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Quantitative measures of disorder in biological oscillations and their implications for bioreactor operation.

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Asia-Pac. J. Chem. Eng. 2007; 2: 640?649
Published online 13 September 2007 in Wiley InterScience
( DOI:10.1002/apj.091
Research Article
Quantitative measures of disorder in biological oscillations
and their implications for bioreactor operation
Pratap R. Patnaik*
Institute of Microbial Technology, Sector 39-A, Chandigarh-160036, India
Received 12 March 2007; Revised 20 July 2007; Accepted 1 August 2007
ABSTRACT: Under realistic conditions, biological oscillations show fluctuations around time-varying deterministic
values. These fluctuations are often created by noise from the environment, and they vary randomly with time. With
continuous flow microbial cultures, the feed stream is a major source of noise. While the detrimental effects of noise
inflow are known, quantitative measures of their effects are not established. Such indexes are important because, apart
from distortions, noise may displace a fermentation into an undesirable state or into chaos. Using Saccharomyces
cerevisiae as an example, four measures of the effect of noise are proposed and their physical implications for
the operation of continuous cultures are discussed. It is shown that, like monotonic cultures studied earlier, noiseaffected oscillating cultures too can be analyzed usefully through their fractal dimensions. ? 2007 Curtin University
of Technology and John Wiley & Sons, Ltd.
KEYWORDS: oscillating cultures; Saccharomyces cerevisiae; external noise; quantitative measures; fractal dimension
Many biological processes exhibit oscillatory behavior
under either natural or controlled conditions. Predator?prey systems (Durrett, 1993), calcium waves (Bootman et al ., 2001) and neuromotor signals observed
through electroencephalograms (Baker et al ., 1999)
are well-known examples of naturally occurring oscillations. Among oscillations that occur under controlled conditions, the most widely studied are those
of continuous cultures of Zymomonas mobilis (Bruce
et al ., 1991) and Saccharomyces cerevisiae (Satroutdinov et al ., 1992), both of which synthesize ethanol.
Between them, S. cerevisiae is the more widely used and
extensively studied. It is important as a model system
for its rich repertoire of oscillating processes, industrially for its product, the absence of endotoxins, the
ease of its downstream processing, and the availability
of noninvasive methods of measurement in continuous
cultures. A paradoxical aspect of both Z. mobilis and
S. cerevisiae cultivations is that while their oscillations
have engaged the attention of research workers, industrial fermentations try to avoid oscillations. Therefore,
the study of oscillatory behavior is important from both
*Correspondence to: Pratap R. Patnaik, Institute of Microbial Technology, Sector 39-A, Chandigarh-160036, India.
? 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
While there are many studies of the mechanisms and
models of oscillating continuous yeast cultures (see
Patnaik, 2003a for a review), they have all focused
on small laboratory-scale bioreactors, where spatial
gradients and fluctuations in the feed stream can usually
be ignored. However, production-scale operations are
often ?corrupted? by such fluctuations or noise, usually
present in an inflow stream. Uncontrolled noise can
seriously harm a cultivation process, creating problems
such as cell lysis, metabolic shifts, changes in the
fermentation pattern, and run-away operation. Some
of these possibilities have indeed also been observed
in a laboratory-scale bioreactor subjected to deliberate
perturbations (Zamamiri et al ., 2001). Therefore, it is
important to be able to analyze and quantify the effect
of noise on oscillating microbial cultures in order to
model, optimize, and control industrial-scale bioreactors
(Rohner and Meyer, 1995).
Quantitative measures of the effect of noise are useful
in determining optimum or safe operating conditions,
and in designing noise filters and automatic control
systems. In view of its importance for both research and
industry, the behavior of a continuous oscillating culture
of S. cerevisiae under the influence of noise in the
feed stream has been analyzed here. To characterize the
effect of noise, three quantitative indices of performance
are proposed. The indices are: (1) degree of distortion
(by noise), (2) degree of recovery (by using a noise
filter), and (3) fractal dimension. They are defined and
their physical significance explained in later sections.
Asia-Pacific Journal of Chemical Engineering
Each index was calculated both for a noise-affected
culture without any intervention and for a culture for
which the noise was ?filtered? as explained below.
