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~V
V Modeling Microbial
Dynamics under
Time-Varying Conditions
Kristel Bernaerts, Els Dens, Karen Vereecken,
Annemie Geeraerd, Frank Devlieghere,
Johan Debevere, and Jan F. Van Impe
CONTENTS
7.1 Introduction
7.2 General Dynamic Modeling Methodology
7.2.1 Basic Elements for Modeling Growth
7.2.2 Basic Elements for Modeling Inactivation
7.3 Example I: Individual-Based Modeling of Microbial Lag
7.3.1 Principles of Individual-Based Modeling
7.3.2 Implementation of Mechanistic Insight into an
Individual-Based Model
7.3.2.1 Modeling Mechanistic Insight on the Temperature
Dependency of Cell Growth
7.3.2.2 Simulation Results
7.3.2.3 Discussion of Results
7.4 Example II. Modeling Microbial Interaction with Product Inhibition
7.4.1 Description of the Inhibition Phenomena
7.4.2 Modeling Microbial Growth with Lactic Acid Production and
Inhibition
7.4.3 Discussion of Results
7.5 Conclusions
Acknowledgments
References
7.1 INTRODUCTION
Predictive food microbiology essentially aims at the quantification of the microbial
ecology in foods by means of mathematical models. 1 These models can then be used
to predict food safety and shelf life, to develop and assist in safety assurance systems
in the food industry (e.g., Hazard Analysis of Critical Control Points), and to establish
2004 by Robin C. McKellar and Xuewen Lu
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exposure studies in the framework of risk assessment (see, e.g., References 2 to 4).
Though challenge testing tends to be the common policy in the food industry,
information on microbial kinetics — in food products — is increasingly consolidated
into mathematical models, which may significantly reduce the number of challenge
tests required to determine, for example, shelf life. In combination with predictive
models for, e.g., heat transfer, and other process variables, and the initial contami-
nation level, these models are essential building blocks in time-saving simulation
studies to optimize and design processing, distribution, and storage conditions (e.g.,
temperature-time regimes) that guard food safety and spoilage (e.g., Reference 5).
In the early years of predictive microbiology, strong preference has been
expressed towards sigmoidal functions that gave a good description of growth curves
obtained under nonvarying environmental conditions. The most commonly used
growth model was probably the modified Gompertz model. 6 Microbial inactivation
at high temperatures — exhibiting a log-linear behavior — could be described as a
first-order decay reaction (see, e.g., Reference 7). Effects of environmental condi-
tions on these primary models (i.e., evolution of cell number as function of time)
are embedded into secondary models (see Chapter 2 and Chapter 3 for more details).
Dynamic primary models capable of (1) dealing with realistic time-varying condi-
tions and (2) including the previous history of the food product in a natural way
have been introduced since the early nineties. 8 ' 9
Besides the need for such dynamic models, it is also clear that real food product
conditions should be taken into account during modeling (e.g., Reference 10). More
(mechanistic) knowledge needs to be built into existing models such that the phys-
iological response of microorganisms and the associated microbial dynamics can be
accurately explained under fluctuating conditions. For example, reliable predictions
for microbial lag phenomena and interaction are lacking nowadays.
In this chapter, the elementary building block for dynamic mathematical models
describing microbial evolution is presented (see Section 7.2). Given this general
expression, (mechanistic) knowledge on the microbial behavior in foods can be
gradually built in to yield a generic model structure describing the microbial dynam-
ics of interest. During this model development process, a continuous trade-off needs
to be made between model complexity and manageability. On the one hand, the
mathematical model should incorporate sufficient (mechanistic) knowledge in order
to generate accurate predictions. Reliable predictions are indispensable to advocate
confidence in predictive microbiology within the food industry. On the other hand,
these mathematical models must remain user-friendly and computationally manage-
able in view of their industrial applicability.
The chapter is organized as follows. Section 7.2 introduces the general dynamic
model building approach. First, this strategy is illustrated for modeling simple growth
and inactivation behavior. However, accurate modeling of microbial dynamics in foods
usually requires more complex model structures. In this respect, (1) the modeling of
microbial lag under time-varying temperature conditions via an individual-based
approach (see Section 7.3) and (2) the modeling of interspecies microbial interactions
mediated by product inhibition (see Section 7.4) are discussed. At the same time, the
fundamentals of microscopic {individual-based) and macroscopic {population level)
modeling are revisited. Section 7.5 summarizes the general conclusions.
