Mathematical analysis of two-phase mass transfer in a batch reactor for the chemical transformation...

Mathematical Analysis of Two-Phase Mass Transfer in a Batch Reactor for the Chemical Transformation of a Steroid

Vincent Pereira, Huseyin Tigli,* and Carl C. Gtytet Department of Chemical Engineering and Applied Chemistry, Columbia University, New Yo& New York 70027

Accepted for publication June 23, 7986

A reactor is described for the conversion of the slightly water-soluble steroid testosterone (TI to 4-androstene- 3,17-dione (4-AD) by enzyme in the presence of excess cofactor. Since the enzyme is subject to substrate inhi- bition, reaction rates are strong functions of aqueous sub- strate concentration. High concentrations of the sub- strate, testosterone, per unit reactor volume are maintained within poly(dimethylsi1oxane) beads that are suspended in the aqueous enzyme solution. Mass transfer (con- trolled by bead size, polymer to water volume ratio, en- zyme loading) is used to control the degree and rate of conversion. The reactor dynamics are predicted over a wide range of reaction conditions. The product steroid is recovered in the polymeric beads from the enzyme so- lution.

INTRODUCTION

The transformation of a substrate such as cholesterol through a series of specific chemical events to end products such as testosterone or other hormone struc- tures is desirable and feasible, but faces serious diffi- culty. The necessary enzymes are active only in aqueous environments, and the steroid moieties are only slightly soluble in water. This leads to a need for large reactors, dilute enzyme solutions having serious separation problems and generally low efficiency in the use of enzymes. In addition? if crude enzyme solutions from only partially purified cell media are used, the sepa- ration problems are greatly increased. To circumvent these problems, Crernonesi et al.'J have demonstrated the feasibility of using aqueous emulsions of a water immiscible solvent in which the steroid can be dis- solved. However, Bhasin and co-workers3 have shown that the water immiscible molecules easily denature enzymes presumably by adsorbing into the hydropho- bic structure of the macromolecule. Buckland and co-

* This paper is taken from the Masters Thesis of Huseyin Tigli,

t To whom all correspondence should be addressed. Columbia University, New York, New York.

workers4 have obtained very high rates of oxidation of cholesterol with whole cells suspended in steroid con- taining organic solvents. More recently, Carrea et al.5 has immobilized enzymes into Sepharose and sus- pended the hydrophilic gels in steroid containing or- ganic solvents. In this case, significant stability of the enzyme in the immobilized state was evident even in the presence of organic solvents.

Bhasin and co-workers3 reported a two phase en- zyme reactor in which the hydrophobic phase was polymeric. They employed lightly crosslinked poly(dimethy1 siloxane), PDMS as a substrate (and product) reservoir. PDMS has a high solubility for ste- roid-Iike molecules because of its similar cohesive en- ergy density. Because of its low glass transition tem- perature this polymer exhibits liquid-like transport properties. These topics have been reviewed by Mi- chaels et aL6 Saunders and co-workers' have extended the application of such PDMS resins to the transfor- mation of diosgenin into a series of increasingly hy- drophobic product steroids.

Partition equilibrium for the steroid reactants and products favor the polymeric PDMS phase over the aqueous region. Thus, the polymer particles act as res- ervoirs for both the reactants and the products. There are a number of advantages. It is easy to separate the beads from the aqueous phase by settling or by filtra- tion. The major portion of the yield resides in the PDMS beads. The diffusive transport within the beads is rea- sonable, and the overall mobility of the steroids be- tween the phases is rapid when the beads are small. The polymer beads are inert and therefore do not cause denaturation of the enzyme by partitioning into the aqueous phase. Since the kinetics of steroid conversion normally follow a complex path as a function of sub- strate concentration, the polymer phase may be used to regulate the concentration of the substrate in the aqueous phase at its optimal level. Because the beads are solid, very little effort is required to maintain a

Biotechnology and Bioengineering, Vol. XXX, Pp. 505-513 (1987) 0 1987 John Wiley & Sons, Inc. CCC 0006-3592/87/040505-09$04.00

dispersion of the system in water. This is in contrast to the problem encountered with low-molecular-weight organic liquids.

