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Multiple Steady States in Continuous Fermentation Processes with Bimodal Growth Kinetics Richard M. Russell and Robert D. fanner’ Chemical Engineering Department, Vanderbilt University, Nashville, Tennessee 3 7235
Continuous fermentation processes are usually designed from batch experimental data. These data are often described by a Monod growth model which, with the transient phase or with inhibition, gives a unimodal or single peaking growth-substraterelationship. In the batch culture of S.cerevisiae (baker’s yeast) grown on glucose limiting media and P. cerevisiae (a bacterium) grown on lysine limiting media, a double peaking or bimodal growthsubstrate relationship has been observed. The bimodal phenomena could be accounted for by either the synchronous growth of microorganisms (a homogeneous culture of cells simultaneously grow and subsequently divide together) or a diauxie effect (one substrate is selected over another in a two substrate medium). This paper deals with the implications of bimodal growth on the design of continuous fermentation processes. From batch culture data for S.cerevisiae grown on chemically defined media both a single and double peaking model were developed. Digital simulation of continuous culture reveals that the unimodal model has one stable operating point, whereas the bimodal model, for residence times below a certain critical value, has two stable operating points. The steady-state operating point achieved in the bimodal model is dependent upon the initial conditions of the cell and substrate concentrations. These results may help explain the occurrence of suboptimal yields in such continuous fermentations as single cell protein production and biological waste treatment processes.
Introduction “The biologically important properties of a biochemical system are the dynamic properties such as oscillations and transitions between steady states, not the steady states themselves.” (Rapp, 1975). Many investigations have been made into the transient properties of the continuous culture of microorganisms. Investigators have applied both theoretical and empirical models of cell growth to dynamic simulation, testing their applicability under unsteady-state conditions. Similarly, this paper shall develop the transient behavior of an observed phenomena, bimodal growth. Bimodal growth is defined here as a double peaking growth curve p ( S ) ,where p is the specific growth rate and (S) is the limiting substrate concentration. Before treating the dynamic arguments concerning this phenomena, however, it is first important to identify its origins. The Occurrence of Bimodal Growth. The most obvious occurrence of bimodal growth is diauxie growth, where one substrate is selected over another in a two-substrate system. This can be seen in the case of E . coli on this mixed medium is shown in Figure 1.The growth rate-time trajectory for this system is depicted as a double peaking curve in Figure 2. If the mixed substrate of glucose and sorbitol is measured as total organic carbon, biological oxygen demand, or some other gross characteristic, the specific growth rate-time trajectory would also be double peaking when the peak separation time is small. If the peak separation time, T , is large, then the low substrate concentration peak of specific growth rate could be suppressed by the large corresponding (XI.Since the substrate concentration is monotonically decreasing in time, the specific growth rate vs. substrate concentration would also have bimodal functionality, as shown in Figure 3, where substrate indicates total substrate for a mixed substrate system. The consecutive reaction kinetics of some fermentations may be treated with a similar argument. An example would be the aerobic culture of yeast on glucose limiting media, which exhibits the Crabtree effect. With glucose repression, growth occurs in two phases. The first utilizes glucose in the
