Journal 0
of Biotechnology,
1992 Elsevier
BIOTEC
Science
22 (1992) Publishers
69-88 B.V.
69 All
rights
reserved
0168-1656/92/$05.00
00668
Chemostat cultures of yeasts, continuous culture fundamentals and simple unstructured mathematical models U. von Stockar and L.C.M. Auberson btstittrt
de G&k
Cltimiqrte,
(Received
Swiss Federal
28 November
btsrirure
1990; revision
of Technology, accepted
Lausanne,
21 April
Switzerland
1991)
Summary
Fundamental aspects of chemostat cultures are reviewed. Using yeast cultures as examples, it is shown that steady states in chemostats may be predicted quantitatively by combining the correct number of unstructured kinetic models with expressions for existing stoichiometric constraints. The necessary number of such kinetic models corresponds to the number of limiting substrates and increases with the number of different metabolic pathways available to the strain. This is demonstrated by an experimental comparison of yeast growth limited by glucose alone for which metabolism is oxidative, and growth doubly limited by both glucose and oxygen, which occurs according to an oxido-reductive metabolism. The steady state data for such experiments can in principle be predicted based on a minimal Correspondence 1015 Lausanne.
ro: U. von Switzerland.
Stockar,
Institut
de GEnie
Chimique,
Swiss
Federal
Institute
of Technology,
Nometrclatwe: ci. concentration of species i, C-mol mm3; CL, dissolved oxygen concentration, mol m-“; D, dilution rate, h-‘; F, volumetric flow rate, m” h-‘; c*, equilibrium concentration for O,, mol rne3: dHT, enthalpy change for combustion of species i to CO?, H,O and NH,, kJ/C-mol; i, element in chemical formula; K, fraction of air in Nz-air mixture; Ki, saturation constant for species i, C-mol m-“; k,a, volumetric mass transfer coefficient, h-‘; m, substrate consumption for maintenance, h-l; qi. specific consumption rate of species i, h-‘; (of”““, maximum specific consumption rate of species i, h- ‘; P, product concentration (e.g. ethanol), C-mol m-” or g I-‘; S, concentration of carbon and energy substrate, C-mol mm3 or g I- ‘: X, concentration of biomass, C-mol mm3 or g I -‘; Yi;j, C-molar yield of i per unit j; yi, reduction degree of species i, definition: Table 1; ai, hydrogenation degree of species i, definition:
Subscripts oxygen; carbon
Table
rate, h-‘: R, aerobicity, see Eq. (10). crab, Crabtree; max. maximum value; N, 0, value in feed stream; ox, oxidative; P, product; red, reductive; suppl, and energy substrate; X, biomass; W, water.
and
I; CL, specific
superscripts:
growth
C, CO,;
nitrogen supply;
source; Q, heat;
0, S,
70
amount of information by a simple stoichiometric model. It represents the overall stoichiometry of growth by a superposition of a fully oxidative and a fully reductive growth reaction and uses the concept of “aerobicity” to characterize the relative importance of the two reactions. Chemostat; Continuous
culture; Dual limitation;
Yeast; Bottleneck metabolism
Introduction The goal of this contribution is to illustrate how a certain number of fundamental aspects of chemostat cultures serve as a basis for understanding continuous culture behavior. This will be done by an analysis of such fundamental aspects in terms of very simple, unstructured and unsegregated models (Fig. 1). The results will be used in order to predict X-D diagrams of chemostat cultures of yeast growing according to oxidative, reductive or mixed metabolisms (Figs. 2-5). A quantitative understanding of chemostat culture behavior necessarily involves a certain amount of mathematical modelling. Models constructed to this effect may be more or less complex, but with respect to the aim pursued by this contribution, simple descriptions are preferable over complex, structured and segregated mathematical models. The former are usually made up of only a small number of equations. In contrast to the latter, which need a very large number of equations, the results of the analysis are much easier to understand and to verify when working with simple models. This is especially the case if the number of equations is so small that one does not even need a computer to solve them. Simple unstructured models also involve only a small number of parameters which often
I
IN
-
OUT
=
CONSUM
a D
kLa(c; Fig.
1. Schematic
drawing
- CL)
@LO
-
-
ACCUM
@(+dci
cd
DCL
+
dt =
qo,X
of a chemostat culture. Equations simple unstructured mathematical
+ %
IOXYWI
show mass models.
balances
used
for
building
71 -a-
-o-
q[wO] 12
Xbk’l
r I 10
9 _ 66-
- 4-
Fig. 2. Chemostat culture of K. fragdis NRRL 1109. O: cell dry weight; n : residual glucose concentration; 0: heat evolution rate; which was measured using special calorimetric techniques in order to characterize the global metabolic activity of the culture (Birou and von Stockar, 1989).
can be measured in experiments that are independent of the continuous culture that the model is supposed to describe. The fact that the values of these parameters can be obtained without having to fit the model to existing chemostat data confers a certain predictive power to the model calculations. x
(9 I-‘)
n
P
(9 I-’ ) A
PO,
(O/o) l
6 -
50
6 II
n
.:
WI
w
A
4 - ’
.
l 4
1
2 K D-’
Fig. 3. Dry biomass concentration function of the relative oxygen
40
-
30
-
20
-
10
I. l
0
-
I
. :. I
3
4
&
#
5
I.
0
6
(h)
(m ), ethanol concentration (A) limitation K/D during continuous and Birou, 1989).
and dissolved culture of
oxygen
K.
fragilis
tension (van
(0) as a Stockar
72
2 L
16 14
-59
i
;
12-
2 $
10 -
c! :
6-
d 0”
6-
Biomass
4-
O+--ctccr
I
0.15
I
0.25
Dilution Fig.
4. Continuous
culture
I
I
0.35
of S. cerec~isirre concentration;
I
I
0.45
Rate
0.55
(h-l)
CBS 426. n : Biomass concentration; a: ethanol concentration.
0:
residual
glucose
Biomass
0
o
0
Ethanol 0
0
0
0
Glucose o!
pa, 0.0
-I
,-
0.2
,-,y,
0.4
0.6
KLA Fig. 5. Dry (A)
biomass as a function
concentration of the relative cere~Goe
; C*/D
0.6
So
( n ), ethanol concentration oxygen limitation (k,a
(Auberson
and von
Stockar,
,-
,“,
q
1.0
1.2
p
1 1.4
(-) (0). c”/DS,)
and residual glucose during continuous
1991; Auberson,
1990).
concentration culture of
S.
73
In consequence, simple unstructured model descriptions will be used in this paper in order to analyze some fundamental aspects of chemostat culture behavior, and to interpret real chemostat data. Based on such simple descriptions, features of four different chemostat cultures of yeasts will be analyzed. The first example is an X-D diagram obtained in highly aerobic conditions with a so-called aerobic respiring yeast (Kfuyueromyces fragilis), (Fig. 2). (This term, introduced by Alexander and Jeffries (1990) means that K. fragilis grows exclusively according to an oxidative metabolism when sufficiently aerated whereas a so-called “aerobic fermenting” (AF) yeast will grow according to a mixed oxido-reductive metabolism at high dilution rates even when p0, is high). The second example involves cultivating K. f,wgih under various degrees of oxygen limitation (Fig. 3). The third case involves an X-D diagram of a so-called “aerobic fermenting” yeast (Sacchnro!?zyce.scere[lisiae), (Fig. 4) and the last example deals with subjecting this type of yeast to oxygen limitation (Fig. 5).
