Journal of Food Quality
Volume 2018 (2018), Article ID 1726761, 10 pages
https://doi.org/10.1155/2018/1726761

Research Article

Modeling the Combined Effects of Temperature, pH, and Sodium Chloride and Sodium Lactate Concentrations on the Growth Rate of Lactobacillus plantarum ATCC 8014

Francieli Dalcanton,¹ Elena Carrasco,² Fernando Pérez-Rodríguez,² Guiomar Denisse Posada-Izquierdo,² Gláucia Maria Falcão de Aragão,³ and Rosa María García-Gimeno²

¹Universidade Comunitária da Região de Chapecó (UNOCHAPECÓ), Av. Senador Atílio Fontana, 591-E, Caixa Postal 1141, 89809-000 Chapecó, SC, Brazil
²Department of Food Science and Technology, University of Córdoba, Agrifood Campus of International Excellence (ceiA3), Campus Rabanales, Edif. C-1, 14014 Córdoba, Spain
³Department of Chemical and Food Engineering, Federal University of Santa Catarina, 88040-900 Florianópolis, SC, Brazil

Correspondence should be addressed to Guiomar Denisse Posada-Izquierdo; bt2poizg@uco.es

Received 10 November 2017; Accepted 23 January 2018; Published 19 February 2018

Academic Editor: Maria Rosaria Corbo

Copyright © 2018 Francieli Dalcanton et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Nowadays, microorganisms with probiotic or antimicrobial properties are receiving major attention as alternative resources for food preservation. Lactic acid bacteria are able to synthetize compounds with antimicrobial activity against pathogenic and spoilage flora. Among them, Lactobacillus plantarum ATCC 8014 has exhibited this capacity, and further studies reveal that the microorganism is able to produce bacteriocins. An assessment of the growth of L. plantarum ATCC 8014 at different conditions becomes crucial to predict its development in foods. A response surface model of the growth rate of L. plantarum was built in this study as a function of temperature (4, 7, 10, 13, and 16°C), pH (5.5, 6.0, 6.5, 7.0, and 7.5), and sodium chloride (0, 1.5, 3.0, 4.5, and 6.0%) and sodium lactate (0, 1, 2, 3, and 4%) concentrations. All the factors were statistically significant at a confidence level of 90%  . When temperature and pH increased, there was a corresponding increase in the growth rate, while a negative relationship was observed between NaCl and Na-lactate concentrations and the growth parameter. A mathematical validation was carried out with additional conditions, demonstrating an excellent performance of the model. The developed model could be useful for designing foods with L. plantarum ATCC 8014 added as a probiotic.

1. Introduction

The use of lactic acid bacteria (LAB) has been widely reported in food fermentation processes of different types of foods such as yoghurt, cheese, and other fermented foods like vegetables, seafood, or meat products [1–3]. However, the increasing demand for healthy foods has conducted LAB research towards the study of their functional properties in foods, revealing that many LAB species, such us Lactobacillus spp., Lactococcus spp., Leuconostoc spp., Streptococcus spp., Pediococcus spp., Oenococcus spp., or Carnobacterium spp., exhibit antimicrobial activity [4–7].

Lactobacillus plantarum is one of the most widespread LAB in the environment. It can be naturally found in the human gastrointestinal tract and it is also encountered in a variety of fermented foods for which stress conditions such as heat, cold, and acidity are common [8, 9]. Lactobacillus plantarum has been recognized for its probiotic characteristics [10, 11] as a member of LAB and extended research has been developed on the inhibitory effect and probiotic properties of L. plantarum [12–16]. Particularly, Lash et al. [15] found that L. plantarum ATCC 8014 had an inhibitory effect upon a wide range of microorganisms including both Gram-negative and Gram-positive bacteria, pointing to bacteriocins as the responsible compounds for such inhibition.

