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Nonisothermal Heterogeneous Reaction in aDenaturable Immobilized Enzyme CatalystINTRODUCTIONIn recent years, considerable attention has been focused on heterogeneous

catalytic reaction in immobilized enzyme catalysts.'-9 Lee and Tsao,I Ollis,*

Rony,3 Sundaram et al.,d and Vieth et al.5 examined the effectiveness factor of

immobilized enzyme catalysts of different geometries. In these works, a simplified

Michaelis-Menten rate equation of zero orfirst order has been employed sothat a closed form ofeffectiveness factor in terms of the Thiele Modulus and otherparameters could be obtained. In reality, such a simplified Michaelis-Menten

rate equation constitutes only a special case of its general form which may not be

able to display the true reaction characteristics of the immobilized enzyme catalysts.

Fink et a1.,6 Miyamoto et al.,? and Moo-Young and Kobayashis investigated

the effectiveness factor by using the complete form of the Michaelis-Menten

rate equation. Horvath and Engasserg considered the effectiveness factor in a

pellicular immobilized enzyme catalyst.

A common assumption made in all the above works is that the enzymaticreaction is carried out under isothermal condition and with a constant enzymeactivity during the whole reaction. This may not be true in a number of real

circumstances because of the exothermic nature of enzymatic reactions.*0-I6

Fora nonisothermal reaction, temperature plays an extremely important role inevaluating the effectiveness factorofan immobilized enzyme catalyst because

the activity of enzyme is rather sensitive to the reaction temperature. It hasbeen widely recognized that above a certain level of temperature, the enzyme

activity decreases significantly.10-' This phenomenon, known as the thermal

inactivation ordenaturation, occurs frequently in many enzymatic reactions.The decreasing enzyme activity tends to place the entire enzymatic reaction in a

transient state rather than a steady state as is generally considered. Thus far

no work has ever taken thermal inactivation into consideration in the investigationof the effectiveness factor of an immobilized enzyme catalyst. The purpose

of this work is to point out that it is important to consider the transient

state when investigating the heterogeneous reactions in an immobilized enzyme

catalyst.

TRANSIENT MATHEMATIC MODELConsider the following enzymatic reaction

ki k

E+ SF=ES-A E + Pk-i1237

@ 1975by John Wiley & Sons, Inc.1238 BIOTECHNOLOGY AND BIOENGINEERING VOL. XVII (1975)

The unsteady state material and energy balances are given by

&,? = k,(? + ?.> - k$E( -AH) exp(- "> (3)bt brzc beS+ k, ROTin which the enzyme concentration, E, is given by

-bE = -k.E exp(- g)bt (4)The initial and boundary conditions for the above equations are:

t= o ; S= 0, T= To, E = EobS bTbrbe

f= o ; - = o , - = oBy imposing appropriate assumptions, eqs. (2) through (7) can be reduced to all

the cases investigated by the previous authors.1-9000.60

040.2

C I I 130 6090 I20

Fig. 1. Effect ofmass transfer Nusselt number on the dimensionless substrate


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concentration with a, = 1.0,K, = 0.35, p2 = 117.5,@, = 18.5,K. = 1.1 X 1046,a2= 0.5, h = 0.5 and (Nu),, = 0.5.

COMMUNICATIONS TO THE EDITOR1239It can be noted that it is more difficult to solve eqs. (2), (3), and (4) than tosolve the corresponding material balance equation for the steady state because

of the highly nonlinear reaction terms involved in these equations. However,

eqs. (2) and (3) can be numerically solved by the implicit Crank-Nicolson finite

difference method," and the fourth order Runge-Kutta method" can be employedto integrate eq. (4). Figures 1, 2, and 3, respectively, show the transient variationsof the dimensionless substrate concentration, enzyme activity, and dimensionless

temperature at the catalyst center (dashed lines) and the catalyst surface(solid line) fora specific example.

Forsteady state enzymatic reaction, the effectiveness factor is defined as theratio of actual reaction rate to the hypothetic reaction rate in the absence ofinternal diffusion resistance.lsJ9 According to the definition, the effectiveness

factor can be written asI

04

Df

80 4

0.2

C030 6090 120

r

Fig. 2. Effect of mass transfer Nusselt number on the enzyme activity.

1240 BIOTECHNOLOGY AND BIOENGINEERING VOL. XVII (1975)1.07r

030 6090 120

rFig. 3. Effect of mass transfer Nusselt number on the dimensionless

temperature.For the steady state reaction, the concentration gradient at the catalyst surfaceis an invariant; however, for the unsteady state, it is time-dependent. Thesubstrate which has already diffused into the particle does not represent the actual

reaction rate because part of the substrate is accumulated inside the particle.

