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

Published online by Cambridge University Press:  28 May 2015

Hao Jiang ,
Xi Chen ,
Yushan Qiu  and
Wai-Ki Ching
Hao Jiang*
Affiliation:
Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
Xi Chen*
Affiliation:
Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
Yushan Qiu*
Affiliation:
Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
Wai-Ki Ching*
Affiliation:
Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
*
Corresponding author. Email: [email protected]
Corresponding author. Email: [email protected]
Corresponding author. Email: [email protected]
Corresponding author. Email: [email protected]
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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.

Keywords

65C2092B05Boolean Networks (BNs)EntropyProbabilistic Boolean Networks (PBNs)genetic regulatory networkssparsitystationary probability distributiontransition probability matrix
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 2 , Issue 4 , November 2012 , pp. 353 - 372
Copyright
Copyright © Global-Science Press 2012

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