Immobilized Microbial Cells

11 Temperature Dependence of the Stability and the Activity of Immobilized Glucose Isomerase J. A. ROELS and R. VAN TILBERG

Downloaded by UNIV OF ROCHESTER on August 31, 2017 | http://pubs.acs.org Publication Date: August 16, 1979 | doi: 10.1021/bk-1979-0106.ch011

Gist-Brocades N.V., Research and Development, P.O. Box 1, 2600 MA Delft, The Netherlands

A number o f microorganisms are capable o f transforming g l u cose i n t o i t s isomer f r u c t o s e by the a c t i o n o f the enzyme glucoseisomerase. This property i s o f p o t e n t i a l commercial s i g n i f i c a n c e as the enzyme can i n p r i n c i p l e be used t o produce a mixture o f glucose and f r u c t o s e using a corn based glucose syrup as a source of raw m a t e r i a l . This mixture i s termed high f r u c t o s e corn syrup (HFCS). HFCS i s considered t o be an important competitor f o r saccharose as a sweetener. In i n d u s t r i a l p r a c t i c e an immobilized form o f glucoseisomerase i s used. G i s t Brocades' immobilized glucoseisomerase, Maxazyme GI-immob c o n s i s t s o f p a s t e u r i z e d whole c e l l s o f Aotinoplanes missourïensis entrapped i n g e l a t i n i n such a way t h a t , a f t e r c r o s s l i n k i n g with glutaraldehyde, the s u b s t r a t e , glucose, and the produ c t , f r u c t o s e , can d i f f u s e more o r l e s s f r e e l y i n t o and out o f the particles. In t h i s paper a mathematical model w i l l be presented d e s c r i b i n g the conversion process i n a f i x e d bed r e a c t o r . The model a l lows the c a l c u l a t i o n o f the temperature dependence o f the i n i t i a l a c i t i v i t y o f the immobilized enzyme. I t a l s o p r e d i c t s the s t a b i l i ty o f t h a t a c t i v i t y as a f u n c t i o n o f the o p e r a t i n g temperature. The model i s o f an approximative nature and the s i m p l i f i c a t i o n s which are introduced allow an a n a l y t i c a l s o l u t i o n o f the equations o f the model. The r e s u l t s o f the t h e o r e t i c a l deductions are v e r i f i e d experimentally. Mathematical

model

Enzyme k i n e t i c s . The k i n e t i c s o f transformation processes c a t a l y s e d by a s i n g l e enzyme are o f t e n described using the Michaelis-Menten equation (1). The d e r i v a t i o n o f t h i s equation i s , however, based on two assumptions. The pseudo steady s t a t e hypothes i s (2_) with r e s p e c t t o the intermediary enzyme-substrate complex i s v a l i d and the reverse r e a c t i o n from product t o s u b s t r a t e can be

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148

IMMOBILIZED MICROBIAL CELLS


n e g l e c t e d . In general the f i r s t assumption i s taken f o r granted i n enzyme c a t a l y s i s . The l a t t e r assumption i s only j u s t i f i e d i f the absolute value o f the free enthalpy change o f the r e a c t i o n i s l a r g e compared to the product o f u n i v e r s a l gas constant and abso l u t e temperature. This presents problems i n the case o f the con v e r s i o n o f glucose to f r u c t o s e , the standard f r e e enthalpy change being -400 J/mole at 60°C; the product o f u n i v e r s a l gas constant and absolute temperature being 2750 J/mole. Fratzke et a l . (3^) performed a mathematical a n a l y s i s based on the f o l l o w i n g k i n e t i c scheme:

G + Ε τψ=* XE ψ=ψ Ε + F i n which G Ε XE F k^, k ^, k^

3

(1)

i s glucose i s the free enzyme i s the intermediary enzyme-substrate complex i s fructose e k i n e t i c constants

a r >

k_2

As can be seen the reverse r e a c t i o n i s i n c l u d e d i n the scheme pro posed by Fratzke e t a l . (3). The r e s u l t s o f t h e i r a n a l y s i s , which again i n c l u d e s the steady s t a t e hypothesis with respect t o the enzyme-substrate com p l e x , i s the f o l l o w i n g k i n e t i c equation: =

r

r

S

T

(C

s, max K

f

+ (C s

i n which r

X

-

s

C ) s

^

X

-

C ) s

s

i s the r a t e o f conversion o f glucose t o f r u c t o s

se (mole/m^s ) r i s the apparent maximal forward r a t e o f the cons, max . _ / , / ^ \ v e r s i o n o f glucose t o f r u c t o s e (mole/nrs) K i s a pseudo Michaelis-Menten constant f o r the s u b s t r a t e (mole/m ) C i s the glucose c o n c e n t r a t i o n (mole/m ) C i s the glucose c o n c e n t r a t i o n corresponding to thermodynamic e q u i l i b r i u m (mole/m^) I f equation (2) i s a p p l i e d t o a conversion process using the f r e e whole c e l l s o f an organism a convenient formulation o f equation (2) i s the f o l l o w i n g : T

T

s

3

3

g

x

g

V =

r

S

f

s, max

κ·

. C„ (C E s

+ (C s

s

-

C*) s y

C*) s

Venkatsubramanian; Immobilized Microbial Cells ACS Symposium Series; American Chemical Society: Washington, DC, 1979.

^

11.

Immobilized Glucose Isomerase

ROELS A N D V A N TKLBURG

149

T

i n which V

i s the maximum s p e c i f i c r a t e o f f r u c t o s e formas, max _ , _ . / , „ * s t i o n (mole/kg organism dry matter ss ) i s the c o n c e n t r a t i o n o f the organism (kg dry matter/m ) The "constants" V and K are r a t h e r complex f u n c t i o n s o f the k i n e t i c constants and the e q u i l i b r i u m glucose c o n c e n t r a t i o n : 3

f

T

s

V*

m

a

= s, max

x

S

.

.k

χ


K

K

=

's

K ^ -

MF "

1

{ ( K

^

MF

+

[E]

0

2

L

(4)

1

MG

K

^G

C

· *>

· s

+

«MF ·

W

( 5 )

i n which Κ

i s the e q u i l i b r i u m constant f o r the conversion o f glucose t o f r u c t o s e (-) K^p i s the Michaelis-Menten constant f o r the conversion of f r u c t o s e to glucose, being equal t o (k_2 + k ) / k _ (mole/m ) i s the Michaelis-Menten constant f o r the conversion o f glucose t o f r u c t o s e , b e i n g equal t o (k_! + k ) / k! (mole/m ) [E] i s the i n t r i n s i c enzyme c o n c e n t r a t i o n p e r u n i t o f mi croorganism dry matter (mole/kg dry matter) Adopting the values o f the k i n e t i c constants given by Fratzke et a l . i t can be shown t h a t f o r the c o n d i t i o n s p r e v a i l i n g i n the f i x e d bed conversion process, C - C i s s m a l l as compared t o K . At an i n i t i a l syrup glucose c o n c e n t r a t i o n o f 3000 moles/m and a r e l a t i v e conversion t o f r u c t o s e o f 45%, K i s o f the order o f 5000 moles/m and C - C* v a r i e s between about 1000 moles/m and 20 moles/m (column i n l e t and column o u t l e t r e s p e c t i v e l y ) . Under these c o n d i t i o n s equation (3) can t o a f a i r degree o f approximation be s i m p l i f i e d t o : 3

2

2

3

2

x

f

s

s

3

T

S

3

3

g

3

r

= Κ (C - C*) s s s i n which Κ i s a pseudo f i r s t order r a t e constant given by: k

( K * + 1) [E] . c

9

Κ = K

[

(

1

+

F

(7)

K

*

(6)

Ç

κ», c " , s

I t i s important t o note t h a t equation (7) i m p l i e s the pseudo f i r s t order r a t e constant t o be a f u n c t i o n o f C and hence o f the i n i t i a l glucose c o n c e n t r a t i o n C Q , Κ f o r m a l l y cannot be consid ered a t r u e k i n e t i c constant. For the purpose o f the present model, d e s c r i b i n g a s i t u a t i o n i n which the i n i t i a l glucose c o n c e n t r a t i o n i s a constant, Κ can be considered to be a constant but i f the r e x

