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Experimental Investigation and Modeling
of Oscillatory Behavior in the Continuous
Culture of Zymomonas mobilis
Andrew J. Daugulis, P. James McLellan, Jinghong Li
Department of Chemical Engineering, Queen’s University, Kingston,
Ontario, Canada K7L 3N6; telephone: (613) 545-2784; fax: (613) 545-6637;
e-mail: daugulis@qucdn.queensu.ca
Received 16 August 1996; accepted 7 February 1997
Abstract: The mechanism causing oscillation in continu-
ous ethanol fermentation by Zymomonas mobilis under
certain operating conditions has been examined. A new
term, ‘‘dynamic specific growth rate,’’ which considers
inhibitory culture conditions in the recent past affecting
subsequent cell behavior, is proposed in this article.
Based on this concept, a model was formulated to simu-
late the oscillatory behavior in continuous fermentation
of Zymomonas mobilis. Forced oscillation fermentation
experiments, in which exogenous ethanol was added at
a controlled rate to generate oscillatory behavior, were
performed in order to obtain estimates for the model
parameters and to validate the proposed model. In addi-
tion, data from a literature example of a sustained oscil-
lation were analyzed by means of the model, and excel-
lent agreement between the model simulation and ex-
perimental results was obtained. The lag in the cells’
response to a changing environment, i.e., ethanol con-
centration change rate experienced by the cells, was
shown to be the major factor contributing to the oscilla-
tory behavior in continuous fermentation of Zymomonas
mobilis under certain operating conditions. © 1997 John
Wiley & Sons, Inc. Biotechnol Bioeng 56: 99–105, 1997.
Keywords: oscillatory behavior; Zymomonas mobilis;
fermentation; ethanol
INTRODUCTION
During the past 15 years,Zymomonas mobilishas been
promoted as a more promising microorganism than yeast for
the industrial production of ethanol.Zymomonas mobilis
possesses higher ethanol tolerance and superior kinetic and
yield characteristics compared to those of yeast (Beavan et
al., 1982). More specifically, it exhibits higher specific rates
of substrate consumption, ethanol production, and a higher
ethanol yield (Rogers et al., 1982).
One drawback associated with the continuous fermenta-
tion of Zymomonas mobilisis the peculiar occurrence of
oscillation or cycling of substrate, biomass, and product
concentrations under certain fermentation conditions (Bruce
et al., 1991; Ghommidh et al., 1989; Jobses et al., 1986; Lee
et al., 1980; Lee et al., 1979). Oscillatory behavior means
that there are periods of time during which there is de-
creased ethanol productivity, and a high level of residual
substrate resulting in unacceptably high substrate losses.
Based on previous work (Li et al., 1995; Li, 1996), the
concept of adynamic specific growth rateis proposed in
this article. The dynamic specific growth rate refers to a
situation in which the cells undergo a growth rate which is
less than or equal to a growth rate characterized by the
instantaneous culture conditions, and reflects (potentially
deleterious) conditions experienced by the cells some time
in the past. Experimental evidence is provided by direct
estimation of dynamic specific growth rates from experi-
mental data and comparisons made with the instantaneous
specific growth rate at the culture conditions. With this
concept, a model is formulated to describe the oscillatory
behavior of continuous ethanol fermentation byZymomonas
mobilis. Forced oscillation fermentation experiments were
performed to validate the proposed model and a literature
example of oscillation in the continuous fermentation of
ethanol was used to test the model.
Model Development
Several earlier models have been proposed to account for
the oscillatory behavior ofZ. mobilis.One model (Ghom-
midh et al., 1989) later expanded to include substrate limi-
tation and product inhibition (Jarzebski, 1992), conceptual-
ized the cell population as consisting of three states: viable
cells, non-viable cells (able to produce ethanol but not to
reproduce), and dead cells. With appropriate mass balance
equations, the model was able to represent the data obtained
by their group during oscillatory culture ofZ. mobilis.Some
experimental support for the presence of viable and non-
viable, or dead cells was provided by plate count and slide
culture estimations, although these assays are time-con-
suming. Another model (Jobses et al., 1986), a structured
two-compartment representation, considered biomass as be-
ing divided into compartments containing specific group-
ings of macromolecules (e.g., protein, DNA, and lipids).
