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

International Journal of Food Microbiology 54 (2000) 171–180
www.elsevier.nl/locate/ijfoodmicro
A combined discrete–continuous model describing the lag phase of
q
Listeria monocytogenes
*
R.C. McKellar , K. Knight
Southern Crop Protection and Food Research Centre – Food Research Program, Agriculture and Agri-Food Canada,
93 Stone Road West, Guelph, Ontario N1G 5C9, Canada
Received 30 April 1999; received in revised form 9 August 1999; accepted 13 November 1999
Abstract
Food microbiologists generally use continuous sigmoidal functions such as the empirical Gompertz equation to obtain the
kinetic parameters specific growth rate (m) and lag phase duration (l) from bacterial growth curves. This approach yields
reliable information on m; however, values for l are difficult to determine accurately due, in part, to our poor understanding
of the physiological events taking place during adaptation of cells to new environments. Existing models also assume a
homogeneous population of cells, thus there is a need to develop discrete event models which can account for the behavior
of individual cells. Time to detection (t ) values were determined for Listeria monocytogenes using an automated
d
turbidimetric instrument, and used to calculate m. Mean individual cell lag times (t ) were calculated as the difference
L
between the observed td
and the theoretical value estimated using m. Variability in tL
for individual cells in replicate wells
was estimated using serial dilutions. A discrete stochastic model was applied to the individual cells, and combined with a
deterministic population-level growth model. This discrete–continuous model incorporating tL
and the variability in tL
(expressed as standard deviation; S.D.
L) predicted a reduced variability between wells with increased number of cells per
well, in agreement with experimental findings. By combining the discrete adaptation step with a continuous growth function
it was possible to generate a model which accurately described the transition from lag to exponential phase. This new model
may serve as a useful tool for describing individual cell behavior, and thus increasing our knowledge of events occurring
during the lag phase. © 2000 Elsevier Science B.V. All rights reserved.
Keywords: Lag; Listeria monocytogenes; Predictive modeling; Bioscreen; Turbidimetric; Discrete; Stochastic; Deterministic
1. Introduction
q
A preliminary account of this work was presented at the 8th
International Symposium on Microbial Ecology, Halifax, August
1998.
*Corresponding author. Tel.: 11-519-829-2400 ext 3005; fax:
11-519-829-2602.
E-mail address: mckellarr@em.agr.ca (R.C. McKellar)
Bacterial growth data is normally analyzed using
an empirical sigmoidal function such as the Gompertz
equation (Gibson et al., 1988; Willocx et al.,
1993). Useful kinetic parameters such as the maximum
specific growth rate (m) and the lag phase
duration (l) can be obtained from the Gompertz
0168-1605/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved.
PII: S0168-1605(99)00204-4

