Choices of regulatory logic class modulate the dynamical regime in random Boolean networks

Genetic network models and statistical properties of gene expression data in knock-out experiments

Random Boolean networks have been proposed as a basic model of genetic regulatory networks for more than four decades. Attractors have been considered as the best way to represent the long-term behaviors of random Boolean networks. Most studies on attractors are made with random topologies. However, the real regulatory networks have been found to be modular or more complex topologies. In this work, we extend classical robustness and entropy analysis of random Boolean networks to random modular Boolean networks. We firstly focus on the robustness of the attractor to perturbations with different parameters. Then, we investigate and calculate how the amount of information propagated between the nodes when on an attractor, as quantified by the average pairwise mutual information. The results can be used to study the capability of genetic information propagation of different types of genetic regulatory networks.

A genetic regulatory network (GRN) is a dynamic system to describe interactions among a large number of different substances in a living biological cell. There have been numerous attempts to model the dynamical behaviour of genetic regulatory networks. Boolean models are a class of discrete models for modelling genetic regulatory networks, which are conditioned on the premise that genes interact with each other through Boolean logic. As mentioned previously, Boolean models include Boolean Networks, Instantaneously Random Probabilistic Boolean Networks and Context-Sensitive Boolean Networks. There are a large number of issues surrounding Boolean models, including Boolean Networks and Probabilistic Boolean Networks. The model inference is the procedure of determining whether the modelled networks are consistent with the given data sample, and choosing one model, from among many, that makes more sense for the data. This paper has reviewed the research into modelling gene regulatory networks, especially upon Boolean models.

After while the “Human Genome Project” proposes, the people complete measures the foreword plan after the multi-gene genome team, also starts to change to these genes and their reciprocity function understanding research. Varieties of gene regulation Boolean networks algorithms have been proposed of the gene expression profiles, however, the problem of noise could always be found in creating a Boolean network. Due to gene expression data are always noisy. In this paper, it show that after the Boolean networks logic function are learned from noisy data, some noise in the Boolean function could be restructure Karnaugh Maps. It could find logic relationships between protein and protein and restructure protein logic networks. It find logic relationship among proteins as well as COGs (clusters of orthogous groups) and build the logic network of protein.

The class of affine Boolean functions is rich enough to express constant bits and dependencies between different bits of different words. For example, the function (x0) ∧ (¬y1) ∧ (x4 ⇔ y7) ∧ (x5 ⇔ ¬y9) is affine and expresses the invariant that the low bit (bit 0) of the variable x is true, that bit 1 of y is false, that the bits 4 and 7 of x and y coincide whereas bits 5 and 9 of x and y differ. This class of Boolean function is amenable to bit-precise reasoning since it satisfies strong chain properties which bound the number of times a system of semantic fixpoint equations need to be reapplied when reasoning about loops. This paper address the key problem of abstracting an arbitrary Boolean function to either a general affine function or a so-called affine function of width 2, when the function is represented as an ROBDD. Novel algorithms are presented for this task: one that manipulates Boolean vectors and another which is inspired by anti-unification. The speed and precision of both algorithms are compared on benchmark circuits, to draw conclusions on the tractability of affine abstraction.

We propose quantum Boolean networks, which can be classified as deterministic reversible asynchronous Boolean networks. This model is based on the previously developed concept of quantum Boolean functions. A quantum Boolean network is a Boolean network where the functions associated with the nodes are quantum Boolean functions. We study some properties of this novel model and, using a quantum simulator, we study how the dynamics change in function of connectivity of the network and the set of operators we allow. For some configurations, this model resembles the behavior of reversible Boolean networks, while for other configurations a more complex dynamic can emerge. For example, cycles larger than 2N were observed. Additionally, using a scheme akin to one used previously with random Boolean networks, we computed the average entropy and complexity of the networks. As opposed to classic random Boolean networks, where “complex” dynamics are restricted mainly to a connectivity close to a phase transition, quantum Boolean networks can exhibit stable, complex, and unstable dynamics independently of their connectivity.

We propose quantum Boolean networks, which can be classified as deterministic reversible asynchronous Boolean networks. This model is based on the previously developed concept of quantum Boolean functions. A quantum Boolean network is a Boolean network where the functions associated with the nodes are quantum Boolean functions. We study some properties of this novel model and, using a quantum simulator, we study how the dynamics change in function of connectivity of the network and the set of operators we allow. For some configurations, this model resembles the behavior of reversible Boolean networks, while for other configurations a more complex dynamic can emerge. For example, cycles larger than 2N were observed. Additionally, using a scheme akin to one used previously with random Boolean networks, we computed the average entropy and complexity of the networks. As opposed to classic random Boolean networks, where “complex” dynamics are restricted mainly to a connectivity close to a phase transition, quantum Boolean networks can exhibit stable, complex, and unstable dynamics independently of their connectivity.

Observability of Boolean networks uniquely determines an initial condition of Boolean networks with a given output sequence. The objective of this study is to simplify the observability determination of Boolean networks and to offer a strategy to reconstruct states of Boolean networks with partial states' measurement. For a given Boolean dynamics and its measurement function, we extracted implicit useful information and deleted the redundant information based on its boolean dynamics and measurement function. Then, the extracted information was transferred to an undirected graph. We derived the necessary and sufficient condition of observability determination based on the corresponding undirected graph. Such necessary and sufficient condition significantly reduced the computational cost comparing with current methods. Further, we proposed a design method to determine the measurement function and reconstruct the unmeasured states. The effectiveness of the method was illustrated with an example. In conclusion, our result established an effective tool to determine the observability for Boolean networks with arbitrary number of nodes and Boolean functions, which further provided a novel way to allocate sensors for partial state measurement and to reconstruct the entire states.