A common feature of the above models is that they are
based upon some physiological quantities which are often
Correspondence to:Andrew J. Daugulis
Contract grant sponsor: Natural Sciences and Engineering Research
Council of Canada
© 1997 John Wiley & Sons, Inc. CCC 0006-3592/97/010099-7
difficult or time consuming to monitor during the fermen-
tation process. Our approach has been more phenomeno-
logical in that macroscopic variables such as ethanol, sub-
strate, and biomass concentrations are the only experimental
quantities which need to be measured. This is not to suggest
that sub-populations do not exist withinZ. mobiliscultures,
particularly in stress situations; however, such stress re-
sponses may be more easily detected through the presence
of filamentation, which has been previously linked to stress
situations experienced by the cells (Daugulis et al., 1985;
Fein et al., 1984), including exposure to high ethanol con-
centrations in a short period of time (Ghommidh et al.,
1989).
Previous work in our laboratory (Li et al., 1995) with
Zymomonas mobilishas shown that the inhibition effect of
ethanol concentration history is insignificant, while the in-
hibition effect of an upward ethanol concentration change
rate is quite intense. Moreover, recognizing that cells are
unable to respond instantaneously to changes in their envi-
ronment, it was postulated that a delay exists between the
time that the cells experience a change, and their metabolic
response. The time delay inhibition effect was modeled as
the effect of the weighted average of ethanol concentration
change rate,Z:
Z~t! = *−`
t
dP/dt|t · cz~t!dt (1)
In Equation (1), the weight for the ethanol concentration
change rate historycz(t) is:
cz(t) 4 b
2 · (t − t) · e−b·(t−t) (2)
wheret is the time in history andt is the current time. The
maximum ofcz(t) occurs att 4 t − 1/b, i.e., 1/b h prior to
the current timet. In other words, the ethanol concentration
change rate which happened at 1/b hours ago has the most
significant influence on current cell performance. The pa-
rameterb indicates the magnitude of the time lag for the
delayed inhibition effect. As noted, this time delay effect
recognizes that cells do not respond instantaneously to
changes in their environment, but require a period of time to
produce a metabolic response.
With Z(t) being defined in Equation (1), it is hypothesized
that the growth of cells at timet is inhibited when the
weighted average of ethanol concentration change rateZ(t)
is positive, while a negativeZ(t) has no effect on the cell
growth. Obviously, such a hypothesis about the effect of
Z(t) leads to a piece-wise function, withZ(t) 4 0 being the
switching point. Usually, a piece-wise continuous function
is not desired in parameter estimation procedures since it is
not suitably differentiable. In order to facilitate the param-
eter estimation procedure, a continuously differentiable hy-
perbolic function,fm (Z(t)), is introduced to approximate the
effect of Z(t) on the cell growth:
fm ~Z~t!! =
1
2 S1 − el · Z~t! − d − e−l · Z~t! + del · Z~t! − d + e−l · Z~t! + dD (3)
l andd are parameters to be estimated from the experimen-
tal data which represent the intensity of the inhibition effect
of the weighted average of ethanol concentration change
rateZ(t). The functionfm (Z(t)) possesses the properties:
fm (Z(t)) ∈ (0,1) for Z(t) ∈ (−`, +`) (4)
fm isthe inhibition factor, and is used to define the dynamic
specific growth ratem(S(t), P(t), Z(t)) in terms of the con-
ventional or ‘‘instantaneous’’ specific growth ratem(S(t),
P(t)):
m(S(t),P(t),Z(t)) 4 m(S(t),P(t) z fm(Z(t)) (5)
The instantaneous specific growth ratem(S(t), P(t)) refers
to the traditional or classic representation of specific growth
rate well established either in textbooks (Bailey and Ollis,
1986) or the literature (Aiba et al., 1968).m(S(t), P(t)) is
instantaneous in the sense that it is strictly a function of the
current values ofS(t) andP(t). The dynamic specific growth
ratem(S(t), P(t), Z(t)) is proposed to include the inhibitory
effect ofZ(t), the weighted average of ethanol concentration
change rate history experienced by cells. The dynamic na-
ture of this specific growth rate arises from the fact thatm
depends onZ, and is associated with the evolution of the
ethanol concentration change rate experienced by the cells.
