Enhancing Logistic Modeling for Diauxic Growth and Biphasic Antibacterial Activity Synthesis by Lactic Acid Bacteria in Realkalized Fed-Batch Fermentations

Abstract

The logistic modeling of diauxic growth and biphasic antibacterial activity (AA) production was enhanced for four lactic acid bacteria (Lactococcus lactis CECT 539, Pediococcus acidilactici NRRL B-5627, Lactobacillus casei CECT 4043, and Enterococcus faecium CECT 410) during realkalized fed-batch fermentations. The improved growth model, also validated for describing the diauxic growth of Mos breed roosters and foals, overcomes a key limitation of the bi-logistic model, which assumes the existence of two distinct populations growing from the start of the culture, each following a different growth profile. In contrast, the improved logistic growth model developed in this study accounts for a single population growing at two rates, offering a fit to the experimental data comparable to that of the commonly used bi-logistic model. The enhanced model for product synthesis accurately describes biphasic AA production, assuming that antibacterial products are synthesized as growth-associated metabolites, depending on the final pH reached in the cultures at each sampling time. Additionally, it is easier to apply than the unmodified or modified differential forms of the Luedeking–Piret model. This study demonstrated, for the first time, the applicability of these two models in describing the diauxic growth and biphasic AA synthesis of LAB.

1. Introduction

Various mathematical models have been employed to describe the kinetics of growth (e.g., Verhulst, Gompertz, Richards, Bertalanffy, Weibull, Monod, etc.) and bacteriocin synthesis (e.g., unmodified and modified forms of the Luedeking–Piret model) in lactic acid bacteria (LAB) during both batch and fed-batch fermentations [1,2,3,4,5,6]. However, for effective monitoring and control of these bioprocesses, it is essential to use models that provide the most accurate description of growth and bacteriocin production kinetics in LAB [1,5], particularly from a microbiological perspective.

The simple-logistic Equation (1) has been commonly used to describe the batch biomass production (X(t)) of LAB in different cultures [4,5,7]:

$$X(t)={\displaystyle \frac{K}{1+a_{X0·}{e}^{{-a}_{X1}·t}}} , \mathrm{being} a_{X0}={\displaystyle \frac{K-X_0}{X_0}},$$

(1)

where K and X₀ are the maximum and the initial biomass concentration (g/L), respectively, a_X1 (hours⁻¹) is the growth rate, and t is the time (hours).

However, Equation (1) fails to accurately describe the lag phase (by either underestimating or overestimating the initial biomass concentration) and the decline phases of growth. To address this limitation, a generalized-logistic Equation (2) was proposed by Edwards and Wilke [7] and DeWitt [8] to better model cell growth:

$$X\left(t\right)= {\displaystyle \frac{K}{1+a_{X0}·e^{\left(-a_{X1}·t-a_{X2}·t^2\dots \dots \dots -a_{Xn}·t^n\right)}}},$$

(2)

where a_X2 (h⁻²) and a_Xn (h⁻ⁿ) are constants in model (2) that determine the influence of the t² (time raised to the power of 2) and tⁿ (time raised to the power of n) terms on the growth behavior being modeled. Thus, a_X2 and a_Xn modulate how time influences biomass production, X(t), in a nonlinear manner based on higher powers of time.

The cell growth and product formation were appropriately described using a particular form of the generalized logistic Equation (2) [9]:

$$X\left(t\right)= {\displaystyle \frac{K}{1+a_{X0}·e^{\left(-a_{X1}·t-a_{X2}·t^2\right)}}}.$$

(3)

The introduction of the term a_X2·t² into the equation may allow the model to better capture more complex growth behaviors than a simple logistic model, including the consideration of the decline in growth toward the end of the culture period [7,8,9]. If the growth experiences either acceleration or deceleration that cannot be captured by a logistic model (1), removing this term (a_X2·t²) could lead to a poor fit.

In the case of a diauxic culture, which involves the growth of a microorganism that can utilize two distinct carbon or nitrogen sources [4,10,11,12], the logistic equation can be modified to reflect this behavior. For example, the diauxic growth of different microbiological [4,10,11,12] and biological (e.g., Mos breed roosters and foals) [5] populations has been described by using the bi-logistic model (4) [5,11,12], mainly due to easy use, simplicity and reproducibility:

$$X\left(t\right)= {\displaystyle \frac{K_1}{1+a_{X0}·e^{-a_{X1}·t}}}+{\displaystyle \frac{K_2}{1+b_{X0}·e^{-b_{X1}·t}}}=X_1\left(t\right)+X_2\left(t\right),$$

(4)

where K₁ and K₂ are considered to be the maximum biomass concentrations (g/L) in the first and second growth phases, respectively, a_X0 (dimensionless) and a_X1 (hours⁻¹), are constants for the first growth phase, and b_X0 (dimensionless) and b_X1 (hours⁻¹) are constants, with the same meaning of a_X0 and a_X1, for the second growth phase. X₁(t) and X₂(t) are the biomass concentrations (g/L) over the time for the population 1 and 2.

Making $a_{X0}=e^{c_1}$; $b_{X0}=e^{c_2}$ gives

$$X\left(t\right)={\displaystyle \frac{K_1}{1+e^{c_1}·e^{-a_{X1}·t}}}+{\displaystyle \frac{K_2}{1+e^{c_2}·e^{-b_{X1}·t}}},$$

(5)

So that

$$X\left(t\right)={\displaystyle \frac{K_1}{1+e^{c_1-a_{X1}·t}}}+{\displaystyle \frac{K_2}{1+e^{c_2-b_{X1}·t}}}.$$

(6)

Expression (6) could be transformed as follows:

$$X\left(t\right)= {\displaystyle \frac{K_1}{1+e^{a_{X1}·\left(\frac{c_1}{a_{X1}}-t\right)}}}+{\displaystyle \frac{K_2}{1+e^{b_{X1}·\left(\frac{c_2}{b_{X1}}-t\right)}}}.$$

(7)

Making K₁ = G_m1; K₂ = G_m2; a_X1 = μ_m1; b_X1 = μ_m2; τ₁ = c₁/a_X1 and τ₂ = c₂/b_X1 gives the expression used by Vázquez et al. [5] to describe the biphasic growth of Mos breed roosters and foals:

$$X\left(t\right)={\displaystyle \frac{G_{m1}}{1+e^{{\mu }_{m1}·\left({\tau }_1-t\right)}}}+{\displaystyle \frac{G_{m2}}{1+e^{{\mu }_{m2}·\left({\tau }_2-t\right)} }}=X_1\left(t\right)+X_2\left(t\right),$$

(8)

where τ (hours), μ_m (hours⁻¹), and G_m (g/L) represent the time required to reach 50% of maximum growth, the specific maximum net growth rate, and the maximum net growth, respectively, and t is the time (hours). The superscripts 1 and 2 refer to the first and second sigmoidal phases of the biphasic growth pattern [5].

To improve the diauxic growth predictions, a generalized bi-logistic model could be used, as proposed by Edwards and Wilke [7] and DeWitt [8] for the monoauxic growth:

$$X\left(t\right)= {\displaystyle \frac{K_1}{1+a_{X0}·e^{\left(-a_{X1}·t-a_{X2}·t^2\right)}}}+{\displaystyle \frac{K_2}{1+b_{X0}·e^{\left(-b_{X1}·t-b_{X2}·t^2\right)}}}=X_1\left(t\right)+X_2\left(t\right).$$

(9)

In fact, the bi-logistic models (4) and (9) are extensions of the logistic models (1) and (3), respectively. These models assume the simultaneous growth of two interacting microbial populations, X₁(t) and X₂(t), from the beginning of the culture (time zero), with their growth limited by factors such as resource availability and interspecies competition. The combined growth of both populations equals the total growth observed in the culture (Figure 1).

However, these assumptions limit the suitability of the bi-logistic model for describing diauxic growth in pure cultures of bacteria or yeast, where growth typically occurs in distinct phases rather than concurrently. Nevertheless, models (4) and (9) may still be valuable for exploring ecological dynamics in microbial communities involving two species and can be applied in studies of microbiology, microbial ecology, and biotechnology.

In contrast to the assumptions of the bi-logistic model, pure cultures of microorganisms involve the growth of a single strain, even when two distinct growth phases are observed (Figure 1). In such cases, the microorganism sequentially utilizes two different carbon or nitrogen sources [4,10,11,12,13,14]. During the first growth phase, the strain consumes the carbon or nitrogen source that is easier to metabolize. Once this nutrient is depleted, the cells enter a transitional phase (TP), characterized by reduced growth and the production of enzymes required to metabolize the second carbon or nitrogen source. Following this phase, the cells resume growth in the second phase by utilizing the remaining nutrient [13,14]. Therefore, the time between the two growth phases, denoted as t_bgp, marks the end of the first growth phase and the beginning of the second, and is located within the transitional phase.

At the end of the second growth phase, the cells enter a stationary phase (SP), characterized by a slowdown in biomass production (Figure 1) and the establishment of a steady state in which cell growth and cell death are balanced. Growth cessation during this phase is typically caused by the depletion of essential nutrients or the accumulation of inhibitory metabolites, both of which restrict biomass production by the growing strain [15,16].

