Mathematical Modeling for Fermentation Systems: A Case Study in Probiotic Beer Production

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This manuscript investigates the kinetic processes of probiotic beer fermentation using the autochthonous yeast strain Saccharomyces cerevisiae PB101 from San Juan, Argentina. It develops a mathematical modeling framework combining the First Order Plus Dead Time (FOPDT) model, logistic growth model, and a modified Luedeking-Piret equation to describe the dynamics of total cells, dead cells, viable cells, and ethanol production. The study validates the yeast’s probiotic potential through simulated gastrointestinal experiments, reporting survival rates of 73.49%–80.17%, and demonstrates its suitability for functional beer production. This study presents an innovative approach to modeling probiotic beer fermentation. However, enhancing methodological transparency, deepening mechanistic discussions, and refining language coherence will elevate its scientific rigor and readability for publication

Commentss for Materials and Methods

1 While "triplicate experiments" are mentioned, the manuscript lacks details on error handling (e.g., incorporation of standard deviations into model parameter fitting). Clarify the experimental replication protocol, data aggregation methods, and statistical tests (e.g., ANOVA or t-tests) to enhance credibility.

2 Model validation relies solely on R² values without residual analysis or cross-validation. Include residual distribution plots or results from an independent validation dataset to demonstrate model generalizability.

3 The parameter estimation process (e.g., time constants τ, delay times t_L/t_D) is described too briefly. Elaborate on optimization algorithms (e.g., least squares, gradient descent), convergence criteria, iteration limits, and sensitivity analysis to improve reproducibility.

4 The fermentation experiments mention "static conditions" (e.g., 18±1°C) but omit monitoring of dissolved oxygen, pH fluctuations, or other potential confounders. Add details on environmental parameter control to eliminate confounding effects.

 

Commentss for Results and Discussion

5 The survival rates are reported quantitatively but lack discussion on whether the final viable cell concentration (e.g., 10⁶ CFU/mL) meets established probiotic dosage thresholds (e.g., 10⁶–10⁸ CFU/g in clinical studies). Reference specific efficacy thresholds and assess practical applicability.

6 While traditional model limitations are noted, the manuscript lacks quantitative comparisons (e.g., RMSE, AIC values) between the proposed models and classical approaches (e.g., original Luedeking-Piret). Include tables or experiments highlighting the improved model’s advantages.

7 The delay between ethanol production and cell growth (t_delay = 2 h) is not linked to metabolic pathways (e.g., time-dependent glycolysis). Incorporate kinetic analyses of metabolic processes to strengthen biological relevance.

8 The manuscript does not address model scalability (e.g., parameter dependence on fermentation scale). Discuss industrial applicability and potential optimizations (e.g., dynamic temperature control’s impact on delay times).

9 Key assumptions (e.g., ignoring substrate inhibition or cellular heterogeneity) are not critiqued. Add a section analyzing how these assumptions affect predictive accuracy and propose future refinements.

 

Mino Commentss for Language and Logic Refinements

10 Inconsistent terms (e.g., "dead cells" vs. "death cells"; undefined acronyms like FOPDT, Line 108). Standardize terminology and spell out acronyms at first mention (e.g., FOPDT: First Order Plus Dead Time).

11 Figures and tables are referenced without highlighting key insights (e.g., "Figure 1 shows good model fit"). Add specific conclusions when citing visuals (e.g., "Figure 1 demonstrates FOPDT’s superior fit (R²=99.33%) over the logistic model (R²=99.09%)").

 

 

Author Response

Dear Reviewer

The manuscript was completely revised according to the suggestions and yourself.