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7.2 GENERAL DYNAMIC MODELING
METHODOLOGY
The elementary dynamic model building block describing microbial dynamics under
batch cultivation within a homogeneous environment consists of the following dif-
ferential equation:
^^ = [i i (N i (t),<N j (t)> i ^<env(t)>,<P(t)>,<phys(t)>,...)-N i (t) (7.1)
with i,j = 1,2,..., n the number of microbial species involved (analogous with
Reference 11). N^t) represents the cell density of species i and (!,•(•) [Ir 1 ] defines its
overall specific evolution rate depending on interactions within and/or between
microbial populations (N t and/or Np respectively), physicochemical environmental
conditions (<env>), microbial metabolite concentrations (<P>), the physiological
state of the cells (<phys>), among others. Microbial proliferation is generated when
|l ; (-) > and microbial decay results from (!,(•) < 0.
Observe that all influencing factors may depend on time themselves. For exam-
ple, temperature may change dynamically with time, and thus acts as an input when
solving the system of differential equations. To describe the time-dependent evolu-
tion of metabolite production and the physiological state of the cells, for example,
additional coupled differential equations are added to the set of differential equations
in 7.1. This is abundantly illustrated throughout the paper.
Within structured food systems, Expression 7.1 describes the local dynamic
behavior of microorganisms. In such case, local inputs are needed. For example,
local temperatures can be computed using heat transfer models. Furthermore, micro-
bial dynamics shall be influenced by spatially varying substrate and nutrient con-
centrations (which may become restricted because of diffusion limitations). Diffu-
sion limitations also cause spatial gradients of metabolic products. In addition, the
need for a valid transport model for microbial growth (i.e., describing spatial colony
dynamics) rises (e.g., Reference 12).
7.2.1 Basic Elements for Modeling Growth
If environmental conditions are constant, the microbial growth curve — the (natural)
logarithm of the cell density as function of time — typically exhibits a sigmoidal
shape consisting of three phases: the lag phase, the exponential phase and the
stationary phase (see Figure 7.1). First, the population needs to adjust to its new
environment. Second, the population attains its maximum specific growth rate char-
acteristic for the specific environment. Third, growth ceases because of, e.g., inhib-
itory effects of metabolites. Eventually, this leads to inactivation.
The overall specific growth rate in Expression 7.1 can be represented by three
factors describing these three phases*:
* The dynamics of a single species are considered and the subscript i can thus be omitted.
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: 1 j,.,,y<\ , : ; ; ;>
. Jr - \
: / \
/ s
/■ \
LAG
PHAS
/ EXPONENTIAL : STATIONARY j
iE /: PHASE : PHASE :
TAIL
time
time
FIGURE 7.1 Left plot: Typical growth curve (full line) at constant environmental conditions.
Right plot: Typical inactivation curve under mild constant processing conditions.
dN(t)
dt
= ^(0-jA ra „O-ji JI «O-iVW
lag
stat
(7.2)
max'
During the exponential phase, the specific growth rate remains constant at |i,
which is the maximum specific growth rate that can be realized within the actual
environment. The dependence on environmental factors such as temperature is typ-
ically incorporated into secondary models (e.g., Reference 13). The first factor |i lag (-)
is introduced to describe the lag behavior and thus needs to embed the gradual
increase of the overall specific growth rate from to |i max . The third factor |i- stat (0
induces the gradual decrease in the specific growth rate towards 0, resulting in the
stationary phase.
Dynamic models in predictive microbiology are reported in, e.g., Baranyi and
Roberts, 14 Baranyi et al., 8 Hills and Mackey, 15 Hills and Wright, 16 McKellar, 17 and
Van Impe et al. 9 ' 18 A well-known dynamic model is the growth model by Baranyi
and Roberts: 14
dN(t)
dt
dQ(t)
dt
Q(f)
1 + 2(0
'V-
max
1-
W)
N
max
N(t)
(7.3)
= \i
max
Q(t)
Recognize the three factors in the right-hand side of the first equation as presented
in Equation 7.2. The first factor, i.e., the so-called adjustment function, describes
the gradual adaptation of the population to attain |l max . Hereto, an additional state
variable Q(t) is introduced into the model [thus Hi ag (<2(0)L This variable denotes the
physiological state of the cells that should augment until the adjustment function
reaches (approximately) its maximum value, namely, 1 . At that point, the exponential
phase starts. The initial value of Q(t) together with the maximum specific growth
rate determines the lag-phase duration. Graphically, |i max corresponds with the slope
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of the log-linear part of the growth curve.* The third factor, i.e., the so-called
inhibition function, causes the growth rate to decrease asymptotically to when the
population density reaches its maximum level /V max [thus [L stat (N(f), N max )].