The environments where L. plantarum can synthetize metabolites with enhanced effects on human health are multiple, including food and gut environment. Thus the assessment of growth and behavior of L. plantarum under food or gut conditions should be the first step of a comprehensive study dealing with probiotic and inhibitory capacity of L. plantarum. In this sense, temperature, pH, and water activity (Aw) are some of the most important controlling factors influencing the behavior of microorganisms in foods [17, 18]. Besides this, foods may contain additives with antimicrobial activity or other functions; for example, sodium lactate (Na-lactate) contributes to the microbial stability of many foods [19–21]. In the gut environment, pH is one of the most important factors affecting the viability of microorganisms [22].

Beside this, the use of predictive microbiology tools is increasing due to their adequacy (readily available and reliable) to assess microorganisms’ kinetics such us growth, survival, inactivation, or cross-contamination [23]. Growth predictive models describe a growth parameter, for example, growth rate, as a function of intrinsic and/or environmental factors [18, 24, 25]. The development of predictive models requires a large amount of microbial growth data, which is tedious and time-consuming to produce through traditional microbial culture techniques [26]. One alternative widely reported by the literature is based on absorbance measurements, resulting in the collection of a huge set of data in a cost-effective manner. This technique is automatic, nondestructive, quick, inexpensive, accurate, and effort-balanced [27–29].

To date, although a number of studies have dealt with the growth modeling of LAB [30] or Lactobacillus spp. [31] in order to assess food spoilage or for fermentation purposes [32, 33], further research is needed to evaluate the development of particular LAB species known to have probiotic/antimicrobial properties, such as Lactobacillus plantarum strain ATCC 8014, under food environment conditions. The aim of the present study was to develop a response surface model to describe the combined effects of temperature, pH, and sodium chloride and sodium lactate concentrations on the growth rate of Lactobacillus plantarum strain ATCC 8014.

2. Material and Methods

2.1. Inoculum

For the preparation of the inoculum of Lactobacillus plantarum ATCC 8014 (Spanish Collection of Strain Types, Valencia, Spain), a freeze-dried strain was recovered and maintained at −18°C in cryovials containing beads and cryopreservative (Microbank™, Richmond Hill, Canada) until utilization. The inoculum was prepared in tubes with 10 mL of MRS broth (Man, Rogosa and Sharpe) (Oxoid Ltd., Hampshire, England), which were incubated at 30°C for 24 h in an incubator with 10% CO₂ atmosphere (Crenesys Instrumentation, Madrid, Spain). Afterwards, 0.1 mL of the culture was transferred to 10 mL of MRS broth and incubated for 18 h under the same conditions presented above. Subsequently, the necessary dilutions were made in MRS broth to obtain an inoculum size of ≈10³ CFU·mL⁻¹ in the test media. Actual inoculum level was estimated by plating appropriate dilutions on MRS Agar (Oxoid Ltd.), followed by incubation at 30°C for 48 h in a 10% CO₂ atmosphere.

2.2. Experimental Design and Data Collection

The behavior of L. plantarum was evaluated under a set of temperatures (4, 7, 10, 13, and 16°C), pH (5.5, 6, 6.5, 7, and 7.5), and sodium chloride (NaCl) (0, 1.5, 3, 4.5, and 6%) and sodium lactate (Na-lactate) (0, 1, 2, 3, and 4%) conditions, comprising the domain of the central composite rotatable design (CCRD), which resulted in 26 combinations that can be observed in Table 1. Also, 54 extra combinations were carried out (Table 2) to allow for model validation. Sodium chloride (Panreac) and sodium DL-lactate solution at 60% (w/w) (Sigma-Aldrich, St. Louis, USA) concentrations were obtained by adding the appropriate amount of these reagents to a series of flasks containing 50 mL MRS. The flasks were autoclaved (121°C for 15 min), and the pH was adjusted with sterile NaOH (5 M) (Panreac) and HCl (5 M) (Panreac) solutions to obtain different combinations of pH. These test media were inoculated with L. plantarum to obtain an approximate concentration of 10³ CFU·mL⁻¹.

Table 1: Observed (Obs) and predicted growth rate (μ, ) by response surface model (RSM) of L.   plantarum for the central composite rotatable design.

