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Deterministic Growth Model of Saccharomyces Cerevisiae, Parameter Identification and Simulation
Copyright © IFAC !'vIode lling an d Control of Biotechnical Processes H elsinki, Finland 1982
DETERMINISTIC GROWTH MODEL OF SACCHAROMYCES CEREVISIAE, PARAMETER IDENTIFICATION AND SIMULATION K.-H. Bellgardt*, W. Kuhlmann** and H.-D. Meyer** *Institut fur Regelungstechnik, Um'versitiit Hannover, D 3000 Hannover, Federal Republic of Germany * *Institut fur Techmsche Chemie, Universitiit Hannover, D 3000 Hannover, Federal Republic of Germany
Abstract. A mathematical model of the growth of SACCHAROMYCES CEREVISIAE H1022 was developed, which can describe oxidative and aerobic fermentative(Crabtreeeffect) growth on glucose, oxidative growth on ethanol, oxygen limited growth on glucose and/or ethanol and the anaerobic growth. A description of the complex growth behaviour with good model accuracy can be obtained with reaction kinetic models which take into consideration the more important metabolic processes in form of balance equations. In this paper a simplification of modelling is obtained by application of Blackman's master-reaction-concept and a quasi steady state approach for cellular metabolic reactions. Most of the parameters of the biological model are advantageously identified with stationary chemostat data by an iterative two-level optimization algorithm. In a following step the parameters which influence the simulated growth lag phases are identified with batch data. Simulations of batch and chemostat cultures show good correspondence of measurements and model predictions if the effectiveness of the respiratory chain (P/O) is not constant but increases with more fermentative growth and reduced oxygen uptake rates. Keywords. Growth models, parameter estimation, convergence of numerical methods, nonlinear systems, chemostat, batch culture, metabolic regulations, Crabtree-effect, Pasteur-effect. I NTRODUCTI ON For some special cases of yeast growth, e.g . pure aerobic growth or growth on a single substrate, empirical models are known, based on an extension of the MONOD-equation (Bergter, Knorre, 1972; Bijkerk, Hall, 1977; Dekkers, deKok, Roels, 1981; Pamment, Hall, 1978; Peringer and others, 1973 and 1974; Toda, Yabe, Yamagata, 1980; Yoon, Klinzing, Blanch, 1977). The generalization of such models to other growth states is not known. Under this point of view models are more successful, which take into consideration chemical reactions of metabolism (Dekkers, deKok, Roels, 1981; Geurts, deKok, Roels, 1980; Hall, Barford, 1981) .
In this paper a deterministic growth model of baker's yeast SACCHAROMYCES CEREVISIAE H1022 on glucose and ethanol as limiting carbon sources is presented. The model shall be used for structure and parameter optimization of continuously operating multi stage stirred tank plants (chemostat cascade). Furthermore the application of the model to an automatic computer control of the process is provided. Hence the model must have a sufficient accuracy and complexity. A by-product of accurate modell ing can be a better insight into the biological processes in living cells. The growth behaviour of this yeast can be roughly subdivided into oxidative growth with high cell yield and high oxygen uptake rates and fermentative growth with ethanol production (Meyenburg, 1969; KUenzi, 1970; Weibel, 1973; Schatzmann, 1975; Fiechter, 1980). Fermentative growth is induced by oxygen limitation or high glucose uptake rates (Crabtree -effect). Every growth state is characterized by a complex cooperation of substrate uptake rate, oxygen uptake rate, cell morphology and of intracellular enzyme pattern, from which as a consequence of metabolic regulation the known diauxic growth in batch cultures and the two growth phases in the aerobi c chemostat follow.
In the following a model of a stirred tank is given first. Then the biological model based on a simplified metabolic reaction scheme and finally simulation results and the used parameter identification procedure are presented. REACTOR MODEL In this section the mathematical model of a stirred tank reactor is given, which is independent of the biological model. The reactor model contains the description of the gas phase and the liquid phase in the 67
68
K.-H. Bellgardt, W. Kuhlmann and H.-D. Meyer
~G(t) =
I
Inlet Gos
Outlel Gos
~ "ni :'., F
aTR
X.
X
E.
E
s.
eTR
GasPhase
1 - XOG E - XCG E
x~G
A
Ovorflow
LiquidPhase
S
0,. c,.
/Av p V
1 - x~G(t) - x~G(t)
II
xCG (t)
=
0,
KlaRvi () E"E OF t
+
x8G I mpeller Speed N
+
+
(ll)
KlaR vi 0' /Av pE VE F
Ko Kl a RV TE
CF(t) Mc pE VE KoKlaRVTE l-x~G-x~G + Mc pE VE CF 1 - x~G(t) - x~G(t)
(12)
C,
Reservoir
Reactor
Fig. 1. Schematic diagram of the stirred tank. tank with an entire balance of oxygen and carbon dioxide, since without that correct mode 11 i ng of oxygen 1i mi ted growth is not possible . Figure 1 shows the schematic diagram of the reactor. It is assumed that gas and liquid phase are ideally mi xed and therefore may be considered as homogeneous. The differential equations of the reactor are (see nomenclature) for the liquid phase (' = d/dt) (1)
S(t) X(t)
~(t)
X(t)
(2)
E(t)
(3)
DF(t)
=
-q02(t) Mo X(t) + D(OFR -OF(t)) + OTR(t) (4)
CF(t)
=
qC02(t) Mc X(t) + D(C fR -CF(t)) - CTR(t) (5)
and for the gas phase , 'E 'A 'T no(t) = no - no - no
(6) (7)
(8)
Exchange of nitrogen between gas phase and liquid phase as well as water and ethanol content of the gas phase are neglected. The mass transfer rates of oxygen and carbon dioxide are given by (Ramm, 1968; Westerterp and others, 1962; Yagi, Yoshida, 1975 and 1977) OTR(t)
K1a (OF x~G(t)
0f(t))
(9)
CTR(t) Since the time constants of the gas phase are small compared to the others a steady state apporach is introduced in Eq. (6) - (8). After application of the gas laws and DALTON's law the following equations, which can be solved iteratively, for the mole fractions in the outlet gas are obtained:
MODEL OF METABOLIC ACTIVITY The biological model of yeast growth permits the determination of metabolic activities as a function of glucose, ethanol and dissolved oxygen concentrations, which are given by the reactor model. Morphological changes of cells or variations during the budding cycle reported by Hall and Barford (1981) , Kuenzi (1970) and Meyenburg (1969) were not considered, so that the model predicts an average behaviour of the culture. The model is restricted to the main metabolic reactions, which are glycolysis via the EmbdenMeyerhof-pathway, removal of the resulting pyruvate by the Tri-Carbon-Acid-cycle (TCC) and ethanol synthesis. In order to simplify the model it is assumed that pyruvate is a direct catabolite of ethanol, and that gluconeogenesis is the reversal of the glycolysis, although in reality ethanol is catabolized to acetyle-CoA and gluconeogenesis is started from the Glyoxylate-cycle . The proposed model structure is qs Sot - S i nt
(13)
r
Sin t -L 2 Pyr + 2 ATP + 2 NADH r BI
Si nt -
( 14)
Cell mass + stored carbohydrates (15)
r Ad
Pyr-Ad +C0 2
(16)
Pyr~Ac +NADH +C0 2
(17)
Ad + NADH ~ EtOH
(18)
r
r
r
Ad ~ Medium
(19)
r
EtOH -1.L. Ad + NADH r BJ
Ac -
(20)
Cell mass
(21)
r
Ac~2C02+ATP+4NADH r
2 ATP+ Sint+ 2NADH2..L..2 Gl r
(22) (23)
NADH + 1/2 O2 --1!l..2 PlO ATP
(24)
ATP -
(25)
r ATP
r
Growth processes
NADH -.!1..... Growth processes and by-products,( 26)
Deterministic Growth Model of Saccharomyces Cerevisia e
Since glucose uptake is a rate limiting step in glycolysis the Michaelis-Menten-Kinetic can be assumed to be valid for the specific glucose uptake rate qs (Becker, Betz, 1972; Sols, 1966; van Stever1inck, 1969). In case of fermentative growth glycerol is a obl igatory hydrogen acceptor. Its formation rate aerobically or anaerobically is proportional to the ethanol production rate. Consumption of carbohydrates and metabolites of the TCC for anabolism is described by r B1 and rB3. The elemental cells is nearly composition of the constant (Dekkers, deKok, Roels, 1981; Weibel, 1973), so that these rates are proportional to the specific growth rate \1. This is not exactly met by r Bl , since the carbohydrates concentration of some stored diminishes at increasing ~ (Kuenzi ,1970). Consumption of NADH for other by-products and biosynthesis and loss of acetaldehyde to the medium ismodelledby r R2 and r Adl . With that the following system of differential equations for the concentrations of metabolites is obtained from the reaction scheme.
