Chemical Engineering Science 58 (2003) 4287–4290www.elsevier.com/locate/ces

Two-parameter model for evaluating e&ectiveness factor forimmobilized enzymes with reversible Michaelis–Menten kinetics

J. L. G/omez∗, A. B/odalo, E. G/omez, J. Bastida, M. F. M/aximoDepartment of Chemical Engineering, University of Murcia, 30071 Murcia, Spain

Received 20 November 2001; received in revised form 27 June 2003; accepted 1 July 2003

Abstract

A two-parameter mathematical model was developed to calculate the e&ectiveness factor for immobilized enzymes in porous sphericalparticles. The model was resolved for reversible Michaelis–Menten kinetics, including simple Michaelis–Menten and product competitiveinhibition kinetics. Since only two dimensionless moduli are involved in the model, the e&ectiveness factor for the three kinetic equationsconsidered can be estimated by using only one generalized graph.? 2003 Elsevier Ltd. All rights reserved.

Keywords: Di&usion; Enzyme; Mathematical; Modeling; E&ectiveness factor; Immobilized enzymes

1. Introduction

Enzymes are usually attached covalently to a porous par-ticle support to increase their thermodynamic stability andfor easy separation from the reaction system so that theycan be reused. Since the catalytic reaction occurs inside theparticles, the reaction rate is a function of the local substrateand product concentrations, and the process is commonlyin<uenced by external and internal di&usional limitations.The internal di&usional e&ects can be quantitatively ex-

pressed by the e&ectiveness factor, �, de=ned as the ratio ofthe actual reaction rate to the rate which would be obtainedif all the enzyme molecules inside the particle were exposedto the same substrate and product concentrations that existat the particle surface (Bailey & Ollis, 1986).Most theoretical models developed for estimating the ef-

fectiveness factor for heterogeneous enzymatic systems arebased on the following assumptions:

• The catalytic particle is a porous sphere with a radius R.• The enzyme is uniformly distributed throughout the wholecatalytic particle.

• Di&usion reaction takes place at a constant temperatureand under steady-state conditions.

∗ Corresponding author. Tel.: +34-968-367-351;fax: +34-968-364-148.

E-mail address: carrasco@um.es (J. L. G/omez).

• The substrate and product di&usion inside the catalyticparticle can be modeled by Fick’s =rst law and e&ectivedi&usivity is the same throughout the particle.

• The enzymatic reaction is monosubstrate and yields onlyone product.

Based on these assumptions, the di&erential mass balancefor substrate and product in spherical coordinates, and theboundary conditions are:

DSr2

ddr

(r2dCSdr

)= VS; r = 0 ⇒ dCS

dr= 0;

r = R⇒ CS = CSR; (1)

DPr2

ddr

(r2dCPdr

)=−VS; r = 0 ⇒ dCP

dr= 0;

r = R⇒ CP = CPR: (2)

The main di&erence between the abundant mathematicalmodels published concerns the enzymatic kinetics used. Thisdetermines the mathematical expression of VS and, obvi-ously, the solution o&ered by those models are restricted tothe enzymatic reactions that are governed by such kinetics.Moreover, the type of resolution method used to solve thedi&erential equations and its degree of complexity dependson the mathematical form of VS .

0009-2509/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0009-2509(03)00307-5

4288 J. L. G'omez et al. / Chemical Engineering Science 58 (2003) 4287–4290

The classical analytical solution for =rst-order kinetics,which provides the e&ectiveness factor value as a hyper-bolic function of the ThiGele modulus, is well known. Forsimple Michaelis–Menten kinetics, the pioneer Engasser andHorvath (1973) proposed a two-parameter model providinggeneralized plots of the e&ectiveness factor as a functionof the dimensionless moduli. Some modi=cations of thismodel were published later (Tuncel, 1999). However, littleattention has been paid to more complex kinetics: reversiblereactions (Paiva & Malcata, 1997), competitive Michaelis–Menten kinetics (Xiu, Jiang, & Li, 2000) or two-substrateenzymatic reactions (Engasser & Hisland, 1979).The three-parameter model developed by the present au-

