From Boolean to probabilistic Boolean networks as models of genetic regulatory networks

2002 ◽  
Vol 90 (11) ◽  
pp. 1778-1792 ◽  
Author(s):  
I. Shmulevich ◽  
E.R. Dougherty ◽  
Wei Zhang
Keyword(s):  
Regulatory Networks ◽  
Boolean Networks ◽  
Genetic Regulatory Networks ◽  
Probabilistic Boolean Networks

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.

Construction of Probabilistic Boolean Networks from a Prescribed Transition Probability Matrix: A Maximum Entropy Rate Approach

2011 ◽  
Vol 1 (2) ◽  
pp. 132-154 ◽  
Author(s):  
Xi Chen ◽  
Wai-Ki Ching ◽  
Xiao-Shan Chen ◽  
Yang Cong ◽  
Nam-Kiu Tsing
Keyword(s):  
Inverse Problem ◽  
Maximum Entropy ◽  
Regulatory Networks ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Boolean Networks ◽  
Genetic Regulatory Networks ◽  
Genomic Research ◽  
Long Run ◽  
Probabilistic Boolean Networks

AbstractModeling genetic regulatory networks is an important problem in genomic research. Boolean Networks (BNs) and their extensions Probabilistic Boolean Networks (PBNs) have been proposed for modeling genetic regulatory interactions. In a PBN, its steady-state distribution gives very important information about the long-run behavior of the whole network. However, one is also interested in system synthesis which requires the construction of networks. The inverse problem is ill-posed and challenging, as there may be many networks or no network having the given properties, and the size of the problem is huge. The construction of PBNs from a given transition-probability matrix and a given set of BNs is an inverse problem of huge size. We propose a maximum entropy approach for the above problem. Newton's method in conjunction with the Conjugate Gradient (CG) method is then applied to solving the inverse problem. We investigate the convergence rate of the proposed method. Numerical examples are also given to demonstrate the effectiveness of our proposed method.

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.

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.

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 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.

Generalized Boolean Networks

2010 ◽  
pp. 429-449 ◽  
Author(s):  
Christian Darabos ◽  
Mario Giacobini ◽  
Marco Tomassini
Keyword(s):  
Biological Networks ◽  
Regulatory Networks ◽  
Dynamical Behavior ◽  
Real Life ◽  
Network Models ◽  
Cell Types ◽  
Boolean Networks ◽  
Genetic Regulatory Networks ◽  
Point Of View ◽  
Scale Free

Random Boolean Networks (RBN) have been introduced by Kauffman more than thirty years ago as a highly simplified model of genetic regulatory networks. This extremely simple and abstract model has been studied in detail and has been shown capable of extremely interesting dynamical behavior. First of all, as some parameters are varied such as the network’s connectivity, or the probability of expressing a gene, the RBN can go through a phase transition, going from an ordered regime to a chaotic one. Kauffman’s suggestion is that cell types correspond to attractors in the RBN phase space, and only those attractors that are short and stable under perturbations will be of biological interest. Thus, according to Kauffman, RBN lying at the edge between the ordered phase and the chaotic phase can be seen as abstract models of genetic regulatory networks. The original view of Kauffman, namely that these models may be useful for understanding real-life cell regulatory networks, is still valid, provided that the model is updated to take into account present knowledge about the topology of real gene regulatory networks, and the timing of events, without loosing its attractive simplicity. According to present data, many biological networks, including genetic regulatory networks, seem, in fact, to be of the scale-free type. From the point of view of the timing of events, standard RBN update their state synchronously. This assumption is open to discussion when dealing with biologically plausible networks. In particular, for genetic regulatory networks, this is certainly not the case: genes seem to be expressed in different parts of the network at different times, according to a strict sequence, which depends on the particular network under study. The expression of a gene depends on several transcription factors, the synthesis of which appear to be neither fully synchronous nor instantaneous. Therefore, we have recently proposed a new, more biologically plausible model. It assumes a scale-free topology of the networks and we define a suitable semi-synchronous dynamics that better captures the presence of an activation sequence of genes linked to the topological properties of the network. By simulating statistical ensembles of networks, we discuss the attractors of the dynamics, showing that they are compatible with theoretical biological network models. Moreover, the model demonstrates interesting scaling abilities as the size of the networks is increased.

Relationships between Models of Genetic Regulatory Networks with Emphasis on Discrete State Stochastic Models

2016 ◽  
pp. 52-79 ◽  
Author(s):  
Randip Pal
Keyword(s):  
Stochastic Models ◽  
Regulatory Networks ◽  
Regulation Of Gene Expression ◽  
Boolean Networks ◽  
Genetic Regulatory Networks ◽  
Discrete State ◽  
Detailed Model ◽  
Stochastic Master Equation ◽  
Differential Equation Models ◽  
Effective Integration

Genetic Regulatory Networks (GRNs) represent the interconnections between genomic entities that govern the regulation of gene expression. GRNs have been represented by various types of mathematical models that capture different aspects of the biological system. This chapter discusses the relationships among the most commonly used GRN models that can enable effective integration of diverse types of sub-models. A detailed model in the form of stochastic master equation is described, followed by it coarse-scale and deterministic approximations in the form of Probabilistic Boolean Networks and Ordinary Differential Equation models respectively.

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