Download Individual-based modelling of growth and migration of Salmonella

International Journal of Food Microbiology 100 (2005) 323 – 333
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Individual-based modelling of growth and migration of
Salmonella enteritidis in hens’ eggs
K. Grijspeerdta,*, J.-U. Kreftb, W. Messensa
a
Ministry of the Flemish Community, Centre of Agricultural Research Gent, Department of Animal Product Quality,
Brusselsesteenweg 370, 9090 Melle, Belgium
b
Botanisches Institut der Universität Bonn, Abteilung Theoretische Biologie, Kirschallee 1, 53115 Bonn, Germany
Received 24 September 2004; accepted 6 October 2004
Abstract
An individual-based model (IbM) was developed to describe the growth and migration of Salmonella enteritidis in hens’
eggs. The Bacteria Simulator (BacSim) environment was used to implement the model; the bacteria are represented by spheres
that grow by nutrient uptake and divide in two daughter cells upon exceeding a certain threshold volume. Motility of the
Salmonella bacteria was described by a run and tumble mechanism. For the sake of simplicity, the bacteria were assumed to
grow exponentially, an appropriate assumption for the initial phase of growth relevant for shelf-life predictions. Both albumen
and yolk were assumed to be homogeneous. The impact of several model parameters (chemotaxis, growth rate, initial
contamination numbers and bacterial swimming speed) was assessed by a sensitivity analysis. The results show that chemotaxis
towards the yolk would have a strong effect on the time needed to reach the vitelline membrane, an aspect that future research
should focus on. The contamination position had less impact on the time to reach the vitelline membrane. The simulation results
illustrate the need for more detailed knowledge on the subject of bacterial migration in hens’ eggs. Our model can easily
incorporate this knowledge when it becomes available.
D 2004 Elsevier B.V. All rights reserved.
Keywords: Individual-based model; Motility; Chemotaxis; Salmonella; Egg; Albumen; Yolk
1. Introduction
Salmonella infections are one of the most widespread microbial infections (Mead et al., 1999). The
* Corresponding author. Tel.: +32 9 2723012; fax: +32 9
2723001.
E-mail addresses: [email protected] (K. Grijspeerdt)8
[email protected] (J.-U. Kreft).
0168-1605/$ - see front matter D 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.ijfoodmicro.2004.10.028
largest share is taken by Salmonella enteritidis
infections, which accounted for 64% of total Salmonella infections in Belgium in 2001 (Ducoffre, 2003).
S. enteritidis infections are mainly associated with
consumption of shell eggs and egg dishes.
Eggs can become contaminated by horizontal
(bacterial penetration through the shell after laying)
or vertical (contamination of the egg contents before
the shell is formed) transmission. Upon vertical
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transmission, both albumen and yolk may become
infected. The predominant site of infection, however,
remains unclear. It was generally believed that the
majority of vertical contaminations occurred in the
albumen (Humphrey et al., 1991), but recently there
has been increasing evidence for contamination of the
yolk (Gast et al., 2002), particularly its membrane
(Gast and Holt, 2001b). Upon horizontal transmission,
the organism crosses the shell and membranes and
gains access to the albumen. The bacteria that succeed
to migrate to the yolk encounter a growth-friendly
environment, whereas albumen evolved to be an
unfavourable growth medium (Hammack et al.,
1993; Braun and Fehlhaber, 1995).
It is the presence of inhibitory substances such as
lysozyme, the shortage in available iron and the
relatively high pH that make albumen an effective
defence against bacteria (Baron et al., 1997). As a
consequence, S. enteritidis will grow poorly or not at
all in the albumen. If the initial contamination takes
place in the albumen, the organisms will only thrive
when they will have reached the yolk. It is not clear if
this occurs by chance or by a chemotactic response to
nutrients that diffuse out of the yolk.
