Nisin Synthesis, Relationship with the Growth of the Producing Strain, Culture pH and Nitrogen Consumption

Nisin, an antibacterial compound produced by Lactococcus lactis strains, has been approved by the US Food and Drug Administration to be used as a safe food additive to control the growth of undesirable pathogenic bacteria. Nisin is commonly described as a pH-dependent primary metabolite, since its production depends on growth and culture pH evolution. However, the relationship between bacteriocin synthesis and the consumption of the limiting nutrient has not been described until now. Therefore, this study aimed to develop a competitive four-dimensional Lotka Volterra-like equation to describe the relationships between culture pH, limiting nutrient (total nitrogen: TN) consumption and production of biomass (X) and nisin (BT) in four series of batch fermentation with L. lactis CECT 539 in diluted whey (DW)-based media. The developed four-dimensional LV-like equation (with a unique set of parameters) could not be used to describe all cultures belonging to each fermentation series. However, the four-dimensional LV-like equation described accurately each individual culture, providing a good description of the relationships between pH, TN, X and BT, higher values for R2 and F-ratios, lower values (&lt; 10%) for the mean relative percentage deviation modulus, with bias and accuracy factor values approximately equal to one.


Introduction
Nisin, a bacteriocin produced by Lactococcus lactis strains, has a wide antibacterial activity against food spoilage and pathogenic bacteria. For this reason, this biomolecule has been recognized by the US Food and Drug Administration as a natural and safe biopreservative in food products, being allowed in the USA and several European Union countries. The advantages of using nisin in foods include the reduction in both the thermal treatment and addition of chemicals to food products and increase in their shelf life [1].
For high nisin production at low cost, it is necessary to know the relationship between the main culture variables, that could be elucidated with the use of appropriate mathematical models. This could also allow a proper monitoring and control of these bioprocesses [2,8].
Different mathematical models have been commonly used to describe the kinetics of growth (e.g., Verhulst, Gompertz, Richards, Bertalanffy, Weibull, Monod) and bacteriocin synthesis (e.g. unmodified and modified forms of the Luedeking and Piret model) by lactic acid bacteria (LAB) in batch fermentations [2][3][4][5][6][7]. Table 1. Initial concentrations (mean ± standard deviations) of total sugars (TS), nitrogen (TN), phosphorous (TP) and proteins (Pr) in culture media prepared with deproteinized diluted whey (DW) and concentrated mussel-processing wastes (CMPW) [14,16]. The data (symbols) corresponding to the four series of batch cultures [14][15][16] and the corresponding predictions (dashed lines) of the developed four-dimensional LV-like equation (1)-(4) are shown in . The values for the constants and the statistical analysis of each equation in each series of cultures are shown in Table 3.  Table 3. Solid lines were obtained by adjusting the fourdimensional LV-like equation to the experimental data corresponding to each individual culture (see parameter values in Table 4). Reproduced with permission from Costas et al. [14], Appl. Microbiol. Biotechnol.; published by Springer Nature, 2016.  Table 3. Solid lines were obtained by adjusting the four-dimensional LV-like equation (1)- (4) to the experimental data corresponding to each individual culture (see parameter values in Table 5). Reproduced with permission from Costas et al. [15], 3-Biotech; published by Springer Nature, 2018.  Table 3. Solid lines were obtained by adjusting the four-dimensional LV-like equation (1)- (4) to the experimental data corresponding to each individual culture (see parameter values in Table 6). Reproduced with permission from Costas et al. [15] Table 3. Solid lines were obtained by adjusting the four-dimensional LV-like equation (1)-(4) to the experimental data corresponding to each individual culture (see parameter values in Table 7).  Table 3. Solid lines were obtained by adjusting the fourdimensional LV-like equation (1)-(4) to the experimental data corresponding to each individual culture (see parameter values in Table 8).