Both cultures have been studied with (1) glucose and
(2) ethanol as the carbon substrates.
To apply optimization and control algorithms, the
distortions caused by noise have to be minimized so that
acceptably noise-free data, revealing the key features
of a system, are recovered from noise-affected values.
This is done by employing filters, which are software
devices that act on noisy data to generate data with
larger signal/noise ratios. Different kinds of noise filters
are available, each with its strengths and weaknesses
(Nelles, 2000). For bioreactors, the extended Kalman
filter (EKF) has been effective in many applications
(Zorzetto and Wilson, 1996; Patnaik, 2005a), even
though there are persuasive arguments in support of
neural filters (Patnaik, 2003b). Unlike a neural filter, an
EKF is intrinsically linked to the biological model and
thus acquires greater physiological fidelity. Therefore,
an EKF was applied to the noise-affected cultures
studied here.
In certain ranges of operating conditions, continuous
cultures of S. cerevisiae exhibit sustained oscillations
with time in the concentrations of biomass, substrate
(glucose or ethanol), product (ethanol), dissolved oxygen and storage carbohydrates, and in the pH, the
oxygen uptake rate (OUR) and carbon dioxide evolution rate (CER). Experimental observations (Satroutdinov et al ., 1992; Keulers et al ., 1996; Zamamiri et al .,
2001) indicate that the oscillations may be controlled
by regulating the dilution rate (in effect, the feed rate
of the main carbon source) and the dissolved oxygen
concentration. Between the two, the dilution rate is preferred as the manipulated variable for bioreactor control
because it is easier to monitor and change and it leads to
superior regulatory control (Dochain and Perrier, 1997).
The complexity and variations of the observed oscillations have made it difficult to formulate mathematical models that are sufficiently accurate and flexible without being too complicated. Modeling efforts
have approached the issue from either of two perspectives. One approach has focused on the reactions
inside the cells and formulated models for their kinetics. Such models offer insight into the metabolic aspects
and enable suitable genetic manipulations for strain
improvement. But they do not consider external influences through transport across the cell walls, flow patterns in a reactor, and the influx of disturbances. Models
that include these features as well as detailed cellular kinetics are often quite complex. Therefore, a second approach combines transport equations with judiciously lumped kinetics. Both kinds of models have
? 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
been reviewed recently (Patnaik, 2003a), where it has
been pointed out that the second approach is more relevant for large nonideal bioreactors. Therefore, a model
of the latter type was used in this study to generate data
mimicking a noise-affected fermentation.
Commercial and practical considerations often restrict
the availability and disclosure of real industrial data.
As an alternative, many investigators (Thompson and
Kramer, 1994; Simutis and Lubbert, 1997; Tian et al .,
2002; Desai et al ., 2006) have relied on simulated data
generated by solving a phenomenologically plausible
model that incorporates experimentally validated kinetics. That approach has been followed here. The cybernetic model of Jones and Kompala (1999) was used
to generate data of oscillating concentration portraits.
They preferred a cybernetic approach so as to overcome some of the weaknesses of mechanistic models
for the time-dependent behavior of many microbial cultures. The cybernetic approach (Dhurjati et al ., 1985)
attributes the ability to learn from experience and to
reach ?intelligent? decisions to living cells. These decisions are usually designed to maximize the probability
of survival of the cells under varying conditions. Therefore, by manipulating the environmental conditions the
cells may be induced to function in a manner that meets
our requirement through their survival. The preferential
synthesis of a desired product may, for instance, meet
both objectives.
In their cybernetic model, Jones and Kompala (1999)
identified three metabolic pathways by which S. cerevisiae may utilize the available carbon sources: glucose
fermentation, ethanol oxidation, and glucose oxidation.
Depending on its past history and the current conditions, the culture may follow either one pathway or two
or more pathways to different extents. They showed that
dynamic competition among the pathways was the driving force for oscillations. By contrast, some mechanistic
models identify this (incorrectly) as a consequence of
oscillatory metabolism. Both types of models, however,
suggest manipulating the dilution rate to control the
nature of the oscillations. The equations of the model
are presented in the Appendix. These equations were
fitted by Jones and Kompala to data from a number
of sources (von Meyenburg, 1973; Satroutdinov et al .,
1992; Keulers et al ., 1996) and the values they obtained
for the parameters are listed in Table 1.
