horvath1974-external and internal diffusion in heterogeneous

BIOTECHNOLOGY AND BIOENGINEERING VOL. XVI, PAGES 904-923 (1974)

External and Internal Diffusion in Heterogeneous Enzymes Systems

CSABA HORVATH and JEAN-MARC ENGASSER, Department of Engineering and Applied Science, Section of Physical Sciences,

School of Medicine, Yale University, New Haven, Connecticut 06620

Summary The intrusion of diffusion in heterogeneous enzyme reactions, which follow

Michaelis-Menten kinetics, is quantitatively Characterized by dimensionless parameters that are independent of the substrate concentration. The effects of these parameters on the overall rate of reaction is illustrated on plots commonly employed in enzyme kinetics. The departure from Michaelis-Menten kinetics due to diffusion limitations can be best aasessed by using Hofstee plots which are also suitable to distinguish between internal and external transport effects. A graphical method is described for the evaluation of the reaction rate BS a function of t,he surface concentration of the substrate from measured data.

INTRODUCTION

The transport of substrate to the catalytic sites has often been found to affect the overall kinetic behavior of immobilized or membrane-bound enzymes. The effect of both external and internal substrate diffusion on the Michaelis-Menten kinetics has been theoretically treated by using different appro ache^.'-^ The purpose of this study is to treat the intrusion of diffusional effects by dimensionless parameters which are independent of the substrate concentration. Such parameters make it convenient to illustrate the effect of diffusional limitations on graphs commonly used in enzyme kinetics.

EFFECT OF EXTERNAL TRANSPORT

When an enzymic reaction takes place at a surface, substrate transport occurs by molecular or convective diffusion. In the following treatment it is assumed that the enzymic reaction per se obeys a Michaelis-Menten kinetic law and the catalytic surface is equiaccessible, i.e., the conditions for substrate transport to any

909

@ 1974 by John Wiley & Sons, Inc.

910 HORVATH AND ENGASSER

point on the surface are identical. At steady state the rate of substrate transport to the surface is equal to the rate of substrate con- sumption by the reaction, i.e.,

where C , and Cb are the surface and bulk concentrations of substrate, respectively, h. the transport coefficient, V,,," the saturation rate per unit surface area, and K m the Michaelis-Menten constant.

By introducing the dimensionless concentration, j3, defined as

P = C/Km (2)

and a dimensionless group which is often referred to as a Damkoehler number, Da, given by

Vmax" Da = __ hK,

eq. (1) can be written in dimensionless form as

(3)

where @ b and are the dimensionless bulk and surface concentrations, respectively.

Equation (4) shows that two limiting cases are possible. When, at a given O b , the Damkoehler number is sufficiently large, goes to zero because the reactivity at the surface is so high that all substrate molecules which can be transported to the surface are immedi- ately converted. As a result the observed reaction rate is equal to the maximum possible rate of external transport, V d i t f n , at a given bulk concentration, i.e.,

v" = vdiff' ' = hCb (5)

In this case the measured reaction rate is virtually independent of the kinetic parameters of the enzymatic reaction.

When Da is very small, the bulk concentration is practically equal t o the surface concentration. Then, the observed reaction rate is equal to the maximum possible rate of the actual chemical reaction, V~i,,", a t a given bulk concentration, thqt is,

DIFFUSION IN ENZYME SYSTEMS 911

The two limiting rates, v d i f f n and Vkin”, therefore, can be considered as expressions for the conductances of the transport process and the reaction, respectively.

When the surface is uniformly accessible and the overall reaction rate depends on both external diffusion and chemical reaction, the two processes take place in series as if they were independent of each other. At steady state both must have the same rate that is determined by both conductances, V d i f f N and V k i n ” . If, however, one of the conductances is much smaller than the other for a given concentration, then the overall rate, V”, which would be measured experiment,ally, is limited by the process of lower cond~ctance.~ In Figure 1 the situation is illustrated by the dependence of Vkin”, V d i f f ” , and the resulting V” on the bulk concentration, for a set of arbitrarily chosen kinetic and diffusional parameters. At sufficiently high concentrations, V,,,”, the saturation value of Vkin”, can always be obtained. At lower concentrations either V d i f f ” or V k i n ” plays the pre- dominant role in determining the overall rate of reaction. The relative effect of these limiting rates depends on the value of Da, i.e., on the relative magnitudes of the initial slopes of V d i f f n and V k i n ” .

