scholarly journals Modular Random Boolean Networks

2011 ◽  
Vol 17 (4) ◽  
pp. 331-351 ◽  
Author(s):  
Rodrigo Poblanno-Balp ◽  
Carlos Gershenson
Keyword(s):  
Chaotic Dynamics ◽  
Regulatory Networks ◽  
Boolean Networks ◽  
Genetic Regulatory Networks ◽  
Random Boolean Networks ◽  
Analytical Results ◽  
Statistical Experiments

Random Boolean networks (RBNs) have been a popular model of genetic regulatory networks for more than four decades. However, most RBN studies have been made with random topologies, while real regulatory networks have been found to be modular. In this work, we extend classical RBNs to define modular RBNs. Statistical experiments and analytical results show that modularity has a strong effect on the properties of RBNs. In particular, modular RBNs have more attractors, and are closer to criticality when chaotic dynamics would be expected, than classical RBNs.

The Robustness and Mutual Information Entropy of Random Modular Boolean Networks

2014 ◽  
Vol 989-994 ◽  
pp. 4417-4420 ◽  
Author(s):  
Nan Zhao ◽  
Bing Hui Guo ◽  
Fan Chao Meng
Keyword(s):  
Mutual Information ◽  
Regulatory Networks ◽  
Boolean Networks ◽  
Genetic Regulatory Networks ◽  
Information Propagation ◽  
Entropy Analysis ◽  
Random Boolean Networks ◽  
Basic Model ◽  
Different Types

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.

The impact of sources and sinks on the dynamics of homogeneous and heterogeneous boolean networks

2021 ◽  
Author(s):  
Luca Agostini
Keyword(s):  
Regulatory Networks ◽  
Boolean Networks ◽  
Genetic Regulatory Networks ◽  
The Other ◽  
Structural Elements ◽  
Sources And Sinks ◽  
Random Boolean Networks ◽  
Dynamical Behaviors ◽  
Other Hand ◽  
The Impact

Random Boolean networks, originally introduced as simplified models for the genetic regulatory networks, are abstract models widely applied for the study of the dynamical behaviors of self-organizing complex systems. In these networks, connectivity and the bias of the Boolean functions are the most important factors that can determine the behavioral regime of the systems. On the other hand, it has been found that topology and some structural elements of the networks such as the reciprocity, self-loops and source nodes, can have relevant effects on the dynamical properties of critical Boolean networks. In this paper, we study the impact of source and sink nodes on the dynamics of homogeneous and heterogeneous Boolean networks. Our research shows that an increase of the source nodes causes an exponentially growing of the different behaviors that the system can exhibit regardless of the network topology, while the amount of order seems to undergo modifications depending on the topology of the system. Indeed, with the increase of the source nodes the orderliness of the heterogeneous networks also increases, whereas it diminishes in the homogeneous ones. On the other hand, although the sink nodes seem not to have effects on the dynamic of the homogeneous networks, for the heterogeneous ones we have found that an increase of the sinks gives rise to an increasing of the order, although the different potential behaviors of the system remains approximately the same.

scholarly journals Dynamical Criticality in Gene Regulatory Networks

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-14 ◽  
Author(s):  
Marco Villani ◽  
Luca La Rocca ◽  
Stuart Alan Kauffman ◽  
Roberto Serra
Keyword(s):  
Regulatory Networks ◽  
Single Gene ◽  
Biological Evolution ◽  
Boolean Networks ◽  
Genetic Regulatory Networks ◽  
Dynamical Regime ◽  
Random Boolean Networks ◽  
Statistical Framework ◽  
Gene Regulatory ◽  
The Relationship

A well-known hypothesis, with far-reaching implications, is that biological evolution should preferentially lead to states that are dynamically critical. In previous papers, we showed that a well-known model of genetic regulatory networks, namely, that of random Boolean networks, allows one to study in depth the relationship between the dynamical regime of a living being’s gene network and its response to permanent perturbations. In this paper, we analyze a huge set of new experimental data on single gene knockouts in S. cerevisiae, laying down a statistical framework to determine its dynamical regime. We find that the S. cerevisiae network appears to be slightly ordered, but close to the critical region. Since our analysis relies on dichotomizing continuous data, we carefully consider the issue of an optimal threshold choice.

Understanding genetic regulatory networks

2003 ◽  
Vol 2 (2) ◽  
pp. 131-139 ◽  
Author(s):  
Stuart Kauffman
Keyword(s):  
Power Law ◽  
Regulatory Networks ◽  
Single Gene ◽  
Chaotic Regime ◽  
Boolean Networks ◽  
Genetic Regulatory Networks ◽  
Power Law Distribution ◽  
Generic Properties ◽  
Wiring Diagram ◽  
Random Boolean Networks

