Journal of Sustainable Bioenergy Systems, 2012, 2, 27-32
http://dx.doi.org/10.4236/jsbs.2012.23004 Published Online September 2012 (http://www.SciRP.org/journal/jsbs)
Using Magnetic Nanoparticles to Eliminate Oscillations in
Saccharomyces cerevisiae Fermentation Processes
Lakshmi N. Sridhar
Chemical Engineering Department, University of Puerto Rico, Mayaguez, Puerto Rico
Received June 4, 2012; revised July 11, 2012; accepted July 23, 2012
This article provides computational evidence to show that functionalized magnetic nanoparticles can eliminate the
wasteful oscillatory behavior in fermentation processes involving Saccharomyces cerevisiae. There has been a consid-
erable amount of work demonstrating the existence of oscillations in fermentation processes. Recently reference 
computationally demonstrated very simple strategies to eliminate the oscillations in the fermentation process. In the
case of the of the Saccharomyces cerevisiae fermentation process it was shown that the addition of a little bit of oxygen
would be successful in eliminatin g the oscillation causing Hopf bifurcation s. The work of [2,3] demonstrated that oxy-
gen mass transfer could be enhanced by using functionalized magnetic nanoparticles. The aim of this work is to incor-
porate the model used by  regarding the enhancement of oxygen mass transfer in the cybernetic Jones Kompala
model  describing the dynamics of the Saccharomnyces cerevisiae fermentation process and demonstrate that using
the functionalized magnetic nanoparticles can by altering the mass transfer coefficient actually succeed in eliminating
the oscillatory behavior that plagues the Saccharomyces cerevisiae fermentation process. This occurs because the oscil-
lation causing Hopf bifurcations are sensitive to the amount of input oxygen and increasing the oxygen mass transfer
coefficient causes the disappearance of the Hopf bifurcation points.
Keywords: Nanoparticles; Fermentation; Oscillations
Production of ethanol from a variety of biomass feed-
stocks, solid and liquid waste resources is a national pri-
ority for a variety of reasons but most importantly for
energy independence and sustain ability. The biggest hur-
dle in the commercialization of any of the processes for
making ethanol is the cost of production. Hence, it is
important to develop strategies to produce clean ethanol
with the least amount of expense avoiding all unnec-
essary expenses and wastage during ethanol production.
Continuous fermentation processes used for ethanol
production has been known to exhibit oscillatory behav-
ior and this has been confirmed both theoretically and
experimentally. Fermentation processes involving both
Saccharomyces cerevisiae and Zymomonas mobilis have
been shown to exhibit oscillatory behavior [5-11]. The
oscillatory behavior was linked to the combination of
substrate excess and product inhibitions by reference 
while reference  conclude that the oscillations take
place at high values of feed substrate concentrations.
References [9,10,12,13] have demonstrated the exis-
tence of oscillatory behavior in con tinuous fermentations
of Saccharomyces cerevisiae. While there have been a lot
of modeling work regarding fermentation processes, the
most sophisticated model available is the Jones Kompala
model . This model has demonstrated the observed
oscillatory behavior [14,15]. Reference  discusses the
effect of oxygen mass transfer on the Saccharomyces
cerevisiae fermentation in a section in their paper titled
“Effect of oxygen mass transfer on yeast oscillations”.
The oscillatory behavior that has been demonstrated
numerically and experimentally by several workers ad-
versely affects the fermentation process and the ethanol
production. Hence one must develop strategies to avoid
or eliminate the oscillations. Reference  recently compu-
tationally demonstrated some techniques for eliminat-
ing oscillations in fermentation processes involving both
Zymomonas mobilis and Saccharomyces cerevisiae. Spe-
cifically, it was shown that in the case of the Saccharo-
myces cerevisiae fermentation that a small increase in the
input oxygen concentration would eliminate the oscilla-
tion causing Hopf bifurcation. This increase in oxygen
mass transfer can also be observed if the mass transfer
coefficient for the oxygen mass transfer can be enhanced.