(°C)
pH
NaCl
(%)
Na-lactate
(%)
μ (h⁻¹)

RSM

7 (−1)
6.0 (−1)
1.5 (−1)
1 (−1)
0.0042
0.0017

13 (+1)
6.0 (−1)
1.5 (−1)
1 (−1)
0.0498
0.0505

7 (−1)
7.0 (+1)
1.5 (−1)
1 (−1)
0.0040
0.0063

13 (+1)
7.0 (+1)
1.5 (−1)
1 (−1)
0.0549
0.0551

7 (−1)
6.0 (−1)
4.5 (+1)
1 (−1)
0
0

13 (+1)
6.0 (−1)
4.5 (+1)
1 (−1)
0.0316
0.0274

7 (−1)
7.0 (+1)
4.5 (+1)
1 (−1)
0
0.0045

13 (+1)
7.0 (+1)
4.5 (+1)
1 (−1)
0.0316
0.0320

7 (−1)
6.0 (−1)
1.5 (−1)
3 (+1)
0.0037
0.0013

13 (+1)
6.0 (−1)
1.5 (−1)
3 (+1)
0.0373
0.0349

7 (−1)
7.0 (+1)
1.5 (−1)
3 (+1)
0.0025
0.0059

13 (+1)
7.0 (+1)
1.5 (−1)
3 (+1)
0.0442
0.0395

7 (−1)
6.0 (−1)
4.5 (+1)
3 (+1)
0
0

13 (+1)
6.0 (−1)
4.5 (+1)
3 (+1)
0.0139
0.0118

7 (−1)
7.0 (+1)
4.5 (+1)
3 (+1)
0
0.0041

13 (+1)
7.0 (+1)
4.5 (+1)
3 (+1)
0.0096
0.0164

4 (−2)
6.5 (0)
3.0 (0)
2 (0)
0
0

16 (+2)
6.5 (0)
3.0 (0)
2 (0)
0.0541
0.0488

10 (0)
5.5 (−2)
3.0 (0)
2 (0)
0.0079
0.0136

10 (0)
7.5 (+2)
3.0 (0)
2 (0)
0.0323
0.0228

10 (0)
6.5 (0)
0 (−2)
2 (0)
0.0210
0.0306

10 (0)
6.5 (0)
6.0 (+2)
2 (0)
0.0033
0.0057

10 (0)
6.5 (0)
3.0 (0)
0 (−2)
0.0202
0.0262

10 (0)
6.5 (0)
3.0 (0)
4 (+2)
0.0047
0.0102

10 (0)
6.5 (0)
3.0 (0)
2 (0)
0.0207
0.0182

10 (0)
6.5 (0)
3.0 (0)
2 (0)
0.0209
0.0182

0.0084

1.0518

1.3690

Real values (coded values) for all variables; average observed in quadruplicate; RMSE: root mean square error; : bias factor; : accuracy factor.

Table 2: Observed (Obs) and predicted growth rate (, ) by response surface model (RSM) of for mathematical validation.

(°C)
pH
NaCl
(%)
Na-lactate
(%)
μ (h⁻¹)

