# Modelling and Multi-Criteria Decision Making for Selection of Specific Growth Rate Models of Batch Cultivation by Saccharomyces cerevisiae Yeast for Ethanol Production

*Saccharomyces cerevisiae*)

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Process Specific

(NH_{4})_{2}SO_{4} | 4.50 g/L |

(NH_{4})_{2}HPO_{4} | 1.90 g/L |

MgSO_{4} 7 H_{2}O | 0.34 g/L |

CaCl_{2} 2 H20 | 0.42 g/L |

FeCl_{3} 6 H20 | 1.50 × 10^{−2} g/L |

ZnSO_{4} 7 H20 | 0.90 × 10^{−2} g/L |

MnSO_{4} 2 H20 | 1.05 × 10^{−2} g/L |

CuSO_{4} 5 H20 | 0.24 × 10^{−2} g/L. |

Myo-inositol | 6.00 × 10^{−2} g/L |

Ca-pantothenate | 3.00 × 10^{−2} g/L |

Thiamine HCl | 0.60 × 10^{−2} g/L |

Pyridoxol HCl | 0.15 × 10^{−2} g/L |

Biotin | 0.30 ×10^{−4} g/L |

Temperature | T = 30 °C |

pH | 5.4 |

Gassing flow rate | Q = 275 L/L/h air |

Stirrer speed at start | N = 800 rpm |

Working volume | 1.5 L |

Glucose | 0.5 g/L |

Time of cultivation | t = 12 h. |

#### 2.2. Kinetic Model of the Batch Processes

_{S/X}and Y

_{S/E}= yield coefficients, g/g.

#### 2.3. Growth Rate Models

_{1}= Monod, M

_{2}= Mink, M

_{3}= Tessier, M

_{4}= Moser, M

_{5}= Aiba, M

_{6}= Andrews, M

_{7}= Haldane, M

_{8}= Luong, M

_{9}= Edward, and M

_{10}= Han-Levenspiel [19,20,21,22,23,24].

_{m}= maximum growth rate, h

^{−1}; K

_{S}= Monod saturation constants for cell growth on glucose, g/L; α = Moser constant; K

_{SI}= inhibition constants for cell growth on glucose, g/L; K = constant in Edward model, g/L; S

_{m}= critical inhibitor concentrations above which the reactions stop, g/L; m, n = constants in the Luong and the Han-Levenspiel models.

#### 2.4. Criteria for Evaluation of the Model Parameters

**x**= vector of estimated parameters in specific growth rate models, $x={[{\mu}_{m},{K}_{S},{K}_{SI},\dots ,{Y}_{S/X},{Y}_{S/E}]}^{T}$; N = number of experiment, N = 12; ${X}_{e}({t}_{j}),{S}_{e}({t}_{j}),{E}_{e}({t}_{j})$ = experimental data; ${X}_{m}({t}_{j}),{S}_{m}({t}_{j}),{E}_{m}({t}_{j})$ = simulation data; and t

_{j}= time partition.

#### 2.5. Criteria for Using the PROMETHEE II Method

- ${C}_{1}=J$ criteria of minimization (4), and the following statistical criteria:
- C
_{2}– statistics λ. The criterion C_{2}was compared to the tabular Fisher coefficient (${F}_{T}^{\lambda}$) with a degree of freedom (M, N − 2). In this way, it was checked whether it met the condition: C_{2}> ${F}_{T}^{\lambda}(M,N-2)$, where M = 3; - Relative error for kinetics variables X, S, and E: ${C}_{3}={S}_{L}^{X}$; ${C}_{4}={S}_{L}^{S}$; ${C}_{5}={S}_{L}^{E}$;
- Fisher coefficient (criteria C
_{6}, C_{7}, and C_{8}) for the kinetics variables X, S, and E: C_{6}= F_{X}; C_{7}= F_{S}; C_{8}= F_{E}. Similarly, the obtained values of C_{3}, C_{4}, and C_{5}were compared with the tabular Fischer coefficient, but for degrees of freedom F_{T}(N − 2, M); - Experimental correlation coefficient R
^{2}for kinetics variables X, S, and E: ${C}_{9}={R}_{X}^{2}$; ${C}_{10}={R}_{S}^{2}$; and ${C}_{11}={R}_{E}^{2}$. The obtained values of C_{9}, C_{10}, and C_{11}were compared to the tabular correlation coefficient with a degree of freedom ${R}_{T}^{2}(N-2)$. Complete formulas of statistical criteria are presented in [27].

_{1}–C

_{8}had to be minimized and C

_{9}–C

_{11}had to be maximized. The alternatives in the PROMETHEE II method were the specific growth models from M

_{1}to M

_{10}.

