Simple Boolean Network Research Articles

Modular construction of Boolean networks

Boolean networks have been used in a variety of settings, as models for general complex systems as well as models of specific systems in diverse fields, such as biology, engineering, and computer science. Traditionally, their properties as dynamical systems have been studied through simulation studies, due to a lack of mathematical structure. This paper uses a common mathematical technique to identify a class of Boolean networks with a “simple” structure and describes an algorithm to construct arbitrary extensions of a collection of simple Boolean networks. In this way, all Boolean networks can be obtained from a collection of simple Boolean networks as building blocks. The paper furthermore provides a formula for the number of extensions of given simple networks and, in some cases, provides a parametrization of those extensions. This has potential applications to the construction of networks with particular properties, for instance in synthetic biology, and can also be applied to develop efficient control algorithms for Boolean network models.

A Boolean approach to bacterial chemotaxis.

Bacterium such as Escherichia coli (E. coli) show biased Brownian motion in different chemical concentration gradients. This chemical sensitive motility or chemotaxis has gained considerable interest among scientists for some remarkable features such as chemo-sensory dynamic range, adaptation, diffusion and drift. A Boolean model of the whole chemotaxis process has been developed in this manuscript. The response of the circuit is in accordance with the experimental results available in the literature, providing indirect validation of the model. This simple Boolean network (BN) can be easily integrated into the paradigm of modular whole cell modelling. Another crucial application is in designing bio-inspired micro-robots to detect certain spatio-temporal chemical signatures.

On dynamical genetic programming: simple Boolean networks in learning classifier systems

Many representations have been presented to enable the effective evolution of computer programs. Turing was perhaps the first to present a general scheme by which to achieve this end. Significantly, Turing proposed a form of discrete dynamical system and yet dynamical representations remain almost unexplored within conventional genetic programming (GP). This paper presents results from an initial investigation into using simple dynamical GP representations within a learning classifier system. It is shown possible to evolve ensembles of dynamical Boolean function networks to solve versions of the well-known multiplexer problem. Both synchronous and asynchronous systems are considered.

Robustness in complex information systems: The role of information “barriers” in Boolean networks

AbstractIn this supposed “information age,” a high premium is put on the widespread availability of information. Access to as much information as possible is often cited as key to the making of effective decisions. While it would be foolish to deny the central role that information and its flow has in effective decision‐making processes, this chapter explores the equally important role of “barriers” to information flows in the robustness of complex systems. The analysis demonstrates that (for simple Boolean networks at least) a complex system's ability to filter out, i.e., block, certain information flows is essential if it is not to be beholden to every external signal. The reduction of information is as important as the availability of information. © 2009 Wiley Periodicals, Inc. Complexity, 2010

Dynamical behavior and influence of stochastic noise on certain generalized Boolean networks

This study considers a simple Boolean network with N nodes, each node’s state at time t being determined by a certain number of parent nodes. The network is analyzed when the connectivity k is fixed or variable. Making use of a Boolean rule that is a generalization of Rule 22 of elementary cellular automata, a generalized formula for providing the probability of finding a node in state 1 at time t is determined. We show typical behaviors of the iterations, and we study the dynamics of the network through Lyapunov exponents, bifurcation diagrams and fixed point analysis. We conclude that the network may exhibit stability or chaos, depending on the underlying parameters. In general high connectivity is associated with a convergence to zero of the probability of finding a node in state 1 at time t . We also study analytically and numerically the dynamics of the network under a stochastic noise procedure. We show that under a smaller probability of disturbing the nodes through the noise procedure the system tends to exhibit more nodes in the same state. For many parameter combinations there is no critical value of the noise parameter below which the network remains organized and above which it behaves randomly.

Growing a vocabulary

This paper describes how simple Boolean Network models can be adapted to explore the way a vocabulary network might grow. The models described have two main parameters — a New Word parameter, which determines the rate at which new words are added to the lexicon, and an independent New Link parameter, which determines the rate at which links form between words. Delaying the application of the New Link parameter by a small amount allows a core of words to establish itself. With these very simple assumptions, a network structure with some of the features of real lexicons can grow itself. The paper describes how the parameter values interact, and how the size of the initial core affects the way the lexicon grows. The paper also speculates about how a more realistic model of lexical growth might be constructed by making the basic processes in the models slightly more complex.