Environmental conditions affecting the outgrowth of microorganisms in food
products are often time-varying. In such case, predictions of the food safety and the
shelf life can be generated by combining a dynamic primary model with a secondary
model relating the typical primary parameters with environmental conditions (e.g.,
[\ max {<env{t)>)). Doing so, it is implicitly assumed that the primary parameters, e.g.,
the maximum specific growth rate, immediately change according to the changing
environmental factors and the secondary model. Consequently, delayed responses
(lag) induced by (sudden) fluctuations of the surrounding environment cannot be
predicted. 19 Furthermore, the cessation of growth is a response to starvation following
exhaustion of nutrients and/or the inhibition by metabolic products. 20 Description of
the inhibition within mixed cultures by, e.g., product formation, cannot be consis-
tently described when using the single model parameter N max (see below).
Section 7.3 and Section 7.4 illustrate how such dynamic growth models (7.2)
can be fine-tuned towards the modeling of microbial lag and growth inhibition.
Eventually, we aim at robust mechanistically inspired models.
7.2.2 Basic Elements for Modeling Inactivation
During mild heat treatment (at constant temperature) microbial inactivation often
shows a non-log-linear behavior characterized by a delayed response {shoulder) and
a resistant population {tailing) (see Figure 7.1, right plot). According to Expression
7.1, a general model structure reads as follows.**
dN{t)
dt
= -k
shoulder
(■)-k m „(-)-k,J-)-N(t)
max
(7.4)
To express the specific microbial inactivation rate the symbol k is commonly used.
On the basis of the mechanistic insight on the occurrence of the shoulder and
tailing phenomenon, 21-24 Geeraerd et al. 25 established the following functions mod-
eling the shoulder and tailing behavior.
dN{t)
dt
dC c {t)
dt
1
l + CW
= -k
max c
CM
k
max
AT
1 res
N(t)
N(t)
(7.5)
* From a mathematical point of view, the adjustment function is exactly equal to 1 only at infinity,
whereas the inhibition function approximates 1 when N(t) « N mm . However, from a numerical point of
view, both factors are 1 during a considerable part of the growth curve. Hence, it can be reasonably said
that during the log-linear part (X max is reached.
** Here too the dynamics of a single species are considered and the subscript i can thus be omitted.
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The first factor in the right-hand side of the first equation models the shoulder of the
inactivation curve. Before first-order inactivation of the population takes place (at a
specific inactivation rate & max ), some critical protective component C c needs to be
inactivated. It is assumed that this occurs according to a first-order relationship (i.e.,
second differential equation in 7.5). The shoulder is obtained by applying a Michae-
lis-Menten-based adjustment function, namely, (1 + C c (f))~ l [thus k AovJdcr (C c (t))].
Starting at a low value, the adjustment function increases towards unity and, at that
point, log-linear inactivation is observed. Analogous with the physiological state Q(t)
in the dynamic growth model 7.3, C c (f) can be interpreted as the physiological state
of the population in the context of inactivation. The tailing phenomenon can be
explained by some resistant subpopulation N res that is unaffected during the (heat)
treatment. This tailing at a residual population 7V res is here modeled by (l-N re JN(t))
[thus k tail (N(t), N ns )]. Note that this residual subpopulation is not necessarily a con-
stant value but may vary when modeling nonthermal inactivation, 26 ' 27 or when sub-
jecting the microbial population to sequences of inactivation treatments. 28
To conclude, observe that the general model structure 7.4 and model 7.5 also
encompass classical log-linear inactivation. In Equation 7.5, log-linear inactivation
is generated by selecting (after identification on experimental data) a very low value
for C c (0) and 7V res , implying the absence of a shoulder and a tail, respectively.