Medium TS (g/L) TN (g/L) TP (g/L) Pr (g/L)
The parameter value is considered statistically significant if its corresponding P-value is lower than 0.05.
Satisfactory results were obtained when the global equation (1) was set to describe the time course of the culture pH in the DW-G series (dashed lines in Figure 1). Thus, the values of RpH 2 (0.9957) and F-ratio (61313.50) were considerably higher, the RPDM value was lower than 10 and both the Bf (0.9977) and Af (1.0058) values were near one. In addition, statistically significant values (P < 0.0001) for both the parameters and global pH equation were obtained (Table 3).
In the case of the DW-TS-TP, DW-MRS and DW-pH series of cultures (Figures 2-5,  Table 3), the results were less satisfactory compared with the DW-G cultures ( Figure 1, Table 3), considering the lower values obtained for RpH 2 (0.9652, 0.9435, and 0.4953) and Fratios (4040.25, 6023.08, and 70.67). Although the RPDM values calculated for the DW-TS-TP, DW-MRS and DW-pH cultures (2.3678, 4.6731, and 3.9241) were lower than 10, and the values of the parameters and equation were statistically significant (P < 0.0001), the predictions of the global pH equation (1) were not always consistent with the experimental pH data (Figures 2-5). In fact, with the exception of the DW-G cultures, the predictions of the global pH equation (1) showed a clear deviation from the experimental data for the DW-TS-TP (Figures 2 and 3), DW-MRS ( Figure 4) and DW-pH cultures ( Figure  5).
The general total nitrogen (TN) consumption equation (2) only satisfactorily described the time course of this nutrient in the fermentation in DW medium supplemented with 15 g glucose/L ( Figure 1) or 37.0 g of TS and 0.43 g TP per L of medium (cultures 9 to 13 in Figure 3). The RPDM value was only lower than 10 in the DW-G cultures (Table 3), but the fit of equation (2) to the experimental TN data was found to be statistically non-significant (P = 1.0000) in all series of fermentation, as occurred with almost all values of the parameters.
In addition, both the general growth (X) equation (3) and nisin (BT) production equation (4) unsatisfactorily described the evolution of both variables in the different series of fermentation (Figures 1-5), except the fermentations in DW medium supplemented with 37.0 g of TS and 0.43 g TP per L of medium (cultures 9 to 13 in Figure  3). Surprisingly, the fit of equation (3) to the experimental biomass data in the series DW-G was statistically significant (P < 0.05) and provided statistically significant values (P < 0.05) for the values of the parameters (Table 3), with a relatively higher value for RX 2 (0.8293) and F-ratio (255.04). However, the RPDM value (12.9625) was higher than 10 and the Bf (0.9058) and Af (1.1523) values were relatively far from one. From the statistical viewpoint, the fitting equation (3) to the experimental data of the series DW-TS-TP, DW-MRS and DW-pH did not always provide statistically significant coefficients, being nonsignificant (P = 1.0000) the fitted equation (3) for the DW-MRS and DW-pH cultures ( Table  3). In these latter cultures, the R 2 coefficient could not be calculated with the statistical software used.
Concerning the use of equation (4), it can be noted that the fitted equations were found to be non-significant (P = 1.0000) for the four series of cultures (Table 3).
These observations suggest that the four-dimensional LV equation could not be used as a general equation to describe the nisin production system in the four series of batch fermentations.
These unsatisfactory results could be related to the different initial compositions of the media used in each series of cultures: DW media supplemented with different initial concentrations of glucose (  (Figures 1-5). In this way, it is well known that the initial culture conditions affect the evolution of these culture variables (pH, TN, X and BT) in a different way [8,[17][18][19]. For example, the pH drop depends on the presence and interaction between some compounds (salts, organic acids, proteins, free amino acids) with buffering capacity in the culture medium [20] and organic acids production by growing cells [17]. TN consumption during fermentation depends on the initial medium composition, mainly the type and concentration of the nitrogen source [18,19] and culture pH [16,19,21]. In fact, the consumption of TN [16] or amino acids [21] in L. lactis strains was maximal when the culture pH reached values between 5.8 and 6.5, and decreased abruptly for high and low pH values.
On the other hand, biomass production depends on different factors including the initial medium composition (concentration and type of nutrients (mainly carbon, nitrogen and phosphorous sources), initial and final pH values in the cultures, pH evolution and production of inhibitory compounds [17]. Nisin synthesis depends not only on the time course of the evolution of biomass concentration, but also on i) the amount of biomass produced, ii) initial concentration and type of nutrient (carbon, nitrogen and phosphorous sources), iii) initial and final pH value, pH evolution and pH drop generated in the cultures [5,8,12,17,19]. So that, the specific effect of these factors on the response variables (culture pH, TN consumption, biomass and nisin production) could be non-synchronous producing a different change in the time course of the latter variables and consequently, in their relationships.
For example, the buffering capacity (BC), which is a measure of the resistance of the culture medium to pH changes, affects biomass and nisin synthesis differently. On the one hand, the increase in BC favors biomass production since the cultures remain longer within the optimum pH range (between 5.8 and 6.5) for nutrient consumption for L. lactis CECT 539 [16,21]. On the other hand, these high pH values inhibit bacteriocin synthesis, which was higher at an optimum pH value of 4.90 in DW medium, due to the need of a low pH value to favor the maturation of the nisin molecule [5,16,22]. The value of this optimum final pH for nisin production depends on the producer strain and composition of culture medium [8,12,[22][23][24][25].
In addition, it has been observed that higher pH drops (rpH) enhance nisin production [12,16,19,25] before the cultures reached an inappropriate pH for survival and cell growth of L. lactis [26]. For example, in the batch cultures conducted in DW medium adjusted to different initial pH values, the highest nutrient (sources of carbon, nitrogen and phosphorous) consumption and biomass concentration were obtained at initial pH values of 6.0 and 6.5. However, the highest nisin concentration was obtained at an initial pH of 7.0, in which the highest final rpH was generated (see Figure 5). Thus, in the specific case of the series of fermentation in the DW media supplemented with glucose ( Figure 1), it can be observed that the evolution of biomass production in each culture was different. The same trend was observed in the batch cultures in DW media supplemented with different levels of glucose and phosphorous (Figures 2 and 3), with nutrients of the MRS broth ( Figure 4) or adjusted to different initial pH values ( Figure 5). In these three series of cultures (DW-TS-TP, DW-MRS and DW-pH), the evolution of culture pH and nisin production was also different.
For these reasons, it is very difficult to develop a general four-dimensional LV-like equation to explain the variations in the time courses of the four variables (culture pH, TN, X and BT). In addition, with the use of a general four-dimensional equation, the effect of different initial culture conditions on the evolution of the four dependent variables could not be explained, leading to a misinterpretation of the kinetics of the cultures.
To solve this problem, we first fitted the four-dimensional LV-like equation (1)-(4) to each individual culture of each series of fermentation to accurately determine how the values of the different parameters change with changes in the initial culture conditions (concentrations of glucose, TP and TS, MRS broth nutrients and pH). Afterward, the fourdimensional LV-like equation (1)-(4) was modified (when this was possible) by including a term for the specific effect of the initial culture conditions (initial concentrations of glucose, TP and TS, MRS nutrients and initial pH) on the evolution of pH, TN, X and BT.

