Macrokinetic model for methylotrophic Pichia pastoris based on stoichiometric balance

Journal of Biotechnology 106 (2003) 53–68

Macrokinetic model for methylotrophic Pichia pastoris based on stoichiometric balance H.T. Ren a , J.Q. Yuan a,b,∗ , K.-H. Bellgardt c a

Department of Automation, Shanghai Jiao Tong University, 1954 Huashan Lu, 200030 Shanghai, PR China b State Key Laboratory of Bioreactor Engineering/ECUST, 200237 Shanghai, PR China c Institut fuer Technische Chemie, University of Hannover, Callinstr. 3, 30167 Hannover, Germany Received 10 February 2003; received in revised form 24 July 2003; accepted 1 August 2003

Abstract A macrokinetic model for Pichia pastoris expressing recombinant human serum albumin is proposed. The model describes the balances of some key metabolites, ATP and NADH, during glycerol and methanol metabolism. In the glycerol growth phase, the metabolic pathways mainly include phosphorylation, glycolysis, tricarboxylic acid cycle, and respiratory chain. In the methanol growth phase, methanol is oxidized to formaldehyde at first. Then, while a part of formaldehyde is oxidized to formate, the rest is condensed with xylulose-5-monophosphate to form glyceraldehyde-3-phosphate, and further assimilated to form cell constituents. The metabolic pathways following glyceraldehyde-3-phosphate were assumed to be similar to those in the glycerol growth phase. Based on the model, the macrokinetic bioreaction rates such as the specific substrate consumption rate, the specific growth rate, the specific acetyl–CoA formation rate as well as the specific oxygen uptake rate are obtained. The specific substrate consumption rate and the specific growth rate are then coupled into a bioreactor model such that the relationship between substrate feeding rates and the main state variables, i.e., the medium volume, the concentrations of the biomass, the substrate, and the product, is set up. Experimental results demonstrate that the model can describe the cell growth and the protein production with reasonable accuracy. © 2003 Elsevier B.V. All rights reserved. Keywords: Pichia pastoris; Recombinant human serum albumin; Metabolic pathways; Stoichiometric model

1. Introduction The methylotrophic yeast Pichia pastoris has been developed as a prominent host for foreign proteins

Abbreviations: AcCoA, acetyl–CoA; FOR, formaldehyde; G3P, glycerol-3-phosphate; GAP, glyceraldehyde-3-phosphate; GLY, glycerol; MeOH, methanol; PYR, pyruvate ∗ Corresponding author. Tel.: +86-21-6293-2031; fax: +86-21-6293-2145. E-mail address: [email protected] (J.Q. Yuan).

because of its competitive features (Brierley et al., 1990; Gellissen, 2000; Slibinskas et al., 2003). In comparison to other eukaryotic cell lines, Pichia expression systems offer many advantages (Cregg et al., 1993; Romanos, 1995): strong methanol-induced alcohol oxidase (AOX1) promoter, lack of endotoxins which makes produced protein suitable for therapeutic use, correct folding, and secretion of foreign proteins into the medium. In the family of yeast, Saccharomyces cerevisiae has obtained widest studies on the metabolic pathways, the

0168-1656/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jbiotec.2003.08.003

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Nomenclature FGly FMeOH FNH3 FO FS KB1 , KB2 , KGly , KMeOH , k1 , k2 , k1 , k2 , a, b mATP MS P P/O rAc rAc1 , rAc2 rAce rAd rATP rB1 , rB2 rE1 , rE2 rG rGAP rGly rGly,M rGly,R rMeOH rNAD rO2 rS rTCC S SEtOH SGly SMeOH SR Suffixmax Suffixmin VF X YATP α ϕ µ ρ ρL–P ρR

H.T. Ren et al. / Journal of Biotechnology 106 (2003) 53–68

glycerol feeding rate (l h−1 ) methanol feeding rate (l h−1 ) ammonia solution feeding rate (l h−1 ) withdrawal rate by sampling (l h−1 ) substrate feeding rate (l h−1 ) model parameters maintenance coefficient for ATP (mol g−1 h−1 ) molecular weight of substrate rHSA concentration (g l−1 ) effectiveness coefficient of oxidative phosphorylation specific acetyl–CoA production rate (mol g−1 h−1 ) specific acetyl–CoA production rate from pyruvate and acetate, respectively (mol g−1 h−1 ) specific acetate production rate (mol g−1 h−1 ) specific acetaldehyde production rate (mol g−1 h−1 ) specific ATP uptake rate (mol g−1 h−1 ) specific bio-synthesis rate of biomass (mol g−1 h−1 ) specific ethanol production and uptake rate, respectively (mol g−1 h−1 ) specific glycolysis rate (mol g−1 h−1 ) specific glyceraldehyde-3-phosphate production rate (mol g−1 h−1 ) specific glycerol uptake rate in the macrokinetic model (mol g−1 h−1 ) specific glycerol uptake rate obtained from Monod model (mol g−1 h−1 ) specific glycerol uptake rate obtained from the regulator model (mol g−1 h−1 ) specific methanol uptake rate (mol g−1 h−1 ) specific NADH uptake rate in respiratory chain (mol g−1 h−1 ) specific oxygen uptake rate (mol g−1 h−1 ) specific substrate uptake rate (mol g−1 h−1 ) specific acetyl–CoA uptake rate (mol g−1 h−1 ) substrate concentration in the medium (g l−1 ) ethanol concentration in the medium (g l−1 ) glycerol concentration in the medium (g l−1 ) methanol concentration in the medium (g l−1 ) substrate concentration in the feed (g l−1 ) maximum or saturation value minimum value volume of medium (l) biomass concentration (g l−1 ) yield coefficient of ATP (g mol−1 ) coefficient of evaporation (h−1 ) the fraction of formaldehyde oxidized to formate specific growth rate (h−1 ) specific product formation rate (h−1 ) specific product formation rate obtained from Luedeking–Piret model (h−1 ) specific product formation rate obtained from the regulator model (h−1 )

