Analysis and practical implementation of a model for combined growth and metabolite production of lactic acid bacteria Karen M. Vereecken, Jan F. Van Impe * BioTeC—Bioprocess Technology and Control, Katholieke Universiteit Leuven, Kasteelpark Arenberg 22, B-3001 Leuven (Heverlee), Belgium Received 16 May 2001; accepted 9 August 2001 Abstract Next to the traditional application of lactic acid bacteria (LAB) as starter cultures for food fermentations, the use of LAB as protective cultures against microbial pathogens and spoilage organisms in other food production processes gains more and more interest. The inhibitory effect of LAB is mainly accomplished through formation of antimicrobial metabolites. In this paper, the model of Nicolaı¨ et al. [Food Microbiol. 10 (1993) 229.], describing cell growth and production of lactic acid, which is the major end-product of LAB metabolism, is investigated. In contrast to classical predictive models, the transition of the expo- nential growth phase to the stationary phase is obtained through the increasing concentrations of undissociated lactic acid [LaH] and decreasing pH in the environment. To describe the variation in time of [LaH] and pH, a novel, robust calculation method is introduced. The model of Nicolaı¨ et al. in combination with the novel method of [LaH] and pH computation is then further applied to an experimental data set of Lactococcus lactis SL05 grown in a rich medium. An accurate description of the measured values of cell concentration, total lactic acid concentration and pH is obtained. D 2002 Elsevier Science B.V. All rights reserved. Keywords: Lactic acid bacteria; Microbial growth; Metabolite production; Modelling 1. Introduction Lactic acid bacteria are traditionally applied as starter cultures for the production of fermented foods. In these products, LAB have two major functions, namely (i) achievement of certain beneficial physico- chemical changes in the food ingredients, e.g., acid- ification, curdling and production of flavour com- pounds, and (ii) inhibition of the outgrowth of microbial pathogens and spoilage organisms. The antimicrobial potential of LAB together with their status as Generally Regarded As Save (GRAS) organisms has motivated a lot of researchers to study a second possible application of LAB in the food indus- try, i.e., as protective cultures, supplemented in mini- mally processed foods (see, e.g., Matilla-Sandholm and Skytta¨, 1996, for an overview). During manufac- turing, these foods only undergo a mild inactivation treatment in order to retain an optimal textural and sensorial quality. LAB thus can serve as an additional 0168-1605/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved. PII: S 0168-1605(01)00641-9 * Corresponding author. Tel.: +32-16-32-19-47; fax: +32-16- 32-19-60. E-mail address: jan.vanimpe@agr.kuleuven.ac.be (J.F. Van Impe). www.elsevier.com/locate/ijfoodmicro International Journal of Food Microbiology 73 (2002) 239–250 biological hurdle to control the level of surviving but undesirable organisms. Both functions of LAB, whether fermentative or protective, are mainly accomplished by the production of microbial metabolites. LAB indeed produce a wide variety of compounds, including low molecular weight metabolites such as CO 2 ,H 2 O 2 , organic acids, alcohols and high molecular weight metabolites, amongst which polysaccharides and bacteriocins. Analogous to other industrial processes, the use of mathematical models in the food industry is gaining more and more attention for process evaluation, opti- misation and design (Walls and Scott, 1997). For mathematical models to be of use in the specif ic subareas of food fermentation a nd biological preser- vation, the following two basic aspects should be taken into account. . In view of the above-m entioned importance of microbial metabolites, an appropriate model should incor porate both growth and produc tion character- istics of LAB. As such, classical predictive growth models, which focus on microbial growth, are not di- rectly applicable. . Several microbial metabolites significantly mod- ify the growth medium. Examples are organic acids , which cause a reduction in pH, and certain poly- saccharides, which augment the