Pyr(t) =[ 2 rs(t) - rAd(t) - rA c(t) ]X(t) h1(t) = [rAd(t) - (l+K AT ) rEl(t) +r E2(t) ]X(t) A~(t) = [rAc(t) - rTCC(t) - KS3~(t)] X(t)
NAi)H (t) = [2rs(t) +rAc(t) +4r Tcc(t) +rE2(t) - 2q02(t) - (1 +KS2 +2 KEG)rEl (t)] X(t) ATP(t) = [(2+R(-rs))r s(t) +rTcc(t) +2P/Oq02(t) - ~(t)/Y ATP -mATP - 2 KEGr El (t)] X(t)
OUR(t) = q02(t) X(t) Mo CPR(t) = [rAd+2rTCC(t) +rAc(t)] X(t) Mc =Qco2(t)X(t)Mc
0
01
R(r) = 1 I'I'e1 rr">< 0
1
69
With this set of equations it has not been possible to simulate yeast growth without further assumptions . Additional predications about the metabolic regulations are necessary to determine the partition of r ' r Ad , r Ac as a function of glucose uptake Srate, ethanol reaction rate and oxygen uptake rate, from which the actual specific growth rate follows. The usual method now is to introduce the unknown reaction rates as complex functions of known rates, metabolites, enzvme or substrate concentations, which are" hardly to confirm with experiments. A simplermodellingof the metabolic regulation is possible on the assumption that the cell strives for the maximum reachable growth rate at given conditions. The metabolic regulation then has the task to limit the rate of fast reactions to give a coordinated overall metabolism without immoderate accumulation of intracellular substances. The slowest steps are then decisive for the whole metabolism. This corresponds to a generalization of Blackman's Master-Reaction-Concept to cell growth (Lapidus, Amundson, 1977). The possible rate limiting steps in this model are the
glucose uptake rate, ethanol uptake rate, (28) oxygen uptake rate, the rates of the respiratory chain or the (29) gluconeogenesis controlled by enzymes and the rate of the starting reaction of the (~) TCC : r Ac . The saturation of the last reaction at high (31 ) gl ucose uptake rates in thi s model is the only reason for the Crabtree-effect. Barford and Hall (1979) stated that saturation of the respiratory chain for another yeast strain occurs. Although this would simplify (32) modelling, here this seems not to be true, because the respiration rate falls considerably below its maximum value when growth is more fermentative. The saturation of r A C coincides with the observation by Meyenburg (1969), that the concentration of (34) pyruvate increases remarkably at the change from oxidative to aerobic fermentative (35) growth, by which the parallel path of r Ac is opened (r Ad ) and ethanol production occurs. (36)
After application of the quasi steady state approach and elimination of r and r TCC the algebraic set of equations Ad KSl - 4K S3
0 0 0
rs KEG
-2 1 -1-2K EG -K S2 rAc 1 P Q02 2+R(-rS) -K S3 - - 1 2- 0 -2K EG YATP 0 r E2 0 -1 0 1 -I-KAT rEl IDekkers, deKok, Roels (1981), Peringer and others (1974), Schatzmann (1975).
5
qs
0 0 mATP
(37)
K.-H. Bellgardt, W. Kuhlrnann and H.-D. Meyer
70
for cellular reactions is obtained. The rate limiting steps are given by the generalized Blackman-Kinetics (38) ethanol limitation, -rE ~KE E (39) oxygen limitation, -Q02 ~Ko OF (40) Crabtree-effect,
E
g
~
and limitation kinetics by the enzyme levels of Q02~1/2rNADmax the respiratory chain, the
gluconeogenesis.
(41)
'f!
(42)
Growth lag phases are modeled by the last two inequalities,which are considered later. At modelling of yeast strains with saturated respiratory chain Eq. (4) must be replaced by r NAD ~ rNADsat' where the saturation level rNADsat is given by the yeast physiology. Now the system (37) can be solved under constraints (38) - (42) by maximizing the specific growth rate ~ with given S,E,OF' rNADmax and rSmax' Since the substrate uptake rate qs' which is the independent variable of the system (37), was introduced asl>1onod-kineticthis model is called Multiple-Monod-Blackman-kinetic. The model behaviour is as follows. If glucose concentration in the fermenter is below 55 mg/l growth is oxidative and only the glucose uptake is rate limiting. Then ethanol can be consumed in addition to the glucose. On a mixed glucose and ethanol substrate the rate of ethanol consumption is given by the remaining capacity of r A ' and therefore the ethanol uptake rate decr~ases with increasing qS' This agrees with experimental results by Geurts, deKok, Roels (1980) and Woehrer, Roehr (1981) .At increased glucose concentrations the Crab tree-effect arises due to saturation of r A and ethanol production starts. The oxygen limitation or anaerobiosis has a similar effect. In this case the actual maximum rates of the TCC and of r A are reached at lower glucose concentrations. Conversely fermentative growth can be shifted to more oxidative by improving the oxygen supply (Pasteureffect). Ethanol growth can be limited by the ethanol concentration rE' the actual possible rate of the respiratory chain or the rate of gluconeogenesis.
0.1
-
~~
0.2
0.3
[llhJ
Qeo]
M
0
E
E
~
0
~Lfl
'" 0
"-
o
[!
~I
~
":i'
.5'
)
~
~
Os
~
~ 'E
I
. ":i'
I ~ ~c:i
i
~
II
'0
._ . ..L_
0.3
Fig. 2. Simulation of an anaerobic chemostat culture of SACCHAROMYCES CEREVISIAE H1022 on 3 % glucose (Schatzmann, 1975).
STEADY STATES IN CHEMOSTAT For simulation of steady states as a function of the dilution rate D the model equations were (1) - (5), (9) - (12), (37) - (42) solved for d/dt = 0 by the Newton-RaphsonIteration-Procedure given in Carnahan, Luther, Wickes (1969) . The identified model parameters are shown in table 1 and simulation results of an anaerobic culture by Schatzmann (1975) at 3 % glucose reservoir concentration in Fig. 2 (lines: simulation, symbols: measurements). At medium and high dilution rates growth is fully described by the Monod-Kinetic of qs' At low dilution rates where the cell mass falls down
a
'"
Fig. 3. Simulation of an aerobic chemostat culture of SACCHAROMYCES CEREVISIAE H1022 on 3 % glucose (von Meyenburg, 1969).
71
Deterministic Growth Model of Saccharomyces Cer e vi s ia e
the influence of energy maintenance metabolismcanbe seen. All intracellular reaction rates are linear functions of the dilution rate which is identical to the specific growth rate ~. Figure 3 shows the simulation of a strict aerobic culture on 3 % glucose (data by Meyenburg (1969)). Below the critical substrate uptake rate qs 't = 0.5 glgh growth is oxidative with crl high growth yield. At higher dilution rates r Ac is saturated and ethanol is produced. The steeper increase of qs is a consequence of lower growth yield due to ethanol production which results in higher glucose concentrations in the reactor. With the reduced q02 growth becomes more fermentative and at ~max
= 0.46 11h rE' r Gl and qC02 are similar to those in the anaerobic case. The simulation results of aerobic and anaerobic growth can not be obtained with the same parameters if they are constant. Provided the parameters are only identified with aerobic chemostat data, anaerobic growth is not correctly described, which means that the maximum specific growth rate would increase from 0.32 11h to 0.42 11h and growth yield from 0.1 to 0.14. If the same model structure is proposed for both cases, not all parameters can remain constant. This conclusion is also confirmed by other authors (Dekkers, deKok, Roels, 1981; Geurts, deKok, 1980; Reringer and others, 1974), who found different values of YATP and PlO at distinct growth states. In case the same energy consumption by growth processes at aerobic and anaerobic growth is assumed the experimental results can be interpreted as a variable effectiveness of the respiratory chain (PlO). Under this assumptions the PlO-ratio as a function of qs' given in Fig.4, was identified together with the other parameters. The obtained values (see Table 1) are YATP = 12 glmolATP and PIO ox = 0.93 at oxidative growth. At fermentative growth the PlO-ratio increases up to PlO max=2.7 which is
identical to that reported by Peringer and others (1974). Although this value is below the theoretical upper bound of three, it exceeds the maximum expected of two for Saccharomyces Cerevisiae if site I in oxidative phosphorylation is not operating (Ohnishi and others, 1966). Whether this is due tomodellingor experimental errors (Barford, Hall, 1979a) or a special attribute to this yeast strain can not be decided so far. In tendency a higher specific oxygen uptake rate at growth under Crabtree-effect would give a lower maximum PlO-ratio. EXTENSION OF THE MODEL TO BATCH CULTURES The diauxic lag phase at beginning of ethanol growth in yeast batch cultures can be explained as either induction of gluconeogenetic enzymes or derepression of oxidative enzymes after the phase of glucose growth with reduced respiratory activity. In this model a simple introduction of the metabolic regulation is possible if the actual maximum reaction rates are subordinated to the actual growth state of the cells, modelled by the ma ximum rates of the respiratory chain rNADmax or gluconeogenesis rSmax' These maximum rates can be obtained as output variables of simple linear systems which have as their inputs the maximum possible rates at the given fermenter conditions rNADopt and r Sopt ' As an example growth dynamics shall be
1-----.---, -.
