Analysis and practical implementation of a model for combined growth and metabolite production of lactic acid bacteria

Analysis and practical implementation of a model for combined
growth and metabolite production of lactic acid bacteria
Karen M. Vereecken, Jan F. Van Impe
*
BioTeC—Bioprocess Technology and Control, Katholieke Universiteit Leuven, Kasteelpark Arenberg 22,
B-3001 Leuven (Heverlee), Belgium
Received 16 May 2001; accepted 9 August 2001
Abstract
Next to the traditional application of lactic acid bacteria (LAB) as starter cultures for food fermentations, the use of LAB as
protective cultures against microbial pathogens and spoilage organisms in other food production processes gains more and more
interest. The inhibitory effect of LAB is mainly accomplished through formation of antimicrobial metabolites. In this paper, the
model of Nicolaı¨ et al. [Food Microbiol. 10 (1993) 229.], describing cell growth and production of lactic acid, which is the
major end-product of LAB metabolism, is investigated. In contrast to classical predictive models, the transition of the expo-
nential growth phase to the stationary phase is obtained through the increasing concentrations of undissociated lactic acid [LaH]
and decreasing pH in the environment. To describe the variation in time of [LaH] and pH, a novel, robust calculation method
is introduced. The model of Nicolaı¨ et al. in combination with the novel method of [LaH] and pH computation is then further
applied to an experimental data set of Lactococcus lactis SL05 grown in a rich medium. An accurate description of the
measured values of cell concentration, total lactic acid concentration and pH is obtained. D 2002 Elsevier Science B.V. All
rights reserved.
Keywords: Lactic acid bacteria; Microbial growth; Metabolite production; Modelling
1. Introduction
Lactic acid bacteria are traditionally applied as
starter cultures for the production of fermented foods.
In these products, LAB have two major functions,
namely (i) achievement of certain beneficial physico-
chemical changes in the food ingredients, e.g., acid-
ification, curdling and production of flavour com-
pounds, and (ii) inhibition of the outgrowth of
microbial pathogens and spoilage organisms.
The antimicrobial potential of LAB together with
their status as Generally Regarded As Save (GRAS)

organisms has motivated a lot of researchers to study a
second possible application of LAB in the food indus-
try, i.e., as protective cultures, supplemented in mini-
mally processed foods (see, e.g., Matilla-Sandholm
and Skytta¨, 1996, for an overview). During manufac-
turing, these foods only undergo a mild inactivation
treatment in order to retain an optimal textural and
sensorial quality. LAB thus can serve as an additional
0168-1605/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved.
PII: S 0168-1605(01)00641-9
*
Corresponding author. Tel.: +32-16-32-19-47; fax: +32-16-
32-19-60.
E-mail address:
(J.F. Van Impe).
www.elsevier.com/locate/ijfoodmicro
International Journal of Food Microbiology 73 (2002) 239–250
biological hurdle to control the level of surviving but
undesirable organisms.
Both functions of LAB, whether fermentative or
protective, are mainly accomplished by the production
of microbial metabolites. LAB indeed produce a wide
variety of compounds, including low molecular
weight metabolites such as CO
2
,H
2
O
2
, organic acids,

alcohols and high molecular weight metabolites,
amongst which polysaccharides and bacteriocins.
Analogous to other industrial processes, the use of
mathematical models in the food industry is gaining
more and more attention for process evaluation, opti-
misation and design (Walls and Scott, 1997). For
mathematical models to be of use in the specif ic
subareas of food fermentation a nd biological preser-
vation, the following two basic aspects should be
taken into account.
.
In view of the above-m entioned importance of
microbial metabolites, an appropriate model should
incor porate both growth and produc tion character-
istics of LAB. As such, classical predictive growth
models, which focus on microbial growth, are not di-
rectly applicable.
.
Several microbial metabolites significantly mod-
ify the growth medium. Examples are organic acids ,
which cause a reduction in pH, and certain poly-
saccharides, which augment the medium solidity. A
model shoul d take into account these modifications
when they considerably affect microbial prolifera-
tion.
In this paper, a model from literature (Nicolaı¨ et al.,
1993) which addresses the two stated aspects, is in-
vestigated. The metabolite taken into consi deration in
this model is lactic acid, being the main end-product
of LAB metabolism. The contributions of the present

