# Revealing Interactions between Temperature and Salinity and Their Effects on the Growth of Freshwater Diatoms by Empirical Modelling

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Organisms and Culture Conditions

^{2}.

#### 2.2. Temperature and Salinity Experiments

^{−}). For consistency, throughout the rest of the paper, total salinity of media will be expressed as electric conductivity. Initial cell densities were separately adjusted to each strain based on a calibration of in vivo chlorophyll fluorescence values measured with a Tecan Infinite 200 Pro microplate reader (Tecan Group Ltd., Männedorf, Switzerland), similarly to the method applied by Albrecht et al. [20] with the following specifications: bottom reading mode with an excitation wavelength of 450 nm and detection at 680 nm. The plates were cultured in the climate chambers as described above for one week at different temperatures ranging from 5 to 28 °C. In vivo chlorophyll fluorescence was measured with the Tecan Infinite 200 Pro plate reader as described above at the same time of day over the seven days of experiments.

#### 2.3. Calculation of Specific Growth Rate

#### 2.4. Empirically Modelling Diatom Growth

#### 2.4.1. Multiplicative and Decoupled Models

#### 2.4.2. Polynomial Regression Equations

#### 2.4.3. Data Fitting

^{2}, p, and AIC). In the data fitting, only the lowest salinity at which the growth did not take place (zero or negative specific growth rate) was considered in order to reduce uncertainties in the analysis. The scenarios with definite confidence intervals for estimated parameters are given in Tables S4–S9, Supplementary Material. For selecting the best model for each species, we used the Akaike Information Criterion [31] as a metric, aiming to balance model complexity (number of parameters) against model fit. The model with the lowest AIC value was considered to provide the best such compromise, i.e., generalizability.

## 3. Results and Discussion

#### 3.1. Variations in the Response to Temperature and Salinity among Diatom Species

#### 3.2. Empirical Modelling of Diatom Growth Considering Temperature and Salinity

#### 3.2.1. Cymbella cf. incurvata

^{2}= 0.97; AIC = −91.06; Table S4; Figure 1A):

^{2}= 0.88; AIC = −62.08; Figure 1B) and the polynomial equation (R

^{2}= 0.82; AIC = −59.42; Figure 1C). Noticeably, estimates of all the parameters in Equation (S13) for C. incurvata were statistically significant (Table S4). According to Equation (ST13), the growth of C. incurvata followed an asymmetric bell-shaped response. The growth rate of this strain increased with increasing temperature, reaching the highest level at the optimum temperature higher than the optimum level of the quadratic portion ($d$ > 0) of 15.5 °C (14.6–16.3 °C). The thermal breadth of this diatom was estimated as 14.8 °C (12.7–16.9 °C; Table S4; Figure 1A). The temperature response contributes to the difference between Equations (ST13), i.e., asymmetric, and (ST12), i.e., symmetric. Between them, the asymmetric response (ST13) explained the variation in the growth of C. incurvata better than the symmetric one (ST12; Table S4). Also, according to Equation (ST13), the conductivity-growth rate followed a bell-shaped response curve. These responses to temperature and conductivity changes produce an evident peak of the growth rate, as displayed in Figure 1A. However, with an optimum conductivity of 0.63 mS/cm, just slightly above the background conductivity (0.43 mS/cm), the growth of C. incurvata was mainly inhibited by salinization at the investigated conditions (Table S4). This explains the relatively good fit of Equations (ST8), (ST9), and (ST10), in which the growth is inhibited by the addition of NaCl at any concentration (Table S4). In addition, based on Equation (ST13), the growth of this diatom strain ceased at conductivity of 4.88 mS/cm (0.88–8.87 mS/cm; Table S4). According to Equation (ST7), which also yields all significant coefficients, the growth rate of this diatom strain was inhibited by 50% at conductivity of 1.81 mS/cm (1.75–1.88 mS/cm; Table S4). However, the analysis using polynomial equations yielded a different trend (Table S4). In particular, according to these equations, the growth of C. incurvata was significantly affected by temperature, but not by salinity (Table S4; Figure 1C); however, the fit of these models was relatively poor when compared to the above-mentioned top candidates (Table S4).

