Biochimie, 68 (1986) 1237- 1243 ©Soci&6 de Chimie biologique/Elsevier, Paris

1237

Short communication

Studies on the kinetics of immobilized enzyme using a recycling enzyme reactor system

Yi-xin LOU*

Laboratory o f Enzyme Engineering, East China Institute o f Chemical Technology, 130, Mei-long Road, Shanghai, China

(Received 27-5-1986, accepted 3-9--1986)

Summary - A method of measuring kinetic parameters of immobilized enzyme with a recycling enzyme reactor system is described. By analyzing the plot of the dimensionless variable Ln(1-X)/X versus the time needed for a unit conversion, t /X , the mechanism of enzymatic reaction can be recognized and then its basic parameters can be evaluated. On the basis of the experimental data measured by P.R. Coulet et al., it has been proposed that the successive degradation of maltodextrins by the collagen membrane-bound amyloglucosidase was a product glucose inhibition reaction and their corresponding constants have been found with this method.

enzyme reactor I apparent kinetics / immobilized enzyme I product inhibition / immobilized amyloglucosidase

R~sum~ - Elude de la cin~tique d 'enzyme fix~e/~ l 'aide d 'un syst~me de r~acteur enzymatique recireulalion. Une mdthode pour mesurer les paramktres cindtiques d'une enzyme fixde ~ l'aide d'un

systbme de rdacteur enzymatique ~ recirculation est ddcrite. L "analyse de la courbe expdrimentale, portant la variable L n ( 1 - X ) / X en fonction du temps mScessaire pour une conversion unitaire, t /X, r~vble le mdca- nisme de la rdaction enzymatique. Ainsi, les parambtres peuvent ~tre calcul#s.

Les valeurs expdrimentales pour l'hydrolyse des maltodextrines mesurdes par P.R. Coulet et al., et expri- redes selon cette mdthode, nous permettent de suggdrer que la ddgradation successive des maltodextrines par l'amyloglucosidase fixde sur une membrane de collagbne est une rdaction d'inhibition par le produit glucose, et ainsi, de calculer ses constantes correspondantes.

reacteur enzymafique I cin~tique apparente I enzyme fix~e I produit inhibition I amyloglucosidase fix~e

Introduction

The recirculation reactor system was used in early studies of heterogeneous catalysis [1,2]. James R. Ford et al. have described in detail the characteris- tics of this system and several of its applications [3]. C. Horvath et aL have also discussed the kine- tic behavior of the hydrolysis of BAEE in a nylon tube by bound trypsin using a recycling system [4]. Theoretically, if a gradientless reactor was used,

such a system could in fact be recognized as a device for measuring basic parameters of immobilized enzyme and a research tool for enzyme engineering.

The purpose of this paper is to develop a method to analyze the reaction mechanisms of immobi- lized enzyme and to evaluate its apparent _kinetic constants K m, maximal specific activity A, and inhibition constants K i. Finally, we use this method to elucidate the hydrolysis of malto- dextrins.

* Address all correspondence to: M. Shu Kang Chen, R6sidence Ecole Centrale de Paris, Ch. D. 221, 2, av. Sully Prudhomme, 92290 Chatenay, Malabry.

1238 Y. Lou

Theory

The simplest recycling enzyme reactor system con- sists of three parts: enzyme reactor, reservoir tank and feed pump (Fig. 1). The enzyme reactor could be a packed bed reactor, a membranous reactor, a hollow fiber reactor or a tubular reactor. But, as a research tool for measuring the basic features of immobilized enzyme, the reactor should ideally be a gradientless differential reactor, i.e., the differ- ence between the inlet and outlet concentrations of the reactor should approach zero (that is, its per pass conversion becomes almost zero). In the gener- al reactor, both the axial mixing of the reaction solution between particles of immobilized enzyme in a packed bed and the nonideal f luid-solid con- tact could disturb the concentration distribution according to the ideal model in a plug flow reac- tor. However, decreasing the volume of the reac- tor and appropriately increasing the recirculatlon flow rate can effectively reduce the influence of these factors.