formation of biomass and the product, ethanol. The second phase utilizes that ethanol for the production of more biomass. Such a phenomenon has also been referred to as diauxie (Mochan and Pye, 1971). Considering the combination of glucose and ethanol as the substrate leads to a growth curve, @(S),with bimodal functionality. An example of this autodiauxie phenomenon, which exhibits a lag phase similar to that shown for mixed substrates in Figure 1,has been observed by Maxon and Johnson (1953) for baker’s yeast. The phenomenon of bimodal or polymodal growth curves can be supported further by the notion of synchronous growth. Synchronous growth is defined here as a homogeneous culture of cells which grow and divide in concert. This can be seen in the work of Fries for S. cerevisiae on glucose limiting media (Reed and Peppler, 1973). Another example of culture synchrony appears in the work of Tanner and Souki (1974). Early transients in the batch culture of P. cereuisiae on lysine limiting media exhibited a bimodal growth curve, p ( t ) , in more than 50 such experiments as exemplified by Figure 4.It is suggested that this early bimodal response is due to culture synchrony stemming from the subculturing steps on inoculum preparation. The work of Dawson (1969) suggests that if cells of the same age are fed to a chemostat, the cells grown continuously may be kept in synchrony. One way to achieve continuous synchronous culture is to allow both the feed and the cell effluent to cycle in phased culture, as shown by Dawson (1972). A third case for bimodal phenomena comes from an analysis of mixed cultures of organisms grown on a single substrate, as suggested by Yang and Humphrey (1975). Consider two organisms A and B with separate growth curves describing changes in identical environments on the same substrate as seen in Figure 5. If the cell concentration measurement did not differentiate between organisms, the bimodal growth curve appearing in Figure 3 would be established, where ( X ) in the specific growth rate term represents total biomass. Gross biomass characterizations are common where mixed cultures of organisms are used, as in biological waste treatment processes. Other experimental evidence supporting bimodal growth characteristics is found in the work of Siege11 and Gaden
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Ind. Eng. Chem. Process Des. Dev., Vol. 17,No. 2, 1978 S * = 6~.1g/ml
8884 -
-
80 76
n6s64SORBITOL
60-
UTILIZATION
56-
52
-
4844-
UTILIZATION
4036 -
32-
20 -
28 TIME, I
Figure 1. Batch culture of E. Coli on a glucose-sorbitol medium (Based on schematic in Aiba et al. (1973)).
24
16-
12
-
08-
0400
T I M E (HRS)
Figure 4. Bimodal specific growth rate for the Leuconostoc Mesenteroides bacteria growing on a lysine limiting substrate.
a
J k
a=
k
1 a 0
(D
Figure 2. Growth rate vs. time for the batch culture of E. coli on glucose-sorbitol medium.
Y
’L
u
w a a
LIMITING SUBSTRATE CONCENTRATION, (S)
Figure 5. Depiction of growth of two different microbial species on a common limiting substrate.
SUBSTRATE CONCENTRATION
Figure 3. Bimodal growth curve exhibiting multiple steady states. (1962) for oxygen limiting yeast culture, assuming that the specific growth rate is proportional to the specific uptake rate of oxygen, which is shown in Figure 6. I t is also interesting to speculate that two parallel pathways contribute to the batch specific growth curve of Lactobacillis delbruekii (Luedeking, 1956),such that a constant p occurs a t several pHs as shown in Figure 7. Conceivably, such curves could lead to an infinite number of steady states in the continuous fermentation.
Dynamic Analysis of Chemostat Culture The analysis of the dynamic behavior of the chemostat has been treated by others and will be only briefly reported here (Yano and Koga, 1969; Harrison and Topiwala, 1972; Koga and Humphrey, 1967). From cell and substrate material balances about a single stirred tank fermentor the differential equations which describe the system can be determined. These are given by d(X)ldt = ( X ) [ p ( S ) D];
(X),= X*
(XI d(S)ldt = D [ ( S ) F- (SI]- -p ( S ) ; (SI,= S*
Y
(1) (2)
where ( X ) = cell concentration (O.D.U. or mg/L), where O.D.U. = optical density units; ( S )= limiting substrate concentration (mg/L); ( D ) = dilution rate (h-l) = feed flow rate divided by the fermentation broth volume; (S)F= feed substrate concentration (mg/L); (S)o= s*,initial substrate concentration; (X), = X*, initial cell concentration; Y = cell
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 2, 1978
0'3
i
159
STABLE OPERATING POINT
I
I
I
02
04
06
OXYGEN TENSION, A t m
Figure 6. Specific uptake rate of oxygen for Saccharomyces cereuisiae (data of Siege11and Gaden (1962)). a
52
SI 0.5
SUBSTRATE CONCENTRATION
a
Figure 8. Steady-state operating points for continuous culture unimodal specific growth rate.
