysis of polysaccharidecontained in agricultural waste [ 11. Industrialization of this enzymatic reaction processwill s&dkanl>y contribute 2Dwmi 2he effecthe U2%222iDT2 of agricultural
resources.
High molecular weight saccharidesare transformed into Iower molecular weight saccharidesby consecutive depolymerization using enzymes. Several researchers have attemptedto construct the mathematicalmodelsthat describe this consecutivedepolymerization kinetics [ 2-71. Fujii et al. [ 31 have proposeda lumped model in which all oligomers andpolymers other than a monomerwereconsideredtogether as one substrate. This model was quite simple and could estimatethe rangeof systemparameters(Km and V,,,,, values and substrateconcentration), but it could not predict the * Corresponding author. Department of Chemical Engineering, Tohoku University, Aoba, Aramaki, Aoba-ku, Sendai 980-8579, Japan. Tel.: + 8122-217-7255; fax: + 81.22-217-7258;e-mail: [email protected] 1369-703X/98/$ - see front matter 0 PIIS1369-703X(98)00003-5
1998 Elsevier
enzyme;
Kinetic
model
mo‘lecu‘larwei@ distributions of the substrateand product. Suga et al. [6] recognized the limitations of the lumped model and proposeda distributed model. In their model, the differential equations following Michaelis-Menten kinetics were formulated for each substratespecieswith the various polymerization degrees.The underlying basisfor the model was more c\osely P‘qwxutative of the actual events present in &se c~~3p~exsystems,but expetimental verification of the theoretical model was not published. Dean and Rollings ] 2] have extended the mathematicalmodelproposedby Suga et al. [S). This model accounted for the fact that a certain number of bondsof the substratewere inaccessibleto hydrolytic action. However, the predictions by their model were inconsistentwith the experimental results. Enzyme is often usedin an immobilized form in industry for the purpose of repeated or continuous uses.When the enzyme is immobilized to a porous support matrix, the intraparticle masstransfer has a great influence on the kinetics. It is therefore necessary to develop comprehensive models accounting quantitatively for the diffusional effects. However, mere is little researchinvolved in the analysis of the
Science S.A. All rights reserved.
202
N. Shibasaki-Kitakawa
et al. /Biochemical
kinetics of consecutive reactions catalyzed by the immobilized enzyme. In this research, the hydrolysis of oligosaccharide using both suspended and immobilized enzymes was carried out experimentally under various conditions. The oligosaccharide of the polymerization degree of 6 was chosen as a model high molecular weight substrate, because the mechanism of consecutive depolymerization was comparatively simple, and the concentration could be easily analyzed. On the basis of the experimental results, a novel mathematical model, which described the hydrolysis kinetics of oligosaccharide using suspended and immobilized enzymes, was constructed.
Engineering
Journal
1 (1998)
enzyme ranged from 1.16 to 2.1 mg dme3, from 1.3 to 2.0 mol dmV3 and from 1.O to 1.5 g, respectively. The uptake amount of the saccharides by the support particle was also measured under the partition equilibrium condition. A weighed amount of support particles was immersed into an aqueous solution of the saccharide. After a 24-h shaking, the saccharide concentration in the supematant solution was measured. Isomaltohexaose and isomaltose were used as the saccharides. The initial substrate concentration and the amount of support particle ranged from 1.5 to 2.0 mol dm-3 and from 3.0 to 4.0 g, respectively.
3. Mathematical 2. Materials
201-209
model
and methods 3.1. Hydrolysis
2.1. Materials Dextranase from PhenisilEium sp. purchased from Sigma was used as enzyme. Isomaltohexaose (Seikagaku), composed of (Y-1,6-linked d-glucose, was used as the oligosaccharide. The Affi-Prep 10 (Bio-Rad Lab.), which is macroporous methacrylate co-polymer bead activated with an N-hydroxysuccinimide ester, was used as a support material for the immobilization of enzyme. Its nominal particle size and pore diameter are 50 pm and 0.1 km, respectively. Other chemicals were of reagent grade and purchased from wako. 2.2. Immobilization
of enzyme
The support particles, Affi-Prep 10, were immersed in distilled water overnight to remove isopropanol used as a storage solution. Five grams of the support particles was added to 30 cm3 dextranase aqueous solution (2.32 mg cme3) and gently agitated at 4°C for 12 h. The immobilized dextranase was then filtered and washed with distilled water. The concentration of unimmobilized enzyme, which remains in the supernatant solution, was measured by a spectrophotometer at 595 nm to determine the immobilized amount of enzyme.
kinetics using the suspended enzyme
The endo-acting enzyme is known to cleave the bonds in the interior of the substrate main chain. Therefore, as shown in Fig. 1, the endo-dextranase cleaves the (Y- 1,6 glycosidic bonds in either position I or II of isomaltohexaose. Isomaltohexaose is hydrolyzed to either two isomaltotrioses or isomaltotetraose and isomaltose. In the latter case, isomaltotetraose is further converted into two isomaltoses. In the kinetic model constructed by Dean and Rollings [ 21, the endo-acting enzyme is considered to evenly cleave the respective bonds in the interior of the substrate main chain. However, the predictions by their model are inconsistent with the experimental results. In this study, a novel kinetic model has been constructed by introducing the selectivity of the enzyme for cleaving the o-1,6 glycosidic bond in position I, defined as a selectivity
2.3. Experimental procedures The rate of oligosaccharide hydrolysis by suspended and immobilized enzymes was investigated at 37°C in a 500-cm3Erlenmeyer flask containing 100 cm3 substrate solution. The flask was shaken in a water bath at the well mixed condition. The reaction was started by adding the enzyme to the substrate solution. The samples werepipetted at specified timeintervals and mixed with a 0.1 M solution of NaOH in a volume ratio of 1: 1 to stop the hydrolysis. The respective saccharide concentrations in the sample solution were determined using an HPLC system equipped with a gel permeation column (NHC18, Hitachi, Japan) and an IR detector (YRU-880mdget, Shimarnura, Japan). The suspended enzyme concentration, the substrate concentration, and the amount of immobilized
Fig. 1. Michaelis-Menten endo-dextranase,
type hydrolysis
process of isomaltohexaose
using
N. Shibasaki-Kitakawa
et al. /Biochemical
coefficient, LY.For the purpose of the simplification of the model, the rate constants for the isomaltotetraose hydrolysis are assumed to be the same as those for the isomaltohexaose hydrolysis (this assumption is also employed by Dean and Rollings [2]). Using Michaelis-Menten type kinetics as shown in Fig. 1 and the selectivity coefficients, LY,the rates of change in the concentration of the respective saccharides are given as: dC 6 =-3klCEC6+k2CES6 dt
Engineering
5
=(1-a)k3CES6+2k3CESl
(5)
dGs4 dt
(6)
=klCEC4-(k2+k3)CES4=0
The total concentration of enzyme in the system is cEm=G+cEs,+Gs,
(7)
Rearranging Eqs. ( 1 )-( 4)) we get: -3vltl,XC6 K,+3C,+C,
dC4
(8)
Vmax[3(i-w6-C4i
dt -03=
K,+3C6+C4 6Vnl&C6
(9) (10)
K,+3C6+C4
[
CCi,calc.
3.2. Hydrolysis
d&s, =3klC,C,-(k,+k,)C,,,=O dt
203
1 2
+ci,exp.)/2
To obtain the calculated values under a certain set of constants, the simultaneous differential equations, Eqs. (8)( 11) , are solved numerically using the Runge-Kutta method. Time step for the numerical calculation is set at 60 s.
where Ci and CES, are the concentrations of the saccharide and the enzyme-substrate complex in the solution, respectively. The subscript i is a counter corresponding to the polymerization degree of the saccharides. k,, k2 and k3 are the rate constants. Further assuming a quasi-steady-state for the concentration of the enzyme-substrate complex, we get:
dt =v4= dC,-
201-209
C,( 0). Thus, the maximum reaction rate per unit weight of enzyme, V,,,‘( = V,,,,,/C,(O)), which is the characteristic value of enzyme, is used for the calculation in the place of VInax*These three constants are estimated using the Simplex parameter seeking method [ 81 to minimize S, which is the sum of square of the relative error between the calculated values of the saccharide concentrations and three sets of experimental data obtained under different reaction conditions, defined as follows:
‘=’
=(l-a)k3CEs,-k,C&+k2CESq
dt -“‘=
1 (1998)
Ci,calc.-Ci,exp.