Balances, stoichiometry
and kinetics in chemostats
Any quantitative model or theory for chemostat cultures is based on balances, such as shown in Fig. 1, for each of the chemical and other entities that enter or leave the growth vessel. In simple model descriptions, the bioreactor contents are usually assumed to be well mixed, and at steady state. The specific consumption or production rate qi for any given compound must therefore be uniform in the whole reactor and invariant with time. Application of these balance principles to the biomass leads to the well known equality of p (= L],) and D. A major issue when constructing quantitative explanations for chemostat data such as shown in Figs. 2-5 concerns the number of major biochemical entities for which a balance, such as is shown in Fig. 1, ought to be written. The question is best dealt with by formulating the conversion occurring in the yeast cultures in terms of a quasi-chemical equation which might read e.g., as follows: Ck,Q,Nsl+
+ Yd,sO2 + G,sNH3
Y;,sCHx,Q2Nx3
+ G,sCH,,O,~N,~
+ Y&CO,
+ G&-LO
(1)
This equation shows 7 major compounds including a carbon and energy substrate, the biomass and a major intermediate product formed in the process, all of which appear as C-molar “chemical” formulae. In order to complete the model, one needs a certain amount of kinetic information. If the Y-values appearing in Eq. (1) were completely independent of each other, if they could assume any set of values, all of the compounds appearing in Eq. (1) could be taken up or produced at any rate and one would need 7 kinetic models for defining the 7 qi values in Fig. 1. There are, however, 4 constraints due to the elemental balances. One would thus generally expect that 3 kinetic models are needed to predict all seven qi values, but additional constraints may further reduce this number.
74
Once these models are identified, the prediction of X-D diagrams is quite straightforward, as will be shown for specific cases in later sections. Key steps involve developing expressions for the biomass yield Yi,, and using them in conjunction with the kinetic models in order to link p to the concentration(s) of the limiting substrate(s) (see e.g. Eqs. 4 and 8). From such a relation, the residual substrate concentration(s) can be predicted as a function of D because of /.L = D. The biomass concentration is then found as: x= Y;,,(s,
- S)
(2)
The specific uptake and consumption rates qi for compounds other than X and S are evaluated by combining the kinetic models with the elemental balances and other constraints. The residual concentrations of these compounds may then be predicted by solving the balances shown in Fig. 1 for the steady state. In the next section, this general philosophy will be used in order to derive model descriptions for interpreting the chemostat experiments shown in Figs. 2-5. Due to space limitations, the emphasis will not always be on predicting all the measured steady state concentrations. Instead, the prediction of the observed apparent yield coefficients of the general stoichiometry appearing in Eq. (1) will be stressed.
An analysis of chemostat cultures of yeasts in terms of simple unstructured
models
Aerobic chemostat cultures of Kluyveromyces fragilis K fragifis is a so-called aerobic-respiring (AR) yeast in the nomenclature introduced by Alexander and Jeffries (1990), which means that the culture grows according to a purely oxidative metabolism even at dilution rates close to wash-out. The corresponding X-D diagram (Fig. 2, Birou and von Stockar, 1989) is therefore relatively simple. The absence of a product (Yi,s = 0) reduces the overall growth stoichiometry to only 6 compounds as shown in Table 1. Due to the 4 elemental balances, the rates at which these compounds are consumed or produced are not independent from each other. By the same token, only one of the 5 remaining stoichiometric coefficients can vary in an independent way. Thus, if one of them is
TABLE 1 Oxidative growth
y,=4+i,-2i,-3i,;
&=fi,-+i,.
known, the other 4 may be computed by solving the elemental balances (Table 1). A further constraint is due to bioenergetic considerations. At moderate to high values of D, where maintenance effects can be neglected, the biomass yield Yiys can be assumed constant and determined by the bioenergetics of the culture. For the aerobic growth of K. frugilis, the five remaining stoichiometric coefficients in Eq. (1) are therefore completely fixed by five constraints. The production and consumption rates of the various entities shown in this equation are therefore totally interlocked: if we know one of these rates we can calculate all the others through the stoichiometry given by Table 1. It is therefore clear that only one of these compounds can be taken up or produced in a kinetically independent way, and that one single kinetic expression suffices to complete the whole model. The next step is to identify the compound whose consumption or production rate will determine all the others, and for which a kinetic expression should consequently be formulated. The answer to the question is linked to fundamental concept of the so called limiting substrate. Since it is usually the carbon and energy substrate that limits growth in chemostat culture, the kinetics of the whole process is governed by the uptake kinetics of this substrate: s 4s = ,iY K, + s The specific growth rate can be calculated by multiplying stoichiometric coefficient:
cL= y;ys
s
qs""
K, + S
Eq. (3) by the respective
(4)
Equation (4) is equivalent to the Monod equation. Since p = D in chemostat cultures, it introduces a direct link between the dilution rate and the residual substrate concentration S which can be seen graphically on Fig. 2. All other specific consumption or production rates may be computed in a similar way and residual concentrations are determined as outlined in the preceding section. At low dilution rates, X is seen to assume lower values (Fig. 21, thus indicating a decrease of Yx,, due to maintenance effects. A consistent way of extending the model to this case consists of the assumption that the growth equation shown in Table 1 remains unchanged, but that an additional consumption of carbon and energy substrate proceeds in parallel: 4s =
(qs&wth
where
+
k)growlh
stoichiometry p
=
y;>
m
(5)
is the specific substrate uptake rate for growth. According to the for growth (Table 11, (qs)growth is related to /.L as follows:
~~s)~row~h
(6)
By substituting (qs)growt,, in Eq. (5) by means of Eq. (6) and dividing by EL, an expression for the apparent growth yield resulting from both oxidative growth and
76
maintenance is obtained:
Equation (7) is known as the linear Herbert-Pirt relation. When the inverse of the apparent biomass yield is computed from Fig. 2 and plotted vs l/D, one obtains indeed a straight line whose slope is m (Birou, 1986; Birou and von Stockar, 1989). The relationship between S and p follows from a combination of Eqs. (7) and (3):
which replaces Monod’s equation (4). Equation (8) can be used to predict S from D. All other specific rates and concentrations may be calculated as indicated before. It is important to stress that growth kinetics in chemostats is governed by the uptake rate of the limiting substrate and not vice versa. Growth is a consequence of substrate .uptake and not the cause. Writing a Monod-type equation for describing the growth kinetics of the culture is thus wrong and leads to an inconsistent description, as already shown by Heijnen (1990). In these cases the substrate uptake kinetics also determines the uptake kinetics of the non-limiting substrates such as oxygen and all growth factors through stoichiometry. This clearly means that these substrates cannot be taken up according to their intrinsic, biological uptake kinetics. The situation is shown for oxygen
qox
C
L
-
kLa
k,a Fig. 6. Kinetics
and stoichiometric
constraints
for the oxygen
+D
c*
L
consumption
rate.
in Fig. 6. The curve represents schematically the intrinsic kinetic model for 0, uptake:
(9) Also shown in this figure are the terms in the balance representing the difference between oxygen influx and outflux from the chemostat culture as shown in Fig. 1. These two terms form a straight line of a slope which is given by the negative sum of k,a + D and which originates on the abscissa from a point determined by the equilibrium oxygen concentration c *. If oxygen were a limiting substrate, the culture would operate at the intersection of the kinetic curve and the straight operating balance line. But since it is the carbon and energy substrate that limits growth, it imposes an oxygen uptake rate on the culture which is lower and given by the horizontal broken line. As can be seen from Fig. 6 the culture is thus forced to leave a large amount of oxygen unused in the culture, thereby giving rise to a considerable dissolved oxygen concentration CL. Fig. 6 can be used in order to develop an operating definition of what one understands by the term limiting substrate: a limiting substrate is one which is taken up according to the intrinsic biological kinetics and for which the culture operates on the respective kinetic curve (Fig. 6). A non-limiting substrate will be taken up at a lower rate dictated by stoichiometric considerations. In Fig. 7 the specific oxygen uptake rate is plotted schematically against the specific carbon and energy substrate uptake rate. The two quantities are linked by a fixed stoichiometry and must therefore form a point on a straight line of a slope equal to the respective stoichiometric coefficient YoFs. If oxygen is supplied in
b uptake
B
SUPPlY 9s
Fig.