8 I s` Q

s s, +

I Fig. 1. Flow sheet of a recycling enzyme reactor system. A: reser- voir tank; B: enzyme reactor; C: feed pump.

If the residence times %, v b are adopted, then

V a V b

and

~'b Vb = ~ =fl (a constant)

Equation l becomes

Sb-Sa = "Ca dSa (2) dt

and thus

Sb dLnS a - - = 1 (3) Sa + vadt

F o r a M i c h a e l i s - M e n t e n t y p e r e a c t i o n

For this reaction

E + S @- ES ~ E + P

the reaction velocity in a reactor can be expressed as

VmS Vb-Km + S

where, V m is the maximal reaction velocity and K~ is the apparent Michaelis constant.

When a feed solution passes through the reac- tor, its concentration will change from S a to S b. Therefore, the residence time in the reactor is

f S b - d S K m S b 1 ~'b Sa Vb -- Vm Ln Sa -~m (Sb-Sa) (4) =j

C a l c u l a t i o n

A material balance in reservoir A gives

dS a Q'Sb-Q'Sa= Va dt (1)

where, Q is the volumetric recirculation flow rate; Sa is the substrate concentration leaving reservoir A (it equals the substrate concentration entering reactor B); S b is the substrate concentration enter- ing reservoir A (it equals the substrate concentra- tion leaving reactor B); V a, V b are the volumes of reservoir A and reactor B, respectively.

Substituting equations 2 and 3 into equation 4, one finds

% = -Km Ln (1 + ~a dLnsa~ za dSa

V m \ dt ] V m dt

In a gradientless reactor, the volumetric flow rate is generally large, the residence times r a and v b are thus very small. And since the per pass conversion Sb-Sa/S a of the reactor is very small, the term dLnSa/dt is without doubt very small. Thus, the following approximation can be adopted

Ln ( l+~adLnSa / dLnSa -- ~a ~ (5)

d t / dt

A p p a r e n t kinetics o f immobi l i zed enzyme 1239

The residence time expression is reduced to

"r b = - - - -

K m dLnS a T a dSa ~'a--

V m dt V m dt

After rearrangement, this equation can be integrat- ed, from t=0 , (Sa)o=So, to t = t , (Sa)t=St, to obtain the substrate concentration expression in the reservoir as follows"

S t 1 +-z-z.. (S t -So) = - [3t Ln ~o Am Km (6)

We define the conversion X in terms of the sub- strate concentration in the reservoir as

S O - S t X = - -

So Thus one obtains the conversion equation

L n ( I - X ) V m [3 t S o - - - + ( 7 )

X K m X K m

By determining experimentally t h e substrate concentration in the reservoir at different times, one can plot out a conversion curve for the varia- bles X and t. Then, after plotting L n ( 1 - X ) / X versus t / X , a straight line in the fourth quadrant will be given for the Michaelis-Menten type reaction. Here, L n ( 1 - X ) / X is a dimensionless variable inferior to 0, and t / X is used to express the time required for a unit conversion. The line and the abscissa intersect at S o / V m [3, and it intersects the extrapolated line of the ordinate in the first quadrant at S o / K m (Fig. 2). Conse- quently, the apparent kinetic constant K m and the maximal reaction velocity V m of immobilized enzyme under the condition of a certain substrate feed concentration and a certain volumetric flow rate can be easily calculated.

F o r inhibi t ion react ions

By analogous deduction, the conversion equations for reactions occurring in the presence of inhibi- tors can be found.