In this report, we focus on a single step: the con- version of testosterone (T) to 4-androstene-3,17-dione (4-AD) in the presence of enzyme and cofactor. The reactant testosterone is originally resident in the PDMS beads. It diffuses into the aqueaus system, wherein the enzyme catalyzes the transformation to the 4-AD prod- uct. The 4-AD is partitioned back into the PDMS beads. Since a constant volume, stirred batch reactor is under consideration, the process is inherently unsteady. Hence, the transient is of far greater interest than the final equilibrium state.

The present two-phase arrangement lacks the sim- plifying condition that at least one of the two phases is kept at constant concentration with respect to either the diffusing reactant or product. Irreversible reactions8 reversible reactions of various s to i ch i~met r i e s ,~ -~~ equilibrium reactions,Iz and consecutive reactions" have been studied with such a transport configuration. In a variation of this configuration, if a solid is dissolved into a reacting liquid, one assumes that the concentra- tion at the solid-liquid interface is constant. In another, if the volume outside a body is large, one often assumes that the outside concentrations are constant at a quasi steady state and thus time-inde~endent.'~ In the pres- ent case, the concentration profiles of reactants and products are constantly changing. For a similar het- erogeneous chemical reactor with two liquid phases,

has worked out an iterative algorithm for ob- taining the interfacial fluxes of the reaction compo- nents in both phases for the case of a fast irreversible reaction. Georgakis and c o - w ~ r k e r s ' ~ and Waterland and co-workers16 have described the solutions to prob- lems in which Michaelis-Menten type enzyme kinetics are coupled with diffusion.

EXPERIMENTAL PROCEDURE

The experimental results used as a basis for the model development reported herein were taken from the the- sis of Bhasin" and the experimental methods are de- scribed in detail by Bhasin and c o - ~ o r k e r s . ~ The batch reactor consisted of a 250-cm3 Erlenmeyer flask with a 3.5-cm magnetic stirring bar that was rotated in all experiments at a constant rate of 80-100 rpm. The

temperature was maintained constant at 22°C. The solid phase consisted of PDMS beads (10 cm3) loaded with testosterone (1.1 mg/cm3). The liquid phase consisted of 50 cm3 aqueous solution of 0.03M Tris-HCL buffer maintained at pH 9 with 0.003M EDTA. To the buffer were added 0.333 units of enzyme (3,17-P-hydroxy- steroid dehydrogenase) and 108.9 mg NAD. To mon- itor the rate of conversion, small aliquots of each phase were withdrawn from the reactor at various times. The analysis of the liquid phase was done by direct injection of the sample into a high-pressure liquid chromato- graph. For analysis of the solid phase, the steroids in the polymer beads had to be extracted with ethyl ace- tate. The solvent was evaporated, the residue was re- dissolved in methanol, then analyzed by high-pressure liquid chromatography (HPLC).3

THEORETICAL DEVELOPMENT

Enzyme Reaction Kinetics

The substrate, testosterone (17P-hydroxyandrost-4- ene-3-one) is transformed to product, 4AD (androst-4- ene-3,17-dione) as shown in Figure 1 by an enzyme 3,17f?-hydroxysteroid dehydrogenase that was re- covered from Pseudomonas Testosteroni. This en- zyme exhibits substrate inhibition at high concentra- tion of reacting steroid. Moreover, a t a critical steroid concentration (C,,), the reaction rate is maximurn.l8 These kinetics are illustrated in Figure 2. To explain this inhibition of enzymatic activity by the substrate, HaldaneI9 put forward the hypothesis that a bimolec- ular complex (containing two substrate molecules per active set of the enzyme) was formed in addition to the activated complex. This concept was later ex- tended by FriedenwaldZ0 and several other mecha- nisms were proposed. The enzyme kinetics reported by Marcus and TalalayI8 were confirmed for the en- zyme used in this work and the rate dependence on substrate composition C is given in eq. (1). The re- action velocity, V (mg conversiodmin cm3), is directly proportional to the kinetic parameter, V,,, (mg/min cm3). Parameter V,,, is proportional to the total en- zyme activity and the cofactor concentration both of which were held constant during the course of a re- action:

0

= @ + NADH + H* + NAD+

0" 4 - A D

04 CiP T

Figure 1. Enzyme transformation of testosterone (T), to 4-androstene-3, 17-dione (4-AD).

506 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 30, SEPTEMBER 1987

I 00.

8 0.

6 0.

4 0.

20.

0. I O - ~ 10-3 10-2 lo-'

CONC EN TR AT1 0 N TESTOSTERONE ( mg/m I

Figure 2. Percent of maximum reaction velocity for conversion of testosterone to 4-AD, as a function of substrate testosterone concentration: C, is the optimum concentration and V,,, is a kinetic parameter proportional to the total enzyme activity (ref. 18).

where Co = 1.73 x mg/cm3 and K = 0.154. The ratio of kinetic parameter Vmax to the observed maxi- mum in the rate of conversion is 1.31. Equation ( 1 ) predicts that the ratio of observed reaction rate to the maximum reaction rate for a given condition is inde- pendent of the absolute concentration of enzyme. In Figure 2, this reaction rate for a given enzyme and cofactor condition is plotted as a function of substrate composition C , the rate being a maximum at substrate composition Co.

Mass Transfer

This system can be adequately described by a com- posite sphere made up of two materials: a central core of PDMS (containing testosterone substrate andlor product), and an outer aqueous shell (containing en- zyme and cofactor). Such a composite sphere with outer radius, a, and inner core of radius, b, is shown in Figure 3 .

If N is the number of polymer beads in the total volume, then R, the ratio of PDMS to liquid volume, is

and

u = us -t uI = 413 r a3 (3)

where

(4) us = 413 r b3

Substituting for eqs. (2)-(4) and rearranging the yields, we have

1 I3

a = [ I + ;] b

The diffusion equation is given by:

ac at

(5 )

Generally a composite sphere may be divided into n shells of thickness E , where j = 1,2, . . . n. Then, by the central difference method, eq. (6) becomes:

ac D - = - [o'-l) c;-, - at je2 2;C; + o'+ 1) C;+J (7)

At the center of the composite sphere, we have due to spherical symmetry,

JCI = o ar

In the central difference form the above equation can be written for j=O as,

PEREIRA, TIGLI, AND GRYTE: CHEMICAL TRANSFORMATION OF STEROIDS 507

c- POLYMER-+ AQUEOUS -

i I

j= I I 0 i- A r

l a

j=k i= m i= n

Figure 3. Mass transfer model of polymeric bead (crosslinked poly(dimethylsi1oxane)) and enzyme containing aqueous phase.

6 ac - (C, -Co) = - € 2 at

(9)

The second boundary condition can be obtained as follows. Let the range 0 = r = a be divided into n equal intervals, each of width Ara and also the range 0 = r = b into k equal intervals, Arb, such that kArb = mAra = b with k, m, and n integers (see Fig. 3). On the boundary surface at r = b we have c k = CmKT, where KT is the partition coefficient (the bar over the concentrations refers to values within the polymeric PDMS phase):

By the central difference method,

- 1 ar 2Arb

- ac j = k - - ck+l - Ck-1 -

By substituting eq. (11) into eq. (lo), and rearranging, we have

On the aqueous side of the interface, the interfacial flux, F, is given by,

and

as well as

Writing eq. (7) at the interface for both the polymer (PDMS) and the aqueous phases, and using eqs. (12) and (15),

4- ( k -k 1) [ c k + i - .(%)I} (16)

aCm D 2AraF - - ___ { ( m

- 1) [T - at m(Ara>’

+ Cm+, - 2mCm + (m + 1)Cm+, (17) 1 1 Eliminating F from the two equations above gives,

1 d c k _ - 2 at

Equation (18) describes the changes in the concentra- tion in the polymer phase at r = b. Equation (9) de- scribes the changes in the concentration at the first shell, and eq. (7) describes the concentration in the succeeding shells in both aqueous and PDMS phase.

