# A Structural Characterisation of the Mitogen-Activated Protein Kinase Network in Cancer

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Complex Network Analysis

**Communities.**Community structure is one of the most studied features of networked systems [19]. Communities in a network are usually described as groups of densely connected nodes with sparse connections to the nodes of other groups.

`greedy_modularity_communities()`. Greedy modularity maximization begins with each node in its own community, then joins the pair of communities that most increases the modularity metric until no such pair exists.

**Modularity**. With the communities identified using greedy modularity maximization, we can additionally compute the resulting modularity of this particular partition of the graph. This will naturally be the maximal modularity value.

`modularity`function to calculate it.

**Small-world coefficient.**Next, we are interested in degrees of small-worldness of the network under study [21]. A small-world network is more clustered with a smaller characteristic path length than degree-preserved random networks. In other words, most nodes can be reached from every other node by a small number of hops or steps. The small-world coefficient that determines how intense this effect is in a network refers to the ratio of clustering coefficient and characteristic path length, which are normalized relative to those of the random networks.

`sigma`function from the

`smallworld`module. Specifically, the original implementation uses a random reference graph, which we deem insufficient, and thus modify the function to use Erdos–Renyi graphs instead.

**Clustering coefficient.**The global clustering coefficient measures the clustering present in the entire network, as opposed to localization around a single node’s neighbourhood.

`triangles`function to calculate t.

**Characteristic path length.**The characteristic path length of a graph $G=(V,E)$, also known as the average path length, is the the average distance between any two pair of nodes in the graph, $l\left(G\right)={\sum}_{u,v\in V}\frac{d(u,v)}{\left|V\right|\left(\right|V|-1)}$, where $d(u,v)$ is the shortest path between u and v. A smaller characteristic path length is linked to a higher efficiency in the transfer of information in the graph [23].

`average_shortest_path_`

`length`function.

**Erdos–Renyi graphs.**A ${G}_{n,p}=(V,E)$ Erdos–Renyi graph [24] is a random graph with $\left|V\right|=n$ vertices, where the presence of each edge $(u,v)\phantom{\rule{0.166667em}{0ex}}\forall u,v\in V$ is determined to be independent and identically distributed with probability p. By the law of large numbers, as the number of nodes in such random graph tends to infinity, the number of generated edges approaches $\left(\right)open="("\; close=")">\genfrac{}{}{0pt}{}{n}{2}$. The likelihood of generating a network G of n vertices and m edges is given by

**Rich-club coefficient.**Afterwards, we turn our attention to the rich-club coefficient of a complex network [25], a feature which describes how well connected a set of hub nodes are to one another and has been shown to influence structural and functional characteristics of networks, including topology, the efficiency of paths and distribution of load [26]. Intuitively, a subnetwork with only rich-club nodes should have more connections than a random network with the same degree and edge distributions.

Algorithm 1 ER graph generation algorithm. | |

Input: n, ${m}_{initial}$, p | |

Output: ${G}_{n,p}$ | |

$V\leftarrow \{{x}_{1},{x}_{2},\cdots ,{x}_{n}\}$ | ▹n nodes |

$E\leftarrow \varnothing $ | |

while$\left|E\right|\ne {m}_{initial}$do | |

$u,v\stackrel{i.i.d\phantom{\rule{4pt}{0ex}}p}{\sim}V$ | |

$E\leftarrow E\cup \left\{\right(u,v\left)\right\}$ | |

end while | |

while ${G}_{n,p}$ not connected do | |

${e}_{discard}\sim E$ | |

$E\leftarrow E\setminus \left\{{e}_{discard}\right\}$ | |

$\mathtt{c}\leftarrow \mathtt{connected}\_\mathtt{components}\left(E\right)$ | ▹ the function is provided from NetworkX |

${c}_{1}\leftarrow \varnothing ,{c}_{2}\leftarrow \varnothing $ | |

while $c1=c2$ do | |

${c}_{1},{c}_{2}\sim c$ | |

end while | |

$u\sim {c}_{1},\phantom{\rule{0.166667em}{0ex}}v\sim {c}_{2}$ | |

$E\leftarrow E\cup \left\{\right(u,v\left)\right\}$ | |

end while |

`rich_club_coefficient`implementation.

#### 2.2. Mapping to Hallmarks of Cancer

## 3. Results

#### 3.1. The MAPK Network in Cancer

#### 3.2. Topological Organisation of the MAPK Network

#### 3.2.1. Degree Distribution

#### 3.2.2. Community Detection

#### 3.2.3. Complex Network Measures

#### 3.3. Relationship with Hallmarks of Cancer

## 4. Discussion and Concluding Remarks

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The Mitogen-Activated Protein Kinase (MAPK) network in cancer, arguably one of the best-established biological networks in cancer, supported by numerous experimental results.

**Figure 3.**Community structure of the MAPK network; the six communities that were detected are highlighted in different colours.

