# Construction Method of Probabilistic Boolean Networks Based on Imperfect Information
## 1 Introduction  
One of the aims in systems biology is to develop a method for modeling, analysis, and control of gene regulatory networks. Control of gene regulatory networks is closely related to therapeutic interventions, and is important toward developing gene therapy technologies [1] in the future.  
In the last decade, a Boolean network (BN) [2] and a probabilistic Boolean network (PBN) [3] have attracted much attention as mathematical models in control of gene regulatory networks. In a BN, dynamics such as interactions between genes are expressed by Boolean functions; that is, gene expression is expressed by a binary value (0 or 1). In a PBN, a Boolean function is randomly chosen at each time among candidates of Boolean functions. For a BN, controllability/reachability analysis [4; 5; 6; 7] and optimal control [8; 9; 10] have been studied. Model reduction has also been studied in [11; 12; 13]. For a PBN, controllability/reachability analysis [14; 15; 16; 17] and optimal control [18; 19; 20; 21; 22] have been studied (see also the survey paper [23]).  
In this paper, we consider the problem of finding a PBN based on imperfect information such as the network structure, the sample mean, and the prescribed Boolean functions. For a BN, such a problem has been studied in [24; 25]. For a PBN, such a problem has been studied in [26; 27; 28]. Comparing the method in [27] with other methods in [26; 28], there is the advantage that the size of matrices manipulated in the procedure is smaller. Hence, we focus on the method in [27]. In this method, a linear programming (LP) problem with no objective function is solved. However, it is appropriate that the objective function is set based on imperfect information. In this paper, a new objective function, which can be set from the prescribed Boolean functions, is proposed. By a numerical example, the effectiveness of the proposed method is presented.  
This paper is organized as follows. In Section 2, as a preparation, the outline of a PBN and its matrix-based representation is explained. In Section 3, the problem studied in this paper is formulated, and its solution method is derived. In Section 4, a numerical example is presented to show the effectiveness of the proposed method. In Section 5, we conclude this paper.  
**Notation:** For the \(n\)-dimensional vector \(x=[\ x_{1}\ \ x_{2}\ \cdots\ x_{n}\ ]^{\top}\) and the index set \(\mathcal{I}=\{i_{1},i_{2},\ldots,i_{m}\}\subseteq\{1,2,\ldots,n\}\), define \([x_{i}]_{i\in\mathcal{I}}:=[\ x_{i_{1}}\ \ x_{i_{2}}\ \cdots\ \ x_{i_{m}}\ ]^{\top}\). For two matrices \(A\) and \(B\), let \(A\otimes B\) denote the Kronecker product of \(A\) and \(B\). In addition, for \(q\) vectors \(y_{1},y_{2},\ldots,y_{q}\) and the index set \(\mathcal{J}=\{j_{1},j_{2},\ldots,j_{p}\}\subseteq\{1,2,\ldots,q\}\), define \(\bigotimes_{j\in\mathcal{J}}y_{j}:=y_{j_{1}}\otimes y_{j_{2}}\otimes\cdots \otimes y_{j_{p}}\). For example, for \(q\) two-dimensional vectors \(z_{1},z_{2},\ldots,z_{q}\) and \(\mathcal{J}=\{1,5\}\), we can obtain \(\bigotimes_{j\in\mathcal{J}}z_{j}=z_{1}\otimes z_{5}=[\ z_{1}^{(1)}z_{5}^{(1)} z_{1}^{(2)}z_{5}^{(2)}\ z_{1}^{(2)}z_{5}^{(1)}\ ]^{\top}\), where \(z_{j}^{(i)}\) is the \(i\)-th element of \(z_{j}\). Finally, let \(1_{m\times n}\) denote the \(m\times n\) matrix whose elements are all one.