The concentration of testosterone in the aqueous phase is governed by the equation:

508 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 30, SEPTEMBER 1987

- I I

The concentration of the product 4-AD, P in the aqueous phase, is given by a similar equation:

where W is the ratio of the molecular weight of the product to the reactant. Proceeding in a manner similar to that shown for the diffusion of reactant, it can be shown that the changes in concentration of the product in the polymer phase at r = b is given by:

(21)

Equations similar to eqs. (7), (8), and (9) can be written for the changes in the concentration of the product in the first shell and succeeding shells in the aqueous and polymer phase, respectively.

RESULTS AND DISCUSSION

Mass transfer out of the PDMS beads in the absence of chemical reaction was examined to test the general behavior of the model and to determine the model pa- rameters from experimental data. The first attempts were made with the solid phase divided into three shells and the aqueous phase into four shells. The model behavior was consistent with what would be physically anticipated. The model parameters that produced the best agreement with experimental results were found to be: = 1.8 x cm2/min; D = 6.0 x cm2/min; and KT = 26.5.

Figure 4 illustrates the model fit. The testosterone concentration reaches partition equilibrium between the two phases in ca. 20 min. Both D and KT, obtained by fitting the experimental data, fall in reasonable ranges. The moiecular diffusivity of testosterone in a lightly crosslinked PDMS was estimated at ca. 3.6 x lo-’ cm2/min by Michaels6 solely from thermodynamic properties. The best model values of 1.8 x low5 cm2/min in PDMS compares well with this estimate. Also the value of KT = 26.5 is within the range found experi- mentally by Bhasin. The partition coefficient for 4-AD was measured to be 13.0.

The magnitude of V,,, depends upon the specific

n 0

lo** 5. 10. 20. 50. 0

1. 2- TIME (MINUTES)

Figure 4. Diffusion of testosterone from polymer to aqueous phase as a function of time (V, = 5 cm3; V , = 25 cm3; polymer bead radius b = 0.0254 cm; initial T loading in polymer = 1 . 1 mp/cm3). The data are for (0, A) experimental results and (-) computed results.

reaction conditions (e.g., units of enzyme, cofactor concentration, and pH). Thus for the model, an ap- proximate Vma, was found by fitting the experimental data in Figures 5 and 6. In Figure 5 the concentration of the reactant testosterone, averaged over each of the phases is plotted. Most evident in Figure 5 is the tes- tosterone concentration in the aqueous phase which was found to be somewhat higher than that predicted from model calculations. This is likely due to the mass balance error in the experimental data. In Figure 5, there are two data points given for the observed tes- tosterone in the solid phase. The lower points are the measured results. However, they do not give exactly a mass balance for the testosterone in the system. The upper points are calculated, forcing the mass balance condition and assuming all of the experimental error

‘*O Po 8 8 % 10-1 t

E “ 1 I- z 4 t- o U W a

0 20 40 60 80 100

TIME ( MINUTES)

Figure 5. Average testosterone concentration in each phase as a function of time. For the experimental results, in the (0) polymer phase and (0) aqueous phase, the conditions are Vr = 50 cm3; Vs = 10 cm3; b = 0.0254 cm; initial loading of T in polymer phase =

1 . 1 mg/cm3. For the (-) computed results, the model parameters are K4--AD = 13 and V,,, = 0.05 mghin.