%
dC,-
Journal
kinetics using the immobilized enzyme
The kinetic model that describes the hydrolysis of the oligosaccharide by the immobilized enzyme mainly consists of the mass balance equations for the liquid and support particle phases. They are formulated on the basis of the following assumptions. ( 1) A perfect mixing is established in the liquid phase. (2) The support particles are homogeneous and spherical, and their sizes are constant. (3) The enzyme is uniformly immobilized inside the support particles. (4) The extraparticle mass transfer resistance is negligible because the reactor is under well mixed condition. (5) There exists the linear partition equilibrium of the saccharides between the liquid phase and the surface of the support particle. clrcRp=
HC,
(14)
(6) The intraparticle mass transfer fluxes of the respective saccharides follow Fick’s law, and the effective diffusion coefficients, Deff, are described using a correlation similar to that reported by Kimura and Nakao [9] for saccharides in aqueous solution, as follows: DFff=A.MiB
(15)
Vm,,[3(1-a)C6+2C41
K,+3C,j+C,
(
K,=
k,+k,
(11)
vnl.3, =k,C,(O)
-9
k,
i.c. t=O; Ci=
C,(0) 0
(i=6) (i=2,3,4)
(12)
In the case of (Y= l/3, the model is identical with that of Dean and Rollings [ 21. There are three unknown constants, K,,,, V,,, and Q, in the model. V,,,,, depends on the initial enzyme concentration,
where A and B are the regression coefficients and M, is the molecular weight of the saccharides. These assumptions give the distribution of the saccharide concentration as shown in Fig. 2. The mass balance equations of the respective saccharides in the liquid phase can be described as: dCi
l-cd(c)
-Tt+
&
i.c. t=O; Ci=
I--E
dt
- E
C,(0) 0
-
(cl,)=0
(161
(i=6) (i=2,3,4)
(17)
204
N. Shibasaki-Kitakawa
et al. /Biochemical
Engineering
Journal
r=R,;
c=HCi
I (1998)
201-209
(26)
where His the partition coefficient. The initial conditions at t = 0 are as follows: liquid
Fig. 2. Saccharide
concentration
distribution
in immobilized
Olr
enzyme
system.
where E is the reactor voidage. The bracketed values (c) and (c> are, respectively, the volumetric average concentration and hydrolysis rate of each saccharide in the support particle determined as:
(18)
C,(O)=0
(27)
There are six unknown constants, H, KF’“, V&‘, aimm, A, B, in the model for the immobilized enzyme system. The maximum reaction rate per unit weight of enzyme, VAT,“’ ( = V$~/C~mm( 0) ), is used for the calculation in the place of V&r similar to the suspended enzyme system. These constants, except for H, are estimated by fitting four sets of experimental results obtained under various conditions using the simplex method [ 81 similarly to the case of the suspended enzyme system. To obtain the calculated results under a certain set of constants, Eqs. ( 16)-( 27)) are approximately formulated with a finite difference scheme and solved numerically using the iterative method. For numerical calculation, the time step is set at 15 s, and the r coordinate in the support particle is divided into 50 points. The partition coefficient, H, is determined from the partition equilibrium experiment.
4. Results and discussion (19) 4.1. Hydrolysis c is the intraparticle local values of the concentration. 6 is the hydrolysis rate given by Eqs. (20)-( 23) similar to that in the suspended enzyme system. -3v;:,“c, -vg= (20) Kzm+3C,+C4 v;;,“[3(1-(u’““)c,-c,1
(21) (22)
V~~~[3(l-crimm)~+2~] -v2= K;mm+3C6+C4
(23)
Here, K F,“” and VA:? are the kinetic constants for the immobilized enzyme system; d”” is also the selectivity coefficient in the immobilized enzyme system. c can be calculated from the mass balance equations for the saccharides in the support particles in the spherical coordinate as:
ac --=Di”’ at
(24)
The spherical symmetry and assumptions 4 and 5 give the following boundary conditions: r=O;
G -0 ar
(25)
kinetics using the suspended enzyme
The experimental and fitted results are shown in Fig. 3. In all of figures, the isomaltohexaose concentration ( Cs) simply decreases with the passage of the reaction time, while the isomaltotetraose ( C,), isomaltotriose ( C,) and isomaltose concentrations (C,) increase, respectively. Finally, isomaltotetraose is also converted into isomaltose, and only isomaltotriose and isomaltose exist in the system. When the initial substrate concentration ( C,( 0) ) decreases from 1.7X10-4moldm-3-liq.inFig.3ato1.3X10-4moldm-3liq. in Fig. 3b, there is a slight difference in the tendencies for the consumption of isomaltohexaose and the formation of isomaltotriose and isomaltose. On the other hand, when the initial concentration of the suspended enzyme (C,( 0)) decreases from 2.1 mg dmP3-liq. in Fig. 3b to 1.16 mg dmP3liq. in Fig. 3c, the consumption rate of isomaltohexaose and the formation rates of isomaltotriose and isomaltose become much smaller as can be seen from the slope of the time courses of the respective saccharide concentrations. For reference, the fitted results by the model proposed by Dean and Rollings [2] are also shown in Fig. 3 by dotted lines. Under any reaction conditions, the calculated results by the present model are in a much better agreement with the experimental values. Therefore, it is valid to introduce the concept of the selectivity of the enzyme for cleaving each IY- 1,6-glycosidic bond of the substrate. The estimated values of the constants are listed in Table 1. The new constant, selectivity coefficient, (Y, is estimated at 0.4 11. cy can be also determined experimentally from the final
N. Shibasaki-Kitakawa
et al. /Biochemical
Table 2 Sensitivity
5,....,....,....,....,....,
27
4
(‘I
Engineering
0
C
-
Present
A
c:
----.
Dean
Journal
1 (1998)
201-209
of model constants in suspended
205
enzyme
and
Rollings’
Model
Constants
Value
S
Deviation (~-.%in~Snlin)
K,
1.01 x lo-4 2.01 x lo+ 4.02 x lO-4
106.2
80.9 94.5
+ 25.3% Minimum + 16.7%
v Inax ’
1.45 x lo-3 2.90x lo-3 5.80X lo-3
123.6 80.9 160.2
+ 52.7% Minimum + 97.9%
a
0.206 0.411 0.822
101.8 80.9 173.4
+ 25.8% Minimum + 114.3%
i;
?3 5 E ‘b
2
X
-1
ti0 0
50
150
100 Time
200
250
[ min 1
5l
ZE L 2 X -1 ci0
0 Time
[ min
1
5 -4 7 E ;3 d b,2 X 1 o’0
0 [ min
1
Fig. 3. Experimental and calculated results in suspended enzyme system. (a) C,(O) =l.7X10d4 mol dm-‘, &(0)=2.1X 10e3 g dme3, (b) C,(O)= 1.3X 10m4 mol dmm3, C,(O)=2.1 x~O-~ g dmm3, (c) C,(O) = 1.7~ 10m4 mol dmv3, C,(O) = 1.16~ 10m3 g dmm3. Table 1 Estimated
To elucidate the sensitivity of each constant, the sum of square of the relative error, S, was calculated by the numerical simulation. Only the value of one constant waschanged from half of the estimated value to a double value without changing other constants. The results are shown in Table 2. The values of S obtained by changing the values of each constant are larger than the minimum values, Sminrby at least 16%. Therefore, the reliability of the model constants obtained in this research is considered to be relatively high. 4.2. Hydrolysis
Time
constants
in suspended
Constants
K, (mol dme3-liq.) V max ’ (mol g-‘-enzyme a(-)
min-‘)
enzyme
system
Model
system Estimated values
Experimental values
2.01 x lo-4 2.90x 1o-3 0.411
a a 0.41-0.42
a Unobtainable.
ratio of C, to C,. The values of (Yobtained from the experiments under various conditions, 0.41-0.42, are almost the same as the estimated u-due by the IW.XM. Tksc v&es iwe greatertianlhne et@ivaknkv&ue, 3 /3,inYn~rnnrM~rz+Er!1 by Dean and Rollings [ 21. Therefore, the endo-dextranase is axm~~eseb~~ ta&y ckayethe. u-~.,6~~~~~~~~tin~~ more interior of the substrate.
