Mathematical modelling of the growth rate and lag time for Listeria monocytogenes
Introduction
Listeria monocytogenes is a well known foodborne pathogen which has been extensively studied since the early 1980s. Numerous studies deal with the growth and survival characteristics of the micro-organism in foods. At the same time, interest in predictive microbiology increased and today several mathematical models are available to describe the effects of environmental factors on growth of micro-organisms. The aim of this work was then to use existing predictive models in order to obtain a global model describing the effect of the environment on the growth parameters of L. monocytogenes and to point out the cases where improvements of these predictive models are required.
Section snippets
Growth data for Listeria monocytogenes
Growth data for L. monocytogenes in microbiological media, dairy products, meats, liquid eggs and seafoods were taken from 74 published papers and from unpublished personal data (Table 1).
Effect of temperature, pH and water activity on μmax
The changes of μmax as a function of temperature, pH, and water activity were described by the cardinal models of Rosso (1998b):where X is temperature, pH or water activity, Xmin is the value below which no growth occurs, Xopt is the value at which μmax is equal to its optimal value μopt(X) (h−1), Xmax is the value above which no growth occurs, and n is a
Effect of growth conditions on lag time
By assuming initially that the ratio of lag time and generation time, that is the work that a cell needs to do to adapt to its environment (Robinson et al., 1998), is constant (not significantly influenced by the growth conditions) for cells in the same initial state, we have: where K is a constant depending on the physiological state of the inoculum.
Model fit
Fits were performed by linear or non-linear regression using the least squares criterion (Box et al., 1978). Estimation of parameters was carried out by minimizing the sum of the squared residuals (SSR) where SSR is defined as follows: where n is the number of data points.
The minimum SSR values were computed with the regress and nlinfit subroutines of matlab 5.2 software (The MathWorks Inc., Natick, MA, USA).
Estimation of model parameters
Data used to estimate cardinal values and MIC-values were taken in papers where the studied environmental factors showed at least three levels and when concomitant variables varied in the same manner for all the levels (complete balanced designs).
Median values of these estimations were chosen to estimate the model parameter values for the whole database since they are less sensitive to outliers than means (Delignette-Muller et al., 1995).
μmax Modelling
Calculated μmax with the global model (Eq. (6)) with the 35 parameters previously described are shown in Fig. 6. The model explains 78.0% of the variability of μmax0.5.
The accuracy factor is frequently used to estimate the average error in growth parameter estimates from models (Baranyi and Roberts, 1995, Baranyi et al., 1996, Baranyi et al., 1999, Ross, 1996, Fernández et al., 1997). This accuracy factor was defined by Baranyi et al. (1999) by the formula: where
Conclusion
The use of existing predictive models allows to explain the main variability of the growth rate of L. monocytogenes in different environmental conditions but the hypothesis of multiplicative effects of environmental factors on μmax leads to a poor fit near the limits of growth of the pathogen which are conditions met in agro-food industry. Furthermore, a great dispersion was observed for some parameter estimations. The coefficients of variation for Tmin, pHmin, aw,min, monolaurin and CO2 MICs,
Acknowledgements
We would like to thank Laurent Rosso for his helpful and critical reading of the manuscript.
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