Additionally, at t = 0 h, the initial biomass concentration (inoculum concentration), as calculated from Equation (9), is given by

$$X\left(t\right)={\displaystyle \frac{K_1}{1+a_{X0}}}+{\displaystyle \frac{K_2}{1+b_{X0}}}=X_1\left(0\right)+X_2\left(0\right).$$

However, this expression incorrectly includes a term for the initial biomass concentration of a second population (X₂(0)). This is an error, as no second population is present at the initial or any stage of fermentation in pure microbial cultures.

On the other hand, the unmodified Luedeking–Piret model (10) is commonly used to describe product formation, P(t), in batch cultures [9,17]:

$${\displaystyle \frac{dP\left(t\right)}{dt}}= \alpha ·{\displaystyle \frac{dX\left(t\right)}{dt}}+\beta ·X\left(t\right),$$

(10)

where α (units of [P]/units of [X]) and β (units of [P]/units of [X]/unit of time) are the growth-associated and non-growth-associated constants, respectively.

The use of model (10) allows for the classification of the product as growth-associated (α ≠ 0, β = 0), non-growth-associated (α = 0, β ≠ 0), or mixed (α ≠ 0, β ≠ 0) metabolites [9,17].

However, this model may not always be suitable for more complex biological systems, where product formation can be influenced by additional factors, denoted as F(t), such as substrate availability, dissolved oxygen concentration, or stress responses (e.g., pH level or the rate of pH decrease, ∆pH) [9,17,18].

Since the production of antibacterial compounds by lactic acid bacteria (LAB) is influenced by the pH time-course and/or the rate of pH decrease, a modified version of the Luedeking–Piret model has been proposed to more accurately represent the synthesis of antibacterial activity by LAB [9,17,18]:

$${\displaystyle \frac{dP\left(t\right)}{dt}}= \left(\alpha ·{\displaystyle \frac{dX\left(t\right)}{dt}}+\beta ·X\left(t\right)\right)·\left(1+\delta ·{\displaystyle \frac{dF\left(t\right)}{dt}}\right),$$

(11)

$$\mathrm{being} {\displaystyle \frac{dF\left(t\right)}{dt}}={\displaystyle \frac{{pH(t}_{n-1})-{pH(t}_n)}{t_n-t_{n-1}}},$$

where δ is a constant (hours) that accounts for the influence of the modulating factor on product synthesis. The terms pH(t_n−1) and pH(t_n) represent the pH values at the previous and current sampling times, respectively, and the difference t_n − t_n−1 denotes the time interval between samplings.

However, models (10) and (11) may not adequately describe the biphasic synthesis of products, such as those observed in fed-batch cultures [4,12,19,20,21,22,23,24,25]. This is because the biphasic increases in both product and biomass synthesis are not always parallel, which could result in a poor fit between growth and product formation. Excluding this consideration can lead to inaccurate predictions in certain fermentation processes, as it significantly alters the dynamics of product formation.

Therefore, the aim of this study is to develop more accurate bi-logistic expressions to describe the diauxic growth and biphasic antibacterial activity (AA) synthesis in fed-batch cultures of Lactococcus lactis ssp. lactis CECT 539 in different culture media. Both models were validated using previously published experimental data from fed-batch cultures of other three LAB strains (Pedicoccus acidilactici NRRL B-5627, Lactobacillus casei ssp. casei CECT 4043, and Enterococcus faecium CECT 410). The improved growth model was also validated by using other experimental data from the diauxic growth of Mos breed roosters and foals.

2. Materials and Methods

2.1. Bacterial Strains

In this study, experimental data on growth and AA production were collected from realkalized fed-batch cultures of four bacterial strains: L. lactis subsp. lactis CECT 539, Lact. casei subsp. casei CECT 4043, and E. faecium CECT 410 from the Spanish Type Culture Collection (CECT), Valencia, Spain, and Pedicoccus acidilactici NRRL B-5627 from the Northern Regional Research Laboratory (NRRL), Peoria, IL, USA. Carnobacterium piscicola CECT 4020 was used as the target bacterium in the antibacterial activity bioassay.

Strains CECT 539, NRRL B-5627, CECT 4043, and CECT 4020 were cultured at 30 °C in MRS (de Man, Rogosa, and Sharpe, Merck, Germany) agar slants or broth, while strain CECT 410 was cultured at 30 °C in Rothe agar (Panreac, Barcelona, Spain) slants or broth.

2.2. Culture Media, Fermentation Conditions and Data Collection

Recent studies have investigated realkalized fed-batch cultures of L. lactis CECT 539, Ped. acidilactici NRRL B-5627, Lact. casei CECT 4043, and E. faecium CECT 410, using various culture media and feeding substrates (

Table 1. Fermentation and feeding media used in the fifteen realkalized fed-batch cultures of Lactococcus lactis subsp. lactis CECT 539.

Fermentation Medium *	Feeding Medium **	Fermentation Name ***	Reference
DW (22.54 g lactose/L)	CW (48.11 g lactose/L) + CL (400 g/L)	I. CECT 539	[4]
MPW (5.33 g glucose/L)	CG (240 g/L)	II. CECT 539
	CMPW (101.33 g glucose/L)	III. CECT 539
DW (22.16 g lactose/L)	CL (400 g/L)	IV. CECT 539	[19]
	CG (400 g/L)	V. CECT 539
DW (22.62 g lactose/L)	CW (51.35 g lactose/L) + CG (400 g/L)	VI. CECT 539	[20]
	CMPW (101.72 g glucose/L) + CG (400 g/L)	VII. CECT 539
DWP (22.61 g lactose/L)	CMPWG (400 g glucose/L) + CW (51.35 g lactose/L)	VIII. CECT 539	[21]
	CMPWGP (400 g glucose/L) + CW (51.35 g lactose/L)	IX. CECT 539
DW (22.62 g lactose/L)	CMPW (101.72 g glucose/L) + CL (400 g/L)	X. CECT 539	[22]
	CMPWGP (400 g glucose/L)	XI. CECT 539
DW25 (20.96 g of TS/L)	CW (53.40 g lactose/L) + CG (400 g/L)	XII. CECT 539	[23]
DW50 (21.92 g of TS/L)	CW (53.40 g lactose/L) + CG (400 g/L)	XIII. CECT 539
DW75 (22.89 g of TS/L)	CW (53.40 g lactose/L) + CG (400 g/L)	XIV. CECT 539
DW100 (24.81 g of TS/L)	CW (53.40 g lactose/L) + CG (400 g/L)	XV. CECT 539

* DW: Diluted whey medium; MPW: Mussel processing waste medium; DWP: DW medium supplemented with KH₂PO₄ (Panreac, Barcelona, Spain) to achieve a total phosphorus (TP) concentration of 0.46 g/L; DW25, DW50, DW75 and DW100: DW medium supplemented with nutrients from MRS broth (excluding glucose and Tween 80) at 25%, 50%, 75%, and 100% (w/v) of the standard concentrations of each component in the complex medium, respectively; TS: total sugars. ** CW: Concentrated whey medium; CL: Concentrated lactose; CG: Concentrated glucose; CMPW: Concentrated MPW medium; CMPWG: CMPW medium supplemented with glucose (Panreac, Barcelona, Spain) to a concentration of 400 g/L; CMPWGP: CMPWG medium supplemented with KH₂PO₄ to a TP concentration of 3.21 g/L. *** CECT: Spanish Type Culture Collection, Valencia, Spain.

and

Table 2. Fermentation and feeding media used in the five realkalized fed-batch cultures of Ped. acidilactici NRRL B-5627, Lactobacillus casei CECT 4043, and E. faecium CECT 410.

Fermentation Medium *	Feeding Medium **	Fermentation Name ***	Reference
MPW (5.33 g glucose/L)	CG (240 g/L)	I. NRRL B-5627	[24]
	CMPW (101.33 g glucose/L)	II. NRRL B-5627
DWYE2 (20.06 g lactose/L)	CWYE2 (48.51 g lactose/L) + CG (400 g/L)	III. NRRL B-5627
	CG (400 g/L)	IV. NRRL B-5627
	CWYE4 (48.51 g lactose/L) + CG (400 g/L)	V. NRRL B-5627
DW (20.54 g lactose/L)	CW (48.11 g lactose/L) + CL (400 g/L)	I. CECT 4043	[25]
	CMPW (101.33 g glucose/L) + CG (310 g/L)	II. CECT 4043
DW (22 g lactose/L)	CL (400 g/L)	I. CECT 410	[12]
	CW (48.11 g lactose/L) + CL (400 g/L)	II. CECT 410

* DW: Diluted whey medium; MPW: Mussel processing waste medium; DWYE2: DW medium supplemented with 2% (w/v) yeast extract. ** CG: Concentrated glucose; CMPW: Concentrated MPW medium; CWYE2: CW medium supplemented with 2% (w/v) yeast extract; CWYE4: CW medium supplemented with 4% (w/v) yeast extract; CW: Concentrated whey medium; CL: Concentrated lactose. *** CECT: Spanish Type Culture Collection, Valencia, Spain; NRRL; Northern Regional Research Laboratory, Peoria, IL, USA.

). These cultures were initiated as batch cultures for 8 h (cultures II and III for L. lactis CECT 539) or 12 h (cultures I, IV–XV for L. lactis CECT 539; cultures I–V for Ped. acidilactici NRRL B-5627; cultures I–II for Lact. casei CECT 4043; and cultures I–II for E. faecium CECT 410), until the culture pH reached its minimum value. After sampling at these respective times, the cultures were realkalized by adding 4 N NaOH (Panreac, Barcelona, Spain) to restore the pH to the initial value of the fermentation medium and were subsequently fed with the appropriate substrate to replenish the initial total sugar concentration. This procedure ensured that, at the beginning of each realkalization and feeding cycle, the fermentation medium consistently maintained the same pH and total sugar concentration [4,12,19,20,21,22,23,24,25].