7.3 EXAMPLE I: INDIVIDUAL-BASED MODELING
OF MICROBIAL LAG
Factors affecting the occurrence and extent of the commonly observed initial (pop-
ulation) lag phase (i.e., a period after inoculation during which cells adapt themselves
to the new environment, see Figure 7.1) can be attributed to the past environment,
the new environment, the magnitude of the environmental change, the rate of the
environmental change, the growth status (e.g., exponential, stationary) of the inoc-
ulated cell culture, and the variability between individual cell lag phases. These
environmental changes may involve nutritional and chemical, as well as physical
changes. Obviously, environmental fluctuations during exponential growth can also
cause lag (i.e., intermediate lag). Large temperature gradients, for example, applied
during the exponential growth phase shall induce an intermediate lag phase observed
as a transient adaptation of the growth rate. 1929
Secondary models describing the relation between the (population) lag-phase
duration and the physicochemical environment are usually based on highly stan-
dardized experiments during which cells are grown to their stationary phase under
optimal growth conditions before being transferred to the new environment, which
is not - deliberately — varied upon the subsequent growth. Such mathematical
models perform well under the conditions that they have been developed for. How-
ever, any deviation within the prehistory of the contaminating population may
seriously alter the lag behavior. 3031 Especially, huge deviations between model
prediction and actual microbial dynamics are observed under time-varying environ-
mental conditions. 32
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Overall, lag phenomena induced by (sudden) environmental changes are insuf-
ficiently explored, and suitable (generic) predictive models are not available. In this
section, a first attempt towards a general model structure that is valid for various
microorganisms and various dynamic temperature conditions — as presented in
Dens 33 — is described. Opposed to the more traditional population modeling
approach, basic mechanistic knowledge concerning the mechanism causing lag at
cell level is embedded into the model. In particular, the theory of cell division is
implemented within an individual-based modeling approach to enable the description
of lag phases that can be induced by sudden temperature rises.
7.3.1 Principles of Individual-Based Modeling
The fundamental unit of bacterial life is the cell, encapsulating action, information
storage and processing as well as variability. It can therefore be appropriate to
construct microbial models in terms of the individual cells. 34 This is the domain
of individual-based modeling. The basic idea behind this approach is that, if it is
possible to specify the rules governing the behavior of the cells, then the global
multicellular behavior can be explained by the interactions between the individual
cell activities. The rules constituting the model reflect the (presumed) behavior
of the individual cells, such as nutrient consumption, biomass growth, cell divi-
sion, movement, differentiation, communication, maintenance, and death. Since
a change in microscopic (individual-based) rules may lead to significantly different
macroscopic (population) behavior, it might be possible to rule out impossible
mechanisms and to learn about the true mechanisms. A very important property
of individual-based models is the fact that they easily allow for differences
between the individuals. This is accomplished by using random variables, drawn
from a certain statistical distribution. The introduction of a range of randomness
and the consideration of a high number of individuals interacting independently
with the environment leads to a good representation of reality and leads to a better
understanding of the cellular metabolism (see, e.g., Reference 35). Spatial effects
can be relatively easily translated into a set of rules. Kreft et al. 34 introduced the
spatial aspect in their model to reproduce the growth of Escherichia coli cells in
a colony.
In general, individual-based models incorporating underlying mechanistic
knowledge of microbial dynamics are widely spread, but are relatively unexplored
in the field of predictive microbiology. The more general modeling approach in
predictive microbiology considers the microbial population as such, i.e., the popu-
lation is described by a single-state variable, namely, N(i). Furthermore, model
parameters are usually assumed to be deterministic, i.e., have one typical value.
When incorporating cell-to-cell variability into population-based models, popula-
tion-related model parameters are considered as random or distributed variables (e.g.,
Reference 36). Individual-based models have the advantage that the cell-to-cell
variability can be incorporated at cell level, i.e., the level from which variability
actually originates. The general concepts of individual-based models and their appli-
cability in the context of predictive microbiology are discussed in Dens. 33
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7.3.2 Implementation of Mechanistic Insight into an
Individual-based Model
7.3.2.1 Modeling Mechanistic Insight on the Temperature
Dependency of Cell Growth
The mechanistic insight into the theory of cell division has been built into an
individual-based model BacSim, originally developed by Kreft et al. 34 In contrast
to the general expression 7.1 describing the evolution of a bacterial population N(t),
biomass growth of the individual cells m{i) is considered and is assumed to occur
exponentially at any time*:
dm(t) / x _ „
— JT = Mmax • «(0 (7-6)
at
This expression forms the elementary building block of the proposed individual-
based model.