Boolean networks are currently receiving considerable attention as a computational scheme for system level analysis and modeling of biological systems. Studying control-related problems in Boolean networks may reveal new insights into the intrinsic control in complex biological systems and enable us to develop strategies for manipulating biological systems using exogenous inputs. This paper considers controllability and observability of Boolean biological networks. We propose a new approach, which draws from the rich theory of symbolic computation, to solve the problems. Consequently, simple necessary and sufficient conditions for reachability, controllability, and observability are obtained, and algorithmic tests for controllability and observability which are based on the Gröbner basis method are presented. As practical applications, we apply the proposed approach to several different biological systems, namely, the mammalian cell-cycle network, the T-cell activation network, the large granular lymphocyte survival signaling network, and the Drosophila segment polarity network, gaining novel insights into the control and/or monitoring of the specific biological systems.

Wherever we see life, we see different kinds of complex networks, reason why they are studied across various fields of science. Random Boolean Networks (RBNs) form a special class in which the links between the nodes and the boolean functions are specified at random. Whereas synchronous RBNs were investigated in detail, there has little been done around their asynchronous counterpart, although there is evidence that most living systems are governed by asynchronous updating. Derrida’s annealed approximation predicts a critical connectivity value of K=2 for synchronous RBNs. We present a similar and original approach for asynchronous RBNs and show that they do not possess such a critical connectivity value. The asynchronous and nondeterministic updating scheme introduces perturbations that reach about 25% of the nodes and thus prevents the networks to become stable. Further, our numerical simulations show that asynchronous RBN tend to amplify small and to reduce big perturbations.

Boolean networks are a popular modeling framework in computational biology to capture the dynamics of molecular networks, such as gene regulatory networks. It has been observed that many published models of such networks are defined by regulatory rules driving the dynamics that have certain so-called canalizing properties. In this paper, we investigate the dynamics of a random Boolean network with such properties using analytical methods and simulations. From our simulations, we observe that Boolean networks with higher canalizing depth have generally fewer attractors, the attractors are smaller, and the basins are larger, with implications for the stability and robustness of the models. These properties are relevant to many biological applications. Moreover, our results show that, from the standpoint of the attractor structure, high canalizing depth, compared to relatively small positive canalizing depth, has a very modest impact on dynamics. Motivated by these observations, we conduct mathematical study of the attractor structure of a random Boolean network of canalizing depth one (i.e., the smallest positive depth). For every positive integer , we give an explicit formula for the limit of the expected number of attractors of length in an -state random Boolean network as goes to infinity.

Two Boolean functions are affine equivalent if one can be obtained from the other by applying an affine transformation to the input variables. For a long time, there have been efforts to investigate the affine equivalence of Boolean functions. Due to the complexity of the general problem, only affine equivalence under certain groups of permutations is usually considered. Boolean functions which are invariant under the action of cyclic rotation of the input variables are known as rotation symmetric (RS) Boolean functions. Due to their speed of computation and the prospect of being good cryptographic Boolean functions, this class of Boolean functions has received a lot of attention from cryptographic researchers. In this paper, we study affine equivalence for the simplest rotation symmetric Boolean functions, called MRS functions, which are generated by the cyclic permutations of a single monomial. Using Pólya’s enumeration theorem, we compute the number of equivalence classes, under certain large groups of permutations, for these MRS functions in any number

In biological systems, the dynamic analysis method has gained increasing attention in the past decade. The Boolean network is the most common model of a genetic regulatory network. The interactions of activation and inhibition in the genetic regulatory network are modeled as a set of functions of the Boolean network, while the state transitions in the Boolean network reflect the dynamic property of a genetic regulatory network. A difficult problem for state transition analysis is the finding of attractors. In this paper, we modeled the genetic regulatory network as a Boolean network and proposed a solving algorithm to tackle the attractor finding problem. In the proposed algorithm, we partitioned the Boolean network into several blocks consisting of the strongly connected components according to their gradients, and defined the connection between blocks as decision node. Based on the solutions calculated on the decision nodes and using a satisfiability solving algorithm, we identified the attractors in the state transition graph of each block. The proposed algorithm is benchmarked on a variety of genetic regulatory networks. Compared with existing algorithms, it achieved similar performance on small test cases, and outperformed it on larger and more complex ones, which happens to be the trend of the modern genetic regulatory network. Furthermore, while the existing satisfiability-based algorithms cannot be parallelized due to their inherent algorithm design, the proposed algorithm exhibits a good scalability on parallel computing architectures.

Driven by the desire to understand genomic functions through the interactions among genes and gene products, the research in gene regulatory networks has become a heated area in genomic signal processing. Among the most studied mathematical models are Boolean networks and probabilistic Boolean networks, which are rule-based dynamic systems. This tutorial provides an introduction to the essential concepts of these two Boolean models, and presents the up-to-date analysis and simulation methods developed for them. In the Analysis section, we will show that Boolean models are Markov chains, based on which we present a Markovian steady-state analysis on attractors, and also reveal the relationship between probabilistic Boolean networks and dynamic Bayesian networks (another popular genetic network model), again via Markov analysis; we dedicate the last subsection to structural analysis, which opens a door to other topics such as network control. The Simulation section will start from the basic tasks of creating state transition diagrams and finding attractors, proceed to the simulation of network dynamics and obtaining the steady-state distributions, and finally come to an algorithm of generating artificial Boolean networks with prescribed attractors. The contents are arranged in a roughly logical order, such that the Markov chain analysis lays the basis for the most part of Analysis section, and also prepares the readers to the topics in Simulation section.

Investigating the use of Boolean networks for the control of gene regulatory networks