Mathematics and Computers in Simulation 65 (2004) 137–145<br />
<br />
<br />
<br />
<br />
Unstructured modelling growth of Lactobacillus acidophilus<br />
as a function of the temperature<br />
L. Bˆaati a,∗ , G. Roux b , B. Dahhou b , J.-L. Uribelarrea a<br />
a<br />
Centre de bioingénierie Gilbert Durand, UMR-CNRS 5504, UR-INRA 792, Institut National des<br />
Sciences Appliquées, Avenue de Rangueil, 31077 Toulouse Cedex 4, France<br />
b<br />
Laboratoire d’Analyse et d’Architecture des Systèmes – CNRS, 7 avenue du Colonel Roche,<br />
31077 Toulouse Cedex 4, France<br />
<br />
<br />
<br />
Abstract<br />
We present modelling software developed under MATLAB in which parameter estimations are obtained by using<br />
non-linear regression techniques. The different parameters appear in a set of non-linear algebraic and differential<br />
equations representing the model of the process. From experimental data obtained in discontinuous cultures a<br />
representative mathematical model (unstructured kinetic model) of the macroscopic behaviour of Lactobacillus<br />
acidophilus has been developed. An unstructured model expressed the specific rates of cell growth, lactic acid<br />
production and glucose consumption for batch fermentation. The model is formulated by considering the inhibition of<br />
growth under sub-optimal culture conditions during Lactobacillus acidophilus fermentation, which is accompanied<br />
by an increase of the maintenance energy. This study permits to predict the cellular behaviour at low growth<br />
temperatures and enables to define the response of the strain to sub-optimal temperature stress.<br />
© 2003 IMACS. Published by Elsevier B.V. All rights reserved.<br />
Keywords: Lactic acid fermentation; Mathematical modelling; Unstructured kinetic model; Software tool<br />
<br />
<br />
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<br />
1. Introduction<br />
<br />
Nowadays computers are simply essential tools for several sorts of research. In applied sciences,<br />
mathematical relationships are used to represent basic mechanisms or global processes. They aim at the<br />
understanding of the mechanisms and of the input–output relationships.<br />
In this study, we present some key parameters acting on the growth of Lactobacillus acidophilus<br />
allowing understanding certain mechanisms of inhibition and limitation, which affect the growth of this<br />
strain. These data will enable us to establish a mathematical model, which reproduces in a satisfactory<br />
way the dynamic behaviour of the studied strain at different temperatures.<br />
∗<br />
Corresponding author. Tel.: +33-5-61-55-94-44; fax: +33-5-61-55-94-00.<br />
E-mail addresses: baati@insa-tlse.fr (L. Bˆaati), roux@laas.fr (G. Roux), dahhou@laas.fr (B. Dahhou), uribelarrea@insa-tlse.fr<br />
(J.-L. Uribelarrea).<br />
<br />
0378-4754/$30.00 © 2003 IMACS. Published by Elsevier B.V. All rights reserved.<br />
doi:10.1016/j.matcom.2003.09.013<br />
138 L. Bˆaati et al. / Mathematics and Computers in Simulation 65 (2004) 137–145<br />
<br />
1.1. Culture conditions<br />
<br />
The fermentations were carried out under strictly anaerobic conditions in 2-l glass fermentors (Stéric<br />
Génie Industriel, Toulouse, France) with pH, temperature and agitation control. For the standard condition<br />
pH, and agitation were set at 6.5, and 250 revolutions min−1 respectively. The bacteria were grown under<br />
a controlled gas environment by flushing both the vessel and the medium with nitrogen. The medium in<br />
the fermentor was aseptically gassed (30 min) immediately before inoculation and maintained under an<br />
N2 atmosphere at a positive pressure of 103 Pa. Cultures in the fermentor were maintained at pH 6.5 by<br />
automatic addition of 10N KOH. Inoculation was at 10%. In this study, three temperatures: 37 ◦ C (the<br />
optimum growth temperature of Lactobacillus acidophilus), 30 and 26 ◦ C were evaluated.<br />
In all these cases, precultures were prepared by incubation of Lactobacillus acidophilus in MRS medium<br />
for 6 h at 37 ◦ C, then washed twice with sterile phosphate buffer (100 mM, pH 6.5) to avoid carryover of<br />
essential nutrients and re-suspended in the same buffer for inoculation.<br />
<br />
1.2. Numerical methods<br />
<br />
Software developed under MATLAB [1] was used. The package is an interactive hierarchical struc-<br />
ture where three principal different actions can be chosen: identification, verification and simulation<br />