7. Stoichiometric
link
between
specific carbon and energy substrate oxygen uptake in obligate aerobes.
consumption
and
specific
7x
excess (supply A) a considerable amount of the supplied oxygen will remain unused. In obligate aerobic cultures one will be able to reverse the situation by limiting the supply of oxygen to level B. This would force the culture to reduce the uptake of S in such a way that the operating point B is again located on the straight line. Since S could not be taken up anymore at its biological uptake kinetic rate, a considerable amount of the substrate would remain unused in the medium. High residual substrate concentrations are indeed observed even at low dilution rate in chemostat cultures which are in fact not limited by the main carbon and energy substrate but by some other usually unknown growth factor. Growth of K. fragilis under oarious degrees of oxygen limitation
The actual result obtained when subjecting a culture of Kluyueromyces fragilis operating at constant dilution rate to an oxygen limitation is shown in Fig. 3 (von Stockar and Birou, 1989). In this experiment the oxygen limitation was introduced by sparging air-nitrogen mixtures through the culture such that the equilibrium oxygen concentration c * in the medium could be varied. The K value appearing on the abscissa indicates the fraction of air in the air-nitrogen mixture (Birou, 1986). It can be seen that the unused excess of oxygen is decreased linearly as a function of the reduction in the oxygen supply. At K/D = 1, the oxygen tension is more or less zero and oxygen becomes a limiting substrate. Since yeasts are not obligate but facultative aerobes, the carbon and energy substrate S is still used to completion even at lower oxygen supplies (not shown) and ethanol appears in the medium. Instead of reducing its substrate uptake in order to stay on the original
qo
SUPPlY uptske
Fig.
8. Stoichiometric
link
between
specific
oxygen
uptake
carbon
and
in facultative
qs
energy
substrate
aerobes.
consumption
and
specific
19 TABLE 2 Reductive growth CH$,,Ns.,
+ YA::NH,
. = (z$,,)-(!LJz Yl;‘;s” rrcll yNS
=
-Ys ---y YP
yx
YP
+
Y~~~CH,,O,,N,,
+ Y$;CH,,OPINpJ
+ Y;.;‘,dCO, + Y;I;“,H,O
-x,)Ygg rrcd
x/s
Y,rLxl c/s rrud Y w/s
yi and ai are defined in Table 1.
stoichiometric line on Fig. 7, the culture metabolizes the excess of substrate according to a reductive mechanism that does not consume oxygen. Therefore, the stoichiometric line is changed as a whole and the culture now operates at point B on Fig. 8. This is a clear case of dual limitation since both the main substrate S and the oxygen will be taken up according to their biological uptake rates. As this dual limitation permits the culture to increase qs, the behavior might be termed a “Pasteur-type” effect (Pasteur, 1861). In order to quantitatively understand the data in Fig. 3 one has to extend the stoichiometry of the model to include the reductive growth mechanism. The easiest way is to represent the overall growth stoichiometry (Eq. 1) as the sum of two parallel biochemical reactions, one being growth at fully oxidative conditions (Table 1) and one being growth at fully reductive conditions (Table 2). Note that both these growth stoichiometries comprise 5 stoichiometric coefficients, 4 of which can be calculated by solving the elemental balances if the biomass yields are known. All the stoichiometric coefficients in such a model can therefore be calculated from the biomass yield for fully oxidative conditions Yiys and the biomass yield for fully anaerobic reductive growth conditions Yi;“s”. These two yield coefficients are true model parameters but they can be measured in independent growth experiments. A complete unstructured model would be obtained if the two stoichiometric equations in Tables 1 and 2 were combined with two kinetic expressions, one for the rate of substrate uptake and one for the rate of oxygen consumption (Eqs. 3 and 9). The overall apparent stoichiometry and all Y values appearing in Eq. (1) could be predicted from there. In order to simplify the calculations further it is convenient to introduce a factor which measures the relative rates between the two equations shown in Tables 1 and 2. This factor has been proposed by von Stockar and Birou (1989) and is defined as follows: 4 k,a c* - (k,a +D)c, nz-= Y&s I (10) ,0X Yo/s x - YxYio;, W%-S)
80
This so-called aerobicity measures the relative rate of oxygen to carbon and energy substrate uptake and is normalized by the respective ratio that one would observe for fully oxidative growth conditions. In Fig. 8 R measures the relative decrease of the slope of the stoichiometric line resulting from the decreasing oxygen supply. For doubly limited cultures such as the one shown in Fig. 3, we now propose an easy approximate estimation technique for 0. Since both the residual substrate concentration and th5 dissolved oxygen tension are very small in such cultures, the aerobicity becomes:
(11) where 1 k,a c* Lzsupp= YA(;” DS”
(12)
4upp indicates the maximum normalized ratio of oxygen to substrate supply. It is very easy to estimate on the basis of just a few independently measurable parameters (biomass yields for fully oxidative growth, kLa, c*, D, S,) and will approximate the real LJ with high accuracy in a doubly limited culture. Based on known values for the aerobicity 0 the apparent overall stoichiometry (Eq. 0, which is the result of the combined effect of the two growth reactions, may be computed as shown earlier (von Stockar and Birou, 1989). Some of the more important stoichiometric coefficients as a function of 0 are given in Table 3. All specific rates, yields and steady state concentrations can be calculated as described in the second section, based on an estimate of R. In order to do this, &,,, was calculated as a function of K/p and Table 3 was used for computing the apparent yields in Eq. 1 (Yo,x>. The results obtained for Y.&, and Yi,, yielded the straight lines on Fig. 9. Also shown on this figure are the experimental yield values recalculated from Fig. 3. Although the value of k,a c* could have been measured in independent experiments, it was found by a best fit technique. As a result, the calculations TABLE 3 Prediction of apparent stoichiometric coefficients in Eq. (1) based on the aerobicity Y&s
= f2y;ys
yi,s
= (l- n,y;,“,d
Y&s
= L!Y&‘;”
Y&s
= 1 - y.$
Y&x
=
Y$ 2.
