298

Biotechnol. Prog. 1992, 8, 298-306

A Model for @-GalactosidaseProduction with a Recombinant Yeast Saccharomyces cerevisiae in Fed-Batch Culture Linawati Hardjito, Paul F. Greenfield,*and Peter L. Lee Chemical Engineering Department, University of Queensland, St. Lucia, Q 4072, Australia

An unstructured model is developed to describe the growth and product formation behavior of a recombinant yeast Saccharomyces cereuisiae using P-galactosidase as a model protein. T h e model shows good agreement with the experimental data over a range of conditions. It also accurately predicts the effect of growth rate on yield coefficient and gene expression. The simplicity and accuracy of the model make it suitable for designing and implementing control and optimization strategies for the production of recombinant proteins a t large scale.

Introduction Simple but accurate models describing the behavior of recombinant microorganisms are required for the control and optimization of recombinant protein production. Yeast is one possible host for producing recombinant proteins since it provides several advantages over the more commonly used Escherichia coEi (Kingsmanet al., 1988).Yeast has a well-developed and well-characterized genetic system and is not pathogenic. I t is able to glycosylate and secrete recombinant proteins and can be readily grown a t large scale. A number of mathematical models have been developed to describe the behavior of the yeast Saccharomyces cereuisiae (Barford, 1981;Sonnleitner and Kappeli, 1986;Bijkerk and Hall, 1987; Briton and Wheals, 1987; Cazzador and Mariani, 1988; Szewczyk, 1989; Alexander and Jeffries, 1990; Barford, 1990) and recombinant variants (Hjortso and Bailey, 1984; Srienc et al., 1986; Sardonini and DiBiasio, 1987; Schwartz et al., 1988; Wittrup and Bailey, 1988; Wittrup et al., 1990). Only a few of these studies included the dynamics of product formation (Coppella and Dhurjati, 1990). Furthermore, this was done via structured models, which are suited to the testing of specific metabolic hypotheses and which have been instrumental in indicating key features of metabolism which should be addressed in any model. At their current stage of development, however, structured models still pose a number of practical difficulties (e.g., parameter estimation) when they are applied to the design and implementation of large-scale control and optimization strategies for recombinant proteins. In this work, a mathematical model describing the behavior of a recombinant yeast harboring a constitutive promoter (CYC1 promoter) for the production of the recombinant protein P-galactosidase is developed by utilizing this previous knowledge. Because the purpose of the model was to design control and optimization strategies, emphasis was placed on developing an unstructured model with sufficient complexity to describe changes in the variables characterizing the external chemical environment. Simulation results and experimental data for a number of fed-batch cultures are presented in order to evaluate the accuracy of the model. Model Development Prior to the introduction of the proposed model, a brief summary of unstructed models describing the growth behavior of both nonrecombinant yeasts and other re-

combinant microorganisms, as well as the dynamics of product formation, is presented. Modeling of Aerobic Growth of S. cerevisiae and Related Yeasts. Toda et al. (1980) developed a model for the aerobic growth of Saccharomyces carlsbergensis in glucose-limited continuous culture. The model assumes dual effects where glucose regulates the flux of both glucose catabolism (respiration and aerobic fermentation) and ethanol utilization in yeast cells. Sonnleitner and Kappeli (1986) developed a model for the growth of the yeast S. cereuisiae on glucose-limited continuous culture. It is based on the fact that glucose degradation proceeds via two pathways (oxidative and fermentative) under conditions of aerobic ethanol formation. Maximum rates of oxidative glucose and ethanol degradation are governed by the respiratory capacity of the cells. A summary of these models is presented in Table I. Modeling of Recombinant Cultures. Ollis and Chang (1982) and Satyagal and Agrawal (1989b) developed mathematical models to describe the behavior of plasmidharboring cells and associated product formation. Both of the models are based on the framework reported by Imanaka and Aiba (1981). The former assumed that product could be growth- or non-growth-associated, while the latter includes plasmid concentration as a dynamic state variable which affects the rate of product formation. Summaries of both models are presented in Table 11. Modeling of the Recombinant Yeast S. cerevisiae. Development of the model is based on the following concepts and assumptions, which reflect the earlier work described above: (1) Cell and culture medium form an unstructured system. (2) Glucose uptake follows Monod kinetics (Sonnleitner and Kappeli, 1986). (3) Glucose transported into the cell is catabolized by both fermentative and oxidative pathways. The fraction of glucose which is fermented depends on the respiratory capacity of the cells, the available oxygen, and the glucose concentration in the culture medium. (4) Aerobic fermentation of the recombinant yeast is indicated by ethanol formation. Since the respiratory capacity of the recombinant yeast is much lower than that of nonrecombinant strains (Gopal et al., 19891, resulting in ethanol production at low concentrations of glucose, it is assumed that only glucose concentration governs ethanol formation. The rate of ethanol formation is therefore