There are some reports on the migration of S.
enteritidis in the albumen. After inoculating the
albumen far from the yolk in situ, Gast and Holt
(2000) found that significant growth required an
inoculum dose of 150 cells and keeping temperature
at 25 8C. With an inoculation dose of 15 cells, growth
was not observed. However, the incubation time of
only 2 days prevents definitive conclusions. In
general, bacteria need to access the yolk before
growth can start. Cogan et al. (2001) observed growth
after 8 days at 20 8C in 7% of whole eggs inoculated
in the albumen near the shell with as few as two cells.
The fraction of contaminated eggs increased up to
50% when the initial inoculation level was increased
to 2500 cells. The temperature also had a significant
impact. Braun and Fehlhaber (1995) found that S.
enteritidis can migrate from the albumen to the yolk
in less than 1 day for 17% of the eggs inoculated with
10 cells/ml albumen and subsequent storage at 20 8C.
After 4 weeks, 72% of the egg yolks were contaminated. Similar results were reported by Baker (1990),
who observed contaminated yolks, albeit not frequently, during storage at 8 8C. It has to be noted that
Braun and Fehlhaber (1995) used buffered peptone
water for the S. enteritidis solution to be injected,
which enhances bacterial growth in albumen. Using a
rather high inoculation level (104 cells) in the albumen
near the shell surface, growth was observed in the
yolk after 2 days of storage at 26 8C (Hammack et al.,
1993). Skipping the albumen layer, Gast and Holt
(2001b) inoculated the vitelline membrane directly
with S. enteritidis and found 6% positive yolk
interiors after 6 h incubation at 25 8C, up to 100%
positive after 24 h. At lower temperatures, the
membrane was less frequently, but still significantly,
penetrated.
Upon vertical transmission, contamination occurs
at very low initial numbers of only a few cells (Cogan
et al., 2001). A fortiori, this will also be true for crossshell contamination, except for cracked eggshells.
Traditional mass transport concepts to model S.
enteritidis migration through the albumen, such as
the growth-diffusion model presented by Grijspeerdt
(2001), are based on a continuous representation of
bacterial mass and are consequently not ideally suited
to describe such low bacterial counts. A discrete
approach appears to be the better choice here.
Individual-based modelling (IbM) of bacterial growth
has been successfully applied for describing colony
and biofilm formation, and is applicable for the whole
range of bacterial densities (Kreft et al., 1998, 2001).
Because these types of model describe the individual
cells, no global population-level assumptions are
necessary, making them utilizable for a broad range
of cases. An IbM was developed to simulate cell
growth and movement within the egg.
2. The model
The model is an extension of the BacSim model of
the growth of bacterial colonies and biofilms (Kreft et
al., 1998, 2001). Bacteria Simulator (BacSim) is an
IbM where each bacterium is simulated as a sphere of
variable volume in a continuous three-dimensional
space.
2.1. Description of egg geometry
Denys et al. (2003) compared four models describing the contours of an egg. The most flexible model
was the four-parameter model of Carter (1974), which
K. Grijspeerdt et al. / International Journal of Food Microbiology 100 (2005) 323–333
was applied to BacSim to define the eggshell. The
yolk was assumed to be spherical, but can be
positioned anywhere inside the egg. In general, the
yolk will be situated close to the centre of the egg, as
this is the most beneficial position to counter bacterial
attacks. However, exceptions occur, especially with
increasing egg age (Board et al., 1994).
2.2. The egg components
Several regions can be defined inside the egg. The
model in this paper simplifies the egg into three basic
compartments: shell, albumen and yolk.
For the simulations presented in this paper, the
shell was considered to be an impermeable barrier for
bacterial movement; hence, no bacteria can escape the
egg. All simulations start from the premise of an
initial contamination within the albumen. This contamination could be due to vertical or horizontal
transmission, in the latter case assuming that the
bacteria have penetrated the shell and shell membranes already.
The albumen is assumed to be homogeneous, as is
the yolk. Apart from viscosity differences, we are not
aware of differences between the thick and thin
albumen that may be relevant for bacterial growth or
migration. From the viewpoint of bacterial motility,
the viscosity of either type of albumen is relatively
low justifying treating it as homogeneous.