Recently, [2,3] investigated the use of functionalized
magnetic nanoparticles with flourinated polymer coating
to enhance oxygen mass transfer in bioreactors. The
opyright © 2012 SciRes. JSBS
L. N. SRIDHAR
flourianted polymer coating actually aids in enhancing
the mass transfer. The question that this paper addresses
is “Can the use of functionalized magnetic nanopartices,
by increasing the mass transfer of oxygen to the Sac-
charomyces cerevisiae fermentation process also elimi-
nate the oscillatory behavior that occurs in these proc-
esses?” The oxygenation which is usually achieved by
sparging air into the fermentation unit can be enhanced
by 1) adding extra oxygen directly to the input stream
and 2) increasing the mass transfer coefficient that per-
tains to the oxygen mass transfer. Reference  compu-
tationally demonstrated that the first strategy was effec-
tive in eliminating the oscillatory behavior in the Sac-
charomyces cerevisiae fermentation. This paper ad-
dresses the second issue.
While oscillatory behavior can cause wastage and af-
fect the ethanol production adversely the constant addi-
tion of oxygen also can be expensive and if this expense
can also be avoided it could be instrumental in reducing
the cost of ethanol production.
The paper is organized as follows. First, the cybernetic
Jones Kompala model  or the Saccharomyces cere-
visiae fermentation is discussed. Then the Olle co-rela-
tion  for the increase in th e mass transfer coefficient is
described and incorporated into the cybernetic model.
The reported cases for the oscillation causing Hopf bi-
furcations are presented and it is shown that when the
mass transfer coefficient is enhanced by the use of the
ferromagnetic nanoparticles, the Hopf bifurcations dis-
appear in these cases.
2. Saccharomyces cerevisiae Model
Reference  have developed a detailed cybernetic
model to represent the Saccharomyces cerevisiae fer-
mentation process. Along three available pathways i,
Glucose fermentation, ethanol oxidation and glucose
oxidation, the cybernetic variables iand i represent
the optimal strategies for enzyme synthesis and activity.
The variables and are given by the equa t ions
while the expressions for th e pathways are given by
with these growth rate equations, the balance equations
 are give n b y
G, E and O represent the concentrations of glucose,
ethanol and dissolved oxygen. i
represents the modi-
fied growth rate constant. i
saturation constants for the carbon substrate and the dis-
solved oxygen for each metabolic pathway. 0 repre-
sents the inlet glucose feed concentration, X is the cell
mass concentration and
karepresents the dissolved
oxygen mass transfer coefficient. Y is the yield coeffi-
represent the enzyme synthesis
and decay rate constants. The stoichiometric coefficients
for the intercellular storage carbohydrate synthesis and
consumption are given by i
. Table 1 giv es the
base values of the variables and constants used. This
model demonstrates the existence of the Hopf bifur-
cations that cause the occurrence of the oscillatory be-
3. Enhancement of Mass Transfer
Coefficient Because of Functionalized
References  and  has observed oxygen transfer en-
hancement in the presence of colloidal dispersions of
magnetized nanoparticles coated with oleic acid and a
polymerizable surfactant. In this work, fermentations in
the presence of nanoparticles were conducted and it was
demonstrated that the Oleic acid coated nanoparticles do
enhance oxygen transfer rates and an empirical co-rela-
tion for the enhanced mass transfer coefficient in the
presence of the nanopartiocles is presented as
Copyright © 2012 SciRes. JSBS
L. N. SRIDHAR 29
NP represents the enhanced mass transfer coeffi-
cient when the nanoparticles are added while
sents the mass fraction of the nanoparticles. C is an ad-
justable parameter which has been determined as 51.4.
This linear relationship holds for mass fractions of
nanoparticles up to
~ 0.01. The value of C was de-
termined from experimental data and Equation (12) is an
empirical one. We will use this relationship to investigate
whether by adding the nanoparticles the oscillatio n caus-
ing Hopf bifurcations will disappear when the enhanced
value of the mass transfer is used. The incorporation of
the enhanced mass transfer coefficient will result in
Equation (9) to be modified as
Oka CO O
4. Numerical Technique Used to Locate the
The program CL_MATCONT [16,17] was used to lo-
cate singularities in the set of ODE that constitute the
Saccharomyces cerevisiae fermentation model (Equa-
tion set 6-11 and equation set 6, 7, 8, 13, 10, 11). For
the resulting equilibrium curve , where
is the continuation parameter, and u the remaining
variables, the defining function is
v and is the tangent vector at
x. Three test functions will be defined as
For the existence of a branch point, 1
are zero for a Hopf bifurcation point and a
limit poin t . * indicates the bialternate product.