During these fermentations, the remaining concentrations of biomass (X(t), in g/L) and antibacterial activity (AA(t), in activity units (AU)/mL), along with culture pH and other variables, were measured at regular intervals. From these measurements, the concentrations of extracted biomass (X_ext(t)) and extracted antibacterial activity (AA_ext(t)), as well as the cumulative concentrations (the sum of remaining and extracted levels) of both variables (ΣX(t) and ΣAA(t)), were calculated [4,12,19,20,21,22,23,24,25].

The values of ΣX(t) and ΣAA(t) obtained from these realkalized fed-batch fermentations were previously used to analyze the kinetic behavior of these variables during the growth of the strains CECT 539, NRRL B-5627, CECT 4043, and CECT 410, as well as to calculate the corresponding yields and efficiencies in the cultures [4,12,19,20,21,22,23,24,25].

In this study, the remaining concentrations of both variables, X (t) and AA(t), in the fermentor were used to refine the bi-logistic growth and product formation models, enabling a more accurate description of growth and AA production by L. lactis CECT 539 in the various realkalized fed-batch cultures [4,19,20,21,22,23].

The applicability of the improved models for describing microbial growth and product formation was further extended by using the remaining concentrations of biomass and AA produced by Ped. acidilactici NRRL B-5627 [24], Lact. casei CECT 4043 [25], and E. faecium CECT 410 [12] in similar realkalized fed-batch cultures.

The preparation of the various substrates used as fermentation and feeding media in the cultures for the four LAB strains has been described in previous works by our research team [4,12,19,20,21,22,23,24,25].

The growth data from the biphasic growth of Mos breed roosters and foals, which were used to validate the applicability of the improved bi-logistic growth model, were extracted from a published article [5] using GetData Graph Digitizer version 2.24 software (2011).

2.3. Biomass and Antibacterial Activity Determinations

The remaining concentrations of biomass, X(t), and antibacterial activity, AA(t), during fed-batch fermentations were determined using methods described in previous works [4,12,19,20,21,22,23,24,25]. AA(t) (in Activity Units (AU)/mL) was quantified as the cumulative effect of bacteriocins (e.g., nisin or pediocin), organic acids (lactic acid and acetic acid), and other inhibitory metabolites (such as butane-2,3-diol and ethanol) produced at each sampling time during the realkalized fed-batch cultures, assessing their impact on the growth of C. piscicola CECT 4020 [4,12,19,20,21,22,23,24,25].

2.4. Models Improvement

2.4.1. Growth Model

Considering Equation (3), the initial biomass concentration (X₀) can be expressed as

$$X_0={\displaystyle \frac{K}{1+a_{X0}}}.$$

(12)

Solving for K in Equation (12) gives

$$K=X_0·(1+a_{X0}).$$

(13)

Substituting K back into model (3) results in

$$X\left(t\right)={\displaystyle \frac{X_0·\left(1+a_{X0}\right)}{1+a_{X0}·e^{\left(-a_{X1}·t-a_{X2}·t^2\right)}}}.$$

(14)

Thus, the numerator of Equation (14) contains the initial biomass, which is a fixed value depending on the inoculum size used, multiplied by the factor (1 + a_X0).

Model (14) serves as the basis for developing an improved bi-logistic model to describe the diauxic growth in pure microbial cultures, based on the following asumptions:

1. The biomass concentration at the time marking the end of the first growth phase and the beginning of the second (t_bgp), denoted as X(t_bgp), is given by

$$X\left(t_{bgp}\right)={\displaystyle \frac{K_1}{1+a_{X0}·e^{\left(-a_{X1}·t_{bgp}-a_{X2}·{t_{bgp}}^2\right)}}}={\displaystyle \frac{K_2}{1+b_{X0}·e^{\left(-b_{X1}·t_{bgp}-b_{X2}·{t_{bgp}}^2\right)}}},$$

(15)

where a_X0, a_X1, and a_X2, are constants corresponding to the first growth phase, and b_X0, b_X1, and b_X2 are constants corresponding to the second growth phase.

The value of K₂ at this point is given by

$$K_2=X\left(t_{bgp}\right)·\left(1+b_{X0}·e^{(-b_{X1}·t_{bgp}-b_{X2}·{t_{bgp}}^2)}\right).$$

(16)

2. In contrast to X₀, which is obtained from Equation (12), the value of X(t_bgp) depends not only on the inoculum size but also on the culture conditions, such as the initial chemical composition of the substrate, pH, temperature, type of fermentation, and other factors. These conditions influence the time course of biomass production during fermentation [4,12,19,20,21,22,23,24,25]. Therefore, it can be hypothesized that X(t_bgp) is reached during the first growth phase as time progresses from 0 to t_bgp, as indicated by Equation (14).

According to Equation (15), the growth in the second growth phase is given by

$$X_2\left(t\right)={\displaystyle \frac{K_2}{1+b_{X0}·e^{\left(-b_{X1}·t-b_{X2}·t^2\right)}}}.$$

(17)

Substituting Equations (14) and (16) into Equation (17) gives

$$X\left(t\right)={\displaystyle \frac{K_1·\left(1+b_{X0}·e^{\left(-b_{X1}·t_{bgp}-b_{X2}·{t_{bgp}}^2\right)}\right)}{\left(1+a_{X0}·e^{\left(-a_{X1}·t-a_{X2}·t^2\right)}\right)·\left(1+b_{X0}·e^{\left(-b_{X1}·t-b_{X2}·t^2\right)}\right)}}.$$

(18)

Thus, the numerator of Equation (18) can be simplified as

$$K_1·\left(1+b_{X0}·e^{(-b_{X1}·t_{bgp}-b_{X2}·{t_{bgp}}^2)}\right)=K_f.$$

In this case, the initial biomass concentration at t = 0 h is

$$X\left(0\right)={\displaystyle \frac{K_f}{\left(1+a_{X0}\right)·\left(1+b_{X0}\right)}}.$$

Finally, as t approaches infinity, the biomass concentration reaches its final value:

$$X\left(t\to \infty \right)= K_f.$$

The main advantage of using model (18) is that it accounts for the existence of a single microbial population whose growth follows a diauxic growth pattern.

Based on the diauxic growth data of the four lactic acid bacteria (Lactococcus lactis CECT 539, Pediococcus acidilactici NRRL B-5627, Lactobacillus casei CECT 4043, and Enterococcus faecium CECT 410), as well as data from Mos breed roosters and foals, it is challenging to precisely determine the t_bgp values [11,12]. To address this, the experimental time interval corresponding to the transitional phase—within which the t_bgp value is located—was estimated by calculating the absolute growth rates, rX(t), between successive experimental time points (t_n−1 and t_n) for each culture, using the following expression:

$$rX\left(t\right)={\displaystyle \frac{dX\left(t\right)}{dt}}= {\displaystyle \frac{X\left(t_n\right)-X\left(t_{n-1}\right)}{t_n-t_{n-1}}}.$$

The values of rX(t) were then plotted against fermentation time (t), as shown in Figures S1–S3. The calculated t_bgp value obtained from model (18) was considered satisfactory if it fell within the experimental time interval corresponding to the transitional phase of growth, as indicated by the red arrows in Figures S1–S3.

2.4.2. Antibacterial Activity Model

For product formation, the following generalized logistic equation is normally used [9]:

$$P\left(t\right)= {\displaystyle \frac{P_m}{\left(1+a_{P0}·e^{{(-a}_{P1}·t-a_{P2}·t^2)}\right)}},$$

(19)

where P_m is the maximum concentration of P (measurements units of P), and a_P0 (dimensionless), a_P1 (time unit⁻¹), and a_P2 (time unit⁻²) are constants in Equation (19).

According to Equation (19), the product appears to increase autocatalytically, similar to the pattern observed in biomass production [26]. However, this is not the case of AA, which is synthesized in a logistic manner by the growing strain, with its production dependent on the final pH during each realkalization and feeding cycle [4,17,24]. Since the biphasic increases in both product and biomass synthesis are not parallel, the following bi-logistic equation was proposed to better describe the time course of AA production:

$$AA\left({\displaystyle \frac{\mathrm{A}\mathrm{U}}{\mathrm{m}\mathrm{L}}}\right)= {\displaystyle \frac{\alpha ·X\left(t\right)·\left(1+\phi ·\left(pHi-pHf\left(t\right)\right)\right)}{\left(1+a_{P0}·e^{{(-a}_{P1}·t-a_{P2}·t^2)}\right)·\left(1+{b_{P0}·e}^{{(-b}_{P1}·t-b_{P2}·t^2)}\right)}},$$

(20)

where α is a proportionality constant (units of AA per units of X), φ is a dimensionless constant accounting for the effect of the pH gradient (the difference between the initial pH (pHi) and the final pH (pHf(t)) at each sampling time on AA production, a_P0, b_P0 (dimensionless), a_P1, b_P1 (in units of time⁻¹), and a_P2, b_P2 (in units of time⁻²) are constants in Equation (20).