#### 2.6. Principles of the PROMETHEE II Method

#### 2.6.1. The Weight

#### 2.6.2. The Preference Function

**Step 1. Determination of Deviations Based on Pair-Wise Comparisons**

_{j}(a, b) denoted the difference between the evaluation of each a and b of each criterion.

**Step 2. Application of the Preference Function**

_{j}(a, b) denoted the preference of alternative a with regard to alternative b on each criterion, as a function of d

_{j}(a, b).

**Step 3. Calculation of an Overall or Global Index**

_{j}was the weight associated with j-th criterion.

**Step 4. Calculation of the Outranking Flow (The PROMETHEE I Partial Ranking)**

^{+}(a) and φ

^{−}(a) denoted the positive outranking flow and negative outranking flow of each alternative, respectively.

**Step 5. Calculation of the Net Outranking Flow / The PROMETHEE II Complete Ranking**

#### 2.6.3. The Software Packages

## 3. Results and Discussion

#### 3.1. Results from Modelling

_{1}–M

_{10}, were developed. The same program also calculated the criteria C

_{j}. The program was developed on Compaq Visual FORTRAN 90. For solving the nonlinear problem (4), we used BCPOL with double precision from IMSL Library of COMPAQ Visual FORTAN 90 [31].

_{m}∈ [0.1, 1.0], h

^{−1}; K

_{S}∈ [0.1, 30.0], g/L; K

_{SI}∈ [10.0, 700.0], g/L; K ∈ [50.0, 100.0], g/L; S

_{m}∈ [50.0, 150.0], g/L; n & m ∈ [0.5, 2.0]; α ∈ [0.9, 2.0]; and Y

_{S/X}& Y

_{S/E}∈ [0.1, 1.0], g/g.

**A**) for the model of Saccharomyces cerevisiae (1)–(3) and for the different specific growth rate models (M

_{1}–M

_{10}) are shown in Table 3.

**А**):

- The criteria C
_{1}changed in the interval C_{1}∈ [0.527, 0.646] × 10^{−3}; - The criteria C
_{2}changed in the interval C_{2}∈ [135.863, 186.356] - The relative errors (criteria C
_{3}, C_{4}, C_{5}) for every kinetic variable were changed in the interval C_{3,4,5}∈ [0.622, 30.456] × 10^{−2}; - The Fisher coefficients (criteria C
_{6}, C_{7}, C_{8}) were changed in the interval C_{6, 7, 8}∈ [1.000, 1.028]; - The correlation coefficient (C
_{9}–C_{11}) was changed in the interval C_{9, 10, 11}∈ [0.998, 1.000].

_{S/X}and Y

_{S/E}are almost even for all of the investigated models. There are small differences in the fourth sign.

_{2}and C

_{6}–C

_{11}were given from statistical tables [32]. The Fisher coefficient for C

_{2}was ${F}_{T}^{\mathsf{\lambda}}(3,10)=3.71$. For criteria C

_{6}–C

_{8}, tabular Fisher coefficients were F

_{T}(10, 3) = 8.79, and for correlation coefficients C

_{9}–C

_{11}, the tabular value was ${R}_{T}^{2}(10)=0.576$. The C

_{2}> ${F}_{T}^{\lambda}$ = 3.71. The Fisher criteria were (C

_{6}, C

_{7}, C

_{8}) < F

_{T}, and the experimental correlation coefficients were (C

_{9}, C

_{10}, C

_{11}) > ${R}_{T}^{2}$.

_{2}, C

_{6}–C

_{11}).

#### 3.2. Application of PROMETHEE II Method

#### 3.2.1. Selection of the Weight

_{1}are the most important. That is why we chose more weight for them (in %), or w

_{1}≅ 28%. The criteria of C

_{2}–C

_{5}are important statistical variables. They show the statistic λ, and the relative error between experimental and simulated results. Their weight should therefore be higher than that of criteria C

_{6}–C

_{11}. For them, we chose six times the weight, or w

_{j}≅ 14%, j = 2, …, 5.

_{6}, C

_{7}, and C

_{8}had approximately equal values and were very close to their minimum. The same applied for the criteria C

_{9}, C

_{10}, and C

_{11}. They were also close to their maximum. For all of them, we chose smaller weights, w

_{j}≅ 3%, j = 6, ..., 11. The sum of all weights fulfilled the condition: Σw

_{j}≅ 100%, j = 1,…,11.

#### 3.2.2. Selection of the Preference Function

_{1}–C

_{5}are important statistical variables. As their value is close to zero, the best part of the results simulated by the models was selected for the experimental data. Very suitable for them was Type VI: Gaussian criterion (Figure 2).

_{6}, C

_{7}, and C

_{8}criteria were close to their minimum, and the preferred function of Type III: V-shape criterion was the most appropriate for them (Figure 3).