medium solidity. A model shoul d take into account these modifications when they considerably affect microbial prolifera- tion. In this paper, a model from literature (Nicolaı¨ et al., 1993) which addresses the two stated aspects, is in- vestigated. The metabolite taken into consi deration in this model is lactic acid, being the main end-product of LAB metabolism. The contributions of the present research can be subdivided in two main parts. A first part, described in Section 3, involves a more theoret- ical analysis of the model. Some mathematical proper- ties of the model are discussed, and an easy-to-use method to take into account the evolution of pH and undissociated lactic acid [LaH] in the medium is pro- posed. In a second part, discussed in Section 4, the model and the newly developed method of pH and [LaH] computation are applied to a case study, namely the growth of Lactococcus lactis SL05 in a rich medium. A part of the results in this paper is also presented in Vereecken and Van Impe (2000). 2. Materials and methods The strain L. lactis SL05, selected by Arilait (France), was kindly provided by ADRIA (France). A fermentation experiment was performed in a 1-l er- lenmeyer flask (Duran Schott), provided with a side- arm. Openings (at the upper end and at the side- arm) were closed with screw caps containing a rubber sep- tum. The medium (500 ml) used was BHIYEG, con- taining Brain Heart Infusion broth 37 g/l (Oxoid), sup- plemented with 3 g/l Yeast Extract (Oxoid) and 18 g/l glucose (Vel). Before inoculation the medium was flushed with N 2 to obtain anaerobic conditions, and pH correction to a value of 6.7 was performed with HCl 4 N. The flask was incubated at 35 °C (Cooling Incubator Series 6000, Termaks) on a rotary shaker (Heidolph Unimax 2010) at 135 rpm. From a frozen stock culture, an inoculation culture was obtained after a first growth period of 24 h at 30 °C and a second period of 16 h at the same temperature (both in tubes containing 5 ml of BHIYEG). The initial cell density was 10 5 cfu/ml. At regular time intervals, samples were taken with a sterile syringe (Norm-Ject, Henke/ Sass/Wolf) and needle (Fine-Ject, Henke/Sass/Wolf) through the lower septum. In these samples, bacterial growth was quantified through determination of cfu/ ml on MRS medium (Oxoid) by means of a spiral plater (Eddy-Jet, IUL Instruments) and pH was deter- mined in the cell suspension with a pH sensor (HI92240, Hanna Instruments). After filtration of the sample to remove the cells (Sarstedt, pore size 0.2 mm) and derivatisation of the lac tic acid present in the filtrate with methanol, lactic acid was measured through gas chromatographic analysis of the methyl ester (separation with a Delsi GC on a 2-m column with 10% carbowax 20 M on chromosorb W-AW 80/ 100, detection with flame ionisation, peak integrator of Trivector). 3. Analysis and adaptation of the model of Nicolaı¨ et al. (1993) Nicolaı¨ et al. (1993) have constructed a dynamic model for the surface growth of lactic acid bacteria on vacuum-packed meats. In the current research, the model is adapted to describe bacterial growth in a li- quid medium. While the original model considers an K.M. Vereecken, J.F. Van Impe / International Journal of Food Microbiology 73 (2002) 239–250240 amount of cells and lactic acid present in a surface liquid layer with known dimensions, the adapted model describes the concentrations of cells and lactic acid in a liquid medium. Microbial growth and lactic acid production are described by the following set of d ifferential equa- tions. dN dt ¼ lN ð1Þ dLaH tot dt ¼ pN ð2Þ with N the cell concent ration [cfu/ml] and LaH tot the total concentration of lactic acid [mM], i.e., produced by the organism plus initially present. In these equa- tions, the specific growth rate l [h À 1 ] and the specific production rate p [10 À 3 mmol h À 1 cfu À 1 ] depend on the concentrations of undissociated lactic acid [LaH] [mM] and pH (or hydrogen ionic concentration [H + ] [mM]) as follows. l ¼ l max ½LaHV½LaH min l ¼ l max expfÀk l ð½LaHÀ½LaH min Þg ½LaH > ½LaH min ð3Þ p ¼ p max ½LaHV½LaH min p ¼ p max expfÀk p ð½LaHÀ½LaH min Þg ½LaH > ½LaH min ð4Þ k l ¼ a þ b ½H þ  2 ð5Þ k p ¼ c þ d ½H þ  2 ð6Þ with l max [h À 