# Phosphorus Threshold for the Growth of Microcystis wesenbergii, Microcystis aeruginosa, and Chlorella vulgaris Based on the Monod Formula

^{*}

## Abstract

**:**

## 1. Modified Monod Model and Its Significance

#### 1.1. Monod Equation

_{max}represents the maximum specific growth rate of microorganisms, and K

_{s}stands for the saturation constant, which corresponds to the substrate concentration when μ equals half of μ

_{max}. It is also known as the half-velocity constant. S represents the concentration of organic substrate, and X is the concentration of microorganisms. The curve of the Monod equation is depicted in Figure 1 [30]. The Monod equation, in its empirical form, is widely utilized in contemporary wastewater biological treatment as a powerful tool to guide indicators, such as organic load, in biological treatment.

#### 1.2. Modified Monod Model

_{syn}represents the synthesized specific growth rate, X

_{a}is the concentration of active bacterial cells (MxL

^{−3}), t is time (T), S is the substrate concentration limiting the growth rate (MSL

^{−3}), $\hat{\mu}$ is the maximum specific growth rate (T

^{−1}), and K is the substrate concentration at which the growth rate is half of the maximum specific growth rate (MSL

^{−3}). Figure 2 illustrates the variation in μ with S and the condition where μ = $\hat{\mu}$/2 when K = S.

^{−1}), and μ

_{dec}denotes the specific growth rate considering decay (T

^{−1}).

_{d}represents the fraction of biodegradable compounds within active bacterial cells. The rate at which active bacterial cells transform into inert cells is determined by the difference between the overall decay rate and the oxidation rate:

_{i}represents the concentration of inert bacterial cells (MxL

^{−3}).

^{−1}; S denotes the concentration of the limiting substrate phosphorus, measured in mg/L; t stands for time, measured in days; X represents algal density; μ

_{max}corresponds to the maximum specific growth rate, expressed in day

^{−1}; K

_{s}is the phosphorus concentration at which algal growth reaches half of the maximum specific growth rate, measured in mg/L; and C represents the specific death rate of algal growth, expressed in day

^{−1}.

## 2. Materials and Methods

#### 2.1. Experimental Materials

#### 2.2. Medium Design

_{2}HPO

_{4}). The specific formulations for the three types of culture media are provided in the Appendix A.

#### 2.3. Experimental Design

#### 2.4. Data Analysis Methods

## 3. Results and Analysis

#### 3.1. Specific Growth Rate of C. vulgaris

^{2}values of the fitting are generally high, indicating a significant linear relationship between the two variables. This situation suggests the validity of the results.

#### 3.2. Growth Rate of M. aeruginosa

#### 3.3. Growth Rate of M. wesenbergii

## 4. Comparison and Discussion Based on the Experiment of Specific Growth Rate of Three Kinds of Algae

#### 4.1. Comparison of the Specific Growth Rates of Three Algal Species

#### 4.2. Fitting of the Modified Monod Model for C. vulgaris

_{max}(0.5711, 0.5658), K

_{s}(0.09321, 0.08543), and C (0.2156, 0.2196). The goodness-of-fit indicators include sum of squared errors (SSE) (0.005515, 0.006661), coefficient of determination (R

^{2}) (0.964, 0.954), adjusted R

^{2}(0.9498, 0.9357), and root-mean-squared error (RMSE) (0.03321, 0.0365).

^{2}values obtained from the fitting indicate that the proposed modified Monod equation effectively reflects the nonlinear relationship between C. vulgaris growth rate and phosphorus nutrient concentration, validating the feasibility of the research hypothesis. This equation can describe the growth of C. vulgaris subjected to limited phosphorus nutrient. Additionally, the threshold phosphorus nutrient concentrations limiting C. vulgaris growth are 0.057 and 0.054 mg/L according to Equation (8). Wu et al. [19] concluded that the threshold for algal growth is a total phosphorus content of 0.059 mg/L, which aligns well with the results of this experiment.

#### 4.3. Fitting of the Modified Monod Model for M. aeruginosa

_{max}(1.838, 1.793), K

_{s}(0.0145, 0.0149), and C (1.336, 1.291). The goodness-of-fit indicators include SSE (0.003874, 0.003697), R

^{2}(0.9837, 0.9843), adjusted R

^{2}(0.977, 0.978), and RMSE (0.02784, 0.02719).

