Oscillatory Behavior Control in Continuous Fermentation Processes

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1 Preprints of the 9th International Symposium on Advanced Control of Chemical Processes The International Federation of Automatic Control June 7-10, 015, Whistler, British Columbia, Canada WeM3.6 Oscillatory Behavior Control in Continuous Fermentation Processes Piotr Skupin Mieczyslaw Metzger Faculty of Automatic Control, Electronics and Computer Science Silesian University of Technology, ul. Akademicka 16, -100 Gliwice, Poland ( Abstract: The continuous fermentation processes involving yeast Saccharomyces cerevisiae or bacterium Zymomonas mobilis often exhibit oscillatory behavior. Because, the oscillatory mode of operation may lead to higher or lower average ethanol concentrations, hence, there is a natural need to control this behavior. The idea presented in this paper is based on the use of a mixture of two substrates that are continuously fed into the reactor chamber and the continuous fermentation process is described by an unstructured mathematical model with a product inhibition on cell growth. The relative contribution of both substrates will be treated as a new control variable. Moreover, it is assumed that the microorganisms exhibit diauxic growth and that the occurrence of the oscillatory behavior is related to a time delay in the response of cells to changes in the environment. From the bifurcation analysis of the system, it is shown that an appropriate ratio of both substrates to the mixture allows for induction and elimination of the oscillatory behavior. Keywords: Bifurcation analysis, Bioreactor, Limit cycles, Nonlinear system, Stability analysis. 1. INTRODUCTION The yeast Saccharomyces cerevisiae or bacterium Zymomonas mobilis are examples of microorganisms that are particularly important in the production of ethanol in continuous fermentation processes. However, these processes are known to be highly complex and nonlinear. Hence, it is not unusual to observe such nonlinear phenomena as multiplicity of steady states (Lei et al., 00), chaotic behavior (Graham et al., 007) or stable limit cycles (Chen et al., 1990; Sridhar, 011). The oscillatory behavior manifests itself as the self-sustained oscillations (SSO) of key process variables (e.g. biomass and product concentrations) for constant feed and culture conditions in some range of dilution rates (Beuse et al., 1999). On one hand, the oscillatory mode of operation can be beneficial, because it may lead to higher average biomass concentrations in comparison to the results obtained in steady-state regime (Bai et al., 009; Balakrishnan and ang, 00). On the other hand, to attenuate the oscillatory changes in the product concentration it is necessary to use a system of surge tanks, which makes the industrial installation more expensive (Bai et al., 009). Moreover, the oscillatory behavior may lead to large amounts of residual (unused) substrate (Bruce et al., 1991; Bai et al., 009). A change of process parameters (e.g. dilution rate D, ph or dissolved oxygen set-point) allows us to choose a desired mode of operation (Parulekar et al., 1986). It is also possible to design controllers that stabilize self-oscillating bioreactors by manipulating the inlet substrate concentration (Wu and Chang, 007). However, many of these parameters have to be kept constant they determine the optimal conditions for the continuous fermentation process. An option is to use a mixture of two substrates. In the literature, one can find examples of experiments that use additional substrates or secreted product to eliminate (or induce) the SSO in continuous fermentation process (Parulekar et al., 1986; Martegani et al., 1990). However, these experiment were carried out for finite portions of substrates or product and the case of continuous feeding of substrates was not taken into account. Interesting results have been presented in (Beuse et al., 1999), where the substrate (galactose) was replaced by glucose during the oscillatory mode of operation for a fixed dilution rate D. The oscillations did not disappear, but there was a change in their frequency and amplitude. The problem of using a mixture of substrates was also discussed in our previous paper (Skupin and Metzger, 013), where two substitutable substrates were continuously fed into the reactor chamber. The contribution of both substrates to the mixture was varied, but the total inlet substrate concentration was kept constant. The results were obtained