kinetics using the immobilized enzyme
The results of the partition equilibrium experiment are shown in Fig. 4. A linear relationship is set up between the equilibrium concentrations of saccharides in the support particle and those in the liquid phase, regardless of the experimental conditions and the polymerization degree of two saccharides. The partition coefficient, H, is determined from the slope of these data using the least squares method, and the value is calculated to be 0.95. The experimental and fitted results are shown in Fig. 5 by symbols and solid lines, respectively. When the initial substrate concentration, C,(O), increases from 1.3 X 10m4 mol drn3-liq. in Fig. 5a to 1.7 X 10V4 mol dm-3-liq. in Fig. 5b and 2.0X 10e4 mol dmP3-liq. in Fig. 5c, there is a slight
N. Shibasaki-Kitakawa
206
et al. /Biochemical
Engineering
Journal
1 (1998)
201-209
(b) i/‘“lii~ll~*.“‘.“‘i”..l’.““‘.” 0
C,
----
model
wrthout
1
I4
I
consldenng
0 __--_____---
200
250
200
250
0 Time
156 [min]
-
200
1250
0
50
-100 Time
and calculated results in immobilized enzyme system. (a) C,(O) = 1.3 X 10e4 mol dm-j, Fig. 5. Experimental w,= I .O g, (c) C,(O) =2.0X 10-4mol dm-‘, W,= l.Og, (d) C,(O) = 1.7~ 10e4 moldm-‘, W,= 1.5 g.
difference in the tendencies for the consumption and the formation of the respective saccharides. On the other hand, when the amount of immobilized enzyme, W,, increases from 1.0 g in Fig. 5b, to 1.5 g in Fig. 5d, the consumption and the formation rates of saccharides become much greater. The model well simulates experimental results obtained under the various reaction conditions. For reference, the fitted results by the model without considering the intraparticle mass transfer resistance are also shown in Fig. 5 by dotted lines. Comparing the two kinds of calculated results, the one with intraparticle mass transfer resistance is in much better agreement with the experimental results, especially for the isomaltotetraose concentration. This is because the converted isomaltotetraose is quickly further transformed into isomaltose inside the support particles due to the intraparticle mass transfer resistance. The distribution of saccharide concentrations in the support particle obtained by the calculation, c, is shown in Fig. 6. All the concentration gradients of the respective saccharides exist only in the neighborhood of the surface of the support particle in Fig. 6a. Each concentration gradient becomes smaller with increasing reaction time, and all the concentrations of the respective saccharides are almost constant in Fig. 6c. The distribution of reaction rates in the particle, G, is also shown in Fig. 7. The negative value of the reaction rate means that the saccharide is net consumed by the reactions, and the positive one means that the saccharide is net formed. All the reactions occur in the neighborhood of the surface of the particle in Fig. 7a similar to the results of the concentration gradients. The reaction rate of C, is turned from the negative
150 [min]
W,=
-
1.0 g, (b) C,(O)
= 1.7 x 10. -4 mol dm- j,
values in Fig. 7a to the positive values in Fig. 7b. Each local reaction rate decreases with increasing reaction time. In Fig. 7c, all the reaction rates are almost zero, and the steady state is considered to be attained. Therefore, only the enzyme immobilized in the neighborhood of the surface of the support particle is considered to effectively act on the hydrolysis of saccharides. The estimated values of the constants are shown in Table 3. The Michaelis constant estimated by this model, Kzrn, is consistent with the values obtained for the various reactions using the immobilized enzyme [ lo]. The kinetic constants for the immobilized enzyme system, KF”’ and VA$“‘, are much greater than those for the suspended enzyme system. This tendency is the same as the results previously reported [ 111. In the model, the immobilized enzyme distribution in the support particle is assumed to be uniform. However, in practice, the enzyme concentration in the neighborhood of the surface of the particle might be much higher than that in the center of the particle [ 12-141. In such a case, the amount of enzyme, which effectively acts on the hydrolysis, is considered to be much larger than that in our model. Consequently, the kinetic constants become smaller than those estimated by our model. This might be one of the reasons for the differences in our kinetic constants between the two systems. The other reason is a difference in the activity of the enzyme between the two systems because of the change in the microscopic environment and enzyme conformation. The estimated value of oimm is almost equal to the experimental value, but it is a little higher than that in the suspended enzyme system. Although the reason is not clear, the change
N. Shibasaki-Kirakawa
51..
I.,
I.
.)I,
*
et al. /Biochemical
.,.,
Engineering
Journal
I (1998)
201-209
207
.,
F4i Cal --2 E: _____. ;3-
f
2.4
E
:
%
oz-
0
0
0
5
.““.“‘I.’
5
10
.‘I
15
.‘.-
20
E
22;
(c)
25
‘: 6’ 5 0
; .
: E -1
x u
.-------------------_______( .
0
;;
-2
ICY’
;
-
-3
’
s--4O5
r [pm1 Fig. 6. Calculated cle. (a) t=lOmin, Table 3 Estimated
20
,4’...“‘..‘....‘....“.” ‘i -; 23 (c)
,
c
:
15
25
“‘.‘..‘.“‘.“‘.“‘..‘.~
T’t -0 . + 3-
10
5
distribution of saccharide concentration (b) t=6Omin, (c) t=2OOmin.
constants in immobilized
enzyme
in support
parti-
system
Constants
Estimated values
Experimental values
PErn (mol dm-3-particle) Vzz’ (mol g-‘-enzyme A (m* s-‘) B(-) drnrn ( - )
8.50X lo-* 7.14x 10-Z 1.66X lo-lo 0.813
a = = a
0.490
0.47-0.50
min-‘)
25 r hml
a Unobtainable.
in the steric configuration of the enzyme caused by the immobilization is considered to be associated with the difference in the selectivity coefficient between the suspended and immobilized enzyme systems. The estimated values of effective diffusivities in the support particle are listed in Table 4 compared with the calculated values from the empirical equation reported by Kimura and Nakao in aqueous solution [9]. In many researches for the immobilized enzyme system, the values of the effective dif-
Fig. 7. Calculated distribution of hydrolysis r=lOmin, (b) r=60min, (c) t=2OOmin.
rate in support
particle.
(a)
Table 4 Estimated values of effective diffusivities in support particle and calculated values from empirical equation reported by Kimura and Nakao [ 91 in aqueous solution Effective (m* s-‘)
diffusivities
In support particle
Dy
6.09~
w pm3 LF2
8.41 x 10-13 1O.6X1O-‘3 14.5 x lo- I3
lo-”
In aqueous
solution
3.19x
lo-‘0 10-l” 4.42 X 10 - ” 5.32 x lo- ”
3.86x
fusivities of substrate or product in the support particle are assumed to be almost equal to those in the aqueous solution [ 15-181. The values in the support particle obtained by this fitting, however, are a few order of magnitude smaller than those in aqueous solution. This tendency is generally observed in many macromolecule media such as ionexchange resins [ 191. The support particle is considered to be a porous matrix swelling with sufficient water because H is almost equal to unity. However, the diffusion in such a
N. Shibasaki-Kitakawa
208 Table 5 Sensitivity
of model constants in immobilized
Constants
Value
h”“”m
4.25X 8.50x 17.0x
pnnv Inax
drnrn
system Deviation (S - &li.~&ni.)
Engineering
C,(O) CZrn( 0) CES,
124.1
+ 16.8%
106.6 122.5
Minimum + 14.9%
ci
122.5
+ 14.9%
G
106.6 124.1 a
Minimum + 16.5% a
106.6 117.9
Minimum + 10.6%
Ci(O)
0.407 0.813 1.626
221.2 106.6 a
+ 107.5% Minimum a
Deff
0.245 0.490 0.980
137.5 106.6 368.7
+ 28.9% Minimum + 245.8%
i
lo-’ lo-’ lo-*
0.83 X lo-I0 1.66X 10-‘O 3.32~ lo-‘”
B
a Calculation
s
3.57 x 1o-2 7.14x lo-* 14.3x lo-*
A
enzyme
et al. /Biochemical
H
impossible.
porous matrix is considered to be limited by the steric hindrance of the network structure in the support particle. The sensitivity of each constant in the immobilized enzyme system was elucidated similarly to the suspended enzyme system. The results are shown in Table 5. The values of S obtained by changing the values of each constant are larger than Smin by at least 15%. Therefore, the reliability of the model constants obtained in this research is considered to be relatively high.
