The effects of the finite size of the network on the evolutionary dynamics of a Boolean network are analyzed. In the model considered, Boolean networks evolve via a competition between nodes that punishes those in the majority. It is found that finite size networks evolve in a fundamentally different way than infinitely large networks do. The symmetry of the evolutionary dynamics of infinitely large networks that selects for canalizing Boolean functions is broken in the evolutionary dynamics of finite size networks. In finite size networks there is an additional selection for input inverting Boolean functions that output a value opposite to the majority of input values. These results are revealed through an empirical study of the model that calculates the frequency of occurrence of the different possible Boolean functions. Classes of functions are found to occur with the same frequency. Those classes depend on the symmetry of the evolutionary dynamics and correspond to orbits of the relevant symmetry group. The empirical results match analytic results, determined by utilizing Polya's theorem, for the number of orbits expected in both finite size and infinitely large networks. The reason for the symmetry breaking in the evolutionary dynamics is found to be due to the need for nodes in finite size networks to behave differently in order to cooperate so that the system collectively performs as well as possible. The results suggest that both finite size effects and symmetry are important for understanding the evolution of real-world complex networks, including genetic regulatory networks.
Boolean networks [1,2,3,4] have been studied extensively over the past three decades.
They consist of a directed graph in which the nodes have binary output states that are determined by Boolean functions of the states of the nodes connected to them with directed in-links. They have applications as models of gene regulatory networks as well as models of physical, social, and economic systems. As “coarse-grained” models of genetic networks they aim to capture the essential features of the dynamical behavior of the real networks while simplifying local gene expression to a binary (on/off) state [5]. Recent work has demonstrated that, despite their simplicity, Boolean networks can indeed describe many of the important features of the dynamics of biological genetic circuits [6,7,8,9]. For example, it has been shown that a Boolean network model of the segment polarity gene regulatory networks that control embryonic segmentation in Drosophila melanogaster can reproduce the wild-type gene expression pattern and ectopic patterns due to various mutants [6].
Motivated by the fact that evolution plays a crucial role in forming the regulatory relations among genes, a number of models that evolve the structure and dynamics of Boolean networks have been studied [10,11,12,13,14,15,16,17,18,19,20,21]. These evolutionary Boolean network (EBN) models generally seek to determine the properties of the networks that result from the evolutionary mechanism being considered. For example, many of the studies have focused on the topology of links of the networks that result from evolutionary mechanisms that rewire the links. Other studies have focused on finding evolutionary mechanisms that result in networks that have dynamics that are robust against various types of perturbations, or that result in networks that are in a “critical” state poised between ordered and “chaotic” dynamical behavior.
Ref. [12], and studied subsequently in Refs. [16,17]. In this model the nodes of a Boolean network compete with each other in a variant of the Minority game [22]. The network evolves by changing the Boolean function of the node that loses the game to a new randomly chosen Boolean function. In the principal variant of the model that has been studied, only the Boolean functions of the nodes evolve, not the links. Although, it should be noted that changing the Boolean functions used by the nodes effectively results in a rewiring of the links [23]. In the original paper on the model, it was shown that the network self-organizes to a statistically steady, nontrivial critical state with this evolutionary mechanism. Later it was discovered that the critical state the network evolves to is highly canalized [16].
Canalization [24] is a type of network robustness known to exist in genetic regulatory networks [25,26,27]. It exists when certain expression states of a subset of the genes that regulate the expression of a gene control the expression of the gene regardless of whether or not the other genes that otherwise affect its expression are being expressed. In this case, the states of the subset of regulatory genes that control the expression of a gene are “canalizing” inputs to the gene. Canalization is thought to be an important property of developmental biological systems because it buffers their evolution, allowing greater underlying variation of the genome and its regulatory interactions before some deleterious variation can be expressed phenotypically [28].
The canalized nature of the evolved steady state was demonstrated by preforming an ensemble of similar simulations [16]. Each simulation in the ensemble involved a different random network and began with a different initial condition. After they were run long enough to allow the networks to evolve to a steady state, the average frequency that the various possible Boolean functions occurred was measured and averaged over the ensemble of simulations. For Boolean networks of size N = 999 nodes in which each node has K = 3 inlinks, it was found that the 256 = 2 2 K possible Boolean functions of three variables organize into 14 different classes in which all of the functions in each class occur with approximately the same frequency. It was then found that the various classes of functions mostly could be distinguished by the fraction of canalizing inputs that their functions have. Moreover, the classes whose functions have larger fractions of canalizing inputs, that is, the ones that are more canalizing, occurred with larger frequency.
The reason that there are 14 different classes of K = 3 Boolean functions is due to the symmetry properties of the evolutionary dynamics [23]. The set of Boolean functions of K inputs maps one-to-one onto the set of configurations of the K-dimensional Ising hypercube in which each of the vertices of the hypercube have a binary state. The different classes correspond to the group orbits of the “Zyklenzeiger” group [29] which is the hyper-cubic, or hyper-octahed
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