0.4
0.3
0.2
0.I 0 0
4
8
12
FERMENTATION TIME, H O U R S
Figure 7. Specific growth rate curves for Lactobacillis Delbrueckii exhibiting constant values at various pH values (data of Luedeking ( 1956)1. yield, d(X)/d(S) (g of cells/g of substrate); Y is nearly constant in the cell growth phase; and p ( S ) = l / ( X ) d(X)/dt, (h-l), specific cell growth rate, typically a function of (S). The highly nonlinear function, p ( S ) , is a function of ( S ) which in turn depends on (X). The function, p , can be linearized by a Taylor series approximation about some point S and to give
x
Using this linearized form of p ( S ) in conjunction with the system of differential equations, eq 1 and 2, the eigenvalues can be determined, and hence the requirements for system stability. For this system the approximation yields two eigenvalues (Harrison and Topiwala, 1972)
(4) The requirement for a stable steady state is that these eigenvalues be negative and real. Since the dilution rate is always positive or zero, XI will always be negative. Xz, however, may be positive if the slope, dp/d(S) is negative, and hence lead to unstable steady states. Returning to the previously discussed double phenomena with this analysis, it can be seen that five steady states exist (Figure 3). States one and three will be stable steady states and states two and four will be unstable. The fifth state (not pictured) is the washout condition and is also stable, as has been demonstrated by Yano and Koga (1969). Illustration of the differences between the bimodal curve and the single peaking curve (Monod growth with inhibition
or the transient phase) can be seen by comparing Figure 3 with Figure 8, respectively. The major difference between these two figures is the additional stable steady state introduced by the second peak, point 3 in Figure 3. The linearization approach to this analysis is useful in gauging the stability of a steady-state solution. However, to examine the transitions between steady-state solutions, it is necessary to account for the nonlinearity of the system. This was accomplished through digital simulation of the equations describing the chemostat using the nonlinear function
dS). The bimodal function p ( S ) , shown in Figure 9, was suggested from batch culture data of S. cereuisiae grown on glucose limiting media at 32 "C and pH 5 (Tanner et al., 1977). Since the chemostat system is described in terms of two variables, (S)and ( X ) , it is convenient to use a phase plane to describe its transient response. A digital computer simulation of eq 1 and 2 is used to generate the phase plane portrait in Figure 10 for various initial conditions p ( S ) .The approximate equations describing the batch curve depicted in Figure 9 and used in the simulation are: for 0 I(S)< 39
+
p1 = ( P M ~ - ~ P T ~ - ~ ) - ~ / ~
0 204
I
-E
t
I E
k
0 W ln 0
nn
1
0
0
\ 1
\
I O 2 0 30 40 50 60 70 8090 100
GLUCOSE C O N C E N T R A T I O N , G R A M MOLES/LITER
Figure 9. Growth-substrate relationship based on batch culture of S. cereuisiae at 32 "C and pH 5.0.
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P
0 0
(X)
10
Figure 11. Phase plane analysis of step changes in dilution rate. CELL CONCENTRATION (OPTICAL DENSITY UNITS)
Figure 10. Phase plane simulated curves for the bimodal specific growth rate curve shown in Figure 9. where 0.204(S) 1.74 (S) C ( T= ~ 0.0085[46.76 - ( S ) ]
+
C(M1 =
point ((X) a 1.5, ( S ) a 90) in its course from ((X) 2 8, (S)= 100) to ((XIa 1/3, ( S ) = 0).