H.T. Ren et al. / Journal of Biotechnology 106 (2003) 53–68

energy balances and modeling (Bellgardt, 1983; Lei et al., 2001; Orlowski and Barford, 1991; Pham et al., 1998; Sonnleitner and Kaeppeli, 1986; von Meyenburg, 1969). However, only few publications on the modeling of P. pastoris may be found in the literature despite its increasing importance in the expression of heterologous proteins (Jahic et al., 2002). On the other hand, the up-to-date knowledge on S. cerevisiae may form a good basis for the kinetic study on P. pastoris because of the similarities of these two strains both in metabolic behaviors and in morphology. This is especially true during the glycerol growth phase of P. pastoris. In the methanol growth phase, methanol is oxidized to formaldehyde by alcohol oxidase at first, and then enters both the dissimilatory pathway and the assimilatory pathway (Cereghino and Cregg, 2000; Jahic et al., 2002; Lueers et al., 1998). In the dissimilatory pathway, formaldehyde is oxidized with the generation of NADH to formate and finally to carbon dioxide. In the assimilatory pathway, formaldehyde is phosphorylated to glyceraldehyde-3-phosphate (GAP). The subsequent metabolic pathways may again be assumed to be similar to those in S. cerevisiae (Cereghino and Cregg, 2000). Therefore, the main frame of the stoichiometric model for S. cerevisiae found in the literature (Bellgardt, 1983, 2000) was used for P. pastoris in this paper. Balances of carbon source, ATP and NADH, have been taken into account in modeling. The bioreaction rates of the model include the specific glycolysis rate of the substrate, the specific growth rate, the specific acetyl–CoA production rate as well as the specific oxygen uptake rate. After coupling the macrokinetic model with a bioreactor model, the relationship between substrate feeding rates and the key state variables, such as the volume of the medium, the concentrations of biomass, protein, and substrates, was set up. Finally, the model was validated by different sets of experimental data.

2. Materials and methods 2.1. Strain and culture conditions The strain P. pastoris GS115 was used in this study. A colony of P. pastoris was inoculated into shake flasks at 30 ◦ C. The culture was grown for

55

12–24 h until the OD600 reached 2–6. Around 5–10% inoculum was used for a 30 l bioreactor (B. Braun, Germany) containing 11 l of batch medium. The cultivation temperature was 30 ◦ C during glycerol growth phase, and was shifted down during methanol growth phase. The pH was measured by pH electrode (Mettler-Toledo, Switzerland), and maintained at 6.5 by the addition of 25% ammonia solution. The dissolved oxygen (DO) was monitored by a DO probe (Mettler-Toledo, Switzerland), and maintained at 30% with a PID cascade controller by regulating the impeller speed between 400 and 1000 rpm. Glycerol and methanol were fed with calibrated peristaltic pumps (Watson 101). The media composition used in this study was the same as in Boze et al. (2001) (all quantities are in g l−1 unless stated differently): • Basal salt solution FM21: CaSO4 ·2H2 O, 1.5; KOH, 6.5; MgSO4 ·7H2 O, 19.5; K2 SO4 , 23.8; H3 PO4 , 85 and 3.5% (v/v). • Trace elements solution PTM1: ZnCl2 , 2; FeSO4 · 7H2 O, 6.5; H3 BO3 , 0.002; CuSO4 ·5H2 O, 0.6; KI, 0.01; MnSO4 ·H2 O, 0.3; H2 SO4 , 96 and 0.2% (v/v); biotin 80 ␮g l−1 . • Inocula medium: glycerol, 10; peptone, 20; yeast extract, 10; KH2 PO4 , 1.2; Na2 HPO4 , 2.29; (NH4 )2 SO4 , 1; MgSO4 ·7H2 O, 1; FeSO4 ·7H2 O, 0.25. • Batch medium: glycerol, 40; biotin, 80 ␮g l−1 ; H3 PO4 , 85 and 2% (v/v); and a predetermined amount of FM21 and PTM1. • Glycerol feeding substrate: 50% (v/v) glycerol plus a predetermined amount of FM21 and PTM1. • Methanol feeding substrate: pure methanol plus a predetermined amount of FM21 and PTM1. 2.2. Analytical methods To determine the biomass concentration, samples were centrifuged at 10,000 rpm for 10 min and washed twice with deionized water, then dried to constant weight at 80 ◦ C with an infrared dryer (Mettler-Toledo Lj-16, Switzerland). Ethanol and methanol concentration were determined with a HP5890 gas chromatograph (Sharp, Japan). Recombinant human serum albumin (rHSA) concentration was determined according to the following procedure:

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2.2.1. Sodium dodecylsulfate–polyacrylamide gel electrophoresis (SDS–PAGE) About 10–20 ␮l of supernatant of each sample was mixed with 20 ␮l of 2× SDS buffer (Bio-Rad) and 10–20 ␮l was subjected to electrophoresis on 8–15% Tris–glycine acrylamide minigels (Bio-Rad) in Tris–glycine–SDS running buffer (Bio-Rad). 2.2.2. Gel staining Gels were stained with Coomassie blue staining reagent (Bio-Rad) for about 4 h, then destined with 40% methanol and 10% acetic acid for about 4 h. 2.2.3. Quantification Quantifying of protein concentrations was carried out by visual inspection of band intensities relative to known standards. 2.3. Cultivation process Cultivation was carried out in two phases: glycerol growth phase and methanol growth phase. The glycerol growth phase was subdivided into two stages, i.e., a 16–20 h batch culture and a 14–16 h fed-batch culture. The methanol growth phase included a 10 h induction stage and a 120–160 h production stage. The majority of rHSA was produced in the production stage. The samples were taken at regular intervals of 2 h in the glycerol growth phase and 4 h in the methanol growth phase, respectively. 3. Simplified metabolic pathways and stoichiometry in P. pastoris Different from the metabolic flux models in which a great number of reactions and large amount of parameters might be included, the macrokinetic model proposed here accounts only for the main metabolic pathways as a first approximation of the metabolic network. Such simplification is necessary because only limited stoichiometric coefficients are available in the literature. 3.1. Metabolic pathways in the glycerol growth phase The simplified metabolic pathways of glycerol are schematically shown in Fig. 1. Glycerol is phos-

phorylated to glycerol-3-phosphate (G3P) by a glycerol kinase at first. Further oxidation of G3P by a FAD-dependent glycerol-3-phosphate dehydrogenase results in dihydroxyacetone phosphate (Gancedo et al., 1968; Nevoigt and Stahl, 1997). Then, pyruvate is formed as the outcome of glycolysis. Pyruvate is further oxidized to acetyl–CoA via pyruvate dehydrogenase. Subsequently, acetyl–CoA enters tricarboxylic acid (TCA) cycle, where many metabolites are produced and used for the synthesis of cellular constituents such as amino acids, nucleic acids as well as cell wall components. Meanwhile, most of energy in the form of ATP and NADH for cell growth and maintenance is yielded in TCA cycle (Ratledge, 2001). Starting from pyruvate, the alcoholic fermentation may be triggered by the limitations of the respiratory capacity and/or the glycolytic flux (Lei et al., 2001; Sonnleitner and Kaeppeli, 1986). In this fermentative bypass, pyruvate is converted to acetaldehyde by pyruvate decarboxylase and further oxidized to ethanol by alcohol dehydrogenase (Inan and Meagher, 2001). Ethanol may also be used as substrate, if the limitations mentioned above are removed (Lei et al., 2001; Nevoigt and Stahl, 1997). In that case, ethanol will be oxidized to acetaldehyde by alcohol dehydrogenase, then to acetate by acetaldehyde dehydrogenase, and finally converted to acetyl–CoA by acetyl–CoA synthetase (Pronk et al., 1996; Vanrolleghem et al., 1996). With respect to cell growth, a small part of biomass was assumed to come from G3P, while the major part came from acetyl– CoA. The respiratory chain has the main function to yield energy and its stoichiometric coefficients are known (Smith et al., 1983; Sonnleitner and Kaeppeli, 1986). In Fig. 1, the stoichiometric balances of ATP– ADP and NADH–NAD are depicted (Lei et al., 2001; Nevoigt and Stahl, 1997; Ratledge, 2001). All the symbols and abbreviations found both in the text body and in the figures are explained in the nomenclature. 3.2. Metabolic pathways in the methanol growth phase The simplified metabolic pathways of methanol are illustrated in Fig. 2. As the first step, methanol-induced alcohol oxidase oxidizes methanol to formaldehyde and hydrogen peroxide (Cereghino and Cregg, 2000; Lueers et al., 1998). Formaldehyde enters both the

H.T. Ren et al. / Journal of Biotechnology 106 (2003) 53–68

Glycerol ATP rGly ADP r Glycerol 3-phosphate B1 Biomass 2ADP 2NAD rG NADH NAD 2NADH 2ATP rE1 rAd Ethanol Acetaldehyde Pyruvate rE2 NAD NAD NADH NAD rAc1 rAce NADH NADH rAc2 Acetyl-CoA Acetate

Phosphorylation

Glycolysis

0.5O2 P/O ADP NADH rNAD P/O ATP

57

ADP

4NAD rTCC

ATP

NAD

H2O

rB2

4NADH CO2

Biomass

Tricarboxylic-acid (TCA) cycle

Respiratory chain

Fig. 1. Metabolic pathways of glycerol in Pichia pastoris.

reducing power in the form of NADH. In the assimilatory pathway, the rest of formaldehyde is metabolized according to the following pathways: condensation of formaldehyde with xylulose-5-monophosphate catalyzed by the peroxisomal enzyme dihydroxyacetone synthase to glyceraldehyde-3-phosphate (GAP),

dissimilatory pathway and the assimilatory pathway (Cereghino and Cregg, 2000; Gellissen, 2000; Shen et al., 1998). In the dissimilatory pathway, a portion of formaldehyde is oxidized to formate and to carbon dioxide by formaldehyde dehydrogenase (FLD) and formate dehydrogenase (FDH), respectively, providing