Mv r RP s
t vi
5. Conclusions
“i
A novel kinetic model for the hydrolysis of oligosaccharide using the suspended enzyme is constructed by introducing the selectivity of the enzyme for cleaving each (Y- 1,6-glycosidic bond of the substrate, defined as the selectivity coefficient, cr. The kinetic constants are estimated by fitting the model with the experimental data obtained under various conditions. The model is extended for the oligosaccharide hydrolysis by the immobilized enzyme by taking into account the intraparticle mass transfer resistance. The kinetic constants and the effective diffusivities of the respective saccharides in the support particle are estimated by fitting the extended model with the experimental data. There are a few orders of magnitude differences in the kinetic properties between the suspended and immobilized enzyme systems. The kinetic model well simulates the experimental results for the two systems.
m
Vmax imm Vmax
WP ff cuimm E
Journal
1 (1998)
201-209
Initial enzyme concentration in solution (mg dm- 3-liq.) Initial enzyme concentration in supportparticle (mg dmp3-particle) Concentration of enzyme-substratecomplex in solution (mol dme3-liq.) Concentration of saccharidein solution (mol dmd3-liq.) Local concentration of saccharidein support particle (mol dm-3-particle) Volumetric average concentration of saccharide in supportparticle ( mol dm - 3-particle) Initial concentration of saccharidein solution (mol dmw3-liq.) Effective diffusivity of saccharide( m2 s- ’ ) Partition coefficient of saccharides( dm3-liq. dm-3-particle) Counter correspondingto polymerization degree of saccharide Rate constant (min- ’ ) Michaelis constantin suspendedenzyme system (mol dme3-liq.) Michaelis constantin immobilized enzyme system (mol dm-3-particle) Molecular weight of saccharide(g mol- ’ ) Radial coordinate (m) Radiusof supportparticle (m) Sum of squareof the relative error between calculated value of saccharideconcentration and experimental data ( - ) Reaction time (min) Hydrolysis rate of saccharidein solution (mol dm-‘-liq. min- ’ ) Local hydrolysis rate of saccharidein support particle (mol dme3-particle min-‘) Volumetric average hydrolysis rate of saccharide in supportparticle (mol dm- ‘-particle min- i ) Maximum rate constant in suspendedenzyme system (mol dmp3-liq. min-‘) Maximum rate constantin immobilized enzyme system (mol dm-3-particle min- ‘) Weight of support particle (g) Selectivity coefficient in suspendedenzyme system( -) Selectivity coefficient in immobilized enzyme system ( - ) Reactor voidage ( - )
References 6. Nomenclature A B
Regression coefficient ( m2 s- ’ ) Regression coefficient ( - )
[ 11 J. Ichita, S. Yamaguchi, M. Kushibiki, H. Ono, K. Hanamatsu, H. Matsue, T. Okuno, K. Miyairi, K. Kazehare, Preparation of pectic oligosaccharides and their application, Shokuhinkogyo 39 ( 1996) 6066.
N. Shibasaki-Kitakawa [2]
[3]
[4]
[5]
[ 61
[7]
[ 81 [ 91
[lo]
[ 1 l]
et al. /Biochemical
SW. Dean III, J.E. Rollings, Analysis and quantification of a mixed exe-acting and endo-acting polysaccharide depolymerization system, Biotechnol. Bioeng. 39 ( 1992) 968-976. M. Fujii, S. Murakami, Y. Yamada, T. Ona, T. Nakamura, Kinetic equation for hydrolysis of polysaccharides by mixed exo- and endoenzyme systems, Biotechnol. Bioeng. 23 (1981) 1393-1398. M. Fujii, M. Shimizu, Synergism of endoenzyme and exoenzyme on hydrolysis of soluble cellulose derivatives, Biotechnol. Bioeng. 28 ( 1986) 878-882. M. Okazaki, M. Moo-Young, Kinetics of enzymatic hydrolysis of cellulose: analytical description of a mechanistic model, Biotechnol. Biocng. 20 (1978) 637-663. K. Suga, G. van Dedem, M. Moo-Young, Degradation of polysaccharides by endo and exo enzymes: a theoretical analysis, Biotechnol. Bioeng. 17 ( 1975) 433-439. M.A. Wheatley, M. Moo-Young, Degradation of polysaccharides by endo- and exoenzymes: dextran-dextranase model systems, Biotechnol. Bioeng. 19 (1977) 219-233. J.A. Nelder, R. Mead, A simplex method for function minimization, Comput. J. 7 (1964) 308-313. S. Kimura, S. Nakao, Design Methods for Membrane Separation Processes. The Membrane Society of Japan, Kitami Syobo, Tokyo, 1985, pp. 3749. C.M. Hooijmans, M.L. Stoop, M. Boon, K.C.A.M. Luyben, Comparison of two experimental methods for the determination of MichaelisMenten kinetics of an immobilized enzyme, Biotechnol. Bioeng. 40 (1992) 16-24. F. Shiraishi, Experimental evaluation of the usefulness of equations describing the apparent maximum reaction rate and apparent michaelis
Engineering
[ 121
[ 131
[ 141
[ 151
[ 161
[ 171
[ 181
[ 191
Journal
1 (1998)
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209
constant of an immobilized enzyme reaction, Enzyme Microb. Technol. 15 (1993) 150-154. S. Blais, R. Lottie, Determination of the intrinsic Michaelis constant of immobilized cu-chymotrypsin, J. Biol. Chem. 268 (1993) 1863718639. R. Scharer, Md.M. Hossain, D.D. Do, Determination of total and active immobilizedenzyme distribution in porous solid supports, Biotechnol. Bioeng. 39 (1992) 679-687. F. Shiraishi, H. Miyakawa, Influences of nonuniform activity distributions on the apparent maximum reaction rate and apparent Michaelis constant of immobilized enzyme reactions, J. Ferment. Bioeng. 77 ( 1994) 224-228. G.V. Cano, A.L. Cabanes, A generalized analysis of internal diffusion of immobilized enzyme in multi-enzyme reactions, Chem. Eng. J. 56 (1994) B61-67. G. Handrikova, V. Stefuca, M. Polakovic, V. Bales, Determination of effective diffusion coefficient of substrate in gel particles with immobilized biocatalyst, Enzyme Microb. Technol. 18 ( 1996) 581-584. R. Lottie, G. Andre, On the use of apparent kinetic constants for immobilized enzyme with noncompetitive substrate inhibition, Enzyme Microb. Technol. 13 (1991) 960-963. A.B. Santoyo, J.L.G. Carrasco, E.G. Gomez, J.B. Rodriguez, E.M. Morales, Transient stirred-tank reactors operating with immobilized enzyme systems: analysis and simulation models and their experimental checking, Biotechnol. Prog. 9 ( 1993) 166-173. I.L. Jones, G. Carta, Ion exchange of amino acids and dipeptides on cation exchange resins with varying degree of crosslinking: 2. Intraparticle transport, Ind. Eng. Chem. Res. 32 (1993) 117-125.