 

Bold letters were made by the reviewers

Italic letters were made by the authors

 

Reviewer: 1

This manuscript investigates the kinetic processes of probiotic beer fermentation using the autochthonous yeast strain Saccharomyces cerevisiae PB101 from San Juan, Argentina. It develops a mathematical modeling framework combining the First Order Plus Dead Time (FOPDT) model, logistic growth model, and a modified Luedeking-Piret equation to describe the dynamics of total cells, dead cells, viable cells, and ethanol production. The study validates the yeast’s probiotic potential through simulated gastrointestinal experiments, reporting survival rates of 73.49%–80.17%, and demonstrates its suitability for functional beer production. This study presents an innovative approach to modeling probiotic beer fermentation. However, enhancing methodological transparency, deepening mechanistic discussions, and refining language coherence will elevate its scientific rigor and readability for publication

Comments for Materials and Methods

1 While "triplicate experiments" are mentioned, the manuscript lacks details on error handling (e.g., incorporation of standard deviations into model parameter fitting). Clarify the experimental replication protocol, data aggregation methods, and statistical tests (e.g., ANOVA or t-tests) to enhance credibility.

Figures 1 and 2 present the experimental results, including the mean values and their respective standard deviations, for PB101 and ethanol kinetics. These results are shown alongside the corresponding mathematical model fits, ensuring that error handling and data variability are properly accounted for.

The significant difference in viable cell counts between commercial yeast and the potentially probiotic yeast PB101 at the end of the process have been added to Table 1. Additionally, the “Data Analysis” section (2.5) in Materials and Methods has been revised in accordance with the reviewer's suggestions, and modifications have been made accordingly.

Lines 426 to 431: In each independent fermentation (2024-2025) triplicate replicates were performed, the results represent the mean of the three determinations together with the corresponding standard deviation. To determine the significance of differences between two groups (US05 and PB101) in each fermentation, the unpaired Student's t-test was used. P-values less than 0.05 were considered statistically significant. InfoStat software version 2020 was used for data analysis.

 

2 Model validation relies solely on R² values without residual analysis or cross-validation. Include residual distribution plots or results from an independent validation dataset to demonstrate model generalizability.

In addition to R², evaluating the accuracy and statistical significance of the proposed models requires key performance metrics such as Lack of Fit (LOF), Mean Squared Error (MSE), F-statistics, and p-value. These parameters provide a comprehensive assessment of model reliability, ensuring that the models accurately represent the experimental data and exhibit strong predictive capabilities.

In Materials and Methods section 2.5 we added:

Lines 434 to 439: Evaluating the accuracy and statistical significance of the proposed models requires the use of key performance metrics such as Lack of Fit (LOF), Mean Squared Error (MSE), F-statistic, and p-value. These parameters provide a comprehensive assessment of model reliability, ensuring an optimal fit to the experimental data and strong predictive capabilities.

In results and discussion, section 3.2 we incorporate:

The models in this work show a strong fit with the experimental data (Tables 2, 3, and 4). To further evaluate their accuracy and statistical significance, key parameters such as the LOF, MSE, F-statistic, and p-value were analyzed. These metrics help assess the reliability of the models, ensuring that they accurately capture experimental trends and provide robust predictive performance.

 

3 The parameter estimation process (e.g., time constants τ, delay times t_L/t_D) is described too briefly. Elaborate on optimization algorithms (e.g., least squares, gradient descent), convergence criteria, iteration limits, and sensitivity analysis to improve reproducibility

The optimization of model parameters was carried out using the coefficient of determination (R²) as the optimization criterion. A hybrid algorithm was developed, integrating the Monte Carlo method with genetic algorithms, originally designed by our research group (Optimization of Parameters Using Monte Carlo Methods and Evolutionary Algorithms. Application to a Trajectory Tracking Controller in Nonlinear Systems). The process began with the generation of an initial set of parameters through 500 Monte Carlo simulations, based on predefined reference values. After evaluating their performance, the 50 best-performing parameter sets were selected to form the first generation of the genetic algorithm. Standard genetic operators—selection, crossover, and mutation—were iteratively applied to further refine the parameter set. To reduce the risk of convergence to local optima, a randomly generated parameter set was introduced into each new generation. The optimization process continued for 30 simulation cycles, ensuring convergence to an optimal solution.

4 The fermentation experiments mention "static conditions" (e.g., 18±1°C) but omit monitoring of dissolved oxygen, pH fluctuations, or other potential confounders. Add details on environmental parameter control to eliminate confounding effects.