S


150



s u i t s are t r a n s l a t e d t o other i n i t i a l glucose concentrations equa t i o n (7) has t o be taken i n t o account. The e f f i c i e n c y f a c t o r f o r an immobilized enzyme. In general the conversion r a t e o f an immobilized enzyme i s lower than t h a t of an equal amount o f the free enzyme. T h i s decreased a c t i v i t y i s caused by d i f f u s i o n a l l i m i t a t i o n s to the r a t e a t which the subtrate i s t r a n s p o r t e d t o the s i t e o f r e a c t i o n i n the immobilized en zyme p a r t i c l e s . In chemical engineering the s u b j e c t o f the i n t e r play between d i f f u s i o n a l l i m i t a t i o n s and chemical k i n e t i c s i n het erogeneous c a t a l y s i s has been e x t e n s i v e l y s t u d i e d . The s t a t e o f the a r t on t h i s s u b j e c t i s described by S a t t e r f i e l d (_4 ). For the case o f a f i r s t order r e a c t i o n i n a s p h e r i c a l p a r t i c l e a r e l a t i o n s h i p between an e f f i c i e n c y f a c t o r , η, and a dimens i o n l e s s number, the so c a l l e d T h i e l e f a c t o r , Φ, can be shown to be given by:

Φ

tanl^

Φ

i n which η i s the r a t i o o f the general conversion i n the p a r t i c l e to the conversion i n absence o f d i f f u s i o n a l l i m i t a tions ( - ) Φ i s the T h i e l e f a c t o r being d e f i n e d as

Φ

=

R

O)

^

P\/D" P

i n which k i s a f i r s t order r e a c t i o n r a t e constant (1/s) D i s the d i f f u s i v i t y o f the r e a c t a n t i n the c a t a l y s t par t i c l e (m /s) Rpthe p a r t i c l e r a d i u s (m) 2

I f the s i m p l i f i e d pseudo f i r s t order equation f o r the conver s i o n r a t e o f glucose to f r u c t o s e , equation ( 6 ) , i s assumed to be s u f f i c i e n t l y accurate, the conversion r a t e o f p a r t i c l e s i n which whole c e l l s are immobilized i s given by: r i n which η

= Κ η C . V si e

s

(10)

i s the e f f i c i e n c y f a c t o r given by equation ( 8 ) , the T h i e l e f a c t o r b e i n g given by (11)

Κ Ό

i s the pseudo f i r s t order r a t e constant given by equa t i o n (7) (1/s) i s the c o e f f i c i e n t o f d i f f u s i o n f o r glucose i n the p a r t i c l e s (m /s) 2


11.

ROELS AND V A N TILBURG


151

C . i s the s u b s t r a t e c o n c e n t r a t i o n a t the p a r t i c l e s i n t e r f a c e (mole/m ) ^ i s the amount o f immobilized enzyme present (m ) 3


The e f f e c t o f the temperature. The o p e r a t i n g temperature a f f e c t s the i s o m e r i z a t i o n process i n two important ways. F i r s t l y the pseudo f i r s t order r a t e constant Κ i s expected to i n c r e a s e with i n c r e a s i n g temperature. In the present treatment the r e l a t i o n s h i p between Κ and temperature w i l l be assumed t o be o f the Arrhenius type: -ΔΗι / RT e

Κ = A

(12)

i n which: A i s a constant (1/s) ΔΗ^ i s the a c t i v a t i o n enthalpy o f the enzyme-catalysedi s o m e r i z a t i o n o f glucose t o f r u c t o s e (J/mole) R i s the u n i v e r s a l gas constant (J/mole K) Τ i s the absolute temperature (K) Secondly the d e a c t i v a t i o n r a t e o f the enzyme a c t i v i t y i s as sumed t o i n c r e a s e with i n c r e a s i n g temperature. I t i s assumed t h a t the pseudo f i r s t order r a t e constant Κ decreases with time accord ing t o : -k t Κ = K e ο d

(13)

i n which Κ i s the i n i t i a l pseudo f i r s t order r a t e constant (1/s) k^ i s the d e a c t i v a t i o n constant (1/day) t i s the o p e r a t i n g time (days) For the temperature dependence o f the d e a c t i v a t i o n constant an Arrhenius r e l a t i o n s h i p i s assumed: -ΔΗ* / RT k

d

i n which A an2

= A e

Z

(14)