The vitelline membrane is not modelled as a
compartment as it is assumed to have no extent,
justified by its thickness of only 11–15 Am (AbouAshour and Edwards, 1970). Once a cell reaches the
vitelline membrane, it sticks to this membrane for a
certain period after which it enters the yolk. This is
equivalent to supposing that the membrane holds back
bacteria initially, but deteriorates after a period of time.
It is left aside whether this is due to bacterial
metabolism or aging effects. Daughter cells that are
created during this hold-up period remain attached to
the membrane as well. This approach is consistent with
the time to membrane breakdown concept used in risk
assessment studies (Baker et al., 1998; World Health
Organization, 2003). Unfortunately, data on the yolk
membrane breakdown time are very diverse (Gast and
Holt, 2001a). Therefore, the time needed for the first
cell to reach the vitelline membrane will be used as a
reference point for most simulation examples.
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2.3. Bacterial growth
It is assumed that cells grow exponentially and do
not die. During the time span of interest bacterial
densities do not reach values high enough to cause
significant deviation from exponential growth. For the
sake of clarity, the lag phase is assumed to be absent,
although the concept of IbM is very suited to include
stochastic growth lag. Although considerable progress
has been made in this field (McKellar and Knight,
2000; McKellar, 2001), information on lag phase
distribution of individual bacteria remains uncertain,
making it difficult to make a reasonable choice at this
moment (Baranyi, 2002). For the same reason, death
kinetics is not included in the model at this stage.
The bacteria grow and divide into two daughter
cells when the volume exceeds a certain threshold
according to the cell division model of Donachie and
Robinson (1987). The growth rate and cell size at
division are varied as described in Kreft et al. (1998),
making this growth model stochastic. This can be seen
in Fig. 2 where a typical growth curve is shown. The
cells clearly grow increasingly asynchronously.
A minimum distance between the cells is maintained by a shoving mechanism, to prevent a closer
packing of cells than physically possible (Kreft et al.,
1998). Note that the importance of shoving is reduced
when dealing with motile bacteria at low densities, as
is the case here.
2.4. Bacterial motility
Flagellated bacteria such as S. enteritidis swim by
rotating their flagellar filaments (Blair, 1995; Purcell,
1997). When these flagella turn counterclockwise they
form a synchronous bundle that pushes the bacterium
forwards; the cell is said to drunT. When turning
clockwise, the bundle comes apart and the cell moves
in an erratic manner; the so-called dtumbleT mode
(Berg, 2000). These two modes alternate and the cell
executes a three-dimensional random walk. In the run
mode, the cell would swim in a straight line were it not
subject to rotational diffusion. In tumble mode, the cell
reorients itself, albeit not completely randomly due to a
forward bias. Tumbles are much shorter than runs, but
they generate much larger changes in direction. The
probability of switching modes is constant per unit
time, and intervals between switching can thus be
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described by a Poisson process. Swimming is superimposed on Brownian movement, which is several
orders of magnitude slower and is therefore negligible.
The bacterial swimming is modelled according to
Berg (1993):
! In the run mode, each step (the displacement of a
cell from one timestep to the next) is a vector two
units long making an angle of 88 with the
preceding vector but rotated around it by a random
angle between 08 and 3608 (see Fig. 6.8 in Berg
(1993)).
! After each run step, there is a probability of 0.1
that the cell switches to the tumble mode.
! In the tumble mode, each step is a vector one unit
long making an angle of 808 with the preceding
vector.
! After each tumble step, there is a probability of 0.3
that the cell reverts to the run mode.
Note that this model of bacterial swimming was
constructed based on experimental observations of
Escherichia coli. As far as we know, there are no such
data available for S. enteritidis, but the above model
can be expected to be a reasonable approximation,
since both bacteria are close relatives and use the
same type of flagellar machinery to swim.
2.5. Chemotaxis
Chemotaxis is the phenomenon that motile bacteria
can bias their random walk in the presence of
repellents or attractants (Armitage, 1999). In case of
the latter, this is accomplished by reducing the
probability of switching to the tumbling mode,
making the straight runs longer (Berg, 1993, 2000).
