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

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 nei-
ther 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