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Scientific
Publications - Work Done by Microbiology Reader Bioscreen C
Journal of Applied Microbiology, 2001, 90 (3), 407-413
Development of a dynamic continuous-discrete-continuous
model describing the lag phase of individual bacterial cells
McKellar, R.C.
ABSTRACT
Aims: A previous model for adaptation and growth of individual
bacterial cells was not dynamic in the lag phase, and could not be used to
perform simulations of growth under non-isothermal conditions. The aim of the
present study was to advance this model by adding a continuous adaptation step,
prior to the discrete step, to form a continuous-discrete-continuous (CDC)
model.
Methods and Results: The revised model uses four parameters: N0,
intial population; Nmax, maximum population; p0, mean
initial individual cell physiological state; SD
p0, standard deviation of the distribution of individual
physiological states. A truncated normal distribution was used to generate
tables of distributions to allow fitting of the CDC model to viable count data
for Listeria monocytogenes grown at 5°C to 35°C. The p0 values
increased with increasing SD p0 and
were, on average, greater than the corresponding population physiological states
(h0); p0 and h0 were equivalent for individual
cells.
Conclusions: The CDC model has improved the ability to simulate the
behaviour of individual bacterial cells by using a physiological state parameter
and a distribution function to handle inter-cell variability. The stages of
development of this model indicate the importance of physiological state
parameters over the population lag concept, and provide a potential approach for
making growth models more mechanistic by incorporating actual physiological
events.
Significance and Impact of the Study: Individual cell behaviour is
important in modelling bacterial growth in foods. The CDC model provides a means
of improving existing growth models, and increases the value of mathematical
modelling to the food industry.
INTRODUCTION
The use of mathematical functions to
describe the growth or death of micro-organisms in food (termed 'predictive
microbiology') has expanded considerably over the last decade (McMeekin et
al. 1993). The focus has been mainly on the control of food-borne pathogens,
and concerted efforts in the UK and USA have resulted in the construction of
models based on extensive databases (Buchanan 1993; McClure et al. 1994).
The initial process in the development of growth models requires fitting of
non-linear functions to viable count data to obtain lag ( )
and specific growth rate ( ).
The most commonly used function for this purpose is the empirical Gompertz
equation (Gibson et al. 1988; Willcocx et al. 1993). However,
there are a number of problems associated with the use of this function,
which include underestimation of
(Miles et al. 1997), and the requirement for experimental data over
the whole growth range (McMeekin et al. 1993; Peleg 1997).In a
series of papers, Baranyi andco-workers (Baranyi et al. 1993; Baranyi and
Roberts 1994, 1995) introduced a non-autonomous differential equation to
model bacterial growth. This new model attributed the
to a shortage of a critical substrate, and proposed an adjustment function
to express the actual growth of the culture relative to the potential growth
(Baranyi et al. 1993). The lag was then a process of adjustment
described by the adjustment function,
 (t),
while the parameter
0 referred to the physiological state of the cells at t=t0
(Baranyi and Roberts 1994, 1995). One of the important outcomes of this
model was that the
and
are related as described in equation [1] (Baranyi and Roberts 1994).
where h0 is a more stable transformation suitable for data
fitting (Baranyi and Roberts 1994, 1995).
A heterogeneous population model (HPM) was developed, in which the
initial physiological state was embodied in a fraction of the population (G0)
which could grow without a
.
It was shown that the Baranyi parameter h0 was related to G0
by equation [2] (McKellar 1997).
The Baranyi model also defined the initial physiological state in terms
of a fraction of the population (Baranyi and Pin 1999).
It has been stated that future improvements in growth models must account
for the behaviour of individual cells (Baranyi 1997). A model has been
proposed which took into account the variation in adaptation or lag time of
individual cells (Buchanan et al. 1997). More recently, Baranyi
described a stochastic model for the lag in which the individual cell lag
times were assumed to be identically distributed independent random
variables (Baranyi 1998). The Bioscreen, an instrument designed for
detecting bacterial growth based on automated turbidimetric measurements,
was used to determine time-to-detection (td) values for cells of
Listeria monocytogenes innoculated individually into wells (McKellar
and Knight 2000). Serial twofold dilutions were used to calculate
,
and individual cell lag times (tL) and a measure of tL
variability (SDL) were determined. A discrete-continuous model
was then developed in which adaptation of individual cells was represented
as discrete events, which, when combined with a continuous logistic growth,
accurately described the transition from lag to exponential phase. In
another report, a method for the determination of tL based on td
was described (Baranyi and Pin 1999). In this study, the authors used the
ratio between varying inoculum sizes and detection level of counts to
develop an ANOVA protocol to determine
and the mean physiological state of the inoculum by minimizing the variance
ratio.
