SIMULA'TION OF A STEROID BIOCONVERSION WITH A MATHEMATICAL MODEL W.
D. MAXON, J. W. C H E N , ' A N D F. R. HANSON
The Upjohn Go., Kalamaroo, Mich.
The oxidation of progesterone to 1 1 a-hydroxyprogesterone by the mold Rhizopus nigricans is typical of steroid bioconversions in that it involves dissolution of a solid substrate, a single-step enzymatic reaction, and precipitation of the product. It also exhibits a physical interaction between substrate and product, in this case a mixed crystal which can b e demonstrated b y x-ray diffraction techniques. Appropriate equations were devised to simulate each of these phenomena. Both c! digital computer and an analog computer were programmed for the simultaneous solution of these equations. The equations contain only two coefficients the value of which is not known a priori and which must b e adjusted to match the simulation to the actual fermentation. One is concerned with solution kinetics, the other with reaction kinetics. With appropriate values of these constants determined b y trial and error, an excellent agreement with experimental data was achieved for bioconversions in which a single addition of powdered substrate was made, or in which substrate was added continuously or incrementally,
described the development of a mathematical model to simulate the 1-dehydrogenation of a steroid by the microorganism Septomyxa uJnis ( 7 ) . I n this case, as with most commercially significant bioconversions, the level of substrate exceeds its rather low solubility. Furthermore, the solubilities of substrate and product exhibit a strong effect on one another. The model took into account this interaction as well as the enzyme kinetics to give good correspondence to actual results. I t was based on the established fact that steroid must be in solution to react. T h e present paper reports on a n examination of another microbiological steroid transformation, the conversion of progesterone to 11a-hydroxyprogesterone by the mold, Rhizopus nigricans.
A
PREVIOUS P A P E R
p 3
t-a Rhizopus
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PROGESTERONE
I lo-HYDROXYPROGESTERONE
This is an example of oxygenation a t carbon-11, another important step on the route to the corticoid hormones. Again if the process is to be economically feasible it is necessary to carry it out a t a level of substrate far exceeding the solubility. T h e factors influencing the course of this bioconversion are the rate of supply of the substrate (by solution), the rate of enzymatic reaction, and the interactions that occur between substrate and product. T h e same factors are important to the 1-dehydrogenation considered previously and are perhaps typical of bioconversions in general. There are significant differences from the former case, however. I t has been again possible to construct a mathematical model that successfully simulates the process. I n view of the complexities of the nonlinear differential equations it was necessary to use computers-both analog and digital-for their
1
Present address, Southern Illinois University, Carbondale, Ill.
solution. Each had its advantages and limitations as described. Methods
Fermentation. The organism Rhizobus nigricans was grown from a vegetative inoculum in a glucose-crude protein medium. The growth and bioconversion were carried out in aerobic, submerged culture in agitated stainless steel fermentors. Progesterone was added as a powder in the amounts and on the schedules indicated, after substantial growth had occurred. Assay. Samples were assayed at intervals for progesterone and 11a-hydroxyprogesterone by descending paper chromatography. The beer was diluted with methanol to solubilize the steroids for analysis. A propylene glycol-toluene two-phase system was used. X-ray diffraction patterns were obtained on a General Electric XRD-5 x-ray diffractometer (c, radiation). The digital computer used was International Business Machines Model 1620. The analog computer was a Heath, extensively modified and expanded in our own laboratories. I t has 15 amplifiers, five servomultipliers, and two X-Y recorders with curve-following equipment. Results and Discussion
A typical example of the course of bioconversion of progesterone to 11a-hydroxyprogesterone by Rhizopus nigricans is shown in Figure 1. After a growth period in the aerated and agitated fermentor, the total quantity of substrate was added (time zero). The rate of disappearance of progesterone increased gradually for about 22 hours then remained constant until about the 50th hour, when conversion abruptly stopped. The increasing rate during the first 22 hours can be attributed to a gradual increase in cell material and thus enzyme concentration. I n certain modified media the dry weight of cells can be easily measured and the correlation with conversion rate shown. The reason for the incomplete conversion is less clear. T h e residual progesterone was about 1470 of the starting material regardless of the amount added ; this proportion was remarkably constant. VOL. 5