Both Satroutdinov and coworkers (1992) and
Keulers et al . (1996) used the same strain of S. cerevisiae (a wild type diploid strain IFO 0233) but different
carbon substrates. Whereas the former authors fed glucose, Keulers et al . supplied ethanol. Since ethanol is
a product of glucose metabolism, this is a crucial difference, whose effect on the impact of noise has been
evaluated here. Once the data have been generated, they
are considered to represent a real bioreactor and the
model then becomes redundant. After the parameters
were determined from the noise-free studies, the model
Asia-Pac. J. Chem. Eng. 2007; 2: 640?649
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
Table 1. Values of the parameters (Jones and Kompala,
kL a
g h?1
g g?1
g g?1
g g?1
g g?1
g g?1
g g?1
g g?1
g l?1
g l?1
g l?1
g l?1
mg l?1
mg l?1
mg l?1
g g?1
g g?1
g g?1
was solved again with 1/f noise added to the flow rate
of the carbon substrate because this kind of noise is the
most prevalent in biological processes (Yates, 1992).
Details of the model are omitted here to conserve space.
It is important to point out that addition of noise to
the inflow stream before the model is solved embeds
the noise in the process. This is more realistic than
adding noise, as done by Desai et al . (2006), to the
concentrations obtained from a solution of the deterministic model. Points on the solution curves for different
concentrations denoted data mimicking a noise-affected
The choice of the sampling interval is an important
aspect of data selection. Although common and convenient, a constant interval may not be a good choice
because a small interval generates an unnecessarily
large number of data points from shallow regions of
a concentration profile whereas too widely spaced data
may omit important variations from steeply changing
regions. These considerations have been discussed by
Simutis et al . (1997) and by Chen and Rollins (2000),
who have referred to two kinds of nonconstant sampling periods. One is random sampling, where the time
interval between any two points varies randomly. While
practical convenience may favor this method in some
plant operations, it is not always an optimum choice
? 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
and may have some of the weaknesses of a constant
sampling period. A more meaningful choice would
require the sampling period to vary according to the
requirements of the process. This results in the second kind of sampling variation, in which the interval
varies inversely as the local gradient of the concentration profile. Thereby, closely spaced data are sampled
from sharply changing regions and widely spaced values from mildly changing regions. This kind of sampling preserves the relevant features without generating
superfluous data; its usefulness has been demonstrated
in previous applications (Patnaik, 2001, 2003b).
The EKF has been shown to be effective in attenuating
the flow of noise into S. cerevisiae fermentations under
both monotonic and oscillating conditions (Zorzetto and
Wilson, 1996; Patnaik, 2005a). A detailed account of
its theory is available in Grewal and Andrews (1993).
Briefly, the basic Kalman filter is a set of mathematical
equations that provides an efficient recursive solution of
the least-squares type. The filter can provide estimations
of past, present, and future states of a system even
when a precise model is not known. This feature is
useful for microbial processes under nonideal (realistic)
conditions because models developed on laboratory
data may become inapplicable or imprecise under the
influence of disturbances and spatial gradients (Gillard
and Tragardh, 1999; Liden, 2002).
The basic Kalman filter addresses the problem of trying to estimate the state x of a discrete-time controlled
process that is governed by the linear difference equation:
xk = Axk ?1 + Buk + wk ?1
with a measurement vector that follows:
zk = Hxk + vk
In these and later equations, lower case letters with
overbars denote vectors while similar capital letters
denote matrices. Scalars do not have overbars. (k ? 1)
is the current instant of time and k is the point one
time-step ahead. wk and vk represent the process noise
and measurement noise respectively.
Previous studies (Montague and Morris, 1994; Rohner
and Meyer, 1995; Gillard and Tragardh, 1999) show
that wk and vk may be represented as white noise with
normal probability distributions:
p(w ) ? N (0, Q)
p(v ) ? N (0, R)
where Q and R are the respective covariance matrices.