The limiting rate, Vkin”, is obtained over the whole concentration range if external diffusion is fast enough to maintain a flat concen-

BULK CONCENTRATION

Fig. 1. Plot of the overall rate of the surface reaction, V”, against the substrate concentration in the bulk solution. In this case, V” is determined by both the limiting rate of the actual enzymic reaction at the surface, Vkin”, and the maximum possible rate of substrate diffusion to the surface, Vdiff”, which are also illustrated.


1.0

0.8

0.2

0.0

1000

0 10 20 30 40 50

4 Fig. 2. Plots of the overall rate of an enzyme catalyzed surface reaction, V”,

normalized to V,.”, against the dimensionless bulk concentration, 06, at different values of Damkoehler number, Da. The effect of diffusion limitations increases with increasing values of Da and results in a decrease of the overall rate of reaction with respect to the kinetically controlled rate, Vkim”/Vmx”, which is obtained at Da = 0.

tration profile. When the transport of substrate is relatively slow, V” departs from VkinN as shown in Figure 2 where the normalized overall reaction rate, V”/V,,x’fl is plotted against pa at different values of Da. As seen, at large Da values the overall rate of reaction is lower than that of the intrinsic enzymic reaction shown for Da -+ 0, because the surface concentration of the substrate is smaller than the bulk concentration due to external diffusion limitations. Such effects are negligible only if Da is smaller than unity. Nevertheless, at sufficiently high concentrations the depletion of the substrate at the surface region does not affect the overall rate significantly, even at large values of Da, so that the saturation rate, V,,,”, can always be attained, at least theoretically.

In order to illustrate the effect of kinetic limitations on the overall reaction rate due to relatively low catalytic activity, V” is normalized to the product hK,, which is proportional to Vdiff”. In Figure 3 this normalized rate is plotted against @ b with Da as the parameter. It is seen that at any given Da value, the departure of V” from Vdjff’’ increases with the bulk concentration because the rate of transport


50

40

30 E

Y C :\ >

20

10

0 0 10 20 30 40 50

4 Fig. 3. Plots of the overall rate of an enzyme catalyzed surface reaction,

normalized to the product of the mass transfer coefficient, h, and K,, against the dimensionless bulk concentration, Ob, a t different values of the Damkoehler number, Da. The effect of kinetic limitations increases with decreasing values of Da and results in a decrease of the overall rate of reaction with respect to the diffusion controlled rate, VdiffnlhK,,,, which is obtained at Da = 00.

increases to a greater extent than the rate of the enzymic reaction and only a fraction of the substrate molecules that could diffuse to the surface are converted. On the other hand, a t any given bb value, V” is approximately equal to Vdiff” only if Da is sufficiently large, i.e., the enzyme activity is high enough to maintain the surface concentration practically zero. Figure 3 shows that at f i b < 5 the reaction is already diffusion controlled when Da > 10, but a t higher bulk concentrations VdifrN is reached only at much higher values of Da.

The interaction between external diffusion and surface reaction can be expressed by the departure of V” from Vkin” or VdiffN in view of the foregoing discussion. For convenience an effectiveness factor, qe is defined by

V” = qeVkin” (7)

which is a quantitative measure of the effect of external diffusion on the overall reaction. Clearly, qs is unity for a kinetically controlled reaction and its value decreases with increasing diffusion limitations. The dependence of q e on p b and Da is shown in Figure 4. When


I

. I

F“

01

I I 10 100 1000

Da

Fig. 4. Graph illustrating the external effectiveness factor, qe, as a function of the Damkoehler number, Da, with the dimensionless bulk concentration, f i b ,

aa the parameter, for an enzymic surface reaction. At sufficiently small concentrations the limiting first order effectiveness factor ee, is reached.

