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178 R.C. McKellar, K. Knight / International Journal of Food Microbiology 54 (2000) 171 –180 the tL is independent of cell number. The substance limited fitting of the discrete–continuous model to of the present study was presented previously (McK- experimental data, and will form the basis of a later ellar, 1998). publication. The present study also reports, for the first time, Baranyi and Pin (1999) have also recently prothe inclusion of a discrete step into the modeling of posed a method for calculating l and m from t d. bacterial growth; all other published models are Their method is based on the biological interpretabased on continuous functions only. This improve- tion of the initial physiological state of the cells, ment is critical to the modeling of single cell where the suitability for growth is represented by a behavior during the adaptation period prior to initia- fraction of the initial cell population. This interpretation of growth, and is facilitated by the use of an tion is similar to the one suggested by McKellar object-oriented programing environment such as (1997) who attributed the potential for growth to a ® ModelMaker . Individual-based models (IBMs) sub-population of the inoculum. The Baranyi and Pin have been used extensively in ecological modeling approach uses an analysis of variance (ANOVA) situations (Grimm, 1999; Lomnicki, 1999), but have method to deal with variability of low cell populayet to be applied in food microbiology. Providing a tions to estimate a value for m. Values for tL are dynamic environment for the simulation of bacterial calculated using the m and the physiological state of growth based on the use of differential rather than the inoculum. In the present study, values for m are explicit equations is also considered important for estimated using a wider range of dilutions than the future development of bacterial growth models reported by Baranyi and Pin, thus minimizing the (Baranyi, 1997). Common software packages which influence of higher variance. are generally used for non-linear regression do not Buchanan et al. (1997) describe the lag phase by have this capability, thus the use of software such as the following equation: ® ModelMaker leads to the development of more tLag 5 ta 1 t m complex models incorporating the multiple steps (5) involved in adaptation and growth. where ta is the time required for the cells to adapt to A theoretical model which accounts for the be- their new environment, and tm is the generation time. havior of individual cells has been suggested by The present model assumes that growth starts imme- Buchanan et al. (1997), who were the first to propose diately after the adaptation step, thus t (this study) L that the transition between lag and exponential is equivalent to t a. phases resulted from biological variation among Baranyi (1998) has recently compared stochastic and deterministic concepts of the lag phase, and has suggested that the l is always less than the t L. This individual cells. These workers provided a theoretical basis for describing the lag phase in terms of individual cells; however, they proposed a simpler, seems reasonable, since increased S.D. L at constant three-phase model for general use which did not tL resulted in a shorter l in the present study (Fig. 7) account for inter-cell variation. The present model and also in the @RISK simulations reported by builds on this foundation by the addition of tur- Buchanan et al. (1997). It is intuitively obvious that bidimetric data which provides evidence for indi- l can only be equal to tL in the special case where vidual cell behavior, and incorporates this variability the cells all adapt simultaneously (e.g., S.D. L50). In into the model as a distinct parameter (S.D. L). the present study using simulated growth curves, l Buchanan et al. (1997) used the risk analysis soft- was greater than tL when determined by the Gom- ware, @RISKE, to simulate the variability between pertz function, and identical to tL in one of two trials cells, but fit only the three-phase linear model to using the HPM. This may be due to the inherent error associated with estimations of l from fitting data with non-linear regression functions. It is also worth mentioning that neither the Gompertz nor the HPM are intended for fitting data derived from distributions of individual cell properties, since neither of these models can account for changes in curvature between lag and exponential phases under experimental data. Since the present model contains a random number generator, it was not possible to fit experimental data using the optimization algorithms ® in ModelMaker , thus optimization must be performed manually. It will be possible, however, to construct tables of distributions with varied S.D. L which can be read into the model. This will allow
