BACKGROUND THEORY
BACTERIAL GROWTH
Bacterial cells reproduce by dividing in two. If growth is not
limited, doubling continues at a constant rate so both the number
of cells and the rate of population increase doubles with each
consecutive time period. For this type of exponential growth,
plotting the natural logarithm of cell number against time produces
a straight line. The slope of this line is the specific growth
rate of the organism, which is a measure of the number of divisions
per cell per unit time. In food, bacteria cannot grow continuously
as the amount of nutrient available will be finite and waste products
will accumulate. In these conditions growth curves tend to be
sigmoidal.
A simple sigmoid growth curve (as shown in Figure
1) can be divided into three parts:
 The lag phase
Initially the number of cells does not increase. On finding
themselves in a new environment, cells will take time to adjust.
They may need to synthesise new enzymes, repair any cell damage
and initiate plasmid or chromosomal replication.
 The exponential phase
This period is characterised by cell doubling. The maximum slope
of the curve will be the specific growth rate of the organism
in that particular environment.
 The stationary phase
Growth will slow down and eventually stop as nutrients become
depleted or inhibitory metabolites build up
MODELLING
The number of pathogenic and spoilage bacteria present in food
has both health and financial implications and is of concern to
everybody from food producers and processors to retailers and
consumers.
Traditional methods for enumerating bacteria are not instantaneous
so you can only gain information on the number of organisms that
were present at the sampling time. It is often desirable to know
the current bacterial population or the number that will be present
in the future if a product is subjected to particular treatments.
Information on initial levels of contamination combined with knowledge
on how bacteria number change allows present and future levels
to be predicted.
Such predictive microbiology is based on the hypothesis that
growth is an intrinsic characteristic of the organism and will
occur reproducibly in the same environment. Thus by measuring
growth once you can tell how a future population will develop
under the same conditions. In its simplest form, these types of
predictions can be made from graphs showing the number of bacteria
measured in samples taken during growth against time. The points
on the growth curve will show a trend allowing the population
at any time to be estimated by interpolation. The trend can also
be described mathematically by a growth function. The mathematical
formula for the applied growth function is called a primary model.
Such a model is only applicable for the environmental conditions
under which the growth curve it describes was measured. Secondary
models describe how the parameters of primary models depend on
environmental factors such as temperature, pH and water activity.
Predictive models of growth usually combine primary and secondary
models. Using these for interpolation, growth against time can
be predicted for any combination of conditions.
A sigmoid growth curve is commonly described by four parameters:
 The natural logarithm of the initial cell concentration (y_{0}).
The initial concentration is a random variable that is specific
to each food and not related to the current environment.
 The time in lag phase (l).
The length of the lag phase will depend on the intrinsic characteristics
of the bacteria, the current environmental conditions and the
physiological history of the cells. Cells that have come from
a different environment or are damaged (for example after heat
treatment or freezing) may require more time to synthesise macromolecules
and repair damage before they can divide than undamaged cells
from a similar environment.
 The maximum specific growth rate (µ_{max}).
This represents the steepest slope of the curve. It is considered
to be an intrinsic characteristic of the organism dependent
on the current environment but not affected by the cell history.
 The natural logarithm of the maximum cell concentration (y_{max}).
The final concentration is of little interest in food safety
as food is spoilt before the maximum bacterial concentration
is reached.

Figure 1. A sigmoid bacterial growth curve can be
characterised by four parameters. 
Although modelling the lag time is very important, to date predictive
modelling has primarily concentrated on growth rates. This is
because the lag phase is technically difficult to study, so quantitative
data is lacking, and lag is more difficult to model as it depends
not only on the current conditions but also on the history, or
initial physiological state, of the cells (See Figure
2). The effect of history can vary from cell to cell so improved
predictions of lag time, and thus growth of bacteria in food,
will require the use of stochastic modelling techniques that are
missing from current predictive microbiology.