This mechanism is extremely fine-tuned, allowing
chemotactic responses to changes in attractant concentration of less than 1% over a concentration range
spanning five orders of magnitude (Stock, 1999). The
tumbling frequency has been reported to decrease up
to five-fold compared to the steady-state frequency
(Staropoli and Alon, 2000) or to virtually zero (Maki
et al., 2000).
It has been postulated that there may be a leakage
of nutrients from the yolk, certainly with increasing
egg age (Lock et al., 1992; Humphrey and Whitehead,
1993), although the evidence to support this hypoth-
esis is contradictory (Abou-Ashour and Edwards,
1970; Messens et al., 2004). Bacteria would then
become attracted towards the yolk, reaching it sooner
and thereby accelerating egg contamination. Chemotaxis towards these hypothetical attractants is implemented in the model by changing the tumbling
probability linearly as a function of the swimming
direction of the cells. When the cell swims straight to
the yolk the tumbling probability is at the minimum
value, when the cell swims parallel to the yolk it is at
the default unstimulated value and when the cell
swims away from the yolk it is at the maximum value.
This is clearly an oversimplification. It is unlikely that
a possible attractor gradient would be uniformly
present in the egg. The possible influence of egg
age and the associated deterioration of the yolk
membrane are also not taken into account. Given the
uncertainty about the possible leakage of yolk
elements into the albumen, and the lack of information on their identity, it is clear that the model here can
only serve to indicate the possible significance of
chemotaxis.
2.6. Model parameters and simulation settings
The simulated temperature was set to 20 8C, and
the egg was supposed to be isothermal. This is a
realistic situation in the EU where most shops store
the eggs at room temperature since there is no legal
requirement to cool eggs during storage. The model
parameter values are summarised in Table 1.
The bacterial swimming speed u has been measured by Schneider and Doetsch (1974) and Atsumi et
al. (1996) for E. coli in solutions within and beyond
the viscosity range found in albumen and yolk
(1.7 10 3 to 4.310 3 Pa s in albumen and
4.810 2 to 0.31 Pa s in yolk, according to Scalzo
Table 1
The model parameters and their default values used for the
simulations
Bacterial swimming speed in albumen
Bacterial swimming speed in yolk
Vitelline membrane hold-up time
Egg length
Egg width
Yolk diameter
Growth rate in albumen (20 8C)
Growth rate in yolk (20 8C)
20 Am/s
4 Am/s
15 h
5.65 cm
4.15 cm
2.82 cm
0.1 1/h
0.75 1/h
K. Grijspeerdt et al. / International Journal of Food Microbiology 100 (2005) 323–333
et al. (1999)). Within both viscosity ranges, the
swimming speed of E. coli is relatively constant.
Increasing the viscosity range from albumen to yolk,
the swimming speed decreases approximately inversely proportional to viscosity.
The hold-up time at the vitelline membrane was set
at 15 h, in accordance with Gast and Holt (2001a). A
typical egg geometry as measured in the lab was used.
The growth parameters were obtained from growth
curves of S. enteritidis at 20 8C in separated albumen
and yolk, respectively (own results, data not shown).
The cell volume was growth rate dependent as
described in Kreft et al. (1998), and the minimal
median volume was 0.410 15 l.
Simulations were carried out using a time step of 1
min for simulating growth and an inner loop of about
0.1 s, depending on the actual swimming speed, was
used for simulating the bacterial swimming. The
results presented are averages of at least 100
repetitions if not mentioned otherwise.
3. Results
3.1. Time evolution
An example of the time evolution of the bacterial
distribution in the egg is shown in Fig. 1. To allow a
2-D visualisation of the 3-D cell locations, the cell
positions are projected on the upper x–y plane, while
preserving the distance to the egg centre. The initial
contamination was one cell, located near the shell
surface at a distance of 8.8 mm from the vitelline
membrane.