I IS_.z

I \ \ \ \

\ \ \ _ _ ,%

\ v,.~

X

Competitive inhibition

E + S ~ ES --+ E + P + I

,Jr EI

The reaction velocity equation can be expressed as

Y b =

VmS

m(l+ 1) +s

where, I is the inhibitor concentration and K i is the inhibition constant. So, the conversion equatioffin the reservoir is

Ln(1-X) V m t3 t S o

X K~n(l+ I)-~i X Km 1 ~ i ' ( + I )

(8)

Hg. 2. AplotofLn(l-X)/Xversust/XforaMichaelis-Menten Under the condition of a certain feed concentra- type reaction, tion and a certain volumetric flow rate, both the

1240 Y. Lou

slope and intercept of the straight line vary with the variation of inhibitor concentration (Fig. 3).

Uncompetitive inhibition

E + S @ E S - - . E + P + I

ESI

The reaction velocity is

VmS

V b - - K m + S (1 + ~ i )

and the conversion equation in the reservoir is

( i) Ln(1-JO Vmfl t S o l+-~i - - = - - ~ ( 9 )

X K m X K m

It is known from the linear equation (9) that, for a certain flow rate, the slope of the line is independent of both the substrate feed con- centrat ion and the inhibitor concentrat ion (Fig. 4).

Noncompetitive inhibition

E + S = ES --* E + P + + I I

E I + S = EIS

Thoreact ion velocity equation is

Vb----- VmS

(Km+S) ( l + l l Ki/

I I

I \ I \ I \ I \ I \ ~x x \ \

IlX I \ I \ I \ \

\ }, \

I \ \ I \ \ I \ \ \ \ ix \ \

\ \ \ \ \ \

\ \ \ X \ \

\ \ \

Fig. 3.3 plots of Ln(1-X)/X versus t/X at different inhibitor Fig. 4. Plots of Ln(I-X)/X versus t/X at different inhibitor concentrations, for a competitive inhibition reaction, concentrations, for an incompetitive inhibition reaction.

Apparent kinetics of immobilized enzyme 1241

and the conversion equation in the reservoir is (Fig. 5)

Ln(1-X) V m [3 t S O - - - t r , - - + - ( 1 0 )

X Km tl + l--- ] X K m k Kd

It is seen in Fig. 5, under the same feed concentra- tion and same flow rate, the slope of the line varies with the variation of inhibitor concentration. All of the lines intersect the extrapolated line of the ordinate in the first quadrant at the same point sdx'~.

If the inhibitor is the product of the reaction, one can derive the conversion equation in the same way as that presented previously.

V b =

and

v~s K m + + S

Ln(1 - X )

X

VmS

+Sol+s(l_ :l Km Kilt \ KiJ

Vm~ t \ Ki ] + ( l l )

It can be seen from this equation that both the slope and the intercept of the line are related to the feed concentration (Fig. 6).

Product competitive inhibition

Provided that the chemical stoichiometric relation of product formation is P=So-S, then:

$.__~o

I t S \

\ \ \ \ \ \ Nx\ ' , \ -

\ \ \ "

\ \ \

Fig. 5. Plots of Ln(I-X)/X versus t/X at different inhibitor concentrations, for a noncompetitive inhibition reaction.

Product noncompetitive inhibition

We have,

VmS Vb -- (I (1 $2

Km +-~i) + 4 SOKi Kml S - ] Ki

/

X

m ± ×

Fig. 6, Plots of Ln(I-X)/X against t/X at different substrate feed concentrations, for a product competitive inhibition reaction.

1242 Y. L o u

and Application and Discussion

Ln(I-X) VmP t +

X X

I m

X + (12)

Obviously, the equation does not yield a linear rela- tionship between Ln(1-X)/Xand t /X , and the rele- vant kinetic parameters can not be calculated simply. However, they can be evaluated using a digital computer.

In conclusion, different enzyme reaction mecha- nisms can give different conversion equations. According to the experimental data of the conver- sion in the reservoir of a recycling system, the mechanism of the reaction can be postulated and the basic parameters of the immobilized enzyme evaluated.