We appreciate the reviewer's insightful comment regarding the importance of detailing environmental parameter control during fermentation. In Section 2.2 (Fermentation Assay), we describe that the fermentation experiments were conducted under static conditions (without agitation) at a controlled temperature of 18 ± 1 °C (lines 178-179). However, we recognize that the continuous monitoring of key parameters, such as dissolved oxygen levels and pH fluctuations, was not explicitly specified in the original manuscript. To address this concern, we have incorporated detailed information on pH and dissolved oxygen monitoring during fermentation. Specifically, the wort was aerated to achieve a dissolved oxygen concentration of 7.8 ppm, following the Analytica Microbiological-EBC method. Additionally, pH measurements were taken at the end of the fermentation process, alongside the determination of alcohol content by volume (% v/v). These additions provide a more comprehensive characterization of the fermentation conditions and help mitigate potential confounding effects, ultimately improving the reproducibility and robustness of the experimental results.

Lines 170- 173: The worts were aerated, reaching 7.8 ppm oxygen, according to the Analytica Microbiological-EBC method [31] and 4,5 g/hl of diammonium phosphate (FERMOPLUS® Dap Free) was added prior pitching

 

Commentss for Results and Discussion

5 The survival rates are reported quantitatively but lack discussion on whether the final viable cell concentration (e.g., 10⁶ CFU/mL) meets established probiotic dosage thresholds (e.g., 10⁶–10⁸ CFU/g in clinical studies). Reference specific efficacy thresholds and assess practical applicability.

Lines 452 to 489: Within the "Results and Discussion" section (3.1), modifications were made according to the reviewer's suggestions.

 

6 While traditional model limitations are noted, the manuscript lacks quantitative comparisons (e.g., RMSE, AIC values) between the proposed models and classical approaches (e.g., original Luedeking-Piret). Include tables or experiments highlighting the improved model’s advantages.

The primary objective of this work is to model viable cells, which is why the modified Luedeking-Piret model, which incorporates viable cells, is the most appropriate for this study. In contrast, the original Luedeking-Piret model uses total cells, making it less suitable for our specific focus on viable cells with probiotic properties.

Including the original Luedeking-Piret model in this work would not only deviate from the main objective but also introduce unnecessary complexity. Moreover, the fit of the original Luedeking-Piret model is suboptimal for our data, as evidenced by the comparison of the two curves and their respective R² values. Presenting these results would detract from the clarity of our work. Therefore, we believe that focusing on the modified Luedeking-Piret model aligns with the core objective of this study, which is to accurately model viable cells.

 

 

Figure 1. Fit to the experimental data using the original Luedeking-Piret model, which yields an R² value of 93.29%.

Below, we present the model proposed in this work.

Figure 2. Fit to the experimental data using the modified Luedeking-Piret model, which yields an R² value of 98.99%.

Figures 1 and 2 reveal a noticeable difference in data fitting. Given this observation and the fact that it is not the main focus of this study, the research group has opted not to include the original Luedeking-Piret model.

 

7 The delay between ethanol production and cell growth (t_delay = 2 h) is not linked to metabolic pathways (e.g., time-dependent glycolysis). Incorporate kinetic analyses of metabolic processes to strengthen biological relevance.

Our study focused on understanding the dynamics of ethanol production in Saccharomyces cerevisiae using a modified Luedeking-Piret model. This model, which incorporates a 2-hour delay (t_delay) in ethanol production, reflects the experimentally observed temporal lag between cell growth and ethanol production. However, we acknowledge that this delay has not been explicitly linked to specific metabolic pathways, such as glycolysis, which is crucial for ethanol production in Saccharomyces cerevisiae. Glycolysis is the metabolic pathway that converts glucose to pyruvate, and this pyruvate is then converted to ethanol through alcoholic fermentation.

To strengthen the biological relevance of the model, we propose the following improvements:

Kinetic analysis of glycolysis and enzyme regulation: In future studies, we could incorporate a detailed kinetic analysis of glycolysis, including the regulation of key enzymes such as hexokinase, phosphofructokinase, and pyruvate decarboxylase. This would allow us to quantify how changes in enzyme activity and substrate availability (such as glucose) affect ethanol production over time.