^{2}values from the fitting indicate that the modified Monod equation proposed in this study (Equation (7)) effectively represents the nonlinear relationship between M. aeruginosa growth rate and phosphorus nutrient concentration. This validates the feasibility of the research hypothesis. The modified Monod model can describe the growth of M. aeruginosa under phosphorus nutrient limitation. Furthermore, the calculated phosphorus nutrient lower threshold for limiting the growth of M. aeruginosa is 0.039 mg/L (or 0.038 mg/L). This result provides a specific reference for nutrient control under oligotrophic conditions, implying that controlling phosphorus concentration in water at approximately 0.038 mg/L can help maintain a dynamic equilibrium in M. aeruginosa growth and thereby prevent the excessive proliferation of algae and occurrence of algal blooms.

#### 4.4. Fitting of the Modified Monod Model for M. wesenbergii

_{max}: (0.7112, 0.6112); K

_{s}: (0.0186, 0.0234); and C: (0.3732, 0.2675). The goodness-of-fit metrics included SSE: (0.001879, 0.001902); R

^{2}: (0.9633, 0.9626); adjusted R

^{2}: (0.9486, 0.9477); and RMSE: (0.01939, 0.0195).

^{2}values obtained from the fitting indicate that the modified Monod equation (Equation (7)) proposed in this study effectively reflects the nonlinear relationship between the growth rate of M. wesenbergii and concentration of phosphorus nutrients, thus validating the feasibility of our research hypothesis. This modified Monod model can describe the growth of M. wesenbergii under phosphorus limitation. Furthermore, Equation (8) allows the calculation of the phosphorus lower threshold that restricts the growth of M. wesenbergii. The thresholds were approximately 0.039 and 0.038 mg/L. This study provides a specific phosphorus limitation threshold as a reference for controlling phosphorus concentration in water. Maintaining the phosphorus concentration in the range of approximately 0.038 mg/L in water bodies potentially maintains the dynamic equilibrium of M. wesenbergii growth, thus preventing the excessive proliferation of algae and occurrence of algal blooms.

#### 4.5. Comparison of Modified Monod Models for C. vulgaris, M. aeruginosa, and M. wesenbergii

## 5. Conclusions

- (a)
- The conventional Monod equation has certain limitations. In this study, we proposed the modified Monod equation $\mu ={\mu}_{max}\frac{S}{{K}_{s}+S}+C$ and designed experiments to validate it. The resulting fit had R
^{2}values of 0.954, 0.964, 0.977, 0.978, 0.9633, and 0.9626, indicating that the modified Monod equation effectively describes algal growth when phosphorus is limited. - (b)
- By using the modified Monod model, we calculated the lower threshold of phosphorus nutrients, which is the phosphorus nutrient concentration at which algal growth reaches a dynamic equilibrium. The calculated values were 0.0565 mg/L for C. vulgaris, 0.0386 mg/L for M. aeruginosa, and 0.0205 mg/L for M. wesenbergii. Controlling phosphorus concentrations at or near these S′ values can theoretically prevent excessive algal proliferation, providing guidance for algal growth control using nutrient limitation.
- (c)
- The S′ of the three algal species followed the order of M. wesenbergii S′ < M. aeruginosa S′ < C. vulgaris S′. M. wesenbergii requires the lowest theoretical phosphorus nutrient concentration for growth, followed by M. aeruginosa, and C. vulgaris requires the highest. The results suggest that cyanobacteria (Microcystis and similar species) have lower phosphorus nutrient thresholds, explaining why algal blooms in China are mainly composed of cyanobacteria and why other algal species do not bloom during cyanobacterial bloom events.
- (d)
- Among the three algal species, M. aeruginosa exhibited the highest maximum specific growth rate, whereas C. vulgaris had the lowest. This result suggests that in natural water bodies with fluctuating phosphorus concentrations, M. aeruginosa would dominate in terms of biomass, followed by M. wesenbergii and C. vulgaris. This observation implies that cyanobacterial biomass tends to be higher than that of green algae, making cyanobacteria the dominant species in freshwater ecosystems.
- (e)
- At high phosphorus concentrations (>2 mg/L), the growth of M. aeruginosa and M. wesenbergii is inhibited to some extent and that of C. vulgaris is unaffected. This result indicates that phosphorus inhibition does not occur in all algal species. The results of this experiment suggest that phosphorus inhibition is evident in cyanobacteria, and future research may explore this phenomenon further by investigating internal phosphorus forms and phosphorus absorption gene sequences in different algal species.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Materials and Methods

