Modular Random Boolean Networks

📝 Abstract

Random Boolean networks (RBNs) have been a popular model of genetic regulatory networks for more than four decades. However, most RBN studies have been made with random topologies, while real regulatory networks have been found to be modular. In this work, we extend classical RBNs to define modular RBNs. Statistical experiments and analytical results show that modularity has a strong effect on the properties of RBNs. In particular, modular RBNs have more attractors and are closer to criticality when chaotic dynamics would be expected, compared to classical RBNs.

💡 Analysis

Random Boolean networks (RBNs) have been a popular model of genetic regulatory networks for more than four decades. However, most RBN studies have been made with random topologies, while real regulatory networks have been found to be modular. In this work, we extend classical RBNs to define modular RBNs. Statistical experiments and analytical results show that modularity has a strong effect on the properties of RBNs. In particular, modular RBNs have more attractors and are closer to criticality when chaotic dynamics would be expected, compared to classical RBNs.

📄 Content

Modular Random Boolean Networks∗ Rodrigo Poblanno-Balp†‡ Carlos Gershenson‡† † Centro de Ciencias de la Complejidad, UNAM, M´exico ‡ Instituto de Investigaciones en Matem´aticas Aplicadas y en Sistemas, UNAM, M´exico October 25, 2018 Abstract Random Boolean networks (RBNs) have been a popular model of genetic regulatory net- works for more than four decades. However, most RBN studies have been made with random topologies, while real regulatory networks have been found to be modular. In this work, we extend classical RBNs to define modular RBNs. Statistical experiments and analytical results show that modularity has a strong effect on the properties of RBNs. In particular, modu- lar RBNs have more attractors and are closer to criticality when chaotic dynamics would be expected, compared to classical RBNs. 1 Introduction Random Boolean networks (RBN) have been a popular model of genetic regulatory networks (GRNs) [25, 26, 16]. Most studies have been made on RBNs with random topologies. Never- theless, it has been found that topologies affect considerably the properties of RBNs. For example, Aldana studied RBNs with a scale-free topology [1], discovering important differences with random ∗A preliminary version of this work was presented at the ALife XII conference in Odense, Denmark on August 20th, 2010. [36] 1 arXiv:1101.1893v2 [nlin.CG] 18 Mar 2011 topologies. In this work, we study the effect of a modular topology in RBNs. We find that mod- ularity changes the properties of RBNs. Given the fact that real GRNs are modular [38, 8, 37] and most RBN studies have been made over random topologies, it is important to understand the differences between random and modular topologies. Modularity plays an important role in evolution [42, 14, 50], since separable functional systems are found at all scales of biological systems [48]. Modularity allows for changes to occur within modules without propagating to other regions and the combination of modules to explore new functions [13]. Thus, the study of modular RBNs is also relevant for understanding the evolution of GRNs. In the next section, classic RBNs are reviewed, together with their dynamical properties and related work. Section 3 presents our model of modular RBNs. Methods and results of statistical experiments follow in Section 4. The discussion in Section 5 reflects on the results and provides an analytical confirmation. Several future research avenues are mentioned to conclude the paper. 2 Random Boolean Networks Random Boolean Networks (RBNs) [25, 26, 16] consist of N nodes with a Boolean state, representing whether a gene is active (“on” or “one”) or inactive (“off” or “zero”). These states are determined by the states of K nodes which can be considered as inputs or links towards a node. Because of this, RBNs are also known as NK networks or Kauffman models [3]. The states of nodes are decided by lookup tables that specify for every 2K possible combination of input states the future state of the node. RBNs are random in the sense that the connectivity (which nodes are inputs of which, see Figure 1) and functionality (lookup tables of each node, see Table 1) are chosen randomly when a network is generated, although these remain fixed as the network is updated each time step. RBNs are discrete dynamical networks (DDNs), since they have discrete values, number of states, and time [53]. They can also be seen as a generalization of Boolean cellular automata [52, 15], where each node has a different neighborhood and rule. RBNs have 2N possible network states, i.e. all possible combinations of Boolean node states. 2 Figure 1: Topology of an example RBN with N = 5, K = 2. Each node has two inputs that determine its state. Since the topology is randomly generated, a node might have several outputs or none at all. Table 1: Example lookup table to update a node z depending on the state of nodes x and y. Lookup tables include all possible combinations of inputs, i.e. 2K rows. Different nodes will have different lookup tables, i.e. Boolean functions. In this example table, the function is the binary XNOR. x(t) y(t) z(t+1) 0 0 1 0 1 0 1 0 0 1 1 1 3 Transitions between network states determine the state space of the RBN. In classic RBNs, the updating is deterministic and synchronous [15]. Since the number of states of the network is finite and the dynamics are deterministic, sooner or later a state will be repeated in theory (in practice, this can take longer than the age of the universe due to the immense state space). When this occurs, the network has reached an attractor, since the dynamics will remain in that subset of the state space. If the attractor consists of only one state, then it is called a point attractor (similar to a steady state), whereas an attractor consisting of several states is called a cycle attractor (similar to a limit cycle). RBNs are dissipative systems, since each state has only one successor, while having the possibility of having several predecessor states

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