East Asian Journal on Applied Mathematics, 1 (2011), pp. 132-154. Construction of Probabilistic Boolean Networks from a Prescribed Transition Probability Matrix: A Maximum Entropy Rate Approach Xi Chen 1, Wai-Ki Ching 1*, Xiao-Shan Chen 2, Yang Cong 1, Nam-Kiu Tsing 11 Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong. 2 School of Mathematical Sciences, South China Normal University, Guangzhou, China. Received 8 March 2010; Accepted (in revised version) 20 September 2010 Available online 7 April 2011 doi:10.4208/eajam.080310.200910a 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. Key words: Boolean networks, conjugate gradient method, genetic regulatory networks, inverse problem, Markov chains, Newton's method, probabilistic Boolean networks, transition-probability matrix. *Corresponding author. Email: dlkcissy@hotmail.com (X. Chen), wching@hkusua.hku.hk (W.-K. Ching),