Methanol rMeOH

Oxidation

NAD

ϕrMeOH

NADH

Formaldehyde

NAD

Formate

ATP

NADH

CO2

rGAP

Phosphorylation

ADP

1/3 Glyceraldehyde-3-phosphate

Glycolysis

rB1

2ADP

NAD

rG NADH

2ATP

Pyruvate rAcNAD

NADH

0.5O2 P/O ADP

Acetyl-CoA NADH

rNAD P/O ATP

H2O

NAD

ADP

rB2

4NAD

rTCC

4NADH

ATP

CO2

Biomass

Respiratory chain Tricarboxylic-acid (TCA) cycle

Fig. 2. Metabolic pathways of methanol in Pichia pastoris.

Biomass

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which then enters the TCA cycle (Cereghino and Cregg, 2000). Similar to the glycerol growth phase, the biomass formation was assumed to come from GAP and acetyl–CoA. The stoichiometric balances of ATP–ADP and NADH–NAD are depicted in Fig. 2 (Cereghino and Cregg, 2000; Gellissen, 2000; Jahic et al., 2002; Lueers et al., 1998; Ratledge, 2001). Veenhuis et al. (1983) pointed out that NADH generated by FLD and FDH serves as the primary energy source for methanol metabolism. On the other hand, Sibirny et al. (1990) stated that the primary role of FLD is to protect the cell from toxic levels of formaldehyde caused by high residual methanol concentration in the medium, while most energy comes from the assimilatory pathway. In this paper, the main carbon source was assumed to be uptaken by the assimilatory pathway. In fact, to achieve higher productivity, the specific growth rate and the residual methanol concentration were controlled at low levels throughout our experiments (see Figs. 9 and 10). Therefore, according to Sibirny et al. (1990), only a small fraction of formaldehyde, denoted by ϕ, should be oxidized to formate. Here, ϕ was set to 0.25. Its true value could only be estimated on the basis of detailed flux data. Nevertheless, inaccurate values of ϕ can actually be compensated by identification of other model parameters. No other bypasses were taken into account in the methanol growth phase due to the low specific growth rates under consideration.

According to our measurements, the residual ethanol concentration was in the range of 0–0.5 g l−1 , see Fig. 8. Therefore, the bypass of ethanol formation was not modeled for simplification. Then, the oxidization of pyruvate to acetyl–CoA is described by: rAc1

→ AcCoA + NADH + CO2 PYR −

(4)

Eq. (5) presents the main flux of acetyl–CoA consumption in TCA cycle. The assimilation of acetyl–CoA to cellular constituents is described with Eq. (6), where rB2 = µKB2 . rTCC

AcCoA −→ 2CO2 + ATP + 4NADH rB2

→ cell material AcCoA −

(5) (6)

Eq. (7) presents the energy balance in the respiratory chain: P 1 rNAD P O2 + NADH + ADP −→ ATP 2 O O

(7)

ATP consumption was assumed to be mainly for the use of cell growth and maintenance (Pirt, 1965): µ rATP = + mATP (8) YATP 3.4. Stoichiometric equations in the methanol growth phase

3.3. Stoichiometric equations in the glycerol growth phase

Similar to those in the glycerol growth phase, the metabolites and energy balances corresponding to Fig. 2 are presented. Eq. (9) describes the oxidization of methanol to formaldehyde.

All substances appearing in the following reaction equations take the unit of mole. The phosphorylation of glycerol is described by

MeOH −−→ FOR

rGly

GLY + ATP − → G3P

(1)

rMeOH

The oxidization of formaldehyde to formate, and further to carbon dioxide is presented as ϕrMeOH

The glycolysis of G3P is formulated with Eq. (2). A small part of G3P is assimilated to cellular constituents, as given in Eq. (3), where rB1 = µKB1 . rG

G3P − → PYR + 2ATP + 2NADH rB1

G3P − → cell material

(2)

FOR −−−→ CO2 + 2NADH

(10)

The phosphorylation of formaldehyde is described by Eq. (11), where three molecules of formaldehyde are consumed in order to produce one net molecule of GAP (Cereghino and Cregg, 2000; Lueers et al., 1998). rGAP

(3)

(9)

FOR + ATP −→ 13 GAP

(11)

H.T. Ren et al. / Journal of Biotechnology 106 (2003) 53–68

The glycolysis of GAP is formulated with Eq. (12). A small part of GAP was assumed to form biomass constituents, as described in Eq. (13) with rB1 = µKB1 . rG

GAP − → PYR + 2ATP + NADH rB1

GAP − → cell material

(12) (13)

The oxidation of pyruvate to acetyl–CoA is presented as: rAc

→ AcCoA + NADH + CO2 PYR −

(14)

The metabolic pathways subsequent to acetyl–CoA and in the respiratory chain were assumed to be the same as those in the glycerol growth phase. It should be pointed out that the values of some model parameters, such as KB1 and KB2 , are carbon source dependent although the same symbols were used in different growth phases.

