# external diffusion effects on the kinetic constants of immobilized enzyme systems

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External Diffusion Effects on the Kinetic Constants ofImmobilized Enzyme SystemsTRANSCRIPT

f. theor. Biol. (1980) 84, 259-279

External Diffusion Effects on the Kinetic Constants of Immobilized Enzyme Systems

SUN BOK LEE AND DEWEY D. Y. RYU

The Korea Advanced Institute of Science, P.O. Box 150, Chung- Ryang-Ri, Seoul, Korea

(Received 29 May 1979, and in revised [orm 29 November 1979)

In order to further our understanding of immobilized enzyme reaction kinetics, the effects of external diffusion on the kinetic constants are studied for various reaction systems. It is shown that the variations of apparent kinetic constants with diftusional limitations are not the same as those of Michaelis-Menten kinetics, although both the inhibition kinetics and the two substrate reaction kinetics can be expressed in the form of simple Michaelis-Menten type by using apparent rate parameters. We found that there will be changes in the apparent kinetic constants depending not only on the types of enzyme reaction kinetics but also on the relative rate of diffusion or flow-through in the microenvironment. For substrate inhibition kinetics, both the apparent maximum reaction rate and the apparent Michaelis constant decrease while the apparent maximum reaction rate for product inhibition kinetics is increased and the apparent Michaelis constant decrease as mass transfer limitation is reduced. In the case of two substrate enzyme reaction kinetics, as the diffusio'nal limitation is reduced the apparent maximum reaction rate increase but the apparent Michaelis constant can increase, decrease or remain nearly constant depending on the values of relative affinity and on the fixed substrate concentrations. The results of theoretical analyses are compared with the experimental data obtained and reported previously, and a very good agreement was found.

1. Introduction

When the enzymes are immobilized, such changes as structural con- formation, microenvironment, the partitioning or electrostatic effect and mass transfer resistance could be brought about (Goldstein, 1976). Among these the effects of mass transfer limitation on the reaction kinetics have been widely studied by many workers from the viewpoint of heterogeneous catalysis.

The effects of internal diffusion in the reaction kinetics have been studied experimentally as well as theoretically (Lee, Fratze, Wun & Tsao, 1976; Moo-Young & Kobayashi, 1972; Engasser & Horvath, 1973). In the case of external diffusion, however, the results of several workers (Lilly etal., 1966;

259 0022-5193/80/100259+21 $02.00/0 9 1980 Academic Press Inc. (London) Ltd.

260 S.B . LEE AND D. D. Y. RYLI

Wilson, Kay & Lilly, 1968a, 1968b; Sharp et al., 1969; Tosa, Mori & Chibata, 1971; Kobayashi & Moo-Young, 1973; Gellf, Thomas, Broun & Kerneves, 1974: Cho & Swaisgood, 1974; Toda, 1975; Hirano, Karube, Matsunaga & Suziki, 1977; Paul, Coulet, Gautheron & Engasser, 1978; Kim, Lee & Ryu, 1978) appear to be contradictory and conflicting.

Most of these authors cited made some endeavors to show the important factors that affect the kinetic constants of immobilized enzymes. Many more works also attempted to analyze theoretically the effects of external diffusion on the kinetic constants. For irreversible Michaelis-Menten kinetics, Hornby, Lilly & Crook (1968), Kobayashi & Moo-Young (1971), Shuler, Aris & Tsuchiya (1972), Hamilton, Stockmeyer & Colton (1973), Kobayashi & Laidler (1974), and Toda (1975) derived the equations for apparent kinetic parameters. For reversible Michaelis-Menten kinetics, Lee, Kim & Ryu (1979) showed that the extent the apparent kinetic :constants changes as a function of superficial velocity depends on the value of l l Vm/Km parameter for forward reaction and for reverse reaction, respectively. For other enzyme reaction kinetics, the effects of external diffusion on the kinetic constant have not been clearly established up to the present.

In view of these developments and some confusion, our study on the effects of external diffusion was undertaken in order to further our under- standing of reaction kinetics of immobilized enzymes. In this paper, the external diffusion effects on the kinetic constants were studied theoretically and the confusion due to conflicting results was clarified.

2. The System

It is assumed that the system is at a steady state, the enzymes are immobilized On a nonporous support material, and the diffusion coefficient is constant.

For the reaction system that a substrate, s, is converted to a product p, the diffusive fluxes can be expressed as

D ds Js = S-~y (I)

4 =Dpd~ (2) with the boundary conditions

s=S, ! p=P

s=S, p=P

aty =0

atY =8

DIFFUSION EFFECTS ON ENZYME KINETICS 261

where the mass fluxes toward the surface of the immobilized enzyme are positive and the origin of the co-ordinate system is at the surface. Under steady state, J, = -Jp =3", and the Equations (1) and (2) yield the following expression

J = kLs (S - -S ) = kreCP-P) (3)

where kLs = Ds/ 8 and kt.p = Dp/ & For the two substrate enzyme reaction system the diffusive flux can be

expressed as

J = ku(S , - S,) = kLi(Si -- 4 ) (4)

which is analogous to equation (3).