8756-7938/92/3008-0298$03.00/00 1992 American Chemical Society and American Institute of Chemical Engineers

Bbtwhnol. Rog., 1992, Vol. 8, No. 4

299

Table I. Summary of Kinetic Models for Aerobic Growth of S. carlsbergensis and S. cerevisiae Proposed by Toda et al. (1980) and Sonnleitner and Kameli (1986) Toda et al. (1980)

Table 11. Summary of Genetic Models Developed by Ollis and Chang (1982) and Satyagal and Agrawal (198913)

Ollis and Chang (1982)

x+ = p+X+(l-8 ) x- = ( p - X - ) + (P+X+j3)

I(s) =

+

(1 26.81S1.313) (1+ 536.2S'.313)

P

b'X+

+ b"p+X+(l-

j3)

Satvaeal and Agrawal(1989b)

x+ = p+x+(1- 8) x- = &+X+ + p-x-

Sonnleitner and Kappeli (1986)" /J

E

p1-p

fig

+ p e = (P"P + pfem) + p e

= y;;Pqre*p;

pfenn

= yt"" I / ~Qfenn

In (2 -e) j3 = 1--; ln2

e =~I-N

" If q Iq d a , where a is oxygen required to oxidize 1mol of glucose, = qoda and then Qresp = ~ 0 2 and 1 ~ qferm 0.0. If q > qozla, then qrMIP Qfenn

Q - qreap-

assumed proportional to the specific growth rate (Agrawal et al., 1989;Alexander and Jeffries, 1990) and is expressed by Monod kinetics. (5) Ethanol accumulated in the medium can be utilized by the cells. Ethanol utilization is inhibited by glucose (Copella and Dhurjati, 1989) while its uptake rate follows modified Monod kinetics as reported by Sonnleitner and Kappeli (1986). (6) Ethanol causes noncompetitive inhibition of growth (Aiba et al., 1968). (7) Growth on glucose and ethanol is quantitatively additive (Todaet al., 1980;Sonnleitner and Kappeli, 1986). (8) 8-Gal production rate follows Monod kinetics described by

This is a modification of that proposed by Ollis and Chang (1982) for growth-associated product formation where it was assumed to be proportional to specific growth rate as follows:

(9) Cellular yield on ethanol is constant and is assumed equal to 0.50 g of cell/g of ethanol. This is close to the constant yields based on ethanol proposed by Toda et al.

(1980) and Copella and Dhurjati (1989) which were equal to 0.56 and 0.53 g of cell/g of ethanol, respectively. The following equations result from a fed-batch system of variable volume (V), material balances on cells ( X ) , substrate (S), ethanol (E),and recombinant protein (P). Empirical models of growth, substrate uptake, ethanol, and recombinant protein production, as described above, are incorporated.