1 ] and p max [10 À 3 mmol h À 1 cfu À 1 ] the maximum specific growth and production rates, respectively, [LaH] min [mM] the minimum inhibitory concentration of undissociated lactic acid, and a, b, c and d parameters. Nicolaı¨ et al. assume a single buffer system present in the medium, consisti ng of a weak acid BuH and its well solvable conjugated salt BuM. For this research, it is assumed that the medium also contains a strong acid AH, allowing to manipulate the initial medium pH (as in the experimental case study, where the initial pH is adapted with HCl—see Section 2). The follow- ing chemical reactions can be written. LaH X K LaH La À þ H þ BuH X K BuH Bu À þ H þ BuM ! Bu À þ M þ AH ! A À þ H þ with dissociation constants K LaH and K BuH . In combi- nation with the associated charge and mass balances, the following two algebraic expressions can be derived. ½H þ ¼ÀBuM a þ AH a þ K LaH LaH tot K LaH þ½H þ  þ K BuH ðBuH a þ BuM a Þ K BuH þ½H þ  ð7Þ ½LaH¼ ½H þ LaH tot K LaH þ½H þ  ð8Þ with BuH a , BuM a and AH a analytical concentrations [mM]. Observe that Eq. (7) is cubic in [H + ]. In Nicolaı¨ et al. (1993), it is proven that only one root of the latter equation can be positive real. It should be mentioned that Nicolaı¨ et al. consid- ered two additional equations to account for the in- fluence of pH on l max and p max . However, the latter dependencies refer to the initial pH, as a medium characteristic, rather than to the hydrogen ions result- ing from microbial metabolism. It is indeed well known that during the exponential growth phase, when a microbial population reaches maximum values for the specific rates, the metabolites accumulate slowly and their influence on growth and production kinetics is small. Therefore, since the medium and associated initial pH is not a variable in the case study presented in this paper, l max and p max can be reason- ably assumed constant. Among microbiologists it is widely accepted that metabolites play a major role in the transition be- tween different growth phases. Microbial metabolites, K.M. Vereecken, J.F. Van Impe / International Journal of Food Microbiology 73 (2002) 239–250 241 whether growth or non-growth associated, can have a specific effect on cell formation. The mutual interfer- ence of microbial cells and products, which is in fact a typical example of intraspecies interactions, can be mathematically translated into a set of coupled differ- ential equations, as presented in Eqs. (1) and (2). The Nicolaı¨ et al. model focuses on the transition from the exponential growth phase, characterise d by a maximum value for the specific growth rate l, to the stationary phase, in whi ch l becomes equal to zero. The decline in the specific growth rate is in this model obtained as a consequence of an increasing amount of lactic acid in the environment. It should be noted here that the model of Nicolaı¨ et al. differs in this respect from the classical predictive growth models (e.g., the modified Gompertz equation, Zwietering et al., 1990, the model of Baranyi and Roberts, 1994). The basic equation behind the classical models is dN dt ¼ lðN ÞN ¼ l max f ðNÞN : ð9Þ In this equation, the specific growth rate l depends upon the cell concentration N and a decrease from a maximum value towards zero—or otherwise said a transition from the exponential growth to the sta- tionary phase—is obtained through a factor f (N), which decreases with increasing N until a maximum population density N max is reached for which f (N), and thus l(N), become equal to zero. When comparing the Nicolaı¨ et al. approach towards modelling of the stationary phase with the classical models, it can be concluded that the model of Nicolaı¨ et al. more closely matches the underlying mechanistic principles of cell growth retardation and ceasing. The metabolite under study in the model of Nicolaı¨ et al. is lactic acid, which inhibits microbial cell growth via two components, hydrogen ions and undissociated lactic acid. Model accuracy and reliability therefore strongly depend upon a proper description of the in- fluence of these two components on the specific rates l and p. Using (i) Eqs. (3), (5) and (8), (ii) the same values for the parameter s [LaH] min , a, b and