Asia-Pac. J. Chem. Eng. 2007; 2: 640?649
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
Since Eqn. (1) applies to linear equations whereas
many fermentation (and other biological) processes
follow nonlinear models, the EKF was developed. It
applies to any nonlinear difference equation of the form:
xk = f(xk ?1 , uk , wk ?1 )
zk = h(xk , vk )
In principle, the EKF solves the problem of determining the current estimates of a set of variables by
expressing them as linear functions centered around the
partial derivatives of the process and measurement functions evaluated at the (known) previous instant of time.
Note that both Eqns (1) and (2) and Eqns (5) and (6), in
pairs, are in discrete form whereas most biological processes are described by continuous models. This is not
an impediment because, in practice, data are sampled at
discrete points in time. Since the EKF allows any arbitrary variation in the sampling interval, this may be varied according to the nature of the process. For instance,
the interval may be made inversely proportional to the
current concentration gradient, thus generating closely
spaced data when the variations are steep and more
widely separated points during mild variations.
As explained before, data simulating a noise-influenced oscillating culture were generated by solving the
Jones and Kompala (1999) model, first without noise
and then with 1/f noise in the feed stream. In an experimental application, the measurement covariance R is
usually measured prior to the operation of the filter since
it relates to the filter and not the process. The process
noise covariance Q is more difficult to determine since
typically we do not have the ability to observe the process we are estimating. So, based on previous studies
(Zorzetto and Wilson, 1996; Soroush, 1998),Q was set
initially to Qd = ([0.0001 . . . . . . ..0.0001]T ) and R to
0.003 I, where I is the identity matrix and Qd is a diagonal matrix. Since the glucose and oxygen feed streams
are the only inflows to the bioreactor, environmental
noise was considered to be present in these two flow
rates, thus making Q a (2*2) matrix. Both Q and R get
updated recursively as shown in Fig. 1.
Apart from its applicability to nonlinear processes,
an important distinction between the EKF and the
basic discrete Kalman filter is that in the former
case the Jacobian Hk in the equation for the Kalman
gain Kk also gets updated with each iteration, thereby
speeding up convergence and improving the accuracy
of estimations. As explained before, data from the
concentration profiles were sampled at intervals of time
inversely proportional to the local gradients and the
tuning of the EKF was updated progressively over
successive intervals.
? 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
Figure 1. Computation procedure of the EKF. In the left
box the ?corrected? state variables x?k?1 at each sampling
instant t = k ? 1 are used to estimate their values at the
next point t = k, i.e. x?. These values are again corrected in
the right box and the results fed back. The process begins
with known initial values, provided from outside as shown.
Under realistic conditions, many biological processes
have a degree of self-similar randomness that may be
characterized by fractal geometry. In some cases the
randomness may be inherent, as in the surface structure
of proteins (Lewis and Rees, 1985), the morphology
of mycelial flocs (Logan and Wilkinson, 1991) and
antibody?antigen interactions (Sadana and Beelaram,
1994). Randomness may also be induced by external influences, mainly noise or disturbances. Recent
work with Escherichia coli (Patnaik, 1999, 2000) has
shown that the randomness introduced by noise entering
through a feed stream can be quantitatively expressed
and interpreted through fractal dimensions calculated
from the concentration-time profiles.
Measurements of microbial fermentations usually
generate data of concentrations or activities as functions
of time. Dubuc et al . (1989) compared a number of
methods to compute fractal dimensions from such timeseries data, and recommended the variation method.
Their work may be consulted for the theory; only the
computer implementation is briefly described here.
Suppose a one-dimensional profile (such as concentration vs time) is discretized into a set of (N + 1) points
?(ti ), i = 0, 1, 2, . . . . . . , N . Choose a set of integers ki ,
i = 1, 2, . . . . . . , imax with imax < N . The values of the
ki are to be so chosen that ?i = ki ? ki ?1 ? ki ?1 and
?i = ki /N 1. Then Dubuc et al . (1989) have shown
that two curves, called the upper envelope u(t) and the
lower envelope b(t), can be defined on either side of the
original fractal curve. For convenience in programming,
we identify the discrete times ti , i = 0, 1, 2, . . . . . . , N
simply by the indices 0, 1, 2, . . . . . . , N . Then the values
of u(t) and b(t) are calculated as follows.
u1 (j ) = max[?(j ? 1), ?(j ), ?(j + 1)]
b1 (j ) = min[?(j ? 1), ?(j ), ?(j + 1)]
Asia-Pac. J. Chem. Eng. 2007; 2: 640?649
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
ui (j ) = max[ui ?1 (j ? ?i ), ui ?1 (j + ?i )]
bi (j ) = min[bi ?1 (j ? ?i ), bi ?1 (j + ?i )]
where i = 2, 3, . . . . . . , imax and j = 1, 2, . . . . . . , N .