4. Modeling 4.1. Model equations for the glycerol growth phase To develop a practically applicable model, a basic assumption has to be introduced (Bellgardt, 1983) for modeling: the intracellular reactions as given in Eqs. (1)–(14) were assumed to be always in quasi-steady state. This implies, there is no significant accumulation of the metabolites in the cell. This assumption, however, does not mean the intrinsic dynamic nature of the cultivation being overlooked. Instead, the reaction rates will follow immediately the change of external conditions as mediated by the substrate uptake rate, or the output of the metabolic regulator as given below. In other words, the steady state is only held within each simulation step, but such steady states may undergo shifting from one state to another depending on the specific substrate uptake rate, which is determined with either Monod model or the regulator model. Such assumption has also been applied successfully in the works of d’Anjou and Daugulis (2001) and Lei et al. (2001). Based on the above assumption and by neglecting the formation/reuse of ethanol (i.e., let rAc =

59

rAc1 ), the macrokinetic model describing the balances of intracellular substances and energy are constructed in matrix form, see Eq. (15), which consists of four sub-balances. From Eqs. (1) to (3), the balance for carbon source is established so that the first line of Eq. (15) is obtained. Similarly, line 4 of Eq. (15) stands for the balance of pyruvate, which results from Eqs. (2) and (4). NADH balance is made according to Eqs. (2) and (4)–(7). From Eq. (2), the specific production rate of NADH in glycolysis is 2rG . From Eqs. (4) and (5), the specific production rate of NADH is (rAc +4rTCC ). On the other hand, rTCC = (rAc − rB2 ) = (rAc − µKB2 ), see Eqs. (4)–(6), so that (rAc +4rTCC ) is equivalent to (5rAc − 4µKB2 ). Finally, the specific turnover rate of NADH in the respiratory chain, rNAD , equals 2rO2 as shown in Eq. (7). Then, one obtains the balance equation for NADH as (2rG −4µKB2 +5rAc −2rO2 = 0) which is exactly line 2 in Eq. (15). ATP balance is made according to Eqs. (1)–(8). From Eqs. (1) and (2), the specific production rate of ATP is (2rG −rGly ). Since rGly = (rG + rB1 ), see Eqs. (1)–(3), (2rG − rGly ) is equivalent to (rG − rB1 ), i.e., (rG − µKB1 ). As described in Eq. (5), the specific production rate of ATP in TCA cycle is rTCC , or equivalently (rAc − µKB2 ), see Eqs. (4)–(6). In the respiratory chain, the specific production rate of ATP is 2P/OrO2 , see Eq. (7). In addition, the specific uptake rate of ATP for the growth and maintenance metabolism of cells is (µ/YATP + mATP ), i.e., Eq. (8). Therefore, the balance equation for ATP is (rG − µKB1 − µKB2 − µ/YATP + rAc + 2P/OrO2 = mATP ), which is line 3 of Eq. (15). 

1

 2   1  1

KB1

0

−4KB2

5

−KB1 − KB2 − 



rGly    0    =  m  ATP 

0

1 YATP

1 −1



 rG  −2   µ      2P   r Ac    O rO2 0 0

(15)

0 In this system of equations, the rates of substrate uptake, rGly , and of ATP-turnover for maintenance, mATP , are known, and therefore, enter the vector of the right

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hand side. Since the coefficient matrix is also known and regular, the system can be solved for the unknown reactions rG , µ, rAc and rO2 . The specific glycerol uptake rate is in principle represented as Monod kinetics: rGly,max SGly rGly,M = (16) KGly + SGly However, it was found that the actual glycerol uptake rate was much lower than rGly,M at the beginning of the batch, despite the high glycerol concentration. Such lag phase could result from the low activity of enzymes responsible for glycerol uptake shortly after inoculation. In the biochemical view, induction and synthesis of enzymes consist of a series of bioreactions. It is possible to describe the transients by a network of reactions including transcription and translation. This leads to biochemically structured models. Then, each of these reactions is usually modeled by enzyme kinetics. Often many of these kinetics are not exactly known so that only simple ones or first order kinetics were assumed in such models even for complicated reactions. Therefore, these models may be useful for the investigation of the principal information flow and action of metabolic regulation. But they are less suited for bioprocess modeling. The biochemically structured models have a too complicated structure and too many unknown parameters which are impossible to be identified from cultivation data only. For the modeling of lag-phases in the course of the cultivation, a model structure as simple as possible and with a minimum number of parameters is most suitable. The metabolic regulator model, proposed and well validated by Bellgardt (1983), is therefore introduced to describe the lag phase. Fig. 3 illustrates the

structure of the regulator model which has usually only three parameters. The dashed frame outlines a first order regulator with rGly as the input and rGly,R as the output, which represents the actual activity of the regulated pathway in analogy to enzyme levels of biochemical structure models. The introduction of the negative feedback by µ is because the enzyme pool may be diluted by the cell growth (Bellgardt et al., 1986). Two parameters k1 and k2 determine the transients of the lag phase. The parameter rGly,min determines the minimum constitutive activity of the regulated pathway. These three parameters are identifiable based on the measurements of biomass concentration. The entire system with input rGly,M , as given by the Monod-type substrate uptake kinetics, and output rGly is called the extended regulator. The switch K is realized by the minimum function rGly = min{rGly,M , rGly,R }. It determines whether the regulated pathway has to be induced or not. Fig. 4 shows the transients of rGly (䊊), the output of the regulator (rGly,R , —) and the output of the Monod model (rGly,M , - - -) for an experiment. This example demonstrates the functions of the regulator: 1. If rGly,R is smaller than rGly,M , then K will shift to the regulator model (rGly = rGly,R ) and the regulator together with the regulated pathway forms a closed control loop. During this induction phase, rGly,R and with it the glycerol uptake rate, are increased by the inherently unstable dynamics of the control loop. This models an accelerating establishment of the enzymes of the pathway. 2. If rGly,R is larger than rGly,M , then K will shift to the Monod model (rGly = rGly,M ) and the regulator will work in servo mode. During this stage, rGly,R will track the output of the Monod model,

Fig. 3. Diagram of extended regulator model for the specific glycerol uptake rate.