research can be subdivided in two main parts. A first
part, described in Section 3, involves a more theoret-
ical analysis of the model. Some mathematical proper-
ties of the model are discussed, and an easy-to-use
method to take into account the evolution of pH and
undissociated lactic acid [LaH] in the medium is pro-
posed. In a second part, discussed in Section 4, the
model and the newly developed method of pH and
[LaH] computation are applied to a case study, namely
the growth of Lactococcus lactis SL05 in a rich
medium.
A part of the results in this paper is also presented
in Vereecken and Van Impe (2000).
2. Materials and methods
The strain L. lactis SL05, selected by Arilait
(France), was kindly provided by ADRIA (France).
A fermentation experiment was performed in a 1-l er-
lenmeyer flask (Duran Schott), provided with a side-
arm. Openings (at the upper end and at the side- arm)
were closed with screw caps containing a rubber sep-
tum. The medium (500 ml) used was BHIYEG, con-
taining Brain Heart Infusion broth 37 g/l (Oxoid), sup-
plemented with 3 g/l Yeast Extract (Oxoid) and 18 g/l
glucose (Vel). Before inoculation the medium was
flushed with N
2
to obtain anaerobic conditions, and
pH correction to a value of 6.7 was performed with
HCl 4 N. The flask was incubated at 35 °C (Cooling
Incubator Series 6000, Termaks) on a rotary shaker

(Heidolph Unimax 2010) at 135 rpm. From a frozen
stock culture, an inoculation culture was obtained after
a first growth period of 24 h at 30 °C and a second
period of 16 h at the same temperature (both in tubes
containing 5 ml of BHIYEG). The initial cell density
was 10
5
cfu/ml. At regular time intervals, samples
were taken with a sterile syringe (Norm-Ject, Henke/
Sass/Wolf) and needle (Fine-Ject, Henke/Sass/Wolf)
through the lower septum. In these samples, bacterial
growth was quantified through determination of cfu/
ml on MRS medium (Oxoid) by means of a spiral
plater (Eddy-Jet, IUL Instruments) and pH was deter-
mined in the cell suspension with a pH sensor
(HI92240, Hanna Instruments). After filtration of the
sample to remove the cells (Sarstedt, pore size 0.2 mm)
and derivatisation of the lac tic acid present in the
filtrate with methanol, lactic acid was measured
through gas chromatographic analysis of the methyl
ester (separation with a Delsi GC on a 2-m column
with 10% carbowax 20 M on chromosorb W-AW 80/
100, detection with flame ionisation, peak integrator of
Trivector).
3. Analysis and adaptation of the model of Nicolaı¨
et al. (1993)
Nicolaı¨ et al. (1993) have constructed a dynamic
model for the surface growth of lactic acid bacteria on
vacuum-packed meats. In the current research, the
model is adapted to describe bacterial growth in a li-

quid medium. While the original model considers an
K.M. Vereecken, J.F. Van Impe / International Journal of Food Microbiology 73 (2002) 239–250240
amount of cells and lactic acid present in a surface
liquid layer with known dimensions, the adapted
model describes the concentrations of cells and lactic
acid in a liquid medium.
Microbial growth and lactic acid production are
described by the following set of d ifferential equa-
tions.
dN
dt
¼ lN ð1Þ
dLaH
tot
dt
¼ pN ð2Þ
with N the cell concent ration [cfu/ml] and LaH
tot
the
total concentration of lactic acid [mM], i.e., produced
by the organism plus initially present. In these equa-
tions, the specific growth rate l [h
À 1
] and the specific
production rate p [10
À 3
mmol h
À 1
cfu
À 1