@ b < 0.1, then q e becomes independent of the concentration, as expected for a first order reaction, and approaches a limiting value, e e , which is related to Da by

1 1 + Da

e, = ~

In Figure 4 the locus of the kinetically controlled reaction is at qc = 1, on the upper horizontal line, whereas the straight lines obtained for qe at high Da values represent the diffusion controlled reaction domain. The border line between the diffusion controlled and the intermediate domain of the effectiveness factor, which is affected by both diffusion and kinetics, is indicated by the broken line. Thus there is a critical Da for any bulk concentration where t,he reaction becomes diffusion controlled for all practical purposes, so that any increase in Da has no significant effect on the overall rate. Since, at a fixed h, the increase in Da is equivalent to an increase in the amount of enzyme, in practice the maximum possible rate can be obtained with some minimum amount of enzyme at- t,ached to the surface.6 The corresponding optimal TI,,,’’ value can


be calculated for a given f i b , h, and K m by evaluating Da at the broken line.

Although the dependence of V” on the bulk concentration is usually very complex as illustrated in Figures 2 and 3, it can be easily treated in the following two limiting cases. At a sufficiently high bulk concentration both V” and Vkinb have the same limiting value, V,,,”, the saturation rate of the enzymic reaction. On the other hand, a t low concentrations V” is a first order reaction with a rate constant V m a x N / ~ e where K~ is defined by

It should be noted that eqs. (3), (8), and (9) can be combined to obtain the following equation:

which expresses the resistance additivity relation for the first order enzyme reaction cum external transport.’

This similarity in the limiting behavior of V“ and v k i n ” , however, does not necessarily mean that the functional relationship between 1”’ and the concentration also has the form of a rectangular hyper- bola that is characteristic of the Michaelis-Menten kinetic law. This is illustrated in Figure 5, which shows plots of the normalized overall reaction rate, V”/Vmnx“, against V”/(Vm,,” x f i b ) . Such plots, like the so-called Hofstee plots used to evaluate kinetic data obtained with soluble enzymes,* yield straight lines when V” obeys the Michaelis-Menten law, tts can be seen for Da + 0 in Figure 5 . How- ever, as a result of diffusion limitations, curves corresponding to increasing values of Da, depart significantly from straight lines, par- ticularly when a wide concentration range is examined. These portions of the plots where the overall reaction rate is essentially determined by the rate of external transport are straight vertical lines.

The assumption that external transport and surface reaction take place in series can be used to obtain thetrue kinetic rate Vkin”(C8) from the measured rate V”(Cb) . When the mass transfer coefficient for the substrate, h, is known, the concentration of the substrate at, the surface can be determined from the bulk concentration and the reaction rate by using the following relationship :

V’’(Cb) = vkin”(C8) = h(cb - Cs) (11)

916

1.0

0.75

x 0

= E > 0.5 =\ >

0.25

0.0 0.0

HORVATW AND ENGASSER

I I I

0.25 0.5 0.75 1.0

VI;IV;axXpb

Fig. 5. Departure from Michaelis-Menten kinetics due to external transport limitations as illustrated by Hofstee type plots at different Damkoehler numbers, Da. The overall surface reaction rate, V”, obtained at different dimensionless bulk concentrations, P b , is normalized to V-,”. In the diffusion controlled reaction domain, i.e., at large values of Da the plot yields vertical straight lines.

A graphical method based on eq. (11) for the determination of Vkin‘‘

is illustrated in Figure 6. First, the measured V” is plotted against the bulk concentration, c b . Then, a t any chosen Ca a straight line with the slope equal to - h is drawn, which represents the flux of the substrate to the surface for all c, values from zero to the given c b .

This straight line intersects the horizontal line, which is drawn to represent the value of V” at this particular C b . The intersection yields both the actual surface concentration, C,, and. the intrinsic rate of reaction, V k i n ” , for that value of c b . Thus, a plot of V k i n ”

against the surface concentration can be constructed by repeating the procedure for a number of bulk concentrations. From this plot the kinetic parameters of V k i n ” can then be determined by standard methods. This graphical procedure is not restricted to Michaelis- Menten kinetics and is quite generally applicable provided the assumption of equiaccessible surface is appropriate and the external transport coefficient is known.