H.T. Ren et al. / Journal of Biotechnology 106 (2003) 53–68

61

Fig. 4. The specific glycerol uptake rate obtained from Monod model (- - -), the regulator model (—) and the extended regulator model (䊊).

see Fig. 4, since the regulator itself is stable. The gap between the solid line (—) and the dashed line (- - -) may be regarded as reserve of enzyme levels. In other words, if more substrate feeding occurs, the enzyme pool is able to uptake more substrate at once within the reserve limit. However, if the substrate feeding is so high that the output of the Monod model becomes greater than rGly,R , then the substrate uptake rate is limited by this value and the regulatory process will start again with further induction. Qualitatively, this is well in coincidence with the observations in bioprocesses. Mathematically, the regulator model is described with Eq. (17). The glycerol uptake rate rGly appearing in the macrokinetic model, see Eq. (15), is obtained according to Eq. (18). drGly,R = k1 (rGly + rGly,min ) + (−k2 − µ)rGly,R dt (17) rGly = min{rGly,M , rGly,R }

(18)

4.2. Model equations for the methanol growth phase The macrokinetic model for the methanol growth phase is presented in Eq. (19), where the four sub-balance equations are obtained from Eqs. (9)–(13);

(5)–(7), (9)–(14); (5)–(8), (9)–(13); (12), (14), respectively. The derivations of these equations are similar to those in the glycerol growth phase. Taking line 1 as an example, (rMeOH = ϕrMeOH + rGAP ) is obtained from Eqs. (9)–(11); then from Eqs. (11)–(13), one obtains (rGAP = 3rG + 3µKB1 ); finally, the sub-balance equation of line 1 is formulated as (3rG /(1 − ϕ) + 3KB1 µ/(1 − ϕ) = rMeOH ).   3 3 0 0 K B1  1−ϕ  1−ϕ    5ϕ + 1  6ϕ   K 5 −2 − 4K B1 B2  1−ϕ  1 − ϕ     1 2P  −1  −3KB1 − KB2 − 1  YATP O  1 0 −1 0     rG rMeOH  µ   0      × (19) =   rAc   mATP  rO2

0

The specific methanol uptake rate is described with Monod kinetics too: rMeOH,max SMeOH rMeOH = (20) KMeOH + SMeOH Since the feeding rate of methanol was very low at the beginning of the methanol growth phase, a lag phase is not taken into account.

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The specific rHSA production rate was assumed to take the form of Luedeking–Piret model (Luedeking and Piret, 1959):

However, the actual specific rHSA production rate was much lower than the output of Eq. (21) during the induction stage. In the induction stage, binding of methanol with the repressor on the AOX1 promoter initiates the expression of rHSA. The dynamics of this process may depend on the concentrations of the messenger RNA or enzymes in the rHSA-pathway, but the exact kinetics remain unclear. Again, an extended regulator model as illustrated in Fig. 5 is introduced to describe the lag-phase, where ρR represents the specific rHSA formation rate obtained from the regulator model, while ρ is the actual specific rHSA formation rate. The model equations corresponding to Fig. 5 are presented dρR = k1 (ρ + ρmin ) + (−k2 − µ)ρR dt

(22)

ρ = min{ρL−P , ρR }

(23)

The initial input of the regulator model, ρmin , together with two other model parameters, k1 and k2 , are identifiable according to rHSA sampling assays. Fig. 6 shows the specific rHSA formation rate ρ (䊊) resulting from the combination of the Luedeking–Piret model (- - -) and the regulator model (—) for an experiment.

ρL-P ρR ρ

-1

(21)

0.0008 ρ , ρR and ρL-P (h )

ρL−P = aµ + b

0.0010

ρR

0.0006 ρL-P

0.0004 ρ=ρL-P

0.0002

ρ

ρ=ρR

0.0000 30 40 50 60 70 80 90 100 110 120 130 140 150 160 Time after methanol feeding (h)

Fig. 6. The specific rHSA formation rate obtained from Luedeking–Piret model (- - -), the regulator model (—) and the extended regulator model (䊊).

dXVF dX dVF =X + VF = µXVF − FO X dt dt dt dSVF dVF dS =S + VF dt dt dt = FS SR − rS MS XVF − F O S dVF dP dPVF =P + VF = ρXVF − FO P dt dt dt

(24)

(25) (26)

4.3. Bioreactor model

The time derivative of VF in Eqs. (24)–(26) is presented as: dVF = FS + FNH3 − FO − αVF (27) dt

Mass balances are applied to obtain the bioreactor model. These are given for the biomass concentration, the substrate concentration as well as the product concentration:

On the right hand of Eq. (27), FS , FNH3 , and FO are feeding rates of substrate, ammonia solution as well as withdrawal rate, respectively. The last term αVF stands for the water evaporation caused by aeration. The evaporation coefficient, α, is typically estimated

Fig. 5. Diagram of the extended regulator model for the specific rHSA production rate.