Engineering
Journal
1 ( 1998)
Biochemical Engineering Journal
201-209
Kinetic model for oligosaccharide hydrolysis using suspended and immobilized enzymes Naomi Shibasaki-Kitakawa a, Benjamas Cheirsilpa a, Ken-Ichiro Iwamura a, Masanori Kushibiki b, Akio Kitakawa ‘, Toshikuni Yonemoto a7* ‘Department
’ Departmeni of Chemical Engineering, Tohoku University, Sendai, Japan hAomori Advanced industrial Technology Center, Aomori, Japan of Materials Science and Engineering, Miyagi National College of Technology, Received
6 October
1997; accepted
13 January
Natori,
Japan
1998
Abstract A novel kinetic model that describes the hydrolysis of oligosaccharide usingthe suspended enzyme has been constructed by introducing the selectivity of the enzyme for cleaving each (Y-1,6 glycosidic bondof the substrate. The Michaelis-Mententype kineticconstants,K,,, and V maxI and the selectivity coefficient, cr, are estimated by fitting the model with the experimental data obtained under various conditions. The
newconstant,a, is estimatedat 0.411,andalmostthe sameasthe experimentalvalues.Themodelhasbeenextendedfor the immobilized enzyme system by taking into account the intraparticle
mass transfer resistance. The model constants are estimated similarly to the case of
the suspended enzymesystem.aimm( LYfor the immobilizedenzymesystem)is a little higherthan that in the suspended enzymesystem. Kmmm and V,,‘-‘, aremuchgreaterthanthosefor the suspended enzymesystem.The estimatedvaluesof the effective diffusivitiesin the supportparticlearea few orderof magnitudesmallerthanthosein aqueoussolution.The modelwell simulates both experimentalresults obtainedin the suspended andimmobilizedenzymesystems. 0 1998ElsevierScienceS.A. All rightsreserved. Keywords:
Hydrolysis
of oligosaccharide;
Suspended
enzyme;
Immobilized
Recently, it hasbeenreported that oligosaccharideshaving physiological activity can be obtained by the enzymatic hydrolysis of polysaccharidecontained in agricultural waste [ 11. Industrialization of this enzymatic reaction processwill s&dkanl>y contribute 2Dwmi 2he effecthe U2%222iDT2 of agricultural
resources.
High molecular weight saccharidesare transformed into Iower molecular weight saccharidesby consecutive depolymerization using enzymes. Several researchers have attemptedto construct the mathematicalmodelsthat describe this consecutivedepolymerization kinetics [ 2-71. Fujii et al. [ 31 have proposeda lumped model in which all oligomers andpolymers other than a monomerwereconsideredtogether as one substrate. This model was quite simple and could estimatethe rangeof systemparameters(Km and V,,,,, values and substrateconcentration), but it could not predict the * Corresponding author. Department of Chemical Engineering, Tohoku University, Aoba, Aramaki, Aoba-ku, Sendai 980-8579, Japan. Tel.: + 8122-217-7255; fax: + 81.22-217-7258;e-mail: [email protected] 1369-703X/98/$ - see front matter 0 PIIS1369-703X(98)00003-5
1998 Elsevier
enzyme;
Kinetic
model
mo‘lecu‘larwei@ distributions of the substrateand product. Suga et al. [6] recognized the limitations of the lumped model and proposeda distributed model. In their model, the differential equations following Michaelis-Menten kinetics were formulated for each substratespecieswith the various polymerization degrees.The underlying basisfor the model was more c\osely P‘qwxutative of the actual events present in &se c~~3p~exsystems,but expetimental verification of the theoretical model was not published. Dean and Rollings ] 2] have extended the mathematicalmodelproposedby Suga et al. [S). This model accounted for the fact that a certain number of bondsof the substratewere inaccessibleto hydrolytic action. However, the predictions by their model were inconsistentwith the experimental results. Enzyme is often usedin an immobilized form in industry for the purpose of repeated or continuous uses.When the enzyme is immobilized to a porous support matrix, the intraparticle masstransfer has a great influence on the kinetics. It is therefore necessary to develop comprehensive models accounting quantitatively for the diffusional effects. However, mere is little researchinvolved in the analysis of the
Science S.A. All rights reserved.
202
N. Shibasaki-Kitakawa
et al. /Biochemical
kinetics of consecutive reactions catalyzed by the immobilized enzyme. In this research, the hydrolysis of oligosaccharide using both suspended and immobilized enzymes was carried out experimentally under various conditions. The oligosaccharide of the polymerization degree of 6 was chosen as a model high molecular weight substrate, because the mechanism of consecutive depolymerization was comparatively simple, and the concentration could be easily analyzed. On the basis of the experimental results, a novel mathematical model, which described the hydrolysis kinetics of oligosaccharide using suspended and immobilized enzymes, was constructed.
Engineering
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1 (1998)
enzyme ranged from 1.16 to 2.1 mg dme3, from 1.3 to 2.0 mol dmV3 and from 1.O to 1.5 g, respectively. The uptake amount of the saccharides by the support particle was also measured under the partition equilibrium condition. A weighed amount of support particles was immersed into an aqueous solution of the saccharide. After a 24-h shaking, the saccharide concentration in the supematant solution was measured. Isomaltohexaose and isomaltose were used as the saccharides. The initial substrate concentration and the amount of support particle ranged from 1.5 to 2.0 mol dm-3 and from 3.0 to 4.0 g, respectively.
3. Mathematical 2. Materials
201-209
model
and methods 3.1. Hydrolysis
2.1. Materials Dextranase from PhenisilEium sp. purchased from Sigma was used as enzyme. Isomaltohexaose (Seikagaku), composed of (Y-1,6-linked d-glucose, was used as the oligosaccharide. The Affi-Prep 10 (Bio-Rad Lab.), which is macroporous methacrylate co-polymer bead activated with an N-hydroxysuccinimide ester, was used as a support material for the immobilization of enzyme. Its nominal particle size and pore diameter are 50 pm and 0.1 km, respectively. Other chemicals were of reagent grade and purchased from wako. 2.2. Immobilization
of enzyme
The support particles, Affi-Prep 10, were immersed in distilled water overnight to remove isopropanol used as a storage solution. Five grams of the support particles was added to 30 cm3 dextranase aqueous solution (2.32 mg cme3) and gently agitated at 4°C for 12 h. The immobilized dextranase was then filtered and washed with distilled water. The concentration of unimmobilized enzyme, which remains in the supernatant solution, was measured by a spectrophotometer at 595 nm to determine the immobilized amount of enzyme.
kinetics using the suspended enzyme
The endo-acting enzyme is known to cleave the bonds in the interior of the substrate main chain. Therefore, as shown in Fig. 1, the endo-dextranase cleaves the (Y- 1,6 glycosidic bonds in either position I or II of isomaltohexaose. Isomaltohexaose is hydrolyzed to either two isomaltotrioses or isomaltotetraose and isomaltose. In the latter case, isomaltotetraose is further converted into two isomaltoses. In the kinetic model constructed by Dean and Rollings [ 21, the endo-acting enzyme is considered to evenly cleave the respective bonds in the interior of the substrate main chain. However, the predictions by their model are inconsistent with the experimental results. In this study, a novel kinetic model has been constructed by introducing the selectivity of the enzyme for cleaving the o-1,6 glycosidic bond in position I, defined as a selectivity
2.3. Experimental procedures The rate of oligosaccharide hydrolysis by suspended and immobilized enzymes was investigated at 37°C in a 500-cm3Erlenmeyer flask containing 100 cm3 substrate solution. The flask was shaken in a water bath at the well mixed condition. The reaction was started by adding the enzyme to the substrate solution. The samples werepipetted at specified timeintervals and mixed with a 0.1 M solution of NaOH in a volume ratio of 1: 1 to stop the hydrolysis. The respective saccharide concentrations in the sample solution were determined using an HPLC system equipped with a gel permeation column (NHC18, Hitachi, Japan) and an IR detector (YRU-880mdget, Shimarnura, Japan). The suspended enzyme concentration, the substrate concentration, and the amount of immobilized
Fig. 1. Michaelis-Menten endo-dextranase,
type hydrolysis
process of isomaltohexaose
using
N. Shibasaki-Kitakawa
et al. /Biochemical
coefficient, LY.For the purpose of the simplification of the model, the rate constants for the isomaltotetraose hydrolysis are assumed to be the same as those for the isomaltohexaose hydrolysis (this assumption is also employed by Dean and Rollings [2]). Using Michaelis-Menten type kinetics as shown in Fig. 1 and the selectivity coefficients, LY,the rates of change in the concentration of the respective saccharides are given as: dC 6 =-3klCEC6+k2CES6 dt
Engineering
5
=(1-a)k3CES6+2k3CESl
(5)
dGs4 dt
(6)
=klCEC4-(k2+k3)CES4=0
The total concentration of enzyme in the system is cEm=G+cEs,+Gs,
(7)
Rearranging Eqs. ( 1 )-( 4)) we get: -3vltl,XC6 K,+3C,+C,
dC4
(8)
Vmax[3(i-w6-C4i
dt -03=
K,+3C6+C4 6Vnl&C6
(9) (10)
K,+3C6+C4
[
CCi,calc.