The present discrete-continuous model (McKellar and Knight 2000) has
several limitations. First, discrete events were defined as distributions of
tL, and not in terms of the initial physiological state of
individual cells. Thus, values for tL must be assigned at t=0 and
cannot be varied after the initiation of simulation. Consequently the model
is not dynamic in the population lag ().
Secondly, it will be necessary to modify the existing model to allow fitting
to experimental data. This is not possible at present due to the use of
random number generator in the earlier version of the model. Therefore, the
purpose of the present study is to make further advances to the
discrete-continuous model to allow resolution of the above limitations. |
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MATERIALS AND METHODS
Models were developed using ModelMaker© Version 3.0.3 (Cherwell
Scientific Publishing Limited, Oxford, UK). Viable count data for L.
monocytogenes at 5°C to 35°C were from a previously published study
(McKellar 1997). Data were fitted to models with ModelMaker© using Marquardt
regression with
2 as the measure of deviation.
The new model developed in this study was based on the
discrete-continuous model previously published (McKellar and Knight 2000). A
continuous adaptation step was added prior to the discrete event to form a
continuous-discrete-continuous (CDC) model as described below.
The model consists of three compartments, two representing non-growing
(NG) or growing (G) cells, and one (pi) representing an array of
values for the initial physiological states of individual cells (i=1, 2, 3, ,
N0) (Fig. 1). The solid line represents the 'flow' of cells to
the G compartment as they become adapted for growth, and the dotted line
represents the influence of the physiological state on this process.
Upon innoculation, cells are initially assigned to the NG compartment and
thus, NG(0)=N0. Since the cells have yet to adapt, there are no
cells in the G compartment and thus, G(0)=0. Cells in the NG compartment do
not grow, thus dNG/dt=0, while cells entering the G compartment start
growing immediately with rate
in response to a logistic function shown in equation [3] (McKellar 1997;
McKellar and Knight 2000)
3">where G is the number of cells in the growing compartment and Nmax
is the maximum cell concentration.
At t0, positive values for individual cell physiological
states are assigned using a truncated normal distribution:
4">where p0 is the mean individual cell physiological state,
and SDp0 is the standard deviation of p0. In this
context, physiological state can be considered equivalent to the 'work to be
done' by the cells during adaptation (Robinson et al. 1998). Thus,
the remaining 'work to be done' by each cell decreases as a function of
.
5">
At each specified time interval, the value of each pi is
tested for the relationship: pi
0. When this relationship becomes true, a discrete event is initiated which
changes the contents of the NG and G compartments as follows:
6">which represents the instantaneous adaptation of a single cell for
growth. Parameters for the complete CDC model are: p0, SDp0,
N0 and Nmax.
TABLES AND FIGURES
Table 1 Effect of
SD p0 on fitting of Listeria
monocytogenes growth data from 5°C with the continuous-di...
Table 2 Fitting
of Listeria monocytogenes growth data using the CDC model
Fig. 1 Structure
of the continuous-discrete-continuous (CDC) model describing the lag phase
and expone...
Fig. 2 Output
from a simulation using the CDC model where the dotted lines represent six
of a total of...
Fig. 3 Output
of a simulation for the adaptation and growth of a single cell using the CDC
model. h0, ...
Fig. 4 Non-linear
regression fit of the CDC model to experimental growth data for Listeria
monocytogen...
Fig. 5 Output
of a simulation with the extended CDC model showing the influence of
temperature cycling...
RESULTS
|
| The results of a sample simulation
with
=0·833
(corresponding to 30°C), p0=1·5 and SDp0=0·6 are shown
in Fig. 2. The dotted lines represent six of a total of 64 pi
values actually simulated in the model; the dashed line represents the log10
of the value in the G compartment (from Fig. 1), and the solid line
represents the log10 of NG + G (from Fig. 1). It should be noted
that p0 was selected as a positive value and thus, pi
values will decrease with time. As a working hypothesis, p0 was
defined in the CDC model as the extent to which stationary phase genes are
expressed. Thus, adaptation can be thought of as a time-dependent reduction
in the concentration of stationary phase gene products during adaptation to
a new growth environment. In the Baranyi model, the physiological state
parameter (h0) is defined as a transformation of the initial
level of a substance critical for growth which increases with time (Baranyi
and Roberts 1994). Since these two definitions appear quite different, it is
necessary to compare them.
