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Applied and Environmental Microbiology, February 2003, p. 1093-1099, Vol. 69, No. 2
0099-2240/03/$08.00+0 DOI: 10.1128/AEM.69.2.1093-1099.2003
Copyright © 2003, American Society for Microbiology. All Rights Reserved.
Research Group of Industrial Microbiology, Fermentation Technology and Downstream Processing, Department of Applied Biological Sciences, Vrije Universiteit Brussel, B-1050 Brussels, Belgium
Received 16 July 2002/ Accepted 20 November 2002
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Recently, there has also been interest in the modeling of beneficial microorganisms deliberately added to food to produce a desired effect. For instance, modeling of the functionality of bacteriocin-producing lactic acid bacteria seems promising for the prediction of bacteriocin bioactivity in foods (12). Bacteriocins are, in general, small peptides or proteins with an antibacterial mode of action towards strains that are closely related to the producer organism, often encompassing spoilage bacteria and food-borne pathogens (6). Bacteriocin-producing strains may be applied as starter cultures or cocultures in the food fermentation industry to obtain more competitive strains and to reduce the risk of the outgrowth of such undesirable bacteria (5). However, some doubt about their industrial application as novel functional starter cultures remains. Strains that display bacteriocin activity under laboratory conditions do not necessarily perform well once they are applied in the food under fermentation conditions (20). Modeling may help to clarify how specific conditions that prevail in the food environment during fermentation influence the performance of bacteriocin-producing starters (16).
As an example, Lactobacillus sakei CTC 494 is of particular interest as a functional bacteriocin-producing starter for sausage fermentation (2, 11). In previous studies, the effects of constant temperature and pH (13), the presence of salt and nitrite (14), and the availability of complex nutrients (15) on the in vitro functionality of L. sakei CTC 494 in (modified) MRS broth were investigated through a modeling approach. Although MRS medium is not a perfect meat simulation medium (4), it permits the study in vitro of the effect of some typical sausage fermentation conditions in a meat peptide environment.
In this study, data from earlier fermentation experiments with L. sakei CTC 494 at constant pH and temperature presented by Leroy and De Vuyst (13) were remodeled using the nutrient depletion model of Leroy and De Vuyst (15). The latter model leads to more accurate fitting of the fermentation data. Moreover, a minimum cell concentration for bacteriocin production [XB] was introduced to take into account induction of bacteriocin production (15). Subsequently, the remodeled biokinetic parameters were used to develop a combined model that is able to predict the effect of temperature, initial pH, and initial nutrient concentration on cell growth, sugar metabolism, and bacteriocin activity of L. sakei CTC 494 in (modified) MRS broth. These factors will further enable the prediction of microbial behavior under typical sausage fermentation conditions.
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Fermentation experiments and sampling.
Fermentations were carried out in a computer-controlled 15-liter laboratory fermentor (BiostatC, B. Braun Biotech International, Melsungen, Germany) containing 10 liters of (modified) MRS broth. Preparation of the fermentor, building of the inoculum, and on-line control of the fermentation process (temperature, pH, and agitation) were performed as described previously (13). Determinations of biomass concentration [X], total lactic acid concentration [L], residual glucose concentration [S], and bacteriocin activity in the cell-free culture supernatant [B] were carried out as described elsewhere (13). Summarizing, biomass (as cell dry mass [CDM]) was determined by gravimetry after membrane filtration, lactic acid and glucose were determined by high-pressure liquid chromatography, and bacteriocin activity was estimated by a twofold critical dilution method. Optical density at 600 nm (Uvikon 923; Kontron Instruments, Milan, Italy) was measured and calibrated against CDM measurements to obtain additional CDM data (15).
Experiments on the 100-ml scale were performed to investigate the effect of temperature and pH and of lactic acid (Merck), acetic acid (Merck), and citric acid (as triammonium citrate; BDM Laboratory Supplies, Poole, United Kingdom) on growth rate inhibition.
Titration of MRS broth.
The pH drop caused by acidification due to growth was predicted by establishing an empirical relationship between the lactic acid concentration and the pH of MRS broth. Lactic acid (Merck) was gradually added to 1 liter of MRS broth with an initial pH of 6.95, and the drop in pH was monitored with a pH meter (pH-526; WTW Measurement Systems, Ft. Myers, Fla.).
Modeling procedure.
In a first step, maximum specific growth rates were determined on a 10-liter as well as 100-ml scale for different conditions of temperature and pH and at different concentrations of lactic, acetic, and citric acids. Determinations were based on linear regression of optical density measurements during the exponential part of the growth curves. The latter experiments permitted us to estimate the parameters of the inhibition functions (equations 6 to 8; see below) using Microsoft Excel (version 97). The concentration of undissociated acid was calculated from the total acid concentration with the equation of Henderson-Hasselbalch.