H.T. Ren et al. / Journal of Biotechnology 106 (2003) 53–68

Initial conditions FS Input FNH3 variables FO

63

Bioreaction rates rS, µ, ρ

Bioreactor model

Macrokinetic model State variables X, S, P, VF

Fig. 7. Combined macrokinetic-bioreactor model.

0.9

90

0.8

70

0.7

60

0.6 SGly

50

X

0.5

40

0.4 FGly

30 20

µ

10

0.2 SEtOH

FNH3

0 0

10

15 Time (h)

20

25

90

-1

X , SGly and SEtOH (gl )

0.0 30

0.9 Exp 2

80

0.8

70

0.7

60

0.6 SGly

50

0.5 X

40

0.4

30

FGly µ

20 10

FNH3

0

5

10

15 Time (h)

SEtOH 20

0.3 0.2 0.1

0

(b)

0.1

µ (h-1) , FGly and FNH3 (lh-1)

(a)

5

0.3

µ (h-1) , FGly and FNH3(lh-1)

80

-1

X, SGly and SEtOH (gl )

Exp 1

25

0.0 30

Fig. 8. Time courses of experiments 1 (a) and 2 (b). Lines: model simulations; symbols: measurements.

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H.T. Ren et al. / Journal of Biotechnology 106 (2003) 53–68

dP FS + FNH3 P + αP = ρX − dt VF

by 0.0012 h−1 for most CSTR type of bioreactors under the aeration rate of 1.0 vvm (authors’ unpublished data). But in this paper it was estimated by 0.0006 h−1 because the waste gas was cooled and the condensate was recycled to the bioreactor. Rearranging Eqs. (24)–(27), the bioreactor model is

(30)

The coupling of the macrokinetic model and the bioreactor model is depicted in Fig. 7.

FS + FNH3 dX = µX − X + αX dt VF

(28)

5. Validation of the model

FS FS + FNH3 dS = SR − r S M S X − S + αS dt VF VF

(29)

Four experiments were carried out to validate the model. Fig. 8 shows the time courses of experiments 1

0.10

120

-1

0.08

80 FMeOH

0.06

-1

60 0.04

FGly

40 µ X 0.1

P 20 0 0

20

40

60

(a)

80 100 Time (h)

0.02

SMeOH

µ

120

140

-1

X , SMeOH and P X 10 (gl )

X

Glycerol growth phase

-1

Methanol growth phase

100

µ (h ) , FGly X 10 and FMeOH (lh )

Exp 3

0.00 160

0.08

100 Methanol growth phase

X , SMeOH and P X 10 (gl-1)

80 Glycerol

X 0.06

growth phase

FMeOH

60

0.04 40 µ X 0.1

FGly P

0.02

20 µ

SMeOH

0 0 (b)

20

40

60

80 100 Time (h)

120

140

µ (h-1) , FGly X 10-1 and FMeOH (lh-1)

Exp 4

0.00 160

Fig. 9. Time courses of experiments 3 (a) and 4 (b). Lines: model simulations; symbols: measurements.

H.T. Ren et al. / Journal of Biotechnology 106 (2003) 53–68

65

Table 1 Model parameters taking fixed values Glycerol growth phase

Methanol growth phase

Kgly (g l−1 )

mATP (mol g−1 h−1 )

KB1 (mol g−1 )

rGly,min (mol g−1 h−1 )

k1 (h−1 )

KMeOH (g l−1 )

mATP (mol g−1 h−1 )

KB1 (mol g−1 )

b

ρmin

k1

0.3

0.0015

0.001

0.0006

0.4

0.3

0.0001

0.0015

0.0008

0.00004

0.15

Table 2 Batch-dependent parameters for Exps. 1–4 Experiment

Glycerol growth phase rGly,max

1 2 3 4

0.0061 0.0057 0.0053 0.0062

(mol g−1 h−1 )

Methanol growth phase KB2

(mol g−1 )

0.01038 0.01462 0.01034 0.00926

i2

(h−1 )

0.20 0.25 0.29 0.23

and 2 where only glycerol growth phase is presented. The cultivation started as a batch run and fed-batch was initiated upon the sudden increase of dissolved oxygen at about 16 h, indicating the exhaustion of glycerol in the batch medium. The cell dry weight was approximately 25 g l−1 by the end of batch culture, and 70 g l−1 by the end of glycerol fed-batch culture. Fig. 9 shows the time courses of experiments 3 and 4, where both glycerol and methanol growth phases are included. During the methanol growth phase, the initial feeding rate of methanol was very low so that cells could adapt to the shifting of the carbon source smoothly. From these figures, satisfactory coincidence between model simulations and measurements are found. In the model Eqs. (15)–(17) and (19)–(22), there are twenty model parameters. Two of them, YATP and P/O, took fixed values of 10.5 g mol−1 and 1.5 mol mol−1 , respectively, the same as those for S. cerevisiae (Yuan and Bellgardt, 1994). Model errors caused by such approximate setting can be compensated by other model parameters identified below. Parameter identification was carried out with Simplex method (Nelder and Mead, 1965). The sensitivity analysis of these parameters was done with the similar method as Beschkov and Velizarov (2000) as well as Claes and Van Impe (2000). Perturbation of each parameter by ±10% around its mean was applied with the sum of square errors between measured and simulated X and P as the objective function for sensitivity testing. It was found that 11

rMeOH,max (mol g−1 h−1 )