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X-Ray Studies. The filter cake from fermentation samples taken at different times was analyzed by x-ray diffraction. Some of the patterns obtained are shown in Figure 2. In the diffraction pattern for a sample withdrawn after 25 hours of bioconversion, peaks similar to those for progesterone crystals were present. More important, however, is the presence of peaks at diffraction angles 28 = 10.lo, 12.5’, and 19.7’, which are not characteristic of either progesterone or 1la-hydroxyprogesterone. With beer sampled at completion of the bioconversion, the pattern showed neither crystalline progesterone nor 11a-hydroxyprogesterone. The only crystal observed was the new component, X. When the final sample was dissolved and recrystallized from acetone, the x-ray diffraction pattern was changed somewhat. Here peaks characteristic of both component X and 11a-hydroxyprogesterone were present. The background reading was reduced. When an artificial mixture of progesterone and 11a-hydroxyprogesterone in 1- t o 4 ratio, the same as in the final fermentation beer, was crystallized from acetone, the same pattern was produced. These results are interpreted to show that component X i s a crystalline form containing both progesterone and 1la-hydroxyprogesterone. Measurement of peak heights permits the rough estimation that it contains one part of progesterone to four parts of 1la-hydroxyprogesterone. The appearance of 1la-hydroxyprogesterone diffraction peaks only after recrystallization indicates that this material precipitates during bioconversion in amorphous form. Thus, as the conversion to 11a-hydroxyprogesterone proceeds, the product material comes out partially as a mixed crystal with progesterone and partially as a n amorphous solid. If it is assumed that the progesterone tied u p in component X is not available for redissolution and attack by the enzyme, the residual 14% of unreacted starting material is explained. Development of Mathematical Model. With this much understanding of the mechanism of this bioconversion, we proceed to the construction of a mathematical model. This would 286
I&EC PROCESS DESIGN A N D DEVELOPMENT
Figure 2.
1
15 IO DIFFRACTION ANGLE,2e
X-ray diffraction patterns
confirm the proposed mechanism. Then the model could be used to predict the effect of certain variables on the course and outcome of the reaction. T h e steps to be mathematically simulated are:
1. Dissolution. S-. [SI 2. Enzymatic reaction. [SI+. [PI 3. Amorphous precipitation. [PI -.P 4. Mixed crystal precipitation. rn [PI [SI + SP where S is concentration of solid substrate [SIis concentration of substrate in solution P is concentration of solid product [PIis concentration of product in solution SP is concentration of solid component X
+
STEP 1. The rate of solution of a solid can be expressed as a transfer coefficient times a transfer area times a driving force dS/dt = -kl’nA([S*]
-
[SI)
where kl’ is the mass transfer coefficient n is the number of particles of average surface area A [S*] is the solubility of the substrate, experimentally determined With powdered substrate the particle size distribution is such that n and A are continuously changing during dissolution. I t is difficult to derive, analytically, a workable expression for this changing situation, but Higuchi (2) has suggested that a n empirical expression may adequately express the observed kinetics: dS/dt = - k S Z ( [ S * ]
- [SI)
(1) where k l is an empirical constant. Higuchi, Rowe, and Heistand have presented a theoretical analysis of this problem ( 3 , 4 )‘
STEP2. Some sim:ple experiments with the bioconversion conducted a t low substrate levels, where all components are in solution, show that first-order kinetics apply to the enzymatic reaction. Thus, d [ S ] / d t(by reaction) = -k2[S] STEPS3 and 4. T h e rate of formation of total product is simply k2[S]. This product is presumed to precipitate immediately when its solubility is reached both as amorphous, P, and mixed crystal, SP. For every mole of solid product formed, R moles of substrate will precipitate with i.t. A reasonable first assumption, later born out by the success of the model, is that R takes the value of the ratio of substrati: to product in solution, [ S ] / [ P ] . Substrate in solution is thus withdrawn by this route a t a rate equal to k2 [SI. [SI/[PI. Experiments show [PI to be constant a t its solubility, [P*], throughout the conversion except for a short period a t the beginning, which was neglected. In some cases, however, such as continuous fermentation, [PI would have to be consi.dered as a variable. Over-all balance on [SI is the summation of its appearance by dissolution of the solid and its disappearance by reaction and coprecipitation : dissolution
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d [ S l / d t = klS*([:j*] - [SI)- kn[S](l
Simultaneous solution of Equations 1 and 2 provides the time course of [SI and S. Additional equations are necessary to determine the values of SA, the total substrate actually determined by the paper chromatcgraphic method used, and PA, the product determined.