Modeling of metabolic transition and gene regulation: We could develop a model that includes the metabolic transition of yeast from the growth phase (where energy is primarily directed towards biomass) to the ethanol production phase (where glucose is converted to ethanol and CO2). This could be achieved by incorporating kinetic equations that describe the activity of key enzymes and the regulation of gene expression, which plays a significant role in the transition of yeast from the growth phase to the ethanol production phase.

We thank the reviewer for their insightful observations and are committed to further investigating the underlying metabolic pathways in future studies to strengthen the biological relevance of the model.

 

8 The manuscript does not address model scalability (e.g., parameter dependence on fermentation scale). Discuss industrial applicability and potential optimizations (e.g., dynamic temperature control’s impact on delay times).

The observation about addressing parameter dependence on fermentation scale and the impact of dynamic temperature control on delay times is highly valuable and aligns with important considerations for industrial applications. However, the primary focus of this study was to develop and validate a mathematical framework for describing the kinetics of probiotic yeast fermentation at a laboratory scale. While the current work does not explicitly address scalability or industrial-scale optimizations, such as dynamic temperature control, we fully acknowledge the importance of these aspects for future research. Scaling up the model and exploring its applicability in industrial settings, including the optimization of parameters under varying fermentation scales and conditions, will be a key direction for subsequent studies. We appreciate the reviewer’s suggestion and will incorporate these considerations into our future work to enhance the model’s relevance and applicability in industrial contexts.

9 Key assumptions (e.g., ignoring substrate inhibition or cellular heterogeneity) are not critiqued. Add a section analyzing how these assumptions affect predictive accuracy and propose future refinements.

The Luedeking-Piret model implicitly considers substrate limitation. However, we did not specifically investigate or delve into the possibility of substrate inhibition, such as that observed at high sugar concentrations in wine fermentations (high osmotic pressure). This is considered important for future studies.

Regarding cellular heterogeneity, we understand this to refer to the different physiological stages of the yeast cells. This aspect was not analyzed in the present study, but we plan to address it in subsequent stages of our research."

Mino Commentss for Language and Logic Refinements

10 Inconsistent terms (e.g., "dead cells" vs. "death cells"; undefined acronyms like FOPDT, Line 108). Standardize terminology and spell out acronyms at first mention (e.g., FOPDT: First Order Plus Dead Time).

We have thoroughly reviewed the entire manuscript and made the necessary corrections, replacing "death cells" with "dead cells" where appropriate. We added the definition of acronyms FOPDT: First Order Plus Dead Time was incorporate for first time in line 120.

 

11 Figures and tables are referenced without highlighting key insights (e.g., "Figure 1 shows good model fit"). Add specific conclusions when citing visuals (e.g., "Figure 1 demonstrates FOPDT’s superior fit (R²=99.33%) over the logistic model (R²=99.09%)").

We have carefully reviewed the manuscript and incorporated additional citations to figures and tables to enhance the clarity and support of our findings.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The manuscript is a strong contribution to the field of probiotic beer production, with a focus on modeling yeast dynamics during fermentation. While the study design and data analysis are generally sound, addressing the specific comments will significantly enhance the clarity, reproducibility, and scientific impact of the manuscript.

Specific comments

1. Introduction:

• The introduction adequately establishes the context of the study but could benefit from a clearer statement of the research objectives. The relationship between the probiotic potential of PB101 yeast and its application in functional beverages needs to be articulated more explicitly at the beginning of the introduction.
• While the manuscript refers to relevant studies (e.g., Saccharomyces cerevisiae var. boulardii), the literature review could be expanded to include more recent research on probiotic yeasts in functional beverages. This would give the reader a more comprehensive understanding of the field and position the study within the current body of knowledge.

2. Methodology:

• The methods for beer fermentation (Section 2.2) could be described in more detail, specifically with regard to temperature, pH, and nutrient levels. These are crucial variables that affect yeast behavior and fermentation outcomes, and their absence makes it harder to replicate the experiments.
• While the manuscript provides good information about the yeast’s survival through simulated gastrointestinal conditions, further clarification is needed on the methodology used for this simulation. Specific details on the conditions of the duodenal and gastric phases would strengthen the reproducibility of this section.