+ Cl- n,y$
- y;;‘,d + (y.$~+y$-y~~s)~
AH; - (I- n)Y,$AH nY~~s+(l-n)Y.g
p*
-AH;
and Yhys may be computed from Yiys and Y.$t according to the relations given in Tables 1 and
81
0.0
0.2
0.4 k,a
0.6 c.*
0.8
1.0
DSO
Fig. 9. C-molar biomass function of the relative
yield (m) and product yield (A) in continuous cultures oxygen limitation, expressed as k,a c*/DS,, (van Stockar
of K. fragilis as a and Birou, 1989).
coincide with the measured values,quite well. Based on the same parameter values, any other yield can be predicted. Details of such calculations will be presented in a separate paper (Auberson and von Stockar, 1991). Aerobic chemostat cultures of S. cerevisiae
An X-D diagram for Saccharomyces cereuisiae is shown in Fig. 4 (Auberson, 1990). The diagram is considerably more complicated than the one shown in Fig. 2. Being a so called aerobic-fermenting (AF) yeast, S. cerevisiae produces ethanol at high dilution rates even when it is well aerated. The appearance of some reductive metabolism at high glucose feeding rates has been linked in earlier times to catabolite repression and to the Crabtree effect (De Deken, 1966; Polakis et al., 1965; Lemoigne et al., 1954). More recently, evidence has accumulated showing that the appearance of ethanol under well aerated conditions may result from an overflow of metabolites into the reductive pathway due to a bottleneck in the respiratory capacity of the yeast (Barford and Hall, 1979; Rieger et al., 1983). Based on this hypothesis, Sonnleitner and Kappeli (1986) presented a predictive mathematical model for continuous cultures of S. cereuisiae. This bottleneck introduces a maximum ceiling equal to q,““” for the uptake of oxygen as shown in Fig. 10. By increasing the dilution rate D in Fig. 4, the specific substrate uptake rate also increases almost proportionally. At low dilution rate the culture will operate on the fully aerobic stoichiometric line (A in Fig. 10). At point B, corresponding to the so-called “critical” dilution rate, the bottleneck is full and the oxygen is now taken up with the maximum rate permitted by the intrinsic biological uptake kinetics. If the dilution rate and thus q, are further augmented, the culture must therefore metabolize the excess supply of S by the reductive pathway, which does not utilize any oxygen. Therefore the ratio of oxygen to
Fig.
10. Stoichiometric uptake
link
in continuous
between cultures
specific of yeast
carbon whose
and
respiratory
energy capacity
consumption is limited
and
specific
by a bottleneck.
substrate consumption must decrease and the whole stoichiometry changes (point C). It is important to note that between B and C the culture is again doubly limited since both S and oxygen are taken up at the maximum rate possible by biology. Nevertheless, large quantities of excess oxygen will remain unused in the medium and the dissolved oxygen tension will be far from zero. A simple procedure for estimating the aerobicity R may also be proposed for this “Crabtree-type” effect by linking 4, and q. as follows: 40 = 4&,,
= s,y;“;sf-J
q. = $I(&
(13)
- S)Y&“/“sO
(14)
By substituting Eqs. (9) and (2) for q. and X, respectively, by noting that S = 0 and C, >> K, in this case, and substituting Y.&, as a function of 0 from Table 3 one obtains: fl = f&Mb
(15)
where a
crab
Y$
ID
(16) &Y
Y&
-
Y ‘OX ( x/s
-
y.p)
The true model parameters appearing in Eq. (16) are qzax, Yi;‘,d and YiFs and can be determined in independent experiments. Y& can be computed from the Yiys as shown in Table 1.
x3
1.0 0.9 0.8
-
0.7
-
7
0.8-
m z1
0.5
2 *
0.4 -
Y’x/s 8 ,.
-
0.3 0.2 -
O.l0.0 0.15
YIP/S I
I
I
0.25
Dilution Fig. 11. Biomass yield ( n ) and ethanol S. cererisiae. Solid lines were obtained
f
I
1
0.35
0.45
0.55
Rate
(h-l)
yield (0) as a function of dilution rate in continuous cultures of by estimating the aerobicity as explained in the text (Auberson and von Stockar, 1991).
Apart from being a simple estimate for the real aerobicity, Eq. (16) also shows the conditions that distinguish aerobic-respiring from “aerobic-fermenting” strains. The Crabtree-type effect shown in Fig. 5 can only be exhibited at 0 < 1. This means that ,:“’ < DYdys/Yiys which is only possible if (17) 9,“” < hi~xydO;ss/y~~s Substituting Y& from Table 1 one finds that only those strains will show a bottleneck in the oxidative metabolism whose q,““” satisfies the following condition: @Tax< Pmux(-Ys- Y;“;,r,)/4Y;ys
(18)
By estimating 0 in terms of L?ncrI,bas a function of dilution rate (Eq. 16) and by applying the relations given in Table 3, the apparent biomass and ethanol yields were predicted as a function of dilution rate and compared to the actually measured value as shown in Fig. 11. As the solid lines show, these predictions, as well as the concentrations shown in Fig. 4, coincided well with the measured values. Growth of S. cerevisiae under various degrees of oxygen limitation
Fig. 5 shows how a culture of an AF yeast reacts when grown at constant supercritical dilution rate and when the oxygen supply is gradually reduced. The x-axis indicates the oxygen supply and is obviously proportional to the criterion %upp. At high levels of oxygen supply the culture seems to be quite insensitive to
SUPPlY B max 90
SUPPlY c
-
px/fex,
nrg
qs “Crasteur” Fig. 12. Stoichiometric specific oxygen uptake
Effect
link between the specific carBoon and energy substrate consumption and the in yeast cultures limited by a respiratory bottleneck and subjected at the same time to an oxygen supply limitation.
an oxygen limitation. Only when k,a CT/D& decreases below 0.5 does the ethanol production slowly increase and the biomass concentration slowly diminish, thereby indicating a shift towards a more reductive metabolism. Fig. 12 explains the effect in terms of the simple model description proposed in this paper. The experiment involves imposing a limitation of the oxygen supply on a culture which is already oxygen limited due to the existence of a bottleneck. Since this involves superimposing a Pasteur type effect on an already existing Crabtree-type behavior, this type of experiment could perhaps be called a “Crasteur’‘-type effect. Even at the high oxygen supply level A in Fig. 12 the culture operates under dual limitations (point A) with an 0 which is already smaller than 1. Neither the oxygen uptake rate nor R changes when the supply is diminished to level B, which explains the insensitivity of the culture at high oxygen supplies. Only when the oxygen supply becomes similar to the maximum oxygen uptake rate or lower, is the culture forced to metabolize a growing part of the substrate through the reductive pathway, thereby decreasing the R and thus the stoichiometric line (point C). In Fig. 13 the experiment is shown to correspond to a gradual reduction in CL* thereby shifting the balance line gradually to the left. It is important to appreciate that a dual limitation exists all along in this experiment and that the oxygen uptake is given by the respective kinetic curve for all points for this experiment. The operating point thus slides along the continuous kinetic curve from the right to the left. This explains the smooth transitions of the concentrations which have been measured. In predicting the observed concentrations and yields flnsUppis calculated for each experimental point according to Eq. (12). The equations in Table 3 are evaluated by using ancrab as long as L& is higher than the former but by using L&,,, as
85
cL
k,= ka+D 6 I.
Fig. 13. Oxygen consumption rate as a function of cl* for the “Crasteur-type”
experiment.
soon as it becomes lower. The predictions become straight lines with sharp kinks. It is obvious that the gradual transition of the concentrations actually observed could also be predicted if the equations were solved by using Eqs. (3) and (9) as the basic kinetic models instead of working with C!SUPPand Oncrab. Fig. 14 summarizes the procedure that is proposed for predicting the quantitative behavior of yeast cultures. Based on a minimal amount of information about the strains and the chemostats, L&, and Oncrabare evaluated according to Eqs. (12) and (16). As long as both of these estimators are above 1, the culture will grow according to a fully oxidative metabolism and the real, metabolic aerobicity will be equal to 1. As soon as either L&, and Oncrabor both become lower than 1, the metabolism will shift to oxido-reductive, and the yields, rates and concentrations may be predicted by observing that the real R will adapt itself to the lower of the two estimators. The four examples presented in this paper correspond to shifts from one region to another in Fig. 14.