Therefore, eq. (8) can not be considered as the effectiveness factor for the unsteady

state case and does not have specific meaning except for representing the variationofsubstrate concentration gradient at the catalyst surface.Nomenclaturea

CCP

DEEo

AEiAEzhsurface area per unit volume of catalyst particledimensionless substrate concentration, S/Soheat capacity offluid

effective diffusivityenzyme concentrationinitial enzyme concentrationactivation energy for enzymatic reaction

activation energy for enzyme inactivationThiele modulus, rgdkoEo/DSoCOMMUNICATIONS TO THE EDITOR1241hr heat transfer coefficient

( -AH) heat generation by enzymatic reactionturnover numberinactivation coefficient of enzyme

mass transfer coefficient

Michaelis-Menten coefficientfrequency factor

thermal conductivitydimensionless inactivation coefficient of enzyme, k,ro2/D

dimensionless Michaelis-Menten coefficient, k,/So


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heat transfer Nusselt number, htro/ktmass transfer Nusselt number, kLaro/Dr radial coordinatero radius of catalyst particle

RR, gas constantSsubstrate concentration

sot time

T temperatureTo initial temperatureGreek Lettersa1

a2

81

82P fluid density

e dimensionless temperature,TITO

'T dimensionless time, t D/ro2+ enzyme activity, E/EO7

.: dimensionless radial coordinate, r/ro

substrate concentration outside the catalyst

ratio of Schmidt number to Prandtl number, kl/pCpDdimensionless heat generation parameter, ( - AH)SO/~C,TOdimensionless activation energy for enzymatic reaction, AEI/R,Todimensionless activation energy for enzyme inactivation, A&/R,l'udimensionless parameter defined by eq. (8)

[Repr intedf rom the Journal of the Amer i canC hemi calS


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G -6-Pw ast henp recipitatebdy additiono f 1.2m ol ofBa(OH)2:t he resul t ings ol id( 502g ) contained9 2o/Bo a G-6-P- lH2O( 0.89 mol ) by enzymat ica ssay.eT his quant i tycorrespondtso a 65%y ieldb asedo n AcP added.T he activitiesof hexokinasaen da cetatek inasew erer ecovereidn theg el in93 and 75 o/oyi eld,r espectivelyT.h e turnovern umberf or ATPduringt her eactionw as) 100;n o effortw asm adet o recover

i t .Three points concerning experimental details deservement ionF. i rst ,t hei ni t ialq uant i t ieosf ATP andM g( l l ) werechosens ucht hatt hec oncent rat ioonf MgATP andM gADPwouldb e well abovet he Michaelisc onstantsfo r the solubleenzymes, le0v ena f terd i lut ionb y theA cP solut ionS. econd,the reactionp roceedesda tisfactorilwy ith AcP having) 80o/opurity. If the purity fell below8 0o/oc,o rnplexatioann d precipitat iono f Mg( l l ) by thep hosphatiem pur i t iesm adei t di f -f icul t to maintaina dequateco ncent rat ionosf MgADP andMgATP in solut iona, ndt roublesomteo isolateB a G-6-Pi nhighp urity.T hird, it wasu sefutl o carryo ut ther eactions ot hataddi t iono f AcP to thes olut ionw aso veralrl ate- l imi t inga nd

AcP wasn everp resenitn the reactionm ixturei n high concent

rat ionsto, minimizes pontaneouhsy drolysiosf AcP wi thconcomitanrte leaseo f phosphate.Compar isono f this preparat iono f G-6-P wi th exist ingchemical tot r enzymat iclm2 ethodsi l lust ratetsh e potent ialof ATP-requiringe nzymatics ynthesisfo r the regioselectivemodificationo f unprotectedw, ater-solublep,o lyfunctionalsubstratesS.i ncet heh exokinasehsa veb roads ubstratesp ecifici ty, rst his sequencseh ouldb e di rect lya ppl icableto thepreparationo f phosphateosf a numbero f others ugars( e.9.,fructosem, annosed, eoxy-o-glucosgel,u cosamineI)n. broaderterms,t hisc onversioens tablishethsa t it rsp racticalt o coupleenzymaticA TP regeneratiowni th ATP-requiringe nzymaticsynthesitso achievela rge-scaoler ganict r ansformat ionRse. -actionsw hichr equirer egeneratioonf ATP from AMP area lsoaccessibules ingt hisr eact ions equenceb,y addinga denylatekinase(A MP:ATP phosphot ransferaEs.e C, . 2.7. 1.3\t o cat -alyzet hec onversioonf AMP andA TP to ADP;5w ew ill provideexampleso f this typeo f reactions equencien the imme-I 977