Next, for each 10-liter fermentation experiment, the equations 1, 2, 13, 17, and 19 of the model (see below) were integrated with the Euler integration technique using Microsoft Excel. The parameters needed for the modeling were estimated by manual adjustment until the best fit was obtained. The lag phase was modeled as a Heaviside function, which forces the specific growth rate to zero during the duration of this phase (3).
In this way, a combined predictive model was built from a set of 16 fermentations at constant pH, including the fermentations presented in previous studies (13, 15). The validation was based on eight independent fermentations performed under a free pH course which were not included in the construction of the model, namely: 20°C, pH0 5.5, 1.0 [CNS]; 22°C, pH0 5.8, 0.7 [CNS]; 25°C, pH0 6.5, 1.0 [CNS]; 28°C, pH0 6.0, 1.0 [CNS]; 28°C, pH0 6.0, 2.0 [CNS]; 30°C, pH0 6.4, 1.0 [CNS]; 31°C, pH0 6.2, 2.5 [CNS]; and 35°C, pH0 6.0, 1.0 [CNS].
Validation parameters mean square error (MSE), correlation coefficient (r2), bias factor, and accuracy factor were calculated as described elsewhere (21, 25). Briefly, the MSE is a measure of variability remaining, mainly due to systematic errors and biological variability. Hence, the lower the MSE, the better the adequacy of the model to describe the data. On the other hand, the higher the value of the regression coefficient (0
r2
1), the better the prediction by the model. The bias factor indicates the structural deviation of a model whereas the accuracy factor indicates how close, on average, predictions are to observations.
Modeling of cell growth. (i) Growth equation.
After the lag phase
(in hours) has finished and the cells have adapted to their new environment, the production of biomass [X] (in grams of CDM per liter) as a function of time t (in hours) is generally related to the specific growth rate µ (per hour) as follows:
![]() | (1) |
By taking into account the specific death rate kd (per hour), meaningful at low pH values and low complex nutrient availability (results not shown), the concentration of living cells [Xv] may be calculated with the following equation:
![]() | (2) |
At the onset of the active growth phase, the cell population increases exponentially as µ is at its maximal value (µmax). However, µ decreases considerably as the cell concentration gets denser. Hence, µ equals the maximum specific growth rate µmax (per hour) multiplied by a self-inhibition function,
i (15):
![]() | (3) |
In turn, µmax depends on the initial conditions that prevail in the microbial environment. Hence, the parameter µmax may be defined as
![]() | (4) |
The value of (µmax)opt (per hour) corresponds with the value of µmax obtained in MRS broth under optimal conditions of temperature and pH and in the absence of inhibitory substances such as lactic acid or buffer components. It was presumed that nutrients were not limiting in the earliest stages of growth.
(ii) Initial growth inhibition.
The inhibition function
0 describes the initial inhibition due to suboptimal temperature (
T) and suboptimal initial pH (
pH 0) conditions. Moreover, inhibition due to the initial presence of lactic acid (
[L]0) and of acetic acid (
[Ac]0) and citric acid (
[Ci]0), originating from the buffer components of MRS broth, has to be taken into account. In analogy with the
concept (25), the inhibition function
0 may be expressed as the combined result of the individual
inhibition functions, presuming that no interaction effects occur among the individual inhibitory actions:
![]() | (5) |
The
functions that describe the growth effect of temperature T (in degrees centigrade) and initial acidity pH0 are cardinal functions based on the maximum, minimum, and optimum values of the studied factor (23):
![]() | (6) |
![]() | (7) |
The inhibition due to the presence of organic acid A, i.e., lactic (L), acetic (Ac), or citric (Ci) acid, is related to the concentration of undissociated organic acid ([HA], in grams per liter) as follows (15):
![]() | (8) |
(iii) Self-inhibition.
Previously, a model was constructed to simulate self-inhibition of a growing L. sakei CTC 494 culture under pH-stat conditions (15). In this paper, the self-inhibition model was extended to take into account the effect of free pH. The self-inhibition function
i (see equation 3) accounts for the depletion of sugar (
[S]), complex nutrients (
[CNS]), and the production of lactic acid (
[L]p):
![]() | (9) |
Growth inhibition due to the depletion of sugar (
[S]) is given by the equation of Monod (15):
![]() | (10) |
The inhibition function
[CNS] may be described by the following three-step function (15):
![]() | (11) |
The inhibition due to lactic acid production was the result of the toxic effect of the undissociated lactic acid molecules produced, the drop of the pH, and the consequent effect on the dissociation degree of the weak acids:
![]() | (12) |
Modeling of lactic acid production.
