# Kinetic Analysis of Gluconacetobacter diazotrophicus Cultivated on a Bench Scale: Modeling the Effect of pH and Design of a Sucrose-Based Medium

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{8}CFU/mL). After preparing dilutions with different bacterial concentrations, they were applied to tomato seedlings, obtaining statistically significant differences in terms of dry weight related to the control treatment without the addition of this bacterium. In that case, the application of G. diazotrophicus exhibited its growth-promoting properties at the early stages of tomato development.

^{12}CFU/mL in flasks with 250 mL working volume for all the media assessed. After the cultivation process, the bacteria had stable growth in nitrogen-free media, but differences were observed in the phosphate solubilization halos in the NBRIP medium, indicating the affectation of the phosphate solubilization ability. Finally, the results demonstrated the influence of the culture medium for the inoculum production on the growth promotion features of G. diazotrophicus when applied to carrot and beet.

## 2. Materials and Methods

#### 2.1. Microorganism

^{8}cells/mL; the resulting suspension was transferred to microtubes that were brought to −80 °C to have reserve material for the different assays.

#### 2.2. Inoculum Preparation

#### 2.3. Cultivation Conditions for Culture Medium Design

#### 2.4. Screening of Medium Components

^{®}(Statpoint Technologies, USA).

#### 2.5. One-Factor Design

#### 2.6. Submerged Cultivation on a Bench Scale

^{8}cells/mL). The inoculation concentration was increased up to 10% (by volume) to speed up bacterial growth and development. The working suspension was stirred at 150 rpm at 30 °C on an orbital shaker. During the evolution of bacterial growth, 10 mL samples were taken every 24 h for 15 days (366 h), in which the concentration of cell biomass was determined by optical density and dry weight.

#### 2.7. Analytical Methods

^{®}membranes using a vacuum pump. The biomass obtained was dried in an oven at 105 °C until constant weight.

## 3. Model Development

^{+}], and a relationship between [H

^{+}] and biomass concentration is provided by the hydrogen model. The hydrogen model is constructed by combining the mass balance for acid concentration, the electroneutrality equation, and the ion dissociations, and applying a simplification. In addition, the proposed model is compared with a basic model in which μ is a function of biomass concentration, but the effect of hydrogen ions is disregarded. The two models are fitted to the measurements of biomass concentration and pH over time from batch submerged culture of G. diazotrophicus performed in a 3 L bioreactor. The proposed model has several advantages. Firstly, it applies to the case of several unknown acid and buffer species. Secondly, the number of parameters to be estimated and the order of the polynomial is smaller than that in the rigorous classic model. Finally, the effect of adding acids and bases can be incorporated, which is an advantage over common simplified models. The procedure for model calibration and calculation of the time courses of pH and concentrations of biomass and substrate is proposed as well.

#### 3.1. Model Formulation

_{2}PO

_{4}and K

_{2}HPO

_{4}.

#### 3.2. Preliminary Modeling and Fitting

^{+}] is disregarded. Considering characteristics C1 to C4, the mass balances for biomass and substrate concentrations are as follows:

^{−1}), Y

_{xs}is the biomass yield coefficient (g biomass/g substrate), and m

_{s}is the maintenance coefficient (g substrate/(g biomass × h)).

_{x}X) was used by Belfares et al. [43]. Accounting for this, the following expression for the specific growth rate was considered:

_{max}is the maximum specific growth rate (h

^{−1}), K is the substrate half-saturation constant (g/L), and K

_{X}is a proportional coefficient related to biomass (L/g).

_{t}is the number of data points; X

_{exp}and S

_{exp}are the measurements of biomass and substrate concentrations, respectively; X

_{mod}and S

_{mod}are the values of biomass and substrate concentrations estimated by the model, respectively; and X

_{max}and S

_{max}are the maximum values of the experimental data of biomass and substrate, respectively [44]. It was found that the term $S/\left(K+S\right)$ is not suitable as the estimate of K exhibited overlarge confidence interval. Then, that term was neglected, but a determination coefficient of 0.0887⫷ 1.0 was obtained for substrate concentration (see Figure A1). Therefore, the effect of carbon source concentration on the specific growth rate was neglected, and further fitting of the substrate model (2) was not performed. This neglection of the effect of the carbon source concentration was also undertaken in the modeling of Gluconobacter japonicus, as reported by Cañete-Rodríguez et al. [45].