thors (B/odalo et al., 1986) could be considered the mostgeneral mathematical model published until now. The modelincludes three dimensionless parameters: �’s, the di&usionmodulus; �, the concentration modulus and , the reversibil-ity e&ect modulus. The model has been successfully usedfor the design of heterogeneous enzymatic reactors of di&er-ent types: batch (B/odalo Santoyo, G/omez Carrasco, G/omezG/omez, Bastida Rodr/Kguez, & Mart/Knez Morales, 1993)packed bed (Manj/on, Iborra, G/omez, G/omez, Bastida, &B/odalo, 1987) and <uidized bed (B/odalo, G/omez, G/omez,Bastida, & M/aximo, 1995).In this work, a new model with only two parameters is

proposed. It has been resolved for reversible Michaelis–Menten kinetics and the results obtained are plotted in onlyone generalized graph of the e&ectiveness factor. The modelalso includes, as particular solutions, the two-parametermodel developed by Engasser and Horvath (1973) for sim-ple Michaelis–Menten and product competitive inhibitionkinetics.

2. Mathematical model

The present model is an improvement based on the pre-viously formulated three-parameter model (B/odalo et al.,1986), since only two parameters are necessary to reachthe solution. Consequently, the di&erential equations for themass balance of substrate and product, in spherical coor-dinates, and the boundary conditions are those derived inthe previous papers (B/odalo et al., 1986), namely, Eqs. (1)and (2).For a general enzymatic mechanism that considers re-

versible Michaelis–Menten kinetics the rate law is

VS =Vm(CS − (CP=Keq))

KM + CS + (KM=KP)CP: (3)

From the di&erential mass balance for substrate and prod-uct (Eqs. (1) and (2)), the following relationship can beestablished:

CP = CPR +DSDP

(CSR − CS): (4)

By substituting in Eq. (3) the value of CP given inEq. (4), the following expression for VS is obtained:

VS =

Vm(1 + 1Keq

DSDP)

(CS −

CPR+DSDP

CPR

Keq+DSDP

)

KM + KMKP

(CPR + DS

DPCSR)+ (1− KM

KPDSDP)CS

: (5)

In this equation the term

CS − CPR + (DS=DP)CPRKeq + (DS=DP)

(6)

represents the driving force of the chemical reaction, sothat VS is annulled when this term becomes zero. Since,in reversible reactions, this happens when the equilibriumconcentration is reached, the substrate concentration in theequilibrium can be inferred as

CSE =CPR + (DS=DP)CPRKeq + (DS=DP)

: (7)

Since the equilibrium constant is

Keq =CPECSE

; (8)

the product concentration when equilibrium is reached isgiven by

CPE = KeqCSE =CPR + (DS=DP)CPR1 + (1=Keq)(DS=DP)

: (9)

By introducing the de=nition of CSE (Eq. (7)) into thekinetic expression (Eq. (5)), the following equation for thereaction rate is obtained:

VS =Vm(1 + 1

KeqDSDP

)(CS − CSE)

KM + KMKP

(CPR + DS

DPCSR)+(1− KM

KPDSDP

)CS: (10)

This expression can be modi=ed to give

VS =Vm(1 + 1

KeqDSDP

)(CS − CSE)

KM + KMKPCPE + CSE +

(1− KM

KPDSDP

)(CS − CSE)

:

(11)

The presence of the term (CS −CSE) in Eq. (11) suggeststhe convenience of using a new de=nition for the dimen-sionless substrate concentration, U :

U =CS − CSECSR − CSE : (12)

The minimum value of U is 0, when CS = CSE , and themaximum value is 1, when CS = CSR.The dimensionless radial coordinate is that used in the

three-parameter model, that is,

�=rR

(13)

J. L. G'omez et al. / Chemical Engineering Science 58 (2003) 4287–4290 4289

and new dimensionless moduli are de=ned as

’=R2Vm

(CSR − CSE)DS

(1 + 1

KeqDSDP

)(1− KM

KPtSDP

) ; (14)

�=KM + KM

KPCPE + CSE

(CSR − CSE)(1− KM

KPDSDP

) : (15)

Introducing the new moduli in the model equations, thedi&erential equation becomes

1�2

dd�

(�2

dUd�

)= ’VU ; (16)

the boundary conditions being

�= 0;dUd�

= 0; �= 1; U = 1 (17)

and the dimensionless reaction rate VU is de=ned as

VU =U

�+ U: (18)