..-........... ......... .-......"............ 0( . --~
- - , .. - -
~
E
•
~
Cl
o
-0
V
V i i
I
1
/
/
-
20 . - .-- - .~
-.- --. -,- -
-
~
...
- ,_.
NI Q
'"o
0
er, -0
Ii
I
J
i
i
ii
i
0.01 0.02 spezif,c Substrate Uptake Rate q, Imol/ghl
Fig. 4.Identified course of the effectiveness of the respiratory chain (PlO ratio in mol ATPlatom 0) at aerobic fermentative growth.
20
Fig. 5. Simulated time course of an aerobic batch culture of SACCHAROMYCES CEREVISIAE H1022 with 3 % initial glucose.
K.-H. Bellgardt, W. Kuhlmann and H.-D. Meye r
72
--~---~ T ~~-·--- ··
-r··- -
.- .
, - -.
,. --,-----,--,--- ---j
o
10 I
.
- ~
~
r ];
~
'£
Vl
0
er :::> o
15
(hi
Fig. 6. Simulated time course of an aerobic batch culture with 3 % initial glucose, oxygen limited growth on ethanol. shown assuming the diauxic lag phase is mainly caused by induction of gluconeogenetic enzymes, although other regulation models can be easily constructed. The model equations for the actual maximum rates of respiratory chain and gluconeogenesis are TNAD rNADmax(t) +rNADmax(t)
=
rNADopt(t)
(43)
TIT2 -r"Smax(t) + (TI+T2) rSmax(t) +rsmax(t) = rSopt(t) _ (44)
where J1(PI) PI -
cost functional, vector of unknown mode 1 parameters, l~(ti) vector of measurements at time t i , ylt- ,PI) model output vector, H 1 - diagonal weighting matrix, I number of measurements, () T transpose of a matrix . With this method the parameters of the regulation model Eq .( 43),(44) and the initial values of the state vector, combined in one vector p = (TNAD,Tl'T 2,X(0),S(0),rNA Dmax (0),KE,Ko)' can be identified with the output vector l(t) = (X(t),S(t),E(t),OUR(t),CPR(t)) T . By application of the numerical optimization procedure by Nelder-Mead (1965) the optimal parameter vector is obtained after approximate 500 iteration steps. This identification method can not be applied to the estimation of the other model parameters. Since the identification problem can only be solved uniquely with simultaneous consideration of oxidative, maximum aerobic and anaerobic glucose growth as well as ethanol growth, one has to pay regard to a great number of output variables and measurements. In most cases the identification procedure did not converge to the optimum. Therefore an interactive two level parameter identification procedure was implemented on a digital computer. The schematic diagram is shown in Fig. 7.
The simulation of an aerobic batch culture on Iteration count i = O. Set starting 3 % glucose is shown in Fig. 5. The first values to the maximum reaction growth phase on glucose is nearly exponential rate vectors (.':ul'.':u2'.':u3'.':u4)O and therefore the specific rates of oxygen uptake and carbon dioxide production remain constant, whereas the maximum specific growth + rate on ethanol has not been reached after 20 h. During the diauxic lag phase and at the t end of the fermentation the oxygen uptake rate Next iteration step i = i • I a: decreases to values of the maintenance metabo Simulate with the actual (£2)i >« olism given by mATP ' It can be seen that the a: ... behaviour during the lag phase is not exactly o modelled by the simple linear systems (43,44) with real poles, since the measured oxygen consumption course is oscillatory. Improve (.':uI'.':u2'.':u~'.':u4Ii The simulation of an oxygen limited batch cula: ... ture is plotted in Fig. 6. During fermentation => the impeller speed is repeatedly changed, which i gives rise to abrupt variations of the dissol- - ved oxygen concentration. After 7. 3 h the growth on ethanol becomes oxygen limited and Compare (.':uj) i and (C:l j I i and the dissolved oxygen concentration falls down minimize J ( 102) to get the below 0.03 mg/l. The linear growth with constant 2i new parameter estimate (~2)i.1 oxygen uptake rate is a consequence of oxygen ...a: o transfer limitation by the reactor. An increase J of impeller speed at 12,13,14 and 15 hours produces higher growth rates. Fig. 7 Schematic diagram of the interactive two-level parameter identification THE PARAMETER IDENTIFICATION PROCEDURE procedure.
!
0-
00-
-r-
:0
~
A usual method for off-line parameter identification with non-linear deterministic models is the minimization of the weighted squared differences of measurements and model outputs, as a function of the parameters. The performance index (cost functional) is then given by L
J1(e) = I(lM(t )-l(ti ,£ I) )Tli (lM(ti ) -l(ti i=1
,El))-+ min
(45)
At the upper level (Operator) vectors of maximum rates r . of the four growth phases are estimated.-uJAt the lower level the cost functional 4
T
•
J2(~) = .I(.!:uj -.!::.!j(~)) li(.!:uj -.!::.!j(.ez))- nnn J=1
where J 2 (E2) cost functional
(46)
Deterministic Growth Hodel of Saccharomyces Cerevisiae
£.z = (qS.a, ,rA"" '~I '~Z'~3 ,KAT>~G' YATP,mATP'P/O" ,P/O.,,), vector of unknown parameters .':u j = (~u' qSu ' q0 z u ' qeo z u ' rE u ) j vector of reaction rates givenby the upper level for different growth states j .':1 j (£.z) vector of reacti on rates computed at the lower level with parameters Pz j index of growth state 1: maximum oxidative glucose growth 2: maximum aerobic glucose growth at Crabtree-effect 3: maximum anaerobic glucose growth 4: maximum ethanol growth is minimized at each iteration step. Then the growth behaviour of batch and che~ostat cultures is simulated with the obtained pa rameters. I n the next i tera t ion step the estimate of the maximum reaction rate vector can be improved at the upper level by comparison of simulation results and measurements. The iteration procedure requires approximately 20 steps which is also the number of necessary simu1ations in contrast with several thousands by use of method (46) without reaching convergence. Since the computational effort at the lower level is small a substantial saving of computer time can be achieved. CONCLUS IONS In this work a dynamic model of the yeast SACCHAROMYCES CEREVISIAE HI022 is proposed by considering a simplified reaction scheme of the matabo1ism. The necessary assumptions about regulation of metabolic pathways are expressed as a Multiple Monod-B1ackmanKinetic. After introduction of an increasing PlO-ratio with decreasing respiratory activity the aerobic, oxygen limited and anaerobic growth on glucose and ethanol can be described. Verifications of some model predictions require further experiments. The simulations indicate that an improved modelling of the diauxic lag phase could be achieved by a linear second order system with conjugate complex poles. The model parameters were identified with chemostat and batch data by application of an interactive two-level procedure. This work was supported by Stiftung Vo1kswagenwerk.