#### 3.3. First Modeling Approach

^{+}] [46]:

_{max}is the maximum biomass concentration achieved during the cultivation (g/L), and f is a constant that defines the degree of inhibition of biomass.

#### 3.4. Second Modeling Approach

^{+}] but disregards the effect of substrate and biomass concentrations, and the hydrogen model that describes the relationship between H

^{+}concentration and biomass concentration. The hydrogen model is obtained by combining the mass balance for acid concentration, the electroneutrality equation, and the equation for the ion dissociation of water, acid, and buffer species.

^{+}] by using an inhibition function [47,48] as follows:

_{max}is the proton concentration at which growth ceases; µ

_{max}, H

_{max}, and d are parameters to be fitted.

_{tot}is the total concentration of acid species, α is the constant of product formation associated with microbial growth, and β is the constant of product formation associated with microbial mass.

_{A}, $\overline{\alpha},\overline{\beta}$, and k

_{0}are constants. The acid ($HA$) is considered weak, so the corresponding dissociation reaction is:

^{+}is the hydrogen ion. For the buffer (${H}_{2}{B}_{u}$), the following dissociation reactions were assumed:

_{b1}and K

_{b2}are dissociation constants of ${H}_{2}{B}_{u}$ and $H{B}_{u}^{-}$. Stating the dissociation equations for acid and H

_{2}O and combining Equations (9) and (10) yields:

_{1}, K

_{2}, K

_{3}, and K

_{4}are constants that depend on the culture medium characteristics and can be estimated by model fitting using measurements of acid (${A}_{tot}$) and pH and the pH definition $pH=-{\mathrm{log}}_{10}[{H}^{+}]$. Equation (12) relates the hydrogen ion concentration [H

^{+}] with the total acid concentration.

^{+}]. In the case that buffer, acid, or base species are considered in addition to HA and H

_{2}B

_{u}, the corresponding ion concentrations must be included in the electroneutrality equation. Since data from acid measurements are not available, Equation (8) must be used, which expresses total acid concentration (A

_{tot}) in terms of biomass concentration (X). Combining Equations (8) and (13) yields the following expression that relates the hydrogen ion concentration [H

^{+}] to the biomass concentration (X):

_{1}, θ

_{2}, θ

_{3}, and θ

_{4}can be obtained by fitting. The following simplification is considered:

^{+}ion as a function of biomass concentration; it is obtained by combining Equations (8) and (12), using a simplification for the relationship between the concentrations of hydrogen ions and acid, what greatly reduces the complexity for computing [H

^{+}]. The hydrogen model for the case when an acid and a base are added is developed in Appendix B.

#### 3.5. Statement of the Fitting Procedure

_{t}is the number of data points, X

_{exp}is the measurement of biomass concentration, X

_{mod}is the value of biomass estimated by the model, and X

_{max}is the maximum value of the experimental data of biomass [53]. The fitting quality is assessed through the determination coefficient (R

^{2}), and the precision of the estimates is evaluated through the width of the 95% asymptotic confidence intervals [54,55,56]. Matlab version 2014 (The MathWorks Inc., Natick, MA, USA) was used for these purposes. In particular, the fmin routine was used for minimization.

^{+}] and biomass. This is performed via the least squares method:

- The parameters of biomass and µ models are estimated on biomass measurements by minimization of SSE (18); in the µ model (6), the [H
^{+}] values are obtained by interpolation of the experimental [H^{+}] data, and the signal $z={{\displaystyle \int}}_{to}^{t}Xdt$ is computed as well, whereas the pH definition is $pH=-{\mathrm{log}}_{10}[{H}^{+}]$ [42]; - The parameters of hydrogen models (17a) and (17b) are estimated on measurements of pH and biomass concentration via least squares, using Equation (19), with the values of $z={{\displaystyle \int}}_{to}^{t}Xdt$ computed in the previous step;
- The parameters of the biomass model (1) and specific growth rate model (6) are estimated on biomass measurements by minimization of the SSE (18); in the specific growth rate model (6), the [H
^{+}] values are obtained by using the hydrogen models (17a) and (17b) fitted in the previous step, instead of using interpolation.