In the caae of a porous enzymic medium with internal diffusion limitations this method yields the overall rate as a function of the surface concentration, which is considered in the following section.

EFFECT OF INTERNAL DIFFUSION

When enzymes are embedded in a porous medium, the substrate has to diffuse inside the pores in order to react a t the catalytic sites. Unlike external transport, internal diffusion proceeds in parallel with the chemical reaction. Thus, the concept of limiting processes in series is no longer applicable and the treatment of the subject is more complex. Our model consists of a membrane containing bound enzymes uniformly distributed. The membrane can be exposed to the substrate solution on one side and sealed on the other. Alternatively, both sides can be open and in contact with the same substrate concentration. At steady state the concentration change in the membrane is described by the following differential equation :

where Deff is the effective diffusivity of the substrate in the membrane, C is the local substrate concentration, Vmadtt is the maximum possible reaction rate per unit volume of the membrane, and x is the distance from the surface. By defining the dimensionless position in the membrane, z, as

(13) x z = - 1

where 1 is the thickness of the membrane, eq. (12) can be written in dimensionless form as

B d28 = 9 2 ~

dz2 1 + B

In eq. (14) 4 is the so-called Thiele modulus given by

Numerical integration of eq. (14) with the boundary conditions

B = B d a t z = O (16)

clSJdz = 0 at z = 1 (17)

and


CONCENTRATION

Fig. 6. Graphical determination of the surface concentration, C,, and the rate of the actual kinetically controlled enzymic reaction a t the surface, Vkin”,

from the. overall rate of reaction V”, measured at various substrate concentrations in the solution, Cb, when the mass transfer coefficient, h, is known. At a chosen value of c b a straight line of slope -h is drawn. The intersection of this line with a horizontal line drawn at the value of V” measured at the same c b

yields both the corresponding surface concentration on the abscissa. and the intrinsic rate of the reaction for this surface concentration on the ordinate. By repeating this procedure for different Cb values a plot of Vkin” against C. can be constructed.

yields the substrate concentration profile in the enzymic membrane and, consequently, the overall rate, V , for the whole enzymic membrane can be calculated. This rate normalized to the maximum possible rate, V,,,, is plotted as a function of the surface concentration with the Thiele modulus as the parameter in Figure 7. It is seen that for 4 < 1 the reaction is essentially kinetically controlled, but a t higher values of 4 the observed rate is lower due to diffusion limitations unless the saturation rate, V,,,, is reached at high concentrations. As expected V is always a function of the kinetic parameters of the reaction even at very high values of cp, in contrast to the case of external diffusion, when the overall rate is independent of the kinetic parameters a t sufficiently high Da values.

The dependence of V on the surface concentration is usually complex as seen in Figure 7. < 0.1, however, the overall reaction rate is first order with the rate constant Vrnax /~ i where ~i is defined by

At small surface concentrations,

Kmcp Ki = tanhcp


I .o

0.0

0.6 E > \ ’ 0.4

0.2

0.0 0 10 20 30 40 50

Ps Fig. 7. Plots of the overall rate of reaction in an enzymic membrane, V ,

normalized to V,,,,,, against the dimensionless concentration of the substrate at the surface, pa, at different values of the ThieIe modulus, Q. The effect of diffusion limitations increases with increasing values of Q and results in a decrease of the overall rate of reaction with respect to the kinetically controlled rate which is obtained at Q = 0.

On the other hand at sufficiently high values of B8, internal diffusion has no effect on the overall rate since V becomes equal to the saturation rate, Vmax. Thus the observed rate shows a limiting behavior similar to that of the Michaelis-Menten scheme. Yet, its functional dependence on the surface concentration is different as illustrated by the plots in Figure 8, which deviate from straight lines when 4 > 1.