R.C. McKellar, K. Knight / International Journal of Food Microbiology 54 (2000) 171 –180 179 the control of the S.D. L parameter. An example of possibility that some cells do not grow has not been this is evident in Fig. 6; the Gompertz function fit to considered. Thus the S.D. L must be considered an a discrete–continuous model simulation shows sys- estimate of the value for single cell variability. In tematic deviations. Systematic differences between l spite of this limitation, the discrete–continuous predicted by several models including the three- model based on estimated tL and S.D. L values gives phase linear model of Buchanan and the Baranyi a good fit to the experimental data. More accurate model have been reported (Buchanan et al., 1997). estimates of single cell variance might be obtained These observations may even suggest that once using improved methods of single cell analysis (e.g., individual cell behavior can be modeled accurately, microscopy). the traditional concept of population lag (l) as a modeling parameter will be of limited further value to predictive microbiology. It is difficult to compare the discrete–continuous Acknowledgements model with the Baranyi model since the latter does The authors would like to thank J. Baranyi for not include a parameter describing the influence of helpful criticism of the manuscript. variation in tL on l. Baranyi (1998) has suggested that the mean population lag [l(N)] increases with lower inoculum, assuming tL values are exponentially distributed. In the present study a normal distribution References was assumed for t L, and with that there was no evidence to support an increased t at lower Baranyi, J., 1997. Simple is good as long as it is enough. Food L Microbiol. 14, 189–192. inocula using either simulations or experimental Baranyi, J., 1998. Comparison of stochastic and deterministic results. Additional simulations using exponential concepts of bacterial lag. J. Theor. Biol. 192, 403–408. rather than normal distributions for tL resulted in Baranyi, J., Pin, C., 1999. Estimating bacterial growth parameters predictions in which t values deviated from linearid by means of detection times. Appl. Environ. Microbiol. 65, ty, and changed depending on the random number 732–736. Baranyi, J., Roberts, T.A., 1994. A dynamic approach to predictseed. These simulations support the suggestion that ing bacterial growth in food. Int. J. Food Microbiol. 23, tL values are normally distributed. Further experi- 277–294. ments are required to establish the correct distribu- Baranyi, J., Roberts, T.A., 1995. Mathematics of predictive food tion. microbiology. Int. J. Food Microbiol. 26, 199–218. It must be emphasized that while the t decreases Breand, S., Fardel, G., Flandrois, J.P., Rosso, L., Tomassone, R., d 1997. A model describing the relationship between lag time with increasing cell number, the tL remains constant and mild temperature increase duration. Int. J. Food Microbiol. (Fig. 1). Similarly, S.D. L is constant, and the appar- 38, 157–167. ent decrease in variability between wells with in- Buchanan, R.L., Whiting, R.C., Damert, W.C., 1997. When is creasing cell numbers per well is a consequence of simple good enough: a comparison of the Gompertz, Baranyi, the majority of the cells having a t close to the and three-phase linear models for fitting bacterial growth L curves. Food Microbiol. 14, 313–326. mean. As numbers of cells per well increase, the Chorin, E., Thuault, D., Cleret, J.J., Bourgeois, C.M., 1997. observed td for each well becomes less dependent on Modelling Bacillus cereus growth. Int. J. Food Microbiol. 38, the td for individual cells, and starts to reflect the 229–234. mean t value. Cuppers, H.G.A.M., Smelt, J.P.P.M., 1993. Time to turbidity d There are some limitations to the use of the measurement as a tool for modeling spoilage by Lactobacillus. J. Ind. Microbiol. 12, 168–171. Bioscreen for indirectly estimating single cell be- Dalgaard, P., Ross, T., Kamperman, L., Neumeyer, K., McMeekin, havior. It is difficult to clearly identify the dilution T.A., 1994. Estimation of bacterial growth rates from turcorresponding to single cells; it is assumed to be the bidimetric and viable count data. Int. J. Food Microbiol. 23, dilution giving the greatest average t . In addition, 391–404. d assuming that cells follow a Poisson distribution, Garthright, W.E., 1991. Refinements in the prediction of microbial growth curves. Food Microbiol. 8, 239–248. some wells which are assumed to have single cells Garthright, W.E., 1997. The three-phase linear model of bacterial might have two or more cells. Wells which show no growth: a response. Food Microbiol. 14, 193–195. growth are assumed to have no cells; however, the Gibson, A.M., Bratchell, N., Roberts, T.A., 1988. Predicting
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