3. Results:

• Tables 2,3 and 4: Numerical parameters for FOPDT and Logistic models are essential, but additional explanations of the parameters and their biological relevance would help readers better interpret these values, especially those less familiar with fermentation modeling.
• The manuscript lacks detailed statistical analysis to support comparisons between the commercial yeast and PB101 yeast populations. Statistical tests (e.g., t-tests or ANOVA) should be applied to ensure that observed differences are significant and not due to experimental variability.

4. Discussion:

• The authors mention a 20-28% reduction in yeast survival during gastrointestinal transit but do not discuss the potential impact of this reduction on the overall probiotic effect. While the survival rates seem adequate, further context on what constitutes a ‘substantial concentration’ for probiotic efficacy would be helpful.
• The models used to represent fermentation dynamics are robust and well-explained, but more discussion is needed regarding the selection of these models. For instance, why was the Modified Luedeking-Piret model chosen for ethanol production, and how does it compare to other models commonly used in fermentation research?
• While the models show a strong fit with the experimental data (R² values of 99.33% for FOPDT and 99.09% for the Logistic model), a more in-depth discussion on the model validation process would be beneficial. For example, did the authors perform any cross-validation or sensitivity analysis to confirm the robustness of the models?

5. Conclusion:

• The conclusion is concise, but it could be strengthened by emphasizing the practical implications of the findings. Specifically, how might the results influence the development of probiotic beers on a commercial scale? Additionally, the authors could provide more insights into future research directions, particularly regarding the scalability of the fermentation process and the long-term stability of probiotic strains in beer.

Author Response

San Juan, March 16^th, 2025

Dear Reviewer 2

Bold letters were made by the reviewers

Italic letters were made by the authors

 

 

The manuscript is a strong contribution to the field of probiotic beer production, with a focus on modeling yeast dynamics during fermentation. While the study design and data analysis are generally sound, addressing the specific comments will significantly enhance the clarity, reproducibility, and scientific impact of the manuscript.

Specific comments

1. Introduction:

The introduction adequately establishes the context of the study but could benefit from a clearer statement of the research objectives. The relationship between the probiotic potential of PB101 yeast and its application in functional beverages needs to be articulated more explicitly at the beginning of the introduction.

We added the following information about the probiotic potential of PB101 yeast

Lines 89-93: Additionally, in laboratory-scale experiments, this yeast exhibited enzymatic activities that promote digestion, such as lipases, proteases, and phytases. It demonstrated antioxidant activity against the DPPH (2,2-diphenyl-1-picrylhydrazyl) radical, reduced cholesterol concentrations in the medium, and produced organic acids of interest, including lactic, acetic, and propionic acids

 

While the manuscript refers to relevant studies (e.g., Saccharomyces cerevisiae var. boulardii), the literature review could be expanded to include more recent research on probiotic yeasts in functional beverages. This would give the reader a more comprehensive understanding of the field and position the study within the current body of knowledge.

We have made modifications and added changes in accordance with the reviewer's suggestions.

Lines 68 to 72: Within functional beverages, dairy products boast the longest established history, commonly utilizing lactic acid bacteria with probiotic functionalities. Lately, fruit, vegetables, legume, cereal, coffee, or tea-based juices are rapidly gaining market prominence. In these beverages, the contribution of yeasts is crucial, providing probiotic properties beneficial to the consumer and placing them in a position of preference [11] [12] [13]

11. Agarbati, A., Canonico, L., Ciani, M., Morresi, C., Damiani, E., Bacchetti, T., & Comitini, F. (2024) Functional Potential of a New Plant-Based Fermented Beverage: Benefits through Non-Conventional Probiotic Yeasts and Antioxidant Properties. Int. J. Food Microbiol. 424, 110857.
12. Amorim, J. C., Piccoli, R. H., & Duarte, W.F. (2018). Probiotic Potential of Yeasts Isolated from Pineapple and Their Use in the Elaboration of Potentially Functional Fermented Beverages. Food Res. Int. 107, 518-527.
13. Ferreira, I., de Sousa Melo, D., Menezes, A. G. T., Fonseca, H. C., de Assis, B. B. T., Ramos, C. L., Maganani, M., Dias D.R. & Schwan, R.F. (2022). Evaluation of Potentially Probiotic Yeasts and Lactiplantibacillus Plantarum in Co-Culture for the Elaboration of a Functional Plant-Based Fermented Beverage. Food Res. Int. 160, 111697.