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The figure illustrates the simultaneous occurrence
of bacterial motility and growth. Once the bacteria
enter the yolk, the total bacterial number rises rapidly.
An alternative view of the same phenomenon can be
seen in the growth curve shown in Fig. 2.
3.2. Sensitivity analysis
Several model parameters (chemotaxis, growth
rate, initial number of cells and linear swimming
speed) were varied individually within a certain range
and the resulting model output was recorded. The
specific growth rate and linear swimming speed were
varied between 50% and +50% of their default
values. For the latter, this is between the reported
values of 10 and 30 Am/s for E. coli (Lowe et al.,
1987). Growth rates in the albumen can vary
considerably (Messens et al., 2004), but the range
studied here is sufficiently wide to capture many
potential cases.
An increasing concentration of a chemoattractant
reduces the tumbling probability from the unstimulated steady state value, causing the cells to swim
longer in the same direction. Increasing the tumbling
probability parameter above the unstimulated value
would imply that the yolk acts as a repellent, which is
implausible. Therefore, as a measure of chemotactic
response towards an assumed yolk derived attractant,
we varied the reduction of the tumbling frequency
from the unstimulated value, in the range from 0%
reduction (no chemotactic response) to 50% reduction. Finally, the initial number of cells was varied
between 1 and 500. For all simulations, the initial
contamination locus was the same as in Fig. 1.
Fig. 1. S. enteritidis migration and growth starting from one initial cell. Indicated are the simulation time (t) in hours, and the total number of
cells (n).
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Fig. 2. Number of cells as function of time in the albumen, yolk and entire egg.
Fig. 3 summarizes the sensitivity of the model
response towards these parameters. Within the parameter ranges studied, it is clear that the presence of
chemotaxis has the most profound impact on the time to
reach the yolk. The initial number of cells is also very
important at lower cell numbers, but for higher
numbers, the effect levels off. Growth rate and bacterial
swimming speed appear to be somewhat less influential.
Fig. 3. Sensitivity analysis of the model responses towards parameter changes. The parameters varied are indicated on the y-axis together with
the baseline values. The range between which the parameters were varied is shown in the scale below each bar. Chemotaxis and tumbling
probability: a positive chemotactic response to an attractant leaking from the yolk would consists in reducing the tumbling probability (thereby
increasing the duration of straight runs) from the steady-state probability of 0.1 in the absence of stimulation (temporal change in attractant
concentration). A faster change of attractant concentration or a stronger response would cause a greater reduction of the tumbling probability.
K. Grijspeerdt et al. / International Journal of Food Microbiology 100 (2005) 323–333
329
Fig. 4. Influence of initial contamination location, starting with one cell, on the time to reach the vitelline membrane. The linear trend is
indicated.
Fig. 5. Simultaneous influence of initial contamination distance from the vitelline membrane and tumbling probability reduction.
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3.3. Influence of initial contamination location
Fig. 4 shows the influence of initial contamination
location on the time to reach the vitelline membrane.
The variation is very large as would be expected from
random walk movement, but there is a slightly
positive correlation between the initial distance from
the vitelline membrane and the required migration
time. Addition of a chemotactic response towards the
yolk diminishes the importance of the contamination
position. This is illustrated by Fig. 5, where the
simultaneous effect of initial distance and chemotaxis
is shown. The data was smoothed with a first-order
negative exponential filter to bring out the overall
trend.
4. Discussion
The newly developed IbM delivers interesting
simulation results. Comparison of model simulations
with experimental data is currently impossible,
because the available data is very diverse, with time
to reach the yolk ranging from less than a day to
several weeks. This diversity points at the difficulty of
in situ measurements in eggs, and the importance of
using correct and standardised procedures. More
efforts should be put in obtaining repeatable, reliable
results.