(Fig. 1). To solve the system of non-linear algebraic and differential equations representing the culture,<br />
the Gauss–Newton method with a mixed quadratic and cubic line search procedure was applied. For<br />
numerical integration low order Runge–Kutta algorithms were used (which checks for integrability, and<br />
thus prevent frequent numerical problems).<br />
For the parameter identification, some modified MATLAB functions as well as newly designed pro-<br />
cedures were employed. Most frequently, the designed variants of Hook-Jeeves and Rosenbrock method<br />
[2] yield the best results for biotechnological problems [3]. As minimisation criterion, the weighted sum<br />
of absolute squared deviations (Eq. (2)) between measured and modelled values of the different state<br />
variables was applied. The optimisation runs were carried out on a multitask Pentium computer.<br />
Fermentation process, which is non-linear, can be modelled by the following dynamics equation:<br />
<br />
˙<br />
X(t) = (X(t), u(t), η(t))<br />
(1)<br />
Y(t) = HX(t)<br />
<br />
where X(t) is the state vector generally including biomass, substrate and product concentrations; Y(t) is<br />
the observation vector which can be measured; u(t) is the input vector which can be used to take into<br />
account the effect of environmental variables; η(t) is the kinetic vector which contain the main biological<br />
parameters of the fermentation reaction. It is known that η(t) is constituted of complex functions of<br />
the state variables and of several biological constant, its expression is different for several fermentation<br />
processes. So the primary task of modelling is to identify which model of η(t) is suited to the real process<br />
and then to determine the corresponding biological constants. The minimisation of the criterion between<br />
the output of the model Ym (t) and the output of the process Y(t):<br />
tF<br />
Ji = min∗ (Y m (t) − Y(t))T Q(Y m (t) − Y(t)) (2)<br />
θi →θi 0<br />
<br />
allows obtaining the best matching parameter vector θi∗ of the model η(t).<br />
L. Bˆaati et al. / Mathematics and Computers in Simulation 65 (2004) 137–145 139<br />
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<br />
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<br />
Fig. 1. Schematic diagram of the architecture software “FERMOD”.<br />
140 L. Bˆaati et al. / Mathematics and Computers in Simulation 65 (2004) 137–145<br />
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To show the capabilities of the software of modelling, we used an example of modelling for Lacto-<br />
bacillus acidophilus. This modelling permits the quantitative description of the dynamic reactions of<br />
the micro-organism as well as the limiting mechanisms of the fermentative processes under sub-optimal<br />
temperature conditions.<br />
<br />
<br />
2. Modelling of Lactobacillus acidophilus<br />
<br />
Lactobacillus acidophilus is a homofermentative lactic acid bacterium, widely studied for its impact in<br />
food biotechnology. This micro-organism is commonly used in frozen or freeze-dried form and is sensi-<br />
tive to cold shock treatment. In a previous study [4], we have shown that sub-optimal growth temperatures<br />
intervene in the resistance of the cells to a freezing process. These results suggest that it is necessary to<br />
control the bacterium behaviour at these temperatures. However, some data remain difficult to be reached<br />
by only the experimental analysis because of the complexity of the biological reaction. Therefore, a math-<br />
ematical modelling coupled with the experimental analysis allows their identification. The experimental<br />
data were used to determine the parameters in the kinetic model. An unstructured model expressed the<br />
specific rates of cell growth, lactic acid production and glucose consumption for batch fermentation.<br />
The effect of the temperature was characterised at temperatures lower than their optimal growth tem-<br />
perature. Our results highlight the two following facts [5]:<br />
1. An inhibition of growth by the produced lactic acid. Phenomenon more accentuated at the lowest<br />
tested temperatures.<br />
2. Existence of an uncoupling “growth–lactic acid production”. This phenomenon is more significant<br />
at sub-optimal growth temperatures. In addition, this uncoupling “growth–lactic acid production”<br />
is accompanied by an increase of the maintenance energy. Tempest et al. [6] already highlighted a<br />
similar freak of uncoupling anabolism and catabolism in the case of a nutritional stress.<br />
