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
Email: lakshmin.sridhar@upr.edu
Received June 4, 2012; revised July 11, 2012; accepted July 23, 2012
ABSTRACT
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 [1]
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 [3] regarding the enhancement of oxygen mass transfer in the cybernetic Jones Kompala
model [4] 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
1. Introduction
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 [8]
while reference [10] 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 [4]. This model has demonstrated the observed
oscillatory behavior [14,15]. Reference [4] 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 [1] 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
C
opyright © 2012 SciRes. JSBS
L. N. SRIDHAR
28
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 [1] 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 [4] or the Saccharomyces cere-
visiae fermentation is discussed. Then the Olle co-rela-
tion [3] 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 [4] 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
r
u v
i
ui
v
i
i
j
j
r
ur
(1)
max
i
i
j
j
r
vr
i
r
(2)
while the expressions for th e pathways are given by
111
1
G
re
K
G
(3)
2
O
EO
222
2
re
K
EK O








(4)
3
333
3O
GO
re
K
GKO








(5)
with these growth rate equations, the balance equations
[4] are give n b y
d
dii
i
X
X
rv DX
t
(6)

112 2
04
12
ddd
ddd
rv rv
GXc
GGDX CX
tYYtt


 




(7)
112 2
112
d
d
rv rv
EDE X
tYY

 


(8)

*33
22
23
23
d
dL
rv
rv
OkaOOX
tYY





(9)
*
d
dii
ijji
j
ii
eS
urve
tKS





(10)


33 3111222
d
dii
i
Crvrvrv CrvC
t

 
(11)
G, E and O represent the concentrations of glucose,
ethanol and dissolved oxygen. i
represents the modi-
fied growth rate constant. i
K
and Oi
K
represent the
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
G
L
karepresents the dissolved
oxygen mass transfer coefficient. Y is the yield coeffi-
cient, while
and
represent the enzyme synthesis
and decay rate constants. The stoichiometric coefficients
for the intercellular storage carbohydrate synthesis and
consumption are given by i
and 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-
havior.
3. Enhancement of Mass Transfer
Coefficient Because of Functionalized
Magnetic Nanoparticles
References [2] and [3] 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

1
L,LNP
K
KC
 (12)
,
L
NP represents the enhanced mass transfer coeffi-
cient when the nanoparticles are added while
K
repre-
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


*22
d1
dLrv
Oka CO O
t

 33
23
23
rv X
YY




,0fu
(13)
4. Numerical Technique Used to Locate the
Singularities
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

,0fu


,Fx
where 1n
x
u
1det
R
v and is the tangent vector at
x. Three test functions will be defined as
x
T
F
v






*
,
un
(14)
2det 2
f
uI
31n
v
0

,u

(15)
(16)
For the existence of a branch point, 1
, while
2
and 3
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
cerevisiae Fermentation
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 [1] 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
rameters in
th
6. Case 1 (Figures 1 and 2)
problem we study is
e Jones Kompala model [4] 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-
particles.
The first Saccharomyces cerevisiae
the problem discussed in [15] In this case the value of
L
ka is 150 and the 0
G va lue is 1 0. Wh en the *
Ovalue
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
cause oscillations.
is 7. Hdle
able 1. Base set of parameters used for the Saccharomyces
e
T
cerevisiae fermentation problem ([4]).
Parameter Valu
G0 10 gm/l
Y1Y3 (0.16,) gg–1
1
, Y2, 0.75, 0.6
,2
,3
,4
*
0.403, 2, 1, 0.95
O7.5 mg/l
*
0.3
0.03
0.7
K1, K2K3 0.05, 0001 , .01, 0.
2
O
K
0.01 mg/l
3
O
K
2.2 mg/l
1, 3
ii
1
2, 0, 10, 0.8
3
,max 1,2,
ii
0.6 44, 0.19, 0.3
Table 2. Problem specifications.
Problem number0
G
Original
required to eliminate
Hopf bifurcation
L
K
a
e valu
1 10 0.006 150
2 11.5225 0.005
3 8.75225 0.0065
Copyright © 2012 SciRes. JSBS
L. N. SRIDHAR
JSBS
30
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)
1 2
(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)
d 4)
evisiae example [14],
7. Case 2 (Figures 3 an
In the second Saccharomyces cer
for a
L
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.
L
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
e*
s a powerful model hat is The Jones Kompala model [4] 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 [18] 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 [4] the oxygen mass
transfer is in the term

*
L
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 [4]. 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
nanoparticl
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 [1]. 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-
tion processes.
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 [1] 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
gen mass
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.
10. Conclusion
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.
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Nomenclature
C: Intracellular Carbohydrate Mass Fraction Storage;
D: Dilution Rate;
V: Volume;
E: Ethanol Concentration;
X: Cell Mass Concentration;
O: Oxygen Concentration;
O*: Oxygen Solubility Limit (7.5);
th
Yi: i Pathway Yield Coefficient.