The effect of internal diffusion on the kinetic behavior of immobilized enzymes in spherical particles has been demonstrated by plot- ting the calculated overall reaction rates according to the schemes known as Lineweaver-Burk and Hofstee plots4. The deviations from straight lines due to the departure from the Michaelis-Menten kinetics have been most pronounced in the latter type of plots, even at relatively low values of the Thiele modulus, in agreement with the earlier suggestiong that the Hofstee plot is more suitable to discern deviations from the straightforward Michaelis-Menten kinetics.

In this study this plot has been found also very sensitive to the effect of internal or external diffusion limitations which manifest

920

1.0

0.75

X

E > 0.5

> \

0.25

0.0

HORVATH AND ENGASSER

I I I /’

0.0 0.25 0.5 0.75 1.0

~ ~ ~ I n , , 4 s

Fig. 8. Departure from Michaelis-Menten kinetics in an enzymic membrane due to diffusion limitations aa illustrated by Hofstee type plots obtained at different Thiele moduli, 9. The overall reaction rate, V , obtained at different surface concentrations, &, is normalized to Vmax. Diffusion limitations at nonzero values of 6 result in sigmoid curves.

themselves by vertical lines and sigmoid curves, respectively, as shown in Figures 5 and 8. Although Lineweaver-Burk plots are used most commonly in the study of immobilized enzymes, the results strongly suggest that Hofstee plots are most appropriate for the investigation of the kinetic behavior of heterogeneous enzyme systems. Such plots are not only more sensitive to the effect of diffusion limitations but also yield information about the nature of the diffusional effect by analyzing the shape of the curves obtained with experimental data. Furthermore, Hofstee plots can also supply valuable information on the combined effect of product inhibition and transport phenomena on the kinetics of heterogeneous enzyme systems.10

The results of this study clearly indicate that the Michaelis-Menten kinetic law is no longer obeyed when transport limitations affect the enzymic reaction ; consequently, there is no K , value to characterize the overall kinetics of the reaction under such conditions. In view


of this fact it seems to be appropriate to review the different ways proposed in the literature to quantitatively express the effect of transport limitations in heterogeneous enzyme kinetics.

When transport phenomena affect the enzymic reaction i t is generally recognized that the overall rate no longer follows the Michaelis- Menten kinetic law. Nonetheless, “apparent K,” values have been widely used and suggested to characterize the intrusion of diffusional effects in the kinetics of the reaction. Furthermore, such apparent K , values have been defined in different ways. In some cases apparent K , values were used to express the limiting first order con- stants,11J2 which are equivalent to K , and K~ discussed above. In other cases the apparent K , was defined as the concentration a t which the rate of the heterogeneous enzyme reaction is half of the saturation rate.13J4

Although both apparent K , values are the same and equal to the actual K , in the absence of diffusion limitations, they are significantly different a t larger values of 4 and Da as shown in Figures 2 and 7 as well as by eqs. (9) and (18). In view of this, the use of the term apparent K , value for the parameters denoted by K in this paper or the substrate concentration yielding the half saturation rate does not appear to be appropriate when the overall reaction does not follow the Michaelis-Menten kinetic law. Nevertheless under certain conditions, e.g., due to some microenvironmental phenomena or the effect of transport limitations and product inhibition in some particular casesl10 the overall kinetics may formally follow this law despite the intrusion of physical and chemical phenomena. Even then the use of an experimentally obtained apparent K , value can be quite dangerous in that i t inspires false confidence and could con- ceivably be used in design under circumstances when the fluid me- chanical conditions are far removed from those employed in the kinetic experiment.

APPENDIX The reviewer pointed out that there is no optimum amount of im-

mobilized enzyme in the external diffusion controlled system as claimed in this paper because the overall reaction rate always increases by increasing V,,,, i.e., the amount of the surface-bound enzyme. The proof provided is as follows.

If only VmaX” is varied i t can be shown t.hat


Since the right-hand side of eq. (19) is always positive, bV”/bV,,,” is always greater than zero, i.e., the rate always increases with V,,,”.