If we define wi =
[ui (j) ? bi (j )],
j =1
i = 1, 2, . . . . . . , imax
then it has been shown that:
ln(wi /ki 2 ) = d ln(1/ki ) + p
where d and p are constants. This equation yields a
straight line whose slope d is the fractal dimension of
the curve.
Satroutdinov et al . (1992) and Keulers et al . (1996)
studied the oscillatory behavior of a continuous fermentation with wild-type diploid strain IFO 0233 of S.
cerevisiae in a Bioflo fermenter (New Brunswick, NJ)
with a working volume of 1350 ml. Satroutdinov et al .
(1992) used glucose feed and observed oscillations in
the concentrations of glucose, ethanol (the product), the
OUR, the CER, and glycogen (the principal storage
carbohydrate). They suggested that oscillations were
regulated by changes in the rate of glycolysis inside
the cells. To test this hypothesis, Keulers et al . (1996)
eliminated glucose by employing ethanol as the carbon
source. Ethanol concentration, the OUR, and the CER
oscillated but in a manner different from those with
glucose feed. In the absence of glucose, glycogen was
not detected but another carbohydrate, trehalose, was
present in small concentrations in both glucose- and
ethanol-fed cultures.
To evaluate the efficiency of the EKF, the cybernetic
model of Jones and Kompala (1999) was solved, as
explained earlier, first without any noise and then with
1/f noise in the feed stream. To obtain data representing different situations, the model containing noise was
solved separately without and with the filter, for both
glucose and ethanol as the carbon substrate in either
case. The dilution rate was maintained at 0.16 h?1 in
all cases. This value was chosen for consistency with
the deterministic results of Jones and Kompala (1999),
who obtained sustained oscillations at this dilution rate.
The effect of a change in dilution rate is discussed
later as part of the present analysis. Figures 2?7 show
the deterministic profiles, the unfiltered noise-affected
profiles, and the filtered profiles for three key variables, each with either glucose or ethanol as the feed
? 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
Figure 2. Oscillations in the concentration of ethanol with
glucose feed at a dilution rate of 0.16 h?1 . The plot without
noise is from Jones and Kompala (1999) while the two
noise-affected plots were obtained in this study.
Figure 3. Oscillations in the concentration of ethanol with
ethanol feed at a dilution rate of 0.16 h?1 . The plots have the
same connotations as in Fig. 2. Note the contrast between
the nature of the plots in Figs 2 and 3, especially the
distortions created by noise in Fig. 3.
stream and at the dilution rates used by the investigators (Satroutdinov et al ., 1992; Keulers et al ., 1996).
Ethanol concentration is an important variable because
it is both the principal product and is utilized by the cells
when there is insufficient glucose, as at low dilution
rates (Satroutdinov et al ., 1992; Keulers et al ., 1996).
Thus, ethanol provides a critical physiological feedback
Asia-Pac. J. Chem. Eng. 2007; 2: 640?649
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
Figure 4. Oscillations in the oxygen uptake rate with glucose
feed at a dilution rate of 0.16 h?1 . The plots have the
same connotations as in Fig. 2. Gas-to-liquid mass transfer
resistance and the low concentrations of dissolved oxygen
contribute to the variations in this figure being sharper than
in Fig. 2.