This equation can be used to characterize the product synthesized as a pH-dependent primary metabolite (α ≠ 0, φ ≠ 0). The value of φ indicates whether the effect of the pH gradient time-course is appropriate for AA synthesis (φ > 0) or not (φ < 0). A positive φ value suggests that the experimental pH gradient values at each sampling time favored the post-translational processing of pre-bacteriocin (with low antibacterial activity) to produce active bacteriocin [27], rather than bacteriocin adsorption onto the producer cell surface [28,29,30,31,32,33,34]. In contrast, a negative φ value indicates that the pH gradient values at each sampling time favored bacteriocin adsorption onto the producer cell surface more than the post-translational processing of pre-bacteriocin to produce its active form.

2.5. Model Parameters Determination and Model Evaluation

The independent experimental data from duplicate fermentations, each with three analytical replicates (six total replicates per experimental point) [4,12,19,20,21,22,23,24,25], were used to develop the growth and AA synthesis models and to calculate the model parameters. The values for the constants were first obtained by minimizing the sum of squared differences between the experimental and model-calculated values, using the nonlinear least-squares (quasi-Newton) method via the Solver tool in Microsoft Excel 2016 (Microsoft Corp., Redmond, WA, USA). Subsequently, the standard errors for each constant, along with the corresponding R², F-values, and p-values for the models, were determined using SigmaPlot^® 15 (Systat Software, Inc., San Jose, CA, USA, 2022).

The goodness-of-fit of the different models was also assessed using the following metrics: the mean relative percentage deviation modulus (RPDM) [22], the sum of squared differences (SSD) between predicted (Y_pi) and experimental (Y_i) data points, as well as the accuracy (A_f) and bias (B_f) factors [5,26]:

$$RPDM= {\displaystyle \frac{100}{n}}{\sum }_{i=1}^n{\displaystyle \frac{\left|Y_i-{Yp}_i\right|}{Y_i}},$$

$$SSD={\sum }_{i=1}^n{\left(Y_i-{Yp}_i\right)}^2,$$

$$A_f={10}^{\sum _{i=1}^n{\displaystyle \frac{\left|log\frac{{Yp}_i}{Y_i}\right|}{n}}},$$

$$B_f={10}^{\sum _{i=1}^n{\displaystyle \frac{log\frac{{Yp}_i}{Y_i}}{n}}},$$

where n is the number of experimental data points.

If the RPDM values are lower than 10% [35,36] and the SSD value is near zero, it could be considered that the fitted models accurately describe the trend observed in the experimental data. The accuracy and bias factors indicate the average deviation between the model predictions and observed results, with their proximity to a value of 1.0 serving as an effective and practical measure of predictive model validity [5,26].

3. Results and Discussion

The accuracy and practical utility of the logistic models (18) and (20), developed in this study for growth and product synthesis, respectively, were initially validated using experimental data from diauxic growth (X) and biphasic antibacterial activity (AA) production in fifteen realkalized fed-batch cultures of L. lactis CECT 539 [4,19,20,21,22,23]. To further demonstrate the applicability of the two developed models in describing growth and AA synthesis, the results from the logistic growth model (18) were compared with those from models (4) and (9), while the results from the logistic AA model (20) were compared with those from models (10) and (11).

3.1. Modeling the Growth of L. lactis CECT 539 in Realkalized Fed-Batch Cultures

Figure 2 shows the experimental data for growth and AA production from the fifteen realkalized fed-batch cultures of L. lactis CECT 539, along with the predictions (lines) for growth and AA synthesis based on the logistic models (4) and (10), respectively. Regarding the biomass production, there was an excellent agreement between the experimental and predicted growth values with model (4), with R² values, F-values, and SSD values ranging from 0.9814 to 0.9993, from 844.27 to 117,949.35, and from 0.006 to 0.598 (Tables S1 and S2). This strong agreement between experimental and predicted values is further supported by RPDM values (1.85–8.26) consistently below 10%, A_f (1.02–1.13), and B_f (0.96–1.00) values close to 1.0 (Tables S1 and S2).

According to the taxonomy of bi-logistic growth curves described by Meyer [11], the biphasic growth curves of fed-batch cultures I, II, III, V, VI, VII, VIII, IX, and X (Figure 2), obtained using model (4), exhibit a profile with two nearly non-overlapping logistic growth phases, referred to as sequential logistics. In contrast, for fed-batch cultures IV, XI, XII, XIII, XIV, and XV, the two growth curves appear superimposed, as the second growth phase is not as clearly defined as the first. This suggests that, in these cultures, the growth kinetics might also be described using logistic model (1).

Figure 3 presents the experimental data for pH at the end of each realkalization and feeding cycle, along with the experimental data for growth and AA production from the fifteen realkalized fed-batch cultures of L. lactis CECT 539. It also shows the predictions for growth and AA synthesis based on logistic models (9) and (11), respectively. Model (9) provided a good fit to the experimental data (Figure 3, Tables S3 and S4), as indicated by relatively high R² values (0.9861–0.9995), F-values (2404.61–144,960.33), low SSD values (0.004–0.551), RPDM values (1.18–8.04) below of 10%, and A_f (1.01–1.12) and B_f (0.97–1.00) values close to 1.0.

Model (18) also produced satisfactory results (Figure 4,

Table 3. Estimated values ± standard errors for the constants of model (18), the determination coefficient (R²), F-Fisher statistic, p-values, sum of squared differences between predicted and experimental data points (SSD), mean relative percentage deviation modulus (RPDM), and the accuracy (A_f) and bias (B_f) factors for the growth of L. lactis CECT 539 in the realkalized fed-batch cultures: I. CECT 539, II. CECT 539, III. CECT 539, IV. CECT 539, V. CECT 539, and VI. CECT 539.

	Cultures
Parameter	I. CECT 539	II. CECT 539	III. CECT 539	IV. CECT 539	V. CECT 539	VI. CECT 539
K1	1.20 ± 0.10	0.59 ± 5.10 × 10−3	1.52 ± 7.10 × 10−3	0.80 ± 0.01	0.86 ± 0.02	1.20 ± 0.03
aX0	46.86 ± 10.35	44.71 ± 16.55	63.92 ± 11.40	24.76 ± 5.66	29.22 ± 9.25	55.03 ± 17.22
aX1	0.23 ± 0.02	0.25 ± 0.06	0.26 ± 0.04	0.23 ± 0.02	0.22 ± 0.03	0.23 ± 0.02
aX2	−7.70 × 10−4 ± 5.80 × 10−5	−1.52 × 10−3 ± 7.00 × 10−4	−9.87 × 10−4 ± 8.14 × 10−6	−1.15 × 10−3 ± 1.00 × 10−4	−1.11 × 10−3 ± 2.00 × 10−4	−1.06 × 10−3 ± 2.00 × 10−4
bX0	0.97 ± 0.09	0.62 ± 0.08	1.77 ± 0.18	0.99 ± 0.23	0.97 ± 0.41	0.70 ± 0.16
bX1	−1.52 × 10−2 ± 2.10 × 10−3	−5.54 × 10−2 ± 4.20 × 10−3	−2.11 × 10−2 ± 2.60 × 10−3	−1.32 × 10−3 ± 4.00 × 10−4	−4.12 × 10−3 ± 7.50 × 10−4	−2.27 × 10−2 ± 5.10 × 10−3
bX2	2.04 × 10−4 ± 1.17 × 10−5	5.61 × 10−4 ± 3.39 × 10−5	2.46 × 10−4 ± 1.54 × 10−5	1.00 × 10−4 ± 3.58 × 10−5	1.07 × 10−4 ± 5.58 × 10−5	2.64 × 10−4 ± 3.49 × 10−5
tbgp	71.97 ± 0.33	57.83 ± 2.89	77.48 ± 7.12	70.63 ± 5.07	65.93 ± 1.56	61.90 ± 2.16
R2	0.9992	0.9911	0.9911	0.9926	0.9924	0.9993
F	170,414.31	6477.24	12,800.73	12,391.34	37,852.02	94,418.86
p	<0.0001	<0.0001	<0.0001	<0.0001	<0.0001	<0.0001
SSD	0.010	0.048	0.659	0.012	0.016	0.005
RPDM	1.50	6.10	7.71	3.68	3.94	1.55
Af	1.02	1.07	1.09	1.04	1.04	1.02
Bf	1.00	0.97	0.98	0.99	0.99	1.00

Parameter values are considered statistically significant when p < 0.05.

,

Table 4. Estimated values ± standard errors for the constants of model (18), the determination coefficient (R²), F-Fisher statistic, p-values, sum of squared differences between predicted and experimental data points (SSD), mean relative percentage deviation modulus (RPDM), and the accuracy (A_f) and bias (B_f) factors for the growth of L. lactis CECT 539 in the realkalized fed-batch cultures: VII. CECT 539, VIII. CECT 539, IX. CECT 539, X. CECT 539, and XI. CECT 539.