The relationship between p0 and h0 for a single
cell is shown in Fig. 3. The value for the initial pi (or p0
when a single cell is being considered) decreases on a natural log scale to
0 at a time corresponding to the individual cell lag time (tL),
giving line 'a', at which point the cell 'reflects back' in response to the
discrete adaptation event to form an exponential growth curve (line 'b'). If
line 'a' is rotated about the x-axis, line 'c' is obtained (dotted
line). This is now reminiscent of the HPM (McKellar 1997), in which the
intercept of the line describing the time-dependent value of the G (growing)
compartment (here represented as lines 'c' and 'b' combined) with the y-axis
denoted the natural log of the initial number of growing cells (G0).
Note that when N0=1, G0 must represent < 1 cell when tL > 0
(Fig. 3). G0 is related to h0 by equation [2] and
thus, for the special case of a single cell (where ln N0=0), h0= ln G0.
It is obvious from Fig. 3 and the above argument that h0 and p0
are identical for a single cell.
It would be useful to fit the CDC model to existing experimental data.
However, the programme cannot optimize parameters using non-linear
regression when a random number generator is included in the model. It is
possible to work around this problem by removing the random number generator
and substituting a series of tables each containing N0 fixed
values drawn from a truncated normal distribution with mean=1 and SDp0
between 0·05 and 0·8. Distributions of values for p0 were then
obtained by multiplying by the tabulated values. Fitting was achieved by
fixing the SDp0 (i.e. selecting one table) and varying the p0;
this process must be essentially repeated for each value of SDp0.
Table 1 shows the effect of varying the SDp0 on the fitted
value for p0 using data for growth of L. monocytogenes at
5°C. For all fittings,
was fixed at the optimized value for this temperature which had been
determined previously (McKellar 1997). Increasing variability in p0
resulted in larger fitted p0 values. However, fitting was better
with larger values as noted by the increasing r 2 and
decreasing
2. The best fit was taken as SDp0=0·6 with the lowest
2.
Table 2 shows the fitting for all seven datasets, with the best SDp0
indicated for each dataset based on
2. Optimal values for
at each temperature were taken from a previous study (McKellar 1997). The
best SDp0 values selected ranged from 0·5 to 0·8. p0
values were calculated from each dataset using an average SDp0 of
0·7, and it was noted that the resulting p0 values were larger
with four of the datasets than the corresponding values for h0
which were calculated from earlier work using equation [2] (McKellar 1997).
Comparison of the mean values for p0 and h0 revealed
that the p0 values were generally larger and had greater
variability.
Figure 4 shows an example of the fit of the CDC model to L.
monocytogenes data at 20°C, with parameter estimates taken from Table 2.
The fitted model describes the transition from lag to exponential phase
quite well.
In order to simulate dynamic behaviour during individual cell adaptation,
was defined as a function of temperature, pH and a w. As
an example, the Gamma model (teGiffel and Zwietering 1999) was used to
describe the effect of the three factors on
.
For this simulation, values for p0 (2·22) and SDp0
(0·7) were the mean values from Table 2, and pH and a w
were fixed at 7·0 and 0·996, respectively. A temperature cycle was achieved
using equation [7]:
7">where Tav is the average temperature (7°C), R is the
temperature range (6°C) and P is the period (8 h). The resulting output
temperature as a function of simulation time is shown in Fig. 5 (broken
line).
The effect of cyclic temperature on adaptation and growth of L.
monocytogenes is also shown in Fig. 5. The pi (four of a
total of 64) are represented by dotted lines, while the log10 of
the total cell number (NG + G) is represented by the solid line. Both the
simulate pi values and growth show the influence of the varying
temperature in a dynamic manner.
DISCUSSION
|
| An earlier version of a model to
describe the adaptation and growth of individual cells was based on the mean
individual cell lag times (tL) and the variability of tL
expressed as standard deviation (SDL) obtained from Bioscreen
turbidimetric data (McKellar and Knight 2000). This model was limited in
that the individual lag times were preset at t0, thus the model
was not dynamic in the lag. The revised model described in the present study
incorporates a continuous adaptation step prior to the discrete shift from
adaptation to exponential growth, and is able to handle non-isothermal
conditions. This provides an improvement over existing dynamic models which
describe adaptation of homogeneous populations of cells (Alavi et al.