Figure 5. Oscillations in the oxygen uptake rate with ethanol
feed at a dilution rate of 0.16 h?1 . The plots have the same
connotations as in Fig. 2. As for ethanol, noise causes large
distortions in the smooth oscillations but an EKF has been
able to restore nearly noise-free behavior.
control. In fact, Keulers et al . (1996) performed experiments with ethanol as the sole carbon source to resolve
the uncertainty in Satroutdinov et al .?s (1992) suggestion that either changes in glycolysis rate caused by
? 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
Figure 6. Oscillations in the carbon dioxide evolution rate
with glucose feed at a dilution rate of 0.16 h?1 . Since carbon
dioxide evolution is a respiratory process whereas ethanol
formation is fermentative, the carbon dioxide concentrations
are much smaller and the differences between ethanol and
glucose substrates are diametrically opposite between Figs 2
and 3 in one pair and Figs 6 and 7 in the other.
Figure 7. Oscillations in the carbon dioxide evolution rate
with ethanol feed at a dilution rate of 0.16 h?1 . The plots
here and in Fig. 6 may be contrasted with those of Figs 2
and 3 as explained for Fig. 6.
switching of glycogen metabolism or feedback regulation of the tricarboxylic acid (TCA) cycle regulated the
oscillations. Since glycolysis is negligible with ethanol
Asia-Pac. J. Chem. Eng. 2007; 2: 640?649
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medium, the observed oscillations supported TCA regulation as the controlling factor.
These inferences about oscillations on glucose and
ethanol encapsulate the more complex connections
within the metabolic framework. Such a summary is
intentional but useful. It is intentional because this
study addresses the macroscopic manifestations of the
effects of disturbances on oscillatory behavior vis-a?-vis
bioreactor operation, and it is not an investigation into
the mechanisms per se. Nevertheless, a brief account
of the mechanistic basis of oscillatory performance is
useful because the occurrence and the type of oscillations depend on the operating conditions (Satroutdinov
et al ., 1992; Jones and Kompala, 1999; Patnaik, 2003a).
While the metabolic ramifications of sustained oscillations, even under noise-free conditions, are not yet
fully understood (Murray et al ., 2001; Zamamiri et al .,
2001), they do reveal that internal feedback controls
have a major role in the regulatory processes (Satroutdinov et al ., 1992; Duboc et al ., 1996; Lloyd, 1998).
These may be monitored and controlled through two
measurable variables, the OUR and the CER, whose
variations are portrayed in Figs 4?7, whereas those of
ethanol are shown in Figs 2 and 3. It may be clarified
here that the time spans of Figs 2, 4, and 6 differ from
those of Figs 3, 5, and 7 because the former set were
generated from the deterministic profiles of Satroutdinov et al . (1992) and the latter from those of Keulers
et al . (1996). This difference does not matter for studies
of the effects of noise since any sufficiently long slice
of time from a persistent oscillating profile provides an
adequate starting basis for such an analysis.
Although the profiles for ethanol, OUR, and CER
are visually similar, there are quantitative differences.
Without noise, Satroutdinov et al .?s (1992) and Keulers
et al .?s (1996) data showed that a change in the carbon
source affected the amplitudes of the oscillations but
not the frequencies. To evaluate the effect of noise on
these variables, three indices may be defined. One is the
span of the amplitude:
a = 100
(maximum amplitude) ? (minimum amplitude)
(minimum amplitude)
The second is the degree of distortion caused by
noise; this may be expressed as a sum of squares of
normalized deviations:
100 d=
(noise-affected value) - (noise-free value) 2
(noise-free value)
? 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
Asia-Pacific Journal of Chemical Engineering
where N is the number of data sampled. The third index
is the degree of recovery of noise-free oscillations from
noise-affected oscillations by using an EKF:
(degree of distortion without an EKF)
? ?(degree of distortion with an EKF) ?
r = 100 ?
(degree of distortion without an EKF)
The values of these indices are presented in Table 2,
and they show interesting trends with respect to the
span of the amplitude. As the span decreases, implying weaker oscillations, the degree of distortion, both
without and with an EKF, increases and the degree of
recovery decreases.
These results pertain to a dilution rate of 0.08?
0.085 h?1 , employed by Satroutdinov et al . (1992) and
Keulers et al . (1996). Since the dilution rate strongly
influences the nature of oscillations (Beuse et al ., 1998;
Jones and Kompala, 1999; Patnaik, 2003a), the effect
of inflow noise at different dilution rates was analyzed.