Concerning the cell cycle of an individual cell, Cooper and Helmstetter 37
observed that, for a constant temperature, a constant time C is needed for the
replication of DNA and a constant time D for cell division. In combination with the
fact that DNA replication is always initiated when the cell attains a certain amount
of biomass 2m c , Donachie 38 derived the following relationship for the amount of
biomass at cell division m.
v d-
m =2m exp(MC+D)) (7.7)
with [l the specific growth rate of the cell biomass (in combination with 7.6, (I
represents |i max ). Following this equation, the cell mass at division (and thus also
the average cell mass of the population) is proportional to the exponent of the product
\i-(C + D). With respect to this equation and based on literature, a number of
hypotheses on the effect of dynamic temperatures on the cell division process (and
thus the overall specific cell-number growth rate) can be formulated:
i. The product \L-(C + D) stays constant for different temperature condi-
tions. This means that temperature variations do not alter the size and
the chemical composition of the cells, as postulated by Cooper. 39 In other
words, the biomass growth rate as well as the population growth rate
will immediately change when imposing temperature changes and no lag
will be observed,
ii. Trueba et al. 40 reported that the average cell volume of E. coli decreases
with decreasing temperatures. Consequently, for these observations, the
product (I- (C + D) depends on temperature as the average cell volume is
proportional to ra d . In case of a temperature increase, for example, the
* The dynamics of a single species are considered and the subscript i can thus be omitted.
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biomass growth rate will instantaneously change but the population num-
ber will lag behind as the average volume for division has increased.
iii. A lag in biomass growth of E. coli induced by sudden temperature shifts
from low to high temperatures has been reported by Ng et al. 41 The authors
assume that cells growing at low temperatures express some damaged
status that needs to be repaired before active growth at high temperatures
can be achieved. This damaged state can be reflected by a limiting con-
centration of one or more enzymes. When passing from a low to a high
temperature, cells first need to increase the concentration of these limiting
enzymes, before they can increase their biomass growth rate.
A (simplified) mathematical translation of this hypothesis reads as
follows:
at m(t) L
with ^ t l = L (L =L, or LJ
dm(i) ' h
with E(f) some critical growth factor, and L the rate at which E is syn-
thesized (after Reference 41). This production rate changes according to
temperature (in a discrete way), i.e., L x and L h are the typical production
rates for low and high temperatures, respectively. For E. coli populations,
the high temperature zone ranges from 20 to 37°C and is also known as
the normal physiological range of E. coli. 41
In conclusion, this hypothesis will predict a lag phase when tempera-
ture variations cross the (lower) boundary of the normal physiological
range.
The temperature dependence in the suboptimal growth temperature range can
be modeled by the square root model of Ratkowsky et al. 42 :
A/
\*^(T(t))=b-(T(t)-T m J (7.8)
For more details on the exact implementation (i.e., parameter values, initial condi-
tions, etc.) of these hypotheses, reference is made to Dens. 33
7.3.2.2 Simulation Results
As a case study, the effect of abrupt shift-up temperatures on the growth of E. coli
is described. The experimental data in Figure 7.2 and Figure 7.3 depict the effect
of a small (i.e., 5°C) and a large (i.e., 20°C) positive temperature shift on the growth
of E. coli, respectively. Full details on the experimental data generation can again
be found in Bernaerts et al. 19 and Dens. 33
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20
18 -
o
16 -
14 -
1 1
1
^*^*>
^^
-
i i
I
40
35
O
30 o^
CD
25 2
CD
Q-
E
20 ®
- 15
10
Time [ h]
FIGURE 7.2 Simulation of the individual-based models proposed in Section 7.3 on experi-
mental data of E. coli (*) submitted to a sudden temperature shift from 22.5 to 27.5°C during
exponential growth (Adapted from Dens, E.J., Predictive Microbiology of Complex Bacte-
rial/Food Systems: Analysis of New Modelling Approaches, Katholieke Universiteit Leuven,
Belgium, 2001). The solid line represents the model prediction using the measured temperature
profile (dashed line). Top: hypothesis (i), middle: hypothesis (ii), bottom: hypothesis (iii).
For each of the temperature shifts, the three hypotheses described in the previous
paragraph have been implemented. It appears from Figure 7.2 that the small temper-
ature increase from 22.5 to 27.5°C does not alter the balanced growth dynamics of
the microorganisms and is properly described in all three cases. On the contrary, cell
density data generated during the larger temperature shift from 15°C (low temperature
range) to 35 °C (high temperature range) induces a lagged growth response that can
be predicted by only hypotheses (ii) and (iii) (see Figure 7.3). In hypothesis (ii), the
lag phase is due to the time needed to increase the cell volume up to the new critical
2004 by Robin C. McKellar and Xuewen Lu
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o
18
16 -
14 -
12 -
14
c
18
16 -
14 -
12 -
14
16
16
Time [ h]
I I
!
