A combined discreteâcontinuous model describing the lag phase of ...

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176 R.C. McKellar, K. Knight / International Journal of Food Microbiology 54 (2000) 171 –180 © Fig. 4. Discrete–continuous model designed with ModelMaker to simulate a complete bacterial growth curve. Fig. 5. Output of discrete–continuous model showing number of cells present in the NonGrowing (s) and Growing (d) compartments, and the total of both compartments (h). (McKellar, 1997). The results in Table 2 show good agreement between the simulated growth curves based on Bioscreen data, and the actual growth curves derived from plate count data. Values for m were lower with the HPM (0.918–1.09) as compared to the Gompertz (1.03–1.26), and were similar for Bioscreen (0.918–1.26) and viable count data (1.03– 1.26). The l values were larger with the Bioscreen as compared to viable counts, with the HPM giving shorter l times than the Gompertz with the exception of Trial D. Fig. 6 shows a comparison between one experimental and one simulated growth curve both fit with the Gompertz function. When discussing lag times, it should be noted that the symbol l is reserved for the population lag phase duration calculated from growth curves (either viable count or number of cells in the model was calculated, the Table 2 resulting curve represents the transition from the lag Kinetic parameters for Listeria monocytogenes determined using phase to the exponential phase. non-linear regression The ability of the discrete–continuous model to a Trial Method Gompertz HPM simulate actual growth curves was further tested by c d comparing the simulated output with growth curves m l m l derived from plate count data. Simulated growth A Bioscreen 1.26 6.55 1.09 5.86 curves were created using the values for t and S.D. B Bioscreen 1.03 4.71 0.918 4.25 L L C Counts 1.03 3.02 0.932 2.92 from Table 1 (with N0 and Nmax being set at 64 and 9 D Counts 1.10 2.22 1.02 2.40 10 cells, respectively), and were fit with the Gom- b Reference Counts 1.04 2.79 0.882 1.89 pertz and the HPM (Table 2, trials A and B). a Heterogeneous population model. Experimental growth curves at 308C were also fit b Data taken from a previous study (McKellar, 1997). with the Gompertz and the HPM (Table 2, trials C c 21 m, specific growth rate (h ). d and D). The reference trial is from a previous study l, Population lag phase duration (h).
R.C. McKellar, K. Knight / International Journal of Food Microbiology 54 (2000) 171 –180 177 Fig. 8. Replicate simulations using the discrete–continuous model Fig. 6. Comparison of simulated (s) and experimental (d) (Fig. 2) with three different random seed numbers for normal growth curves fit with the Gompertz function. (open symbols) and exponential (closed symbols) distributions. t d values were simulated for a total of 20 wells at each of 1, 2, 4, 8, 16 or 32 cells per well, with tL and S.D. L values taken from trial A (Table 1). simulated) using a non-linear curve fitting routine, while the symbol tL refers to the mean individual cell lag times determined using the Bioscreen. or an exponential (Baranyi, 1998) distribution for 20 The influence of the S.D. at constant t on the l replicate wells each containing 1, 2, 4, 8, 16 or 32 L L is shown in Fig. 7. When the S.D. is small (open cells. A different random number seed was used for L circles) the adaptation is rapid, as all the cells adapt each replicate simulation. tL and S.D. L values (nor- at approximately the same time. As the S.D. L mal distributions) were taken from trial A in Table 1, increases to a maximum (open triangles), the number and tL values for the exponential distributions were of cells adapting earlier than the majority increases. adjusted lower in order to separate the two data sets. These cells quickly dominate, and the result is a Fig. 8 shows that, with an exponential distribution, shorter l. the td values deviate from linearity when the number The influence of distribution type was also ex- of cells per well ,4. Mean individual cell td values amined. Replicate simulations were performed with (1 cell per well) varied considerably depending on the discrete–continuous model using either a normal the random number seed. In contrast, simulations using normal distributions were linear over the whole range of cell numbers, and random number seed had little apparent effect. 4. Discussion Fig. 7. Effect on S.D. L on apparent population lag phase duration (l), where S.D. L is: 0.0 (s); 0.3 (d); 0.6 (h); 0.9 (j); and 1.2 (^). A discrete–continuous model has been developed which improves our understanding of the behavior of cells during the period of adaptation generally referred to as the ‘‘lag phase’’. In this model the lag phase is described by two parameters, the mean individual cell lag time, t L, and the standard deviation of the variation between tL values, S.D. L. When these two parameters are used with m, N 0, and N max, a complete growth curve can be simulated. It should be noted that a key assumption of this model is that
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