] depend
on the concentrations of undissociated lactic acid
[LaH] [mM] and pH (or hydrogen ionic concentration
[H
+
] [mM]) as follows.
l ¼ l
max
½LaHV½LaH
min
l ¼ l
max
expfÀk
l
ð½LaHÀ½LaH
min
Þg ½LaH > ½LaH
min
ð3Þ
p ¼ p
max
½LaHV½LaH
min
p ¼ p
max
expfÀk
p
ð½LaHÀ½LaH
min
Þg ½LaH > ½LaH

min
ð4Þ
k
l
¼ a þ
b
½H
þ

2
ð5Þ
k
p
¼ c þ
d
½H
þ

2
ð6Þ
with l
max
[h
À 1
] and p
max
[10
À 3
mmol h
À 1

cfu
À 1
]
the maximum specific growth and production rates,
respectively, [LaH]
min
[mM] the minimum inhibitory
concentration of undissociated lactic acid, and a, b, c
and d parameters.
Nicolaı¨ et al. assume a single buffer system present
in the medium, consisti ng of a weak acid BuH and its
well solvable conjugated salt BuM. For this research,
it is assumed that the medium also contains a strong
acid AH, allowing to manipulate the initial medium
pH (as in the experimental case study, where the initial
pH is adapted with HCl—see Section 2). The follow-
ing chemical reactions can be written.
LaH X
K
LaH
La
À
þ H
þ
BuH X
K
BuH
Bu
À
þ H

þ
BuM ! Bu
À
þ M
þ
AH ! A
À
þ H
þ
with dissociation constants K
LaH
and K
BuH
. In combi-
nation with the associated charge and mass balances,
the following two algebraic expressions can be
derived.
½H
þ
¼ÀBuM
a
þ AH
a
þ
K
LaH
LaH
tot
K
LaH

þ½H
þ

þ
K
BuH
ðBuH
a
þ BuM
a
Þ
K
BuH
þ½H
þ

ð7Þ
½LaH¼
½H
þ
LaH
tot
K
LaH
þ½H
þ

ð8Þ
with BuH
a

, BuM
a
and AH
a
analytical concentrations
[mM]. Observe that Eq. (7) is cubic in [H
+
]. In Nicolaı¨
et al. (1993), it is proven that only one root of the latter
equation can be positive real.
It should be mentioned that Nicolaı¨ et al. consid-
ered two additional equations to account for the in-
fluence of pH on l
max
and p
max
. However, the latter
dependencies refer to the initial pH, as a medium
characteristic, rather than to the hydrogen ions result-
ing from microbial metabolism. It is indeed well
known that during the exponential growth phase,
when a microbial population reaches maximum values
for the specific rates, the metabolites accumulate
slowly and their influence on growth and production
kinetics is small. Therefore, since the medium and
associated initial pH is not a variable in the case study
presented in this paper, l
max
and p
max

can be reason-
ably assumed constant.
Among microbiologists it is widely accepted that
metabolites play a major role in the transition be-
tween different growth phases. Microbial metabolites,
K.M. Vereecken, J.F. Van Impe / International Journal of Food Microbiology 73 (2002) 239–250 241
whether growth or non-growth associated, can have a
specific effect on cell formation. The mutual interfer-
ence of microbial cells and products, which is in fact
a typical example of intraspecies interactions, can be
mathematically translated into a set of coupled differ-
ential equations, as presented in Eqs. (1) and (2).
The Nicolaı¨ et al. model focuses on the transition
from the exponential growth phase, characterise d by a
maximum value for the specific growth rate l, to the
stationary phase, in whi ch l becomes equal to zero.
The decline in the specific growth rate is in this model
obtained as a consequence of an increasing amount of
lactic acid in the environment. It should be noted here
that the model of Nicolaı¨ et al. differs in this respect
from the classical predictive growth models (e.g., the
modified Gompertz equation, Zwietering et al., 1990,
the model of Baranyi and Roberts, 1994). The basic
equation behind the classical models is
dN
dt
¼ lðN ÞN ¼ l
max
f ðNÞN : ð9Þ
In this equation, the specific growth rate l depends