Implications of Bimodal Growth on Chemostat Culture Subject to Various Types of Disturbances Change in Dilution Rate. Koga and Humphrey (1967) have treated the problem of chemostat response to step changes in dilution rate for Monod type growth kinetics of the form
and for 39 5 ( S ) 5 100 !J2
=
(FM2-4 -k C(T2-4)-1’4
where O.l9[(S) - 38.591 0.767 [(S) - 38.591 C(T2 = 0.006[100 - (s)]
C(M2 =
+
The feed substrate concentration is 100 g/L and Y ,the yield, is selected to be at 10?/. The constant yield is verified in Figure 10 by the straight line joining all three stable steady-state cperating points (these points are circled). The dilution rate used in this analysis is 0.1 h-l as indicated in Figure 10. It is useful at this point to compare this phase plane to one established for the single peaking growth curve, Monod growth with substrate inhibition or transient phase, developed by Yano and Koga (1969). Except for the presence of an additional stable steady state, for the double peaking curve, the two phase planes have many trajectory shapes in common. An important feature of the bimodal phase plane portrait not in the unimodal phase plane, however, is the presence of unusual pathways of some trajectories in their approach to the middle stable steady-state point. These features are due to the close proximity of the middle stable state to the second unstable point (points 3 and 2, respectively, in Figure 3). The “Sshaped” separatrix, as defined by Lin and Segal (1974), dividing the trajectories leading to the lowest stable steady-state point ((X), 1 10, (S),= 2) from those converging on the = 6, (S),= 40) passes middle stable steady-state point ((X, a 6.5, ( S ) z 35) in its through that unstable saddle point ((X) course from ((X) z 13.5, ( S ) = 100) to ((X) = 2.4, (S)= 0). Similarly, the “upper” “bent arm” separatrix dividing the trajectories leading to the upper or washout stable steady state ((X) = 0 , (S) = 100) from those converging on the middle stable steady-state point passes through that unstable saddle
where pmaxis the maximum specific growth rate, and K , is the saturation constant. Their analysis shows that for these changes no extrema would be observed in S ( t )or X ( t ) . As can be seen in Figure 11, both the final and initial steady-state points of the phase plane lie on the constant yield line. The bimodal phase plane also predicts no extrema for this type of disturbance since the trajectories on and near the constant yield line are linear. Change in Feed Substrate Concentration. This problem has also been treated by Koga and Humphrey (1967) for the Monod model of cell growth and by Yano and Koga (1969) for Monod growth with substrate inhibition type kinetics. These earlier studies have shown that extrema can exist in the substrate trajectories, but for this type of perturbation no extrema will be present in X ( t ) . Returning to the unusual curvature of trajectories approaching the middle stable steady state, in light of the previous work, provides an interesting departure from the Monod-Monod/inhibition model studies. The line starting a t (X) = 2, (S) = 0 in Figure 10 is a trajectory where two extrema are present for the cell concentration trajectory and one for the substrate concentration trajectory. Yano and Koga (1969), using the unimodal curve (Monod growth with inhibition), have identified only one extrema in X ( t ) . However, if this extrema is experienced, their system will be driven into the washout condition, for there is not central stable steadystate solution. The two extrema response of X ( t ) in chemostat culture subject to feed substrate concentration changes is found in the work of Gilley and Bungay (1967). Here step changes in feed substrate concentrations produced damped oscillatory responses in X ( t ) . Koga and Humphrey (1967) have demonstrated this phenomena for systems with large maintenance requirements. However the dilution rates used in the work
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 2, 1978
of Gilley and Bungay (1967) were very close to the maximum specific growth rate and would tend to make the applicability of large maintenance requirements highly suspect. Other experimental support of these results may be indicated in the work of Koplove and Cooney (1975). E. coli was cultured continuously on glucose limiting media and subjected to step changes in feed glucose concentrations. The transient response shows a reduction in cell yield from an initial steady-state value of 0.14 g of cells/g of glucose to 0.10 g of cells/g of glucose, 24 h after the perturbation for the presumed new steady state. Possibly this finding indicates the presence of multiple steady states, since the culture exhibited biphasic growth. Shock Loading. The last type of disturbance to be treated is the shock loading of a chemostat culture with respect to its limiting nutrient. Using the bimodal phase plane, it can be seen that large instantaneous changes in chemostat substrate concentrations can lead to a steady state which is different than the original. Thus, it is possible, in some circumstances, to obtain sustained suboptimal yields in systems subject to shock loading.
Conclusions A case has been presented for the occurrence of two stable steady states in continuous fermentations resulting from bimodal growth rate kinetics. Such kinetics are exhibited in batch systems. This type of approach to the analysis of chemostat dynamics may lead to new insights and understanding of continuous fermentation processes. Recently, for example, Aris and Humphrey (1976) have shown that many possible steady state configurations can occur when two organisms compete for a common substrate. Acknowledgment We gratefully acknowledge Professor Rutherford B. Aris's perceptive comments on the original manuscript. In partic-
161
ular, we are pleased to include his observation of the separatrices associated with Figure 10.