~cmb < 1 nonb < nsuppl r-1 ---------nauppl> 1 ncnb > 1 I”-‘l
nsuppl
of Biotechnology,
1992 Elsevier
BIOTEC
Science
22 (1992) Publishers
69-88 B.V.
69 All
rights
reserved
0168-1656/92/$05.00
00668
Chemostat cultures of yeasts, continuous culture fundamentals and simple unstructured mathematical models U. von Stockar and L.C.M. Auberson btstittrt
de G&k
Cltimiqrte,
(Received
Swiss Federal
28 November
btsrirure
1990; revision
of Technology, accepted
Lausanne,
21 April
Switzerland
1991)
Summary
Fundamental aspects of chemostat cultures are reviewed. Using yeast cultures as examples, it is shown that steady states in chemostats may be predicted quantitatively by combining the correct number of unstructured kinetic models with expressions for existing stoichiometric constraints. The necessary number of such kinetic models corresponds to the number of limiting substrates and increases with the number of different metabolic pathways available to the strain. This is demonstrated by an experimental comparison of yeast growth limited by glucose alone for which metabolism is oxidative, and growth doubly limited by both glucose and oxygen, which occurs according to an oxido-reductive metabolism. The steady state data for such experiments can in principle be predicted based on a minimal Correspondence 1015 Lausanne.
ro: U. von Switzerland.
Stockar,
Institut
de GEnie
Chimique,
Swiss
Federal
Institute
of Technology,
Nometrclatwe: ci. concentration of species i, C-mol mm3; CL, dissolved oxygen concentration, mol m-“; D, dilution rate, h-‘; F, volumetric flow rate, m” h-‘; c*, equilibrium concentration for O,, mol rne3: dHT, enthalpy change for combustion of species i to CO?, H,O and NH,, kJ/C-mol; i, element in chemical formula; K, fraction of air in Nz-air mixture; Ki, saturation constant for species i, C-mol m-“; k,a, volumetric mass transfer coefficient, h-‘; m, substrate consumption for maintenance, h-l; qi. specific consumption rate of species i, h-‘; (of”““, maximum specific consumption rate of species i, h- ‘; P, product concentration (e.g. ethanol), C-mol m-” or g I-‘; S, concentration of carbon and energy substrate, C-mol mm3 or g I- ‘: X, concentration of biomass, C-mol mm3 or g I -‘; Yi;j, C-molar yield of i per unit j; yi, reduction degree of species i, definition: Table 1; ai, hydrogenation degree of species i, definition:
Subscripts oxygen; carbon
Table
rate, h-‘: R, aerobicity, see Eq. (10). crab, Crabtree; max. maximum value; N, 0, value in feed stream; ox, oxidative; P, product; red, reductive; suppl, and energy substrate; X, biomass; W, water.
and
I; CL, specific
superscripts:
growth
C, CO,;
nitrogen supply;
source; Q, heat;
0, S,
70
amount of information by a simple stoichiometric model. It represents the overall stoichiometry of growth by a superposition of a fully oxidative and a fully reductive growth reaction and uses the concept of “aerobicity” to characterize the relative importance of the two reactions. Chemostat; Continuous
culture; Dual limitation;
Yeast; Bottleneck metabolism
Introduction The goal of this contribution is to illustrate how a certain number of fundamental aspects of chemostat cultures serve as a basis for understanding continuous culture behavior. This will be done by an analysis of such fundamental aspects in terms of very simple, unstructured and unsegregated models (Fig. 1). The results will be used in order to predict X-D diagrams of chemostat cultures of yeast growing according to oxidative, reductive or mixed metabolisms (Figs. 2-5). A quantitative understanding of chemostat culture behavior necessarily involves a certain amount of mathematical modelling. Models constructed to this effect may be more or less complex, but with respect to the aim pursued by this contribution, simple descriptions are preferable over complex, structured and segregated mathematical models. The former are usually made up of only a small number of equations. In contrast to the latter, which need a very large number of equations, the results of the analysis are much easier to understand and to verify when working with simple models. This is especially the case if the number of equations is so small that one does not even need a computer to solve them. Simple unstructured models also involve only a small number of parameters which often
I
IN
-
OUT
=
CONSUM
a D
kLa(c; Fig.
1. Schematic
drawing
- CL)
@LO
-
-
ACCUM
@(+dci
cd
DCL
+
dt =
qo,X
of a chemostat culture. Equations simple unstructured mathematical
+ %
IOXYWI
show mass models.
balances
used
for
building
71 -a-
-o-
q[wO] 12
Xbk’l
r I 10
9 _ 66-
- 4-
Fig. 2. Chemostat culture of K. fragdis NRRL 1109. O: cell dry weight; n : residual glucose concentration; 0: heat evolution rate; which was measured using special calorimetric techniques in order to characterize the global metabolic activity of the culture (Birou and von Stockar, 1989).
can be measured in experiments that are independent of the continuous culture that the model is supposed to describe. The fact that the values of these parameters can be obtained without having to fit the model to existing chemostat data confers a certain predictive power to the model calculations. x
(9 I-‘)
n
P
(9 I-’ ) A
PO,
(O/o) l
6 -
50
6 II
n
.:
WI
w
A
4 - ’
.
l 4
1
2 K D-’
Fig. 3. Dry biomass concentration function of the relative oxygen
40
-
30
-
20
-
10
I. l
0
-
I
. :. I
3
4
&
#
5
I.
0
6
(h)
(m ), ethanol concentration (A) limitation K/D during continuous and Birou, 1989).
and dissolved culture of
oxygen
K.
fragilis
tension (van
(0) as a Stockar
72
2 L
16 14
-59
i
;
12-
2 $
10 -
c! :
6-
d 0”
6-
Biomass
4-
O+--ctccr
I
0.15
I
0.25
Dilution Fig.
4. Continuous
culture
I
I
0.35
of S. cerec~isirre concentration;
I
I
0.45
Rate
0.55
(h-l)
CBS 426. n : Biomass concentration; a: ethanol concentration.
0:
residual
glucose
Biomass
0
o
0
Ethanol 0
0
0
0
Glucose o!
pa, 0.0
-I
,-
0.2
,-,y,
0.4
0.6
KLA Fig. 5. Dry (A)
biomass as a function
concentration of the relative cere~Goe
; C*/D
0.6
So
( n ), ethanol concentration oxygen limitation (k,a
(Auberson
and von
Stockar,
,-
,“,
q
1.0
1.2
p
1 1.4
(-) (0). c”/DS,)
and residual glucose during continuous
1991; Auberson,
1990).
concentration culture of
S.
73
In consequence, simple unstructured model descriptions will be used in this paper in order to analyze some fundamental aspects of chemostat culture behavior, and to interpret real chemostat data. Based on such simple descriptions, features of four different chemostat cultures of yeasts will be analyzed. The first example is an X-D diagram obtained in highly aerobic conditions with a so-called aerobic respiring yeast (Kfuyueromyces fragilis), (Fig. 2). (This term, introduced by Alexander and Jeffries (1990) means that K. fragilis grows exclusively according to an oxidative metabolism when sufficiently aerated whereas a so-called “aerobic fermenting” (AF) yeast will grow according to a mixed oxido-reductive metabolism at high dilution rates even when p0, is high). The second example involves cultivating K. f,wgih under various degrees of oxygen limitation (Fig. 3). The third case involves an X-D diagram of a so-called “aerobic fermenting” yeast (Sacchnro!?zyce.scere[lisiae), (Fig. 4) and the last example deals with subjecting this type of yeast to oxygen limitation (Fig. 5).