PEREIRA, TIGLI, AND GRYTE: CHEMICAL TRANSFORMATION OF STEROIDS 509

t 'I lo2

$- I-

n 0

E t 1 1 I 1 1 1 1 1 1 1 1

0 20 40 60 00 100 TIME ( M I N U T E S )

Figure 6. Average 4-AD concentration in each phase as a function of time. Experimental results are shown for the (0) polymer phase and (B) aqueous phase. (Conditions and model parameters are the same as in Fig. 5).

lay in the measurement of testosterone concentration in the solid phase. In Figure 6, similar data are given that compare the experimental and computed 4-AD concentration in both aqueous and PDMS phase. The agreement between model and experiment must be considered acceptable given the uncertainties in the physical parameters of the system. Moreover since all parameters are now determined, this reactor system can be simulated of over a much wider set of condi- tions.

Because of substrate inhibition it is desirable to con- trol the testosterone concentration in the aqueous so- lution by simultaneous variation in the size of the PDMS beads, by the amount of enzyme (value of V,,,) and the solid-liquid ratio, R. In Figure 7, the total product yield (evaluated after 90 min) is given for different enzyme levels as a function of the bead radius. For each level of V,,,, there exists a bead size at for which the product is a maximum. Under these conditions the interrelationship of reaction rate and mass transfer maintains the substrate composition within the aqueous phase near to point at which the enzyme is most re- active. As the bead size gets smaller, the amount of enzyme required to maintain this maximum in rate in- creases. For very large solid particles, the mass trans- fer becomes rate determining and the maximum be- comes insignificant.

That this effect on the rate is the result of the sub- strate concentration in the aqueous phase is given in Figure 8. For these data the magnitude of V,,, is 1.6 x mg/min/cm3. From Figure 7, an optimum par- ticle radius is observed to be ca. 0.0508 cm. In Figure 8, we see the level of aqueous substrate remains con- stant in the absence of chemical reaction. For very small beads, the mass transfer is high and the substrate concentration is above C, giving a low rate of product formation. If the particles have large radii, e.g. 0.07 cm, then mass transfer is slow and the substrate com- position is lower than the optimal value, C,. Conse- quentially, the yield of product is reduced. For polymer particles of ca. 0.0508 cm, the rate of mass transfer

6-01

1 \ OPTIMUM ( m g / m i n em31 \. "ma,

A% 1.6 I O - ~

A: 1.0 I O - ~

BEAD RADIUS ( c m )

Figure 7. cm'; initial loading of T in polymer phase = 1 . 1 mg/cm3; a11 at indicated values of V,,,.

Total 4-AD yield as a function of bead radius. The conditions are V , = 10 cm3; V L = 50

510 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 30, SEPTEMBER 1987

- - E

E

\

0

Y

w v)

I a n (r W l-

9 z w z 0 K W

0

W I-

ti L

I "It

/- NO REACTION

BEAD RADIUS 0-0127 c m d

105 I 1 I I I 1 I 0 15 30 45 60 75 90 105

TIME (MINUTES 1

Figure 8. Average testosterone concentration in the aqueous phase as a function of time. The conditions are V,,, = 1.6 x mg/min cm'; V,\ = 10 cm3; V , = 50 cm3; initial loading of T in polymer phase = I . I mg/cm3; all at indicated values of bead radius, b.

maintains the aqueous testosterone concentration near the optimal level; as a result, conversion is most rapid. One concludes that it is desirable to have the substrate pass through the optimal aqueous substrate concen- tration at a point equal to one-half the residence time in the reactor. Under these circumstances, the enzy-

Table I. Testosterone conversion as a function of reactor parameters.

matic reaction proceeds at or near its maximum level over the largest period of time. It is also suggested that a CFSTR could be designed using this principal so as to maintain the reactor at the optimal kinetic point.