Obs
RSM

6.0
0
1
0.0689
0.0918

7.0
0
1
0.0785
0.0963

6.5
0
3
0.0637
0.0476

6.5
1.0
2
0.0660
0.0711

6.5
2.0
1
0.0703
0.0715

6.5
2.0
4
0.0281
0.0366

6.0
3.0
0
0.0700
0.0696

7.0
3.0
0
0.0646
0.0742

6.0
3.0
4
0.0271
0.0232

7.0
3.0
4
0.0237
0.0278

6.5
4.0
1
0.0552
0.0492

6.5
4.0
3
0.0282
0.0260

6.5
5.0
2
0.0326
0.0263

6.0
6.0
0
0.0364
0.0359

7.0
6.0
0
0.0284
0.0405

7.0
0
0
0.0563
0.0744

7.0
0
4
0.0499
0.0432

6.5
0.5
3
0.0499
0.0448

6.5
2.5
2
0.0484
0.0373

6.5
3.0
0
0.0515
0.0490

6.5
3.0
4
0.0147
0.0178

6.5
3.5
2
0.0396
0.0296

7.0
6.0
0
0.0248
0.0283

7.0
0
0
0.0222
0.0409

7.0
0
4
0.0196
0.0249

6.0
1.0
1
0.0226
0.0281

7.0
1.0
1
0.0211
0.0327

6.0
2.0
3
0.0176
0.0160

7.0
2.0
3
0.0091
0.0206

7.0
3.0
2
0.0127
0.0205

6.5
4.0
0
0.0173
0.0220

6.5
4.0
1
0.0202
0.0181

7.0
0
0
0.0031
0.0074

7.0
0
4
0.0040
0.0066

6.5
0.5
3
0.0046
0.0042

6.5
2.5
2
0.0038
0.0032

6.5
3.0
0
0.0033
0.0033

6.5
3.5
2
0.0024
0.0026

6.5
3.0
4
0
0.0025

7.0
0
0
0
0

6.0
0
1
0
0

7.0
0
1
0
0

6.5
0
3
0
0

7.0
1.0
0
0
0

6.5
1.0
2
0
0

6.0
1.0
4
0
0

6.5
2.0
1
0
0

6.5
2.0
4
0
0

6.0
3.0
0
0
0

7.0
3.0
0
0
0

6.5
4.0
1
0
0

6.5
5.0
2
0
0

6.0
6.0
0
0
0

7.0
6.0
0
0
0

0.0080

1.0863

1.1808

Average observed in quadruplicate; RMSE: root mean square error; : bias factor; : accuracy factor.

To obtain L. plantarum growth data, a Bioscreen C analyzer instrument (Labsystem, Helsinki, Finland) was employed. The equipment was set to collect absorbance (abs) measurements at 600 nm at adequate time intervals. Bioscreen C plates’ wells were filled up with 250 μL of the inoculated test media, and the analyses were performed until the stationary phase of L. plantarum was reached. Four wells per condition were filled up with inoculated media and two wells per condition were filled up with sterile media to serve as control.

2.3. Growth Rates Estimation

2.3.1. Growth Curve Fitting

L. plantarum growth data obtained through Bioscreen C (abs data) were loaded onto a Microsoft Excel spreadsheet and analyzed to calculate the growth rate for each test condition. Thus, a three-phase linear model [34] was applied to describe the growth of the microorganisms, expressed as Log₁₀ CFUmL⁻¹ versus time, being the exponential phase modeled by

where is the decimal logarithm of the cell concentration at time t (CFU·mL⁻¹), is the initial cell concentration (CFU·mL⁻¹), is the maximum growth rate (h⁻¹), which will be called only growth rate in this study, and is time (h).

Bioscreen C is limited to indirectly measure cell concentrations in the range of about 10⁷ to 10⁹ CFU·mL⁻¹. In this study, as the Bioscreen C plates were inoculated with almost 10³ CFU·mL⁻¹ per well, only a part of the exponential phase was detected, because the adaptation phase (lag phase) recorded by the equipment actually corresponds to the lag phase together with part of the exponential phase (including counts from 10³ up to 10⁷ CFU·mL⁻¹). Therefore, (1) cannot be applied directly and some considerations were established. First of all, a correlation between absorbance and concentration (CFU·mL⁻¹) of L. plantarum was determined by means of a calibration curve (see (2)).

where is the decimal logarithm of the cell concentration at time , abs is the absorbance, and and are equation parameters.

In (1), was replaced by the expression of (2), and the parameter was isolated, as shown in (3), which can be applied to any pair of points in the exponential phase.

where log₁₀, , , and have the same meaning as in (1), abs, , and have the same meaning as in (2), and the subscript letters and correspond to any time points of the exponential phase.

From (3), the parameter μ was isolated, as can be seen in (4). This equation involves abs data, related time h, and initial concentration of L. plantarum.

where log₁₀, , , and have the same meaning as in (1), abs and have the same meaning as in (2), and the subscript letters and have the same meaning as in (3).