Becausefm lies between 0 and 1, Equation (5) states that
the dynamic specific growth ratem(S(t), P(t), Z(t)) assumes
a value in the range between 0 andm(S(t), P(t)), depending
upon the value offm (Z(t)). When the weighted average of
ethanol concentration change rate is negative or zero, cor-
responding to a sustained downward ethanol concentration
change rate trend or a trend of constant ethanol concentra-
tion, the dynamic specific growth ratem(S(t), P(t), Z(t)) will
reach a value identical to the instantaneous specific growth
rate m(S(t), P(t)). When an upward ethanol concentration
change rate trend occurs (Z(t) > 0), the dynamic specific
growth ratem(S(t), P(t), Z(t)) will only be able to reach a
fraction fm (Z(t)) of the instantaneous specific growth rate
m(S(t), P(t)). For convenience and simplicity in reporting
our results, the variable namesS, PandZ will be used to
representS(t), P(t) andZ(t) in the following sections.
The instantaneous specific growth ratem(S,P) in Equa-
tion (5) is modeled using the product and substrate inhibi-
tion formula of Veeramallu and Agrawal (1990):
m~S,P! =
mmaxSS1 − S PPmaD
aDS1 − S P − PobPmb − PobD
bD
Ks + S+
S~S− Si!
Ki − Si
(6)
where
P − Pob
Pmb − Pob
= 0 if P ø Pob
S− Si = 0 if Sø Si
P − Pob
Pmb − Pob
= 1 if P . Pmb
100 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 56, NO. 1, OCTOBER 5, 1997
The specific production rate Qp is modeled using the litera-
ture formula (Lee et al., 1983; Luong, 1985),
Qp = QPmax ·
S
Kmp + S
· (1 − (
P
Pme
)a) (7)
In order to be able to undertake modeling for both batch and
continuous fermentation, the complete differential equation
model for the cutlure ofZ. mobiliswas formulated as:
dX
dt
= ~m~S,P,Z! − D − s~t! · De! · X (8)
dS
dt
= −1/YP/S · Qp · X + D · Sf
− ~s~t! · De + D! · S (9)
dP
dt
= Qp · X + s~t! · De · Pe
− ~s~t! · De + D! · P (10)
dZ
dt
= b · ~W − Z! (11)
dW
dt
= b · ~Qp · X + s~t! · De · Pe
− ~s~t! · De + D! · P − W! (12)
Note that this model also has provision to account for ex-
ternally added ethanol, in order to be able to model the
forced oscillation experiments described later. The variable
s(t) is either 0 or 1 depending on whether exogenous etha-
nol is being added (i.e., whether the ethanol feed pump is off
or on). This ethanol can be added at a dilution rateDe and
at a concentrationPe.
The first three equations in the model [eqs. (8)–(10)] are
established from mass-balance considerations in the fer-
mentor. Equations (11) and (12) result from differentiating
Z(t) [eq. (1)] with respect tot, whereW(t) is an intermediate
variable which is also a weighted average of the previous
ethanol concentration change rate with weightcw(t),
W~t! = *−`
t
dP/dt|t · cw~t!dt (13)
cw~t! = b · e
−b · ~t − t! (14)
As will be seen, the above model [eqs. (1)–(14)] has been
used to simulate batch and forced oscillation fermentation
operations. It has also been used to simulate a literature
example of a sustained oscillation fermentation.
The parameters in the proposed model have been ob-
tained from several sources. The intrinsic kinetic parameters
for Z. mobilis were obtained from the literature, and are
shown in Table I. The operating parameters during continu-
ous and forced oscillation fermentation for substrate feed
(Sf), dilution rate (D), ethanol dilution rate (De) and exog-
enous ethanol concentration (Pe) were fixed at 20 g/L, 0.084
h−1, 0.015 h−1, and 75% (v/v), respectively. The ethanol
yield was determined experimentally to be 0.495 g/g for
conventional continuous fermentation and 0.46 g/g for the
forced oscillation fermentations. The remaining parameters
(b, l, d, a, b, Qpmax, and a) were obtained by multi-
response nonlinear regression estimation using the experi-
mental data (Li, 1996).
