# Evolving Always-Critical Networks

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## Abstract

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## 1. Introduction

^{th}century, or to molecular biology in the years which followed the end of the Second World War. On the other hand, progress is slow when there is an imbalance between the richness of data and that of theories. Nowadays, in fundamental physics, there are various elegant and sophisticated theories that seem far from possible experimental verification in the foreseeable future, while in biology, the reverse is true, since there is an overabundance of data, with very few general principles by which to interpret them. Therefore, such general concepts would be extremely valuable, and we will focus on one interesting candidate, which will be called the “criticality” principle (CP) in this paper.

## 2. Boolean Models of Gene Regulatory Networks

- models of specific genetic circuits in specific organisms, including, e.g., the mammalian cell cycle, the T-cell differentiation, the yeast apoptosis, and others. We will not refer to individual papers here, but to the Cell Collective repository [21] where several models of this kind are available, including, e.g., the mammalian cell cycle, the T-cell differentiation, yeast apoptosis, the Arabidopsis thaliana cell cycle, and many others. These networks are formed on the basis of detailed biological information, which can come from static or dynamic knowledge. One limitation is that our present knowledge of any genetic circuit is limited, so the model might miss important links, or create spurious ones. Moreover, the specific networks comprise a number of variables which ranges from about ten to a few hundred variables, while the overall gene network of the organism is composed of thousands or tens of thousands of variables. In order to take this into account, models of this kind include genes which represent the effects of the rest of the network (and of the external environment), i.e., nodes whose values are given from outside the model (they are not regulated by what happens in the network)
- “generic” models, which do not try to describe any particular organism, but rather, aim at understanding general properties, common to different organisms [2,3,12,22,23]. The questions that can be addressed by these models are mostly related to how different properties scale with the size of the network, or with its topology, the type of gene–gene relationships that are possible, etc. In these models, randomness plays a key role. There may be some constraints based upon the available biological knowledge, but the main way to uncover generic properties is to allow a number of model choices to vary at random, and to collect the results of a large ensemble of cases.

**N**nodes, each having an associated Boolean variable, x

_{i}, i = 1,...,

**N**, and a Boolean function f

_{i}(x

_{i1},...,x

_{ik}) which depends on k other nodes.

**X**= (x

_{1}...x

_{N}) denote the global state of the network. The value at time t + 1 of the i-th node, i.e., x

_{i}(t+1), is determined by the value of f

_{i}(x

_{i1},...,x

_{ik}) computed at the previous time step, i.e., x

_{i}(t+1)=f

_{i}(x

_{i1}(t),...,x

_{ik}(t)

- the topology and the Boolean functions are fixed in time
- each node receives exactly k distinct inputs from k other nodes chosen at random with uniform probability among the remaining n − 1 nodes (avoiding self-loops and multiple connections)
- the Boolean functions are defined by choosing at random, for each of the 2
^{k}entries of the truth table, the value 1 with probability p (called the bias). The Boolean function associated with a node is chosen independently from those of other nodes - the updating is synchronous, so
**X**(t + 1) is a deterministic function of**X**(t)

## 3. Evolving Boolean Networks

## 4. Genetic Algorithms and the Case Studies

#### 4.1. The Genetic Algorithms

#### 4.2. The Case Studies

- the fraction of active genes on attractors is at least a predetermined fraction of the total (“fitness1”)
- ○
- we therefore maximize the function $f\left(x\right)=\{\begin{array}{c}{\mathsf{\Phi}}_{1}\text{}iff\text{}{\mathsf{\Phi}}_{1}\le \theta \\ \theta \text{}iff\text{}{\mathsf{\Phi}}_{1}\theta \end{array}\text{}$. Where not explicitly indicated, θ = 0.8.
- ○
- the fraction Φ
_{1}of active genes is calculated on the states of the attractors found using 10,000 initial conditions

- the fraction of UP genes in an “avalanche” (the fraction of nodes whose activation increases when hit by an avalanche) is at least a predetermined fraction of the total affected nodes (“fitness2”)
- ○
- we maximize the function $f\left(x\right)=\{\begin{array}{c}{\mathsf{\Psi}}_{1}\text{}iff\text{}{\mathsf{\Psi}}_{1}\le \delta \\ \delta \text{}iff\text{}{\mathsf{\Psi}}_{1}\delta \end{array}$. Where not explicitly indicated, δ = 1.0.
- ○
- the fraction Ψ
_{1}of UP genes is calculated based on the total of the affected nodes. For an RBN to have a nonzero fitness, it is required that at least 10 avalanches can be carried out in it. In practice, this constraint has always been overcome, and therefore, should not particularly influence the results.

