**DOI:**10.1128/AEM.69.2.1093-1099.2003

## ABSTRACT

The use of bacteriocin-producing lactic acid bacteria for improved food fermentation processes seems promising. However, lack of fundamental knowledge about the functionality of bacteriocin-producing strains under food fermentation conditions hampers their industrial use. Predictive microbiology or a mathematical estimation of microbial behavior in food ecosystems may help to overcome this problem. In this study, a combined model was developed that was able to estimate, from a given initial situation of temperature, pH, and nutrient availability, the growth and self-inhibition dynamics of a bacteriocin-producing *Lactobacillus sakei* CTC 494 culture in (modified) MRS broth. Moreover, the drop in pH induced by lactic acid production and the bacteriocin activity toward *Listeria* as an indicator organism were modeled. Self-inhibition was due to the depletion of nutrients as well as to the production of lactic acid. Lactic acid production resulted in a pH drop, an accumulation of toxic undissociated lactic acid molecules, and a shift in the dissociation degree of the growth-inhibiting buffer components. The model was validated experimentally.

Predictive microbiology is frequently applied in the area of food microbiology to develop and apply mathematical models to simulate the responses of undesirable microorganisms to specified environmental variables (8, 10, 18, 22, 24). In the meat industry, for example, several of these models have been successfully developed in the area of risk assessment and microbial shelf-life studies, focusing on the outgrowth, toxin production, or inactivation of harmful microorganisms (1, 19).

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 [*X _{B}*] 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.

## MATERIALS AND METHODS

Microorganisms and media.*L. sakei* CTC 494, a producer of the bacteriocin sakacin K (11), and *Listeria innocua* LMG 13568, an indicator for the estimation of sakacin K activity, were maintained as described previously (13). Fermentation experiments were performed in (modified) MRS broth. Modifications of MRS broth involved changes in the total amount of the complex nutrient source (CNS), i.e. bacteriological peptone (Oxoid, Basingstoke, United Kingdom), meat extract (Lab Lemco powder; Oxoid), and yeast extract (Merck, Darmstadt, Germany).

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, pH_{0} 5.5, 1.0 [CNS]; 22°C, pH_{0} 5.8, 0.7 [CNS]; 25°C, pH_{0} 6.5, 1.0 [CNS]; 28°C, pH_{0} 6.0, 1.0 [CNS]; 28°C, pH_{0} 6.0, 2.0 [CNS]; 30°C, pH_{0} 6.4, 1.0 [CNS]; 31°C, pH_{0} 6.2, 2.5 [CNS]; and 35°C, pH_{0} 6.0, 1.0 [CNS].

Validation parameters mean square error (MSE), correlation coefficient (*r*^{2}), 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 ≤ *r*^{2} ≤ 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 λ (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:
$$mathtex$$\[d[X]/dt{=}{\mu}[X]\ {\ }{\ }if\ t{\geq}{\lambda}\]$$mathtex$$(1)

By taking into account the specific death rate *k _{d}* (per hour), meaningful at low pH values and low complex nutrient availability (results not shown), the concentration of living cells [

*X*] may be calculated with the following equation: $$mathtex$$\[d[X_{v}]/dt{=}({\mu}\ {-}k_{d})\ [X]\ {\ }{\ }if\ t{\geq}{\lambda}\]$$mathtex$$(2)

_{v}At the onset of the active growth phase, the cell population increases exponentially as μ is at its maximal value (μ_{max}). However, μ decreases considerably as the cell concentration gets denser. Hence, μ equals the maximum specific growth rate μ_{max} (per hour) multiplied by a self-inhibition function, γ_{i} (15):
$$mathtex$$\[{\mu}{=}{\mu}_{max}\ {\gamma}_{i}\]$$mathtex$$(3)

In turn, μ_{max} depends on the initial conditions that prevail in the microbial environment. Hence, the parameter μ_{max} may be defined as
$$mathtex$$\[{\mu}_{max}\ {=}\ ({\mu}_{max})_{opt}\ {\gamma}_{0}\]$$mathtex$$(4)

The value of (μ_{max})_{opt} (per hour) corresponds with the value of μ_{max} 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 γ_{0} describes the initial inhibition due to suboptimal temperature (γ_{T}) and suboptimal initial pH (γ_{pH 0}) conditions. Moreover, inhibition due to the initial presence of lactic acid (γ_{[L]0}) and of acetic acid (γ_{[Ac]0}) and citric acid (γ_{[Ci]0}), originating from the buffer components of MRS broth, has to be taken into account. In analogy with the γ concept (25), the inhibition function γ_{0} may be expressed as the combined result of the individual γ inhibition functions, presuming that no interaction effects occur among the individual inhibitory actions:
$$mathtex$$\[{\gamma}_{0}{=}\ {\gamma}_{T}\ {\gamma}_{pH_{0}}\ {\gamma}_{[L]_{0}}\ {\gamma}_{[Ac]_{0}}\ {\gamma}_{[Ci]_{0}}\]$$mathtex$$(5)

