1. Introduction
Lactic Acid Bacteria (LAB) refer to a group of microorganisms that share certain morphological, physiological, and metabolic characteristics. They have the peculiarity of producing lactic acid from several carbohydrates through a process known as microbial fermentation. They are normally found in cultures such as milk, whey, and pickles [
1,
2]. The Danish microbiologist Orla-Jensen said that “
True lactic acid bacteria are a large group of cocci and Gram positive, immobile, spore free bacilli, which produce lactic acid in the fermentation of sugar” [
3]. Currently, LAB have a primary role in the food industry, as they are used to acidify and preserve food. Further, they contribute to texture, taste, smell, and scent development in all kinds of fermented foods [
4,
5].
In the industry, most fermentation is carried out through batch culture. This means that only a culture medium with the necessary nutrients is added to the fermenter, and this is incubated for the growth of bacteria and the production of its primary metabolite. During the time of incubation, nothing other than oxygen is added [
6]. Hence, the bacterial growth follows an asymmetric cell division (ACD), which is a conserved mechanism evolved to generate cellular diversity. A key principle of ACD is the establishment of distinct sibling cell fates by mechanisms linked to mitosis [
7]. ACD also occurs in the development and physiology of unicellular organisms ranging from bacterial species to yeasts and flagellates [
8,
9]. In these organisms, ACD underlies replicative aging as a means of maintaining the immortality of the mitotically proliferating population [
10].
LAB are very demanding microorganisms and need a set of growing factors, including sugars such as carbohydrates, as well as amino acids and vitamins. Therefore, LAB can only grow in a medium that supplies these nutrients [
3]. The interest in the physiology of LAB has been studied by its industrial importance and potential use of genetic engineering in strain optimization. For example, in the particular case of
Lactococcus lactis spp.
lactis, a minimal growth medium should contain glucose, acetate, vitamins, and amino acids [
11]. Milk is the medium that contains all these nutrients. For this reason, LAB are used as starter cultures for the preparation and preservation of dairy products, such as acidified milk, yogurt, butter, cream, and cheese [
1]. In batch fermentation, bacteria cannot grow exponentially indefinitely. The latter due to bacteria depleting nutrients as it grows, changes in the chemical composition of the medium
, and toxic compounds that are accumulated. The characteristic curve of bacterial growth is composed by four phases: (i) lag, (ii) exponential, (iii) stationary, and (iv) death. Lactic acid production occurs during the reproduction of LAB; the accumulation of this and other organic acids decrease the
of the medium, which made culture conditions become more selective; hence, the more acid-tolerant bacterium will prevail.
The total number of cells in fermentation are usually represented by a logarithmic scale, and normally, at the end of this process, one can see values from
up to
CFU/mL (Cell-Forming Units per milliliter) [
2]. In addition to nutritional requirements, temperature is one of the most important factors in LAB growth. Furthermore, there is an optimal temperature and other environmental conditions for which these microorganisms present a higher growth rate [
4]. LAB are acid-tolerant and may grow either at
values as low as
or as high as
. Usually, most of LAB strains grow at
values between 4 and
[
1].
In microbiology, several mathematical models have been developed to predict the behavior of microorganisms on the influence of different factors in culture. This branch is known as predictive microbiology, and it aims to study the response of bacteria to environmental factors that can be controlled, such as temperature,
, and water volume, among others. In this sense, a tool may be formulated to ensure both quality and safety of products to estimate their useful lifetime and to make decisions regarding the composition and design of fermented products [
12]. Mathematical models that relate biomass production and time are called primary models, while those that describe the relationship between primary models parameters and environmental conditions are known as secondary models [
13]. Some of these models are discussed below.