dSA/dt =: dS/dt
+ k2[S]. [ S ] / [ P * ]
dPA/dt = k2[S]
(3)
(4)
Equation 3 ignores the contribution of soluble substrate, which in the present case is negligible. Solution of these equations is not easy to accomplish. Because of their nonlinea.rity, a computer must be used. This was done with both analog and digital types. Different values of kl and kl were used until a good fit to the data was achieved. T h e value of k l : the solution rate coefficient, for this simulation was 3.0. This constant. affects the shape of the curve for only a few hours before comp:.etion of the conversion and cannot be very precisely evaluated by this trial and error procedure. T h e value of k?, the rnzyme titer coefficient, used was 16.5 after 22 hours. Before that time, the enzyme titer was assumed to rise exponentially from zero and an appropriate equation for k2 was usecl instead of a constant value: k z =: 16.5eO.OW1-22) T h e value of k~ coiild be determined independently by measuring the reaction rate in solution for a sample of cells. This was done in one case and found to give correct results. I n the general cases, however, this procedure is tedious and the empirical selection of kp is preferred. Simulation of Batch i4ddition. Tracings of both the analog and digital solutions are shown in Figure 1. Both fit the data very well. The analog computer has the advantage of great speed, running off a set of curves in less than a minute. The
very large drop in [SImakes necessary the complication of changing scaling factors at several points during the last portions of the simulation. This complication is responsible for the lack of precision at the very end. T h e digital computer is much more precise, and the trouble with the scaling factor is eliminated. I t is rather slow, however. The solution involves Taylor series approximations with four terms and a rather fine calculation mesh (0.002 hr.). Three hours of computer time are required to complete one set of curves. However, no extensive effort was made to improve the numerical methodology in order to reduce this time. T h e digital type can be conveniently used a t widely separated times because it is easy to reprogram while the analog must be repatched each time, a more tedious job. The analog computer used, while adequate for the job a t hand, was nearing its maximum capability with some of the simulations attempted. Figure 1 shows that the dissolved substrate is a sensitive variable: which theoretically could be checked by experiment. However, the difficulties involved in measuring a small amount of dissolved material in the presence of a large amount of insoluble material prevented making this verification in most cases. Simulation of Continuous Feed. The batch feed bioconversions always end with the same percentage of residual substrate, regardless of enzyme titer or substrate level (within limits). This is found experimentally and is confirmed by the model. T o reduce this residual the model predicts that it is necessary to reduce the amount of dissolved substrate at steady-state. Figure 3 shows the experimental results of such a run and the digital computer simulation of the run. The match is good. The program was modified by including a factor, F, for the feed rate in Equation 1. Thus, dS/dt = F
- RiS*([S*] - [SI)
The value of k l used was 18.0. This is higher than the figure of 3.0 used for the batch feed, but the batch value was not very precise. Furthermore, k l represents a different thing in a run where fresh substrate is being continuously added than when it is all added initially. For a given value of
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Experimental d a t a Simulation
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S,solid substrate in the fermentor the continuous feed system would be expected to have more fine (freshly added) particles, thus a higher solution rate and a higher value of k l . For this simulation, a value of kz = 30.0, the enzyme titer coefficient, was used. This is a higher value than was used for the batch feed and is constant for the entire run. These differences are explained by the facts that certain improvements had been made in the fermentation conditions and a different type of fermentor was used. Figure 4 shows the experimental results and the computer simulation of another run, identical except for the feed rate. T h e rather unusual schedule was brought about by mechanical complications. T h e computer, however, predicted the results rather well, with no changes in k l or kz. Simulation of Incremental Addition. Having examined the behavior of this process with both batchwise and continuous addition of substrate, the authors tried the case where feed was added incrementally. While continuous addition reduces the residual substrate, it involves the continuous handling of a powdered solid, which might be difficult for plant-scale operations. I n Figure 5, the results of an incremental addition procedure are shown. Substrate was added in an initial quantity, then after 5 hours in hourly increments. The digital computer was used for the simulation. I t would have been difficult to provide a step function input of sufficient accuracy with the analog computer. The same equations as before were employed, except for the modification in feed schedule. The values of k l and kz were the same as for the continuous feed case. A good check between simulation and experiment was achieved. The slightly higher residual substrate for the incremental case as compared with the continuous case is seen in both simulation and experiment.