Random Boolean networks (RBM) were introduced about 35 years ago as first crude models of genetic regulatory networks. RBNs are comprised of N on–off genes, connected by a randomly assigned regulatory wiring diagram where each gene has K inputs, and each gene is controlled by a randomly assigned Boolean function. This procedure samples at random from the ensemble of all possible NK Boolean networks. The central ideas are to study the typical, or generic properties of this ensemble, and see 1) whether characteristic differences appear as K and biases in Boolean functions are introducted, and 2) whether a subclass of this ensemble has properties matching real cells.Such networks behave in an ordered or a chaotic regime, with a phase transition, ‘the edge of chaos’ between the two regimes. Networks with continuous variables exhibit the same two regimes. Substantial evidence suggests that real cells are in the ordered regime. A key concept is that of an attractor. This is a reentrant trajectory of states of the network, called a state cycle. The central biological interpretation is that cell types are attractors. A number of properties differentiate the ordered and chaotic regimes. These include the size and number of attractors, the existence in the ordered regime of a percolating ‘sea’ of genes frozen in the on or off state, with a remainder of isolated twinkling islands of genes, a power law distribution of avalanches of gene activity changes following perturbation to a single gene in the ordered regime versus a similar power law distribution plus a spike of enormous avalanches of gene changes in the chaotic regime, and the existence of branching pathway of ‘differentiation’ between attractors induced by perturbations in the ordered regime.Noise is serious issue, since noise disrupts attractors. But numerical evidence suggests that attractors can be made very stable to noise, and meanwhile, metaplasias may be a biological manifestation of noise. As we learn more about the wiring diagram and constraints on rules controlling real genes, we can build refined ensembles reflecting these properties, study the generic properties of the refined ensembles, and hope to gain insight into the dynamics of real cells.

scholarly journals Adaptive Local Information Transfer in Random Boolean Networks

2017 ◽  
Vol 23 (1) ◽  
pp. 105-118 ◽  
Author(s):  
Taichi Haruna
Keyword(s):  
Information Processing ◽  
Information Transfer ◽  
Regulatory Networks ◽  
Mean Field Theory ◽  
Mean Field ◽  
Boolean Networks ◽  
Local Information ◽  
System Level ◽  
Adjustment Process ◽  
Random Boolean Networks

Living systems such as gene regulatory networks and neuronal networks have been supposed to work close to dynamical criticality, where their information-processing ability is optimal at the whole-system level. We investigate how this global information-processing optimality is related to the local information transfer at each individual-unit level. In particular, we introduce an internal adjustment process of the local information transfer and examine whether the former can emerge from the latter. We propose an adaptive random Boolean network model in which each unit rewires its incoming arcs from other units to balance stability of its information processing based on the measurement of the local information transfer pattern. First, we show numerically that random Boolean networks can self-organize toward near dynamical criticality in our model. Second, the proposed model is analyzed by a mean-field theory. We recognize that the rewiring rule has a bootstrapping feature. The stationary indegree distribution is calculated semi-analytically and is shown to be close to dynamical criticality in a broad range of model parameter values.

CHAOTIC MEAN FIELD DYNAMICS OF A BOOLEAN NETWORK WITH RANDOM CONNECTIVITY

2007 ◽  
Vol 18 (09) ◽  
pp. 1459-1473 ◽  
Author(s):  
MALIACKAL POULO JOY ◽  
DONALD E. INGBER ◽  
SUI HUANG
Keyword(s):  
Regulatory Networks ◽  
Boolean Network ◽  
Mean Field ◽  
Period Doubling ◽  
Structural Heterogeneity ◽  
Boolean Networks ◽  
Analytical Treatment ◽  
Random Boolean Networks ◽  
The Mean ◽  
Parameter Values

Random Boolean networks have been used as simple models of gene regulatory networks, enabling the study of the dynamic behavior of complex biological systems. However, analytical treatment has been difficult because of the structural heterogeneity and the vast state space of these networks. Here we used mean field approximations to analyze the dynamics of a class of Boolean networks in which nodes have random degree (connectivity) distributions, characterized by the mean degree k and variance D. To achieve this we generalized the simple cellular automata rule 126 and used it as the Boolean function for all nodes. The equation for the evolution of the density of the network state is presented as a one-dimensional map for various input degree distributions, with k and D as the control parameters. The mean field dynamics is compared with the data obtained from the simulations of the Boolean network. Bifurcation diagrams and Lyapunov exponents for different parameter values were computed for the map, showing period doubling route to chaos with increasing k. Onset of chaos was delayed (occurred at higher k) with the increase in variance D of the connectivity. Thus, the network tends to be less chaotic when the heterogeneity, as measured by the variance of connectivity, was higher.

scholarly journals Steady-State Analysis of Genetic Regulatory Networks Modelled by Probabilistic Boolean Networks

2003 ◽  
Vol 4 (6) ◽  
pp. 601-608 ◽  
Author(s):  
Ilya Shmulevich ◽  
Ilya Gluhovsky ◽  
Ronaldo F. Hashimoto ◽  
Edward R. Dougherty ◽  
Wei Zhang
Keyword(s):  
Markov Chain ◽  
Steady State ◽  
Markov Chains ◽  
Regulatory Networks ◽  
Boolean Networks ◽  
Genetic Regulatory Networks ◽  
Long Run ◽  
Probabilistic Boolean Networks