3.2. Hydrolysis
d&s, =3klC,C,-(k,+k,)C,,,=O dt
203
1 2
+ci,exp.)/2
To obtain the calculated values under a certain set of constants, the simultaneous differential equations, Eqs. (8)( 11) , are solved numerically using the Runge-Kutta method. Time step for the numerical calculation is set at 60 s.
where Ci and CES, are the concentrations of the saccharide and the enzyme-substrate complex in the solution, respectively. The subscript i is a counter corresponding to the polymerization degree of the saccharides. k,, k2 and k3 are the rate constants. Further assuming a quasi-steady-state for the concentration of the enzyme-substrate complex, we get:
dt =v4= dC,-
201-209
C,( 0). Thus, the maximum reaction rate per unit weight of enzyme, V,,,‘( = V,,,,,/C,(O)), which is the characteristic value of enzyme, is used for the calculation in the place of VInax*These three constants are estimated using the Simplex parameter seeking method [ 81 to minimize S, which is the sum of square of the relative error between the calculated values of the saccharide concentrations and three sets of experimental data obtained under different reaction conditions, defined as follows:
‘=’
=(l-a)k3CEs,-k,C&+k2CESq
dt -“‘=
1 (1998)
Ci,calc.-Ci,exp.
%
dC,-
Journal
kinetics using the immobilized enzyme
The kinetic model that describes the hydrolysis of the oligosaccharide by the immobilized enzyme mainly consists of the mass balance equations for the liquid and support particle phases. They are formulated on the basis of the following assumptions. ( 1) A perfect mixing is established in the liquid phase. (2) The support particles are homogeneous and spherical, and their sizes are constant. (3) The enzyme is uniformly immobilized inside the support particles. (4) The extraparticle mass transfer resistance is negligible because the reactor is under well mixed condition. (5) There exists the linear partition equilibrium of the saccharides between the liquid phase and the surface of the support particle. clrcRp=
HC,
(14)
(6) The intraparticle mass transfer fluxes of the respective saccharides follow Fick’s law, and the effective diffusion coefficients, Deff, are described using a correlation similar to that reported by Kimura and Nakao [9] for saccharides in aqueous solution, as follows: DFff=A.MiB
(15)
Vm,,[3(1-a)C6+2C41
K,+3C,j+C,
(
K,=
k,+k,
(11)
vnl.3, =k,C,(O)
-9
k,
i.c. t=O; Ci=
C,(0) 0
(i=6) (i=2,3,4)
(12)
In the case of (Y= l/3, the model is identical with that of Dean and Rollings [ 21. There are three unknown constants, K,,,, V,,, and Q, in the model. V,,,,, depends on the initial enzyme concentration,
where A and B are the regression coefficients and M, is the molecular weight of the saccharides. These assumptions give the distribution of the saccharide concentration as shown in Fig. 2. The mass balance equations of the respective saccharides in the liquid phase can be described as: dCi
l-cd(c)
-Tt+
&
i.c. t=O; Ci=
I--E
dt
- E
C,(0) 0
-
(cl,)=0
(161
(i=6) (i=2,3,4)
(17)
204
N. Shibasaki-Kitakawa
et al. /Biochemical
Engineering
Journal
r=R,;
c=HCi
I (1998)
201-209
(26)
where His the partition coefficient. The initial conditions at t = 0 are as follows: liquid
Fig. 2. Saccharide
concentration
distribution
in immobilized
Olr
enzyme
system.
where E is the reactor voidage. The bracketed values (c) and (c> are, respectively, the volumetric average concentration and hydrolysis rate of each saccharide in the support particle determined as:
(18)
C,(O)=0
(27)
There are six unknown constants, H, KF’“, V&‘, aimm, A, B, in the model for the immobilized enzyme system. The maximum reaction rate per unit weight of enzyme, VAT,“’ ( = V$~/C~mm( 0) ), is used for the calculation in the place of V&r similar to the suspended enzyme system. These constants, except for H, are estimated by fitting four sets of experimental results obtained under various conditions using the simplex method [ 81 similarly to the case of the suspended enzyme system. To obtain the calculated results under a certain set of constants, Eqs. ( 16)-( 27)) are approximately formulated with a finite difference scheme and solved numerically using the iterative method. For numerical calculation, the time step is set at 15 s, and the r coordinate in the support particle is divided into 50 points. The partition coefficient, H, is determined from the partition equilibrium experiment.
4. Results and discussion (19) 4.1. Hydrolysis c is the intraparticle local values of the concentration. 6 is the hydrolysis rate given by Eqs. (20)-( 23) similar to that in the suspended enzyme system. -3v;:,“c, -vg= (20) Kzm+3C,+C4 v;;,“[3(1-(u’““)c,-c,1
(21) (22)
V~~~[3(l-crimm)~+2~] -v2= K;mm+3C6+C4
(23)
Here, K F,“” and VA:? are the kinetic constants for the immobilized enzyme system; d”” is also the selectivity coefficient in the immobilized enzyme system. c can be calculated from the mass balance equations for the saccharides in the support particles in the spherical coordinate as:
ac --=Di”’ at
(24)
The spherical symmetry and assumptions 4 and 5 give the following boundary conditions: r=O;
G -0 ar
(25)
kinetics using the suspended enzyme
The experimental and fitted results are shown in Fig. 3. In all of figures, the isomaltohexaose concentration ( Cs) simply decreases with the passage of the reaction time, while the isomaltotetraose ( C,), isomaltotriose ( C,) and isomaltose concentrations (C,) increase, respectively. Finally, isomaltotetraose is also converted into isomaltose, and only isomaltotriose and isomaltose exist in the system. When the initial substrate concentration ( C,( 0) ) decreases from 1.7X10-4moldm-3-liq.inFig.3ato1.3X10-4moldm-3liq. in Fig. 3b, there is a slight difference in the tendencies for the consumption of isomaltohexaose and the formation of isomaltotriose and isomaltose. On the other hand, when the initial concentration of the suspended enzyme (C,( 0)) decreases from 2.1 mg dmP3-liq. in Fig. 3b to 1.16 mg dmP3liq. in Fig. 3c, the consumption rate of isomaltohexaose and the formation rates of isomaltotriose and isomaltose become much smaller as can be seen from the slope of the time courses of the respective saccharide concentrations. For reference, the fitted results by the model proposed by Dean and Rollings [2] are also shown in Fig. 3 by dotted lines. Under any reaction conditions, the calculated results by the present model are in a much better agreement with the experimental values. Therefore, it is valid to introduce the concept of the selectivity of the enzyme for cleaving each IY- 1,6-glycosidic bond of the substrate. The estimated values of the constants are listed in Table 1. The new constant, selectivity coefficient, (Y, is estimated at 0.4 11. cy can be also determined experimentally from the final
N. Shibasaki-Kitakawa
et al. /Biochemical
Table 2 Sensitivity
5,....,....,....,....,....,
27
4
(‘I
Engineering
0
C
-
Present
A
c:
----.
Dean
Journal
1 (1998)
201-209
of model constants in suspended
205
enzyme
and
Rollings’
Model
Constants
Value
S
Deviation (~-.%in~Snlin)
K,
1.01 x lo-4 2.01 x lo+ 4.02 x lO-4
106.2
80.9 94.5
+ 25.3% Minimum + 16.7%
v Inax ’
1.45 x lo-3 2.90x lo-3 5.80X lo-3
123.6 80.9 160.2
+ 52.7% Minimum + 97.9%
a
0.206 0.411 0.822
101.8 80.9 173.4
+ 25.8% Minimum + 114.3%
i;
?3 5 E ‘b
2
X
-1
ti0 0
50
150
100 Time
200
250
[ min 1
5l
ZE L 2 X -1 ci0
0 Time
[ min
1
5 -4 7 E ;3 d b,2 X 1 o’0
0 [ min
1
Fig. 3. Experimental and calculated results in suspended enzyme system. (a) C,(O) =l.7X10d4 mol dm-‘, &(0)=2.1X 10e3 g dme3, (b) C,(O)= 1.3X 10m4 mol dmm3, C,(O)=2.1 x~O-~ g dmm3, (c) C,(O) = 1.7~ 10m4 mol dmv3, C,(O) = 1.16~ 10m3 g dmm3. Table 1 Estimated
To elucidate the sensitivity of each constant, the sum of square of the relative error, S, was calculated by the numerical simulation. Only the value of one constant waschanged from half of the estimated value to a double value without changing other constants. The results are shown in Table 2. The values of S obtained by changing the values of each constant are larger than the minimum values, Sminrby at least 16%. Therefore, the reliability of the model constants obtained in this research is considered to be relatively high. 4.2. Hydrolysis
Time
constants
in suspended
Constants
K, (mol dme3-liq.) V max ’ (mol g-‘-enzyme a(-)
min-‘)
enzyme
system
Model
system Estimated values
Experimental values
2.01 x lo-4 2.90x 1o-3 0.411
a a 0.41-0.42
a Unobtainable.