	Cultures
Parameter	VII. CECT 539	VIII. CECT 539	IX. CECT 539	X. CECT 539	XI. CECT 539
K1	1.35 ± 0.09	2.18 ± 0.30	2.28 ± 0.03	1.30 ± 4.69 × 10−5	1.70 ± 0.01
aX0	65.76 ± 6.04	90.29 ± 0.01	85.61 ± 35.21	45.56 ± 10.78	63.81 ± 16.05
aX1	0.21 ± 0.10	0.23 ± 0.01	0.26 ± 0.02	0.22 ± 0.02	0.21 ± 0.01
aX2	−2.17 × 10−4 ± 3.00 × 10−5	−8.85 × 10−4 ± 2.00 × 10−4	−1.06 × 10−3 ± 1.00 × 10−4	−3.50 × 10−4 ± 5.00 × 10−5	−9.31 × 10−4 ± 7.15 × 10−5
bX0	1.06 ± 0.27	0.61 ± 0.10	0.86 ± 0.109	1.57 ± 0.19	0.90 ± 0.10
bX1	−2.18 × 10−2 ± 3.30 × 10−3	−3.28 × 10−2 ± 3.48 × 10−3	−1.14 × 10−2 ± 3.00 × 10−3	−8.54 × 10−3 ± 2.60 × 10−3	−1.00 × 10−2 ± 3.30 × 10−3
bX2	2.39 × 10−4 ± 2.01 × 10−5	3.19 × 10−4 ± 2.24 × 10−5	1.63 × 10−4 ± 2.01 × 10−5	1.47 × 10−4 ± 1.27 × 10−5	2.19 × 10−4 ± 2.60 × 10−5
tbgp	60.42 ± 3.13	95.84 ± 0.28	71.92 ± 0.94	71.79 ± 0.18	50.78 ± 0.73
R2	0.9987	0.9968	0.9926	0.9987	0.9981
F	78,111.71	24,760.71	12,577.51	86,485.63	32,646.16
p	<0.0001	<0.0001	<0.0001	<0.0001	<0.0001
SSD	0.043	0.083	0.201	0.028	0.030
RPDM	2.87	3.17	4.68	2.36	3.34
Af	1.03	1.03	1.06	1.03	1.04
Bf	0.98	0.99	0.97	0.99	0.98

Parameter values are considered statistically significant when p < 0.05.

and

Table 5. Estimated values ± standard errors for the constants of model (18), the determination coefficient (R²), F-Fisher statistic, p-values, sum of squared differences between predicted and experimental data points (SSD), mean relative percentage deviation modulus (RPDM), and the accuracy (A_f) and bias (B_f) factors for the growth of L. lactis CECT 539 in the realkalized fed-batch cultures: XII. CECT 539, XIII. CECT 539, XIV. CECT 539, and XV. CECT 539.

	Cultures
Parameter	XII. CECT 539	XIII. CECT 539	XIV. CECT 539	XV. CECT 539
K1	2.11 ± 0.19	2.45 ± 0.01	3.06 ± 0.20	3.97 ± 0.33
aX0	33.43 ± 2.35	39.02 ± 4.36	53.84 ± 3.95	72.91 ± 13.72
aX1	0.36 ± 0.05	0.42 ± 0.08	0.41 ± 1.40 × 10−3	0.41 ± 0.01
aX2	−1.60 × 10−3 ± 2.00 × 10−4	−1.88 × 10−3 ± 6.79 × 10−6	−1.83 × 10−3 ± 6.00 × 10−6	−1.75 × 10−3 ± 3.00 × 10−4
bX0	2.50 ± 0.07	2.23 ± 0.08	1.52 ± 0.11	3.77 ± 0.05
bX1	8.53 × 10−3 ± 6.00 × 10−4	1.18 × 10−2 ± 8.00 × 10−4	7.13 × 10−3 ± 1.30 × 10−3	4.80 × 10−3 ± 3.00 × 10−4
bX2	4.79 × 10−5 ± 2.91 × 10−6	2.32 × 10−5 ± 3.97 × 10−6	4.08 × 10−5 ± 1.12 × 10−5	2.34 × 10−6 ± 6.35 × 10−7
tbgp	95.77 ± 1.02 × 10−2	95.64 ± 4.28	120.37 ± 5.12	158.13 ± 3.47
R2	0.9966	0.9961	0.9971	0.9955
F	29,868.49	30,138.42	46,630.97	32,385.67
p	<0.0001	<0.0001	<0.0001	<0.0001
SSD	0.053	0.063	0.050	0.098
RPDM	2.16	2.19	1.55	1.98
Af	1.02	1.02	1.01	1.02
Bf	1.00	1.00	1.00	1.00

Parameter values are considered statistically significant when p < 0.05.

). The R² values (0.9911–0.9993), F-values (6477.24–170,414.31), SSD (0.005–0.659), RPDM (1.50–7.71), and A_f (1.01–1.09), B_f (0.97–1.00) and p (<0.0001) values supported the model’s suitability for describing the diauxic growth of L. lactis CECT 539 in realkalized fed-batch cultures in different culture media.

To better assess the efficacy of growth models (4), (9), and (18) in describing the diauxic growth pattern observed across the fifteen cultures of L. lactis CECT 539, the values of RPDM, SSD, A_f, and B_f for each culture were comparatively presented in Figure S4. As shown, the three models yielded similar results, with RPDM values consistently below 10%, SSD values lower than 0.70, and A_f and B_f values very close to 1.0.

These findings indicate that all three models provide accurate and reliable descriptions of the experimental diauxic growth data. The low RPDM and SSD values reflect strong agreement between model predictions and observed data, while A_f and B_f values close to unity suggest the absence of systematic deviations or biases in the fits. Collectively, these results demonstrate that the three models are robust in capturing the overall biphasic growth dynamics of L. lactis CECT 539, and that their predictive performance is consistent across different culture conditions. However, from a practical perspective, the use of models (4) and (9) to describe the biphasic growth of a pure bacterial or yeast culture is limited, since they assume the existence of two populations growing simultaneously from the start of the culture (Figure 1), as previously described [5,12]. In contrast, model (18) addresses this limitation by capturing the biphasic growth dynamics of a single population, thereby offering a more realistic and biologically consistent representation of biphasic growth in pure cultures.

Moreover, when applied to Ped. acidilactici NRRL B-5627, Lact. casei CECT 4043, and E. faecium CECT 410 cultures under similar fermentation conditions, model (18) also yielded satisfactory results (Figure 5, Figure 6 and Figure 7,

Table 6. Estimated values ± standard errors for the constants of model (18), the determination coefficient (R²), F-Fisher statistic, p-values, sum of squared differences between predicted and experimental data points (SSD), mean relative percentage deviation modulus (RPDM), and the accuracy (A_f) and bias (B_f) factors for the growth of Ped. acidilactici NRRL-B 5627 in the five realkalized fed-batch cultures: I. NRRL B-5627, and II. NRRL B-5627, III. NRRL B-5627, IV. NRRL B-5627, and V. NRRL B-5627.

	Cultures
Parameter	I. NRRL B-5627	II. NRRL B-5627	III. NRRL B-5627	IV. NRRL B-5627	V. NRRL B-5627
K1	0.71 ± 3.70 × 10−3	1.00 ± 6.00 × 10−3	2.58 ± 0.66	1.95 ± 4.20 × 10−3	2.49 ± 4.19 × 10−3
aX0	34.93 ± 6.73	40.22 ± 17.52	81.05 ± 3.60 × 10−3	76.35 ± 10.96	39.13 ± 10.02
aX1	0.56 ± 0.03	0.40 ± 0.05	1.61 ± 4.00 × 10−4	0.50 ± 0.01	0.39 ± 0.04
aX2	4.46 × 10−3 ± 3.00 × 10−4	−2.84 × 10−3 ± 4.00 × 10−4	5.67 × 10−3 ± 1.17 × 10−5	2.01 × 10−3 ± 9.93 × 10−4	1.67 × 10−2 ± 6.99 × 10−3
bX0	4.37 ± 0.33	0.96 ± 0.09	3.11 ± 0.20	0.81 ± 0.04	3.68 ± 0.07
bX1	1.80 × 10−2 ± 2.90 × 10−3	−2.83 × 10−2 ± 3.90 × 10−3	6.67 × 10−3 ± 1.60 × 10−3	4.44 × 10−3 ± 2.00 × 10−3	1.14 × 10−2 ± 4.26 × 10−4
bX2	1.67 × 10−4 ± 2.62 × 10−5	5.29 × 10−4 ± 3.70 × 10−5	7.10 × 10−5 ± 8.52 × 10−6	1.47 × 10−4 ± 1.74 × 10−5	4.20 × 10−5 ± 2.42 × 10−6
tbgp	32.58 ± 0.27	53.78 ± 0.38	94.83 ± 0.06	74.17 ± 0.40	71.04 ± 0.11
R2	0.9884	0.9866	0.9852	0.9940	0.9954
F	5725.28	5656.88	7266.46	3121.57	7140.52
p	<0.0001	<0.0001	<0.0001	<0.0001	<0.0001
SSD	0.055	0.065	0.620	0.044	0.275
RPDM	5.85	5.17	4.23	3.63	2.08
Af	1.06	1.10	1.04	1.04	1.02
Bf	1.02	1.00	1.00	0.97	1.00

Parameter values are considered statistically significant when p < 0.05.

and

Table 7. Estimated values ± standard errors for the constants of model (18), the determination coefficient (R²), F-Fisher statistic, p-values, sum of squared differences between predicted and experimental data points (SSD), mean relative percentage deviation modulus (RPDM), and the accuracy (A_f) and bias (B_f) factors for the growth of Lact. casei CECT 4043 (realkalized fed-batch cultures: I. CECT 4043, and II. CECT 4043), and E. faecium CECT 410 (realkalized fed-batch cultures: I. CECT 410, and II. CECT 410).