172 R.C. McKellar, K. Knight / International Journal of Food Microbiology 54 (2000) 171 –180
function; however, there are some limitations to the Some of the fundamentals of this approach have
use of this type of function. For example, it can be been discussed by McMeekin et al. (1993). These
shown mathematically that the Gompertz rate is authors and others emphasized the limits of this
always the maximum rate and occurs at an arbitrary method, the most severe of which is the fact that OD
point of inflection (Garthright, 1991, 1997). The methods are comparative only, and cannot be used to
Gompertz equation tends to overestimate m as it fits predict viable counts unless some attempt at calia
sigmoidal curve to a straight line. In addition, the l bration is made (McMeekin et al., 1993; Baranyi and
calculated with the Gompertz is always at a defined Roberts, 1995). McClure et al. (1993) used a simple
point relative to the upper and lower asymptotes quadratic equation to relate OD to viable counts.
(Garthright, 1991, 1997). Thus, empirical equations Dalgaard et al. (1994) used two equivalent methods
have a limited ability to enhance our knowledge for calibration: one in which stationary phase cells
concerning the physiological stages of bacterial were diluted to the appropriate OD, and the other in
adaptation to new environment and subsequent which samples for OD and viable count were taken
growth. during growth. Predicted generation times were
It has been suggested that connecting the behavior lower with viable count data (Dalgaard et al., 1994),
of a single cell to that of the whole population is the and this factor has been taken into account in later
next stage in developing a more mechanistic ap- studies (Miles et al., 1997). Similar methods have
proach to predictive food microbiology (Baranyi, been used to relate turbidimetric and viable count
1997). While there have been some attempts to data (Chorin et al., 1997).
develop mechanistic models for bacterial growth In some studies, the Gompertz equation was fitted
(Baranyi and Roberts, 1994, 1995; Hills and Wright, directly to OD data; however, no data was available
1994; Hills and Mackey, 1995), these models tend to
7
at below the minimum detectable OD (ca. 10 cfu/
view bacteria as a homogeneous population, and ml) thus the estimates for l and m should be
there have been few attempts to model bacterial questioned (Hudson, 1994; Hudson and Mott, 1994).
adaptation and growth on the basis of single cells. A form of calibration was achieved by relating l
Recently, a model was proposed in which the using OD measurements to that determined with
bacterial population was divided into non-growing viable counts by a regression equation (Hudson and
and growing cells (McKellar, 1997). This model was Mott, 1994). McMeekin et al. (1993) have discussed
expressed in the form of differential equations, and the correct way to fit the Gompertz function to %
the behavior of the two types of cells was modeled transmittance data, and this method has been used to
independently. Buchanan et al. (1997) have proposed calculate generation times (Neumeyer et al., 1997).
a model which takes into account the variation in Other studies have been carried out without any
adaptation (or lag) time of individual cells. Simula- apparent calibration (Huchet et al., 1995). l values
tions with this model gave rise to ‘‘traditional’’ have been estimated from OD data by extrapolation
growth curves; however, these authors did not pro- of the exponential portion of the curve back to the
vide experimental evidence for their model. More initial cell numbers (Breand et al., 1997); however,
recently, Baranyi and Pin (1999) and Baranyi (1998) this method may be inaccurate since the growth rate
have proposed a lag model based on behavior of estimated from the OD data may be lower than that
individual cells.
obtained during the period of maximum growth
Construction of models using viable count data is (McMeekin et al., 1993).
time consuming and expensive, and researchers have Interestingly, the time to detection (t
d) approach
explored other, more rapid, methods for accumulat- has not been used to any great extent. The td
for a
ing sufficient data for modeling. One of the simplest turbidimetric instrument can be defined as the time
method is the use of optical density (OD), where required for an initial measurable increase in OD.
growth can be related to the increase in turbidity of a The difference between td
for serial two-fold dilu-
bacterial culture. This method lends itself particu- tions gives the doubling time, from which the m can
larly well to automation, and a number of studies be determined (Cuppers and Smelt, 1993). The l can
have used automated turbidmetric instruments such be calculated subsequently by the difference between
as the Bioscreen (McClure et al., 1993; Huchet et al., the predicted td
based on the m, and the observed td
1995). (Cuppers and Smelt, 1993). This method was used to