These results suggest that growth characterisation of Lactobacillus acidophilus at low temperatures<br />
must take account of factors leading to the dysfunction of the regulation systems of the intracellular pH<br />
(pHin ) in the presence of strong concentrations of lactic acid. Indeed, at low growth temperatures, cells<br />
need to develop mechanisms supporting the maintenance of pH constant at high temperatures also and<br />
of a normal turgor pressure [7–9].<br />
<br />
2.1. Construction of the model<br />
<br />
The fermentative process dynamics are modelled starting from the material balances obtained for each<br />
macroscopic element of the biological reaction in the case of a discontinuous culture. The model proposed<br />
in this work is composed of the following three differential equations.<br />
Biomass (X):<br />
dX<br />
= µX (3)<br />
dt<br />
Product (P):<br />
dP<br />
= νP X (4)<br />
dt<br />
L. Bˆaati et al. / Mathematics and Computers in Simulation 65 (2004) 137–145 141<br />
<br />
Substrate (S):<br />
dS<br />
= −qS X (5)<br />
dt<br />
where µ (h−1 ), νP (g g−1 h−1 ) and qS (g g−1 h−1 ), respectively, represent the specific biomass growth rate,<br />
the specific lactic acid production rate and the specific substrate consumption rate.<br />
<br />
2.2. Modelling of the growth<br />
<br />
The evolution of the growth rate must take account of inhibition by the lactic acid and the variation of<br />
the energy of maintenance as function of the temperature:<br />
<br />
µ=0 if Θ > Θm<br />
(6)<br />
µ = ανP − m if Θ < Θm<br />
where m is a maximum temperature beyond which there is no more growth. We assume that the<br />
average maintenance varied hyperbolically with temperature until a certain limiting temperature (m ).<br />
This maximum temperature of growth was experimentally identified [5] and was fixed in the model with<br />
an aim of decreasing the complexity of the parametric identification. This value is set at Θm = 45 ◦ C.<br />
• The evolution of the coefficient of maintenance energy (g g−1 h−1 ) according to the temperature of<br />
growth (Θ, ◦ C) follows an hyperbolic pattern which can be described by the following equation:<br />
Θm − Θ<br />
m=δ (7)<br />
β + (Θm − Θ)<br />
One used the model established by Wijtzes et al. [10] but one adapted to our set of problems. Indeed,<br />
our objective is to study the behaviour of the strain under extreme conditions of growth, in particular<br />
the low temperatures.<br />
• To take account of the residual substrate at low temperatures of growth we used a model of the Monod’s<br />
type [11]:<br />
S<br />
m=δ (8)<br />
Ka + S<br />
<br />
If one takes account of the two effects the variation of the maintenance energy can be expressed by the<br />
following equation:<br />
Θm − Θ S<br />
m=δ (9)<br />
β + (Θm − Θ) Ka + S<br />
The preceding remarks enable us to write the final law governing the cell multiplication if the temperature<br />
is lower than the maximum temperature of growth Θm :<br />
Θm − Θ S<br />
µ = ανP − δ (10)<br />
β + (Θm − Θ) Ka + S<br />
The terms α (yield of biomass on lactate), β (constant of affinity) and δ (maximal maintenance) are<br />
constants, Ka (g l−1 ) is the substrate catabolic constant of affinity of the non-proliferating cells.<br />
142 L. Bˆaati et al. / Mathematics and Computers in Simulation 65 (2004) 137–145<br />
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2.5<br />
<br />
<br />
2.0<br />
(νpmax)0.5 = 0.085 T – 1.19<br />
(νp max)0.5 (g g-1 h-1)<br />
<br />
<br />
<br />
<br />
1.5<br />
<br />
<br />
1.0<br />
<br />
<br />
0.5<br />
<br />
<br />
0.0<br />
20 24 28 32 36 40<br />
Temperature (˚C)<br />
<br />
Fig. 2. Variation of νP0.5max according to the temperature of growth.<br />
<br />
<br />
<br />
2.3. Modelling of the lactic acid production<br />
<br />
The results obtained during the characterisation of the temperature effect on the metabolism of Lac-<br />
tobacillus acidophilus, highlight the variation of the maximum specific lactic acid production rate [5].<br />
As shown in Fig. 2, this variation can be expressed by an equation (Eq. (11)) of the same type as that of<br />
Bélehradek [12] proposed to describe the evolution of the maximum growth rate according to the growth<br />
temperature:<br />
<br />
νPmax = (Kb Θ − Kc )2 (11)<br />
<br />
• The evolution of the specific lactate production rate (νP ) as function of the lactate concentration was<br />
described using an exponential type function [13]:<br />
<br />
νP = νPmax e−KP P (12)<br />