## 4. Results and Discussion

#### 4.1. Cultivation of Gluconacetobacter Diazotrophicus on a Laboratory Scale

#### 4.2. Key Components of the Culture Medium

_{4})

_{2}SO

_{4}as a nitrogen source to promote the activation of bacterial reproduction in the liquid medium; it also has salts that favor its development. According to the results obtained regarding the effect of the eight components of the medium on biomass production, no statistically significant differences (p < 0.05) were detected between the nutrients evaluated. That is, there were no significant differences in biomass production when the different components of the medium were varied within the ranges predefined in the Plackett–Burman experimental design. Considering the linear interaction model implicit in this type of design, this outcome implies that it is not possible to select a group of nutrients whose variability influences the formation of new cells of G. diazotrophicus differently (to a greater degree) than the rest of the components of the medium.

#### 4.3. Definition of Carbon Source Concentration

#### 4.4. Bench-Scale Submerged Cultivation

^{8}CFU/mL, unlike the present work, in which G. diazotrophicus reached a count of about 7 × 10

^{8}CFU/mL at 294 h. For bacterial growth, sucrose hydrolysis is required, which may imply an increase in the duration of the submerged culture compared to the use of glucose as a carbon source [21]. However, it is important to consider the use of sucrose instead of glucose for the design of culture media for the production of G. diazotrophicus since it allows for the cost reduction of industrial culture media. For these purposes, the use of low-cost agro-industrial by-products such as molasses or starch hydrolyzates will be addressed in future work.

#### 4.5. Fitting of the Kinetic Model

#### 4.5.1. Fitting of First Approach Models

#### 4.5.2. Fitting of Second Approach Models

^{+}] and X was obtained, as can be noticed from the modeling results (Figure 5 and Figure 6) and the corresponding determination coefficients (Table 6). This supports the assumptions made, including a single acid-like behavior and a single buffer-like behavior. For the description of [H

^{+}] as a function of X and ${{\displaystyle \int}}_{to}^{t}Xdt$, the parameter θ

_{3}of model (17a) and θ

_{1}and θ

_{2}of model (17b) exhibited overlarge confidence intervals. Thus, the following simplified hydrogen models were applied, which achieved adequate confidence intervals:

^{+}] as a function of X are shown in Figure 5a, and the corresponding simulation of the time course of [H

^{+}] is presented in Figure 5c. The simulation of [H

^{+}] as a function of ${{\displaystyle \int}}_{to}^{t}Xdt$ is shown in Figure 5b. A high correspondence between simulated and experimental values was obtained.

#### 4.6. Comparison of Model Fitting via First and Second Approaches

_{max}and H

_{max}using the hydrogen models (Equations (21) and (22)) agree. This is because a satisfactory representation of [H

^{+}] is achieved by both hydrogen models (see Figure 5).

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Overview of Scientific Literature on Modeling Approaches for the Relationship between Hydrogen Ions, Acids, Bases, and Buffer

- The Wiener models [24].

_{2}PO

_{4}were used by Antwi et al. [47], and the pH model was fitted for each. However, the model does not allow accounting for the future addition of bases or acids. The empiric model of Ellouze et al. [65] is a modified logistic model representation of the relationship between total acid concentration and pH. This approach was used for the culture of acid lactic bacteria. However, the model does not allow accounting for the future addition of bases or acids.

## Appendix B. Equation of pH for the Case of External Addition of Acids/Bases/Buffer Compounds

^{+}]:

_{a}− C

_{b}are provided by McAvoy et al. [32]:

^{+}]/dt, the dĒ/dt term is a function of the acid and base flowrates (F

_{a}and F

_{b}), thus providing an input–output relationship with a relative degree of one. Thus, a control law can be defined to control the hydrogen ion concentration by manipulating the acid and base flowrates.

## Appendix C. Fitting of Substrate Concentration

**Figure A1.**Preliminary modeling for G. diazotrophicus batch fermentation. Experimental data are represented by filled circles; the solid lines are calculated by the model proposed. S: substrate concentration. Graph built using Matlab (The MathWorks Inc., Natick, MA, USA).