_{9}, C

_{10}, and C

_{11}were close to their maximum. For them, the preferred function was Type V: V-shape with indifference criterion (Figure 4).

#### 3.2.3. The Software Packages

_{6}) was the highest ranked one. The mathematical model of the batch process of Saccharomices cerevisiae with the Andrews model is shown below:

_{m}= 0.410 h

^{−1}; K

_{S}= 7.919, g/L; K

_{SI}= 249.365, g/L; Y

_{S}

_{/X}= 0.173, g/g; Y

_{S}

_{/E}= 0.507, g/g.

_{E}and S

_{E}were the experimental data for biomass and glucose concentration.

## 4. Conclusions

## Conflicts of Interest

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**Figure 1.**Simulation and experimental results for ten specific growth rates: (

**a**) Biomass concentration; (

**b**) Glucose concentration; (

**c**) Ethanol concentration.

**Figure 7.**Experimental and simulation data with Andrews model: (

**a**) Biomass and glucose concentration; (

**b**) Ethanol concentration.

**Figure 8.**Residuals for models M

_{1}–M

_{10}: (

**a**) Residuals for biomass concentration; (

**b**) Residuals for glucose concentration; (

**c**) Residuals for ethanol concentration.

Model | Equation | Model | Equation |
---|---|---|---|

M_{1} | $\mu (S)=\frac{{\mu}_{m}\text{}S}{{K}_{S}+S}$ | M_{6} | $\mu (S)=\frac{{\mu}_{m}\text{}S}{({K}_{S}+S)(1+S/{K}_{SI})}$ |

M_{2} | $\mu (S)=\frac{{\mu}_{m}\text{}{S}^{2}}{{K}_{S}+{S}^{2}}$ | M_{7} | $\mu (S)=\frac{{\mu}_{m}\text{}S}{{K}_{S}+S+{S}^{2}/{K}_{SI}}$ |

M_{3} | $\mu (S)={\mu}_{m}\text{}\left(1-\mathrm{exp}\left(-S/{K}_{SI}\right)\right)$ | M_{8} | $\mu (S)=\frac{{\mu}_{m}\text{}S\text{}}{{K}_{S}+S}\text{}{\left(1-S/{S}_{m}\right)}^{\text{}n}$ |

M_{4} | $\mu (S)=\frac{{\mu}_{m}\text{}{S}^{\alpha}}{{K}_{S}+{S}^{\alpha}},\text{}\alpha 0$ | M_{9} | $\mu (S)=\frac{{\mu}_{m}\text{}S}{{K}_{S}+S+\left(1+S/K\right)\text{}\left({S}^{2}/{K}_{SI}\right)\text{}}$ |

M_{5} | $\mu (S)=\frac{{\mu}_{m}\text{}S}{{K}_{S}+S}\text{}\mathrm{exp}\left(-\frac{S}{{K}_{SI}}\right)$ | M_{10} | $\mu (S)=\frac{{\mu}_{m}\text{}S\text{}{\left(1-S/{S}_{m}\right)}^{n}}{S+{K}_{S}\text{}{\left(1-S/{S}_{m}\right)}^{\text{}m}}$ |

Model | µ_{m} | K_{S} | K_{SI} | K | S_{m} | n | m | α | Y_{S/X} | Y_{S/E} |
---|---|---|---|---|---|---|---|---|---|---|

M_{1} | 0.350 | 6.026 | – | – | – | – | – | – | 0.173 | 0.507 |

M_{2} | 0.294 | 25.448 | – | – | – | – | – | – | 0.175 | 0.506 |

M_{3} | 0.292 | – | 6.907 | – | – | – | – | – | 0.174 | 0.506 |

M_{4} | 0.312 | 10.184 | – | – | – | – | – | 1.422 | 0.174 | 0.507 |

M_{5} | 0.852 | 20.000 | 50.000 | – | – | – | – | – | 0.172 | 0.505 |

M_{6} | 0.410 | 7.919 | 249.365 | – | – | – | – | – | 0.173 | 0.507 |

M_{7} | 0.392 | 7.490 | 287.081 | – | – | – | – | – | 0.173 | 0.507 |

M_{8} | 0.771 | 20.000 | – | – | 107.239 | 1.500 | – | – | 0.174 | 0.506 |

M_{9} | 0.376 | 6.982 | 671.647 | 81.473 | – | – | – | – | 0.173 | 0.507 |

M_{10} | 0.691 | 19.452 | – | – | 69.423 | 1.027 | 0.988 | – | 0.174 | 0.506 |

Model | C_{1} × 10^{−3} | C_{2} | C_{3} × 10^{−2} | C_{4} × 10^{−2} | C_{5} × 10^{−2} | C_{6} | C_{7} | C_{8} | C_{9} | C_{10} | C_{11} |
---|---|---|---|---|---|---|---|---|---|---|---|