The γ functions that describe the growth effect of temperature *T* (in degrees centigrade) and initial acidity pH_{0} are cardinal functions based on the maximum, minimum, and optimum values of the studied factor (23):
$$mathtex$$\[{\gamma}_{T}{=}\frac{(T{-}T_{min})^{2}(T{-}T_{max})}{(T_{opt}{-}T_{min})[(T_{opt}{-}T_{min})(T{-}T_{opt}){-}(T_{opt}{-}T_{max})(T_{opt}{+}T_{min}{-}2T)]}\]$$mathtex$$(6)
$$mathtex$$\[{\gamma}_{pH_{0}}\ {=}\ \frac{(pH_{0}\ {-}\ pH_{min})(pH_{0}\ {-}\ pH_{max})}{(pH_{0}\ {-}\ pH_{min})(pH_{0}\ {-}\ pH_{max})\ {-}\ (pH_{0}\ {-}\ pH_{opt})^{2}}\]$$mathtex$$(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):
$$mathtex$$\[{\gamma}_{[A]}{=}(1{-}\frac{[HA]}{[HA]_{max}})^{n}\]$$mathtex$$(8)
with *n* a fitting exponent and [*HA*]_{max} the maximum value of [*HA*] that still allows growth.

(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 γ_{i} (see equation 3) accounts for the depletion of sugar (γ_{[S]}), complex nutrients (γ_{[CNS]}), and the production of lactic acid (γ_{[L]p}):
$$mathtex$$\[{\gamma}_{i}\ {=}\ {\gamma}_{[S]}\ {\gamma}_{[CNS]}\ {\gamma}_{[L]_{p}}\]$$mathtex$$(9)

Growth inhibition due to the depletion of sugar (γ_{[S]}) is given by the equation of Monod (15):
$$mathtex$$\[{\gamma}_{[S]}{=}\frac{[S]}{[S]{+}K_{S}}\]$$mathtex$$(10)
with *K _{S}* being the Monod constant for substrate

*S*(in grams per liter) and [

*S*] the residual glucose concentration (in grams per liter). The value of

*K*was set to 0.01 g/liter (15).

_{S}The inhibition function γ_{[CNS]} may be described by the following three-step function (15):
$$mathtex$$\[\ \begin{array}{ll}{\gamma}_{[CNS]}{=}1&if\ [X]{\leq}[X_{1}]\\{\gamma}_{[CNS]}{=}1{-}I_{1}\ ([X]{-}[X_{1}])&if\ [X_{1}]{<}[X]{\leq}[X_{2}]\\{\gamma}_{[CNS]}{=}(1{-}I_{1}\ ([X_{2}]{-}[X_{1}]){-}I_{2}\ ([X]{-}[X_{2}])\ &if\ [X]{>}[X_{2}]\end{array}\]$$mathtex$$(11)
with [*X*_{1}] and [*X*_{2}] being the critical biomass concentrations for nutrient inhibition (in grams of CDM per liter) and *I*_{1} and *I*_{2} the inhibition slopes (15). The values of [*X*_{1}], [*X*_{2}], *I*_{1}, and *I*_{2} were independent of the pH but were a function of the temperature and the initial complex nutrient concentration.

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: $$mathtex$$\[{\gamma}_{[L]_{p}}{=}\frac{{\gamma}_{[L]}}{{\gamma}_{[L]_{0}}}\ \frac{{\gamma}_{[Ac]}}{{\gamma}_{[Ac]_{0}}}\ \frac{{\gamma}_{[Ci]}}{{\gamma}_{[Ci]_{0}}}\ \frac{{\gamma}_{pH}}{{\gamma}_{pH_{0}}}\]$$mathtex$$(12)

Modeling of lactic acid production.The residual sugar concentration may be calculated from the biomass production with the equation of Pirt (13, 14):
$$mathtex$$\[d[S]/dt{=}{-}1/Y_{X/S}\ d[X]/dt{-}m_{S}\ [X_{v}]\]$$mathtex$$(13)
with *Y _{X/S}* being the cell yield coefficient (in grams of CDM per gram of glucose) and

*m*the cell maintenance coefficient (in grams of glucose per gram of CDM per hour). For

_{S}*L. sakei*CTC 494,

*Y*and

_{X/S}*m*were shown to be dependent on the temperature and pH (13). In analogy with the γ concept (see above), the latter parameters may be described as follows: $$mathtex$$\[Y_{X/S}{=}(Y_{X/S})_{opt}\ {\alpha}_{T}\ {\alpha}_{pH}\]$$mathtex$$(14) $$mathtex$$\[m_{S}{=}(m_{S})_{opt}\ {\beta}_{T}\ {\beta}_{pH}\]$$mathtex$$(15) where (