Modelling the evolution of genetic regulatory networks

An evolutionary model of genetic regulatory networks is developed, based on a model of network encoding and dynamics called the Artificial Genome (AG). This model derives a number of specific genes and their interactions from a string of (initially random) bases in an idealized manner analogous to that employed by natural DNA. The gene expression dynamics are determined by updating the gene network as if it were a simple Boolean network. The generic behaviour of the AG model is investigated in detail. In particular, we explore the characteristic network topologies generated by the model, their dynamical behaviours, and the typical variance of network connectivities and network structures. These properties are demonstrated to agree with a probabilistic analysis of the model, and the typical network structures generated by the model are shown to lie between those of random networks and scale-free networks in terms of their degree distribution. Evolutionary processes are simulated using a genetic algorithm, with selection acting on a range of properties from gene number and degree of connectivity through periodic behaviour to specific patterns of gene expression. The evolvability of increasingly complex patterns of gene expression is examined in detail. When a degree of redundancy is introduced, the average number of generations required to evolve given targets is reduced, but limits on evolution of complex gene expression patterns remain. In addition, cyclic gene expression patterns with periods that are multiples of shorter expression patterns are shown to be inherently easier to evolve than others. Constraints imposed by the template-matching nature of the AG model generate similar biases towards such expression patterns in networks in initial populations, in addition to the somewhat scale-free nature of these networks. The significance of these results on current understanding of biological evolution is discussed.

Asynchronous random Boolean network model based on elementary cellular automata rule 126

This paper considers a simple Boolean network with N nodes, each node's state at time t being determined by a certain number k of parent nodes, which is fixed for all nodes. The nodes, with randomly assigned neighborhoods, are updated based on various asynchronous schemes. We make use of a Boolean rule that is a generalization of rule 126 of elementary cellular automata. We provide formulas for the probability of finding a node in state 1 at a time t for the class of asynchronous random Boolean networks (ARBN) in which only one node is updated at every time step, and for the class of generalized ARBNs (GARBN) in which a random number of nodes can be updated at each time point. We use simulation methods to generate consecutive states of the network for both the real system and the models under the various schemes. The results match well. We study the dynamics of the models through sensitivity of the orbits to initial values, bifurcation diagrams, and fixed point analysis. We show, both theoretically and by example, that the ARBNs generate an ordered behavior regardless of the updating scheme used, whereas the GARBNs have behaviors that range from order to chaos depending on the type of random variable used to determine the number of nodes to be updated and the parameter combinations.

On the properties of cycles of simple Boolean networks

We study two types of simple Boolean networks, namely two loops with a cross-link and one loop with an additional internal link. Such networks occur as relevant components of critical K=2 Kauffman networks. We determine mostly analytically the numbers and lengths of cycles of these networks and find many of the features that have been observed in Kauffman networks. In particular, the mean number and length of cycles can diverge faster than any power law.

Random Boolean network model exhibiting deterministic chaos.

This paper considers a simple Boolean network with N nodes, each node's state at time t being determined by a certain number of parent nodes, which may vary from one node to another. This is an extension of a model studied by Andrecut and Ali [Int. J. Mod. Phys. B 15, 17 (2001)]], who consider the same number of parents for all nodes. We make use of the same Boolean rule as Andrecut and Ali, provide a generalization of the formula for the probability of finding a node in state 1 at a time t, and use simulation methods to generate consecutive states of the network for both the real system and the model. The results match well. We study the dynamics of the model through sensitivity of the orbits to initial values, bifurcation diagrams, and fixed point analysis. We show that the route to chaos is due to a cascade of period-doubling bifurcations which turn into reversed (period-halving) bifurcations for certain combinations of parameter values.

A proposal regarding the mechanism which underlies lineage choice during hematopoietic differentiation

We present a simple Boolean Network model of a genetic regulatory network which can account for the differentiation of hemopoietic cells. In this model while decision choice is stochastic, the probability of differentiating along a specific pathway is determined by the gene expression state of the cell when the signal to differentiate appears. This probability can be altered by the acute and chronic needs of the individual. Mutations disrupt normal maturation. The paper also presents a larger Boolean Network model which can be used to study normal and leukemic stem cell attractor states.