Table 1 Identified Parameters of Chemostat Growth 0.018 mol/gh qs 0.4 g/l Ks 0.0048 mo1/g KSI 0.39 mol/mol Ksz 0.0044 mol/g Ks 3 0.071 mol/mol KEG 0.17 mol/mol Kr 0.93 mol ATP P 0" P/O. a, 2.7 atoiiilf YATP 11.96 g/mol 0.0003 mo1/gh mAJP r Ac • a x 0.0034 mol/gh
Table 2 Additional Identified Parameters for Batch Cultures TNad 4.0 h 2.4 h TI Tz 2.8 h 0.016 mol/gh qs 0.0053 mol/g Ks 1 0.05 mol/mol KEG 0.08 mol/mol KAT mATP 0.0025 mol/gh 1.1 l/gh KE 12.8 l/mg h Ko
73
NOMENCLATURE Ac Ad
Acetyl-CoA (-concentration), mo1/1. Acetaldehyde (-concentration), mo1/1. CF dissolved COz-concentration, mg/1. CIe maximum COz-saturation concentration at pure COz-atmosphere, mg/1 . carbon dioxide COz CPR COz-production rate, mg/1 h. CTR COz-transfer rate, mg/1 h. D dilution rate (= F/V), 1/h. E ethanol concentration, g/l. EtOH ethano 1 (-concentrat ion), mol 11 . relative Ad-excretion rate. relative rates of bio-synthesis. = 0.8, ratio of COz- and Oz-diffusioncoefficients in the medium. B1ackman constant of ethanol, 1/g h. relative glycero1e production rate. B1ackman constant of oxygen, 1/mg h. saturation constant of glucose, g/l. volumetric absorption coefficient, 1/h. ATP-maintenance coefficient, mo1/g h. molecular weight, g/mo1. particle quantity, mol. particle flow, mo1/h. oxygen dissolved oxygen concentration, mg/1. maximum Oz-saturation concentration at pure Oz-atmosphere, mg/1. OTR oxygen transfer rate, mg/1 h. OUR oxygen uptake rate, mg/1 h. p pressure, bar. Pyr pyruvate (-concentration), mo1/1. PlO effectiveness of oxidative phosphorylation, mol ATP/atomO. specific COz-production rate, mo1/g h. specific Oz-uptake rate, mo1/g h. specific glucose uptake rate, mo1/g h. qSmax maximum value of qs, mo1/g h. specific Ac-formation rate, mo1/g h. rAcmax saturation rate of rAc, mo1/g h. rAd specific Ad-formation rate, mo1/g h. rAIP ATP consumption for growth, mo1/g h. specific rates for bio synthesis, mo1/g h. = rEl+rEZ, EtOH reaction rate, mo1/gh. specific EtOH production rate, mo1/g h. specific EtOH uptake rate, mo1/g h. = 2rGl,glycero1e~roduction rate,mo1/gh. specific substrate consumption for glycero1e formation, mo1/g h. specific rate of respiratory chain, mo1/g h. rs specific rate of glycolysis, mo1/g h. rlee specific rate of the TCC, mo1/g h. R gas constant, J/mo1 K. RQ = qeoz/qoz, respiratory quotient. S glucose concentration, g/l. Sext glucose in the medium, g/l. Sint glucose in the cells, g/l. T temperature, K. Tl,T z ,T NAD time constants, h. Y liquid phase volume, 1. VE inlet gas stream, 1/h. VG gas phase volume, 1. x mole fractions of 02, C02, nitrogen. X bio mass concentration (dry weight),g/l. YAIP ATP yield coefficient, g/mo1. ~ specific growth rate, 1/h.
K.-H. Bellgardt, W. Kuhlamnn and H.-D. Meyer
74
C Carbon dioxide. E Ethanol. N Nitrogen. o Oxygen. R Reservoir. S Glucose. Superscripts A Outlet gas stream. E Inlet gas stream. T Transfer gas phase/liquid phase.
Subscripts
REFERENCES Barford, J.P. and Hall, R.J. (197~). An exa~-. ination of the Crabtree-effect ln S.cerevlslae. . Journ. of Gen. Mikrobiol., 114,267. Barf ord, J.P. and Hall, R.J.(1979). Investlgation of the significance of carbon and redox balances to the measurements of gaseous metabolism of S.cerevisiae. Biotech.Bioeng., 21, 609. Barford, J.P. and Hall, R.J.(19S1). A mathematical model for the aerobic growth of S.cerevisiae with saturated respiratory capacity. Biotech.Bioeng. ,23, 1735. Becker,J.U. and Betz, A.(1972). Membran transport as controlling pacemaker of glycolysis in Saccharomyces carlsb. Biochem. Biophys. Acta, 274, 5S4. Bergter,F. and Knorre, W.A. (1972). Co~putersimulation von Wachstum und Produktblldung bei S. cerevisiae. Zeitschr. f.allg. Mikrobiologie, 12,S, 613. Bijkerk, A.H.E. and Hall, R.J. (1977). A mechanistic model of the aerobic growth of S.cerevisiae. Biotech.Bioen~.,19, 267. Carnahan, B., ~uther, H:A. and Wlckes,J.O. (1969). Applled numerlcal methods, 319, John Wiley & Sons, N.Y. Dekkers,J.G.E., deKok,H. and Roels,J.A.(1981) Energetics.of S.cerevi~iae CBS426.~0~par~son of anaeroblC and aeroblc glucose llmltatlon. Biotech. Bioeng. 23, 1023. . Fiechter, A.(1980). Regulatory aspects ln yeast metabolism. In M.Moo-Young (Ed.), Advances in Biotechnology, Vol.1, Pergamon Press, Toronto, p. 261. Fredrickson, A.G. and Tsuchiya, H.M.(1977). Microbial kinetics and dynamics. In L.Lapidus and N.R. Amundson (Ed.), Chemical reactor theory,Prentice-Hall ,Inc., New Jersey, p.405. Geurts,T.G.E.; deKok,H.; Roels,J.A. (1980). A quantitative description of the growth of S.cerevisiae CBS 426 on a mixed substrate of glucose and ethanol. Biotech.Bioeng.,22, 2031. Hall,R.J. and Barford, J.P.(1981) Simulation of the integration of the energy metabolism and the cell cycle of S.cerevisiae. Biotech.Bioen 23, 1763. Kuenzi, M.(1970 . Ober den Reservekohlenhydratstoffwechsel von S. cerevisiae. Dissertation, ETH-Zurich. Meyenburg,K.(1969) Katabolitrepression und S~rossungszyklus von S.cerevisieae, Dlssertation ETH-Zurich. Nelder,R. and Mead, J.A. (1965). A simplex method for function minimization. The Computer Journal, 7, 30S.
., J
Ohnishi,T.; Sottocasa,G.; Ernster,L.(1966). Bull.Soc.Chem.Biol.,4S,l1S9 Pamment,N.B. ;Hall ,R.J.; Barford,J.P. (197S). Mathematical modelling of Lag phases in microbial growth.Biotech. Bioeng. ,20,349 Peringer, P.; Blachere, H.; Corrieu, G. Lane, A.G. (1973). Mathematical model of the the kinetics of S. cerevisiae. Biotech. Bioeng. Symp. No. 4, 27. Peringer, P.; Blachere, H.; Corrieu, G.; Lane, A. G. (1974). A generalized ~athe matical model for the growth kinetics of S. cerevisiae with experimental determination of parameters. Biotech. Bioeng., 16,
Ra~~:V.M. (196S). Absor~tion of gases.
Israel Program for SClent.Transl. ,Jerusalem Schatzmann,H. (1975). Anaerobes \~achstum von S.cerevisiae. Dissertation, ETH-Zurich. Sols, A.(1966). Symo. on Some Aspects of Yeast Metabolism. Dublin, 1965. Blackwell Sci. Publ., Oxford, p. 47. van Steveninck. J. (1969). The mechanism of transmembran glucose transport in yeast. Archives of Biochem. and Bio hys., 130,244. Toda, K.; Yabe, J.; Yamagata, T. SO Kinetics of biphasic growth of yeast in continuous and fpdbatch-cultures. Biotech.Bioeng., 22, lS05. Weibel, E.K. (1973). Zur Energetik von S. cerevisiae. Dissertation, ETH-Zurich. Westerterp, K.R.; van Dierendonck, L.L.; de Kraa, J.A. (1962). Chem.Ing.Science,18. Woehrer, l~. and Roehr, M. (1981). Regulatory aspects of baker's yeast metabolism in aerobic fed-batch-cultures. Biotech.Bioeng., 23, 567. Vagi, H. and Yoshida, F. (1975).Enhancement factor for oxygen absorption into fermentation broth. Biotech.Bioeng., 17, 1083. Yagi, H. and Yoshida, F. (1977). Desorption of carbon dioxide from fermentation broth. Biotech.Bioeng., 19, SOl. Yoon, H.; Klinzing G.; Blanch, H.W. (1977). Competition for mixed substrates by microbial populations. Biotech.Bioeng. ,19, 1193.
