# Optimization-Based Approaches to Control of Probabilistic Boolean Networks
## 2 Probabilistic Boolean Networks  
In this section, we introduce a probabilistic Boolean network (PBN). Consider the following PBN:  
\[\left\{ \begin{array}{l}x_{1}(k+1)=f^{(1)}(x(k),u(k)),\\ x_{2}(k+1)=f^{(2)}(x(k),u(k)),\\ \vdots\\ x_{n}(k+1)=f^{(n)}(x(k),u(k)),\end{array} \right. \tag{1}\]  
where \(x=[x_{1}\ x_{2}\ \cdots\ x_{n}]^{\top}\in\{0,1\}^{n}\) is the state (e.g., the expression of genes), and \(u=[u_{1}\ u_{2}\ \cdots\ u_{m}]^{\top}\in\{0,1\}^{m}\) is the control input (e.g., the expression of genes), i.e., the value of \(u\) can be arbitrarily determined, \(k\in\{0,1,2,\dots\}\) is the discrete time. The function \(f^{(i)}:\{0,1\}^{n}\times\{0,1\}^{m}\rightarrow\{0,1\}^{1}\) is a given Boolean function consisting of logical operators such as AND (\(\wedge\)), OR (\(\vee\)), XOR (\(\oplus\)), and NOT (\(\
eg\)). In deterministic Boolean networks, \(x(k+1)\) is uniquely determined for given \(x(k)\) and \(u(k)\). In PBNs, the candidates of \(f^{(i)}(x(k),u(k))\) are given, and for each \(x_{i}\), selecting one Boolean function is probabilistically independent at each time. The candidates of \(f^{(i)}(x(k),u(k))\) are denoted by \(f^{(i)}_{j}(x(k),u(k)),j=1,2,\dots,l(i)\), and the probability that \(f^{(i)}_{j}(x(k),u(k))\) is selected is denoted by  
\[c^{(i)}_{j}=P\left(f^{(i)}(x(k),u(k))=f^{(i)}_{j}(x(k),u(k))\right)\in[0,1].\]  
Then, the relation  
\[\sum_{j=1}^{l(i)}c^{(i)}_{j}=1 \tag{2}\]  
must be satisfied. First, we present a biological example of (deterministic) Boolean networks (BNs). **Example 1**. : _Consider the following BN of an apoptosis network (see also Figure 1):_  
\[\left\{ \begin{array}{l}x_{1}(k+1)=-x_{2}(k)\wedge u(k),\\ x_{2}(k+1)=-x_{1}(k)\wedge x_{3}(k),\\ x_{3}(k+1)=x_{2}(k)\lor u(k),\end{array} \right. \tag{3}\]  
_where \(x_{1}\) is the concentration level (high or low) of the inhibitor of apoptosis proteins (IAP), \(x_{2}\) is the concentration level of the active caspase 3 (C3a), and \(x_{3}\) is the concentration level of the active caspase 8 (C8a). The control input \(u\) corresponds to the concentration level of the tumor necrosis factor (TNF, a stimulus). This model is introduced in [46], and is a simplified version of an apoptosis network model in [47]. In this model, \(x_{2}(k)=0\) implies cell survival, and \(x_{1}(k)=0,x_{2}(k)=1\) imply cell death._  
Next, we present a simple example of PBNs. Figure 1: Simplified model of an apoptosis network. **Example 2**: _Consider the PBN with three states and single control input. Boolean functions and probabilities are given by_  
\[f^{(1)} = \left\{\begin{array}{ll}f^{(1)}_{1}=x_{3}(k)\wedge u(k),&c^{(1)}_ {1}=0.8,\\ f^{(1)}_{2}=\
eg x_{3}(k),&c^{(1)}_{2}=0.2,\\ f^{(2)}=f^{(2)}_{1}=x_{1}(k)\wedge\
eg x_{3}(k),&c^{(2)}_{1}=1.0,\\ f^{(3)}=\left\{ \begin{array}{ll}f^{(3)}_{1}=x_{1}(k)\wedge\
eg x_{2}(k),&c^{ (3)}_{1}=0.7,\\ f^{(3)}_{2}=x_{2}(k)\wedge u(k),&c^{(3)}_{2}=0.3,\end{array} \right.\]  
_where \(l(1)=2\), \(l(2)=1\) and \(l(3)=2\) hold, and we see that the relation (2) is satisfied. Next, consider the state trajectories. Then, for \(x(0)=[ \begin{array}{cccc}0&0&0\end{array} ]^{\top}\) and \(u(0)=1\), we can obtain_  
\[P\left(x(1)=[ \begin{array}{cccc}0&0&0\end{array} ]^{\top}\mid x(0)=[ \begin{array}{cccc}0&0&0\end{array} ]^{\top}\right) = 0.8,\] \[P\left(x(1)=[ \begin{array}{cccc}1&0&0\end{array} ]^{\top}\mid x(0)= [ \begin{array}{cccc}0&0&0\end{array} ]^{\top}\right) = 0.2.\]  
_In this example, the cardinality of the finite state set \(\{0,1\}^{3}\) is given by \(2^{3}=8\), and we can obtain the state transition diagram of Figure 2 by computing the transition from each state under \(u(0)=0\). In Figure 2, the number assigned to each node denotes \(x_{1}\), \(x_{2}\), \(x_{3}\) (elements of the state), and the number assigned to each arc denotes the transition probability from some state to other state. Note here that, for simplicity, the state transitions from only \(x(k)=[ \begin{array}{cccc}0&0&0\end{array} ]^{\top}\), \([ \begin{array}{cccc}0&0&1\end{array} ]^{\top}\), \([ \begin{array}{cccc}0&1&0\end{array}