Modeling the Batch Nisin Production System in Individual Cultures Corresponding to Each Series of Fermentation
When the four-dimensional LV-like equation (1)-(4) was used to describe the relationships between the four response variables (pH, TN, X and BT) in each individual culture, both equations and values of the parameters were statistically significant (P < 0.050), with R 2 and F-values considerably higher, and Bf and Af values ~ 1 (Tables 4-8).
In addition, the predictions of the four-dimensional LV-like equation (1)-(4) for each response variable (solid lines  were in perfect agreement with the corresponding experimental data. This indicates that the developed four-dimensional LV-like equation (1)-(4) is consistent and robust to accurately describe the trend observed in the experimental data of culture pH, TN, X and BT.
The results obtained for each series of fermentation are discussed below. Table 4 shows the parameter values as well as the statistical analysis obtained when the four-dimensional LV-like equation (1)-(4) was fitted to the experimental data of culture pH, TN, X and BT in the DW-G cultures.

Series of fermentation DW-G
In this case, all values of the parameters in the equations (1)-(4) were significant (P < 0.05) and considerably higher values for RpH 2 (between 0.9886 and 0.9916), RTN 2 (between 0.9938 and 0.9991), RX 2 (0.9999) and RBT 2 (between 0.9927 and 0.99829) were obtained. The Bf and Af values calculated for equations (1)-(3) were ~ 1 and the RPDM values were considerably lower than 10%. However, in the case of nisin production, the values of Bf (between 0.8207 and 0.9475) and Af (between 1.0756 and 1.2185) obtained using equation (4) were the farthest from one and the RPDM values corresponding to the cultures in DW media supplemented with 15, 20, and 25 g glucose/L were slightly higher than 10% (Table  4). Table 4. Statistically significant (P < 0.05) parameter values (as estimates  confidence intervals) calculated with the four-dimensional LV-like equation (1)-(4) for each individual culture of the series of fermentation DW-G.