The residual sugar concentration may be calculated from the biomass production with the equation of Pirt (13, 14):
![]() | (13) |
concept (see above), the latter parameters may be described as follows:
![]() | (14) |
![]() | (15) |
pH and ßpH are isomorphic with equation 7, and the function
T is isomorphic with equation 6. The required minimum and maximum values of temperature and pH are set equal to the limits for cell growth as determined with equations 6 and 7, respectively. The function ßT, which describes how the maintenance coefficient increases with increasing temperatures towards a plateau value, is given by
![]() | (16) |
The total lactic acid concentration (in grams per liter) may be calculated from sugar consumption as follows:
![]() | (17) |
Modeling of pH drop.
After experimentally establishing the relationship between lactic acid concentration and pH, the pH drop in MRS broth induced by lactic acid production may be modeled with the following equation:
![]() | (18) |
![]() View larger version (16K): [in a new window] |
FIG. 1. Titration of MRS broth with externally added lactic acid. [L0] is the hypothetical lactic acid concentration needed to obtain an initial pH0, [L] is the amount of lactic acid produced by L. sakei CTC 494, and [Lth] is the theoretical lactic acid concentration needed to obtain a well-defined fermentation pH. Symbols represent experimental data. The solid line is according to the model (equation 18).
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![]() | (19) |
In a previous study, kB was shown to be dependent on the temperature and pH (13). Interestingly, a ceiling value, (kB)max, of this parameter was noticed. Moreover, the complex nutrients concentration had a considerable influence on kB (15). Hence, kB was related to the temperature, pH, and complex nutrient concentration as follows:
![]() | (20) |
T an inhibition function for suboptimal temperatures, and
[CNS] an inhibition function for suboptimal complex nutrient availability.
T is isomorphic with equation 6 and
[CNS] is a fractal function of the complex nutrient concentration (see Results).
The parameter kinact was a function of temperature and pH only (13):
![]() | (21) |
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functions for temperature, pH, and lactic acid inhibition are displayed in Fig. 2. The values of the parameters needed in equations 6 to 8 are summarized in Table 1. From the growth inhibition studies in function of the organic acid concentrations, it was found that the most toxic acid, in its undissociated form, was citric acid ([HCi]max = 0.4 g per liter), followed by lactic acid ([HL]max = 2.8 g per liter), and acetic acid ([HAc]max = 7.1 g per liter).
![]() View larger version (13K): [in a new window] |
FIG. 2. Influence of temperature (a), pH (b), and undissociated lactic acid concentration (c) on growth inhibition of L. sakei CTC 494 in MRS broth. Solid symbols indicate experiments on the 10-liter scale, and open symbols indicate experiments on the 100-ml scale. Error bars indicate standard deviations. Solid lines are according to the model (equations 6 to 8).
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View this table: [in a new window] |
TABLE 1. Kinetic data on the parameters for cell growth and sugar consumption (µmax, YX/S, and mS) concerning the influence of pH, temperature, and the presence of undissociated organic acids for L. sakei CTC 494 in modified MRS broth
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![]() | (22) |
![]() | (23) |
Based on the entire set of pH-stat fermentations, the specific death rate kd (per hour) was expressed as an empirical function of the pH and the complex nutrient concentration [CNS]:
![]() | (24) |
(ii) Modeling of lactic acid production.
The biokinetic parameters YX/S and mS obtained in equation 13 were reestimated according to the nutrient depletion model of Leroy and De Vuyst (15). The values of (YX/S)opt and (mS)opt were set at 0.46 g of CDM (g of glucose)-1 and 0.95 g of glucose (g of CDM)-1 h-1, respectively. An overview of the coefficients needed to calculate mS and YX/S is given in Table 1.
(iii) Modeling of pH drop.
An experimental relationship between lactic acid concentration and pH was established, indicating that 12 g of lactic acid per liter was required to achieve a final pH of 4.0. The fit parameters a1 and a2 for the modeling of the pH drop were estimated experimentally at 0.45 and 0.18 liter g-1, respectively (Fig. 1).
(iv) Modeling of bacteriocin activity.
The specific bacteriocin production kB was related to the temperature, pH, and complex nutrient concentration according to equation 20. The function f(pH), estimated for pH-stat experiments at different pH values (pH 4.5 to 6.5) but at the same temperature (30°C), displayed a discontinuity at pH 5.37 (13):
![]() | (25) |
The ceiling value of (kB)max was estimated at 2.6 MAU (g of CDM)-1. The minimum, optimum, and maximum temperature values for bacteriocin production were estimated to be 3, 24, and 32°C, respectively.
[CNS] is the normalized inhibition function for the description of the influence of the complex nutrient source on the specific bacteriocin production. The latter factor was determined at 25°C and a controlled pH of 5.5 as follows (Fig. 2):
![]() | (26) |
The parameter kinact was a function of temperature and pH only, the kinetic coefficients K0, k1, and k2 needed to calculate kinact (see equation 21) were estimated at 4 x 10-7 liters (g of CDM)-1 h-1, 0.115°C-1, and 1.427, respectively.