Additional insight supporting the concept of dynamic
specific growth ratem(S, P, Z) can be obtained by estimat-
ing this quantity directly from the experimental data ob-
tained from batch, conventional continuous, and forced os-
cillation experiments. Using finite differences to represent
dX/dt,the dynamic specific growth ratem(S, P, Z) at timeti
can be calculated directly from the experimental data using
Equation (8) as follows:
m~Si , Pi , Zi! =
SXi − Xi−1ti − ti−1 D
Xi
+ D + s~t! · De (15)
where, for batch fermentationD 4 0, De 4 0; for conven-
tional continuous fermentation or if the ethanol feeding
pump is off during forced oscillation fermentationDe 4 0.
The subscripti refers to theith sample in the experimental
data. Because all the effects ofS, P,andZ on the fermen-
tation should be reflected in the experimental data which are
used directly in Equation (15), them(Si , Pi , Zi) calculated by
this equation provides information about the dynamic spe-
cific growth rate as a function of these quantities. If there
was no memory effect the estimated values should match
the instantaneousm(S,P) calculated atSi , Pi within a rea-
sonable range of experimental error. However, if the esti-
mated specific growth rates deviated significantly from the
instantaneous values, this would support the concept of a
dynamic specific growth rate. Furthermore, the estimates of
m are not dependent on any of the parameters described in
the preceding paragraph.
MATERIALS AND METHODS
The microorganism, the medium composition, and the ex-
perimental procedure were described in previous work (Li
et al., 1995). Beginning with batch and conventional con-
tinuous fermentation, forced oscillation fermentation ex-
periments were subsequently conducted during which an
oscillatory ethanol concentration trajectory was created by a
pump turning on and off which pumped ethanol into the
Table I. Literature values of parameters used in the modeling of oscil-
latory behavior.
Model parameter
Veeramallu and
Agrawal (1990)
Lee and
Rogers (1983)
mmaxh
−1 0.41
Pob (g/L) 50.0
Pma (g/L) 217.0
Pmb (g/L) 108.0
Pme(g/L) 127.0
Ks (g/L) 0.5
Ki (g/L) 200.0
Si (g/L) 80.0
Kmp (g/L) 0.5
DAUGULIS, MCLELLAN, AND LI: OSCILLATORY BEHAVIOR OF ZYMOMONAS MOBILIS 101
fermentor. Two such sequential fermentations (batch, fol-
lowed by continuous, followed by forced oscillation) were
conducted, one with an ethanol oscillation period of 40 h,
and the other with a period of 50 h. The feeding substrate
concentration was intentionally kept at a low level, 20 g/L
glucose, to ensure that the ethanol converted from glucose
would not affect the imposed ethanol concentration trajec-
tory significantly. With this imposed oscillatory ethanol
concentration trajectory, the fermentation was monitored by
measuring the ethanol, biomass, and residual glucose con-
centrations.
RESULTS
Forced Oscillation Fermentation
Two forced oscillation experiments were undertaken, with
the first being conducted in order to impose ethanol oscil-
lations on the fermentation system from which estimates for
the model parameters could be obtained. With these esti-
mates, the model was used to simulate the data obtained
from the second forced oscillation experiment (with a dif-
ferent ethanol oscillation period) to help determine how
wellthe model predictions provided a fit to the experimental
data. In both cases batch fermentation was followed by
conventional continuous fermentation to obtain a stable
steady state situation before the initiation of the forced os-
cillations.
Figure 1 shows the results of the first forced oscillation
experiment (40 h oscillation period), along with the model
predictions for this run. As can be seen, the imposed ethanol
oscillations caused corresponding oscillations in the cell and
residual substrate concentrations. At time zero of the simu-
lation (the onset of the batch fermentation), the weighted
average of ethanol concentration change rates was assumed
zero, i.e.,Z(0) 4 0.0,W(0) 4 0.0 which was a reasonable
assumption provided that the inoculum was in the stationary
phase before being added to the fermentor. As might be
expected, the model simulations closely matched the experi-
mental data for this run.