_{S}*Y*)

_{X/S}_{opt}and (

*m*)

_{S}_{opt}are the maximum values of

*Y*and

_{X/S}*m*, respectively, that can be obtained for

_{S}*L. sakei*CTC 494 in MRS broth. The functions α

_{pH}and β

_{pH}are isomorphic with equation 7, and the function α

_{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 β

_{T}, which describes how the maintenance coefficient increases with increasing temperatures towards a plateau value, is given by $$mathtex$$\[{\beta}_{T}{=}\frac{1}{1{+}y\ e^{{-}zT}}\]$$mathtex$$(16) where

*y*and

*z*(per degree centigrade) are two fitting coefficients.

The total lactic acid concentration (in grams per liter) may be calculated from sugar consumption as follows:
$$mathtex$$\[d[L]/dt{=}{-}Y_{L/S}\ d[S]/dt\]$$mathtex$$(17)
with *Y _{L/S}* being the yield coefficient for the production of lactic acid (in grams of lactic acid per gram of glucose). The value of

*Y*is not significantly influenced by the environment and equals 0.99 g of lactic acid per g of fermentable sugar.

_{L/S}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:
$$mathtex$$\[pH\ {=}\ \frac{pH_{i}{+}a_{1}[L_{th}]}{1{+}a_{2}[L_{th}]}\]$$mathtex$$(18)
where *a*_{1} and *a*_{2} are fit parameters (in liters per gram) and [*L*_{th}] is the theoretical lactic acid concentration needed to obtain a well-defined fermentation pH, starting from an initial pH_{i} of 6.9 (Fig. 1). In practice, this [*L*_{th}] is equal to the hypothetical lactic acid concentration [*L*_{0}] needed to reduce the pH from pH_{i} to the initial pH of the fermentation (pH_{0}) plus the amount of lactic acid produced by *L. sakei* CTC 494 during the fermentation ([*L*]).

Modeling of bacteriocin activity.The bacteriocin activity in the cell-free culture supernatant [*B*], expressed in mega-arbitrary units (MAU) per liter, is given by
$$mathtex$$\[d[B]/dt{=}k_{B}\ d[X]/dt{-}k_{inact}\ [X]\ [B]\ {\ }{\ }if\ [X]{\geq}[X_{B}]\]$$mathtex$$(19)
where *k _{B}* is the specific bacteriocin production (in MAU per gram of CDM),

*k*

_{inact}is the apparent bacteriocin inactivation rate (in liters per gram of CDM per hour), and [

*X*] is the minimum biomass concentration for bacteriocin production, below which the value of

_{B}*k*was equal to zero (in grams of CDM per liter).

_{B}In a previous study, *k _{B}* was shown to be dependent on the temperature and pH (13). Interestingly, a ceiling value, (

*k*)

_{B}_{max}, of this parameter was noticed. Moreover, the complex nutrients concentration had a considerable influence on

*k*(15). Hence,

_{B}*k*was related to the temperature, pH, and complex nutrient concentration as follows: $$mathtex$$\[\begin{array}{ll}k_{B}{=}f\ (pH)\ {\delta}_{T}\ {\delta}_{[CNS]}&if\ f(pH)\ {\delta}_{T}{\leq}(k_{B})_{max}\\{=}(k_{B})_{max}\ {\delta}_{[CNS]}&if\ f(pH)\ {\delta}_{T}{>}(k_{B})_{max}\end{array}\]$$mathtex$$(20) with

_{B}*f*(pH) being a function of the pH, δ

_{T}an inhibition function for suboptimal temperatures, and δ

_{[CNS]}an inhibition function for suboptimal complex nutrient availability. δ

_{T}is isomorphic with equation 6 and δ

_{[CNS]}is a fractal function of the complex nutrient concentration (see Results).

The parameter *k*_{inact} was a function of temperature and pH only (13):
$$mathtex$$\[k_{inact}{=}K_{0}\ e^{{-}k_{1}T}\ e^{{-}k_{2}pH}\]$$mathtex$$(21)
where *K*_{0} (in liters per gram of CDM per hour), *k*_{1} (per degree centigrade), and *k*_{2} are fitting coefficients.

## RESULTS

Estimation of model parameters. (i) Modeling of cell growth.The value of (μ_{max})_{opt} for *L. sakei* CTC 494 corresponding with optimal growth was estimated at 0.94 h^{−1} (equation 4). The γ 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).