Initial glucose concentrations (g/L) in the DW medium
The parameter value is considered statistically significant if its corresponding P-value is lower than 0.05.
This was probably because nisin production is quantified by a photometric biossay using an indicator strain and consequently, the experimental error in determining nisin titers could be greater than those of the analytical methods used in pH, total nitrogen and biomass measurements. So that, in the latter cultures, the differences between the experimental and predicted nisin values during the first 9 h of fermentation were low ( Figure  1), but the experimental nisin data in this interval, used as the denominator in the equation (8) were also considerably low, increasing the RPDM value (Table 4).
Concerning the equation parameters, it can be noted that the values of αBC and αpH,X in equation (1) decreased with the increase in the initial concentration of the carbon source (G0) in the medium. This is mainly due to the inhibition that increasing glucose concentrations produced on the growth of L. lactis CECT 539 and consequently on lactic acid production [14], causing a gradual reduction in the pH drop in the culture media ( Figure  1). Since the cultures reached almost the same final pH level (between 4.73 and 4.92), the values of K1 only varied between 4.945 and 4.996 ( Table 4).
The values of αTN,X and K2 in equation (2) did not show a significant variation (mean values of 0.296  0.010 and 0.238  0.009, respectively), indicating that the TN consumption for the growth was similar in the six glucose-supplemented cultures. In addition, slight increases in the values of αTN,BT (from 0.006 to 0.0014) and K3 (from 0.479 to 0.558) were observed, due to the different nisin titers (22.48, 20.64, 18.35, 15.60, 23.40, and 10.70 BU/mL) produced in each culture ( Figure 1). So that, the high variability observed in the αTN,BT, and K3 values compared to those of αTN,X and K2 was due to the higher variations in the nisin concentrations synthesized compared to the levels of biomass produced ( Figure  1).
Equation (3) also provides an appropriate description of biomass production in each culture. In this case, the values of αX,TN decreased due to the above-mentioned growth inhibition produced by the carbon source, that led to a reduction in the amounts of TN consumed. However, the efficiency of TN consumption for nisin production (α*X,BT) did not change, since the reduction in growth was proportional to that observed in nisin synthesis, because this bacteriocin was produced in this culture as a pH-dependent primary metabolite [14]. The value of αX,pH decreased for initial concentrations of glucose higher than 10 g/L, while the K4 value decreased with the increase in glucose supplementation, as a consequence of the decrease in biomass production and the increase in the final pH values in the cultures (Figure 1).
A constant value was obtained for the αBT,X, α*BT,X and αBT,pH in equation (4) for the different cultures. This corroborates the affirmation that bacteriocin was produced as a pH-dependent primary metabolite and indicates that the TN consumption for biomass production was proportional to the production of bacteriocin and biomass (Table 4). Decreasing values were obtained for αBT,TN, and K5 due to the reduction in nisin titers with an increase in the initial concentration of the carbon source in the medium.
From the detailed observation of the series of fermentation DW-G (Figure 1), it can be noted that the increase in the initial glucose levels (G0) produced an increase the final culture pH and TN concentration, and a decrease in the final biomass and nisin levels, but each variable evolved in the same way in the different cultures. In fact, the rates of culture pH (rpH = dpH/dt) and TN (rTN = dTN/dt) decrease and biomass (rX = dX/dt) and nisin (rBT = dBT/dt) production in the glucose-supplemented culture media exhibited an exponential decrease in comparison with the respective rates in the culture in the unsupplemented culture ( Figure 6).
Then, when the modified four-dimensional equation (8) Figures 2 and 3. As observed before for the series of fermentation DW-G, the DW-TS-TP cultures were satisfactorily described using this procedure (Tables 5 and 6).
The values of αBC in equation (1) were similar (between 0.010 and 0.014) and the highest αpH,X value was obtained under the optimum condition (OC), in which the highest pH drop (difference between the initial and final pH value) was observed (Figures 2 and 3). The calculated values for K1 varied between 4.525 and 5.363 (Tables 5 and 6), which are in perfect agreement with the range of final pH values (between 4.53 and 5.37) obtained in the cultures (Figures 2 and 3).
The highest values of αTN,X and αTN,BT in equation (2) were obtained in the culture performed under the optimum conditions (TS = 22.6 g/L, TP = 0.46 g/L), in which the highest amounts of TN were consumed (0.23 g/L), while a constant K2 and K3 varied slightly in accordance with the initial levels of TS and TP in the culture media (Figures 2 and 3). This suggests that the total nitrogen source consumption depended on the initial composition of the fermentation medium, as indicated before [2,5,[14][15][16].
The values of αX,TN and K4 in equation (3) varied as a function of the initial media composition, but the values of α*X,BT and αX,pH were constant in all cultures (Tables 5 and  6), indicating that the competition between the biomass and bacteriocin by the nitrogen source and the effect of pH on the growth were similar in the different fermentations.
As observed in the series of fermentation DW-G, the values of αBT,X, α*BT,X, and αBT,pH in eq. (4) were constant for the different cultures (Tables 5 and 6), meanwhile the values calculated for αBT,TN, and K5 depended on the initial TS and TP concentrations in the different culture media. Table 5. Statistically significant (P < 0.05) parameter values (as estimates  confidence intervals) calculated with the four-dimensional LV-like equation (1) Table 6. Statistically significant (P < 0.05) parameter values (as estimates  confidence intervals) calculated with the four-dimensional LV-like equation (1)   In the series of cultures DW-TS-TP, the effects of initial TS and TP concentrations on both growth and bacteriocin production were described by empirical quadratic equations [15]. However, the inclusion of terms for explaining these effects in the four-dimensional LV-like equation (1)-(4) could contribute to obtaining a general equation for describing the evolution of the four response variables (pH, TN, X and BT). However, this approach has several drawbacks, since too large equations could be obtained, and information about the true relationship between the dependent variables (pH(t), TN(t), X(t), BT(t)) and the own essence of the LV relationships would be lost.
In fact, the rates of culture pH drop (rpH), total nitrogen consumption (rTN), and biomass (rX) and nisin (rBT) production in the different experiments (1-14) did not show a clear dependence on changes in initial TS and TP concentrations (Figure 8).

