# Stability of Signaling Pathways during Aging—A Boolean Network Approach

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## Abstract

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## 1. Introduction

## 2. Results

#### 2.1. Experimental Settings

#### 2.1.1. Data Processing

#### 2.1.2. Inferring Boolean Networks from Binarized Time-Series Data

#### 2.2. Stability Measure of Boolean Networks

#### Analysis of Reconstructed Boolean Networks

#### 2.3. Boolean Functions

#### 2.4. Network Stability

## 3. Discussion

## 4. Conclusions

## 5. Materials and Methods

#### 5.1. Data

#### 5.2. Boolean Networks

#### 5.3. Inferring Boolean Networks

#### 5.3.1. Binarization of Time-Series Data

#### 5.3.2. Inferring Boolean Functions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Schematic representation of a Boolean network approach to investigate stability changes in aging signaling networks. First, the time-series data is binarized and reduced using the BASC A algorithm of the R-package BiTrinA [20]. The resulting time-series data is split into two age groups (young (n = 7) and aged (n = 8)) and used to infer Boolean networks using the R-package BoolNet [21]. In the next step, the stability of the resulting Boolean networks is investigated by perturbation experiments. The best-fit algorithm can return a number of different Boolean functions for each gene in the network. From these possible functions 1000 synchronous Boolean networks are created for each age group by randomly drawing one of the inferred Boolean functions for each gene. Next, randomly generated states ($x\left(t\right)$) are perturbed using bitflips (${x}^{\prime}\left(t\right)$). The normalized Hamming distance ($H(x,{x}^{\prime})$) of the successor states $x(t+1)$ and ${x}^{\prime}(t+1)$ and $x(t+5)$ and ${x}^{\prime}(t+5)$ of $x\left(t\right)$ and ${x}^{\prime}\left(t\right)$ is computed. This is repeated for 1000 random states, the successor states of 1000 random states and random attractor state following 1000 random states with random bitflips. Finally, the mean normalized Hamming distance of these 3000 tests for each of the 1000 networks of each phenotype is compared.

**Figure 2.**Network wiring of reconstructed Boolean networks, showing one of the possible combinations of the reconstructed functions which were drawn. (

**A**) shows a network representing the young phenotype and (

**B**) the aged phenotype.

**Figure 3.**(

**A**) shows the mean of the number of inputs of all Boolean functions of the young and aged phenotype Boolean networks as a bar plot. The standard deviations are included as error bars. (

**B**) The boxplot shows the average, normalized Hamming distance between the successor states $t+1$ and $t+5$ of 1000 random states, the successor states of 1000 random states, attractor states following on 1000 random states and their perturbed versions for 1000 random combinations of inferred Boolean functions of the young and aged phenotype (Wilcoxon rank sum test $p<2.2\times {10}^{-16}$ for all robustness comparisons).

**Figure 4.**Schematic representation of one gene in the gene expression data (NCBI GEO ID GSE362). In the experiments muscle samples from 15 healthy humans of different age (21–75) were taken. The samples were arranged in ascending order by age to form a time-series. Samples of all humans between 21–27 years represent the young phenotype. The samples of all humans between 67–75 years represent the aged phenotype.

**Table 1.**Overview over the measured normalized Hamming distances of the young and aged phenotypes starting from random initial states, random successor states and random attractor states compared to perturbed networks after one and after five state transitions.

After One State Transition | After Five State Transitions | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Young Phenotype | Aged Phenotype | Young Phenotype | Aged Phenotype | |||||||||

Min | Max | Mean | Min | Max | Mean | Min | Max | Mean | Min | Max | Mean | |

random initial state | 0.041 | 0.056 | 0.048 | 0.044 | 0.065 | 0.053 | 0.028 | 0.151 | 0.063 | 0.040 | 0.201 | 0.085 |

random successor state | 0.041 | 0.057 | 0.048 | 0.045 | 0.067 | 0.055 | 0.027 | 0.143 | 0.064 | 0.034 | 0.209 | 0.087 |

random attractor state | 0.036 | 0.057 | 0.047 | 0.037 | 0.064 | 0.054 | 0.004 | 0.144 | 0.060 | 0.007 | 0.211 | 0.083 |

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**MDPI and ACS Style**

Schwab, J.D.; Siegle, L.; Kühlwein, S.D.; Kühl, M.; Kestler, H.A.
Stability of Signaling Pathways during Aging—A Boolean Network Approach. *Biology* **2017**, *6*, 46.
https://doi.org/10.3390/biology6040046

**AMA Style**

Schwab JD, Siegle L, Kühlwein SD, Kühl M, Kestler HA.
Stability of Signaling Pathways during Aging—A Boolean Network Approach. *Biology*. 2017; 6(4):46.
https://doi.org/10.3390/biology6040046

**Chicago/Turabian Style**

Schwab, Julian Daniel, Lea Siegle, Silke Daniela Kühlwein, Michael Kühl, and Hans Armin Kestler.
2017. "Stability of Signaling Pathways during Aging—A Boolean Network Approach" *Biology* 6, no. 4: 46.
https://doi.org/10.3390/biology6040046