# Robustness and Complexity of Directed and Weighted Metabolic Hypergraphs

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## Abstract

**:**

## 1. Introduction

## 2. Metabolic Networks as Hypergraphs

#### 2.1. Hypergraphs Definition

#### 2.2. Metabolic Hypergraphs

#### 2.3. Literature Background

#### 2.4. Dataset

## 3. Measurements

#### 3.1. Hypergraph Communicability

#### 3.2. Hypergraph Search Information

## 4. Results and Discussion

#### 4.1. Exploring the E. coli Core Model: A Practical Example

_{2}is produced via oxygenic photosynthesis. In those organisms, O

_{2}should be part of the strongly connected component.

#### 4.2. Robustness and Complexity across Organisms

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. BiGG Models

**Table A1.**BiGG models and their numbers of metabolites and reactions. The directed hypergraph constructed from the model has a number of nodes equal to the number of metabolites and a number of hyperedges bigger than the number of reactions because of the presence of reversible reactions.

Organism | BiGG Model | Metabolites | Reactions | Hyperedges |
---|---|---|---|---|

Saccharomyces cerevisiae S288C | iND750 | 1059 | 1266 | 1702 |

Pseudomonas putida KT2440 | iJN746 | 907 | 1054 | 1415 |

Plasmodium cynomolgi strain B | iAM_Pc455 | 907 | 1074 | 1563 |

e_coli_core | e_coli_core | 72 | 95 | 141 |

Staphylococcus aureus subsp. aureus USA300_TCH1516 | iYS854 | 1335 | 1453 | 1872 |

Mycobacterium tuberculosis H37Rv-1 | iNJ661 | 825 | 1022 | 1293 |

Mycobacterium tuberculosis H37Rv-2 | iEK1008 | 998 | 1224 | 1500 |

Clostridium ljungdahlii DSM 13528 | iHN637 | 698 | 773 | 988 |

Yersinia pestis CO92 | iPC815 | 1552 | 1960 | 2507 |

Shigella dysenteriae Sd197 | iSDY_1059 | 1888 | 2529 | 3172 |

Escherichia coli str. K-12 substr. MG1655 | iJR904 | 761 | 1075 | 1329 |

Lactococcus lactis subsp. cremoris MG1363 | iNF517 | 650 | 730 | 979 |

Helicobacter pylori 26695 | iIT341 | 485 | 554 | 737 |

Homo sapiens | iAB_RBC_283 | 342 | 469 | 645 |

Homo sapiens2 | iAT_PLT_636 | 738 | 1008 | 1455 |

Plasmodium falciparum 3D7 | iAM_Pf480 | 909 | 1083 | 1576 |

Escherichia coli BL21(DE3) | iEC1356_Bl21DE3 | 1918 | 2730 | 3376 |

Synechococcus elongatus PCC 7942 | iJB785 | 768 | 843 | 1064 |

Plasmodium berghei | iAM_Pb448 | 903 | 1067 | 1554 |

Trypanosoma cruzi Dm28c | iIS312 | 606 | 519 | 806 |

Staphylococcus aureus subsp aureus N315 | iSB619 | 655 | 729 | 945 |

Thermotoga maritima MSB8 | iLJ478 | 570 | 652 | 852 |

Methanosarcina barkeri str. Fusaro | iAF692 | 628 | 690 | 900 |

Clostridioides difficile 630 | iCN900 | 885 | 1222 | 1455 |

Plasmodium vivax Sal-1 | iAM_Pv461 | 909 | 1078 | 1570 |

Bacillus subtilis | iYO844 | 990 | 1250 | 1589 |

Synechocystis sp. PCC 6803 | iJN678 | 795 | 862 | 1086 |

Geobacter metallireducens GS-15 | iAF987 | 1109 | 1281 | 1642 |

Acinetobacter baumannii AYE | iCN718 | 888 | 1013 | 1436 |

Salmonella enterica | STM_v1_0 | 1802 | 2528 | 3133 |

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**Figure 1.**An example of a metabolic network mapped into a hypergraph with edge-dependent vertex weight. In (

**a**), we present a small network composed of three reactions and five metabolites. The first reaction ${r}_{1}$ is reversible and is represented with the double arrow. In (

**b**), we show the corresponding stoichiometry matrix. Reactants are negative and products are positive. Note that we need to split the reversible reaction into two irreversible reactions ${r}_{1}^{+}$ and ${r}_{1}^{-}$ to write it in matrix form. This stoichiometry matrix is the weighted incidence matrix of the hypergraph with edge-dependent vertex weights shown in (

**c**). For the sake of visualization, only the hyperedge ${r}_{1}^{+}$ is shown. The hyperedge ${r}_{1}^{-}$ is just the same but with the opposite sign. Note that weights are both positive and negative, meaning that the hypergraph is directed. Indeed, we separate the head and tail of each hyperedge with a dashed line.

**Figure 2.**Access vs. hide information for reactions (

**a**) and metabolites (

**b**). Reactions are colored differently according to the pathway they belong to. Note that the y axis is cut for visualization purposes. Metabolites are divided into compartments; c stands for cytosol compartment and e for extracellular space.

**Figure 3.**Reactions’ average communicability for the e_coli_core model. A simplified Escher map is used as a background to help with the visualization. For a more accurate version of the map, visit [57].

**Figure 4.**The robustness measured as the natural connectivity ${\overline{\lambda}}^{V}$ of 30 different BiGG models. The organisms resistant to antibiotics are shown in different colors. The models are ordered with increasing robustness.

**Figure 5.**The complexity measured as the average search information ${\sigma}^{V}=\frac{{S}^{V}}{{log}_{2}N}$ of 30 different BiGG models. The models are ordered with increasing complexity, and the y axis is zoomed in for visualization purposes.

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**MDPI and ACS Style**

Traversa, P.; Ferraz de Arruda, G.; Vazquez, A.; Moreno, Y.
Robustness and Complexity of Directed and Weighted Metabolic Hypergraphs. *Entropy* **2023**, *25*, 1537.
https://doi.org/10.3390/e25111537

**AMA Style**

Traversa P, Ferraz de Arruda G, Vazquez A, Moreno Y.
Robustness and Complexity of Directed and Weighted Metabolic Hypergraphs. *Entropy*. 2023; 25(11):1537.
https://doi.org/10.3390/e25111537

**Chicago/Turabian Style**

Traversa, Pietro, Guilherme Ferraz de Arruda, Alexei Vazquez, and Yamir Moreno.
2023. "Robustness and Complexity of Directed and Weighted Metabolic Hypergraphs" *Entropy* 25, no. 11: 1537.
https://doi.org/10.3390/e25111537