<br />
• According to our results the growth stops before exhaustion of glucose for the cultures at low temper-<br />
atures. The consumption of the substrate is expressed by the following equation:<br />
S<br />
νP = νPmax (13)<br />
KS + S<br />
<br />
By taking into account the two effects previously described, the specific lactic acid production rate can<br />
be written in the following form:<br />
S<br />
νP = νPmax e−KP P (14)<br />
KS + S<br />
<br />
In this equation νPmax (g g−1 h−1 ) is the maximum specific lactic acid production rate, the terms Kb and<br />
Kc are two constants, KS (g l−1 ) is the substrate anabolic constant of affinity of the proliferating cells and<br />
KP (g l−1 ) is the product constant of inhibition.<br />
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2.4. Modelling of the glucose consumption<br />
<br />
The results obtained show that the production of the lactic acid is proportional to the consumption of<br />
glucose and that the yield (YP ) corresponding to the relationship between the lactic acid production rate<br />
and the substrate consumption rate is practically constant [5]. In addition, our results show the implication<br />
of glucose in the formation of the biomass at an optimal temperature of growth (37 ◦ C) thus supporting<br />
the increase in the biomass yield (YX ). The expression of the specific catabolic substrate consumption<br />
rate (qS ) must take into account these two effects and can thus be written in the following form:<br />
νP ανP<br />
qS = + (15)<br />
YP YX<br />
<br />
2.5. Simulation results<br />
<br />
We have used experimental data for a fermentation performed in semi-synthetic medium at different<br />
growth temperatures (26 and 37 ◦ C). On the basis of these two experiments and the software package<br />
“FERMOD”, we have obtained the 10 parameters values of the previous developed model (Eqs. (10),<br />
(11), (14) and (15)). In order to validate the modelling, we did a third experiment at 30 ◦ C to compare<br />
these experimental results and the simulation results obtained with the parameters values determined by<br />
non-linear regression for the other two temperatures (Table 1). The result representing these experimental<br />
data and the data resulting from the model are presented in Fig. 3.<br />
The model developed in this study describes satisfactorily the kinetic behaviour of the strain for the<br />
discontinuous cultures carried out at various growth temperatures. The evolution of the growth rate<br />
must take account of inhibition by the produced lactic acid and the variation of the maintenance energy<br />
according to the temperature. The values of the model are specific for the bacterial strain. The numerical<br />
values of the kinetic parameters of the model are shown in Table 2. The parametric identification is<br />
illustrated by a good correlation between the experimental values and those given by the model. The<br />
errors remains lower than 6%, whatever the modelled element (Table 1). This value is comparable with<br />
the precision of experimental measurements inherent to such fermentation.<br />
• The limitation of the growth following the exhaustion of glucose is checked what justifies the low value<br />
of KS , found.<br />
• The coupling between the cell growth and the lactic acid production observed under 37 ◦ C is less and<br />
less significant for 30 and 26 ◦ C; this was confirmed by the simulation results.<br />
• Lactobacillus acidophilus is homofermentative strain on all studied growth temperatures. In comparison<br />
with the profiles of concentrations in substrate and lactic acid provided by model, YP (0.98 g g−1 ) is<br />
also justified.<br />
Table 1<br />
The model checking criteria<br />
Biomass (g l−1 ) Lactic acid (g l−1 ) Glucose (g l−1 )<br />
Temperature (◦ C) 26 37 26 37 26 37<br />
Real average error (%) 4.48 5.04 5.61 2.53 5.7 2.99<br />
Absolute average error 0.03 0.11 0.73 0.40 1.12 0.56<br />
Absolute maximum error 1.15 0.25 1.15 0.77 2.24 1.4<br />
144 L. Bˆaati et al. / Mathematics and Computers in Simulation 65 (2004) 137–145<br />
<br />
3 20<br />
<br />
(A) (B) (C)<br />
16<br />
Glucose, Lactate (g/l)<br />
2<br />
Biomass (g/l)<br />
<br />
<br />
<br />
<br />
12<br />
<br />
<br />
8<br />
1<br />
<br />
4<br />
<br />
<br />
0 0<br />
0 5 10 0 5 10 15 20 25 300 10 20 30 40 50 60 70<br />
Time (h) Time (h) Time (h)<br />
<br />
Fig. 3. Experimental and modelled data for a fermentation performed in semi-synthetic medium at different growth temperatures<br />