Since a number of variables have to be compared across
a range of dilution rates, it is cumbersome to use the
indices a, d, and r defined above. Fractal indices offer a
more convenient way to characterize disorder in biological oscillations (Pincus, 1991; Patnaik, 1999, 2000). For
each dilution rate, the concentration plot of each variable yields one fractal dimension, computed according
to the variation method described above (Dubuc et al .,
1989). These dimensions have been plotted in Fig. 8.
At small dilution rates, the fractal dimensions are close
to the unit value of a smooth curve. This is expected
since there is little influx of noise. As the dilution rate
increases, so does the influx of noise; consequently, the
fractal dimensions also increase and approach the limiting value of 1.75, which is characteristic of 1/f noise.
Consistent with the inferences from the indices a, d, and
r above, strong oscillations are less perturbed by noise
than are weak oscillations, resulting in smaller fractal
dimensions and a slower approach to the limiting value.
Table 2.
Span of the amplitude, degree of
noise-induced disorder, and degree of recovery by EKF
for different variables.
Degree of
OUR (G)a
Ethanol (G)
Ethanol (E)
Span of
Degree of
G = glucose feed; E = ethanol feed.
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Asia-Pacific Journal of Chemical Engineering
the inflow of noise may also drive a smoothly oscillating process to chaotic behavior (Patnaik, 2005b). Such
changes obviously have serious implications for cultivation processes, thus emphasizing the importance of
quantitative measures of the effects of noise on microbial oscillations.
Figure 8. Variations in the fractal dimensions for different
variables over a range of dilution rates and with glucose and
ethanol, separately, as the substrate. 1?Ethanol (Ethanol);
2?CER (Glucose); 3?CER (Ethanol); 4?OUR (Ethanol);
5?Ethanol (Glucose); 6?OUR (Glucose). The names in
brackets denote the feed streams.
These trends pose two interesting practical questions
for large-scale cultivations. First, does noise affect the
metabolic basis of deterministic oscillations? Secondly,
how best can the smaller sensitivities to disturbances
and the better filtering efficiencies for large oscillations
be balanced against the greater complexity of their control policies? Although measurements of intracellular
metabolites under noise-affected conditions have not yet
been reported, results from experiments in the absence
of noise suggest that with glucose as the medium the
oscillations are caused by changes in glycolytic flux
(Satroutdinov et al ., 1992), whereas with ethanol these
are driven by flux changes in the TCA cycle (Keulers
et al ., 1996). Within either system, the oscillations of
different metabolites differ in their amplitudes, phases,
and time constants (Duboc et al ., 1996; Keulers et al .,
1996; Wittmann et al ., 2005).
We recall at this point that at low glucose concentrations S. cerevisiae switches to ethanol as its main carbon
source, and reverts to glucose again when its level rises
sufficiently (Satroutdinov et al ., 1992; Duboc et al .,
1996; Beuse et al ., 1998; Jones and Kompala, 1999).
Thus, the driving force and the mechanistic basis of the
observed oscillating outputs may shift over a length of
time. This possibility and the fact that under a particular
set of conditions both oscillating states and nonoscillating stationary states can exist (Zamamiri et al ., 2001)
imply that noise can displace a culture from an oscillating state to a stationary state or the other way. Indeed,
? 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
Under realistic conditions representative of large bioreactors, oscillating microbial cultures in fed-batch and
continuous operations are affected by noise carried by
the feed streams. Mild noise distorts the oscillations,
thereby rendering it difficult to identify the true features
of the process. Stronger noise may drive a fermentation
into another metabolic state, indicated by changes in the
oscillating characteristics or by a shift to a nonoscillating state. More seriously, noise may even cause smooth
oscillations to degenerate into chaos.
Therefore, for optimal operation of a cultivation
process, it is important to have good estimates of the
disorder induced on the oscillations by the influx of
noise. In this study, the effects of noise on an oscillating
continuous culture of S. cerevisiae were investigated,
both without any filter and with an EKF to reduce the
inflow of noise. Four indexes of the effect of noise were
introduced: (1) the span of the amplitude, (2) the sum
of squares of normalized deviations, (3) the degree of
recovery, and (4) the fractal dimension. It was observed
that weak oscillations are distorted more and recover
poorly upon removal (or filtering) of the noise.