Probabilistic Boolean networks (PBNs) have recently been introduced as a promising class of models of genetic regulatory networks. The dynamic behaviour of PBNs can be analysed in the context of Markov chains. A key goal is the determination of the steady-state (long-run) behaviour of a PBN by analysing the corresponding Markov chain. This allows one to compute the long-term influence of a gene on another gene or determine the long-term joint probabilistic behaviour of a few selected genes. Because matrix-based methods quickly become prohibitive for large sizes of networks, we propose the use of Monte Carlo methods. However, the rate of convergence to the stationary distribution becomes a central issue. We discuss several approaches for determining the number of iterations necessary to achieve convergence of the Markov chain corresponding to a PBN. Using a recently introduced method based on the theory of two-state Markov chains, we illustrate the approach on a sub-network designed from human glioma gene expression data and determine the joint steadystate probabilities for several groups of genes.

Evolving Boolean Networks with Structural Dynamism

2012 ◽  
Vol 18 (4) ◽  
pp. 385-397 ◽  
Author(s):  
Larry Bull
Keyword(s):  
Regulatory Networks ◽  
Boolean Model ◽  
Boolean Networks ◽  
Genetic Regulatory Networks ◽  
Short Article ◽  
Evolutionary Innovation ◽  
Current Cell ◽  
Cell Life ◽  
Node Connectivity ◽  
Asynchronous Updating

This short article presents an abstract, tunable model of genomic structural change within the cell life cycle and explores its use with simulated evolution. A well-known Boolean model of genetic regulatory networks is extended to include changes in node connectivity based upon the current cell state to begin to capture some of the effects of transposable elements. The evolvability of such networks is explored using a version of the NK model of fitness landscapes with both synchronous and asynchronous updating. Structural dynamism is found to be selected for in nonstationary environments with both update schemes and subsequently shown capable of providing a mechanism for evolutionary innovation when such reorganizations are inherited. This is also found to be the case in stationary environments with asynchronous updating.

ON CONSTRUCTION OF STOCHASTIC GENETIC NETWORKS BASED ON GENE EXPRESSION SEQUENCES

2005 ◽  
Vol 15 (04) ◽  
pp. 297-310 ◽  
Author(s):  
WAI-KI CHING ◽  
MICHAEL M. NG ◽  
ERIC S. FUNG ◽  
TATSUYA AKUTSU
Keyword(s):  
Gene Expression ◽  
Regulatory Networks ◽  
Expression Patterns ◽  
Boolean Networks ◽  
Genetic Regulatory Networks ◽  
Efficient Estimation ◽  
Estimation Methods ◽  
Series Data ◽  
Model Parameters ◽  
Proposed Model

Reconstruction of genetic regulatory networks from time series data of gene expression patterns is an important research topic in bioinformatics. Probabilistic Boolean Networks (PBNs) have been proposed as an effective model for gene regulatory networks. PBNs are able to cope with uncertainty, corporate rule-based dependencies between genes and discover the sensitivity of genes in their interactions with other genes. However, PBNs are unlikely to use directly in practice because of huge amount of computational cost for obtaining predictors and their corresponding probabilities. In this paper, we propose a multivariate Markov model for approximating PBNs and describing the dynamics of a genetic network for gene expression sequences. The main contribution of the new model is to preserve the strength of PBNs and reduce the complexity of the networks. The number of parameters of our proposed model is O(n2) where n is the number of genes involved. We also develop efficient estimation methods for solving the model parameters. Numerical examples on synthetic data sets and practical yeast data sequences are given to demonstrate the effectiveness of the proposed model.

scholarly journals Minimum complexity drives regulatory logic in Boolean models of living systems

2021 ◽  
Author(s):  
Ajay Subbaroyan ◽  
Olivier C. Martin ◽  
Areejit Samal
Keyword(s):  
Biological Networks ◽  
Boolean Functions ◽  
Regulatory Networks ◽  
Statistical Physics ◽  
Boolean Networks ◽  
Average Sensitivity ◽  
Boolean Models ◽  
Complexity Measures ◽  
Random Boolean Networks ◽  
Update Rules

The properties of random Boolean networks as models of gene regulation have been investigated extensively by the statistical physics community. In the past two decades, there has been a dramatic increase in the reconstruction and analysis of Boolean models of biological networks. In such models, neither network topology nor Boolean functions (or logical update rules) should be expected to be random. In this contribution, we focus on biologically meaningful types of Boolean functions, and perform a systematic study of their preponderance in gene regulatory networks. By applying the k[P] classification based on number of inputs k and bias P of functions, we find that most Boolean functions astonishingly have odd bias in a reference biological dataset of 2687 functions compiled from published models. Subsequently, we are able to explain this observation along with the enrichment of read-once functions (RoFs) and its subset, nested canalyzing functions (NCFs), in the reference dataset in terms of two complexity measures: Boolean complexity based on string lengths in formal logic which is yet unexplored in the biological context, and the average sensitivity. Minimizing the Boolean complexity naturally sifts out a subset of odd-biased Boolean functions which happen to be the RoFs. Finally, we provide an analytical proof that NCFs minimize not only the Boolean complexity, but also the average sensitivity in their k[P] set.

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