*
/ , .
1 "
/
/
i. . , .
/
/
' i
.^mjr.
-
t - '
T -* -
/
i
■ i
18 20
Time [ h]
22
I I
1
1
*
/
*
.
/
/
/*
-
I
/
/
■■/■■■ ■ ■^-*
/*
-
r . m —
i i
-ft.jL/*'
i
■
18 20
Time [ h]
22
40
35
-I 30
- 25
- 20
15
10
24
40
-| 35
- 30
o
CD
2
CD
Q.
E
CD
O
- 25 £
20
15
10
2
CD
Q.
E
CD
24
FIGURE 7.3 Simulation of the individual-based models proposed in Section 7.3 on experi-
mental data of E. coli (*) submitted to a sudden temperature shift from 15 to 35 °C during
exponential growth (Adapted from Dens, E.J., Predictive Microbiology of Complex Bacte-
rial/Food Systems: Analysis of New Modelling Approaches, Katholieke Universiteit Leuven,
Belgium, 2001). The solid line represents the model prediction using the measured temperature
profile (dashed line). Top: hypothesis (i), middle: hypothesis (ii), bottom: hypothesis (iii).
mass at division. Biomass growth exhibits an immediate rate adjustment whereas
cell number shows lag behavior. In hypothesis (iii), the lag phase is reproduced at
the level of biomass growth and propagates into the cell number evolution.
7.3.2.3 Discussion of Results
Individual-based modeling yields an excellent tool to integrate mechanistic knowl-
edge at the level of the individual cell behavior into a model structure. Simulations
with the individual -based model can then explain the population dynamics.
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In this example, three cell mechanisms describing the effect of dynamic tem-
peratures could be extracted from literature references. Two of the three hypotheses
could describe both a small and a large temperature shift equally well. Given only
population density measurements, it is therefore impossible to discriminate between
the established models. At this point, additional (more advanced) measurements are
needed to further establish the model structure. Such more advanced measurements
can be biomass weight, DNA concentration, RNA concentration, protein concentra-
tion, etc. In other words, the revised modeling example clearly points out the two-
way interaction between model building and data generation. Besides the selection
of essential measurements, this two-way interaction embraces the design of infor-
mative experiments, i.e., the selection of appropriate {dynamic) input conditions (see,
e.g., References 19 and 43) or {static) treatment combinations (see, e.g., References
44 and 45).
A disadvantage of the individual-based modeling approach is that the models
may become relatively complex and computationally tedious. However, the obtained
mechanistic knowledge can eventually form a sound basis for population-based
models (which are more easily manageable).
7.4 EXAMPLE II. MODELING MICROBIAL
INTERACTION WITH PRODUCT INHIBITION
In this section, the interaction of lactic acid bacteria (antagonist) with pathogenic
bacteria (target) is discussed and modeled. Information given has been extracted
from Vereecken et al. 46-47 and Vereecken and Van Impe. 48
7.4.1 Description of the Inhibition Phenomena
During the fermentation process of lactic acid bacteria, lactic acid is produced
(biological process). This lactic acid released into the medium will dissociate and
lower the medium pH (chemical process). Both the undissociated lactic acid con-
centration ([LaH]) and the decreased pH (~[H + ]) have an inhibitory effect on micro-
organisms. In the first place, the lactic acid production will cause the inhibition of
the bacterium growth itself. The cessation of growth observed as the stationary phase
can thus be attributed to a self-induced inhibitory effect. In addition, this lactic acid
production affects neighboring microorganisms. Pathogenic bacteria, like Yersinia
enterocolitica (see Figure 7.4), can be very sensitive to this inhibitory compound. 24
The increasing lactic acid concentration will cause an early termination of the growth
process. For this reason, lactic acid bacteria can be exploited as a natural antimicro-
bial agent within (fermented) food products or as a protective culture.