In Table I , the conditions for various simulations are given in which bead diameter, polymer-to-aqueous- volume ratio, initial testosterone loading, and enzyme kinetics are varied. The total product as a function of time is plotted in Figure 9. From these data the average rate is determined over the initial 60 min of reaction and this rate together with the fraction of initial tes- tosterone converted is compared in Table I with the optimal rate for the specific enzyme conditions. Both rate and yield decrease as the bead diameter is de- creased for a given initial loading (see Cases A, B, and K). Small particles have efficient mass transfer into solution and as a result the enzyme kinetics operates at a high substrate concentration that give increasingly slow conversion rates. Case M gives a low rate under similar conditions because of the large particle size which holds the substrate concentration at a value lower than the optimal value. The highest rates and conver- sions are found in Case F in which small beads (giving rapid mass transfer) are coupled with high concentra- tions of enzyme (that keeps the substrate near the op- timal concentration). The lowest rate and yield are obtained when high initial substrate loadings cause a high substrate concentration in the aqueous phase and thus low kinetic rates.

From Figure 9 we see that the mass transfer and reaction kinetics are coupled and reactor conditions can be determined by suitable combinations of polymer particle, initial loading, and enzyme amount. In addi- tion to this control of reaction conditions, the product steroid is recovered in the bead and can be recovered in high concentration after a simple mechanical sepa- ration of the beads from the enzyme solution. As in-

Initial steroid Reaction Observed Bead radius, Polymer-to- Volume loading velocity, conversion Observed

b aqueous ratio, aqueous T O Vmax V L ratea Observed fraction of Case (cm) R (cm') (mg/cm3) (mg/min) (mg/min) yieldb optimal rateC

0.0254 0.0550 0.0127 0.1OOo 0.0254 0.0254 0.0254 0.005 0.0254

0.2 0.2 0.2 0.2 0.085 0.085 0.140 0.20 0.20

50 50 50 50

118 1 I8 71 50 50

1.1 1.1 1 . 1 1 .1 1.1 1 . 1 1.1 1 . 1 3.3

0. I6 0. I6 0. I6 0.16 0. I9 0.38 0.23 0.40 0.16

0.05 0.068 0.045 0.037 0. I05 0.130 0.108 0. I36 0.018

0.27 0.37 0.236 0.218 0.570 0.710 0.600 0.75 0.03

0.409 0.557 0.367 0.301 0.724 0.445 0.616 0.445 0.147

Note: all data refer to a polymer phase volume of 10 cm3. Computed from yield after 60 min reaction. Fraction of initial loading of steroid converted in 60 min reaction. Observed conversion rate divided by maximum enzyme rate for given V,,, and liquid volume

PEREIRA, TIGLI, AND GRYTE: CHEMICAL TRANSFORMATION OF STEROIDS 51 1

c m E

I- 0 3

0

Y

n a n

a J

l- 0 I-

- 0 15 30 45 60 75 90 105 120 135 150

TIME (MINUTES)

Figure 9. Total product 4-AD as a function of time for different batch reactor conditions identified in Table I: (El) Case A, (m) Case B, (0) Case K, (PI) Case M, (A) Case C, (0) Case D, (0) Case E, (A) Case F, (8) Case H

dicated by Bha~ in ,~ these polymeric reservoirs, in con- trast to low-molecular-weight organics, have minimal effect on the stability of the enzymes themselves. It is interesting to note that these polymer-based, two-phase enzyme reactors produce at a given activity ca. 8 x

mg steroidmin in each cm3 reactor volume. Sin- gle-phase reactors having similar enzyme activity would require a much larger total volume to maintain the substrate at the optimal concentration.