from simulation and bifurcation analysis of the unstructured mathematical model with variable yield coefficient. However, the model was simplified and did not incorporate a product (ethanol) inhibition on cell growth. Moreover, it was assumed that both substrates were utilized simultaneously, while, in fermentation processes, a diauxic growth pattern is frequently observed as well (Egli, 1995). Hence, in contrast to the previous study, this paper describes the continuous flow bioreactor, which is fed with a mixture of two substitutable substrates including product inhibition effects on the growth of microorganisms. The relative contribution of both substrates will be treated as a new control variable. Moreover, it is assumed that the occurrence of the oscillatory behavior is related to a time delay in the response of cells to changes in the environment. First, basic properties of the unstructured fermentation model with the product inhibition for a single limiting substrate are presented. Then, assuming the diauxic growth pattern, a mixture of two substrates is used to induce (or eliminate) the oscillatory behavior. Copyright 015 IFAC 1115

2 IFAC ADCHEM 015 June 7-10, 015, Whistler, British Columbia, Canada. CONTINUOUS FERMENTATION PROCESS (SINGLE LIMITING SUBSTRATE) In unstructured mathematical models of the continuous fermentation processes, the entire population of microorganisms is described by a single variable biomass concentration. In order to derive an unstructured model of the system, following assumptions are used: the bioreactor is well-mixed and the culture volume is constant (Vconst), the growth of microorganisms is described by the Monod equation with an additional product inhibition term, the rate of product formation is described by the Luedeking-Piret equation (Luedeking and Piret, 1959), some fraction of the substrate is used to support basic life functions of microorganisms (Pirt, 1965). From mass balances on substrate, biomass and product, the system equations are as follows: ds d dp µ ( S,P) D( Sin S ) mx (1) D + µ ( S,P) () DP + µ ( S,P) p + m p (3) where: S in, S inlet and outlet substrates concentrations, respectively [g/l], biomass concentration [g/l], P product (ethanol) concentration [g/l], D dilution rate [1/h], biomass yield coefficient [g/g], p product yield coefficient [g/g], m x maintenance coefficient related to nongrowth-associated maintenance [1/h], m p constant coefficient related to non-growth-associated product formation [1/h]. Finally, µ(s,p) is a specific growth rate [1/h], which is described by the following relation: µ ms K i µ ( S, P) () S + K s P + Ki where: µ m maximum specific growth rate [1/h], K s halfsaturation constant [g/l], K i inhibitory constant [g/l]. In the next paragraphs of this section, the basic properties of the bioreactor system (1)-() will be given. A similar system was considered in (Ajbar, 001; Ajbar and Fakeeha, 00; Nelson et al., 009). In contrast to the analysis presented in (Nelson et al., 009), the bioreactor system (1)-() has a more general form including Luedeking-Piret equation and maintenance coefficient, but does not include death rate terms. Moreover, their model has not been investigated in terms of the existence of periodic solutions as shown in (Ajbar, 001; Ajbar and Fakeeha, 00). In contrast to the analysis presented in (Ajbar, 001; Ajbar and Fakeeha, 00), the influence of time delay effects on the existence of periodic solution is presented. Additionally, we do not introduce dimensionless variables in order to have a physical interpretation of the results..1 Equilibrium points Our considerations will be limited to the set Ω{(S,,P): S 0, 0, P 0}, because only non-negative concentrations have physical meaning. By setting the derivatives in (1)-(3) equal to zero, two equilibrium points corresponding to washout and no washout states are obtained. The washout equilibrium is: S S 1 in, 1 0, P 1 0 (5) The second equilibrium point (no washout state) can be found from the following relations: D A ( Sin + A S ) µ ms K i D + mxx/s, A > 0 S + K s x/s Dp + mp ( Sin S ) ( D + mx ) ( Dp + m p ) D (6) DP Moreover, S >0 for D>0. The Jacobian matrix for the system (1)-() at the washout equilibrium (5) is: J ( S1, 1,P1 ) D 0 0 µ µ ( S1,P1 ) m µ ( S1,P1 ) D 0 ( S,P ) + m D 1 1 p p x 0 and the corresponding characteristic equation: ( + D) ( λ + D µ ( S 1,P 1 )) 0 λ (8) Hence, the condition for the local stability of the washout equilibrium (5) is: D c <D, Dc µ ( S,P1 ) µ S S + K in s (7) m in 1 (9) where: