# Multi-Objective Optimization of Microalgae Metabolism: An Evolutive Algorithm Based on FBA

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## Abstract

**:**

## 1. Introduction

## 2. Results

## 3. Discussion

## 4. Materials and Methods

#### 4.1. Proposed Evolutionary Approach Based on NSGAII

#### 4.1.1. Multi-Objective Optimization Model for FBA

#### 4.1.2. Evolutionary Approach for MOFBA

#### 4.2. Case of Study: Metabolic Network of Chlamydomonas reinhardtii

#### 4.3. Experimental Design

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ACCOA | Acetyl-coenzyme A |

CIT | Citrato |

OAA | Oxaloacetate |

PEP | Phosphoenol pyruvate |

T3P | Dihydroxyacetone phosphate and 3-phosphoglycerate |

PYR | Pyruvate |

PROT | Protein |

F6P | Fructose-6-phosphate |

G6P | Glucose-6-phosphate |

CARB | Carbohydrates |

FBA | Fluxes balance analysis |

CO_{2} | Carbon dioxide |

NSGAII | Genetic Algorithm of Non-Dominated Classification |

MO-FBA | Multi-objective flux balance analysis |

MO-FVA | Multi-objective flux variability analysis |

MOEAs | Multi-objective Evolutionary Algorithms |

SBX | Simulated Binary Crossover |

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**Figure 2.**Pareto approximation for configuration ${C}_{0}$ with respect to the objectives $({v}_{10},{v}_{14},{v}_{18})$.

**Figure 3.**Pareto approximation for configuration ${C}_{0}$ with respect to the plane formed by objectives $({v}_{10},{v}_{14})$.

**Figure 4.**Pareto approximation for configuration ${C}_{0}$ with respect to the plane formed by objectives $({v}_{10},{v}_{18})$.

**Figure 5.**Pareto approximation for configuration ${C}_{0}$ with respect to the plane formed by objectives $({v}_{14},{v}_{18})$.

**Figure 6.**The distribution fluxes of objective function. Subfigure (

**a**) shows the optimization of the ${v}_{14}$ flux using FBA; subfigures (

**b**–

**d**) correspond to different fluxes distributions obtained from NSGAII optimizing ${v}_{14}$, ${v}_{10}$ and ${v}_{18}$ simultaneously.

Euclidean Distance to Ideal Point | |||||
---|---|---|---|---|---|

Config. | ${\mathcal{Q}}^{\mathbf{NSGAII}}$ | ${\mathcal{Q}}_{{\mathit{O}\mathit{b}\mathit{j}}_{\mathbf{1}}}^{\mathbf{FBA}}$ | ${\mathcal{Q}}_{{\mathit{O}\mathit{b}\mathit{j}}_{\mathbf{2}}}^{\mathbf{FBA}}$ | ${\mathcal{Q}}_{{\mathit{O}\mathit{b}\mathit{j}}_{\mathbf{3}}}^{\mathbf{FBA}}$ | $|{\mathit{F}}_{\mathbf{0}}|$ |