Community | Genes |
---|---|

C1 | PPP2CA, MAP2K1, PI3K, RASA1, RAF, GAB1, FOS, RSK |

MAP3K13, SOS1, ELK1, ERK | |

C2 | GADD45, SMAD2, ATF2, AP1, JUN, TAK1, JNK, MTK1 |

FOXO3, TGFBR1 | |

C3 | P14, RPS6KB1, MAX, MYC, MDM2, PTEN, AKT, P21 |

PDK1, P54 | |

C4 | SPRYD1, FRS2, GRB2, PRKCA, PLCG1, FGFR3, EGFR |

C5 | CREB, P38, MSK, DUSP1, TAOK1, BCL2, ATM |

C6 | EGF, VEGF, HIF1 |

**Table 2.**Complex network measures of the MAPK network. To study the topological organisation, well-established measures were computed. The analysis confirms a highly non-regular organisation with regard to the characteristic path length and the modularity, as compared to regular ER networks and randomised networks of an identical degree distribution. No significant difference is observed with regard to the clustering-coefficient or the small-worldness.

Measure | Value | ER Network Value | Randomized Network Value |
---|---|---|---|

char path | 3.928 | 2.969 (std: 0.073) | 2.941 (std: 0.060) |

modularity | 0.472 | 0.431 (std: 0.023) | 0.425 (std: 0.020) |

clustering coefficient | 0.001 | 0.001 (std: 0.03) | 0.001 (std: 0.0002) |

small-worldness | 0.993 | 0.986 (std: 0.298) | 1.173 (std: 0.305) |

**Table 3.**Rich-club coefficient value for each degree found in the network, normalized using 1000 randomly generated reference networks with the same degree distribution as the original network as reference. The rich-club coefficients at high degrees are lower than expected for reference networks, indicating that the MAPK network’s high-degree nodes avoid connections among each other.

Degree | Normalized Value | Z-Score | p-Value |
---|---|---|---|

2 | 0.981 | −1.325 | 0.092 |

3 | 0.956 | −0.936 | 0.175 |

4 | 0.602 | −2.342 | 0.009 |

5 | 0.189 | −2.754 | 0.003 |

6, 7, 8 | 0.0 | −2.401 | 0.008 |

Hallmark | Shorthand |
---|---|

Self-sufficiency in growth signals | H1 |

Insensitivity to anti-growth signals | H2 |

Evading apoptosis | H3 |

Limitless replicative potential | H4 |

Sustained angiogenesis | H5 |

Tissue invasion and metastasis | H6 |

Reprogramming energy metabolism | H7 |

Evading immune response | H8 |

**Table 5.**Identified communities of the MAPK network in cancer entail network modules in accordance with specific hallmarks of cancer. Individual network communities are associated with subsets of hallmarks. The (relative) prominence of the hallmarks in individual communities is quantified using the MOP value. ⋆ indicates that the probability of observing the frequency of associations with a given hallmark is statistically significant ($p<0.05$ for one-sided binomial test). Overall, the network topology reflects a compartmentalised and modular organisation with regard to the enabling mechanisms of cancer.

Community | Example GO Descriptions | Hallmarks (MOP) |
---|---|---|

C1 | reg. of apoptosis, | H3 (1.5), H6 (4.5⋆), H7 (1.13) |

phosphorus metabolic process, | ||

chemotaxis | ||

C2 | regulation of apoptosis, | H3 (1.5), H7 (2.63⋆) |

pos. reg. of metabolic process | ||

C3 | response to extracellular stimulus, | H2 (6.0), H4 (6.0⋆) |

reg. of cell cycle | ||

reg. of cell cycle arrest | ||

C4 | embryonic organ development, | H1 (4.8⋆), H7 (0.75) |

tissue development, | ||

phosphate metabolic process | ||

C5 | reg. of apoptosis | H3 (3.0⋆), H7 (1.5⋆) |

reg. of programmed cell death, | ||

phosphate metabolic process | ||

C6 | reg. of cell proliferation | H1(1.2), H5 (5.15⋆) |

pos. reg. of angiogenesis | ||

primitive heopoiesis |

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## Share and Cite

**MDPI and ACS Style**

Chatzaroulas, E.; Sliogeris, V.; Victori, P.; Buffa, F.M.; Moschoyiannis, S.; Bauer, R.
A Structural Characterisation of the Mitogen-Activated Protein Kinase Network in Cancer. *Symmetry* **2022**, *14*, 1009.
https://doi.org/10.3390/sym14051009

**AMA Style**

Chatzaroulas E, Sliogeris V, Victori P, Buffa FM, Moschoyiannis S, Bauer R.
A Structural Characterisation of the Mitogen-Activated Protein Kinase Network in Cancer. *Symmetry*. 2022; 14(5):1009.
https://doi.org/10.3390/sym14051009

**Chicago/Turabian Style**

Chatzaroulas, Evangelos, Vytenis Sliogeris, Pedro Victori, Francesca M. Buffa, Sotiris Moschoyiannis, and Roman Bauer.
2022. "A Structural Characterisation of the Mitogen-Activated Protein Kinase Network in Cancer" *Symmetry* 14, no. 5: 1009.
https://doi.org/10.3390/sym14051009