D c is a critical dilution rate and if the condition (9) holds then all the biomass is washed out of the reactor. For 0<D<D c the equilibrium point (5) is a saddle point (all eigenvalues are real, both of them are negative and one is positive) (Hirsch et al., 00). Indeed, for any initial condition with (0)0 [g/l] (no microorganisms) each trajectory of the system (1)-() tends to the washout equilibrium point. In turn, the Jacobian matrix for the no washout equilibrium (6) is: J D p p D ( S,,P ) 0 (10) D m D + m where the derivatives with asterisks are calculated for S,,P, and the corresponding characteristic equation is: ( ) 3 λ + a λ + a λ + a 0 (11) 1 where: a a p 1 + D 3 D p x p + D p D + m m a3 D D p + Dmx Dm p x p Copyright 015 IFAC 1116

3 IFAC ADCHEM 015 June 7-10, 015, Whistler, British Columbia, Canada From the Routh-Hurwitz criterion, the conditions for the local stability of (6) (no washout state) are: a 1 >0, a 3 >0 and a 1 a >a 3 (1) Since, />0 and /<0 for S>0 and P 0, the first two inequalities in (1) are satisfied for >0 and 0<D<D c. By performing some symbolic calculations, it can be shown that the third inequality in (1) is satisfied for D>0.. The existence of periodic solutions One of the conditions for the occurrence of the Hopf bifurcation in the system (1)-() is that the Jacobian matrix (10) has a pair of pure imaginary eigenvalues and the remaining eigenvalue is negative (Hassard et al., 1981). In the presented case, this is possible if the characteristic equation (11) has a form: (λ +ω )(λ+a 0 )0 (13) where: ω, a 0 >0. This corresponds to the following condition on a 1, a and a 3 in (11): a 1 >0, a 3 >0 and a 1 a a 3 (1) Since, a 1 >0 and a 3 >0 for 0<D<D c, the last equality in (1) cannot be fulfilled for D>0. Hence, the system (1)-() does not exhibit the oscillatory behavior. The same conclusion was drawn in (Ajbar and Fakeeha, 00) for a more general form of the specific growth rate µ(s,p), which was monotonic with respect to the product concentration. From the biological point of view, it is believed that the oscillatory behavior in continuous fermentation processes is related to a delayed response of the microorganisms to changes in their environment as shown, for instance, in the recent work by Bai and Zhao (01). As has been suggested in (Jobses et al., 1986; Bai and Zhao, 01), the secreted ethanol inhibits the formation of an intracellular metabolite and the metabolite influences the specific growth rate. In other words, the inhibitory effect of the ethanol on the specific growth rate is delayed (indirect inhibition). This delay can be modeled as a pure time delay in the specific growth rate () for the product concentration. The other possibility is to approximate the time delay effect by defining n auxiliary variables (Mocek et al., 005). These variables are linked by a set of n linear differential equations and can be interpreted as a signal transduction in yeast cells (Roland et al., 00; Mocek et al., 005). Hence, for sufficiently large n, the time delay effect can be described as follows: C( s) P( s) e n T s 0 1 n (15) st 0 + where: P(s), C(s) Laplace transforms of the ethanol concentration in the reactor medium and the auxiliary variable representing time delay effect, respectively, T 0 time delay, n N, s C. In further investigations, it assumed that n, since the th order element provides a satisfactory approximation. Hence, the description of the bioreactor system (1)-() with a delayed response to changes in ethanol concentration (15) is as follows: ds d dp µ ( S, C) D( Sin S ) mx (16) D + µ ( S, C) (17) DP + µ ( S, C) p + m p (18) d C ( 1) ( ) ( 3) f ( C,C,C,C,T0,P) (19) µ ms K i µ ( S, C) (0) S + K s C + K i where f is a function of C and its time derivatives C (i) (i1,,3) corresponding to the linear system (15) for n. Treating the dilution rate D as a bifurcation parameter, it can be shown that for sufficiently large time delay T 0 there exist a stable limit cycle for some range of dilution rates. This will be presented in the results section, when the bioreactor is fed with a single limiting substrate. 