Mathematical modelling of the growth rate and lag time for Listeria monocytogenes

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Abstract

Growth data for Listeria monocytogenes were collected from the literature and a global model built with existing secondary models describing independently the effects of environmental factors on the growth rate and lag time was based on these data. The growth rates calculated with this model were consistent with the published ones but the fit was poor near the limits of growth of the micro-organism. The model was also less accurate to describe the lag time. It seems then that reliable predictions of the growth rate of L. monocytogenes could be obtained in a wide range of growth conditions, but models should take into account interactions between environmental factors. Furthermore, it is necessary to better model the lag phase duration and particularly to model the effect of the history of the inoculum on the subsequent lag time.

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in microbiological media, dairy products, meats, liquid eggs and seafoods were taken from 74 published papers and from unpublished personal data (Table 1).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):μmaxopt(X)·CMn(X)CMn(X)=0,X≤Xmin(X−Xmax)·(X−Xmin)n(Xopt−Xmin)n1·(Xopt−Xmin)·(X−Xopt)−(Xopt−Xmax(n−1)·Xopt+Xmin−n·T,Xmin<X<Xmax0,X≥Xmaxwhere 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:lagTg=K 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: SSR=i=1nvalue(i)observedvalue(i)fitted2where 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: Af=expE(lnxfittedlnxobserved)2 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.

References (140)

  • C. Bégot et al.

    Recommendations for calculating growth parameters by optical density measurements

    J. Microbiol. Methods

    (1996)
  • R.R. Beumer et al.

    Growth of Listeria monocytogenes on sliced cooked meat products

    Food Microbiol.

    (1996)
  • H. Blom et al.

    Addition of 2.5% lactate and 0.25% acetate controls growth of Listeria monocytogenes in vacuum-packed, sensory-acceptable servelat sausage and cooked ham stored at 4°C

    Int. J. Food Microbiol.

    (1997)
  • S. Bréand et al.

    A model describing the relationship between lag time and mild temperature increase duration

    Int. J. Food Microbiol.

    (1997)
  • S. Bréand et al.

    A model describing the relationship between regrowth lag time and mild temperature increase for Listeria monocytogenes

    Int. J. Food Micrbiol.

    (1999)
  • N. Chen et al.

    Relationship between water activity, salts of lactic acid, and growth of Listeria monocytogenes in a meat model system

    J. Food Prot.

    (1992)
  • D.E. Conner et al.

    Growth, inhibition, and survival of Listeria monocytogenes as affected by acidic conditions

    J. Food Prot.

    (1990)
  • P. Dalgaard

    Modelling of microbial activity and prediction of shelf life for packed fresh fish

    Int. J. Food Microbiol.

    (1995)
  • P. Dalgaard et al.

    Estimation of bacterial growth rates from turbidimetric and viable count data

    Int. J. Food Microbiol.

    (1994)
  • M.L. Delignette-Muller

    Relation between the generation time and the lag time of bacterial growth kinetics

    Int. J. Food Microbiol.

    (1998)
  • M.L. Delignette-Muller et al.

    Accuracy of microbial growth predictions with square root and polynomial models

    Int. J. Food Microbiol.

    (1995)
  • F. Denis et al.

    Antibacterial activity of the lactoperoxidase system on Listeria monocytogenes in trypticase soy broth: UHT milk and French soft cheese

    J. Food Prot.

    (1989)
  • W.J. Dorsa et al.

    Low temperature growth and thermal inactivation of Listeria monocytogenes in precooked crawfish tail meat

    J. Food Prot.

    (1993)
  • L.L. Duffy et al.

    Growth of Listeria monocytogenes on vacuum-packed cooked meats: effects of pH, aw, nitrite and ascorbate

    Int. J. Food Microbiol.

    (1994)
  • G. Duffy et al.

    The effect of aeration, initial inoculum and meat microflora on the growth kinetics of Listeria monocytogenes in selective enrichment broths

    Food Microbiol.

    (1994)
  • J. Dufrenne et al.

    The effect of previous growth conditions on the lag phase time of some foodborne pathogenic micro-organisms

    Int. J. Food Microbiol.

    (1997)
  • Y.-H. Duh et al.

    Modeling the effect of temperature on the growth rate and lag time of Listeria innocua and Listeria monocytogenes

    J. Food Prot.

    (1993)
  • M.A. El-Shenawy et al.