Figure 2. The number of cells present at a given
time will depend not only on the maximum specific growth rate
but also on the lag time which is historydependent. These
growth curves are from replicate experiments, except that
the inoculation was prepared differently and led to different
physiological states of the primary culture. The maximum specific
growth rates are the same while the lag periods are different
because the latter parameter depends on the history of the
cells. 
LAG TIMES OF INDIVIDUAL CELLS
Hundreds or thousands of organisms need to be present before
they can be detected by standard counting method thus growth curves
show the population lag time. Each individual cell will have its
own lag time before it can start to multiply and it is the joint
effect of all these individual lag times that make the lag time
of the population growth. The BACANOVA project aims to look at
the variability of the lag times of individual cells that make
up the population lag.
The reasons for this are:
1.The lag of the population can be deduced from the lag of
individual cells but not vice versa.
The total population at any time will be the sum of the bacteria
present initially and the offspring they have produced in that
time. Knowing the growth curves of the subpopulations generated
by individual cells, a population growth curve can be constructed
of the total population for any initial number of bacteria. However,
population growth curves cannot be used to obtain information
on the growth of individual bacteria. If a population consists
of cells with differing lag times, the contribution each cell
makes to the final cell number will not be equal; the subpopulations
derived from cells that start to multiply fastest will make a
greater contribution to the total population than those from the
slow growing cells. As the contribution from each cell to the
final population is not the same, one cannot use observations
made at the population level to deduce the distribution of lag
times amongst individual cells, or even the average lag time.
Consider two bacterial populations with individual cell lag time
distributions A and B (Figure 3).
The means of the lag times of the cells in each population are
the same but the lag times of the two populations are markedly
different (but not their maximum specific growth rates).

Figure 3. 
The reason the population growth curves differ lies in the left
hand tail of the distribution. Well adapted cells, with shorter
lag times, will dominate growth so increasing the number of these
cells reduces the population lag time. On the other hand, if the
difference in the distributions appears mainly at the right hand
tail, as in the case of distributions C and D (the
mirror images of A and B), the two generated growth
curves are indistinguishable, as shown in the second pair of plots.
In addition, knowing the number of cells with long lag times
can be important when trying to determine the number of bacteria
present in a sample. Cells with long lag may remain undetected
by standard viable count procedures. This will lead to the probability
of survival being underestimated.
2.When there are low numbers of cells, the distribution of
the "time to a specific concentration" will be more
closely related to the distribution of the lag phase of individual
cells than lag of large populations.
This is particularly important for pathogens that can be present
at very low levels. Consider a food where a small amount of contaminated
sample has found its way into a large batch of product. When the
food is divided into packets each packet will contain very low
numbers of the contaminating organism. If the packs contain a
single cell, the time before numbers start increasing in the pack
will be the same as the lag phase of the cell in that pack and
the distribution of times to a specific concentration will be
the same as the distribution of lag times for individual cells.
In this case, the probability of cell concentration reaching an
infective dose by the time of consumption is the same as the distribution
of individual lag times.
3.By determining the distribution of lag phase you may be
able to determine the prehistory of the cells and improve modelling.
Current predictive models for bacterial growth in foods are based
only on the conditions in the food and its environment, ignoring
the initial physiological state of the bacterial cells. The past
of a cell, such the conditions under which it was grown and any
treatments like heating or freezing to which it has been subjected,
will effect the ability of the cell to grow. If the variability
of individual cells within a microbial population reflects the
history of those cells, it may be possible to use that variability
as an indication of cell history. Combining the effects of history
with those of the actual environment would lead to improved models.
By understanding how the process history can effect the distribution
of lag times, it may also be possible to adjust the historical
treatments to improve processing. Figure 3 illustrates how small
changes to the individual cell lag time distributions can alter
the lag time of the population. Understanding and quantifying
this phenomenon should help to optimise the microbiological quality
and safety of food.