upon the cell concentration N and a decrease from a
maximum value towards zero—or otherwise said a
transition from the exponential growth to the sta-
tionary phase—is obtained through a factor f (N),
which decreases with increasing N until a maximum
population density N
max
is reached for which f (N),
and thus l(N), become equal to zero.
When comparing the Nicolaı¨ et al. approach
towards modelling of the stationary phase with the
classical models, it can be concluded that the model of
Nicolaı¨ et al. more closely matches the underlying
mechanistic principles of cell growth retardation and
ceasing.
The metabolite under study in the model of Nicolaı¨
et al. is lactic acid, which inhibits microbial cell growth
via two components, hydrogen ions and undissociated
lactic acid. Model accuracy and reliability therefore
strongly depend upon a proper description of the in-
fluence of these two components on the specific rates l
and p. Using (i) Eqs. (3), (5) and (8), (ii) the same
values for the parameter s [LaH]
min
, a, b and K
LaH
as
mentioned in Nicolaı¨ et al. (1993), (iii) a value of
0.6920 [h
À 1

] for the parameter l
max
(which corre-
sponds with the value of l
max
at the optimal pH of 5.74
for the species Lactobacillus delbrueckii considered in
the cited paper), and (iv) an initial pH of 6.3, the
specific growth rate is visualised in Fig. 1 in two and
three dimensions. The different lines in the two-dimen-
sional plots represent intersections of the surface in
Fig. 1a parallel to the [LaH]-axis (i.e., lines of equal
values of pH) (Fig. 1b) or parallel to the pH-axis (i.e.,
lines of equal values of [LaH]) (Fig. 1c). From Fig. 1b,
it can be seen that, after a period in which l remains at
a constant value l
max
, it decreases with increasing
[LaH], which is to be expected from a microbiological
viewpoint. On the other hand, Fig. 1c reveals that the
specific growth rate increases— although no t very
pronounced—with increasing [H
+
] (or decreasing
pH) which is unacceptable in the suboptimal range
of pH. The same conclusio ns can be drawn when
considering the partial derivatives of  with respect
to [LaH] and [H
+
]. If [LaH] > [LaH]

min
the following
relations apply
@l
@½LaH
¼Àl
max
k
l
exp Àk
l
ð½LaHÀ½LaH
min
Þ
ÈÉ
ð10Þ
@l
@½H
þ

¼ 2l
max
ð½LaHÀ½LaH
min
Þ
À
b=½H
þ

3

Á
 exp Àk
l
ð½LaHÀ½LaH
min
Þ
ÈÉ
: ð11Þ
If [LaH] V [LaH]
min
, @l/@[H
+
] and @l/@[LaH] are of
course equal to zero. When all model parameters are
positive, @l/@[LaH] is negative, involving a decrease
in l with decreasing [LaH], and @l/@[H
+
] is positive,
implying an increase in l with decreasing pH.
To assure that the specific growth rate decreases
with increasing [LaH] and [H
+
], relations (10) and
(11) should be negative and thus the following
inequalities must be satisfied.
b < 0 ð12Þ
k
l
¼ a þ
b

½H
þ

2
> 0 ð13Þ
While a should in deed be a pos itive parameter, b
cannot be positive. Analogous conditions can be de-
rived for the parameters c and d which appear in the
expression of the specific production rate. When
applying the model to experimental data, as in Section
4, one should take into account these four constraints
K.M. Vereecken, J.F. Van Impe / International Journal of Food Microbiology 73 (2002) 239–250242
on the parameter values a, b, c and d in order to avoid
unrealistic model predictions.
Next to the expressions relating the specific growth
rate and the specific production rate to [LaH] and pH,
the accuracy of latter input values is also of major im-
portance. Therefore, an adequate calculation method
for the actual values of [LaH] and pH at each time
point during growth is necessary. According to classi-
cal chemical laws, the pH is fully determined when the
total amount of lactic acid in the medium and the
buffering ca pacity of the medium are known. The
amount of undissociated lactic acid is then further
related to the total amount of lactic acid LaH
tot
and
the pH by the lactic acid chemical equilibrium. As
explained above, the approach of Nicolaı¨ et al. is based
on these chemical principles and assumes a single

buffer compound BuH
a
/BuM
a
present in the medium.
For the present research, the precise interdependency
between the three factors LaH
tot
, pH and [LaH] is
further investigated. Fig. 2 represents hereby a number
of two-dimensional plots in which the relations
between each couple of the variables LaH
tot
, pH and
[LaH] are visualised. By simulating cell growth and
lactic acid formation making use of model Eqs. (1) –
(8), time-dependent courses of LaH
tot
, pH and [LaH]
are obtained. Next, through elimination of time, plots
of [LaH] versus LaH
tot
(Fig. 2a and d), pH versus
[LaH] (F ig. 2b and e) and pH versus LaH
tot
(Fig. 2c
and f) are constructed. The arrows indicate the direc-
tions which are followed during the simulated fermen-
Fig. 1. Specific growth rate  as function of [LaH] and pH (relations (3) and (5) with l
max