P.R. Coulet et aL have used an enzyme-collagen membranous reactor for the hydrolysis of malto- dextrins into glucose at a high flow rate in recycling experiments [5]. We have their permission to use their original data for the kinetic study.

In, a typical experiment (Q = 42 l/h), the volume of the solution recirculated was 3000 ml. The extent of conversion of maltodextrins into glucose in the reservoir was measured with a glucose-enzyme electrode (Fig. 7). The inner liquid volume of the reactor was 115 ml and the enzymatic surface avail- able was 1.6 m 2.

According to the method described above, we could calculate from Fig. 7 the values of Ln(1- X)/Xand t / X i n the case of the substrate feed con- centrations: 226 g/l, 189 g/l and 95 g/l, respec- tively, and thus obtained the plots of L n ( I - X ) / X versus t / X (Fig. 8). These plots are straight lines with the different slopes which are inversely pro- portional to the feed concentrations. Consequently, we drew the conclusion that there existed a product competitive inhibition in the reaction system. We propose the mechanism of maltodextrin hydrolysis by amyloglucosidase-collagen membrane as follows:

ioo X

75 •

°i 42 I. h "~

! | !

2.5 50 75

(ho rs)

t

X

-0.5

-t.O

-2.

-2.$

25 ~o 75 i

± X

Fig. 7. Hydrolysis of maltodextrins in recycling experiments Fig. 8. Plots of Ln(l-X)/Xagainst t / X a t three maltodextrin feed (from Pierre R. Coulet et aL 1980). concentrations 266 g/l ( ~ ) , 189g/1 ( × × ), 95 g/l ( -'- e ).

A p p a r e n t kinetics o f immobi l i zed enzyme 1243

S n + E ES n ~ E + S n _ l + G

Jt ESn_1 ~ E + S n _ 2 + G

Jt ESn_ 2 ~ E+Sn_ 3 +G

E + G ~ EG

where G = p r o d u c t glucose; Sn, Sn_ I . . . . . . . =maltodextrins and their various degradation intermediates. The conversion equation (11) for product competitive inhibition can be obtained using the deductive method presented above.

If Y=the slope of equation (11), we obtain ar~bther linear equation:

I a

1 _ K m K m __1 So (13) Y Vmfl Vmfl g i

and thus from (Fig. 9), the product glucose inhibi- tion constant :

the intercept of the line = 6.35 × 10 -2 M. K i - the slope of the line

According to the intercept of the line in Fig. 8, one finds K r n - l . 4 7 × 1 0 - I M" V m = 3 . 1 9 x l 0 -2 M/min. Finally, the maximal specific activity A o f the enzyme membrane was

V m × the inner volume of the reactor

.4 - = 229 mU/cm 2. the available surface

of enzyme-membrane

-¢0

-20

-30

-zto

-50

0.5 £0 ¢.~7 i 1

± ¥

Fig. 9. Plots of I /Y versus S o.

v

so Acknowledgements

This work was supported by the LBTM-CNRS, Univer- site Claude Bernard (Lyon I). The author wishes to thank Prof. D.C. Gautheron for comments and discussions, and Prof. P.R. Coulet for kindly providing their origi- nal experimental data.

References

1 Anderson R.B. (1968) in: Experimental Method in Catalytic Research (R.B. Anderson, ed.), Academic Press, New York, p. 17

2 Levenspiel O. (1962) Chemical Reaction Engineering, John Wiley & Sons, New York, p. 452

3 Ford J.R., Lambert A.H., Cohen W. & Chambers R.P. (1972) Biotechnology and Bioengineering Symposium 3, 267-284

4 Horvath C. & Solomon B.A. (1974) Enzyme Engi- neering (Pye E.K. & Wingard L.B. Jr., eds.) Vol. 2, Plenum Press, New York, p. 470

5 Coulet P.R., Paul F., Dupret D. & Gautheron D.C. (1980) Enzyme Engineering (Weetall H.H. & Royer G.P., eds) Vol. 5, Plenum Press, New York, p. 231