In (4) it is assumed that no lag phase occurs. Nevertheless, this is not true, especially at limiting conditions. For this reason, (4) was modified to include lag phase, as observed in

where log₁₀, , , and have the same meaning as in (1), abs and have the same meaning as in (2), the subscript letters and have the same meaning as in (3), and is the lag phase (h).

The lag phase was calculated on the basis of the relation between and given in (6), as supported by many authors [35–39].

where is a constant which is microorganism-specific at the same preculture conditions.

The constant was obtained by applying (6) to kinetic results ( and ) from a previous experiment carried out with L. plantarum cultivated in MRS broth at different temperatures. Combining (5) and (6), the expression to calculateλ is given by

where log₁₀, , and have the same meaning as in (1), b has the same meaning as in (2), and are and , respectively, which have the same meaning as in (3), and has the same meaning as in (6).

2.3.2. Response Surface Model Development

The response surface model (RSM) was used to describe the statistical significance of different variables (temperature, pH, NaCl, and Na-lactate) on the L. plantarum growth rate through a central composite rotatable design (CCRD). RSM is an empirical statistical technique employed for multiple regression analysis using quantitative data obtained from properly designed experiments to solve multivariable equations simultaneously. In a system involving four significant independent variables, the mathematical relationship of the response of these variables can be approximated by a second degree polynomial equation (see (8)) [40, 41].

where is the predicted value, is the average coefficients, , , , and are the independent variables, , , , and are the linear coefficients, , , , and are the quadratic coefficients, and , , , , , and are the cross product coefficients.

2.4. Model Validation

After the establishment of RSM, 54 additional conditions for model validation were selected randomly within the domain of the model, which were homogeneously distributed among the five investigated temperatures (Table 2), in order to evaluate the predictive capacity of the proposed model.

2.5. Evaluation Criteria of Model Performance

The experimental data obtained by CCRD and additional conditions were treated with the software STATISTICA 6.0 (StatSoft™, Tulsa, OK, USA). The goodness of fit of the RSM was expressed by the determination coefficient , and its statistical significance was determined by analysis of variance (ANOVA) ( test).

To evaluate the predictive accuracy of the RSM, the following evaluation criteria were employed: root mean square error (RMSE) (see (9)), bias factor (see (10)), and accuracy factor (see (11)) [42].

where is the growth rate, obs is the observed value, pred is the predicted value, is the number of experimental data, and is the number of model parameters.

3. Results and Discussion

3.1. Counts Data versus Absorbance Data

The relationship between cell concentration ( (CFU·mL⁻¹)) and absorbance data () of L. plantarum is given by (12). Absorbance represents only that of bacterial suspension, as absorbance of blanks was subtracted.

The parameter (i.e., 9.3152) obtained in this equation was used for calculating the lag phase and growth rate, according to (7) and (5), respectively. The lag phase was calculated considering the value estimated in a previous experiment (results are shown in Table 3) by (6) ( average = 0.41). In (7), and values corresponded to pairs of consecutive abs data versus time, within the range of abs values of 0.04 to 0.2. These low values were selected to make sure that abs data do not correspond to the stationary phase of the microorganism, especially at limiting conditions.

Table 3: Kinetic parameters (λ and μ) of experiments from previous data of Lactobacillus plantarum and calculated .

  (°C)
λ (h)
μ (h⁻¹)

30.2
0.0112
0.34

10.8
0.0334
0.36

6.5
0.0717
0.47

3.6
0.1151
0.41

1.8
0.2475
0.45

The growth rates obtained (μ Obs) (see (5)) from CCRD (Table 1) were used to build the response surface model.

3.2. Response Surface Model

Through the results of the CCRD (Table 1), it was possible to determine the regression coefficients and identify the variables showing the greatest influence on L. plantarum growth rate. Table 4 shows only the factors that were statistically significant , as well as their respective regression coefficients. The analysis of variance (ANOVA) is shown in Table 5.