However, the simulations allow identifying some
trends. In the absence of growth, the average time to
traverse 8.8 mm of albumen by a cell (6 days) would
probably be too long for the cell to survive in the
hostile albumen environment. However, as soon as
some growth and/or a limited chemotaxis towards the
yolk are present, the required average migration time
drops dramatically. There is a limited fraction of cases
in which the vitelline membrane is reached faster,
even in the absence of growth. In 1% of the replicate
simulations, the time to reach the vitelline membrane
is 12 h or less. These are the worst-case scenarios for a
risk analysis.
Whenever the bacteria have penetrated the yolk,
the total number of cells increases rapidly (Fig. 2),
justifying the use of the yolk membrane breakdown
concept. However, it should be noted that the number
of bacteria in the albumen should not be neglected in
the framework of a risk assessment. Baker et al.
(1998) mentioned an infectious dose as low as 10 cells
when dealing with susceptible persons.
The time to reach the vitelline membrane is
significantly influenced by the initial contamination
level when this is smaller than 100 cells. Cogan et al.
(2001) illustrated that at such small inoculation levels,
growth in albumen also occurs less frequently than
when using larger inocula. This indicates that cells
will suffer considerable damage by the defence
mechanisms of the albumen and only a few cells will
survive the initial phase after contamination. Note that
for the results presented in Fig. 3 all cells were
assumed to be in the same area, initially. When
locations would be spread out more randomly, the
average migration time would be lower.
The contamination location plays a role, albeit less
pronounced than chemotaxis (Fig. 3). There is no
general consensus on the most important locus of
contamination of eggs in the case of vertical transmission, both contaminations in the yolk and in the
albumen have been reported (Gast et al., 2002). There
is also evidence that a significant number of contaminations take place at the outside of the vitelline
membrane (Gast and Holt, 2001a). When horizontal
contamination occurs, it is obvious that the contamination of the egg internals starts in the albumen.
The simulations with reduced tumbling switching
probability pointed at the potential importance of
chemotactic responses. Lock et al. (1992) found that
bacteria migrate preferentially towards the yolk. In
their set-up, the egg was broken first, so the question
remains if the vitelline membrane was kept totally
intact. A more rigorous implementation of chemotaxis
would require more information on what is leaking
out of the yolk, in what quantities, at what rate and
where exactly. Information on the diffusion of leaked
compounds into the albumen would also be required.
The simulation of the diffusion of these compounds
would carry a high computational cost but is
straightforward (Kreft et al., 2001). Leaked components from the yolk could also enhance growth in the
albumen (Humphrey and Whitehead, 1993), which in
turn would lead to higher levels of cells in the
albumen and a shorter time to reach the yolk.
Swimming speed and growth rate appeared less
influential on the model simulations within the range
studied. Recently, Cogan et al. (2004) presented
experimental evidence that motility is an important
K. Grijspeerdt et al. / International Journal of Food Microbiology 100 (2005) 323–333
factor for growth of Salmonella enterica serovars in
eggs because it allows the bacteria to reach the yolk
and subsequently grow to high levels. These findings
illustrate the importance of including bacterial motility for modelling bacterial behaviour in hens’ eggs.
A series of assumptions were made in the model due
to shortcomings in the current knowledge on bacterial
migration in eggs. A major assumption is the homogeneity of the albumen. Albumen is a hostile environment to bacteria due to several reasons, but there is
little knowledge about the distribution and timevarying nature of these factors in the albumen. The
same applies to the possibility of nutrient leaking from
the yolk. There is a lot of speculation on this subject,
but no direct evidence is available. The common belief
is that yolk leakage is absent in fresh eggs, and would
only occur after about 3 weeks of storage as a result of
alterations of the vitelline membrane. This conclusion
was based on observations that S. enteritidis did not
grow in the albumen during the first 3 weeks
(Humphrey and Whitehead, 1993). Another explanation points at the ageing effect on the inhibitory
properties of the albumen, although several authors
have reported growth of S. enteritidis in fresh albumen
(Schoeni et al., 1995; Baron et al., 1997; Cogan et al.,
2001). Messens et al. (2004) did not detect an
enhanced growth of Salmonella in albumen of older
eggs, either separated from the yolk or in situ.