Series of fermentation DW-MRS
In this series of fermentation (Figure 4), the values of αBC and αpH,X in equation (1) decreased with the increase in the initial MRS nutrients concentration in the medium (Table 7), due to the increase in the buffering capacity of the MRS nutrients-supplemented DW media [16]. The values of K1 varied slightly between 4.693 and 4.931 (Table 7)  In equation (3), αX,TN decreased, suggesting that the nitrogen source was not consumed in parallel with the initial TN concentration in the medium. As observed in the series of fermentation DW-TS-TP, the values of α*X,BT and αX,pH were constant in all cultures (Table 7). Table 7. Statistically significant (P < 0.05) parameter values (as estimates  confidence intervals) calculated with the four-dimensional LV-like equation (1)   The values of K4 increased with the increase in MRS nutrients addition, in a perfect agreement with the increase in the final biomass concentrations (0.482, 0.582, 0.621, 0.759, 0.818, and 0.833 g/L) in the DW-MRS media ( Figure 4).
As occurred in the two previous series of fermentation DW-G and DW-TS-TP, constant values for the parameters αBT,X, α*BT,X and αBT,pH were obtained in eq. (4) in the different cultures (  (Figure 4).
In this series of cultures, the increase in the MRS nutrients addition to the DW medium affected both the evolution of the cultures and the final concentrations of biomass and nisin obtained (Figure 4), as well as the rates rpH, rTN, rX and rBT (Figure 9). The rates of culture pH decrease exhibited a transition from exponential decay-shaped curves (in the DW25, DW50, DW75 media) to bell-shaped curves (in the DW100 and DW125 media), meanwhile the rTN, rX and rBT profiles showed bell-shaped curves. Although the rpH decreased exponentially with the increase in the initial MRS nutrients concentration ([Nut]0) in the medium (upper left part of Figure 9), the rTN, rX and rBT profiles did not show an appreciable relationship (linear, quadratic, sigmoidal, etc.) with [Nut]0.
Therefore, in this case, it is difficult to develop a general four-dimensional LV-like equation (e.g. equations (8)-(11)) describing the evolution of pH, TN, X and BT as a function of the initial concentrations of MRS nutrients.

Series of fermentation DW-pH
The results obtained in these series of cultures are shown in Table 8 and the predictions of equations (1)-(4) are shown as solid lines in Figure 5. In these fermentations, the lowest values of αBC were calculated for the cultures performed at pH values lower than 6.0, in which the lowest growth and pH decrease were observed ( Figure 5). In contrast, the highest αBC values were obtained in the cultures conducted at pH values ≥ 6.0 ( Table  8), in which the highest growths and pH decreases were observed ( Figure 5).