    Sodium benzoate inhibits growth of or inactivates Listeria monocytogenes

    J. Food Prot.

    (1988)
  • M.A. El-Shenawy et al.

    Inhibition and inactivation of Listeria monocytogenes by sorbic acid

    J. Food Prot.

    (1988)
  • J.P. Erickson et al.

    Behavior of psychrotrophic pathogens Listeria monocytogenes, Yersinia enterocolitica, and Aeromonas hydrophila in commercially pasteurized eggs held at 2, 6.7 and 12.8°C

    J. Food Prot.

    (1992)
  • J.M. Farber et al.

    Predictive modelling of the growth of Listeria monocytogenes in CO2 environments

    Int. J. Food Microbiol.

    (1996)
  • P.S. Fernández et al.

    Predictive model of the effect of CO2, pH, temperature and NaCl on the growth of Listeria monocytogenes

    Int. J. Food Microbiol.

    (1997)
  • P.M. Foegeding et al.

    Heat resistance and growth of Listeria monocytogenes in liquid whole egg

    J. Food Prot.

    (1990)
  • S.M. George et al.

    Predictive models of the effect of temperature, pH and acetic and lactic acids on the growth of Listeria monocytogenes

    Int. J. Food Microbiol.

    (1996)
  • F.H. Grau et al.

    Occurence, numbers, and growth of Listeria monocytogenes on some vacuum-packaged processed meats

    J. Food Prot.

    (1992)
  • F.H. Grau et al.

    Aerobic growth of Listeria monocytogenes on beef lean and fatty tissue: equations describing the effects of temperature and pH

    J. Food Prot.

    (1993)
  • P.C. Houtsma et al.

    Modelling growth rates of Listeria innocua as a function of lactate concentration

    Int. J. Food Microbiol.

    (1994)
  • J.A. Hudson et al.

    Growth of Listeria monocytogenes, Aeromonas hydrophila and Yersinia enterocolitica in pâté and comparison with predictive models

    Int. J. Food Microbiol.

    (1993)
  • J.A. Hudson et al.

    Growth of Listeria monocytogenes, Aeromonas hydrophila and Yersinia enterocolitica on cold-smoked salmon under refrigeration and mild temperature abuse

    Food Microbiol.

    (1993)
  • J.A. Hudson et al.

    Growth of Listeria monocytogenes, Aeromonas hydrophila and Yersinia enterocolitica on cooked beef under refrigeration and mild temperature abuse

    Food Microbiol.

    (1993)
  • J.A. Hudson et al.

    Growth of Listeria monocytogenes, Aeromonas hydrophila, and Yersinia enterocolitica on vacuum and satured carbon dioxide controlled atmosphere-packaged sliced roast beef

    J. Food Prot.

    (1994)
  • P.S. Ita et al.

    Intracellular pH and survival of Listeria monocytogenes Scott A in tryptic soy broth containing acetic, lactic, citirc, and hydrochloric acids

    J. Food Prot.

    (1991)
  • C. Johansen et al.

    The combined inhibitory effect of lysozyme and low pH on growth of Listeria monocytogenes

    J. Food Prot.

    (1994)
  • D.L. Marshall et al.

    Growth of Listeria monocytogenes at 10°C in milk preincubated with selected Pseudomonads

    J. Food Prot.

    (1988)
  • P.J. McClure et al.

    Predictive modelling of growth of Listeria monocytogenes The effects on growth of NaCl, pH, storage temperature and NaNO2

    Int. J. Food Microbiol.

    (1997)
  • R.C. McKellar et al.

    Factors influencing the survival and growth of Listeria monocytogenes on the surface of Canadian retail wieners

    J. Food Prot.

    (1994)
  • R.C. McKellar et al.

    Modelling the influence of temperature on the recovery of Listeria monocytogenes from heat injury

    Food Microbiol.

    (1997)
  • A.J. Miller

    Combined water activity and solute effects on growth and survival of Listeria monocytogenes Scott A

    J. Food Prot.

    (1992)
  • D.A. Nolan et al.

    Minimal water activity levels for growth and survival of Listeria monocytogenes and Listeria innocua

    Int. J. Food Microbiol.

    (1992)
  • D.-H. Oh et al.

    Effect of pH on the minimum inhibitory concentration of monolaurin against Listeria monocytogenes

    J. Food Prot.

    (1992)
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