2

i s a constant i s the a c t i v a t i o n enthalpy o f the enzyme d e a c t i v a t i o n process

Model f o r f i x e d bed i s o m e r i z a t i o n . In i n d u s t r i a l p r a c t i c e immobilized glucoseisomerase i s o f t e n a p p l i e d t o the i s o m e r i z a t i o n of glucose i n a f i x e d bed r e a c t o r . Under the assumptions about the k i n e t i c s presented above the c o n s t r u c t i o n o f a simple mathematical model f o r t h i s process i s q u i t e s t r a i g h t f o r w a r d . A balance equation f o r glucose over an i n f i n i t e s i m a l s l i c e o f the f i x e d bed (see f i g u r e 1) can be formula ted as follows : V dC - -K ÎC - C > · Π d s s s

ε

~ )

A


(15)


152


V (m/s> C 0 (mole/m 3

3

s

H

V

Figure 1.

Model for fixed-bed isomerization

(

m /S) 3

CsE (mole/m )


3

11.

ROELS AND

VAN

TILBURG

i n which V i s the flow r a t e o f the glucose syrup through f i x e d bed (m3/s) ε i s the p o r o s i t y o f the bed (-) ^ A i s the f i x e d bed cross s e c t i o n (m ) h i s the height coordinate (m) The boundary c o n d i t i o n s f o r equation (15) are : C C


i n which C

s

= C

n

sO

= C ^ s sE

153


the

h = 0 (16) h = H

i s the glucose c o n c e n t r a t i o n i n the syrup e n t e r i n g the column (mole/m^) C ^ i s the glucose c o n c e n t r a t i o n i n the isomerized syrup l e a v i n g the column (mole/m ) The s o l u t i o n o f (15) with the boundary c o n d i t i o n s according to (16) i s given by: g 0

S

3

„ _ Κ η( 1 - ε) A .Η - In ((C - C*) / (C ρ - C*)) so s sE s

,

.

V

Several assumptions are i m p l i c i t i n the d e r i v a t i o n presented here; two o f the important ones are : - the r e s i s t a n c e f o r mass t r a n s f e r to the p a r t i c l e s can be ne glected - the syrup flows through the column i n i d e a l p l u g flow Equation (17) gives the r e l a t i o n s h i p between the flow r a t e , V, which r e s u l t s i n conversion to an e x i t glucose c o n c e n t r a t i o n C £ as a f u n c t i o n o f Κ and η. As Κ was assumed to decrease with ope r a t i n g time (equation (13)) and η i s , through the T h i e l e f a c t o r , a f u n c t i o n o f K,the product Kn w i l l decrease with time. The de r i v a t i o n presented above i s only v a l i d i f the pseudo steady s t a te hypothesis i s invoked with respect to the change o f Κ with time. Κ i s assumed to be v i r t u a l l y constant during the residence time o f the syrup i n the column. I f t h i s assumption i s not made the simple d i f f e r e n t i a l equation (15) has to be r e p l a c e d by a s e t of simultaneous p a r t i a l d i f f e r e n t i a l equations i n time and height and the s o l u t i o n becomes by no means t r i v i a l . The pseudo steady s t a t e hypothesis with respect to Κ i s , however, q u i t e reasonable as the residence time i n the column i s o f the order o f one hour and the time constant o f the d e a c t i v a t i o n process i s o f the order o f 1 t o 100 days. Equation (17) can thus be used to s p e c i f y the flow v e l o c i t y at any moment during the operating time i f the mo mentary value o f Κ i s c a l c u l a t e d according to equation (13). s