R.C. McKellar, K. Knight / International Journal of Food Microbiology 54 (2000) 171 –180 173
estimate individual cell lag times (Pin and Baranyi, t (h), that is, the time required for the Bioscreen to
d
1998). Given the limitations and inherent inac- record a 0.05 increase in optical density from t .
0
curacies of the calibration method, the td
approach Duplicate wells for each dilution were used for the
would seem to be the only valid one.
preparation of a standard curve of log OD against
The purpose of the present study was to (1) obtain cfu/ml. For the determination of m, five replicate
data on the lag phase experienced by single cells of wells were used for each dilution, with 20 replicate
Listeria monocytogenes using the Bioscreen and (2) wells being used for dilutions which were close to 0
develop a discrete–continuous model which com- cfu/ml.
bines cell adaptation as a property of individual cells Viable cells were enumerated for each two-fold
(discrete activity) with a continuous model for dilution by spread plating 0.1 ml of appropriate serial
bacterial growth.
dilutions in duplicate onto tryptic soy agar (TSA;
Difco Labs.). The plates were incubated at 308C for
48 h and colonies were counted using a Quebec
Colony Counter (Model 15; American Optical, Buf-
2. Materials and methods
2.1. Strains and culture conditions
Dynamic models were created using
©
ModelMaker Version 3.0 (Cherwell Scientific Pub-
lishing, Oxford, UK).
The Gompertz and heterogeneous population
(HPM) (McKellar, 1997) models were fit to growth
data (obtained either from actual or simulated cell
®
counts) using Scientist (Micromath Scientific Soft-
ware, Salt Lake City, UT, USA). A modified Powell
algorithm was used to minimize the sum of squared
deviation between observed data and model calcula-
tions. Initial parameter estimates were obtained using
simplex optimization. Differential equations were
solved numerically by the method of Runge–Kutta,
since the software does not require analytical forms
of equations.
Prism Version 2.0 (GraphPad Software for Intuitive
Science, San Diego, CA, USA) was used to
create plots.
Listeria monocytogenes Scott A (human clinical
isolate) was obtained from the culture collection at
the Food Research Program (Guelph, Canada). The
culture was grown for 24 h at 308C in tryptic soy
broth (TSB; Difco Labs., Detroit, MI, USA). Stock
cultures were prepared in TSB plus 15% glycerol
(BDH, Toronto, Canada) and were frozen in 0.3-ml
aliquots in cyrovials at 2 258C.
The contents of one cyrovial was transferred to 10
ml of TSB, incubated for 24 h at 308C in a shaking
waterbath (New Brunswick Scientific, Edison, NJ,
USA) at 1500 rpm. The culture was transferred (1%)
to 10 ml fresh TSB and incubated under the same
conditions. The resulting culture was used as the
inoculum for experiments. API Listeria spp. Identifi-
cation Strip (BioMerieux Canada, St. Laurent,
Canada) was used to confirm the identity of the
culture.
2.2. Bioscreen growth experiments
falo, NY, USA).
2.3. Modeling
Serial two-fold dilutions of the inoculum were 3. Results
made using fresh TSB to obtain a range of dilutions
5
representing approximately 10 to 0 cfu/ml. From Kinetic parameters describing bacterial growth can
each of the two-fold dilutions, 0.35 ml was trans- be determined from turbidity data (Cuppers and
ferred to wells of a Bioscreen plate (Labsystems, Smelt, 1993). Plots of td
obtained from serial dilu-
Helsinki, Finland). The filled plates were placed in tions of the original inoculum against ln cfu/ml (Fig.
the Bioscreen (Labsystems) at an incubation temslope
1) gave straight lines, and m was calculated from the
perature of 308C. Measurements were taken using a
by Eq. (1):
wide band filter, with pre-shaking at medium intensity
for 10 s prior to OD reading; measurements were
1
m 52 ]]
taken every 4 min for 25 h. Results were reported as
Slope
(1)

174 R.C. McKellar, K. Knight / International Journal of Food Microbiology 54 (2000) 171 –180
Table 1
Kinetic parameters for Listeria monocytogenes determined using
a
the Bioscreen
Trial t m t S.D. % Growth
d L L
(n520)
A 20.35 1.04 5.86 0.783 60
B 21.32 0.876 4.12 0.814 65
a 21
t
d, Time to detection (h); m, specific growth rate (h ); t
L,
mean individual cell lag phase duration (h); S.D.
L, standard
deviation of the mean individual lag phase duration; % growth,
percent of wells showing growth.
single cell per well can be calculated from a Poisson
Fig. 1. Determination of specific growth rate (m) and lag phase
duration (t
L) for Listeria monocytogenes using time to detection
distribution:
(t ) data obtained from the Bioscreen. Experimental data (d),
2b i
d
e b
simulated data (j). P(X 5 i) 5 ]]
i!
(2)
where P(X5i) is the probability of finding i cells in
It is also possible to calculate m from cell counts a randomly chosen well, and b is the expected value
obtained by using either quadratic (McClure et al., of that cell number.
1993) or cubic (Stephens et al., 1997) calibration Using the observation that 12/20 or 60% of wells
curves to convert Bioscreen absorbance data; how- contain one or more cells, the value of b may be
ever, this method was not employed in the present calculated from the following equation:
study.
Calculation of m using a serial dilution method is
independent of the absolute numbers of cells present.
2b
P(X . 0) 5 1 2 e 5 0.6 (3)
However, it is more difficult to calculate the in- Substituting b (0.916) in Eq. (2), it is possible to
dividual cell lag phase duration (t
L). It was assumed calculate the probability of finding one (37%) or two
that the dilution giving the largest td
was equal to ln (17%) cells per well. Thus, as many as five of the 12
cfu/well50 (Fig. 1). The calculated m was used in wells showing growth could have arisen from more
the HPM to predict the time required to detect than one cell. This suggests that the S.D.
L
values
growth from a defined number of cells where the must be considered only as estimates for single cells.
6
detection limit of the Bioscreen is 3.510 cfu/well. A more direct method (such as microscopic examina-
The detection limit was confirmed by means of a tion) is needed to obtain accurate distributions of
calibration curve (data not shown). Fig. 1 shows that single cell tL
values.
©
simulated values for t underestimated the ex- The simulation software, ModelMaker , was used
d
perimental td
by an amount equivalent to t
L. Note to develop a combined discrete–continuous model
that for each dilution, tL
was constant, thus was which can account for the behavior of individual
independent of cell numbers. Replicate values of t cells, and is described in the diagram in Fig. 2. Note
L
were calculated from 20 wells by subtracting the that the various blocks in Fig. 2 are of different
simulated value for td
from the replicate experimen- shape depending on their function: compartment
tal values, and the resulting mean tL
and standard blocks are rectangular, and the values change with
deviations (S.D. ) are given in Table 1 for two trials. time according to user-defined differential equations;
L
S.D.
L
values are based on ,20 wells giving growth; variable blocks have rounded ends, and values are
in the two trials reported here 12 and 13 wells, calculated at each time interval according to userrespectively,
showed growth.
defined explicit equations; defined value blocks have
The S.D.
L
values in trial A (Table 1) are based on pointed ends, and values are assigned at t0
or at
the supposition that all 12 wells showing growth particular times during the simulation; independent
arise from a single cell. The probability of finding a
event blocks are hexagonal, and are activated at a