Literature Cited Aiba, S., Humphrey, A. E., Miilis, N. F., "Biochemical Engineering," Academic Press, New York, N.Y., 1973. Aris, R., Humphrey, A. E., Biotech. Bioeng., 19, 1375 (1977). Dawson, P. S. S., "Continuous Synchronous Culture-The Production of Extracellular Enzymes by Continuous Phased Culture of Bacillus subtilus, '' in Proceedings, IV International Fermentation Symposium, Fermentation Technology Today, p 121, G. Terui, Ed., The Society for Ferrnertation Technology, Kyoto, Japan, 1972. Dawson, P. S. S., "Continuous Phased Culture-Experimental Technique," in "Continuous Cultivation of Microorganisms," p 71, i. Malek, Ed., Academia, Prague, 71 1969. Gilley, J. W., Bungay, H. R., 3rd., Biotech. Bioeng., 9, 617 (1967). Harrison, D. E. F.. Topiwala, H. E., Adv. Biochern. Eng., 3, 167 (1972). Koga, S., Humphrey, A. E., Biotech. Bloengin., 9, 375 (1967). Koplove, M. H., Cooney, C. L., "Enzymatic Response to Perturbations in Continuous Culture," First Chemical Congress of the North American Continent, Mexico City, Mexico, Dec 1975. Lin, C. C., Segal. L. A., "Mathematics Applied to Deterministic Problems in the Natural Sciences, Macmillan, New York, N.Y., 1974. Luedeking, R., Ph.D. Dissertation, University of Minnesota, 1956. Maxon, W. D., Johnson, M. J., lnd. Eng. Chem., 45 (2), 2554 (1953). Mochan, E., Pye, K. E., "Kinetics of the Transition from Glucose to Ethanol Utilization in Yeast Cultures," 162nd National Meeting of the American Chemical Society, Washington, D.C., Sept 1971. Rapp, P., Biosystems, 7 ( l ) ,92 (1975). Reed, G., Peppler, H. J., "Yeast Technology," p 71, The Avi Publishing Company, Inc., Westport, Conn., 1973. Tanner, R. D., Souki, N. T., Russell, R. M., Biotech. Bioeng., 19, 27 (1977). Tanner, R. D., Souki, N. T., "On Reducing the Lysine Microbiological Assay Time," 168th National Meeting of the American Chemical Society, Atlantic City, N.J., Sept 1974. Yano, R. D., Humphrey, A. E., Biotech. Bioeng., 17, 1211 (1975). Yano, R. D., Koga, S., Biotech. Bioeng., 11, 139 (1969).
Received for review J a n u a r y 31, 1977 Accepted N o v e m b e r 7 , 1 9 7 7 Presented a t the D i v i s i o n of M i c r o b i a l a n d B i o c h e m i c a l Technology, 1 7 2 n d N a t i o n a l M e e t i n g of t h e A m e r i c a n C h e m i c a l Society, S a n Francisco, Calif., Aug 31, 1976.
GASP IV and the Simulation of Batch/Semicontinuous Operations: Single Train Process Brad W. Overturf, Glntaras V. Reklaitis," and John M. Woods School o f Chemical Engineering, Purdue University, West Lafayette, lndiana 47907
The GASP IV combined discrete-continuous stimulation language provides a powerful modelling tool for simulating batch/semicontinuous plants. This paper discusses the salient features of GASP IV and shows its application to a single train process consisting of a batch unit, two semicontinuous units, storage tanks, and stochastic variables. The model of the process is based on the work of Davis and Kermode. The GASP IV model allows the simulation of multiple batch runs to be carried out and shows that the operating policy derived previously is unsatisfactory in the long run.
Introduction Batch and semicontinuous processes are representative of a considerable portion of the chemical industry and are used to produce some of the highest unit price products. Yet, they have been very much neglected in the computer oriented chemical engineering literature especially when compared to the intense interest shown in computer aided analysis of steady-state continuous processes. This neglect has occurred for various reasons, the four major of which are the following: 0019-7882/78/1117-0161$01.00/0
(1) Continuous processes are generally of larger scale. Therefore, the potential payoff for computer-aided process studies is thought to be larger. (2) Batch/semicontinuous processes are usually multiproduct facilities and are considerably more difficult to analyze since they inherently involve competing demands for these facilities. (3) Batch and semicontinuous processes are by their very nature flexible. They are built because this flexibility is needed and for this reason tend to be subjected to frequent changes and modifications. 0 1978 American Chemical Society