154


Experimental The parameters o f the model and t h e i r temperature dependences were estimated on the b a s i s o f f i x e d bed experiments on a l a b o r a t o r y s c a l e . The enzyme used was Gist-Brocades immobilized glucose isomerase, Maxazyme GI-immob. The glucose "syrup" was a 45% w/w s o l u t i o n o f c r y s t a l l i n e dextrose i n d i s t i l l e d water c o n t a i n i n g 3 mM o f MgS0 and 100 ppm S 0 . The pH o f the syrup was 7.5. The f i x e d bed enzyme r e a c t o r was a jacketed g l a s s column provided with a f i l t e r p l a t e and c o n t a i n i n g about 30 ml o f bulk volume o f the immobilized enzyme. The conversion was determined p o l a r o m e t r i c a l l y as w e l l as by means o f high performance l i q u i d chromotography. The range o f o p e r a t i n g temperatures was 55 to 75°C. Two types o f expe riments were performed. Three s e t s o f experiments were performed at constant flow r a t e o f the s u b s t r a t e s o l u t i o n , the percentage glucose isomerized being determined as a f u n c t i o n o f o p e r a t i n g time. One set o f experiments was conducted at constant f r a c t i o n a l conversion to f r u c t o s e , the flow r a t e b e i n g determined as a func t i o n o f operating time. The d u r a t i o n of the experiments v a r i e d from 5 days at the h i g h e s t temperature t o more than 100 days at the lowest temperature. The parameters o f the model were estimated from the e x p e r i mental data using a non l i n e a r m u l t i v a r i a t e curve f i t t i n g t e c h n i que. In t h i s process the temperature dependence o f the d i f f u s i o n c o e f f i c i e n t f o r glucose was assumed to be s m a l l i n the range o f temperatures s t u d i e d . The e q u i l i b r i u m constant K was assumed t o be given by: f


4

2

x

K* = 28.8

exp (-1100 / RT)

(18)

This equation i s based on data reported by Fratzke et a l . (_3). The d i f f u s i o n c o e f f i c i e n t f o r glucose i n the immobilized en zyme p a r t i c l e s , ID , was estimated to be 6.7x10" (m2/s). A r e a sonable value i f i t i s compared with the value o f 8 . 8 x l 0 (m /s) obtained by V e l l e n g a (5^) f o r a d i f f e r e n t k i n d o f immobilized g l u coseisomerase. The dependence o f the estimated values o f the i n i t i a l value of the pseudo f i r s t order r a t e constant, KQ, on temperature was i n t e r p r e t e d i n terms o f an Arrhenius r e l a t i o n s h i p . In f i g u r e 2 the Arrhenius p l o t f o r K , s t a n d a r d i z e d , w i t h i n each s e t o f e x p e r i ments, with respect t o the i n i t i a l value o f the pseudo f i r s t order r a t e constant a t 65°C, KQ 5 5 , i s shown. The a c t i v a t i o n enthalpy o f the r e a c t i o n i s estimated'to be 79x10 J/mole. This can be compa red with a value reported by Fratzke et a l . being 70x10 J/mole. The dependence o f the experimental values o f the d e a c t i v a t i o n constant on temperature i s shown i n an Arrhenius p l o t i n f i g u r e 3. The a c t i v a t i o n enthalpy o f the d e a c t i v a t i o n r e a c t i o n i s estimated to be 2 0 ^ 1 0 J/mole. This agrees w e l l with e a r l i e r r e s u l t s o f Fratzke et a l . ( 3 ) , 204X10 J/mole and N i e l s en ( Ο , 197xl0 J/mole. 11

- 1 1

0

3

3

3

3


3

2

ROELS A N D V A N TILBURG



11.

Figure 2.

Arrhenius relationship for fixed-bed initial activity. (Φ) Constant version, ( Δ , • , O) constant flow.


156


From f i g u r e s 2 and 3 i t i s a l s o c l e a r that the r e s u l t s o f the constant flow experiments agree w e l l with those o f the con s t a n t conversion experiment. This may be considered an i n d i c a t i o n that the assumption o f n e g l i g a b l e d i f f u s i o n a l r e s i s t a n c e f o r g l u cose t r a n s p o r t t o the p a r t i c l e s i s c o r r e c t . In t a b l e I the k i n e t i c constants and t h e i r temperatures de pendences f o r the present immobilized enzyme and some other r e l e vant data have been summarized.