DETERMINISTIC GROWTH MODEL OF SACCHAROMYCES CEREVISIAE, PARAMETER IDENTIFICATION AND SIMULATION K.-H. Bellgardt*, W. Kuhlmann** and H.-D. Meyer** *Institut fur Regelungstechnik, Um'versitiit Hannover, D 3000 Hannover, Federal Republic of Germany * *Institut fur Techmsche Chemie, Universitiit Hannover, D 3000 Hannover, Federal Republic of Germany
Abstract. A mathematical model of the growth of SACCHAROMYCES CEREVISIAE H1022 was developed, which can describe oxidative and aerobic fermentative(Crabtreeeffect) growth on glucose, oxidative growth on ethanol, oxygen limited growth on glucose and/or ethanol and the anaerobic growth. A description of the complex growth behaviour with good model accuracy can be obtained with reaction kinetic models which take into consideration the more important metabolic processes in form of balance equations. In this paper a simplification of modelling is obtained by application of Blackman's master-reaction-concept and a quasi steady state approach for cellular metabolic reactions. Most of the parameters of the biological model are advantageously identified with stationary chemostat data by an iterative two-level optimization algorithm. In a following step the parameters which influence the simulated growth lag phases are identified with batch data. Simulations of batch and chemostat cultures show good correspondence of measurements and model predictions if the effectiveness of the respiratory chain (P/O) is not constant but increases with more fermentative growth and reduced oxygen uptake rates. Keywords. Growth models, parameter estimation, convergence of numerical methods, nonlinear systems, chemostat, batch culture, metabolic regulations, Crabtree-effect, Pasteur-effect. I NTRODUCTI ON For some special cases of yeast growth, e.g . pure aerobic growth or growth on a single substrate, empirical models are known, based on an extension of the MONOD-equation (Bergter, Knorre, 1972; Bijkerk, Hall, 1977; Dekkers, deKok, Roels, 1981; Pamment, Hall, 1978; Peringer and others, 1973 and 1974; Toda, Yabe, Yamagata, 1980; Yoon, Klinzing, Blanch, 1977). The generalization of such models to other growth states is not known. Under this point of view models are more successful, which take into consideration chemical reactions of metabolism (Dekkers, deKok, Roels, 1981; Geurts, deKok, Roels, 1980; Hall, Barford, 1981) .
In this paper a deterministic growth model of baker's yeast SACCHAROMYCES CEREVISIAE H1022 on glucose and ethanol as limiting carbon sources is presented. The model shall be used for structure and parameter optimization of continuously operating multi stage stirred tank plants (chemostat cascade). Furthermore the application of the model to an automatic computer control of the process is provided. Hence the model must have a sufficient accuracy and complexity. A by-product of accurate modell ing can be a better insight into the biological processes in living cells. The growth behaviour of this yeast can be roughly subdivided into oxidative growth with high cell yield and high oxygen uptake rates and fermentative growth with ethanol production (Meyenburg, 1969; KUenzi, 1970; Weibel, 1973; Schatzmann, 1975; Fiechter, 1980). Fermentative growth is induced by oxygen limitation or high glucose uptake rates (Crabtree -effect). Every growth state is characterized by a complex cooperation of substrate uptake rate, oxygen uptake rate, cell morphology and of intracellular enzyme pattern, from which as a consequence of metabolic regulation the known diauxic growth in batch cultures and the two growth phases in the aerobi c chemostat follow.
In the following a model of a stirred tank is given first. Then the biological model based on a simplified metabolic reaction scheme and finally simulation results and the used parameter identification procedure are presented. REACTOR MODEL In this section the mathematical model of a stirred tank reactor is given, which is independent of the biological model. The reactor model contains the description of the gas phase and the liquid phase in the 67
68
K.-H. Bellgardt, W. Kuhlmann and H.-D. Meyer
~G(t) =
I
Inlet Gos
Outlel Gos
~ "ni :'., F
aTR
X.
X
E.
E
s.
eTR
GasPhase
1 - XOG E - XCG E
x~G
A
Ovorflow
LiquidPhase
S
0,. c,.
/Av p V
1 - x~G(t) - x~G(t)
II
xCG (t)
=
0,
KlaRvi () E"E OF t
+
x8G I mpeller Speed N
+
+
(ll)
KlaR vi 0' /Av pE VE F
Ko Kl a RV TE
CF(t) Mc pE VE KoKlaRVTE l-x~G-x~G + Mc pE VE CF 1 - x~G(t) - x~G(t)
(12)
C,
Reservoir
Reactor
Fig. 1. Schematic diagram of the stirred tank. tank with an entire balance of oxygen and carbon dioxide, since without that correct mode 11 i ng of oxygen 1i mi ted growth is not possible . Figure 1 shows the schematic diagram of the reactor. It is assumed that gas and liquid phase are ideally mi xed and therefore may be considered as homogeneous. The differential equations of the reactor are (see nomenclature) for the liquid phase (' = d/dt) (1)
S(t) X(t)
~(t)
X(t)
(2)
E(t)
(3)
DF(t)
=
-q02(t) Mo X(t) + D(OFR -OF(t)) + OTR(t) (4)
CF(t)
=
qC02(t) Mc X(t) + D(C fR -CF(t)) - CTR(t) (5)
and for the gas phase , 'E 'A 'T no(t) = no - no - no
(6) (7)
(8)
Exchange of nitrogen between gas phase and liquid phase as well as water and ethanol content of the gas phase are neglected. The mass transfer rates of oxygen and carbon dioxide are given by (Ramm, 1968; Westerterp and others, 1962; Yagi, Yoshida, 1975 and 1977) OTR(t)
K1a (OF x~G(t)
0f(t))
(9)
CTR(t) Since the time constants of the gas phase are small compared to the others a steady state apporach is introduced in Eq. (6) - (8). After application of the gas laws and DALTON's law the following equations, which can be solved iteratively, for the mole fractions in the outlet gas are obtained:
MODEL OF METABOLIC ACTIVITY The biological model of yeast growth permits the determination of metabolic activities as a function of glucose, ethanol and dissolved oxygen concentrations, which are given by the reactor model. Morphological changes of cells or variations during the budding cycle reported by Hall and Barford (1981) , Kuenzi (1970) and Meyenburg (1969) were not considered, so that the model predicts an average behaviour of the culture. The model is restricted to the main metabolic reactions, which are glycolysis via the EmbdenMeyerhof-pathway, removal of the resulting pyruvate by the Tri-Carbon-Acid-cycle (TCC) and ethanol synthesis. In order to simplify the model it is assumed that pyruvate is a direct catabolite of ethanol, and that gluconeogenesis is the reversal of the glycolysis, although in reality ethanol is catabolized to acetyle-CoA and gluconeogenesis is started from the Glyoxylate-cycle . The proposed model structure is qs Sot - S i nt
(13)
r
Sin t -L 2 Pyr + 2 ATP + 2 NADH r BI
Si nt -
( 14)
Cell mass + stored carbohydrates (15)
r Ad
Pyr-Ad +C0 2
(16)
Pyr~Ac +NADH +C0 2
(17)
Ad + NADH ~ EtOH
(18)
r
r
r
Ad ~ Medium
(19)
r
EtOH -1.L. Ad + NADH r BJ
Ac -
(20)
Cell mass
(21)
r
Ac~2C02+ATP+4NADH r
2 ATP+ Sint+ 2NADH2..L..2 Gl r
(22) (23)
NADH + 1/2 O2 --1!l..2 PlO ATP
(24)
ATP -
(25)
r ATP
r
Growth processes
NADH -.!1..... Growth processes and by-products,( 26)
Deterministic Growth Model of Saccharomyces Cerevisia e
Since glucose uptake is a rate limiting step in glycolysis the Michaelis-Menten-Kinetic can be assumed to be valid for the specific glucose uptake rate qs (Becker, Betz, 1972; Sols, 1966; van Stever1inck, 1969). In case of fermentative growth glycerol is a obl igatory hydrogen acceptor. Its formation rate aerobically or anaerobically is proportional to the ethanol production rate. Consumption of carbohydrates and metabolites of the TCC for anabolism is described by r B1 and rB3. The elemental cells is nearly composition of the constant (Dekkers, deKok, Roels, 1981; Weibel, 1973), so that these rates are proportional to the specific growth rate \1. This is not exactly met by r Bl , since the carbohydrates concentration of some stored diminishes at increasing ~ (Kuenzi ,1970). Consumption of NADH for other by-products and biosynthesis and loss of acetaldehyde to the medium ismodelledby r R2 and r Adl . With that the following system of differential equations for the concentrations of metabolites is obtained from the reaction scheme.
