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.

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.

On Generating Optimal Sparse Probabilistic Boolean Networks with Maximum Entropy from a Positive Stationary Distribution

2012 ◽  
Vol 2 (4) ◽  
pp. 353-372 ◽  
Author(s):  
Hao Jiang ◽  
Xi Chen ◽  
Yushan Qiu ◽  
Wai-Ki Ching
Keyword(s):  
Maximum Entropy ◽  
Stationary Distribution ◽  
Sparse Matrix ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Boolean Networks ◽  
Efficiency And Effectiveness ◽  
Probabilistic Boolean Networks ◽  
Transition Probability Matrices ◽  
Probability Matrices

Abstract.To understand a genetic regulatory network, two popular mathematical models, Boolean Networks (BNs) and its extension Probabilistic Boolean Networks (PBNs) have been proposed. Here we address the problem of constructing a sparse Probabilistic Boolean Network (PBN) from a prescribed positive stationary distribution. A sparse matrix is more preferable, as it is easier to study and identify the major components and extract the crucial information hidden in a biological network. The captured network construction problem is both ill-posed and computationally challenging. We present a novel method to construct a sparse transition probability matrix from a given stationary distribution. A series of sparse transition probability matrices can be determined once the stationary distribution is given. By controlling the number of nonzero entries in each column of the transition probability matrix, a desirable sparse transition probability matrix in the sense of maximum entropy can be uniquely constructed as a linear combination of the selected sparse transition probability matrices (a set of sparse irreducible matrices). Numerical examples are given to demonstrate both the efficiency and effectiveness of the proposed method.

On Construction of Sparse Probabilistic Boolean Networks

2012 ◽  
Vol 2 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Xi Chen ◽  
Hao Jiang ◽  
Wai-Ki Ching
Keyword(s):  
Objective Function ◽  
Stationary Distribution ◽  
Transition Probability ◽  
Additional Term ◽  
Control Policy ◽  
Transition Probability Matrix ◽  
Boolean Networks ◽  
Entropy Rate ◽  
Sparse Solution ◽  
Probabilistic Boolean Networks

AbstractIn this paper we envisage building Probabilistic Boolean Networks (PBNs) from a prescribed stationary distribution. This is an inverse problem of huge size that can be subdivided into two parts — viz. (i) construction of a transition probability matrix from a given stationary distribution (Problem ST), and (ii) construction of a PBN from a given transition probability matrix (Problem TP). A generalized entropy approach has been proposed for Problem ST and a maximum entropy rate approach for Problem TP respectively. Here we propose to improve both methods, by considering a new objective function based on the entropy rate with an additional term of La-norm that can help in getting a sparse solution. A sparse solution is useful in identifying the major component Boolean networks (BNs) from the constructed PBN. These major BNs can simplify the identification of the network structure and the design of control policy, and neglecting non-major BNs does not change the dynamics of the constructed PBN to a large extent. Numerical experiments indicate that our new objective function is effective in finding a better sparse solution.

CONTROL OF STATIONARY BEHAVIOR IN PROBABILISTIC BOOLEAN NETWORKS BY MEANS OF STRUCTURAL INTERVENTION

2002 ◽  
Vol 10 (04) ◽  
pp. 431-445 ◽  
Author(s):  
ILYA SHMULEVICH ◽  
EDWARD R. DOUGHERTY ◽  
WEI ZHANG
Keyword(s):  
Steady State ◽  
Regulatory Networks ◽  
Dynamical Behavior ◽  
Optimal Solution ◽  
Boolean Networks ◽  
Underlying Structure ◽  
State Behavior ◽  
Efficient Manner ◽  
Long Run ◽  
Probabilistic Boolean Networks

Probabilistic Boolean Networks (PBNs) were recently introduced as models of gene regulatory networks. The dynamical behavior of PBNs, which are probabilistic generalizations of Boolean networks, can be studied using Markov chain theory. In particular, the steady-state or long-run behavior of PBNs may reflect the phenotype or functional state of the cell. Approaches to alter the steady-state behavior in a specific prescribed manner, in cases of aberrant cellular states, such as tumorigenesis, would be highly beneficial. This paper develops a methodology for altering the steady-state probabilities of certain states or sets of states with minimal modifications to the underlying rule-based structure. This approach is framed as an optimization problem that we propose to solve using genetic algorithms, which are well suited for capturing the underlying structure of PBNs and are able to locate the optimal solution in a highly efficient manner. Several computer simulation experiments support the proposed methodology.

scholarly journals On the evolution of corruption patterns in the post-communist countries

Equilibrium ◽  
2015 ◽  
Vol 10 (1) ◽  
pp. 33 ◽  
Author(s):  
Andrzej Cieślik ◽  
Łukasz Goczek
Keyword(s):  
Transition Probability ◽  
Transition Probability Matrix ◽  
Markov Transition Probability ◽  
Long Run ◽  
Markov Transition ◽  
Corruption Index ◽  
Communist Countries ◽  
Control Of Corruption ◽  
Over Time

In this paper, we study the evolution of corruption patterns in 27 post-communist countries during the period 1996-2012 using the Control of Corruption Index and the corruption category Markov transition probability matrix. This method allows us to generate the long-run distribution of corruption among the post-communist countries. Our empirical findings suggest that corruption in the post-communist countries is a very persistent phenomenon that does not change much over time. Several theoretical explanations for such a result are provided.

scholarly journals On the long-run sensitivity of probabilistic Boolean networks

2009 ◽  
Vol 257 (4) ◽  
pp. 560-577 ◽  
Author(s):  
Xiaoning Qian ◽  
Edward R. Dougherty
Keyword(s):  
Boolean Networks ◽  
Long Run ◽  
Probabilistic Boolean Networks

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.

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