K LaH as mentioned in Nicolaı¨ et al. (1993), (iii) a value of 0.6920 [h À 1 ] for the parameter l max (which corre- sponds with the value of l max at the optimal pH of 5.74 for the species Lactobacillus delbrueckii considered in the cited paper), and (iv) an initial pH of 6.3, the specific growth rate is visualised in Fig. 1 in two and three dimensions. The different lines in the two-dimen- sional plots represent intersections of the surface in Fig. 1a parallel to the [LaH]-axis (i.e., lines of equal values of pH) (Fig. 1b) or parallel to the pH-axis (i.e., lines of equal values of [LaH]) (Fig. 1c). From Fig. 1b, it can be seen that, after a period in which l remains at a constant value l max , it decreases with increasing [LaH], which is to be expected from a microbiological viewpoint. On the other hand, Fig. 1c reveals that the specific growth rate increases— although no t very pronounced—with increasing [H + ] (or decreasing pH) which is unacceptable in the suboptimal range of pH. The same conclusio ns can be drawn when considering the partial derivatives of  with respect to [LaH] and [H + ]. If [LaH] > [LaH] min the following relations apply @l @½LaH ¼Àl max k l exp Àk l ð½LaHÀ½LaH min Þ ÈÉ ð10Þ @l @½H þ  ¼ 2l max ð½LaHÀ½LaH min Þ À b=½H þ  3 Á  exp Àk l ð½LaHÀ½LaH min Þ ÈÉ : ð11Þ If [LaH] V [LaH] min , @l/@[H + ] and @l/@[LaH] are of course equal to zero. When all model parameters are positive, @l/@[LaH] is negative, involving a decrease in l with decreasing [LaH], and @l/@[H + ] is positive, implying an increase in l with decreasing pH. To assure that the specific growth rate decreases with increasing [LaH] and [H + ], relations (10) and (11) should be negative and thus the following inequalities must be satisfied. b < 0 ð12Þ k l ¼ a þ b ½H þ  2 > 0 ð13Þ While a should in deed be a pos itive parameter, b cannot be positive. Analogous conditions can be de- rived for the parameters c and d which appear in the expression of the specific production rate. When applying the model to experimental data, as in Section 4, one should take into account these four constraints K.M. Vereecken, J.F. Van Impe / International Journal of Food Microbiology 73 (2002) 239–250242 on the parameter values a, b, c and d in order to avoid unrealistic model predictions. Next to the expressions relating the specific growth rate and the specific production rate to [LaH] and pH, the accuracy of latter input values is also of major im- portance. Therefore, an adequate calculation method for the actual values of [LaH] and pH at each time point during growth is necessary. According to classi- cal chemical laws, the pH is fully determined when the total amount of lactic acid in the medium and the buffering ca pacity of the medium are known. The amount of undissociated lactic acid is then further related to the total amount of lactic acid LaH tot and the pH by the lactic acid chemical equilibrium. As explained above, the approach of Nicolaı¨ et al. is based on these chemical principles and assumes a single buffer compound BuH a /BuM a present in the medium. For the present research, the precise interdependency between the three factors LaH tot , pH and [LaH] is further investigated. Fig. 2 represents hereby a number of two-dimensional plots in which the relations between each couple of the variables LaH tot , pH and [LaH] are visualised. By simulating cell growth and lactic acid formation making use of model Eqs. (1) – (8), time-dependent courses of LaH tot , pH and [LaH] are obtained. Next, through elimination of time, plots of [LaH] versus LaH tot (Fig. 2a and d), pH versus [LaH] (F ig. 2b and e) and pH versus LaH tot (Fig. 2c and f) are constructed. The arrows indicate the direc- tions which are followed during the simulated fermen- Fig. 1. Specific growth rate  as function of [LaH] and pH (relations (3) and (5) with l max = 6.9200  10 À 1 [h À 1 ], [LaH] min = 1.333 [mM], a = 5.7600  10 À 2 [mM À 1 ], b = 9.9000  10 À 7 [mM]). (a) Three-dimensional representation; (b) l as function of [LaH] with the different lines representing lines of equal pH values; (c) l as function of pH with the different lines representing lines of equal [LaH] values. K.M. Vereecken, J.F. Van Impe / International Journal