1999). The new model uses p0 and SDp0 as parameters,
so it is necessary to develop the means to estimate these values. It has
been assumed that the mean of the individual physiological states is a
function of the mean of the individual cell lag times detected by the
Bioscreen (Equation [8]).
8">
The Baranyi model also relates the mean of the individual physiological
states to the mean of the individual lag times (Baranyi and Pin 1999). This
approach presumes, however, that the rate of adaptation of each cell is
equal to
,
and does not take into account any distribution of
.
Further work is necessary to determine if the variability of
for individual cells is sufficiently great to invalidate the use of equation
[8].
Another alternative is to fit experimental viable count data with the CDC
model to estimate values for p0 and SDp0. This has
been attempted, and while reasonable estimates can be obtained, the
influence of the selected SDp0 on p0 makes it
difficult to obtain a unique solution for any single dataset. In addition,
few datasets have sufficient points over the region of transition from lag
to exponential growth and thus slight improvements in r 2
or
2 values may have little real meaning. It is also unlikely that
stochastic compartmental modelling techniques can be applied to improve
fitting, since these approaches are designed to track the movement of
'particles' between compartments separated by location or state (Tolley
et al. 1978; Godfrey 1983). In the present system, bacterial cells
cannot be directly monitored after the stochastic transition since growth
occurs, and it would be impossible to determine which cells had actually
adapted and which were the product of subsequent growth. Thus, it appears
necessary to develop the means to independently measure initial
physiological states, as has previously been been suggested (Baranyi et
al. 1995).
As a working hypothesis, it was suggested here that p0 might
be a measure of the extent to which an individual cell has expressed
stationary phase genes. Thus, during the lag phase, expression of these
genes would be down-regulated and the gene products removed. This is
analgous with the idea of h0 as the 'work to be done' by the
cells in preparation for growth, as proposed by Robinson et al. (1998).
Tentative support for this working hypothesis can be found in a recent study
by Jordan et al. (1999), working with acid tolerance in Escherichia coli
O157:H7. These workers showed that when stationary phase cells (or
exponential phase cells in which an acid tolerance response (ATR) had been
induced) were reinoculated into fresh medium at neutral pH, a lag phase was
induced which closely followed the loss of acid tolerance. Baranyi and
Roberts (1994) have defined physiological state as the concentration of a
substance critical for growth which increases during the lag phase. This
view, and the proposed definition of p0 as described above,
appear disparate. However, p0 and h0 were shown to be
equivalent for single cells. These two approaches are not incompatible, and
it is likely that adaptation during the lag phase actually involves both the
synthesis and removal of important cellular constituents.
Results presented here suggest that cells may be unable to initiate
growth until specific proteins required for survival in the stationary phase
have been removed, and removal of these proteins may reflect the change in p0.
The development of improved models will require a more complete
understanding of physiological changes occurring during the lag phase.
However, at present, our ability to monitor such physiological changes in
cell populations, or even single cells, is limited. It was recently reported
that expression of stationary phase genes in Salmonella typhimurium
under the control of rpoS can be monitored in real-time using a
luminescent construct (Thompson et al. 1999). This may provide an
opportunity for further research to attempt to link an actual physiological
process with parameters representing physiological state.
The present study has also questioned the definition and even the
relevance of
,
the population lag. The Baranyi model defines
in terms of the mean physiological state of the inoculum using equation [9]
(Baranyi 1998).
This approach does not seem to account sufficiently for individual cell
variability, which the present study and previous work (McKellar and
Knight 2000) have shown to be important, since growth of a culture will be
influenced primarily by the first cells to adapt. Thus models describing
individual cell behaviour must have an SD parameter. The CDC model does not
calculate
directly. However, simulated growth curves using independently determined
physiological state parameters can be fitted with other non-linear models
such as the Gompertz or the HPM (McKellar 1997) to obtain a value for
.
Knowledge of the value of
may be of limited use to modellers anyway if the
can be descrbed more accurately in terms of cell physiological states. In
fact, the
could be considered as an artificial phase of growth, but was traditionally
modelled since quantitiation was relatively easy using conventional
non-linear regression methods. As pointed out in an earlier work (McKellar
1997), some cells must adapt and commence exponential growth prior to the
end of the
.
Thus, the
should not be considered a 'safe' period wherein no important physiological
changes have taken place. |
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