Pyr(t) =[ 2 rs(t) - rAd(t) - rA c(t) ]X(t) h1(t) = [rAd(t) - (l+K AT ) rEl(t) +r E2(t) ]X(t) A~(t) = [rAc(t) - rTCC(t) - KS3~(t)] X(t)
NAi)H (t) = [2rs(t) +rAc(t) +4r Tcc(t) +rE2(t) - 2q02(t) - (1 +KS2 +2 KEG)rEl (t)] X(t) ATP(t) = [(2+R(-rs))r s(t) +rTcc(t) +2P/Oq02(t) - ~(t)/Y ATP -mATP - 2 KEGr El (t)] X(t)
OUR(t) = q02(t) X(t) Mo CPR(t) = [rAd+2rTCC(t) +rAc(t)] X(t) Mc =Qco2(t)X(t)Mc
0
01
R(r) = 1 I'I'e1 rr">< 0
1
69
With this set of equations it has not been possible to simulate yeast growth without further assumptions . Additional predications about the metabolic regulations are necessary to determine the partition of r ' r Ad , r Ac as a function of glucose uptake Srate, ethanol reaction rate and oxygen uptake rate, from which the actual specific growth rate follows. The usual method now is to introduce the unknown reaction rates as complex functions of known rates, metabolites, enzvme or substrate concentations, which are" hardly to confirm with experiments. A simplermodellingof the metabolic regulation is possible on the assumption that the cell strives for the maximum reachable growth rate at given conditions. The metabolic regulation then has the task to limit the rate of fast reactions to give a coordinated overall metabolism without immoderate accumulation of intracellular substances. The slowest steps are then decisive for the whole metabolism. This corresponds to a generalization of Blackman's Master-Reaction-Concept to cell growth (Lapidus, Amundson, 1977). The possible rate limiting steps in this model are the
glucose uptake rate, ethanol uptake rate, (28) oxygen uptake rate, the rates of the respiratory chain or the (29) gluconeogenesis controlled by enzymes and the rate of the starting reaction of the (~) TCC : r Ac . The saturation of the last reaction at high (31 ) gl ucose uptake rates in thi s model is the only reason for the Crabtree-effect. Barford and Hall (1979) stated that saturation of the respiratory chain for another yeast strain occurs. Although this would simplify (32) modelling, here this seems not to be true, because the respiration rate falls considerably below its maximum value when growth is more fermentative. The saturation of r A C coincides with the observation by Meyenburg (1969), that the concentration of (34) pyruvate increases remarkably at the change from oxidative to aerobic fermentative (35) growth, by which the parallel path of r Ac is opened (r Ad ) and ethanol production occurs. (36)
After application of the quasi steady state approach and elimination of r and r TCC the algebraic set of equations Ad KSl - 4K S3
0 0 0
rs KEG
-2 1 -1-2K EG -K S2 rAc 1 P Q02 2+R(-rS) -K S3 - - 1 2- 0 -2K EG YATP 0 r E2 0 -1 0 1 -I-KAT rEl IDekkers, deKok, Roels (1981), Peringer and others (1974), Schatzmann (1975).
5
qs
0 0 mATP
(37)
K.-H. Bellgardt, W. Kuhlrnann and H.-D. Meyer
70
for cellular reactions is obtained. The rate limiting steps are given by the generalized Blackman-Kinetics (38) ethanol limitation, -rE ~KE E (39) oxygen limitation, -Q02 ~Ko OF (40) Crabtree-effect,
E
g
~
and limitation kinetics by the enzyme levels of Q02~1/2rNADmax the respiratory chain, the
gluconeogenesis.
(41)
'f!
(42)
Growth lag phases are modeled by the last two inequalities,which are considered later. At modelling of yeast strains with saturated respiratory chain Eq. (4) must be replaced by r NAD ~ rNADsat' where the saturation level rNADsat is given by the yeast physiology. Now the system (37) can be solved under constraints (38) - (42) by maximizing the specific growth rate ~ with given S,E,OF' rNADmax and rSmax' Since the substrate uptake rate qs' which is the independent variable of the system (37), was introduced asl>1onod-kineticthis model is called Multiple-Monod-Blackman-kinetic. The model behaviour is as follows. If glucose concentration in the fermenter is below 55 mg/l growth is oxidative and only the glucose uptake is rate limiting. Then ethanol can be consumed in addition to the glucose. On a mixed glucose and ethanol substrate the rate of ethanol consumption is given by the remaining capacity of r A ' and therefore the ethanol uptake rate decr~ases with increasing qS' This agrees with experimental results by Geurts, deKok, Roels (1980) and Woehrer, Roehr (1981) .At increased glucose concentrations the Crab tree-effect arises due to saturation of r A and ethanol production starts. The oxygen limitation or anaerobiosis has a similar effect. In this case the actual maximum rates of the TCC and of r A are reached at lower glucose concentrations. Conversely fermentative growth can be shifted to more oxidative by improving the oxygen supply (Pasteureffect). Ethanol growth can be limited by the ethanol concentration rE' the actual possible rate of the respiratory chain or the rate of gluconeogenesis.
0.1
-
~~
0.2
0.3
[llhJ
Qeo]
M
0
E
E
~
0
~Lfl
'" 0
"-
o
[!
~I
~
":i'
.5'
)
~
~
Os
~
~ 'E
I
. ":i'
I ~ ~c:i
i
~
II
'0
._ . ..L_
0.3
Fig. 2. Simulation of an anaerobic chemostat culture of SACCHAROMYCES CEREVISIAE H1022 on 3 % glucose (Schatzmann, 1975).
STEADY STATES IN CHEMOSTAT For simulation of steady states as a function of the dilution rate D the model equations were (1) - (5), (9) - (12), (37) - (42) solved for d/dt = 0 by the Newton-RaphsonIteration-Procedure given in Carnahan, Luther, Wickes (1969) . The identified model parameters are shown in table 1 and simulation results of an anaerobic culture by Schatzmann (1975) at 3 % glucose reservoir concentration in Fig. 2 (lines: simulation, symbols: measurements). At medium and high dilution rates growth is fully described by the Monod-Kinetic of qs' At low dilution rates where the cell mass falls down
a
'"
Fig. 3. Simulation of an aerobic chemostat culture of SACCHAROMYCES CEREVISIAE H1022 on 3 % glucose (von Meyenburg, 1969).
71
Deterministic Growth Model of Saccharomyces Cer e vi s ia e
the influence of energy maintenance metabolismcanbe seen. All intracellular reaction rates are linear functions of the dilution rate which is identical to the specific growth rate ~. Figure 3 shows the simulation of a strict aerobic culture on 3 % glucose (data by Meyenburg (1969)). Below the critical substrate uptake rate qs 't = 0.5 glgh growth is oxidative with crl high growth yield. At higher dilution rates r Ac is saturated and ethanol is produced. The steeper increase of qs is a consequence of lower growth yield due to ethanol production which results in higher glucose concentrations in the reactor. With the reduced q02 growth becomes more fermentative and at ~max
= 0.46 11h rE' r Gl and qC02 are similar to those in the anaerobic case. The simulation results of aerobic and anaerobic growth can not be obtained with the same parameters if they are constant. Provided the parameters are only identified with aerobic chemostat data, anaerobic growth is not correctly described, which means that the maximum specific growth rate would increase from 0.32 11h to 0.42 11h and growth yield from 0.1 to 0.14. If the same model structure is proposed for both cases, not all parameters can remain constant. This conclusion is also confirmed by other authors (Dekkers, deKok, Roels, 1981; Geurts, deKok, 1980; Reringer and others, 1974), who found different values of YATP and PlO at distinct growth states. In case the same energy consumption by growth processes at aerobic and anaerobic growth is assumed the experimental results can be interpreted as a variable effectiveness of the respiratory chain (PlO). Under this assumptions the PlO-ratio as a function of qs' given in Fig.4, was identified together with the other parameters. The obtained values (see Table 1) are YATP = 12 glmolATP and PIO ox = 0.93 at oxidative growth. At fermentative growth the PlO-ratio increases up to PlO max=2.7 which is
identical to that reported by Peringer and others (1974). Although this value is below the theoretical upper bound of three, it exceeds the maximum expected of two for Saccharomyces Cerevisiae if site I in oxidative phosphorylation is not operating (Ohnishi and others, 1966). Whether this is due tomodellingor experimental errors (Barford, Hall, 1979a) or a special attribute to this yeast strain can not be decided so far. In tendency a higher specific oxygen uptake rate at growth under Crabtree-effect would give a lower maximum PlO-ratio. EXTENSION OF THE MODEL TO BATCH CULTURES The diauxic lag phase at beginning of ethanol growth in yeast batch cultures can be explained as either induction of gluconeogenetic enzymes or derepression of oxidative enzymes after the phase of glucose growth with reduced respiratory activity. In this model a simple introduction of the metabolic regulation is possible if the actual maximum reaction rates are subordinated to the actual growth state of the cells, modelled by the ma ximum rates of the respiratory chain rNADmax or gluconeogenesis rSmax' These maximum rates can be obtained as output variables of simple linear systems which have as their inputs the maximum possible rates at the given fermenter conditions rNADopt and r Sopt ' As an example growth dynamics shall be
1-----.---, -.