= (P,+ P,)X

F -+

F - S) s = -qx + v(Si

P'px-? F

V=F

(3) (4)

(6) (7)

Biotechnol. Prog., 1992, VOI. a, NO. 4

300 SmaxS

4=4s +

s

amax'

cy=-

cy,

+s

P m a xS

P=-

P,+S

Compared to models already published, the model developed here predicts variable cell yield implicitly instead of using separate but constant yield coefficients (q p and q,ip)It . excludes the oxygen effect on ethanol formation since ethanol is formed at very low glucose concentrations due to the respiratory capacity of the recombinant yeast being much lower than that of nonrecombinant strains (Gopal et al., 1989). In addition, the model assumes that the cells form a nonsegregated system because the fraction of plasmid-containing cells remained constant over the fermentation cycle. To cover the possibility that the plasmid copy number is unstable, the proposed model is extended to include plasmid loss. Ethanol inhibition of its own uptake is neglected because of the relatively low ethanol concentrations reached in the batch or fed-batch system (Toda et al., 1980; Sonnleitner and Kappeli, 1986; Copella and Dhurjati, 1989).

Parameter Estimation Kinetic parameters were determined using a nonlinear regression technique developed by Powell (Kuester and Mize, 1973) in which the sum of the square error (SSE) is minimized, where SSE is defined as follows:

where Yij is the experimental value of the predicted variable Y . ji'% is the predicted value which is determined by solving eqs 3-12 using the Runge-Kutta Gill method with given initial conditions X ( t = 0) = X,;S(t = 0) = So; E(t = 0) = E,; P(t = 0) = Po; V ( t = 0) = V,. Y.j is the average experimental value for the j t h variable. The parameter n represents the number of samples while 1 refers to the number of variables used to determine a particular parameter. The algorithm which was applied proceeds as follows. (1)Determination of pg,max, Ks,g,qmax,a n d qs. These parameters were determined from the experimental data for cell (X) and substrate (S) concentrations and volume (V) using eqs 3,4, 7 , 8 , and 10 by initially assuming pe = 0 for each fed-batch culture. For this analysis, eq 8 becomes

K , , was estimated first using a predetermined value of I L ~which , ~ was ~ ~assumed equal to 0.45 h-' (Alexander and Jeffries, 1990),qmmwas assumed to be 3.0 g g-' h-' (Sonnleitner and Kappeli, 1986), and qa was assumed equal to 0.1 g L-' (Sonnleitnert and Kappeli, 1986). After K,,gwas obtained from each of the fed-batch cultures, the average value of this parameter was taken to determine the next parameter which was pg,max (using the predetermined values for the parameters qmax and 4 , and the newly determined value for Following this, qmax was determined using the predetermined value for qsand the

newly determined values for K , , and pg,". Lastly, the newly determined parameters for Ks,g,pg,", and qmaxwere used to determine 9,. k,and Ki(2) Determination of a m a x , as, ,x+ a,/ These parameters were determined from the experimental data for cell (X), substrate ( S ) ,and ethanol ( E ) concentrations and volume (V) using eqs 3, 4, 5, 7, 8, 9, 10, and 11. The parameters cy, and Ks,ewere determined simultaneously by assuming amax= 0.7 h-l, Ne," = 0.11 h-' (Toda et al., 1980), Ki = 0.1 g L-' (Sonnleitnert and Kappeli, 1986), and k = 0.03 L g-'. Next, pe," and Ki were determined simultaneously. Finally, cy, and k were determined simultaneously. (3) Determination of pmax a n d pa. Experimental data for cell (X), substrate (S),ethanol ( E ) , and 0-gal ( P ) concentrations and volume (V) were used to determine these parameters using eqs 3-12. Both parameters were determined simultaneously. After all parameters were obtained, the estimation of each parameter was repeated one by one until the value of SSE defined by eq 13 was minimized. Materials and Methods Plasmid and Organism. The plasmid pLG669-z (Guarante and Ptashne, 1981) and the yeast S. cerevisiae strain YN124(cir+alpha ura 3-52 tryp 1-289 gal 2) were used in this experiment. The plasmid contains markers for ampicillin resistance in E. coli and for uracil auxotrophy in yeast and a section of the E. coli 0-galactosidase gene, under control of the yeast CYCl promoter from S. cerevisiae. Culture Media. A nonselective, yeast extract peptone glucose medium (YPG), as reported by Barnett et al. (1983), was used in this experiment. It was modified by the addition of 15 mM citrate-phosphate buffer giving a pH of 5.2. An inoculum was grown in a selective medium to obtain a high initial fraction of plasmid-containing cells. The selective medium was a yeast nitrogen base (Difco) with the addition of tryptophan to a final concentration of 20 mg L-I. During growth of the microorganism, the temperature was maintained at 30 "C, while pH was maintained within the range of 4.0-5.2. Cell Concentration. Cell concentration was followed by measuring the optical density (OD) and dry weight of the cell suspension. The OD was measured with a Hitachi spectrophotometer Model 100-60 at 600 nm. The dry weight cell concentration was determined by filtering 10 mL of a sample through 0.2-pm filter paper, which was washed with distilled water and then dried a t 100 "C overnight. The correlation between dry weight cell concentration and OD6m nm was found to be