E v a l u a t i o n o f the model The model was a p p l i e d t o estimate the t h e o r e t i c a l r e l a t i o n ship between the f i x e d bed i n i t i a l flow v e l o c i t y r e s u l t i n g i n 45% of glucose b e i n g isomerized t o f r u c t o s e and the ope r a t i n g temperature f o r an immobilized enzyme d e f i n e d by the char a c t e r i s t i c s given i n t a b l e I . The r e s u l t s o f t h i s e v a l u a t i o n , i n terms o f the i n i t i a l flow v e l o c i t y r e l a t i v e to that a t 65°C, are shown i n an Arrhenius p l o t i n f i g u r e 4. The experimental r e s u l t s used i n the parameter e s t i m a t i o n are a l s o shown. The t h e o r e t i c a l r e l a t i o n s h i p i s shown t o be d e f i n i t e l y non l i n e a r i n an Arrhenius plot. To o b t a i n a b e t t e r understanding o f the i n f l u e n c e o f the ac t i v i t y o f the f r e e enzyme on t h i s r e l a t i o n s h i p c a l c u l a t i o n s were performed f o r a f r e e enzyme a c t i v i t y twice as w e l l as h a l f o f that o f the reference s i t u a t i o n s p e c i f i e d i n t a b l e I . Table I .

Parameter values f o r reference s i t u a t i o n

K

0,65

=

K

0,T

=

k

d

=

3 .0 χ 10

3

(1/s) ί exp (

9

oo m 4 .83 χ 10

1 .51

κ

10

Ό

= 6 .7 κ Ι Ο "

R Ρ

= 6 χ 10

-4

3 0

1 1

9500x ^—)

exp (-

2

7 ° )

(1/s)

(1/day)

2

(m /s)

(m)

The r e s u l t i n g Arrhenius p l o t s are shown i n f i g u r e 5. As can be seen the i n i t i a l f r e e enzyme a c t i v i t y a f f e c t s the Arrhenius p l o t . T h i s , i n combination with the already mentioned non l i n e a r i t y o f the Arrhenius p l o t , shows that d i r e c t e s t i m a t i o n o f the a c t i v a t i o n enthalpy o f r e a c t i o n from an Arrhenius p l o t o f the i n i t i a l flow rel o c i t y cannot be r i g h t . This i s a l s o i l l u s t r a t e d i n f i g u r e 6; the



ROELS A N D V A N

Figure 3.

TiLBURG


Arrhenius rehtionship for deactivation constant. (Φ) Constant conver sion, (Δ, • , O) constant flow.


157



158

Figure 4. Experimental data and theoretical relationship between fixed-bed initial flow velocity and temperature. (Φ) Constant conversion, ( Δ , O) con stant flow.



ROELS A N D V A N

Figure 5.


159

Characteristics of the Arrhenius plot of fixed-bed initial flow velocity for different activities of the free enzyme

lnV

Figure 6.