Validation of the model.
A series of fermentations were performed to test the validity of the proposed model (Fig. 3). In a first step, simulations were made to predict cell growth, sugar consumption, lactic acid production, pH drop, and bacteriocin activity for eight different sets of fermentation conditions (i.e., different conditions of temperature, initial pH, and complex nutrient concentration). Next, the fermentations were performed experimentally to monitor the relevant experimental data.
![]() View larger version (37K): [in a new window] |
FIG. 3. Prediction of cell growth ( , or when calculated from OD measurements), glucose consumption ( ), lactic acid production ( ), pH drop ( ), and bacteriocin activity ( ) by L. sakei CTC 494 in (modified) MRS broth at 20°C, pH0 5.5, 1.0 [CNS] (a); 22°C, pH0 5.8, 0.7 [CNS] (b); 25°C, pH0 6.5, 1.0 [CNS] (c); 28°C, pH0 6.0, 1.0 [CNS] (d); 28°C, pH0 6.0, 2.0 [CNS] (e); 30°C, pH0 6.4, 1.0 [CNS] (f); 31°C, pH0 6.2, 2.5 [CNS] (g); and 35°C, pH0 6.0, 1.0 [CNS] (h). Symbols represent experimental data; solid lines are predicted by the model. dag, decagram.
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FIG. 4. Correlation between experimental and predictive values of µmax, the maximum biomass concentration [X]max, and the maximum bacteriocin activity [B]max. Circles represent the entire data set, and squares represent the validation experiments only.
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View this table: [in a new window] |
TABLE 2. Validation data for µmax, the maximum biomass concentration [X]max, and the maximum bacteriocin activity in the cell-free culture supernatant [B]max of the entire data set and of the validation data set only
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Yet, information on the prediction of beneficial properties of desirable and healthy microbes in food ecosystems is scarce. Moreover, studies that report on beneficial microbes frequently make use of empirical models based on polynomial regression and surface response methodologies (7, 8). The main disadvantages are the difficult physiological interpretation of the parameters used, the nondynamic and nonindependent behavior of the mathematical relationships established, and the nonmechanistic behavior of the equations used.
The modeling approach presented in this study permitted to obtain a reliable simulation of the functional behavior of L. sakei CTC 494 as a potential novel sausage starter culture under selected conditions of pH, temperature, and complex nutrient availability. The model for cell growth is significantly more complex than classical growth models such as the logistic equation or the Gompertz equation (26). However, it offers considerable advantages over classical growth models, such as the ability to describe growth under free pH and a more mechanistic insight into the effect of the environment on growth inhibition. The combined model makes a clear distinction between the individual effects of each environmental factor that has been included in the model and is able to quantify these effects separately. Moreover, as has been demonstrated previously with the nutrient depletion model (15), a closer agreement with the experimental data was observed when using this extended nutrient depletion model, compared to fitting with the logistic model used by Leroy and De Vuyst (13, 14). As a result, some shifts in parameterization were observed. For instance, the modeled minimum, optimum, and maximum values of pH and temperature for cell growth were slightly different from the ones found by Leroy and De Vuyst (13). This was partially due to the extension of the data set but mainly to the substitution of the original logistic growth equation with the nutrient depletion model, which resulted in a recalculation of the µmax values (15). They were, however, still in agreement with the physiologically determined growth limits (i.e., 3.9 < pH < 8.6 and 0 < T < 44°C).
The predictive performance of the model was evaluated. The predictions for the values of µmax were excellent, whereas the predictions for [X]max and [B]max were still satisfactory. Moreover, the predicted lines of cell growth, sugar consumption, lactic acid production, pH drop, and bacteriocin activity were in good agreement with the experimental data.
The validated predictive model permits us to simulate bacterial behavior under a defined set of environmental conditions. In this way, the effect of changes in fermentation technology may be studied. For example, a temperature between 20 and 30°C is required for good bacteriocin production by L. sakei CTC 494. At higher temperatures, bacteriocin activity levels would decrease considerably. In contrast, cell growth and acidification are faster at temperatures above 30°C.
Despite the fact that a fermentor is a very different environment than a real sausage, it is assumed that bacterial growth and bacteriocin activity in a sausage take place in the water phase or at least near the surface of the meat particles. Also, a fermentor is a powerful device to study the kinetics of microbial behavior, making use of a simulation medium. Although MRS broth is distinct from a real food environment, the information obtained in this paper may be very useful for the selection of suitable starter cultures for a particular fermentation process and is a first step in the optimization of food fermentation processes and technology as well. Some other factors that will have to be taken into account are the presence of spices in the sausage batter, interactions with the background microflora, diffusion limitations in the meat matrix, substrate and oxygen gradients, and bacteriocin activity losses due to adsorption to fat and meat particles.
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