	Cultures
Parameter	I. CECT 4043	II. CECT 4043	I. CECT 410	II. CECT 410
K1	0.56 ± 5.66 × 10−4	0.86 ± 6.32 × 10−2	0.53 ± 1.16 × 10−2	0.93 ± 3.48 × 10−3
aX0	19.48 ± 1.03	53.88 ± 6.33	51.29 ± 18.99	10.76 ± 3.05 × 10−6
aX1	1.65 × 10−2 ± 6.20 × 10−3	0.24 ± 2.01 × 10−2	0.12 ± 5.14 × 10−3	7.73 ± 3.43 × 10−2
aX2	3.14 × 10−2 ± 2.59 × 10−4	−6.00 × 10−4 ± 2.81 × 10−5	−5.12 × 10−4 ± 8.84 × 10−6	3.07 × 10−2 ± 2.16 × 10−5
bX0	2.09 ± 0.06	2.48 ± 6.42 × 10−2	0.24 ± 0.05	11.30 ± 1.01
bX1	3.65 × 10−2 ± 5.50 × 10−4	−2.98 × 10−4 ± 3.88 × 10−5	−5.69 × 10−2 ± 9.42 × 10−3	7.71 × 10−3 ± 1.19 × 10−3
bX2	−9.80 × 10−5 ± 3.71 × 10−6	1.44 × 10−5 ± 1.02 × 10−6	3.65 × 10−4 ± 4.63 × 10−5	2.85 × 10−5 ± 4.23 × 10−6
tbgp	48.35 ± 0.16	138.65 ± 2.13	88.84 ± 1.86 × 10−2	120.34 ± 2.18 × 10−2
R2	0.9981	0.9848	0.9973	0.9933
F	32,471.24	14,792.41	16,743.02	13,551.61
p	<0.0001	<0.0001	<0.0001	<0.0001
SSD	0.001	0.086	0.008	0.290
RPDM	1.33	5.39	4.53	4.87
Af	1.01	1.06	1.05	1.07
Bf	1.00	0.99	0.97	1.00

Parameter values are considered statistically significant when p < 0.05.

).

Additionally, to extend the efficacy of the model (18) to other biological systems, experimental data corresponding to the biphasic growth of Mos breed roosters and foals [5] were also used (

Table 8. Estimated values ± standard errors for the constants of model (18), the determination coefficient (R²), F-Fisher statistic, p-values, sum of squared differences between predicted and experimental data points (SSD), mean relative percentage deviation modulus (RPDM), and the accuracy (A_f) and bias (B_f) factors for the growth of Mos breed roosters and foals.

	Cultures
Parameter	Mos Breed Roosters	Foals
K1	6.60 ± 1.35 × 10−2	131.69 ± 0.80
aX0	2.74 ± 1.31 × 10−2	34.03 ± 3.31
aX1	7.46 × 10−3 ± 1.11 × 10−4	3.30 × 10−2 ± 8.46 × 10−4
aX2	−3.69 × 10−6 ± 5.04 × 10−7	−2.66 × 10−5 ± 2.23 × 10−6
bX0	38.97 ± 1.03	0.17 ± 1.61 × 10−3
bX1	7.19 × 10−2 ± 5.01 × 10−4	−1.86 × 10−2 ± 2.54 × 10−3
bX2	−1.69 × 10−4 ± 2.68 × 10−6	3.43 × 10−5 ± 1.09 × 10−5
tbgp	167.54 ± 5.25	321.38 ± 2.44
R2	0.9961	0.9868
F	20,610.75	7691.90
p	<0.0001	<0.0001
SSD	0.186	1072.62
RPDM	5.37	6.67
Af	1.08	1.03
Bf	1.03	1.01

Parameter values are considered statistically significant when p < 0.05.

, Figure 8). Good agreements was found between the model predictions and experimental data, with R², F-ratio, RPDM, SSD, A_f, B_f and p values of 0.9961, 20,610.75, 5.37, 0.186, 1.08, 1.03, and <0.0001, respectively (in case of Mos breed roosters), and 0.9868, 7691.90, 6.67, 1072.62, 1.03, 1.01, and <0.0001, respectively (in case of foals). These results were comparable to those obtained using the bi-logistic model (4) to describe the growth of both animals [5]. For example, the values for R², F-ratio, and A_f, and B_f for Mos breed roosters were 0.9979, 1125.5, 1.06, and 1.02, respectively; while for foals, the corresponding values were 0.9918, 346.0, 1.05, and 0.99, as reported by Vázquez et al. [5].

In all cases, the transition point (t_bgp) calculated using model (18) fell within the experimental time interval corresponding to the transitional phase of growth observed in the corresponding realkalized fed-batch cultures of the four LAB strains (Figures S1 and S2), as well as in the biphasic growth of Mos breed roosters and foals (Figure S3). The identification of t_bgp as the point marking the end of the first phase and the beginning of the second is crucial, as it allows the model to accurately capture the dynamics of diauxic growth. This also recognizes that biomass is not solely a function of the inoculum but also depends on the duration and conditions of the first phase of growth.

As observed in Figure 8, the growth curve of Mos breed roosters displays a profile with two superimposed growth phases. In contrast, the growth of foals follows two sequential logistic phases (although without showing the second stationary phase), and consequently, model (18) predicted a clear bi-logistic growth pattern (Figure 8).

These results confirm the effectiveness, accuracy, and practical utility of model (18) for modeling diauxic growth in both microbiological and biological systems. Moreover, the model provides predictions that are as accurate as those from the bi-logistic models (4) and (9), while avoiding their erroneous assumption of two parallel-growing populations—where observed growth is merely the sum of two independent growth curves.

3.2. Modeling the Production of Antibacterial Activity

To highlight the advantages of the antibacterial activity (AA) model (20) developed in this study for describing the experimental biphasic production of AA by L. lactis CECT 539 in realkalized fed-batch cultures, its predictions were compared with those of models (10) and (11). Subsequently, model (20) was validated using experimental data on biphasic AA production by Ped. acidilactici NRRL B-5627, Lact. casei CECT 4043, and E. faecium CECT 410.

As expected, model (10) did not consistently yield satisfactory results in describing AA production, particularly for cultures II, III and VI (Figure 2, Tables S5 and S6). This model yielded R², F, RPDM, SSD, A_f and B_f values ranging from 0.8815 to 0.9952, 1006.22 to 17,754.71, 3.78 to 34.80, 89.04 to 996.82, 1.04 to 1.43, and 0.86 to 1.01, respectively. These limitations are attributed to the absence of a specific term accounting for effect of the time-course of pH, the pH drop, or final pH, factors known to influence bacteriocin synthesis and, consequently, AA production [4,17,24,27,28,29].

In contrast, the predictions of model (11) were substantially more accurate than those of model (10), particularly for cultures II and III (Figure 2 and Figure 3, Tables S5–S8). The R², F, RPDM, SSD, A_f, and B_f values obtained using model (11) ranged from 0.9237 to 0.9956, 1461.01 to 58,611.40, 3.57 to 14.45, 32.71 to 754.35, 1.03 to 1.30, and 0.91 to 1.01, respectively (Tables S7 and S8). These improvements emphasize the importance of incorporating the pH time-course or pH drop effect into the Luedeking–Piret model to more accurately describe AA synthesis in realkalized fed-batch cultures.

Figure 4 presents the experimental data for AA production (squares), along with the predictions (lines) from the developed logistic model (20) for the fifteen realkalized fed batch cultures of L. lactis CECT 539. As observed, there is excellent agreement between the experimental and predicted values, with R², F, and SSD, values ranging from 0.9793 to 0.9986, 2368.62 to 20,107.32, and 4.17 to 382.53, respectively, and p-values consistently lower than 0.0001 (

Table 9. Estimated values ± standard errors for the constants of model (20), determination coefficient (R²), F-Fisher statistic, p-values, sum of squared differences between predicted and experimental data points (SSD), mean relative percentage deviation modulus (RPDM) values, as well as the accuracy (A_f) and bias (B_f) factors for the AA synthesis by L. lactis CECT 539 in the realkalized fed-batch cultures: I. CECT 539, II. CECT 539, III. CECT 539, IV. CECT 539, V. CECT 539, and VI. CECT 539.