TiLBURG

0

Overall characteristics of the Arrhenius plot of fixed-bed initial flow velocity



160


c h a r a c t e r i s t i c s o f the Arrhenius p l o t o f i n i t i a l flow v e l o c i t y are shown f o r a broad temperature range. The slope o f the Arrhenius p l o t corresponds to h a l f the a c t i v a t i o n enthalpy o f the a c t i v i t y of the free enzyme at high temperatures and i s equal to the slope corresponding t o the a c t i v a t i o n enthalpy a t low temperatures. In the intermediary temperature range the slope g r a d u a l l y decreases with i n c r e a s i n g temperature. In f i g u r e 7 the t h e o r e t i c a l r e l a t i o n s h i p between the i n i t i a l flow v e l o c i t y r e s u l t i n g i n a f i x e d percentage o f conversion and the a c t i v i t y o f the f r e e enzyme i s shown. The r e l a t i o n s h i p i s l i n ear i f the f r e e enzyme has a very low a c t i v i t y o r , a l t e r n a t i v e l y , i f the p a r t i c l e r a d i u s i s very s m a l l or the d i f f u s i v i t y o f glucose very high (low T h i e l e f a c t o r ) . At high T h i e l e f a c t o r s (enzyme a c t i v i t y very h i g h , radius l a r g e , d i f f u s i v i t y low) a square r o o t r e l a t i o n s h i p i s obtained. To show the relevance o f both types o f behaviour to the immobilized enzyme used i n the present i n v e s t i g a t i o n i t s expected behaviour at temperatures o f 50 and 80°C i s shown i n f i g u r e 7. The present model a l s o allows c a l c u l a t i o n o f the r e l a t i o n ship between flow v e l o c i t y at constant r e l a t i v e conversion and o p e r a t i n g time i n a f i x e d bed. In the past l i n e a r as w e l l as exp o n e n t i a l functions have been proposed f o r t h i s p r o p e r t y . In f i gure 8 the expected r e l a t i o n s h i p s according to the present model, l i n e a r decay and exponential decay are shown. The r e s u l t s of one r e p r e s e n t a t i v e experiment have a l s o been shown. The present theory r e s u l t s i n a decay curve which i s between the l i n e a r and exponential r e l a t i o n s h i p s . The assumption o f l i n e a r decay i n volves an underestimation o f the a c t i v i t y h a l f l i f e ; the assumpt i o n o f exponential decay, however, overestimates the h a l f l i f e of the i n i t i a l flow v e l o c i t y . In f i g u r e 9 the t h e o r e t i c a l r e l a t i o n s h i p between the h a l f l i f e o f the i n i t i a l flow v e l o c i t y at 45% r e l a t i v e conversion and temperature i s shown i n an Arrhenius p l o t . The r e s u l t s obtained i n the experiments used i n the parameter e s t i m a t i o n are a l s o shown. The o v e r a l l c h a r a c t e r i s t i c s o f the Arrhenius p l o t o f flow vel o c i t y h a l f l i f e are shown i n f i g u r e 10. The p l o t i s l i n e a r with a slope corresponding to the d e a c t i v a t i o n a c t i v a t i o n enthalpy o f the f r e e enzyme a t low temperatures (low T h i e l e f a c t o r s ) . The h a l f l i f e o f the immobilized enzyme a c t i v i t y i s equal to t h a t o f the f r e e enzyme. In the intermediary range o f T h i e l e f a c t o r s or temperatures the slope slowly decreases then i n c r e a s e s again to the slope corresponding to the d e a c t i v a t i o n a c t i v a t i o n enthalpy o f the f r e e enzyme. At high T h i e l e f a c t o r s or high temperatures the slope again corresponds to the d e a c t i v a t i o n a c t i v a t i o n enthalpy but the a c t i v i t y h a l f l i f e o f the immobilized enzyme i s twice t h a t o f the f r e e enzyme. From the foregoing i t i s c l e a r t h a t the i n i t i a l flow v e l o c i ty h a l f l i f e o f the immobilized enzyme i s determined not only by the d e a c t i v a t i o n p r o p e r t i e s o f the f r e e enzyme but a l s o by the


ROELS A N D V A N

TiLBURG



11.

0

5

10

15 t i m e (days)

Figure 8.

Decay of the fixed-bed velocity at constant relative conversion: (O) experimental.


161



162


ROELS A N D V A N TILBURG



11.

Figure 10.