#### 3.2.2. Nitzschia linearis

^{2}= 0.59; AIC = −133.49; Figure 2B):

^{2}/mS

^{2}) being the salinity effect factor.

^{2}= 0.53; AIC = −130.43; Figure 2A; Table S5) and polynomial equations (Table S5; Figure 2C). Both Equations (ST14) and (ST15) depict a bell-shaped response of growth rate to both temperature and salinity. According to the best Equation (ST15), the thermal breadth for N. linearis becomes narrower with increasing salinity (Table S5). Similar interactions were predicted with the second ranking model (Equation (ST17); AIC = −131.76). The inclusion of this interactive effect improved the predictive potential. This is shown by the lower AIC of Equation (ST15) compared to Equation (ST5), which uses the same functional forms apart from the interaction (Table S5). With Equation (ST15), the maximum growth rate of 0.59 (1/d) was reached at 15.9 °C (14.7–17.1 °C; Table S5) and 2.44 mS/cm (2.02–2.86 mS/cm; Table S5). The significant influence of temperature and salinity was also found with the non-interactive polynomial equation (Equation (4)) as given in Table S5.

#### 3.2.3. Cyclotella meneghiniana

^{2}= 0.80; AIC = −43.90; Figure 3B; Table S6). According to this equation, the maximum growth rate of C. meneghiniana (0.89; 1/d) was reached at 17.3 °C (13.9–20.8 °C) and 0.69 mS/cm (0.58–0.81 mS/cm). Compared to the estimations for these coefficients, high uncertainty is included in the estimate for the salinity effect factor (${k}_{S}$) and the thermal breadth ($w$; Table S6). Different estimates were obtained by combining the Arrhenius equation for temperature responses with various salinity functions (Table S6). According to the best non-interactive equation (Equation (ST25)) (R

^{2}= 0.52; AIC = −34.29; Figure 3A; Table S6), the growth rate of C. meneghiniana increased with increasing temperature. An opposite pattern was found with Equation (ST24), which describes the salinity–growth rate with a Michaelis–Menten function (Table S6). Equation (ST24) yielded a maximum conductivity of 1.38 mS/cm (1.06–1.71 mS/cm; Table S6).

#### 3.2.4. Melosira varians

^{2}= 0.57; AIC = −51.79; Figure 4B; Table S7):

^{2}= 0.21; AIC = −42.61; Figure 4A; Table S7). With this equation, the maximum growth rate of M. varians (0.44; 1/d) was obtained in the medium without addition of NaCl and at 15.5 °C (9.5–21.5 °C) (Figure 4A). The growth of this diatom was linearly inhibited by NaCl at any concentration. In addition, the growth of M. varians ceased at the temperature outside the range of 15.5 ± 9.4 °C (thermal breadth: 18.8 °C; Table S7). In contrast to the influence of temperature and salinity estimated by multiplicative equations, non-significant effects were predicted by polynomial equations (Table S7).

#### 3.2.5. Ulnaria acus

^{2}= 0.74; AIC = −94.70; Figure 5B) with all significant coefficients (Table S8):

^{2}) is the temperature effect factor describing the change in the growth rate, with increasing temperature below or above the optimum (${T}_{opt}$; °C); ${S}_{opt}$ and ${S}_{max}$ (mS/cm) are the optimum and maximum conductivity for the growth, respectively; and a

_{ST}(cm/mS) represents the interactive effect. Also, the second (Equation (ST17), AIC = −90.30) and third (Equation (ST33), AIC = −89.33) ranking model had interactive effects, whilst the best non-interactive model (Equation (ST18)) yielded poor performance (AIC = −88.67; Table S8).

^{2}= 0.67; AIC = −88.67; Table S8; Figure 5A), the conductivity–growth curve exhibits asymmetric responses, i.e., at conductivity above the optimum, the growth rate decreases with a higher slope compared to the change at conductivity below the optimum. Higher uncertainty is included in the estimation with this equation as two effect factors are required to simulate the asymmetric response (Table S8). Statistically significant estimates of all coefficients could be obtained when the number of coefficients was reduced (Equations (ST5), (ST9), (ST10), (ST19), and (ST25); Table S8). However, these equations display different temperature- and conductivity-growth responses. Contrasting with a common bell-shaped response curve (Equations (ST5), (ST10), (ST9), and (ST25)), Equation (ST9) exhibits a linear inhibition of the growth rate by salinization. Equations (ST10) and (ST25) display an exponential increase in the growth rate with increasing temperature (monotonic response), while the other equations show the opposite trend at temperatures above the optimum (bell-shaped response). In contrast to such influences of temperature and salinity predicted with these equations, the growth rate of U. acus was estimated to be not significantly affected based on polynomial equations (Table S8; Figure 5C).