 

 

1. Methodology:

The methods for beer fermentation (Section 2.2) could be described in more detail, specifically with regard to temperature, pH, and nutrient levels. These are crucial variables that affect yeast behavior and fermentation outcomes, and their absence makes it harder to replicate the experiments.

We added more details according to reviewer´s suggestions:

Lines 165- 170: A total of two beer fermentations were conducted during the years 2024 and 2025. 165 Inoculated 1x10⁷ cells/mL of previously activated yeast in 2000 mL Erlenmeyer flasks with 166 1400 mL of brewer's wort Kolsch style (initial conditions: 1046 OG/ 11.5 °P initial, 20 IBU, 167 pH 5.2). The worts were aerated, reaching 7.8 ppm oxygen, according to the Analytica 168 Microbiological -EBC method [31] and 4,5 g/hl of diammonium phosphate 169 (FERMOPLUS® Dap Free) was added prior pitching.

 

While the manuscript provides good information about the yeast’s survival through simulated gastrointestinal conditions, further clarification is needed on the methodology used for this simulation. Specific details on the conditions of the duodenal and gastric phases would strengthen the reproducibility of this section.

Lines 191 to 219: More details were added for clarity and reproducibility of the methodology used, considering the reviewer's suggestion.

 

1. Results:

Tables 2,3 and 4: Numerical parameters for FOPDT and Logistic models are essential, but additional explanations of the parameters and their biological relevance would help readers better interpret these values, especially those less familiar with fermentation modeling.

Additional explanations of the parameters and their biological relevance have been incorporated into Tables 2, 3, and 4. Further details can also be found in Sections 2.4.1, 2.4.2, and 2.4.3, as well as in their respective 'Methodology for Parameter Estimation' subsections.

 

The manuscript lacks detailed statistical analysis to support comparisons between the commercial yeast and PB101 yeast populations. Statistical tests (e.g., t-tests or ANOVA) should be applied to ensure that observed differences are significant and not due to experimental variability.

We added statistical test used in Section 2.5 (lines 439 to 445) and discussed in Section 3.1 (Lines 474 to 477) and table 1

 

1. Discussion:

The authors mention a 20-28% reduction in yeast survival during gastrointestinal transit but do not discuss the potential impact of this reduction on the overall probiotic effect. While the survival rates seem adequate, further context on what constitutes a ‘substantial concentration’ for probiotic efficacy would be helpful.

Thank you for your comments. Since both reviewers observed and made suggestions for this section, we have rewritten Section 3.1 (Lines 452 – 489).

 

The models used to represent fermentation dynamics are robust and well-explained, but more discussion is needed regarding the selection of these models. For instance, why was the Modified Luedeking-Piret model chosen for ethanol production, and how does it compare to other models commonly used in fermentation research?

The primary objective of this work is to model viable cells, which is why the modified Luedeking-Piret model, which incorporates viable cells, is the most appropriate for this study. In contrast, the original Luedeking-Piret model uses total cells, making it less suitable for our specific focus on viable cells with probiotic properties.

Including the original Luedeking-Piret model in this work would not only deviate from the main objective but also introduce unnecessary complexity. Moreover, the fit of the original Luedeking-Piret model is suboptimal for our data, as evidenced by the comparison of the two curves and their respective R² values. Presenting these results would detract from the clarity of our work. Therefore, we believe that focusing on the modified Luedeking-Piret model aligns with the core objective of this study, which is to accurately model viable cells.

 

 

Figure 1. Fit to the experimental data using the original Luedeking-Piret model, which yields an R² value of 93.29%.

Below, we present the model proposed in this work.