3. CONTINUOUS FERMENTATION PROCESS (MITURE OF SUBSTRATES) Since, the bioreactor can be fed with a mixture of two substrates, the system equations (16)-(0) have to be extended. It is assumed that the oscillatory behavior is observed for each single limiting substrate. This is consistent with experimental data obtained in (Beuse et al., 1999) in continuous culture of S. cerevisiae for two single limiting substrates: glucose and galactose. Moreover, it is assumed that the substrate of inlet concentration S in is fed into the reactor chamber at the expense of the substrate of inlet concentration S in1. The contribution of the individual substrates to the mixture will be represented by the parameter r [0,1] and treated as a new control variable. In this case the mathematical model can be derived from mass balances on both substrates, biomass and product: ds1 µ 1( S1,S,C) D( Sin1( 1 r) S1 ) mx1 (1) 1 ds µ ( S1,S,C) D( Sinr S ) mx () d D + µ ( µ 1,µ ) (3) dp DP + ( µ 1 ( S1,S,C) p 1 + µ ( S1,S,C) p ) + m p () d C ( 1) ( ) ( 3) f ( C,C,C,C,T0,P) (5) where: S 1, S are outlet concentrations of substrates [g/l], S in1, S in are inlet concentrations of substrates [g/l], biomass concentration [g/l], P product concentration [g/l], µ i (S 1,S,C) i-th specific growth rate [1/h] (i1,), i i-th biomass yield coefficient [g/g] (i1,), pi i-th product yield coefficient [g/g] (i1,), µ the overall specific growth rate [1/h] (can be dependent on the individual specific growth rate µ i ). For simplicity reasons, the system includes only one approximating equation (5) to describe time delay effects in each individual specific growth rate µ i (i1,). Copyright 015 IFAC 1117

4 IFAC ADCHEM 015 June 7-10, 015, Whistler, British Columbia, Canada The choice of an appropriate form of the overall specific growth rate µ is based on the assumption that the growth of biomass is possible for each substrate fed separately (the substrates are substitutable). As mentioned in the introduction section, the biomass growth in the presence of two (or more) limiting substrates is more complex, because the substrates can be utilized either simultaneously or sequentially (according to the diauxic growth pattern) (Egli, 1995; Narang et al., 1997). In the continuous flow bioreactor, the diauxic growth pattern is observed when there is a simultaneous utilization of both limiting substrates for small dilution rates (D D trans ) and for D>D trans the only utilized substrate is a substrate on which microorganisms can grow faster (Egli, 1995). The parameter D trans is called a transition dilution rate. The substrate supporting the faster growth of microorganisms will be referred to as the more preferred one, or in other words, more preferred by microorganisms and the latter one the less preferred substrate. In the batch mode (D0), the substrates are utilized sequentially, starting from the substrate that supports the fastest growth. In order to model the diauxic growth pattern, the overall specific growth rate is described by the generalized Monod equation (oon et al., 1977) with additional product inhibition terms: ( µ 1,µ ) µ 1( S1,S,C) + µ ( S1,S,C) (6) µ µ m1s1 K i1 µ 1( S1,S,C) S1 + K s1 + as C + K (7) i1 µ ms K i µ ( S1,S,C) S + K s + a1s1 C + K (8) i where: a 1 K s /K s1, a K s1 /K s are dimensionless coefficients representing the inhibition effects of the individual substrates (they allow to model the diauxic growth pattern), K ij (j1,) product inhibition coefficients [g/l]. operation (D0) (Figure 1). In this case, the equation (5) was omitted and CP. The experimental results (Figure 1) reveal a pure diauxic growth pattern, where the preferred substrate is glucose. In turn, the time delay (T 0 0[h]) was chosen to obtain the oscillatory behavior for each single limiting substrates (r0 and r1) for some range of dilution rates as observed, for instance, in (Beuse et al.,1999). All the parameter values have been listed in Table 1. Table 1. Parameter values µ m1 K s1 1 p1 K i1 m x1 m p µ m K s p K i m x T Further investigations of the bioreactor system (1)-(8) are based on numerical simulations of the model equations in MATLAB including detection of the SSO according to the algorithm described in (Skupin, 010) and on numerical bifurcation analysis in PPAut software (Ermentrout, 00).. RESULTS It is assumed that both inlet substrates concentrations are equal and S in1 S in 35[g/L]. The dilution rate D is a bifurcation parameter and Figures and 3 presents the steady state diagrams of the product concentrations, when the bioreactor is fed only with the more (r0) or with the less preferred substrate (r1). Fig.. Product (ethanol) concentration versus dilution rate D for the more preferred substrate (r0) Fig. 1. Comparison of the model prediction (continuous lines) with experimental data for: glucose (S 1 ), galactose (S ), biomass (), ethanol (P). The initial concentrations were: S 1 (0)S (0)10[g/L], (0)0.06[g/L], P(0)0.