Description of the stochastic model for simultaneous growth of Listeria monocytogenes and LAB (stochastic LmLAB-model)

Fig. 1. User interface for the stochastic LmLAB-model.

Input variables:

The stochastic LmLAB-model includes different input variables for product characteristics, storage conditions, Listeria monocytogenes (Lm) and lactic acid bacteria (LAB) (Fig. 1). Of the 12 environmental parameters, the performance of the stochastic LmLAB model has been successfully validated for the effect of temperature, salt (NaCl), pH, phenol, acetic acid and lactic acid (Mejlholm et al. 2014).

Product characteristics and storage conditions

• Salt (NaCl), pH, CO₂, phenol (smoke components), nitrite, acetic acid, benzoic acid, citric acid, diacetate, lactic aicd and sorbic acid:  To evaluate the effect of the different product characteristics on the potential growth of L. monocytogenes and LAB it is recommended to use gamma distributions. Shape and scale parameters to be used as input for the stochastic LmLAB-model can be estimated by fitting of measured product characteristics to gamma distributions. Alternatively, shape and scale parameters can be estimated from mean values and standard deviations of the relevant product characteristics using the "Gamma distribution calculator" included as part of the stochastic LmLAB model (Fig. 1). Using this calculator the shape and scale parameters are calculated according to Equation 1 below. Example: For a mean value of 3.74 % NaCl and a standard deviation of 0.54 % NaCl the calculated shape and scale parameter will be 48.0 and 0.0780, respectively (Table 1, Fig. 2). As default model settings, gamma distributions for salt (NaCl), pH, phenol, acetic acid and lactic acid are used as input for the stochastic LmLAB-model based on measurement and fitting of product characteristics for 56 samples of vacuum-packed cold-smoked and marinated salmon collected by a Danish seafood processor during a period of 15 months (Mejlholm et al. 2014). Shape parameters and scale parameters for salt (NaCl), pH, phenol, acetic acid and lactic acid are shown in Table 1 and as examples gamma distributions for pH and phenol are shown in Fig. 3. For product characteristics not being relevant for the examined product normal distributions with mean values and standard deviations of 0 (i.e. no effect on the predicted growth of L. monocytogenes and LAB) should be chosen in order to be able to run the stochastic LmLAB-model (Table 1). Thus, for the default version of the stochastic LmLAB-model normal distributions with mean values and standard deviations of 0 has been chosen for CO₂, nitrite, benzoic acid, citric acid, diacetate and sorbic acid as these product characteristics were not relevant for the 56 samples of vacuum-packed cold-smoked and marinated salmon used to evaluate the performance of the model. 
• Storage temperature (°C): The storage temperature is assumed to follow a normal distribution. As default a mean value of 5.0 °C and a standard deviation of 1.0 °C is used for the storage temperature.
• Storage period (hours): The desired storage period in hours can be chosen from the drop-down menu (between  120 and 1200 hours ~ between 5 and 50 days). A default value of 720 hours (30 days) is used for the storage period.

Equation 1. Calculation of shape and scale parameters to be used as input for gamma distributions.

Table 1. Default settings for product characteristics and storage conditions included in the stochastic LmLAB-model.

^a In the water phase of the product

^b Gamma distribution with shape parameter α and scale parameter β

^c Normal distribution with mean value and standard deviation

Fig. 2. By pressing the gamma button it is possible to enter shape and scale parameters. The chosen distribution can also be changed e.g. from gamma to normal.

Fig. 3. Gamma distributions for pH and phenol in cold-smoked and marinated salmon (n = 56).

Listeria monocytogenes (Lm)

• Initial conc. of Lm (cfu/g): The initial cell concentration of L. monocytogenes is assumed to follow a log-normal distribution. As default a mean value of 1 cfu/g and a standard deviation of 0 cfu/g is used for the initial cell concentration of L. monocytogenes.

• Lm lag time (hours): The lag time for L. monocytogenes is assumed to follow a log-normal distribution. As default a mean value of 0 hours and a standard deviation of 0 hours is used corresponding to no lag time for L. monocytogenes. To include the lag time, input values should be obtained from the output section of the model by using the following procedure: Enter the relevant product characteristics and storage conditions into the input section of the model → go to the output section and press Calc. for "Lm lag time" → enter the calculated values for the lag time (mean value and standard deviation) into "Lm lag time" of the input section of the model. Now the model includes the lag time for L. monocytogenes when "Conc. of Lm following storage" and "Conc. of Lm following storage (with the effect of microbial interaction between Lm and LAB) are predicted.

• RLT for Lm: The lag time for L. monocytogenes is calculated using the relative lag time (RLT) concept (Eqn. 2). Based on our results we recommend that a RLT of 3.0 (default value) is used for naturally contaminated samples of cold-smoked and marinated fish products (Mejlholm et al., 2014). The desired RLT can be chosen from the drop-down menu (1.5, 3.0, 4.5 or 6.0).  

Equation 2. Calculation of lag time using the relative lag time (RLT) concept.

Lactic acid bacteria (LAB)

• Initial conc. of LAB (cfu/g): The initial cell concentration of LAB is assumed to follow a log-normal distribution. As default a mean value of 10 cfu/g and a standard deviation of 0 cfu/g is used for the initial cell concentration of LAB.
• Nmax (log cfu/g): The maximum population density of LAB during storage. The desired Nmax can be chosen from the drop-down menu (7.0, 7.5, 8.0, 8.5 or 9.0 log(cfu/g)). As default a Nmax value of 8.5 log (cfu/g) is used.