The Gompertz equation, formulated in 1825, is one of the most used in predictive microbiology. It is based on an exponential relationship between the growth rate and the density of a population. This equation was formulated to describe the law of human mortality. However, years later, it was adapted and re-parametrized for its use in microbiology [
14]. Giraud et al. [
15] observed that
variations resulted in a decrease in the growth rate when measuring the fermentation performance of
L. plantarum at a controlled
between
and
. Baranyi et al. [
16] applied a non-autonomous differential equation to describe the dynamics of growing bacterial cultures. Based on more than 500 growth curves, the statistical properties of this equation were compared to the Gompertz approach, which is the most commonly used in food microbiology. After these results, Baranyi and Roberts [
17] proposed a growth model where a single variable represents the physiological state of cells. The lag phase period is determined by the value of this variable at inoculation and by the post-inoculation environment. The model was able to describe bacterial growth in an environment where factors, such as temperature and
, change over time. Nicolai et al. [
18] constructed a dynamical model for the growth of LAB in vacuum-packed meat. The model was divided into two parts: one part describing the fermentation and the other describing the
evolution in the liquid surface layer. These models were given as two differential equations and two algebraic equations, respectively. Passos et al. [
19] developed an unstructured model to describe bacterial growth, substrate utilization, and lactic acid production by
L. plantarum in cucumber juice. They also developed an equation that relates the specific mortality rate and the sodium chloride
concentration. Drosinos et al. [
20] formulated an empirical model to describe the growth and production of bacteriocin by
Leuconostoc mesenteroides under different conditions of
and temperature. Further, a De Man, Rogosa, and Sharpe (MRS) broth was used as a growth medium in the fermenter. Vázquez and Murado [
21] proposed a model based on the re-parametrized logistic equation to describe fermentation kinetics of lactic acid production by
Lactococcus lactis and
Pediococcus acidilactici in a batch system. Da Silva et al. [
22] studied the growth of
L. plantarum,
W. viridescens, and
L. sakei under different isothermal culture conditions at temperatures of 4, 8, 12, 16, 20, and 30 ∘C. They determined that LAB growth was strongly influenced by the culture temperature and created new models that allowed them to predict growth at temperatures ranging from 4 to 30 ∘C. Dalcanton et al. [
23] built a response model of the growth rate of
L. plantarum as a function of temperature,
, and concentrations of
and sodium lactate
.
Despite the existence of several mathematical models in the literature that describe the behavior of LAB during the fermentation process in its three phases, i.e., lag, exponential, and stationary phases, they do not include the death phase in these models. In batch fermentation, after the stationary state, the depletion of nutrients in the culture medium occurs, and therefore, the death phase begins and should be considered in these models. By not taking the death phase into account, a problem arises when trying to fit real-life experimental data with the models usually used in predictive microbiology. Therefore, the objective of this research is to formulate a time-variant mathematical model that describes the growth of LAB, including its death phase in batch fermentation.
The remainder of this paper proceeds as follows. In
Section 2, we present the real-life experimental data concerning biomass growth for eight LAB strains; we formulate our cubic mathematical and explain each parameter; and the most important kinetic models identified in the literature are explored. In
Section 3, a statistical analysis is performed to establish which model better fits the experimental data, and these results are illustrated by means of several numerical simulations. Finally, discussions are described in
Section 4, and conclusions are given in
Section 5.
3. Results
In order to validate our model, a statistical analysis was performed to calculate and compare the descriptive capacity of the cubic model (
14) and models (
15)–(
17). The statistic used was the coefficient of determination
, which measures the goodness of fit of a non-linear model according to experimental data. Its calculation is made based on the sum of squared residuals
, and the explanatory capacity of a model is better the closer
is to 1, see Section 11-8.2 [
26].
Figure 3,
Figure 4,
Figure 5 and
Figure 6 illustrate the dynamics of the cubic model and its comparison with analyzed models from the literature concerning the experimental data of eight LAB strains,
. One can see in
Figure 3 that the cubic model describes the four expected growth phases in batch fermentation, while the Gompertz model, see
Figure 4; the Baranyi model, see
Figure 5; and Vázquez-Murado model, see
Figure 6; only describe up to the stationary phase, which caused the
value to be higher for these models, and therefore, the results concerning
and
indicate a poorer fit. The lag phase time was around 5 to 9 h. The maximum biomass growth was reached between 30 and 36 h for each of the strains, and from that time, the death phase begins.