The effect of complex nutrient availability and temperature on the nutrient depletion kinetics was quantified as follows:
$$mathtex$$\[\ \begin{array}{l}{[}X_{1}]\ =\ 0.35\ a_{T}\ [CNS]\\{[}X_{2}{]}\ =\ 0.83\ a_{T}\ [CNS]\\I_{1}\ =\ 1.44\ (a_{T}\ [CNS])^{-1}\\I_{2}\ =\ 0.14\ (a_{T}\ [CNS])^{-1}\end{array}\]$$mathtex$$(22)
where [CNS] represents the concentration of complex nutrient source, expressed as the factor by which the concentration of the complex nutrient source of standard MRS broth was multiplied, and *a _{T}* is a dimensionless temperature correction factor. Based on the individual fitting of the values of [

*X*

_{1}], [

*X*

_{2}],

*I*

_{1}, and

*I*

_{2}of the pH-stat fermentations for different temperatures in the range from 20 to 37°C, the latter was empirically modeled as a linear function: $$mathtex$$\[a_{T}{=}1{-}0.016\ (T{-}20)\]$$mathtex$$(23)

Based on the entire set of pH-stat fermentations, the specific death rate *k _{d}* (per hour) was expressed as an empirical function of the pH and the complex nutrient concentration [CNS]:
$$mathtex$$\[k_{d}{=}3\ (pH)^{{-}9}\ ([CNS])^{{-}3}\]$$mathtex$$(24)

(ii) Modeling of lactic acid production.The biokinetic parameters *Y _{X/S}* and

*m*obtained in equation 13 were reestimated according to the nutrient depletion model of Leroy and De Vuyst (15). The values of (

_{S}*Y*)

_{X/S}_{opt}and (

*m*)

_{S}_{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

*m*and

_{S}*Y*is given in Table 1.

_{X/S}(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 *a*_{1} and *a*_{2} 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 *k _{B}* 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): $$mathtex$$\[\begin{array}{ll}f(pH){=}35\ (pH\ {-}\ 4.60)\ (5.45\ {-}\ pH)&if\ pH\ {<}\ 5.37\\{=}\ 2.34&if\ pH\ {\geq}\ 5.37\end{array}\]$$mathtex$$(25)

The ceiling value of (*k _{B}*)

_{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. δ

_{[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): $$mathtex$$\[{\delta}_{[CNS]}\ {=}\ 0.27\ [CNS]/(0.10\ {+}\ 0.19\ [CNS]^{2})\]$$mathtex$$(26)

The parameter *k*_{inact} was a function of temperature and pH only, the kinetic coefficients *K*_{0}, *k*_{1}, and *k*_{2} needed to calculate *k*_{inact} (see equation 21) were estimated at 4 × 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.

The performance of the predictive model was estimated by comparing the observed experimental data with the predicted ones (Fig. 4). In Table 2, a validation overview is shown for the total data set as well as for the data of the validation experiments only. The regression coefficient for the entire data set was high for μ_{max} (0.974) and [*X*]_{max} (0.957), but somewhat lower for [*B*]_{max} (0.920). The bias factor was close to 1 for μ_{max}, [*X*]_{max}, and [*B*]_{max}, indicating that the observed values were, on an average, not systematically lower or higher than the predicted values. The accuracy factors for μ_{max}, [*X*]_{max}, and [*B*]_{max} indicated that the observed and predicted values for the entire data set differed, on average, by 5, 10, and 17%, respectively. The low accuracy of bacteriocin activity predictions was inherent to the twofold dilution method that was used to measure bacteriocin activity levels. At higher activity levels, uncertainty about the bacteriocin activity increased.

## DISCUSSION

Predictive microbiology is a field of increasing importance. Predictive models help us to understand when foods become spoiled or what risk is associated with certain food products for outbreaks of food-borne pathogens or food poisoning. Alternatively, predictive models will be indispensable to estimate the beneficial effects, such as bacteriocin production, exopolysaccharide formation, aroma development, and probiotic effects, of deliberately added starter cultures or cocultures in both fermented foods (e*.*g*.*, cheese and sausage) and nonfermented food products (e*.*g*.*, modified-atmosphere-packaged cooked meat products). This will be of particular importance in view of the food-health relationship (natural and healthy products, functional foods, etc*.*) put forward by both the consumer and the food processor.

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 μ_{max} 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 μ_{max} 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.

## ACKNOWLEDGMENTS

We acknowledge financial support from the Research Council of the Vrije Universiteit Brussel, the Fund for Scientific Research—Flanders (FWO), the Institute for the Promotion of Innovation by Science and Technology in Flanders (IWT), in particular the STWW project “Functionality of novel starter cultures in traditional fermentation processes,” and the European Commission (grant FAIR-CT97-3227). F.L. was supported by a grant from the IWT (Ph.D. bursary) and the FWO (postdoctoral fellowship). *L. sakei* CTC 494 was kindly provided by M. Hugas (Institut de Recerca i Tecnología Agroalimentáries, Centre de Tecnología de la Carn, Monells, Spain).

## FOOTNOTES

- Received 16 July 2002.
- Accepted 20 November 2002.

- American Society for Microbiology