Using Magnetic Nanoparticles to Eliminate Oscillations in Saccharomyces cerevisiae Fermentation Processes

doi:10.4236/jsbs.2012.23004

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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[email protected]

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

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 [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

u v

 (1)

max



(2)

while the expressions for th e pathways are given by

111







(3)

222

EK O











 (4)

333

GKO













(5)

with these growth rate equations, the balance equations

[4] are give n b y

dii

rv DX

t

 (6)



112 2

ddd

rv rv

GXc

GGDX CX

tYYtt





 





(7)

112 2

112

rv rv

EDE X

tYY





 





(8)



*33

OkaOOX

tYY











(9)

dii

ijji

urve

tKS











 (10)







33 3111222

dii

Crvrvrv CrvC



 

 (11)

G, E and O represent the concentrations of glucose,

ethanol and dissolved oxygen. i



represents the modi-

fied growth rate constant. i

and Oi

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

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

L. N. SRIDHAR 29



L,LNP



 (12)

NP represents the enhanced mass transfer coeffi-

cient when the nanoparticles are added while



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

dLrv

Oka CO O



 33

rv X













,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





1det



v and is the tangent vector at

x. Three test functions will be defined as













(14)

2det 2



31n













(15)

(16)

For the existence of a branch point, 1



, while



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

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

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

cerevisiae fermentation problem ([4]).

Parameter Valu

G0 10 gm/l

Y1Y3 (0.16,) gg–1

, Y2, 0.75, 0.6



0.403, 2, 1, 0.95

O7.5 mg/l



0.3



0.03



0.7

K1, K2K3 0.05, 0001 , .01, 0.

0.01 mg/l

2.2 mg/l





1, 3



 1

2, 0, 10, 0.8





,max 1,2,



 0.6 44, 0.19, 0.3

Table 2. Problem specifications.

Problem number0

Original



required to eliminate

Hopf bifurcation

e valu

1 10 0.006 150

2 11.5225 0.005

3 8.75225 0.0065

L. N. SRIDHAR

JSBS

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

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 [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



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.

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);

Yi: i Pathway Yield Coefficient.