While offering quantitative measures of disorder
introduced by noise, these inferences also suggest
that the common preference for fermentations with
monotonic profiles may not be good under noiseinfluenced conditions. Operational difficulties also discourage strongly oscillating cultures. Thus, the optimal
choice involves a balance between the sensitivity to
noise for monotonic profiles and weak oscillations, on
the one hand, and the difficulties of controlling more
robust fermentations with strong oscillations.
kL a
intracellular storage carbohydrate concentration (g l?1 )
dilution rate (h?1 )
key enzyme concentration for i-th pathway (g
g?1 biomass)
ethanol concentration in the bioreactor (g l?1 )
glucose concentration in the bioreactor (g l?1 )
glucose concentration in the feed stream (g
l?1 )
oxygen mass transfer coefficient (h?1 )
Asia-Pac. J. Chem. Eng. 2007; 2: 640?649
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
Michaelis constant for i-th pathway (g l?1 )9
KO2 , KO3 oxidative pathway oxygen saturation constants (mg l?1 )
dissolved oxygen concentration in the bioreactor (mg l?1 )
dissolved oxygen solubility limit (mg l?1 )
biomass growth rate on i-th pathway (h?1 )
carbon substrate concentration for i-th pathSi
way (g l?1 )
elapsed time (h)
cybernetic variable controlling key enzyme
synthesis for i-th pathway (?)
cybernetic variable controlling key enzyme
activity for i-th pathway (?)
biomass concentration in the bioreactor (g
l?1 )
yield coefficient for i-th pathway (g biomass
g?1 substrate)
礽 ,max
specific enzyme synthesis rate (h?1 )
constitutive enzyme synthesis rate (g h?1 )
specific enzyme degradation rate (h?1 )
stoichiometric coefficient for i-th carbon substrate (?)
stoichiometric coefficients for storage carbohydrate synthesis and degradation (?)
specific growth rate of biomass on i-th substrate (h?1 )
maximum specific growth rate on i-th substrate (h?1 )
The pathways are not mutually exclusive and, at a
given instant, the organism may follow two or more
pathways at different rates. Each pathway is controlled
by a key enzyme ei , with synthesis rate ui and activity
vi , which follow:
ui = (A4)
vi =
max rj
With Eqns (A1) ? (A5), the mass balances for a
continuous flow bioreactor may be written as follows:
(ri vi )-D X
r1 v1 r3 v3
= (G0 ? G)D ?
? ?4 C
r1 v1 r2 v2
= ?DE + ?1
r2 v 2
r3 v3
= kL a(O ? O) ? ?2
+ ?3
X (A9)
= ?ui
Ki + Si
? ? (rj vj ) + ? ? ei + ? ?
The model of Jones and Kompala (1999)
Depending on the prevailing conditions, S. cerevisiae
may follow any one of three metabolic pathways. The
rate of growth ri along each pathway follows modified
Monod kinetics, as given below.
Glucose fermentation
r1 = �e1
Ethanol oxidation
r2 = �e2
Glucose oxidation
r3 = �e3
K2 + E
K3 + G
K1 + G
KO2 + O
KO3 + O
? 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
= ?3 r3 v3 ? (?1 r1 v1 + ?2 r2 v2 )C
(rj vj )C
Inclusion of the term ? ? in the enzyme synthesis
Eqn (A10) is based on study by Turner and Ramkrishna
(1988), who have shown its importance in predicting the
induction of enzymes that have been repressed for long
durations. The specific growth rates thus also include
? ? in the model:
礽,max + ?
礽 = 礽,max
? + ??
Equation (A11) expresses the rate of change of
internal storage carbohydrates that are an integral part
of the metabolism (Satroutdinov et al ., 1992; Duboc
et al ., 1996).
Asia-Pac. J. Chem. Eng. 2007; 2: 640?649
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
The ?i are the stoichiometric coefficients for different
substrates Si , and ?i are similar coefficients for carbohydrate synthesis and consumption by the cells. Jones and
Kompala (1999) may be consulted for a full discussion
of the model. A point not clarified there is the identification of S1 , S2 , and S3 . Reference to Eqns (A1)?(A3)
shows that S1 = G, S2 = E , and S3 = G. This identification is needed to solve the model.
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