of Food Microbiology 73 (2002) 239–250 243 Fig. 2. Two-dimensional relations between the variables LaH tot , [LaH] and pH. (a, b, c): variation of the initial pH (pH 0 ). (d, e, f): variation of the buffer amount (BuH a /BuM a ). K.M. Vereecken, J.F. Van Impe / International Journal of Food Microbiology 73 (2002) 239–250244 tations. In each figure, a reference curve is realised (full line) by means of the parameter values used in the work of Nicolaı¨ et al. to construct Fig. 3(1), p. 234 in their paper. Furthermore, to comprehend the influence of the buffer amount BuH a /BuM a and initial pH, two additional curves are generated in each plot for differ- ent levels of these factors, higher or lowe r than the ones used in the reference curve, each level being marked on the different plots. It should be noted here that for an individual plot, relating two variables out of three to each other, the one left out of consideration is not a constant. For example, in the plot of pH versus [LaH] (Fig. 2b and e), the total lactic acid concen- tration gradually increases when traversing the curves from left to right. On the basis of the six plots, the following con- clusions can be drawn. . From the [LaH] versus LaH tot plots (Fig. 2a and d), it can be seen that when the initial pH is equal to 3.86 (which corresponds to a hydrogen ionic concen- tration equal to the dissociation constant of lactic acid) (Fig. 2a) or when no buffer BuH a /BuM a is present (Fig. 2d), the curve is more or less a straight line passing through the origin. In case of a higher initial pH and/or in the presence of buffer, the curve initially increases very slowly and approaches a straight line for larger LaH tot values. The slope of this straight line is always the same. Furthermore, when the pH and/or buffer amount further increase, the intersection of this line with the LaH tot -axis increases, or otherwise said, the straight line is shifted to the right. . The pH versus [LaH] plots (Fig. 2b and e) are monotonically decreasing curves in which no inflec- tion points occur. The initial pH determines the in- tersection point with the pH-axis (Fig. 2b) while the buffer amount determines the deviation from the initial pH at a given (non-zero) [LaH] value (Fig. 2e). . Finally, the pH versus LaH tot plots (Fig. 2c and f) are monotonically decreasing curves with a variable number of inflection points. The intersection with the pH-axis is self-evidently the initial pH. As already mentioned, a main objective of the pre- sent research is the application of the model of Nicolaı¨ et al. to experimental data. This implies attributing numerical values to the buffer system BuH a /BuM a and its associated dissociation constant K BuH (relation (7)). This is, however, not straightforward for two reasons. Firstly, the medium used in the present case study is a rich and nutritious, yet undefined commer- cial growth medium, which presumably contains more than one substance with buffering capacity. If—or, to what extent—the buffering substances can be merged together into one o verall buffering compound is uncertain. Secondly, Eq. (7), in which the positive Fig. 3. (a) Representation of relation (14). A reference curve a, b, c (bold line) approximates the LaH tot -axis for small LaH tot values and a straight line with slope a for larger LaH tot values (thin full lines). Curves for different parameter values are indicated. (b) Representation of relation (15). K.M. Vereecken, J.F. Van Impe / International Journal of Food Microbiology 73 (2002) 239–250 245 real root needs to be selected and used to calculate pH and [LaH]—give n the experimentally measured pro- file of LaH tot —is from a mathematical viewpoint a rather complex expression on the basis of which the unknown values BuH a , BuM a and K BuH are to be identified. For these reasons, in this research an alternative method for pH and [LaH] computation is proposed, which can be used in combination with all models of the general class (Eqs. (1) and (2)). The method is inspired by Fig. 2a –e discussed above. . To represent the relation between [LaH] and LaH tot (Fig. 2a and d), the following equation, inspired by Dabes-kinetics (Van Impe et al., 1994), is proposed. ½LaH¼aLaH tot À ab 