Xm(dry weight cell, g L-l) = -0.16 + 0.170D6,

nm

(14)

Glucose Measurement. Glucose concentrations were measured by an HPLC method (Bio-Rad Aminex HPX87H column). Samples were prepared by filtration through a 0.2-pm filter before analysis. The solvent was 0.008 N sulfuric acid, at a flow rate of 0.6 mL min-l, while detection was effected with a differential refractometer. Temperature was maintained at 65 "C. Plasmid Stability a n d @-GalactosidaseAssays. The plasmid stability was analyzed by a plating technique. Cells were serially diluted and spread onto X-gal plates. The plates were incubated at 30 "C, and the fraction of plasmid-containing cells was obtained by counting blue (plasmid-containing) and white (plasmid-free) colonies. In this context, the plasmid-free colonies refer to colonies that were grown on X-gal plates and produced no color.

Biotechnd. Rog., 1992, Vol. 8, No. 4

a01

Table 111. Kinetic Parameters of the ProDosed Model. value value 0.45 0.08 2.80 f 0.20 0.13 f 0.06 0.028 f 0.006 0.05 f 0.04 0.89 f 0.22 0.10 =k 0.05 0.18 f 0.08 0.61 f 0.07 0.46 f 0.12 0.005 f 0.002 0.02 f 0.01 ~~

~~

*

95% confidence intervals are also shown. 0.8 0.8,

_J

0.4

-iLL!l Y

0.2

0

0 2

-5

-6

0

6

10

16

20

0

6

10

16

Cl

3

8

s

20 0.0

m e ( h )

b e ( h )

Figure 2. Comparison between model simulation and experimental data for fed-batch culture with Si = 97.0 g L-l, So = 1.0 g L-l, feed rate = 12.7 mL h-l for the first 4 h after which it was increased to 17.5 mL h-l. Symbols: -, simulated data; 0, experimental data (95% confidence interval). Time zero corresponds to the start of the fed-batch culture. 2,

--6

0

6 10 w e ( h )

15

20

-5

0

6 10 m e ( h )

15

20-

Figure 1. Comparison between model simulation and experimental data for fed-batch culture with Si = 20 g L-l, So = 1.0 g L-l, and feed rate = 12.7 mL h-l for the first 3 h after which it was increased to 17.5 mL h-l. Symbols: -, simulated data; 0, experimental data (95% confidence interval). Time zero corresponds to the start of the fed-batch culture.

