Unstructured modelling growth of Lactobacillus acidophilus as a function of the temperature

Unstructured modelling growth of Lactobacillus acidophilusas a function of the temperature: 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.
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Unstructured modelling growth of Lactobacillus acidophilus as a function of the temperatureMathematics 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, FranceAbstract We present modelling software developed under MATLAB in which parameter estimations are obtained by usingnon-linear regression techniques. The different parameters appear in a set of non-linear algebraic and differentialequations representing the model of the process. From experimental data obtained in discontinuous cultures arepresentative mathematical model (unstructured kinetic model) of the macroscopic behaviour of Lactobacillusacidophilus has been developed. An unstructured model expressed the specific rates of cell growth, lactic acidproduction and glucose consumption for batch fermentation. The model is formulated by considering the inhibition ofgrowth under sub-optimal culture conditions during Lactobacillus acidophilus fermentation, which is accompaniedby an increase of the maintenance energy. This study permits to predict the cellular behaviour at low growthtemperatures 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 tool1. 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 theunderstanding of the mechanisms and of the input–output relationships. In this study, we present some key parameters acting on the growth of Lactobacillus acidophilusallowing understanding certain mechanisms of inhibition and limitation, which affect the growth of thisstrain. These data will enable us to establish a mathematical model, which reproduces in a satisfactoryway 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: baati@insa-tlse.fr (L. Bˆaati), roux@laas.fr (G. Roux), dahhou@laas.fr (B. Dahhou), uribelarrea@insa-tlse.fr(J.-L. Uribelarrea).0378-4754/$30.00 © 2003 IMACS. Published by Elsevier B.V. All rights reserved.doi:10.1016/j.matcom.2003.09.013138 L. Bˆaati et al. / Mathematics and Computers in Simulation 65 (2004) 137–1451.1. Culture conditions The fermentations were carried out under strictly anaerobic conditions in 2-l glass fermentors (StéricGénie Industriel, Toulouse, France) with pH, temperature and agitation control. For the standard conditionpH, and agitation were set at 6.5, and 250 revolutions min−1 respectively. The bacteria were grown undera controlled gas environment by flushing both the vessel and the medium with nitrogen. The medium inthe fermentor was aseptically gassed (30 min) immediately before inoculation and maintained under anN2 atmosphere at a positive pressure of 103 Pa. Cultures in the fermentor were maintained at pH 6.5 byautomatic addition of 10N KOH. Inoculation was at 10%. In this study, three temperatures: 37 ◦ C (theoptimum 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 mediumfor 6 h at 37 ◦ C, then washed twice with sterile phosphate buffer (100 mM, pH 6.5) to avoid carryover ofessential 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 struc-ture 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. Fornumerical integration low order Runge–Kutta algorithms were used (which checks for integrability, andthus prevent frequent numerical problems). For the parameter identification, some modified MATLAB functions as well as newly designed pro-cedures 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 sumof absolute squared deviations (Eq. (2)) between measured and modelled values of the different statevariables 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) isthe observation vector which can be measured; u(t) is the input vector which can be used to take intoaccount the effect of environmental variables; η(t) is the kinetic vector which contain the main biologicalparameters of the fermentation reaction. It is known that η(t) is constituted of complex functions ofthe ...