Microorganisms, Culture Media and Inoculum Preparation
In this work, Lactococcus lactis CECT 539 and Carnobacterium piscicola CECT 4020 were used as the nisin-producing strain and target bacterium (in the nisin activity bioassay), respectively. Both strains were obtained from the Spanish Type Culture Collection (CECT) and cultured at 30 ºC in MRS (de Man Rogosa and Sharpe) agar slants or broth.
Diluted whey (DW) and concentrated mussel-processing waste (CMPW) were used to prepare the different culture media (Table 1). Sterilization (121 ºC/15 min) of these substrates led to the precipitation of a protein fraction that interfered with biomass measurements. For this reason, the precipitated material was removed by acidification of the DW and CMPW substrates to pH 4.5 with 5 N HCl, heating (121 ºC/15 min) and centrifugation (12,000 × g for 15 min) [22,23].
Given that the nisin-producing strain is not an amylolytic bacterium, the glycogen contained in the CMPW was enzymatically hydrolyzed to produce a glucose-containing substrate as described in Costas et al. [27].
To prepare the different fermentation substrates, DW medium was supplemented with the following nutrients: i) different amounts of glucose to obtain 5, 10, 15, 20, and 25 g glucose/L of medium (series of fermentation DW-G) [14], ii) different volumes of CMPW medium (101.33 g glucose/L) and amounts of KH2PO4 to obtain initial total sugars and phosphorous concentrations between 22.61 and 51.35 g/L, and 0.24 and 0.63 g/L, respectively (series of fermentation DW-TS-TP) [15], and iii) MRS broth nutrients at 25, 50, 75, 100, and 125% (w/v) of their standard concentrations in the complex substrate to produce the DW25, DW50, DW75, DW100, and DW125 media (series of fermentation DW-MRS) [16]. In the latter cultures, glucose and Tween 80 were not added to the DW medium because the addition of these compounds did not improve nisin production [16]. In these three series of fermentation, control cultures in unsupplemented DW medium were performed to obtain data for the comparisons [14][15][16].
The fourth series of batch cultures was conducted in DW100 medium adjusted to different initial pH values: 4.5, 5.0, 5.5, 6.0, 6.5, 7.0 or 7.5 (series of fermentation DW-pH) [16]. Tables 1 and 2 show the mean compositions of the resulting culture media used in this work.
To prepare the preculture, cells of L. lactis CECT 539 were inoculate from MRS agar slants to sterile MRS broth (10 mL) and incubated at 30 ºC for 12 h with shaking at 200 Second, the values of the parameters of the four-dimensional LV equation were continuously adjusted to describe the time course of pH(t), TN(t), X(t) and BT(t) of each individual culture in different series of fermentation.
Before being used to fit the four-dimensional LV-like equation, the experimental data of the culture pH and remaining concentrations of total nitrogen, biomass and nisin [14][15][16] were smoothed using the following logistic equations (5-7): For the culture pH (Q(t) = pH(t)) and total nitrogen (Q(t) = TN(t)) decrease, we modified the logistic decline equation presented by Goudar et al. [29] as follows: ( ) = − ( 0 + 1 · + 2 · 2 ) For biomass (X(t)) production, the logistic equation presented by Goudar et al. [29] was used by considering that the death cell rate was zero: Being = and = − For nisin (BT(t)) synthesis, we modified the logistic decline equation [29] as follows: ( ) = 1 + · (− · ) − 1 + Being = In an attempt to obtain a general equation for describing the batch nisin production system, the four-dimensional LV-like equation (1)-(4) was modified by including a term (δi = exp(-J·F0)) to explain the effect of the initial culture conditions (F0: concentrations of glucose, TS and TP, and MRS broth nutrients, or culture pH) on the evolution of the dependent variables pH(t), TN(t), X(t) and BT(t). So that, the four-dimensional LV-like equation (1)-(4) was modified as follows: Where n is the number of experimental data, Yexpi is the experimental value and Ypredi is the value predicted by the model. Values of R 2 ≥ 0.95, RPDM < 10% [5], and Bf and Af close to 1 [30] are indicative that the corresponding equation was accurately fitted to the experimental data.

K4
Theoretical maximum biomass concentration (g/L) affected by the competition between biomass and nisin production for the nitrogen source and pH time course

BT(t)
Nisin concentration (BU/mL) over the time further knowledge is provided about the relationship between the main culture variables (culture pH, total nitrogen consumption and the synthesis of biomass and bacteriocin) involved in nisin production, which is usually difficult to explain. However, a general four-dimensional LV-equation (with a unique set of parameters) could only be appropriately developed to describe each series of fermentation when the change in the initial nutrient composition or pH in the media produced a clearly observable effect (e.g., linear, quadratic or exponential) on the evolution of the rates of culture pH decrease, TN consumption and production of biomass and nisin.