KB2 (mol g−1 )

a

k2 (h−1 )

0.0016 0.0012

0.0123 0.0170

0.48 0.63

0.13 0.10

parameters were less sensitive to the perturbation. Therefore, constant values were used in the model as shown in Table 1. On the other hand, the other seven parameters were quite sensitive to the perturbation, so that they were classified as batch-dependent parameters. The identification results of these parameters for the four experiments are presented in Table 2.

6. Discussion and conclusion Based on the analysis of metabolic pathways of glycerol and methanol, a macrokinetic model for P. pastoris is established. Experimental results demonstrate that the cell growth and the heterologous protein formation can be described by the combined macrokinetic and bioreactor model with reasonable accuracy. Relatively small deviation of the identified parameters (see Table 2) among different experiments indicates the robustness of the model. According to the model simulations and measurements, the macroscopic yield coefficients of biomass on glycerol and methanol are about 0.45 and 0.15 g g−1 , respectively, as listed in Table 3 which are well in accordance with those reported in the literature (Loewen et al., 1997). These results may lead to the conclusion that the very simplified structure of the metabolic network the model is based on covers the pathways with major importance during the cultivation. Nevertheless, after sufficient experimental data become available, a more

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H.T. Ren et al. / Journal of Biotechnology 106 (2003) 53–68

Table 3 Biomass yield on glycerol and methanol Experiment

Yield on glycerol (g g−1 )

Yield on methanol (g g−1 )

1 2 3 4

0.46 0.42 0.47 0.45

0.15 0.15

Mean

0.45

0.15

sumption and carbon dioxide production were not investigated due to the lack of data. Should those data become available, more adequate flux mapping and more intensive model validation could be expected. The valid horizon of the proposed model is limited to the case where the specific growth rate in the methanol growth phase is lower than 0.008 h−1 , which seems to be the upper limit of it on methanol for the given expression system. Fig. 10 shows the transients of experiment 5 where a linearly increasing methanol feeding profile was applied but an unexpected over feeding happened. The specific growth rate was between 0.0085 and 0.009 h−1 in the whole methanol growth phase. Until 80 h, the simulated residual methanol and biomass concentrations follow the measurements quite well without adjusting the batch-dependent parameters, but such coincidence failed afterwards. The measurements revealed a fast accumulation of methanol up to 10 g l−1 at 140 h. After 140 h, product degradation occurred and cells shifted to fast growth as if they were “awaked” from a stagnant state. The accumulated methanol was completely uptaken at 170 h. Moreover, fast growth of cells extended another 10 h. During these 10 h, the yield coefficient of biomass on methanol was about 0.3 g g−1 , double of the normal value. Whether this was at cost of rHSA or the metabolic pathways had shifted to wild type is a question for further study.

detailed consideration of metabolism and physiology could surely improve the model and deepen our understanding of cell growth and product synthesis. For example, parameter P/O was set to constant value in modeling. But actually, P/O may change depending on the growth phases or flux rates of keynote metabolites (Fontaine et al., 1997; Kadenbach, 2003). Principally, a variation of P/O may be taken into account in the modeling after the relationship between P/O and its influencing factors becomes clear. Moreover, only key metabolites have been taken into account in the stoichiometric model. The neglect of possible bypasses and byproducts gives room for further improvement of the model. However, the simplified model should be sufficiently accurate for the purpose of model-based feeding control, as indicated by our other experiments (data not shown). The balance model for oxygen con-

Exp 5 100 15

-1

X and P X 10 (gl )

X 80 µ

60

10 P

40 FMeOH

5

20

SMeOH

0 0

20

40

60

80 100 Time (h)

120

140

160

µ X 103 (h-1) , SMeOH (gl-1) and FMeOHX102 (lh-1)

20

120

0 180

Fig. 10. Time course of an extraordinary operated experiment (experiment 5).

H.T. Ren et al. / Journal of Biotechnology 106 (2003) 53–68

However, the toxic effect of high residual methanol concentration (Kobayashi et al., 2000; Sarramegna et al., 2002; Yamane et al., 1976) is not obvious for the given strain. If the toxic effect did exist, it seems to be also reversible according to the data after 140 h shown in Fig. 10. For enhancing protein productivity, the methanol sensor-based feeding strategy has been studied (Guarna et al., 1997; Hong et al., 2002; Zhou et al., 2002). However, in our expression system, the residual methanol concentration is practically undetectable, if the specific growth rate is controlled to stay below 0.008 h−1 . That means, the model-based feeding strategy, which is under study in the laboratory of our industrial partner, will become a promising alternative for the optimal control of productivity.

Acknowledgements The authors gratefully acknowledge the support of the Natural Science Foundation of China (Grant No. 60174024) and the National High Technology Research and Development Program of China (Grant No. 2001AA413110). Earlier study of this work was supported by the Alexander von Humboldt Foundation/Germany. The referees of this paper are acknowledged for their valuable comments for improving the manuscript.

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