= 6.9200 Â 10
À 1
[h
À 1
], [LaH]
min
= 1.333 [mM],
a = 5.7600 Â 10
À 2
[mM
À 1
], b = 9.9000 Â 10
À 7
[mM]). (a) Three-dimensional representation; (b) l as function of [LaH] with the different
lines representing lines of equal pH values; (c) l as function of pH with the different lines representing lines of equal [LaH] values.
K.M. Vereecken, J.F. Van Impe / International Journal of Food Microbiology 73 (2002) 239–250 243
Fig. 2. Two-dimensional relations between the variables LaH
tot
, [LaH] and pH. (a, b, c): variation of the initial pH (pH
0
). (d, e, f): variation of the buffer amount (BuH
a
/BuM
a
).
K.M. Vereecken, J.F. Van Impe / International Journal of Food Microbiology 73 (2002) 239–250244
tations. In each figure, a reference curve is realised
(full line) by means of the parameter values used in the
work of Nicolaı¨ et al. to construct Fig. 3(1), p. 234 in
their paper. Furthermore, to comprehend the influence

of the buffer amount BuH
a
/BuM
a
and initial pH, two
additional curves are generated in each plot for differ-
ent levels of these factors, higher or lowe r than the
ones used in the reference curve, each level being
marked on the different plots. It should be noted here
that for an individual plot, relating two variables out of
three to each other, the one left out of consideration is
not a constant. For example, in the plot of pH versus
[LaH] (Fig. 2b and e), the total lactic acid concen-
tration gradually increases when traversing the curves
from left to right.
On the basis of the six plots, the following con-
clusions can be drawn.
.
From the [LaH] versus LaH
tot
plots (Fig. 2a and
d), it can be seen that when the initial pH is equal to
3.86 (which corresponds to a hydrogen ionic concen-
tration equal to the dissociation constant of lactic acid)
(Fig. 2a) or when no buffer BuH
a
/BuM
a
is present
(Fig. 2d), the curve is more or less a straight line

passing through the origin. In case of a higher initial
pH and/or in the presence of buffer, the curve initially
increases very slowly and approaches a straight line for
larger LaH
tot
values. The slope of this straight line is
always the same. Furthermore, when the pH and/or
buffer amount further increase, the intersection of this
line with the LaH
tot
-axis increases, or otherwise said,
the straight line is shifted to the right.
.
The pH versus [LaH] plots (Fig. 2b and e) are
monotonically decreasing curves in which no inflec-
tion points occur. The initial pH determines the in-
tersection point with the pH-axis (Fig. 2b) while the
buffer amount determines the deviation from the
initial pH at a given (non-zero) [LaH] value (Fig. 2e).
.
Finally, the pH versus LaH
tot
plots (Fig. 2c and f)
are monotonically decreasing curves with a variable
number of inflection points. The intersection with the
pH-axis is self-evidently the initial pH.
As already mentioned, a main objective of the pre-
sent research is the application of the model of Nicolaı¨
et al. to experimental data. This implies attributing
numerical values to the buffer system BuH