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**Figure 1.**Time profile of G. diazotrophicus ATCC 49037 growth (

**a**) and correlation between biomass concentration measured by dry weight and optical density (

**b**). Cells were grown on LGI-P medium in 500 mL flasks at 150 rpm and 30 °C. The optical density of the initial inoculum was 1.0, corresponding to about 10

^{8}cells/mL. OD: optical density at 600 nm; X: biomass concentration measured by dry weight. Graphs built by using MS Excel (Microsoft Corporation, Redmon, IL, USA).

**Figure 2.**Production of G. diazotrophicus ATCC 49037 biomass for different sucrose concentrations in the culture medium. Biomass concentration values are the mean of 10 replicates. Standard deviations of the ten replicates are shown as vertical error bars. X: cell biomass concentration. Graph built by using MS Excel (Microsoft Corporation, Redmon, IL, USA); statistical package used to analyze the obtained data: Statgraphics

^{®}(Statpoint Technologies, Warrenton, VA, USA).

**Figure 3.**Time profile of substrate, cell biomass, and ammonium nitrogen concentrations (

**a**) and pH (

**b**) in the culture medium during batch submerged cultivation of G. diazotrophicus ATCC 49037 in a 3 L stirred-tank bioreactor. AN: ammonium nitrogen; TC: total carbohydrates; X: cell biomass. Values of biomass and ammonium nitrogen concentrations are multiplied by 10. Graphs built using Matlab (The MathWorks Inc., Natick, MA, USA).

**Figure 4.**Results of the first modeling approach for the kinetics of G. diazotrophicus batch cultivation. Experimental data are represented by filled circles; the solid line is calculated by the model proposed. Graph built using Matlab (The MathWorks Inc., Natick, MA, USA).

**Figure 5.**Fitting of the hydrogen models for the kinetics of G. diazotrophicus batch cultivation. Experimental data are represented by filled circles; the solid lines are calculated by the model proposed. (

**a**) Effect of biomass concentration (X) on the concentration of hydrogen ions [H

^{+}], with fitting using linear hydrogen model (Equation (21)). (

**b**) Effect of ${I}_{x}={{\displaystyle \int}}_{to}^{t}Xdt$ on the concentration of hydrogen ions, with fitting using a nonlinear hydrogen model (Equation (22). (

**c**) Time course of the concentration of hydrogen ions, with fitting using both hydrogen models. Graphs built using Matlab (The MathWorks Inc., Natick, MA, USA).

**Figure 6.**Comparison of the results of the first and second modeling approaches for the kinetics of G. diazotrophicus batch fermentation. Experimental data are represented by filled circles; the solid and dashed lines are calculated by the model proposed. The second approach involves the linear hydrogen model (Equation (21)) and a nonlinear hydrogen model (Equation (22)). Graph built using Matlab (The MathWorks Inc., Natick, MA, USA).