Figure 2. Fit to the experimental data using the modified Luedeking-Piret model, which yields an R² value of 98.99%.

Figures 1 and 2 reveal a noticeable difference in data fitting. Given this observation and the fact that it is not the main focus of this study, the research group has opted not to include the original Luedeking-Piret model.

The same applies when attempting to use the model to fit viable cells, as detailed in the Introduction, lines 94 to 103. For this reason, this study focuses on the developed models rather than comparing them with others.

 

While the models show a strong fit with the experimental data (R² values of 99.33% for FOPDT and 99.09% for the Logistic model), a more in-depth discussion on the model validation process would be beneficial. For example, did the authors perform any cross-validation or sensitivity analysis to confirm the robustness of the models?

In addition to R², evaluating the accuracy and statistical significance of the proposed models requires key performance metrics such as Lack of Fit (LOF), Mean Squared Error (MSE), F-statistics, and p-value. These parameters provide a comprehensive assessment of model reliability, ensuring that the models accurately represent the experimental data and exhibit strong predictive capabilities.

In Materials and Methods section 2.5 we added:

Lines 434 to 439: Evaluating the accuracy and statistical significance of the proposed models requires the use of key performance metrics such as Lack of Fit (LOF), Mean Squared Error (MSE), F-statistic, and p-value. These parameters provide a comprehensive assessment of model reliability, ensuring an optimal fit to the experimental data and strong predictive capabilities.

In results and discussion, section 3.2 we incorporate:

The models in this work show a strong fit with the experimental data (Tables 2, 3, and 4). To further evaluate their accuracy and statistical significance, key parameters such as the LOF, MSE, F-statistic, and p-value were analyzed. These metrics help assess the reliability of the models, ensuring that they accurately capture experimental trends and provide robust predictive performance.

 

1. Conclusion:

The conclusion is concise, but it could be strengthened by emphasizing the practical implications of the findings. Specifically, how might the results influence the development of probiotic beers on a commercial scale? Additionally, the authors could provide more insights into future research directions, particularly regarding the scalability of the fermentation process and the long-term stability of probiotic strains in beer.

We appreciate the valuable suggestions from the reviewer and have modified the conclusions and included future projections that strengthen this initial work.

Lines 728-748: In conclusion, the probiotic yeast Saccharomyces cerevisiae PB101 demonstrated effective brewing fermentation capabilities, achieving parameters comparable to those of the control strain. After simulated gastrointestinal exposure, PB101 showed high survival rates, suggesting its ability to reach significant concentrations at the probiotic site of action.  

Logistic and First-Order Plus Dead Time (FOPDT) models were employed to fit experimental data on viable cell populations, achieving accuracy levels of 99.33% and 99.09%, respectively. The modified Luedeking-Piret equation indicated that ethanol production is more closely related to the cell growth rate than to cell count.

The study's results indicate that the probiotic yeast fermentation process using S. cerevisiae PB101 has significant potential for scaling up to commercial production of probiotic beers, given its ability to survive harsh gastrointestinal conditions and maintain viability during storage. The developed mathematical models provide valuable tools for optimizing fermentation processes and ensuring the viability of probiotic strains in the final product, potentially enhancing efficiency and quality of functional beers.

Further research is needed to evaluate the scalability of the fermentation process in large-scale industrial settings, considering variations in wort composition and fermentation temperatures. Stability studies are essential to determine the shelf life of probiotic beers, and clinical trials should be conducted to validate the health benefits of S. cerevisiae PB101, particularly regarding gut health, thus providing scientific evidence for its use in functional beverages.

 

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript has provided satisfactory responses to the comments received, and I agree to accept it in its current form.

Author Response

We are pleased to inform you that the reviewer agrees with the previous revisions made to the manuscript. We appreciate your attention to this matter and look forward to your response.