#### 3.2.6. Navicula gregaria

^{2}= 0.86; AIC = −229.82; Figure 6A; Table S9):

^{2}) are the temperature effect factors at temperature below and above the optimum, respectively; and ${k}_{S}$ (cm

^{2}/mS

^{2}) is the salinity effect factor.

^{2}= 0.86; AIC = −219.61; Figure 6B; Table S9) or the non-interactive polynomial Equation (4) (R

^{2}= 0.84; AIC = −222.78; Figure 6C; Table S9). Moreover, all the best formulation forms display a non-interactive norm that is consistent with the data fitting with polynomial equations (Table S9).

#### 3.3. Temperature and Salinity Tolerance of Freshwater Diatoms

#### 3.4. Ambiguity in Empirically Modelling Interactive Effects under Various Environmental Conditions

#### 3.4.1. Diversity of Interaction Types

#### 3.4.2. Ambiguity in Result Interpretation

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Components and Final Concentration in Culture Medium | Stock Solution | Addition Per Litre of Culture Medium |
---|---|---|

1. HEPES (1.00 mM) | 238.10 g/L dH_{2}O | 1.0 mL |

2. Ca(NO_{3})_{2} × 4 H_{2}O (0.21 mM) | 100.00 g/L dH_{2}O | 0.5 mL |

3. MgSO_{4} × 7 H_{2}O (0.203 mM) | 20.00 g/L dH_{2}O | 2.5 mL |

4. K_{2} HPO_{4} × 3 H_{2}O (13.20 µM) | 5.00 g/L dH_{2}O | 0.6 mL |

+ NaNO_{3} (0.35 mM) | 50.00 g/L dH_{2}O | |

+ Na_{2}CO_{3} (0.19 mM) | 32.00 g/L dH_{2}O | |

5. H_{3}BO_{3} (16 µm) | 1.00 g/L dH_{2}O | 1 mL |

6. Vitamin Solution | 1 mL | |

Vitamin B12 (0.15 nM) | 0.20 mg/L dH_{2}O | |

Biotin (4.10 nM) | 1.00 mg/L dH_{2}O | |

Thiamine-HCl (0.30 µM) | 100.00 mg/L dH_{2}O | |

Niacinamide (0.80 nM) | 0.10 mg/L dH_{2}O | |

pH of the Vitamin Solution should be around pH 7 | ||

7. Trace Metals | 1 mL | |

7.1. Preparation of Trace Metal Solution | ||

Na_{2}EDTA × 2 H_{2}O: 4.36 g | ||

FeCl_{3} × 6 H_{2}O: 3.15 g | ||

Dissolve in 1000 mL dH_{2}O, then add 1 mL of Primary Trace Metals each (see below). | ||

Primary Trace Metals are stored frozen as 1 mL aliquots. | ||

7.2. Primary Trace Metals | ||

7.2.1. K_{2}CrO_{4} | 0.194 g/100 mL dH_{2}O | |

7.2.2. CoCl_{2} × 6 H_{2}O | 1.00 g/100 mL dH_{2}O | |

7.2.3. CuSO_{4} × 5 H_{2}O | 0.25 g/100 mL dH_{2}O | |

7.2.4. MnCl_{2} × 4 H_{2}O_{5} | 18.00 g/100 mL dH_{2}O | |

7.2.5. Na_{2}MoO_{4} × 2 H_{2}O | 1.89 g/100 mL dH_{2}O | |

7.2.6. NiSO_{4} × 6 H_{2}O | 0.27 g/100 mL dH_{2}O | |

7.2.7. H_{2}SeO_{3} | 0.13 g /100 mL dH_{2}O | |

7.2.8. Na_{3}VO_{4} | 0.184 g /100 mL dH_{2}O | |

7.2.9. ZnSO_{4} × 7 H_{2}O | 2.20 g/100 mL dH_{2}O | |

8. Na_{2}SiO_{3} × 9 H_{2}O (0.50 mM)–optional | 28.42 g/L dH_{2}O | 5 mL |

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**Figure 1.**Simulation of the growth of Cymbella incurvata based on experimental data according to non-interactive multiplicative Equation (ST13) ((