5. Results: Elimination of the Hopf
Bifurcation Points in Saccharomyces
In this section, examples of Saccharomyces cerevisiae
fermentation problems where the Ho pf bifurcation poin ts
disappear when the mass transfer coefficients are en-
hanced because of the nanoparticles are presented. In all
the cases D is the bifurcation parameter. The same prob-
lems were investigated by  and the oscillatio n causing
Hopf bifurcation points were eliminated there by in-
creasing the input oxygen. The value of O* in all the
air. In all the cases studied it was seen that the addition of
a small amount of the ferromagnetic nanoparticles results
in the disappearance of the Hopf bifurcation.
Table 1 gives the base values for the pa
cases was 7.5 which is the concentration of the oxygen in
6. Case 1 (Figures 1 and 2)
problem we study is
e Jones Kompala model  and Table 2 provides the
additional problem specifications and contains the mass
fraction of the ferromagnetic nanoparticles required to
eliminate the oscillation causing Hopf bifurcations. Ta-
ble 3 provides the concentration values of the Hopf bi-
furcations before the addition of the ferromagnetic nano-
The first Saccharomyces cerevisiae
the problem discussed in  In this case the value of
ka is 150 and the 0
G va lue is 1 0. Wh en the *
5 mg/l there are 2opf points, 5 neutral sads and
one limit point. This is indicated in Figure 1. When the
mass fraction of the added ferromagnetic nanoparticles is
0.006 the Hopf bifurcations disappear. This is shown in
Figure 2. Here 3 neutral saddles and a limit point remain.
Neutral saddles are not bifurcation points and do not
is 7. Hdle
able 1. Base set of parameters used for the Saccharomyces
cerevisiae fermentation problem ().
G0 10 gm/l
Y1Y3 (0.16,) gg–1
, Y2, 0.75, 0.6
0.403, 2, 1, 0.95
K1, K2K3 0.05, 0001 , .01, 0.
2, 0, 10, 0.8
0.6 44, 0.19, 0.3
Table 2. Problem specifications.
required to eliminate
1 10 0.006 150
2 11.5225 0.005
3 8.75225 0.0065
Copyright © 2012 SciRes. JSBS
L. N. SRIDHAR
ble 3. Ctration values at theifurcation points before ferromagnetic nanoparticles ar e added.
Problem nanopartices addition
Taoncen Hopf b
No. of Hopf points before Hopf bifurcation co-ordinates (X, C, G, E, O, e1, e2, e3, D)
(4.917289 0.0287940 0.119159)
(5.978401 0.734092 0.004877 0. 004366 0.001034 0.039255 0.044905 0.314754 0.097338)
938 0.027608 0.029763 0.000793 0.074316 0.14 50 99 0.14
2 1 (6.873118 0.732206 0.006475 0.006061 0.001337 0. 038620 0.045410 0.316961 0.121043)
3 1 (4.706247 0.148703 0.038663 0.124465 0.001940 0. 065938 0.120793 0.215518 0.159153)
evisiae example ,
7. Case 2 (Figures 3 an
In the second Saccharomyces cer
kavalue of 225, a 0
G value of 11.5, and an
Ovalue is 7.5 mg/l, we get a Hopf bifurcation point,
two neutral saddles and a limit point. This is shown in
Figure 3. When the mass fraction of the ferromagnetic
nanoparticles is 0.005, the Hopf bifurcation disappears.
The new curve is shown in Figure 4 where two neutral
saddles and a limit point remain.
8. Case 3 (Figures 5 and 6)
In the third Saccharomyces cerevisiae example, for a
Figure 1. Hopf bifurcation for case 1 when no ferromag-
netic nanoparticles are added.
kavalue of 225, a 0
G valu of 8.75, and an Ovalue
is 7.5 mg/l, we get one Hopf bifurcation point, four neu-
tral saddles and a limit point. This is shown in Figure 3.
When the mass fraction of the ferromagnetic nanoparti-
cles is 0.0065, the Hopf bifurcation disappears, as shown
in Figure 6 and only the limit point remains.
9. Discussion of Results
s a powerful model hat is The Jones Kompala model  i
used to model the growth dynamics of Saccharomyces
cerevisiae and predicts not only the oscillatory behavior
but also the observed variations in the oscillations over a
wide rang of dilution rates. Reference  experimen-
tally studied the effect of oxygen mass transfer on oscil-
latory behavior in fermentation processes. In the cyber-
netic model of Jones and Kompala  the oxygen mass
transfer is in the term
kaO O and therefore en-
hancing the mass transfer coefficient would therefore
have the same effect as increasing the input oxygen.