#### Appendix A.1. Experimental Materials

Lab Equipment | Lab Equipment |
---|---|

BG11 medium (Changde Det Bio-Tech Co., Ltd.) | Photobioreactor |

Centrifuge | Centrifuge tubes |

Autoclave | Count Star cell counter |

Count Star counting chamber | Analytical balance |

1000 mL volumetric flasks | 1000 mL volumetric flasks |

500 mL Erlenmeyer flasks | Breathable membrane caps |

Aseptic workbench | Alcohol lamp |

Glass rods | Graduated cylinders |

Wash bottles | Beakers |

Pipettes | Wide-mouth bottles |

Wide-mouth bottles (brown) |

#### Appendix A.2. Medium Design

#### Appendix A.2.1. Initial Culture Medium

Drug Name | Dosage per Liter of Medium |
---|---|

NaNO_{3} | 1.5 g |

K_{2}HPO_{4} | 0.04 g |

MgSO_{4}·7H_{2}O | 0.075 g |

CaCl_{2}·2H_{2}O | 0.036 g |

Citric acid | 0.006 g |

Ferric ammonium citrate | 0.006 g |

EDTANa_{2} | 0.001 g |

Na_{2}CO_{3} | 0.02 g |

A5 | 1 ml |

Drug Name | Usage per Liter of A5/mg |
---|---|

H_{3}BO_{3} | 2.86 |

MnCl_{2}·4H_{2}O | 1.86 |

ZnSO_{4}·7H_{2}O | 0.22 |

NaMoO_{4}·2H_{2}O | 0.021 |

CuSO_{4}·5H_{2}O | 0.08 |

Co(NO_{3})_{2}·6H_{2}O | 0.05 |

#### Appendix A.2.2. Phosphate-Deficient Medium

_{2}HPO

_{4}).

#### Appendix A.2.3. Phosphate Concentration Gradient Medium

_{2}HPO

_{4}). The specific procedure is as follows: The standard BG11 medium consists of five stock solutions (Stock1...5), and their specific components are listed in Table A4. The concentration of phosphorus in the medium is determined by the amount of K

_{2}HPO

_{4}added to Stock2 solution. Different concentrations of experimental Stock 2 solutions, denoted as a1, a2, a3...a8, were prepared using potassium hydrogen phosphate as the phosphorus source, with specific addition amounts as shown in Table A5.

Stock Solution | Preparation Method |
---|---|

Stock1 | 0.30 g C_{6}H_{8}O_{7}, 0.30 g C_{6}H_{8}FeNO_{7}, 0.050 g EDTANa_{2}, Dissolve and make up to 100 mL in a volumetric flask. |

Stock2 | 30.0 g NaNO_{3}, 0.78 g K_{2}HPO_{4}, 1.50 g MgSO_{4}·7H_{2}O, Dissolve and make up to 1000 mL in a volumetric flask. |

Stock3 | 1.90 g CaC1_{2}·2H_{2}O, Dissolve and make up to 100 mL in a volumetric flask. |

Stock4 | 2.00 g Na_{2}CO_{3}, Dissolve and make up to 100 mL in a volumetric flask. |

Stock5 | 2.860 g H_{3}BO_{3}, 1.8100 g MnCl_{2}·4H_{2}O, 0.2220 g ZnSO_{4}·7H_{2}O, 0.3910 g Na_{2}MoO_{4}, 0.0790 g CuSO_{4}·5H_{2}O, 0.0490 g Co(NO_{3})_{2}·6H_{2}O, Dissolve and make up to 1000 mL in a volumetric flask. |

P gradient Stock Solution | Amount of K_{2}HPO_{4} to Be Added (in Grams) |
---|---|

a1 | 0.01 |

a2 | 0.02 |

a3 | 0.04 |

a4 | 0.06 |

a5 | 0.1 |

a6 | 0.3 |

a7 | 0.6 |

a8 | 0.78 |

Phosphorus Concentration Gradient Culture Medium | P Concentration (mg/L) |
---|---|

A1 | 0.03557 |

A2 | 0.07114 |

A3 | 0.14228 |

A4 | 0.2134 |

A5 | 0.3557 |

A6 | 1.0671 |

A7 | 2.1342 |

A8 | 2.7745 |

#### Appendix A.3. Experimental Design

## Appendix B

#### Appendix B.1. Specific Growth Rate of M. aeruginosa

#### Appendix B.2. Specific Growth Rate of M. wesenbergii

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Phosphorus Concentration (mg/L) | Specific Growth Rate (day ^{−1}) | Fitted R-Squared Value (R²) |
---|---|---|