**A**); R

^{2}= 0.97; AIC = −91.06), interactive multiplicative Equation (ST2) ((

**B**); R

^{2}= 0.88; AIC = −62.08), and polynomial Equation (4) ((

**C**); R

^{2}= 0.82; AIC = −59.42).

**Figure 2.**Simulation of the growth of Nitzschia linearis based on experimental data according to non-interactive multiplicative Equation (ST14) ((

**A**); R

^{2}= 0.53; AIC = −130.43), interactive multiplicative Equation (ST15) ((

**B**); R

^{2}= 0.59; AIC = −133.49), and polynomial Equation (4) ((

**C**); R

^{2}= 0.51; AIC = −128.74).

**Figure 3.**Simulation of the growth of Cyclotella meneghiniana based on experimental data according to non-interactive multiplicative Equation (ST25) ((

**A**); R

^{2}= 0.52; AIC = −34.29), interactive multiplicative Equation (ST15) ((

**B**); R

^{2}= 0.80; AIC = −43.90), and polynomial Equation (4) ((

**C**); R

^{2}= 0.58; AIC = −34.25).

**Figure 4.**Simulation of the growth of Melosira varians based on experimental data according to non-interactive multiplicative Equation (ST9) ((

**A**); R

^{2}= 0.21; AIC = −42.61), interactive multiplicative Equation (ST11) ((

**B**); R

^{2}= 0.57; AIC = −51.79), and polynomial Equation (5) ((

**C**); R

^{2}= 0.45; AIC = −45.37).

**Figure 5.**Simulation of the growth of Ulnaria acus based on experimental data according to non-interactive multiplicative Equation ST18 ((

**A**); R

^{2}= 0.67; AIC = −88.67), interactive multiplicative Equation ST35 ((

**B**); R

^{2}= 0.74; AIC = −94.70), and polynomial Equation (4) ((

**C**); R

^{2}= 0.51; AIC = −80.87).

**Figure 6.**Simulation of the growth of Navicula gregaria based on experimental data according to non-interactive multiplicative Equation (ST26) ((

**A**); R

^{2}= 0.86; AIC = −229.82), interactive multiplicative Equation (ST52) ((

**B**); R

^{2}= 0.83; AIC = −219.61), and polynomial Equation (4) ((

**C**); R

^{2}= 0.84; AIC = −222.78).

Strain | Best Model | Symbol (Unit) | Model Parameter | AIC |
---|---|---|---|---|

Cymbella cf. incurvata | $\mu =\left(c\times {e}^{d\times T}\times \left(1-{\left(\frac{T-z}{w/2}\right)}^{2}\right)\right)\times \left(1-{\left(\frac{S-{S}_{opt}}{{S}_{max}-{S}_{opt}}\right)}^{2}\right)$ | $z$ (°C) | Optimum temperature of the quadratic portion; when $d$ = 0, this value is identical to the optimum temperature of the whole curve (i.e., growth reaches the maximum rate, and the thermal response is symmetric) | −91.06 |

$w$ (°C) | the thermal breadth (i.e., the range over which diatoms grow) | |||

${S}_{opt}$ (mS/cm) | Optimum conductivity | |||

${S}_{max}$ (mS/cm) | Maximum conductivity above which growth ceases | |||

Nitzschia linearis | $\mu ={\mu}_{max}\times \left(1-{\left(\frac{T-{T}_{opt}}{w\times \left(1+{a}_{ST}\times S\right)/2}\right)}^{2}\right)\times {e}^{-{k}_{S}\times {\left(S-{S}_{opt}\right)}^{2}}$ | ${\mu}_{max}$ (1/d) | Maximum growth rate reached at the optimum temperature $({T}_{opt}$; °C) and the optimum conductivity $({S}_{opt}$; mS/cm) | −133.49 |