With model parameters estimated from all phases of the
experiment, simulations were run in order to examine the
ability of the model to fit the data for the second experiment
(50 h oscillation period), as shown in Figure 2. Similar
responses were obtained for the cell and substrate concen-
trations and, as can be seen, the model was able to predict
all phases of the experiment exceedingly well.
The dynamic specific growth ratem(S, P, Z) calculated
from the experimental data obtained during the first forced
oscillation fermentation experiment using Equation (15) is
plotted against the ethanol concentration in Figure 3, to-
gether with the instantaneous specific growth rate curve
calculated by Equation (6) using the necessary estimated
parameters (i.e., a4 0.3332, b4 1.14), and the necessary
parameters from the literature as given in Table I. It can be
seen that the instantaneous specific growth rate curve pro-
vides an upper boundary for the dynamic specific growth
rates as expected. The dynamic specific growth ratesm(S, P,
Z) are found in an area either adjacent to or below the
instantaneous specific growth rate curve depending on the
value of the weighted average of ethanol concentration
change rateZ. When the weighted average of ethanol con-
centration change rateZ is positive, the dynamic specific
growth rate is less than the instantaneous specific growth
rate by a factor defined by the functionfm(Z) [eq. (3)–(5)]
and it falls in an area below the instantaneous specific
growth rate curve. When the weighted average of ethanol
concentration change rateZ is zero or negative,fm(Z) ap-
proaches 1.0, and consequently the dynamic specific growth
rate approaches the instantaneous specific growth rate. As a
result, the dynamic specific growth rate falls in an area
adjacent to the instantaneous specific growth rate curve.
Figure 3 provides a graphical illustration of the distinction
between the instantaneous specific growth rate and the dy-
namic specific growth rate as determined experimentally.
The difference is ascribed to the earlier ethanol concentra-
tion change rate experienced by the cells.
Figure 1. (a) Ethanol and glucose concentrations for forced oscillations having an imposed oscillation period of 40 h. For the model predictions, the model
parameters were estimated to be:b 4 0.1732,l 4 0.4557,d 4 0.5559,a 4 5.498,a 4 0.3332,b 4 1.14. (b) Ethanol and biomass concentrations for
forced oscillations having an imposed oscillation period of 40 h. For the model predictions, the model parameters were the same as for (a).
102 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 56, NO. 1, OCTOBER 5, 1997
Modeling Literature Data
The literature data published by Ghommidh et al., 1989,
were examined to see if they supported the hypotheses and
model developed in this paper. The operating conditions in
their experiment wereSf 4 200 g/l andD 4 0.05 h
−1, with
a different strain ofZ. mobilis,NRRC B-14022, having been
used. Since the residual glucose concentrations were not
reported in their work, an overall product yield coefficient
Yp/s was assumed (0.495), which is the same value as was
obtained in our experimental work. The estimated param-
eters and the model simulation for their experimental data
are shown in Figure 4. As can be seen, good agreement
between the simulation and their experimental data is ob-
tained.
As was done in the previous experimental analysis,m(S,
P, Z) andm(S, P) were estimated from the data of Ghom-
midh et al., and are plotted against ethanol concentration as
shown in Figure 5. The characteristics of this plot are con-
sistent with those seen in Figure 3, and support the proposed
concept of dynamic specific growth rate.
CONCLUSION
It had been previously demonstrated (Li et al., 1995) that the
effect of ethanol concentration history on the fermentative
capability ofZ. mobilisis negligible while the upward etha-
nol concentration change rate has a significant inhibitory
effect on this organism. In the current work, these experi-
mental observations were incorporated into a model for
simulating/predicting dynamic behavior in continuous fer-
mentations ofZ. mobilis. The concept of a dynamic specific
growth ratem(S, P, Z), incorporating the effect of inhibitory
conditions experienced at an earlier time by the cells, has
been proposed, and a distinction made between this term
and the traditional specific growth rate, here called the in-
stantaneous specific growth ratem(S, P). A model consist-
ing of ordinary differential equations was formulated, in-
corporating the newly proposed concept of dynamic specific
growth ratem(S, P, Z).