2ðb À cÞ h ðLaH tot þ bÞ À ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðLaH tot þ bÞ 2 À 4ðb À cÞLaH tot q i ð14Þ A graphical representation of this relation for different parameter values can be found in Fig. 3a. As can be seen from this figure, the three parameters in this expression are all easy interpretable: (i) a equals the slope of the straight lines with whom the different curves in Fig. 2a and d nearly coincide for larger LaH tot values, (ii) b equals the intersection point of this straight line wi th the LaH tot -axis, and (iii) c de- termines the transition from the slowly increasing to the constant slope region of the curves. From Fig. 2a and d, it can be easily seen that the parameters a and c are in fact constan t for all the curves shown and b is the only parameter that varies for different values of initial pH or buffer amount. . To represent the relation between pH and [LaH] (Fig. 2b and e), a second expression, also based on the Dabes-kinetics, is developed. pH ¼ 1 2a 1 c 1 h ðb 1 À 2c 1 Þ½LaH À ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 2 1 ½LaH 2 þ 4a 1 b 2 1 c 1 ½LaH q i þ pH 0 ð15Þ Parameters a 1 , b 1 and c 1 have a similar interpretation as a, b and c, respectively, in Eq. (14), while pH 0 symbolises the initial pH. A graphical representation of expression (15) is depicted in Fig. 3b. Combination of the two proposed relations (14) and (15) enables to calculate [LaH] and pH values starting from increasing (experimentally measured) LaH tot values. In contrast to Eqs. (7) and (8), all parameters in rel ation s (14) an d (15) h ave a straightforw ar d interpretation in connection with the curves of Fig. 2. Of course, it should be kept in mind that the novel method also assumes, although more implicitly than the method of Nicolaı¨ et al., a single buffer compound or a single buffer-like behaviour of the total of all buffering substances. In the next section, where the proposed method will be tested on real experimental data, some indications on the validity of this under- lying assumption will be presented. In summary, the proposed model consists out of Eqs. (1) –(6) completed with Eqs. (14) and (15). 4. Practical implementation In this section, the model o f Nicolaı¨ et al., in combination with the newly developed method of pH and [LaH] calculation, will be applied to the experimental case study described in Section 2. The lactic acid bacterium and the growth medium used are selected carefully in order to be compliant with two basic aspect s of the model of Ni colaı¨ et al. Firstly, the bacterial strain has a homofermentative metabolism and, as such, lactic acid is the only end-product that has to be taken into account. Secondly, the rich growth medium with high glucose content assures that no substrate limitation occurs and the influence of substrate concentration on growth and production kinetics can be neg lected. Application of the adapted model of Nicolaı¨ et al. to the experimental data set of cell concentration N, total lactic acid concentration LaH tot and pH consists out of two major steps, namely, (i) determination of medium related parameters (i.e., the parameters of Eqs. (14) and (15)), and (ii) determination of growth and production related parameters (i.e., the parameters of Eqs. (1) –(6)). This will be explained in detail in the further text. 4.1. Determination of medium related parameters Eqs. (14) and (15) constitute the newly developed method of pH and [LaH] calculation and their param- K.M. Vereecken, J.F. Van Impe / International Journal of Food Microbiology 73 (2002) 239–250246 eters can be determined on the basis of the LaH tot and pH meas urements, without considering the cell c on- centration data. At each measurement instant t,a couple of [LaH tot (t), pH(t)] values is available. Start- ing from these [LaH tot (t), pH(t)] couples, the estima- tion procedure is as follows. (1) Since both expressions (14) and (15) involve the undissociated lactic acid concentration [LaH], which is not measured experimentally, a value for [LaH](t)is calculated for each [LaH tot (t), pH(t)] couple, by mak- ing use of Eq. (8). (2) Next, the computed values of [LaH] are plotted versus LaH tot or versus pH at matching time instants. Fig. 