While one of the most novel ways to increase the mass
transfer coefficient is to use ferromagnetic nanoparticles
this paper computationally incorporates the effect of the
mass transfer enhancement in the cybernetic model of
Jones and Kompala . In the three cases, discussed the
mass fraction of the nanoparticles required was 0.006,
0.005 and 0.0065. It is interesting to note that the usage
of a very minute amount of nanoparticles is effective in
eliminating the oscillatio ns in the fermentation processes.
Having to use such a minute amount of the nanoparticles
Figure 2. Hopf bifurcation in case 1 dispppears for a nano-
particle mass fraction of 0.006.
Figure 3. Hopf bifurcation in case 2.
Copyright © 2012 SciRes.
L. N. SRIDHAR 31
Figure 4. Hopf bifurcation in case 2 eliminated because of
es (mass fraction = 0.005).
Figure 5. Hopf bifurcation in case 3.
Figure 6. Hopf bifurcation in case 3 eliminated as a result of
adding nanoparticles (mass fraction 0.0065).
would be more economical than having to continuously
supply extra oxygen to elimi nate the oscillation s . The
results discussed in this paper should motivate experi-
mentalists to try to use nanoparticles to eliminate un-
wanted and wasteful oscillations that occur in fermenta-
The Hopf bifurcations that occur in the fermentation
processes are extremely sensitive to the change in the
mass transfer coefficient. And a small change in the mass
transfer coefficient. is enough to remove the Hopf bifur-
cations and that is the main focus of this paper. Refer-
ence  actually demonstrated computationally that in-
creasing oxygen would remove the oscillation causing
Hopf bifurcations. Oxygen can be increased directly or
by increasing the mass transfer coefficient. Nanoparticles
definitely increase the mass transfer (either by increasing
kL or by increasing the area and therefore they have the
ability to remove the oscillations that plague the fermen-
tation process and that is the main focus of this paper.
While more advanced and sophisticated co-relations con-
necting the mass transfer coefficient to the amount of
nanoparticles can indeed be developed, the facts are 1
tic Model of the
cs of the Saccharomyces cerevisiae in
Journal of Biotechnol-
Hopf bifurcations can indeed disappear as a result of the
increase of oxygen mass transfer and 2) the oxy
transfer coefficient can be enhanced by using a minute
amount of nanoparticles. The computational demonstra-
tion of these facts is important to guide the experimen-
talists to perform the necessary experiments to validate
the fact that a small amount of nanoparticles can remove
the oscillation causing Hopf bifurcations.
The effect of using ferromagnetic nanoparticles on the
oxygen mass transfer coefficient in the cybernetic model
for the Saccharomyces cerevisiae fermentation process
has been studied. In particular, it is shown that using a
minute amount of the ferromagnetic nanoparticles would
be effective in the elimination of the unwanted oscilla-
tory behavior in the fermentation process.
 L. N. Sridhar, “Elimination of Oscillations in Fermenta-
tion Processes,” AICHE Journal, Vol. 57, No. 9, 2011, pp .
 B. Olle, S. Bucak, T. C. Holmes, L. Bromberg, A. Hatton
and D. I. C. Wang, “Enhancement of Oxygen Mass
Transfer Using Functionalized Magnetic Nanoparticles,”
Industrial & Engineering Chemistry Research, Vol. 45,
No. 12, 2006, pp. 4355-4363.
 B. Olle, “Mechanistic Modeling of Increased Oxygen
Transport Using Functionalized Magnetic Fluids in Bio-
reactors,” PhD Thesis, Massachusetts Institute of Tech-
nology, Cambridge, 2006.
 K. D. Jones and D. S. Kompala, “Cyberne
Batch and Continuous Cultures,”
ogy, Vol. 71, No. 1-3, 1999, pp. 105-131.
 K. J. Lee, D. E. Tribe and P. L. Rogers, “Ethanol Produc-
tion by Zymomo nas mobilis in Continuous Culture at High
Copyright © 2012 SciRes. JSBS
L. N. SRIDHAR
Copyright © 2012 SciRes. JSBS
l. 2, No. 8, 1980
pp. 339-344. doi:10.1007/BF00138666
Glucose Concentrations,” Biotechnology Letters, Vol. 1,
No. 10, 1979, pp. 421-426.