R.C. McKellar, K. Knight / International Journal of Food Microbiology 54 (2000) 171 –180 175
©
Fig. 2. Discrete–continuous model designed with ModelMaker to simulate td
and standard deviation of individual cell lag times (S.D.
L)
based on kinetic parameters derived from Bioscreen data.
pre-defined time; and component event blocks are variation in individual cell lag times, t
L, whereas
square, and are activated in response to other com- S.D. refers to the variation between wells observed
ponents in the model.
with any defined number of cells per well.
When the model run is initiated, the AssignLag The model described in Fig. 2 was extended to
block (an independent event block activated at t
0)
allow the simulation of a complete growth curve.
uses a random number generator based on a trun- The discrete adaptation function was retained, and
cated (positive values only) normal distribution with combined with the HPM to provide the continuous
mean tL and S.D.
L
from Table 1 to assign tL
values growth function (Fig. 4). In this model, the Adaptato
each of up to 64 cells. These values are stored in tion block moves one cell from the NonGrowing to
the defined value Triggers block. The Adaptation the Growing compartment at each of the Trigger
block reads these values, and adds a single cell to the times. This preserves the initial number of cells in
Growing compartment at each time corresponding to the model. The blocks to calculate the log of the cell
an individual cell t
L. Once in the Growing compart- numbers are provided to assist in the visualization of
ment, cells start growing immediately according to a the growth curve.
logistic equation (McKellar, 1997):
The output of this model for a total of 64 cells is
shown in Fig. 5. Note that values of the the Growing
dG G compartment are not shown where the number of
] 5 Gm S1 2 ]] D (4)
dx N max
cells was zero. These results show that the simulated
values for the NonGrowing cells decreased as the
where G is the number of cells in the Growing numbers of Growing cells increased. When the total
compartment and Nmax
is the maximum population
density.
The LogGrowing block calculates the log cfu, and
at each time increment the component event Monitor
block tests to see if the value of LogGrowing has
6
exceeded the detection limit (defined as 3.510
cfu/well). The defined value TimetoDetection block
holds the calculated t
d.
The model in Fig. 2 was used to simulate values
for td
corresponding to up to 32 cells per well. The
simulated S.D. values were derived from a total of
20 simulations for each of 1, 2, 4, 8, 16 or 32 cells
per well. Fig. 3 shows that both the simulated td
and
S.D. values are in close agreement with the ex- Fig. 3. Comparison of t
d
(s) and S.D.
L
(h) from experimental
perimental findings. Note that S.D. refers to the data (open symbols) or simulated data (closed symbols).
L

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 calcu-
lated 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

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

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
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rather than normal distributions for tL
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