Output variables:

The stochastic LmLAB-model includes different output variables for L. monocytogenes (Lm) and lactic acid bacteria (LAB) (Fig. 1). To run the simulations the calc button besides each of the outputs should be pressed. For each simulation 10.000 Monte-Carlo repetitions are carried out.

• Conc. of Lm following storage (log cfu/g): Predicts the cell concentration of L. monocytogenes following storage.
• Conc. of Lm following storage (with the effect of microbial interaction between Lm and LAB (log cfu/g): Predicts the cell concentration of L. monocytogenes following storage including the effect of microbial interaction between the pathogen and LAB. Microbial interaction between L. monocytogenes and LAB is modelled using the classical Jameson effect (Giménez and Dalgaard, 2004).
• Lm lag time (hours): Calculates the lag time for L. monocytogenes to be used as input variable for the model if the importance of lag time on the cell concentration of the pathogen should be included (See explanation above).
• Lm Psi-value: Calculates the Psi-value for L. monocytogenes. The Psi-value provides a measure of the distance between  a given set of environemental parameters and the growth boundary of L. monocytogenes (Le Marc et al., 2002; Mejlholm and Dalgaard, 2009). The growth boundary is defined by a Psi-value of 1.0, and predicted growth and no-growth responses has Psi-values lower and higher than 1.0, respectively.
• Conc. of LAB following storage (log cfu/g): Predicts the concentration of LAB following storage.
• LAB Psi-value: Calculates the Psi-value for LAB. 

Importantly:  Number formats ending with 'm' or 'u' should be interpreted as 10^-3 or 10^-6 (Example: 7.5m = 7.5 * 10^-3 and 7.5u = 7.5 * 10^-6)

Example on how to use the stochastic LmLAB-model:

1. Input variables: Storage period → 840 hours, RLT for Lm → 3 and Nmax → 8.5 log (cfu/g) (Fig. 4). Other input variables see Table 2

Table 2. Input variables for example

^a Calculated using the Gamma distribution calculator of the stochastic LmLAB-model (See Fig. 1)

Fig. 4. Stochastic LmLAB-model with input variables (See Table 2 for details).

2. Calculate the lag time for L. monocytogenes (Lm lag time): Output → Lm lag time → press the Calc button → choose "Table" → "Statistics" for Lm lag time (Fig. 5).  

Fig. 5. Statistics on calculated lag times of Listeria monocytogenes (Select "Table" → "Statistics").

3. Enter the calculated lag time for L. monocytogenes into the input section of the model: Input → Lm lag time → enter mean and standard deviation (488 and 274.1 hours, respectively) for the lag time.
4. Predict the concentration of L. monocytogenes following storage (with the effect of microbial interaction between Lm and LAB): Output → Conc. of Lm following storage (with the effect of microbial interaction between Lm and LAB) → press the Calc button → Different model outputs are available when choosing "Table" or "Graph" (See Fig. 6 and 7 for examples). 

Fig. 6. Statistics on predicted concentrations of Listeria monocytogenes following storage. With an initial cell concentration of 0 log (cfu/g) for Listeria monocytogenes the highest predicted cell concentration following storage is 2.085 log (cfu/g) and the mean value is 0.1214 log (cfu/g) (Select "Table" → "Statistics").

Fig. 7. Predicted concentration of Listeria monocytogenes following storage for each of the 10.000 Monte-Carlo repetitions (Select "Graph" → "Sample").

5. Calculate the Psi-value for L. monocytogenes: Output → Lm Psi-value → press the Calc button → Different model outputs are available when choosing "Table" or "Graph" (See Fig. 8 and 9 for examples). 

Fig. 8. Statistics on calculated Psi-values for Listeria monocytogenes (Select "Table" → "Statistics").

Fig. 9. Cumulative probability plot of Psi-values for Listeria monocytogenes showing that approximately 12% of the 10.000 Monte-Carlo repetitions have a Psi-value higher than 1.0 (Select "Graph" → "Cumulative Probability").

References

Mejlholm, O., Bøknæs, N. and Dalgaard, P. (2014). Development and evaluation of a stochastic model for potential growth of Listeria monocytogenes in naturally contaminated lightly preserved seafood. Food Microbiol. http://dx.doi.org/10.1016/j.fm.2014.06.006

Mejlholm, O. and Dalgaard, P. (2015). Modelling the simultaneous growth of Listeria monocytogenes and lactic acid bacteria in seafood and mayonnaise-based seafood salads. Food Microbiol. 46, 1-14.

Mejlholm, O. and Dalgaard, P. (2007b). Modeling and predicting the growth of lactic acid bacteria in lightly preserved seafood and their inhibiting effect on Listeria monocytogenes. J. Food Prot. 70 (11), 2485-2497.

Mejlholm, O. and Dalgaard, P. (2009). Development and validation of an extensive growth and growth boundary model for Listeria monocytogenes in lightly preserved and ready-to-eat shrimp. J. Food Prot. 72 (10), 2132-2143.

Mejlholm, O. and Dalgaard, P. (2013). Development and validation of an extensive growth and growth boundary model for psychrotolerant Lactobacillus spp. in seafood and meat products. Int. J. Food Microbiol. 167 (2), 244-260.

Giménez, B. and Dalgaard, P. (2004). Modelling and predicting the simultaneous growth of Listeria monocytogenes and spoilage micro-organisms in cold-smoked salmon. J. Appl. Microbiol. 96 (1), 96-109.

Le Marc, Y., Huchet, V., Bourgeois, C. M., Guyonnet, J. P., Mafart, P. and Thuault, D. (2002). Modelling the growth kinetics of Listeria as a function of temperature, pH and organic acid concentration. Int. J. Food Microbiol. 73, 219-237.