A liquid assay as reported by Miller (1972) using a modification of the method proposed by Guarante and Ptashne (1981) was used to measure @-galactosidase expression. An aliquot of cell culture was diluted in Zbuffer (1 L contains 16.1 g of NazHPOp7H20, 5.5 g of NaH~POpH20,0.75g of KCl, 0.246 g of MgSOp7H20, and 2.7 mL of 2-mercaptoethanol) to a final volume of 1.0 mL and the cells were lysed by addition of 50 pL of chloroform and 20 pL of 0.1% SDS. After 5 min of preincubation, 2-nitrophenyl8-galactosidase(ONPG) was added and the enzyme was assayed a t 28 OC. One 8-galactosidase unit is defined as the amount of the enzyme which hydrolyzes 1 nmol of ONPG/min a t 28 OC (Miller, 1972). Fed-Batch Culture. Fed-batch culture experiments were carried out using an LH 500 series 1-L vessel (LH Fermentation, Stoge Poges, U.K) with 750 mL of working volume. The initial volume was 500 mL, the initial glucose concentration (when feeding was started) varied from 0.30 to 0.80 g L-l, the volumetric feed rate varied from 13 to 60 mL h-1, and the feed glucose concentration varied from 20 to 150 g L-I. A peristaltic pump (LKB 1601 Ultraferm module) was used to feed the medium which was stirred a t 750 rpm. Air was supplied a t 3.0-3.5wm. The dissolved oxygen tension (DOT) was maintained above 50% of the saturated air value. Feeding of the medium was started after 8 h of incubation.

Results and Discussion The kinetic parameters of the proposed model were determined from data from 8 fed-batch experiments (after

a

8

I I i

0.1

I

*.

I

100.w1

(IL-') U comcmhtion

Rdlctd w

0.i

0.01

.,

I'

-3

P

10

(IL-1) concentrntlon

Redlcbd

1

pndicbd ethanol concmhtion

1

a3

10

1M)

( lo00 uulb d - 1 ) pndiotrd P-,d conwnhtion

Figure 3. Plot of residual versus predicted variables for eight fed-batch cultures. Symbols: 0, FB-1; V,FB-2; 0 ,FB-3; A,FB4; 0 , FB-5; V, FB-6; FB-7; A, FB-8. UTL is upper tolerance limit, and LTL is lower tolerance limit. The limits show that with 95% confidence, the interval will contain 95% of the residuals.

feeding was started). Results of the fitted kinetic parameters are tabulated in Table 111. To test the accuracy of the proposed model, plots comparing predicted and experimental data with the 95 9% confidence intervals, a statistical Z-test, and an F-testwere generated. A Z-test was performed to test the randomness of the residuals (n > 301, while an F-test was performed to test the variance differences which resulted from modeling and experimental errors, respectively.

Biotechnol. prog., 1992, VOI.

302

Table V. Lack of Fit Analysis for the Proposed Modela

Table IV. Z-Test Result (a’= 0.05) variable cell glucose ethanol P-gal

Zcomuuted

0.69 4.93 1.72 1.37

Z”12

1.96 1.96 1.96 1.96

Ho accepted accepted accepted accepted

Typical results comparing experimental and predicted data are shown in Figures 1and 2. A Z-test was performed on the hypothesis (Ho) that the average of the residuals (Ao) = 0 against the alternative hypothesis (HI) that the average of the residuals (Ao) # 0, for each residual variable independently. The decision criterion was to reject Ho if (15) A plot of the residual values versus the predicted variables can be seen in Figure 3, while the results of the Z-test are shown in Table IV. An F-test was performed on the hypothesis (Ho) that the variance resulting from the modeling error (u,*) is equal to the variance resulting from experimental error ( ue2) against the alternative hypothesis (HI) that am2is greater than ue2. The decision criterion was to reject Ho if

where sm2and se2 are the sample variances from the modeling error and experimental error, respectively; CY’ is type I error probability; u, and ue are the degrees of freedom from the model and experiment, respectively. The variances sm2and se2can be estimated as follows: I

n ‘ z e , - yj)’ j=l

u, =

1’- np

(17)

’m

n’ is the number of replications which was equal to 3 (triplicate samples were taken a t each time); I’ is the number of samples from the whole set of runs (FB-1 to FB-8) which was equal to 37. The results of the F-test are shown in Table V. Figure 3 shows that only a few of the residuals fall outside the tolerance limits. From Figures 1and 2 and the other results (not shown), it is clear that the proposed model predicts well the growth of the recombinant yeast S. cereuisiae and the associated P-gal production. Even though some of predicted points do not fall within the 95% confidence intervals of the experimental data, the Z-test shows that the residuals of the predicted variables are randomly distributed. Table V shows that the modeling errors associated with the cell, glucose, ethanol, and fl-gal concentration variables are about 4, 4, 1.6, and 2.4 times the experimental error, respectively (shown by comparing the SSE values). The greater variances of the modeling error are due in part to the range of conditions of the fed-batch cultures used to validate the model. The glucose feed concentration (Si)and volumetric feed rate (F)varied from 20 to 150 g L-l and 13 to 60 mL h-l, respectively. From the above analysis, it can thus be concluded that the