..-........... ......... .-......"............ 0( . --~
- - , .. - -
~
E
•
~
Cl
o
-0
V
V i i
I
1
/
/
-
20 . - .-- - .~
-.- --. -,- -
-
~
...
- ,_.
NI Q
'"o
0
er, -0
Ii
I
J
i
i
ii
i
0.01 0.02 spezif,c Substrate Uptake Rate q, Imol/ghl
Fig. 4.Identified course of the effectiveness of the respiratory chain (PlO ratio in mol ATPlatom 0) at aerobic fermentative growth.
20
Fig. 5. Simulated time course of an aerobic batch culture of SACCHAROMYCES CEREVISIAE H1022 with 3 % initial glucose.
K.-H. Bellgardt, W. Kuhlmann and H.-D. Meye r
72
--~---~ T ~~-·--- ··
-r··- -
.- .
, - -.
,. --,-----,--,--- ---j
o
10 I
.
- ~
~
r ];
~
'£
Vl
0
er :::> o
15
(hi
Fig. 6. Simulated time course of an aerobic batch culture with 3 % initial glucose, oxygen limited growth on ethanol. shown assuming the diauxic lag phase is mainly caused by induction of gluconeogenetic enzymes, although other regulation models can be easily constructed. The model equations for the actual maximum rates of respiratory chain and gluconeogenesis are TNAD rNADmax(t) +rNADmax(t)
=
rNADopt(t)
(43)
TIT2 -r"Smax(t) + (TI+T2) rSmax(t) +rsmax(t) = rSopt(t) _ (44)
where J1(PI) PI -
cost functional, vector of unknown mode 1 parameters, l~(ti) vector of measurements at time t i , ylt- ,PI) model output vector, H 1 - diagonal weighting matrix, I number of measurements, () T transpose of a matrix . With this method the parameters of the regulation model Eq .( 43),(44) and the initial values of the state vector, combined in one vector p = (TNAD,Tl'T 2,X(0),S(0),rNA Dmax (0),KE,Ko)' can be identified with the output vector l(t) = (X(t),S(t),E(t),OUR(t),CPR(t)) T . By application of the numerical optimization procedure by Nelder-Mead (1965) the optimal parameter vector is obtained after approximate 500 iteration steps. This identification method can not be applied to the estimation of the other model parameters. Since the identification problem can only be solved uniquely with simultaneous consideration of oxidative, maximum aerobic and anaerobic glucose growth as well as ethanol growth, one has to pay regard to a great number of output variables and measurements. In most cases the identification procedure did not converge to the optimum. Therefore an interactive two level parameter identification procedure was implemented on a digital computer. The schematic diagram is shown in Fig. 7.
The simulation of an aerobic batch culture on Iteration count i = O. Set starting 3 % glucose is shown in Fig. 5. The first values to the maximum reaction growth phase on glucose is nearly exponential rate vectors (.':ul'.':u2'.':u3'.':u4)O and therefore the specific rates of oxygen uptake and carbon dioxide production remain constant, whereas the maximum specific growth + rate on ethanol has not been reached after 20 h. During the diauxic lag phase and at the t end of the fermentation the oxygen uptake rate Next iteration step i = i • I a: decreases to values of the maintenance metabo Simulate with the actual (£2)i >« olism given by mATP ' It can be seen that the a: ... behaviour during the lag phase is not exactly o modelled by the simple linear systems (43,44) with real poles, since the measured oxygen consumption course is oscillatory. Improve (.':uI'.':u2'.':u~'.':u4Ii The simulation of an oxygen limited batch cula: ... ture is plotted in Fig. 6. During fermentation => the impeller speed is repeatedly changed, which i gives rise to abrupt variations of the dissol- - ved oxygen concentration. After 7. 3 h the growth on ethanol becomes oxygen limited and Compare (.':uj) i and (C:l j I i and the dissolved oxygen concentration falls down minimize J ( 102) to get the below 0.03 mg/l. The linear growth with constant 2i new parameter estimate (~2)i.1 oxygen uptake rate is a consequence of oxygen ...a: o transfer limitation by the reactor. An increase J of impeller speed at 12,13,14 and 15 hours produces higher growth rates. Fig. 7 Schematic diagram of the interactive two-level parameter identification THE PARAMETER IDENTIFICATION PROCEDURE procedure.
!
0-
00-
-r-
:0
~
A usual method for off-line parameter identification with non-linear deterministic models is the minimization of the weighted squared differences of measurements and model outputs, as a function of the parameters. The performance index (cost functional) is then given by L
J1(e) = I(lM(t )-l(ti ,£ I) )Tli (lM(ti ) -l(ti i=1
,El))-+ min
(45)
At the upper level (Operator) vectors of maximum rates r . of the four growth phases are estimated.-uJAt the lower level the cost functional 4
T
•
J2(~) = .I(.!:uj -.!::.!j(~)) li(.!:uj -.!::.!j(.ez))- nnn J=1
where J 2 (E2) cost functional
(46)
Deterministic Growth Hodel of Saccharomyces Cerevisiae
£.z = (qS.a, ,rA"" '~I '~Z'~3 ,KAT>~G' YATP,mATP'P/O" ,P/O.,,), vector of unknown parameters .':u j = (~u' qSu ' q0 z u ' qeo z u ' rE u ) j vector of reaction rates givenby the upper level for different growth states j .':1 j (£.z) vector of reacti on rates computed at the lower level with parameters Pz j index of growth state 1: maximum oxidative glucose growth 2: maximum aerobic glucose growth at Crabtree-effect 3: maximum anaerobic glucose growth 4: maximum ethanol growth is minimized at each iteration step. Then the growth behaviour of batch and che~ostat cultures is simulated with the obtained pa rameters. I n the next i tera t ion step the estimate of the maximum reaction rate vector can be improved at the upper level by comparison of simulation results and measurements. The iteration procedure requires approximately 20 steps which is also the number of necessary simu1ations in contrast with several thousands by use of method (46) without reaching convergence. Since the computational effort at the lower level is small a substantial saving of computer time can be achieved. CONCLUS IONS In this work a dynamic model of the yeast SACCHAROMYCES CEREVISIAE HI022 is proposed by considering a simplified reaction scheme of the matabo1ism. The necessary assumptions about regulation of metabolic pathways are expressed as a Multiple Monod-B1ackmanKinetic. After introduction of an increasing PlO-ratio with decreasing respiratory activity the aerobic, oxygen limited and anaerobic growth on glucose and ethanol can be described. Verifications of some model predictions require further experiments. The simulations indicate that an improved modelling of the diauxic lag phase could be achieved by a linear second order system with conjugate complex poles. The model parameters were identified with chemostat and batch data by application of an interactive two-level procedure. This work was supported by Stiftung Vo1kswagenwerk.