7.4.2 Modeling Microbial Growth with Lactic Acid
Production and Inhibition
In contrast to the classical approach, Equation 7.3, where the stationary phase is
modeled as function of Nj{t) and N max , growth inhibition emerges from lactic acid
production, which is therefore explicitly incorporated into the model structure:
2004 by Robin C. McKellar and Xuewen Lu
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10
L. sakel, monoculture
10
2 ■
1
10
9 '
: Q<3D
$
rt 6
A
- " y
V
5 10 15 20 25 30 35
time [h]
L. sakel+ Y. enterocolltlca coculture
0.08 ■
0.06 ■
X
CD
10 15 20 25 30 35 40
time [h]
Y. enterocolltlca monoculture
0.04 ■
0.02 ■
20
time [h]
40
0.08 ■
^ 0.06 ■
X
CD
0.04 ■
0.02 •
20
time [h]
0.1
it****.
20
time [h]
6.5
FIGURE 7.4 Description of experimental data of Lactobacillus sakei (o) and Yersinia entero-
colitica (0) grown in mono- and coculture with the dynamic model structures (Equation 7.9
and Equation 7.10 in combination with 7.11 and 7.12) presented in Section 7.4 (Adapted
from Vereecken, K.M. and Van Impe, J.F., Int. J. Food Microbiol., 73(2/3), 239, 2002 [x refers
to cell numbers below detection limit]). The total lactic acid concentration [LaH] tot (A) and
pH (*) are depicted in the right-hand plots. The dissociation kinetics of the applied medium
have been computed according to Wilson et al. 49 (Observe that the inactivation of Y. entero-
colitica cannot be predicted by the model structure [dashed line].)
2004 by Robin C. McKellar and Xuewen Lu
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dN.(t)
dt
= V lag (-)-\L m J-)-V T „„MLaHUH + ])-N i (t)
LaH,H
(7.9)
This general expression describes the growth characteristics of both target and
antagonist.* The growth-related lactic acid production — particularly by the antag-
onist — requires an additional coupled differential equation:
d[LaH], o ,.(t)
dt
= n(-)-N i (t)
(7.10)
with 7i(-) the specific lactic acid production rate of the antagonistic bacterium (i).
Note that [LaH] tot refers to the total lactic acid concentration, i.e., the sum of the
undissociated and dissociated lactic acid concentration. In case multiple lactic acid
producing strains are present, the overall growth rate of each strain will be affected
by the sum of all [LaH] tot , concentrations.
To describe the chemical process of lactic acid dissociation in complex media,
several methods inspired by traditional chemical laws are available (e.g., References
48 and 49). Given the medium, the process of lactic acid dissociation can be fully
identified irrespective of the microbial model. Observe that [LaH] and [H + ] vary
with time and are determined by the lactic acid producing strain and the dissociation
properties of [LaH] tot in the growth medium.
Several inhibitory functions can be proposed for \i + ([LaH], [H + ]). On the
LaH,H
basis of a rigorous model structure evaluation, Vereecken et al. translated the
inhibitory effect of undissociated lactic acid and the proton concentration (pH) into
the following equation:
M
LaH,H
(
1-
V
[LaH]
[LaH]
\
a
max J
=
f
1-
[H + ]
\
-
V
[/n
3
max /
when [LaH] < [LaH]
and [H + ]<[H + ]
max
'max
when [LaH] > [LaH]
or[H + ]>[H + ]
max
■max
(7.11)
with [LaH] max the lactic acid concentration at which growth ceases, [H + ] max the proton
concentration associated with the minimum pH for growth, and a and P some small
positive values. The inhibition terms have no effect on the microbial dynamics as
long as the undissociated lactic acid concentration and proton concentration remain
well below their inhibitory value. In such cases, both functions are approximately
equal to 1 . When [LaH] and [H + ] become significant as time proceeds, either function
evolves towards and growth stagnates.
To complete the model structure, the specific lactic acid production rate needs
to be mathematically modeled. Combining the traditional linear law with the concept
*
|: The subscript i thus refers to either the antagonist or the target.
2004 by Robin C. McKellar and Xuewen Lu
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of metabolism inhibitory concentrations, 7i(-) in Equation 7.10 can be represented
as follows:
"0 = y MW| -V l (-)+Y mi ([LaHUH + ]) (7.12)
4 * ' v -^ '
growth maintenance
with Y LaWN the yield coefficient [mmol cfir 1 ], |i 7 (-) the overall specific growth rate
(embracing the terms [l lag , |l max , and |J L «//,// + within Equation 7.9), and Y mi ([LaH],
[H + ]) the maintenance coefficient [mmol cfu -1 fr 1 ]. The first factors present the
growth-related production. The maintenance coefficient assures the observed pro-
duction of [LaH] tot during the first hours of the stationary phase. This maintenance-
related production also ceases when some inhibitory proton or undissociated lactic
acid concentration is reached. 50
The general model structure consisting of the coupled differential Equation 7.9
and Equation 7.10 yields accurate prediction for monocultures as well as mixed-
culture growth. This is illustrated for experimental data of Lactobacillus sakei and
Y. enterocolitica in Figure 7.4. More details on parameter values and the practical
model implementation are available in References 46 to 48.