3. Theoretical Examination of the Mass Transfer Effect on the Kinetic Constants

(A) METHOD OF ANALYSIS

Some workers defined the apparent Michaelis constant as that cor- responds to a value at whicl~ the rate of enzyme reaction is half of the maximum reaction rate (Goldman, Kedem & Katchalski, 1971; Sundaram et al., 1972; Kobayashi & Laidler, 1974), although it cannot describe the overall reaction rate exactly (Horvath & Engasser, 1974; Kobayashi & Laidler, 1974). In general, for the homogeneous enzyme reaction system, the maximum reaction rate is defined as the reaction rate at a high substrate concentration or S-~ co and the Michaelis constant is the substrate concen- tration at which the reaction rate is half the maximal. If these concepts are applied to the heterogeneous system, then we can define that

VJtrn(app ) = lim R (5) $~o0

I !

gm(app) ---- S la t R =0.5 V:,.pp, (6)

where R represents the reaction rate of immobilized enzymes. This type of analysis was found to be very useful if the trends or variations of kinetic constants with diffusional limitations are to be determined, especially for the complex enzyme reaction system.

It is now convenient to introduce the following dimensionless variables,

R s v" , O r = - - /2 ,=

= V'., K'.,.' kLsK'~

where, ~" represents the dimensionless reaction rate, o" the dimensioniess substrate concentration, and Ix the mass transfer modulus.

262 s .B . LEE AND D. D. Y. RYLI

Equations (5) and (6) becomes

a*= lim ~" (7) o- -~ oO

/3* = r s o* (8)

where a* and/3* represent the dimensionless apparent maximum reaction rate and the apparent Michaelis constant, respectively.

We will now examine the effects of diffusional limitations on the kinetic parameters for various enzyme reactions by using equations (7) and (8). The purpose of this examination is to find how a* and/3* can be affected as the mass transfer modulus, it, changes.

(B) ONE SLIBSTRATE ENZYME REACTION SYSTEMS

( i) Michaelis-Menten kinetics

When the enzymes are immobilized, the enzyme reaction takes place at the surface and the reaction rate for this heterogeneous enzyme reaction system can be expressed as

v 'g g = K" +-----ff (9)

where, R is the observed global rate of heterogeneous reaction per unit area. Equation (9) may be rewritten in terms of measurable bulk substrate concentration instead of surface concentration

/ / V,nS

R =K~ +S" (10)

From equation (3), S = S - ( J / k~) and substitution into equation (9) yields

~" = (11) 1+o. - ~rtx

in dimensionless form since the mass transfer flux, J, will be equal to the rate of reaction, R, under steady state. Equation (11) yields

= ~ {( 1 +/.t + o.) - [(1 + tt + o.)2 _ 4bto.] 1/2} (12a)

= 2o'{(1 +/z +o-)+[(1 +/z + o.)2- 4/zo.]1/2}-1 (12b)

since 0-< s r-< 1. From equation (12), for Michaelis-Menten kinetics

a*=l : (13) /3* = 1+0.5/~ (14)

DIFFUSION EFFECTS ON ENZYME KINETICS 263

From these equations one finds that the apparent maximum reaction rate, a*, is constant and the apparent Michaelis constant, /~*, increases with diffusional limitations (i.e. with increasing t~). An expression similar to equation (14) was previously defined as apparent Michaelis constant by Kobayashi & Laidler (1974), although the relationship holds only for the one substrate Michaelis-Menten type enzyme reaction system.

(ii) Substrate inhibition

The substrate inhibition kinetics expressed as

v 'g R=

K" +S+ ~ ' S /Ki~

can be rewritten in a dimensionless form

(15)

where,

~" = 1 +,7-(tL +Ks(o"- ~'tz) 2 (16)

K" K,=

K~,"

Equation (16) yields a cubic equation for ~" and its analytical solution is complex. !n Fig. 1 the Lineweaver-Burk plot is shown at Ks = 0.05 using the result of computer simulation. There is little change in K~', with the variation of t* while V~, and K~, vary similarly to those of Michaelis-Menten kinetics.

(iii) Product inhibition

There are, in general, three types of product inhibition, that is, competi- tive, non-competitive, and anti-competitive inhibitions (Laidler & Bunting, 1973). For competitive product inhibition kinetics,

v 'g R Kin(1 +WK,p)+g (17)

can be expressed as

o ' -6 ~" = (18)

(1 + ~: -t- gd~') + o- - ~'tz

where

P K" k~ ~ = ~',.p, andKp =K~p kLp