$w$ (°C) | Thermal breadth that can be affected by salinity depending on the interaction coefficien ${a}_{ST}$ (cm/mS) | |||

${k}_{S}$ (cm^{2}/mS^{2}) | Salinity effect factor | |||

Cyclotella meneghiniana | $\mu ={\mu}_{max}\times \left(1-{\left(\frac{T-{T}_{opt}}{w\times \left(1+{a}_{ST}\times S\right)/2}\right)}^{2}\right)\times {e}^{-{k}_{S}\times {\left(S-{S}_{opt}\right)}^{2}}$ | See above | See above | −43.90 |

Melosira varians | $\mu =\left({\mu}_{20}\times {\theta}^{T-20}\right)\times \left(1-{\left(\frac{S-{S}_{opt}}{{S}_{max}\times \left(1+{a}_{ST}\times T\right)-{S}_{opt}}\right)}^{2}\right)$ | ${\mu}_{20}$ (1/d) | Growth rate at 20 °C | −51.79 |

${S}_{opt}$ (mS/cm) | Optimum conductivity | |||

${S}_{max}$ (mS/cm) | Maximum conductivity above which growth ceases | |||

Ulnaria acus | $\mu ={\mu}_{max}\times {e}^{-{k}_{T}\times {\left(T-{T}_{opt}\times \left(1+{a}_{ST}\times S\right)\right)}^{2}}\times \left(1-{\left(\frac{S-{S}_{opt}}{{S}_{max}-{S}_{opt}}\right)}^{2}\right)$ | ${k}_{T}$ (1/°C^{2}) | Temperature effect factor describing the change in the growth rate with increasing temperature below or above the optimum $({T}_{opt}$; °C) | −94.70 |

${S}_{opt}$ (mS/cm) | Optimum conductivity for growth | |||

${S}_{max}$ (mS/cm) | Maximum conductivity for growth | |||

${a}_{ST}$ (cm/mS) | Represents the interactive effect | |||

Navicula gregaria | $\mu ={\mu}_{max}\times \left\{\begin{array}{c}{e}^{-{k}_{T1}\times {\left(T-{T}_{opt}\right)}^{2}}forT\le {T}_{opt}\\ {e}^{-{k}_{T2}\times {\left(T-{T}_{opt}\right)}^{2}}forT{T}_{opt}\end{array}\right.\times {e}^{-{k}_{S}\times {\left(S-{S}_{opt}\right)}^{2}}$ | ${\mu}_{max}$ (1/d) is | Maximum growth rate obtained at the optimum temperature ${T}_{opt}$ (°C) and the optimum conductivity ${S}_{opt}$ (mS/cm) | −229.82 |

${k}_{T1}$$\mathrm{and}{k}_{T2}$ (1/°C^{2}) | Temperature effect factors at temperature below and above the optimum, respectively | |||

${k}_{S}$ (cm^{2}/mS^{2}) | Salinity effect factor. |

**Table 2.**Temperature and salinity tolerance of the freshwater diatoms estimated with the best equation.

Strain | Temperature Response | Salinity Response | ||||
---|---|---|---|---|---|---|

Optimum Temperature (°C) | Thermal Breadth (°C) | Temperature Tolerance Range (°C) | Optimum Conductivity (mS/cm) | Maximum Conductivity (mS/cm) | Half-Saturation Conductivity (mS/cm) | |

Cymbella cf. incurvate | 15.8 (15.3–16.3) | 14.8 (12. 7–16.9) | 15.5 ± 7.4 | 0.63 (0.35–0.90) | 1.81 (1.75–1.88) | 0.81 (0.01–1.61) |

Nitzschia linearis | 15.9 (14.7–17.1) | 31.8 (24.1–39.4) | 15.9 ± 15.9 | 2.44 (2.02–2.87) | 5.29 (4.84–5.74) | 4.60 (1.17–8.04) |

Cyclotella meneghiniana | 17.3 (13.9–20.8) | 59.6 (0–121.0) | 17.3 ± 29.8 | 0.69 (0.58–0.81) | 1.38 (1.06–1.71) | |

Melosira varians | 1.20 (1.06–1.34) | 6.41 (3.81–8.90) | ||||

Ulnaria acus | 17.1 (12.2–22.0) | 28.1 (11.7–44.4) | 17.1 ± 14.1 | 1.08 (0.79–1.37) | 2.29 (2.02–2.56) | |

Navicular gregaria | 18.4 (17.3–19.4) | 19.3 (17.9–20.6) | 18.4 ± 9.7 | 2.12 (0.29–3.96) | 9.01 (6.65–11.37) |

**Table 3.**Illustration of various types of interactive effects visualized by a comparison of changes in the shape of a reaction norm.