Reviewer 2 Report

Comments and Suggestions for Authors

Dear Authors,

Thank you for your continued efforts in addressing the reviewers’ comments. Your revisions have improved the manuscript; however, several aspects still require further clarification:

• The Introduction would benefit from a clearer definition of the study objectives. It would be helpful if you could explicitly state the specific aims of the study, ideally in either the first or last paragraph of the Introduction, to provide a more concise overview for the reader.
• Additional information was provided regarding the simulation, but the exact pH values, temperature, and duration of each simulation phase (stomach, duodenum) were not fully specified. These details are crucial for reproducibility and should be included for clarity. If they are already present, please make sure they are clearly stated.
• The revised section on yeast survival is much improved. However, I had previously requested context regarding what is considered a sufficient concentration for probiotic efficacy. Could you please provide a reference or a threshold value (such as the minimum viable cell count required for a probiotic effect)? This would enhance the robustness of the discussion.
• You have included LOF, MSE, F-statistics, and p-values as evaluation criteria. However, I had specifically inquired about additional validation steps, such as cross-validation or sensitivity analysis of the model to data variations. Could you briefly comment on whether these procedures were performed, or provide a justification for their exclusion if they were deemed unnecessary?

Looking forward to your revised manuscript.

Author Response

San Juan, March 22^nd, 2025

Dear Reviewer 2

Bold letters were made by the reviewers

Italic letters were made by the authors

 

 Thank you for your continued efforts in addressing the reviewers’ comments. Your revisions have improved the manuscript; however, several aspects still require further clarification:

The Introduction would benefit from a clearer definition of the study objectives. It would be helpful if you could explicitly state the specific aims of the study, ideally in either the first or last paragraph of the Introduction, to provide a more concise overview for the reader.

We appreciate your suggestion. We have modified the last paragraph to explicitly present the main objective of our work.

Lines 125- 134: The primary objective of this study is to apply a novel and comprehensive modeling approach to describe probiotic beer production. This approach integrates the logistic growth model, the FOPDT model, and a modified Luedeking-Piret equation [15]. The models incorporate two state variables: total cells and dead cells. Additionally, they include delay functions to account for the lag phase and cell death, both essential for accurately modeling microbial kinetics. Experimental data from controlled microfermentations quantified biomass growth, dead phase dynamics, and metabolite production rates. These factors are particularly relevant for autochthonous probiotic strains, whose lag phases are influenced by environmental conditions [27]. Modeling these dynamics is necessary to achieve a final cell concentration that ensures health benefits.

 

Additional information was provided regarding the simulation, but the exact pH values, temperature, and duration of each simulation phase (stomach, duodenum) were not fully specified. These details are crucial for reproducibility and should be included for clarity. If they are already present, please make sure they are clearly stated.

We have reviewed the manuscript and have added the requested changes where appropriate (Lines 191 to 213, Section 2.3).

 

The revised section on yeast survival is much improved. However, I had previously requested context regarding what is considered a sufficient concentration for probiotic efficacy. Could you please provide a reference or a threshold value (such as the minimum viable cell count required for a probiotic effect)? This would enhance the robustness of the discussion.

We would like to clarify that both in the introduction section (lines 51 to 58) and in the results and discussion section (lines 480 to 493), we have stated that it is currently not acceptable to discuss a fixed, single dose. Instead, dosing should depend on the desired effect or property. Nevertheless, we have added a new sentence at the end of the section to address your comments:

Lines 493 -496: While a minimum concentration of 1x 10⁶ cells/mL was classically considered necessary for probiotic efficacy, current research indicates this is an oversimplification. The effective dose now appears to be dependent on the desired effect/property, the target outcome, and the specific probiotic microorganism employed.

 

You have included LOF, MSE, F-statistics, and p-values as evaluation criteria. However, I had specifically inquired about additional validation steps, such as cross-validation or sensitivity analysis of the model to data variations. Could you briefly comment on whether these procedures were performed, or provide a justification for their exclusion if they were deemed unnecessary?

In response of your comment, we confirm that a sensitivity analysis was conducted.

Lines 440- 445:  In this work, the global sensitivity of the models was explored by varying each parameter within a range of ± 50% of its estimated value, while keeping the other parameters constant. The effect of these variations was measured through the relative error of the model. A parameter was defined as sensitive and well-estimated if a variation greater than 5% generated an increase of more than 5% in the discrepancy between the simulation and the experimental data.

Author Response File: Author Response.pdf