a
/BuM
a
and its associated dissociation constant K
BuH
(relation
(7)). This is, however, not straightforward for two
reasons. Firstly, the medium used in the present case
study is a rich and nutritious, yet undefined commer-
cial growth medium, which presumably contains more
than one substance with buffering capacity. If—or, to
what extent—the buffering substances can be merged
together into one o verall buffering compound is
uncertain. Secondly, Eq. (7), in which the positive
Fig. 3. (a) Representation of relation (14). A reference curve a, b, c (bold line) approximates the LaH
tot
-axis for small LaH
tot
values and a
straight line with slope a for larger LaH
tot
values (thin full lines). Curves for different parameter values are indicated. (b) Representation of
relation (15).
K.M. Vereecken, J.F. Van Impe / International Journal of Food Microbiology 73 (2002) 239–250 245
real root needs to be selected and used to calculate pH
and [LaH]—give n the experimentally measured pro-
file of LaH
tot
—is from a mathematical viewpoint a
rather complex expression on the basis of which the

unknown values BuH
a
, BuM
a
and K
BuH
are to be
identified.
For these reasons, in this research an alternative
method for pH and [LaH] computation is proposed,
which can be used in combination with all models of
the general class (Eqs. (1) and (2)). The method is
inspired by Fig. 2a –e discussed above.
.
To represent the relation between [LaH] and
LaH
tot
(Fig. 2a and d), the following equation,
inspired by Dabes-kinetics (Van Impe et al., 1994),
is proposed.
½LaH¼aLaH
tot
À
ab
2ðb À cÞ
h
ðLaH
tot
þ bÞ
À

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðLaH
tot
þ bÞ
2
À 4ðb À cÞLaH
tot
q
i
ð14Þ
A graphical representation of this relation for different
parameter values can be found in Fig. 3a. As can be
seen from this figure, the three parameters in this
expression are all easy interpretable: (i) a equals the
slope of the straight lines with whom the different
curves in Fig. 2a and d nearly coincide for larger
LaH
tot
values, (ii) b equals the intersection point of
this straight line wi th the LaH
tot
-axis, and (iii) c de-
termines the transition from the slowly increasing to
the constant slope region of the curves. From Fig. 2a
and d, it can be easily seen that the parameters a and c
are in fact constan t for all the curves shown and b is the
only parameter that varies for different values of initial
pH or buffer amount.
.
To represent the relation between pH and [LaH]

(Fig. 2b and e), a second expression, also based on the
Dabes-kinetics, is developed.
pH ¼
1
2a
1
c
1
h
ðb
1
À 2c
1
Þ½LaH
À
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b
2
1
½LaH
2
þ 4a
1
b
2
1
c
1
½LaH
q

i
þ pH
0
ð15Þ
Parameters a
1
, b
1
and c
1
have a similar interpretation
as a, b and c, respectively, in Eq. (14), while pH
0
symbolises the initial pH. A graphical representation
of expression (15) is depicted in Fig. 3b.
Combination of the two proposed relations (14) and
(15) enables to calculate [LaH] and pH values starting
from increasing (experimentally measured) LaH
tot
values. In contrast to Eqs. (7) and (8), all parameters
in rel ation s (14) an d (15) h ave a straightforw ar d
interpretation in connection with the curves of Fig. 2.
Of course, it should be kept in mind that the novel
method also assumes, although more implicitly than
the method of Nicolaı¨ et al., a single buffer compound
or a single buffer-like behaviour of the total of all
buffering substances. In the next section, where the
proposed method will be tested on real experimental
data, some indications on the validity of this under-
lying assumption will be presented.

In summary, the proposed model consists out of
Eqs. (1) –(6) completed with Eqs. (14) and (15).
4. Practical implementation
In this section, the model o f Nicolaı¨ et al., in
combination with the newly developed method of
pH and [LaH] calculation, will be applied to the
experimental case study described in Section 2. The
lactic acid bacterium and the growth medium used are
selected carefully in order to be compliant with two
basic aspect s of the model of Ni colaı¨ et al. Firstly, the
bacterial strain has a homofermentative metabolism
and, as such, lactic acid is the only end-product that
has to be taken into account. Secondly, the rich
growth medium with high glucose content assures
that no substrate limitation occurs and the influence of
substrate concentration on growth and production
kinetics can be neg lected.
Application of the adapted model of Nicolaı¨ et al.
to the experimental data set of cell concentration N,
total lactic acid concentration LaH
tot
and pH consists
out of two major steps, namely, (i) determination of
medium related parameters (i.e., the parameters of
Eqs. (14) and (15)), and (ii) determination of growth
and production related parameters (i.e., the parameters
of Eqs. (1) –(6)). This will be explained in detail in the
further text.
4.1. Determination of medium related parameters
Eqs. (14) and (15) constitute the newly developed