7.4.3 Discussion of Results
The model building strategy described in this example starts from the identification
of main phenomena determining the dynamics of the microbial system. The derived
general model structure allows the stationary phase to be described in a natural
(mechanistically sound) and consistent way. Moreover, the mechanistically inspired
model structure can easily describe both single species and multiple species dynam-
ics (with interaction).
To conclude, the present example illustrates how microbial growth on itself may
cause a dynamic change of the environmental conditions, e.g., by the production of
metabolites.
7.5 CONCLUSIONS
Dynamic mathematical models allow for a consistent computation of the impact of
different steps associated with the production, distribution, and retailing of a food
(characterized by time- varying conditions) on microbial dynamics. Moreover, the
intrinsic properties of microbial evolution such as growth-related product formation
and inhibition can be easily integrated and predicted.
Examples given in this paper illustrate how we can learn from predictive mod-
eling based on biological and physical ideas. The individual-based modeling
approach, for instance, serves as an excellent tool to test generic cell mechanisms
with respect to the observed population behavior. However, such a modeling
approach with an increased level of detail demands more advanced measurements
at the cell or population level or both.
2004 by Robin C. McKellar and Xuewen Lu
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In view of expanding the applicability of predictive models, researchers must
be encouraged to aim at an increased generality — and thus transferability — of
model structures. For example, the complete model structure established in Section
7.4 can describe the individual behavior of lactic acid bacteria as well as the
inhibitory mechanism in the presence of pathogenic or spoilage bacteria. This exam-
ple also illustrates that cell density measurements are not always sufficient to estab-
lish complex model structures. Components interfering with the microbial dynamics,
such as metabolite formation, should be identified, measured, and built into the
model structure. Given this increased (experimental) knowledge on the microbial
dynamics, we can aim at more robust mechanistically inspired models yielding a
high predictive quality.
In this respect, it ought to be stressed that model builders can learn (more) from
dynamic experimental data. Microbial dynamics under realistically time-varying
conditions are not necessarily observable from (commonly available) static data. In
the first example (see Section 7.3), the application of time-varying temperature
profiles revealed the induction of an intermediate lag phase during the exponential
growth of E. coli.
When extrapolating model structures established on static experimental data to
more realistic dynamic conditions, e.g., combination of processing steps, model
predictions may fail to describe the microbial evolution accurately. Stephens et al., 51
for example, observed that slow heating rates applied during inactivation of Listeria
monocytogenes induced thermotolerance. Predictions using an inactivation model
developed on static experiments (not taking into account the magnitude of heating
rate) systematically overestimate the effect of the applied heat treatment. Future
research should thus pay attention to dynamic model development using dynamic
experimental data. In such cases only, complementary effects of dynamic conditions
or subsequent treatments can be properly incorporated within the model structure.
Observe that synergetic effects form the basic principles within the hurdle technology
(see, e.g., Reference 52), which is often addressed in the food industry.
Overall, model improvement aims at an increased predictive accuracy. However,
striving for this increased modeling accuracy, one must always keep an eye on the
model structure complexity. In this respect, it must always be clearly specified for
which purpose the model is being developed. An important challenge for the future
is therefore the search for a satisfactory trade-off between predictive power and
manageability of mathematical models: When is simple good enough? (after Refer-
ence 53).
ACKNOWLEDGMENTS
This research was supported by the Research Council of the Katholieke Universiteit
Leuven, the Fund for Scientific Research - Flanders (FWO), the European Com-
mission, the Belgian Program on Interuniversity Poles of Attraction, and the Second
Multi -Annual Scientific Support Plan for a Sustainable Development Policy, initiated
by the Belgian State, Prime Minister's Office for Science, Technology and Culture.
2004 by Robin C. McKellar and Xuewen Lu
1237_C07.fm Page 259 Wednesday, November 12, 2003 1:04 PM
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