M_{1} | 0.646 | 186.356 | 0.886 | 2.439 | 25.365 | 1.001 | 1.002 | 1.028 | 1.000 | 1.000 | 0.998 |

M_{2} | 0.618 | 137.607 | 1.456 | 30.465 | 24.037 | 1.001 | 1.005 | 1.023 | 1.000 | 1.000 | 0.998 |

M_{3} | 0.559 | 135.863 | 0.859 | 11.191 | 24.331 | 1.001 | 1.001 | 1.025 | 1.000 | 1.000 | 0.998 |

M_{4} | 0.583 | 136.275 | 0.767 | 15.136 | 24.667 | 1.000 | 1.001 | 1.026 | 1.000 | 1.000 | 0.998 |

M_{5} | 0.580 | 137.627 | 2.750 | 7.020 | 22.437 | 1.005 | 1.013 | 1.017 | 1.000 | 1.000 | 0.998 |

M_{6} | 0.566 | 136.928 | 0.790 | 8.501 | 24.328 | 1.001 | 1.002 | 1.025 | 1.000 | 1.000 | 0.998 |

M_{7} | 0.603 | 151.831 | 0.622 | 5.076 | 24.886 | 1.001 | 1.002 | 1.026 | 1.000 | 1.000 | 0.998 |

M_{8} | 0.527 | 136.028 | 1.938 | 15.997 | 23.037 | 1.003 | 1.002 | 1.021 | 1.000 | 1.000 | 0.998 |

M_{9} | 0.614 | 158.768 | 0.717 | 4.715 | 25.012 | 1.000 | 1.001 | 1.027 | 1.000 | 1.000 | 0.998 |

M_{10} | 0.529 | 138.508 | 2.505 | 20.325 | 22.429 | 1.003 | 1.000 | 1.020 | 1.000 | 1.000 | 0.999 |

Criteria | Min Max | Type of Criteria | Parameters | Criteria | Min Max | Type of Criteria | Parameters |
---|---|---|---|---|---|---|---|

C_{1} | min | VI | σ_{1} = 0.125 | C_{6,}C_{7}, and C_{8} | min | III | p_{6} = 0.003 |

C_{2} | σ_{2} = 16.607 | p_{7} = 0.008 | |||||

C_{3} | σ_{3} = 0.790 | p_{8} = 0.007 | |||||

C_{4} | σ_{4} = 8.496 | C_{9,}C_{10}, and C_{11} | max | V | q_{j} = 5 × 10^{−5}; p_{j} = 1 × 10^{−3}, j = 9, …11 | ||

C_{5} | σ_{5} = 1.044 |

Rank | Model | φ | φ^{+} | φ^{−} |
---|---|---|---|---|

1 | M_{6} | 0.0996 | 0.1622 | 0.0626 |

2 | M_{3} | 0.0821 | 0.1547 | 0.0726 |

3 | M_{8} | 0.0402 | 0.1933 | 0.1531 |

4 | M_{5} | 0.0303 | 0.2237 | 0.1934 |

5 | M_{4} | 0.0284 | 0.1381 | 0.1097 |

6 | M_{7} | 0.0256 | 0.1505 | 0.1249 |

7 | M_{10} | −0.0066 | 0.2087 | 0.2153 |

8 | M_{9} | −0.0135 | 0.1521 | 0.1656 |

9 | M_{2} | −0.1336 | 0.1078 | 0.2414 |

10 | M_{1} | −0.1523 | 0.1346 | 0.2869 |

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## Share and Cite

**MDPI and ACS Style**

Petrov, M.
Modelling and Multi-Criteria Decision Making for Selection of Specific Growth Rate Models of Batch Cultivation by *Saccharomyces cerevisiae* Yeast for Ethanol Production. *Fermentation* **2019**, *5*, 61.
https://doi.org/10.3390/fermentation5030061

**AMA Style**

Petrov M.
Modelling and Multi-Criteria Decision Making for Selection of Specific Growth Rate Models of Batch Cultivation by *Saccharomyces cerevisiae* Yeast for Ethanol Production. *Fermentation*. 2019; 5(3):61.
https://doi.org/10.3390/fermentation5030061

**Chicago/Turabian Style**

Petrov, Mitko.
2019. "Modelling and Multi-Criteria Decision Making for Selection of Specific Growth Rate Models of Batch Cultivation by *Saccharomyces cerevisiae* Yeast for Ethanol Production" *Fermentation* 5, no. 3: 61.
https://doi.org/10.3390/fermentation5030061