Determination of Kinetic Constants in Enzyme Reactor Systems by Transformation of Variables INTRODUCTION Immobilized enzymes have been widely used as industrial catalysts and as components of various analytical devices. They have also facilitated the use of continuous-flow enzyme reactor systems. The method that has been used so far for the determination of kinetic constants in the continuous-flow enzyme reactor system is not very satisfactory and inconvenient although many have reported on reasonably satisfactory methods for soluble enzymes or batch reactor systems where Michaelis-Menten reaction kinetics a p p l i e ~ . ~ When . ~ the immobilized enzymes are employed in continuous-flow reactors, the method used for soluble enzymes is not applicable. In this paper, a convenient method of determining the kinetic constants using transformation of rate-equation variables is reported and the usefulness of this method is illustrated.

THEORETICAL CONSIDERATIONS Continuous Stirred-Tunk Reactor (CSTR)

The performance equation can be obtained from the material balance, which takes into account a given component within a volume element of the system. At steady state, the material balance equation for the CSTR becomes S,Q

=

S f Q + RV

(1)

where, So and S , represent the substrate concentration at the inlet and effluent streams, respectively, Q is the flow rate of the fluid, R is the reaction rate in the reactor, and V is the volume of reaction mixture. By rearranging eq. (1): R = (So - S f ) / 7 =

SOX/T

(21

where T is the mean residence time and X is the fractional conversion. Using this expression, the performance equations can be derived for various enzyme reaction kinetics as shown in Table I. On the other hand, it is worth noting that the reaction rate is equal to the productivity of the reactor as shown in eq. ( 2 ) , and that the fractional conversion and the reaction rate are evaluated at effluent stream conditions. In the CSTR, the reaction rate can be replaced by the productivity of the reactor and the concentration terms can be replaced by the effluent concentration, which is the same as the conditions within the reactor. In view of these facts, we can use the initial reaction rate equations directly as the performance equations for the CSTR by transforming the initial reaction rate and initial substrate and product concentrations into: v' = R = S , X / r

S' =

s,=

S0(1 -

X)

and

P'

= (So +

P o ) - S'

=

Po + S O X

Biotechnology and Bioengineering. Vol. X X I , Pp. 232Y-2336 (lY7Y) @ 1Y7Y John Wiley & Sons, Inc.

0006-3SY2/7Y/0021-232Y$01.00

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BIOTECHNOLOGY A N D BIOENGINEERING VOL YXI (1979)

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COMMUNICATIONS TO THE EDITOR

where u ' , S',and P' represent the transformed variables of reaction rate and concentrations of substrate and product for CSTR, respectively. Using these expressions the performance equations for various kinetics can also be derived from the transformed rate equations, which are identical to those derived from eq. (2) as shown in Table I. Plug-Flow Reactor ( P F R ) As in the case of the CSTR, the performance equation can be obtained from the material balance equation. In the PFR, the composition of fluid varies from point to point along the flow paths and the material balance must be determined for a differential volume element, d V . At steady state, the material balance becomes QsIv

=

(6)

Q s I v + w + R dV

which gives -dS

R=--

dr '

s = SoatT = o

and

(7) S=Slatr=7

Since the performance equation for the PFR yields a differential equation while that for the CSTR yields a difference equation, the performance equations can be obtained by integrating eq. (7): they are summarized in Table 11. Instead of employing the integrated performance equation, the transformation of variables may be introduced similarly to that of CSTR, such as

Jn

and P*

=

( S o + P o ) - S* = ( S o + P o ) + S,X/ln(l

-

X)

(10)

where, u * , S*, and P* represent the transformed variables for the PFR. Since the composition of fluid varies along the flow paths in the PFR, the transformed reaction rate and transformed concentrations are expressed as the mean reaction rate or the productivity of the reactor and the logarithmic mean concentrations in the reactor, respectively. These are comparable to the transformed variable for the CSTR system in which the contents are well mixed and uniform throughout. Using eqs. (%( 10). the performance equations for various enzyme kinetics can be derived from the transformed rate equations. In the case of the PFR, however, there are some deviations between the performance equations derived from transformed rate equations and those derived from eq. (7). For Michaelis-Menten-type and competitive product inhibition kinetics, the transformed rate equations are identical to the performance equations. For substrate inhibition, noncompetitive and anticompetitive product inhibition kinetics, the trans-

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BIOTECHNOLOGY AND BIOENGINEERING VOL. XXI (1979)

COMMUNICATIONS T O THE EDITOR

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BIOTECHNOLOGY AND BIOENGINEERING VOL. XXI (1979)

formed rate equations deviate slightly from the performance equations owing to the approximate solution for the transformed Concentration. An exact expression for the concentration term for linear reaction system is