Balances, stoichiometry
and kinetics in chemostats
Any quantitative model or theory for chemostat cultures is based on balances, such as shown in Fig. 1, for each of the chemical and other entities that enter or leave the growth vessel. In simple model descriptions, the bioreactor contents are usually assumed to be well mixed, and at steady state. The specific consumption or production rate qi for any given compound must therefore be uniform in the whole reactor and invariant with time. Application of these balance principles to the biomass leads to the well known equality of p (= L],) and D. A major issue when constructing quantitative explanations for chemostat data such as shown in Figs. 2-5 concerns the number of major biochemical entities for which a balance, such as is shown in Fig. 1, ought to be written. The question is best dealt with by formulating the conversion occurring in the yeast cultures in terms of a quasi-chemical equation which might read e.g., as follows: Ck,Q,Nsl+
+ Yd,sO2 + G,sNH3
Y;,sCHx,Q2Nx3
+ G,sCH,,O,~N,~
+ Y&CO,
+ G&-LO
(1)
This equation shows 7 major compounds including a carbon and energy substrate, the biomass and a major intermediate product formed in the process, all of which appear as C-molar “chemical” formulae. In order to complete the model, one needs a certain amount of kinetic information. If the Y-values appearing in Eq. (1) were completely independent of each other, if they could assume any set of values, all of the compounds appearing in Eq. (1) could be taken up or produced at any rate and one would need 7 kinetic models for defining the 7 qi values in Fig. 1. There are, however, 4 constraints due to the elemental balances. One would thus generally expect that 3 kinetic models are needed to predict all seven qi values, but additional constraints may further reduce this number.
74
Once these models are identified, the prediction of X-D diagrams is quite straightforward, as will be shown for specific cases in later sections. Key steps involve developing expressions for the biomass yield Yi,, and using them in conjunction with the kinetic models in order to link p to the concentration(s) of the limiting substrate(s) (see e.g. Eqs. 4 and 8). From such a relation, the residual substrate concentration(s) can be predicted as a function of D because of /.L = D. The biomass concentration is then found as: x= Y;,,(s,
- S)
(2)
The specific uptake and consumption rates qi for compounds other than X and S are evaluated by combining the kinetic models with the elemental balances and other constraints. The residual concentrations of these compounds may then be predicted by solving the balances shown in Fig. 1 for the steady state. In the next section, this general philosophy will be used in order to derive model descriptions for interpreting the chemostat experiments shown in Figs. 2-5. Due to space limitations, the emphasis will not always be on predicting all the measured steady state concentrations. Instead, the prediction of the observed apparent yield coefficients of the general stoichiometry appearing in Eq. (1) will be stressed.
An analysis of chemostat cultures of yeasts in terms of simple unstructured
models
Aerobic chemostat cultures of Kluyveromyces fragilis K fragifis is a so-called aerobic-respiring (AR) yeast in the nomenclature introduced by Alexander and Jeffries (1990), which means that the culture grows according to a purely oxidative metabolism even at dilution rates close to wash-out. The corresponding X-D diagram (Fig. 2, Birou and von Stockar, 1989) is therefore relatively simple. The absence of a product (Yi,s = 0) reduces the overall growth stoichiometry to only 6 compounds as shown in Table 1. Due to the 4 elemental balances, the rates at which these compounds are consumed or produced are not independent from each other. By the same token, only one of the 5 remaining stoichiometric coefficients can vary in an independent way. Thus, if one of them is
TABLE 1 Oxidative growth
y,=4+i,-2i,-3i,;
&=fi,-+i,.
known, the other 4 may be computed by solving the elemental balances (Table 1). A further constraint is due to bioenergetic considerations. At moderate to high values of D, where maintenance effects can be neglected, the biomass yield Yiys can be assumed constant and determined by the bioenergetics of the culture. For the aerobic growth of K. frugilis, the five remaining stoichiometric coefficients in Eq. (1) are therefore completely fixed by five constraints. The production and consumption rates of the various entities shown in this equation are therefore totally interlocked: if we know one of these rates we can calculate all the others through the stoichiometry given by Table 1. It is therefore clear that only one of these compounds can be taken up or produced in a kinetically independent way, and that one single kinetic expression suffices to complete the whole model. The next step is to identify the compound whose consumption or production rate will determine all the others, and for which a kinetic expression should consequently be formulated. The answer to the question is linked to fundamental concept of the so called limiting substrate. Since it is usually the carbon and energy substrate that limits growth in chemostat culture, the kinetics of the whole process is governed by the uptake kinetics of this substrate: s 4s = ,iY K, + s The specific growth rate can be calculated by multiplying stoichiometric coefficient:
cL= y;ys
s
qs""
K, + S
Eq. (3) by the respective
(4)
Equation (4) is equivalent to the Monod equation. Since p = D in chemostat cultures, it introduces a direct link between the dilution rate and the residual substrate concentration S which can be seen graphically on Fig. 2. All other specific consumption or production rates may be computed in a similar way and residual concentrations are determined as outlined in the preceding section. At low dilution rates, X is seen to assume lower values (Fig. 21, thus indicating a decrease of Yx,, due to maintenance effects. A consistent way of extending the model to this case consists of the assumption that the growth equation shown in Table 1 remains unchanged, but that an additional consumption of carbon and energy substrate proceeds in parallel: 4s =
(qs&wth
where
+
k)growlh
stoichiometry p
=
y;>
m
(5)
is the specific substrate uptake rate for growth. According to the for growth (Table 11, (qs)growth is related to /.L as follows:
~~s)~row~h
(6)
By substituting (qs)growt,, in Eq. (5) by means of Eq. (6) and dividing by EL, an expression for the apparent growth yield resulting from both oxidative growth and
76
maintenance is obtained:
Equation (7) is known as the linear Herbert-Pirt relation. When the inverse of the apparent biomass yield is computed from Fig. 2 and plotted vs l/D, one obtains indeed a straight line whose slope is m (Birou, 1986; Birou and von Stockar, 1989). The relationship between S and p follows from a combination of Eqs. (7) and (3):
which replaces Monod’s equation (4). Equation (8) can be used to predict S from D. All other specific rates and concentrations may be calculated as indicated before. It is important to stress that growth kinetics in chemostats is governed by the uptake rate of the limiting substrate and not vice versa. Growth is a consequence of substrate .uptake and not the cause. Writing a Monod-type equation for describing the growth kinetics of the culture is thus wrong and leads to an inconsistent description, as already shown by Heijnen (1990). In these cases the substrate uptake kinetics also determines the uptake kinetics of the non-limiting substrates such as oxygen and all growth factors through stoichiometry. This clearly means that these substrates cannot be taken up according to their intrinsic, biological uptake kinetics. The situation is shown for oxygen
qox
C
L
-
kLa
k,a Fig. 6. Kinetics
and stoichiometric
constraints
for the oxygen
+D
c*
L
consumption
rate.
in Fig. 6. The curve represents schematically the intrinsic kinetic model for 0, uptake:
(9) Also shown in this figure are the terms in the balance representing the difference between oxygen influx and outflux from the chemostat culture as shown in Fig. 1. These two terms form a straight line of a slope which is given by the negative sum of k,a + D and which originates on the abscissa from a point determined by the equilibrium oxygen concentration c *. If oxygen were a limiting substrate, the culture would operate at the intersection of the kinetic curve and the straight operating balance line. But since it is the carbon and energy substrate that limits growth, it imposes an oxygen uptake rate on the culture which is lower and given by the horizontal broken line. As can be seen from Fig. 6 the culture is thus forced to leave a large amount of oxygen unused in the culture, thereby giving rise to a considerable dissolved oxygen concentration CL. Fig. 6 can be used in order to develop an operating definition of what one understands by the term limiting substrate: a limiting substrate is one which is taken up according to the intrinsic biological kinetics and for which the culture operates on the respective kinetic curve (Fig. 6). A non-limiting substrate will be taken up at a lower rate dictated by stoichiometric considerations. In Fig. 7 the specific oxygen uptake rate is plotted schematically against the specific carbon and energy substrate uptake rate. The two quantities are linked by a fixed stoichiometry and must therefore form a point on a straight line of a slope equal to the respective stoichiometric coefficient YoFs. If oxygen is supplied in
b uptake
B
SUPPlY 9s
Fig.