ratio of C, to C,. The values of (Yobtained from the experiments under various conditions, 0.41-0.42, are almost the same as the estimated u-due by the IW.XM. Tksc v&es iwe greatertianlhne et@ivaknkv&ue, 3 /3,inYn~rnnrM~rz+Er!1 by Dean and Rollings [ 21. Therefore, the endo-dextranase is axm~~eseb~~ ta&y ckayethe. u-~.,6~~~~~~~~tin~~ more interior of the substrate.
kinetics using the immobilized enzyme
The results of the partition equilibrium experiment are shown in Fig. 4. A linear relationship is set up between the equilibrium concentrations of saccharides in the support particle and those in the liquid phase, regardless of the experimental conditions and the polymerization degree of two saccharides. The partition coefficient, H, is determined from the slope of these data using the least squares method, and the value is calculated to be 0.95. The experimental and fitted results are shown in Fig. 5 by symbols and solid lines, respectively. When the initial substrate concentration, C,(O), increases from 1.3 X 10m4 mol drn3-liq. in Fig. 5a to 1.7 X 10V4 mol dm-3-liq. in Fig. 5b and 2.0X 10e4 mol dmP3-liq. in Fig. 5c, there is a slight
N. Shibasaki-Kitakawa
206
et al. /Biochemical
Engineering
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1 (1998)
201-209
(b) i/‘“lii~ll~*.“‘.“‘i”..l’.““‘.” 0
C,
----
model
wrthout
1
I4
I
consldenng
0 __--_____---
200
250
200
250
0 Time
156 [min]
-
200
1250
0
50
-100 Time
and calculated results in immobilized enzyme system. (a) C,(O) = 1.3 X 10e4 mol dm-j, Fig. 5. Experimental w,= I .O g, (c) C,(O) =2.0X 10-4mol dm-‘, W,= l.Og, (d) C,(O) = 1.7~ 10e4 moldm-‘, W,= 1.5 g.
difference in the tendencies for the consumption and the formation of the respective saccharides. On the other hand, when the amount of immobilized enzyme, W,, increases from 1.0 g in Fig. 5b, to 1.5 g in Fig. 5d, the consumption and the formation rates of saccharides become much greater. The model well simulates experimental results obtained under the various reaction conditions. For reference, the fitted results by the model without considering the intraparticle mass transfer resistance are also shown in Fig. 5 by dotted lines. Comparing the two kinds of calculated results, the one with intraparticle mass transfer resistance is in much better agreement with the experimental results, especially for the isomaltotetraose concentration. This is because the converted isomaltotetraose is quickly further transformed into isomaltose inside the support particles due to the intraparticle mass transfer resistance. The distribution of saccharide concentrations in the support particle obtained by the calculation, c, is shown in Fig. 6. All the concentration gradients of the respective saccharides exist only in the neighborhood of the surface of the support particle in Fig. 6a. Each concentration gradient becomes smaller with increasing reaction time, and all the concentrations of the respective saccharides are almost constant in Fig. 6c. The distribution of reaction rates in the particle, G, is also shown in Fig. 7. The negative value of the reaction rate means that the saccharide is net consumed by the reactions, and the positive one means that the saccharide is net formed. All the reactions occur in the neighborhood of the surface of the particle in Fig. 7a similar to the results of the concentration gradients. The reaction rate of C, is turned from the negative
150 [min]
W,=
-
1.0 g, (b) C,(O)
= 1.7 x 10. -4 mol dm- j,
values in Fig. 7a to the positive values in Fig. 7b. Each local reaction rate decreases with increasing reaction time. In Fig. 7c, all the reaction rates are almost zero, and the steady state is considered to be attained. Therefore, only the enzyme immobilized in the neighborhood of the surface of the support particle is considered to effectively act on the hydrolysis of saccharides. The estimated values of the constants are shown in Table 3. The Michaelis constant estimated by this model, Kzrn, is consistent with the values obtained for the various reactions using the immobilized enzyme [ lo]. The kinetic constants for the immobilized enzyme system, KF”’ and VA$“‘, are much greater than those for the suspended enzyme system. This tendency is the same as the results previously reported [ 111. In the model, the immobilized enzyme distribution in the support particle is assumed to be uniform. However, in practice, the enzyme concentration in the neighborhood of the surface of the particle might be much higher than that in the center of the particle [ 12-141. In such a case, the amount of enzyme, which effectively acts on the hydrolysis, is considered to be much larger than that in our model. Consequently, the kinetic constants become smaller than those estimated by our model. This might be one of the reasons for the differences in our kinetic constants between the two systems. The other reason is a difference in the activity of the enzyme between the two systems because of the change in the microscopic environment and enzyme conformation. The estimated value of oimm is almost equal to the experimental value, but it is a little higher than that in the suspended enzyme system. Although the reason is not clear, the change
N. Shibasaki-Kirakawa
51..
I.,
I.
.)I,
*
et al. /Biochemical
.,.,
Engineering
Journal
I (1998)
201-209
207
.,
F4i Cal --2 E: _____. ;3-
f
2.4
E
:
%
oz-
0
0
0
5
.““.“‘I.’
5
10
.‘I
15
.‘.-
20
E
22;
(c)
25
‘: 6’ 5 0
; .
: E -1
x u
.-------------------_______( .
0
;;
-2
ICY’
;
-
-3
’
s--4O5
r [pm1 Fig. 6. Calculated cle. (a) t=lOmin, Table 3 Estimated
20
,4’...“‘..‘....‘....“.” ‘i -; 23 (c)
,
c
:
15
25
“‘.‘..‘.“‘.“‘.“‘..‘.~
T’t -0 . + 3-
10
5
distribution of saccharide concentration (b) t=6Omin, (c) t=2OOmin.
constants in immobilized
enzyme
in support
parti-
system
Constants
Estimated values
Experimental values
PErn (mol dm-3-particle) Vzz’ (mol g-‘-enzyme A (m* s-‘) B(-) drnrn ( - )
8.50X lo-* 7.14x 10-Z 1.66X lo-lo 0.813
a = = a
0.490
0.47-0.50
min-‘)
25 r hml
a Unobtainable.
in the steric configuration of the enzyme caused by the immobilization is considered to be associated with the difference in the selectivity coefficient between the suspended and immobilized enzyme systems. The estimated values of effective diffusivities in the support particle are listed in Table 4 compared with the calculated values from the empirical equation reported by Kimura and Nakao in aqueous solution [9]. In many researches for the immobilized enzyme system, the values of the effective dif-
Fig. 7. Calculated distribution of hydrolysis r=lOmin, (b) r=60min, (c) t=2OOmin.
rate in support
particle.
(a)
Table 4 Estimated values of effective diffusivities in support particle and calculated values from empirical equation reported by Kimura and Nakao [ 91 in aqueous solution Effective (m* s-‘)
diffusivities
In support particle
Dy
6.09~
w pm3 LF2
8.41 x 10-13 1O.6X1O-‘3 14.5 x lo- I3
lo-”
In aqueous
solution
3.19x
lo-‘0 10-l” 4.42 X 10 - ” 5.32 x lo- ”
3.86x
fusivities of substrate or product in the support particle are assumed to be almost equal to those in the aqueous solution [ 15-181. The values in the support particle obtained by this fitting, however, are a few order of magnitude smaller than those in aqueous solution. This tendency is generally observed in many macromolecule media such as ionexchange resins [ 191. The support particle is considered to be a porous matrix swelling with sufficient water because H is almost equal to unity. However, the diffusion in such a
N. Shibasaki-Kitakawa
208 Table 5 Sensitivity
of model constants in immobilized
Constants
Value
h”“”m
4.25X 8.50x 17.0x
pnnv Inax
drnrn
system Deviation (S - &li.~&ni.)