In the rigorous theoretical sense that is true because VdirfN as well as v k i n ” are fictitious reaction rates. In theory Vdirr” would only be reached at Da = CQ to obtain zero surface concentration. Never- theless Figure 3 shows that in practice Vdirf” can be obtained in a wide bulk concentration range already at Da = 100. Then C , and, according to eq. (19), bV”/bV,,,“ are practically zero and a further increase in Da by increasing V,,,” does not yield a higher reaction rate.

Similarly, Vkin is never reached theoretically because Da and the effectiveness factor is never zero and unity, respectively. Accord- ing to our analysis, however, the reaction rate equals Vkin” in practice when Da < 0.01.

This conflict of asymptotic limits, which are never obtained theoretically and limiting rates which are encountered in practice, is inherent to the Michaelis-Menten kinetic scheme also. For that case, if only the concentration is varied, it can be shown that

As seen, bV/bC is always greater than zero so that the saturation . -

V,,,, is never reached theoretically.

Nomenclature substrate concentration substrate concentration in the bulk fluid substrate concentration at the interface substrate diffusivity in external solution effective substrate diffusivity in the enzymic medium Damkoehler number mass transfer coefficient for the substrate Michaelis-Menten constant thickness of membrane overall reaction rate overall reaction rate per unit surface area maximum rate of substrate transport per unit surface area at a given bulk concentration of the substrate maximum reaction rate per unit surface area at a given surface concentration of the substrate saturation reaction rate according to Michaelis-Menten kinetics saturation rat,e uer unit area of catalyst surface VI%3X’’

V,,,’” saturation rate per unit volume of catalyst I z

internal distance from the membrane surface dimensionless internal distance from the membrane stirface


Greek Let.ters

j3 dimensionless substrate concentration j3b

C.

rlC external effectiveness factor K~

~i

+ Thiele modulus

dimensionless substrate concentration in the bulk fluid dimensionless substrate concentration at the interface external effectiveness factor for first order reaction

kinetic parameter for first order enzyme reaction with external transport limitations kinetic parameter for first order enzyme reaction with internal diffusion limitations

This work was supported by grants No. R01 GM 21084 and No. RR-356 from J.-M.E. is a the National Institutes of Health, U.S. Public Health Service.

recipient of a Yale University Fellowship.

References 1. E. Katchalski, I. H. Silman, and R. Goldman, Advan. Enzymol., 34, 445

2. M. Moo-Young and T. Kobayashi, Can. J. Chem. Eng., 50, 162 (1972). 3. D. J. Fink, T. Y. Na, and J. S. Schultz, Biotechnol. Bioeng., 15,879 (1973). 4. J. M. Engasser and C. Horvath, J. Theor. Biol., 42, 137 (1973). 5. D. E. Rosner in Annual Review of Materials Science, Vol. 2, R. A. Huggins,

Ed., Annual Reviews, Inc., Palo Alto, Calif., 1972, pp. 573-606. 7. D. A. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical

Kinetics, Plenum Press, New York, 1969, pp. 53-57. 8. B. H. J. Hofstee, Science, 116, 329 (1952). 9. J. E. Dowd and D. S. Riggs, J. Biol. Chem., 240, 863 (1965).

(1971).

10. J. M. Engasser, Doctoral thesis, Yale University, 1974. 11. W. E. Hornby, M. D. Lilly, and E. M. Crook, Biochem. J., 107,669 (1968). 12. P. S. Bunting and K. J. Laidler, Biochemistry, 11,4477 (1972). 13. P. V. Sundaram, E. K. Pye, T. M. S. Chang, V. H. Edwards,

A. E. Humphrey, N. 0. Kaplan, E. Katchalsky, Y. Levin, M. D. Lilly, G. Manecke, K. Mosbach, A. Patchornik, J. Porath, H. H. Weetall, and L. B. Wingard, Jr., in Enzyme Engineering (BiolechTol. Bioeng. Symp. No. 3) L. B. Wingard, Jr., Ed., Wiley-Interscience, New York, 1972, pp. 15-18.

14. R . Goldman, 0. Kedem, and E. Katchalski, Biochemistry, 10, 165 (1971).

Accepted for Publication January 14, 1974

horvath1974-external and internal diffusion in heterogeneous

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