7. Stoichiometric
link
between
specific carbon and energy substrate oxygen uptake in obligate aerobes.
consumption
and
specific
7x
excess (supply A) a considerable amount of the supplied oxygen will remain unused. In obligate aerobic cultures one will be able to reverse the situation by limiting the supply of oxygen to level B. This would force the culture to reduce the uptake of S in such a way that the operating point B is again located on the straight line. Since S could not be taken up anymore at its biological uptake kinetic rate, a considerable amount of the substrate would remain unused in the medium. High residual substrate concentrations are indeed observed even at low dilution rate in chemostat cultures which are in fact not limited by the main carbon and energy substrate but by some other usually unknown growth factor. Growth of K. fragilis under oarious degrees of oxygen limitation
The actual result obtained when subjecting a culture of Kluyueromyces fragilis operating at constant dilution rate to an oxygen limitation is shown in Fig. 3 (von Stockar and Birou, 1989). In this experiment the oxygen limitation was introduced by sparging air-nitrogen mixtures through the culture such that the equilibrium oxygen concentration c * in the medium could be varied. The K value appearing on the abscissa indicates the fraction of air in the air-nitrogen mixture (Birou, 1986). It can be seen that the unused excess of oxygen is decreased linearly as a function of the reduction in the oxygen supply. At K/D = 1, the oxygen tension is more or less zero and oxygen becomes a limiting substrate. Since yeasts are not obligate but facultative aerobes, the carbon and energy substrate S is still used to completion even at lower oxygen supplies (not shown) and ethanol appears in the medium. Instead of reducing its substrate uptake in order to stay on the original
qo
SUPPlY uptske
Fig.
8. Stoichiometric
link
between
specific
oxygen
uptake
carbon
and
in facultative
qs
energy
substrate
aerobes.
consumption
and
specific
19 TABLE 2 Reductive growth CH$,,Ns.,
+ YA::NH,
. = (z$,,)-(!LJz Yl;‘;s” rrcll yNS
=
-Ys ---y YP
yx
YP
+
Y~~~CH,,O,,N,,
+ Y$;CH,,OPINpJ
+ Y;.;‘,dCO, + Y;I;“,H,O
-x,)Ygg rrcd
x/s
Y,rLxl c/s rrud Y w/s
yi and ai are defined in Table 1.
stoichiometric line on Fig. 7, the culture metabolizes the excess of substrate according to a reductive mechanism that does not consume oxygen. Therefore, the stoichiometric line is changed as a whole and the culture now operates at point B on Fig. 8. This is a clear case of dual limitation since both the main substrate S and the oxygen will be taken up according to their biological uptake rates. As this dual limitation permits the culture to increase qs, the behavior might be termed a “Pasteur-type” effect (Pasteur, 1861). In order to quantitatively understand the data in Fig. 3 one has to extend the stoichiometry of the model to include the reductive growth mechanism. The easiest way is to represent the overall growth stoichiometry (Eq. 1) as the sum of two parallel biochemical reactions, one being growth at fully oxidative conditions (Table 1) and one being growth at fully reductive conditions (Table 2). Note that both these growth stoichiometries comprise 5 stoichiometric coefficients, 4 of which can be calculated by solving the elemental balances if the biomass yields are known. All the stoichiometric coefficients in such a model can therefore be calculated from the biomass yield for fully oxidative conditions Yiys and the biomass yield for fully anaerobic reductive growth conditions Yi;“s”. These two yield coefficients are true model parameters but they can be measured in independent growth experiments. A complete unstructured model would be obtained if the two stoichiometric equations in Tables 1 and 2 were combined with two kinetic expressions, one for the rate of substrate uptake and one for the rate of oxygen consumption (Eqs. 3 and 9). The overall apparent stoichiometry and all Y values appearing in Eq. (1) could be predicted from there. In order to simplify the calculations further it is convenient to introduce a factor which measures the relative rates between the two equations shown in Tables 1 and 2. This factor has been proposed by von Stockar and Birou (1989) and is defined as follows: 4 k,a c* - (k,a +D)c, nz-= Y&s I (10) ,0X Yo/s x - YxYio;, W%-S)
80
This so-called aerobicity measures the relative rate of oxygen to carbon and energy substrate uptake and is normalized by the respective ratio that one would observe for fully oxidative growth conditions. In Fig. 8 R measures the relative decrease of the slope of the stoichiometric line resulting from the decreasing oxygen supply. For doubly limited cultures such as the one shown in Fig. 3, we now propose an easy approximate estimation technique for 0. Since both the residual substrate concentration and th5 dissolved oxygen tension are very small in such cultures, the aerobicity becomes:
(11) where 1 k,a c* Lzsupp= YA(;” DS”
(12)
4upp indicates the maximum normalized ratio of oxygen to substrate supply. It is very easy to estimate on the basis of just a few independently measurable parameters (biomass yields for fully oxidative growth, kLa, c*, D, S,) and will approximate the real LJ with high accuracy in a doubly limited culture. Based on known values for the aerobicity 0 the apparent overall stoichiometry (Eq. 0, which is the result of the combined effect of the two growth reactions, may be computed as shown earlier (von Stockar and Birou, 1989). Some of the more important stoichiometric coefficients as a function of 0 are given in Table 3. All specific rates, yields and steady state concentrations can be calculated as described in the second section, based on an estimate of R. In order to do this, &,,, was calculated as a function of K/p and Table 3 was used for computing the apparent yields in Eq. 1 (Yo,x>. The results obtained for Y.&, and Yi,, yielded the straight lines on Fig. 9. Also shown on this figure are the experimental yield values recalculated from Fig. 3. Although the value of k,a c* could have been measured in independent experiments, it was found by a best fit technique. As a result, the calculations TABLE 3 Prediction of apparent stoichiometric coefficients in Eq. (1) based on the aerobicity Y&s
= f2y;ys
yi,s
= (l- n,y;,“,d
Y&s
= L!Y&‘;”
Y&s
= 1 - y.$
Y&x
=
Y$ 2.
