Temperature Dependence of the Stability and the Activity of

GI-immob consists of pasteurized whole cells of Aotinoplanes mis- sourïensis ..... (1/s). K 0,T = 4 oo m 9 ί. 9500x .83 χ 10 exp (. ^—). (1/s) k ...
0 downloads 0 Views 1MB Size
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

979 Sufiifify t i b i a l Society

1155 16th St. N. W. Washington, D. C. Microbial 20036 Cells Venkatsubramanian; Immobilized ACS Symposium Series; American Chemical Society: Washington, DC, 1979.

148

IMMOBILIZED MICROBIAL CELLS

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

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

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

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

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

150

IMMOBILIZED MICROBIAL CELLS

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

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

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

11.

ROELS AND V A N TILBURG

Immobilized Glucose Isomerase

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

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

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

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

(15)

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

152

IMMOBILIZED MICROBIAL CELLS

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 )

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

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

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

i n which C

s

= C

n

sO

= C ^ s sE

153

Immobilized Glucose Isomerase

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

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

154

IMMOBILIZED MICROBIAL CELLS

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

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

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

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

3

2

ROELS A N D V A N TILBURG

Immobilized Glucose Isomerase

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

11.

Figure 2.

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

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

156

IMMOBILIZED MICROBIAL CELLS

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.

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

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

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

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

ROELS A N D V A N

Figure 3.

TiLBURG

Immobilized Glucose Isomerase

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

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

157

IMMOBILIZED MICROBIAL CELLS

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

158

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

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

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

ROELS A N D V A N

Figure 5.

Immobilized Glucose Isomerase

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

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

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

160

IMMOBILIZED MICROBIAL CELLS

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

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

ROELS A N D V A N

TiLBURG

Immobilized Glucose Isomerase

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

11.

0

5

10

15 t i m e (days)

Figure 8.

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

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

161

IMMOBILIZED MICROBIAL CELLS

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

162

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

ROELS A N D V A N TILBURG

Immobilized Glucose Isomerase

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

11.

Figure 10.

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

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

163

IMMOBILIZED MICROBIAL CELLS

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

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:

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

11.

ROELS AND V A N TILBURG

165

Immobilized Glucose Isomerase

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

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

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

166

IMMOBILIZED MICROBIAL CELLS

-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 :

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

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)



(24)

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

]