method of pH and [LaH] calculation and their param-
K.M. Vereecken, J.F. Van Impe / International Journal of Food Microbiology 73 (2002) 239–250246
eters can be determined on the basis of the LaH
tot
and
pH meas urements, without considering the cell c on-
centration data. At each measurement instant t,a
couple of [LaH
tot
(t), pH(t)] values is available. Start-
ing from these [LaH
tot
(t), pH(t)] couples, the estima-
tion procedure is as follows.
(1) Since both expressions (14) and (15) involve the
undissociated lactic acid concentration [LaH], which is
not measured experimentally, a value for [LaH](t)is
calculated for each [LaH
tot
(t), pH(t)] couple, by mak-
ing use of Eq. (8).
(2) Next, the computed values of [LaH] are plotted
versus LaH
tot
or versus pH at matching time instants.
Fig. 4 displays both plots for the experimental values
of the case study (circles).
(3) Based on these plots, the parameters a, b and c
of expression (14) and a
1

, b
1
, c
1
and pH
0
of expression
(15) are identified. Simulations of relations (14) and
(15) with the estimated parameter values for the case
study are presented in Fig. 4 as full lines. It can be
concluded that a realistic description of the plots is
obtained. Moreover, the shape of the plots corresponds
very well with the ones displayed in Fig. 2, providing
support for the assumption of a single buffer-like
behaviour as stated above.
(4) Finally, given the estimated values of a, b, c, a
1
,
b
1
, c
1
and pH
0
, prediction of any time dep endent
course of pH (and [LaH]) is possible starting from an
experimentally measured LaH
tot
profile. Fig. 5 depicts
the experimental values of pH as a function of time,

together with a prediction based on the new method
and the experimental LaH
tot
values, linearly interpo-
lated between two subsequent observations. It can be
seen that a combination of relations (14) and (15) with
the estimated parameter s provides a satisfactory pre-
diction of the measured pH values.
Fig. 5. Variation of pH in function of time for the experimental case
study; ‘6’: experimental data; ‘—’: simulation based on relations
(14) and (15) and the experimental LaH
tot
values, linearly
interpolated.
Fig. 4. [LaH] versus LaH
tot
(a) and pH versus [LaH] (b) for the experimental case study; ‘6’: experimental data; ‘—’: description with relation
(14) (a) or relation (15) (b).
K.M. Vereecken, J.F. Van Impe / International Journal of Food Microbiology 73 (2002) 239–250 247
It should be noted that the procedure described
here has to be accomplished only once for a given
medium and initial pH to calibrate the param eters in
the Eqs. (14) and (15). Indeed, since the parameters a,
b, c, a
1
, b
1
and c
1
only depend on the medi um char-

acteristics and the parameter pH
0
only on the initial
pH, they are suited for every growth experiment in this
particular medium and initial pH. After calibration,
prediction of the time dependent course of pH (and
[LaH]) is always possible when a LaH
tot
profile only
is at hand.
4.2. Determination of growth and production related
parameters
To identify the parameters of Eqs. (1)–(6), describ-
ing cell growth and metabolite production, the com-
plete data set (including cell, lactic acid and pH
measurements) is necessary. Obviously, use is made
of relations (14) and (15) and the determined values for
a, b, c, a
1
, b
1
, c
1
and pH
0
to determine the input values
[LaH](t)and[H
+
](t ) of the specific growth and
production rates (Eqs. (3)– (6)). As mentioned above,

the variation in time of pH is fully determined when
the LaH
tot
profile is available and the param eters a, b,
c, a
1
, b
1
, c
1
and pH
0
are known. Therefore, only N and
LaH
tot
values are taken into account in the cost
criterion during the calibration procedure. Further-
more, the constraints (12) and (13) on the parameter
values a, b, c and d, specified in the previous section,
have been taken into account. Table 1 lists the opti-
Table 1
Parameter values for the model of Nicolaı¨ et al. (1993) for L. lactis
SL05 in BHIYEG medium at a temperature of 35 °C and an initial
pH of 6.7
Parameter Units Value
l
max
[h
À 1
] 1.4160