	Cultures
Parameter	I. CECT 539	II. CECT 539	III. CECT 539	IV. CECT 539	V. CECT 539	VI. CECT 539
α	1216.73 ± 57.13	16.40 ± 0.15	7.85 ± 0.35	38.68 ± 0.20	757.00 ± 1.89	135.86 ± 0.91
φ	0.30 ± 4.81 × 10−2	0.45 ± 7.69 × 10−2	0.23 ± 5.13 × 10−2	0.66 ± 7.36 × 10−2	3.31 ± 3.40 × 10−2	1.98 ± 0.94
aP0	71.50 ± 2.61	1.97 × 10−2 ± 3.58 × 10−3	−0.51 ± 3.17 × 10−2	2.17 ± 0.24	214.51 ± 1.38	22.39 ± 8.96
aP1	3.44 × 10−3 ± 4.97 × 10−4	−0.21 ± 3.81 × 10−2	5.25 × 10−2 ± 0.97 × 10−2	6.81 × 10−3 ± 4.60 × 10−4	8.11 × 10−3 ± 1.08 × 10−4	6.23 × 10−3 ± 3.05 × 10−4
aP2	−7.36 × 10−6 ± 1.42 × 10−6	3.93 × 10−3 ± 1.6 × 10−3	1.46 × 10−3 ± 8.01 × 10−4	1.05 × 10−4 ± 4.78 × 10−6	3.62 × 10−5 ± 7.65 × 10−7	−2.77 × 10−5 ± 2.16 × 10−6
bP0	−2.39 × 10−3 ± 1.44 × 10−4	−8.13 × 10−7 ± 1.36 × 10−7	−2.60 × 10−2 ± 4.15 × 10−4	−0.26 ± 3.20 × 10−3	−0.79 ± 0.16	−0.65 ± 0.22
bP1	−1.34 ± 5.80 × 10−3	−0.38 ± 2.59 × 10−3	−6.61 × 10−2 ± 2.49 × 10−4	−0.16 ± 9.94 × 10−4	−1.29 × 10−2 ± 8.35 × 10−4	−5.03 × 10−2 ± 8.48 × 10−3
bP2	7.35 × 10−2 ± 4.00 × 10−4	2.93 × 10−3 ± 3.66 × 10−5	3.92 × 10−4 ± 2.44 × 10−6	7.30 × 10−3 ± 7.12 × 10−5	2.68 × 10−3 ± 6.88 × 10−4	6.53 × 10−3 ± 2.20 × 10−3
R2	0.9903	0.9876	0.9833	0.9979	0.9986	0.9930
F	4403.83	2368.62	3111.67	12,841.35	19,306.78	3950.01
p	<0.0001	<0.0001	<0.0001	<0.0001	<0.0001	<0.0001
SSD	131.34	34.60	99.11	4.28	4.17	72.56
RPDM	3.83	6.66	6.92	1.16	1.00	3.09
Af	1.04	1.07	1.08	1.01	1.01	1.03
Bf	1.00	0.96	0.97	1.00	1.00	1.01

Parameter values are considered statistically significant when p < 0.05.

,

Table 10. Estimated values ± standard errors for the constants of model (20), determination coefficient (R²), F-Fisher statistic, p-values, sum of squared differences between predicted and experimental data points (SSD), mean relative percentage deviation modulus (RPDM) values, as well as the accuracy (A_f) and bias (B_f) factors for the AA synthesis by L. lactis CECT 539 in the realkalized fed-batch cultures: VII. CECT 539, VIII. CECT 539, IX. CECT 539, X. CECT 539, and XI. CECT 539.

	Cultures
Parameter	VII. CECT 539	VIII. CECT 539	IX. CECT 539	X. CECT 539	XI. CECT 539
α	1099.32 ± 28.45	1187.59 ± 218.14	34.90 ± 3.97	21.47 ± 1.11	31.29 ± 2.96
φ	0.57 ± 3.07 × 10−2	0.60 ± 0.20	0.30 ± 9.08 × 10−2	0.44 ± 5.63 × 10−2	8.23 × 10−2 ± 2.70 × 10−3
aP0	65.63 ± 0.78	70.89 ± 1.52	1.12 ± 2.52 × 10−2	4.03 × 10−2 ± 1.46 × 10−2	0.87 ± 4.81 × 10−2
aP1	5.61 × 10−4 ± 6.05 × 10−5	1.21 × 10−3 ± 3.64 × 10−5	1.04 × 10−2 ± 1.83 × 10−4	−0.13 ± 8.30 × 10−3	1.46 × 10−2 ± 2.26 × 10−3
aP2	6.06 × 10−6 ± 7.51 × 10−8	2.15 × 10−6 ± 2.09 × 10−7	2.15 × 10−6 (NS)	1.66 × 10−3 ± 4.26 × 10−5	−9.97 × 10−7 ± 1.83 × 10−8
bP0	−2.33 × 10−3 ± 1.43 × 10−4	−1.15 × 10−3 ± 7.47 × 10−4	−3.00 × 10−3 ± 5.11 × 10−4	0.23 (NS)	−2.98 × 10−3 (NS)
bP1	−1.34 ± 5.46 × 10−3	−2.00 ± 1.41 × 10−2	−2.09 ± 1.32 × 10−2	1.72 × 10−2 (NS)	2.17 (NS)
bP2	7.35 × 10−2 ± 4.45 × 10−4	0.13 ± 1.18 × 10−3	0.14 ± 1.10 × 10−3	0.22 (NS)	0.14 (NS)
R2	0.9979	0.9958	0.9965	0.9943	0.9793
F	20,107.32	7060.46	9541.68	6870.30	9211.53
p	<0.0001	<0.0001	<0.0001	<0.0001	<0.0001
SSD	146.59	254.11	239.51	152.59	392.53
RPDM	2.55	4.15	2.64	4.85	9.14
Af	1.03	1.04	1.03	1.06	1.14
Bf	1.01	1.01	1.00	0.99	0.95

Parameter values are considered statistically significant when p < 0.05; otherwise, they are considered not significant (NS).

and

Table 11. Estimated values ± standard errors for the constants of model (20), determination coefficient (R²), F-Fisher statistic, p-values, sum of squared differences between predicted and experimental data points (SSD), mean relative percentage deviation modulus (RPDM) values, as well as the accuracy (A_f) and bias (B_f) factors for the AA synthesis by L. lactis CECT 539 in the realkalized fed-batch cultures: XII. CECT 539, XIII. CECT 539, XIV. CECT 539, and XV. CECT 539.

	Cultures
Parameter	XII. CECT 539	XIII. CECT 539	XIV. CECT 539	XV. CECT 539
α	37.31 ± 1.22	41.62 ± 1.16	43.45 ± 0.69	47.71 ± 0.50
φ	−5.82 × 10−3 ± 2.36 × 10−2	−6.95 × 10−2 ± 1.93 × 10−2	−6.03 × 10−2 ± 1.80 × 10−2	−0.10 ± 1.35 × 10−2
aP0	1.61 ± 0.88	1.91 ± 0.55	2.04 ± 0.47	2.33 ± 0.23
aP1	1.03 × 10−2 ± 4.65 × 10−3	0.11 ± 2.28 × 10−2	0.13 ± 1.58 × 10−2	0.14 ± 1.39 × 10−2
aP2	1.62 × 10−2 ± 3.86 × 10−3	3.07 × 10−3 ± 1.17 × 10−3	6.20 × 10−4 (NS)	−6.09 × 10−4 ± 3.50 × 10−6
bP0	−1.73 × 10−3 (NS)	−1.19 × 10−3 (NS)	−1.19 × 10−3 (NS)	−1.16 × 10−3 (NS)
bP1	21.02 (NS)	21.02 (NS)	21.02 (NS)	21.02 (NS)
bP2	1.33 (NS)	1.33 (NS)	1.33 (NS)	1.33 (NS)
R2	0.9962	0.9948	0.9946	0.9971
F	7807.78	6386.46	6208.53	8595.60
p	<0.0001	<0.0001	<0.0001	<0.0001
SSD	85.80	134.63	186.07	145.41
RPDM	2.24	2.41	2.28	1.89
Af	1.02	1.02	1.02	1.02
Bf	1.00	1.00	1.00	1.00

Parameter values are considered statistically significant when p < 0.05; otherwise, they are considered not significant (NS).

). This strong correlation is further supported by RPDM values ranging from 1.00 to 9.14, consistently below 10%, as well as A_f (1.01 to 1.14) and B_f (0.95 to 1.01) values close to 1.0 (Table 9, Table 10 and Table 11). Notably, the predictions of model (20) improved the fit of the experimental AA data compared to model (11), particularly for cultures II, III, IV, V, VI, IX, X, and XI (Figure 3 and Figure 4, Tables S7 and S8 and Table 9, Table 10, Table 11).

This suggests that model (11) may not always be suitable for describing biphasic AA production by L. lactis CECT 539 in realkalized fed-batch cultures. This is likely due to the numerical integration of the integral form of model (11), which can introduce estimation errors in the AA levels, resulting in larger discrepancies between the experimental and calculated values. In contrast, model (20) provides a direct fit to the experimental data and does not require integration. Based on this, it can be emphasized that model (20) is easier to use compared to the integral forms of models (10) and (11).

Additionally, when the values of RPDM, SSD, A_f, and B_f for each culture were compared (Figure S5), it was evident that the application of model (20) consistently reduced RPDM values, which remained below 10%, and lowered SSD values, while also yielding A_f and B_f values very close to 1.0 compared with models (10) and (11) (Figure S5, Tables S5–S8 and Table 9, Table 10, Table 11). These results highlight the superior accuracy and reliability of model (20) in describing AA synthesis.

Moreover, regarding model (20), positive values of the φ parameter were observed in cultures I–XI of L. lactis CECT 539 (Table 9 and Table 10), suggesting that the corresponding pH gradient trajectories favored AA production [27]. A positive φ reflects a condition in which the decrease in pH during growth enhances the activity of the biosynthetic pathways involved in AA formation, likely by maintaining enzyme stability and catalytic efficiency within their optimal pH range. In contrast, negative φ values were obtained for cultures XII–XV, suggesting that in these cases the pH trajectories shifted away from the optimal range, thereby reducing the stimulatory effect on AA synthesis and potentially impairing enzyme activity or metabolic fluxes related to AA production [28,29,30,31,32,33,34].