o

;].TOHG-6-P

(ATP\cH,]co2-\

ADP

ooll ll -ocHrcoPi\ - - - tJAcP\LcH,:t-:0\*'r'H3P0,

Journal of the American Chemical Society / 99:7 / March 30,diatef uture.T he goods tabilityo f the immobilizede nzymes,andt he easeo f theirr ecoverys, uggesttsh at theses ynthesiasn dregeneratiosnc hemessh ouldh aveb roada pplicabilityin preparativeorganicc hemistryI. 2Referencesa nd Notes


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(1) Supportedb y the NationalS cienceF oundation(R ANN),G rantN o. Gl34284.(2) Estinntedc osts,$ /nrole:A TP,2 ,000:N AD+,2 ,500;N ADH.1 8,000;N ADP+,60,000;N ADPH2, 50,000.(3) J. B. Jones,D . Perlmana, ndC . J. Sih,E d.," Applicationso f BiochemicalSysternsin OrganicC hemistryP, arts1 and2 ", Wiley,N ew York, N.Y.,'1976.(4) C. J. Suckl inga ndK . E. Suckl ingC, hem.S oc.B ev. ,3,3 87 (1974) .(5) C. R.G ardneer t al.i n "EnzymeE ngineerin2g", ,K.E. foe andL . B.W ingard,

Ed.,P lenumP ress,N ew York,N .Y.,1 974,p 209;G . M. Whitesidese t al.,ibid., 9 217.(6) Enzymesw ere immobilizedb y additionto a solutionc ontaininga polymerizingmixtureo f acrylamideN, ,M-methylenebisacrylamiadned, t he l*hydroxysuccinimidaec livee stero f methacrylica cid, 10s beforeg el formation.The procedure used is a modification of that described (G. M.Whitesidese t al., MethodsE nzymol.i,n press).l mmobilizationy ieldsw ere35o/o lor hexokinase, and 40o/o for acetate kinase. The enzymes werecommerciapl reparation(sS igma)a, nd were usedw ithoutp urificationt:h eirspecifica ctivities(p molm in-1m g-1)were:h exokinas(ef romy east)4, 20;acetatek inase( fromE . colr)f ollowinga ctivationw ithd ithiothreito3l,0 0.(7) Diammonium acetyl phosphate was prepared in a separate step by reactionof ketenew itha nhydroups hosphorica cid,f ollowedb y neutralizatioann d

2367precipitation with anhydrous ammonia: G. M. Whitesides, M. Siegel, andP. Garrett, J. Ag Ctem., 40, 2516 (1975). The material used was 80-85%pure, with ammonium acetate and acetamide as the principal impurities.The acetyl phosphate solution was maintained at 0 "C before addition tominimize hydrolysis.

(8) Preliminary experiments indicated that, under simulated reactor conditions,the rate ol G-6-P formation was faster at pH 6.7 than at higher pH wherethe soluble enzymes would be expected to be more active (A. Sols, G.delaFuente, C. D. Villar-Palasi, and C. Ascensio, Biochem. Biophys. Acta,30, 92 (1958); l. A. Rose, M. Grunberg-Manago, S. T. Korey, and S. Ochoa,J. Biol. Chem., 2'11,737 (1954)).(9) H. U. Bergmeyer, Ed., "Methods of Enzymatic Analysis", Verlag ChemieWeinheim, Academic Press, New York and London, 1974, p 1238.('10) Hexokinase, A. Sols et al., Biochim. Biophys. Acta, 30,92 (1958); acetatekinase, C. A. Janson and W. W. Cleland, J. Biol. Chem., 249,2567 (19741,R. S. Langer, C. R. Gardner, B. K. Hamilton, and C. K. Colton, AIChE J., i npress.(11) H. A. Lardy and H. O. L. Fischer, Biochem. Prep.,2,39 (1952).(12) W. A. Wood and B. L. Horecker, Biochem. Prep.,3,71 (1953).(13) M. Dixon and E. C. Webb, "Enzymes", 2nd ed, Academic Press, New York,N.Y., 1964, p 216, and references cited therein.(14) G. M. Whitesides, in ref 3, part 2, Chapter Vll.

Alfred Pollak, Richard L. Baughn, George M. Whitesides*Deport ment of C he mi st r yM as s ac hus et t s I nst it ut e of Tec h nol og1:

Cambridge,M assachusetts0 2 I 39Receiued December 7. I976

mass generration

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