Engineering
C,(O) CZrn( 0) CES,
124.1
+ 16.8%
106.6 122.5
Minimum + 14.9%
ci
122.5
+ 14.9%
G
106.6 124.1 a
Minimum + 16.5% a
106.6 117.9
Minimum + 10.6%
Ci(O)
0.407 0.813 1.626
221.2 106.6 a
+ 107.5% Minimum a
Deff
0.245 0.490 0.980
137.5 106.6 368.7
+ 28.9% Minimum + 245.8%
i
lo-’ lo-’ lo-*
0.83 X lo-I0 1.66X 10-‘O 3.32~ lo-‘”
B
a Calculation
s
3.57 x 1o-2 7.14x lo-* 14.3x lo-*
A
enzyme
et al. /Biochemical
H
impossible.
porous matrix is considered to be limited by the steric hindrance of the network structure in the support particle. The sensitivity of each constant in the immobilized enzyme system was elucidated similarly to the suspended enzyme system. The results are shown in Table 5. The values of S obtained by changing the values of each constant are larger than Smin by at least 15%. Therefore, the reliability of the model constants obtained in this research is considered to be relatively high.
Mv r RP s
t vi
5. Conclusions
“i
A novel kinetic model for the hydrolysis of oligosaccharide using the suspended enzyme is constructed by introducing the selectivity of the enzyme for cleaving each (Y- 1,6-glycosidic bond of the substrate, defined as the selectivity coefficient, cr. The kinetic constants are estimated by fitting the model with the experimental data obtained under various conditions. The model is extended for the oligosaccharide hydrolysis by the immobilized enzyme by taking into account the intraparticle mass transfer resistance. The kinetic constants and the effective diffusivities of the respective saccharides in the support particle are estimated by fitting the extended model with the experimental data. There are a few orders of magnitude differences in the kinetic properties between the suspended and immobilized enzyme systems. The kinetic model well simulates the experimental results for the two systems.
m
Vmax imm Vmax
WP ff cuimm E
Journal
1 (1998)
201-209
Initial enzyme concentration in solution (mg dm- 3-liq.) Initial enzyme concentration in supportparticle (mg dmp3-particle) Concentration of enzyme-substratecomplex in solution (mol dme3-liq.) Concentration of saccharidein solution (mol dmd3-liq.) Local concentration of saccharidein support particle (mol dm-3-particle) Volumetric average concentration of saccharide in supportparticle ( mol dm - 3-particle) Initial concentration of saccharidein solution (mol dmw3-liq.) Effective diffusivity of saccharide( m2 s- ’ ) Partition coefficient of saccharides( dm3-liq. dm-3-particle) Counter correspondingto polymerization degree of saccharide Rate constant (min- ’ ) Michaelis constantin suspendedenzyme system (mol dme3-liq.) Michaelis constantin immobilized enzyme system (mol dm-3-particle) Molecular weight of saccharide(g mol- ’ ) Radial coordinate (m) Radiusof supportparticle (m) Sum of squareof the relative error between calculated value of saccharideconcentration and experimental data ( - ) Reaction time (min) Hydrolysis rate of saccharidein solution (mol dm-‘-liq. min- ’ ) Local hydrolysis rate of saccharidein support particle (mol dme3-particle min-‘) Volumetric average hydrolysis rate of saccharide in supportparticle (mol dm- ‘-particle min- i ) Maximum rate constant in suspendedenzyme system (mol dmp3-liq. min-‘) Maximum rate constantin immobilized enzyme system (mol dm-3-particle min- ‘) Weight of support particle (g) Selectivity coefficient in suspendedenzyme system( -) Selectivity coefficient in immobilized enzyme system ( - ) Reactor voidage ( - )
References 6. Nomenclature A B
Regression coefficient ( m2 s- ’ ) Regression coefficient ( - )
[ 11 J. Ichita, S. Yamaguchi, M. Kushibiki, H. Ono, K. Hanamatsu, H. Matsue, T. Okuno, K. Miyairi, K. Kazehare, Preparation of pectic oligosaccharides and their application, Shokuhinkogyo 39 ( 1996) 6066.
N. Shibasaki-Kitakawa [2]
[3]
[4]
[5]
[ 61
[7]
[ 81 [ 91
[lo]
[ 1 l]
et al. /Biochemical
SW. Dean III, J.E. Rollings, Analysis and quantification of a mixed exe-acting and endo-acting polysaccharide depolymerization system, Biotechnol. Bioeng. 39 ( 1992) 968-976. M. Fujii, S. Murakami, Y. Yamada, T. Ona, T. Nakamura, Kinetic equation for hydrolysis of polysaccharides by mixed exo- and endoenzyme systems, Biotechnol. Bioeng. 23 (1981) 1393-1398. M. Fujii, M. Shimizu, Synergism of endoenzyme and exoenzyme on hydrolysis of soluble cellulose derivatives, Biotechnol. Bioeng. 28 ( 1986) 878-882. M. Okazaki, M. Moo-Young, Kinetics of enzymatic hydrolysis of cellulose: analytical description of a mechanistic model, Biotechnol. Biocng. 20 (1978) 637-663. K. Suga, G. van Dedem, M. Moo-Young, Degradation of polysaccharides by endo and exo enzymes: a theoretical analysis, Biotechnol. Bioeng. 17 ( 1975) 433-439. M.A. Wheatley, M. Moo-Young, Degradation of polysaccharides by endo- and exoenzymes: dextran-dextranase model systems, Biotechnol. Bioeng. 19 (1977) 219-233. J.A. Nelder, R. Mead, A simplex method for function minimization, Comput. J. 7 (1964) 308-313. S. Kimura, S. Nakao, Design Methods for Membrane Separation Processes. The Membrane Society of Japan, Kitami Syobo, Tokyo, 1985, pp. 3749. C.M. Hooijmans, M.L. Stoop, M. Boon, K.C.A.M. Luyben, Comparison of two experimental methods for the determination of MichaelisMenten kinetics of an immobilized enzyme, Biotechnol. Bioeng. 40 (1992) 16-24. F. Shiraishi, Experimental evaluation of the usefulness of equations describing the apparent maximum reaction rate and apparent michaelis
Engineering
[ 121
[ 131
[ 141
[ 151
[ 161
[ 171
[ 181
[ 191
Journal
1 (1998)
201-209
209
constant of an immobilized enzyme reaction, Enzyme Microb. Technol. 15 (1993) 150-154. S. Blais, R. Lottie, Determination of the intrinsic Michaelis constant of immobilized cu-chymotrypsin, J. Biol. Chem. 268 (1993) 1863718639. R. Scharer, Md.M. Hossain, D.D. Do, Determination of total and active immobilizedenzyme distribution in porous solid supports, Biotechnol. Bioeng. 39 (1992) 679-687. F. Shiraishi, H. Miyakawa, Influences of nonuniform activity distributions on the apparent maximum reaction rate and apparent Michaelis constant of immobilized enzyme reactions, J. Ferment. Bioeng. 77 ( 1994) 224-228. G.V. Cano, A.L. Cabanes, A generalized analysis of internal diffusion of immobilized enzyme in multi-enzyme reactions, Chem. Eng. J. 56 (1994) B61-67. G. Handrikova, V. Stefuca, M. Polakovic, V. Bales, Determination of effective diffusion coefficient of substrate in gel particles with immobilized biocatalyst, Enzyme Microb. Technol. 18 ( 1996) 581-584. R. Lottie, G. Andre, On the use of apparent kinetic constants for immobilized enzyme with noncompetitive substrate inhibition, Enzyme Microb. Technol. 13 (1991) 960-963. A.B. Santoyo, J.L.G. Carrasco, E.G. Gomez, J.B. Rodriguez, E.M. Morales, Transient stirred-tank reactors operating with immobilized enzyme systems: analysis and simulation models and their experimental checking, Biotechnol. Prog. 9 ( 1993) 166-173. I.L. Jones, G. Carta, Ion exchange of amino acids and dipeptides on cation exchange resins with varying degree of crosslinking: 2. Intraparticle transport, Ind. Eng. Chem. Res. 32 (1993) 117-125.