${C}_{0}$ | 7.16 | 10.12 | 10 | 10 | 349 |

${C}_{1}$ | 8.07 | 10.12 | 10 | 10 | 158 |

${C}_{2}$ | 11.56 | 14.23 | 14.14 | 14.14 | 2501 |

${C}_{3}$ | 11.56 | 14.23 | 14.14 | 14.14 | 1701 |

${C}_{4}$ | 7.12 | 10.12 | 10 | 10 | 217 |

${C}_{5}$ | 8.34 | 10.12 | 10 | 10 | 359 |

${C}_{6}$ | 8.19 | 14.23 | 10 | 10 | 617 |

${C}_{7}$ | 10 | 10.12 | 14.14 | 10 | 53 |

${C}_{8}$ | 10 | 10.12 | 14.14 | 10 | 68 |

${C}_{9}$ | 10 | 14.27 | 10 | 10 | 147 |

${C}_{10}$ | 8.25 | 14.31 | 10 | 10 | 397 |

${C}_{11}$ | 8.24 | 14.31 | 10 | 10 | 279 |

${C}_{12}$ | 7.13 | 10.12 | 10 | 10 | 218 |

${C}_{13}$ | 0 | 0 | 0 | 0 | 125 |

${C}_{14}$ | 8.16 | 10 | 10 | 14.14 | 1821 |

${C}_{15}$ | 8.16 | 10 | 10 | 14.14 | 1646 |

${C}_{16}$ | 0 | 0 | 0 | 0 | 189 |

${C}_{17}$ | 0 | 0 | 0 | 0 | 216 |

${C}_{18}$ | 9.98 | 10 | 10 | 10 | 160 |

${C}_{19}$ | 0 | 0 | 0 | 0 | 88 |

${C}_{20}$ | 0 | 0 | 0 | 0 | 171 |

${C}_{21}$ | 8.20 | 10.06 | 10.06 | 10 | 202 |

${C}_{22}$ | 7.13 | 10.12 | 10.12 | 10 | 325 |

${C}_{23}$ | 7.13 | 10.12 | 10.12 | 10 | 465 |

${C}_{24}$ | 0 | 0.06 | 0.06 | 0 | 81 |

${C}_{25}$ | 0 | 0 | 0 | 0 | 125 |

${C}_{26}$ | 8.17 | 10 | 10 | 14.14 | 1597 |

${C}_{27}$ | 8.18 | 10 | 10 | 14.14 | 1199 |

${C}_{28}$ | 0 | 0 | 0 | 0 | 175 |

${C}_{29}$ | 0 | 0 | 0 | 0 | 280 |

${C}_{30}$ | 0 | 0 | 0 | 0 | 391 |

${C}_{31}$ | 8.27 | 10 | 10 | 10 | 276 |

${C}_{32}$ | 7.17 | 10 | 10 | 10 | 261 |

${C}_{33}$ | 0 | 0 | 0 | 0 | 122 |

${C}_{34}$ | 0 | 0 | 0 | 0 | 55 |

${C}_{35}$ | 0 | 0 | 0 | 0 | 90 |

${C}_{36}$ | 0 | 0 | 0 | 0 | 25 |

${\mathcal{Q}}_{{\mathit{O}\mathit{b}\mathit{j}}_{1}}^{\mathbf{FBA}}$ | ${\mathcal{Q}}_{{\mathit{O}\mathit{b}\mathit{j}}_{2}}^{\mathbf{FBA}}$ | ${\mathcal{Q}}_{{\mathit{O}\mathit{b}\mathit{j}}_{3}}^{\mathbf{FBA}}$ | ${\mathcal{Q}}_{{\mathit{O}\mathit{b}\mathit{j}}_{1}}^{\mathbf{NSGAII}}$ | ${\mathcal{Q}}_{{\mathit{O}\mathit{b}\mathit{j}}_{2}}^{\mathbf{NSGAII}}$ | ${\mathcal{Q}}_{{\mathit{O}\mathit{b}\mathit{j}}_{3}}^{\mathbf{NSGAII}}$ | ${\mathcal{Q}}_{\mathbf{Euclid}}^{\mathbf{NSGAII}}$ | ||
---|---|---|---|---|---|---|---|---|

BY OBJECTIVE | ${v}_{10}$ | 0 | 10 | 0 | 0 | 9.99 | 3.93 | 5.099 |

${v}_{14}$ | 10.12 | 0 | 10 | 10.11 | 0.045 | 6.68 | 5.019 | |

${v}_{18}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

EUCLIDEAN | 10.12 | 10 | 10 | 10.119 | 10 | 7.49 | 7.15 | |

OBJECTIVE | ${v}_{10}$ | 10 | 0 | 10 | 10 | $3.8\times {10}^{-9}$ | 6.60 | 4.90 |

