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

Published online by Cambridge University Press:  28 May 2015

Xi Chen ,
Wai-Ki Ching ,
Xiao-Shan Chen ,
Yang Cong  and
Nam-Kiu Tsing
Xi Chen*
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
Xiao-Shan Chen*
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou, China
Yang Cong*
Affiliation:
Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
Nam-Kiu Tsing*
Affiliation:
Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
*
Corresponding author. Email: dlkcissy@hotmail.com
Corresponding author. Email: wching@hkusua.hku.hk
Corresponding author. Email: cxs333@21cn.com
Corresponding author. Email: congyang0305@yahoo.com.cn
Corresponding author. Email: nktsing@hku.hk
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Abstract

Modeling 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.

Keywords

Boolean networksconjugate gradient methodgenetic regulatory networksinverse problemMarkov chainsNewton's methodprobabilistic Boolean networkstransition-probability matrix
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 1 , Issue 2 , May 2011 , pp. 132 - 154
Copyright
Copyright © Global-Science Press 2011

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