0.0356 | −0.0347 | 0.7798 |

0.0711 | −0.0251 | 0.9716 |

0.1423 | 0.1696 | 0.7126 |

0.2134 | 0.1802 | 0.6222 |

0.3557 | 0.2363 | 0.7709 |

1.0671 | 0.3167 | 0.9039 |

2.1342 | 0.3301 | 0.9112 |

2.7745 | 0.3271 | 0.8985 |

Phosphorus Concentration (mg/L) | Specific Growth Rate (day ^{−1}) | Fitted R-Squared Value (R²) |
---|---|---|

0.0356 | −0.0286 | 0.3596 |

0.0711 | −0.0244 | 0.8491 |

0.1423 | 0.178 | 0.6019 |

0.2134 | 0.1824 | 0.6214 |

0.3557 | 0.2375 | 0.7763 |

1.0671 | 0.3154 | 0.8986 |

2.1342 | 0.3144 | 0.8899 |

2.7745 | 0.3226 | 0.8954 |

Phosphorus Concentration (mg/L) | Specific Growth Rate (day ^{−1}) | Fitted R-Squared Value (R ^{2}) |
---|---|---|

0.0356 | −0.0397 | 0.9486 |

0.0711 | 0.2308 | 0.9238 |

0.1423 | 0.3048 | 0.9411 |

0.2134 | 0.3562 | 0.9434 |

0.3557 | 0.4339 | 0.9675 |

1.0671 | 0.4863 | 0.9846 |

2.1342 | 0.5108 | 0.9773 |

2.7745 | 0.4842 | 0.9873 |

Phosphorus Concentration (mg/L) | Specific Growth Rate (day ^{−1}) | Fitted R-Squared Value (R ^{2}) |
---|---|---|

0.0356 | −0.0369 | 0.7486 |

0.0711 | 0.2317 | 0.8806 |

0.1423 | 0.3047 | 0.9338 |

0.2134 | 0.3564 | 0.9386 |

0.3557 | 0.4339 | 0.9613 |

1.0671 | 0.4873 | 0.9793 |

2.1342 | 0.5065 | 0.9783 |

2.7745 | 0.4876 | 0.9815 |

Phosphorus Concentration (mg/L) | Specific Growth Rate (day ^{−1}) | Fitted R-Squared Value (R ^{2}) |
---|---|---|

0.0356 | 0.0955 | 0.3495 |

0.0711 | 0.1891 | 0.8780 |

0.1423 | 0.2416 | 0.9411 |

0.2134 | 0.2907 | 0.9814 |

0.3557 | 0.3231 | 0.9864 |

1.0671 | 0.3099 | 0.9765 |

2.1342 | 0.3535 | 0.9946 |

2.7745 | 0.3123 | 0.9907 |

Phosphorus Concentration (mg/L) | Specific Growth Rate (day ^{−1}) | Fit R-Squared Value (R ^{2}) |
---|---|---|

0.0356 | 0.1019 | 0.4399 |

0.0711 | 0.1926 | 0.8860 |

0.1423 | 0.2455 | 0.9493 |

0.2134 | 0.2868 | 0.9843 |

0.3557 | 0.3259 | 0.9890 |

1.0671 | 0.3136 | 0.9722 |

2.1342 | 0.3621 | 0.9929 |

2.7745 | 0.3177 | 0.9846 |

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## Share and Cite

**MDPI and ACS Style**

Guo, Y.; Fu, W.; Xiong, N.; He, J.; Zheng, Z.
Phosphorus Threshold for the Growth of *Microcystis wesenbergii*, *Microcystis aeruginosa*, and *Chlorella vulgaris* Based on the Monod Formula. *Water* **2023**, *15*, 4249.
https://doi.org/10.3390/w15244249

**AMA Style**

Guo Y, Fu W, Xiong N, He J, Zheng Z.
Phosphorus Threshold for the Growth of *Microcystis wesenbergii*, *Microcystis aeruginosa*, and *Chlorella vulgaris* Based on the Monod Formula. *Water*. 2023; 15(24):4249.
https://doi.org/10.3390/w15244249

**Chicago/Turabian Style**

Guo, Yansen, Wenrui Fu, Nan Xiong, Jian He, and Zheng Zheng.
2023. "Phosphorus Threshold for the Growth of *Microcystis wesenbergii*, *Microcystis aeruginosa*, and *Chlorella vulgaris* Based on the Monod Formula" *Water* 15, no. 24: 4249.
https://doi.org/10.3390/w15244249