4 displays both plots for the experimental values of the case study (circles). (3) Based on these plots, the parameters a, b and c of expression (14) and a 1 , b 1 , c 1 and pH 0 of expression (15) are identified. Simulations of relations (14) and (15) with the estimated parameter values for the case study are presented in Fig. 4 as full lines. It can be concluded that a realistic description of the plots is obtained. Moreover, the shape of the plots corresponds very well with the ones displayed in Fig. 2, providing support for the assumption of a single buffer-like behaviour as stated above. (4) Finally, given the estimated values of a, b, c, a 1 , b 1 , c 1 and pH 0 , prediction of any time dep endent course of pH (and [LaH]) is possible starting from an experimentally measured LaH tot profile. Fig. 5 depicts the experimental values of pH as a function of time, together with a prediction based on the new method and the experimental LaH tot values, linearly interpo- lated between two subsequent observations. It can be seen that a combination of relations (14) and (15) with the estimated parameter s provides a satisfactory pre- diction of the measured pH values. Fig. 5. Variation of pH in function of time for the experimental case study; ‘6’: experimental data; ‘—’: simulation based on relations (14) and (15) and the experimental LaH tot values, linearly interpolated. Fig. 4. [LaH] versus LaH tot (a) and pH versus [LaH] (b) for the experimental case study; ‘6’: experimental data; ‘—’: description with relation (14) (a) or relation (15) (b). K.M. Vereecken, J.F. Van Impe / International Journal of Food Microbiology 73 (2002) 239–250 247 It should be noted that the procedure described here has to be accomplished only once for a given medium and initial pH to calibrate the param eters in the Eqs. (14) and (15). Indeed, since the parameters a, b, c, a 1 , b 1 and c 1 only depend on the medi um char- acteristics and the parameter pH 0 only on the initial pH, they are suited for every growth experiment in this particular medium and initial pH. After calibration, prediction of the time dependent course of pH (and [LaH]) is always possible when a LaH tot profile only is at hand. 4.2. Determination of growth and production related parameters To identify the parameters of Eqs. (1)–(6), describ- ing cell growth and metabolite production, the com- plete data set (including cell, lactic acid and pH measurements) is necessary. Obviously, use is made of relations (14) and (15) and the determined values for a, b, c, a 1 , b 1 , c 1 and pH 0 to determine the input values [LaH](t)and[H + ](t ) of the specific growth and production rates (Eqs. (3)– (6)). As mentioned above, the variation in time of pH is fully determined when the LaH tot profile is available and the param eters a, b, c, a 1 , b 1 , c 1 and pH 0 are known. Therefore, only N and LaH tot values are taken into account in the cost criterion during the calibration procedure. Further- more, the constraints (12) and (13) on the parameter values a, b, c and d, specified in the previous section, have been taken into account. Table 1 lists the opti- Table 1 Parameter values for the model of Nicolaı¨ et al. (1993) for L. lactis SL05 in BHIYEG medium at a temperature of 35 °C and an initial pH of 6.7 Parameter Units Value l max [h À 1 ] 1.4160 p max [10 À 3 mmol h À 1 cfu À 1 ] 7.2302  10 À 8 [LaH] min [mM] 3.8417  10 À 3 a [mM À 1 ] 3.3449  10 À 1 b [mM] À 2.4118  10 À 7 c [mM À 1 ] 3.8389  10 À 1 d [mM] À 1.0000  10 À 10 Fig. 6. L. lactis SL05 in BHIYEG medium at a temperature of 35 °C and an initial pH of 6.7; ‘6’: experimental data; ‘—’: description with the adapted model of Nicolaı¨ et al. K.M. Vereecken, J.F. Van Impe / International Journal of Food Microbiology 73 (2002) 239–250248 [...]