According to our numerical simulations, one can observe that the proposed cubic model (
14) and the other primary models (
15)–(
17) have a good adjustment in the lag phase, as is shown by each solution and the corresponding experimental data. Concerning the exponential phase, the Baranyi model is slightly above the others. However, this model has a lower growth rate, which causes a delay when reaching the maximum concentration, approximately at 48 h, while other models reach this value at around 30 or 36 h. Regarding the maximum biomass growth, the three primary models remain in the stationary phase, while the experimental data goes to the death phase. Therefore, the
of these models is higher than the cubic model, as this one better fits the observed data in this last phase.
Table 2 shows the results of
and
for the cubic, Gompertz, Baranyi, and Vázquez-Murado models with respect to the experimental data illustrated in
Figure 1.
Now, concerning strain 1, the cubic model presented an
of
and, consequently, a higher value of
given by
compared to Gompertz, Baranyi and Vázquez-Murado models, which obtained an
equal to
,
, and
and an
equal to
,
, and
, respectively. For strain 2, the highest
value was for the cubic model with
, followed closely by the Vázquez-Murado and Gompertz models with
and
, respectively, while the Baranyi model had a result of
. In the particular case of strain 3, all models had a lower adjustment when comparing them with the other seven strains. However, the best fit was for the cubic model with an
of
, while the Gompertz model obtained a value of
, Baranyi obtained
, and Vázquez-Murado
. For strain 4, very similar
values were obtained for Gompertz, and Vázquez-Murado models,
and
, respectively, while the cubic model obtained a result of
, and Baranyi got a value of
. In strain 5, the cubic model had an
value of
, which was higher than the Gompertz
, Baranyi
, and Vazquez-Murado
. Strain 6 had a better fit for all models. The Vázquez-Murado model had an
of
, Gompertz
, cubic model
, and the lowest value was for Baranyi, with
. These higher
values are due to fact that the experimental data for this strain have fewer outliers, as seen in
Figure 1, and its death phase is lower compared to the other strains. Strain 7 had a better fit for the cubic model, with an
of
, than the Gompertz
, Baranyi
, and Vázquez-Murado
models. Finally, as in most cases, the cubic model had the best fit for strain 8 with an
value of
, while Gompertz and Vázquez-Murado obtained, respectively,
and
. In general, the cubic model had higher values of
than the rest of the models, except in strain 6, where the Vázquez-Murado model had the better fit to the experimental data. The latter was due to the data not yet presenting the death phase as the other strains in the 48 h of the experiment.
Furthermore, the Akaike Information Criterion
test was performed, which allows us to determine which model better fits the observed data. To calculate this value, the goodness of fit between predictions of models and experimental data is considered through the
while penalizing models that have a greater number of parameters due to these becoming more complex for its practice [
27].
Table 3 shows results for every mathematical model.
By taking the latter into account, the model that better fits the experimental data is the one with the lowest value. For strain 1, the lowest AIC was for the cubic model with a value of , while Gompertz, Baranyi, and Vázquez-Murado models obtained values of , , and , respectively. In strain 2, the cubic model had the lowest , with a value of , and the Gompertz model and the Vázquez-Murado model had similar results, and respectively, while the Baranyi had a value of . The cubic model fitted better to the experimental data of strain 3 with an of , followed by Gompertz with , Vázquez-Murado with , and finally, Baranyi with . Again, for strain 4, according to the values, the cubic model () was better than the Gompertz (), Baranyi (), and Vázquez-Murado () models. In strain 5, the best value was for the cubic model (), while Gompertz obtained , Baranyi , and Vázquez-Murado . In strain 6, very close values were obtained for Vazquez-Murado and Gompertz , but the Vázquez-Murado model was slightly better. The cubic model had a value of . On the other hand, Baranyi model had a lower fit with an of . For strain 7, the lowest value was obtained for the cubic model , while Gompertz had a value of , Baranyi , and Vázquez-Murado . Finally, regarding strain 8, the cubic model had the better fit to the experimental data based on its value (), and Baranyi had the worst fit (), while Gompertz and Vázquez-Murado obtained results of and , respectively.
After analyzing the statistical results, it is evident that our formulated cubic primary model better fits the experimental data because it was the one that presented the lowest
in general and therefore had higher values for the
in all LAB strains but number 6, which had a higher value with the Vázquez-Murado model
. However, the result for the cubic model was of
, which is still a good value concerning this parameter. Therefore, based on the results for the
,
, and the
, we can establish that the cubic primary model (
14) better represents the overall dynamics of the experimental data for the eight LAB strains under study in this research.