Eq. (16) describes the whole di&usion-reaction process onan enzymatic porous catalyst. When Eq. (16) is integrated,the radial pro=le of U is obtained. From this pro=le, thee&ectiveness factor can be evaluated as

�= 3(�+ 1)∫ 1

0

U�+ U

�2 d�: (19)

The relationships between the new dimensionless moduli,’ and �, and the moduli used in the three-parameter model(�’s, � and ) (B/odalo et al., 1986) are:

’=�′S

1− ; �=� + 1−

and between U and S:

U =S − 1− :

3. Results and discussion

The present model is a non-linear boundary value prob-lem, therefore numerical calculus must be used to resolveit. Eqs. (16) and (19) were resolved using the same numer-ical procedure previously described (B/odalo et al., 1986),simply adapting it for two parameters. The described resultswere obtained with a maximum error of ±0:1%. The ra-dial pro=les for U and the corresponding e&ectiveness fac-tor values were obtained for the following range of moduli:0:016’6 10; 000 and 16 �6 5000.In Fig. 1, the e&ectiveness factor is plotted against ’ with

� as the parameter. When modulus ’ is smaller than 1, � ispractically unity for all � values, so that di&usion does nota&ect the rate of reaction. The e&ectiveness factor value de-creases quite rapidly as’ increases, approaching zero at high’ values, which correspond to internal di&usion-controlled

0

0.2

0.4

0.6

0.8

1

0.01 0.1 1 10 100 1000 10000ϕ

η

α =α = 1 1α =α = 2 2α =α = 10 10α =α = 20 20α = 100α = 100α = 200α = 200α = 1000α = 1000α = 2000α = 2000

Fig. 1. Generalized graph of the e&ectiveness factor as a function of ’and � dimensionless moduli. Reversible Michaelis–Menten, competitiveproduct inhibition and simple Michaelis–Menten kinetics.

processes. Additionally, for a constant value of ’, the e&ec-tiveness factor increases with increasing values of �.The exact values of the e&ectiveness factor for a =rst-order

reaction, as a function of the Thi/ele modulus, were com-pared with those obtained from the numerical solution ofthe two-parameter model proposed in this work. From Eqs.(14) and (15), the relationship between the dimensionlessmoduli ’ and � and the Thi/ele modulus, �, can be easilyderived, for =rst-order kinetics:

�=

√’�:

The error between the predicted � values and the exactvalues is lower than ±0:1% for a wide range of the Thi/elemodulus.

4. Conclusion

A detailed analysis of the kinetic equation, taking into ac-count the concentration values in the equilibrium, allows usto de=ne a new dimensionless substrate concentration, U . Italso permits us to reduce from three to two the dimensionlessmoduli necessary for the resolution of the di&usion-reactionmodel for immobilized enzyme systems in porous spheri-cal particles, obeying reversible Michaelis–Menten kineticswith product inhibition e&ects.

Notation

CP product concentration inside the spherical particleCPE equilibrium product concentrationCPR local product concentration at particle surfaceCS substrate concentration inside the spherical particleCSE equilibrium substrate concentrationCSR local product concentration at particle surfaceDP e&ective product di&usivity inside the particleDS e&ective substrate di&usivity inside the particleKeq equilibrium constantKM Michaelis constantKP competitive product inhibition constant

4290 J. L. G'omez et al. / Chemical Engineering Science 58 (2003) 4287–4290

r radial coordinate of the particleR radius of the particleS dimensionless substrate concentration, de=ned as

CS /CSR for the three-parameters model (B/odaloet al., 1986)

U dimensionless substrate concentration de=ned inEq. (12) for the two-parameter model

Vm maximum reaction rate per unit of catalytic particlevolume

VS local reaction rate per unit of catalytic particlevolume

VU dimensionless reaction rate as de=ned in Eq. (18)

Greek letters

� dimensionless modulus de=ned in Eq. (15)� dimensionless modulus de=ned in the three-

parameter model (B/odalo et al., 1986) dimensionless modulus de=ned in the three-

parameter model (B/odalo et al., 1986)� e&ectiveness factor’ dimensionless modulus de=ned in Eq. (14)� ThiGele modulus�′S dimensionless modulus de=ned in the three-

parameter model (B/odalo et al., 1986)

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