Overall characteristics of the Arrhenius plot of initial flow velocity half life


163



164

p r o p e r t i e s o f the p a r t i c l e s and i n p a r t i c u l a r the p a r t i c l e r a d i u s and the glucose d i f f u s i v i t y i n the p a r t i c l e s . The s i g n i f i c a n c e o f d i f f u s i o n a l l i m i t a t i o n t o the h a l f l i f e o f the immobilized enzyme used i n the present i n v e s t i g a t i o n i s i l l u s t r a t e d i n f i g u r e 11 where the r a t i o o f the a c t i v i t y h a l f l i f e o f the immobilized en zyme t o the estimated a c t i v i t y h a l f l i f e o f the free enzyme i s shown as a f u n c t i o n o f temperature. The immobilized enzyme a c t i v i ty h a l f l i f e i s h i g h e r by 10 t o 60% than the h a l f l i f e o f the free enzyme a c t i v i t y . One o f the i m p l i c a t i o n s o f the remarks made above i s that e s t i m a t i o n o f the a c t i v a t i o n enthalpy o f the d e a c t i v a t i o n o f the free enzyme from an Arrhenius p l o t o f the flow v e l o c i t y h a l f l i f e o f the immobilized enzyme i n a f i x e d bed i s , i n g e n e r a l , not j u s t i f i e d . One remark s t i l l needs t o be made. Most experiments were per formed a t a constant flow r a t e through the f i x e d bed and the de crease i n the f r a c t i o n o f glucose isomerized was observed as a f u n c t i o n o f time. In the foregoing d i s c u s s i o n we r e f e r r e d t o the h a l f l i f e o f the i n i t i a l flow v e l o c i t y a t constant r e l a t i v e con v e r s i o n and t h i s i s d e f i n i t e l y d i f f e r e n t from the r e l a t i v e conver s i o n h a l f l i f e a t constant flow r a t e . In general the h a l f l i f e o f the l a t t e r property i s c o n s i d e r a b l y longer than that o f the former. The present model furthermore p r e d i c t s that the h a l f l i f e o f the r e l a t i v e conversion a t constant flow r a t e i s dependent on the i n i t i a l glucose c o n c e n t r a t i o n and the i n i t i a l r e l a t i v e conversion. On i n s p e c t i o n o f equation (17) i t i s c l e a r that the h a l f l i f e o f the i n i t i a l flow v e l o c i t y a t constant r e l a t i v e conversion i s equal t o the h a l f l i f e o f the property In ( ( C o - C*) / ( C - C*)) a t con stant flow r a t e and our c a l c u l a t i o n s o f the i n i t i a l flow v e l o c i t y h a l f l i f e from the constant flow experiments were based on t h i s equivalency. Figure 9 shows that the h a l f l i f e estimates from the constant flow experiments obtained i n t h i s way do indeed agree with those obtained d i r e c t l y from constant conversion experiments. A commercially i n t e r e s t i n g property o f an immobilized enzyme i s i t s p r o d u c t i v i t y , the t o t a l cumulative amount o f syrup conver ted, during the time i t i s used f o r p r o d u c t i o n . In p r i n c i p l e the r e l a t i o n s h i p between amount o f syrup converted and o p e r a t i n g time can be obtained by i n t e g r a t i o n o f the equation f o r the flow v e l o c i t y r e s u l t i n g i n a given percentage o f r e l a t i v e conversion o f glucose (equation (17)) with r e s p e c t t o time i . e . : s

s E

t V ( t ) dt

(19)

ο i n which V ( t ) i s given by equation (17), the values o f Κ i s c a l c u l a t e d as a f u n c t i o n o f o p e r a t i n g time from equation (13) and the e f f i c i e n c y f a c t o r i s c a l c u l a t e d from equation ( 8 ) . The T h i e l e f a c t o r t o be s u b s t i t u t e d i n equation (8) i s given as a f u n c t i o n o f time by the f o l l o w i n g equation:


11.

ROELS AND V A N TILBURG

165



18h

1.4 h

1.2r

10

55 Figure 11.

60

65

70

75

80

T°C

Theoretical relationship on ratio between activity half life of free and immobilized enzyme


166


-i k t Φ = Φ e ο d

i n which Φ

(20)

i s the i n i t i a l T h i e l e f a c t o r being given by:

ο

ο

p

(21)

Ό

y

By combination o f equations ( 2 1 ) , (20) and (13) i t can be shown that the momentary value o f Κ can a l s o be w r i t t e n as :


2

Φ Κ = Κ ^ ο ,ζ Φ ο

(22)

Equation (19) can now with the a i d o f equations ( 1 7 ) , (21) and (22) be evaluated t o : t p

( t )

=

Γ i

P ( l - ε) . A.H

R

2

ρ

l

n

c

so c

s E

c

3

|

- C

1

φ

t a n h

*

_ U

d

t

(

2

3

)

φ

x

Equation (23) s t i l l presents a r a t h e r complex problem i f s t r a i g h t f o r w a r d i n t e g r a t i o n i s attempted. I t can however be sim p l i f i e d considerably by the f o l l o w i n g manipulations F i r s t i t i s recognized that the f o l l o w i n g r e l a t i o n s h i p h o l d s :

dt = 4 r · (1Φ αΦ

From equation (20)

dΦ

(24)

i s c a l c u l a t e d t o be

]

Immobilized Microbial Cells

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