${v}_{14}$ | 0.48 | 10.6 | 0.6 | 0.48 | 10.55 | 3.91 | 5.58 | |

${v}_{18}$ | 10 | 10 | 10 | 10 | 10 | 10 | 10 | |

FLUXES | ${v}_{1}$ | 10 | 10 | 10 | 10 | 10 | 10 | 10 |

${v}_{2}$ | 10 | 10 | 10 | 10 | 10 | 10 | 10 | |

${v}_{3}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

${v}_{4}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

${v}_{5}$ | 10 | 10 | 10 | 10 | 10 | 10 | 10 | |

${v}_{6}$ | 10 | 10 | 10 | 10 | 10 | 10 | 10 | |

${v}_{7}$ | 0 | 10 | 0 | 0 | 9.99 | 3.93 | 5.099 | |

${v}_{8}$ | 10 | 0 | 10 | 10 | $3.8\times {10}^{-9}$ | 6.60 | 4.90 | |

${v}_{9}$ | 10 | 0 | 10 | 10 | $3.8\times {10}^{-9}$ | 6.60 | 4.90 | |

${v}_{10}$ | 10 | 0 | 10 | 10 | $3.8\times {10}^{-9}$ | 6.60 | 4.90 | |

${v}_{11}$ | 0.24 | 10.3 | 0.3 | 0.24 | 10.27 | 3.65 | 5.34 | |

${v}_{12}$ | 0.48 | 10.6 | 0.6 | 0.48 | 10.55 | 3.91 | 5.58 | |

${v}_{13}$ | 0.48 | 10.6 | 0.6 | 0.48 | 10.55 | 3.91 | 5.58 | |

${v}_{14}$ | 0.48 | 10.6 | 0.6 | 0.48 | 10.55 | 3.91 | 5.58 | |

${v}_{15}$ | 0.24 | 0.3 | 0.3 | 0.24 | 0.27 | 0.26 | 0.24 | |

${v}_{16}$ | 0.24 | 0.3 | 0.3 | 0.24 | 0.27 | 0.26 | 0.24 | |

${v}_{17}$ | 0.24 | 0.3 | 0.3 | 0.24 | 0.27 | 0.26 | 0.24 | |

${v}_{18}$ | 10 | 10 | 10 | 10 | 10 | 10 | 10 |

Encode Solution w | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Decision Variables | Objectives | ||||||||||

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

$L{B}_{{v}_{1}^{M}}$ | ${\Delta}_{{v}_{1}^{M}}$ | $L{B}_{{v}_{2}^{M}}$ | ${\Delta}_{{v}_{2}^{M}}$ | $L{B}_{{v}_{3}^{M}}$ | ${\Delta}_{{v}_{3}^{M}}$ | $L{B}_{{v}_{1}^{b}}$ | ${\Delta}_{{v}_{1}^{b}}$ | $L{B}_{{v}_{2}^{b}}$ | ${\Delta}_{{v}_{2}^{b}}$ | $L{B}_{{v}_{3}^{b}}$ | ${\Delta}_{{v}_{3}^{b}}$ |

5 | 0.50 | 0 | 0.75 | 2 | 0.80 | 10 | 1.00 | 2.5 | 0.50 | 7.5 | 0.10 |

**Table 4.**Reactions derived from the metabolic network Figure 7.

Name | Formula | Name | Formula |
---|---|---|---|

v_{1} : | –> acetate | v_{10} : | PROT –> |

v_{2} : | acetate –> ACCOA | v_{11} : | T3P <=> F6P |

v_{3} : | acetate –> CIT | v_{12} : | F6P <=> G6P |

v_{4} : | CIT –> | v_{13} : | G6P –> CARB |

v_{5} : | ACCOA –> OAA | v_{14} : | CARB –> |

v_{6} : | OAA <=> PEP + CO_{2} | v_{15} : | E4P + X5P –> F6P + T3P |

v_{7} : | PEP <=> T3P | v_{16} : | –> E4P |

v_{8} : | PEP –> PYR | v_{17} : | –> X5P |

v_{9} : | PYR –> PROT | v_{18} : | CO_{2}–> |

**Table 5.**Experiment’s additional configurations of the reactions fluxes apart from ${C}_{0}=\{{v}_{10},{v}_{14},{v}_{18}\}$.

No. | Configuration | No. | Configuration | No. | Configuration |
---|---|---|---|---|---|

${C}_{1}$ | $\{{v}_{10},{v}_{14},{v}_{2}\}$ | ${C}_{13}$ | $\{{v}_{10},{v}_{18},{v}_{2}\}$ | ${C}_{25}$ | $\{{v}_{14},{v}_{18},{v}_{2}\}$ |

${C}_{2}$ | $\{{v}_{10},{v}_{14},{v}_{3}\}$ | ${C}_{14}$ | $\{{v}_{10},{v}_{18},{v}_{3}\}$ | ${C}_{26}$ | $\{{v}_{14},{v}_{18},{v}_{2}\}$ |