S* =

dr

S dr/[

and S* becomes the logarithmic mean concentration for a first-order reaction and becomes the arithmetic mean concentration for a negative first-order reaction. For both the substrate inhibition and noncompetitive product inhibition kinetics, the differences between performance equations and transformed rate equations are the arithmetic mean, 1 - (X/2), and the logarithmic mean, Xiln(1 - X ) , respectively. As shown in Table 111 there is about a 1% error when the fractional conversion is 30% and only about a 5% error at 55% conversion. For the substrate and product inhibition kinetics the fractional conversion may be reduced due to the high substrate concentration or the initial product concentration. We can safely and reliably use these transformed rate equations up to 55% fractional conversion to determine the kinetic constants for the PFR system. In the case of anticompetitive product inhibition kinetics, the relative errors become greater than those for substrate inhibition and noncompetitive product inhibition kinetics as shown in Table IV. The error is about 6% when the fractional conversion is 30% and becomes 10% at 45% conversion. In this case, if we introduce another transformed product concentration,

P* = P, iISOX

(12)

which is the arithmetic mean of product concentration instead of the logarithmic mean values, the transformed rate equation for anticompetitive product inhibition kinetics becomes identical to the performance equation. We may then use eq. (12) only for the anticompetitive product inhibition kinetics as an expedient method. DISCUSSION Since the forms of the transformed rate equations are identical to those of the initial reaction rate equations, the method of determining kinetic constants illustrated TABLE 111 Relative Errors between Transformed Rate Equation and Performance Equation for Substrate Inhibition and Noncompetitive Product Inhibition Kinetics in a PFR

X 0.10 0.20

0.30 0.40 0.50 0.55 0.60

.~ X

In(l - X ) 0.949 0.896 0.841 0.783 0.72 I 0.689 0.655

Relative error (I -

AX)

0.950 0.900 0.850 0.800 0.750 0.725 0.700

(%)

0.1

0.4 1.1

2.1 3.9 5.0 6.4

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COMMUNICATIONS TO THE EDITOR TABLE IV Relative Errors between Transformed Rate Equation and Performance Equation for Anticompetitive Product Inhibition Kinetics in a PFR

X

1 +-

0.10 0.20 0.30 0.40 0.45 0.50 0.55

Relative error

X In(1

-

X)

0.05 1 0.104 0.159 0.217 0.247 0.279 0.311

IX

(%)

0.050 0.100 0.150 0.200 0.225 0.250 0.275

I .7 3.7 5.9 8.5 9.9 11.5 13.2

in this paper can easily be applied to any other method that is used for evaluating the kinetic constants in the initial reaction rate equations by proper transformation of variables as shown in eqs. (3)-(5), (8)-(lo), and (12). As an example, the determination methods of kinetic constants for Michaelis-Menten kinetics are illustrated in Table V. Previously, we reported the advantages of S 0 X / 7 v s . ln(1 - X ) / T plot which is basically identical to the transformed Eadie-Hofstee plot.' Similarly, the substrate and product inhibition kinetics can be analyzed by a graphical method using transformed rate equations. Also the results for PFR systems can be used for batch reactors by only replacing the mean residence time. 7 , by the reaction time, t .

TABLE V Determination Methods of Kinetic Constants in a CSTR and a PFR System for Michaelis-Menten Kinetics Type of plot Method Conventional

CSTR

PFR

SOX vs. X/(l - X ) S O X vs. In(1

-

Slope Intercept X)

Km

Vrn~

Km v,,

1 v,n

Transformed LineweaverBurka EadieHofsteeb a

_F

1

soxvs. S0(l - X ) sox 1 x -vs.-7 71-x

7

s,x vs.

-In(l - X ) SOX

I X v S . --In(]

l / v ' vs. 11s' for CSTR, and I / v * vs. 1/S* for PFR. u ' vs. v ' / S . for CSTR, and u* vs. v * / S * for PFR.

-

X)

-K,"

V,

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BIOTECHNOLOGY AND BIOENGINEERING VOL. XXI (1979) References

I . S. B. Lee and D. D. Y. Ryu, Biotechnol. Bioeny., in press. 2. M. D. Lilly, W. E. Hornby, and E. M. Crook, Biochem. J . , 100, 718 (1966). 3. S . P. O’Neill, P. Dunnill, and M. D. Lilly, Biotechnol. Bioeng., 13, 337 (1971). 4. K. J . Laidler and P. S . Bunting, The Chemical Kinetics of Enzyme Action, 2nd ed. (Clarendon, Oxford, 1973). 5 . I. H. Segel, Enzyme Kinetics (Wiley-Interscience, New York, 1975).

SUNBOK LEE DEWEYD. Y . RYU The Korea Advanced Institute of Science P.O. Box 150, Chong-Ryang Ri Seoul, Korea Accepted for Publication May 25, 1979