bioconversion process. This understanding of the mechanism is one important value obtained from this approach. Another important value is the ability to predict the outcome of changes in operating conditions without complete experimentation. Figure 6 shows the expected results of continuous feed runs in which enzyme titer constant (kz) is changed, and the feed rate is not. The analog computer was used for the simulation because of its speed. Easily seen is the large effect that the activity of the enzyme has on the outcome. For batch addition, the residual is 10 mmoles per liter. This is reduced to about 6 for continuous feed a t kz = 20, to about 2 for k z = 40. The value of attempting to raise the enzyme titer is evident. Figure 7 shows the result of variations in feed rate ( F ) a t constant kp. The effect is much smaller. Lengthening the bioconversion from 35 to 55 hours decreases residual substrate from about 6 to only about 4. At this point, one is tempted to conclude that the mathematical model will accurately predict the behavior of the F = 1.38 (hourly increments) k l = 18.0 k2= 30.0
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The close correspondence of the actual experimental data with the computer calculated values for the several cases shown here, and for many more that have been run but not presented, demonstrates that the mathematical model is a reasonably close approximation to the true mechanism of this
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Acknowledgment
The x-ray diffraction patterns were obtained and interpreted by John W. Shell, Allergan Pharmaceuticals, Inc., Santa Ana, Calif. The analog computer was constructed, programmed, and operated by Clayton D. Alway. Technical assistance was provided by John R. Marsh. W. I . Higuchi, University of Michigan, gave us many helpful suggestions.
Substrate I
IO
20
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30 40 TIME, t(hours)
P
1
50
60
Figure 7. Analog computer simulation of continuous feed bioconversions with different feed rates bioconversion with all manner of feeding procedures. Closer examination, however, shows that there are limitations to this conclusion. T h e equation used for dissolution rate kinetics was : dS/a’t = klS2([S* J -
[SI)
Sz is used to approximate empirically the effect of changing particle size as the solid substrate dissolves. A certain value of k1 was appropriate when all the starting material was added initially. I n the continuous case another value was appropriate because for a given concentration of solid substrate in the fermentor, the continuous feed system would be expected to have more fine (freshly added) particles, thus a higher solution rate and a higher value of k1. Although still another value of kl would be expected in the incremental case, the value used for continuous operation still gave good results. Nevertheless, an empirical constant such as kl must be used with discretion. Experimental data must be taken to confirm computer simulations whenever conditions are changed markedly. To eliminate the doubt caused by the use of an empirical expression for dissolution rate, we have changed the model so that the batch value of kl would be employed directly in the incremental case. This we envisioned being carried to the
Nomenclature
A F kl kl’ kz m
= = = = = =
n
=
P
=
[PI = [P*] = PA =
S [SI
= =
[S*] =
SA
SP
= =
t
=
R
=
average surface area of a substrate particle feed rate of substrate empirical dissolution rate constant mass transfer coefficient enzymatic first-order reaction rate constant number of product molecules per substrate molecule in component X number of substrate particles concentration of solid product (1 la-hydroxyprogesterone) concentration of product in solution [PI a t saturation product by analysis concentration of solid substrate (progesterone) concentration of substrate in solution [SIa t saturation substrate by analysis concentration of solid component X time number of substrate molecules that precipitate with each product molecule
literature Cited (1) Chen, J. W., Hills, F. J., Koepsell, H. J., Maxon, W. D.,
IND.ENG.CHEM.PROCESS DESIGN DEVELOP. 4, 421-5 (1965).
( 2 ) Higuchi, W. I., University of Michigan, Ann Arbor, Mich.,
1962, personal communication. (3) Higuchi, W. I., Heistand, E. N., J . Pharm. Sci. 5 2 , 62-71 (1963). (4) Higuchi, \V. I., Rowe, E. L., Heistand, E. N., Zbid., 5 2 , 162164 (1963).
RECEIVED for review September 16, 1965 ACCEPTED December 30, 1965
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