 K. J. Lee, M. L. Skotnicki, D. E. Tribe and P. L. Rogers,
“Kinetic Studies on a Highly Productive Strain of Zymo-
monas mobilis,” Biotechnology Letters, Vo,
 I. M. L. Jobse. C. A. M. Luyben
6, 1986, pp. 868-877.
s, G. T. C. Egbertsa, K
and J. A. Roels, “Fermentation Kinetics of Zymomonas
mobilis at High Ethanol Concentrations; Oscillations in
Continuous Cultures,” Biotechnology and Bioengineering,
Vol. 28, No.
 C. Ghommidh, J. Vaija, S. Bolarinwa and J. M. Navarro,
“Oscillatory Behavior of Zymomonas mobilis in Con-
tinuous Cultures: A Simple Stochastic Model,” Biotech-
nology Letters, Vol. 11, No. 9, 1989, pp. 659-664.
 L. J. Bruce, D. B. Axford, B. Ciszek and J. A. Daugulis,
“Extractive Fermentation by Zymomonas mobilis and the
Control of Oscillatory Behavior,” Biotechnology Letters,
Vol. 13, No. 4, 1991, pp. 291-296.
 L. Perego, J. M. Cabral, S. Dias, L. H. Koshimizu, M. R.
De Melo Cruz, W. Borzani and M. L. R. Vairo, “Influ-
ence of Temperature, Dilution Rate and Sugar Concentra-
tion on the Establishment of Steady-State
Ethanol Fermentation of Molasses,” Bio in Continuous
mass, Vol. 6, No.
3, 1985, pp. 247-256. doi:10.1016/0144-4565(85)90044-7
 A. Mulchandani and B. Volesky, “Modelling of the Ace-
tone-Butanol Fermentation with Cell Retention,” Cana-
dian Journal of Chemical Engineering, Vol. 64, No. 4,
nchronization of Sac-
wn in Continuous Culture, II.
1986, pp. 625-631.
 C. Strassle, B. Sonnleitner and A. A. Fiechter, “A Predic-
tive Model for the Spontaneous Sy
charomyces cerevisiae Gro
Experimental Verification,” Journal of Biotechnol ogy, Vol.
9, No. 3, 1989, pp. 191-208.
 H. K. Von Meyenberg, “Stable Synchrony Oscillations in
Continuous Culture of Saccharomyces cerevisiae under
Glucose Limitation,” In: B. Chance, E. K. Pye, A. K. Shosh
and B. Hess, Eds., Biological and Biochemical Oscilla-
tors, Academic Press, New York, 1973, pp. 411-417.
 Y. Zhang and M. A. Henson, “Bifurcation Analysis of
Continuous Biochemical Reactor Models,” Biotechnology
Progress, Vol. 17, No. 4, 2001, pp. 647-660.
 D. J. W. Simpson, D. S. Kompala and J. D. Meiss, “Dis-
continuity Induced Bifurcations in a Model of Saccharo-
myces cerevisiae,” Mathemati
No. 1, 2009, pp. 40-49. cal Biosciences, Vol. 218,
 A. Dhooge, W. Govearts and A. Y. Kuznetsov, “MAT-
CONT: A Matlab Package for Numerical Bifurcation
Analysis of ODEs,” ACM Transactions on Mathematical
Software, Vol. 29, No. 2, 2003, pp. 141-164.
 A. Dhooge, W. Govaerts, Y. A. Kuznetsov, W. Mestrom,
A. M. Riet and CL_MATCONT, “A Continuation Tool-
box in Matlab,” 2004.
 S. J. Parulekar, G. B. Se mones, M. J. Rolf, J. C. Lievense
and H. C. Lim, “Introduction and Elimination of Oscil-
lations in Continuous Cultures of Saccharomyces cere-
visiae”, Biotechnology and Bioengineering, Vol. 28, No.
5, 1986, pp. 700-710. doi:10.1002/bit.260280509
C: Intracellular Carbohydrate Mass Fraction Storage;
D: Dilution Rate;
E: Ethanol Concentration;
X: Cell Mass Concentration;
O: Oxygen Concentration;
O*: Oxygen Solubility Limit (7.5);
Yi: i Pathway Yield Coefficient.