(af

= 0.05)

source cell LOF error g1ucose LOF error ethanol LOF error P-gal

LOF

error

Sm2 =

a, NO. 4

SSE

MSE

Fcomputed

H O

34.35 8.83

25 74

0.12 1.37

11.4

rejected

1.03 0.35

25 74

0.04 0.005

8.0

rejected

19.96 12.41

25 74

0.80 0.17

4.7

rejected

115.95 47.64

25 74

4.64 0.64

7.25

rejected

(om, v,) = 1.7. Abbreviations: LOF,lack of fit; Fcritical or SSE, sum of squared error; MSE, mean squared error; u, degrees of freedom.

proposed model describes the behavior of the system in a statistically acceptable manner. The model also predicts a variable yield in a simpler manner compared to others (Toda et al., 1980;Pyun et al., 1989; Agrawal et al., 1989). The growth yield on glucose (YXls) can be derived from the ratio of specific growth rate (pg)to the specific glucose uptake rate ( q ) , while the ethanol yield (Ye/s) can be derived from the ratio of specific ethanol production rate (a)to the specific glucose uptake rate ( 4 ) . A plot of YXisand Y+ versus growth rate is presented in Figure 4a. A related plot of Yxisand Ye,, versus glucose concentration in the medium is presented in Figure 4b. Panels a and b of Figure 4 show that the yield coefficient varies with the specific growth rate or glucose concentration in the medium. A yield of about 0.15 g of cell/g of glucose is reached when the specific growth rate increases to about 0.40 h-l. This agrees with the experimental results of Von Meyenberg (1969), Aiba et al. (19761, and Copella and Dhurjati (19891, where the yield is between 0.14 and 0.15 g g-’ when the specific growth rate (p)is greater than 0.40 h-l. The critical specific growth rate (Rieger et al., 1983) resulting from the proposed model is between 0.10 and 0.14 h-l, which results in a cell yield of between 0.40 and 0.45 g g-l. This critical specific growth rate is lower than the critical specific growth rate of the host cell (Von Meyenberg, 1969;Aibaet al., 1976;CopellaandDhurjati, 1989). However, it is close to the critical dilution rate reported by Gopal et al. (1989) and Da Silva and Bailey (1991). Gopal et al. (1989) examined heterologous protein expression in S. cereuisiae harboring plasmid pMA27 and pMA91-66 and showed that the critical dilution rate occurred between 0.12 and 0.15 h-l. Da Silva and Bailey (1991) reported that, in a continuous culture of S. cereuisiae harboring pLGSD5, cell yields of 0.50, 0.31, and 0.20 (g g-l) correlated with dilution rates of 0.10,0.20, and 0.26 h-l, respectively. Figure 4b shows that ethanol production is associated with glucose concentrations of about 20 mg L-l. Copella and Dhurjati (1989)reported that ethanol production takes place when the concentration of glucose is greater than 50 mg L-l. The earlier occurrence of the critical specific growth rate for a recombinant yeast compared to that for nonrecombinant strains suggests that part of the respiratory system may also be affected as a consequence of protein overexpression, leading to a lower overall respiratory capacity compared to the host strain (Gopal et al., 1989). This is the likely consequence of competition

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303

0.6 a , . I

it 5

1\

A model for beta-galactosidase production with a recombinant yeast Saccharomyces cerevisiae in fed-batch culture.

An unstructured model is developed to describe the growth and product formation behavior of a recombinant yeast Saccharomyces cerevisiae using beta-ga...
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