Table 4: Regression coefficient results from the results of the central composite rotatable design (CCRD).

Model term
Regression coefficient
Standard error

value

Mean/interaction
0.018
0.001
15.889
<0.0001

1 T (L)
0.015
0.001
12.837
<0.0001

2 pH (L)
0.002
0.001
1.929
0.0688

3 NaCl (L)
−0.006
0.001
−5.227
<0.0001

4 Na-lactate (L)
−0.004
0.001
−3.357
0.0033

1 L by 3 L
−0.005
0.001
−3.649
0.0017

1 L by 4 L
−0.004
0.001
−2.611
0.0172

(L): linear factor. Statistics calculated with a degree of freedom = 19.

Table 5: Analysis of variance (ANOVA) of the predictive model obtained for growth rate according to the CCRD.

Source of variations
Sum of squares
Df
Mean square
test
value

Regression
0.007726
6
0.001288
37.88
<0.00001

Residual
0.000646
19
0.000034

Total
0.008372
25

Degree of freedom. test is the calculated value. Determination coefficient = 92.28%; = 2.11 (tabulated).

From Table 4, it is observed that temperature was the most important factor on the growth rate, followed by the concentrations of NaCl and of Na-lactate and by the pH. As expected, when temperature and pH increased, there was a corresponding increase in the growth rate (positive regression coefficient), whereas when the concentration of NaCl and Na-lactate increased in the culture medium, the growth rate decreased (negative regression coefficient). In addition, the interactions between temperature and NaCl and temperature and Na-lactate were statistically significant.

The statistical significance of the model was checked by an test (ANOVA) (Table 5). As the test value was 37.88, 17.95 times higher than the tabulated (2.11; ), it can be concluded that the model is predictive, having high significance , and the percentage of variation explained by the model was 92.28%.

Through the results presented above it was possible to establish the response surface model as a function of statistically significant regression coefficients. The model for L. plantarum growth rate fitted in the coded factors is given by

The predictive model (see (13)) describes how the combined effects of temperature, pH, NaCl, and Na-lactate influence the L. plantarum growth rate, considering when . Figure 1 shows the response surface plots for the growth rate of L. plantarum, as a function of temperature with pH, NaCl, and Na-lactate.

(a)

(b)

(c)

Figure 1: Response surfaces for the growth rate (μ, h⁻¹) of L. plantarum, as a function of temperature and pH (a), temperature and NaCl (b), and temperature and Na-lactate (c) (temperature and salts are given in °C and %, resp.).

Analyzing firstly the response surface of Figure 1(a), it is possible to verify that under these conditions pH showed a small influence on the L. plantarum growth rate and that at the higher values of this variable the microorganism growth was slightly higher. On the other hand, temperature was the factor that showed the largest effect on the response, as can be seen in Figure 1. At higher temperatures, the growth rate was higher. At the lowest temperature (4°C) and at some conditions at 7°C, there was no growth of microorganism during the studied period (six months), showing that the increase of this variable directly influenced the development of microorganism. The analysis between temperature and NaCl concentration had a significant influence on the decrease of L. plantarum growth rate (Figure 1(b)). The same behavior can be observed in the interaction between temperature and Na-lactate (Figure 1(c)), and in the presence of small concentrations of these compounds, the decrease in growth rate is already noticeable.

García-Gimeno et al. [28] evaluated the effect of temperature (20 and 28°C), pH (4–7), and NaCl (0–6%) on the kinetic parameters of L. plantarum growth curves using RSM. All variables studied showed a significant effect on the growth rate. These results are in agreement with the results presented in this study, although the authors did not find a significant interaction between temperature and NaCl.

Devlieghere et al. [43] established a predictive model to evaluate the effect of temperature (4, 8, and 12°C), water activity, and dissolved CO₂ on L. sakei growth in brain heart infusion (BHI) broth. This model was extended with Na-lactate (0–3%) in BHI and validated in meat products [44]. These authors found that the interaction between Na-lactate and temperature was significant for the growth rate and observed that the addition of Na-lactate extended the products’ shelf life, especially at lower temperatures. According to Drosinos et al. [45], the addition of 2% Na-lactate inhibits LAB growth in modified MRS and meat products, without causing changes in the flavor of the latter. Moreover, its inhibitory activity may increase when combined with other antimicrobials, such as NaCl and Na-acetate. In this study, the combined effect of the salts showed a greater inhibition on the growth rate when compared with the effect of each salt individually, as found in our study.