${C}_{3}$ | $\{{v}_{10},{v}_{14},{v}_{4}\}$ | ${C}_{15}$ | $\{{v}_{10},{v}_{18},{v}_{4}\}$ | ${C}_{27}$ | $\{{v}_{14},{v}_{18},{v}_{2}\}$ |

${C}_{4}$ | $\{{v}_{10},{v}_{14},{v}_{5}\}$ | ${C}_{16}$ | $\{{v}_{10},{v}_{18},{v}_{5}\}$ | ${C}_{28}$ | $\{{v}_{14},{v}_{18},{v}_{2}\}$ |

${C}_{5}$ | $\{{v}_{10},{v}_{14},{v}_{6}\}$ | ${C}_{17}$ | $\{{v}_{10},{v}_{18},{v}_{6}\}$ | ${C}_{29}$ | $\{{v}_{14},{v}_{18},{v}_{2}\}$ |

${C}_{6}$ | $\{{v}_{10},{v}_{14},{v}_{7}\}$ | ${C}_{18}$ | $\{{v}_{10},{v}_{18},{v}_{7}\}$ | ${C}_{30}$ | $\{{v}_{14},{v}_{18},{v}_{2}\}$ |

${C}_{7}$ | $\{{v}_{10},{v}_{14},{v}_{8}\}$ | ${C}_{19}$ | $\{{v}_{10},{v}_{18},{v}_{8}\}$ | ${C}_{31}$ | $\{{v}_{14},{v}_{18},{v}_{2}\}$ |

${C}_{8}$ | $\{{v}_{10},{v}_{14},{v}_{9}\}$ | ${C}_{20}$ | $\{{v}_{10},{v}_{18},{v}_{9}\}$ | ${C}_{32}$ | $\{{v}_{14},{v}_{18},{v}_{2}\}$ |

${C}_{9}$ | $\{{v}_{10},{v}_{14},{v}_{11}\}$ | ${C}_{21}$ | $\{{v}_{10},{v}_{18},{v}_{11}\}$ | ${C}_{33}$ | $\{{v}_{14},{v}_{18},{v}_{2}\}$ |

${C}_{10}$ | $\{{v}_{10},{v}_{14},{v}_{12}\}$ | ${C}_{22}$ | $\{{v}_{10},{v}_{18},{v}_{12}\}$ | ${C}_{34}$ | $\{{v}_{14},{v}_{18},{v}_{2}\}$ |

${C}_{11}$ | $\{{v}_{10},{v}_{14},{v}_{13}\}$ | ${C}_{23}$ | $\{{v}_{10},{v}_{18},{v}_{13}\}$ | ${C}_{35}$ | $\{{v}_{14},{v}_{18},{v}_{2}\}$ |

${C}_{12}$ | $\{{v}_{10},{v}_{14},{v}_{15}\}$ | ${C}_{24}$ | $\{{v}_{10},{v}_{18},{v}_{15}\}$ | ${C}_{36}$ | $\{{v}_{14},{v}_{18},{v}_{2}\}$ |

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**MDPI and ACS Style**

Briones-Baez, M.F.; Aguilera-Vazquez, L.; Rangel-Valdez, N.; Martinez-Salazar, A.L.; Zuñiga, C.
Multi-Objective Optimization of Microalgae Metabolism: An Evolutive Algorithm Based on FBA. *Metabolites* **2022**, *12*, 603.
https://doi.org/10.3390/metabo12070603

**AMA Style**

Briones-Baez MF, Aguilera-Vazquez L, Rangel-Valdez N, Martinez-Salazar AL, Zuñiga C.
Multi-Objective Optimization of Microalgae Metabolism: An Evolutive Algorithm Based on FBA. *Metabolites*. 2022; 12(7):603.
https://doi.org/10.3390/metabo12070603

**Chicago/Turabian Style**

Briones-Baez, Monica Fabiola, Luciano Aguilera-Vazquez, Nelson Rangel-Valdez, Ana Lidia Martinez-Salazar, and Cristal Zuñiga.
2022. "Multi-Objective Optimization of Microalgae Metabolism: An Evolutive Algorithm Based on FBA" *Metabolites* 12, no. 7: 603.
https://doi.org/10.3390/metabo12070603