N. | Component | Amount for 1 L | Concentration (g/L) |
---|---|---|---|

1 | White sugar | 100 g | 100 |

2 | K_{2}HPO_{4} (10% solution) | 2 mL | 0.02 |

3 | KH_{2}PO_{4} (10% solution) | 6 mL | 0.06 |

4 | MgSO_{4}·7H_{2}O (10% solution) | 2 mL | 0.02 |

5 | CaCl_{2}·2H_{2}O (1% solution) | 2 mL | 0.002 |

6 | Na_{2}MoO_{4}·2H_{2}O (0,1% solution) | 2 mL | 0.0002 |

7 | FeCl_{3}·6H_{2}O (1% solution) | 1 mL | 0.001 |

8 | (NH_{4})_{2}SO_{4} | 1 g | 1 |

Coded Factor | Factor | Experimental Range (g/L) | |
---|---|---|---|

Low Level (−1) | High Level (+1) | ||

Z_{1} | Sucrose | 70 | 130 |

Z_{2} | K_{2}HPO_{4} | 0.005 | 0.035 |

Z_{3} | KH_{2}PO_{4} | 0.045 | 0.075 |

Z_{4} | MgSO_{4}·7H_{2}O | 0.005 | 0.035 |

Z_{5} | CaCl_{2}·2H_{2}O | 0.0005 | 0.0035 |

Z_{6} | Na_{2}MoO_{4}·2H_{2}O | 0.00005 | 0.00035 |

Z_{7} | FeCl_{3}·6H_{2}O | 0.0001 | 0.0019 |

Z_{8} | (NH_{4})_{2}SO_{4} | 0.1 | 1.9 |

Z_{9}–Z_{11} | Dummy factors | - | - |

N. | Coded Factors | Biomass (g/L) ^{1} | Standard Deviation | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Z_{1} | Z_{2} | Z_{3} | Z_{4} | Z_{5} | Z_{6} | Z_{7} | Z_{8} | Z_{9} | Z_{10} | Z_{11} | |||

1 | +1 | −1 | +1 | +1 | +1 | +1 | +1 | −1 | −1 | −1 | −1 | 0.83 | 0.20 |

2 | +1 | +1 | −1 | +1 | −1 | +1 | −1 | +1 | −1 | +1 | −1 | 1.08 | 0.32 |

3 | −1 | +1 | +1 | −1 | −1 | +1 | +1 | +1 | −1 | −1 | +1 | 1.10 | 0.23 |

4 | −1 | +1 | +1 | +1 | +1 | −1 | −1 | −1 | −1 | +1 | +1 | 1.17 | 0.11 |

5 | −1 | −1 | +1 | −1 | +1 | +1 | −1 | +1 | +1 | +1 | −1 | 1.10 | 0.04 |

6 | −1 | −1 | −1 | +1 | −1 | +1 | +1 | −1 | +1 | +1 | +1 | 1.42 | 0.63 |

7 | +1 | −1 | +1 | +1 | −1 | −1 | −1 | +1 | +1 | −1 | +1 | 1.11 | 0.20 |

8 | −1 | +1 | −1 | +1 | +1 | −1 | +1 | +1 | +1 | −1 | −1 | 1.28 | 0.19 |

9 | −1 | −1 | −1 | −1 | −1 | −1 | −1 | −1 | −1 | −1 | −1 | 1.32 | 0.22 |

10 | +1 | +1 | −1 | −1 | +1 | +1 | −1 | −1 | +1 | −1 | +1 | 1.35 | 0.43 |

11 | +1 | +1 | +1 | −1 | −1 | −1 | +1 | −1 | +1 | +1 | −1 | 1.15 | 0.17 |

12 | +1 | −1 | −1 | −1 | +1 | −1 | +1 | +1 | −1 | +1 | +1 | 1.19 | 0.24 |

^{1}Mean of three replicates.

Source | Chi-Squared | Degrees of Freedom | t-Value | Mean Squared Deviation | p-Value |
---|---|---|---|---|---|

Z_{1} | 1.370723 | 1 | 2.032245 | 7.097078 | 0.2416877 |

Z_{2} | 0.272593 | 1 | 2.032245 | 7.212022 | 0.6015979 |

Z_{3} | 2.604274 | 1 | 2.032245 | 6.965699 | 0.1065760 |

Z_{4} | 1.261836 | 1 | 2.032245 | 7.108559 | 0.2613038 |

Z_{5} | 0.002253 | 1 | 2.032245 | 7.240039 | 0.9621434 |

Z_{6} | 1.766220 | 1 | 2.032245 | 7.055222 | 0.1838505 |

Z_{7} | 0.361454 | 1 | 2.032245 | 7.202788 | 0.5476996 |

Z_{8} | 0.012265 | 1 | 2.032245 | 7.239003 | 0.9118151 |

Contrast | Difference | +/− Limits | Lower Limit | Higher Limit |
---|---|---|---|---|