p
max
[10
À 3
mmol h
À 1
cfu
À 1
] 7.2302 Â 10
À 8
[LaH]
min
[mM] 3.8417 Â 10
À 3
a [mM
À 1
] 3.3449 Â 10
À 1
b [mM] À 2.4118 Â 10
À 7
c [mM
À 1
] 3.8389 Â 10
À 1
d [mM] À 1.0000 Â 10
À 10
Fig. 6. L. lactis SL05 in BHIYEG medium at a temperature of 35 °C and an initial pH of 6.7; ‘6’: experimental data; ‘—’: description with the
adapted model of Nicolaı¨ et al.
K.M. Vereecken, J.F. Van Impe / International Journal of Food Microbiology 73 (2002) 239–250248
mised parameter values. The specified conditions for

the parameters a, b, c and d are satisfied for the es-
timated parameter values. Fig. 6 shows the cell con-
centration, total concentration of lactic acid and pH as
a funct ion of time. Displaying the curves in this way
reveals that LaH
tot
and pH are almost constant for a
long period while the cells are growing, followed by a
rather sudden increase and decrease respectively when
a cell concentration of approximately 10
7
cfu/ml is
reached. During these changes, the cells pass from the
exponential growth phase to the station ary growt h
phase. As mentioned in Section 3, this transition is
in the model of Nicolaı¨ et al. described as a conse-
quence of the increasing concentrations of [LaH] and
[H
+
] in the medium. From the three plots in Fig. 6, it
can be seen that the adapted model of Nicolaı¨ et al.
provides an accurate description of the three variables
N, LaH
tot
and pH.
5. Conclusions
The main contributions of this paper can be sum-
marised as follows.
.
The model of Nicolaı¨ et al. (1993), which consists

out of two coupled differential equations describing
cell growth and lactic acid production of LAB, is eval-
uated from a mat hematical point of view.
In contrast to classical predi ctive growth models,
the specific growth rate m and the specific production
rate p in this model explicitly depend on the concen-
trations of undissociated lactic acid [LaH] and hydro-
gen io ns [H
+
], originating from the lactic acid
produced by the micro-organism. Transition of the
exponential growth phase towards the stationary phase
is described as a consequence of increasing amounts of
[LaH] and [H
+
] in the environment.
Next, some constraints on the parameter values,
which should be taken into account for realistic model
predictions, are highlighted.
Further, a novel, robust method for calculation of
the variation in time of [LaH] and pH values, due to
the lactic acid production, is introduced.
.
Next to the mathematical analysis, the model and
the newly developed procedure of [LaH] and pH
computation are applied to an experimental case study,
namely the growth of L. lactis SL05 in a rich medium.
During the experiment, the cell concentration N, the
total lactic acid concentration LaH
tot

and the pH are
monitored. In a first step, the parameters which only
depend on the growth medium used and the initial pH,
are estimated by means of the experimentally meas-
ured values of LaH
tot
and pH. Secondly, the remaining
parameters are estimated on the basis of the complete
data set, including the cell concentration data. As such,
an accurate description of N, LaH
tot
and pH as a func-
tion of time is obtained.
Future research will focus on the experimental va-
lidation of the computation method of [LaH] and pH
for different growth media and different values for the
initial pH. Moreover, alternative model structures
which describe both cell growth and metabolite pro-
duction will be investigated.
Acknowledgements
This research has been supported by the Research
Council of the Katholieke Universiteit Leuven as part
of projects OT/99/24 and COF/98/008, the Institute for
the Promotion of Innovation by Science and Technol-
ogy in Flanders (IWT), the Fund for Scientific Re-
search-Flanders (FWO) as part of project G.0267.99,
the Belgian Program on Interuniversity Poles of At-
traction, initiated by the Belgian State, Prime Minis-
ter’s Office for Science, Technology and Culture, and
the European Commission as part of project EU-FAIR-

CT97-3129. The scientific responsibility is assumed
by its authors.
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