Forced oscillation fermentations were performed in order
to obtain experimental data of oscillatory behavior, to esti-
mate model parameters, and to test the proposed model. It
was demonstrated that the proposed model is capable of
fitting the experimental data well, and of predicting the
dynamic behavior of experiments conducted under different
operating conditions. The model was further tested by simu-
lating a literature example of another sustained oscillation
Figure 3. Instantaneous specific growth rate [from eq. (6)], and estimated
dynamic specific growth rates [from experimental data and eq. (15)] for the
forced oscillation experiment having an oscillation period of 40 h.
Figure 4. Ethanol and biomass concentrations (from the data of Ghom-
midh et al., 1989), and our model predictions. Model parameters were
estimated to be:b 4 0.03178,l 4 7.586,d 4 0.515,a 4 11.25,a 4
0.2038,b 4 1.715.
Figure 2. (a) Ethanol and glucose concentrations for forced oscillations having an imposed oscillation period of 50 h. For the model predictions, the model
parameters were estimated to be:b 4 0.1732,l 4 0.4557,d 4 0.5559,a 4 5.498,a 4 0.3332,b 4 1.14. (b) Ethanol and biomass concentrations for
forced oscillations having an imposed oscillation period of 50 h. For the model predictions, the model parameters were the same as for (a).
DAUGULIS, MCLELLAN, AND LI: OSCILLATORY BEHAVIOR OF ZYMOMONAS MOBILIS 103
fermentation ofZ. mobilis,and good agreement between the
simulation and the experimental data was obtained. The
proposed concept of a dynamic specific growth rate was
illustrated by plotting the experimentally obtained dynamic
specific growth rate, together with the estimated instanta-
neous specific growth rate against ethanol concentration.
The instantaneous specific growth rate curvem(S, P) pro-
vides an upper boundary for the dynamic specific growth
rate m(S, P, Z), with m(S, P, Z) falling in an area either
below or adjacent to them(S, P) curve depending on the
value ofZ.
Recent work has focused on generating overdamped con-
tinuous, damped oscillation and sustained oscillation fer-
mentation data in our laboratory. The validity of the model
proposed in this paper will now be tested against these
additional data, which represent the spectrum of modes of
continuous operation possible with this organism. The stress
response of the cells (i.e., filamentation) to high ethanol
concentration change rates is also being investigated, along
with the relationshipof these morphological changes tom
andQp values.
We are grateful to the School of Graduate Studies and Research,
Queen’s University, for student support for one of us (JHL).
NOMENCLATURE
a instantaneous specific growth rate parameter
b instantaneous specific growth rate parameter
D dilution rate (h−1)
De exogenous ethanol dilution rate (h
−1)
fm inhibition factor for dynamic specific growth rate
Ki substrate inhibition parameter for specific growth rate (g/L)
Ks substrate saturation parameter for specific growth rate (g/L)
Kmp substrate saturation parameter for ethanol production rate
(g/L)
P ethanol concentration (g/L)
Pe exogenous ethanol concentration (g/L)
Pma ethanol inhibition threshold for instantaneous specific growth
rate (g/L)
Pmb maximum ethanol inhibition threshold for cell growth (g/L)
Pme maximum ethanol inhibition threshold for ethanol production
(g/L)
Pob ethanol inhibition threshold for instantaneous specific growth
rate (g/L)
Qp specific ethanol production rate (g/g-h)
Qpmax specific ethanol production rate (g/g-h)
S substrate (glucose) concentration (g/L)
Si lower threshold for substrate inhibition of specific growth
rate (g/L)
t time (h)
X biomass concentration (g/L)
W first-order weighted average ethanol concentration change
rate (g/L-h)
Yp/s yield coefficient of ethanol vs. glucose (g/g)
Z second-order weighted average ethanol concentration change
rate (g/L-h)
Greek Symbols
a ethanol inhibition exponent for ethanol production rate
b weighted history parameter for ethanol concentration change
rate
d ethanol concentration change rate inhibition factor parameter
l ethanol concentration change rate inhibition factor parameter
s variable in equations 8, 9, 10, 12 and 15 to account for
exogenous ethanol addition
cw weighted average of previous ethanol concentration change
rate (h−1)
m(S,P) instantaneous specific growth rate (h−1)
m(S,P,Z) dynamic specific growth rate (h−1)
mmax instantaneous specific growth rate at zero instantaneous etha-
nol concentration (h−1)
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