+ Cl- n,y$
- y;;‘,d + (y.$~+y$-y~~s)~
AH; - (I- n)Y,$AH nY~~s+(l-n)Y.g
p*
-AH;
and Yhys may be computed from Yiys and Y.$t according to the relations given in Tables 1 and
81
0.0
0.2
0.4 k,a
0.6 c.*
0.8
1.0
DSO
Fig. 9. C-molar biomass function of the relative
yield (m) and product yield (A) in continuous cultures oxygen limitation, expressed as k,a c*/DS,, (van Stockar
of K. fragilis as a and Birou, 1989).
coincide with the measured values,quite well. Based on the same parameter values, any other yield can be predicted. Details of such calculations will be presented in a separate paper (Auberson and von Stockar, 1991). Aerobic chemostat cultures of S. cerevisiae
An X-D diagram for Saccharomyces cereuisiae is shown in Fig. 4 (Auberson, 1990). The diagram is considerably more complicated than the one shown in Fig. 2. Being a so called aerobic-fermenting (AF) yeast, S. cerevisiae produces ethanol at high dilution rates even when it is well aerated. The appearance of some reductive metabolism at high glucose feeding rates has been linked in earlier times to catabolite repression and to the Crabtree effect (De Deken, 1966; Polakis et al., 1965; Lemoigne et al., 1954). More recently, evidence has accumulated showing that the appearance of ethanol under well aerated conditions may result from an overflow of metabolites into the reductive pathway due to a bottleneck in the respiratory capacity of the yeast (Barford and Hall, 1979; Rieger et al., 1983). Based on this hypothesis, Sonnleitner and Kappeli (1986) presented a predictive mathematical model for continuous cultures of S. cereuisiae. This bottleneck introduces a maximum ceiling equal to q,““” for the uptake of oxygen as shown in Fig. 10. By increasing the dilution rate D in Fig. 4, the specific substrate uptake rate also increases almost proportionally. At low dilution rate the culture will operate on the fully aerobic stoichiometric line (A in Fig. 10). At point B, corresponding to the so-called “critical” dilution rate, the bottleneck is full and the oxygen is now taken up with the maximum rate permitted by the intrinsic biological uptake kinetics. If the dilution rate and thus q, are further augmented, the culture must therefore metabolize the excess supply of S by the reductive pathway, which does not utilize any oxygen. Therefore the ratio of oxygen to
Fig.
10. Stoichiometric uptake
link
in continuous
between cultures
specific of yeast
carbon whose
and
respiratory
energy capacity
consumption is limited
and
specific
by a bottleneck.
substrate consumption must decrease and the whole stoichiometry changes (point C). It is important to note that between B and C the culture is again doubly limited since both S and oxygen are taken up at the maximum rate possible by biology. Nevertheless, large quantities of excess oxygen will remain unused in the medium and the dissolved oxygen tension will be far from zero. A simple procedure for estimating the aerobicity R may also be proposed for this “Crabtree-type” effect by linking 4, and q. as follows: 40 = 4&,,
= s,y;“;sf-J
q. = $I(&
(13)
- S)Y&“/“sO
(14)
By substituting Eqs. (9) and (2) for q. and X, respectively, by noting that S = 0 and C, >> K, in this case, and substituting Y.&, as a function of 0 from Table 3 one obtains: fl = f&Mb
(15)
where a
crab
Y$
ID
(16) &Y
Y&
-
Y ‘OX ( x/s
-
y.p)
The true model parameters appearing in Eq. (16) are qzax, Yi;‘,d and YiFs and can be determined in independent experiments. Y& can be computed from the Yiys as shown in Table 1.
x3
1.0 0.9 0.8
-
0.7
-
7
0.8-
m z1
0.5
2 *
0.4 -
Y’x/s 8 ,.
-
0.3 0.2 -
O.l0.0 0.15
YIP/S I
I
I
0.25
Dilution Fig. 11. Biomass yield ( n ) and ethanol S. cererisiae. Solid lines were obtained
f
I
1
0.35
0.45
0.55
Rate
(h-l)
yield (0) as a function of dilution rate in continuous cultures of by estimating the aerobicity as explained in the text (Auberson and von Stockar, 1991).
Apart from being a simple estimate for the real aerobicity, Eq. (16) also shows the conditions that distinguish aerobic-respiring from “aerobic-fermenting” strains. The Crabtree-type effect shown in Fig. 5 can only be exhibited at 0 < 1. This means that ,:“’ < DYdys/Yiys which is only possible if (17) 9,“” < hi~xydO;ss/y~~s Substituting Y& from Table 1 one finds that only those strains will show a bottleneck in the oxidative metabolism whose q,““” satisfies the following condition: @Tax< Pmux(-Ys- Y;“;,r,)/4Y;ys
(18)
By estimating 0 in terms of L?ncrI,bas a function of dilution rate (Eq. 16) and by applying the relations given in Table 3, the apparent biomass and ethanol yields were predicted as a function of dilution rate and compared to the actually measured value as shown in Fig. 11. As the solid lines show, these predictions, as well as the concentrations shown in Fig. 4, coincided well with the measured values. Growth of S. cerevisiae under various degrees of oxygen limitation
Fig. 5 shows how a culture of an AF yeast reacts when grown at constant supercritical dilution rate and when the oxygen supply is gradually reduced. The x-axis indicates the oxygen supply and is obviously proportional to the criterion %upp. At high levels of oxygen supply the culture seems to be quite insensitive to
SUPPlY B max 90
SUPPlY c
-
px/fex,
nrg
qs “Crasteur” Fig. 12. Stoichiometric specific oxygen uptake
Effect
link between the specific carBoon and energy substrate consumption and the in yeast cultures limited by a respiratory bottleneck and subjected at the same time to an oxygen supply limitation.
an oxygen limitation. Only when k,a CT/D& decreases below 0.5 does the ethanol production slowly increase and the biomass concentration slowly diminish, thereby indicating a shift towards a more reductive metabolism. Fig. 12 explains the effect in terms of the simple model description proposed in this paper. The experiment involves imposing a limitation of the oxygen supply on a culture which is already oxygen limited due to the existence of a bottleneck. Since this involves superimposing a Pasteur type effect on an already existing Crabtree-type behavior, this type of experiment could perhaps be called a “Crasteur’‘-type effect. Even at the high oxygen supply level A in Fig. 12 the culture operates under dual limitations (point A) with an 0 which is already smaller than 1. Neither the oxygen uptake rate nor R changes when the supply is diminished to level B, which explains the insensitivity of the culture at high oxygen supplies. Only when the oxygen supply becomes similar to the maximum oxygen uptake rate or lower, is the culture forced to metabolize a growing part of the substrate through the reductive pathway, thereby decreasing the R and thus the stoichiometric line (point C). In Fig. 13 the experiment is shown to correspond to a gradual reduction in CL* thereby shifting the balance line gradually to the left. It is important to appreciate that a dual limitation exists all along in this experiment and that the oxygen uptake is given by the respective kinetic curve for all points for this experiment. The operating point thus slides along the continuous kinetic curve from the right to the left. This explains the smooth transitions of the concentrations which have been measured. In predicting the observed concentrations and yields flnsUppis calculated for each experimental point according to Eq. (12). The equations in Table 3 are evaluated by using ancrab as long as L& is higher than the former but by using L&,,, as
85
cL
k,= ka+D 6 I.
Fig. 13. Oxygen consumption rate as a function of cl* for the “Crasteur-type”
experiment.
soon as it becomes lower. The predictions become straight lines with sharp kinks. It is obvious that the gradual transition of the concentrations actually observed could also be predicted if the equations were solved by using Eqs. (3) and (9) as the basic kinetic models instead of working with C!SUPPand Oncrab. Fig. 14 summarizes the procedure that is proposed for predicting the quantitative behavior of yeast cultures. Based on a minimal amount of information about the strains and the chemostats, L&, and Oncrabare evaluated according to Eqs. (12) and (16). As long as both of these estimators are above 1, the culture will grow according to a fully oxidative metabolism and the real, metabolic aerobicity will be equal to 1. As soon as either L&, and Oncrabor both become lower than 1, the metabolism will shift to oxido-reductive, and the yields, rates and concentrations may be predicted by observing that the real R will adapt itself to the lower of the two estimators. The four examples presented in this paper correspond to shifts from one region to another in Fig. 14.
~cmb < 1 nonb < nsuppl r-1 ---------nauppl> 1 ncnb > 1 I”-‘l
nsuppl