Since bacterial levels in the simulations are far
below asymptotic growth conditions, the assumption
of exponential bacterial growth is justified. No lag
phase was assumed for the simulations presented in
this paper. IbMs are particularly suited to incorporate
a stochastic lag phase concept as described by Baranyi
(2002). Given the large uncertainties associated with
stochastic lag, this was not yet added to the model.
However, the trends that emerge from the simulation
result can be expected to remain valid after inclusion
of a lag phase.
Since the egg was assumed to be isothermal,
thermotaxis was not considered although swimming
bacteria are likely to be susceptible to temperature
gradients (Shi et al., 1993; Eisenbach, 2001). Thermotaxis could be an important factor when considering
horizontal transmission just after lay. The warm egg
cools off causing an underpressure in the egg, possibly
sucking inwards bacteria present on the shell (Bruce
and Drysdale, 1994). When the bacteria then swim
331
preferably towards the warmer parts of the cooling
egg, i.e. the centre, the migration time could be
significantly reduced. Furthermore, temperature gradients will also lead to convective flow (Denys et al.,
2003), which superimposes on bacterial swimming.
Possible interactions between the bacteria are not
included in the current model. Cogan et al. (2001)
have observed that S. enteritidis cells benefit from the
presence of other cells and put forward two possible
reasons. First, the combined enterochelin activity of a
Salmonella population might outcompete ovotransferrin for iron. In this context, West and Buckling
(2003) described a synergistic effect when communities of related bacteria produce siderophores, relating
in higher growth rates. Second, the death of some cells
in the albumen can allow others to utilise them as an
energy source or a source of iron (cryptic growth).
The presence of both mechanisms would lead to
spreading patterns where the cells would be more
clustered than in our simulations. An IbM would
allow implementing interactions much more easily
than continuous models. However, no evidence for the
occurrence of these mechanisms in eggs could be
found in the literature. The same applies for interaction between different microbial cultures.
The results show that IbM is a promising technique
to model the issue of bacterial migration in eggs. The
main drawback of the methodology is the computational requirements. On a current desktop computer
(Athlon XP 2000+, 1.5 GB RAM) it takes several
days to reach a bacterial load of about 1.4106 cells/
egg (of which 104 cells are in the albumen) starting
from one cell. Bacterial growth in eggs is characterised by a large variability. To allow convenient
comparison of results, Clay and Board (1992)
introduced the concept of generalised growth. Generalised growth is said to occur when the bacterial level
reaches 104 cfu/ml (equivalent to about 35104 cfu/
albumen) or 106 cfu/ml (about 50106 cfu/egg) when
inoculating separated albumen or whole eggs, respectively. Levels close to generalised growth can already
be reached by IbM simulations. Extending simulations to higher bacterial numbers would not contribute
much to the issue of food safety. Another approach
when higher cell densities need to be simulated could
be the combination of IbM and continuous methods.
Once the IbM-simulated cell densities reach a certain
threshold level, a continuous model could take over
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using the final IbM output as initial state. A similar
concept has been applied by McKellar (2001) when
developing a continuous-discrete-continuous dynamic
model to describe the lag phase of individual cells.
Future developments of the model will include a
refinement of the bacterial growth model with the
inclusion of a lag phase and death kinetics and the
introduction of more stochastic elements in both the
bacterial growth and swimming models as well as in
the egg model (membrane breakdown times, yolk
location).
5. Conclusions
IbM can be applied for modelling bacterial migration in eggs. Because of the low initial numbers of
contamination, it is definitely better suited than
continuous diffusion-type models to deal with this
problem, and offers a level of flexibility and detail that
is difficult to obtain by continuous methods. The
model can already indicate some trends, but additional
information on bacterial migration in eggs will have to
be collected in order to further refine the model.
Acknowledgements
The authors are grateful for financial support
from the European Community, project QLK5CT-2001-01606.
An extended abstract of this paper has been
published in the proceedings of the 4th International
Conference on predictive modelling in foods (Grijspeerdt et al., 2003).
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