# Biological Networks Entropies: Examples in Neural Memory Networks, Genetic Regulation Networks and Social Epidemic Networks

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## Abstract

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## 1. Introduction

## 2. Material and Methods

#### 2.1. The Dynamical Tools

#### 2.1.1. Definition of the Notion of Attractor

- (i)
- $A=LoB\left(A\right)$, where the composed operator $LoB$ is obtained by applying the basin operator B and then the limit operator L to the set A
- (ii)
- There is no set A’ containing strictly A, shadow-connected to A and verifying (i)
- (iii)
- There is no set A” strictly contained in A verifying (i) and (ii)

#### 2.1.2. Potential Dynamics

#### 2.1.3. Hamiltonian Dynamics

#### 2.1.4. Potential-Hamiltonian Dynamics

#### 2.2. Examples of Dissipative Energy

#### 2.2.1. Discretization of the Continuous Potential System with Block-Parallel Updating

- interaction matrix W: ${w}_{ij}$ represents the action of the gene j on gene i. W plays the role of a discrete Jacobian matrix. We can consider the associated signed matrix S, with ${s}_{ij}=1$, if ${w}_{ij}>0$, ${s}_{ij}=-1$, if ${w}_{ij}<0$ and ${s}_{ij}=0$, if ${w}_{ij}=0$,
- updating matrix U: ${u}_{ij}=1$, if j is updated before or with i, else ${u}_{ij}=0$,
- trajectory matrix F: ${f}_{bc}=1$, where b and c are two states of E, if and only if $b=f(c,1)$, else ${f}_{bc}=0$.

- in the case of W, the most frequent dependence is called the Hebbian dynamics: if the vectors ${\left\{{x}_{i}\left(s\right)\right\}}_{s<t}$ and ${\left\{{x}_{j}\left(s\right)\right\}}_{s<t}$ have a correlation coefficient ${\rho}_{ij}\left(t\right)$, then ${w}_{ij}(t+1)={w}_{ij}\left(t\right)+h{\rho}_{ij}\left(t\right)$, with $h>0$, corresponding to a reinforcement of the absolute value of the interactions ${w}_{ij}$ having succeeded in inhibiting or activating their target gene i: in case where, for $s<t$, ${x}_{j}\left(s\right)$ remained equal to one, that leads to increase the ${x}_{i}\left(s\right)$’s, if the ${w}_{ij}\left(s\right)$’s were positive, and conversely to decrease the ${x}_{i}\left(s\right)$’s , if the ${w}_{ij}\left(s\right)$’s were negative,
- in the case of U, we can have an autonomous (in time) clock based on the behaviour of r chromatin clock genes having indices $1,\dots ,r$, with three possible behaviours:
- if $y\left(t\right)={\prod}_{i=1,\dots ,r}{x}_{i}\left(t\right)=1$, then the rule (1) is available
- if $y\left(t\right)=0$ and ${\sum}_{s=t,\dots ,t-c}y\left(s\right)>0,x(t+1)=x\left({s}^{*}\right)$, where ${s}^{*}$ is the last time between $t-c$ and t, where $y\left({s}^{*}\right)=1$
- if $y\left(t\right)=0$ and ${\sum}_{s=t,\dots ,t-c}y\left(s\right)=0$, then $x(t+1)=0$ (by exhaustion of the pool of expressed genes).

- (i)
- there are two chromatin clock genes involved in a regulon, i.e., the minimal network having only one negative circuit and one positive circuit reduced to an autocatalytic loop
- (ii)
- there are three morphogens corresponding to the apex and to axillary buds involved as nodes of a 3-switch network fully connected, with all interactions negative except three autocatalytic loops at each vertex
- (iii)
- there are three inhibitory interactions from the auto-catalysed node of the regulon on the nodes of the 3-switch, as indicated on Figure 2 Top left.

#### 2.2.2. Discrete Lyapunov Function (Neural Network)

## 3. Results

#### 3.1. Attractor Entropy and Neural Networks

#### 3.2. Dynamic Entropy and Genetic Networks

#### 3.3. Centrality of Nodes and Social Networks

## 4. Discussion

#### 4.1. The Notion of Entropy Centrality

#### 4.2. The Mathematical Problem of Robustness and the Notion of Global Frustration

**Proposition**

**1.**

**Proposition**

**2.**

## 5. Conclusions and Perspectives

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Top left : Attractor A is invariant for the composed operator LoB. Top right: Shadow trajectory between x and y. Bottom: State of the attractor A returning to A after a perturbation in the attraction basin B(A).

**Figure 2.**Top left: interaction graph of a network made of a 3-switch (system made of three genes fully inhibited except the auto-activations) representing morphogens linked to a regulon representing chromatin clock genes. Top middle: the updating graph corresponding to a block-parallel dynamics ruling the network. Top right: a part of the trajectory graph exhibiting a limit-cycle of period 12 having internally a cycle of period four for the chromatin clock genes. Bottom: updating graphs corresponding successively (from the left to the right) to the parallel, sequential and block-sequential dynamics.