When the pH time courses from all fifteen cultures of L. lactis CECT 539 are analyzed collectively (Figure 9), it becomes evident that the pH values at the end of each realkalization and feeding cycle in cultures XII–XV exhibited a smoother increase than those observed in cultures I–XI. This difference is likely due to the lower buffering capacity of the unsupplemented diluted whey (DW) used as the fermentation medium in cultures I–XI, compared to the MRS nutrient–supplemented DW medium used in cultures XII–XV [23].

However, the final pH values in cultures IV and V were higher than those in cultures XII and XIII after the third realkalization and feeding cycle, and higher than those in cultures XIV and XV after the seventh cycle (84 h) (Figure 9). This was likely due to the low nutrient content of the unsupplemented DW medium and CMPWGP [19,20,21,22], which progressively limited the growth of L. lactis CECT 539 and, consequently, its ability to recover from acidic conditions (Figure 3). As a result, AA synthesis in cultures IV and V increased from 0.43 to 23.58 and 28.37 AU/mL, respectively, at rates of 0.655 and 0.788 AU/mL/h. However, from 36 h of incubation until the end of fermentation (168 h), AA synthesis continued at slower rates of 0.147 and 0.185 AU/mL/h, reaching final AA concentrations of 19.41 and 24.42 AU/mL, respectively. This slowdown in AA production in cultures IV and V from the third realkalization and feeding cycle suggests that the final pH values progressively deviated from the optimal pH for antibacterial compound synthesis [27,28,29,30,31,32,33,34].

The applicability of the logistic model (20) was also validated using experimental data from the biphasic AA production of Ped. acidilactici NRRL B-5627 (Figure 5), Lact. casei CECT 4043 (Figure 6), and E. faecium CECT 410 (Figure 7) during realkalized fed-batch fermentations in different culture media. As shown, fitting model (20) to the corresponding experimental AA data yielded satisfactory results (Figure 5, Figure 6 and Figure 7), with statistically significant parameters and consistent model performance in all cases (

Table 12. Estimated values ± standard errors for the constants of model (20), determination coefficient (R²), F-Fisher statistic, p-values, sum of squared differences between predicted and experimental data points (SSD), mean relative percentage deviation modulus (RPDM) values, as well as the accuracy (A_f) and bias (B_f) factors for the AA synthesis by Ped. acidilactici NRRL-B 5627 in the five realkalized fed-batch cultures: I. NRRL B-5627, and II. NRRL B-5627, III. NRRL B-5627, IV. NRRL B-5627, and V. NRRL B-5627.

	Cultures
Parameter	I. NRRL B-5627	II. NRRL B-5627	III. NRRL B-5627	IV. NRRL B-5627	V. NRRL B-5627
α	430.84 ± 2.09	215.07 ± 19.31	2.69 ± 2.48 × 10−2	144.47 ± 0.33	76.55 ± 0.55
φ	4.37 × 10−2 ± 3.98 × 10−3	0.89 ± 0.14	25.65 ± 0.24	−0.25 ± 1.29 × 10−3	0.12 ± 7.84 × 10−3
aP0	−0.32 ± 9.33 × 10−3	−4.64 × 10−4 ± 1.45 × 10−4	−0.99 ± 9.82 × 10−3	−0.45 (NS)	−1.97 × 10−2 ± 5.73 × 10−3
aP1	−3.76 × 10−2 ± 1.23 × 10−3	−0.24 ± 6.15 × 10−3	27.13 ± 3.82 × 10−6	1.67 (NS)	−0.14 ± 7.69 × 10−3
aP2	1.20 × 10−3 ± 2.77 × 10−5	2.50 × 10−3 ± 2.04 × 10−5	9.43 × 10−2 ± 1.16 × 10−7	4.09 × 10−5 (NS)	2.04 × 10−3 ± 4.85 × 10−5
bP0	1.25 × 10−2 (NS)	−0.25 (NS)	9.73 × 10−3 (NS)	−0.33 ± 1.81 × 10−3	−0.65 ± 0.25
bP1	37.76 ± (NS)	37.76 (NS)	37.76 (NS)	−1.42 × 10−2 ± 1.05 × 10−4	37.76 ± 1.54 × 10−6
bP2	12.81 (NS)	12.81 (NS)	12.81 (NS)	1.80 × 10−4 ± 1.24 × 10−6	12.81 ± 4.05 × 10−5
R2	0.9971	0.9893	0.9652	0.9986	0.9942
F	20,120.08	5104.55	1079.16	32,637.69	8564.28
p	<0.0001	<0.0001	<0.0001	<0.0001	<0.0001
SSD	2199.96	10,427.44	5885.46	193.91	2431.36
RPDM	1.62	3.73	5.39	1.07	3.65
Af	1.02	1.04	1.05	1.01	1.02
Bf	1.00	0.98	1.00	1.00	1.00

Parameter values are considered statistically significant when p < 0.05; otherwise, they are considered not significant (NS).

and

Table 13. Estimated values ± standard errors for the constants of model (20), determination coefficient (R²), F-Fisher statistic, p-values, sum of squared differences between predicted and experimental data points (SSD), mean relative percentage deviation modulus (RPDM) values, as well as the accuracy (A_f) and bias (B_f) for the AA synthesis by Lact. casei CECT 4043 (realkalized fed-batch cultures: I. CECT 4043, and II. CECT 4043), and E. faecium CECT 410 (realkalized fed-batch cultures: I. CECT 410, and II. CECT 410).

	Cultures
Parameter	I. CECT 4043	II. CECT 4043	I. CECT 410	II. CECT 410
α	13.66 ± 9.98 × 10−2	29.49 ± 1.32	0.38 ± 3.41 × 10−2	1.11 ± 1.10 × 10−2
φ	−6.75 × 10−2 ± 7.92 × 10−3	8.93 × 10−2 ± 1.06 × 10−2	0.57 ± 0.13	−1.93 × 10−2 ± 7.19 × 10−3
aP0	1.24 ± 0.15	7.79 ± 2.26	0.10 ± 3.22 × 10−2	−0.44 ± 6.87 × 10−2
aP1	5.28 × 10−2 ± 2.72 × 10−3	1.15 × 10−2 ± 3.63 × 10−3	−0.11 ± 6.25 × 10−3	−1.01 × 10−3 ± 3.49 × 10−6
aP2	−1.37 × 10−4 ± 1.85 × 10−5	1.20 × 10−4 ± 1.45 × 10−5	1.37 × 10−3 ± 2.82 × 10−5	1.73 × 10−4 ± 4.25 × 10−5
bP0	0.75 ± 0.11	7.15 × 10−4 ± 3.85 × 10−5	2.12 (NS)	16.66 (NS)
bP1	7.21 × 10−2 ± 1.28 × 10−5	−8.99 × 10−2 ± 1.32 × 10−3	7.21 × 10−2 (NS)	7.21 × 10−2 (NS)
bP2	0.22 ± 1.46 × 10−8	2.59 × 10−4 ± 5.92 × 10−6	0.22 (NS)	0.22 (NS)
R2	0.9963	0.9895	0.9960	0.9934
F	7573.89	6288.56	8814.40	8650.07
p	<0.0001	<0.0001	<0.0001	<0.0001
SSD	0.575	111.35	0.01	0.32
RPDM	2.33	11.13	4.35	2.52
Af	1.02	1.13	1.04	1.03
Bf	0.99	0.95	1.01	1.00

Parameter values are considered statistically significant when p < 0.05; otherwise, they are considered not significant (NS).

).

These results indicate that model (20), developed in this study, adequately describes the biphasic AA synthesis by four LAB strains in realkalized fed-batch cultures.

However, when designing or refining a bi-logistic model, the nature of the experimental data plays a crucial role, as it influences the values of R², F, RPDM, SSD, A_f and B_f. High experimental error in measurements—such as that observed in AA synthesis by Lact. casei CECT 4043 in culture II (right part of Figure 6), E. faecium CECT 410 in culture II (right part of Figure 7), and foal growth (right part of Figure 8)—can impact model accuracy (Table 8, Table 12 and Table 13). Additionally, an insufficient number of data points to clearly capture biphasic product formation may reduce the statistical significance of model parameters. This limitation is evident in AA production by L. lactis CECT 539 in cultures X–XV (Table 10 and Table 11), Ped. acidilactici NRRL B-5627 in cultures I–IV (Table 12), and E. faecium CECT 410 in cultures I and II (Table 13). In these cultures, non-significant values were obtained for the parameters corresponding to either the first or second phase of AA production.

4. Conclusions

Overall, the results obtained in this study demonstrate that the proposed modeling approach is robust and provides a solid framework for understanding diauxic growth in microbial cultures. The successful validation of the logistic growth and AA production equations using experimental data from four LAB strains suggests that both models could serve as valuable tools in microbiology, microbial ecology, and biotechnology.

The developed logistic growth Equation (18) is logically structured to incorporate the transition between growth phases, as well as the influence of time and cultivation conditions on biomass development. Notably, this model enables the estimation of the transition time (t_bgpt), which marks the end of the first growth phase and the beginning of the second. This approach provides a reliable method for describing diauxic growth in other microbiological and biological systems by adapting the logistic model to account for two distinct growth phases. From a practical perspective, model (18) offers a more accurate representation of biphasic growth in pure bacterial cultures, eliminating the flawed assumption—common in bi-logistic models—that two distinct populations grow in parallel and independently.

Furthermore, the application of the biphasic AA model (20) provides a simpler and more effective way to describe product formation compared to the differential forms of both the unmodified and modified Luedeking–Piret models.