## 5. Results

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### A.1. Free GA

**Figure A1.**(

**a**) Probability occurrence for each Boolean function, in the case of random and evolved RBN-the bars for each measure indicate the standard deviation of the distribution). Boolean functions are ordered by increasing bias. (

**b**) The same of (

**a**), by using the four canalization classes; F fixed functions; C1 canalizing function in one input channel and C2 in two input channels; R, the XOR and NOT XOR functions. (

**c**) Average population bias, and average fitness. In random BN the number of active genes is on average equal to the bias of the Boolean functions: this property seems maintained by the BN evolved through the free GA. (

**d**) Frequency for each Boolean function in the case of random (red line) and evolved RBN: the Boolean functions are categorized by using their bias. In (

**a**) and (

**b**) parts of the figure the variables deviating from independence at level of 0.02 (stronger evidence) are highlighted by red circles, and those deviating at 0.05 level (weaker evidence) by blue circles.

#### A.2. Balanced GA

**Figure A2.**(

**a**) Frequency of each Boolean function, in the case of random and evolved RBN-the bars for each measure indicate the standard deviation of the distribution. Boolean functions are ordered based on the presence of a zero or a one in correspondence of the input consisting of a double “one”. (

**b**) The same of (

**a**), by using the four canalization classes; F fixed functions; C1 canalizing function in one input channel and C2 in two input channels; R, the XOR and NOT XOR functions. (

**c**) A table illustrating the strategy implemented by the balanced GA-the table shows only the trend of growth or decrease of the Boolean functions whose frequency has significantly changed. (

**d**) Probability occurrence for each Boolean function in the case of random (red line) and evolved RBN: the Boolean functions are categorized by using their bias. In (

**a**) and (

**b**) parts of the figure the variables deviating from independence at level of 0.02 (stronger evidence) are highlighted by red circles, and those deviating at 0.05 level (weaker evidence) by blue circles.

- the frequency of FALSE function is decreased
- ○
- this function tends to introduce zeros-and is therefore to be removed

- the frequency of AND function is increased, and that of its opposite NAND is decreased
- ○
- when both parent nodes are turned on, the downstream node remains turned on, and the opposite has not to happen

- the frequency of COPY A and COPY B is increased, whereas that of their opposite NOT A and NOT B is decreased
- ○
- at steady state in these systems there are many activated nodes: in such situation the downstream nodes copy this high activity-the opposite has not to happen

- the frequency of OR is increased, and that of its opposite NOR is decreased
- ○
- it is sufficient that one of the parent nodes is active to activate-the opposite has not to happen

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**Figure 1.**Average or maximum values of some indices, as the 15 evolutionary processes of each GA version-free and balanced, respectively GAf and Gab-progress. (

**a**) The maximum of the 15 fitness maxima, and the minimum of the 15 minima, for the two versions of the GA, in cases of populations made of RBNs with k = 2 and initial bias = 0.5, which corresponds, on average, to a critical dynamic regime. (

**b**) The average Derrida coefficient of free GA and balanced GA (RBNs with k = 2); note that the balanced GA succeeds in maintaining the critical dynamic regime. Similar considerations can be made in the case of RBNs with k = 3 (data not shown).

**Figure 2.**Average or maximum values of some indices, as the 15 evolutionary processes of each GA version-free and balanced, respectively GAf and Gab-progress. (

**a**) The maximum of the 15 fitness maxima, and the minimum of the 15 minima, for the two versions of the GA, in case of populations made of RBNs with k = 2 and initial bias = 0.5, which corresponds on average to a critical dynamic regime. (

**b**) The average Derrida coefficient of free GA and balanced GA (RBNs with k = 2); note that the balanced GA succeeds in maintaining the critical dynamic regime. Similar considerations can be made in the case of RBNs with k = 3 (data not shown).