**Figure 3.**Top: Logic neural network with local transition rules ⊕, ∨ and ∧. Bottom: Discrete trajectory graph in the state space $E={\left\{0.1\right\}}^{5}$ with indication of the values of $64L$, where L is the Lyapunov function (in red).

**Figure 4.**Top: asymptotic phase shift between two close isochrons in false colors from 0 to $2\pi $, when an instantaneous perturbation is made on the Wilson-Cowan oscillator. Bottom left: perturbations of same intensity made at two different phases ${\phi}_{1}$ and ${\phi}_{2}$ on the limit-cycle of the Wilson-Cowan oscillator. Bottom right: value of the period of the limit-cycle in false colors depending on the values of the parameters ${\tau}_{x}$ and ${\tau}_{y}$ from 3.5 to 7.

**Figure 5.**The interaction graph of the iron regulatory network, whose interactions can be activatory (+) or inhibitory (−), such as those of microRNAs, like miR-485.

**Figure 6.**Top left: social graphs related to the friendship relationships between pupils (overweight or obese in red, not obese in blue) of a French high school in 5th and 4th classes, corresponding to ages from 11 to 13 years. Top right: analogue graph for corresponding classes in a Tunisian high school in Tunis. Middle: histograms of the number of friends for pupils from French (left) and Tunisian (right) high schools. Bottom left: mean weight (in black, surrounded by the 95%-confidence interval in green) of pupils coming back to an acceptable “normality”, due to a preventive education of 10% of the betweenness central nodes obese, calculated for two sub-populations of tolerance h = 1 (top) and h = 0 (bottom). Bottom right: percentage of obese in these two sub-populations.

**Figure 7.**Comparison between two classical types of centrality in the graph of the Tunisian high school between eigenvector (left) and total degree (right) centralities (node size is proportional to its centrality).

**Figure 8.**Top: representation of the whole graph of the French high school. The size of the nodes corresponds to their centrality in-degree (left), eigenvector (middle) and total degree (right). Bottom: threshold for a therapeutic education leading back to the normal weight state the N obese individuals having the entropic centrality maximum: after stabilization of the social network dynamics, we get all individuals overweight or obese in red (left) if N = 20 and all individuals normal in green (right) if N = 21, this number constituting the success threshold of the education.

**Figure 9.**Description of the attractors of circuits of length 8 for which Boolean local transition functions are either identity or negation.

**Figure 10.**Frustrated pairs of nodes belonging to positive circuits of length 2 in the genetic network controlling the flowering of Arabidopsis thaliana. The network evolves by diminishing the global frustration until the attractor (here a fixed configuration, whose last changes are indicated in red) on which the frustration remains constant.

**Table 1.**Recapitulation of the attractors of the iron regulatory network of Figure 5, for the parallel updating mode, with the list of expressed (in state 1) and not expressed (in state 0) genes and their relative attraction basin sizes, obtained by simulating in parallel updating mode a threshold Boolean network having a constant absolute value of non zero weights equal to one and a threshold equal to zero, from all the possible initial configurations.

Order | Gene | Fixed Point | Fixed Point 2 | Limit-Cycle 1 | Limit-Cycle 2 | |||
---|---|---|---|---|---|---|---|---|

1 | TfR1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |

2 | FPN1a | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

3 | C-Myc | 0 | 1 | 0 | 0 | 0 | 0 | 0 |

4 | Notch | 0 | 0 | 1 | 1 | 1 | 1 | 1 |

5 | GATA-3 | 0 | 0 | 1 | 1 | 1 | 1 | 1 |

6 | IRP | 0 | 0 | 0 | 1 | 0 | 0 | 1 |

7 | Ft | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

8 | Fe | 0 | 0 | 0 | 0 | 1 | 0 | 1 |

9 | MiR-485 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

10 | CiRs-7 anti-sense | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Relative Attraction Basin Size | 512/1024 | 256/1024 | 216/1024 | 40/1024 |

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**MDPI and ACS Style**

Demongeot, J.; Jelassi, M.; Hazgui, H.; Ben Miled, S.; Bellamine Ben Saoud, N.; Taramasco, C. Biological Networks Entropies: Examples in Neural Memory Networks, Genetic Regulation Networks and Social Epidemic Networks. *Entropy* **2018**, *20*, 36.
https://doi.org/10.3390/e20010036

**AMA Style**

Demongeot J, Jelassi M, Hazgui H, Ben Miled S, Bellamine Ben Saoud N, Taramasco C. Biological Networks Entropies: Examples in Neural Memory Networks, Genetic Regulation Networks and Social Epidemic Networks. *Entropy*. 2018; 20(1):36.
https://doi.org/10.3390/e20010036

**Chicago/Turabian Style**

Demongeot, Jacques, Mariem Jelassi, Hana Hazgui, Slimane Ben Miled, Narjes Bellamine Ben Saoud, and Carla Taramasco. 2018. "Biological Networks Entropies: Examples in Neural Memory Networks, Genetic Regulation Networks and Social Epidemic Networks" *Entropy* 20, no. 1: 36.
https://doi.org/10.3390/e20010036