Table 1 Identified Parameters of Chemostat Growth 0.018 mol/gh qs 0.4 g/l Ks 0.0048 mo1/g KSI 0.39 mol/mol Ksz 0.0044 mol/g Ks 3 0.071 mol/mol KEG 0.17 mol/mol Kr 0.93 mol ATP P 0" P/O. a, 2.7 atoiiilf YATP 11.96 g/mol 0.0003 mo1/gh mAJP r Ac • a x 0.0034 mol/gh
Table 2 Additional Identified Parameters for Batch Cultures TNad 4.0 h 2.4 h TI Tz 2.8 h 0.016 mol/gh qs 0.0053 mol/g Ks 1 0.05 mol/mol KEG 0.08 mol/mol KAT mATP 0.0025 mol/gh 1.1 l/gh KE 12.8 l/mg h Ko
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NOMENCLATURE Ac Ad
Acetyl-CoA (-concentration), mo1/1. Acetaldehyde (-concentration), mo1/1. CF dissolved COz-concentration, mg/1. CIe maximum COz-saturation concentration at pure COz-atmosphere, mg/1 . carbon dioxide COz CPR COz-production rate, mg/1 h. CTR COz-transfer rate, mg/1 h. D dilution rate (= F/V), 1/h. E ethanol concentration, g/l. EtOH ethano 1 (-concentrat ion), mol 11 . relative Ad-excretion rate. relative rates of bio-synthesis. = 0.8, ratio of COz- and Oz-diffusioncoefficients in the medium. B1ackman constant of ethanol, 1/g h. relative glycero1e production rate. B1ackman constant of oxygen, 1/mg h. saturation constant of glucose, g/l. volumetric absorption coefficient, 1/h. ATP-maintenance coefficient, mo1/g h. molecular weight, g/mo1. particle quantity, mol. particle flow, mo1/h. oxygen dissolved oxygen concentration, mg/1. maximum Oz-saturation concentration at pure Oz-atmosphere, mg/1. OTR oxygen transfer rate, mg/1 h. OUR oxygen uptake rate, mg/1 h. p pressure, bar. Pyr pyruvate (-concentration), mo1/1. PlO effectiveness of oxidative phosphorylation, mol ATP/atomO. specific COz-production rate, mo1/g h. specific Oz-uptake rate, mo1/g h. specific glucose uptake rate, mo1/g h. qSmax maximum value of qs, mo1/g h. specific Ac-formation rate, mo1/g h. rAcmax saturation rate of rAc, mo1/g h. rAd specific Ad-formation rate, mo1/g h. rAIP ATP consumption for growth, mo1/g h. specific rates for bio synthesis, mo1/g h. = rEl+rEZ, EtOH reaction rate, mo1/gh. specific EtOH production rate, mo1/g h. specific EtOH uptake rate, mo1/g h. = 2rGl,glycero1e~roduction rate,mo1/gh. specific substrate consumption for glycero1e formation, mo1/g h. specific rate of respiratory chain, mo1/g h. rs specific rate of glycolysis, mo1/g h. rlee specific rate of the TCC, mo1/g h. R gas constant, J/mo1 K. RQ = qeoz/qoz, respiratory quotient. S glucose concentration, g/l. Sext glucose in the medium, g/l. Sint glucose in the cells, g/l. T temperature, K. Tl,T z ,T NAD time constants, h. Y liquid phase volume, 1. VE inlet gas stream, 1/h. VG gas phase volume, 1. x mole fractions of 02, C02, nitrogen. X bio mass concentration (dry weight),g/l. YAIP ATP yield coefficient, g/mo1. ~ specific growth rate, 1/h.
K.-H. Bellgardt, W. Kuhlamnn and H.-D. Meyer
74
C Carbon dioxide. E Ethanol. N Nitrogen. o Oxygen. R Reservoir. S Glucose. Superscripts A Outlet gas stream. E Inlet gas stream. T Transfer gas phase/liquid phase.
Subscripts
REFERENCES Barford, J.P. and Hall, R.J. (197~). An exa~-. ination of the Crabtree-effect ln S.cerevlslae. . Journ. of Gen. Mikrobiol., 114,267. Barf ord, J.P. and Hall, R.J.(1979). Investlgation of the significance of carbon and redox balances to the measurements of gaseous metabolism of S.cerevisiae. Biotech.Bioeng., 21, 609. Barford, J.P. and Hall, R.J.(19S1). A mathematical model for the aerobic growth of S.cerevisiae with saturated respiratory capacity. Biotech.Bioeng. ,23, 1735. Becker,J.U. and Betz, A.(1972). Membran transport as controlling pacemaker of glycolysis in Saccharomyces carlsb. Biochem. Biophys. Acta, 274, 5S4. Bergter,F. and Knorre, W.A. (1972). Co~putersimulation von Wachstum und Produktblldung bei S. cerevisiae. Zeitschr. f.allg. Mikrobiologie, 12,S, 613. Bijkerk, A.H.E. and Hall, R.J. (1977). A mechanistic model of the aerobic growth of S.cerevisiae. Biotech.Bioen~.,19, 267. Carnahan, B., ~uther, H:A. and Wlckes,J.O. (1969). Applled numerlcal methods, 319, John Wiley & Sons, N.Y. Dekkers,J.G.E., deKok,H. and Roels,J.A.(1981) Energetics.of S.cerevi~iae CBS426.~0~par~son of anaeroblC and aeroblc glucose llmltatlon. Biotech. Bioeng. 23, 1023. . Fiechter, A.(1980). Regulatory aspects ln yeast metabolism. In M.Moo-Young (Ed.), Advances in Biotechnology, Vol.1, Pergamon Press, Toronto, p. 261. Fredrickson, A.G. and Tsuchiya, H.M.(1977). Microbial kinetics and dynamics. In L.Lapidus and N.R. Amundson (Ed.), Chemical reactor theory,Prentice-Hall ,Inc., New Jersey, p.405. Geurts,T.G.E.; deKok,H.; Roels,J.A. (1980). A quantitative description of the growth of S.cerevisiae CBS 426 on a mixed substrate of glucose and ethanol. Biotech.Bioeng.,22, 2031. Hall,R.J. and Barford, J.P.(1981) Simulation of the integration of the energy metabolism and the cell cycle of S.cerevisiae. Biotech.Bioen 23, 1763. Kuenzi, M.(1970 . Ober den Reservekohlenhydratstoffwechsel von S. cerevisiae. Dissertation, ETH-Zurich. Meyenburg,K.(1969) Katabolitrepression und S~rossungszyklus von S.cerevisieae, Dlssertation ETH-Zurich. Nelder,R. and Mead, J.A. (1965). A simplex method for function minimization. The Computer Journal, 7, 30S.
., J
Ohnishi,T.; Sottocasa,G.; Ernster,L.(1966). Bull.Soc.Chem.Biol.,4S,l1S9 Pamment,N.B. ;Hall ,R.J.; Barford,J.P. (197S). Mathematical modelling of Lag phases in microbial growth.Biotech. Bioeng. ,20,349 Peringer, P.; Blachere, H.; Corrieu, G. Lane, A.G. (1973). Mathematical model of the the kinetics of S. cerevisiae. Biotech. Bioeng. Symp. No. 4, 27. Peringer, P.; Blachere, H.; Corrieu, G.; Lane, A. G. (1974). A generalized ~athe matical model for the growth kinetics of S. cerevisiae with experimental determination of parameters. Biotech. Bioeng., 16,
Ra~~:V.M. (196S). Absor~tion of gases.
Israel Program for SClent.Transl. ,Jerusalem Schatzmann,H. (1975). Anaerobes \~achstum von S.cerevisiae. Dissertation, ETH-Zurich. Sols, A.(1966). Symo. on Some Aspects of Yeast Metabolism. Dublin, 1965. Blackwell Sci. Publ., Oxford, p. 47. van Steveninck. J. (1969). The mechanism of transmembran glucose transport in yeast. Archives of Biochem. and Bio hys., 130,244. Toda, K.; Yabe, J.; Yamagata, T. SO Kinetics of biphasic growth of yeast in continuous and fpdbatch-cultures. Biotech.Bioeng., 22, lS05. Weibel, E.K. (1973). Zur Energetik von S. cerevisiae. Dissertation, ETH-Zurich. Westerterp, K.R.; van Dierendonck, L.L.; de Kraa, J.A. (1962). Chem.Ing.Science,18. Woehrer, l~. and Roehr, M. (1981). Regulatory aspects of baker's yeast metabolism in aerobic fed-batch-cultures. Biotech.Bioeng., 23, 567. Vagi, H. and Yoshida, F. (1975).Enhancement factor for oxygen absorption into fermentation broth. Biotech.Bioeng., 17, 1083. Yagi, H. and Yoshida, F. (1977). Desorption of carbon dioxide from fermentation broth. Biotech.Bioeng., 19, SOl. Yoon, H.; Klinzing G.; Blanch, H.W. (1977). Competition for mixed substrates by microbial populations. Biotech.Bioeng. ,19, 1193.