... can be seen that the adapted model of Nicolaı et al ¨ provides an accurate description of the three variables N, LaHtot and pH 5 Conclusions The main contributions of this paper can be summarised as follows The model of Nicolaı et al (1993), which consists ¨ out of two coupled differential equations describing cell growth and lactic acid production of LAB, is evaluated from a mathematical point of. .. Vereecken, J.F Van Impe / International Journal of Food Microbiology 73 (2002) 239–250 mised parameter values The specified conditions for the parameters a, b, c and d are satisfied for the estimated parameter values Fig 6 shows the cell concentration, total concentration of lactic acid and pH as a function of time Displaying the curves in this way reveals that LaHtot and pH are almost constant for a long period... LaHtot and pH Secondly, the remaining parameters are estimated on the basis of the complete data set, including the cell concentration data As such, an accurate description of N, LaHtot and pH as a function of time is obtained Future research will focus on the experimental validation of the computation method of [LaH] and pH for different growth media and different values for the initial pH Moreover, alternative... contrast to classical predictive growth models, the specific growth rate m and the specific production rate p in this model explicitly depend on the concentrations of undissociated lactic acid [LaH] and hydrogen ions [H + ], originating from the lactic acid produced by the micro-organism Transition of the exponential growth phase towards the stationary phase is described as a consequence of increasing amounts... [LaH] and pH computation are applied to an experimental case study, namely the growth of L lactis SL05 in a rich medium During the experiment, the cell concentration N, the 249 total lactic acid concentration LaHtot and the pH are monitored In a first step, the parameters which only depend on the growth medium used and the initial pH, are estimated by means of the experimentally measured values of LaHtot... amounts of [LaH] and [H + ] in the environment Next, some constraints on the parameter values, which should be taken into account for realistic model predictions, are highlighted Further, a novel, robust method for calculation of the variation in time of [LaH] and pH values, due to the lactic acid production, is introduced Next to the mathematical analysis, the model and the newly developed procedure of. .. growth of lactic acid bacteria in vacuum-packed meat Food Microbiology 10, 229 – 238 Van Impe, J.F., Nicolaı, B., Vanrolleghem, P., Spriet, J., De Moor, B., ¨ Vandewalle, J., 1994 Optimal control of the penicillin G fedbatch fermentation: an analysis of the model of Heijnen et al Optimal Control Applications and Methods 15 (1), 13 – 34 Vereecken, K.M., Van Impe, J.F., 2000 Experimental validation of a 250... cells are growing, followed by a rather sudden increase and decrease respectively when a cell concentration of approximately 107 cfu/ml is reached During these changes, the cells pass from the exponential growth phase to the stationary growth phase As mentioned in Section 3, this transition is in the model of Nicolaı et al described as a conse¨ quence of the increasing concentrations of [LaH] and [H... (FWO) as part of project G.0267.99, the Belgian Program on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister’s Office for Science, Technology and Culture, and the European Commission as part of project EU-FAIRCT97-3129 The scientific responsibility is assumed by its authors References Baranyi, J., Roberts, T .A. , 1994 A dynamic approach to predicting bacterial growth in... alternative model structures which describe both cell growth and metabolite production will be investigated Acknowledgements This research has been supported by the Research Council of the Katholieke Universiteit Leuven as part of projects OT/99/24 and COF/98/008, the Institute for the Promotion of Innovation by Science and Technology in Flanders (IWT), the Fund for Scientific Research-Flanders (FWO) as . Analysis and practical implementation of a model for combined growth and metabolite production of lactic acid bacteria Karen M. Vereecken, Jan F. Van Impe * BioTeC—Bioprocess Technology and. of Trivector). 3. Analysis and adaptation of the model of Nicolaı¨ et al. (1993) Nicolaı¨ et al. (1993) have constructed a dynamic model for the surface growth of lactic acid bacteria on vacuum-packed meats growth rate and the specific production rate to [LaH] and pH, the accuracy of latter input values is also of major im- portance. Therefore, an adequate calculation method for the actual values of [LaH]