Canalization in the Critical States of Highly Connected Networks of Competing Boolean Nodes

📝 Abstract

Canalization is a classic concept in Developmental Biology that is thought to be an important feature of evolving systems. In a Boolean network it is a form of network robustness in which a subset of the input signals control the behavior of a node regardless of the remaining input. It has been shown that Boolean networks can become canalized if they evolve through a frustrated competition between nodes. This was demonstrated for large networks in which each node had K=3 inputs. Those networks evolve to a critical steady-state at the boarder of two phases of dynamical behavior. Moreover, the evolution of these networks was shown to be associated with the symmetry of the evolutionary dynamics. We extend these results to the more highly connected K>3 cases and show that similar canalized critical steady states emerge with the same associated dynamical symmetry, but only if the evolutionary dynamics is biased toward homogeneous Boolean functions.

💡 Analysis

Canalization is a classic concept in Developmental Biology that is thought to be an important feature of evolving systems. In a Boolean network it is a form of network robustness in which a subset of the input signals control the behavior of a node regardless of the remaining input. It has been shown that Boolean networks can become canalized if they evolve through a frustrated competition between nodes. This was demonstrated for large networks in which each node had K=3 inputs. Those networks evolve to a critical steady-state at the boarder of two phases of dynamical behavior. Moreover, the evolution of these networks was shown to be associated with the symmetry of the evolutionary dynamics. We extend these results to the more highly connected K>3 cases and show that similar canalized critical steady states emerge with the same associated dynamical symmetry, but only if the evolutionary dynamics is biased toward homogeneous Boolean functions.

📄 Content

arXiv:1106.2574v1 [q-bio.MN] 13 Jun 2011 Canalization in the Critical States of Highly Connected Networks of Competing Boolean Nodes Matthew D. Reichl1, 2 and Kevin E. Bassler1, 2 1Department of Physics, University of Houston, Houston, Texas 77204-5005, USA 2Texas Center for Superconductivity, University of Houston, Houston, Texas 77204-5002, USA (Dated: June 5, 2018) Canalization is a classic concept in Developmental Biology that is thought to be an important feature of evolving systems. In a Boolean network it is a form of network robustness in which a subset of the input signals control the behavior of a node regardless of the remaining input. It has been shown that Boolean networks can become canalized if they evolve through a frustrated competition between nodes. This was demonstrated for large networks in which each node had K = 3 inputs. Those networks evolve to a critical steady-state at the boarder of two phases of dynamical behavior. Moreover, the evolution of these networks was shown to be associated with the symmetry of the evolutionary dynamics. We extend these results to the more highly connected K > 3 cases and show that similar canalized critical steady states emerge with the same associated dynamical symmetry, but only if the evolutionary dynamics is biased toward homogeneous Boolean functions. PACS numbers: 89.75.Fb, 87.23.Kg, 05.65.+b, 89.75.Hc I. INTRODUCTION Boolean networks were originally proposed as models of genetic regulatory networks and are now widely used as models of self-regulatory behavior in biological, physical, social, and engineered systems [1–4]. They are designed to capture essential features of the complex dynamics of real networks by “coarse-graining” that assumes that dy- namical state of each node is Boolean, or simply on/off [5]. For example, in a model of a genetic regulatory sys- tem each node corresponds to a gene and its Boolean dynamical state refers to whether or not the gene is cur- rently being expressed. The regulatory interactions be- tween nodes are described by a directed graph in which the Boolean (output) state of each node is determined by a function of the states of the nodes connected to it with directed in-links. It has been shown that, despite their simplicity, Boolean networks capture many of the important features of the dynamics of real self-regulating networks, including biological genetic circuits [6–9]. Perhaps the most notable feature of Boolean networks is that they have two distinct phases of dynamical behav- ior. These two phases are called “frozen” and “chaotic”, and in random Boolean networks there is a continuous phase transition between them[10–12]. The two phases can be distinguished by how a perturbation in the net- work spreads with time: in the frozen phase a perturba- tion decays with time, while in the chaotic phase a per- turbation grows with time [10, 13]. In networks in which the states of the nodes are updated synchronously, the two phases can also be distinguished by the distribution of network’s attractor periods [3, 14]. When the updates are synchronous the system always settles onto a dynam- ical attractor of finite period. In the frozen phase the distribution of attractor periods is sharply peaked with a mean that is independent of the number of nodes N. In the chaotic phase the distribution of attractor periods is also sharply peaked, but with a mean that grows as exp(N). In the “critical” state, at the boundary between the two phases, the distribution of attractor periods is broad, described by a power-law [14–17]. Many naturally occurring, as well as engineered, self- regulating network systems develop through some sort of evolutionary process. Motivated by this fact, a num- ber of models that evolve the structure and dynamics of Boolean networks have been studied [17–32]. These evo- lutionary Boolean network (EBN) models generally seek to determine the properties of networks that result from the evolutionary mechanism being considered. For exam- ple, some studies have explored evolutionary mechanisms that result in networks that have dynamics that are ro- bust again various types of perturbations, or that result in networks that are in a critical state. One example of an EBN is the model of competing Boolean nodes first introduced in Ref. [17], and later studied in Refs. [23, 24, 26]. In this model, the Boolean functions of the nodes evolve through a frustrated com- petition for limited resources between nodes that is a variant of the Minority game [33]. In the original pa- per on the model, it was shown that the network self- organizes to a nontrivial critical state with this evolu- tionary mechanism. Later it was discovered that this critical state is highly canalized [23]. Canalization [34] is a type of network robustness, and is a classic idea in de- velopmental biology. Recently, experiments have demon- strated its existence in genetic regulatory networks [35– 37]. It occurs when certain expression states of o

This content is AI-processed based on ArXiv data.