4. Discussion
In predictive microbiology, primary models based on sigmoidal functions are usually used. Among the most common models, we have the Gompertz [
14], the Baranyi [
17], and the Vázquez-Murado [
21] models. Nonetheless, the latter only describe the first three growth phases, i.e., lag phase, exponential growth, and the stationary state. According to Buchanan et al. [
28] and Garre et al. [
29], these phases are sufficient to fit experimental data for growth models. However, according to Mandigan et al. [
6], in batch fermentation, the bacterial population presents a death phase due to nutrient depletion in the culture medium. Therefore, when adjusting these models to observed data in this kind of fermentation, values of greater discrepancy are found in the death phase between the experimental data and approximated values. Chatterjee et al. [
30] incorporated the death phase into the Gompertz model to better describe the behavior of
E. coli and
S. aureus. Studies by Bevilacqua et al. [
31] were focused on demonstrating the importance of the death phase of microorganisms and its impact on the shelf life of food. Solano et al. [
32] evaluated the capacity of five models, including the Gompertz, Baranyi, and Vázquez-Murado, to predict acid production in lactic fermentation of fishing products, and they found that models with the lowest residual variance were those of Gompertz and Baranyi. Further, this study shows that the Vázquez-Murado model failed to give an adequate adjustment for lactic fermentation. In the same way, Zwietering et al. [
33] managed to describe the behavior of
L. plantarum in an MRS medium with the Gompertz model, while the Vázquez-Murado logistic model was not suitable to accurately approximate the experimental data. In contrast, Kedia et al. [
34] obtained a good fit with the Vázquez-Murado model for
L. reuteri,
L. plantarum, and
L. acidophilus in oat fractions. Da Silva et al. [
22] studied the growth of
L. plantarum,
W. viridescens, and
L. sakei in vacuum-packed meats, and, as well as Baty and Delignette [
35] who studied different growth models for various LAB, all authors in these two works found that the Baranyi model has a slightly better fit than the Gompertz model.
According to the
and
values, it can be established that the cubic model has a better fit to the experimental data with average values of
and
. There was not enough difference between the average values for the Gompertz model with
and
, while the Vázquez-Murado had average results of
and
. Regarding the Baranyi model, the average results were as follows
and
. Further, the overall adjustment to the experimental data for the four primary models under study is illustrated in
Figure 3,
Figure 4,
Figure 5 and
Figure 6.
In this research, the four primary models presented a good adjustment in the lag phase and reached the maximum growth registered in the experimental data. However, the maximum growth rate value had a greater impact on the adjustment of the exponential growth phase, in which the cubic model was the one that had the better fit. Nonetheless, in order to get a good fit with primary models, it is necessary to measure the largest amount of experimental data with the greatest possible continuity, which will ensure a better prediction by the models. In the batch fermentation process performed for this work, the eight LAB strains showed outliers, which could be due to the method applied for the measurements. Therefore, the experimental data had to be smoothed to calculate all required parameters for each mathematical model. It is important to consider that outliers are present in all real-life data measurements, such as biomass growth.
In the cubic model, the parameter
also influences the death phase of the biomass. The latter because a higher growth rate implies that the maximum biomass concentration will be reached, and due to the inherit dynamics of this kind of function, after reaching this maximum point, the biomass begins to decline. It is important to consider that if this mathematical model is simulated at time values further than 70 h, biomass will take negative values, which is not biologically feasible in real-life scenarios. However, this can be neglected because the fermentation process to produce acidified milk generally takes around 48 h, and it is not necessary to predict the behavior of this variable beyond this time, as this is the final desired product. Predictive models aim to ensure food quality and provide a tool that supports decision-making on fermentation design. Therefore, based on the results and as it is illustrated in
Figure 3,
Figure 4,
Figure 5 and
Figure 6, a decision can be made to perform a fermentation process under the same culture conditions only up to 36 h, i.e., when the maximum biomass growth is reached for this type of fermented milk.