15–30 | −0.22601 | 0.132147 | −0.358157 | −0.093863 |

15–60 | −0.14268 | 0.132147 | −0.274827 | −0.010533 |

30–75 | 0.32433 | 0.132147 | 0.192183 | 0.456477 |

30–90 | 0.13634 | 0.132147 | 0.004193 | 0.268487 |

30–105 | 0.31333 | 0.132147 | 0.181183 | 0.445477 |

30–120 | 0.35267 | 0.132147 | 0.220523 | 0.484817 |

30–135 | 0.29701 | 0.132147 | 0.164863 | 0.429157 |

30–150 | 0.187 | 0.132147 | 0.054853 | 0.319147 |

45–75 | 0.20699 | 0.132147 | 0.074843 | 0.339137 |

45–105 | 0.19599 | 0.132147 | 0.063843 | 0.328137 |

45–120 | 0.23533 | 0.132147 | 0.103183 | 0.367477 |

45–135 | 0.17967 | 0.132147 | 0.047523 | 0.311817 |

60–75 | 0.241 | 0.132147 | 0.108853 | 0.373147 |

60–105 | 0.23 | 0.132147 | 0.097853 | 0.362147 |

60–120 | 0.26934 | 0.132147 | 0.137193 | 0.401487 |

60–135 | 0.21368 | 0.132147 | 0.081533 | 0.345827 |

75–90 | −0.18799 | 0.132147 | −0.320137 | −0.055843 |

75–150 | −0.13733 | 0.132147 | −0.269477 | −0.005183 |

90–105 | 0.17699 | 0.132147 | 0.044843 | 0.309137 |

90–120 | 0.21633 | 0.132147 | 0.084183 | 0.348477 |

90–135 | 0.16067 | 0.132147 | 0.028523 | 0.292817 |

120–150 | −0.16567 | 0.132147 | −0.297817 | −0.033523 |

Modeling Approach and Model | Parameter and R^{2} | Value |
---|---|---|

First modeling approach; biomass model (1) and µ model (20) | µ_{max} | 0.0093 ± 0.0017 h^{−1} |

X_{max} | 0.3578 ± 0.0503 g/L | |

R^{2} (for biomass concentration over time) | 0.9364 | |

Second modeling approach; hydrogen model (21) | θ_{1} | 0.0665 ± 0.0093 |

θ_{2} | 158.5 ± 9.1 | |

R^{2} (for H^{+} concentration as a function of biomass concentration) | 0.9841 | |

Second modeling approach; hydrogen model (22) | θ | 29,449,609 ± 1,652,515 |

R^{2} (for H^{+} concentration as a function of the integral of biomass concentration) | 0.9811 | |

Second modeling approach; biomass model (1) and µ model (23), with [H^{+}] provided by the hydrogen model (21) | µ_{max} | 0.0076 ± 0.0012 h^{−1} |

H_{max} | 0.00184 ± 0.00032 mol/L (corresponds to pH_{min} = 2.735) | |

R^{2} (for biomass concentration over time) | 0.9364 | |

Second modeling approach; biomass model (1) and µ model (23), with [H^{+}] provided by the hydrogen model (22) | µ_{max} | 0.0078 ± 0.0010 h^{−1} |

H_{max} | 0.00181 ± 0.00027 mol/L | |

R^{2} (for biomass concentration over time) | 0.9357 |

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## Share and Cite

**MDPI and ACS Style**

Restrepo, G.M.; Rincón, A.; Sánchez, Ó.J.
Kinetic Analysis of *Gluconacetobacter diazotrophicus* Cultivated on a Bench Scale: Modeling the Effect of pH and Design of a Sucrose-Based Medium. *Fermentation* **2023**, *9*, 705.
https://doi.org/10.3390/fermentation9080705

**AMA Style**

Restrepo GM, Rincón A, Sánchez ÓJ.
Kinetic Analysis of *Gluconacetobacter diazotrophicus* Cultivated on a Bench Scale: Modeling the Effect of pH and Design of a Sucrose-Based Medium. *Fermentation*. 2023; 9(8):705.
https://doi.org/10.3390/fermentation9080705

**Chicago/Turabian Style**

Restrepo, Gloria M., Alejandro Rincón, and Óscar J. Sánchez.
2023. "Kinetic Analysis of *Gluconacetobacter diazotrophicus* Cultivated on a Bench Scale: Modeling the Effect of pH and Design of a Sucrose-Based Medium" *Fermentation* 9, no. 8: 705.
https://doi.org/10.3390/fermentation9080705