**Figure 3.**Average or maximum values of some indices, as the 10 evolutionary processes progress. (

**a**) The maximum of the 10 fitness maxima, and the minimum of the 10 minima, for the two versions of the GA, in case of populations made of RBNs with k = 2 and initial bias = 0.5. (

**b**) The same for RBNs with k = 3 and initial bias = 0.21. It can be noted that in the case of BNs with k = 3, the task is more difficult than with systems with k = 2, with the former systems having a higher connectivity, a fact that leads to many more feedback loops which must be be controlled. (

**c**) The average Derrida coefficient of free GA and balanced GA (RBNs with k = 2). (

**d**) The same for RBNs with k = 3 and initial bias = 0.21. (

**e**) Average population bias of free GA and balanced GA (RBNs with k = 2). (

**f**) The same for RBNs with k = 3 and initial bias = 0.21.

**Table 1.**The general scheme of the used GAs (#G denotes the number of elements in set G). Steps 6 and 7 highlight the differences between free and balanced GA [52].

Step | Step Description | |
---|---|---|

1 | create network topology (it will be the same for all the networks for all generations) | |

2 | create the first population G of networks (Boolean functions are generated at random with bias p) | |

3 | for each network in G, compute the fitness | |

4 | select the set E of the individuals with the highest fitness, which will be passed unaltered to the next generation (elitism) | |

5 | select parents for the individuals of the new generation, with probabilities proportional to their fitness (their number being equal to #G−#E) | |

6 | generate the set G’ by applying single-point crossover to the selected parents in G with a given probability (otherwise parents pass unmodified in the new population) | BALANCED GA. generate the set G’ by applying single-point crossover to the selected parents in G with a given probability (otherwise, parents pass unmodified to the new population). The crossover cutoff point is chosen among those that least change the overall bias of the resulting individuals |

7 | generate the set G” by applying single-point mutation to individuals in G’ with a fixed (small) probability | BALANCED GA. generate the set G” by applying single-point mutation to individuals in G’ with a fixed (small) probability. Add some mutations in order to maintain the children’s bias close to the parents’ bias |

8 | generate the new population of networks G = E∪G” | |

9 | if the termination condition has not been met, return to step 3 | |

10 | End |

**Table 2.**The parameter values used during simulations. ** In order to characterize the systems under analysis, when necessary (searching for attractors, determination of the Derrida parameter), we used 10,000 initial conditions.

System | Parameter | Value |
---|---|---|

GA | number of generations | 100 |

GA | number of RBNs in population | 100 |

GA | crossover probability | 0.7 |

GA | mutation probability per node | 0.02 |

GA | number of individuals in elite E | 3 |

RBN | number of nodes per RBN | 50 |

RBN | number of inputs per node * | 2 (3) |

RBN | average initial bias in initial population ^{1} | 0.5 (0.79) (0.21) |

RBN | number of initial conditions per RBN ^{2} | 10,000 |

^{1}The RBNs can have 2 or 3 input links for each node. In the case of 2 links, we always used bias = 0.5, while with 3 connections, we used bias = 0.21 or bias = 0.79 (both bias in RBNs ensure critical dynamic regimes).

^{2}In order to characterize the systems under analysis, when necessary (searching for attractors, determination of the Derrida parameter), we used 10,000 initial conditions.

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Villani, M.; Magrì, S.; Roli, A.; Serra, R. Evolving Always-Critical Networks. *Life* **2020**, *10*, 22.
https://doi.org/10.3390/life10030022

**AMA Style**

Villani M, Magrì S, Roli A, Serra R. Evolving Always-Critical Networks. *Life*. 2020; 10(3):22.
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**Chicago/Turabian Style**

Villani, Marco, Salvatore Magrì, Andrea Roli, and Roberto Serra. 2020. "Evolving Always-Critical Networks" *Life* 10, no. 3: 22.
https://doi.org/10.3390/life10030022