Sarmento [46] established a RSM with NaCl, Na-lactate, garlic, and interactions between NaCl and polyphosphate and polyphosphate and garlic for the evaluation of L. plantarum growth in MRS broth at 35°C. Based on these results, new formulations for smoked sausage and mortadella were proposed in order to extend the shelf life of these products.

The obtained response surface model (see (13)) was used to predict values of growth rate, as shown in Table 1 (μ, RSM). The table also shows the statistical factors that indicate the average deviations between observed and predicted values. The low value for the RMSE (0.0084) showed the acceptable prediction ability of the model. Ross [42] introduced measures of model performance called Bf and Af. Ideally, predictive models would have Af = Bf = 1 but, typically, the accuracy factor increases by 0.10–0.15 (10–15%) for every variable included in the model [47]. Thus, in this study, with four variables (temperature, pH, NaCl, and Na-lactate), a value of Af = 1.4–1.6 could be expected. As can be seen in Table 1, Af = 1.3690, demonstrating that the model can be used to predict L. plantarum growth rate with enough accuracy. The Bf value of 1.0518 confirms the good fit of the model proposed, being closer to 1. As Bf is greater than 1, it can be withdrawn that, on average, the model overestimates μ; thus predictions are fail-safe [42, 47].

3.3. Mathematical Validation

There are two steps in evaluating a predictive model [48]. The first one is to ensure that the model accurately describes the data from which it has been generated and the second step should be the comparison of the prediction with other data, also called model validation. For mathematical validation, 54 extra conditions were carried out within the domain of the model, and the observed and the predictive values by the response surface model in these conditions are shown in Table 2. The table shows the same error criteria described in Table 1.

The statistical indices calculated for the additional conditions (Table 2) showed values similar to those obtained by the experimental design (Table 1). The low value of RMSE indicates the good model prediction ability. The Af value (1.1808) was lower when compared with the value of the experimental design (1.3690), having only 18% of variation between predicted and observed values. The Bf value was closer to 1, indicating minimal differences between the predicted and observed data. The results obtained in this study were similar to those described by other authors for validation of the model. García-Gimeno et al. [49] obtained values of Bf = 1.09 and Af = 1.27 in the validation of their RSM developed for E. coli, and Zurera-Cosano et al. [50] showed values of Bf = 1.03–1.07 and Af = 1.17–1.20 for validation of RSM of L. mesenteroides growth rate, in aerobic and anaerobic conditions.

4. Conclusion

The response surface model presented in this paper showed significant effects of temperature, NaCl, Na-lactate, pH, and the interactions between temperature and NaCl and temperature and Na-lactate on L. plantarum growth rate, where the temperature was the most decisive factor, followed by the concentrations of NaCl and Na-lactate and by the pH. When temperature and pH increased, there was a corresponding increase in the growth rate, whereas when NaCl and Na-lactate concentrations increased, the growth parameter decreased. The excellent performance of the model was confirmed through mathematical validation by statistics indices RMSE, Bf, and Af factors.

Gaining knowledge on the extent of growth of L. plantarum ATCC 8014 in foods becomes crucial when not only food additives but also metabolites with inhibitory capacity produced by L. plantarum ATCC 8014 (e.g., bacteriocins) are part of the antimicrobial strategy against pathogenic and spoilage flora in foods.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors gratefully acknowledge the Capes Foundation (Brazil) for the financial support of Francieli Dalcanton for her Ph.D. studies and both the Department of Food Science and Technology of the University of Córdoba (Córdoba, Spain) and the Andalusian Government (Spain) for funding the experimental work (Grant no. P08-CTS 3620).

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