Interaction ID | Type of Interaction | Interactive Effects | Simulation |
---|---|---|---|

A | Stressor 1—modulated slope of the response to stressor 2 | Salinization increased or decreased the slope of the temperature–growth curve, which is accompanied by the narrowed or broadened thermal breadth, respectively. Similar effects can be exerted by temperature increases on the conductivity–growth curve. | |

B | Stressor 1—shifted optimum level of stressor 2 | Salinization increases the optimum temperature of the bell-shaped temperature–growth curve | |

C | Combination of A and B | The temperature–growth curve is modulated both vertically and horizontally by salinization. Salinization narrows the thermal breadth. Both the optimum temperature and the maximum growth rate are lowered. |

**Table 4.**Various types of interactions revealed for freshwater diatoms. The best model for each strain with respect to AIC value marked with double asterisk; models within 6 AIC units from the latter (relative likelihood compared to best model < 5%) with a single asterisk.

Equation | Interaction ID | AIC | |||||
---|---|---|---|---|---|---|---|

Cymbella incurvata | Nitzschia linearis | Cyclotella meneghiniana | Melosia varians | Ulnaria acus | Navicula gregaria | ||

ST7 | A | −52.53 | −33.02 | ||||

ST15 | A | −133.49 ** | −43.90 ** | −173.43 | |||

ST17 | A | −131.76 * | −90.30 * | ||||

ST34 | A | −88.75 * | |||||

ST11 | A | −56.91 | −51.79 ** | ||||

ST21 | A | −120.46 | −32.72 | −42.93 | |||

ST31 | A | −43.25 | |||||

ST22 | A | −117.83 | −78.93 | ||||

ST23 | A | −118.88 | −87.97 * | ||||

ST37 | A | −88.43 * | |||||

ST39 | A | −88.09 * | |||||

ST27 | A | −42.19 | |||||

ST29 | A | −44.10 | |||||

ST28 | A | −42.85 | |||||

ST30 | A | −44.78 | |||||

ST32 | A | −41.02 | |||||

ST36 | A | −87.56 | |||||

ST38 | A | −89.24 * | |||||

ST41 | A | −86.34 | |||||

ST43 | A | −84.32 | |||||

ST51 | A | −211.67 | |||||

ST52 | A | −219.61 ** | |||||

ST33 | B | −89.72 * | |||||

ST35 | B | −94.70 ** | |||||

ST2 | C | −62.08 ** | |||||

ST4 | C | −60.63 * |

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**MDPI and ACS Style**

Le, T.T.Y.; Becker, A.; Kleinschmidt, J.; Mayombo, N.A.S.; Farias, L.; Beszteri, S.; Beszteri, B.
Revealing Interactions between Temperature and Salinity and Their Effects on the Growth of Freshwater Diatoms by Empirical Modelling. *Phycology* **2023**, *3*, 413-435.
https://doi.org/10.3390/phycology3040028

**AMA Style**

Le TTY, Becker A, Kleinschmidt J, Mayombo NAS, Farias L, Beszteri S, Beszteri B.
Revealing Interactions between Temperature and Salinity and Their Effects on the Growth of Freshwater Diatoms by Empirical Modelling. *Phycology*. 2023; 3(4):413-435.
https://doi.org/10.3390/phycology3040028

**Chicago/Turabian Style**

Le, T. T. Yen, Alina Becker, Jana Kleinschmidt, Ntambwe Albert Serge Mayombo, Luan Farias, Sára Beszteri, and Bánk Beszteri.
2023. "Revealing Interactions between Temperature and Salinity and Their Effects on the Growth of Freshwater Diatoms by Empirical Modelling" *Phycology* 3, no. 4: 413-435.
https://doi.org/10.3390/phycology3040028