[g/L]. Knowing that the oscillatory behavior was observed in continuous culture of S. cerevisiae for two single limiting substrates: glucose and galactose (Beuse et al.,1999), the parameter values for the bioreactor system (1)-(8) were obtained by fitting the model using least squares method. The experimental data taken from (Huisjes et al., 01) for the mixture of glucose and galactose in the batch mode of The thin continuous lines are stable branches and the broken lines are unstable ones. The white circles represent a point for which DD c and for D>D c the bioreactor content is washed of the reactor. As can be clearly seen, for some range of dilution rates there exist a stable limit cycle that occurs between the two Hopf points (represented by the black filled in squares). In this range, the grey lines are the maximum and minimum ethanol concentrations and the black dots are the average ethanol concentrations obtained from numerical simulations of the system (1)-(8). The next two figures (Figure and 5) show the steady state diagrams, when the bioreactor is fed with a mixture of substrates for two chosen ratios, with predominance of the more (r0.) and the less (r0.8) preferred substrate. The notations and symbols are Copyright 015 IFAC 1118

5 IFAC ADCHEM 015 June 7-10, 015, Whistler, British Columbia, Canada the same as in Figures and 3. The obtained results show that an appropriate value of the new control variable r (the relative contribution of both substrates) allows for induction or elimination of the oscillatory behavior. Dc S in1 Sin1( 1 r) µ m1 rsinµ m + ( 1 r) + K s1 + asinr rsin + K s + a1s in1( 1 r) It is easy to notice that in the predominance of the less preferred substrate the oscillatory region is much smaller. Most probably, this results from the different preference of substrates. Fig. 3. Product (ethanol) concentration versus dilution rate D for the less preferred substrate (r1) Fig. 6. The regions of the SSO (stable limit cycle), steady states (no washout state) and the washout state on the parameter plane D-r The analysis of the average concentrations of residual substrates versus dilution rate D for r (0,1) shows that the both substrates are utilized simultaneously for small dilution rates (for D<D trans ), while for higher dilution rates, the growth takes place on the more preferred substrate (Figure 6). 5. CONCLUDING REMARKS Fig.. Steady state diagram for the mixture of two substrates (predominance of the more preferred substrate, r0.) Fig. 5. Steady state diagram for the mixture of two substrates (predominance of the less preferred substrate, r0.8) This is well seen on the parameter plane D-r (Figure 6) indicating three qualitatively different regions: no washout state (white), washout state (dark grey) and oscillatory (light grey). For high dilution rates (D>D c ) the washout occurs and the product concentration drops to zero. By setting the derivatives in (1)-(5) equal to zero, it is possible to find the equilibrium point corresponding to the washout state. From the local stability analysis of this point, it can be shown that the critical dilution rate is: The presented analysis was carried out with the use of the unstructured mathematical model of the continuous fermentation process with a product inhibition. First, for the single limiting substrate, conditions on the dilution rate D for the local stability of the bioreactor system were derived. It was shown that the delayed response of the microorganisms to changes in the ethanol concentration in the bioreactor medium is a crucial factor in the occurrence of the SSO. Then, the mathematical model was extended to the case of a mixture of two substitutable substrates assuming that the substrates were utilized sequentially (the diauxic growth pattern). Based on the numerical bifurcation analysis, it was shown that the introduction of the new control variable r allowed for a qualitative change of system behavior (induction or elimination of the oscillations). Moreover, the proposed control variable r is well-suited for supervisory control systems where process operators decide on the value of r, which can be changed in a piecewise constant manner. ACKNOWLEDGEMENTS This work was supported by the National Science Centre under grant No. 01/05/B/ST7/00096 and by the Ministry of Science and Higher Education under grant BKM-UiUA. REFERENCES Ajbar, A. (001). Periodic Behaviour of a Class of Unstructured Kinetic Models for Continuous Copyright 015 IFAC 1119

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