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