A combined discrete–continuous model describing the lag phase of ...

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