Boolean networks have been successfully used in modelling gene regulatory networks. In this paper we propose a reduction method that reduces the complexity of a Boolean network but keeps dynamical properties and topological features and hence it makes the analysis easier; as a result, it allows for a better understanding of the role of network topology on the dynamics. In particular, we use the reduction method to study steady states of Boolean models.
Models of biochemical networks are frequently high-dimensional and complex. Reduction methods that preserve important dynamical properties are therefore essential in their study. Interactions between the nodes in such networks are frequently modeled using a Hill function, $x^n/(J^n+x^n)$. Reduced ODEs and Boolean networks have been studied extensively when the exponent $n$ is large. However, the case of small constant $J$ appears in practice, but is not well understood. In this paper we provide a mathematical analysis of this limit, and show that a reduction to a set of piecewise linear ODEs a…
In the applications of Boolean networks to modeling biological systems, an important computational problem is the detection of the fixed points of these networks. This is an NP-complete problem in general. There have been various attempts to develop algorithms to address the computation need for large size Boolean networks. The existing methods are usually based on known algorithms and thus limited to the situations where these known algorithms can apply. In this paper, we propose a novel approach to this problem. We show that any system of Boolean equations is equivalent to one Boolean equati…
We consider two approaches to modelling of cell metabolism of 6-mercaptopurine, which is one of the important chemotherapy drugs used for treating of acute lymphocytic leukemia: kinetic ordinary differential equations and random Boolean networks, and analyse their interplay with respect to taking into account ATP concentration as a key parameter of switching between different pathways. It is shown that Boolean networks, which allow for avoiding complexity of general kinetic modelling, preserve an opportunity to the reproduction of the principal switching mechanism. To keep a detailed quantitat…
Many biological systems, such as metabolic pathways, exhibit bistability behavior: these biological systems exhibit two distinct stable states with switching between the two stable states controlled by certain conditions. Since understanding bistability is key for understanding these biological systems, mathematical modeling of the bistability phenomenon has been at the focus of researches in quantitative and system biology. Recent study shows that Boolean networks offer relative simple mathematical models that are capable of capturing these essential information. Thus a better understanding o…
Finite dynamical systems (e.g. Boolean networks and logical models) have been used in modeling biological systems to focus attention on the qualitative features of the system, such as the wiring diagram. Since the analysis of such systems is hard, it is necessary to focus on subclasses that have the properties of being general enough for modeling and simple enough for theoretical analysis. In this paper we propose the class of AND-NOT networks for modeling biological systems and show that it provides several advantages. Some of the advantages include: Any finite dynamical system can be written…
Boolean networks is a well-established formalism for modelling biological systems. A vital challenge for analysing a Boolean network is to identify all the attractors. This becomes more challenging for large asynchronous Boolean networks, due to the asynchronous updating scheme. Existing methods are prohibited due to the well-known state-space explosion problem in large Boolean networks. In this paper, we tackle this challenge by proposing a SCC-based decomposition method. We prove the correctness of our proposed method and demonstrate its efficiency with two real-life biological networks.
Boolean networks have long been used as models of molecular networks and play an increasingly important role in systems biology. This paper describes a software package, Polynome, offered as a web service, that helps users construct Boolean network models based on experimental data and biological input. The key feature is a discrete analog of parameter estimation for continuous models. With only experimental data as input, the software can be used as a tool for reverse-engineering of Boolean network models from experimental time course data.
Boolean networks are special types of finite state time-discrete dynamical systems. A Boolean network can be described by a function from an n-dimensional vector space over the field of two elements to itself. A fundamental problem in studying these dynamical systems is to link their long term behaviors to the structures of the functions that define them. In this paper, a method for deriving a Boolean network's dynamical information via its disjunctive normal form is explained. For a given Boolean network, a matrix with entries 0 and 1 is associated with the polynomial function that represents…
Discrete modelling frameworks of Biological networks can be divided in two distinct categories: Boolean and Multi-valued. Although Multi-valued networks are more expressive for qualifying the regulatory behaviours modelled by more than two values, the ability to automatically convert them to Boolean network with an equivalent behaviour breaks down the fundamental borders between the two approaches. Theoretically investigating the conversion process provides relevant insights into bridging the gap between them. Basically, the conversion aims at finding a Boolean network bisimulating a Multi-val…
Boolean networks are discrete dynamical systems where each automaton has its own Boolean function for computing its state according to the configuration of the network. The updating mode then determines how the configuration of the network evolves over time. Many of updating modes from the literature, including synchronous and asynchronous modes, can be defined as the composition of elementary deterministic configuration updates, i.e., by functions mapping configurations of the network. Nevertheless, alternative dynamics have been introduced using ad-hoc auxiliary objects, such as that resulti…
Continuous-time systems with switch-like behaviour occur in chemical kinetics, gene regulatory networks and neural networks. Networks with hard switching, as a limiting case of smooth sigmoidal switching, retain the richest possible range of behaviors but are mathematically more tractable. The form of an underlying discrete (fractional-linear) map encodes information on existence, stability and exact periods of periodic orbits. In richly connected structures with four or more variables, aperiodic behaviour can occur. We investigate a simple 4-dimensional example with Boolean interaction terms …
We study the properties of the distance between attractors in Random Boolean Networks, a prominent model of genetic regulatory networks. We define three distance measures, upon which attractor distance matrices are constructed and their main statistic parameters are computed. The experimental analysis shows that ordered networks have a very clustered set of attractors, while chaotic networks' attractors are scattered; critical networks show, instead, a pattern with characteristics of both ordered and chaotic networks.
On the Robustness of NK-Kauffman Networks Against Changes in their Connections and Boolean Functions
NK-Kauffman networks { \cal L}^N_K are a subset of the Boolean functions on N Boolean variables to themselves, Λ_N = {ξ: \IZ_2^N \to \IZ_2^N }. To each NK-Kauffman network it is possible to assign a unique Boolean function on N variables through the function Ψ: { \cal L}^N_K \to Λ_N. The probability { \cal P}_K that Ψ(f) = Ψ(f'), when f' is obtained through f by a change of one of its K-Boolean functions (b_K: \IZ_2^K \to \IZ_2 ), and/or connections; is calculated. The leading term of the asymptotic expansion of { \cal P}_K, for N \gg 1, turns out to depend on: the probability to extract the t…