(A) 37 ◦ C, (B) 30 ◦ C and (C) 26 ◦ C (the experimental data (symbols) for lactic acid (䊏), glucose (䉲) and biomass (䊉); modelling<br />
results are represented by lines).<br />
<br />
<br />
Table 2<br />
Numeric values of the different parameters obtained by modelling<br />
Description Parameter Value<br />
a −1<br />
Anabolic constant of affinity KS (g l ) 0.001<br />
Catabolic constant of affinityb Ka (g l−1 ) 6<br />
Constant of inhibition KP (g l−1 ) 0.157<br />
Coefficient of regression Kb 0.085<br />
νPmax (Θ = 0 ◦ C) Kc 1.19<br />
Yield of biomass on lactate α (g g−1 ) 0.172<br />
Constant of affinity β (◦ C) 10<br />
Maximal maintenance δ (g g−1 h−1 ) 0.122<br />
Lactic acid yield YP (g g−1 ) 0.98<br />
Biomass yield YX (g g−1 ) 0.9<br />
a<br />
Constant of affinity of the proliferating cells for the substrate.<br />
b<br />
Constant of affinity of the non-proliferating cells for the substrate.<br />
<br />
<br />
3. Conclusion<br />
<br />
The goal of this study was twofold: first built a global model able to reproduce satisfactorily the dy-<br />
namic behaviour of the strain under several environmental conditions of temperature; second shown the<br />
capability of a software package developed in our laboratory (FERMOD). From the experimental data<br />
obtained in discontinuous cultures a mathematical model representative of the macroscopic behaviour<br />
of Lactobacillus acidophilus for various temperatures of growth has been developed. This unstructured<br />
kinetic model whose parameters were identified using FERMOD makes it possible to describe correctly<br />
the observed phenomena such as cell multiplication, the consumption of glucose, the lactic acid pro-<br />
duction in discontinuous cultures carried out at various temperatures of growth. Thus, this model can<br />
contribute effectively to the implementation of strategies of command allowing of the productivities at<br />
L. Bˆaati et al. / Mathematics and Computers in Simulation 65 (2004) 137–145 145<br />
<br />
low temperature to improve the rate of survival of the strain after processing of deep freezing [5]. This<br />
model takes account of inhibition by lactic acid, principal phenomenon intervening in lactic fermentation<br />
at levels of temperatures lower than the optimal temperature of growth. It will be interesting to evaluate<br />
the validity of this model under other conditions of culture (different medium, temperatures higher than<br />
the optimal temperature of growth).<br />
<br />
<br />
References<br />
<br />
[1] MATHWORKS: MATLAB User’s Guide, The MathWorks Inc., 1991.<br />
[2] S.S. Rao, Optimization: Theory and applications, Wiley Eastern Ltd., New Delhi, India, 1979.<br />
[3] L. Edelstein-Keshet, Mathematical Models in Biology, McGraw-Hill, New York, 1988.<br />
[4] L. Bˆaati, C. Fabre-Gea, D. Auriol, Ph. Blanc, Study of the cryotolerance of Lactobacillus acidophilus: effect of culture and<br />
freezing conditions on the viability and cellular protein levels, Int. J. Food Microbiol. 59 (2000) 241–247.<br />
[5] L. Bˆaati, Incidence de l’itinéraire technologique sur la valeur d’usage de Lactobacillus acidophilus: Recherche de marqueurs<br />
physiologiques. Thèse de doctorat de l’INSA, Toulouse, France, 11 Décembre 2000.<br />
[6] D. Tempest, D. Neijssel, The status of YATP and maintenance energy as biologically interpretable phenomena, Ann. Rev.<br />
Microbiol. 38 (1984) 459–486.<br />
[7] E. Kashket, Bioenergetics of lactic acid bacteria: cytoplasmic pH and osmotolerance, FEMS Microbiol. Rev. 46 (1987)<br />
233–244.<br />
[8] W. Konings, R. Otto, Energy transduction and solute transport in streptococci. Antonie van Leeuwenhoek (1983) 247–257.<br />
[9] J. Russel, G. Cook, Energetics of bacterial growth: balance of anabolic and catabolic reactions, Microbiol. Rev. 59 (1995)<br />
48–62.<br />
[10] T. Wijtzes, J.C. De Wit, J.H. Huis In’t Veld, K. Van’t Reit, M.H. Zwietering, Modeling bacterial growth of Lactobacillus<br />
curvatus as a function of acidity and temperature, Appl. Environ. Microbiol. 61 (1995) 2533–2539.<br />
[11] J. Monod, Recherches sur la croissance des cultures bactériennes, Hermann and Cie, Paris, 1942.<br />
[12] D.A. Ratkowsky, J. Oley, T.A. McMeekin, A. Ball, Relationship between temperature and growth rate of bacterial cultures,<br />
J. Bacteriol. 149 (1) (1982) 1–5.<br />
[13] M. Taniguichi, N. Kotani, Kobayashi, High-concentration cultivation of lactic acid bacteria in fermentor with cross-flow<br />
filtration, J. Ferment. Technol. 65 (1987) 179–184.<br />