CONCLUSIONS

PDMS is a suitable reservoir material for the release of substrate steroids into an aqueous enzyme solution. The product steroid diffuses back in high concentration to the PDMS particles. These PDMS particles do not denature the enzyme. Such polymer particles can be designed (size and solid liquid ratio) so that the sub- strate composition retains its optimal value for maxi- mum rate of reaction. As the enzyme concentration increases, the particle size decreases for equivalent rate of reaction. This scheme has two principal ad- vantages over single-phase batch reactions: 1) the aqueous volume is small thus enzyme quantity is small per unit conversion, and 2) the steroid product is iso- lated a high concentration in a phase that is mechan- ically separable from the enzyme broth.

NOMENCLATURE outer radius of composite sphere (cm) radius of the polymer particles (cm) concentration in aqueous (polymeric) phase (mgfcm') bulk aqueous concentration concentration for which kinetics is at maximum diffusivity of solutes in aqueous (polymeric) phase (cm2/min) solute flux (mg/cm2 min) enzyme rate parameter [see eq. (1)l partition coefficient for reactant and product integer denoting last shell in polymer phase integer denoting first shell in aqueous phase number of beads ratio of the volume of polymeric phase/volume of aqueous phase time (min) velocity of reaction (mgfmin) enzyme kinetic parameter [see eq. (l)] (mgfmin cm') volume aqueous (polymeric) phase (cm') thickness of shell in aqueous (polymeric) phase (cm) thickness of shell (cm) radial coordinate (cm)

References

1. P. Cremonesi, G. Carrea, G. Sportoletti, and E. Antonini, Arch.

2. P. Cremonesi, G. Carrea, L. Ferrara, and E. Antonini, Bio- Biochern. Biophys., 159, 7 (1973).

rechnol. Bioeng., 17, 1101 (1975).

51 2 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 30, SEPTEMBER 1987

3. D. P. Bhasin, J. F. Studebaker, and C. C. Gryte, Biotechnol.

4. B. C. Buckland, P. Dunnill, and M. D. Lilly, Biotechnol. Bioeng.,

5. G. Carrea, F. Colombi, G. Mazzola, P. Cremonesi, and E. An-

6. A. S. Michaels, P. S. L. Wong, R. Prather, and R. Gale, AIChE

7. R. P. Saunders, R. Hardman, and P. S. J. Cheetham, Biotechnol.

8. G. Froment and K. Bischoff, Chemical Reactor Analysis and

9. C. J. Huang and C. H. Kuo, AZChE J., 11,901 (1965). 10. R. M. Secor and I. A. Beutler, AZChE J., 13, 365 (1967). 11. K. Onide, E. Sada, T. Kolayashi, and M. Fujine, Chem. Eng.

12. D. R. Olander, AIChE J., 6, 233 (1960).

Bioeng., 18, 1777 (1976).

17, 815 (1975).

tonini, Biotechnol. Bioeng., 21, 39 (1979).

J., 21, 1073 (1975).

Bioeng., 27, 825 (1985).

Design (Wiley, New York, 1979).

Sci.. 25, 753, 761 (1970).

13. H. S. Carslaw, Conduction ofHeat in Solids (Wiley, New York,

14. V. Rod, Chem. Eng. J., 7, 137 (1974). 15. C. Georgakis, P. C. H. Chan, and R. Aris, Biotechnol. Bioeng.,

17, 99 (1975). 16. L. R . Waterland, A. S. Michaels, and C. R. Robertson, AIChE

J., 20, 50 (1974). 17. D. H. Bhasin, “Interactions of hydrophobic species with hy-

drophilic environments,” M. S. thesis, Columbia University, New York, NY, 1976.

18. P. I. Marcus and P. Talalay, Proc. Roy. Soc London B, 144, 116 (1955).

19. J. B. S. Haldane, The Enzymes (Longmans Green, London, 1930).

20. J. S. Friedenwald and G. Maenguin-Davies, The Mechanism of Enzyme Action (Johns Hopkins Press, Baltimore, 1954).

1949).

PEREIRA, TIGLI, AND GRYTE: CHEMICAL TRANSFORMATION OF STEROIDS 51 3